12 February 1998
Physics Letters B 419 Ž1998. 1–6
The proton and neutron distributions in Na isotopes:
the development of halo and shell structure
J. Meng, I. Tanihata, S. Yamaji
The Institute of Physical and Chemical Research (RIKEN), Hirosawa 2-1, Wako-shi, Saitama 351-01, Japan
Received 3 September 1997
Editor: W. Haxton
Abstract
The interaction cross sections for A Na q 12 C reaction are calculated using Glauber model. The continuum HartreeBogoliubov theory has been generalized to treat the odd particle system and take the continuum into account. The theory
reproduces the experimental result quite well. The matter distributions from the proton drip line to the neutron drip line in
Na isotopes have been systematically studied and presented. The relation between the shell effects and the halos has been
examined. The tail of the matter distribution shows a strong dependence on the shell structure. The neutron N s 28 closure
shell fails to appear due to the coming down of the 2 p 3r2 and 2 p1r2 . The development of the halo was understood as
changes in the occupation in the next shell or the sub-shell close to the continuum limit. The central proton density is found
to be decreasing near the neutron drip line, which is due to the proton-neutron interaction. However the diffuseness of the
proton density does not change for the whole Na isotopes. q 1998 Elsevier Science B.V.
PACS: 21.10.Gv; 21.60.-n; 24.10.Cn; 25.45.De
Keywords: Relativistic mean field; Pairing; Continuum; Halo; Shell structure
The recent developments in the accelerator technology and the detection techniques all around the
world have changed the nuclear physics scenario. It
is now possible to produce and study the nuclei far
away from the stability line – so called ’’EXOTIC
NUCLEI’’. Experiments of this kind have casted
new light on nuclear structure and novel and entirely
unexpected features appeared: e.g. the neutron halo
in 11 Li w1x and neutron skin w2x as the rapid increase
in the measured interaction cross sections in the
neutron-rich light nuclei. The extreme proton and
neutron ratio of these nuclei and physics connected
with these low density matter have attracted more
and more attentions in nuclear physics as well as
other fields such as astrophysics.
With the exotic matter distribution near the drip
line, a lot of questions are still open, e.g., the relation
between the halo and the shell effect , the difference
about the proton and neutron distribution on the
stability line and away from the stability line. How is
the halo formed ? Are there a rapid change from the
normal nuclear density to the halo density or a
gradual change in the particle number ? As the
matter distribution is not measurable directly , series
of experiment at different incident beam energy are
necessary in order to determine the density distribution of both proton and neutron model-independently. Among all the experiments carried out so far,
Na isotopes provide a good opportunity to study the
density distributions over a wide range of neutron
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 3 8 6 - 5
2
J. Meng et al.r Physics Letters B 419 (1998) 1–6
numbers w3x. Although theoretically, lots of works
have been reported with either the non-relativistic
Hartree-Fock or relativistic mean field, but the pairing property is always neglected or simply treated by
the BCS approximation, e.g., see w4,5x. For the pairing correlation, the contribution from the continuum
are essential for the description of the drip line
nuclei. We report in this letter a systematic study of
nuclear density distributions in Na isotopes within
the Relativistic Mean Field Theory ŽRMF., with the
pairing and the blocking effect for odd particle system properly described by Hartree-Bogoliubov theory in coordinate representation, and try to answer
some of the general questions in these low-density
nuclear matter.
Definitely it is necessary to have a self-consistent
theory to describe directly the cross section, so that
the usual model dependent way for extracting the
matter distribution could be avoided. As discussed in
great detail in a recent review article w6x and in the
references given therein, one has applied so far
rather different techniques to describe halo phenomena in light nuclei, as for instance the exact solution
of few-body equations treating inert sub-clusters as
point particles, or the density dependent Hartree-Fock
method in a localized mean field taking into account
all the particles in a microscopic way, or a full shell
model diagonalizations based on oscillator spaces
with two different oscillator parameters for the coreand the halo-particles. Recently, a fully self-consistent calculation within the Relativistic HartreeBogoliubov ŽRHB. theory in coordinate space for the
description of the chain of Lithium isotopes ranging
from 6 Li to 11 Li was reported w7x. It combines the
advantages of a proper description of the spin-orbit
term with those of full Hartree-Bogoliubov theory in
the continuum, which allows in the canonical basis
the scattering of Cooper pairs to low lying resonances in the continuum. Excellent agreement including binding energy, matter radius, and matter
distribution with the experimental data was obtained
without any modification of the neutron 1 p 1r2 level
like the former works Žsee w7x and the references
therein..
We first study the isospin dependence of the
density distribution and the ground state properties
of the Na nuclei within RHB. Then the cross sections
based on Glauber model calculations with the den-
sity obtained from RHB are directly compared with
the experimentally determined ones. As the theory
used here is fully microscopic and parameter free, it
gives consistent description of the proton and neutron distribution, and the development of proton and
neutron halo or skin could be examined.
The basic ansatz of the RMF theory is a Lagrangian density whereby nucleons are described as
Dirac particles which interact via the exchange of
various mesons and the photon. The mesons considered are the scalar sigma Ž s ., vector omega Ž v . and
iso-vector vector rho Ž r .. The latter provides the
necessary isospin asymmetry w8x. The scalar sigma
meson moves in a self-interacting field having cubic
and quartic terms with strengths g 2 and g 3 respectively. The Lagrangian then consists of the free
baryon and meson parts and the interaction part with
minimal coupling, together with the nucleon mass M
and ms Ž gs ., mv Ž g v . and mr Ž g r . the masses
Žcoupling constants. of the respective mesons:
L s c Ž i Euy M . c q
1
2
1
Em sE ms y U Ž s .
1
1
y Vmn V mn q mv2 vm v m y Rmn R mn
4
2
4
1
1
q mr2 rm r m y Fmn F mn y gs csc
2
4
yg v cvu c yg r cru tc y ec Auc
Ž 1.
where a non-linear scalar self-interaction UŽ s . of
the s meson has been taken into account w9x.
For the proper treatment of the pairing correlations and for correct description of the scattering of
Cooper pairs into the continuum in a self-consistent
way, one needs to extend the present relativistic
mean-field theory to a continuum RHB theory w10x.
Using Green’s function techniques it has been shown
in Ref. w11x how to derive the RHB equations from
such a Lagrangian:
ž
h
yD )
D
yh )
U
V
/ž /
k
s Ek
U
V
ž /
,
Ž 2.
k
Ek are quasi-particle energies and the coefficients
Uk Ž r . and Vk Ž r . are four-dimensional Dirac spinors.
h is the usual Dirac Hamiltonian
h s a p q g v v q b Ž M q gs s . y l
Ž 3.
J. Meng et al.r Physics Letters B 419 (1998) 1–6
containing the chemical potential l adjusted to the
proper particle number and the meson fields s and
v determined as usual in a self-consistent way from
the Klein Gordon equations in no-sea-approximation.
The pairing potential D in Eq. Ž2. is given by
1
D a b s Ý Vapbpc d k c d
Ž 4.
2 cd
It is obtained from the pairing tensor k s U ) V T and
the one-meson exchange interaction Vapbpc d in the
pp-channel. As in Ref. w7x Vapbpc d in Eq. Ž4. is the
density dependent two-body force of zero range:
V0
V Ž r 1 ,r 2 . s
Ž 1 q P s . d Ž r 1 y r 2 . Ž 1 y r Ž r . rr 0 . .
2
Ž 5.
The RHB equations Ž2. for zero range pairing
forces are a set of four coupled differential equations
for the HB quasi-particle Dirac spinors UŽ r . and
V Ž r .. They are solved by the shooting method in a
self-consistent way as w7x. With a step size of 0.1 fm
and using proper boundary conditions the above
equations are solved in a spherical box of radius
R s 25 fm. As shown in w12x, for these relatively
lighter nuclei R s 25 fm give quite accurate result.
Since we use a pairing force of zero range Ž5. we
have to limit the number of continuum levels by a
cut-off energy. For each spin-parity channel 20 radial
wave functions are taken into account, which corresponds roughly to a cut-off energy of 120 MeV for
R s 25 . After fixing the cut-off energy and the box
radius R, the strength V0 of the pairing force Ž5. for
both the neutrons and protons is fixed by a similar
calculation of Gogny force D1S w13x by reproducing
the corresponding pairing energy y 12 Tr Dk i as Ref.
w7x. For r 0 we use the nuclear matter density 0.152
fmy3 . The ground state <C ) of the even particle
system is defined as the vacuum with respect to the
quasi-particle: bn <C )s 0, <C )s Łn bn < y ) ,
where < y ) is the bare vacuum. For odd system,
the ground state can be correspondingly written as:
<C ) m s bm† Łn / m bn < y ) , where m is the level
which is blocked. The exchange of the quasiparticle
creation operator bm† with the corresponding annihilation operator bm means the replacement of the
column m in the U and V matrices by the corresponding column in the matrices V ) , U ) w15x.
3
A systematic set of calculations have been carried
out for all the nuclei in Na isotopes with mass
number A ranging from 17 to 45. We have employed
in the calculations the non-linear Lagrangian parameter set NLSH which was widely used for the description of all the medium and heavy nuclei, particularly
drip line nuclei w16x.
The calculated binding energies EB and the interaction cross sections with the Glauber Model are
presented in Fig. 1. The calculated binding energies
EB are in good agreement with the empirical values
w17x for the radioactive isotopes, which are our current interests. The resonance states of 17 Na and 18 Na
Žwith a positive Fermi energy. are exactly reproduced. 19 Na is bound but unstable against the proton
emission, reproducing the experimental observation.
The neutron drip-line nucleus has been predicted to
Fig. 1. Upper part: The interaction cross sections sI of A Na
isotopes on a carbon target at 950 A MeV: the open circles are the
result of RHB and the available experimental data Ž As 20y
23,25y32. are given by solid circles with their error-bar. The
dashed line is a simple extrapolation based on the RHB calculation for 28y31 Na. Lower part: Binding energies for Na isotopes,
the convention is the same as the upper part, but the RHB result
for particle unstable isotopes are indicated by triangle.
4
J. Meng et al.r Physics Letters B 419 (1998) 1–6
relatively fast increase has been predicted. As we
have seen here, the measured cross section shows
similar behavior with that of RHB.
As the density, which is obtained from a fully
microscopic and parameter free model, is well supported by the experimental cross sections and binding energies, we proceed to examine the density
distributions of the whole isotopes and study the
relation between the development of halo and shell
effect within the model.
The density distributions for both the proton and
the neutron are given in Fig. 2 . As seen in the upper
part of Fig. 2 , the change of the neutron density is
as follows: for the nucleus with less number of
neutrons Ž N ., the density at the center is low and it
spreads only to some smaller distance. With the
increase of N, the density near the center increases
due to the occupation of the 2 s1r2 level, so does the
Fig. 2. The neutron Župper. and proton Žlower. density distributions in Na isotopes. The same figures but in logarithm scale are
given as inserts to show the tail part of the density distribution.
be 45 Na in the present model. The difference between the calculations and the empirical values for
the stable isotopes is from the deformation, which
has been neglected here.
To compare the cross section directly with experimental measured values, the densities rn, p Ž r . of the
target 12 C and the Na isotopes obtained from RHB
Žsee Fig. 2. were used. The cross sections were
calculated in Glauber model by using the free nucleon-nucleon cross section w14x for the proton and
neutron respectively. The cross sections for reaction
of Na isotopes at 950 A MeV on 12 C have been
compared with the experimental values w3x in the
upper part of Fig. 1. The agreement between the
calculated results and measured ones are fine. The
cross section below 22 Na changes only slightly with
the neutron number, which means the proton density
has played important role to remedy the contribution
of less neutron. From 25 Na to the neutron drip line, a
gradual increase of the cross section has been observed. After 32 Na, although no data exist yet, a
Fig. 3. The same as Fig. 2 but for matter density distribution. The
upper part is given in logarithm scale Žthe radius is labeled at the
top of the figure. and the radius r 0 at which r Ž r 0 . s10y4 for
different isotopes is given as inserts.
J. Meng et al.r Physics Letters B 419 (1998) 1–6
development of neutron radius. In the lower part of
Fig. 2 , the proton density shows different behavior.
The surface is more or less unchanged because of the
Coulomb Barrier, but with the increase of N, the
density of the center decreases due to the slight
increasing at the tail. This is considered to be due to
the attractive proton neutron interaction. But as the
density must be multiplied by a factor 4p r 2 before
the integration in order to give the fixed Z, the big
change in the center does not influence the outer part
of the proton distribution very much. This result is
fully consistent with the recent experiment on the
charge-changing cross section w18x.
The matter densities for the even N Na isotopes
are given in Fig. 3. the shortest tail in the total
density occurs for 23y25 Na, the most stable ones. For
either the proton or the neutron rich side, the tail
density increases monotonically. The tail Ž r ) 10 fm.
of the proton rich nuclei is mainly due to the contribution of the proton and that of the neutron drip-line
is mainly due to the contribution of the neutron.
Compared with the neutron-rich isotopes, the proton
distribution with less N has higher density at the
5
center, lower density in the middle Ž2.5 - r - 4.5fm.,
a larger tail in the outer Ž r ) 4.5 fm. part.
Next we will examine the density distribution for
the neutron rich side. It is interesting to connect the
matter distribution with the level distribution Žsee
Fig. 4.: after 25 Na, as it is a sub-closure shell for the
1d 5r2 , then the neutrons are filled in the 2 s1r2 and
1d 3r2 . So the tail of the density for 27 Na is two
order of magnitude larger than 25 Na at r s 10 fm,
while the tail of the density from 27 Na to 31 Na is
very close to each other. But as more neutrons are
filled in, the added neutrons are filled in the next
shell 1 f 7r2 , 2 p 3r2 and 2 p 1r2 . So again two order of
magnitude’s increase has been seen from 31 Na to
33
Na, and then a gradual increase after 33 Na. So it
becomes clear that the rapid increase in the cross
section is connected with the filling of neutrons in
the next shell or sub shell. In the inserts in Fig. 3 :
the radius r 0 at which r Ž r 0 . s 10y4 fmy3 is given
as a function of the mass number to see the relation
between the shell effect and matter distribution more
clearly. It is very interesting to see a slight decrease
of r 0 from proton drip-line to 25 Na. The tail of 25 Na
Fig. 4. Left part: Single particle energies for neutrons in the canonical basis as a function of the mass number. The dashed line indicates the
chemical potential. Right part: The occupation probabilities in the canonical basis for 35 Na.
6
J. Meng et al.r Physics Letters B 419 (1998) 1–6
is the smallest. From 26 Na to 31 Na, one sees a almost
constant r 0 . After a jump from 31 Na to 32 Na, a rapid
increasing tendency appears again.
In Fig. 4 , the microscopic structure of the single
particle energies in the canonical basis w15x is given.
In the left panel of Fig. 4, the single particle levels in
the canonical basis for the isotopes with an even
neutron number are shown. Going from A s 19 to
A s 45 we observe a big gap above the N s 8,
N s 20 major shell , and N s 14 sub-shell. The
N s 28 shell for stable nuclei fails to appear, as the
2 p 3r2 and 2 p 1r2 come so close to 1 f 7r2 . When
N G 20, the neutrons are filled to the levels in the
continuum or weakly bound states in the order of
1 f 7r2 , 2 p 3r2 , 2 p 1r2 , and 1 f5r2 . In the right part, the
occupation probabilities in the canonical basis of all
the neutron levels below E s 10 MeV have been
given for 35 Na to show how the levels are filled in
nuclei near the drip line. The importances of careful
treatment of the pairing correlation, of treating properly the scattering of particle pairs to higher lying
levels, are noted in the figure.
Summarizing our investigations, the development
of a proton skin as well as neutron skin has been
systematically studied with a microscopic RHB
model, where the pairing and blocking effect have
been treated self-consistently. A systematic set of
calculations for the ground state properties of nuclei
in Na isotopes is presented using the RHB together
with standard Glauber theory. The RHB equations
are solved self-consistently in coordinate space so
that the continuum and the pairing have been better
treated. The calculated binding energies are in good
agreement with the experimental values. A Glauber
model calculation has been carried out with the
density obtained from RHB. A good agreement has
been obtained with the measured cross sections for
12
C as a target and a rapid increase of the cross
sections has been predicted for neutron rich Na
isotopes beyond 32 Na. The systematics of the proton
and neutron distribution are presented. After systematic examination of the neutron, proton and matter
distributions in the Na nuclei from the proton dripline to the neutron drip-line, the connection between
the tail part of the density and the shell structure has
been found. The tail of the matter distribution is not
so sensitive to how many particles are filled in a
major shell. Instead it is very sensitive to whether
this shell has occupation or not. The physics behind
the skin and halo has been revealed as a spatial
demonstration of shell effect: simply the extra neutrons are filled in the next shell and sub-shell. This is
in agreement with the mechanism observed so far in
the halo system but more general. As the 1 f 7r2 is
very close to the continuum, the N s 28 close shell
for stable nuclei fails to appear due to the coming
down of the 2 p 3r2 and 2 p 1r2 levels. Another important conclusion here is that, contrary to the usual
impression, the proton density distribution is less
sensitive to the proton and neutron ratio. Instead it is
almost unchanged from the proton drip-line to the
neutron drip-line. Similar conclusion has been obtained recently by charge change reaction experimently w18x. The influence of the deformation, which
is neglected in the present investigation, is also
interesting to us, more extensive study by extending
the present study to deformed cases are in progress
References
w1x I. Tanihata et al., Phys. Rev. Lett. 55 Ž1985. 2676; I.
Tanihata, Prog. Part. and Nucl. Phys. 35 Ž1995. 505.
w2x G.D. Alkhazov et al., Phys. Rev. Lett. 78 Ž1997. 2313.
w3x T. Suzuki et al., Phys. Rev. Lett. 75 Ž1995. 3241.
w4x X. Campi et al., Nucl. Phys. A 251 Ž1975. 193.
w5x S. Patra, C. Praharaj, Phys. Lett. B 273 Ž1991. 13.
w6x M.V. Zhukov et al., Phys. Rep. 231 Ž1993. 150.
w7x J. Meng, P. Ring, Phys. Rev. Lett. 77 Ž1996. 3963.
w8x P. Ring, Prog. Part. and Nucl. Phys. 37 Ž1996. 193.
w9x J. Boguta, A.R. Bodmer, Nucl. Phys. A 292 Ž1977. 413.
w10x J. Dobaczewski, H. Flocard, J. Treiner, Nucl. Phys. A 422
Ž1984. 103.
w11x H. Kucharek, P. Ring, Z. Phys. A 339 Ž1991. 23.
w12x J. Meng, P. Ring, Phys. Rev. Lett., to appear.
w13x J.F. Berger, M. Girod, D. Gogny, Nucl. Phys. A 428 Ž1984.
32c.
w14x L. Ray, Phys. Rev. C 20 Ž1979. 1857.
w15x P. Ring, P. Schuck, The Nuclear Many-body Problem,
Springer Verlag, Heidelberg, 1980.
w16x M. Sharma, M. Nagarajan, P. Ring, Phys. Lett. B 312 Ž1993.
377.
w17x G. Audi, A.H. Wapstra, Nucl. Phys. A 565 Ž1993. 1.
w18x O. Bochkarev et al., submitted to Z. Phys.
12 February 1998
Physics Letters B 419 Ž1998. 7–13
Pairing reduction and rotational motion
in multi-quasiparticle states
G.D. Dracoulis a , F.G. Kondev a , P.M. Walker
a
a,b
Department of Nuclear Physics, RSPhysSE, Australian National UniÕersity, Canberra ACT, 0200, Australia
b
Department of Physics, UniÕersity of Surrey, Guildford, Surrey, GU2 5XH, UK
Received 28 March 1997; revised 8 September 1997
Editor: J.-P. Blaizot
Abstract
Calculations of pairing energies in multi-quasiparticle states using the Lipkin-Nogami prescription predict a discrete
reduction in pairing with a geometric dependence on seniority. An abrupt transition from superfluid to normal motion is
therefore not expected, even with a large number of orbits blocked. Recent experimental results on rotational bands
associated with multi-quasiparticle intrinsic states allow comparisons to be made for a subset of orbitals as a function of
seniority. Approximate agreement is obtained between the observed moments-of-inertia and calculated values supporting the
view that for the highest seniority states identified so far, the pairing persists at nearly half of the full value. q 1998 Elsevier
Science B.V.
PACS: 21.10.Re; 21.60.Ev; 21.90.q f
Keywords: Lipkin-Nogami pairing; Multi-quasiparticle states; Rotational properties; Moments-of-inertia
A seminal advance in the understanding of nuclear structure was the recognition that superfluidity,
caused by pairing correlations, could account for the
low collective moments-of-inertia found for rotational bands in deformed nuclei w1–4x. These fell far
short of the values expected for rigid bodies with the
deformations implied by other nuclear properties.
The analogy between nuclei and superconducting
metals described by BCS theory w5x was recognised
to be valuable but incomplete, since nuclei are finite;
the pairing correlation involves only a few particles
compared to the many involved in macroscopic systems; and the single-particle levels near the Fermi
surface are sparse, and distinctly non-uniform, reflecting the shell structure which controls the stability at deformed shapes. The average energy gain per
interaction yG Ž; 20rA MeV. is of the same order
as the energy spacing at the Fermi surface. Further,
because of the different character of the orbitals at
the surface, the interactions may well be configuration-dependent.
Notwithstanding these qualifications, early predictions w6x were that sharp phase transitions would
occur if the nucleus were subjected to high-frequency
rotation. The rotation acts to uncouple pairs of particles and destroy the correlations, much like the
quenching of superconductivity by the application of
magnetic fields. The analogous critical angular momentum Ic was predicted to be ; 12–20 ". More
sophisticated calculations gave marginally higher spin
estimates w7x. Breaking and mutual alignment of
individual pairs was certainly observed as high-spin
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 5 6 - 1
8
G.D. Dracoulis et al.r Physics Letters B 419 (1998) 7–13
studies developed w8,9x, but a first-order phase transition was not.
Evidence for the more subtle effects of reduction
of pairing induced by high-frequency rotation has
since been pursued vigorously and in depth Žsee, for
example w10x and the review of Shimizu et al. w11x.
but clear manifestations have been elusive w12x. One
difficulty has been that the collective bands whose
properties are examined as probes are often not well
characterised in configuration. It has been accepted
that configuration-dependent pairing is responsible
for differences in band-crossing frequencies w13x and
evidence has also been presented recently that the
stepwise pairing reduction that is expected on the
basis of more realistic multi-quasiparticle calculations is reflected in the level densities at high spin,
and then manifested in the shape of the statistical
g-ray spectra produced by transitions through states
in the continuum w14x.
The complementary approach reported in this letter is the study of the rotational bands based on
well-defined, multi-quasiparticle states. Such intrinsic states can be characterised in terms of the number
of unpaired particles Žthe seniority., and the specific
orbitals those particles occupy. The emphasis is on
high seniority, rather than high rotational frequency,
Fig. 1. Predicted neutron pairing for intrinsic states of different
seniority in 178 W.
but using the rotation as an essential attribute. The
rotational properties of the multi-quasiparticle bands
will depend on the pairing which is predicted to be
sensitive to the blocking of orbitals near the Fermi
surface since the correlation in nuclei is mediated by
only a few particles. For uniform level densities, the
pairing gap was initially estimated w15x to depend on
the seniority n and the average level spacing, d, as
D , D0 Ž D0 y n = d . . Since both proton and neutron pairing values Ž D0 . are about 1 MeV and the
average spacing is about 300 keV, on that basis,
pairing should have collapsed for a critical seniority
of 3 or 4 in each nucleon type Žanalogous to the
critical spin Ic .. Only recently have sufficient data
become available to confront systematically such
predictions.
Representative neutron pairing gaps calculated
with the Lipkin-Nogami method Žsee for example
w16,17x. which avoids the premature collapse inherent in BCS calculations when applied to nuclei are
shown in Fig. 1, for seniority up to six. ŽThis formalism includes an additional Lagrange multiplier, l2
so that D q l2 is the quantity to be compared with
odd-even mass differences w16x. The calculations use
the prescription given by Nazarewicz et al. w18x to
treat the Fermi level and pairing gaps self-consistently, to include particle-number conservation,
and to have blocked states removed for multi-quasiparticle configurations. Values for specific configurations and specific nuclei are given in Table 1. For
these calculations a space of 3 oscillator shells giving 64 levels Ž128 states. with about 34 active
particles was used, taken from the Nilsson scheme
with minor adjustments of the single-particle energies near the Fermi surface consistent with evaluation of the 1-quasiparticle levels in similar calculations in this region w20x. A more extensive set of
basis states was used for evaluation of very high
seniorities. ŽNote that in Ref. w14x uniform level
densities are used as input to particle-number projected calculations which lead to similar results for
stepwise reduction, albeit with less fluctuation.. In
contrast to the early estimates, the neutron pairing in
Fig. 1 shows discrete successive reductions with an
approximately geometric dependence,
(
Dn , 0.75 = Dny2
Ž 1.
so that even with six orbitals blocked near the Fermi
G.D. Dracoulis et al.r Physics Letters B 419 (1998) 7–13
surface the pairing remains at ; 40% of the unblocked value. Similar results were obtained for the
protons.
The key factor enabling the approach to be taken
here of examining multi-quasiparticle band properties, is the existence of a favourable region of the
nuclear chart where both the intrinsic Ži.e. bandhead.
and rotational states are close enough to the yrast
line for both to be observed. The structures can then
be defined because the spin and parity of the bandhead is often only obtainable from a restricted or
even unique combination of orbitals near the Fermi
surface; the in-band decay properties can differenti-
9
ate between protons and neutrons and between specific orbitals; and band spacings indicate whether
component particles which suffer Coriolis effects are
present.
This was one imperative for recent measurements
aimed at identifying bands and multi-quasiparticle
states in nuclei near N ; 104 and Z ; 74, a traditional region for the observation of isomers of highK, obtained by the summation of the projections, V i
of individual nucleons. These compete with collective rotation of lower-seniority configurations as a
means of carrying angular momentum. Many welldefined states of increasing seniority and with asso-
Table 1
Configurations and Gap Parameters for multi-quasiparticle states
Nuclide
Kp
178
Ž0q .
6q
7y
13y
15q
18y
21y
22y
25q
28 Žy.
45r2y
179
W
W
Ref.
Configuration a
1665
1738
3525
3653
4878
5312
5626
6571
8147
w24x
w24x
w23x
w23x
w24x
w24x
w24x
w24x
w23x
w23,24x
n Ž 52 72 .
y H
n Ž 72 72 .
y H
q q
n Ž 72 72 . p Ž 52 72 .
y H
q y
n Ž 72 72 .p Ž 72 92 .
y H
q q y I
n Ž 72 72 .p Ž 52 72 92 12 .
y y H H
q y
n Ž 52 72 72 92 .p Ž 52 92 .
y y H H
q y
n Ž 52 72 72 92 .p Ž 72 92 .
y y H H
q q y I
5
7
7
9
5
n Ž 2 2 2 2 .p Ž 2 72 92 12 .
y y H H
q y y I
5
7
7
9
7
n Ž 2 2 2 2 .p Ž 2 92 112 12 .
4922
w26x
n Ž 72
Eexp
ŽkeV.
y y
y H H
7
2
H
9
2
.p Ž 52
7
2
9
2
1
2
.
8y
14y
20y
27y
186
1370
2770
5293
w19x
w19,22x
w22x
w22x
n Ž 72 .p Ž 92 .
H
q q y
n Ž 72 .p Ž 52 72 92 .
y y H
q q y
n Ž 52 72 72 .p Ž 52 72 92 .
y y H H H
q q y
n Ž 52 72 52 72 92 .p Ž 52 72 92 .
178
Ta
7y
9y
15y
0
392
1468
w20x
w20x
w20x
16q
22q
1892
3134
w20x
w20x
n Ž 72 .p Ž 72 .
H
y
n Ž 92 .p Ž 92 .
y y q
y
n Ž 52 72 92 .p Ž 92 .
q
q q y
n Ž 92 .p Ž 52 72 92 .
y H H
y
n Ž 72 72 92 .p Ž 92 .
y H H
q q y
7
7
9
5
n Ž 2 2 2 .p Ž 2 72 92 .
9r2y
25r2q
37r2q
31
1318
2640
w21x
w21x
w21x
p Ž 92 .
y H
y
n Ž 72 92 .p Ž 92 .
y H
q q y
7
9
5
n Ž 2 2 .p Ž 2 72 92 .
a
b
855
616
617
617
617
617
446
446
446
446
1083
1083
1083
876
876
612
861
876
612
620
511
609
587
587
401
323
899
620
620
620
574
589
402
589
403
403
861
856
856
593
856
593
705
497
497
830
830
582
y
Ta
179
Dp b
ŽkeV.
q q y I
176
y
Dn b
ŽkeV.
q
y
Ta
n : 12 y w521x, 52 y w512x, 72 y w514x, 72 H w633x, 92 H w624x; p : 12 I w541x, 52 q w402x, 72 q w404x, 92 y w514x, 112 y w505x.
. values corrected as in w17x; Gn s 19.65, Gp s 22.7 for 178 W and 179 W; Gn s 18.6, Gp s 21.4 for tantalum cases.
10
G.D. Dracoulis et al.r Physics Letters B 419 (1998) 7–13
ciated rotational bands, have thus been observed
w19–26x. A selected group is listed in Table 1. Most
of the orbits active near the Fermi surface are included, extending to an eight-quasiparticle state
Ž n 4p 4 . in 178 W . Many involve one or two of the
i 13r2 neutrons Ž7r2qw 633x and 9r2qw 624x. and the
h 9r2 proton Ž1r2yw 541x. which are strongly Coriolis-mixed and therefore show rotational alignment.
In cases where alignment is not important, one
would deduce the moment-of-inertia I from the
observed energy spacings between states of spin I in
the rotational bands and the rotor formula, E
2
s 2"I Ž I Ž I q 1. y K 2 .. The nucleus, however, is
more properly described as particles coupled to a
rotor Ža system of ‘‘quantum gyroscopes’’ w27x. particularly if those additional particles have large intrinsic spins, and consequently feel Coriolis and
centrifugal forces strongly. The particle and collective motions can be approximately separated by a
transformation of the experimentally observed quantities to the rotating frame to obtain the total aligned
angular momentum referred to the rotation axis, I x ,
and the rotational frequency " v , using standard
prescriptions w28x.
Plots of I x against the rotational frequency for the
main bands in 178 W, are shown in Fig. 2. With
increasing seniority the curves become more linear
and increase in magnitude at any given frequency,
qualitatively consistent with the addition of more
aligned angular momentum as the number of particles increases. However, a. the effective momentof-inertia remains well below that for a rigid body
Žthe extreme case on average if pairing had collapsed
w27x. and b. the curves do not cross a ‘‘barrier’’
exemplified by the results for the 25q band.
If no other effects were present, the net alignment
i given by the difference between I x at a given
rotational frequency and that for a collective reference core
i s I x Ž v . y Ž I 0 v q I 1 v 3 . ref .
Ž 2.
would be the sum of the aligned angular momenta of
the components. However, blocking of the pairing
correlations by the component particles affects both
the collective rotation and alignment itself, so additivity fails.
Fig. 2. Total aligned angular momenta as a function of rotational
frequency for selected bands indicated by their K p values in 178 W
and one in 179 W. ŽSee Table 1 for the configurations..
The consequence in 178 W is an apparent saturation of the net alignment implied by the ‘‘barrier’’ in
I x noted above. Consideration of the configurations
involved Žall contain one or two i 13r2 neutrons but
those of higher seniority including the 25q state in
178
W also contain the h 9r2 proton., leads to the
deduction that there is a specific loss of additivity
with reduced pairing. In effect,
i tot - Ý Ž i Ž p h 9r2 . q i Ž n i 13r2 .
qi Ž other particles. .
and;
i tot ™ i Ž p h 9r2 . .
Ž 3.
The justification for this assertion rests partly with
estimates of the dependence of alignment on pairing
for the key particles as shown in simplified particlerotor calculations in Ref. w29x. Those calculations
show that the Coriolis effects, and therefore the
alignment, for the h 9r2 particle persists, despite reduced pairing, because the proton Fermi level is
close to the V s 1r2y orbital, whereas the i 13r2
neutron alignment drops because the neutron Fermi
level is higher in the shell, between the V s 7r2q
and 9r2q orbitals.
G.D. Dracoulis et al.r Physics Letters B 419 (1998) 7–13
This interpretation is supported by the results for
the multi-quasiparticle bands in the tantalum isotopes
which do not contain the h 9r2 proton. Several pairs
of these are compared in Fig. 3. The absence of the
h 9r2 proton allows underlying changes to be revealed – the 15y, n 3p state in 178 Ta for example, is
obtained by adding the 6q, n 2 component to the 9y,
np state. Despite the extra particles, the apparent
alignment is lower; the same situation prevails for
the 22q, n 3p 3 band and the 16q, n 3p configuration,
the former having an additional 6q, p 2 component
but less apparent alignment than the latter. Similar
results are seen for 179 Ta. The implication is that the
fall in alignment caused by reduced pairing is greater
than the effect of an underlying increase in the
collective moment-of-inertia, which when using the
reference appropriate for the low-seniority states,
would lead to an apparent increase in i.
To evaluate these effects precisely would require
access to a many-particles-plus-rotor model which
treated consistently the successive removal of particles from a ‘‘core’’, while continuing to define the
coupled motion, a model which has yet to be developed. The approach taken here is to fit each band
over a limited spin region Žwhere there are no alignment gains . with the generalised Harris formula
I x Ž v . s i q I 0 v q I 1 v 3 and then examine the parameters. Coriolis anti-pairing leads mainly to the
v 3 term Žsee w30x and references therein. so that as
Fig. 4. The moment-of-inertia parameter I 0 deduced from a fit to
rotational bands using the modified Harris parameterisation given
in the text, plotted against the values using the Migdal formula
with pairing values calculated for each configuration. The bars on
the upper border indicate the expected dependence of the momentof-inertia for equal proton and neutron seniority Žgiving a total
2= n X . and a geometric pairing dependence.
pairing decreases the expectation is that I 1 will
decrease and I 0 will increase, leading to the more
linear and higher curves seen for 178 W in Fig. 2. The
parameters extracted in the tantalum cases show
more scatter Žas can be seen from the preliminary
results of Ref. w31x. hence they have been evaluated
setting I 1 s 0, which is equivalent to extraction of
the local dynamic moment-of-inertia w11x.
We have combined the calculated values of the
proton and neutron pairing gaps from Table 1 for
each configuration, in the two-fluid formula using
the prescription of Migdal w4x with the proton and
neutron contributions separated, so that
I tot s
I
Fig. 3. Net aligned angular momenta evaluated using a common
reference given by I 0 s 34 MeVy1 " 2 and I 1 s 70 MeVy3 " 4 .
11
N
A
I n Ž Dn . q
p,n s I r i g
ž
1y
Z
A
I
p
Ž Dp . ;
ln Ž x q '1 q x 2
x'1 q x
2
.
/
Ž 4.
with x s d 2"Dvp 0, n where d is the deformation parame-
12
G.D. Dracoulis et al.r Physics Letters B 419 (1998) 7–13
ter, for comparison with the extracted values of I 0 .
This formalism has been successfully applied to
describe the moments-of-inertia of ground-state bands
at low spin w32x.
The values are compared for cases of different
seniority in Fig. 4, on a range which extends to the
rigid-body value, I rig . ; 83 MeVy1 " 2 . The additional scale on the upper border of Fig. 4 shows the
dependence of the moment-of-inertia obtained by
taking equal numbers of proton and neutrons to give
total seniority 2 = n X and inserting the estimate of
Eq. Ž1. in Eq. Ž4.. There is qualitative agreement
between the observed moments and those predicted,
despite the approximate formulation and the restricted range of states over which the experimental
parameters can be defined. Some notable exceptions
such as the 16q band in 178 Ta fall above the diagonal, even though the 22q band, which has a related
configuration, does not. It is apparent that the moments-of-inertia obtained in this way partly conform
to the expectation that an increase would follow
seniority. ŽAlternative methods of extracting I 0 or
use of exact pairing estimates for the 178 W cases w24x
lead to similar values.. They are consistent with
persistence of the pairing at about half of the unperturbed value, as given by the calculations.
The mechanism behind the survival of the pairing
correlation is that as orbitals near the Fermi surface
are blocked, the correlation spreads away from the
surface, involving more remote orbitals, thus negating the simple estimates originally used to predict
complete quenching. Further insights into these
mechanisms can be gained from the shell model
calculations of Zelevinsky et al. w33x which predict
similar discrete reductions in pairing asymptoting to
minimum Žnon-zero. values controlled by the dynamical fluctuations Žimplicitly contained in the present calculations. and thermal fluctuations which
would not arise in a strict mean-field description.
While the implication is also that yet higher seniority
states would not show marked decreases in pairing,
experimental confirmation is of importance given
that the present limits are at the point where the
pairing predicted is of the same magnitude as the
dynamical values w11x.
G.D.D. is grateful to Witek Nazarewicz, Neil Rowley, Olivier Burglin and Vladimir Zelevinsky for
discussions and communications on various aspects
of this work.
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12 February 1998
Physics Letters B 419 Ž1998. 14–18
Magnetic moments of Dqq and Vy from QCD sum rules
Frank X. Lee
Nuclear Physics Laboratory, Department of Physics, UniÕersity of Colorado, Boulder, CO 80309-0446, USA
Received 28 July 1997
Editor: W. Haxton
Abstract
QCD sum rules for the magnetic moments of Dqq and Vy are derived using the external field method. They are
analyzed by a Monte Carlo based procedure, using realistic estimates of the QCD input parameters. The results are
consistent with the measured values, despite relatively large errors that can be attributed mostly to the poorly-known vacuum
susceptibility x . It is shown that a 30% level accuracy can be achieved in the derived sum rules, provided the QCD input
parameters are improved to the 10% level. q 1998 Elsevier Science B.V.
PACS: 13.40.Em; 12.38.Lga; 11.55.Hx; 14.20.G; 02.70.Lg
The QCD sum rule method w1x is a powerful tool
in revealing the deep connection between hadron
phenomenology and QCD vacuum structure via a
few condensate parameters. The method has been
successfully applied to a variety of problems to gain
a field-theoretical understanding into the structure of
hadrons. Calculations of the nucleon magnetic moments in the approach were first carried out in Refs.
w2x and w3x. They were later refined and extended to
the entire baryon octet in Refs. w4–7x. On the other
hand, the magnetic moments of decuplet baryons
were less well studied within the same approach.
There were previous, unpublished reports in Ref. w8x
on Dqq and Vy magnetic moments. The magnetic
form factor of Dqq in the low Q 2 region was
calculated based on a rather different technique w9x.
In recent years, the magnetic moment of Vy has
been measured with remarkable accuracy w10x: mVy
s y2.02 " 0.05 m N . The magnetic moment of Dqq
has also been extracted from pion bremsstrahlung
w11x: mDqqs 4.5 " 1.0 m N . The experimental information provides new incentives for theoretical
scrutiny of these observables.
In this letter, we present an independent calculation of the magnetic moments of Dqq and Vy in
the QCD sum rule approach. The goal is two-fold.
First, we want to find out the applicability of the
method as an alternative way of understanding the
measured mDqq and mV y, and hope to gain some
insights into the internal structure of these baryons
from a nonperturbative-QCD perspective. Second,
we want to achieve some realistic understanding of
the uncertainties involved in such a determination by
employing a Monte Carlo based analysis procedure.
This will help find possible ways for improvement.
The calculation is algebraically more involved than
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 3 1 2 - 9
F.X. Lee r Physics Letters B 419 (1998) 14–18
the octet case since one has to deal with the more
complex spin structure of spin-3r2 particles, but
presents no conceptual difficulties.
Consider the two-point correlation function in the
QCD vacuum in the presence of a constant background electromagnetic field Fmn :
Pa b Ž p . s i d 4 x e i pP x ²0 < T ha Ž x . hb Ž 0 . 4 < 0:F ,
H
Ž 1.
where ha is the interpolating field for the propagating baryon. The subscript F means that the correlation function is to be evaluated with an electromagnetic interaction term: LI s yAm J m, added to the
QCD Lagrangian. Here Am is the external electromagnetic potential and is given by AmŽ x . s
y 12 Fmn x n in the fixed-point gauge, and J m s
e q qg mq the quark electromagnetic current. The magnetic moments can be obtained by considering the
linear response of the correlation function to the
external field. The action of the external field is
two-fold: it couples directly to the quarks in the
baryon interpolating fields, and it also polarizes the
QCD vacuum. The latter can be described by the
introduction of vacuum susceptibilities.
The interpolating field is constructed from quark
fields, and has the quantum numbers of the baryon in
question. For Dqq and Vy, they are given by
qq
haD Ž x . s e a b c Ž u aT Ž x . Cga u b Ž x . . u c Ž x . ,
y
haV Ž x . s e a b c Ž s aT Ž x . Cga s b Ž x . . s c Ž x . ,
Ž 2.
merous tensor structures, not all of them independent. The dependencies can be removed by ordering
the gamma matrices in a specific order Žwe chose
pˆga gm gn gb where the hat notation means pˆ ' p mgm ..
After a lengthy calculation, a sum rule involving
only the magnetic moment can be isolated at one of
the structures: ga Fmn smn pb . Here we give the final
results, for Dqq:
9
28
e u L4r27E1 M 4 q 563 e u bL4r27 y 76 e u x a 2 L12r27
y 47 e u k Õ a 2 L28r27
1
M2
y 141 e u Ž 4k q j . a 2 L28r27
q 14 e u x m 20 a 2 Ly2r27
s l˜ 2B
ž
mB
M2
1
M
2
2
Ž 4.
where ua is the Rarita-Schwinger spin-vector.
The QCD sum rules are derived by calculating the
correlator in Ž1. using Operator-Product-Expansion
ŽOPE., on the one hand, and matching it to a phenomenological representation, on the other. The calculation is similar in spirit to the octet case Žsee Ref.
w4x, for example., but with the added complexity of
spin-3r2 structures. A direct evaluation led to nu-
1
M2
q 121 e u m20 a2 L14r27
2
q A eyM B r M ,
/
1
M4
Ž 5.
and for Vy:
9
28
e s L4r27E1 M 4 y 157 e s ff m s x aLy12r27E0 M 2
q 563 e s bL4r27 y 187 e s fm s aL4r27
y 289 e s ff Ž 2 k q j . m s aL4r27
y 67 e s f 2fx a2 L12r27 y 47 e s f 2k Õ a2 L28r27
y 141 e s f 2f Ž 4k q j . a2 L28r27
Ž 3.
where C is the charge conjugation operator, and the
superscript T means transpose. The ability of a
interpolating field to annihilate the ground state
baryon into the QCD vacuum is described by a
phenomenological parameter l B Žcalled current coupling or pole residue., defined by the overlap
²0 < ha < Bps : s l B ua Ž p, s . ,
15
q 14 e s f 2fx m20 a2 Ly2r27
y 289 e s fm s m 20 aLy10r27
q 121 e s f 2 m20 a2 L14r27
s l˜ 2B
ž
mB
M2
M2
1
M2
1
M2
1
M2
1
M4
2
2
q A eyM B r M .
/
1
Ž 6.
In these equations, the magnetic moment m B is given
in particle’s natural magnetons. The various symbols
are defined as follows. The condensates are represented by a s yŽ2p . 2 ² uu:, b s ² g c2 G 2 :, ² ug c s P
Gu: s ym20 ² uu:, and the coupling l˜ B s Ž2p . 2l B .
The factors e u s 2r3 and e s s y1r3 are quark
charges in units of electric charge. The vacuum
16
F.X. Lee r Physics Letters B 419 (1998) 14–18
susceptibilities x , k and j are defined by ² qsmn q :F
' e q x ² qq : Fmn , ² qg c Gmn q :F ' e q k ² qq : Fmn , and
² qg c emnrl G rlg 5 q :F ' ie q j ² qq : Fmn . The parameters f s ² ss :r² uu: s ² sg c s P Gs :r² ug c s P Gu:
and f s x srx s k srk s j srj account for flavor
symmetry breaking of the strange quark. Possible
violation of the four-quark condensate is considered
by the parameter k Õ as defined in ² uuuu: s k Õ² uu:2 .
The anomalous dimension corrections of the various
operators are taken into account via the factor L s
2
2
.rlnŽ m2rLQCD
.x,
w a s Ž m2 .ra s Ž M 2 .x s wlnŽ M 2rLQCD
where m s 500 MeV is the renormalization scale and
LQC D is the QCD scale parameter. As usual, the pure
excited state contributions are modeled using terms
on the OPE side surviving M 2 ™ ` under the assumption of duality, and are represented by the
factors EnŽ x . s 1 y eyx Ý n x nrn! with x s w 2rMB2
and w an effective continuum threshold. The parameter A accounts for all contributions from the transitions caused by the external field between the ground
state and the excited states. Such contributions are
not exponentially suppressed relative to the ground
state double pole and must be included. The presence
of such contributions is a general feature of the
external field technique.
Let us note in passing that since Vy and Dqq
are simply related by the interchange of quark flavors u l s, one can verify that sum rule Ž6. reduces
to sum rule Ž5. if one sets e s ™ e u , m s s 0, f s 1,
and f s 1.
To analyze the sum rules, we use a Monte Carlo
based procedure first developed in Ref. w12x. The
basic steps are as follows. First, the uncertainties in
the QCD input parameters are assigned. Then, randomly-selected, Gaussianly-distributed sets are generated, from which an uncertainty distribution in the
OPE can be constructed. Next, a x 2 minimization is
applied to the sum rule by adjusting the phenomenological fit parameters. This is done for each QCD
parameter set, resulting in distributions for phenomenological fit parameters, from which errors are
derived. Usually, 100 such configurations are sufficient for getting stable results. We generally select
1000 sets which help resolve more subtle correlations among the QCD parameters and the phenomenological fit parameters.
The Borel window over which the two sides of a
sum rule are matched is determined by the following
two criteria. First, OPE conÕergence: the highest-dimension-operators contribute no more than 10% to
the QCD side. Second, ground-state dominance:
excited state contributions should not exceed more
than 50% of the phenomenological side. The first
criterion effectively establishes a lower limit, the
second an upper limit. Those sum rules which do not
have a valid Borel window under these criteria are
considered unreliable and therefore discarded.
The QCD input parameters and their uncertainty
assignments are given as follows w13x. The condensates are taken as a s 0.52 " 0.05 GeV 3, b s 1.2 "
0.6 GeV 4 , and m 20 s 0.72 " 0.08 GeV 2 . For the
factorization violation parameter, we use k Õ s 2 " 1
and 1 F k Õ F 4. The QCD scale parameter is restricted to LQCD s 0.15 " 0.04 GeV. The vacuum
susceptibilities have been estimated in studies of
nucleon magnetic moments w2–4x, but the values
vary in a wide range depending on the method used.
Here we take their median values with 50% uncertainties: x s y6.0 " 3.0 GeVy2 , k s 0.75 " 0.38,
and j s y1.5 " 0.75. Note that x is almost an
order of magnitude larger than k and j , and is the
most important of the three. The strange quark parameters are placed at m s s 0.15 " 0.02 GeV, f s
0.83 " 0.05 and f s 0.60 " 0.05 w5x. These uncertainties are assigned conservatively and in accord
with the state-of-the-art in the literature. While some
may argue that some values are better known, others
may find that the errors are underestimated. In any
event, one will learn how the uncertainties in the
QCD parameters are mapped into uncertainties in the
phenomenological fit parameters.
To illustrate how well a sum rule works, we first
2
2
cast it into the subtracted form, P S s l˜ 2B m B eyM B r M ,
then plot the logarithm of the absolute value of the
two sides against the inverse of M 2 . In this way, the
right-hand side will appear as a straight line whose
slope is yMB2 and whose intercept with the y-axis
gives some measure of the coupling strength and the
magnetic moment. The linearity Žor deviation from
it. of the left-hand side gives an indication of OPE
convergence, and information on the continuum
model and the transitions.
To extract the magnetic moments, a two-stage fit
was performed. First, the corresponding chiral-odd
mass sum rule, as obtained previously in Ref. w13x,
was fitted to get the mass MB , the coupling l˜ 2B and
F.X. Lee r Physics Letters B 419 (1998) 14–18
17
Table 1
Monte Carlo analysis of the QCD sum rules for the magnetic
moments of Dqq and Vy. The third column represents the
percentage contribution of the excited states and transitions to the
phenomenological side at the lower end of the Borel region Žit
increases to 50% at the upper end.. The second row for each sum
rule corresponds to results with reduced, uniform 10% errors
assigned to all the QCD input parameters. The uncertainties were
obtained from consideration of 1000 QCD parameter sets. In the
event that the resultant distribution is not Gaussian, the median
and asymmetric deviations are reported.
Sum rule
Region
ŽGeV.
Cont w
Ž%. ŽGeV.
Ž5. for Dqq 0.765 to 1.47
0.765 to 1.47
Ž6. for Vy 0.747 to 1.66
0.747 to 1.66
9
8
7
6
A
ŽGeVy2 .
mB
Ž mN .
q3.68
1.65 0.53"0.77 3.60y3
.55
q1.40
1.65 0.35"0.37 4.13y1
.18
q1.12
2.30 y0.15"0.14 y1.25y1 .17
2.30 y0.10"0.05 y1.49q0.40
y0 .49
Fig. 1. Monte Carlo fits of the magnetic moment sum rules Ž5.
and Ž6.. Each sum rule is searched independently. The solid line
corresponds to the ground state contribution, the dotted line the
rest of the contributions ŽOPE minus continuum minus transition..
The error bars are only shown at the two ends for clarity.
the continuum threshold w 1. Then, MB and l˜ 2B were
used in the magnetic moment sum rule for a threeparameter fit: the transition strength A, the continuum threshold w 2 , and the magnetic moment m B .
Note that w 1 and w 2 are not necessarily the same.
We impose a physical constraint on both w 1 and w 2
requiring that they are larger than the mass, and
discard QCD parameter sets that do not satisfy this
condition. The above procedure is repeated for each
QCD parameter set until a certain number of sets are
reached. In the actual analysis of sum rules Ž5. and
Ž6., however, we found that such a full search was
unsuccessful: the search algorithm consistently returned w 2 either zero or smaller than MB . This
signals insufficient information in the OPE to completely resolve the spectral parameters. To proceed,
we fixed w 2 at w1 , which is a reasonable and
commonly adopted choice in the literature, and
searched for A and m B .
Fig. 1 shows the match for the sum rules Ž5. and
Ž6.. The extracted results are given in Table 1.
Relatively large errors in the magnetic moments are
found, approaching 100%. But the sign and order of
magnitudes are unambiguous when compared to the
measured values. The situation is consistent with a
previous finding on g A w14x regarding three-point
functions. It is interesting to observe that the distribution of errors is not uniform throughout the Borel
window, with the largest errors at the lower end
where the power corrections are expected to become
more important. The quality of the match deteriorates for Vy in this region, signaling probably insufficient convergence of the OPE. The match for Dqq
is good in the entire Borel region, despite the large
uncertainties. To gain some idea on how the uncertainties change with the input, we also analyzed the
sum rules by adjusting the error estimates individually. We found large sensitivities to the susceptibility
x . In fact, most of the errors came from the uncertainty in x . We also tried with reduced error estimates on all the QCD input parameters: 10% relative
errors uniformly. The results are given in Table 1 as
a second entry. It leads to about 30% accuracy on the
Table 2
Comparisons of magnetic moments from various calculations: this work ŽQCDSR., lattice QCD ŽLatt. w15x, chiral perturbation theory
Ž x PT. w16x, light-cone relativistic constituent quark model ŽRQM. w17x, simple non-relativistic constituent quark model ŽNQM., chiral
quark-soliton model Ž x QSM. w18x. All results are expressed in units of nuclear magnetons.
Baryon
Exp.
QCDSR
Latt
x PT
RQM
NQM
x QSM
qq
4.5 " 1.0
y2.024 " 0.056
4.13 " 1.30
y1.49 " 0.45
4.91 " 0.61
y1.40 " 0.10
4.0 " 0.4
–
4.76
y2.48
5.56
y1.84
4.73
y2.27
D
Vy
18
F.X. Lee r Physics Letters B 419 (1998) 14–18
magnetic moments. Further improvement of the accuracy by reducing the input errors is beyond the
capability of these sum rules as it will lead to
unacceptably large x 2rNDF w12x. For that purpose,
one would have to resort to finding sum rules that
depend less critically on x and have better convergence properties.
A comparison with those from other calculations
and the experimental data is compiled in Table 2.
The results with 10% errors from the QCD sum rule
method are used in the comparison. They are consistent with data within errors, although the central
value for Vy is somewhat underestimated. The result for Dqq is consistent with other calculations,
while for Vy it is closer to lattice QCD calculations.
In conclusion, we have demonstrated that the
magnetic moments of Dqq and Vy can be understood from the QCD sum rule approach, despite
large errors that can be traced to the uncertainties the
QCD input parameters. A 30% accuracy can be
achieved in the derived sum rules with improved
estimates of the QCD input parameters, preferably
on the 10% accuracy level. In particular, large sensitivities to the vacuum susceptibility x are found.
Better estimate of this parameter is needed. Extension of the calculations to other members of the
decuplet appears straightforward, and is under way
w19x. There we hope to address the issues involved in
greater detail.
It is a pleasure to thank D.B. Leinweber for providing an original version of his Monte Carlo analysis
program and for helpful discussions. This work was
supported in part by US DOE under Grant DEFG03-93DR-40774.
References
w1x M.A. Shifman, A.I. Vainshtein, Z.I. Zakharov, Nucl. Phys. B
147 Ž1979. 385, 448.
w2x B.L. Ioffe, A.V. Smilga, Phys. Lett. B 133 Ž1983. 436; Nucl.
Phys. B 232 Ž1984. 109.
w3x B.L. Ioffe, A.V. Smilga, Phys. Lett. B 129 Ž1983. 328.
w4x C.B. Chiu, J. Pasupathy, S.L. Wilson, Phys. Rev. D 33
Ž1986. 1961.
w5x J. Pasupathy, J.P. Singh, S.L. Wilson, C.B. Chiu, Phys. Rev.
D 36 Ž1986. 1442.
w6x S.L. Wilson, J. Pasupathy, C.B. Chiu, Phys. Rev. D 36
Ž1987. 1451.
w7x C.B. Chiu, S.L. Wilson, J. Pasupathy, J.P. Singh, Phys. Rev.
D 36 Ž1987. 1553.
w8x V.M. Belyaev, preprint ITEP-118 Ž1984.; ITEP report Ž1992.,
unpublished.
w9x V.M. Belyaev, preprint CEBAF-TH-93-02, hep-phr9301257.
w10x N.B. Wallace et al., Phys. Rev. Lett. 74 Ž1995. 3732.
w11x A. Bosshard et al., Phys. Rev. D 44 Ž1991. 1962.
w12x D.B. Leinweber, Ann. of Phys. ŽN.Y.. 254 Ž1997. 328.
w13x F.X. Lee, preprint CU-NPL-1147, hep-phr9707332.
w14x F.X. Lee, D.B. Leinweber, X. Jin, Phys. Rev. D 55 Ž1997.
4066.
w15x D.B. Leinweber, T. Draper, R.M. Woloshyn, Phys. Rev. D
46 Ž1992. 3067.
w16x M.N. Butler, M.J. Savage, R.P. Springer, Phys. Rev. D 49
Ž1994. 3459.
w17x F. Schlumpf, Phys. Rev. D 48 Ž1993. 4478.
w18x H.C. Kim, M. Praszalowicz, K. Goeke, hep-phr9706531.
w19x F.X. Lee, in preparation.
12 February 1998
Physics Letters B 419 Ž1998. 19–24
Effects of collective expansion on light cluster spectra in
relativistic heavy ion collisions
Alberto Polleri a , Jakob P. Bondorf a , Igor N. Mishustin
a
a,b
The Niels Bohr Institute, BlegdamsÕej 17, DK-2100 Copenhagen Ø, Denmark
The KurchatoÕ Institute, Russian Scientific Center, Moscow 123182, Russia
b
Received 30 May 1997; revised 3 September 1997
Editor: J.-P. Blaizot
Abstract
We discuss the interplay between collective flow and density profiles, describing light cluster production in heavy ion
collisions at very high energies. Calculations are performed within the coalescence model. We show how collective flow can
explain some qualitative features of the measured deuteron spectra, provided a proper parametrization of the spatial
dependence of the single particle phase space distribution is chosen. q 1998 Elsevier Science B.V.
PACS: 25.75 -q; 25.75 -Ld
Keywords: Relativistic heavy ion collisions; Coalescence model; Clusters; Transverse flow; Effective temperature
In the course of a relativistic heavy ion collision a
hot and dense fireball is created in the interaction
region. Due to the high internal pressure it expands
and cools down, finally disintegrating into hadrons.
The emergence of a collective flow can be considered as a signature that actually an extended piece of
hot and dense matter is formed. Among the products
of the reactions a few light nuclei and antinuclei
have been observedŽsee, for instance, the Quark
Matter ’96 proceedings w1x and references therein., a
very surprising fact for such high collision energies.
A common scenario employed to explain these observations is based on the coalescence model Žfor a
review, see w2x.. A large amount of data on composite particle spectra is accumulated now at intermediate collision energies w3x, where the multifragmentation of nuclei is a most striking phenomenon. Although some trends in the fragment spectra are similar to light cluster spectra in relativistic collisions,
the mechanism of cluster production is in general
very different in these two energy domains. In particular, the coalescence picture does not really work for
intermediate mass fragments.
In this letter we examine the effect of collective
expansion on the final spectra of clusters, assuming
that they are produced via coalescence. We also
underline how their measurements can give an additional information on the latest stages of a relativistic
nuclear collision.
In the attempt of describing light cluster production in heavy ion reactions at very high energies, one
encounters a somewhat subtle problem. Because these
composite objects, typically d, d, t and 3 He, are very
loosely bound, they can only be formed at the very
late stage of the reaction. This is because the system
is then quite dilute and interactions with the environment are therefore rare, preventing the formed clusters from breaking-up. On the other hand, it is
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 5 5 - X
A. Polleri et al.r Physics Letters B 419 (1998) 19–24
20
known that light nuclei cannot be formed by scattering nucleons in free space, even when the process is
gentle enough, simply as a consequence of energymomentum conservation. The formation of a bound
state requires the presence of a third body which
carries away an amount of energy equal to the
cluster binding energy. It is also clear that the system
cannot be arbitrarily dilute and there must be a
density around which the formation process is optimized.
In the present study we assume that the production process is governed by two distinct factors. The
early stages of the reaction and therefore the way
particles are produced and emitted are parametrized
via a many-body phase space distribution. This is
what we call source function and it represents the
probability that A nucleons are emitted at a given
phase space point. The source function is taken at a
sufficiently late time, such that the conditions previously discussed are fulfilled. The probability that
these particles form a bound state, is taken as the
overlap between the cluster and the A-nucleon wave
functions. This framework has been quite commonly
used and it is well described in w4x.
Assuming that nucleons are emitted uncorrelated,
one can factorize the generic A-body distribution as
a product of single particle ones. Denoting a phase
space point and the corresponding measure as
3
x i s Ž r i , pi . ,
dx i s
3
d r i d pi
Ž 2p .
3
,
i s 1, . . . , A ,
Ž 1.
we can write the phase space distribution of mass-A
clusters as
A
fAŽ r , p. s
H is1
Ł dx
i
f Ž x i . PA Ž x 1 , . . . , x A ;r , p . .
Ž 2.
This formula expresses the fact that, among all AA
particle states, represented by Ł is1
f Ž x i ., some can
become bound with a probability PA . The integration
goes over all phase space points, where particles are
emitted. The formation probability PA is obtained by
squaring a corresponding quantum-mechanical amplitude, as done in w5,6x. Below we adopt an approximation motivated by comparing the ranges of varia-
tion in phase space of two factors, the single particle
distribution f and the formation probability PA .
Obviously the first quantity has a much bigger range
of variation than the second one, especially when
considering a large and hot system. This allows us to
set r i s r and pi s prA, for i s 1, . . . , A, and writing the general formula for the phase space distribution of mass-A clusters in the form w7x
f A Ž r , p . s f Ž r , prA .
A
.
Ž 3.
This expression is the starting point of our subsequent analysis.
Let us now specify the shape of the nucleon
distribution in phase space. We assume that the
system is in local thermal equilibrium, characterized
by a temperature T0 , considered to be constant
throughout the whole fireball at the freeze-out stage
of the reaction. We also assume that particles are
subject to a collective velocity field, often also named
collective flow. At very high energies it is generated
by the partial transparency of nuclei, along the longitudinal Žbeam. direction, and by the pressure created
in the hot overlap zone, in the transverse direction.
Since the dynamics in these two directions is very
different, we disregard possible correlations and represent the collective velocity field as a sum of two
independent contributions,
z Ž r . s zLŽ r L . q zT Ž r T . ,
Ž 4.
where r L s z e z and r T s x e x q y e y . The nucleon
momenta in a local frame k obey a thermal distribution with temperature T0 . The transformation to a
global frame is made with a boost of velocity zŽ r ..
It is well known that the longitudinal dynamics is
highly relativistic, while the transverse expansion, in
the first approximation, can be considered non-relativistic, at least for nucleons. Therefore, the nucleon
momentum in the global frame can be written as the
sum of thermal and flow components as p T s k T q
m zT Ž r T .. In the following discussion we ignore all
issues related to the longitudinal dynamics, focusing
attention on the transverse direction. The transverse
velocity field is parametrized as
zT Ž r T . s Õ f
rT
ž /
R0
a
eT ,
Ž 5.
where Õ f and R 0 are the strength and scale parameters of flow and the power-law profile is character-
A. Polleri et al.r Physics Letters B 419 (1998) 19–24
ized by the exponent a . In building the phase space
distribution, we follow Ref. w8x. Assuming cylindrical symmetry we represent the nucleon density in a
factorized form
r Ž r . s N n LŽ r L . nT Ž r T . ,
Ž 6.
such that n L and nT are normalized to 1 in the
respective domains. Using Ž3. we can now calculate
the cluster phase space distribution function. Because
of Ž4. and Ž6., it also factorizes into longitudinal and
transverse parts, namely
f A Ž r , p . s f L Ž r L , pL . f T Ž r T , p T . .
Ž 7.
The transverse contribution for clusters of mass
number A is therefore given by
1
2
f T Ž r T , p T . s Ž 2p .
Ž 2p MT0 .
= ey
Ž p T yM z T . 2
nAŽ rT . ,
2 M T0
Ž 8.
where M s A m is the cluster mass and
n A Ž r T . s NT Ž A . nT Ž r T .
A
Ž 9.
is the transverse part of the cluster density, with the
normalization factor NT Ž A.. Position and momentum
of particles, completely uncorrelated in a purely
thermal system, are now partially linked due to the
presence of collective flow.
The transverse momentum spectrum of clusters is
obtained by integrating expression Ž7. over the whole
volume and around a particular value pL of the
longitudinal momentum 1. The p T-spectrum for clusters of mass number A can be written in the form
dNA
< s nA Ž pL . SA Ž p T . ,
Ž 10 .
dpT2 pL
where nAŽ pL . is the total number of clusters of mass
A produced at pL and
SA Ž p T . s d 2 r T
H
1
Ž 2p MT0 .
ey
Ž p T yM z T . 2
2 M T0
nAŽ rT .
Ž 11 .
is the p T-dependent part of the momentum spectrum.
This last factor is quite interesting. If flow were
1
In a relativistic formulation, a more familiar notation in terms
of rapidity y instead of pL would appear, without anyway
affecting our discussion on transverse spectra.
21
absent, the integral would give directly the Boltzmann factor, but the present case is, in general, more
complicated, and a numerical treatment is needed.
The most common parametrization used in the literature combines a gaussian profile for the nucleon
density w9x with a linear profile Ž a s 1. for collective
flow Žsee w8,10x, especially in relation with source
parametrizations in interferometry studies.. Only this
choice allows for an analytical solution, which is the
Boltzmann distribution
SA Ž p T . s
1
Ž 2p MT) .
p T2
ey 2 M T ) ,
Ž 12 .
but now with the modified effective temperature
Žslope parameter.
T) s T0 q m Õ f2 .
Ž 13 .
At first sight, this result looks appealing, but it
actually contradicts both intuition and experiment.
What is wrong in the previous expression is the
dependence of the slope T) only on m but not on
M, as one would expect also by looking at the slopes
extracted from measured spectra w11x. When performing the integral in Ž11., one notices an interesting feature. From Ž9. one sees that the density of
clusters of mass A is proportional to the A-th power
of the nucleon density. In the case of a gaussian
profile, one can see that the A-cluster density shrinks
towards the central region. This is easy to understand, since it is clearly more probable to make a
cluster where there are many particles than on the
tail, where there are only a few. Together with this,
we choose a linear flow profile. This is parametrized
in Ž5. defining Õ f as the flow strength at the surface
of the density distribution, characterized by the scale
parameter R 0 . What happens with the gaussian profile is that the actual size of the cluster density has a
smaller radius, thereby picking up a smaller value for
the flow velocity at the surface, as compared to the
case of single nucleons. This effect exactly cancels
the A-dependence of M in Ž13.. The other extreme
would be to take a uniform density with a sharp
surface at a given radius. Any power of this function
would give the same profile, with the same radius. In
other words it is equally probable to have clusters
everywhere in the region with non-zero density. As a
consequence we expect in this case that the slope
parameter will depend on M, since the flow velocity
A. Polleri et al.r Physics Letters B 419 (1998) 19–24
22
at the surface is the same for all clusters. This is not
the whole story. The flow profile could have a
smaller exponent Ž a s 1r2, for example.. Indications of such a behaviour have been observed in
microscopic models of heavy ion collisions such as
RQMD w12,13x. Now some dependence of T) on M
would appear, even for a gaussian density profile.
Let us now look more closely at the interplay
between flow and density profiles. It is clear that
they cannot be considered independently because the
density shape at a given time during expansion is the
result of the particle motion characterized by the
collective velocity field. The information about the
profiles of density and collective velocity, can be
extracted, in principle, from the energy spectra of
different clusters w3x. Unfortunately this is not an
easy task because of the sensitivity of energy spectra
to all kinds of corrections w14,15x. We prefer a more
global analysis where the effective temperature Žslope
parameter. is extracted from the A-dependence of
the mean transverse energy 2
2
² ET :A s
H
2
d r T d pT
Ž 2p .
2
p T2
2M
f T Ž r T , pT . .
Ž 14 .
For a classical Boltzmann gas at temperature T0 we
have ² ET :A s T0 , where the usual factor 3r2 has
changed to 2r2 since we consider only the transverse degrees of freedom. In the present case we
have instead
² ET :A s ² ET :thA q ² ET :flA ,
Ž 15 .
where the first term corresponds to the purely thermal, Boltzmann gas, while the second contribution
arises due to the presence of flow ŽIt vanishes if we
set Õ f s 0.. We define the effective temperature via
T) s ² ET :A s T0 q Law nx Ž A . M Õ f2 .
Law nx Ž A . s 12 d 2 r T
H
rT
ž /
R0
2a
nAŽ rT . .
Ž 17 .
It can be calculated analytically for the two interesting cases of gaussian and box profiles for the density, for all values of a . In the first case one obtains
LaGauss Ž A . s
2 ay1
Aa
G Ž a q 1. ,
Ž 18 .
where G is Euler’s Gamma function. One can readily see that for a linear flow profile Ž a s 1. the
coefficient is equal to 1rA and it exactly cancels the
A factor carried by M in Ž16.. For lower powers of
a the situation changes one maintains a weak A-dependence. In the case of a box profile we obtain
LaBox Ž A . s
1
2aq2
,
Ž 19 .
independent of A, as we expected after our previous
discussion. Choosing parameters according to Table
1, we illustrate the results in Fig. 1, where the
effective temperature is plotted as a function of mass
number A. The higher curves represent the extreme
case of a box-shaped density which gives the
strongest A-dependence of T) . The other extreme,
as pointed out previously, is the gaussian density
with linear flow, which gives an A-independent T) .
Fig. 1 suggests that the choice of a gaussian profile
for the density and a flow profile with a s 1r2 give
the best agreement with the measured values of the
slopes.
There is another interesting feature regarding cluster spectra, which can be experimentally measured.
Ž 16 .
The coefficient in front of the flow term depends on
the flow parameter a , is a functional of the transverse density and is a function of the cluster mass
number A. Different choices of density and flow
2
profiles will result in a different A-dependence. The
explicit expression for this coefficient is
This approach was further developed in w16x, including in the
analysis also the variance of the energy distribution.
Table 1
Parameters used to perform the calculations. They are not chosen
in order to fit the data but only to show the qualitative beaviour in
the following figures.
Density
a
Õf
T0 ŽMeV.
R 0 Žfm.
Box
1r2
1
0.63
0.72
140
140
8
8
Gauss
1r2
1
0.48
0.34
120
140
8
8
A. Polleri et al.r Physics Letters B 419 (1998) 19–24
23
In the early days of heavy ion physics the proportionality relation between cluster spectra and the
corresponding powers of single particle spectra was
quite well established w2x. In recent experiments at
much higher energies a momentum dependence in
the proportionality constant BA was observed w17x.
Namely, BA increases with increasing transverse momentum. In the present analysis we compare the
transverse momentum spectrum of clusters of mass
number A with the Ath power of the single particle
spectrum,
A
dNA
dpT2
< p s bA Ž pL , p T .
L
dN
pT
ž / 0
d
<
2 pL
,
Ž 20 .
A
A
where the first factor does not coincide with the
usually quoted BA because it is calculated for a
small window around pL . We therefore discuss the
p T-dependence in this factor. Again, we perform the
calculations with box and gaussian profiles, taking
a s 1, 1r2. Using Ž10. we obtain
bA Ž pL , p T . s c A Ž pL .
SA Ž p T .
SA Ž p TrA .
A
,
Ž 21 .
where c A is a normalization factor which gives the
order of magnitude of bA , but does not affect its
p T-dependence. In Fig. 2 we show various plots of
b 2 , using the parameters from Table 1. Although
microscopic simulations are able to more or less
Fig. 1. Effective temperature T) as a function of mass number A.
The top curve for each choice of density profile corresponds to
a s 1r2, while the lower is for a s 1. The straight lines could
perfectly coincide if a more accurate choice of Õ f were made.
This is evident from Ž16., due to the A-independence of Law nx for
the box profile.
Fig. 2. b 2 factor as a function of transverse momentum. The top
curve for each choice of density profile corresponds to a s 1r2,
while the lower is for a s 1. The curve labelled ‘‘Surface’’
corresponds to integration over a spherical shell from R 0 r2 to
R 0 , in order to simulate surface emission.
reproduce this feature w18x, it is instructive to understand how it arises within the simple picture presented above. This behaviour is a pure manifestation
of collective flow, which only cluster measurements
can reveal. This effect depends on the relation between flow and density as we discussed above,
resulting in turn in different shapes of momentum
spectra for clusters and single nucleons. We emphasize again that the linear velocity profile and the
gaussian shape for the density distribution are in
contradiction with the p T-dependence of bA ŽIn this
case both cluster and single particle spectra have the
same slope.. Also the choice a s 1r2 does not help
much, suggesting that a better understanding of the
density shape is necessary. Therefore we indicate in
Fig. 2 that surface formation of clusters, at a slightly
earlier time with respect to the complete disintegration of the system, could improve our scenario. This
is done by performing the spatial integration over a
spherical shell from R 0r2 to R 0 and is equivalent to
having a density with a depleted central region, as
suggested in w19x. The actual situation is clearly a
combination of this early surface emission and final
bulk disintegration and a consistent implementation
of this aspect, together with a proper description of
time evolution, is the subject of our current study.
In summary, we have shown that a suitable implementation of collective flow can account for important qualitative features of light cluster spectra, measured in heavy ion collisions at very high energies,
even though more has to be done to build a consis-
24
A. Polleri et al.r Physics Letters B 419 (1998) 19–24
tent and quantitative description of the late expansion stage. The observed A-dependence of the slope
parameters and the p T-dependence of the coalescence coefficients impose serious constraints on the
spatial profiles of the collective velocity and the
particle density at the freeze-out stage. The most
common parametrizations for both flow and density
profiles fail to reproduce these features. Quantitative
conclusions will be possible in the near future when
cluster spectra for large and symmetric collision
systems will be available.
Acknowledgements
We would like to thank Ian Bearden and the
experimental heavy ion group at the Niels Bohr
Institute for the pleasant collaboration. This work
was supported in part by I.N.F.N. ŽItaly., the Carlsberg Foundation ŽDenmark. and the EU-INTAS grant
94-3405.
References
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12 February 1998
Physics Letters B 419 Ž1998. 25–29
Evidence for Dy components in nuclei
C.L. Morris a , J.D. Zumbro a,b, J.A. McGill a,1, S.J. Seestrom a , R.M. Whitton a ,
C.M. Reidel a,c , A.L. Williams d,2 , M.R. Braunstein e,3, M.D. Kohler e,4 , B.J. Kriss e,
S. Høibraten
˚ e,5, J. Ouyang e,6 , R.J. Peterson e, J.E. Wise e,7
a
b
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
MIT-Bates Linear Accelerator Center, Middleton, MA 01949, USA
c
UniÕersity of Minnesota, Minneapolis, MN 55455, USA
d
UniÕersity of Texas at Austin, Austin, TX 78712, USA
e
UniÕersity of Colorado, Boulder, CO 80309, USA
Received 22 November 1996; revised 5 November 1997
Editor: J.P. Schiffer
Abstract
Ratios of double-differential cross sections for incident 500-MeV pions are presented for Žpq,pqp. and Žpq,pyp.
reactions on 12,13 C, 90 Zr, and 208 Pb. A comparison with intra-nuclear cascade-model calculations suggest that the outgoing
pion spectra show a feature consistent with quasi-free scattering from the Dy component of the nuclear ground state wave
function. q 1998 Elsevier Science B.V.
PACS: 25.80.Hp; 24.80.-x; 21.65.q f
Meson models of nuclear binding predict both
excess pions and excited nucleons in nuclear matter
1
Present address: MS 9000, SSCL, 2550 Beckleymeade Ave,
Dallas, TX 75237.
2
Present address: The Johns Hopkins Oncology Center, Division of Radiation Oncology, Baltimore, MD 21287.
3
Present address: Department of Physics, Central Washington
University, Ellensburg, WA 98926.
4
Present address: Department of Physics, Bridgewater State
University, Bridgewater, MA 02324.
5
Present address: FFIVM, P.O. Box 25, Kjeller, Norway N2007.
6
Present address: Department of Physics, Boston University,
Boston, MA 02215.
7
Present address: Department of Physics, Northwestern University, Evanston, IL 60208.
w1x. Recently, the absence of an anti-quark excess
due to excess pions, inferred from Drell-Yan experiments w2x, has cast doubt on the validity of this
model. In the present paper we suggest a new method
for measuring the the Dy component of the nuclear
wave function, and present results which are in
agreement with the predictions of meson models of
nuclear binding and apparently contradict this interpretation of the Drell-Yan results.
In the 1960s, theoretical investigations w3,4x first
suggested nucleon resonances might play a role in
nuclear structure. Although calculations that use the
D-N mass difference and the size of nuclear-matrix
elements as their inputs have suggested virtual Ds
should exist in the nuclear-wave function at the
few-percent level w5,6x, they have proven difficult to
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 5 3 - 6
26
C.L. Morris et al.r Physics Letters B 419 (1998) 25–29
detect despite significant experimental effort. A measurement of these wave-function components would
provide important constraints on models of nuclear
binding and on the nature of short-range interactions
in the nuclear medium w1x.
Experimental signatures of virtual Ds are difficult
to find for at least two reasons. First, since virtual
Ds are made in NN collisions, it is reasonable to
expect the probability for finding Ds to go up as the
nuclear density squared. However, most strongly-interacting probes scatter before reaching centralnuclear densities and thus are not sensitive to the
peak probabilities. Second, most searches have relied
on detecting knocked-out Ds. In these experiments it
has proven difficult to distinguish Ds that are
knocked out of the nucleus from those that are
created in the reaction w7,8x.
A recent experiment looking for the Dqq probability in 9 Be has reported a positive result by detecting the residual 8 He where the the Dqq s were
knocked out with a 1-GeV proton beam w9x. Backgrounds from nucleon knockout were eliminated by
the change of two units of charge associated with the
knockout of one baryon. Knockout of a D was
distinguished from D creation by analyzing the momentum spectrum of the residuals. A distinctive
low-momentum component of the spectrum was associated with Dqq knockout. Unfortunately, this
technique only measures the small fraction of the
Dqq probability in which the Dqq is coupled to
particle-stable states of the residual nucleus, and
cannot be used in heavy nuclei because of the difficulties in detecting the heavy recoil. Although several new signatures have been suggested using both
pion w10x and electron beams w11,12x, the utility of
these techniques has not yet been demonstrated.
In this Letter we propose a more general method
of measuring D wave-function components, by using the reaction DyŽ pq,py .p as a probe to search
for Dys in nuclei, and we show some initial results
suggesting a positive signal using this method. Some
diagrams that can contribute to the DCX reaction are
shown in Fig. 1. If only nucleonic degrees of freedom are considered, a featureless outgoing-pion energy spectrum is expected because of the washed-out
kinematic signatures of a two-step reaction. This is
the process shown in Fig. 1a. However, if one allows
pionic and delta degrees of freedom in the nucleus,
Fig. 1. Diagrams that can contribute to the Žpq,py p. reaction.
other possibilities emerge. Specifically, since the
isospin of the D is 3r2, it exists in four charge
states. The beam pq can transfer two units of
charge to the Dy in a single step as shown in Fig.
1c. This leads to the expectation of a significant
enhancement in the cross section for DCX on delta
components of the nuclear-wave function with respect to the two-step background. In this case the
spectrum is expected to show the quasi-free ŽQF.
signature of two-body scattering, i.e., a broad peak
near v s q 2r2m, where v is the energy loss in the
reaction, q is the momentum transfer, and m is the
mass of the recoiling particle, since all of the momentum is transferred to one particle. The recoiling
particle is expected to be emitted near the angle
expected from free kinematics, but this angle will be
smeared due to the initial momentum distribution of
the D.
The suggestion of using DCX as a probe of Ds in
the nuclear wave function is not new. Previously,
DCX on Ds was suggested as a mechanism that
might explain the unexpected large cross sections for
resonance energy DCX between non-analog 0q states
w13x. Subsequently, it was pointed out that in first
order the spinless pion cannot cause the double-spin
flip necessary at 08 in the absence of distortions w14x.
C.L. Morris et al.r Physics Letters B 419 (1998) 25–29
And indeed calculations of D charge-exchange scattering from the nucleus show it is a more likely
process w14x.
Coincident protons from the Žpq,py p. DCX reaction at resonance energies were measured in an
attempt to improve the signal-to-noise ratio by using
the angular correlation of QF scattering. No signature of Ds was observed w15x. In the present work a
higher-energy pion beam Ž500 MeV. is used, so that
the mean free path is longer and higher densities are
probed. The larger coupling to the D provided by the
N ) component of the interaction is also expected to
increase the D signal at 500 MeV. Additionally, the
energies of both the pion and the proton are measured so that missing-energy cuts can be used to
reduce the second-order backgrounds.
The experiment was performed using the P 3 East
pion channel at the Clinton P. Anderson Meson
Physics Facility. Scattered pions from both the
Žpq,pq p., NCX, and Žpq,py p., DCX, reactions
were measured using the LAS w16x spectrometer at
angles of 408 and 508. Protons were detected in a
plastic-BGO phoswich detector mounted in the direction of the recoil assuming free p-p scattering. Protons into an angular cone of about 78 were detected
with a solid angle of about 50 msr. This angular cut
accepts a full width in transverse momentum of
Fig. 2. Top: Spectra of the time difference Ž D t. between the
phoswich detector and a pion detected in the spectrometer. Left:
NCX; Right: DCX. Bottom: Energy ŽE. Õs. energy loss Ž D E. in
the phoswich detector gated by D t. The total kinetic energy is the
sum of E and D E. The most prominent band in E-D E corresponds
to protons. Protons were identified using cuts on both Ž D t. and
E-D E.
27
Fig. 3. Pion energy-loss spectra measured with an incident energy
of 500 MeV and an outgoing angle of 508. Solid: inclusive;
dotted: gated by coincident protons; chaindash: gated by coincident protons and less than 30 MeV of missing energy. The solid
red histograms are the result of INC calculations described in the
text. Left: NCX; Right: DCX multiplied by 10.
about 60 MeVrc for protons emitted at the QF peak.
A clear signature of coincident protons was obtained
by measuring time differences between the LAS and
the proton detector ŽTOF., and by using the energy
and energy-loss ŽE-D E. signals from the phoswich
detector. The quality of the particle identification is
shown in Fig. 2, where TOF and E-D E spectra are
displayed for both NCX and DCX. The phoswich
detector allowed measurement of proton energies up
to 200 MeV. Above this energy the protons passed
through the detector and reliable particle identification could not be made.
The result of requiring the detection of protons at
the free-recoil angle in coincidence with pions on the
pion-energy loss spectra is shown in Fig. 3. Here
NCX spectra are shown on the left and DCX spectra
are shown on the right: with no cuts; with a coincident proton; and with the additional requirement of
less than 30 MeV of missing energy in the residual
nucleus. The cuts can be seen to reduce the background under the QF spectrum in the NCX spectra
by nearly a factor of 100. In the DCX spectrum a
peak in pion-energy loss results when low-missing
energies are required. The choice of a 30-MeV missing energy cut was driven by the experimental observation that larger missing energy cuts increased the
background under the energy loss energy peak while
28
C.L. Morris et al.r Physics Letters B 419 (1998) 25–29
not significantly increasing the cross section. The
presence of a peak near the QF energy in the DCX
pion energy-loss spectrum Žnote the semi-logarithmic
scale. is suggestive of the proposed QF D-knockout
mechanism.
Some other potential sources of this signal are
shown in Fig. 1a ŽSEQ. and Fig. 1b ŽDINT.. The
relative scales of the central parts of these diagrams
can be established by comparing forward-angle
ground-state DCX cross sections on T s 0 and T s 1
isotopic pairs such as 16,18 O ŽRefs. w17x.. Here the
DINT process is expected to be an important contribution to the cross section on the T s 0 target,
whereas SEQ is expected to dominate the cross
section on the T s 1 target w14x. In contrast with the
nearly equal cross sections at resonance energies
w18x, the T s 1 cross section is more than 10 times
larger than the T s 0 cross section at 500 MeV. This
large ratio leads to the conclusion that the central
SEQ interaction is much stronger than DINT at 500
MeV.
Fig. 3 shows a comparison of the coincidence
DCX and NCX cross sections with results of a
calculation Žred histogram. using an intranuclearcascade code w19x. This model has been shown to
give a good description of inclusive scattering of
500-MeV pions from carbon w20x and should provide
a good description of both incoherent SEQ and
quasi-free pion-production mechanisms. The calculation has approximately the same missing energy,
coplanarity, and opening angle cuts as the experimental data. The cross section scale has been adjusted by normalizing the calculation to fit the exclusive-NCX data. The model gives a good description
of the shape of the NCX-coincidence cross sections
but under predicts the DCX-cross section by more
than a factor of 5 in the region of interest.
We have measured cross sections for the
Žpq,py p. reaction on the isotopic pair 12,13 C in an
attempt to evaluate the importance of the SEQ process, which couples a coherent-SCX reaction with
QF SCX. This process would not be predicted by the
semi-classical calculation described above. We emphasize the need for a coherent SCX in order to
maintain the two-body correlation expected of QF
scattering. If this mechanism were important, one
would expect to see a large difference in cross
section Žan order of magnitude. on this isotopic pair
Table 1
Measured cross section ratios and extracted values of PD
Target
12
C
C
12
C
90
Zr
208
Pb
13
Angle
Ždeg.
s DC X r s NCX
Ž=10 3 .
PD
Ž%.
40
40
50
50
50
0.97Ž17.
1.64Ž16.
2.5Ž4.
2.8Ž9.
2.7Ž10.
1.5Ž3.
1.9Ž2.
3.1Ž6.
2.3Ž8.
1.7Ž6.
because in 13 C there is an intermediate-analog state
available for coherent SCX, whereas there is not in
12
C. In the data displayed in Table 1 one can observe
an enhancement in 13 C of only a factor of 1.7Ž4.,
consistent with the number of Dys expected, as
shown below.
An estimate of the probability for preexisting Ds
can be obtained from the data by integrating the
cross sections in the peak and making the ratio Ž R .
between DCX and NCX cross sections. This should
cancel distortion factors and leaves:
R s PD
ANDy
s Ž pqq p ™ pqq p .
ZND
s Ž pqq Dy™ pyq p .
ž /ž
kD
/ž /
kF
2
.
Ž 1.
where PD is the sum of the probabilities for all D
charge states, NDyrND is the number of Dys over
the sum of Ds in all charge states, Z and A are the
proton and nucleon numbers, and account for the
number of protons Žmeasured in NCX. per nucleon,
kD characterizes the momentum spread of the D part
of the wave function, and k F is the momentum
spread of the nucleons. Arguments given in w11x have
been generalized from A s 3 to give:
NDy
ND
3
s
4
ž
1q
Z
q
Ny1
Z Ž Z y 1.
N Ž N y 1.
y1
/
,
Ž 2.
where N is the neutron number of the nucleus. The
relative probabilities for each of the D-charge states
are assumed to be given by sums of isospin Clebsch–Gordan coefficients multiplied by the appropriate number of T s 1 paired nucleons: N Ž N y 1.,
NZr2, and ZŽ Z y 1. for proton-proton, neutron-proton, and proton-proton pairs, respectively. Finally, an
additional factor arises from the larger-momentum
spread expected for virtual-D components of the
C.L. Morris et al.r Physics Letters B 419 (1998) 25–29
wave function, kD , when compared with nucleons
k F . We have used measured branching ratios w21x for
N ) Ž1520. ™ p q N and N ) Ž1520. ™ p q DŽ1232.,
corrected for two-body phase space with the appropriate isospin Clebsch-Gordan coefficients to estimate the cross-section ratio in Eq. Ž1.. In the absence
of any better knowledge we have assumed both cross
sections have the same angular dependence. The
increased momentum spread decreases the acceptance for p-p coincidences as Ž kDrk F . 2 . The value
of k F has been taken to be 200 MeVrc and kD has
been taken to be 400 MeVrc, the peak of the
momentum distribution predicted for the pion excess
calculated in ŽRef. w1x.. Both of these are large when
compared with the transverse momentum acceptance
of the experiment of about 60 MeVrc.
The measured cross section ratios extracted using
a 30-MeV missing energy gate and resulting D
probabilities for all targets and angles are given in
Table 1. The fluctuations of a factor of two in the
extracted PD are likely to be a reflection of uncertainties in extracting PD from the data. We see that
the extracted values of PD are somewhat smaller
than the expected 4% w1x. This may reflect an error
in the assumed momentum distribution or relatively
more D than nucleon spectral strength extended
beyond the 30-MeV missing energy cut used in
extracting the cross sections.
In conclusion, data for the Žpq,py p. reactions on
12,13
C, 90 Zr, and 208 Pb shown an enhancement in the
kinematic region appropriate for a two body reaction
which is not predicted by INC calculations. We
suggest this is due to quasi-free knockout of preexisting Dqq and that this reaction may provide a new
technique for measuring virtual D components of the
nuclear wave function. The resulting probabilities are
demonstrated to be near the values expected from
pionic models of nuclear binding.
29
Acknowledgements
We would like to thank Professor W.R. Gibbs,
Dr. G.T. Garvey, and Dr. M.B. Johnson for helpful
discussions. This work has been supported in part by
the United States Department of Energy and the
Robert A. Welch Foundation.
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12 February 1998
Physics Letters B 419 Ž1998. 30–36
Isolation of the true degree of freedom and normalizable
wave functions for the general type V cosmology
T. Christodoulakis, G. Kofinas, E. Korfiatis, A. Paschos
Nuclear and Particle Physics Section , Physics Department, UniÕersity of Athens, Panepistimiopolis, Athens 15771, Greece
Received 28 April 1997; revised 23 October 1997
Editor: R. Gatto
Abstract
The quantization of the most general Type V geometry Žwith all six scale factors as well as the shift vector present. is
considered. The information carried by the linear constraints is used to reduce the Wheeler–DeWitt equation Žarising from a
valid Hamiltonian found earlier., which initially included six variables, to a final PDE in three variables, getting rid of three
redundant variables Žgauge degrees of freedom.. The full space of solutions to this equation is presented. In trying to
interpret these wave functions, we are led through further consideration of the action of the automorphism group on the
configuration space, to a final reduction to the one and only true degree of freedom, i.e. the only independent curvature
invariant of the slice t s constant. Thus, a normalizable wave function in terms of the true degree of freedom is obtained.
q 1998 Elsevier Science B.V.
1. It is well known that in order to quantize
gravity in a non-perturbative way one has to realize
the following steps w1x:
Ži. define the basic operators gˆmn Ž x,t ., pˆ mn Ž x,t .
and the canonical commutation relations they satisfy;
Žii. define quantum operators HˆmŽ x,t . whose classical counterparts are the constraint functions
HmŽ x,t .;
Žiii. define the quantum states C w g x as the common null eigenvectors of HˆmŽ x,t ., i.e. these satisfying HˆmC w g x s 0. As a consequence one has to
check that HˆmŽ x,t . form a closed algebra under the
basic Canonical Commutation Relations ŽCCR.;
Živ. find the states and define the inner product in
the space of these states.
It is fair to say that the full program has not yet
been carried out, although partial steps have been
made w2x.
In this work we apply these steps to the Bianchi
Type V homogeneous cosmological model and not
to the full quantum gravity. Since this way we only
quantize a finite number of degrees of freedom we
are in the quantum cosmology approximation. The
difference of the present work from previous investigations on quantum cosmology is twofold: on one
hand, usually only up to three gravitational degrees
of freedom Žchosen in some sense arbitrarily. were
considered at the classical level Žsay, the three scale
factors of some anisotropic Bianchi type model. and
thus quantized, while we allow for all six gab Ž t . Žthe
time components of the spatial metric with respect to
the invariant basis one-forms., as well as the shift
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 1 9 - 6
T. Christodoulakis et al.r Physics Letters B 419 (1998) 30–36
vector N a Ž t . to appear at the classical level, thus
considering the quantization of the most general
Bianchi Type V model; on the other hand the canonical treatment of the Type V model Žother than FRW
with k s y1. has been inaccessible in the past for
Quantum Cosmology, since it belongs to the Class B
cosmologies having no known valid Lagrangian
andror Hamiltonian. The construction of such a
valid Hamiltonian w3x allows us to fully quantize the
model. An analogous treatment has recently been
given for the general Type II cosmology w4x, but the
non-scalar character of the known Lagrangian has
prohibited us from isolating the corresponding one
true degree of freedom, i.e. the only curvature invariant.
2. The Type V isometry group is characterised by
1
1
2
2
the structure constants C13
s yC31
s 1, C23
s yC32
s 1 Žall others are zero..
Classically, Einstein’s Field equations follow from
the Hamiltonian H s N m Hm q N 0 H0 , According to
w3x the linear constraint functions HmŽ t . for the Class
B Bianchi Type models should be given by Hm '
a
Cbm
ga r p br . Consequently in Type V they become:
H1 s yg 1 m p 3 m , H2 s yg 2 m p 3 m , H3 s g 1 m p 1 m q
g 2 m p 2 m . ŽObviously greek indices run from 1 to 3..
The quadratic constraint is given by H 0
s 12 Qabgd p abp gd q RrF , where
gˆab ,pˆ mn 4 s damnb s 12 Ž damdbn q dan dbm . .
t
s
F s 18 Ž Cta
Csb
g ab . ,
r
Qabgd s 21 F 1r3 Ž Cuzz C´b
ga r g ´uggd
r
qCuzz C´a
gb r g ´uggd q Cuzz C´dr ggr g ´uga b
r
qCuzz C´g
gdr g ´uga b . q Ž gag gbd q ga d gbg . F
y 73 gab ggd F
q
2 Cmad
Cnda g mn
Ž 1.
Žii. We define the quantum counterparts of the
classical constraints Hm and H0 as w6x:
Hˆ1 s yg 1 m
Ĥ3 s g 1 m
2
and R s Clbm Cuat ga b g ulg tn
m
b nl
4Cmn
Cbl
g .
From a geometric point of view one is thus led to
identify these spacetime solutions of Einstein’s equations. Therefore, the number of degrees of freedom
for a homogeneous spacetime must be determined by
the space of solutions to Einstein’s vacuum equations modulo the spatial diffeomorphisms which map
each group invariant triad to itself modulo a possible
global Ži.e. position independent. linear mixing. As
is well known, the above given linear constraints
generate such spatial diffeomorphisms w5x. For most
Bianchi Types there exist additional generators of
such spatial diffeomorphisms to be discussed later.
Hence, at the moment, we expect at the classical
Žand consequently at the quantum. level a reduction
of the degrees of freedom Žvariables. from 6 Žthe six
gab . to 3 Žsince the constraints are 3..
We now realize the previously mentioned steps:
Ži. We define as basic operators gˆab s ga b , pˆ ab
s yi EgEab satisfying the Canonical Commutation Relations:
E
Eg 3 m
,
Hˆ2 s yg 2 m
E
3
31
Eg 1 m
E
Eg 3 m
,
E
qg2m
Eg 2 m
.
Ž 2a .
ŽOne may also consider them to be vector fields
generating on the configuration space generating automorphic motions as already mentioned..
Care must be taken so that the derivative with
respect to non diagonal gab ’s be multiplied by a
factor 1r2, as it is evident from Ž1..
Ĥ0 s y 12 =c q RrF ,
Ž 2b .
q
It is critical for this work to note that in the
geometric description of general relativity two spacetimes are regarded as physically indistinguishable if
they are isometric, i.e. if there exists a diffeomorphism from one spacetime to the other, which maps
the metric of the first to the metric of the second.
where =c s '1Q EgEab 'Q Qa bgd EgEgd q 15 R , is the conformal Laplacian, w7x, corresponding to our choice of
Ž t s a b . is the Ricci
factor ordering, R s 15
32 Cta Csb g
scalar of the supermetric Q abgd and Q s < Q abgd <.
Žiii. The quantum states c w gab x are defined as the
common null eigenvectors of the quantum constraints Ž2a., Ž2b.. It is straightforward to check that,
3
2
32
T. Christodoulakis et al.r Physics Letters B 419 (1998) 30–36
given Ž1. the quantum constraints Ž2. satisfy the
following commutation relations:
Hˆ0 , Hˆ0 s 0
Ž 3a .
Hˆ0 , Hˆa s 0
Ž 3b .
Hˆa , Hˆb s y2Cagb Hˆg .
Ž 3c .
The first of these is trivial in the Žfinite number of
degrees of freedom. quantum cosmology approximation; it is only included to remind us of the analogous non-trivial relation of full quantum gravity.
Eqs. Ž3c. are precisely the integrability conditions for
the system of linear Eqs. Ž2a. to possess non-zero
solutions. Using the Jacobi identities one can easily
see that every scalar Ži.e., having all the frame
indices contracted. combination of gab Ž t . and Cagb
is annihilated by the linear constraints. In three
dimensions there are only up to three independent
combinations of the kind w8x.
Živ. Our first task is to solve the system of the
linear constraints for C :
ĤmC s 0 .
Ž 4.
As mentioned above, Eqs. Ž3c. guarantee the existence of solutions to these equations; in fact, Frobenious theorem informs us that the general solution to
Ž4. will be an arbitrary function C of three combinations of gab Ž t .. Using the method of characteristics
w9x we find that C is of the form
The next thing to do is solve the quadratic constraint equation for C :
Ĥ0C s 0 .
Ž 6.
Eqs. Ž3b. imply that indeed the solution Ž5. is compatible with Ž6., i.e. if one inserts Ž5. into Ž6. and
uses the chain rule, all the terms become functions of
the three arguments of Ž5.. Then, Ž6. will become a
PDE in these variables.
We therefore define as new variables the following:
q1 s
q3 s
g 11
q2 s
,
g 12
2
g 11 g 22 y g 12
g
g 12
g 22
,
t
s
s 14 Ctm
Csn
g mn .
Now, using the chain rule, we reduce Eq. Ž6. from
coordinates gab to coordinates q i Ž i s 1–3.. The
result is that C should satisfy:
Ž q1 .
2
Ž q 1 y q 2 . C 11 q q 2 Ž q 1 y q 2 . C 22 y
q2
q 2 Ž q 1 y q 2 .Ž 2 q 2 y q 1 . C 12 q 2
2 q3
q
3
C3 q
ž
3
4
768
y
Ž q3.
2
/
q1
q3
3
C 33
Ž q1 y q 2 . C1
q2
Cs0
Ž 7.
2
C s C Ž g 11 rg 12 ,g 12rg 22 , Ž
2
g 11 g 22 y g 12
. rg . .
Ž 5.
Since wave functions Ž5. are annihilated by the Hˆm
seen as constraints, they are invariant under the finite
motions induced by the Hˆm seen as vector fields on
the configuration space; as explained in the appendix, these linear transformations of gab are automorphisms Žcalled inner automorphisms. of the Lie
algebra of the isometry group and can be seen as
generated by the spatial diffeomorphisms. This explains why C is invariant under these diffeomorphisms. The earlier mentioned reduction caused by
the linear constraints is thus encoded in Ž5..
where, for example, C 33 s E qE 3ECq 3 .
Separating variables via C Ž q 1 , q 2 , q 3 . s
X Ž q 1,q 2 .Y Ž q 3 . we get the following equations:
Ž q1 .
q2
2
Ž q 1 y q 2 . X11 q q 2 Ž q 1 y q 2 . X 22
1
q 2Ž q y q
2
.Ž 2 q
2
yq
1
. X12 q 2
q1
q2
Ž q 1 y q 2 . X1 s cX
Ž 8.
2
Ž q 3 . Y¨y 2 q 3 Y˙q
ž
2304
3 2
Žq .
9
y
4
/
Y s 3cY ,
Ž 9.
T. Christodoulakis et al.r Physics Letters B 419 (1998) 30–36
where c is a separation constant and a dot denotes
differentiation of Y with respect to q 3.
In order to solve Ž8. we effect the following
transformation:
us
1
q 1q 2
,
q2
Õs
q1
.
Ž 10 .
Eq. Ž8. is then transformed into a separable form and
thus we get two ODE’s:
u 2 U XX q uU X y c1U s 0
Ž 11a.
4Õ Ž 1 y Õ . V XX q 2 Ž 1 y 2 Õ . V X y
c
ž
/
y 4 c1 V
1yÕ
s0 ,
Ž 11b.
1Ž
2Ž
where U s UŽ u., V s V Ž Õ ., X Ž q u,Õ .,q u,Õ .. s
UŽ u.V Ž Õ . and c1 is another separation constant.
The general solution will be the product of the
solutions of Ž9., Ž11a., Ž11b.. Eq. Ž9. is a Bessel
equation. Eq. Ž11a. is an Euler equation. As for Eq.
Ž11b., it can, via the substitution V Ž Õ . s Ž1 y
Õ . Ž1q '1q4 c . ZŽ Õ ., be transformed to
1
4
Õ Ž 1 y Õ . ZXX q
1
2
y Ž 1 q a q b . ZX y abZ s 0 ,
Ž 12 .
1
4
where a:s Ž1 q '1 q 4 c y 4 c1 . and b:s 14 Ž1
q '1 q 4 c q 4 c1 ..
This is a hypergeometric equation with well
known general solution. From all the above we find:
(
(
c
C Ž q 1 ,q 2 ,q 3 . s a Ž q 1q 2 . ' 1 q b Ž q 1q 2 .
'
2
Ž q 3 . C¨ y 2 q 3C˙ q
q e'Õ F Ž a q 12 ,b q 12 , 32 ;Õ .
3
=Ž q
2
gJ
2
q hY
y
9
q3 c
48
y
9
2
q3 c
q3
,
ž
2304
3 2
9
y
4
Žq .
/
Cs0 .
Ž 14 .
This may be transformed to a Bessel equation with
general solution
48
( ž /
( ž /
.
We denote their generators by EŽai.b Ž i s 1,2.. Corresponding to these outer automorphisms there exist
further spatial diffeomorphisms under which the wave
function must be invariant. Thus, some combination
of q 1 , q 2 , q 3 still represent ‘‘gauge’’ degrees of
freedom. Therefore solution C as appears in Ž15. is
not a ‘‘faithful representation’’ of the physical system in the following sense: There exist different
triplets of values of q 1 , q 2 , q 3 Žand consequently
different values of C . corresponding to physically
indistinguishable spacetimes. In order for C to really
represent the geometry, and not depend on the coordinate system, it must be annihilated by the 5 vector
fields XŽ i. s lŽai.b ga m EgEbm , where lŽai.b s Ž CŽar . b , EŽaj.b .
Žobviously, the three Hm’s, whose action has already
been taken care of in arriving at Ž5., are included in
the XŽ i.’s, since CŽar . b are the structure constant
matrices of the Type V symmetry group generating
the inner automorphisms..
Hence, for the Type V case, the number of independent variables of the wave function is further
reduced from 3 to 1. This is assured by the closure of
the algebra of the Hr ’s and the H0 with these
additional constraints Žsee appendix.. Since one may
verify that q 3 is annihilated by the all the XŽ i. , i.e.
XŽ i. q 3 s 0, while q 1 , q 2 are not, we conclude that q 3
is the one remaining variable for C . That is to say
that q 3 is the only physical degree of freedom of the
system.
Substituting C s C Ž q 3 . in Ž7. we obtain the ODE:
y c1
= dF Ž a,b, 12 ;Õ .
3
33
3
q3
C Žq
3
3
.sŽq .
2
48
ž /
ž /
AJ 3'2 y
q3
2
Ž 13 .
is the general solution to Ž7., where a,b,d,e, g,h are
constants.
For the type V case, apart from the inner automorphisms, the automorphism group contains two
extra elements, the so called outer automorphisms.
q BY 3'2 y
2
48
q3
.
Ž 15 .
This solution has already appeared in w3x. However,
there is a huge difference between the two derivations. In w3x, C s C Ž q 3 . was imposed as a simplifying ansatz in order to solve the Wheeler–DeWitt
T. Christodoulakis et al.r Physics Letters B 419 (1998) 30–36
34
equation. Here, C s C Ž q 3 . is shown to be the necessary condition in order to isolate the physical
content of the theory.
We now turn to the question of measure on the
space of solutions. In order to complete step ŽIV., we
have to find an appropriate inner product for the
construction of a Hilbert space out of the solutions.
Due to the hyperbolic character of Ž7. we cannot
expect the solutions C Ž q 1 ,q 2 ,q 3 . to be normalizable
with respect to all the arguments under any measure.
One may consider as a ‘‘natural’’ choice the square
root of the determinant of the induced metric. However, this choice does not make the induced operator
Hermitian. The induced contravariant metric on the
one dimensional submanifold parametrized by q 3 s
3
Eq3
a
b
Cam
Cbn
g mn is EgE qab Eg
Qa bgd A Ž q 3 . 7r2 . Hence the
gd
natural induced measure is proportional to Ž q 3 .y7 r4 ,
under which the operator in Ž14., is not Hermitian.
The unique measure that makes the operator acting
on C in Ž14. Hermitian is m Ž q 3 . s Ž q13 . . Consequently the probability density is
4
P Ž q 3 . s m Ž q 3 . <C Ž q 3 . < 2 .
Ž 16 .
To obtain normalizability, P Ž q 3 . must go to zero
when q 3 ™ `, so we choose B s 0. Therefore, the
normalized wave function is
3 2pi
1
2
4
'
C Žq
3
. se
18
3
3
Žq .
2
J 3'2
2
3
48
ž /
q3
.
Ž 17 .
For q ™ 0 or `, C Ž q ™ 0, while for q 3 ™ 0,
P Ž q3. f
3.
'2
8p
cos 2
ž
48
q
3
y
Ž 3'2 q 1 . p
4
/
.
Ž 18 .
This exhibits an oscillatory character near zero. An
increasing sequence of maxima is also present. The
total maximum occurs for q 3 ( 13.9. Note that since
q 3 is essentially the Ricci 3-curvature invariant Ž R s
6 q 3 ., it has a dimension of inverse length square, i.e.
ŽPlanck Length.y2 .
3. Discussion. The discovery of a valid Hamiltonian for the type V geometry w3x has enabled us to
consider its quantization. The treatment is the most
general possible, since we allowed for all six gab ’s
as well as the shift vector to appear at the classical
level. At the quantum level we have adopted the
point of view that the linear constraints must be
imposed as conditions on the wave function, much
like the quadratic constraint, rather than be solved
classically. It is well known that the two approaches
give physically different results w10x. We believe that
the equal treatment of all constraints has an advantage over solving some of them at the classical level.
More importantly, the realization of first class constraints as conditions on the wave function ŽDirac’s
programme w1x. preserves the covariance of the ensuing quantum theory under constant transformations
of the configuration space variables w11x.
The hyperbolicity of the ensuing Eq. Ž7. Žafter
imposing the linear constraints. indicates that some
combinations of the q i ’s still represent gauge degrees of freedom; indeed from a geometrical point of
view a general type V three-space is completely
characterized by its only independent curvature invariant, namely q 3 defined above the Eq. Ž7. w8bx.
The scalar form of our Hamiltonian action enables us
to dispose off q 1, q 2 by imposing the generators of
outer automorphisms as supplementary conditions on
our states, instead of just making the Žallowable.
ansatz C s C Ž q 3 .. Of course, the ultimate justification for this final reduction is, as explained in w8bx,
and briefly recalled in the appendix, the identification of the action on gab ’s of the entire automorphism group to the action induced on gab ’s by
particular spatial diffeomorphisms ŽAutomorphism
Inducing Diffeomorphisms.. Our final states Ž17. are
chosen so that they are normalisable with respect to
the measure that makes the reduced operator Ž14.
Hermitian.
Acknowledgements
The authors would like to acknowledge the valuable comments and especially the assistance of the
referee in finding the general solution to Eq. Ž11b..
This work is part of the 1995 PENED program
‘‘Quantum and Classical Gravity – Black Holes’’
Žno. 512. supported by the General Secretariat for
the Research and Technology of the Hellenic Department of Industry, Research and Technology. Two of
T. Christodoulakis et al.r Physics Letters B 419 (1998) 30–36
35
the authors ŽG. Kofinas and A. Paschos. were partially supported by the Hellenic Fellowship Foundation ŽI.K.Y...
where sai Ž f . and Lab are the matrices inverse to
si a Ž f . and Pab respectively. The necessary and
sufficient condition for this set of PDE’s to have a
solution is:
Appendix A
r
a
2 sai Ž f . s kg Ž x . sj d Ž x . Cgd
Lra y Cmn
LgmLdn s 0 ,
For the sake of completeness we briefly recall the
arguments linking the automorphism group to
G.C.T.’s:
Consider the spatial part ds 2 s gab si a Ž x . =
bŽ .
sj x dx i dx j of the line element of the general
homogeneous spacetime with s ’s the invariant basis
one-forms appropriate for some given Bianchi
three-space. The homogeneity of this line element is
of course preserved under any G.C.T. of the form:
x i ™ x˜ i s f i Ž x . .
Ž A.1 .
Under such a transformation ds simply becomes:
Ž ds 2 '. ds˜2 s gab s˜ma Ž x˜ . s˜nb Ž x˜ . dx˜ m dx˜ n ,
Ž A.2 .
where the basis one-forms are supposed to transform
in the usual way:
sma˜ Ž x˜ . s si a Ž x .
E xi
.
Ž A.3 .
E x̃ m
To find the change in form induced on the line-element by ŽA.1. we express ŽA.2. in terms of the old
basis one-forms si a Ž x˜ . Žat the new point.. There is
always a non singular matrix Pba Ž x˜. connecting s̃
and s , i.e.
s˜ma Ž x˜ . s Pma Ž x˜ . smm Ž x˜ . .
Ž A.4 .
Using this matrix P we can write the line element
ŽA.2. in the form:
ds˜2 s gab Pma Ž x˜ . smm Ž x˜ . Pnb Ž x˜ . snn Ž x˜ . dx˜ m dx˜ n .
Ž A.5 .
Ž A.8 .
which is satisfied if and only if L Žand thus also P .
is a Lie Algebra Automorphism.
Thus, for every Bianchi group the automorphisms
m
a
are defined by the relation Lma Cbg
s Lbr Lgs Crs
. Subm
stituting the components of Cbg for the Type V case
we find that the general form for the matrices Lbr is:
a
Lbr s d
0
b
e
0
c
f ,
1
0
where a,b,c,d,e, f g R.
m
The generators lbr satisfy the relation lma Cbg
s
r a
s a
lb Crg q lg Cbs . The general form for the matrices
lbr is
a
lbr s d
0
b
e
0
c
f .
0
0
The vector fields corresponding to lbr will then be
XŽ i. ' lŽri. m grn EgEmn . Three of them are the well known
a
momentum constraints Hr ' Crm
gan EgEmn . The remaining two additional independent vector fields Ej '
´Žrj. m grn EgEmn are the generators of the outer automorphisms. It is straightforward to check that the following first class algebra holds:
Ha , Hb 4 s 12 Cagb Hg , EŽ i. , Hr 4 s 12 lŽmi. r Hm ,
EŽ i. , EŽ j. 4 s CŽ Ži.Žk .j. EŽ k . .
X
i
If the functions f defining the transformation are
such that the matrix P does not depend on the
spatial point, then there is a well defined, non trivial
action of these transformations on gab .
gmn ™ g˜mn s Pma Pnbga b .
Exj
s sai Ž f . Lba sj b Ž x . ,
References
Ž A.6 .
With the use of ŽA.3. and ŽA.4. the requirement that
P does not depend on the spatial point places the
following differential restrictions on the f i ’s:
E f iŽ x.
Ž A.9 .
Ž A.7 .
w1x P.A.M. Dirac, Can. J. Math. 2129 Ž1950.; Proc. Roy. Soc.
London A 246 Ž1958. 333; Phys. Rev. 114 Ž1959. 924;
Lectures on Quantum Mechanics, Yeshiva University, New
York, 1965.
w2x C.J. Isham, Canonical Quantum Gravity and the Problem of
Time, Proceedings of the NATO Advanced Summer Institute, Recent Problems in Mathematical Physics, 1992.
36
T. Christodoulakis et al.r Physics Letters B 419 (1998) 30–36
w3x T. Christodoulakis, E. Korfiatis, A. Paschos, Phys. Rev. D 54
Ž1996. 2691.
w4x T. Christodoulakis, G. Kofinas, E. Korfiatis, A. Paschos,
Phys. Lett. B 390 Ž1997. 55.
w5x A. Ashtekar, J. Samuel, Class. Quan. Grav. 8 Ž1991. 2191;
R.T. Jantzen, Commun. Math. Phys. 64 Ž1979. 211.
w6x K.V. Kuchar, P. Hajicek, Phys. Rev. D 41 Ž1990. 1091; K.V.
Kuchar, P. Hajicek, Jour. Math. Phys. 31 Ž1990. 1723.
w7x T. Christodoulakis, E. Korfiatis, J. Math. Phys. 33 Ž1992.
2863.
w8x T. Christodoulakis, E. Korfiatis, Simultaneous Hamiltonian
Treatment of Class A Space-Times and Reduction of Degrees of Freedom at the Quantum Level, preprint UArNPPS-
17, June 1994; T. Christodoulakis, E. Korfiatis, Kuchar’s
Physical Variables and Automorphism Inducing Diffeomorphisms in Class A Cosmologies, Univ. of Athens, Preprint-6,
1996.
w9x P.R. Garabedian, Partial Differential Equations, Chap. 2,
Chelsea, New York, 1986.
w10x A. Ashtekar, G.T. Horowitz, Phys. Rev. D 26 Ž1982. 3342;
K.V. Kuchar, M.P. Ryan, in: H. Sato, T. Nakamura ŽEds..,
Gravitational Collapse and Relativity, World Scientific, Singapore, 1986.
w11x T. Christodoulakis, E. Korfiatis, Phys. Lett. B 256 Ž1991.
484.
12 February 1998
Physics Letters B 419 Ž1998. 37–39
Diffeomorphism invariance of geometric descriptions
of Palatini and Ashtekar gravity 1
Yan Luo 2 , Ming-Xue Shao 3, Zhong-Yuan Zhu
4
Institute of Theoretical Physics, Academia Sinica, Beijing 100080, PR China
Received 15 July 1997; revised 10 October 1997
Editor: P.V. Landshoff
Abstract
In this paper, we explicitly prove the presymplectic forms of the Palatini and Ashtekar gravity to be zero along gauge
orbits of the Lorentz and diffeomorphism groups, which ensures the diffeomorphism invariance of these theories. q 1998
Elsevier Science B.V.
PACS: 04.20.Cv; 04.20.Fy
Keywords: Diffeomorphism invariance; Presymplectic forms; Palatini; Ashtekar
Geometric description and quantization w1x is the
global generalization of ordinary Hamiltonian canonical description and quantization. This formalism has
been shown to provide an natural way to investigate
global and geometrical properties of physical systems with geometrical invariance, such as ChernSimons theory w2x, anyon system w3x, and so on. But
the traditional descriptions of geometrical and canonical formalism of classical theories are not manifestly covariant, because from the beginning one has
to explicitly choose a ‘‘time’’ coordinate to define
the canonical conjugate momenta and the initial data
of systems. Several year’s ago, Witten w4x and Zuck1
This work is supported by NSF of China, Pan Den Plan of
China and LWTZ-1298 of Chinese Academy of Sciences.
2
E-mail: [email protected].
3
E-mail: [email protected].
4
E-mail: [email protected].
erman w5x and Crnkovic w6x et al. suggested a manifestly covariant geometric description, where they
took the space of solutions of the classical equations
as phase space. This definition is independent of any
special time choice so that is manifestly covariant.
Then they used this description to discuss Yang-Mills
theory, string theory and general relativity etc. Recently, Dolan and Haugh w7x used Crnkovic and
Witten’s method to deal with the Ashtekar’s gravity.
They investigated the problems related to the complex nature. But a thorough discussion needs to
prove the vanishing of components of presymplectic
form v tangent to the diffeomorphism and Lorentz
group orbits, as Crnkovic and Witten emphasized in
Ref. w6x for the case of general relativity. Essentially
they pointed this proof is the most delicate point in
their treatment. Therefore this short paper is devoted
to complete this proof for cases of Palatini and
Ashtekar gravity.
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 6 8 - 8
Y. Luo et al.r Physics Letters B 419 (1998) 37–39
38
The first order action of Palatini with the tetrads
and Lorentz connections as its configuration space
variables is given in w8x:
where a is valued in local Lorentz Lie algebra
which means a will vanish at infinity, so we obtain
the transformations of d e Kc and dv bI J along the
Lorentz group orbits,
SP Ž e, v . s 12 R aI Jb e Ia e Jb ed 4 x ,
d e Kc ¨ d e Kc s d e Kc y a K J e Jc ,
H
Ž 1.
where e Ia ’s are tetrads and e is the determinant of
e aI . The curvature of the Lorentz connections v aI J is
defined to be R aI Jb s Ea v bI J y E b v aI J q w v a , v b x I J.
Here ‘‘a,b,c,d, . . . ’ ’ stand for Riemannian indices
and ‘‘I, J, K, L, . . . ’’ stand for the internal SO Ž3,1.
indices. The variations of the action with respect to
v aI J and e Ia give the equations of motion:
IJ
c
e Ic R cb
y 12 R cMdN e M
e Nd e bJ s 0,
Ž 2.
E b e Ia q v b I J e Ja q G bac e Ic s 0,
Ž 3.
X
X
dv bI J ¨ d v bI J s dv bI J q , b a I J .
Ž 9.
From Ž8., and keeping only the terms up to first
order of a ,
X
V yVsn V
s
HS, a
b
IJ
n d Ž e Ia e Jb .
y e Ia e Jb e cKd e Kc
4 ed Sa .
Ž 10 .
IK
2, w c dv bI Jx e Ic s yR cb
d Ž e Ic e a K . e a J ,
Ž 6.
where we have used Eqs. Ž4., Ž7. and the antisymmetry of IJ in a I J. Clearly the integrand of right hand
side of Eq. Ž10. is of the form , b Ž X a b . e with X a b ,
an antisymmetric tensor which makes the integral to
be a surface integral at infinity. Since we restrict the
local transformations in limited region, the surface
integral vanishes identically. So, the presymplectic
form Ž8. is degenerate along Lorentz group orbits.
Next, we concentrate on the proof of the degeneracy of V along diffeomorphism directions. The diffeomorphism transformations are in the form
E xc
a
a
a
a
c
b
x ¨ y s x q j , eK Ž x . ¨ eK Ž y .
.
E yb
From the second equation of motion Ž3. and noticing
, b j c s E b j c q G bcd j d, one obtains
, b d e Ia q dv b I J e Ja q dG bac e Ic s 0.
Ž 7.
e Kc ¨ e Kc y , b j c e kb y v d K J e Jc j d
in which G bac is the Christoffel. The second equation
of motion Ž3. can be written in the form:
, b e Ia s 0,
Ž 4.
in which , is torsion-free connection on both
space-time and internal indices. By using Eq. Ž4., the
first equation of motion Ž2. becomes
R a b s 0,
Ž 5.
which is just the Einstein field equation in vacuum.
The tangent vectors of the solution space satisfy the
linearized equations of motions,
From the action Ž1., we get the presymplectic form
and similarly,
IJ
Vs
dv bI J n d
HS
Ž
e Ia e Jb e
. d Sa ,
Ž 8.
v bI J ¨ v bI J q , d v bI Jj d q v dI J , b j d y w v d , v b x j d ,
so that
X
where S is the space-like supersurface in space-time
manifold. Obviously the presymplectic form V is
independent of the choice of the space-like supersurface S and invariant under Riemannian coordinate
and Lorentz transformations. But as pointed in w6x,
we need to prove the degeneracy along gauge orbits
of Lorentz and diffeomorphism groups and, in fact,
the proof is not trivial.
Under infinitesimal Lorentz transformations, the
tetrads and the connections transform as
e Kc ¨ e Kc y a K J e Jc ,
v bI J ¨ v bI J q , b a I J ,
d e Kc ¨ d e Kc s d e Kc y , b j c e kb y v d K J e Jc j d ,
X
dv bI J ¨ d v bI J
s dv bI J q , d v bI Jj d q v dI J , b j d
IJ
yw vd , v b x j d .
Ž 11 .
The presymplectic V can be rewritten in the form
Vs
a
HS j ed S ,
a
j a s dv bI J n ye Ia e Jb e cKd e Kc q d e Ia e Jb q e Iad e Jb .
Ž 12 .
Y. Luo et al.r Physics Letters B 419 (1998) 37–39
From Ž11. and Ž12. and keeping only the terms up to
first order of j , after tedious calculations, we obtain
X
j a y j a s n j a s n j1a q n j2a q n j3a ,
n j1a s , d ydv bI J e Jd e Ia n j b
qdv bI J e Jb n Ž e Iaj d y e Idj a .
yv bI J e Jd e Iaj b e cK n d e Kc
q v bI Jj b n Ž e Iad e Jd y e Idd e Ja . ,
n j2a s R bI Jd
1
2
Ž e Jb d e Id y e Id e b M e c J d e Mc . n j a
c
qe Id e a M e c J d e M
n j b y e Ia e Jb e cKd e Kc n j d
q d e Ia e Jb n j d ,
n j3a s ydG bac n e Jc e Lb v dJLj d ,
Ž 13 .
where we have used Eqs. Ž3., Ž6., Ž7. and the
antisymmetry of IJ in v I J. From the deformation of
IJ
the equations of motion e Ic R cb
s 0, we have n j2a s 0.
Obviously due to the symmetry of b c in dG bac and
the antisymmetry of JL in v dJL , n j3a s 0. So, there
only leaves with
n Vs
a
1
HSn j ed S .
a
Like the integrand of right hand side of Ž10., n j1a is
again of the form , d X d a with X d a being an antisymmetric tensor, one gets
n Vs
HSE Ž eX
d
da
. d Sa s H eX d adSd a .
Ž 14 .
ES
If assuming j l has compact support or, more
generally, is asymptotic at infinity to a killing vector
field Ža more detailed discussion on boundary conditions can be found in w9x., n V obviously vanishes
which ends our proof of the degeneracy of the
presymplectic form Ž8. of Palatini gravity along the
39
directions of diffeomorphism transformations. If denote Z the solution space of equations of motion, G1
the diffeomorphisms group and G 2 the Lorentz group,
the presymplectic form Ž8. is a well defined symplectic form on the moduli space ZrG1rG 2 which
means that the system has constraints corresponding
to diffeomorphisms transformation and local Lorentz
transformation.
The same procedures of proof is suitable for
Ashtekar gravity w10,8x. Since the only difference is
that in Ashtekar’s case tetrads and connections are
complex and self-dual Žor anti-self-dual. which does
not change the proof, so that we arrive at the conclusion that the diffeomorphism invariance of above
geometrical description is also correct for Ashtekar
gravity.
References
w1x N.M.J. Woodhouse, Geometric Quantization, Clarendon
Press, Oxford, 1992; J. Snitycki, Geometric Quantization and
Quantum Mechanics, Springer-verlager, 1980.
w2x S. Axelrod, S.D. Pietra, E. Witten, J. Differential Geometry
33 Ž1991. 787.
w3x Y. Yu, Z.Y. Zhu, H.C. Lee, J. Math. Phys. 34 Ž1993. 988.
w4x E. Witten, Nucl. Phys. B 276 Ž1986. 291.
w5x G. Zuckerman, Yale university preprint print-89-0321.
w6x C. Crnkovic, E. Witten, in: S.W. Hawking, W. Israel ŽEds..,
Three Hundred Years of Gravitation, Cambridge University
Press, 1987, p. 676.
w7x B.P. Dolan, K.P. Haugh. Class. Quantum Grav. 14 Ž1997.
477.
w8x A. Ashtekar, Lectures on Nonpertubertive Canonical Gravity,
in: Advanced Series in Astrophysics and Cosmology, vol. 6,
World Scientific, Singapore, 1992.
w9x A. Ashtekar, L. Bombelli, O. Reula, in: M. Francaviglia, D.
Holm ŽEds.., Mechanics, Analysis and Geometry: 200 Years
after Lagrangian, North-Holland, Amsterdam, 1991, p. 417.
w10x A. Ashtekar, Phys. Rev. Lett. 57 Ž1986. 2244; Phys. Rev. D.
36 Ž1987. 1587.
12 February 1998
Physics Letters B 419 Ž1998. 40–48
Canonical quantum statistics of Schwarzschild black holes
and Ising droplet nucleation
H.A. Kastrup
1
Institute for Theoretical Physics, RWTH Aachen, 52056 Aachen, Germany
Received 13 October 1997; revised 13 November 1997
Editor: P.V. Landshoff
Abstract
Recently it was shown ŽH.A. Kastrup, Phys. Lett. B 413 Ž1997. 267. that the imaginary part of the canonical partition
function of Schwarzschild black holes with an energy spectrum En s s'n EP ,n s 1,2, . . . , has properties which – naively
interpreted – leads to the expected unusual thermodynamical properties of such black holes ŽHawking temperature,
Bekenstein-Hawking entropy etc... The present paper interprets the same imaginary part in the framework of droplet
nucleation theory in which the rate of transition from a metastable state to a stable one is proportional to the imaginary part
of the canonical partition function. The conclusions concerning the emerging thermodynamics of black holes are essentially
the same as before. The partition function for black holes with the above spectrum was calculated exactly recently Žsee the
above mentioned reference... It is the same as that of the primitive Ising droplet model for nucleation in 1st-order phase
transitions in 2 dimensions. Thus one might learn about the quantum statistics of black holes by studying that Ising model,
the exact complex free energy of which is presented here for negative magnetic fields, too. q 1998 Elsevier Science B.V.
PACS: 04.70.-s; 04.70.Dy; 64.60.My; 64.60.Qb
1. Introduction
In a recent paper w1x I discussed properties of the
canonical partition function for a quantum energy
spectrum
En s s'n EP ,
n s 1,2, . . . ,
EP s c 2 c "rG ,
'
s s O Ž 1. ,
`
Ž 1.
where the n-th level has the degeneracy
dn s g n ,
g)1 .
Ž 2.
The main interest in this spectrum comes from the
1
E-mail: [email protected].
many, differently justified, proposals Žsee the corresponding quotations in Ref. w1x. that a quantized
Schwarzschild black hole might have such a spectrum! The associated canonical partition function
ZŽ t , x . s
Ý e n t ey 'n x ,
ns0
t s ln g ,
x s bs EP ,
Ž 3.
converges only for < g < F 1, whereas one is interested
in its properties for g ) 1. However, the function
ZŽ g, x . can be continued analytically into the com-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 6 0 - 3
H.A. Kastrupr Physics Letters B 419 (1998) 40–48
plex g-plane by means of the integral representation
w1x
x
ZŽ gset, x . s
x
s
2'p
du
H
2'p 1
dt
`
H0
`
t
3r2
eyx
2
eyx
2
ln
rŽ4 ln u.
3r2
1
uyg
u
1
rŽ4 t .
1 y e Ž ty t .
,
Ž 4.
Ž 5.
which exhibits a branch cut of ZŽ g, x . from g s 1 to
g s q`. Approaching the cut from above yields a
complex Z, with an imaginary part
Zi Ž t , x . s
'p x
2 t 3r2
eyx
2
rŽ4 t .
Ž 6.
and the principal value integral
Zr Ž t , x . s p.v.
1
1
`
H dt Z Ž t , x . 1 y e
p 0
i
ty t
Ž 7.
as its real part.
Amazingly one gets essentially all the expected
thermodynamical properties of a black hole, including the Hawking temperature, if one – naively –
uses the imaginary part Zi above for their derivation.
This is strange and provocative and needs further
analysis. If such an approach can be substantiated,
one will have an indication which quantum states of
a black hole could be responsible for its unusual
thermodynamical properties!
In the following I shall try to put the properties
mentioned into a certain perspective and especially
discuss their relationship to other approaches Žwhich
was not done in Ref. w1x., namely the euclidean path
integral Žsemiclassical. quantisation of black holes
by Hawking and others and the theory of droplet
nucleation associated with metastable states.
The paper is organized as follows: In chapter 2
the properties of the model are briefly summarized
and its theoretical and physical potentials and limitations sketched. Parts of it recall well-known material
which serves to have the discussion more or less
self-contained. Chapter 3 discusses the relationship
of the model to Hawking’s euclidean path integral
approach to the thermodynamics of black holes,
where he also gets a purely imaginary partition
function w2x! Chapter 4 deals with the nucleation of
droplets occuring when a system changes from a
metastable state to a stable one, e.g. in 1st-order
41
phase transitions or in the decay of ’’false’’ ground
states. Langer w3x was the first one to relate the
imaginary part of the canonical partition function to
the transition rate of such processes. Later Gross,
Perry and Yaffe w4x interpreted the imaginary part of
the semiclassical euclidean path integral – i.e. the
canonical partition function – for the Schwarzschild
black hole Ž’’instanton’’. in this context, thus giving
a different interpretation of the same quantity Hawking employed.
In chapter 5 the relation of the partition function
Ž3. to that of the primitive Ising droplet model for
nucleation in 2 dimensions is discussed and it is very
easy to see that both are exactly the same. The
integral representation Ž4. therefore provides an exact free energy expression for this model, too, which
as far as I know, is new w5x. Chapter 6 presents some
conclusions.
2. Scope of the model
In the background of trying to push the quantum
analysis – including its quantum statistics – of the
simple isolated Schwarzschild gravitational system
as far as possible are the following considerations:
The quantized radiation emitted by a ’’classical’’
Schwarzschild black hole has a thermal ŽPlanck.
distribution governed by the ŽHawking. temperature
w6x
k B TH ' by1
H s
c3"
8p GM
,
Ž 8.
where M is the mass of the black hole. Assuming
that this temperature of the radiation arises from its
thermal contact with the black hole of the same
temperature, the immediate question was – and still
is –, how does the black hole gets this temperature?
As the temperature Ž8. is proportional to Planck’s
constant the quantum theory of gravity has to play a
decisive role in its understanding. Especially one
would like to identify the microscopic quantum gravitational degrees of freedom which yield the corresponding quantum statistics and its macroscopic
thermodynamics.
Many attempts have been made to achieve this
and I shall mention only a few of them in the
42
H.A. Kastrupr Physics Letters B 419 (1998) 40–48
following. The most active and enthusiastic reseach
in this area presently takes place in connection with
string theory. I shall say nothing about that in the
following and refer to the very recent review by
Horowitz w7x.
In view of the lack of a generally accepted quantum theory of Einstein’s General Relativity it seems
difficult to extract those microscopic gravitational
degrees of freedom within this framework. However,
the situation is not quite es hopeless as it might
appear. The reason is that Einstein’s theory for a
rotationally symmetric isolated gravitational system,
Schwarzschild gravity, can consistently be quantized,
namely by first identifying its classical observables
in the sense of Dirac by solving the constraints
associated with the gauge Ždiffeomorphims. degrees
of freedom and then quantizing the remaining
gauge-independent physical degrees of freedom afterwards. This has been done by Thiemann and
myself w8x in the framework of Ashtekar’s formalism
and briefly afterwards by Kucharˇ w9x in the geometrodynamical framework, the results being
equivalent. They have been briefly summarized in
Ref. w10x.
The essential point is that this system classically
has just one canonical pair of ŽDirac. observables,
namely its mass M and as the canonically conjugate
quantity of M a time functional T w g; S x Žof the
metric g . which describes the difference tqy ty in
proper time t of two observers at the two Žasymptotic. ends of the 1-dimensional spacelike hypersurface S which, together with the 2-dimensional
spheres S 2 , provides a time slicing of the 4-dimensional manifold. Formally the quantity
ber, e.g. g s 4p for the time period DH of the
euclidean section w2x of the complex Schwarzschild
manifold.
Having eliminated the Žinfinite. gauge degrees of
freedom classically one can now quantise the remaining physical degrees of freedom w8–10x. The
extremely simple Schrodinger
equation for this sys¨
tem,
d s T w g ; S x y Ž tqy ty .
Introducing such boundary conditions is easier than
justifying them by physical arguments. In any case it
means that the system is represented by the plane
waves only for the time D after which it is abruptly
terminated, e.g., by forming of the horizon. This may
happen, e.g., because the state described by the plane
wave is a metastable one - see below. There are
several physical time scales D s g R Src associated
with a black hole all of which w10x have g ; O Ž1., so
that
Ž 9.
is the constant Dirac observable.
Thus, for an observer at the asymptotic end where
r ™ q` on the Schwarzschild manifold with the
time t ' tq there are – in the framework of this
model – only two quantities available in order to
describe the system: its Mass M and his proper time
t ! Everything else has to be expressed by them Žand
some fundamental constants like G,c,",k B .. Examples are: the Schwarzschild radius R S s 2 MGrc 2 or
any multiple thereof, the area A s 4p R 2S of the
horizon, but also any time interval D associated with
the system, i.e. D s g R Src, where g is some num-
i" Et f Ž t . s Mc 2f Ž t . ,
Ž 10 .
has the plane wave solutions
i
2
f Ž M ,t . s x Ž M . ey " M c t ,
Ž 11 .
where M is continuous and ) 0. So at first sight
there is no such spectrum like Ž1.. However, one can
get it by changing the boundary conditions.
My doing so in Ref. w10x was stimulated by the
paper of Bekenstein and Mukhanov w11x, in which
they discussed properties of the spectrum Ž1. with
s 2 s ln grŽ4p ., g s 2. Already in 1974 Bekenstein
w12x discussed this spectrum as arising from a BohrSommerfeld type quantisation of the area of the
horizon Ž A A n" .. Since then many authors have
proposed such a spectrum Žsee Refs. w3–23x in Ref.
w1x..
One knows from elementary quantum mechanics
how to make the continuous momentum in plane
waves discontinuous by periodic boundary conditions in space. Applying corresponding boundary
conditions in time Žwith period D . to the plane
waves Ž11. yields
c 2 M D s 2p " n ,
En s
(
n s 1,2, . . . .
Ž 12 .
p
g
'n EP ' s'n EP .
Ž 13 .
H.A. Kastrupr Physics Letters B 419 (1998) 40–48
3. Euclidean path integral as canonical partition
function of a Schwarzschild black hole
First I want to recall how Hawking’s path integral
approach to the canonical partition function in a
semiclassical approximation leads to a purely imaginary partition function, too:
Shortly after Hawking’s discovery that the quanta
emitted by a collapsing black hole have a thermal
distribution governed by the temperature Ž8. Gibbons
and Hawking w13x employed the relationship between
the canonical partition function of a system and the
corresponding euclidean path integral in order to
tackle the canonical quantum statistics of a black
hole itself. For pure gravity that partition function Z
is formally given by
Z s D w g x eyI E w g xr " ,
H
IE s
1
16p G
Hd
4
xR Ž g . Ž g .
1r2
q surface terms,
Ž 14 .
where IE is the euclidean action of pure gravity Žfor
more details of the path integral approach to quantum gravity see the reprint collection in Ref. w2x.. In
principle one has to take into account all field configurations gmn which are periodic in euclidean time
with period " b and which obey appropriate spatial
boundary conditions. In practice the path integral has
been evaluated only semiclassically: One puts
(
gmn s gmn q hmn ,
Ž 15 .
(
where gmn is a classical solution of the euclidean
field equations compatible with the boundary conditions just mentioned and the hmn denote Žsmall.
quantum fluctuations around the classical solution.
Inserting Ž15. into the action Ž14. and keeping only
2 .
terms up to order O Ž hmn
gives the semiclassical
approximation
IEscl s IEcl g q IEŽ2. ,
(
( 1r2
IEŽ2. s d 4 x Ž g .
H
Ž hA Ž g( . h . ,
Ž 16 .
(
where AŽ g . is a second order differential operator
(
depending on the classical metric gmn .
Gibbons and Hawking evaluated IEcl for the euclidean Schwarzschild metric and obtained
IEcl w Schwarzschildx s 12 bH Mc 2 ,
Ž 17 .
43
where the period " bH follows from the requirement
that the euclidean Schwarzschild section has no conical singularities. The associated partition function
ZScl s eyb H M c
2
r2
2
2
s eyE P b H rŽ1 6p .
Ž 18 .
gives the desired thermodynamics of black holes,
especially the Bekenstein-Hawking entropy
A
S BH rk B s 2 , l P2 s G"rc 3 .
Ž 19 .
4 lP
The approach appears to become ’’unconventional’’,
if one includes the term IEŽ2. for which the path
integral Ž14. is a Gaussian one which – formally – is
proportional to the inverse square root of the product
of the eigenvalues of the operator A Žs
ŽdetŽ A..y1 r2 .. For the Schwarzschild case the term
IEŽ2. has been evaluated by Gibbons and Perry w14x. A
crucial point is that the operator A has just one
negative eigenvalue w4x which contributes a factor i
to the partition function. The combined contribution
to the partition function ZSscl of the classical euclidean Schwarzschild solution and the fluctuations
around it is given by w4,14x
ZSscl s iZ g ZGHP ,
ZGH P s
Z g s ep
1
V
2 64p
2
3 3
lP
V rŽ c 3 " 3 b H3 .
Ž a bH .
,
212r45 yE P2 b H2 rŽ16 p .
e
,
Ž 20 .
where V is the spatial volume of the system and a is
a ’’cutoff’’ associated with the z-function regularization of detŽ A.. ŽThe index ’’GHP’’ stands for ’’Gibbons, Hawking, Perry’’.. The factor Z g is the same
(
as one would obtain for flat space Ž gmn s dmn .. It
represents the gas of free thermal gravitons surrounding the black hole.
Except for the spatial volume factor the structure
of ZGH P in Eq. Ž20. is very similar to that of Zi in
Eq. Ž6. above: If s 2 s trŽ4p . – see Ref. w1x – then
the exponentials in both cases are equal for b s bH .
The factors with the powers of b – which in Eq.
Ž20. comes from short-distance fluctuations around
the classical solution – are different in both cases.
However this is not surprising because both approaches are so different. These factors may give rise
to logarithmic corrections w1x to the BekensteinHawking entropy Ž19. which is dominated by the
exponential. I shall come back to this in the next
H.A. Kastrupr Physics Letters B 419 (1998) 40–48
44
chapter.
Hawking has argued Žsee Ref. w2x, ch. 15.8. that
the partition function Ž20. has to be imaginary in
order for the density of states N Ž E . in the Laplace
transforms
`
ZŽ b . s
NŽ E. s
yb E
H0 dEN Ž E . e
1
,
qi`
H dbZŽ b . e
2p i yi`
bE
Ž 21 .
tto be real if Z cl from Eq. Ž18. is inserted into the
last integral and which exists only if the contour of
integration is rotated by pr2! Because of this Hawking has concluded w2x that the canonical ensemble is
inappropriate for the description of black holes.
Recently Hawking and Horowitz have discussed
w15x a more refined version of the relationship between the classical euclidean gravitational action Ž14.
– especially its boundary terms – and the entropy of
the system. This work is related to that of York and
collaborators w16x who suggested to improve the
approach of Gibbons and Hawking w13x and Gross,
Perry and Yaffe w4x by employing a more microcanonical framework. Related is also the work of
Iyer and Wald w17x.
In any case, the imaginary parts ZGH P of Eq. Ž20.
or Zi of Eq. Ž6., respectively, do seem to yield
interesting though unconventional thermodynamical
properties, if one treates them as if they were ’’normal’’ partition functions. As they imply negative
heat capacities they signal instabilities of the system.
This leads us to a closely related interpretation of
the imaginary part of a partition function, namely of
being associated with the decay of a metastable state
into stable one in the form of nucleation:
tion. Coleman and Callen w18x discussed this approach in the context of instanton solutions in the
euclidean version of otherwise lorentzean field theories and their role for the decay of ’’false’’ Žmetastable. vacua. The methods involved have found wide
applications in different fields of physics w19x.
The main procedure is very similar to the one
described above for the euclidean section of the
Schwarzschild solution: The instantons Žor
’’bounces’’. are classical solutions of the ’’euclideanized’’ field equations satisfying appropriate
boundary conditions of a certain path integral. The
quantum Žor thermal. fluctuations around these classical solutions leave the system stable if all the
eigenvalues of the second-variation operator A Žsee
above. are positive. However, if one of the eigenvalues is negative, then the classical solution is not
related to a minimum of the euclidean action integral, but only to a saddle point and the fluctuations
can drive the system from its metastable state into a
more stable lower one.
The transition rate is essentially determined by
I Ž Z scl .rZ0 , where Z0 is the real part of the canonical partition function Žwhich represents the metastable
state. and by the absolute value of the negative
eigenvalue.
It is this framework in which Gross, Perry and
Yaffe w4x interpreted the contribution Ž20. of the
euclidean Schwarzschild solution Ž’’Schwarzschild
instanton’’.:
Eq. Ž20. represents the ’’1-instanton’’ contribution to the partition function. The next step is a
dilute-gas approximation: Neglecting the ’’interactions’’ between the instantons, the contribution of N
of them to the grand partition function ZG is
Zg
4. Imaginary part of partition functions and
droplet nucleation
In a series of very influential papers w3x Langer
discussed the imaginary part of partition functions
for systems with a 1st-order phase transition where
the transition from a metastable state to a stable one
is initiated by the ’’nucleation’’ of droplets. Langer
suggested that the nucleation transition rate is proportional to the imaginary part of the partition func-
1
N!
N
Ž iZGHP . .
Ž 22 .
Summing over all N gives
ZG s Z g e i Z GH P
Ž 23 .
and the grand canonical potential
C Ž b ,V , a . s ln ZG s b pV ,
dC s yUd b q pb dV y Nd a ,
Ž 24 .
H.A. Kastrupr Physics Letters B 419 (1998) 40–48
Ž a ' ymb , m : chemical potential, p: pressure. here
has the form
C s ln Z g q iZGHP ,
Ž 25 .
whereas the general structure is w3,19x
C s ln Z0 q i I Ž Z scl . rZ0 .
Ž 26 .
Because of the special form of Z scl in Ž20. the factor
Z g , representing the metastable graviton gas Ži.e.
Z g s Z0 . drops out in Eq. Ž26.!
According to Langer w3x Žand others w18,19x. the
rate G of transition per unit volume from the
metastable state Žhere the graviton gas. to the stable
state Žhere the black hole. is given by
<k <
Gs
p
<k <
I Žc . s
p
I Ž Z sclrZ0 . ,
c s CrV ,
Ž 27 .
were k is the single negative eigenvalue mentioned
above Žthat a large class of systems with possible
negative eigenvalues has just one of them was proven
by Coleman w20x..
The factor < k < in Eq. Ž27. is a dynamical one,
depending on nonequilibrium properties of the system w3,19x. As to more recent evaluations of < k < for
relativistic systems see Refs. w21x.
The relation between the entropy Ž19. and the
exponential factor expŽyb 2rŽ16p .. in the transition rate Ž27. associated with Eq. Ž25. is perhaps not
so obvious anymore. However, in the case of nucleation one writes w3,19x
I Ž Z scl . rZ0 s eyb Ž F
F scl s y
1
b
scl
yF 0 .
ln I Ž Z scl . ,
,
F0 s y
1
b
ln Z0
Ž 28 .
and interprets Fˆ s F scl y F0 as the excess free energy of the critically large droplet. In our case we
have Z0 s Z g and therefore FˆS s yŽ1rb .ln ZGHP . In
this sense ZGH P represents the partition function of
the bare black hole, i.e. it describes the thermodynamics of the black hole without the surrounding
graviton gas.
Thus, the more sophisticated nucleation picture
essentially leads to the same thermodynamical properties of Schwarzschild black holes as the ’’naiÕe’’
use [1] of the imaginary part of the partition function!
45
Still, it seems desirable to have a more systematic
analysis of the relationship between the two approaches.
Page w22x has given a detailed analysis of the rate
Ž27. for the transition of a gas into a black hole. The
picture that black holes are formed by nucleation
from a gas was already discussed by Gibbons and
Perry w23x.
5. Schwarzschild black hole quantum statistics
and Ising droplet nucleation in 2 dimensions
One of the most amazing parts of the whole
analysis presented here is that the quantum partition
function Ž3. associated with a Schwarzschild black
hole is the same as that of the Žprimitive. Ising
droplet model for nucleation Žin 1st-order phase transitions. in 2 dimensions w24x:
Suppose a d-dimensional Ž d G 2. lattice of N
Ising spins below the critical temperature Tc to be in
a positive magnetic field H so that almost all the
spins are up Žq1.. If one then slowly turns the
magnetic field negative, the system comes into a
metastable state in which the total magnetization is
still positive, before finally becoming negative, too.
A simple model to describe the transition into that
stable state is the following: Assume that Žsmall.
droplets containing l down-spins form, in a background of up-spins. The droplets are supposed to be
noninteracting Ždilute-gas approximation. and the
number n l of droplets of size l to be given by
n l s Neyb e l , l s 1, . . . .
Ž 29 .
The droplet formation energy e l is assumed to consist of two Žcompeting. terms, the bulk energy 2 Hl
and a surface energy term f l Ž dy1.r d , so that
e l s 2 Hl q f l Ž dy1.r d .
Ž 30 .
The interesting part of the canonical partition function ZN is the finite sum
ZN rN s Ý eyb 2 H ly bf l
Ž dy 1.r d
.
Ž 31 .
l
Letting N go to ` and employing the grand canonical ensemble in exactly the same way as for the
instanton gas above yields the following grand
canonical potential c per spin
`
c Ž b,H . s
Ý ey2 l H b eyl
ls0
Ž dy 1.r d
fb
.
Ž 32 .
H.A. Kastrupr Physics Letters B 419 (1998) 40–48
46
The theoretical investigations of the function c have
focussed mainly on its behaviour for negative H
Žwhere the sum no longer converges!. in the neighbourhood of H s 0. It is clear that c must have a
singularity at H s 0, the so-called ’’condensation
point’’. Many approximate analytical calculations and
rigorous estimates w25x suggested that c has an
essential singularity there, a supposition supported
by numerical calculations w26x.
Now for d s 2 the series Ž32. is just the same as
Ž3. which has been summed exactly in Ref. w1x by
using Lerch’s observation w27x that the relation
ey 'n x s
2
< x<
yx 2 Õ 2 r4yn r Õ 2
'p H0 dÕe
< x<
s
`
dt
`
H
2'p 0
t
3r2
6. Conclusions
eyx
2
rŽ4 t .ynt
Ž 33 .
turns the series into a geometrical one which can be
summed under the integral sign and then continued
analytically. We only have to put t s y2 b H, x s fb
and then get from Eq. Ž6. the exact result
I Ž c . Ž b , H - 0.
s
'p
4'2
of t s 0 for d s 3, too! I shall only indicate what to
expect: As that partial differential equation is invariant under the scale transformation t ™ l t, x ™ l2r3 x,
there should exist solutions of the type f Ž y ., y s
x 3rt 2 . Inserting this gives for f Ž y . the ordinary diff.
eq. 27f XXX q Ž4 q 54ry . f XX q 6Ž1ry q 1ry 2 . f X s 0.
For t ™ 0, y ™ `, there is the approximate solution
f XX ; expŽyŽ4r27. y . which agrees with the form of
the expected essential singularity as to small negative H in 3 dimensions w25x. However, more analysis
is certainly necessary here.
fby1r2 < H <y3 r2 eyŽ f
2
b .rŽ8 < H <.
.
Ž 34 .
Thus, there is indeed an essential singularity at H s
0. Instead of < H <y3 r2 in front of the exponential
factor the approximation methods of Langer w3x and
Gunther
et al. w25x yield the factor < H <. The modified
¨
method used by Harris w26x gives the correct < H <-dependence!
The real part of c is to be calculated by means of
the principal value Ždispersion. integral Ž7. Žif the
sum Ž32. starts with l s 1 etc. corresponding changes
have to be made w1x..
In Ref. w1x I pointed out that the real part Ž7. can
become negative for small x. The droplet nucleation
picture might help to understand the Žphysical?.
background of this.
For a recent discussion of metastability in the
2-dimensional Ising model itself see Ref. w28x.
It was already stressed in Ref. w1x that the series
Ž3. obeys the heat equation E t Z s Ex2 Z. The corresponding partial differential equation for c in d s 3
dimensions is E t2c s yEx3c . This ought to help
analysing the behaviour of c in the neighbourhood
It appears that there can be hardly any doubt that
the square root spectrum Ž1. is closely related to the
expected thermodynamics of black holes. It is therefore important to understand its physical meaning
and the theoretical background better. Because it
implies that the area A of the horizon is proportional
to " one should probably look here for an interpretation w11x. This corresponds to other recent investigations which stress the importance of bifurcate horizons and the associated surfaces for understanding
the thermodynamics of black holes w29x.
Furthermore, the discussion above shows that a
’’naive’’ and a droplet nucleation interpretation, respectively, of the imaginary part of a partition function provide just two different aspects of the thermodynamical properties of metastable systems.
Similarly important is the physical understanding
of the degeneracy Ž2. which is equally essential for
the derivation of the thermodynamics. In the Ising
droplet model above g is replaced by the factor
exp y b H which represents the influence of the
driÕing external Žnegative. magnetic field. The same
role can have a Žpositive. chemical potential w3,19x.
Bekenstein and Mukhanov w11x used information theoretical arguments to put g s 2. However, the arguments above go through with any g ) 1. The physical background of the degeneracies Ž2. probably
needs further understanding. ŽThe ’’driving field’’
behind g ) 1 is most likely the gravitational attraction which leads to the black hole..
In addition it will be interesting to see whether
and how the above considerations can be extended to
H.A. Kastrupr Physics Letters B 419 (1998) 40–48
the thermodynamics of the Reissner-Nordstrøm
model w30x where investigations similar to those of
Refs. w8,9x for the Schwarzschild case exist w31x.
Very exciting is the possibility to map the quantum statistics of a Schwarzschild black hole onto the
statistics of the 2-dimenionsal Ising model for droplet
nucleation. This might allow to learn from a known
field of physics for an unknown one. Finally it is
surprising that the pursuit of black hole physics leads
to an exact mathematical solution for an old condensed-matter problem.
7. Note added in response to a question of the
referee
If one considers a Schwarzschild black hole in
d q 1 space-time dimensions w32x, d G 3, then its
Schwarzschild radius R S is proportional to M 1rŽ dy2.,
where M is the mass of the system. Thus, if we
again assume the time interval D in Eq. Ž12. above
to be proportional to R S , then 'n in Eq. Ž13. is
replaced by nŽ dy2.rŽ dy1. and the resulting partition
function is the same as that of the Ising droplet
model for nucleation in d y 1 dimenions Žsee Eq.
Ž32..!
Acknowledgements
I am very much indebted to Malcolm Perry for
drawing my attention to Refs. w3x and w4x.
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Phenomena, vol. 8, Academic Press, London etc., 1983, p.
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w26x
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12 February 1998
Physics Letters B 419 Ž1998. 49–56
Fermionic fields in the d-dimensional anti-de Sitter spacetime
R.R. Metsaev
Department of Theoretical Physics, P.N. LebedeÕ Physical Institute, Leninsky prospect 53, 117924 Moscow, Russia
Received 15 August 1997
Editor: P.V. Landshoff
Abstract
Arbitrary spin free massless fermionic fields corresponding to mixed symmetry representations of the SO Ž d y 1. compact
group and propagating in even d-dimensional anti-de Sitter spacetime are investigated. Free wave equations of motion,
subsidiary conditions and the corresponding gauge transformations for such fields are proposed. The lowest eigenvalues of
the energy operator for the massless fields and the gauge parameter fields are derived. The results are formulated in
SO Ž d y 1,2. covariant form as well as in terms of intrinsic coordinates. q 1998 Elsevier Science B.V.
In view of the aesthetic features of anti-de Sitter
field theory a interest in this theory was periodically
renewed Žsee w1x- w8x and references therein.. One of
the interesting directions of this theory is the higherspin massless field theory. At present there is reason
for revival of interest in higher-spin massless fields.
Recently it was discovered w9x that the consistent
equations of motion of interacting higher-spin massless fields in four dimensional Ž d s 4. anti-de Sitter
spacetime could be set up with the help of higher-spin
superalgebras Žfor review see w7x.. These equations
for the case of higher spacetime dimensions d ) 4
have been generalized in w10x. Because these equations are formulated in terms of wavefunctions which
depend on usual spacetime coordinates and certain
twistor variables it is not clear immediately what
kind of fields they describe. In w9x, it was established
that in four dimensions Ž d s 4. they describe unitary
dynamics of massless fields of all spins in anti-de
Sitter spacetime.
The long-term motivation of our investigation is
to provide an answer to question: do the higher
dimensional theories of Ref. w10x describe higher-spin
massless fields? The first step in such an investigation is a description of free equations of motion for
arbitrary spin massless fields. In contrast to completeness of description for d s 4 Žw1,2,8,11x- w13x.,
not much is known on the higher-spin fermionic
massless fields for d ) 4 even at the level of free
fields, unless considerations are restricted to totally
Žanti.symmetric representations Žw14x- w16x.. The present paper is a sequel to our paper w17x where
description of bosonic massless fields of all spins in
anti-de Sitter spacetime of arbitrary dimensions was
developed. The case of half-integral spins, presented
here is a necessary step in our study of massless
fields of all spins for arbitrary d. With higher-spin
theories in mind we also hope that our results may
be of wider interest, especially to anti-de Sitter supergravity theories.
Let us first formulate the main problems we solve
in this letter. A positive-energy lowest weight irreducible representation of the SO Ž d y 1,2. group denoted as DŽ E0 ,m., is defined by E0 , the lowest
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 4 6 - 9
R.R. MetsaeÕr Physics Letters B 419 (1998) 49–56
50
eigenvalue of the energy operator, and by m s
Ž m1 , . . . mn ., n ' d y2 2 , which is the weight of the
unitary representation of the SO Ž d y 1. group. The
m i are half integers for fermions. For the case of
SO Ž3,2. group it has been discovered w11x that
fermionic massless fields propagating in four dimensional anti-de Sitter spacetime are associated with
DŽ s q 1, s . i.e. spin-s anti-de Sitter massless particle
takes lowest value of energy equal to E0 s s q 1. In
w16x for the case of arbitrary d it was found that
E0 s s q d y 3 for totally symmetric massless representation m s y m s Ž s,1r2, . . . ,1r2., s G 3r2, and E0
s d y sX y 1r2 for the totally antisymmetric one
m a s s Ž3r2, . . . ,3r2,1r2 . . . 1r2. Žwhere the 3r2
occurs sX F n times in this sequence.. The cases
m s y m and m a s are very special: there are many
representations corresponding to arbitrary m Žmixed
symmetry representations.. In this paper we construct free equations of motion and gauge transformations for massless fields with arbitrary m and
determine the E0 . Here we restrict ourselves to the
case of even spacetime dimension d.
Let us describe our conventions and notation. We
describe the anti-de Sitter spacetime as a hyperboloid
hA B y A y B s y1 , hA B s Ž y,q, . . . ,q,y . ,
A, B s 0,1, . . . ,d y 1,d q 1 ,
Ž 1.
in d q 1- dimensional pseudo-Euclidean space with
metric tensor hA B . The indices A, B are raised and
lowered by h A B and hA B respectively. In what follows to simplify our expressions we will drop the
metric tensor hA B in scalar products. The generators
J A B of the SO Ž d y 1,2. group satisfy the commutation relations
wJ
AB
,J
CD
BC
x sh J
AD
s 0, = A y A s d and satisfies the commutation relations
= A, y B su AB ,
A form for M A B depends on the realization of the
representations. We will use the tensor realization of
representations. As the carriers for DŽ E0 ,m. we use
tensor-spinor field
C
AŽ m.
sC
Xn
X
A11 , . . . , A 1m 1 , . . . , An1 , . . . , Anm
,
mXi ' m i y
1
2
,
defined on the hyperboloid Ž1.. The C has one
spinor index which we do not show explicitly. By
definition, C AŽ m. is a tensor-spinor field whose
d q 1 spacetime indices A have the structure of the
Young tableaux Ž YT . corresponding to the irreps of
the SO Ž d y 1,2. group labeled by mX s Ž mX1 . . . mXn ..
In what follows we use the notation Y TŽ mX . to
indicate such YT. At the same time the m is the
weight of a representation of the SO Ž d y 1. group,
and the m i satisfy the inequalities
m1 G . . . G mn G 1r2 .
Ž 2.
In the language of YT, the mXi indicates the number
of boxes in the i-th row of Y TŽ mX .. Thus given an
arbitrary lowest weight massless representation
DŽ E0 ,m. we assume that a covariant description can
be formulated with the field C AŽ m.. To avoid cumbersome tensor expressions we introduce n creation
and annihilation operators a lA and a lA Ž l s 1, . . . n ,
n s Ž d y 2.r2. which satisfy
a iA ,a jB s h A Bd i j ,
a iA <0: s 0
and construct a Fock space vector
mXl
n
il
l
1
1
Ł a lA C A , . . . , A
<C : ' Ł
q 3 terms .
w = A ,= B x s yL A B .
mX 1
1 ,...,
Xn
An1 , . . . , Anm
<0: .
Ž 3.
ls1 i ls1
As is usual, we split J A B into an orbital part L A B
and a spin part M A B : J A B s L A B q M A B. The realization of L A B in terms of differential operators
defined on the hyperboloid Ž1. is:
For a realization of this kind, M A B has the form
M AB sMX AB qs
where
AB
,
n
E
L
AB
A
B
B
sy = yy =
A
,
A
= 'u
AB
E yB
,
u AB 'h AB qy A y B ,
The tangent derivative = A has the properties y A= A
MX ABs
Ý Ž a lAa lB y a lBa lA . ,
ls1
s
AB
1
s
4
Ž g Ag B y g Bg A .
and g A are the gamma matrices: g A ,g B 4 s 2h A B.
R.R. MetsaeÕr Physics Letters B 419 (1998) 49–56
Throughout of the paper, unless otherwise specified,
the indices i, j,l,n run over 1, . . . , n . For these indices we drop the summation over repeated indices.
If the C AŽ m. is associated with Y TŽ mX . then the
<C : satisfies
< :
´ i j a i j <C : s 0 ,
Ž a i i y mXi . <C : s 0 , ay
ij C s0 ,
Ž 4.
where in Ž4. and below we use the notation
a i j ' a iA a jA
,
A A
ay
i j' a i a j
,
A A
aq
i j' a i a j
,
51
Introducing
a i j ' a iA a jA ,
A A
ay
i j' a i a j ,
A A
aq
i j' a i a j
and using the constraints Ž7. and the equalities
y
ay
i j s a i j y yi y j ,
a i j s a i j y yi y j ,
Ž 9.
the constraints Ž4. can be rewritten in the form which
we use in what follows:
< :
´ i j a i j <C : s 0 .
Ž a i i y mXi . <C : s 0 , ay
ij C s0 ,
Ž 10 .
ij
and ´ s 1Ž0. for i - jŽ i G j .. The 1st equation in
Ž4. tells us that a i occurs mXi times on the right hand
side of Eq. Ž3.. Tracelessness of C AŽ m. is reflected
in the 2nd equation in Ž4.. The 3rd equation in Ž4. is
a consequence of the following Young’s symmetrization rule we use: Ži. first we perform alternating with
respect to indices in all columns, Žii. then we perform symmetrization with respect to indices in all
rows Žfor more explanations see w19x.
Being the carrier for the DŽ E0 ,m., the field <C :
satisfies the equation
1
Ž 5.
Ž Q y ² Q : . <C : s 0 , Q ' J A B J A B
2
and allows the following subsidiary covariant constraints
=n <C : s 0 , Ž divergenelessness.
Ž 6.
<
:
yn C s 0 , Ž transversality.
Ž 7.
gn <C : s 0 ,
Ž 8.
n s 1, . . . . n . In Ž5. Q is the second order Casimir
operator of the soŽ d y 1,2. algebra while ² Q : is its
eigenvalue for DŽ E0 ,m.
² Q : s yE0 Ž E0 q 1 y d .
n
y Ý m l Ž m l y 2 l q d y 1. ,
ls1
where E0 is the lowest eigenvalue of i J dq 10 . The
² Q : can be calculated according to the well known
procedure w18x. In Ž6.-Ž8. and below we use the
notation
=n ' a nA= A , =n ' = A a nA ,
a nA ' u A Ba nB ,
yn ' y A a nA ,
gn ' g
A
a nA
,
a nA ' u A Ba nB .
yn ' y A a nA ,
gn ' g
A
a nA
For E0 corresponding to the massless case a solution
of Eqs. Ž5.-Ž8. decomposes into a physical massless
representations and some additional representations
corresponding to pure gauge fields Ži.e. the situation
is similar to the d s 4 case w12x..
Now rewriting the expression for Q as
1
Q s y= 2 q M A B L A B q M A B M A B ,
2
= 2 ' = A= A
and taking into account the easily derived equalities
Žusing Eqs. Ž6.-Ž8. and Ž10..
s
AB
L A B <C : s yu=u <C : ,
M X A B M X A B <C :
n
s y2 Ý mXl Ž mXl y 2 l q d q 1 . <C : ,
ls1
n
M X A B L A B <C : s 2
Ý m l <C : ,
X
ls1
n
M X A Bs
AB<
C : s y Ý mXl <C : ,
ls1
2
=u s = y yu=u , where =u ' g A= A , yu ' g A y A , =u , yu 4
s d, we can rewrite Eq. Ž5. as
1
1
=u 2 y E0 q
E0 q y d <C : s 0 .
Ž 11 .
2
2
ž ž
2
/ž
//
To define E0 corresponding to massless fields we
construct gauge invariant equations of motion for
<C :. In order to formulate gauge transformations we
use the gauge parameters fields whose spacetime
indices correspond to the YT which one can make by
removing one box from the Y TŽ mX .. Thus the most
general gauge transformations we start with are
dŽ n. <C : ; =n < L n : q yn < R n : q gn < Sn : ,
.
Ž 12 .
where the gauge parameters fields < L n :, < R n : and
R.R. MetsaeÕr Physics Letters B 419 (1998) 49–56
52
< Sn : are associated with Y TŽ mXŽ n. ., and i-th component of the mXŽ n. is equal to mXiŽ n. s mXi y d i n . The
Y TŽ mXŽ n. . is obtained by removing one box from
n-th row of the Y TŽ mX .. We assume that only those
< L n :, < R n : and < Sn : are non-zero whose mXŽ n. satisfy
the inequalities
X
X
m1Ž n. G . . . G mn Ž n. G 0 .
Ž 13 .
X
yi < L n : s 0 ,
g i < Ln : s 0 ,
Ž 14 .
and constraints obtained from Ž14. by replacing L
™ R and L ™ S. Since < L n :, < R n : and < Sn : correspond to Y TŽ mXŽ n. ., they satisfy the constraints
< :
Ž a i i y mXiŽ n. . < L n : s 0 , ay
i j Ln s 0 ,
´ i j ai j < Ln : s 0 ,
Ž 15 .
and those which are obtainable from Ž15. by replacing L ™ R and L ™ S. Using Ž9. and Ž14., the
constraints Ž15. can be rewritten in the form similar
to Ž10.
w a i j , a n l x s d jn a i l y d i l a n j ,
y
y
ay
i j , a n l s d jn a i l q d i n a jl ,
where Dn ' J X A Ba nA y B. The usage of Di , Di , a "
ij
and a i j is advantageous since all of them commute
with yn and yn .
Žii. using Ž14., Ž16. one can verify that the 1st and
2nd constraints from Ž10. are invariant with respect
to Ž18. i.e. the equations Ž a i i y mXi . dŽ n. <C : s 0 and
< :
ay
i j dŽ n. C s 0 are fulfilled. The invariance requirement of 3rd constraint from Ž10. Ži.e. ´ i j a i j dŽ n. <C :
s 0. is the most difficult point. Fortunately, this part
of analysis is similar to that of the bosonic case
Žw17x.. Our result for dŽ n. is:
dŽ n. <C : s Dn < L n : q Gn < Sn : ,
n
Dn '
´ i j a i j < Ln : s 0 .
Now we should find such dŽ n. <C : and E0 that the
constraints and the equations of motion Ž6.-Ž8., Ž10.,
Ž11. be invariant with respect to gauge transformations. We proceed in the following way.
Ži. from the invariance requirement of Ž7. Ži.e.
yi dŽ n. <C : s 0. and Ž12., Ž14. one gets
n
Ž 17 .
ls1
By substituting Ž17. into Ž12. we obtain
dŽ n. <C : ; Dn < L n : q Ž g a n . < Sn : ,
n g S ( mX ) ,
Ž 18 .
where Žg a n . ' g A a nA and
n
Dn ' =n q
Ý Ž yyl a n l q aqn l yl . .
Ý
Ž 19 .
ls1
In Ž19. we add by hand the 2nd term in the sum. Due
to Ž14. the gauge transformations Ž18. are unaffected
Gn '
Ý Pn l Ž g a l . ,
Ž 20 .
ls1
where
Pn l s
Ý
js0
j
n
ny1
Ž 16 .
Ý a l n < Ll : q yu < Sn : .
n
Pn l D l ,
ls1
< :
Ž a i i y mXiŽ n. . < L n : s 0 , ay
i j Ln s 0 ,
< Rn: s y
ay
i j , Dn s d i n Dj q d jn Di ,
a i j , Dn s d jn Di ,
X
Given the m , the set of those n whose mŽ n. satisfy
Ž13. will be referred to as S ( mX ), while the number
of such n will be referred to as n X . Thus we have n X
nontrivial gauge transformations. We impose on the
< L n :, < R n : and < Sn : the constraints similar to Ž6.-Ž8.
=i < L n : s 0 ,
by this term. The Dn introduced in such a way
allows the representation Dn s a nA y B J X A B, where
J X A B ' L A B q M X A B, which is very convenient in
practical calculations. Some useful commutation relations are
Ž y.
j
Ý
dn l jq 1 d l1 l Ł
l 1 . . . l jq1 s1
is1
´ l i l iq 1
ll i n
ll n ' m l y m n q n y l q 1 .
a l iq 1 l i ,
Ž 21 .
In Ž21. the a l iq 1 l i are ordered as follows:
a l jq 1 l j . . . a l 2 l 1 . As an illustration of Ž20. we write
down dŽ n. <C : for n s 1,2,3 assuming that such an n
belongs to S Ž mX .:
dŽ1. <C : s D 1 < L1 : q Ž g a1 . < S1 : ,
dŽ2. <C : s D 2 < L2 : q Ž g a 2 . < S2 :
a 21
y
Ž D 1 < L 2 : q Ž g a1 . < S2 : . ,
l12
dŽ3. <C : s D 3 < L3 : q Ž g a 3 . < S3 :
a 31
y
Ž D 1 < L 3 : q Ž g a1 . < S3 : .
l13
a 32
y
Ž D 2 < L 3 : q Ž g a 2 . < S3 : .
l23
a 32 a 21
q
Ž D 1 < L 3 : q Ž g a1 . < S3 : . .
l13 l23
R.R. MetsaeÕr Physics Letters B 419 (1998) 49–56
53
Žiii. from the invariance requirement of Eq. Ž8.
Ži.e. g i dŽ n. <C : s 0. we get the equation for < L n :
and < Sn :
Then making use of unitarity condition one proves
w17x that for the level-k Young tableaux the E0
should satisfy the inequality
Ž =u y Ž m n y n . yu . < L n : q Ž d q 2 Ž m n y n . . < Sn : s 0
E0 G m k y k y 2 q d .
X
X
Ž 22 .
and from the invariance requirement of Eq. Ž6. Ži.e.
=i dŽ n. <C : s 0. we get the equation
Ž= 2 y Ž m n y n . Ž m n y n y 1 q d . . < Ln :
X
X
q Ž =u q Ž mXn y n q d . yu . < Sn : s 0 .
Ž 23 .
Finally from the invariance requirement of Eq. Ž11.
Ži.e. Ž=u 2 y . . . . dŽ n. <C : s 0. and from Eqs. Ž22., Ž23.
we derive the equation for E0
1
1
E0 q
E0 q y d
2
2
ž
/ž
X
/
X
s Ž m n y n y 1. Ž m n y n y 1 q d . ,
n g S ( mX ) ,
Ž 24 .
whose solutions read as
E0ŽŽ1.n. s m n y n y 2 q d ,
Ž2.
E0Ž
n. s n q 1 y m n ,
n g S Ž mX . .
Ž 25 .
As seen from Ž25. there exists an arbitrariness of E0
parametrized by subscript n which labels gauge
transformations and by superscripts Ž1.,Ž2. which
label two solutions of quadratic Eq. Ž24.. Because
the values of E0 have been derived by exploiting
gauge invariance we can conclude that the gauge
invariance by itself does not uniquely determine the
physical relevant value of E0 Žin this regard the
situation is similar to the d s 4 case, see w1x.. To
choose physical relevant value of E0 we exploit the
unitarity condition, that is: 1. hermiticity Ži J A B . † s
i J A B ; 2. the positive norm requirement. The resulting procedure is the same that has been previously
encountered in the bosonic case w17x. Therefore we
do not give deal of technical details and formulate
the result.
Given Y TŽ mX . let k, k s 1 . . . n , indicates maximal number of upper rows which have the same
number of boxes. We call such Young tableaux the
level-k Y TŽ mX .. For the case of level-k Young
tableaux the inequalities Ž2. can be rewritten as
m1 s . . . s m k ) m kq1 G m kq2 G . . . G mn G 1r2 .
Ž 26 .
Ž 27 .
Comparing Ž25. with Ž27. we conclude that only
E0ŽŽ1.nsk . satisfies the unitarity condition. Thus anti-de
Sitter fermionic massless particles described by
level-k Y TŽ mX . takes lowest value of energy equal to
E0 s m k y k y 2 q d .
Ž 28 .
Note that it is gauge transformation with n s k
Ž20. that leads to relevant E0 , i.e. given level-k
Y TŽ mX . only the gauge transformation dŽ k . respects
the unitarity. Therefore only the dŽ k . will be used in
what follows. From now on we use letter k to
indicate level of Y TŽ mX .. Our result for E0 Ž28.
cannot be extended to cover the case mX s 0 because
the E0 obtained are relevant only for gauge fields.
This case should be considered in its own right and
the relevant value E0 Žfor mX s 0. s Ž d y 1.r2 can
be obtained by using requirement of conformal invariance Žsee w21x.. The case mX s 0, d s 4 has been
investigated in w22x.
Up to now we analysed second-order equations of
motion for C and gauge invariance of these equations. Now we would like to derive first-order equations of motion and corresponding gauge transformations. To do that we use the relations
=u 2 s k 2 y d k ,
k's
AB
LA B ,
k s yu=u ,
Ž 29 .
and rewrite equation of motions Ž11. as follows:
ž
k y E0 y
1
2
/ž
k q E0 q
1
2
/
y d <C : s 0 .
Thus we could use the following first-order equations of motion:
ž
ž
k q E0 q
k y E0 y
1
2
1
2
/
y d <C : s 0 ,
Ž 30 .
/˜
Ž 31 .
<C : s 0 .
Due to relation
ž
k q E0 q
1
2
/
ž
y d <C : s yu k y E0 y
1
2
/u
y <C :
it is clear that <C : and <C˜ : are related by <C : s
R.R. MetsaeÕr Physics Letters B 419 (1998) 49–56
54
yu <C˜ :, i.e. Eqs. Ž30. and Ž31. are equivalent. We will
use the equation of motion given by Ž30.. Now we
should verify gauge invariance of the first-order Eq.
Ž30. with respect to gauge transformations Ž20.. It
turns out that the invariance requirement of Ž30. with
respect to Ž20. leads to
< Sk : s 0 .
Ž 32 .
Thus because of Ž26. and Ž32. the final form of
gauge transformations is
n
ky1
dŽ k . <C : s
Ž y.
Ý
js0
j
d k l jq 1
Ý
l 1 . . . l jq1 s1
j
=Ł
is1
´ l i l iq 1 a l iq 1 l i
k q 1 y li
n
Dl1 < L k : .
Ž 33 .
= y mk y
1
2
//
y k yu < L k : s 0
Ž 34 .
which can also be rewritten in terms of k and E0L
ž
k q E0L q
1
2
/
y d < Lk : s 0 ,
Ž 35 .
where we introduce lowest energy value for < L k ::
E0L s E0 q 1
E0L s m k y k y 1 q d .
g A y B J A B s =u q
1
Ý Ž g l yl y ylg l . q 2 dyu ,
Ž 37 .
ls1
The equations of motion for L can be obtained from
Ž22. and Ž32.
žu ž
Note that the equations of motion are written in
terms of the operator k Žsee Eq. Ž29.. introduced in
w20x while constructing the equation of motion for
the field associated with the representation
DŽ E0 ,1r2.. The k is expressible in terms of the
orbital momentum L A B Žsee Eq. Ž29... Now we
would like to rewrite our equations of motion in
terms of complete angular momentum J A B. We
believe that such a formulation will form a good
basis for establishing an action leading to the equations of motion under consideration. To this end let
us first multiply Eqs. Ž30. and Ž35. by yu . Then
making use of equalities
Ž 36 .
With the values for E0L at hand we are ready to
provide an answer to the question: do the gauge
parameter fields meet the masslessness criteria? Because the inter-relation between of spin m and energy value E0 for massless field is given by Ž28. we
should express the E0L in terms of m L and k L,
where k L is a level of Y TŽ mX L .. Due to relations
k L s k y 1, m kLL s m k y d k1 we transform Ž36. to
E0L s m kLL y k L y 2 q d q d k1 .
Comparing this relation with Ž28. we conclude that
only for k ) 1 the gauge parameters are massless
fields while for k s 1 they are massive fields.
Thus we have constructed equations of motion
Ž30. which respect gauge transformations Ž33., where
the gauge parameter fields L satisfy the constraints
Ž14., Ž16. and equations of motion Ž35.. The relevant
values of E0 and E0L are given by Ž28. and Ž36..
a nA y B J A B s Dn q
1
2
a nAu A Bg B yu ,
Ž 38 .
and the constraints Ž7., Ž8., Ž14. we get the desired
form for equations of motion:
ž
ž
g A y B J A B y E0 y
ž
ž
g A y B J A B y E0L y
dy1
2
dy1
2
/ u/
/ u/
y <C : s 0 ,
Ž 39 .
y < Lk : s 0 ,
Ž 40 .
where E0 and E0L are given by Ž28. and Ž36..
Because the term Dl 1 < L k : from gauge transformations Ž33. can be expressed as
ž
D l < L k : s a lA y
aCl u C Dg D
2 E0 q 3 y d
/
g A y B J A B < Lk : ,
Ž 41 .
it is seen that the gauge transformations Ž33. are also
expressible in terms of J A B. Note that in deriving of
Ž40. and Ž41. it is necessary to use the equations of
motion for < L k : Ž35. and the relevant values of E0
and E0L given by Ž28. and Ž36..
Now we would like to transform our results to
intrinsic coordinates, in terms of covariant derivatives and vierbein fields. Let x m, m s 0,1, . . . ,d y 1
be the intrinsic coordinates for anti-de Sitter spacetime and let y A Ž x . be imbedding map, where y A Ž x .
satisfy Ž1.. The relationship between d q 1dimensional tensor-spinor field C A . . . and the usual
R.R. MetsaeÕr Physics Letters B 419 (1998) 49–56
Rarita-Schwinger tensor-spinor field cm . . . is given
by Žsee w12x.
cm . . . Ž x . s My1 Ž x . ymA . . . C
A...
Ž y. ,
and now the gauge transformation Ž33. is rewritten
as
dŽ k . < c :
where ymA ' Em y A and M is an 2 d r2 = 2 d r2 matrix
defined by
M Em M s
1
2
vma bsa b q
1
2
emaga
Pl ' a lb e bm ymA a lA ,
Ž 43 .
ls1
where a lb are new creation operator w a ia,a bj x s d i jh a b ,
h a b s Žy,q, . . . ,q .. Note that the < c : is actually a
generating function for tangent space RaritaSchwinger field: ca . . . s e am . . . cm . . . . Now making
use of Ž42. the equations of motion Ž30. can be
rewritten in terms of < c : as follows
ž
Dm L ' Em q
1
2
2
/
<c :s0 ,
vma b M a b ,
n
M ab '
Ý Ž a laa lb y alb a la . q s a b .
Ž 44 .
ls1
In order to transform the gauge transformation to
intrinsic coordinates we introduce the gauge parameter < l k : by analogy with Ž43.
n
< l k : s My1
X
Ł Plm lŽ k . < L k : ,
ls1
d k l jq 1
´ l i l iq 1 Ž a l iq 1 a l i .
k q 1 y li
a lb1 e bm Dm L q
ž
1
2
/
g b < lk : ,
Ž 46 .
Ž 42 .
X
1yd
Ý
l 1 . . . l jq1 s1
j
,
j
a b
Ł Plm l <C : ,
g a e am Dm L q E0 q
js0
is1
s a b s Žg ag b y g bg a .r4, where a,b s 0,1, . . . ,d y 1
are the tangent space indices and g a are the gamma
matrices g a,g b 4 s 2h a b. The ema and vma b are the
vierbein and Lorentz connection of anti-de Sitter
spacetime. Concrete representation for matrix M
may be found in Žw12x.. As usual it is convenient to
introduce generating function for Rarita-Schwinger
tensor-spinor field
n
Ý
Ž y.
=Ł
My1 Ž yu Em yu . M s emaga ,
< c : s My1
n
ky1
s
y1
55
Ž 45 .
where Ž a i a j . ' ha b a a . Note that the < l n : satisfies
the equation of motion which is obtainable from Ž44.
by making there the substitutions < c : ™ < l k : and
E0 ™ E0L. The constraints Ž6., Ž8. take the form
1
a nb e bm Dm L q g b < c : s 0 , a nb g b < c : s 0 , Ž 47 .
2
n s 1, . . . , n , and similar constraints for < l k : are
obtainable from Ž47. by making there the substitution < c : ™ < l k :. The constraints Ž10. transform to
Ž Ž a i a i . y mXi . < c : s 0 , Ž a i a i . < c : s 0 ,
ž
/
´ i j Ž ai a j . < c : s 0 ,
Ž 48 .
i, j s 1, . . . , n , while similar constraints for < l k : are
obtainable from Ž48. by making there the substitutions < c : ™ < l k : and mXi ™ mXiŽ k . .
In conclusion let us summarize the results of this
letter. For massless fermionic field <C : of arbitrary
spin labeled by m we have constructed: Ži. free wave
equations of motion in SO Ž d y 1,2. covariant form
Ž30. as well as in terms of intrinsic coordinates Ž44.;
Žii. corresponding gauge transformations Ž33., Ž46.,
subsidiary conditions Ž6.-Ž8., Ž10., Ž16., Ž47., Ž48.
and equations of motion for gauge parameter Ž35.
field written also in both forms; Žiii. the new representation for equations of motion Ž39.,Ž40. and gauge
transformations Žsee Ž33.,Ž41.. in terms of the generators of the anti-de Sitter group SO Ž d y 1,2.; Živ.
lowest energy values for massless fermionic field
Ž28. and for gauge parameter field Ž36.. Also we
have demonstrated that the gauge parameter for
massless field associated with level k s 1 Young
tableaux is a massive field while for k ) 1 the gauge
parameter is a massless field.
The author would like to thank Prof. B. de Wit
for hospitality at Institute for Theoretical Physics of
Utrecht University where a part of this work was
56
R.R. MetsaeÕr Physics Letters B 419 (1998) 49–56
carried out. This work was supported in part by
INTAS contracts No. CT93-0023 and No.96-538, by
the Russian Foundation for Basic Research, Grant
No.96-01-01144, and by the NATO Linkage, Grant
No.931717.
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12 February 1998
Physics Letters B 419 Ž1998. 57–61
Non-renormalizable terms and M theory during inflation
David H. Lyth
Department of Physics, UniÕersity of Lancaster, Lancaster LA1 4YB, UK
Received 13 October 1997
Editor: M. Dine
Abstract
Inflation is well known to be difficult in the context of supergravity, if the potential is dominated by the F term.
Non-renormalizable terms generically give < V XX < ; VrM 2 , where V Ž f . is the inflaton potential and M is the scale above
which the effective field theory under consideration is supposed to break down. This is equivalent to <h < ; Ž M PlrM . 2 ) 1
where M Pl s Ž8p G .y1r2 , but inflation requires <h < - 0.1. I here point out that all of the above applies also if the D term
dominates, with the crucial difference that the generic result is now easily avoided by imposing a discrete symmetry. I also
point out that if extra spacetime dimensions appear well below the Planck scale, as in a recent M-theory model, one expects
M < M Pl , which makes the problem worse than if M ; M Pl . q 1998 Elsevier Science B.V.
1. To achieve slow-roll inflation, the potential
V Ž f . must satisfy the flatness conditions e < 1 and
<h < < 1, where w1x
e'
1
2
M Pl2
X
Ž V rV .
2
Ž 1.
h ' M Pl2 V XXrV
Ž 2.
and M Pl s Ž8p G .y1 r2 s 2.4 = 10 18 GeV is the reduced Planck mass. When these are satisfied, the
time dependence of the inflaton f is generally given
by the slow-roll expression 3 Hf˙ s yV X , where H ,
1
y2
3 M Pl V is the Hubble parameter during inflation.
On a given scale, the spectrum of the primordial
curvature perturbation, thought to be the origin of
structure in the Universe, is given by
(
d H2 Ž k . s
1
150p
2
V
M Pl4
e
Ž 3.
The right hand side is evaluated when the relevant
scale k leaves the horizon. On large scales, the
COBE observation of the cmb anisotropy corresponds to
V 1r4re 1r4 s .027M Pl s 6.7 = 10 16 GeV
Ž 4.
The spectral index of the primordial curvature
perturbation is given by
n y 1 s 2h y 6 e
Ž 5.
A perfectly scale-independent spectrum would correspond to n s 1, and observation already demands
< n y 1 < - 0.2. Thus e and h have to be Q 0.1 Žbarring a cancellation. and this constraint will get tighter
if future observations move n closer to 1. Many
models of inflation predict that this will be the case,
some giving a value of n completely indistinguishable from 1.
Usually, f is supposed to be charged under at
least a Z2 symmetry f ™ yf , which is unbroken
during inflation. Then V X s 0 at the origin, and
inflation typically takes place near the origin. As a
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 9 6 - 2
D.H. Lyth r Physics Letters B 419 (1998) 57–61
58
result e negligible compared with h , and n y 1 s
2h ' 2 M Pl2 V XXrV. We assume that this is the case in
what follows. If it is not, the nonrenormalizable
terms generically give both <h < ; 1 and e ; 1 at a
generic point in field space, making model-building
even more tricky.
2. In supergravity, the tree-level potential is the
sum of an F-term and a D-term w2x,
V s VF q VD
Ž 6.
Supergravity is a non-renormalizable field theory,
whose lagrangian presumably contains an infinite
number of non-renormalizable terms. Taking the theory to be an effective one, holding up to some scale
M, the coefficients of the non-renormalizable terms
are expected to be generically of order 1, in units of
M. Hopefully, they can be calculated if one understands what goes on at scales above M. The most
optimistic view is to take M to be the scale above
which field theory itself breaks down, due to gravitational effects such as the quantum fluctuation in the
spacetime metric. It is not unreasonable to take this
view in the context of inflation, if the inflaton is a
gauge singlet, and we adopt it here. 1 With four
spacetime dimensions, this means that M s M Pl . If
more dimensions open up well below the Planck
scale, the answer is less obvious Žsee below..
The non-renormalizable terms respect the gauge
symmetries possessed by the renormalizable theory,
but they need not respect its global symmetries. On
the contrary, it is generally felt that the usual continuous global symmetries Žbuilt out of UŽ1.’s acting
on the phases of the complex fields. will not generically be respected, at least if M is the scale at which
gravitational effects spoil field theory. This viewpoint derives in part from superstring theory, but that
theory also suggests that discrete subgroups of the
global symmetries Žbuilt out of ZN ’s. will occur. By
imposing suitable discrete symmetries, one can ensure that a global continuous symmetry is approxi-
mately preserved at field values < M w3x. This is
important, because several cases are known where an
approximate global UŽ1. is phenomenological desirable. The best known case is Peccei-Quinn symmetry, which must be preserved to high accuracy in
order to keep the axion sufficiently light.
Some time ago w4x, it was emphasized that non-renormalizable terms will contribute to VF , generically
giving < VFXX < ; VFrM 2 . 2 Most models of inflation
have the F-term dominating Ž F-term inflation. and
then the non-renormalizable terms in VF give
<h < ; M Pl2 rM 2
Ž 7.
This generic result is too big, since slow-roll inflation per se requires <h < < 1, and the observational
bound on n requires <h < - 0.1
A little later w7x it was pointed out that the problem may disappear if instead VD dominates Ž D-term
inflation.. It was also pointed out that D-term hybrid
inflation occurs quite naturally, if the lagrangian
contains a Fayet-Illiopoulos term. ŽWithin the
single-field paradigm, D-term inflation is essentially
impossible, because the inflaton then has to be
charged under the relevant UŽ1. and the gauge couplings spoil inflation unless they are unreasonably
small w8x.. Several authors w9–15x have since studied
models of D-term inflation, always under the same
assumption that non-renormalizable terms can be
ignored in that case.
Here I point out that non-renormalizable terms
also contribute to VD , through the gauge kinetic
function. They generically give <h < ; Ž M Pl rM . 2 in
that case also, but in contrast with the case of the F
term this generic result is easy to evade by imposing
a discrete symmetry. The reason is that the gauge
kinetic function appearing in the D term is holomorphic, in contrast with the Kahler
potential which
¨
appears in the F term.
I consider the usual model of D-term hybrid
inflation w7,9–12,14,15x. There is just one nonvanishing D field, which contains a Fayet-Illiopoulos term with coefficient j . The fields charged under
the relevant local UŽ1. have been driven to zero Žor
1
For fields charged under the Standard Model gauge interactions one should use M s MGU T ,10 16 GeV in the low energy
theory, and for hiddenrsecluded sector fields below the condensation scale L one should use M s L.
2
The result holds actually for the second derivative in any field
direction, as was noted a long time ago w5,6x.
D.H. Lyth r Physics Letters B 419 (1998) 57–61
anyhow sufficiently small values. along with VF
Then
V , VD ,
j 2g2
2
Re fy1
Ž 8.
where g is the gauge coupling of the relevant UŽ1.,
and the gauge kinetic function f is a holomorphic
function of all of the complex scalar fields fn . If we
regard g 2 as fixed, f must be invariant under all
internal symmetries.
At a given point in field space, one can choose
f s 1 corresponding to canonical normalization of
the gauge field of the relevant UŽ1.. This point is
conveniently taken to be the origin, defined as the
fixed point of the usual internal symmetries. Then,
along say the f 1 direction,
1rf s 1 q l My2f 12 q PPP
59
strong interaction. It has to be preserved to high
accuracy in order to keep the axion sufficiently light,
and it appears that discrete symmetries of the model
ensure this. 3 Of course, this means that the quadratic
term in f is killed. This model has the virtue of
making contact with both Peccei-Quinn symmetry
and the Standard Model. On the negative side, it so
far lacks a definite origin for the Fayet-Illiopoulos
term; in contrast with the other models of D-term
inflation, a direct origin in the superstring seems
impossible since V 1r4 is extremely low.
3. Let us briefly recall the situation for the F term
w4x. At least at tree level,
2
VF s e K r M Pl
=
Ž 9.
nm
Ý Ž Wn q My2
ž Wm q My2
Pl WK n . K
Pl WK m /
nm
There is no linear term, unless f 1 is a singlet under
all of the internal symmetries that are unbroken
during inflation. The quadratic term is allowed if the
only symmetry is f 1 ™ yf 1. Then, if inflaton is
f s '2 Re f 1 , it gives a contribution h s
lŽ M Pl rM . 2 , with l generically of order 1.
The offending term is forbidden if there is a ZN
symmetry Ž f 1 ™ expŽ i a . f 1 with a s 2prN . with
N G 3. It is also forbidden if there is a global UŽ1.
symmetry, corresponding to arbitrary a . In all of the
models of D-term inflation proposed so far, such a
global UŽ1. is present as an R symmetry, with
W A f 1. In the simplest case,
Here, W is the superpotential Žholomorphic in the
fields. and K is the Kahler
potential Ža non-singular
¨
function of the fields and their complex conjugates..
A subscript n denotes ErEfn , and a subscript m
denotes ErEfm . Also, K n m is the matrix inverse of
K n m . Only the combination G ' K q ln < W < 2 is physically significant.
Canonically normalizing the fields at Žsay. the
origin corresponds to
W s cf 1 f 2 f 3
ŽAny linear term can be absorbed into W.. Then,
Ž 10 .
where f 2 and f 3 oppositely charged under the
relevant gauge UŽ1.. One indeed needs to forbid
terms in W of the form f 12fn Ž n / 1. because they
would generate a quartic term in the inflaton potential and spoil inflation. But to achieve this it is
enough to have the symmetry f 1 ™ yf 1 Žacting on
W as an R symmetry.. I am here pointing out that
this symmetry is not enough to forbid the disastrous
quadratic term in f ; it has to be promoted to Z2 N
with N ) 1, or to the full UŽ1.. One hopes that the
superstring will ultimately determine which discrete
symmetry actually holds, if any.
In one model of D-term inflation w15x, the global
Ž
U 1. is the Peccei-Quinn symmetry, whose non-perturbative breaking ensures the CP invariance of the
< <2
y3My2
Pl W
Ž 11 .
K s Ý < fn < 2 q O Ž fn3 .
Ž 12 .
n
K n m s dn m q My2 Ý l n < fn < 2 q PPP
Ž 13 .
n
In the last expression, the terms not displayed are
linear and higher; the quadratic term displayed is a
particularly simple one, which cannot be forbidden
by any of the usual symmetries.
Now make again the assumption that f s
Re f 1r '2 ; as we remark in a moment, the opposite
assumption that f corresponds instead to the phase
3
so.
In an earlier version of this paper I stated that they will not do
D.H. Lyth r Physics Letters B 419 (1998) 57–61
60
of f 1 would give something dramatically different.
For simplicity, also set e K s 1 corresponding to
small field values. Then, one can identify some
contributions to V XX ,
V XX s My2
V y < W1 < 2 q My2
Pl
izable terms. A more promising avenue is to suppose
that K and W have very special forms w7,21,22x,
corresponding roughly to versions of no-scale supergravity. In this case, it may be justified to use the
limit of renormalizable global supersymmetry.
Ý l n < Wn < 2 q PPP
n
Ž 14 .
In the first bracket, V comes from differentiating e K ,
and the coefficient of y< W1 < 2 is the sum of q2
coming from the first term in the bracket of Eq. Ž11.,
< <2
and y3 coming from the y3My2
Pl W . The three
terms displayed will give a contribution
h s 1 y a q b Ž M Pl rM .
2
Ž 15 .
with generically < b < ; 1.
It was pointed out in w4x if W s L2f 1 during
inflation, then a s 1. ŽIt is easy to construct models
of inflation where this is exactly w4,16x or approximately w17x true.. In that case, one need only require
< b < < 1, corresponding to an accidental suppression
of the other non-renormalizable contributions. Some
authors w17–19x have taken the view that this is an
improvement on the generic situation, where one
needs in addition the accident < a y 1 < < 1.
Notice that Eq. Ž11. contains M Pl , as distinct
from the scale M above which field theory is supposed to break down. Taking M Pl to infinity with M
fixed converts supergravity into a non-renormalizable globally supersymmetric theory. More usually,
one considers the limit where M Pl and M go to
infinity together, corresponding to a renormalizable
globally supersymmetric theory. It is clear from the
form of Eqs. Ž4. and Ž15. that, during inflation,
neither of these prescriptions should be used without
justification. As we have just seen though, the use of
non-renormalizable global supersymmetry can be
justified if the superpotential is linear.
If b ; 1 Žthe generic case. the only way of evading the result <h < ; Ž M Pl rM . 2 is to suppose that f
is the pseudo-Goldstone boson of a global symmetry
w20x acting on Žsay. f 1. Then all < fn < are fixed
during inflation, and if the symmetry is broken only
by W and not by K, the non-renormalizable terms in
the latter need cause no problem w4x, though one still
has the problem of understanding why a global
continuous symmetry is respected by non-renormal-
4. Finally, I note that the scale M might not be as
big as M Pl if additional space dimensions open up
below M Pl . I have in mind a specific example, where
the underlying theory is an M theory w23x. A field
theory is valid below some scale Q - M Pl , and it is
attractive to choose Q ; 10y2 M Pl ; ))10 16.5 GeV
to account for the apparent unification of the gauge
couplings. This field theory lives in effectively five
spacetime dimensions, until we get down to some
still lower scale. If the fifth dimension makes no
significant difference, the scale M of the non-renormalizable terms will be M s Q ; 10y2 M Pl . Otherwise M will presumably be lower, though with two
mass scales in the underlying theory the concept of a
single scale M may not even be useful. As yet
nothing has been worked regarding these questions,
even in the specific example mentioned.
If M is indeed of order )) 10 16.5 GeV, a fieldtheory model of inflation will not make sense w24x if
V 1r4 ; 10 16.5 GeV, the maximum allowed by the
COBE normalization Eq. Ž4.. A four-dimensional
field theory model will not make sense unless V 1r4
is even lower. However, many models have been
proposed with low V 1r4 .
Of these, the simplest is the D-term model, with
the slope of the potential coming from the 1-loop
correction w9,10x. In that case COBE normalization
requires w11,12x
V 1r4
Ž g 2r2.
1r4
s j s 3 = 10 15 GeV
'
Ž 16 .
This might be low enough to justify the use of
four-dimensional field theory. Also, if j comes the
superstring, it will be a loop suppression factor times
M 2 , which might be compatible with the M theory
model just mentioned. Again, this has yet to be
investigated.
The most promising F-term model is probably
that of w25x. Starting at the scale f s M with the
generic V XX Ž f . , Ž M PlrM . 2 V, renormalization
D.H. Lyth r Physics Letters B 419 (1998) 57–61
group equations are invoked to run this quantity to
zero at a scale f < M, where inflation occurs. In
this case one can have V 1r4 as low as Mm s where
m s s 100 GeV, amply justifying the use of four-dimensional field theory. As given, this model takes
M s M Pl , and it is unclear whether the change M ;
M Pl r100 would make an important difference.
(
Acknowledgements
I thank Toni Riotto for useful comments on a first
draft of this paper. This work is partially supported
by grants from PPARC and from the European Commission under the Human Capital and Mobility programme, contract No. CHRX-CT94-0423.
References
w1x For a review of this material concerning inflation, see A.R.
Liddle, D.H. Lyth, Phys. Rep. 231 Ž1993. 1.
w2x For reviews of supersymmetry and supergravity, see H.P.
Nilles, Phys. Rep. 110 Ž1984. 1; H.E. Haber, G.L. Kane,
Phys. Rep. 117 Ž1985. 75; D. Bailin, A. Love, Supersymmetric Gauge Field Theory and String Theory, IOP, Bristol,
1994.
w3x A useful review of these questions is given by M. Dine,
hep-thr9207045.
w4x E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewart, D.
Wands, Phys. Rev. D 49 Ž1994. 6410.
61
w5x M. Dine, W. Fischler, D. Nemeschansky, Phys. Lett. B 136
Ž1984. 169.
w6x G.D. Coughlan, R. Holman, P. Ramond, G.G. Ross, Phys.
Lett. B 140 Ž1984. 44.
w7x E.D. Stewart, Phys. Rev. D 51 Ž1995. 6847.
w8x J.A. Casas, C. Munoz, Phys. Lett. B 216 Ž1989. 37; J.A.
Casas, J.M. Moreno, C. Munoz, M. Quiros, Nucl. Phys. B
328 Ž1989. 272.
w9x P. Binetruy, G. Dvali, Phys. Lett. B 388 Ž1996. 241.
w10x E. Halyo, Phys. Lett. B 387 Ž1996. 43.
w11x R. Jeannerot, hep-phr9706391.
w12x G. Dvali, A. Riotto, hep-phr9706408.
w13x J.A. Casas, G.B. Gelmini, hep-phr9706439.
w14x D.H. Lyth, A. Riotto, hep-phr9707273.
w15x M. Bastero-Gil, S.F. King, hep-phr9709502.
w16x G. Dvali, Q. Shafi, R. Schaefer, Phys. Rev. Lett. 73 Ž1994.
1886.
w17x K.-I. Izawa, T. Yanagida, hep-phr9608359.
w18x A.D. Linde, A. Riotto, hep-phr9703209, to be published in
Phys. Rev. D.
w19x S. Dimopoulos, G. Dvali, R. Rattazzi, hep-phr9705348.
w20x K. Freese, J. Frieman, A.V. Olinto Phys. Rev. Lett. 65
Ž1990. 3233; F.C. Adams et al. 1993, Phys. Rev. D 47
Ž1993. 426; L. Knox, A. Olinto, Phys. Rev. D 48 Ž1993. 946;
W.H. Kinney, K.T. Mahanthappa, Phys. Rev. D 52 Ž1995.
5529; D 53 Ž1996. 5455; J. Garcia-Bellido, A. Linde, D.
Wands, astro-phr9605094.
w21x M.K. Gaillard, H. Murayama, K.A. Olive, Phys. Lett. B 355
Ž1995. 71.
w22x M.K. Gaillard, H. Murayama, D.H. Lyth, in preparation.
w23x E. Witten, Nucl. Phys. B 471 Ž1996. 135.
w24x This observation has been made by T. Banks, M. Dine,
hep-thr9609046.
w25x E.D. Stewart, hep-phr9703232.
12 February 1998
Physics Letters B 419 Ž1998. 62–72
D-brane interaction in the type IIB matrix model
Bhabani Prasad Mandal 1, Subir Mukhopadhyay
2
Institute of Physics, SachiÕalaya Marg, Bhubaneswar-751005, India
Received 17 September 1997
Editor: L. Alvarez-Gaumé
Abstract
We calculate the potential between bound states of D-branes of different dimension in IIB matrix model upto one loop
order and find nice agreement with the open string calculations in short and large distance limit. We also consider the
scattering of bound states of D-branes, calculate the scattering phase shift and analyze the effective potential in different
limits. q 1998 Elsevier Science B.V.
1. Introduction
Recently, a non-perturbative formulation Žmatrix theory. w1x has been proposed for the M-theory at infinite
momentum frame. Making use of the fact, that at infinite momentum frame all other degrees of freedom with
finite momentum get decoupled, the M-theory has been shown to be described by that of a system of large
number of D-particles. The theory is given by a large-N super Yang-Mills ŽSYM. theory in Ž0 q 1. dimension.
A number of issues have been tested such as supergraviton spectrum, D-brane interaction and various kinds of
compactifications w1–7x.
This model has inspired another large-N super Yang-Mills theory in Ž0 q 0. dimension w8x which is proposed
to describe the non perturbative physics of type IIB string. It has its origin in the fact that a large-N reduced
SYM theory describes string at a proper double scaling limit w9x. However, a closer look reveals that this is the
theory of D-instanton Žapart from a chemical potential term in the action. and one can use a similar infinite
momentum frame argument to arrive at this theory w10x. This theory, by compactification on one circle and a
subsequent wick rotation leads to the BFSS matrix model w1x. The massless spectrum and world volume theories
of D-branes can be recovered from this type II B matrix model w8,9,11x.
In this paper, we shall study the interaction of the D-branes in the latter formulation which, among other
things, serves as a consistency check for the type IIB matrix theory and helps to understand the relation between
these two matrix theories.
1
Address after November 97: Theory group, Saha Institute of Nuclear Physics, 1rAF Bidhannagar, Calcutta-64, India. E-mail:
[email protected].
2
E-mail: [email protected].
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 2 1 - 4
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
63
The D-branes are the BPS states of the string theory which couple to the RR fields. Usually a p D-brane
couple to a Žp q 1. form potential. By virtue of the interaction term HC n tr Ž e Ž i F . ., a p-brane with electro-magnetic field turned on in its world volume can interact with a brane of lower dimension and thus can form a
non-threshold bound states w7x. These composite objects with large electric or magnetic field turned on in its
world volume, i.e. a D-brane coupled to a large number of lower dimensional D-branes, come out naturally in
the matrix theory as the classical solutions of the equation of motion. So from Matrix theory one can study the
interactions between these D-branes. Since the Matrix theory describes the interaction mediated through the
open string, it should reproduce the short distance behavior. Further, due to the presence of a large number of
lower dimensional D-branes it should also reproduce the long distance limit.
In the BFSS theory w1x, the interaction among D-branes has been studied in one loop order of the matrix
theory leading to a satisfactory agreement with the open string calculation upto one loop both at short and long
distance limit w5–7x. In the IKKT theory w8x, the long distance limit of the interaction potential for some
configurations breaking half of the SUSY, has been compared with the result obtained from the Born-Infeld
action and nice agreement has been reached w13x. The short distance limit has also been analyzed in w14,15x but
the lower dimensional branes have not been considered.
Here we shall study the potential and the scattering phase shift from the IIB matrix theory and compare the
results with those obtained from the open and the closed string calculations.
The plan of this paper is as follows. In the next section we will discuss some preliminaries concerning the
IKKT model. Then in the two following sections the potential energy and the scattering between two D-branes
of odd dimensions will be discussed. Finally we will conclude discussing our results and possible further
extensions.
2. Preliminary
In this section, we describe the necessary preliminaries required for the study of interaction among D-branes
in the type IIB matrix model w8x. Let us start with the action which is obtained by reduction of ten dimensional
Euclidean SYM theory with SUŽ N . gauge group reduced to zero dimension.
1
S s a y Tr Am , An
4
½
2
1
y
2
Tr cG m Am , c
ž
/
5
qbN
Ž 2.1 .
where Am and c are gauge fields in the SUŽ N . adjoint and so we can consider them to be N = N Hermitian
matrices. The parameter N is considered to be a dynamical variable, a and b are related to the string coupling.
The third term implies that the sum over N in partition function should be taken with the weight factor eyb N .
Later the sum over N is replaced by a double scaling limit w9x.
This action can be viewed either as obtained from type IIB theory by taking the Schield gauge w8x or as the
action for N D-instantons w10x apart from the last term. It has the manifest Lorentz invariance and N s 2
supersymmetry. The SUSY transformations are given by
d Ž1.c s
i
2
Am , An G mne ,
d Ž1.Am s eGm c ,
d Ž2.c s j ,
d Ž2.Am s 0.
Ž 2.2 .
The equation of motion is given by
A m , Am , An
s0
Am , Ž G mc . a s 0
Ž 2.3 .
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
64
The solution of the equation of motion correspond to non-perturbative states in IIB theory. In particular, those
cases for which w Am , An x s c y number, are interesting as they preserve half of the supersymmetry and are
interpreted as BPS states. The odd p-branes, being BPS states in type IIB theory, are associated with this type of
solutions.
So a plausible choice for the p-D-brane solution is
Amcl s Bm ,
for 0 F m F p,
cs0
Amcl s 0 otherwise
Ž 2.4 .
where Bm’s are the N = N Hermitian matrices satisfying w Ba , Bb x s yic a b I, and a,b s 0, PPP p with c a b is a c
number.
A consistent supersymmetry algebra is yet to be constructed Žan attempt was made in w15x. but the central
charge of a p-brane can be taken as w4,14,15x
A i 2 , A i 3 PPP A i py 1 , A i p
Zi 1 , PPP i p s A 0 , A i 1
Ž 2.5 .
which is consistent with this solution. But this means that the p-branes contain non zero electric or magnetic
flux and so it actually correspond to a large number of lower dimensional branes coupled to it.
In general, this solutions represent D-branes of infinite extension. However, we would like to deal with finite
ones for the time being. So we wrap the p-brane on a torus of dimensions L i . For the sake of convenience of the
calculation, let us make use of the Lorentz transformation to cast the c a b in a block diagonal form with
eigenvalues v i ’s. Then the eigenvalues, related to L i by the relation
2pv i s
L2 iy2 L2 iy1
Ž 2.6 .
Ni
represents the amount of flux linked with the respective planes.
At the end of the calculation we will take the infinite limit for the L i ’s and Ni keeping v i finite. So we can
consider the Bm’s to be Hermitian operator on some Hilbert space and the matrix index can be taken to be
continuous. Further, we, use the Schrodinger
representation for the Hilbert space operators Bi and write
¨
B2 iy2 s i v i E i ;
B2 iy1 s qi
i , j s 1,2 PPP
Ž 2.7 .
where E i ' ErE qi and qi represent conjugate operators on the Hilbert space. Finite values of L i ’s imply that the
eigenvalues of qi are distributed in wy L2q , L2q x.
We will consider the interaction between p-branes upto one loop. Summing over the fluctuations and
considering only the quadratic terms, the one loop effective action of the matrix model can be written in a
compact form w8x
Ws
1
2
Tr ln Ž P 2dmn y 2 iFmn . y
1
4
Tr ln P 2 q
ž
i
2
Fmn G mn
/ž
1 q G 11
2
/
y Tr ln Ž P 2 .
Ž 2.8 .
where Pm and Fmn are operators acting on the space of Hermitian matrices and are defined by
Pm X s Amcl , X ,
Fmn X s i
Amcl , Ancl , X
Ž 2.9 .
In other words Pm and Fmn are the adjoint representation of Am and Fmn respectively.
Here we will restrict ourselves to the one loop order. We will get the classical solution corresponding to the
configuration and obtain the effective action from Ž2.8.. This analysis can be extended to higher loop orders but
that will involve w16x explicit Feynman diagrams and the effective action will be much more complicated.
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
65
3. Potential between branes
In this section we will consider the potential between two branes of dimensions p and q respectively. A
configuration of two parallel branes of same dimension is known to saturate the BPS limit and so the potential is
zero. For p / q in the generic case it breaks the supersymmetry and so the potential is nonzero. All these have
been investigated by usual D-brane analysis w12,17,18x. However, in matrix model, instead of getting pure
D-branes, the classical solutions represent D-branes with electric and magnetic fluxes turned on in its world
volume, which, as stated earlier, represent configurations of D-branes of different dimensions w3,6,7x. Our aim is
to carry out similar kind of investigation in the IIB matrix model framework.
We shall consider the two brane configuration as the classical solution of the equations of motion Ž2.3.. Then
we will sum over all fluctuations around it upto one loop order using Ž2.9. and get the interaction potential as
described in Ref. w8x.
Let us start with the two branes of p and q dimensions parallel to the X 1 PPP p and X 1 PPP q planes and place
them at a distance "br2 along X qq 1 respectively.
The classical configuration is given by,
Bi
0
ž
Ai s
A qq 1 s
0
,
Bi
/
b 1
2 0
for 0 F i F p;
0
;
y1
ž
/
Ai s
ž
Bi
0
0
,
0
/
for p q 1 F i F q ;
A i s 0 for i ) q q 1,
Ž 3.1 .
where Bi ’s are the Hermitian operators on Hilbert spaces mentioned earlier.
¨
So the B’s in the Scrodinger
representation are given by B2 iy2 s i vE i , B2 iy1 s qi for 1 F i F l with the
value of v is given by Ž2.6..
In order to get the potential we have to sum over the fluctuations using Ž2.8. around this classical solution
Ž3.1.. The first thing we need is the adjoint representation of the operators corresponding to Am given by the
action on an arbitrary Hermitian matrix according to the Ž2.9.. This is given by
P2 iy2
P2 iy2
for
Pqq 1
ž
ž
ž
X
Y†
Y
X
s i v i Ž E i1 q E i2 .
Z
Y†
Y
,
Z
X
Y†
i v i Ž E i1 q E i2 . X
Y
s
Z
i vi E 2 Y †
i vi E 1Y
pq3
2
X
Y†
/
/
FiF
ž
ž
0
/
,
ž
X
Y†
P2 iy1
Y
X
s Ž qi1 y qi2 .
Z
Y†
/
ž
X
Y†
ž
Y
s
Z
/
ž
Y
,
Z
/
for 1 F i F
Ž qi1 y qi2 . X
qi1 Y
yqi2 Y †
0
/
pq1
2
,
qq1
2
Y
0
s br2 †
Z
Y
/
/
P2 iy1
ž
Y
,
Z
/
Ž 3.2 .
where each entry is an infinite dimensional matrix e.g. X s X Ž q 1 ,q 2 . and also P1 s i v 1 E 1 d Ž q11 y q 2 y 1.,
q1 s q 1d Ž q11 y q12 . etc. are infinite dimensional matrices.
The adjoint representation of the Fmn can be obtained by using Ž2.9. from which we see that the only Fi,iq1
components have non-trivial actions. Further they act only on the matrices of the form
correspond to a non-zero eigenvalue. The eigenvalues are v i with k - i F l.
Similarly the eigenvalues of P 2 can be obtained as
k
Es2
0
Y†
Y
0
/ which
l
Ý Ž pi2 q q˜i2 . q Ý Ž pi2 q qi2 .
is1
ž
kq1
Ž 3.3 .
;
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
66
where w pi ,q j x s yi v i d i j and other commutators are vanishing. It looks like Hamiltonian of k free particles and
l y k oscillators.
Collecting all these three we get the interaction potential from the Ž2.8.. After Simplification it becomes
Ws
1
D
Ý Tr ln Ž E y 2 a i . q Tr ln Ž E q 2 a i . y 2 2y D Ý Tr ln Ž a i li . q Ž 4 y D . ln E
2
1
Ž 3.4 .
l i4
where l i s "1, D s l y k and a i s v kqi .
In order to compare this with the existing result it is better to write the potential in the form of a proper time
integral. Using the identity w14x,
ds
ln a s y
H s exp Ž ysa.
Ž 3.5 .
we can write the potential as
Ws
D
ds
ys E
H s Tre
½
D
Ý cosh Ž 2 s v i . y 4 Ł cosh Ž s v i . q Ž 4 y D .
is1
1
5
,
Ž 3.6 .
where E is given by Ž3.3..
The trace can be evaluated in a straightforward manner and is given by
Treys E s
N 2k
ž /
L0 PPP L p 2 s
l
k
p
exp Ž yb 2 s .
Ł Ž 2sinh v i s . y1
Ž 3.7 .
iskq1
Using this we can get
l
k
W s Cp Ł
is1
1
ds
<2pv i < 2
Hs
exp Ž yb 2 s .
Ý
k
p
ž /
l
cosh Ž 2 s v i . y 4 Ł cosh Ž s v i . q 4 y D
P
kq1
iskq1
Ž 3.8 .
l
2s
Ł
2sinh Ž v i s .
iskq1
where C p is the volume of the hyperplane which is common to both the branes.
Note that this expression is almost identical to the expression of the phase shift obtained in the scattering of
D-particle and 4-branes in w7x. Only difference is that they consider the scattering while here we are considering
the static potential. This confirms the T-duality between the two matrix models.
As mentioned earlier this represents the potential between p and q-branes with stack of lower dimensional
branes attached to these. In order to compare this with the result obtained from open string calculation upto one
loop we consider the specific brane configurations.
For the sake of brevity let us consider the 3-brane and 1-brane configuration. So we put q s 3 and p s 1.
The 3-brane has a magnetic field turned on in its world volume on the X 23 plane with strength HC 23 F ' v 2 .
Besides, both the 3 and the 1-brane has an electric field in their world volume with HC 01 F ' v 1. So the 3-brane,
considered in this matrix model actually represents a bound state of a 3-brane,
L 0 L1
2pv 1
L2 L3
2pv 2
D-strings along the X 1 and
D-instantons and similarly the 1-brane represents a bound state of a 1-brane with
L 0 L1
2pv 1
D-instantons.
In this case the potential is reduced to
W s L1
1
<2pv 1 <
ds
2
2
p
H s exp Ž yb s . ž 2 s / P
cosh Ž 2 s v 2 . y 4cosh Ž s v 2 . q 3
2sinh Ž v 2 s .
Ž 3.9 .
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
67
Note that the contribution from the common field always comes through the free particle term in the P 2
eigenvalue Ž3.3. and so always gives rise to an overall factor. The tachyonic instability w19x, like that in the
BFSS model w7x is also present for very small separation.
We consider the potential for the above configuration already calculated from open string theory upto one
loop. A 3-brane with electric field F1 along X 1 and a magnetic field F2 along X 23 plane interacts with a
1-brane with the same electric field in its world volume through the term HC n tr Ž e i F ..
The potential is given by w12x
As
L Ž 1 q F12 .
2p
ds eyb
Hs
2
s
4p s
B=J,
Ž 3.10 .
where B and J are the contribution from the bosonic and the fermionic term respectively and the prefactor takes
1
care of the common electric field. Also we have taken a X s
.
2p
B and J are given by
Bs
1
2
J s Ž yf 24Q 2 Ž i e s . q f 34Q 3 Ž i e s . q f 44Q4 Ž i e s . .
y1
fy4
1 Q1 Ž ie s. ,
Ž 3.11 .
where e is related to the magnetic field by tanŽpe . s F2
The matrix model describes the interaction through the open string and so let us consider the short distance
limit of this potential.
If we write e i s p2 y pv i where tanŽpe i . s Fi and consider the contribution of the massless modes only the
in the limit of large fields i.e. small v the potential can be written as
W s L1
1
ds
<2pv 1 <
2
2
p
H s exp Ž yb s . ž 2 s / P
cosh Ž 2 s v 2 . y 4cosh Ž s v 2 . q 3
Ž 3.12 .
2sinh Ž v 2 s .
which is precisely the result obtained from the matrix model.
The comparison also makes it clear that, at least at the short distance limit, the bosonic contribution comes
from the treys E term while the Žy1. F of the R sector contribution comes from the second term in the braces of
Ž3.6.. It is interesting that how the latter will flip sign in the case of the brane-antibrane potential.
We can now calculate the long range potential from the matrix model calculation for p, q-branes and
compare with that obtained in string theoretic calculation. A straightforward approximation for the large value
of b in the expression for potential in Ž3.8. leads to the expression
1
pk 2
2
l
Ý
v i4 y Ý v 12 v j2
iskq1
i-j
l
2
lyk
Ł
G Ž 4 y l . b 2 ly8 ' V Ž b .
Ž 3.13 .
vi
iskq1
Now we will see how this potential depends on b for the specific cases .
of 3-brane and 5-brane interaction, V Ž b . s
p4
1
8
3 4
F23
b
ž /
ž /
For 1-brane and 3-brane interaction, V Ž b . s
p4
1
8
F453 b 4
between 1-brane and 5-brane. In that case V Ž b . s
which is a repulsive potential. Similarly for the case
. Something interesting happens for the interaction
p Ž v 22 y v 32 .
16 v 2 v 3
2
V Ž b. s
p3
ž /Ž
b
4
2
F23
y F452 .
3
3
F23
F45
. and the potential
vanishes for F23 s F45 . This is because when the magnetic fields in the X 23 and X 34 planes are equal the
configuration becomes a BPS state and so there is no force between them.
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
68
Other D-brane bound states can be analysed in a similar manner and those also lead to a similar kind of
agreement. Also one can consider the D-branes oriented orthogonally to each other. Further all these results also
match with those obtained from BFSS model which is an evidence for the duality between the matrix theories.
4. Scattering of branes
In this section, we will consider the interaction between two moving branes of different dimensions. Again
such a configuration is not a BPS state and breaks the supersymmetry. We will calculate the phase shift of one
brane Žof lower dimension. while passing by the other in an eikonal approximation w6,7x in the matrix model and
then compare with the results obtained by D-brane technique w17,18x.
Let us consider two branes of dimension p and q parallel to the X 1 PPP p and X 1 PPP q hyperplanes moving
parallel to X qq 1 with a relative velocity Õ at a relative separation b along X qq 2 axis. The classical solutions
corresponding to this configuration can easily be obtained by boosting the static solutions.
These solutions are:
A0 s
ž
B0 cosh e
0
A qq 1 s
ž
0
;
B0 cosh e
/
B0 sinh e
0
Ai s
0
;
B0 sinh e
/
ž
Bi
0
A qq2 s
0
,
Bi
/
b 1
2 0
for 1 F i F p;
0
;
y1
ž
/
A i s 0,
Ai s
ž
Bi
0
0
,
0
/
for i G q q 3.
for p - i F q ;
Ž 4.1 .
where Bi ’s have the usual meaning mentioned earlier, Õ ' tanh e and e is the boosting angle.
Now we consider the action of the solution on Hermitian matrices in the adjoint representation. This time the
representations are more complicated than the previous one due to the presence of boost. The explicit forms are
P0
ž
X
Y†
Pqq 1
Ppq 1
Ppq 2
Pi
ž
ž
ž
ž
Y
X
s i v 1cosh e Ž E 11 q E 12 .
Z
Y†
/
ž
1
Y
,
Z
/ ž
2
X
Y†
Ž E1 qE1 . X
Y
s i v 1 sinh e
Z
y Ž E 11 y E 12 . Y †
X
Y†
Y
s i v k 1q1
Z
X
Y†
Y
s
Z
X
Y†
Y
s 0,
Z
/
ž
ž
1
2
Ž E kq
1 . q E kq1 . X
1
E kq1
Y
2
†
E kq
1Y
0
1
2
Ž qkq
1 y q kq1 . X
1
qkq1
Y
2
†
yqkq
1Y
0
X
Y†
Y
X
s Ž q11 y q12 .
Z
Y†
/
Ž E 11 y E 12 . Y
y Ž E 11 q E 12 . Z
ž
/
/
/
P1
/
,
/
Y
,
Z
ž
/
/
,
,
Pqq2
ž
X
Y†
Y
0
sb †
Z
Y
/ ž
Y
,
Z
/
for i G q q 3,
Ž 4.2 .
where q 1 and q 2 are the first and the second arguments of the infinite dimensional matrices.
We evaluate the action of Fi j in the adjoint representation and it has the non-zero action only on the matrices
Y.
F1, qq1
ž
X
Y†
Y
0
s 2 v 1 sinh e
Z
Y†
/
ž
Y
;
0
/
F2 iy2,2 i
ž
X
Y†
Y
0
s yi v i
Z
yY †
/
ž
Y
,
0
/
for i s k q 1 PPP q
Ž 4.3 .
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
69
Therefore Fi j gives nontrivial results only when it acts on Y. So we plug in the above expressions in the Eq.
Ž3.8. and get the phase shift to be
`
ReW s
H0
ds
s
l
ž
l
Ý cosh Ž 2 s v i . y 2 1y D 2cosh Ž 2 s v 1 sinh e . Ł 2cosh Ž s v i .
Treys E cosh Ž 2 s v 1 sinh e . q
kq1
kq1
q3 y D .
Ž 4.4 .
where D s l y k.
The eigenvalues of P 2 in the adjoint representation can be obtained in a straightforward manner which again
represents the Hamiltonian of some free particles and some harmonic oscillators.
k
E s 2 p 12 cosh2e q 2
l
Ý Ž pi2 q q˜i2 . q Ý
is2
pi2 q qi2 q 2 p 12 sinh2e q qi2 q b 2
Ž 4.5 .
iskq1
where w pi ,q j x s yi v i d i j for k - i, j F l and all other commutators vanish.
Now we calculate Treys E as
ys E
Tre
s
N 2 ky1 L1
p
p
2
ž /
L0 L1 PPP L p 2 s
P
eyb
1
cosh e
2
s
Ž 4.6 .
l
2sinh Ž s v 1 sinh e .
Ł
2sinh Ž 2 s v i .
iskq1
Substituting this expression for Trace in 3.6 we get
k
ReW s yVp 2pv i Ł
i
1
ds
`
Ž 2pv i .
2
H0
s
eyb
2
s
p
p
ž /
2
Ž 4.7 .
2s
So the expression of the phase shift can be written as
yp
ReW s yVp 2
2
yp
k
v1 Ł
is1
1
`
v i2
H0
ds
s
yb 2 S
e
Ž 4p s .
2
l
cosh Ž 2 s v 1 sinh e . q
l
Ý cosh Ž 2 s v i . y 2 1y D 2cosh Ž s v 1 sinh e . Ł 2cosh Ž s v i . q 3 y D
kq1
kq1
l
cosh e 2sinh Ž 2 s v 1 sinh e .
Ł
Ž 4.8 .
2sinh Ž s v i .
iskq1
where Vp represents the space time volume of the p-brane.
In order to compare with the open string result it is better to consider a specific configuration. We consider a
simple one – in between a 3-brane and a 1-brane. So we take p s 1 and q s 3. Also as in the last section the
3-brane has non-zero magnetic field along the X 23 plane and each of the branes has an electric field in X 1
direction.
Then the phase shift gets reduced to
y1
ReW s y
V1
4'2
v1
1
v 12
`
H0
ds
s
yb 2 s
e
Ž 4p s .
2
2cosh Ž 2 s v 1 sinh e . q 2cosh Ž 2 s v 2 . y 8cosh Ž s v 1 sinh e . cosh Ž s v 2 . q 4
2cosh e sinh Ž 2 s v 1 sinh e . 2sinh Ž s v 2 .
Ž 4.9 .
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
70
It is difficult to obtain the scattering phase shift between two D-brane configurations with electric field in
their world volume from the open string calculation. The presence of the electric field and the velocity mixes up
the coordinates and it is complicated to get a basis in which the energy momentum tensor can be diagonalized.
But we have seen, in the case of the potential and also in the cases of scattering with a magnetic field, that the
matrix model result gives the correct short distance behavior. So we can demand that the matrix model gives the
phase shift for branes with electric fields in world volume, at least at the limit of infinite field strength.
In the case of scattering of p, q-branes we can also calculate the long range potential from matrix model
calculations. This can be obtain from Ž4.8. by considering large b approximation.
l
tanh e
1
Ł
ai
iskq1
l
Ý a4i y 2 Ý a2i a2j
isk
ak Ł ai b
i-j
'V s Ž b.
7y q
Ž 4.10 .
where a i s v i for k q 1 F i - l and a k s v 1 sinh e . This is the effective potential responsible for the scattering.
In the case of two branes of same dimension it gets reduced to
V s Ž b, e . s v 12 sinh2e tanh2e b qy 7
Ž 4.11 .
2
In the low velocity limit the leading order term is V Ž b,Õ . s
p Õ
4
2 7yp
F01
b
. So it reproduces the correct leading order
behavior of the potential. For electric field becomes infinite, the boundary conditions becomes Dirichlet and so
the potential vanishes.
Now let us consider the specific cases: For 1-brane and 3-brane scattering case,
s
V Ž b, e . s
tanh e Ž v 12 sinh2e y v 22 .
v 1 v 22 sinh e
Therefore for sinh e s
V Ž b,Õ . ;
p F01
2 4
F34
b
F01
F34
ž
2
by4 .
again the potential will vanish giving rise to BPS state. In the low velocity limit,
1y
2
F34
2
F01
2
Õ
/
.
Similar interpretation will hold good for the case of 3-brane scattering with 5-brane.
For p s 1, q s 5 the potential that corresponds to the phase shift is given by
V Ž e ,b . ;
tanh e
v 12 sinh e
=
1
v 23 v 32
Ž v 1 sinh e y v 2 q v 3 . Ž v 1 sinh e q v 2 y v 3 . Ž v 1 sinh e y v 2 y v 3 .
= Ž v 1 sinh e q v 2 q v 3 . = by2
So for sinh e s "
Ž v2 y v3.
v1
,"
Ž v2 q v3.
v1
Ž 4.12 .
the potential will vanish.
In the low velocity limit it becomes
v1
2
V Ž b,Õ . s
v 12 Õ 2 y Ž v 2 y v 3 .
3 2
Ž v1v2 v .
v 12 Õ 2 y Ž v 2 q v 3 .
2
It is interesting that here the potential changes its sign twice with the increase of the velocity.
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
71
The matrix model results, as mentioned earlier, is expected to reproduce the short distance behavior. For
these cases of bound states of D-branes this should be true at least in the limit of the large electromagnetic field
which we have checked in the case of the potential. On the basis of the presence of large field we can similarly
argue w6,7x that the long distance limit should also be reproduced reliably. It will be interesting to check these
scattering of the bound states of D-branes Žwith electric field. in the closed string theory.
5. Discussion
In this paper, we have discussed the potential between D-branes with electromagnetic fields turned on in its
world volume in the matrix model framework. This sort of configuration correspond to the stack of lower
dimensional branes attached to it and thus forming a non-threshold bound state. Such a configuration, usually,
does not saturate the BPS bound but we have seen that they correspond to the classical solutions of the matrix
model which breaks half of the supersymmetry. The fact that the number of the lower dimensional branes is
infinity makes this possible.
The interaction of these kind of objects have been discussed in w12x from the open string calculation. We
have seen that our result agrees with those obtained from open string calculation at the short distance limit. This
is expected since the matrix model takes care of the massless modes of the open strings and the open strings are
known to dominate the dynamics at the short distance limit. We have matched our result with the long distance
limit also which is dominated by the closed strings. This is again due to the fact that the electromagnetic fluxes
are very large.
Apart from getting agreement with the string theory, we have also noted the similarity between the potential
between the odd and the even dimensional branes obtained from the two different matrix models which is an
evidence for the duality between the two matrix models. This is also expected since they describe the IIA and
IIB theories and the dynamics are also of the D-particles and the D-instantons which are T-dual to each other.
We have also calculated the scattering phase shift between branes of odd dimensions with electromagnetic
fields. Due to the presence of the electric fields i.e the D-instantons attached to the branes, this type of
scattering is difficult to study in the open string calculation. On the other hand it is easier to study it in the
matrix model framework. On the basis of other agreements we can propose the result to be consistent with the
true short distance behavior at the limit of large field. The long distance limit is also interesting and for a few
choices of velocities the potential obtained from the phase shift vanishes, signaling existence of BPS saturation.
A similar investigation in the closed string theory may be interesting.
Finally, we have considered the trivial classical solution of the matrix model. There are non-trivial solutions,
such as instanton solutions which correspond to other D-brane configurations and the the dynamics of these kind
of objects can be studied in a similar manner. Also there are anti branes whose interaction with the branes may
be studied in this framework. However as mentioned earlier in such a case the fermionic contribution should flip
its sign and it is not apparent how it will come about. By using the T-duality argument in the matrix model one
can study the interaction between the D-branes and the NS 5-branes of type IIB theory in away similar to that
used in w20x. Also it will be interesting if the pure D-brane or a D-brane with finite flux turned on can be
constructed and studied in this matrix model framework.
Acknowledgements
It is a plesure to thank Dr. A. Kumar for useful discussions and for careful reading of the manuscript. We
would also like to thank K. Roy for useful discussions and suggestions.
72
B.P. Mandal, S. Mukhopadhyayr Physics Letters B 419 (1998) 62–72
References
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w6x G. Lifschytz, S.D. Mathur, Supersymmetry and membrance interaction in M Žatrix. theory, hep-thr9612087.
w7x G. Lifschytz, Four-brane and six-brane interactions in M Žatrix. theory, hep-thr9612223.
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w9x B. Sethiapalan, Fundamental strings and D-strings in the type IIB matrix Model, hep-thr9703133.
w10x A.A. Tseytlin, Phys. Rev. Lett. 78 Ž1997. 1864, hep-thr9612164.
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model, hep-thr9711040.
w12x G. Lyfshitz, Comparing D-branes to black branes, hep-thr9604156; probing bound states of D-branes, hep-thr9610125.
w13x I. Chepelov, A.A. Tseytlin, Interaction of type IIB D-branes from D-instanton, hep-thr9705120.
w14x A. Fayyazuddin, Y. Makenko, P. Olesen, D.J. Smith, K. Zarembo, Nucl. Phys. B 499 Ž1997. 159, hep-thr9703038.
w15x I. Chepelov, Y. Makeenko, K. Zarembo, Properties of D-branes in matrix model of IIB superstring, hep-thr9701151; A. Fayyazuddin,
D.J. Smith, Mod. Phy. Lett. A 12 Ž1997. 1447.
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12 February 1998
Physics Letters B 419 Ž1998. 73–78
Spin-orbit interaction from Matrix Theory
Per Kraus
1
California Institute of Technology, Pasadena, CA 91125, USA
Received 13 November 1997
Editor: M. Dine
Abstract
We study the leading order spin dependence of graviton scattering in eleven dimensions, and show that the results
obtained from supergravity and from Matrix Theory precisely agree. q 1998 Elsevier Science B.V.
PACS: 11.25.-w
Keywords: D-branes; Matrix Theory; Spin-orbit interaction
1. Introduction
There are by now a substantial number of checks
of the correspondence between Matrix Theory w1x
and eleven dimensional supergravity. Impressive
though they are, these checks have probed only a
very limited part of the structure of supergravity.
Among the most striking of the checks are the
successful computations of the Õ 4rr 7 and Õ 6rr 14
terms arising in the scattering of two gravitons w1–4x.
The results can be thought of as probing the cubic
and quartic interaction vertices of supergravity.
However, these calculations effectively average over
the polarizations of the scattered gravitons, and so
are sensitive only to the magnitude of the vertices
and not to their tensor structures. In this work we
will try to see whether the tensor structure comes out
correctly by studying the leading order spin dependence of graviton scattering.
1
Work supported in part by DOE grant DE-FG03-92-ER40701
and by a DuBridge fellowship.
E-mail: [email protected].
There have recently been some string theory analyses of the spin dependence of D0-brane scattering,
or equivalently, graviton scattering in compactified
M theory. w5x used a series of duality transformations
to map the problem to one involving fundamental
strings, and w6x approached the problem through the
boundary state formalism. Here we proceed somewhat differently, by finding the linearized metric of a
spinning D0-brane and studying the the action of a
D0-brane probe moving in that background. Since
we will be working with the linearized theory, we
will only pick up terms of first order in the spin,
whereas w5,6x found the higher order contributions as
well. In the present approach, by solving the full
field equations it should be straightforward to recover the extra terms, but we will not attempt that
here.
Our approach has the advantage that it can be
extended to include contributions which are of higher
orders in the gravitational coupling. The problem
with the other methods is that to compare with
Matrix Theory beyond the lowest order one must
consider not standard IIA theory, but rather the
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 5 0 1 - 3
P. Krausr Physics Letters B 419 (1998) 73–78
74
theory resulting from compactifying M theory along
a null direction w7x. In our framework this can easily
be implemented by lifting the solution to eleven
dimensions and then recompactifying along a null
direction. It is less clear how to proceed within the
framework of w5,6x.
On the Matrix Theory side, we will compute one
loop contributions to the effective action in the presence of both bosonic and fermionic background
fields. The fermions encode the spin of the D0-brane
in precisely the right way to reproduce the supergravity result. The term we compute is part of the
supersymmetric completion of the bosonic Ž Fmn . 4
terms which arise at one loop. It would be nice to
demonstrate that supersymmetry is sufficient to fix
the coefficients of the fermionic terms.
Before proceeding to the calculations, let us mention that in principle an efficient way of approaching
the supergravity side of the problem would be through
use of the supersymmetric Born-Infeld action w8–10x.
The coupling of a supersymmetric D0-brane to target
space fields has been worked out in w9,10x. However,
the results there are presented in terms of superfields,
whereas to apply them to the problem we are studying one really needs to work out their component
forms.
where v 8 is the area of the unit 8-sphere. It is
convenient to introduce the quantity
Q0 s
16p GN
7v 8
E
h 00
s
h iEj s
8 r7
1 Q0
8 r7
h 0Ei s y
h iEj s
16p GN T0
8v8
r7
16p GN T0
56 v 8 r 7
h 0Ei s y
di j
8p GN x k J k i
v8
r9
.
Ž 1.
r9
2 T0
.
Ž 2.
Now we transform to the string metric using,
S
E
gmn
s e f r2 gmn
.
To linear order, the dilaton takes the same value as it
does in the unspinning case,
ey f s 1 y
3 Q0
4 r7
.
Thus we find
g iSj s 1 q
E
h 00
s
.
di j
7 Q0 x k J k i
2. Spin-orbit potential from supergravity
with hmn s diagŽy,q,q PPP q., the needed formulas are
2T0
7 Q0
ž
E
E
s hmn q hmn
,
gmn
15
Note that we are working in string units Ž2pa X s 1..
Then,
S
g 00
sy 1y
We would like to compute, in ten dimensions, the
linearized metric produced by a D0-brane of mass T0
and angular momentum J i j. It is easiest to begin by
considering the Einstein metric; formulas for the
linearized metric are then found in w11x. Writing
T0 s
ž
g 0Si s y
Q0
2 r7
Q0
2 r7
/
/
di j
7 Q0 x k J k i
2 T0
r9
.
Ž 3.
S
g 00
, g iSj of course take their standard values, while
S
g 0 i gives the spin contribution.
Actually, the metric that we want is obtained from
the one above by dropping the 1’s in the parentheses
S
of g 00
, g iSj . This is because we want a solution
corresponding to the theory obtained by compactifying eleven dimensional supergravity along a null
direction. Such a solution can be generated by lifting
the metric Ž3. to eleven dimensions using the standard formulas for spacelike compactification, and
then returning to ten dimensions by compactifying a
null direction. The result is precisely to remove the
1’s just mentioned. In fact, this procedure is unecessary in the present context, as it only affects terms of
P. Krausr Physics Letters B 419 (1998) 73–78
higher power in velocity than the spin-orbit term, but
we mention it here for completeness. From now on
we drop the 1’s and refer to the resulting metric as
simply gmn .
Next, we consider the action of a D0-brane probe
moving in this background:
S0 s yT0 dt eyf y gmn X m˙ X n˙ y Cm X m˙ .
½ (
H
5
Ž 4.
To compute the potential we need to know the value
of the RR gauge field Cm . We take
C0 s y
Q0
r7
;
Ci s
7 Q0 x k J k i
4 T0
r9
.
The form of C0 is the conventional one. The value
for Ci could be arrived at by determining the magnetic moment of the D0-brane and using the standard
formula for the resulting magnetic field. Instead we
have chosen the coefficient 7r4 so as to cancel a
term linear in velocity coming from the expansion of
tive action of a system of two D0-branes. Before
doing any explicit computation, dimensional analysis
and the systematics of the loop expansion allow one
to write the general form of the effective action
w4,12x,
Sl ; dt r 4y 3 l f
H
S0 s yT0 dt 1 y 12 z P z
H
ž
Õ
r
2
c
,
r
3r2
/
.
Ž 6.
Here Sl denotes the l’th loop contribution. Note that
at the one loop level the action includes the term
Õ 3c 2rr 8 , and that this term has the correct Õ and r
dependence to match onto the spin-orbit term of Ž5..
We would like to check whether the numerical coefficient and the tensor structure similarly agree.
We will be following the conventions of w3x, and
some details of the calculation which we omit can be
found there. The Matrix Theory action, including
gauge fixing and ghost terms, is
S s dt Tr
H
(y gXX˙ ˙ . The linear term is known to be absent
Žfrom the calculations of w5,6x, for example.. At any
rate, this term is irrelevant as far as the coefficient of
the spin-orbit term is concerned.
Now, inserting the fields into S0 and expanding in
powers of velocity we find
75
½
T0
2
Fmn F mn y i c Du c q T0 D mAm
ž
qSghost ,
2
/
5
Ž 7.
where m , n s 0 . . . 9, Am s Ž A, X i ., and
D mAm s yE t A q B i , X i
F0 i s E t X i q w A, X i x ,
Fi j s X i , X j
Dt c s E t c q w A, c x ,
Di c s w X i , c x ;
Ž 8.
i
q
7 Q0 Ž x i J i j Õ j . Ž z P z .
r9
8 T0
y
Q0 Ž z P z .
8
B is the bosonic background field. The fluctuations
about the background will be denoted by Y i ,
2
r7
i
X sB q
q PPP 4
½
s dt yT0 q
H
y
T0
2
r9
q
15 Ž z P z .
16
2
r7
q PPP 4 .
Yi
(T
.
0
We will be studying a system of two D0-branes, so
all fields take values in the Lie algebra of UŽ2.. In
terms of the UŽ2. generators we write
zPz
105 Ž x i J i j Õ j . Ž z P z .
16T0
i
As
Ž 5.
The third term gives the spin-orbit term which we
would like to reproduce from Matrix Theory.
3. Spin-orbit potential from Matrix Theory
On the Matrix Theory side the calculation proceeds by evaluating the quantum mechanical effec-
i
2
i
i
Ž A 0 1 q A a s a . , X i s Ž X 0i 1 q X ai s a .
2
Ž 9.
Ž c 1 q ca s a .
2 0
For the background fields, we will give nonzero
values to
cs
B31 s Õt ,
B32 s b and c 3 .
The values given to Bai correspond to two D0-branes
moving with relative velocity Õ along the x 1 direc-
P. Krausr Physics Letters B 419 (1998) 73–78
76
tion, and separated by distance b along the x 2
direction.
The fermionic background c 3 gives fermionic
expectation value "c 3r2 to each of the two D0branes. However, one is free to shift the background
by an amount proportional to 1 since the UŽ1. part of
the action decouples from the SUŽ2. part. Thus the
setup equally well applies to the case where the two
D0-branes have fermionic expectation values c 3 and
0. The latter picture corresponds to the supergravity
configuration we are trying to describe.
Now it is straightforward but tedious to expand
out the action in terms of the field components
defined above. We work in Euclidean space t ™
it , A ™ yiA. The action takes the schematic form
2
2
2
˙ q c 3 Yc q c 3 A c
S ; Ž A . q Ž Y . q Ž c . q BAY
addition to the massive fields just described, there
are massless fields which play no role in the following discussion.
Given these quadratic actions, we can work out
propagators. For the bosonic fields we define
DB Žt 1 ,t 2 N m2 . as the solution to
Ž yEt 2 q m2 . DB Žt 1 ,t 2 N m2 . s d Ž t 1 y t 2 . .
1
Note that m is allowed to be time dependent. Then
we find
X
² Sy Ž t 1 . Sq Ž t 2 . : s DB Ž t 1 ,t 2 N r 2 q 2 Õ .
²Ty Ž t 1 . Tq Ž t 2 . : s DB Ž t 1 ,t 2 N r 2 y 2 Õ . .
where PPP indicates terms cubic and quartic in
fluctuations which won’t contribute to our analysis.
Our strategy will be to treat the first four terms
exactly and the last two terms perturbatively. That is,
in terms of Feynman diagrams, the first four terms
supply the propagators and the last two will supply
the vertices. Let us first study the mass spectrum by
considering the propagator terms. One finds the following mass eigenstates:
Then
Y "n s
S "s
'2
Ž n s 2 PPP 9 .
Y11 " iY21 . iA1 q A 2
T "s
c "s
'2
Y11 " iY21 . iA1 y A 2
'2
c1 " ic2
'2
m2 s r 2
1
q m . DF Ž t 1 ,t 2 N m . s d Ž t 1 y t 2 . .
² cq Ž t 1 . cy Ž t 2 . : s DF Ž t 1 ,t 2 N Õt 1 g 1 q bg 2 . .
Ž 15 .
In fact, DF can be related to DB by
DF Ž t 1 ,t 2 N Õt 1 g 1 q bg 2 .
s Ž Et 1 q Õt 1 g 1 q bg 2 . DB Ž t 1 ,t 2 N r 2 y Õg 1 . .
Ž 16 .
c 3T Pq DF Ž Õt 1 g 1 q bg 2 . Py c 3
b
s
2
m sr y2Õ
2
c 3Tg 1 g 2 c 3 DB Ž r 2 q Õ .
c 3T Py DF Ž Õt 1 g 1 q bg 2 . Pq c 3
m s Õtg 1 q bg 2 ,
Ž 11 .
where r 2 s b 2 q Ž Õt . 2 . Here, by ‘‘mass eigenstate’’
we mean that the action takes the form
i dt 12 fq Ž Et 2 y mf2 . fy
b
s y c 3Tg 1 g 2 c 3 DB Ž r 2 y Õ .
2
9
Ý c 3Tgn DF Ž Õt 1 g 1 q bg 2 . gn c 3
ns2
H
3b
s
and
Ž 14 .
It will turn out that we won’t need the full structure
of DF , but only part of it. The formulas we will need
are
m2 s r 2 q 2 Õ
2
Ž 13 .
We similarly define DF by
Ž yEt
Y1n " iY2n
X
² Yyn Ž t 1 . Yqn Ž t 2 . : s DB Ž t 1 ,t 2 N r 2 . d n n
Ž 10 .
q PPP ,
Ž 12 .
i dt cqT Ž Et y mc . cy
H
for the case of bosons and fermions respectively. In
2
c 3Tg 1 g 2 c 3 DB Ž r 2 q Õ . y DB Ž r 2 y Õ . ,
Ž 17 .
where P "s Ž1 " g 1 .r2, and we have suppressed the
P. Krausr Physics Letters B 419 (1998) 73–78
t dependence. These relations are easily derived
upon recalling c 3Tc 3 s c 3Tg i c 3 s 0, which follows
from the grassmann property of c 3 and the symmetry of g i .
Now we can work out the fermionic dependence
of the one loop effective action. For this, we need to
first find the c 3 Yc and c 3 A c terms in Sfermi s
yiHdt Tr c Du c . We find
Sfermi s y
i
q'2
1
'2
Ž 18 .
Hdt 1 dt 2
= ² Yyn Ž t 1 . Yqn Ž t 2 . : c 3Tgn² cq Ž t 1 . cy Žt 2 . :gnX c 3
y2² Sy Ž t 1 . Sq Ž t 2 . : c 3T Pq² cq Ž t 1 . cy Ž t 2 . : Py c 3
y2²Ty Ž t 1 . Tq Ž t 2 . : c 3T Py² cq Ž t 1 . cy Ž t 2 . : Pq c 3 4 .
Ž 19 .
Using our previous results for the propagators we
obtain
b
T0
c 3Tg 1 g 2 c 3 dt 1 dt 2 3 DB Ž t 1 ,t 2 N r 2 .
H
= DB Ž t 1 ,t 2 N r 2 q Õ . y DB Ž t 1 ,t 2 N r 2 y Õ .
y DB Ž t 1 ,t 2 N r 2 q 2 Õ . DB Ž t 1 ,t 2 N r 2 q Õ .
q DB Ž t 1 ,t 2 N r 2 y 2 Õ . DB Ž t 1 ,t 2 N r 2 y Õ . 4 .
It is evident that the terms linear and quadratic in
velocity will cancel out in the above expression. To
evaluate the Õ 3 term we need to expand out the
propagators and compute the integrals. After doing
the t 2 integral the result will take the form
bÕ 3c 3Tg 1 g 2 c 3 dt 1
H
b
½
Õ2
q
48 b 4
1
Ž b 2 q Õ 2t 12 .
9r2
1
2
aÕ
y
4 b2
1 q b <t 2 <
Ž 9a 2 y 6 . Ž 1 q b < t 2 < .
.
Given this fact, it is easier to proceed by evaluating
the t 2 integral with t 1 s 0 and then restoring the t 1
96 b 6
Ž 15 a 2 y 30 . Ž 1 q b <t 2 < .
q Ž 6 a 2 y 24 . b 2 <t 2 < 2 q Ž a 2 y 14 . b 3 <t 2 < 3
y4b 4 <t 2 < 4
5 q PPP
Plugging this expansion into Sso and doing the t 2
integral gives
Sso s y
X
Sso s y
eyb <t 2 <
s
y
Sq cyT Py c 3
The spin-orbit interaction is found by expanding
e i S fermi to quadratic order in c 3 and taking the vacuum expectation value. This gives
T0
DB Ž 0,t 2 N b 2 q a Õ .
a Õ3
Sy c 3T Pq cqy
y'2 Ty c 3T Py cqq '2 Tq cyT Pq c 3 4 .
Sso sy
dependence afterwards. The expansion of the bosonic
propagator is
q Ž 3 a 2 y 6 . b 2 <t 2 < 2 y 4 b 3 <t 2 < 3
n T
n T
y c 3 gn cqq Yq cy gn c 3
( Hdt Y
T0
77
105
32T0
H
dt 1
bÕ 3c 3Tg 1 g 2 c 3
r9
.
Ž 20 .
Now we can compare with the result from supergravity. Transforming back to Minkowski space, we
find that the spin-orbit terms from Ž5. and Ž20. agree
provided
Ž x iJ i j Õ j . Ž z P z.
i bÕ 3c 3Tg 1 g 2 c 3
sy
.
Ž 21 .
2
r9
r9
Does this equivalence make sense? To see that it
does we need to recall the expression for the angular
momentum operator of Matrix Theory. Starting from
the action Ž7., the operator which generates rotations
in the transverse space is the sum of a bosonic piece
and a fermionic piece. If we work in the rest frame
of the source D0-brane – the one carrying the
fermionic expectation value – then the bosonic contribution to the angular momentum of the source
vanishes. The fermionic piece is the standard expression for the angular momentum of a spinor field,
i
J i j s cg i g j c .
2
Recalling that c is Majorana, and that the relative
velocity and separation of the D0-branes are along
the x 1 and x 2 axes respectively, we find that Ž21. is
satisfied. Thus we have verified that supergravity
and Matrix Theory agree as to the leading spin
dependence of the scattering amplitude.
78
P. Krausr Physics Letters B 419 (1998) 73–78
Acknowledgements
I am grateful to M. Becker, E. Keski-Vakkuri,
and J. Schwarz for helpful discussions. I would also
like to thank M. Serone for helping me to correct an
error in an earlier version of the manuscript.
w5x
w6x
w7x
w8x
References
w1x T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys.
Rev. D 55 Ž1997. 5112, hep-thr9610043.
w2x M.R. Douglas, D. Kabat, P. Pouliot, S.H. Shenker, Nucl.
Phys. B 485 Ž1997. 85, hep-thr9608024.
w3x K. Becker, M. Becker, A Two Loop Test of M Žatrix. theory,
hep-thr9705091.
w4x K. Becker, M. Becker, J. Polchinski, A. Tseytlin, Higher
w9x
w10x
w11x
w12x
Order Graviton Scattering in M Žatrix. Theory, hepthr9706072.
J.A. Harvey, Spin Dependence of D0-brane Interactions,
hep-thr9706039.
J. Morales, C. Scrucca, M. Serone, A Note on Supersymmetric D-brane Dynamics, hep-thr9709063.
L. Susskind, Another Conjecture About M Žatrix. theory,
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Ž1997. 311, hep-thr9610249; Nucl. Phys. B 495 Ž1997. 99,
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M. Cederwall, A. von Gussich, B. Nilsson, P. Sundell, A.
Westerberg, Nucl. Phys. B 490 Ž1997. 179, hep-thr9611159.
E. Bergshoeff, P.K. Townsend, Nucl. Phys. B 490 Ž1997.
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Matrix Theory, hep-thr9708063.
12 February 1998
Physics Letters B 419 Ž1998. 79–83
The threebrane soliton of the M-fivebrane
P.S. Howe 1, N.D. Lambert 2 , P.C. West
3
Department of Mathematics, King’s College, London WC2R 2LS, UK
Received 10 October 1997
Editor: P.V. Landshoff
Abstract
We discuss the supersymmetry algebra of the M theory fivebrane and obtain a new threebrane soliton preserving half of
the six-dimensional supersymmetry. This solution is dimensionally reduced to various D-p-branes. q 1998 Elsevier Science
B.V.
1. Introduction
The dynamics of the M theory fivebrane are given
by an interacting Ž2,0. tensor multiplet containing a
self-dual three
tensor h m n p , Ž m,n, p s 0, . . . ,5. five
X
scalars X b Ž bX s 1X , . . . ,5X . and sixteen fermions Qai ,
Ž a s 1, . . . ,4, i s 1, . . . ,4.. In this paper we shall
seek a supersymmetric string soliton solution to the
M theory fivebrane equation’s of motion w1,2x. We
will use the six-dimensional covariant field equations
of motion derived in w1x. Alternative formulations of
the fivebrane were given in w3x.
The most general form for the Ž2,0. supersymmetry algebra in six dimensions is
Qai ,Qbj 4 s h i j Ž g m . ab Pm q Ž g m . a b Zmi j
q Ž g m n p . ab Zmi jn p .
ij
Ž 1.1 .
Here h is the invariant tensor of USpŽ4. ( SpinŽ5.,
Pm is the momentum, Zmi j is in the symmetric 5 of
1
E-mail: [email protected].
E-mail: [email protected].
3
E-mail: [email protected].
2
SpinŽ5. and Zmi jn p is self-dual and in the anti-symmetric 10 of SpinŽ5.. It is possible to add to Ž1.1. a
five form central charge in the 5 of SpinŽ5., however, due to the self-duality constraint it contains no
additional degrees of freedom. In fact the left hand
side can be thought of as a sixteen by sixteen
symmetric matrix and therefore has 16 =2 17 s 136 degrees of freedom. Similarly the right hand side contains 6 q 5 = 6 q 10 = 10 s 136 degrees of freedom
and hence there can be no additional independent
terms w4x. Since the p-form charges of p-dimensional extended objects can give rise the additional
central charges in Ž1.1. w5x, we may expect to find
onebrane and threebrane solitons of the M-fivebrane
equations preserving half of the spacetime supersymmetry. We will shortly return to the interpretation of
the five form central charge.
There are in fact two other arguments which also
lead to the appearance of onebranes and threebranes
on the M-fivebrane worldvolume. The first is from
the M theory interpretation in eleven dimensions.
Here one has two possible configurations which
preserve a quarter of the eleven-dimensional supersymmetry; a membrane intersecting a fivebrane over
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 3 3 - 0
P.S. Howe et al.r Physics Letters B 419 (1998) 79–83
80
a onebrane w6,7x or two fivebranes intersecting over a
threebrane w8x. In terms of the worldvolume of the
fivebrane these configurations will appear as onebrane and threebrane solitons respectively, preserving half of the six-dimensional supersymmetry.
Another reason one may expect only one form
and three form central charges comes from the interpretation of these configurations as the effective field
outside of a D-p-brane w9x of the self-dual string
theory. Here we will assume that the notion of a
D-p-brane Žas a Ž p q 1.-dimensional hyperplane in
spacetime where open strings end. can be extended
to an analogous object of the six-dimensional selfdual string theory. In this case the supersymmetries
preserved by the modified boundary conditions on
the self-dual string are those for which
e L s G 0 . . . Gp e R ,
Ž 1.2 .
where the D-brane lies in the x 0 , . . . , x p plane and
e L and e R are the left and right handed supersymmetry generators on the self-dual string worldsheet respectively. Since e L and e R are of the same six-dimensional chirality it is easy to check that Ž1.2. is
consistent if and only if p s 1,3,5. In this case there
again appears the possibility of a D-fivebrane on the
worldvolume and hence a five form central charge in
Ž1.1.. This would be analogous the the D-ninebrane
of the type IIB string and carry no physical degrees
of freedom. It does however point to the possibility
of performing an orientifold projection to a self-dual
Ž1,0. string theory in six dimensions, in analogy with
the construction of the type I string as an orientifold
of the type IIB string w9x. The analogy with type IIB
string theory also suggests that the six-dimensional
self-dual string may possess a sort of D-instanton or
y1-brane. However we shall see below that this is
not the case.
This Ž1,0. string theory can be understood within
M theory if we suppose that there is some kind of
ninebrane. It would then be possible for a ninebrane
to intersect a fivebrane over a fivebrane Ži.e. a
fivebrane-ninebrane bound state.. 4 This would give
rise to a five form central charge on the fivebrane.
The ninebrane would then have four directions transverse to the fivebrane, indicated by four of the five
scalars. By choosing which four scalars of the five
on the worldvolume to use one obtains a multiplet of
fivebranes in the 5 of SpinŽ5., in agreement with the
algebra Ž1.1.. This construction of the Ž1,0. self-dual
string has appeared in w10x, in connection with the
fivebrane of the E8 = E8 heterotic string.
In w11x a family of supersymmetric onebrane solitons on the M-fivebranes equations of motion were
found, transforming under the 5 of SpinŽ5.. In this
paper we shall seek a family of supersymmetric
threebrane solitons in the 10 of SpinŽ5. and thereby
complete the p-brane spectrum of the M-fivebrane.
The threebrane of the M-fivebrane is also of some
interest in its own right. It has apeared in connection
with the counting of entropy in near extremal black
holes w12x and is also related to the low energy
Seiberg-Witten effective action of N s 2, D s 4
Yang-Mills theory w13x. The precise relation between
the dynamics of the threebrane and the Seiberg-Witten effective action is the subject of a forthcoming
paper w14x. In the next section we will obtain multithreebrane solutions and describe their elementary
properties. In the final section we will use the threebrane solitons to obtain Ž p y 1.-branes and Ž p y 2.branes on the D-p-branes of type II string theory.
2. The threebrane soliton
In this paper we denote the six-dimensional coordinates of the fivebrane by hatted variables m,n
ˆ ˆs
0,1,2, . . . ,5. In addition all indices are raised and
lowered with respect to the flat Euclidean metric
unless explicitly indicated otherwise. Let us look for
a threebrane in the plane x 0 , x 1 , x 2 , x 3. We let unhatted variables refer to the transverse coordinates of
the threebrane i.e. x 4 , x 5. We will assume all fields
to depend only on the transverse coordinates.
To find
X
a threebrane we dualise one scalar X 1 to a five-form
G5
X
Gmˆ nˆ pˆ qˆ rˆ s e mˆ nˆ pˆ qˆ rˆ sˆ E ŝ X 1 .
Ž 2.1 .
Our ansatz is then
X
4
In fact some state of this kind must exist since there is a type
IIA eightbrane-fourbrane bound state preserving eight supersymmetries, indeed we shall construct it below.
G 0123 m s e m n E n X 1 ' Õm ,
X
X 2 sf ,
hm n p s 0 ,
Ž 2.2 .
P.S. Howe et al.r Physics Letters B 419 (1998) 79–83
where e m n is the volume element
on Xthe transverse
X
X
space and the other scalars X 3 , X 4 , X 5 are constant.
With this ansatz the equations of motion of the
fivebrane simplify greatly. We introduce the metric
g mˆ nˆ
1 3= 3
X
X
X
dm n q Em X 1En X 1 q Em X 2En X 2
dQb j s y
1
1
2
'y det g
X
0
gk
where 1 3= 3 this the unit matrix in three dimensions.
Since the three form is zero the only equations of
motion we need to solve for are w1x
X
X
g m n=m=n X 1 s g m n=m=n X 2 s 0 ,
Ž 2.4 .
where = is the Levi-Civita connection of g mˆ nˆ .
As with the string soliton found in w11x it is
helpful now to determine the condition that half of
the supersymmetry is preserved by the soliton. It is
instructive to first consider the linearised supersymmetry
d 0 Qb j s e a i
ž
1
2
j
Ž g m . ab Ž g bX . i Em X b
X
y 16 Ž g m n p . ab d i j h m n p .
j
X
s 12 e a i Ž g m . ab Ž g bX . i Em X b ,
Ž 2.5 .
for h m n p s 0. Inserting our ansatz in to d 0 Qb j s 0
one finds
1
2
ai
=
½
4
Ž g . ag Ž g 1 .
X
j
g
l
j
g
q Ž g 1X 2X . k d l jd bg E4 f q Ž g 1X 2X . l Ž g 45 . b Õ4
5.
Ž 2.6 .
1
1 q Ž Ef .
2
E 2f s 0 ,
Ž 2.10 .
where E 2f s d m nEm En f and Ž Ef . 2 s d m nEm fEn f . Furthermore it follows from the Bogomoln’yi
condition
X
X
Ž2.7., that Ž2.10. is solved by any X 1 , X 2 . One can
also check from Ž2.10. that G5 is a closed form.
Thus the solution corresponding to N threebranes
located at yI Ž I s 0, . . . , N y 1. has the general form
G 0123 m s "Em f ,
Ny1
Ý
Q I ln < x y yI < ,
Ž 2.11 .
where f 0 and Q I are constants. Clearly this solution
has bad asymptotic behaviour unless ÝQ I s 0, however one can still define the charge of a single
threebrane
Qs
Thus if we set
Õm s "Em f ,
Ž 2.7 .
we find that the solution will be invariant under
supersymmetries which satisfy
b
d j
Is0
d kjd bg Õ5 q Ž g 1X 2X . k Ž g 45 . b E 5 f
j
l
Here we have set h m n p s 0 and we refer the reader
to w11x for a detailed description of the matrices E
and u. Clearly Ž2.9. vanishes if d 0 Qb j s 0, provided
that the preserved supersymmetries are e b j s
e 0a i Eabi j, where e 0a i satisfies Ž2.8..
If we substitute the Bogomol’nyi condition Ž2.7.
into the metric Ž2.3. it is now an easy matter to see
that the equation of motion becomes
f s f0 q
k
i
X
=e a i Ž Ey1 . a i Em X b Ž g m . gd Ž g bX . k Ž u . l b .
Ž 2.9 .
,
Ž 2.3 .
0s e
preserved to all orders we recall the expression
obtained in w11x for the full non-linear supersymmetry
0
y1
s 0
81
e 0a i Ž g 1X 2X . i Ž g 45 . a s .e 0b j .
Ž 2.8 .
Ž2.7. is equivaNote that the Bogomol’nyi condition
X
X
lent to the statement that X 1 q iX 2 is an Žanti-.holomorphic function of x 4 q ix 5 for the minus Žplus.
sign in Ž2.7.. To verify that the supersymmetries are
1
2p
HS wG s "Q
1
`
5
0
,
Ž 2.12 .
where S`1 is the transverse circle at infinity and w is
the flat six-dimensional Hodge star. The presence of
the conformal factor in Ž2.10. indicates that the
equations of motion are satisfied even at the points
where the solution is ill behaved and hence no
sources are needed.
Let us consider the zero modes of a single threebrane soliton. Clearly there are two bosonic zero
modes y 0m describing the location of the threebrane
P.S. Howe et al.r Physics Letters B 419 (1998) 79–83
82
in the transverse space. Now consider the three-form
h 3 s db 2 s 0. The closed two form b 2 has a gauge
symmetry b 2 ™ b 2 q dA1. However, because of
Poincare´ invariance of the threebrane along
x 0 , x 1, x 2 , x 3, there are vector fields A1 whose indices tangent to the threebrane do not vanish at
infinity, corresponding to so-called large gauge
transformations. Following the standard treatment of
soliton zero-modes Žsee for example w15x., we must
interpret these components of A1 as zero modes of
the threebrane. Thus there is also a four dimensional
vector zero mode living on the threebrane worldvolume. 5 The fermionic zero modes come from the
broken supersymmetries and hence there are eight of
them and in four dimensions they are necessarily
non-chiral. Thus we find an N s 2, D s 4 vector
multiplet of zero modes on the threebrane.
Ž5., by
Since the scalars transform under Spin
X
X
choosing an arbitrary pair of scalars Ž X a , X b . with
aX / bX for our solution, we obtain a multiplet of
threebranes transforming as a 5 =2 4 s 10 dimensional
representation of SpinŽ5., in agreement with the
algebra Ž1.1.. In the M theory interpretation, where
the threebrane represents the intersection of two
fivebranes, these scalars point along the two directions of the external fivebrane which are transverse
to the worldvolume of the fivebrane we are considering.
We can also see that there is no BPS instanton
configuration or y1-brane. For in this case weX
would again
have only
two non-trivial scalars, X 1
X
X
2
1
and X , with Em X interpreted as the field strength
of the y1-brane. However, what made the threebrane supersymmetric was the fact that in the two-dimensional transverse space the two scalars Žor more
precisely their field strengths. are dual to each other.
For a y1-brane, where the transverse space is the
six-dimensional Euclidean worldvolume of the fivebrane, there is no such condition. In fact one could
always perform a SpinŽ5. rotation to a configuration
5
The same reasoning also implies that there is a two dimensional vector zero mode on the self-dual string soliton worldvolume. This was ignored in w11x because it carries no degrees of
freedom.
with just one scalar. Clearly such a configuration
could not preserve any supersymmetry.
3. Discussion
In this paper we have discussed the one half
supersymmetric BPS states of the M theory fivebrane and in particular we obtained a new threebrane
soliton. We also determined the zero modes of resulting p-branes. We note that the zero modes can be
obtained by the dimensional reduction of a D s 6,
N s Ž1,0. super-Maxwell multiplet to p q 1 dimensions. We would like to conclude with a brief discussion of the dimensional reduction of the threebrane
to D-p-brane worldvolumes.
It was shown in w1x that the fivebrane equations of
motion can be dimensionally reduced to those of the
D-fourbrane. Thus the threebrane soliton can be
double dimensionally reduced to a twobrane on the
D-fourbrane. By T-duality these can be extended to
Ž p y 2. solitons on a D-p-brane Žwe will restrict our
attention to p G 2.. These Ž p y 2.-branes have exactly the same form as Ž2.11., only with G5 now
replaced by a p-form G 0 . . . Ž py2. m s "Em f . These
solutions represent two intersecting D-p-branes over
a Ž p y 2.-brane. These solutions have appeared before in w16,17x.
The above discussion only works for p - 8 since
the D-eightbrane only has one scalar field on its
worldvolume. Therefore the naive extrapolation of
the soliton Ž2.11. which involves two scalars cannot
work. Instead the vector field on the eightbrane must
become non-zero. More precisely consider two Dfourbranes intersecting over a twobrane Žsay one in
the x 0 , x 1 , x 2 , x 3 , x 4 plane and the other in the
x 0 , x 1 , x 2 , x 5 , x 6 plane.. Now T-dualise along
x 5, x 6 , x 7, x 8 to obtain an eightbrane in the x 0 , . . . , x 8
plane, containing a fourbrane in the x 0 , x 1 , x 2 , x 7, x 8
plane. We expect that this configuration is actually
described by an instanton in the four transverse
dimensions to the fourbrane; x 3 , x 4 , x 5, x 6 . To see
this explicitly let us construct the vector field A m in
this transverse space. Now, in order to have applied
T-duality we must set A 5 s A 6 s 0 Žor at least they
must be X pure gauge.. Consequently,
if we reinterpret
X
A 3 s X 1 and A 4 s X 2 , we find that the Bogomol’nyi
P.S. Howe et al.r Physics Letters B 419 (1998) 79–83
condition Ž2.7. Ži.e. holomorphicity. is equivalent to
self-duality of the gauge field A m .
One could also directly dimensionally reduce the
threebrane soliton Ž2.11. to a threebrane on the
D-fourbrane worldvolume. This leads to a domain
wall on the fourbrane and is associated with a cosmological constant, which can be obtained by a
Sherk-Schwarz type of dimensional reduction invoking one of the scalars. The form of this solution
should also be clear to the reader, the only modification being that there is only one transverse dimension so that the logarithms in Ž2.11. are replaced by
linear functions. This threebrane soliton represents a
D-fourbrane intersecting with an NS-fivebrane.
Clearly this solution can also be T-dualised to other
D-p-branes intersecting with an NS-fivebrane over a
Ž p y 1.-brane.
Similarly to the above case this only works for
p - 7. To obtain a sevenbrane one must T-dualise in
the transverse space of the NS-fivebrane. This leads
to a Kaluza-Klein like fivebrane Ži.e. a geometry
which is R 6 = M Taub - NUT ., bound to the sevenbrane,
which is wrapped around one compact and one
non-compact dimension of M Taub - NUT . It is impossible to T-dualise again to obtain an eightbrane configuration, since there is no suitable Taub-Nut geometry
with the Žlocal. form R 2 = T 2 . This is reflected by
the fact that there is no way to obtain a cosmological
constant in the nine-dimensional Maxwell action by
performing a Sherk-Schwarz reduction from ten dimensions.
Acknowledgements
We would like to thank G. Papadopolous and
H.A. Chamblin for discussions.
83
References
w1x P.S. Howe, E. Sezgin, P.C. West, Phys. Lett. B 399 Ž1997.
62, hep-thr9702008.
w2x P.S. Howe, E. Sezgin, Phys. Lett. B 394 Ž1997. 49, hepthr9611008.
w3x M. Perry, J.H. Schwarz, Nucl. Phys. B 489 ŽŽ1997. 47,
hep-thr9611065; M. Aganagic, J. Park, C. Popescu, J.H.
Schwarz, Worldvolume action of the M-theory fivebrane,
hep-thr9701166; I. Bandos, K Lechner, A. Nurmagambetov,
P. Pasti, D. Sorokin, M. Tonin, Covariant action for the super
fivebrane of M-theory, hep-thr9701149.
w4x P.K. Townsend, P-brane Democracy, hep-thr9507048.
w5x J.A. de Azcarraga, J.P. Gauntlett, J.M. Izquierdo, P.K.
Townsend, Phys. Rev. Lett. 63 Ž1989. 2443.
w6x P.K. Townsend, Phys. Lett. B 373 ŽŽ1995. 68, hepthr9512062.
w7x A. Strominger, Phys. Lett. B 383 ŽŽ1995. 44, hepthr9512059.
w8x G. Papadopoulos, P.K. Townsend, Phys, Lett. B 384 Ž1996.
86, hep-thr9605146.
w9x J. Polchinski, Phys. Rev. Lett. 75 Ž1995. 4724, hepthr9510017.
w10x O. Ganor, A. Hanany, Nucl. Phys. B 474 ŽŽ1996. 122,
hep-thr9602120.
w11x P.S. Howe, N.D. Lambert, P.C. West, The Self-Dual String
Soliton, hep-thr9709014.
w12x I. Klebanov, A. Tseytlin, Nucl. Phys. B 479 ŽŽ1996. 319,
hep-thr9607107.
w13x E. Witten, Solutions of Four Dimensional Field Theories via
M Theory, hep-thr97003166.
w14x P.S. Howe, N.D. Lambert, P.C. West, Classical M-Fivebrane
Dynamics and Quantum N s 2 Yang-Mills, hep-thr9710034.
w15x J.A. Harvey, Magnetic Monopoles, Duality and Supersymmetry, hep-thr9603086.
w16x G.W. Gibbons, Born-Infeld Particles and Dirichlet p-branes,
hep-thr9709027.
w17x C.G. Callan, J.M. Maldecena, Brane Death and Dynamics
From the Born-Infeld Action, hep-thr9708147.
12 February 1998
Physics Letters B 419 Ž1998. 84–90
Solitonic d y 5 brane and vortex defects on the world-sheet
Kazuo Ghoroku
1
Department of Physics, Fukuoka Institute of Technology, Wajiro, Higashiku, Fukuoka 811-02, Japan
Received 15 August 1997; revised 3 October 1997
Editor: M. Dine
Abstract
We examine the behavior of the vortex defects on the world-sheet when a solitonic d y 5 brane is existing in the
d-dimensional target space, and it is seen that the vortices of any charge should be dissociated to the free-gas phase when
they approach to the brane. The meanings of this fact are addressed through the analysis of the renormalization group
equations on the world sheet. q 1998 Elsevier Science B.V.
1. Introduction
The rapid developments in the recent string theories have made clear the importance of the ŽD.pbranes in understanding the non-perturbative aspect
of string- and field-theories. So it is meaningful to
examine the characteristic properties of p-branes from
various viewpoints. Here, we examine the availability of the brane-solution obtained from the low-energy effective-action in terms of the world-sheet
action of the closed string propagating in this background configuration. In the world-sheet action, the
sine-Gordon term is introduced to represent the vortices on the world-sheet. It is possible to examine the
stability of the background configuration through the
behavior of the vortices on the world-sheet. This idea
can be considered as the developed version of w1x,
where the radius of a compactified space of S 1 has
been studied. In the case of D-brane background, it
has been shown that the recoil effect of the D-brane
can be expressed by the appropriate terms in the
world-sheet action w2,3x. As for the vortex on the
1
E-mail: [email protected].
world-sheet, there has been an interesting proposal
w4x that the sine-Gordon term might represents the
appearance and disappearance of the virtual D-branes.
Here we consider the d y 5 brane, whose availability as a background configuration is examined
here, as a prototype of the solitonic solution of the
low-energy effective action in d-dimension. The solution can be obtained in a similar form to the
5-brane solution of the 10d supergravity w5x. This
brane can be expected as an elementary object in the
strong coupling limit from the S-duality like the
magnetic monopole solution in 4d QCD. The sineGordon term introduced in the world-sheet should
receive a modification from the requirement of the
conformal invariance w6,7x of the world-sheet action.
This modification and the renormalization group
properties of the world-sheet action gives an important clue to examine the availability of the brane
configuration in the target space, and this analysis
would shed some light on the problem of the vacuum
configuration of the target space.
In the noncritical target-space dimension, the
world-sheet action contains the Liouville field or the
conformal mode of two dimensional gravity quan-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 7 9 - 2
K. Ghorokur Physics Letters B 419 (1998) 84–90
tized in the conformal gauge. In Refs. w4,8x, the
sine-Gordon term is written by the Liouville field.
However, there is no reason to take this assignment
in our case. In fact, any bosonic field on the worldsheet is useful to represents the sine-Gordon potential, and it could lead to the the desirable vortex
free-energy. Here the sine-Gordon field is not assigned to the Liouville field but to some other scalar
field. The Liouville field is instead used to represent
a physical scale in the 2d curved surface as in w6x in
order to derive the renormalization group equations
of the parameters of the world-sheet action. When
the sine-Gordon field is assigned to one of the
coordinates transverse to the d y 5 brane, we can
show that the vortices becomes gaseous and the
sine-Gordon potential is relevant near the surface of
the brane even if how large the vortex charge is. In
other words, d y 5 brane configuration in this region
should be changed to a more appropriate form in
which the vortex effects are reflected.
In the next section, the solitonic d y 5 brane is
derived from the low-energy effective target-space
action. Then, vortices are introduced in Section 3 in
terms of the sine-Gordon action. This is done such
that the theory is conformally invariant. In Section 4,
the renormalization group analysis of the parameters
are studied, and the summary and discussions are
given in the final section.
85
The parameter k is remained finite since noncritical
dimension d is considered. For bosonic case, it is
written as, k s Ž25 y c .r3, where c represents the
central charge of the theory. The tachyon ŽT. term is
necessary here since this part is assigned to the
sine-Gordon potential in the world sheet action,
which is introduced in the next section.
The solution of Ž1. is obtained by solving the
following equations of motion,
= 2 T y 2=F= T s 12 ÕX Ž T . ,
k
2
= 2F y 2 Ž =F . s y q 12 Õ Ž T . y 16 H32 ,
2
Ž 3.
R M N y 2=M=N F s y=M T=N T y 14 HKML H N K L ,
Ž 5.
E M Ž 'G ey2 F H M N K . s 0,
Ž 6.
Ž 4.
where the indices M, N, PPP run over d-dimensions,
0–d y 1. The d-dimensional coordinates are denoted
as X M s Ž x m, y m ., where m s 0–d y 5 and m s d
y 4–d y 1. Here, x m denotes the coordinates of the
world volume of d y 5 brane, and y m are the transverse ones to the brane. Further, the ’’time’’ x 0 is
identified with the Liouville field or the conformal
mode Ž r . on the world sheet.
We solve the above equations in the Euclidean
metric by assuming T s 0 and taking the following
Ansatz,
¨
ds 2 s dmn dx mdx n q e 2 BŽ y.dm n dy m dy n ,
2. d I 5 solitonic brane
H m nl s
In d s 10, N s 1 supergravity, 5-brane solitonic
solution is given in w5x. Considering this solution as a
prototype of the soliton, we extend it to the case of
the non-critical dimension, d / 10.
Our starting point is the following low energy
effective action w11,12x of d-dimension,
St s
1
4p
Hd
d
X'G ey2F R y 4 Ž =F .
2
qŽ = T . q Õ Ž T . q
1
2 = 4!
2
H32 y k ,
Ž 1.
where H3 is the field strength of the 2-form potential
Ž BM N . given in Ž11. and
2
1
6
3
Õ Ž T . s y2T q T q PPP .
Ž 2.
2 e m nls
(det g
Ž 7.
EsF Ž y . ,
Ž 8.
ij
where y s dm n y m y n , the distance from the brane.
And g i j represents the metric of the transverse four
dimensional space. Tangential components of H3 ,
whose indices include at least one of the tangential
directions denoted by m , are zero.
Before solving our equations, we remind the supersymmetric 5-brane solution for the case of d s 10
Ž k s 0.. It has been obtained w5x in the following
form,
(
F Ž y . s B Ž y . s Fp , exp Ž 2Fp . s e 2 f 0 q
QD
y2
,
Ž 9.
Ž7., Ž8.. Here, f 0 is a
under the above Ansatze
¨
86
K. Ghorokur Physics Letters B 419 (1998) 84–90
constant and Q D denotes the ’’magnetic’’ charge,
which is supported by H3 , of the 5-brane.
Here we must find a solution for d / 10 and
k / 0. For the case of k s 0, the extended solution
to the diverse dimension Ž d / 10. has been given in
w9x, and the solution is written by one harmonic
function similar to Ž9.. However, in the case of
k / 0, the asymptotic form of the d y 5 brane solution should take the linear dilaton vacuum. So we
must modify at least the dilaton part in the solutions
obtained in w5,9x. The simplest solution is obtained
by changing the configuration of the flat inner space
x m, m s 0–d y 5, of the brane to the linear dilaton
form. Then our solution is obtained in the following
form,
F s F 0 q Fp Ž y . , F 0 s 12 Q r ,
B Ž y . s Fp Ž y . ,
Ž 10 .
where Fp Ž y . is the one given in Ž9., and Q s 'k .
Here we give some comments on this solution. Ži.
It may be possible to consider a more general form
for F 0 in terms of the constants cm as follows,
F 0 s S cm x mr2 with S cm2 s k . However, we do not
take this general form here since we are considering
a theory, in which the dilaton should have its asymptotic form of Ž10.. This is naturally derived from the
quantization of the string theory at noncritical dimension. Žii. B Ž y . is different from F by F 0 . Then
this solution would not satisfy the requirement of the
supersymmetry, which demands that the solution
must be written by one common function. So the
d y 5 brane given here would not keep a supersymmetry. Žiii. Due to the linear dilaton part F 0 , it
seems impossible to obtain a ’’fundamental’’ string
soliton-solution, which can be obtained from the
combined action of St ŽŽ1.. and the two dimensional
source action Ž S2 . embedded in the d-dimension. For
k s 0, this solution can be found even if d / 10
w9,13x.
3. Vortices and monopoles
Next, we introduce the vortices on the world-sheet
from the viewpoint of the path-integral formulation
of the world-sheet action. The action, which is responsible for the target space action Ž1., can be
written in the form of the following non-linear sigma
model,
Ss
1
2
4p
ž
Hd z'gˆ
= gˆ a b q i
1
2
Ž G M N Ž X . q BM N Ž X . .
e ab
'ĝ
/
Ea X MEb X N
q RˆF Ž X . q T Ž X . ,
Ž 11 .
where X M are assigned as in the previous section,
X M s Ž x m, y m .. The theory on the sheet is quantized
in the conformal gauge, gmn s e 2 r gˆmn , and the conformal mode Ž r ., which is alive through the quantum measure, is assigned as the ’’time’’, x 0 s r .
The vortex configuration on the world-sheet can
be introduced as a topological defect w14x through a
scalar field, say X Õ Ž z .. The effective action of such
vortices can be represented by the sine-Gordon potential, cosŽ pX Õ ., where p denotes the vorticity or
the vortex charge. Then the gas of vortices, whose
total charge is zero, is expressed by the following
lagrangian,
1
2
2
Ž E X Õ . q lcos Ž pX Õ . ,
Ž 12 .
for the flat surface. Here l is a parameter for the
vortex gas and it will be discussed in the following.
From the viewpoint of the conformal field theory,
the same partition function with the one for the
vortex gas can be derived w10x in terms of the
following potential,
lcos Ž q X Ž z . y X Ž z .
.,
where we notice that the usual scalar field has the
plus combination, X Ž z, z . s X Ž z . q X Ž z .. The corresponding configuration to the minus combination
is called as the monopole w10x which is dual to the
vortex in the sense that the vortex charge p and the
monopole charge q satisfies the Dirac quantization
condition, pq s 2p n, where n is some integer. This
monopole defect is also considered in w4x as the
representation of the appearance and disappearance
of virtual D-branes. We will briefly comment on this
monopole below.
First, we consider the case of vortex which can be
included in the above action Ž11. by making an
assignment of X Õ to the one of X M in Ž11. and
K. Ghorokur Physics Letters B 419 (1998) 84–90
adding the sine-Gordon potential of Ž12. which is
written by the assigned coordinate field. But it should
be modified due to the quantization of the 2d surface, so our first task is to find the modified sineGordon potential. Since this newly added potential
can be assigned to the tachyon part, T Ž X . of Ž11.,
then its modified form can be obtained by solving
the Eq. Ž3.. This is corresponding to obtaining a
conformal invariant action under the condition that
the added sine-Gordon term is a perturbation to a
conformal invariant vacuum, which is described here
by the d y 5 brane. According to the procedure of
w6x, we assume that the modification for the sineGordon potential can be expressed by the dressed
factor, which is given as follows,
TÕ s le gr cos Ž pX . ,
Ž 13 .
where the original factor e 2 r is replaced by e gr .
Then from the Eq. Ž3., we obtain the value of g by
assuming l being small and linearizing the equation
with respect to T s TÕ .
In solving Ž3., we should notice the following
points; Ži. The assignments of X Õ to the one of X M
are separated into the following two groups, Ža. the
one of x m Ž m / 0. Žthe inner space of the d y 5
brane. and Žb. the one of y m Žthe transverse space..
Although we need a modification of the potential in
both cases, the second case Žb. is more interesting as
shown below. Žii. Secondly, the d y 5 brane background given above must be considered in writing
explicitly Ž3..
The linearized form of Ž3. with respect to T is
written as
Ž G M NE M E N y QE 0 q 2 . T s 0,
ž
(
/
Next, consider the second case Žb., where X is
assigned to one of y m. In this case, the coefficient of
the differential operator depends on y, so we solve
the equation by assuming the constancy of y.
Namely, we consider the problem on a special surface in the target d-dimensional space, where the
distance Ž y . from the d y 5 brane is fixed. Then we
obtain the following solution,
(
g s 12 Q y Q 2 q 4 p˜ 2 y 8 ,
ž
Ž 16 .
/
where
p˜ 2 s ey2 F p Ž y. p 2 ,
Ž 17 .
and Fp Ž y . is given in Ž9.. In this case, the critical
vortex-charge varies with y, and it is given by
QD
pcr2 s 2e 2F p Ž y. s 2 1 q 2 ,
Ž 18 .
y
ž
/
which approaches to infinity when y goes to zero,
just on the d y 5 brane. This fact implies that all of
the vortices of any charge are in the plasma phase
near the d y 5 brane. Namely, the vortex and antivortex pair dissociate to the free gas even if how
large the vortex charge is.
Next, we consider the case of monopoles. Since
the monopole charge Ž q . and the vortex charge Ž p .
should satisfy the Dirac condition, pq s 2p n where
n is the integer, the monopole is expected to be in
dipole phase if the vortex is in the gaseous phase.
This expectation is true for the flat background. We
examine this point in the case of the presence of the
d y 5 brane in the background. Then, we solve Ž14.
by replacing T by
X
Ž 14 .
where we have used Ž7. and Ž10., and the term like
E BE T has cancelled out due to the characteristic
form of the d y 5 brane.
First, we consider the case of Ža. where X being
assigned to x m. Substituting Ž13. to Ž14., then we
obtain
g s 12 Q y Q 2 q 4 p 2 y 8 ,
87
Ž 15 .
which is the same one obtained in the linear dilaton
vacuum previously w6x for the sine-Gordon model
coupled to the 2d gravity. At this order of the
approximation, the d y 5 brane does not affect on g .
Then the Kosteritz-Thouless ŽKT. transition point,
p 2 s 2 which is given in w6x, is not changed.
Tm s lX e g r cos Ž p X Ž z . y X Ž z .
..
Ž 19 .
In this case, we must define the operation of E 2 on
Tm in Ž13.. Since E 2 is the laplacian in the flat space,
it might be replaced by the Virasoro operator, L 0 q
L 0 , defined for a free scalar field X. According to
the usual operator formalism,
x
1
X Ž z . s y ik ln z q i Ý
a m zym ,
Ž 20 .
2
m
m/0
XŽ z. s
x
2
y ik ln z q i
1
Ý
m/0
m
a m zym ,
Ž 21 .
`
L0 s L0 s
Ý
:a m aym : ,
Ž 22 .
ms1
where x is the center of mass of X, and k can be
88
K. Ghorokur Physics Letters B 419 (1998) 84–90
taken as zero except for the Liouville field. Since we
do not assign X to the Liouville field here, k is
taken zero. Then we can solve Ž14., and we obtain g X
in the same form with Ž15. and Ž16. for the two
kinds of assignments respectively. This result implies
that the dual phase relation of monopole and the
vortex is broken near the brane since both critical
charges become infinite near the d y 5 brane and
both defects would be in the plasma phase near the
d y 5 brane. Then we can say that the d y 5 brane
dissociate all the topological defects, which are in a
dipole phase, into the free-gas phase.
4. Renormalization group analysis
In order to understand the above result more
deeply, we examine here the behaviors of the parameters, l and p, of the vortex through the renormalization group analysis. The renormalization group
equations of these parameters are obtained according
to the method in w6,7x. It is as follows. First, the
effective action on the world sheet is separated into
the conformally exact part and a small perturbation
which is characterized by a small parameter Žhere l..
To restore the conformal invariance, which is broken
by the small perturbation, the first exact solution
must be modified order by order. Namely the new
exact solution can be expanded in the power series
of l. After getting the effective action at some order
of l, the renormalization group equations are obtained by shifting the conformal field, r , by a constant in the world sheet action and absorbing this
shift in the parameters. For the first order approximation, the effective action can be obtained by solving
Eqs. Ž3. – Ž6..
Strictly speaking, the configuration of d y 5 brane
given here is not an exact one since the starting
action Ž1. is an approximate target space action
where higher derivative terms and the massive modes
are abbreviated. As an exact solution, we consider
here the linear dilaton vacuum, GM N s d M N ,F s F 0
and others are zero. So the d y 5 brane solution is
expanded around this vacuum configuration in powers of Q D ry 2 , which is assumed small, and further
add the sine-Gordon term as a small perturbation.
Therefore, there are two small parameters in this
case, l and Q D ry 2 . For the sake of the brevity, we
consider the case of Q D ry 2 – l2 hereafter.
First, we consider the case of the assignment,
X Õ s x 1. Far away from the brane, where the relation
Q D ry 2 – l2 is satisfied, we can expand Gmn , F , H
and T as follows,
QD
GM N s d M N q l2 h M N q 2 d Mm d Nm q PPP ,
Ž 23 .
y
QD
F s F Ž0. q l2F Ž2. q 2 q PPP ,
Ž 24 .
2y
T s lŽ T Ž0. q lT Ž1. q PPP . ,
Ž 25 .
s
y
H m n l s ye m n l s 4 Q D q PPP ,
Ž 26 .
y
where PPP denotes the higher order terms. Since H
is the order of 1ry 3 , this field can be neglected
hereafter in solving the equations. Since T 2 is the
lowest power in the Eqs. Ž4. and Ž5., it follows that
the lowest order corrections to Gmn and F are of the
order O Ž l2 .. Then, we make an Ansatz,
¨
h M N s d M1 d N1 h Ž r . ,
Ž 27 .
w
x
as in 6 . The reason of this setting is that the lowest
order correction should appear as the renormalization
of the field x 1 because of its self-interaction through
the sine-Gordon potential. Then, the equations of
O Ž l2 . are obtained as follows,
Q
Ž 28 .
Ž E 2 y 2 QE 0 . F Ž2. q 4 E 0 h s yT02 ,
2
2 E 02F Ž2. y 12 E 02 h s Ž E 0 T0 . ,
Ž 29 .
2 E 1 E 0F Ž2. s E 0 T0 E 1T0 ,
1
2
Ž E 02 y QE 0 . h y 2 E 12F Ž2. s y Ž E 1T0 .
E 0 E iF Ž2. s E 1 E iF Ž2. s E i E jF Ž2. s 0,
Ž E 2 y QE 0 q 2 . T1 s 14 T02 ,
2
,
Ž 30 .
Ž 31 .
Ž 32 .
Ž 33 .
where the higher order terms like 1ry 3 are neglected. As a result, these equations are equivalent to
the one obtained under the assumption of Q D ry 2 <
l2 since the terms depending on Q D cancel or are
neglected as the higher order terms.
Rescaling T by factor four, these equations are
solved near g –0, and we obtain
1
1
F Ž2. s 256
cos Ž 2 pf . q
rqOŽg . ,
Ž 34 .
32 Q
p2
rqOŽg . ,
16Q
T1 s 321 Ž 1 y cos2 pf . q O Ž g . .
hs
Ž 35 .
Ž 36 .
K. Ghorokur Physics Letters B 419 (1998) 84–90
As a result, the effective action is obtained up to
O Ž l2 . near g s 0 where p s '2 . Then, the renormalization group equations are obtained as mentioned above through a shift r ™ r q 2 dlra . Here,
dl denotes a small scale of the length and a represents the dressed factor for the cosmological constant. And, a is determined by solving Eq. Ž14. with
T s m2 expŽ a r .. Denoting as p s '2 q e , we obtain
l̇ s
4'2
aQ
el ,
e˙ s
'2 p 2
16 a Q
2
l ,
Ž 37 .
where a˙ means the derivative, a˙ s ydardln l. From
these results, we can see that the sine-Gordon term is
irrelevant for e ) 0 where the vortex is in dipole
phase. On the other hand, it becomes relevant in the
gaseous phase where e - 0. And l becomes large in
the infrared limit and the coordinate of the target
space assigned to X Õ would be fixed at some value
of the periodic copies.
For the case of the assignment, X Õ s y m Ž m s d
y 4., the Ansatz
¨ Ž27. for the perturbation of the
metric is replaced by,
h M N s d Mdy 4d Ndy 4 h Ž r . ,
Ž 38 .
and other fields are expanded in the same way with
the former case. The resultant equations of motion
and the solutions are equivalent to the above case
except for the fact that the role of the suffix 1 is
replaced by d y 4 in this case, and we obtain the
same renormalization group equations of l and e
with the ones given in Ž37..
As mentioned in the previous section, the vortices
of any charge, which is large enough to bind them to
the dipole pairs in the flat background configuration,
are in the gaseous phase Ž e - 0. near the brane when
we assigned as X Õ s y m. As a result, the sine-Gordon
term becomes relevant there. This fact means that the
d y 5 brane configuration should be modified so that
the sine-Gordon term can be irrelevant even if we
consider it near the brane. An alternative implication
is to modify the background manifold by assuming
the large value of fixed point of l. In this case, we
can consider a possibility that the coordinate y m Žthe
coordinate in the sine-Gordon potential. might be
fixed at some value, where the potential is minimum.
This phenomenon seems to imply that the effective
central charge of the noncritical string would be
89
suppressed by the d y 5 brane, since one of y m does
not contribute to the central charge.
In order to understand this analysis more widely,
it is interesting to consider an extended form of the
sine-Gordon potential such as
dy1
TÕ s le gr
Ý
cos Ž pi y i . ,
Ž 39 .
isdy4
and to see in what circumstances this potential becomes relevant or not. It is easy to perform the same
analysis given above with this potential, but we can
not obtain a simple result of the renormalization
group equations since new counter terms are necessary even in the first order of the approximation in
this case. Then, it seems to be necessary to extend
the approximation of the analysis to more higher
orders in order to arrive at a meaningful result. This
is out of our scope here.
5. Summary and discussions
The solitonic d y 5 brane-solution of the low-energy effective action is considered in the d-dimensional target space as an extension of the 5-brane of
the 10d supergravity. From the standpoint of noncritical string, the availability of the d y 5 brane-solution is examined through the renormalization group
analysis of the world-sheet action, where the sineGordon term is introduced to represent the vortex
defects on the world-sheet.
Our result implies that the brane-solution should
be modified near the surface of the brane since the
sine-Gordon term becomes relevant there. However,
it is available in the region far from the surface of
the brane. There is an implication for the search for a
more accurate configuration near the brane. If the
fixed point in the gaseous phase of the vortex was
realized at large value of l, then there is a possibility that the allowed region of one of the coordinate
y i Žthe transverse direction of the d y 5 brane. might
be restricted to the region around the minimum of
the sine-Gordon potential. This means that at least
one scalar field on the world-sheet becomes effectively a constant and one central charge vanishes as a
result. If this idea is meaningful, it is interesting to
extend the form of the sine-Gordon potential such
that it includes many coordinates. Analysis in this
direction will be given in the near future.
90
K. Ghorokur Physics Letters B 419 (1998) 84–90
Finally we comment on the possibility of the
sine-Gordon term as a realistic, condensed form of
the tachyon part of a noncritical string as suggested
from a different origin w15x, where the condensed
tachyon has a different form. In this case, the string
fields propagate in the potential which is periodic in
some direction. Then the spectrum of the field shows
a band structure as we can see in the case of an
electron within a crystal. It is interesting if we can
observe such a evidence in some field’s spectrum in
our macroscopic world.
Acknowledgements
The author thanks to the members of the high-energy group of Kyushu University for useful discussions and comments.
References
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387 Ž1996. 483.
w4x J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, hepthr9704169; hep-thr9706125.
w5x C.G. Callan Jr., J.A. Harvey, A. Strominger, Nucl. Phys. B
359 Ž1991. 611; A. Strominger, Nucl. Phys. B 343 Ž1990.
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w7x C. Schmidhuber, Nucl. Phys. B 404 Ž1993. 1833.
w8x B. Ovrut, S. Thomas, Phys. Lett. B 257 Ž1991. 292.
w9x M.J. Duff, J.X. Lu, Nucl. Phys. B 416 Ž1994. 301; M.J.
Duff, R.R. Khuri, J.X. Lu, Phys. Rep. 259 Ž1995. 213.
w10x B. Ovrut, S. Thomas, Phys. Rev. D 43 Ž1991. 1314.
w11x A. Cooper, L. Susskind, L. Thorlacius, Nucl. Phys. B 363
Ž1991. 132.
w12x S.R. Das, B. Sathiapalan, Phys. Rev. Lett. 56 Ž1986. 2664;
C. Itoi, Y. Watabiki, Phys. Lett. B 198 Ž1987. 486; A.A.
Tseytlin, Phys. Lett. B 264 Ž1991. 311.
w13x G.T. Horowitz, A.A. Tseytlin, Phys. Rev. D 50 Ž1994. 5204.
w14x J. Zinn-Justin, Quantum field theory and critical phenomena,
Oxford Science Pub., 1989.
w15x K. Ghoroku, hep-thr9612201, FIT-HE-96-82, to be published in Classical and Quantum Gravity.
12 February 1998
Physics Letters B 419 Ž1998. 91–98
Geometry, D-branes and N s 1 duality in four dimensions. II
Changhyun Ahn
1
Asia Pacific Center for Theoretical Physics (APCTP) 207-43 Cheongryangri-dong, Dongdaemun-gu, Seoul 130-012, South Korea
Received 22 May 1997
Editor: M. Dine
Abstract
We study N s 1 dualities in four dimensional supersymmetric gauge theories in terms of wrapping D 6-branes around
3-cycles of Calabi-Yau threefolds in type IIA string theory. We generalize the recent work of geometrical realization for the
models which have the superpotential corresponding to an A k type singularity, to various models presented by Brodie and
Strassler, consisting of D kq2 superpotential of the form W s Tr X kq1 q Tr XY 2. We discuss a large number of representations for the field Y, but with X always in the adjoint Žsymmetric. wantisymmetricx representation for SUŽ SO .w Sp x gauge
groups. q 1998 Elsevier Science B.V.
1. Introduction and geometrical setup
String theory interprets many nontrivial aspects of
four dimensional supersymmetric field theory by exploiting T-duality on the local model for the compactification manifold. There are two approaches to
describe this local model.
Approach I. is to consider the local description as
purely geometric structure of compactification manifold together with D-branes wrapping around cycles
w1–3x. The compactification of F-theory on elliptic
Calabi-Yau ŽCY. fourfolds from 12 dimensions leads
to N s 1 supersymmetric field theories in four dimensions. It has been studied in w1x, for the case of
pure SUŽ Nc . Yang-Mills gauge theory, that the gauge
symmetry can be obtained in terms of the structure
of the D 7-brane worldvolume. By adding some
numbers of D 3-branes and bringing them near the
1
E-mail: [email protected].
complex 2-dimensional surface Žwhich is the compact part of D 7-brane worldvolume., the local string
model gives rise to matter hypermultiplets in the
fundamental representation with pure SUŽ Nc . YangMills theory w2x. Moreover, Seiberg’s duality w4x for
the N s 1 supersymmetric field theory can be
mapped to T-duality exchanging the D 3-brane charge
and D 7-brane charge. For the extension of SO Ž Nc .
and SpŽ Nc . gauge theories w5,6x coupled to matter,
the local string models are type IIB orientifolds, for
which T-duality symmetry applies, with D 7-branes
on a curved orientifold 7-plane w3x.
Approach II. which is T-dual to the approach I is
to interpret N s 1 duality for SUŽ Nc . gauge theory
as D-brane description together with NS 5-branes in
a flat geometry w7x according to the approach of w8x.
Extension of this to the case of SO Ž Nc . and SpŽ Nc .
gauge theories with flavors was presented in w9x by
adding an orientifold 4-plane. The generalization to
the construction of product gauge groups SUŽ Nc . =
SUŽ NcX . with matter fields is given in w10,11x by
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 7 8 - 0
92
C. Ahn r Physics Letters B 419 (1998) 91–98
suspending two sets of D 4-branes between three NS
5-branes. See also the recent paper w12x dealing with
SO Ž Nc . = SpŽ NcX . product gauge group.
Ooguri and Vafa have considered in w13x that
N s 1 duality can be embedded into type IIA string
theory with D 6-branes, partially wrapped around
three cycles of CY threefold, filling four dimensional
spacetime. They discussed in the spirit of w2,3x what
happens to the wrapped cycles and studied the relevant field theory results when a transition in the
moduli of CY threefolds occurs. Furthermore, they
reinterpreted the configuration of D-branes in the
presence of NS 5-branes w7x as purely classical geometrical realization.
It is natural to ask how a number of known field
theory dualities which contain additional field contents arise from two different approaches we have
described so far. Recently, it was obtained in w14x
according to the approach I that one can generalize
the approach of w13x to various models, consisting of
one or two 2-index tensors and some fields in the
defining representation Žfundamental representation
for SUŽ Nc . and SpŽ Nc ., vector representation for
SO Ž Nc .., presented earlier by many authors w15–18x
and study its geometric realizations by wrapping D
6-branes about 3-cycles of CY threefolds. On the
other hand, very recently by introducing a multiple
of coincident NS 5-branes and that of NS’ 5-branes,
it was argued w19x that SUŽ Nc . gauge theory with
one or two adjoints superfields in addition to the
fundamentals can be obtained. Moreover, SO Ž Nc .
gauge group with an adjoint field and Nf vectors
was described in terms of a multiple of coincident
NS 5-branes and a single NS’ 5-brane with orientifold 4-plane Žsimilarly SpŽ Nc . with a traceless antisymmetric tensor and Nf flavors by adding orientifold 6-plane.. See also the relevant paper w20x
analyzing the brane configurations associated with
field theories in various dimensions.
In this paper, we apply the approach I to the
various models presented by Brodie and Strassler
w21x, consisting of D kq 2 superpotential with SU,SO
and Sp gauge groups along the line of w14x. We
discuss a large number of representations for the
field Y, but with X always in the adjoint Žsymmetric. wantisymmetricx representation for SUŽ SO .w Sp x
gauge groups. The superpotentials are given by W s
Tr X kq 1 q Tr XY 2 in the three cases while those in
other two cases have different forms that will appear
later.
We will review the main geometrical setup of
w13,14x in the remaining part of this section. Let us
start with the compactification of type IIB string
theory on the CY threefold leading to N s 2 supersymmetric field theories in 4 dimensions. Suppose
various D 3-branes wrapping around a set of three
cycles of CY threefold. It is known w22x that whenever the integration of the holomorphic 3-form on
the CY threefold around three cycles takes the form
of parallel vectors in the complex plane, such a D
3-brane configuration allows us to have a BPS state.
Then after we do T-dualizing the 3-spatial directions
of three torus T 3 we obtain type IIA string theory
with D 6-branes, partially wrapping around three
cycles of CY threefold, filling 4 dimensional spacetime. We end up with N s 1 supersymmetric field
theories in 4 dimensions.
The local model of CY threefold can be described
by w23,13x five complex coordinates x, y, xX , yX and z
satisfying the following equations:
x 2 q y 2 s Ł Ž z y ai . ,
i
x X 2 q yX 2 s Ł Ž z y b j .
j
)
where each of C ’s is embedded in Ž x, y .-space and
Ž xX , yX .-space respectively over a generic point z.
This describes a family of a product of two copies of
one-sheeted hyperboloids in Ž x, y .-space and
Ž xX , yX .-space respectively parameterized by the z-coordinates. For a fixed z away from a i and bj there
exist nontrivial S 1 ’s in each of C ) ’s corresponding
to the waist of the hyperboloids. Notice that when
z s a i or z s bj the corresponding circles vanish as
the waists shrink. Then we regard 3 cycles as the
product of S 1 = S 1 cycles over each point on the
z-plane, with the segments in the z-plane ending on
a i or bj . When we go between two a i ’s Ž bj ’s.
without passing through bj Ž a i . the 3 cycles sweep
out S 2 = S 1. On the other hand, when we go between
a i and bj the 3 cycle becomes S 3. We will denote
the 3-cycle lying over between a i and bj by w a i ,bj x
and also denote other cycles in a similar fashion.
In order to study SO Ž Nc . and SpŽ Nc . gauge theories, as done in w13,14x consider the local model of
the CY threefold given by
x 2 q y 2 s y Ł Ž z y a i . Ž z y aX . ,
i
xX 2 q yX 2 s yz
C. Ahn r Physics Letters B 419 (1998) 91–98
where a i ’s and aX are real numbers with a1 - a2 PPP - a k - 0 - aX . It is easily checked that the S 2 =
S 1 associated with w a iy1 ,a i x for i - k is realized
either by real values for x, y, xX , yX , z or by purely
imaginary values for x, y, xX , yX but real values for z.
Also note that the S 3 associated with w a k ,0x is realized by taking the real values of x, y, xX , yX and z
while the S 3 associated with w0,aX x is realized by
taking the imaginary values of x, y, xX and yX and
real values for z. In next section we will describe our
main results by exploiting this geometrical setup. We
will start with by writing down the configurations of
ordered points in the real axis of z-plane to various
models we are concerned with.
2. Geometrical realization of N s 1 duality
Let us consider particular N s 1 supersymmetric
field theories and see how their N s 1 dualities arise
from the configurations of ordered points after the
transition in moduli space of CY threefolds.
2.1. SU(Nc ) with two adjoint fields and Nf fundamental flaÕors [24,21]:
We study supersymmetric Yang-Mills theory with
gauge group SUŽ Nc . coupled to two chiral matter
superfields X and Y which transform under the
adjoint representation of SUŽ Nc ., Nf fundamental
chiral multiplets Q i and Nf antifundamental multiplets Q˜ i˜ where i,i˜s 1, PPP , Nf . The superpotential is
Tr X kq 1 q Tr XY 2 where k is odd. We will discuss
every steps for this case in detail while the other
cases will be explained very concisely. Let us consider the configuration of points ordered as
X
X
1
2
ž a ,b ,b ,b ,b ,b , PPP ,b
b
,a /
1
1
2
3
X
3 Ž ky1 .
,bXX1 ,bXX2 , PPP ,
2
XX
3 Ž ky1 .
2
2
where it is understood that when k s 1 the only
points of b 1 ,b 2 and b 3 appear between a1 and a2 .
Suppose that we wrap Ni D-branes around the cycle
w a1 ,bi x, NjX D-branes around the cycle w a1 ,bjX x and NlXX
D-branes around w a1 ,bXXl x for each i, j and l such that
3 Ž ky1 .
Ý3is1 Ni q Ýjs12
Ž NjX q NjXX . s Nc . Each of them has
93
Nf D-branes around the cycles w bi ,a2 x,w bjX ,a2 x and
w bXXl ,a 2 x for i, j and l respectively.
We now proceed the case of k s 1 for simplicity
and discuss how the N s 1 duality is realized geometrically by D-brane picture from our above proposed configuration. There are two points a1 and a2
along the real part of z-plane where the first C )
degenerates and three points b 1 ,b 2 and b 3 along the
real axis between a1 and a2 where the second C )
degenerates. Now we have five ordered special points
Ž a1 ,b 1 ,b 2 ,b 3 ,a2 . along the real axis. Then the three
cycle w a1 ,b 1 x lying between a1 and b 1 is S 3 and the
three cycle w b 1 ,b 2 x lying between b 1 and b 2 is
S 1 = S 2 . Thus the three cycle w a1 ,b 2 xis a bouquet of
S 3 and S 1 = S 2 joined together at z s b 1. We wrap
Ni D-branes around the three cycle w a1 ,bi x for each
i s 1,2,3 such that Ý3is1 Ni s Nc Žbecause in the limit
of b 1 ™ b 3 and b 2 ™ b 3 this system should be consistent with Nc D-branes around the cycle w a1 ,b 3 x.
and Nf D-branes around the three cycle w bi ,a 2 x for
each i where we assume that 3 Nf G Nc .
Now we would like to move to other point in the
moduli of CY threefolds and end up with the configuration in which the degeneration points are along
the real z-axis except that the orders are changed
from Ž a1 ,b 1 ,b 2 ,b 3 ,a2 . to Ž b 1 ,b 2 ,b 3 ,a1 ,a 2 .. As done
in w13,14x, first of all we push the point b 1 up along
the imaginary direction since we have the freedom to
turn on a Fayet-Iliopoulos ŽFI. D term. Then Ž N1 q
N2 q N3 . of D-branes connect directly between
Ž a1 ,b 2 . and Ž N2 q Nf q N3 y N1 y N2 y N3 . of Dbranes go between Ž b 1 ,b 2 .. We continue to move b 1
along the negative real axis, pass the x-coordinate of
a1 and push down it to the real axis. At this moment,
the Ž Nf y N1 . D-branes which were between Ž b 1 ,b 2 .
decompose to Ž Nf y N1 . D-branes between Ž b 1 ,a1 .
and Ž Nf y N1 . D-branes between Ž a1 ,b 2 . which
amounts to the decomposition of the three cycle
w b 1 ,b 2 x into a bouquet of two 3-cycles of S 3. The
Ž N1 q N2 q N3 . D-branes which were going between
Ž a1 ,b 2 . will recombine with the newly generated
Ž Nf y N1 . D-branes ending up with the total of Ž Nf q
N2 q N3 . D-branes along w a1 ,b 2 x cycle. Similarly we
do push the point b 2 in turn and move it between b 1
and a1. Then Ž Nf q N2 q N3 . of D-branes connect
directly between Ž a1 ,b 3 . and Ž2 Nf q N3 y Nf y N2 y
N3 . of D-branes go between Ž b 2 ,b 3 .. We can see that
the Ž Nf y N2 . D-branes which were between Ž b 2 ,b 3 .
94
C. Ahn r Physics Letters B 419 (1998) 91–98
decompose to Ž Nf y N2 . D-branes between Ž b 2 ,a1 .
and Ž Nf y N2 . D-branes between Ž a1 ,b 3 .. The Ž Nf q
N2 q N3 . D-branes which were going between Ž a1 ,b 3 .
do recombine with the new Ž Nf y N2 . D-branes ending up with the total of Ž2 Nf q N3 . D-branes along
w a1 ,b 3 x cycle. Of course, there are Ž2 Nf y N1 y N2 .
D-branes between Ž b 2 ,a1 . due to the two contributions from Ž Nf y N1 . D-branes between Ž b 1 ,a1 .
Žwhich can be decomposed into again Ž Nf y N1 .
D-branes between Ž b 1 ,b 2 . and those between Ž b 2 ,a1 ..
and Ž Nf y N2 . D-branes between Ž b 2 ,a1 .. Finally, we
push the point b 3 and move it between b 2 and a1.
Then Ž2 Nf q N3 . of D-branes connect directly between Ž a1 ,a 2 . and Ž3 Nf y N3 y 2 Nf . of D-branes go
between Ž b 3 ,a2 .. It is easy to see that the Ž Nf y N3 .
D-branes which were between Ž b 3 ,a 2 . decompose to
Ž Nf y N3 . D-branes between Ž b 3 ,a1 . and Ž Nf y N3 .
D-branes between Ž a1 ,a 2 .. Then the Ž2 Nf q N3 . Dbranes which were going between Ž a1 ,a 2 . will recombine with the new Ž Nf y N3 . D-branes allowing
us to get the total of 3 Nf D-branes along w a1 ,a2 x
cycle. There exist Ž3 Nf y N1 y N2 y N3 . D-branes
wrapping around w b 3 ,a1 x coming from Ž2 Nf y N1 y
N2 . D-branes between Ž b 2 ,a1 . Žthat will be decomposed into the same number of D-branes between
Ž b 2 ,b 3 . and Ž b 3 ,a1 .. and Ž Nf y N3 . D-branes between Ž b 3 ,a1 ..
The final configuration by putting all together is a
configuration of points ordered as Ž b 1 ,b 2 ,b 3 ,a1 ,a 2 .
with Ž Nf y N1 . D-branes wrapped around w b 1 ,b 2 x,
Ž2 Nf y N1 y N2 . D-branes wrapped around w b 2 ,b 3 x,
Ž3 Nf y Ý3is1 Ni . D-branes wrapping around w b 3 ,a1 x
and 3 Nf D-branes wrapped around w a1 ,a 2 x. Notice
that the number of D-branes along the cycle w b 3 ,a 2 x
in the original configuration of Ž a1 ,b 1 ,b 2 ,b 3 ,a2 . in
the moduli space of CY threefolds are the same of
those along the cycle w a1 ,a 2 x after we moved the
points b 1 ,b 2 and b 3 . In the limit b 1 ™ b 3 and b 2 ™
b 3 , this is exactly the dual magnetic description of
the original theory. The gauge group w24,21x is
SUŽ Nc˜ . s SUŽ3 Nf y Nc . since Ý3is1 Ni s Nc . This is a
marginal deformation of k s 3 duality of w15x clarifying the correspondence between D 3 and A 3 type
singularities. In addition to the Nf flavors of dual
quarks q i ,q˜i˜ and adjoint dual superfields X andY we
have three singlet chiral superfields Mi which interact with the dual quarks through the superpotential in
the magnetic theory.
We expect that for general value of k, the above
procedure can be done similarly. The final configuration after all the bi ,bjX and bXXl ’s are moved to the left
of a1 keeping the order of them we get is a configuration of points ordered as
X
X
1
2
ž b ,b ,b ,b ,b , PPP ,b
1
2
3
X
3 Ž ky1 .
,bXX1 ,bXX2 , PPP ,bXX 3Ž ky1. ,
2
2
a1 ,a2 .
with Ž Nf y N1 . D-branes wrapped around w b 1 ,b 2 x,
Ž2 Nf y N1 y N2 . D-branes wrapped around w b 2 ,b 3 x,
3 Ž ky 1 .
PPP ,
3 Ž ky 1 .
ŽŽ3 k y 1 . N f y Ý 3is 1 Ni y Ý js 12
y1
Ý ls 12
NlXX .
D-branes
around
NjX y
w b XX3 Ž ky 1 .
,
y1
2
3 Ž ky1 .
bXX3Ž ky1. x, Ž3kNf y Ý3is1 Ni y Ýjs12
Ž NjX q NjXX .. D-
2
branes wrapped around w bXX 3Ž ky1. ,a1 x and 3kNf D2
branes wrapped around w a1 ,a 2 x. In the limit of
bi ,bjX ,bXXl ™ bXX 3Ž ky1. , the gauge group SUŽ Nc˜ . s
2
SUŽ3kNf y Nc . appears. In this case there are also 3k
singlet fields, Mi j Ž i s 1, PPP ,k and j s 1,2,3. coupled to the magnetic quarks.
2.2. SO(Nc ) with two symmetric tensors and Nf Õectors (Sp(Nc ) with two antisymmetric tensors and Nf
flaÕors) [21]:
We discuss supersymmetric Yang-Mills theory
with gauge group SO Ž Nc . where the fields X and Y
are in the Nc Ž Nc q1. y1 traceless symmetric tensor rep2
resentation of SO Ž Nc . and Nf fields Q i are in the Nc
dimensional vector representation of SO Ž Nc . Ž i s
1, PPP , Nf .. The superpotential is Tr X kq 1 q Tr XY 2
where k is odd. Let us study the configuration of
points ordered as
Ž a1 ,a2 ,a3 ,aX1 ,aX2 , PPP ,aXky1 ,aXX1 ,aXX2 , PPP ,aXXky1 ,0,aX .
where aXi and aXXj are present for k G 3. Suppose we
consider N2i . 2 D-brane charges around the cycle
w a i ,0x, N2j . 2 D-brane charges around the cycle w aXj ,0x
and N2l . 2 D-brane charges around w aXXl ,0x for each
XX
Ž X
.
i, j and l such that Ý3ls1 Ni q Ý ky1
js1 Nj q Nj s Nc .
X
XX
C. Ahn r Physics Letters B 419 (1998) 91–98
There are also D-brane charges on the w0,aX x cycle of
Nf
3 Nf
2 for each i and 4 for each j and l after the action
of the orientifolding on the D-branes.
Let us first analyze the simplest case for the case
of k s 1. That is the configuration points of ordered
as Ž a1 ,a2 ,a3 ,0,aX .. We wrap N1 D-branes around
w a1 ,0x, N2 D-branes around w a 2 ,0x and N3 D-branes
around w a 3 ,0x. Each of them has Nf D-branes around
w0,aX x such that Ý3i Ni s Nc which can be understood
that the number of D-branes on the cycle w a 3 ,0x
should be Nc as a1 and a 2 get close to a 3 . Now the
conjugation Ž x, y, x X , yX z . ¨ Ž x ) , y ) , x X ) , yX ) , z ) .,
together with exchange of left- and right-movers on
the world sheet, will produce an orientifolding of the
above configuration. The conjugation preserves the
above equation for a i and aX real. In view of type I’
theory, these D-branes must be counted as N2i and N2f
D-branes after orientifolding since the orientifolding
leaves these cycles invariant. Therefore we get Dbrane charges of the w a i ,0x cycle of N2i . 2. Here the
factor .2 is due to a contribution from the orientifold plane in addition to the physical D-branes. The
upper sign corresponds to the SO Ž Nc . gauge group
and the lower one does the SpŽ Nc . gauge group.
By passing the point a 3 through the point 0
directly along the real axis due to the fact that this
operation should keep the orientifolding, the D-brane
charge gets changed to the value of N2f y N23 " 2 on
w0,a3 x where we assumed Nf G N3 . 4. Next we
move the point a2 to the positive real axis. The
D-brane charge changes to the value of y N22 " 2 on
w0,a2 x. The final configurations after we move a1 are
given by the Žy N21 " 2. D-brane charge on w0,a1 x,
Žy N22 " 2. D-brane charge on w0,a2 x, Ž N2f y N23 " 2.
D-brane charge around w0,a3 x, N2f D-brane charge on
w a 3 ,aX x and two N2f D-brane charges on w0,aX x. Recombining the last two N2f D-branes into the D-branes
around w0,a 3 x cycle and taking the limit of a1 ™ a3
and a2 ™ a3 , it leads to 32N f y N21 y N22 y N23 q 6 that
shows our expression for the magnetic dual group
w21x for k s 1 case, SO Ž N˜c . s SO Ž3 Nf y Nc q 12.
since there is no orientifold plane for a i Ž i s 1,2,3. )
0 and all these D-brane charges are physical Dbranes. There exist also 32N f D-branes on w a 3 ,aX x. The
fields X and its dual X are massive and can be
integrated out. This gives a quartic superpotential for
Y and its magnetic dual Y which are related to A 3
type singularity appeared in w16,18x. For the case of
95
Sp group with two antisymmetric tensors and Nf
flavors, by recognizing that in the convention of w18x
the symplectic group whose fundamental representation is 2 Nc dimensional as SpŽ Nc . and a flavor of it
has two fields in the fundamental representation
therefore 2 Nf fields we obtain 2 N˜c s 3Ž2 Nf . y 2 Nc
y 12 which gives rise to the following dual description SpŽ N˜c . s SpŽ3 Nf y Nc y 6..
For general value of k, the final configurations,
after we moved all the a i ,aXj and aXXl ’s to the right of
the position of zero by successively doing similar
things for the previous case, are given by the Žy N2i
" 2. D-brane charge on w0,a i x, the Žy N2j " 2. Dbrane charge on w0,aXj x, the Žy N2l " 2. D-brane
1
f
Ž 3
charge on w0,aXXl x for 1 F l F k y 2, Ž 3kN
2 y 2 Ý is1 Ni
XX
ky 1 Ž X
q Ý js1 Nj q Nj .. " 2Ž2 k q 1.. D-brane charge
f
around w0,aXXky1 x and 3kN
D-brane charge on
2
X
XX
f
w a ky 1 ,a x. Here the factor of 3kN
comes from the
2
Ž
.
3 Nf
contributions of the sum of 2 and 3 2 k y2 N f . In the
X
XX
4
limit of a i ,aXj ,aXXl ™ aXXky1 , we get the magnetic dual
group, SO Ž N˜c . s SO Ž3kNf y Nc q 4Ž2 k q 1... By
similar reasoning for the counting of dimension and
number of fields of k s 1, we obtain the magnetic
dual gauge group for symplectic group as SpŽ N˜c . s
SpŽ3kNf y Nc y 2Ž2 k q 1...
2.3. SO(Nc ) with a symmetric tensor and an adjoint
field and Nf Õectors (Sp(Nc ) with an antisymmetric
tensor and an adjoint field and Nf Õectors) [21]:
We analyze supersymmetric Yang-Mills theory
with gauge group SO Ž Nc . where the field X is in the
N c Ž N c q1 .
y1 dimensional symmetric tensor represen2
tation of SO Ž Nc ., the adjoint field Y is in the Nc Ž Nc y1.
2
dimensional antisymmetric tensor of SO Ž Nc . and Nf
flavors Q i are in the Nc dimensional vector representation of SO Ž Nc . Ž i s 1, PPP , Nf .. The superpotential
is Tr X kq 1 q Tr XY 2 where k is odd. Let us consider
the configuration of points ordered as
Ž b1 ,a1 ,aX1 ,aX2 , PPP ,aXky1 ,aXX1 ,aXX2 , PPP ,aXXky1 ,0,aX . .
After orientifolding, the net D-brane charge of w a1 ,0x
cycle becomes N20 . 2 and that of w0,aX x cycle is N2f
and we bring other D-branes to the left hand side of
the point a1 in the real axis whose D-brane charges
of w b 1 ,0x cycle are N1 and that of w0,aX x cycle are Nf .
C. Ahn r Physics Letters B 419 (1998) 91–98
96
X
There are also Ž N2j . 2. D-brane charges around the
cycle w aXj ,0x and Ž N2l . 2. D-brane charges around
w aXXl ,0x for each j and l such that N0 q 2 N1 q
XX
1Ž X
.
Ý ky
js1 Nj q Nj s Nc . There are also D-brane charges
X
on the w0,a x cycle of 34N f for each j,l after the action
of the orientifolding on the D-branes.
For simplicity, we will start with the case of
k s 3 ŽRemember that for the case of k s 1, the
configuration for the magnetic theory was yN1 Dbrane charge around the cycle w0,b 1 x, 32N f y N20 " 2
on w0,a1 x and 32N f on w a1 ,aX x. We have seen that this
duality was given in w17,18x already. which is the
configuration of points ordered as Ž b 1 ,a1 ,aX1 ,aX2 ,aXX1 ,
aXX2 ,0,aX .. We push the point b 1 ,a1 ,aXi and aXXj ’s along
the real axis to the right and pass the point 0. In
order to count the number of D-branes wrapping
around cycles we use the D-brane charge conservation and the orientation of the D-branes. Then we get
the Žy N21 " 2. D-brane charge on w0,a1 x, PPP , the
Žy N21 " 2. D-brane charge on w0,aXX1 x, Ž 7N2 f y 12 Ž2 N1
q Ý2js1Ž NjX q NjXX .. " 10. D-brane charge around
w0,aXX2 x and 7N2 f D-brane charge on w aXX2 ,aX x. Next we
move the point b 1 along the real axis from negative
to positive values. The D-brane charge on w0,b 1 x is
yN0 due to the orientation. The Nf D-branes which
were going between Ž0,aX . can be decomposed into
Nf D-branes between Ž0,aXX2 . and Nf D-branes between Ž aXX2 ,aX .. Therefore the final picture we end up
with is that there are yN0 D brane charge on Ž0,b 1 .,
Žy N21 " 2. D-brane charge on w0,a1 x, PPP , Ž 92N f
y 12 Ž N0 q 2 N1 q Ý2js1Ž NjX q NjXX .. " 10. D-brane
charges on Ž0,aXX2 . and 9N2 f on Ž aXX2 ,aX .. In the limit of
b 1 ,a1 ,aXi ,aXXj ™ aXX2 , the magnetic dual group can be
written as SO Ž N˜c . s SO Ž9Nf y Nc q 20.. By twicing
the Nf and Nc and dividing by two which leads to
9 Ž2 N f .y2 N c y20
, we get SpŽ N˜c . s SpŽ9Nf y Nc y 10. for
XX
XX
2
the symplectic group.
For the general value of k G 5, the final configuration is that there are yN0 D brane charges on
Ž0,b 1 ., Žy N21 " 2. D-brane charge on w0,a1 x, PPP ,
Žy N2i " 2. D-brane charge on w0,aXi x, Žy N2j " 2.
f
D-brane charge on w0,aXXj x for 1 F j F k y 2, Ž 3kN
2
X
XX
1
ky
1
y 2 Ž N0 q 2 N1 q Ý js1 Ž Nj q Nj .. " 2Ž2 k y 1.. Df
brane charges on Ž0,aXXky 1 . and 3kN
on Ž aXXky1 ,aX .. In
2
X
XX
XX
the limit of b 1 ,a1 ,a i ,a j ™ a ky1 , the dual theory has
gauge group, SO Ž N˜c . s SO Ž3kNf y Nc q 8 k y 4. and
SpŽ N˜c . s SpŽ3kNf y Nc y 4 k q 2..
X
XX
2.4. SU(Nc ) with an adjoint field, a symmetric and
conjugate symmetric tensors and Nf fundamental
flaÕors (SU(Nc ) with an adjoint field, an antisymmetric and conjugate antisymmetric tensors and Nf fundamental flaÕors) [21]:
The field X is in the adjoint representation of
SUŽ Nc ., Y and Y˜ are Nc Ž Nc q1. symmetric tensor and
2
N c Ž N c q1 .
conjugate symmetric tensor representations
2
of SUŽ Nc . respectively and there are Nf fundamental
multiplets Q i and Nf antifundamental multiplets Q˜ i˜
where i,i˜s 1, PPP , Nf . The superpotential is Tr X kq 1
q Tr XYY˜ where k is odd. Let us consider the
configuration of points ordered as
X
X
1
2
ž b ,b ,b , PPP ,b
1
X
3 Ž ky1 .
2
,bXX1 ,bXX2 , PPP ,bXX 3Ž ky1. ,
2
X
a,0,a . .
Now we continue to repeat the procedure we have
done so far for the case of ordered configuration as
Ž b 1 ,bX1 ,bX2 ,bX3 ,bXX1 ,bXX2 ,bXX3 ,a,0,aX . when we consider k s
3 case ŽRecall that when k s 1, the configuration
was yN1 D-brane charge on w0,b 1 x and 32N f y N20 " 2
on w0,ax and 32N f on w a,aX x. This duality was discussed in w18x.. After orientifolding, D-brane charges
are N1 around w b 1 ,0x, N20 . 2 around w a,0x, NiX around
w biX ,0x and NjXX around w bjXX ,0x. Each of them produces
Nf
D-brane charge around w0,aX x except that N1
2
D-brane on w0,b 1 x does Nf D-brane charge where
N0 q 2 N1 q 2Ý3is1Ž NiX q NiXX . s Nc . The final configuration is given by yN1 D-brane charge on
Ž0,b 1 ., PPP , Ž 9N2 f y N20 y N1 y Ý3is1Ž NiX q NiXX . " 2.
D-brane charge on w0,ax and 9N2 f D-brane charge on
w a,aX x. Finally we see that the dual theory has the
gauge group SUŽ N˜c . s SUŽ9Nf y Nc q 4.. On the
other hand, when we consider antisymmetric and its
conjuagte tensors with flavors one we get SUŽ N˜c . s
SUŽ9Nf y Nc y 4..
For the general value of k, after orientifolding
D-brane charges are N1 around w b 1 ,0x, N20 . 2 around
w a,0x, NiX around w biX ,0x and NjXX around w bjXX ,0x. The
final configuration after we move all the b 1 ,biX ,bjXX
and a to the positive real axis of z-plane is given by
N0
f
yN1 D-brane charge on Ž0,b 1 ., PPP , the Ž 3kN
2 y 2
3 Ž ky1 .
y N1 y Ýis12
Ž NiX q NiXX . " 2. D-brane charge on
C. Ahn r Physics Letters B 419 (1998) 91–98
f
w0,ax and 3kN
w Xx
2 D-brane charge on a,a . We arrive at
the magnetic dual group for SU, SUŽ N˜c . s SUŽ3kNf
q 4 y Nc . and that for SUŽ Nc . with an antisymmetric flavor and Nf fundamental flavors is SUŽ N˜c . s
SUŽ3kNf y 4 y Nc ..
2.5. SU(Nc ) with an adjoint field, an antisymmetric
tensor and a conjugate symmetric tensor [21]:
In this case the field X is in the adjoint representation of SUŽ Nc ., the field Y is in the Nc Ž Nc y1.
97
D-branes wrapped around w a1 ,a 2 x. In the limit b 1 ,biX
Ž
.
™ bXky 1 , the gauge group SUŽ N˜c . s SUŽ 3 k m f q m˜ f y
2
Nc . appears.
Acknowledgements
I thank Kyungho Oh for helpful discussions and
Dept. of Physics, Hanyang Univ. for the hospitality
where part of this work was done.
2
representation, the field Y˜ in the Nc Ž Nc q1. representa2
tion and m f Ž m
˜ f . fields Q i Ž Q˜ i˜. in the Žanti.fundamental representation. The superpotential is given by
TrŽ XX˜ . kq 1 q Tr XYY˜ where k is odd or even. Let us
consider the configuration of points ordered as
Ž a1 ,b1 ,bX1 , PPP ,bXky1 ,a2 . .
We wrap N1 D-branes around the cycle w a1 ,b 1 x, NiX
D-branes around the cycle w a1 ,biX x for each i s
3Ž m f q m
˜ f.
1 X
1,2, PPP ,k y 1 such that N1 q Ý ky
is1 Ni s Nc ,
2
Ž
.
D-branes around the cycle w b 1 ,a 2 x and 3 m f q m̃ f D2
branes around the cycle w biX ,a2 x for each i. We
expect that the above procedure of case 1. can be
applied similarly, for example, k s 2. We wrap N1
D-branes around the three cycle w a1 ,b 1 x and N1X
D-branes around w a1 ,bX1 x such that N1 q N1X s Nc and
3 Ž m f q m̃ f .
D-branes around the cycle w b 1 ,a 2 x and
2
w bX1 ,a2 x respectively. The final configuration after we
move b 1 and bX1 to the left of a1 in the configuration
of point ordered as Ž a1 ,b 1 ,bX1 ,a 2 . we get is a configuration of points ordered as Ž b 1 ,bX1 ,a1 ,a 2 . with
Ž 3Ž m f q m˜ f . y N1 . D-branes wrapped around w b 1 ,bX1 x
2
and Ž3Ž m f q m
˜ f . y N1 y N2 . D-branes wrapped
around w bX1 ,a1 x, 3Ž m f q m
˜ f . D-branes wrapped around
w a1 ,a 2 x. In the limit b 1 ™ bX1 , the gauge group becomes SUŽ N˜c . s SUŽ3Ž m f q m
˜ f . y Nc ..
For general value of k, the final configuration is a
configuration of points ordered as Ž b 1 ,bX1 , PPP ,
Ž
.
bXky 1 ,a1 ,a 2 . with Ž 3 m f q m˜ f y N1 . D-branes wrapped
2
around w b 1 ,bX1 x, Ž3Ž m f q m
˜ f . y N1 y N2 . D-branes
Ž
.
X
X
wrapped around w b 1 ,b 2 x, PPP ,Ž 3 k m f q m˜ f y Ý kis1 Ni .
2
D-branes wrapped around w bXky 1 ,a1 x, and
3kŽm f q m
˜ f.
2
References
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w2x M. Bershadsky, A. Johansen, T. Pantev, V. Sadov, C. Vafa,
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in String Theory, hep-thr9702014.
w8x A. Hanany, E. Witten, Type IIB Superstrings, BPS
Monopoles, and Three-Dimensional Gauge Dynamics, hepthr9611230.
w9x N. Evans, C. Johnson, A. Shapere, Orientifolds, Branes, and
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w11x A. Brandhuber, J. Sonnenschein, S. Theisen, S. Yankielowicz, Brane Configuration and 4D Field Theory Dualities,
hep-thr9704044.
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Groups from Brane Configurations, hep-thr9704198.
w13x H. Ooguri, C. Vafa, Geometry of N s1 Dualities in Four
Dimensions, hep-thr9702180.
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Four Dimensions I, hep-thr9704061.
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Schwimmer, Brane Dynamics and N s1 Supersymmetric
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12 February 1998
Physics Letters B 419 Ž1998. 99–106
Dualities in 4D theories with product gauge groups
from brane configurations
Radu Tatar
1
UniÕersity of Miami, Department of Physics, Coral Gables, FL, USA
Received 21 September 1997; revised 13 October 1997
Editor: M. Dine
Abstract
We study brane configurations which correspond to N s 1 field theories in four dimensions. By inverting the order of the
NS 5-branes and D6-branes, a check on dualities in four dimensional theories can be made. We consider a brane
configuration which yields electricrmagnetic duality for gauge theories with SO Ž Nc1 . = SpŽ Nc 2 . product gauge group. We
also discuss the possible extension to any alternating product of SO and Sp groups. The new features arising from the
intersection of the NS 5-branes on the orientifold play a crucial role in our construction. q 1998 Elsevier Science B.V.
1. Introduction
Three years ago, the study of non-perturbative
effects in supersymmetric gauge theories has received a big impact from the work of Seiberg and
collaborators w1,2x. They studied the nonperturbative
description of N s 1 supersymmetric gauge theories
for many gauge groups with flavors in different
representations.
After the electricrmagnetic dualities for N s 2
theories were considered in w11x, Seiberg conjectured
the electric-magnetic duality for N s 1 gauge theories w12x.
The duality was formulated for gauge theories
with gauge group SUŽ Nc . and Nf flavors in fundamental representation, the dual being a theory with
gauge group SUŽ Nf y Nc . and Nf flavors in the
1
E-mail: [email protected].
fundamental representation. Following this example,
an intense effort was made to obtain dualities for
other gauge groups with flavors in all possible representations. w3–10x
More recently, the connection between string dualities and field theory dualities has become more
and more evident. The most important tools are the
D-branes, which describe solitonic defects where the
open string can end w14x.
Because the open string can end on them, they
have a world-sheet Abelian gauge field. When we
put N D-branes on top of each other, the system
behaves like a UŽ N . gauge theory. By T-duality, we
can navigate between lower and higher dimensional
D-branes. Besides D-branes, NS fivebranes are also
important in obtaining the field theory dualities from
string theories.
The first study of brane configuration came in the
paper of Hanany and Witten w16x. They studied the
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 3 9 - 1
100
R. Tatarr Physics Letters B 419 (1998) 99–106
brane configuration which gives N s 2 in 4 dimensions. Their construction is equivalent, by a T-duality transformation, to the geometric singularity approach of w17–19x. The main idea is to compactify
type IIB superstring on K 3 = R 5,1, K 3 being viewed
as an elliptic fibration over the complex plane and
having singularities where one of the cycles in the
fibre shrinks to zero. As shown in w17,18x, by moving in the fibration between 2 singularities, the cycle
covers a holomorphic curve which has the property
of breaking half of the supersymmetry. So we can
wrap a D-brane partially on this holomorphic curve
and partially along other noncompact direction in
space time. In w18,19x it was proved that by taking
the T-dual one obtains at singularities 2 NS fivebranes carrying magnetic charge. The D p-brane
which was wrapped around the holomorphic curve in
the original picture becomes now a D p y 1-brane
with 1 compact direction, being stretched between 2
NS fivebranes.
For p s 4, the result is a 3 q 1 worldvolume
stuck between two NS fivebranes, with one compact
direction, which is actually a 3 dimensional field
theory. For Nc D3-branes on top of each other, one
obtains a 3 dimensional theory with gauge group
UŽ Nc .. If we insert Nf D5-branes between the two
NS fivebranes, they describe a UŽ Nf . global theory
because their worldvolume is noncompact in all directions. By changing the positions of branes in
spacetime, very interesting connections have been
obtained between Coulomb phase, Higgs phase and
the mirror symmetry.
In their very important paper, Elitzur, Giveon and
Kutasov w15x have used a configuration of branes
which gives a theory with 4 supercharges in 4 dimensions. They have checked the Seiberg duality for
the group UŽ Nc . with Nf flavors in the fundamental
representation. The construction was generalized by
other authors to other configurations. Particularly
interesting is the case of the SO and Sp groups,
where the string theory becomes non-orientable and
we have to add orientifolds w14,23,24x. More interesting results for brane configurations corresponding to
three, four and five dimensional field theories have
been obtained in w28x. The brane configurations were
also discussed from the point of view of strongly
coupled string theory ŽM theory. and very nice connections with Seiberg-Witten spectral curves have
been obtained in w13,23x. In the present work, we try
to generalize the work of w23x and w24x to the case of
product gauge group SO Ž Nc1 . = SpŽ Nc2 ., using results from the above papers and the result of w25x and
w27x for the product gauge groups SUŽ Nc1 . =
SUŽ Nc2 .. The final result will be compared with w8x.
In section two we will review some important
aspects of the known dualities.
In section three we will discuss a new duality
given by brane configurations and we will see that
the conjectures made in w23,24x are verified in this
case.
We will conclude by making some comments.
2. Some dualities from brane configurations
2.1. Brane configurations in the oriented case
If one compactifies type IIA on a Calabi-Yau
manifold, a configuration which preserves N s 1
supersymmetry in 4 dimensions is obtained. The
Calabi-Yau manifold is viewed as a double elliptic
fibration. Take the cycles of the two tori living in the
fiber in the Ž89. and Ž45. directions, wrap D-branes
on the holomorphic curves determined by moving
these 2 cycles between two singularity points and
make the T dualities with respect to the Ž45. directions and Ž89. directions to obtain two NS fivebranes
with different orientations. 2
This is the configuration considered in w15x and
consist in:
1. NS fivebrane with worldvolume Ž x 0 , x 1 , x 2 , x 3 ,
4
x , x 5 .,
2. NS’ fivebrane with worldvolume Ž x 0 , x 1 , x 2 , x 3,
8
x , x 9 .,
3 . D4 four branes with worldvolum e
Ž x 0 , x 1 , x 2x, 6x.3 ,,
4. D6 sixbranes with worldvolume Ž x 0 , x 1 , x 2 , x 3 ,
7
x , x 8 , x 9 ..
The 6-th direction is the compact direction for the
D4 branes which is stretched in that direction be-
2
This connection between the geometric singularity picture and
the intersecting brane picture was very beautiful explained by
Prof. Ooguri in his excellent set of lectures held at Spring School
in String Theory-Trieste 1997.
R. Tatarr Physics Letters B 419 (1998) 99–106
tween the NS and NS’ fivebranes. As explained in
w15x this brane configuration keeps 1r8 of the original SUSY, so it gives N s 1 in four dimensions. The
NS and NS’ branes have to coincide in the 7-th
direction, otherwise the supersymmetry will be further broken. In w15x one of the two NS branes was
moved in the 7-th direction with the cost of breaking
the UŽ Nc . gauge group given by Nc D4 branes living
between NS and NS’. This is the Fayet-Iliopoulos
mechanism which protects us against supersymmetry
breaking.
By moving the NS’ brane to the left of the NS
brane considered to be fixed, the authors of w15x
obtained the dual description. The main point was
the conservation of linking number Žmagnetic charge.
which, as Hanany and Witten first showed in w16x
leads to the appearance of a new D4 brane everytime
NS’ and a D6 branes change their positions. The
dual description obtained after all these interchanges
describes a theory with gauge group UŽ Nf y Nc . and
with the same number of flavors.
In the case of oriented string, the work of w15x
was generalized in w25x and w27x for the case of
product groups UŽ Nc1 . = UŽ Nc2 ..
101
gauge symmetries. By taking one NS to be stuck on
the orientifold and moving all other NS branes from
the left to the right of the fixed one, a dual configuration is obtained and the result agrees with the ones of
field theory. The main point is that during the transition from electric to dual theory, the NS branes have
to intersect and at the intersection point we are in
strong coupling limit and some interesting phenomena happen. The fact we are in the strong coupling
limit can be seen both from the point of view of
string theory Žthe dilaton blows up there, so the
string coupling constant becomes infinite. and field
theory Žthe gauge coupling constant is proportional
with the inverse of the distance between the two NS
branes we are talking about..
Another important aspect of the presence of the
orientifold is that the possible flavor symmetry is Sp
when the gauge group is SO and SO when the
gauge group is Sp w23x. The origin of this difference
is the different sign of V 2 when it acts on D4 and
D6 branes, a fact that is T45789-dual to the situation of
w21,22x where the action of V 2 had different signs
when acting on D5 and D9 branes.
2.2. Nonorientable case
3. A new duality
For nonorientable strings, the gauge group which
appears when one puts D-branes on top of each other
is SO Ž N . or SpŽ N .. A new feature of the nonorientable theories is the appearance of orientifolds.
These are generalizations of the orbifolds, having a
supplementary worldsheet symmetry besides the
space-time symmetry. Because of the orientifold, the
D-branes are forced to appear in pairs. The orientifold is a BPS state so it breaks half of the supersymmetry. To avoid a supplementary breaking of
SUSY in the presence of an orientifold, the authors
of w23x and w24x considered the case of an orientifold
which is parallel with the D branes. We can have 2
types of orientifolds:
- O4 which are parallel and have the same worldvolume as the D4 branes.
- O6 which are parallel and have the same worldvolume as the D6 branes.
As a function of the orientifold charge, the configurations considered in w23,24x can have SO or Sp
We will consider in this paper the case of an O4
orientifold.
As we stated before, the orientifold is parallel
with the D4 brane and the D4 brane is the only brane
which is not intersected by the O4 orientifold. The
orientifold gives a spacetime reflection
Ž x 4 , x 5, x 7, x 8 , x 9 . ™ Žyx 4 ,y x 5,y x 7,y x 8 ,y x 9 .
which are all noncompact directions so the field
line can go to infinity.
On directions where the orientifold is a point, any
object which is extended along them will have a
mirror copy of itself. The NS branes has a mirror in
x 4 , x 5 directions, NS’ has a mirror in x 8 , x 9 and D6
has one in x 7, x 8 , x 9 directions. These objects and
their mirrors enter only once, taking both would be
overcounting. In our discussion, we will make a
difference between the number of branes when we
count branes and their mirrors and the number of
physical branes when we count only branes without
R. Tatarr Physics Letters B 419 (1998) 99–106
102
their mirrors. We will specify at any moment if we
are referring to branes or physical branes.
Another important aspect of the orientifold is its
charge, given by the charge of H Ž6. s dAŽ5. coming
from RR sector. In the natural normalization, where
the D4 brane carries unit charge, the charge of the
O4 plane is "1, for yV 2 s "1 in the D4 brane
sector.
We now introduce the electric theory. The gauge
group is SO Ž Nc1 . = SpŽ Nc2 . with 2 Nf 1 flavors in the
vector representation of SO group and 2 Nf 2 flavors
in the fundamental representation of Sp group. In
brane language, this corresponds to three NS branes.
As discussed in w27x, it is not sufficient to have only
perpendicular branes like NS and NS’ i.e. in Ž x 4 , x 5 .
and Ž x 8 , x 9 . directions. We need branes at different
angles in Ž x 4 , x 5, x 8 , x 9 . directions. Denote them by
A, B and C from right to left Žso A is in the far right.
on the compact x 6 direction. Then we have the
following orientation: the B brane is oriented at zero
degree, i.e in Ž x 4 , x 5 . direction, the brane A is
oriented at an angle u 1 with respect to B and C is
rotated at an angle u 2 with respect to B. The angles
u1 and u 2 are not arbitrary. The N s 1 theory is
obtained from N s 2 supersymmetric theory when
we give mass to the adjoint fields of the N s 2
theory. As explained in w27x, the angles u 1 and u 2
are just given by:
m1 s tan Ž u 1 . ,
m 2 s tan Ž u 2 .
Ž 1.
where m1 ,m 2 are the masses for the adjoint fields,
one for SO groups and the other one for Sp group,
which are integrated out when we go from N s 2
theory to N s 1 theory.
From right to left we have: Nc1 D4 branes between A and B, Nc2 D4 branes between B and C .
As a function of the orientifold projection, we have
two sectors, between A and B we have symmetric
O4 projection and between B and C we have antisymmetric O4 projection. Therefore the number of
physical branes between A and B is Nc1r2 and the
number physical branes between B and C is Nc2 . As
discussed in w23x, the sign of the AŽ5. charge flips as
one passes a NS fivebrane. If the sign of AŽ5. is
chosen to be positive between A and B, it will be
negative between B and C. For this reason the gauge
group product SO = Sp or Sp = SO Žif we choose
the sign to be negative between A and B. is the only
possibility, meaning that SO = SO or Sp = Sp cannot exist. With our choice of sign, the gauge group is
SO Ž Nc1 . = SpŽ Nc2 .. Between A and B we have 2 Nf 1
D6 branes which intersect the Nc1 D4 branes and
between B and C we have 2 Nf 2 D6 branes which
intersect the Nc2 D4 branes. Here we count the
number of branes, i.e. the number of branes plus
their mirrors. If we talk about the number of physical
branes, we have Nf 1 physical D6 branes between A
and B and Nf 2 physical branes between B and C.
Strings stretching between the Nf 1 physical D6branes and the Nc1r2 physical D4 branes are the
chiral multiplets in the vector representation of
SO Ž Nc1 .. Strings stretching between the Nf 2 physical
D6 branes and the Nc2 physical D4 branes are the
chiral multiplets in the fundamental representation of
SpŽ Nc2 . group. The Nf 1 physical D6 branes are
parallel with the A brane and the Nf 2 physical D6
branes are parallel with the C brane so there exist
chiral multiplets which correspond to the motion of
D4 branes in between the NS and D6 branes, as
discussed in w27x. These states are precisely the chiral
mesons of the dual theory.
When one considers strings stretched between the
Nc1r2 and Nc2 physical D4 branes, a field X in the
Ž Nc1 ,2 Nc2 . representation of product gauge group is
obtained.
For the field X, there is a superpotential in the
theory which truncates the chiral ring. This superpotential is deduced as follows: start with an N s 2
theory with gauge group SO Ž Nc1 . = SpŽ Nc2 ., Nf 1 hypermultiplets Q i charged under SO Ž Nc1 . and Nf 2
hypermultiplets QX i charged under SpŽ Nc2 . and the
adjoint fields X 1 , X 2 of respectively SO and Sp
groups. Then we write the superpotential of the
theory which has terms like
W s l1 QX1 Q q l2 QX X 2 QX q XX1 X q XX 2 X
Ž 2.
Breaking the N s 2 supersymmetry, we set l1 s l2
s 0, we give the adjoint fields X 1 , X 2 masses m1 ,m 2
and we integrate them out. What remains is a superpotential:
Wsy
1
2
ž
1
1
q
m1
m2
/
Tr X 4
Ž 3.
R. Tatarr Physics Letters B 419 (1998) 99–106
So the superpotential of N s 1 supersymmetric theory goes like W s Tr X 4 and this truncates the
chiral ring. As we discussed before the masses of the
adjoint fields are directly connected with the angle of
rotation on the NS fivebrane in Ž x 4 , x 5, x 8 , x 9 . plane.
as in Eq. Ž1..
Now we go to the magnetic theory.
We show that the result is a theory with the gauge
group SO(N˜c1 . = SpŽ N˜c2 . with N˜c1 s 4 Nf 2 q 2 Nf 1 y
2 Nc2 and N˜c2 s 2 Nf 1 q Nf 2 y Nc1r2. The anomaly
cancellation for the SpŽ Nc2 ., requiring Nf 2 q Nc1 to
be even, ensures that N˜c2 is an integer.
Like in w27x, first move all the physical Nf 1
physical D6 branes Žplus their mirrors if we talk
about the total number of branes. to the left of all NS
branes. They are intersecting both B and C NS
branes. Using the linking number conservation argument, it results that each physical D6 brane has two
physical D4 branes on its right after transition.
By moving all Nf 2 physical D6 branes to the right
past, they intersect both B and A and the conservation of linking number tells that each physical D6
brane has two physical D4 branes on its left.
Because the NS branes are trapped at the spacetime orbifold fixed points Žwhich form the orientifold plane., they cannot avoid the intersection so
they have to meet and there is a strong coupling
singularity. When each one of B and C actually
meets A, such a singularity appears. From the field
theory point of view, such a non smooth behavior
was expected because for SpŽ Nc . and SO Ž Nc . groups
there is a phase transition.
In w23x, the effect of such a singularity was deduced to be the appearance or disappearance of two
D4 branes. In w24x the transition was shown to be
smooth when the linking number in both sides of any
NS brane is the same.
For SO groups, the procedure is to put two D4
branes on top of the orientifold plane and break the
other Nc1 y 2 D4 branes, entering in a Higgs phase.
For Sp groups a pair of D4 branes and anti-D4 brane
plus their mirrors were created, the antifour-branes
cancelling the charge difference along the orientifold.
103
Let us see how the transition to magnetic theory
works. Remember that the initial configuration is,
from right to left:
the NS brane A, to its left the NS brane B
(between them the O4 projection being symmetric SO), and to the left of B being the NS brane C
(between B and C the O4 projection being antisymmetric - Sp).
First move C to the right of B. In w23x language,
two D4 branes must disappear because we have Sp
group. Between B and C branes we have a deficit of
two D4 branes. Now move C to the right of A. When
C passes A, 2 D4 branes appear between A and C
because we have an SO group, so we have now a
deficit of 2 D4 branes between B and A and no
deficit between A and C. In this moment the configuration is as follows, from right to left:
the NS brane C, to its left the NS brane A
Žbetween them the O4 projection being symmetric SO . and to the left of A being the NS brane B
Žbetween A and B the O4 projection being antisymmetric - Sp and there is a deficit of 2 D4 branes..
We want to move now B to the right of A in order
to arrive to the magnetic picture. We encounter a
new phenomenon here. Between B and A we have
antisymmetric O4 projection, so after this transition,
there is another deficit of 2 D4 branes between A
and B Žfrom left to right.. But the D4 branes which
were before between B and A are changing the
orientation after B comes to the right of A so we
have actually a deficit of 2 D4 branes with one
orientation and 2 D4 branes with another orientation
which are thus cancelling each other i.e. the addition
of their physical charges gives 0. So there is no
supplement or deficit of D4 branes. Remembering
that between A and C there was no supplement or
deficit of D4 branes, it results that there are no D4
branes which appear or disappear in the transition
from the electric to magnetic theory.
In w24x language, for a smooth transition between
B and C, we need to create 2 pairs of D4 branes and
anti D4 branes, the anti-fourbranes neutralize the
charge difference along the orientifold. In the same
way as before, the 2 D4 branes to be put on top of
R. Tatarr Physics Letters B 419 (1998) 99–106
104
the orientifold when C passes A and B passes A
annihilate the anti four branes. The two D4 branes
which were on top of the orientifold came from the
Nc1 fourbranes connecting A and B therefore leaving
Nc1 y 2 fourbranes between A and B. So the two D4
branes which remain after their anti-branes vanish
add to the Nc1 y 2. Therefore, by smoothing the
transition we did not create any D4 or anti D4
branes, as expected from the field theory calculation.
The final picture is the following, from left to
right:
Nf 1 physical D6 branes connected by 2Nf 1 physical D4 branes with A. Between A and B we haÕe N˜c2
physical D4 branes, between B and C we haÕe
Ñc1 r 2 physical D4 branes and to the right of C we
haÕe Nf 2 physical D6 branes, connected by 2Nf 2
physical D4 branes with C (plus their mirrors).
Here, we have counted only the physical D branes.
Between A and B we have antisymmetric O4-projection, so we are forced to place an even number of D4
branes, the number of physical D4 branes being N˜c2 .
Between the Nf 1 physical D6 branes and A we have
2 Nf 1 physical D4 branes because the Nf 1 physical
D6 branes have passed two NS branes ŽB and C. so
this is the correct number. The same for the 2 Nf 2
physical D6 branes which are between the Nf 2 physical D6 branes and C.
We use the linking number of A to calculate N˜c2
and we apply the formula:
l NS s 12 Ž R D6 y L D6 . q Ž L D4 y R D4 .
qQ Ž O4 . Ž LO4 y R O4 .
Ž 4.
where Ž L, R . D6 ŽŽ L, R . O4 .wŽ L, R . D4 x is the number of
physical D6 branes ŽO4 planes. wphysical D4 branesx
to the left or right of the NS fivebrane for which we
are calculating the linking number. Here QŽO4. is
the charge of the O4 plane. In the original picture,
the A brane sees an O4 plane of charge y y 1 on its
left Žbecause of the symmetric O4 projection between A and B. and an O4 plane of charge q1 on
its right. In the final picture, the A brane sees an O4
plane of charge q1 on its right Žbecause of the
antisymmetric O4 projection. and an O4 plane of
charge y1 to its left. So the contribution of the O4
plane to the linking number is the same in the initial
and in the final configurations. Therefore the conservation of the linking number is the same as the
conservation of the physical charge. In the original
picture the charge is yNf 1r2 y Nf 2r2 q Nc1r2
where we used the numbers of physical D branes. In
the magnetic picture the charge is yNf 1r2 q Nf 2r2
y N˜c2 q 2 Nf 1. We have counted only the contributions of the physical branes. Making the above physical charges equal, we obtain
Ñc2 s 2 Nf 1 q Nf 2 y Nc1r2.
Ž 5.
For the B brane, in the original and the final pictures
it sees an O4 plane of charge y1 to its left and an
O4 plane of charge q1 to its right. Therefore the
conservation of the linking number is the same as the
conservation of the physical charge. This conservation between the initial and the final configurations
gives Žagain we consider the number of physical
branes.:
yNf 2r2 q Nf 1r2 q Nc2 y Nc1r2
s N˜c2 y N˜c1r2 y Nf 1r2 q Nf 2r2
Ž 6.
We obtain
Ñc1 s 4 Nf 2 q 2 Nf 1 y 2 Nc2 .
Ž 7.
The values for N˜ci ,i s 1,2 coincide with the ones
obtained in w8x. Žfor k s 0 case where 2 k q 1 corresponds to the number of NS branes which are moving. In this paper we considered the case of a single
NS brane which is moving..
From the brane configuration discussed above, the
field content of the theory is:
- gauge group SO(N˜c1 . = SpŽ N˜c2 .,
- Nf 2 fields in the vector representation of
SO Ž N˜c1 .,
- Nf 1 fields in the fundamental representation of
SpŽ N˜c2 .,
- a field Y in the (N˜c1 ,2 N˜c2 . representation of the
product gauge group.
- the chiral mesons of the dual theory which
appear as discussed before and which haÕe the same
form as in [8].
We will make an additional check of our result.
Change the overall sign of V 2 so the gauge group
R. Tatarr Physics Letters B 419 (1998) 99–106
becomes now SpŽ Nc1 . = SO Ž Nc2 .. Now we have Nc1
physical D4 branes between A and B because of the
antisymmetric O4 projection and Nc2r2 physical D4
branes between B and C because of the symmetric
O4 projection. We have 2 Nf 1 flavors in the fundamental representation of the Sp group and 2 Nf 2 in
the vector representation of the SO group.
When strings stretch between the Nc1 and Nc2r2
physical branes, the corresponding field X is in the
Ž2 Nc1 , Nc2 . representation and a superpotential W s
Tr X 4 appears. Going to the dual, the same manipulations as above give us the following brane configuration Žfrom left to right.:
Nf 1 physical D6 branes connected by 2Nf 1 physical D4 branes with A, N˜c2 r 2 physical D4 branes
connecting A and B, N˜c1 physical D4 branes connecting B and C and Nf 2 physical D6 branes at the
right of C which are connected by 2Nf 2 physical D4
branes with C.
Again we have 2 Nf 1 physical D4 branes to the
left of A and 2 Nf 2 physical D4 branes to the right of
C because the Nf 1 physical D6 branes are passing B
and C and the Nf 2 physical D6 branes are passing B
and A.
During the transition, after C passes B and A, the
intermediary configuration is similar with the one
obtained in the SO Ž Nc1 . = SpŽ Nc2 . case, after moving C to the right of B and A. This configuration is,
from right to left: the NS brane C, the NS brane A
Žwith antisymmetric O4 projection between C and A.
and the NS brane B Žwith symmetric O4 projection
between B and A and 2 supplementary D4 branes
between B and A.. When we move B to the right of
A, there are 2 supplementary D4 branes which appear between A and B but the previous supplementary D4 branes are changing the orientation so they
are cancelling each other. Therefore there is no
supplement or deficit of D4 branes.
We want to see which is the dual gauge group.
Again the charges of the O4 plane to the left and to
the right of each NS brane is the same in the original
and in the dual theory so the conservation of the
linking number is the same as the conservation of the
physical charge. For A, the physical charge is
yNf 1r2 y Nf 2r2 q Nc1 in the original theory and
yNf 1r2 q Nf 2r2 y N˜c2r2 q 2 Nf 1 in the magnetic
105
theory. So we obtain N˜c2 s 4 Nf 1 q 2 Nf 2 y 2 Nc1. The
same condition for B would give N˜c1 s 2 Nf 2 q Nf 1
y Nc2r2. From the field theory point of view, the
initial theory can be viewed as SO Ž Nc2 . = SpŽ Nc1 .
with 2 Nf 2 Ž2 Nf 1 . vector Žfundamental. flavors and
the final theory can be viewed as SO Ž N˜c2 . = SpŽ N˜c1 .
with 2 Nf 1Ž2 Nf 2 . vector Žfundamental. flavors. We
see that the above obtained formulas for the magnetic gauge group agree with Ž5., Ž7. for the choice
of the electric gauge groups and flavor representations.
This construction can be generalized to any product of gauge groups, but we have to put them in
alternating order i.e. SO Ž Nc1 . = SpŽ Nc2 . = SO Ž Nc3 .
= SpŽ Nc4 . = . . . . By changing the overall sign of
V 2 we can start with a Sp group from right to left.
For a product of more than 2 gauge groups, there
are two cases:
- when there is an even number of gauge groups
in the product, the effects of SO and Sp projections
will cancel each other so in the overall picture of the
dual no D4 branes appear or disappear. The result is
similar to the one obtained by w27x with the modifications that are to be done when one considers nonorientable string theory. Taking their results, we
modify N˜c to 2 N˜c and Nc to 2 Nc anytime we talk
about the Sp gauge groups, obtaining the dual for the
alternating product of SO and Sp gauge groups. The
argument that we use for the product of 2 gauge
groups applies also here. So one has to be careful
when changing the positions of 2 NS branes connected by supplementary D4 branes or having a
deficit of D4 branes between them.
- when there is an odd number of gauge groups in
the product, we need to create or to annihilate D
branes in the overall picture Žor to put D4 branes or
anti D4 branes on the top of the orientifold in order
to make a smooth transition..
The final result should be checked by field theory
methods.
4. Conclusions
In this paper, by changing brane positions in
space time, we managed to check the duality for a
theory with SO = Sp gauge groups and matter in
R. Tatarr Physics Letters B 419 (1998) 99–106
106
vector and fundamental representation. We also discussed the generalization to the case of products of
more than 2 gauge groups.
A very nice check of the results of brane configuration picture would be to obtain the same results in
the geometric singularity picture, where dualities
have been checked for simple SU, SO and Sp groups,
but not for gauge group products w18,20x. Also, a lot
of results obtained in field theory dualities remain to
be rederived and verified from brane configurations.
Acknowledgements
We would like to thank Hiroshi Ooguri for his
very interesting set of lectures at Trieste-97, which
helped to clarify our understanding. We would like
to thank David Kutasov and Clifford V. Johnson for
very important comments on the manuscript and for
many explanations and Orlando Alvarez for advices.
We would like to express our gratitude to the organizers of Spring School 1997 ŽTrieste. for their
intense effort to organize an extremely interesting
school.
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12 February 1998
Physics Letters B 419 Ž1998. 107–114
Instability of effective potential for non-abelian toroidal D-brane
Shin’ichi Nojiri
a
a,1
, Sergei D. Odintsov
b,c,2
Department of Mathematics and Physics, National Defence Academy, Hashirimizu, Yokosuka 239, Japan
b
Tomsk Pedagogical UniÕersity, 634041 Tomsk, Russia
c
Dep.de Fisica, UniÕersidad del Valle, AA25360 Cali, Colombia
Received 21 June 1997; revised 22 September 1997
Editor: P.V. Landshoff
Abstract
We calculate the one-loop effective potential for the toroidal non-abelian D-brane in the constant magnetic SUŽ2. gauge
field. The study of its properties shows that the potential is unbounded below. This fact indicates the instability of the
non-abelian D-brane in the background under consideration like the instability of chromomagnetic vacuum in SUŽ2. gauge
theory. q 1998 Elsevier Science B.V.
PACS: 04.50.q h; 4.60.-m; 11.25.-w
One of the key ingredients in the study of the dynamics of D s 11 M-theory w1x is D-brane w2–4x which
represents the solitonic solution of string theory. The effective theories of D-branes have been the subject of
much recent study Žsee w5x and references therein..
The study of the effective potential in theory of extended objects in one-loop or large d-approximation may
clarify the quantum properties of these objects. For string case, large d approximation has been developed in
Ref. w7x as systematic expansion for the effective action in powers of 1rd. The static potential may be then
obtained by studying the saddle point equations for the leading order term. The similar program may be realized
for membranes w8,9x, p-branes w9x and abelian D-branes w10x.
The non-abelian generalization of DBI-theory has been recently presented in Ref. w6x. That is the purpose of
the present note to look to the properties of the effective potential in the non-abelian toroidal D-brane in the
1
2
E-mail: [email protected].
E-mail: [email protected], [email protected].
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 2 8 5 - 9
S. Nojiri, S.D. OdintsoÕr Physics Letters B 419 (1998) 107–114
108
constant magnetic SUŽ2. field. We find that the correspondent potential shows the instability in the background
under discussion.
The action which corresponds to the non-abelian generalization of Born-Infeld D-brane is given by w6x
S s k dz
H
pq1
STr
(ž
=
)
det d m n y
ž
ž
det Gi j q Di X m d m n y
i
g
w X m, X nx
/
i
g
y1
w X m, X nx
/
Dj X n q
Fi j
g
/
.
Ž 1.
Here we consider the action in the Euclidean signature and m,n s p q 1, p q 2, . . . ,d y 1 Ž d is the space-time
dimension., i, j s 0,1,2, . . . , p. Gi j is the metric on the D-brane world volume, X m belong to the adjoint
representation of the non-abelian algebra, Fi j is the field strength of the algebra and g is the coupling constant.
det is the determinant with respect to the indices i, j. STr is the trace about the matrix which is the
representation of the algebra, after all it should be symmetrized with respect to Fi j , Di X m and w X m , X n x.
In this letter, we concentrate, for simplicity, on case that p s 2 Žmembrane. and the non-abelian algebra is
SUŽ2.. We also assume the metric Gi j is given by Gi j s R i R j d i j Žhere the sum convention is not used and R i ’s
are constants and we normalize R 0 s 1.. The coordinates satisfy the periodic boundary condition Žtoroidal
D-brane.:
X m Ž z 0 ,z 1 ,z 2 , . . . ,z p . s X m Ž z 0 q T ,z 1 ,z 2 , . . . ,z
p
. s X m Ž z 0 , z 1 q 1, z 2 , . . . , z p .
s X m Ž z 0 , z 1 , z 2 q 1, . . . , z p . s . . . s X m Ž z 0 , z 1 , z 2 , . . . , z p q 1 . .
Ž 2.
i
Therefore R i Ž i / 0. is the period with respect to the world volume coordinate z . We consider the one-loop
effective potential in the non-abelian constant magnetic background:
A1B s 12 As 1 ,
A 2B s 12 As 2 ,
Fi Bj s 12 f iBj s 3 ,
B
f 12
s yi w A1 , A 2 x s 12 A 2s 3 ,
X B m s0 .
Ž 3.
Here s a ’s Ž a s 1,2,3. are Pauli matrices. in the following, we abbreviate the index m of X m for the simplicity
in the notation.
The effective potential is defined by w7x
V s y lim
T™`
1
T
ln D XeyS .
Ž 4.
H
Here we treat the gauge field as a classical external field. The quantum contribution from the gauge fields will
be taken into account below.
Let decompose Di X in the components
Di X ' E i X y i w A i , X x s 12 Di X as a s 12 Ž E i X a q e a b cA bi X c . s a .
Ž 5.
We expand the gauge field and the field strength around the non-abelian magnetic background Ž3. as follows:
A1 ™ 12 As 1 q 12 A1a s a ,
A 2 s 12 As 2 q 12 A a2 s a ,
F0 i s 12 E 0 A ai s a ,
F12 s 12 A 2s 3 q 12 F2as a .
Ž 6.
Here we have chosen the A 0 s 0 gauge and
1
F12
s E 1 A12 y E 2 A11 q A21 A32 y A31 Ž A q A22 . ,
F122 s E 1 A22 y E 2 A21 q A31 A12 y Ž A q A11 . A32 ,
3
F12
s E 1 A32 y E 2 A31 q A Ž A11 q A22 . q A11 A22 y A21 A12 .
Ž 7.
S. Nojiri, S.D. OdintsoÕr Physics Letters B 419 (1998) 107–114
109
First we should note
STr Di XDj X Ž
ž
2n
F12B
/s
.
STr Fi j Fk l Ž F12B .
2n
ž /
3
2n
ž /
2 Ž 2 n q 1.
1
Ý
2
Di X a Dj X a
as1,2
2n
A
Ý Fiaj Fkal .
2 Ž 2 n q 1. 2
as1,2
Therefore if f Ž x . is an arbitrary even Ž f Žyx . s f Ž x .. function, we find
ž
/
Fi3j Fk3l q
2n
A2
1
3
Di X Dj X q
2
A
s 12
2n
A2
1
2
ž /
2
A2
ž /
1
2
STr Ž Di XDj Xf Ž F12B . . s f
2
A
ž /
A2
1
Di X 3 Dj X 3 q
Ž 8.
A2
H0 2
f Ž x . dx
Di X a Dj X a
Ý
as1,2
A2
1
2
f Ž x . dx Ý Fiaj Fkal .
Ž 9.
2
A2 0
as1,2
Then expanding the action Ž1. in the background Ž3. up to the terms quadratic with respect to the fields, we
obtain
STr Ž Fi j Fk l f Ž
F12B
.. s
1
2
ž /
f
Fi3j Fk3l q
H
(
(
S s k d z 3 STr det Ž Gi j q Di XDj X q Fi j . s k d z 3 2
H
H
R4 q
A2
4g2
q LX Ž X m . q L A Ž A i . q . . .
Ž 10 .
LX Ž X m . s 12
(
R4 q
A4
R2
2
Ž D0 X 3 . q
4g2
(
2
q
1
4
Ý
as1,2
gR 2
q
A2
(
A2
R2
2
R2
(
4
A
R4 q
2A
g
q
2A
' A2
ln
2
ln
2
3
01
žŽF
.
2
gR 4
q
2A
R4 q
q
2g
½Ž D X
3 2
.
1
R
2
1
R2
3
q F02
2
A2
R4 q
q
2g
A2
R4 q
q
2g
2
/ / qB
2
Ž F123 .
2
2
A4
4g2
q
R4 q
q
2g
2
5
A4
4g2
/5
Ž D0 X a .
2
2
(
/5
/5
Ý
as1,2
2
Cˆ Ž D 0 X a . q Dˆ Ž D 1 X a . q Ž D 2 X a .
Ý
2
4
,
as1,2
1
A4
4g
A2
5q
4
½ ž (
½ ž (
1
q Ž D2 X 3 .
2
2
Ž D1 X a . q Ž D 2 X a . 4
4g2
2
.
4g2
R2
A4
/ /q
3 2
½ ž (
/5
q Ž D2 X 3 .
.
A
1
1
ln
2
2
3 2
3
q F02
01
žŽF
4
4g
gR 2
q
4g2
R q
½ ž (
1
ln
' AˆŽ D 0 X 3 . q Bˆ
LA Ž Ai . s
A4
R4 q
4
½Ž D X
4
R4 q
A
Ž F123 .
4
2
4g2
Ý Ž Ž F01a .
2
a
q F02
.
as1,2
2
/
2
Ý Ž F12a .
as1,2
2
a 2
a
q F02
01
½C Ž Ž F
2
.
.
a 2
12
/ qD Ž F . 5 .
2
Ž 11 .
S. Nojiri, S.D. OdintsoÕr Physics Letters B 419 (1998) 107–114
110
Here we assume R i s R Ž i / 0. and . . . denotes the higher order terms with respect to X m , A i and total
derivative terms.
We now consider the contribution from X m to the one-loop effective potential. Eq. Ž11. can be rewritten as
follows:
2
LX Ž X m . s Aˆ Ž E 0 X 3 . q
qCˆ
Bˆ
Aˆ
½
2
Ž E1 X 3 . q Ž E2 X 3 .
2
5
q
2 A 2 Dˆ
Aˆ
Ž X 3.
Dˆ
2
A 2 Bˆ
2
2
2
Ý Ž E 0 X a . q Cˆ Ž E 1 X a . q Ž E 2 X a . 4 q Cˆ Ž X a . 2
as1,2
q Ž Bˆ q Dˆ . A Ž X 2E 1 X 3 y X 3E 1 X 2 y X 1E 2 X 3 q X 3E 2 X 1 . q total derivative terms .
By diagonalizing Ž12. with respect to X a, we find the following expression for the effective potential:
VX s VX0 q VXqq VXy
(
Ž d y 3 . 2p
VX0 s
2
VX"s
Dˆ
Cˆ
Ý
2
ž
ž
n1 , n 2sy`
n1 , n 2sy`
n12 q n 22 q
Bˆ
`
Ž d y 3 . 2p
q
`
Ý
ž
Dˆ
Aˆ
ˆ ˆ Ž 2Cˆ 2 q Aˆ2 .
2 A 2 BD
Aˆ2 Cˆ 2
q
Cˆ
/Ž
A 2 Bˆ
Dˆ
n12 q n 22
ˆ ˆ q 2 Bˆ 2
y8 BD
q
ˆˆ
AC
1
/
2
s
. qA
/
2
ž
Ž d y 3 . 2p
Dˆ
2
Cˆ
2 Dˆ
Aˆ
Bˆ
q
Ž n12 q n22 . q
Cˆ
A 2 Bˆ
( ž /
/ ½ž /
"
E
Aˆ
ˆ ˆ y AB
ˆ ˆ.
A 4 Ž 2 DC
Aˆ2 Cˆ 2
2
Dˆ
y
Ž n12 q n22 .
Cˆ
1
¶•
ß
2
Ž 13 .
Ž 14 .
Dˆ
Bˆ
Ž 12 .
2
2
1
2
.
Ž 15 .
In Ž14., we have used zeta function regularization Žsee w11x for an introduction. and EŽ q . is defined by
1
`
EŽ q. '
Ý 2
n12 q n22 q q
4
n1 , n 2s1
1
s yq 2 y
p
3
3
q2 y
8
p
1
q2
Ý ky1 Ý d 2
d<k
ks1
ž
Ý
ky1 K 1 Ž 2p k q . q q 4
'
ks1
`
q2
3
`
1q
q
d
2
`
ks1
1
/ žp(
2
K1 2 k 1 q
q
d2
/
3
Ý ky 2 Ky 3 Ž 2p k'q .
.
2
Ž 16 .
Since Kn Ž z . ; eyz when < z < ™ `, we find
p
EŽ q. ™y
3
q2
Ž 17 .
3
when q ™ q` and
f T Ž 1,1 .
E Ž 0. s
2p
Here
Ž 18 .
1
`
f T Ž 1,1 . s 2p
Ý
n1 , n 2sy`
Ž n12 q n22 .
2
s y1.438 PPP .
Ž 19 .
S. Nojiri, S.D. OdintsoÕr Physics Letters B 419 (1998) 107–114
111
Note that EŽ q . is monotonically decreasing function therefore always negative when q G 0. Note that, in the
limit A ™ 0, Eq. Ž13. reproduces the result for p-brane in w12x, up to a factor 3 which corresponds to dimension
of the SUŽ2. group:
VX0 s
Ž d y 3 . 2p f T Ž 0,0 .
3
2
.
R
Ž 20 .
In order to calculate the contribution from the quantum gauge field, first we solve the constraint E i A ai s 0
obtained from the A 0 s 0 gauge condition non-locally Žlocally in the momentum space. by introducing a scalar
field A a
Ž A1a , Aa2 . s '
1
Ž E 2 Aa ,y E 1 Aa . Ž D s E 12 q E 22 . ,
yD
Ž 21 .
and further redefining A0 , Al and At as
A0 ' AA3 ,
Al '
C
Ž E 1 A1 q E 2 A 2 . ,
'y D
At '
1
1
'y D Ž E 2 A y E 1 A 2 . .
Ž 22 .
Then we can rewrite Eq. Ž11. up to total derivative terms as follows
2
2
2
L A s Ž E 0 A0 . q Ž E 0 A t . q Ž E 0 A l . q
q
D2
ž
C
yD . q
2 Ž
A2B2
C
/
2
ž
D2A 2
A
A2
yD . q
2 Ž
D2
2
Ž At . q
B2
C
2
Ž yD . Ž Al .
/
2
Ž A0 . q
2 AŽ B 2 q D2 .
AC
A0'y D At
2
Ž 23 .
The appearance of mass-like term for gauge field is seen in above equation. That is contrary to the abelian case
w10x where such term does not appear. After diagonalizing Ž23. with respect to A0 and At , we find the
contribution from the gauge field to the one-loop potential is given by
VA s VA0 q VAqq VAy
VA0 s
VA"'
D
f T Ž 0,0 .
C
`
1
Ý
'2
n1 , n 2sy`
2
q2
ž
B D
A4
2
ž
B2
A2
2
B D
q
C4
D2
q
C2
2
4
/
Ž n12 q n22 . q
4
2
ž
B qD q4B D
q
A2 C 2
D2
A2
B2
q
C2
/ ½ž
A2"
2
/
A 2 Ž n12 q n 22 . q
ž
B2
A2
D
D2
y
2
A2
C2
B
y
2
C2
2
/Ž
n12 q n22 .
1
2
/ 5
2
2
1
2
A4
.
Ž 24 .
We now consider the asymptotic behavior of the one-loop potential. First we should note that
ˆ Cˆ ™
A,
R2
2
,
ˆ Dˆ ™ 12 ,
B,
A2 , C 2 ™ 14 ,
B2 , D2 ™
1
4 R2
Ž 25 .
S. Nojiri, S.D. OdintsoÕr Physics Letters B 419 (1998) 107–114
112
when R ™ ` and
Aˆ™
A2
4g
, Bˆ ™
gR 2
A2 ™
A2
B2 ™
,
2 A2
gR 2
, Cˆ ™
g
gR 2
, Dˆ ™
8g
C2™
,
2 A2
A2
A2
ln
gR 2
A2
2A
gR 2
ln
2
A2
gR 2
,
D2 ™
g
2A
ln
2
A2
gR 2
.
Ž 26 .
Therefore the asymptotic behavior of the potential is given by
VX0 ™
2p Ž d y 3 . E Ž A 2 .
2
R
2p Ž d y 3 . E Ž 2 A 2 .
VXq™
2
VA0 ™
R
f T Ž 1,1 .
R
,
VA"™
VXy™
,
2p E Ž A 2 .
2p Ž d y 3 . E Ž A 2 .
2
R
,
R
Ž 27 .
when R ™ ` and
3
VX0 ™
VX"™
VA0 ™
VAq™
VAy™
2p Ž d y 3 . 2'2 gR
2p Ž d y 3 . 2 gR
A2
2
1
R
(
ln
A2
2
A
ln
A2
gR 2
E
0
ln
A2
;y
2p Ž d y 3 . 2'2 p gR
3 A2
2
ln
A2
gR 2
gR 2
`
Ý
n1 , n 2sy`
½
n12 q n 22 q A 2 "
ž Ž n12 q n22 .
2
q A4
/
2
A2
2
0
ln
A2
gR 2
1
1
2
gR 2
A2
2
5
f T Ž 0,0 .
1
3
2
R
E
ž
2 A2
3
ln
1
f T Ž 1,1 .
4p'2
R
A2
gR
2
/
(
;y
3
2
p
3R
ž
2 A2
3
ln
A2
gR
2
3
/
2
Ž 28 .
when R ™ 0. Therefore the asymptotic behavior of the total one-loop potential
(
VT s 2 k
R4 q
A4
2 g2
q VX q VA
Ž 29 .
is dominated by the classical term Žthe first term in Ž34. when R ™ `:
V ™ 2 kR 2
Ž 30 .
S. Nojiri, S.D. OdintsoÕr Physics Letters B 419 (1998) 107–114
113
Fig. 1. The rough behavior of the effective potential V as a function of R.
and dominated by VAq
(
V™y
3
2
p
3R
ž
2 A2
3
ln
A2
gR 2
3
/
2
Ž 31 .
at small R. This tells the one-loop potential is unstable near R ; 0. Not exact but rough behavior of the
potential is given in Fig. 1.
We may also consider the static potential in the large d approximation as in Ref. w10x. It is difficult, however,
to formulate it consistently with the definition of STr.
In summary, we have shown that one-loop potential for SUŽ2. non-abelian toroidal D-brane is not stable. It is
in close analogy with instability of chromomagnetic vacuum in SUŽ2. gauge theory w13x. It is much likely that in
order to get the stability Žagain like in case of usual gauge theory w14x. one has to consider another gauge group
and Žor. another background field.
Acknowledgements
We thank A. Sugamoto for useful remarks, A. Tseytlin for valuable comments and referee for pointing out
mistake in original version of this work. We are also grateful to T. Muta and whole Particle Physics Group at
Hiroshima University for kind hospitality during the completing of this work.
References
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C. Hull, P.K. Townsend, Nucl. Phys. 438 Ž1995. 109.
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Ž1996. 7187; M. Douglas, hep-thr9512077; A. Tseytlin, Nucl. Phys. B 469 Ž1996. 51; A. Bytsenko, S. Odintsov, L.N. Granda, Phys.
Lett. B 387 Ž1996. 282; M. Abou-Zeid, C. Hull, hep-thr9704021; M. Aganagic, C. Popescu, J.H. Schwarz, hep-thr9610249; M.B.
Green, C.M. Hull, P.K. Townsend, Phys. Lett. B 382 Ž1996. 65; A. Hashimoto, W. Taylor, hep-thr9703217.
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Odintsov, Class. Quant. Grav. 9 Ž1992. 391.
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w12x I.L. Buchbinder, S.D. Odintsov, I.L. Shapiro, Effective Action in Quantum Gravity, IOP Publishing, Bristol and Philadelphia, 1992.
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12 February 1998
Physics Letters B 419 Ž1998. 115–122
Instantons at angles
G. Papadopoulos, A. Teschendorff
DAMTP, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK
Received 11 September 1997; revised 14 November 1997
Editor: P.V. Landshoff
Abstract
We interpret a class of 4k-dimensional instanton solutions found by Ward, Corrigan, Goddard and Kent as four-dimensional instantons at angles. The superposition of each pair of four-dimensional instantons is associated with four angles
which depend on some of the ADHM parameters. All these solutions are associated with the group SpŽ k . and are examples
of Hermitian-Einstein connections on E 4 k . We show that the eight-dimensional solutions preserve 3r16 of the ten-dimensional N s 1 supersymmetry. We argue that under the correspondence between the BPS states of Yang-Mills theory and
those of M-theory that arises in the context of Matrix models, the instantons at angles configuration corresponds to the
longitudinal intersecting 5-branes on a string at angles configuration of M-theory. q 1998 Elsevier Science B.V.
1. Introduction
Instantons are Yang-Mills configurations which
are characterized by the first Pontryagin number, n ,
and obey the antiself-duality condition 1 wF s yF,
where F is the field strength of a gauge potential A,
the star is the Hodge star and ww s 1 for Euclidean
signature 4-manifolds. One key property of this condition is that together with the Bianchi identities,
=w M FN Lx s 0, it implies the field equations of the
Yang-Mills theory. At first, a large class of instanton
solutions was found using the ansatz of w1,2x. Later a
1
We have chosen the antiself-duality condition in the definition
of the instanton but our results can be easily adapted to the
self-duality one.
systematic classification of the solutions of the antiself-duality conditions was done in w3x, which has
become known as the ADHM construction. Subsequently, many generalizations of the self-duality condition beyond four dimensions have been proposed
and for some of them ADHM-like constructions
have been found w4–6x.
The understanding of non-perturbative aspects of
various superstring theories involves the investigation of configurations which have the interpretation
of intersecting branes. In the context of the effective
theory, these are extreme solutions of various supergravity theories w7x. Such solutions are constructed
using powerful superposition rules from the ‘elementary’ brane solutions that preserve 1r2 of the spacetime supersymmetry Žfor recent work see w8,9x and
references within.. One feature of the intersecting
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 7 0 - 6
116
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
branes is that the intersections can occur at angles in
such a way that the configurations preserve a proportion of supersymmetry w10x. Solutions of various
supergravity theories that have the interpretation of
intersecting branes at angles have been found in
w11–16x.
More recently, a non-perturbative formulation of
M-theory and string theory was proposed using the
Matrix models w17–20x. These involve the study of
certain Yang-Mills theories on a torus which is dual
to the compactification torus of M-theory. The U-duality groups w21x of IIA superstring theory are naturally realised in the context of matrix models w22x.
As a result, there is a correspondence between the
BPS states of these Yang-Mills theories w23x and
those of M-theory and superstring theories. In the
low energy effective theory, the BPS states of the
latter can be described by various classical solitonic
solutions of various supergravity theories. This indicates that there is a correspondence between BPS
states of Yang-Mills theories with the solitonic solutions of supergravity theories. Since in some cases,
the solitonic BPS states of Yang-Mills theories can
be described by classical solutions that carry the
appropriate charges, the above argument suggests
that there is a correspondence between solitonic solutions of supergravity theory with solitonic solutions
of Yang-Mills theory. Although this correspondence
is established for a compactification torus of finite
size, we expect that the correspondence between the
solutions of the Yang-Mills theory and supergravity
theory will persist in the various limits where the
torus becomes very large or very small. One example
of such correspondence is the following: The longitudinal five-brane of the Matrix model is described
either as the five-brane solution of D s 11 supergravity superposed with a pp-wave and compactified
on a four-torus or as an instanton solution of the
Yang-Mills theory Žsee for example w22,23x. on the
dual four-torus. ŽThe pp-wave carries the longitudinal momentum of the five-brane.. Ignoring the size
of the compactifying torus, one can simply say that
there is a correspondence between longitudinal fivebranes and instantons. One purpose of this letter is to
demonstrate the correspondence between BPS solutions of the Yang-Mills theories with BPS solutions
of the supergravity theories with another example. In
D s 11 supergravity theory, there is a configuration
with the interpretation of intersecting five-branes at
angles w11x on a string. This configuration is associated with the group SpŽ2.; the proportion of supersymmetry preserved is directly related to the number
of singlets in the decomposition of the spinor representation in eleven dimensions under SpŽ2.. Superposing this configuration with a pp-wave along the
string directions corresponds to a Matrix theory configuration with the interpretation of intersecting longitudinal fiÕebranes at angles. Now using the correspondence between the longitudinal fivebranes and
the Yang-Mills instantons mentioned above, one
concludes that there should exist solutions of YangMills theory that have the interpretation of instantons at angles. Moreover since the instanton configuration must preserve the same proportion of supersymmetry as the supergravity one, it must also be
associated with the group SpŽ2..
To superpose Yang-Mills BPS configurations, like
for example Yang-Mills instantons or monopoles, at
angles, we embed them as solutions of an appropriate higher-dimensional Yang-Mills theory. Such solutions will be localized on a subspace of the associated spacetime. Then given such a configuration we
shall superpose it with another similar one which,
however, may be localized in a different subspace.
We require that the solutions of the Yang-Mills
equations which have the interpretation of BPS configurations at angles to have the following properties: Ži. They solve a BPS-like equation which reduces to the standard BPS conditions of the original
BPS configurations in the appropriate dimension Žii.
if the solution which describes the superposition can
be embedded in a supersymmetric theory, then it
preserves a proportion of supersymmetry.
In this letter, we shall describe the superposition
of four-dimensional instantons. Since instantons are
associated with four-planes, it is clear that the configuration that describes the superposition of two
instantons at an angle is eight-dimensional; in general the configuration for k instantons lying on linearly independent four-planes is 4k-dimensional. It
turns out that the appropriate BPS condition in 4 k
dimensions necessary for the superposition of fourdimensional instantons has being found by Ward w5x.
Here we shall describe how this BPS condition is
associated with the group SpŽ k .. This allows us to
interpret a class of the 4 k-dimensional instanton
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
solutions of Corrigan, Goddard and Kent w6x as
four-dimensional instantons at angles. We shall find
that there four angles associated with the superposition of each pair of two four-dimensional instantons
which depend on the ADHM parameters. The eightdimensional solutions preserve 3r16 of the N s 1
ten-dimensional supersymmetry and correspond to
the intersecting longitudinal fivebranes at angles
configuration of the eleven-dimensional supergravity.
2. Tri-Hermitian-Einstein connections
The BPS condition of Ward w5x is naturally associated with the group SpŽ k .. For this, we first observe that the curvature
FM N s E M A N y E N A M q w A M , A N x ,
Ž 1.
of a connection A on E 4 k can be thought as an
element of L2 ŽE 4 k . with respect to the space indices. Now L2 ŽE 4 k . s soŽ4 k . where soŽ4 k . is the
Lie algebra of SO Ž4 k .. Decomposing soŽ4 k . under
SpŽ k . P SpŽ1., we find that
L2 Ž E 4 k . s sp Ž k . [ sp Ž 1 . [ l20 m s 2 ,
Ž 2.
where we have set L1 ŽE 4 k . s l m s under the decomposition of one forms under SpŽ k . P SpŽ1., l20 is
the symplectic traceless two-fold anti-symmetric irreducible representation of SpŽ k . and s 2 is the
symmetric two-fold irreducible representation of
SpŽ1.. The integrability condition of w5x can be summarized by saying that the components of the curvature along the last two subspaces in the decomposition Ž2. vanish. This implies that the space indices of
the curvature take values in spŽ k .. It is convenient to
write this BPS condition in a different way. Let
Ir ;r s 1,2,3.4 be three complex structures in E 4 k
that obey the algebra of imaginary unit quaternions,
i.e.
Ir Is s ydr s q e r st It .
Ž 3.
The BPS condition then becomes
Ž Ir .
P
M
Ž Ir .
R
N FP R
a
b s FM N
a
b
,
Ž 4.
117
Žno summation over r ., i.e. the curvature is Ž1,1.
with respect to all complex structures. We shall refer
to the connections that satisfy this condition as triHermitian-Einstein connections. We remark that this
is precisely the condition required by Ž4,0. supersymmetry on the curvature of connection of the
Yang-Mills sector in two-dimensional sigma
models 2 .
It is straightforward to show that Ž4. reduces to
the antiself-duality condition in four dimensions 3
and that it implies the Yang-Mills field equations in
4k dimensions. Moreover any solution of Ž4. is an
example of an Hermitian-Einstein Žor Hermitian
Yang-Mills. connection 4 Žsee for example w28,29x..
The latter are connections for which Ži. the curvature
tensor is Ž1,1. with respect to a complex structure
and Žii.
g ab Fab s 0 ,
Ž 5.
where g is a hermitian metric. We remark that the
above two conditions for Hermitian-Einstein connections on R 2 n imply that the non-vanishing components of the curvature are in suŽ n.. ŽThe condition
Ž5. together with the Bianchi identity imply the
Yang-Mills field equations.. The connections that
satisfy Ž4. are Hermitian-Einstein with respect to
three different complex structures.
3. One-angle solutions
To interpret some of the solutions of w5x and w6x as
four-dimensional instantons at angles, we first use an
appropriate generalization of the ’t Hooft ansatz. For
2
For the application of the instantons of w6x in sigma models
see w27,24–26x.
3
In fact it reduces to either the self-duality or the antiself-duality condition depending on the choice of orientation of the fourmanifold.
4
For k )1, the Hermitian-Einstein BPS condition is weaker
than that of Ž4..
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
118
this, we write E 4 k s E 4 m E k , i.e. we choose the
coordinates
This is a harmonic-like equation and a solution is
y M 4 s x i m ; i s 1, . . . ,k ; m s 0,1,2,3 4 ,
fs1q
Ž 6.
r2
4k
on E . Then we introduce three complex structures
on E 4 k
Ž Ir . i m
jn s Ir
m
n
dij ,
Ž 7.
where Ir ; r s 1,2,34 are three complex structures on
E 4 associated with the Kahler
forms
¨
v 1 s dx 0 n dx 1 q dx 2 n dx 3 ,
v 2 s ydx 0 n dx 2 q dx 1 n dx 3 ,
v 3 s dx 0 n dx 3 q dx 1 n dx 2 ,
Ž 8.
respectively. Choosing the volume form as e s dx 0
n dx 1 n dx 2 n dx 3, these form a basis of self-dual
2-forms in E 4 . We also write the Euclidean metric on
E 4 k as
ds 2 s dmn d i j dx i m dx jn .
Ž 9.
Similarly, the curvature two-form is
FM N s Fi m jn .
Ž 10 .
Writing
Fi m jn s Fw mn x Ž i j. q FŽ mn . w i j x ,
Ž 11 .
we find that Ž4. implies that
1
2
Fw mn x Ž i j. e mn rs s yFw rs x Ž i j. ,
FŽ mn . w i j x s dmn f i j ,
Ž 12 .
where f i j is a k = k antisymmetric matrix. Next, we
shall further assume that the gauge group is SUŽ2. s
SpŽ1. and write the ansatz
A i m s i Sm nE i n log f Ž x . ,
Ž 13 .
where the matrix two-form Smn is the ’t Hooft
tensor, i.e.
Smn s 12 v r mn s r ,
Ž 14 .
and sr ;r s 1,2,34 are the Pauli matrices. After some
computation, we find that the field strength F of A
in Ž13. satisfies the first equation in Ž12. provided
that
1
f
Ei P Ej f s 0 .
Ž 15 .
Ž pi x i m y a m .
2
,
Ž 16 .
where p4 s Ž p 1 , p 2 , . . . , p k . are k real numbers, a m
is the centre of the harmonic function and r 2 is the
parameter associated with the size of the instanton.
More general solutions can be obtained by a linear
superposition of the above solution for different
choices of p4 and a m leading to
fs1q Ý
Ý
p4 a
r a2 Ž p 4 .
Ž pi x i m y a m Ž p 4 . .
2
.
Ž 17 .
Finally, it is straightforward to verify that the curvature of the connection Ž13. with f given as in Ž17.,
satisfies Ž4..
To investigate the properties of the above solutions, let us suppose that the harmonic function f
involves only one p4 as in Ž16.. In this case it is
always possible, after a coordinate transformation, to
set p4 s 1,0, . . . ,04 . Then f becomes a harmonic
function on the ‘first’ copy of E 4 in E 4 k . The
resulting configuration is that derived from the ansatz
of w1,2x on E 4 . In particular, the instanton number is
equal to the number of centres a m4 of the harmonic
function. In this case the singularities at the centres
of the harmonic function can be removed with a
suitable choice of a gauge transformation w30x. Next,
let us suppose that our solution involves two linearly
independent vectors p4 , say p4 s p4 and p4 s
q4 . If we take < pi x i < ™ `, then the configuration
reduces to the solution associated with a harmonic
function that involves only q4 . As we have explained above, this is the instanton solution of w1,2x
on an appropriate 4-plane in E 4 k . Similarly if we
take < qi x i < ™ `, then the configuration reduces to the
solution associated with the harmonic function that
involves only p4 . We can therefore interpret this
solution as the superposition of two four-dimensional
instantons at angles in E 4 k . The angle is
cos u s
pPq
< p< < q<
.
Ž 18 .
As it may have been expected from the non-linearity
of the gauge potentials in Ž4., this superposition is
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
non-linear even in the case for which p4 and q4 are
orthogonal. The solution appears to be singular at the
positions of the harmonic function. For simplicity let
us consider two instantons at angles with instanton
number one. In this case we have two centres. The
singular set is defined by the two codimension-fourplanes
im
m
pi x y a s 0 ,
qi x i m y b m s 0 ,
Ž 19 .
8
in E . We have investigated the singularity structure
of the solution when p4 and q4 are orthogonal
using a method similar to w30x. We have found that
these singularities can be removed everywhere apart
from the intersection of the two singular sets. A
similar observation has been made in w6x. To conclude, let us briefly consider the general case. The
number of four dimensional instantons involved in
the configuration is equal to the number of linearly
independent choices for p4 . Their instanton number
is equal to the number of centres associated with
each choice of p4 . The general solution describes
the superposition of these four-dimensional instantons at angles.
4. Four-angle solutions
The ADHM ansatz of w6x describes solutions with
more parameters than those found in the previous
section using the ’t Hooft ansatz. Here we shall
interpret some of these parameters as angles. In fact
we shall find that there are four angles between each
pair of planes associated with these solutions all
determined in terms of ADHM parameters. For simplicity, we shall take SpŽ1. as the gauge group. The
ADHM ansatz is
A s Õ†dÕ ,
Ž 20 .
where Õ is an Ž l q 1. = 1 matrix of quaternions
normalized as Õ† Õ s 1, and l is an integer which is
identified with the ‘total’ instanton number Žsee w6x..
The matrix Õ satisfies the condition
Õ†D ' Õ† Ž a q bi x i . s 0 ,
Ž 21 .
where a,bi are Ž l q 1. = l matrices of quaternions;
we have arranged the 4k coordinates into quaternions
x i ;i s 1, PPP ,k 4 ŽE 4 k s H k .. The connection Ž20.
119
satisfies the condition Ž4. provided that the matrices
a†a,b†i bj ,a† bi are symmetric as matrices of quaternions.
A convenient choice of matrices a,bi leads to
Ž D . 0 n s ya0 l n ,
Ž D . n m s Ž pni† x i y a n . dn m ,
Ž 22 .
where l n ;n s 1, . . . , l 4 are real numbers, and pn i ;n
s 1, . . . , l ;i s 1, . . . ,k 4 and a0 ,a n ;n s 1, . . . , l 4 are
quaternions. The solutions discussed in the previous
section using the ’t Hooft ansatz correspond to
choosing pn i ;n s 1, . . . , l ,i s 1, . . . ,k 4 to be real
numbers. For Õ, we find
a 0 y1
f ,
Ž Õ.0 sy
< a0 <
y1
†
Ž Õ . n s y< a0 < l n Ž x i . pn i y a†n
fy1 ,
Ž 23 .
where f is the normalization factor
f 2 s 1 q < a0 < 2
l2n
l
Ý
ns1
< pn† i x i y a n < 2
.
Ž 24 .
We remark that the sum in the normalization factor
is over the centres a n ;n s 1, . . . , l 4 . It is clear from
the form of the normalization factor that this solution
is associated with l codimension-four-planes in E 4 k .
The equations of these planes are
pn† i x i y a n s 0 ,
Ž 25 .
for n s 1, . . . , l . The four normalized normal vectors
to these planes are
1
Nn s
< pn <
Ž pni .
†
E
E xi
,
Ž 26 .
in quaternionic notation, where < pn < 2 s d i j Ž pni . † pnj.
Next let us consider two such codimension-four-plane
determined say by p s p 1 and q s p 2 and with
associated normal vectors N and N˜ , respectively.
The angles associated with these planes are given by
the inner product of their normal vectors, so
†
†
cos Ž u . s Ž N˜ . N s
di j Ž q i . p j
,
< p< < q<
Ž 27 .
where cosŽ u . is a quaternion in the obvious notation.
There are four angles because cosŽ u . has four components. These four angles are all different for a
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
120
generic choice of p,q and independent from the
choice of gauge fixing for the residual symmetries of
the ADHM construction w6x. It is worth mentioning
that each pair of four-dimensional instantons are
superposed at SpŽ2. angles. For this we observe that
the normal vectors Ž26. of the two codimensionfour-planes span an eight-dimensional subspace in
E 4 k and their coefficients are quaternions. So, they
can be related with an SpŽ2. rotation Žafter choosing
a suitable basis in E 4 k .. As a result the two fourplanes spanned by the normal vectors in E 8 are at
SpŽ2. angles. Such four-planes are parameterized by
the four-dimensional coset space SpŽ2.rSO Ž4.; this
explains the presence of four angles in Ž27.. It also
suggests that there may be a generalization of the
solutions of w11x which describe the superposition of
five-branes and KK-monopoles depending on four
angles.
A naive counting of the dimension of the moduli
of the 4k-dimensional solutions which takes account
of the dimension of moduli of 4-dimensional instantons involved in the superposition and their associated angles does not reproduce the dimension of the
moduli space in w6x. However, all their solutions
have a decaying behaviour at large distances consistent with the interpretation that they are four-dimensional instantons at angles.
3r32 of spacetime supersymmetry. This is in agreement with the proportion of supersymmetry preserved by the intersecting five-branes on a string at
angles and superposed with a pp-wave solution of
D s 11 supergravity. However there is a difference.
In the case of two orthogonal intersecting five-branes
on a string with a pp-wave superposed the proportion
of the supersymmetry preserved is 1r8 but this is
not the case for the configuration of two orthogonal
instantons. To see this, first observe that the solutions of the previous two sections will preserve 1r4
of the N s 1 ten-dimensional supersymmetry if the
components of the curvature are in the spŽ1. [ spŽ1.
subalgebra of spŽ2.. However this is not so because
a direct calculation reveals that the mixed components Fm1, n 2 of the curvature tensor do not vanish.
Further we remark that one might have thought that
it is possible to superpose two four-dimensional instantons Bm and Ca with gauge group G localised at
two orthogonal four-planes in E 8 by simply setting
A s Ž B,C . and requiring that the gauge group of the
new connection is again G. However, this is not a
solution of the BPS condition Ž4. for a non-abelian
gauge group in eight-dimensions. This is because Ž4.
is non-linear in the connection. However the above
linear superposition is a solution if the gauge groups
of B and C are treated independently, i.e. if the
gauge group of A is G = G.
5. Supersymmetry
6. Concluding remarks
Among the above Yang-Mills configurations on
E 4 k only the instantons at angles in E 8 and the
instantons on E 4 are solutions of ten-dimensional
supersymmetric Yang-Mills theory preserving a proportion of supersymmetry. The supersymmetry condition is
FM N G
MN
es0 ,
Ž 28 .
where e is the supersymmetry parameter. In the
four-dimensional case the instantons preserve 1r2 of
the supersymmetry. In the eight-dimensional case
condition Ž4. implies that FM N is in spŽ2., an argument similar to that in w11x can be used to show that
the instantons at angles in E 8 will preserve 3r16 of
supersymmetry. As a solution of the Matrix theory,
the eight-dimensional instantons at angles preserve
We have interpreted some of the 4k-dimensional
instantons of w5x and w6x as superposition of four-dimensional instantons at angles. We have shown that
there four angles associated with the superposition
of two four-dimensional instantons which depend on
the ADHM parameters. These solutions are examples
of Hermitian-Einstein connections in 4k dimensions
and the eight-dimensional ones preserve 3r16 of the
N s 1 ten-dimensional supersymmetry. Because of
the close relation between the self-duality condition
and the BPS equations for magnetic monopoles we
expect that one can generalize the above construction
to find solutions of Yang-Mills equations on E 3 k
which have the interpretation of monopoles at angles. In addition, the Yang-Mills configurations of
G. Papadopoulos, A. Teschendorffr Physics Letters B 419 (1998) 115–122
w6x and the HKT geometries described in w31x combined give new examples of two-dimensional supersymmetric sigma models with Ž4,0. supersymmetry.
These provide consistent backgrounds for the propagation of the heterotic string w32x. Another application of the instantons of w6x is in the context of
D-branes. In IIA theory, they are associated with the
D-brane bound state of a 0-brane within a 4-brane
within an 8-brane, and in the IIB theory they are
associated with the D-brane bound state of a D-string
within a D-5-brane within a D-9-brane.
There are many other ways to superpose two or
more four-dimensional instantons. Here we have described the superposition using the condition that the
components of the curvature F of the Yang-Mills
connection are in spŽ k .. Another option is to use the
Hermitian-Einstein condition associated with suŽ2 k .
which we have already mentioned in section two. In
eight dimensions this will lead to configurations
preserving 1r8 of the N s 1 ten-dimensional supersymmetry. In addition in eight dimensions one can
allow the components of F to lie in spinŽ7.. This
will result in superpositions of instantons preserving
1r16 of the N s 1 ten-dimensional supersymmetry.
In fact, some spinŽ7. instanton solutions have already been found w33x. It will be of interest to see
whether they can be interpreted as four-dimensional
instantons at angles. Since SpŽ2. is a subgroup of
SpinŽ7., the solutions of w6x are also examples of
spinŽ7. instantons, albeit of a particular type. There
are more possibilities in lower dimensions, for example the G 2 instantons in seven dimensions w34,35x
which may have a similar interpretation.
7. Note added
While we were revising our work, the paper w36x
by N. Ohta and J-G. Zhou appeared in which the
relation between supergravity configurations and
Yang-Mills ones is also discussed. However, they
use constant Yang-Mills configurations on the eight
torus.
Acknowledgements
We would like to thank P. Goddard and A. Kent
for helpful discussions. A.T. thanks PPARC for a
121
studentship. G.P. is supported by a University Research Fellowship from the Royal Society.
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12 February 1998
Physics Letters B 419 Ž1998. 123–131
M-theory model-building and proton stability
John Ellis a , Alon E. Faraggi b, D.V. Nanopoulos
c
a
Theory DiÕision, CERN, CH-1211, GeneÕa, Switzerland
Institute for Fundamental Theory, Department of Physics, UniÕersity of Florida, GainesÕille, FL 32611, USA
c
Center for Theoretical Physics, Dept. of Physics, Texas A & M UniÕersity, College Station, TX 77843-4242, USA
Astroparticle Physics Group, Houston AdÕanced Research Center (HARC), The Mitchell Campus, Woodlands, TX 77381, USA
Academy of Athens, Chair of Theoretical Physics, DiÕision of Natural Sciences, 28 Panepistimiou AÕenue, Athens 10679, Greece
b
Received 9 September 1997
Editor: R. Gatto
Abstract
We study the problem of baryon stability in M theory, starting from realistic four-dimensional string models constructed
using the free-fermion formulation of the weakly-coupled heterotic string. Suitable variants of these models manifest an
enhanced custodial gauge symmetry that forbids to all orders the appearance of dangerous dimension-five baryon-decay
operators. We exhibit the underlying geometric Žbosonic. interpretation of these models, which have a Z2 = Z2 orbifold
structure similar, but not identical, to the class of Calabi-Yau threefold compactifications of M and F theory investigated by
Voisin and Borcea. A related generalization of their work may provide a solution to the problem of proton stability in M
theory. q 1998 Elsevier Science B.V.
There has recently been important progress towards a better understanding of the underlying nonperturbative formulation of superstring theory. The
picture which emerges is that all the superstring
theories in ten dimensions, as well as 11-dimensional
supergravity, which were previously thought to be
distinct, are in fact different limits of a single fundamental theory, often referred to as M Žor F. theory
w1x. To elevate this new mathematical understanding
into contact with experimentally-oriented physics is
a rewarding challenge. Different directions may be
followed in pursuing this endeavour. On the one
hand, one may look ab initio for generic phenomenological properties which may characterize the
fundamental M Žor F. theory. Or, on the other hand,
one may adapt the technologies that have been developed for the analysis of realistic classes of heterotic string solutions in four dimensions w2x, and
explore the extent to which they may apply in the
context of M and F theory.
Following the first line of thought, one of the
issues in superstring phenomenology that received a
great deal of attention is the problem of superstring
gauge coupling unification. As is well known, the
discrepancy between the Grand Unification scale of
around 2 = 10 16 GeV estimated by extrapolating
naively from the measurements at LEP and elsewhere w3x and the estimate of around 4 = 10 17 GeV
found in weakly-coupled heterotic string theory w4x
may be removed if the Theory of Everything is M
theory in a strong-coupling limit, corresponding to
an eleventh dimension that is considerably larger
than the naive Planck length w5x. This scenario would
explain naturally why the value of sin2u W measured
at accelerators is in good agreement with minimal
supersymmetric GUT predictions, a feature not
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 2 9 4 - X
124
J. Ellis et al.r Physics Letters B 419 (1998) 123–131
shared by generic string models with extra particles
at intermediate scales, large Planck-scale threshold
corrections, or different Kac-Moody levels for different gauge group factors w6x.
This economic strong-coupling solution to the
reconciliation of the minimal supersymmetric GUT
and string unification scales may be desirable for
resolving other phenomenological issues, such as
stabilizing the dilaton vacuum expectation value and
selecting the appropriate vacuum point in moduli
space w7x. However, as is only too often the case,
closing the door for one genie may open a door for
another. In this case, we fear that the problems
associated with proton stability will resurface and in
fact worsen.
This has been a prospective problem for a quantum theory of gravity ever since the no-hair theorems
were discovered and it was realized that non-perturbative vacuum fluctuations could engender baryon
decay w8x, in the absence of any custodial exact
Žgauge. symmetry. This problem became particularly
acute with the advent of supersymmetric GUTs,
when it was realized that effective dimension-five
operators of the form
QQQL
Ž 1.
could induce rapid baryon decay. Operators of this
form could be generated either by the exchange of
GUT particles w9x or by quantum gravity effects w10x.
Specifically, in the context of string theory, such an
operator could in general be induced by the exchange of heavy string modes. In this case, the
coefficient of the operator ŽEq. Ž1.. would be suppressed by one inverse power of the effective string
scale M. In the perturbative heterotic string solutions
studied heretofore, this scale is of the order M ; 10 18
GeV, whilst in the proposed non-perturbative M-theory solution to the string-scale gauge-coupling unification problem this scale would be of the order
M ; 10 16 GeV. Thus, the magnitude of the effective
dimension-five operator ŽEq. Ž1.. may increase by
; two orders of magnitude. As proton stability
considerations severely restrict the magnitude of such
operators, and as the general expectation is that this
kind of operator is abundant in a generic superstring
vacuum, it would seem that the M-theory resolution
of the problem of string-scale gauge-coupling unification, we have re-introduced a far more serious
problem, namely that of baryon decay.
A full M- Žor F-.theory solution to this problem
lies beyond technical reach at this time. However,
we believe that useful insight into this problem may
be obtained by examining perturbative heterotic string
models in four dimensions that possess some realistic properties, identifying symmetries that guarantee
the absence of dangerous dimension-five operators to
all orders in string perturbation theory, and then
investigating the possibility of elevating such models
to a full non-perturbative M- Žor F-.theory formulation.
For this purpose, we choose to investigate the
three-generation superstring models w11–16x derived
in the free-fermion formulation w17x. This construction produces a large number of three-generation
models with different phenomenological characteristics, some of which are especially appealing. This
class of models corresponds to Z2 = Z2 orbifold
compactification at the maximally-symmetric point
in the Narain moduli space w18x. The emergence of
three generations is correlated with the underlying
Z2 = Z2 orbifold structure. Detailed studies of specific models have revealed that these models may
explain the qualitative structure of the fermion mass
spectrum w19x and could form the basis of a realistic
superstring model. We refer the interested reader to
several review articles which summarize the phenomenological studies of this class of models w2x.
For our purposes here, let us recall the main
structures underlying this class of models. In the
free-fermion formulation w17x, a model is defined by
a set of boundary condition basis vectors, together
with the related one-loop GSO projection coefficients, that are constrained by the string consistency
constraints. The basis vectors, b k , span a finite additive group, J s Ý k n k b k where n k s 0, PPP , Nz k y 1.
The physical states in the Hilbert space of a given
sector a g J , are obtained by acting on the vacuum
with bosonic and fermionic operators and by applying the generalized GSO projections. The UŽ1.
charges QŽ f . corresponding to the unbroken Cartan
generators of the four-dimensional gauge group,
which are in one-to-one correspondence with the
UŽ1. currents f ) f for each complex fermion f, are
given by:
Q Ž f . s 12 a Ž f . q F Ž f .
Ž 2.
J. Ellis et al.r Physics Letters B 419 (1998) 123–131
125
where a Ž f . is the boundary condition of the worldsheet fermion f in the sector a , and Fa Ž f . is a
fermion-number operator that takes the value q1 for
each mode of f, and the value y1 for each mode of
f ) , if f is complex. For periodic fermions, which
have a Ž f . s 1, the vacuum must be a spinor in
order to represent the Clifford algebra of the corresponding zero modes. For each periodic complex
fermion f, there are two degenerate vacua < q :, < y :,
annihilated by the zero modes f 0 and f 0) , respectively, and with fermion numbers F Ž f . s "1.
Realistic models in this free-fermionic formulation are generated by a suitable choice of boundarycondition basis vectors for all world-sheet fermions,
which may be constructed in two stages. The first
stage consists of the NAHE set w11,20x of five
boundary condition basis vectors, 1,S,b 1 ,b 2 ,b 3 4 . After generalized GSO projections over the NAHE set,
the residual gauge group is SO Ž10. = SO Ž6. 3 = E8
with N s 1 space–time supersymmetry 1. The
space–time vector bosons that generate the gauge
group arise from the Neveu–Schwarz sector and
from the sector 1 q b 1 q b 2 q b 3 . The Neveu–
Schwarz sector produces the generators of SO Ž10. =
SO Ž6. 3 = SO Ž16.. The sector 1 q b 1 q b 2 q b 3 produces the spinorial 128 of SO Ž16. and completes the
hidden-sector gauge group to E8 . The vectors b 1 , b 2
and b 3 correspond to the three twisted sectors in the
corresponding orbifold formulation, and produce 48
spinorial 16-dimensional representations of SO Ž10.,
sixteen each from the sectors b 1 , b 2 and b 3 .
The second stage of the basis construction consists of adding three more basis vectors to the NAHE
set, corresponding to Wilson lines in the orbifold
formulation, whose general forms are constrained by
string consistency conditions such as modular invariance, as well as by space-time supersymmetry. These
three additional vectors are needed to reduce the
number of generations to three, one from each of the
sectors b 1 , b 2 and b 3 . The details of the additional
basis vectors distinguish between different models
and determine their phenomenological properties.
The residual three generations constitute representations of the final observable gauge group, which
can be SUŽ5. = UŽ1. w11x, SO Ž6. = SO Ž4. w13x or
SUŽ3. = SUŽ2. = UŽ1. 2 w12,14,16x. In the former two
cases, an additional pair of 16 and 16 representations
of SO Ž10. is obtained from the two basis vectors that
extend the NAHE set. The electroweak Higgs multiplets are obtained from the Neveu–Schwarz sector,
and from a sector that is a combination of the two
basis vectors which extend the NAHE set. This
combination has the property that X R P X R s 4, and
produces states that transform solely under the observable-sector symmetries. Massless states from this
sector are then obtained by acting on the vacuum
with fermionic oscillators with frequency 1r2. Details of the flavor symmetries differ between models,
but consist of at least three UŽ1. symmetries coming
from the observable-sector E8 . Additional flavor UŽ1.
factors arise from the complexification of real
world-sheet fermions, corresponding to the internal
manifold in a bosonic formulation. The models typically contain a hidden sector in which the final
gauge group is a subgroup of the hidden E8 , and
three matter representations in the vectorial 16 of
SO Ž16., which arise from the breaking of the hidden
E8 to SO Ž16. 2 .
The cubic and higher-order terms in the superpotential are obtained by evaluating the correlators
1
The vector S in this NAHE set is the supersymmetry generator, and the superpartners of the states from a given sector a are
obtained from the sector Sq a .
2
In general, the models also contain exotic massless states that
arise from the breaking of the non-Abelian SO Ž10. symmetry at
the string level.
A N ; ² V1f V2f V3b PPP VN : ,
Ž 3.
where Vi f Ž Vi b . are the fermionic Žscalar. components of the vertex operators, using the rules given in
w21x. Generically, correlators of the form ŽEq. Ž3..
are of order O Ž g Ny 2 ., and hence of progressively
higher orders in the weak-coupling limit. One of the
UŽ1. factors in the free-fermion models is anomalous, and generates a Fayet–Ilioupolos term which
breaks supersymmetry at the Planck scale. The
anomalous UŽ1. is broken, and supersymmetry is
restored, by a non-trivial VEV for some scalar field
that is charged under the anomalous UŽ1.. Since this
field is in general also charged with respect to the
126
J. Ellis et al.r Physics Letters B 419 (1998) 123–131
other anomaly-free UŽ1. factors, some non-trivial set
of other fields must also get non-vanishing VEVs
V , in order to ensure that the vacuum is supersymmetric. Some of these fields will appear in the
nonrenormalizable terms ŽEq. Ž3.., leading to effective operators of lower dimension. Their coefficients
contain factors of order VrM, which may not be
very small, particularly in the context of the M-theory resolution of the unification-scale problem.
This technology has previously been used to study
the issue of proton decay in the context of SUŽ5. =
UŽ1. and SUŽ3. = SUŽ2. = UŽ1. 2 type models w22x.
Here we study this issue using two specific freefermion models w14,15x as case studies. In both
models, the color-triplet Higgs multiplets from the
Neveu–Schwarz sectors are projected out by a superstring doublet–triplet splitting mechanism, so that
conventional GUT-scale dimension-five operators are
absent. Whilst the model of w14x contains one pair of
color-triplet Higgs fields from the sector b 1 q b 2 q a
q b , in the model of w15x the Higgs color triplets
from this sector are projected out by the generalized
GSO projections. The two models also contain exotic
color triplets from sectors that arise from the SO Ž10.
Wilson-line breaking, and carry lepton numbers
"1r2.
Examining the superpotential terms in the first
model w14x, we find the following non-renormalizable terms at order N s 6
q
Q3 Q 2 Q 2 L3F45Fy
2 and Q 3 Q1 Q1 L 3F45F 1
Ž 4.
Thus, dangerous dimension-five operators arise in
4
this model if either of the sets of fields F45 ; Fy
2 or
F45 ; Fq
4 gets a VEV in the cancellation of the
1
anomalous UŽ1. D-term equations. Even if we can
choose flat directions such that these order N s 6
terms are suppressed, higher-order terms can still
generate the dangerous operators. If the suppression
factor ² f :rM ; 1r10, the dimension-five terms are
suppressed by ; 10 Ny 2 in each successive order.
The order N s 6 terms would certainly lead to proton decay at a rate that contradicts experiment, and
the same would be true of many higher-order operators in the proposed M-theory context.
Next we turn to the model of w15x, which introduces a new feature. In this model, the observable-
sector gauge group formed by the gauge bosons from
the Neveu–Schwarz sector alone is
SU Ž 3 . = SU Ž 2 . = U Ž 1 . C = U Ž 1 . L = U Ž 1 . 1,2,3,4,5,6 .
Ž 5.
However, in this model there are two additional
gauge bosons from the vector combination 1 q a q
2g w15x, where the vector 2g has periodic boundary
c o n d itio n s fo r th e in te rn a l fe rm io n s
c 1, PPP ,5,h 1,h 2 ,h 3 , f 1, PPP ,4 4 and anti-periodic boundary conditions for all the remaining world-sheet
fermions:
Ž6.
This model also has two new gauge generators,
whose gauge bosons are singlets of the non-Abelian
group, but carry UŽ1. charges. Referring to these
two generators as T ", we can define the linear
combination
T 3 ' 14 U Ž 1 . C q U Ž 1 . L q U Ž 1 . 4 q U Ž 1 . 5
qU Ž 1 . 6 q U Ž 1 . 7 y U Ž 1 . 9
Ž 7.
such that the three generators T ",T34 together form
the enhanced symmetry group SUŽ2.. Thus, the original observable symmetry group ŽEq. Ž5.. has been
enhanced to
SU Ž 3 . C = SU Ž 2 . L = SU Ž 2 . cust = U Ž 1 . C X = U Ž 1 . L
= U Ž 1 . 1,2,3 = U Ž 1 . 4X ,5X ,7XX
Ž 8.
and the remaining UŽ1. combinations which are
orthogonal to T 3 are given by
U Ž 1 . C ' 13 U Ž 1 . C y 12 U Ž 1 . 7 q 12 U Ž 1 . 9
X
U Ž 1. 4 ' U Ž 1. 4 y U Ž 1. 5
X
U Ž 1. 5 ' U Ž 1. 4 q U Ž 1. 5 y 2 U Ž 1. 6
X
U Ž 1 . 7 ' U Ž 1 . C y 53 U Ž 1 . 4 q U Ž 1 . 5 q U Ž 1 . 6
XX
qU Ž 1 . 7 y U Ž 1 . 9 .
Ž 9.
The weak hypercharge can still be defined as the
linear combination 13 UŽ1. C q 12 UŽ1.L . However, as
the UŽ1. C symmetry is now part of the extended
J. Ellis et al.r Physics Letters B 419 (1998) 123–131
SUŽ2. gauge group, UŽ1. C is given as a linear combination of the generators above
1
3
½
U Ž 1 . C s 25 U Ž 1 . C q
5
X
16
T 3 q 35 U7XX
5
.
Ž 10 .
Since the weak hypercharge is not orthogonal to the
enhanced SUŽ2. symmetry, it is convenient to define
a new linear combination of the UŽ1. factors:
U Ž 1 . Y X ' U Ž 1 . Y y 18 T 3
s 12 U Ž 1 . L q
5
24
U Ž 1. C
y 18 U Ž 1 . 4 q U Ž 1 . 5 q U Ž 1 . 6 q U Ž 1 . 7
yU Ž 1 . 9 ,
Ž 11 .
so that the weak hypercharge is expressed in terms
of UŽ1. Y X as
U Ž 1 . Y s U Ž 1 . Y X q 12 T 3
´
3
Qe .m .s TL3 q Y s TL3 q Y X q 12 Tcust
. Ž 12 .
The final observable-sector gauge group is therefore
SU Ž 3 . C = SU Ž 2 . L = SU Ž 2 . cust = U Ž 1 . Y X
= seven other U Ž 1 . factors 4 .
127
Even if we break the custodial SUŽ2.cust by, e.g., the
VEV of the right-handed neutrino and its complex
conjugate, the higher-order terms will not be invariant under the combined symmetries SUŽ2.cust and
UŽ1.L . A computerized search for all possible operators that might lead to proton decay confirms that
such terms do not arise at any order in this model.
This model therefore provides an example how
the proton decay problem may be resolved in a
robust way: even if the string unification scale is
lowered to the minimal supersymmetric GUT scale,
as proposed in the M-theory strong-coupling solution
to the string gauge-coupling unification problem,
such a model can evade the proton decay constraints
to all orders in perturbation theory. However, we
recognize that this approach does not encompass
strictly non-perturbative string effects which may
appear in a direct M- or F-theory construction. On
the other hand, we also note that enhanced gauge
symmetries appear in many M- and F-theory constructions w23x, and may play a role analogous to that
played by the enhancement in the above model.
Such enhanced symmetries arise frequently in the
free-fermion models, whenever there is a combination of the basis vectors which extends the NAHE
set:
Ž 13 .
The remaining seven UŽ1. factors must be chosen as
linear combinations of the previous UŽ1. factors so
as to be orthogonal to the each of the other factors in
ŽEq. Ž13...
Up to UŽ1. charges, the massless spectrum from
the Neveu–Schwarz sector and the sector b 1 q b 2 q
a q b is the same as in the previous model. However, because of the enhanced symmetry, the spectrum from the sectors bj is modified. The sectors
bj [ 1 q a q 2g . produce the three light generations,
one from each of the sectors bj Ž j s 1,2,3., as
before. However, the gauge enhancement noted
above has the important corollary that only the leptons, L,e Lc , NLc 4 , transform as doublets of the enhanced SUŽ2.cust gauge group, whilst the quarks,
Q,u cL ,d Lc 4 , are SUŽ2.cust singlets. Therefore, terms of
the form QQQL are not invariant under the enhanced
SUŽ2.cust gauge group. Furthermore, such terms are
forbidden to all orders of non-renormalizable terms,
as can be verified by an explicit computerized search.
X s na a q nb b q ng g
Ž 14 .
for which X L P X L s 0 and X R P X R / 0. Such a combination may produce additional space–time vector
bosons, depending on the choice of generalized GSO
phases. For example, in the flipped SUŽ5. model of
w24x, in addition to the gauge bosons from the
Neveu–Schwarz sector and the sector I s 1 q b 1 q
b 2 q b 3 , additional space–time vector bosons are
obtained from the sectors b 1 q b4 " a [ I. In this
case, the hidden-sector SUŽ4. gauge group, arising
from the gauge bosons of the NS [ I sectors, is
enhanced to SUŽ5.. This particular enhancement does
not modify the observable gauge sector, and does
nothing to forbid dangerous higher-order operators.
However, other, more interesting, cases may exist.
The type of enhancement depends not only on the
boundary-condition basis vectors, but also on the
discrete choices of GSO phases. For example, in the
model of w14x, the combination of basis vectors
J. Ellis et al.r Physics Letters B 419 (1998) 123–131
128
X s b 1 q b 2 q b 3 q a q b q g q Ž I . has X L P X L s
0, and thus may give rise to additional space–time
vector bosons. All the extra space–time vector bosons
are projected out by the choice of generalized GSO
projection coefficients. However, with the modified
GSO phases
tors b 1 , b 2 and b 3 correspond to the three twisted
sectors of these orbifold models. To see this correspondence, we add to the NAHE set the basis vector
Ž17.
a
a
1
1
c g ™ yc g ,c
™ yc
b
b
ž / ž /
ž /
ž /
g
g
and c
™ yc
,
b
b
ž/
ž/
Ž 15 .
additional space–time vector bosons are obtained
from the sector b 1 q b 2 q b 3 q a q b q g q Ž I .. In
addition, the sector b 1 q b 2 q b 3 q a q b q g q Ž I .
produces the representations 31 q 3y1 of SUŽ3.H ,
where one of the UŽ1. combinations is the UŽ1. in
the decomposition of SUŽ4. under SUŽ3. = UŽ1.. In
this case, the hidden-sector SUŽ3.H gauge group is
extended to SUŽ4.H . If instead we take the modified
phases
1
1
g ™ yc g ,
c g ™ yc g ,c
a
a
ž /
ž / ž /
ž /
Ž 16 .
then the sector 1 q a q b q g produces two additional space–time vector bosons which enhances one
of the UŽ1. factors to SUŽ2.. These examples further
illustrate the point that this type of enhancement is
common in realistic free-fermion models, which
makes it interesting to explore further in connection
with the proton stability problem.
We now explore the possibility of a connection
between the type of models discussed above and the
vacua of M Žand F. theory. At present, a direct
connection between known features of the non-perturbative formulation of M theory and the above
realistic free-fermion models, with a full basis consisting of eight vectors, is not yet possible. However,
it is nevertheless possible to make observations suggesting a possible connection of these models to the
type of M- Žand F-.theory compactifications which
have been discussed in the literature w25x.
We start by studying in more detail the geometric
interpretation of the five-basis-vector NAHE set,
1,S,b 1 ,b 2 ,b 3 4 that underlies the realistic free
fermionic models. This set corresponds to a Z2 = Z2
orbifold compactification of the weakly-coupled
ten-dimensional heterotic string, and the basis vec-
with the following choice of generalized GSO projection coefficients:
X
X
X
C b s yC
sC
s q1 .
S
1
j
ž /
ž / ž /
Ž 18 .
This set of basis vectors produces models with an
SO Ž4. 3 = E6 = UŽ1. 2 = E8 gauge group and N s 1
space–time supersymmetry. The matter fields include 24 generations in 27 representations of E6 ,
eight from each of the sectors b 1 [ b 1 q X, b 2 [ b 2
q X and b 3 [ b 3 q X. Three additional 27 and 27
pairs are obtained from the Neveu–Schwarz [ X
sector.
The subset of basis vectors
1,S, X , I s 1 q b1 q b 2 q b 3 4
Ž 19 .
generates a toroidally-compactified model with N s
4 space–time supersymmetry and SO Ž12. = E8 = E8
gauge group. The same model is obtained in the
geometric Žbosonic. language by constructing the
background fields which produce the SO Ž12. Narain
lattice w18,26x, taking the metric of the six-dimensional compactified manifold to be the Cartan matrix
of SO Ž12.:
2
y1
0
gi j s
0
0
0
y1
2
y1
0
0
0
0
y1
2
y1
0
0
0
0
y1
2
y1
y1
0
0
0
y1
2
0
0
0
0
y1
0
2
0
Ž 20 .
and the antisymmetric tensor
bi j
°g
¢yg
s~0
ij
ij
;
i ) j,
;
;
i s j,
i - j.
Ž 21 .
J. Ellis et al.r Physics Letters B 419 (1998) 123–131
129
When all the radii of the six-dimensional compactified manifold are fixed at R I s '2 , it is easily seen
that the left- and right-moving momenta
q b 1 q b 2 q b 3 ,b 1 ,b 2 4 , projecting out the 16 [ 16
from the sector X by taking
PRI , L s m i y 12 Ž Bi j " Gi j . n j e iI )
c
Ž 22 .
reproduce all the massless root vectors in the lattice
of SO Ž12., where in ŽEq. Ž22.. the e i s e iI 4 are six
linearly-independent vectors normalized: Ž e i . 2 s 2.
The e iI ) are dual to the e i , and e i) P e j s d i j .
Adding the two basis vectors b 1 and b 2 to the set
ŽEq. Ž19.. corresponds to the Z2 = Z2 orbifold model
with standard embedding. The fermionic boundary
conditions are translated in the bosonic language to
twists on the internal dimensions and shifts on the
gauge degrees of freedom. Starting from the Narain
model with SO Ž12. = E8 = E8 symmetry w18x, and
applying the Z2 = Z2 twisting on the internal coordinates, we then obtain the orbifold model with SO Ž4. 3
= E6 = UŽ1. 2 = E8 gauge symmetry. There are sixteen fixed points in each twisted sector, yielding the
24 generations from the three twisted sectors mentioned above. The three additional pairs of 27 and 27
are obtained from the untwisted sector. This orbifold
model exactly corresponds to the free-fermion model
with the six-dimensional basis set 1,S, X, I s 1 q b 1
q b 2 q b 3 ,b 1 ,b 2 4 . The Euler characteristic of this
model is 48 with h11 s 27 and h 21 s 3.
This Z2 = Z2 orbifold, corresponding to the extended NAHE set at the core of the realistic free
fermionic models, differs from the one which has
usually been examined in the literature w25x. In that
orbifold model, the Narain lattice is SO Ž4. 3 , yielding
a Z2 = Z2 orbifold model with Euler characteristic
equal to 96, or 48 generations, and h11 s 51, h 21 s 3.
In more realistic free-fermion models, the vector
X is replaced by the vector 2g ŽEq. Ž6... This
modification has the consequence of producing a
toroidally-compactified model with N s 4 space–
time supersymmetry and gauge group SO Ž12. =
SO Ž16. = SO Ž16.. The Z2 = Z2 twisting breaks the
gauge symmetry to SO Ž4. 3 = SO Ž10. = UŽ1. 3 =
SO Ž16.. The orbifold twisting still yields a model
with 24 generations, eight from each twisted sector,
but now the generations are in the chiral 16 representation of SO Ž10., rather than in the 27 of E6 . The
same model can be realized with the set 1,S, X, I s 1
X
X
™ yc
.
I
I
ž /
ž /
Ž 23 .
This choice also projects out the massless vector
bosons in the 128 of SO Ž16. in the hidden-sector E8
gauge group, thereby breaking the E6 = E8 symmetry to SO Ž10. = UŽ1. = SO Ž16.. This analysis confirms that the Z2 = Z2 orbifold on the SO Ž12. Narain
lattice is indeed at the core of the realistic free
fermionic models.
We can now examine whether some connection
with M- Žand F-.theory compactifications can be
contemplated, in view of the extensive literature on
Z2 = Z2 orientifolds of M and F theory w25x. In
particular, these interesting papers have examined in
detail the Z2 = Z2 orbifold model with h11 s 51 and
h 21 s 3. This model is precisely the Z2 = Z2 orbifold
model obtained by twisting the SO Ž4. 3 Narain lattice. In this compactification, the six-dimensional
compactified space is the direct product of three
simple two-tori, i.e., ŽT2 . 3. This model has been
investigated extensively in w25x in connection with
M- and F-theory compactifications on special classes
of Calabi–Yau threefolds that have been analyzed by
Voisin w27x and Borcea w28x. They have been further
classified by Nikulin w29x in terms of three invariants
Ž r,a, d ., in terms of which Ž h1,1 ,h 2,1 . s Ž5 q 3r y
2 a,65 y 3r y 2 a.. Within this framework, the
Ž h1,1 ,h 2,1 . s Ž51,3. Z2 = Z2 orbifold model coincides
with the Voisin–Borcea model with Ž r,a, d . s
Ž18,4,0.. The dual relations between compactifications of this manifold on M-, F-theory compactifications and type IIB orientifolds have been demonstrated in w25x.
The task ahead is clear. Naively, the Ž27,3. Z2 =
Z2 orbifold would correspond to a Voisin–Borcea
model with Ž r,a, d . s Ž14,10,0.. However, such a
Voisin–Borcea model is not Žyet. known to exist.
The problem is that in the Voisin–Borcea models the
factorization of the six-dimensional manifold as a
product of three disjoint manifolds is essential. However, our Ž27,3. Z2 = Z2 model, being an orbifold of
a SO Ž12. lattice, is an intrinsically T 6 manifold,
and, as such, factorization a` la Voisin-Borcea is not
130
J. Ellis et al.r Physics Letters B 419 (1998) 123–131
possible. Therefore, the first step in trying to connect
realistic free-fermion models to M- and F-theory
compactifications is to construct the Calabi–Yau
threefolds which correspond to the Z2 = Z2 orbifold
on the SO Ž12. lattice. Although the relevant manifolds, or their Landau–Ginzburg potential realizations, are not yet known Žat least not to us., we
believe that they are not fundamentally different in
nature from, or intrinsically more difficult than, the
corresponding orientifolds related to the Z2 = Z2 orbifold on SO Ž4. 3 lattice.
We now recap the current status of the effort to
connect M- and F-theory compactifications to relevant phenomenological data. On the one hand, we
have the appealing free-fermion models, in which
one can address in detail many relevant phenomenological questions, and which provide promising candidates for a realistic superstring model. We have
examined in detail in this paper the issue of proton
stability, and shown how enhanced gauge symmetries which prevent fast proton can arise. Thus, we
have exhibited a robust solution to the problem of
the proton lifetime, which is of crucial relevance for
M-theory compactifications.
As a step towards the elevation of these ideas to
M and F theory, we have discussed the orbifold
correspondences of these models. At the core of the
realistic free-fermion models there is a Z2 = Z2 orbifold on an SO Ž12. lattice with Ž h1,1 ,h 2,1 . s Ž27,3..
This is not the standard Z2 = Z2 orbifold that has
been discussed extensively in the literature, the more
familiar one being that with Ž h1,1 ,h 2,1 . s Ž51,3..
Nevertheless, the existence of duality relations between the Ž51,3. Z2 = Z2 orbifold and M- and F-theory compactifications leads us to expect the existence of similar relations for the Ž27,3. Z2 = Z2
orbifold. If the relevant Calabi-Yau threefold or its
Landau–Ginzburg realization can indeed be found,
the connection of M and F theory to relevant low-energy data would have made a major step forward,
particularly with regard to proton stability. Such
progress is not out of sight.
Acknowledgements
We are pleased to thank Julie Blum and Cumrun
Vafa for discussions. This work was supported in
part by the Department of Energy under Grants No.
DE-FG-05-86-ER-40272 and DE-FG05-91-GR40633.
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12 February 1998
Physics Letters B 419 Ž1998. 132–138
Compactifications of F-theory on Calabi-Yau threefolds
at constant coupling
Changhyun Ahn 1, Soonkeon Nam
2
Department of Physics and Research Institute for Basic Sciences, Kyung Hee UniÕersity, Seoul 130-701, South Korea
Received 10 February 1997; revised 26 March 1997
Editor: M. Dine
Abstract
Generalizing the work of Sen, we analyze special points in the moduli space of the compactification of the F-theory on
elliptically fibered Calabi-Yau threefolds where the coupling remains constant. These contain points where they can be
realized as orbifolds of six torus T 6 by Z m = Z nŽ m,n s 2,3,4,6.. At various types of intersection points of singularities, we
find that the enhancement of gauge symmetries arises from the intersection of two kinds of singularities. We also argue that
when we take the Hirzebruch surface as a base for the Calabi-Yau threefold, the condition for constant coupling corresponds
to the case where the point like instantons coalesce, giving rise to enhanced gauge group of SpŽ k .. q 1998 Published by
Elsevier Science B.V.
Our understanding of nonperturbative aspects of
N s 2 supersymmetric gauge theories w1x in four
dimensions progressed very much. Deeper understanding of the gauge dynamics comes from the
study of brane probes in string theory, where there
are also a lot of exciting developments in the conjectured string dualities w2x. Among these, the heteroticrtype II duality has been studied in detail w3x.
In fact, it has been extended to the F-theoryrheterotic duality w4x where the heterotic strings compactified on a two torus T 2 is dual to F-theory in eight
dimensions compactified on K 3 which admits an
elliptic fibration. F-theory w5x is defined as the compactifications of type IIB string in which the com-
1
2
E-mail: [email protected].
E-mail: [email protected].
plex coupling changes over the base. The K 3 surface
which is a fiber space where the base is one dimensional complex projective space CP 1 and torus as
the fiber is represented by y 2 s x 3 q f Ž z . x q g Ž z .
where z is the coordinate of the base CP 1 and f and
g are the polynomials of degree 8, 12 respectively in
z. This describes a torus for each point on the base
CP 1 labelled by the coordinate z.
Extension of this to the compactification down to
six dimensions is interesting for various reasons.
First of all, although the string theory answer is
rather trivial, the classical field theory on the 3-brane
which produces the answer is not so trivial and has
been considered in Ref. w6x. Secondly, the orbifold
limit of Calabi-Yau threefoldŽCY3. itself is an interesting object to study. So far the examples which has
been considered are the Voisin-Borcea models w7x
which are listed in w4x and are the product of a
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 7 7 - 9
C. Ahn, S. Nam r Physics Letters B 419 (1998) 132–138
two-torus T 2 and K 3 divided by a Z 2 symmetry;
they correspond to type IIB compactification with a
space-independent coupling constant. It would therefore be interesting to study other examples of constant couplings and the physics of the gauge theories
arising from the string theories.
One has the duality w4x between F-theories compactified on CY threefold and heterotic strings on
K 3. For example, the SO Ž32. heterotic string compactified on a K 3 surface was discussed in Ref. w8x.
One of the interesting results is that one can obtain
nonperturbative SpŽ1. extra gauge group when an
instanton shrinks to zero size. Furthermore, when k
instantons collapse at the point in the K 3, the SpŽ1. k
factor is replaced by the enhanced gauge symmetry
of SpŽ k .. These results are also reproduced in Ref.
w9x in the context of dual theory of F-theory on an
elliptically fibered CY3. On the other hand, E8 = E8
heterotic string compactified on a K 3 has been
studied in Ref. w4x where only extra massless tensor
multiplets appear as instantons shrink down to zero
size. An aspect of generic pointlike instantons for
both SO Ž32. string and E8 = E8 string has been
analyzed in Refs. w9,10x. Further compactification to
four dimensions corresponds to the type IIrheterotic
string duality considered in Ref. w3x.
Sen w11x has shown a precise relation conjectured
in Ref. w4x between the F-theory on a smooth elliptic
K 3 manifold and a type IIB orientifold on T 2. Using
the orbifold limit of K 3, i.e. for the simplest case of
T 4rZ 2 where the coupling is constant over the base,
new insight into the K 3 compactification was obtained. The points of enhanced gauge symmetries in
the F-theory corresponds to those of enhanced global
SO Ž8. symmetries in the Seiberg-Witten gauge theory w1x. It has been found in Ref. w12x further that
there exist other points for which the coupling as
p
constant i.e., t s i or t s e . At these special points,
K 3 becomes the orbifolds of four torus T 4rZ m
where m s 3,4,6 with the base, T 2rZ m . At these
orbifold points, a singularity analysis shows that
exceptional gauge group symmetries appear. Nontrivial superconformal field theories for E6,7,8 type
singularities has been discussed w13x in the context of
Seiberg-Witten gauge theory. Very recently, generalizing the work of Sen w11x, the field theory of
3-brane probes w14x in a compactification of F-theory
on a six torus T 6 by Z 2 = Z 2 w15x with hodge
i
133
number Ž h11 ,h21 . s Ž51,3. was considered in Ref.
w6x. This has an interpretation in terms of multiple
3-branes probes on an F-theory orientifold as was
discussed in Ref. w16x.
In this paper, we do the following two things.
First, we analyze special points in the moduli space
of the compactification of the F-theory on elliptically
fibered CY3’s where the coupling remains constant,
along the lines of Refs. w11,12x. This is rather
straightforward and can be realized as other orbifolds of six torus T 6rZ m = Z n where m,n s 2,3,4,6.
At various types of intersection points between G,GX
s SO Ž8., E6,7,8 types of singularities, we find that the
enhancement of gauge symmetries arises from the
intersection of two types of singularities, different
from the naively expected gauge symmetry of G =
GX . We find that the naive gauge symmetries get
modified due to the interplay of the singularities. In
the second part of this paper, we consider the case
when the base for the CY3 is a Hirzebruch surface
Fn and realize that the property of the constancy of
coupling leads to exactly the coalescence of pointlike
instantons for SO Ž32. heterotic string.
Let us first consider the compactification of F-theory on elliptically fibered CY3 where the coupling is
constant over the base. It has been found in Refs.
w4,17x that the CY3 can be described as an elliptic
fibration in the Weierstrass form y 2 s x 3 q f Ž z,w . x
q g Ž z,w .. z and w are the coordinates on the base
CP 1 = CP 1 and f and g are the polynomials of
degree 8, 12 respectively in each of them. Notice
that there exists an exchange symmetry when we
exchange the two CP 1 ’s and simultaneously the
coefficient of the term, z l w k is exchanged with that
of z k w l in each of the terms. The modular parameter
t Ž z,w . of the fiber can be written in terms of the
invariant j function given by
3
j Ž t Ž z ,w . . s
4 Ž 24 f Ž z ,w . .
D Ž z ,w .
3
,
Ž 1.
where the discriminant is DŽ z,w . s 4Ž f Ž z,w .. 3 q
27Ž g Ž z,w .. 2 . From now we will consider only the
cases in which f and g are factorized. That is,
f Ž z,w . s a f 1Ž z . f 2 Ž w . and g Ž z,w . s g 1Ž z . g 2 Ž w .
where a is a constant. Note that jŽt Ž z,w .. blows up
at the zeroes of the discriminant.
C. Ahn, S. Nam r Physics Letters B 419 (1998) 132–138
134
The one solution for the case of constant modulus
by rescaling y and x and setting the overall coefficient to be 1 has been found in Ref. w6x. Thus we get
4
4
2
2
f 1Ž z . s Ł Ž z y z i . ,
f 2 Ž w . s Ł Ž w y wi . ,
is1
is1
4
4
3
3
g 1Ž z . s Ł Ž z y z i . ,
g 2 Ž w . s Ł Ž w y wi . ,
is1
is1
Ž 2.
where z i ’s and wi ’s are constants. This special compactification corresponds to a configuration where
the 24 7-branes are grouped into 4 sets of 6 coincident 7-branes located at the points, z 1 , z 2 , z 3 , z 4 . There
exists an SLŽ2,Z . monodromy around each of fixed
points z i ’s. The same is true at the points w s wi
because the base is simply a product of the CP 1 ’s. It
is obvious that we have SO Ž8. singularities at z s z i
and w s wi . The spacetime theory is an N s 1 supersymmetric theory whose field content was found in
Refs. w15,18x. For example, the open string sectors
lead to SO Ž8. gauge group for each 7-branes coming
from two Z 2 factors for a total enhanced gauge
symmetries Ž SO Ž8.. 4 = Ž SO Ž8.. 4 w18x.
Now we continue on to carry out the same procedure for other various subspaces of the moduli space
on which the elliptic fiber remains constant modulus.
As pointed out in Ref. w12x, in the limit of a ™ 0, we
p
get jŽt Ž z,w .. s 0 from which t Ž z,w . s e . The
polynomials are given by
i
3
three of them together deforming the CP 1 to T 2rZ 3 .
For each point w s wi on the second CP 1 , there is a
deficit angle of p all four of them deforming CP 1 to
T 2rZ 2 . This is related to orientifold of F-theory on
T 6rZ 3 = Z 2 . In F-theory, F4 gauge symmetry corresponds to the ‘generic’ E6 singularity in the sense
that the condition on the polynomial of g Ž z,w . splits
the double zeroes of it in the E6 gauge symmetry.
Furthermore, G 2 gauge symmetry corresponds to the
‘generic’ SO Ž8. singularity with different constraint
on the polynomial g Ž z,w . w10x. Near a zero at z 1 the
singular fiber is of E6 type. On the other hand,
SO Ž8. type of singularity appears near a zero at w 1.
For simplicity, at the intersection points between
these two ‘generic’ singularities the corresponding
gauge group is simply the product of F4 = G 2 w20x. It
is clear that there are no extra enhanced gauge
symmetry factor because that one blowup of the base
gives rise to II singularity 3 on the exceptional
divisor w20x, leading to non-gauge group. Thus the
full enhanced gauge symmetry group is Ž F4 . 3 = Ž G 2 . 4
by resolving the singularity for each point of z s z i
and w s wi . From an exchange symmetry between
the two CP 1 factors we have mentioned before we
can proceed similarly for the case of compactification F-theory on T 6rZ 2 = Z 3 .
When the 12 zeroes of g 2 Ž w . has coalesced into 3
identical ones of order 4 each and those of g 1Ž z . are
given as before in Eq. Ž3., then we have the following:
f 1 s 0,
3
4
f 1 Ž z . s 0,
g 1Ž z . s Ł Ž z y z i . ,
3
g 1Ž z . s Ł Ž z y z i . ,
is1
4
2
3
4
is1
4
f 2 s 0,
4
g 2 Ž w . s Ł Ž w y wi . ,
is1
3
f 2 Ž w . s Ł Ž w y wi . ,
Ž 5.
g 2 Ž w . s Ł Ž w y wi . ,
is1
is1
Ž 3.
where the discriminant is given by
3
where the 12Ž12. zeroes of g 1Ž z .Ž g 2 Ž w .. coalesce
into 3Ž4. identical ones of order 4Ž2. each. In this
case, the discriminant, DŽ z,w . takes the form of
3
D Ž z ,w . s 27 Ł Ž z y z i .
is1
4
8
Ł Ž w y wj . 6 .
Ž 4.
js1
The singularity type from Tate’s algorithm w19x at a
zero of the discriminant gives rise to the enhancement of gauge symmetries w10x. Each point z s z i on
the first CP 1 factor carries a deficit angle of 32p all
D Ž z ,w . s 27 Ł Ž z y z i .
is1
3
8
Ł Ž w y wj . 8 .
Ž 6.
js1
Of course, each point w s wi on the second CP 1
factor has a deficit angle of 32p all three of them
together deforming the CP 1 to T 2rZ 3 . This corresponds to F-theory orientifold on T 6rZ 3 = Z 3 . Each
3
We keep the notations of In , II, III, IV, PPP for the types of
fiber in Kodaira’s classification of singularities. ŽSee Refs. w22,4x.
C. Ahn, S. Nam r Physics Letters B 419 (1998) 132–138
singular fiber over the fixed point z 1 , z 2 , z 3 and
w 1 ,w 2 ,w 3 is of E6 type. According to the requirement of the elliptic fibration on the blowup surface
having CY, the relation for the blown up surface
restricts to the possible resolutions and satisfies CY
condition i.e. the sum of coefficients 4 of two intersecting singular types in Kodaira’s list w23x is the
coefficient of singular type on the exceptional divisor plus 1 w20x. The first blowup of the base leads to
IV singularity on the exceptional divisor which will
produce SUŽ3. gauge group. The next blowups in
turn appear in the intersection of IV = IV ) , known
as dual for which the sum of the coefficients of
IV = IV ) is always 1. So this intersection does not
produce the enhancement of gauge group. For the
intersection points of two E6 singularities, for example, at z s z 1 and w s w1 , we get the gauge group of
E6 = SUŽ3. = E6 by an extra SUŽ3. factor w20x.
Therefore the total enhancement of gauge symmetry
group is given by Ž E6 . 3 = Ž SUŽ3.. 3 = Ž E6 . 3.
If the 12 zeroes of g 2 Ž w . has coalesced into 3
zeroes of order 5, 4, 3 each and those of g 1Ž z . are
the same as before like Ž5., it is easy to see that we
have the following:
f 1 s 0, f 2 s 0,
3
4
g 1Ž z . s Ł Ž z y z i . ,
is1
5
4
3
g 2 Ž w . s Ž w y w1 . Ž w y w 2 . Ž w y w 3 . .
Ž 7.
1
Each point w s wi on the second CP factor has a
deficit angle of 53p , 43p and p all together deforming
the CP 1 to T 2rZ 6 . We can describe this point as the
F-theory orientifold on T 6rZ 3 = Z 6 . In this case
naive expectation is that the enhanced gauge symmetry group is the product of Ž F4 . 3 = Ž E8 = E6 = G 2 ..
It is sometimes stated that this is not allowed because they violate the CY conditions w20x. However,
one may blow up a transversal intersection curves of
IV ) = II ) fibers without violating CY condition
w21x. We do not need further blow up in this case
because the CY is already smooth. The resolutions
for IV ) = II ) include II, I0) , IV, I0 . The transverse
135
IV ) = II intersection is of the form y 2 s x 3 q s 4 t
where s s 0 gives the IV ) curve and t s 0 does the
II curve. This leads to F4 gauge symmetry by analyzing the monodromy around t s 0 in the curve
IV ) ŽSee the appendix of Ref. w24x.. The IV line cuts
the I0) line and II line. Each of these intersections
induces monodromy within the IV fiber exchanging
two of the three rational curves. It turns SUŽ3. into
SUŽ2. type singularity w21x. Therefore, Ž F4 . 3 = Ž G 2
= SUŽ2. = SUŽ3.. = Ž E8 = E6 = G 2 . gauge symmetry appears. We can do the similar analysis for the
gauge group Ž E8 = E6 = G 2 . = Ž F4 . 3 by exchanging
z with w.
Suppose that we go to the special point where the
each point z s z i Ž w s wi . on the firstŽsecond. CP 1
factor has a deficit angle of 53p , 43p and p all together
deforming the CP 1 to T 2rZ 6 which indicates F-theory on T 6rZ 6 = Z 6 . The 12 zeroes of g 1Ž z . and
g 2 Ž w . coalesce into 3 ones of order 5, 4, 3 each, and
f 1 s f 2 s 0. The naive result for the enhanced gauge
symmetry group is Ž E8 = E6 = G 2 . 2 which violates
the CY conditions. In the case of intersection of
I0) = II ) allows us to have the resolution of II and
IV type singularities for which the sum of the coefficients of them are less than 1.
Let us consider the case in which for each point
w s wi on the second CP 1, there is a deficit angle of
p all four of them deforming CP 1 to T 2rZ 2 , while
the first CP 1 factor remains unchanged. Putting this
together, we find
5
f 1 Ž z . s 0,
4
is1
3
g 2 Ž w . s Ł Ž w y wi . ,
is1
Ž 8.
where from the type of singularities we get the
enhanced gauge symmetry group is Ž E8 = E6 = G 2 .
= Ž SO Ž8.. 4 naively which is again not allowed due
to the CY conditions by intersecting of II ) and I0)
with the similar argument of the above.
Another possibility is as follows:
3
3
2
f 1Ž z . s Ž z y z1 . Ž z y z 2 . Ž z y z 3 . ,
f 2 Ž w . s Ł Ž w y wi . ,
is1
g 1 Ž z . s 0,
4
2
The coefficients a i for each type of singularity are listed in
Ref. w23x: they are given by 12n Ž In ., 16 Ž II ., 14 Ž III ., 13 Ž IV ., 12 q 12n Ž In) .,
2
Ž ) ., 34 Ž III ) ., 56 Ž II ) . where we denote the corresponding sin3 IV
gularity inside the bracket.
3
4
2
f 2 Ž w . s Ł Ž w y wi . ,
4
4
4
g 1Ž z . s Ž z y z1 . Ž z y z 2 . Ž z y z 3 . ,
3
g 2 Ž w . s Ł Ž w y wi . ,
is1
Ž 9.
which corresponds to t s i from jŽt Ž z,w .. s 13824.
C. Ahn, S. Nam r Physics Letters B 419 (1998) 132–138
136
This time it can be easily checked that the discriminant is given by
D Ž z ,w .
4
9
9
s 4 Ž z y z1 . Ž z y z 2 . Ž z y z 3 .
6
Ł Ž w y wi . 6 .
is1
Ž 10 .
The singular fiber over each fixed points z 1 , z 2 is of
E7 type. The other singular fiber over z 3 is of SO Ž8.
type. At the intersection points near z s z 1 and
w s w 1 , the gauge group appears to be E7 = SUŽ2.
= SO Ž8. enhanced by an extra SUŽ2. factor by the
fact that the first blowup for this intersection appears
singular type III on the exceptional divisor using the
CY condition again. The intersection of III = III )
leads to a I0 type singularity which does not produce
the enhanced gauge symmetry. Hence, Ž E7 = E7 =
SO Ž7.. = Ž SUŽ2.. 2 = Ž SO Ž8.. 4 gauge symmetry appears there.
Finally, we have the case when the 8 zeroes of
f 1Ž z .Ž f 2 Ž w .. coalesce into 3 ones of order 3, 3 and 2,
and g 1 s g 2 s 0 For the intersection of III ) = III ) ,
the first blow up gives rise to I0) singularity on the
exceptional divisor by using the CY condition as
discussed above. The arguement for the next intersection of III ) = I0) are given in the previous paragraph. Then the five resolutions correspond to
I0 , III, I0) , III, I0 in which there are three possibilities
for the type of I0) , i.e. SO Ž8.,SO Ž7. or G 2 depending on whether the singularity is split, semi-split or
non-split w10x. There are three cases: no factorization
in the polynomial x 3 q f Ž z,w . x q g Ž z,w . corresponds to non-split case, a product of three linear
factors does split case, a product of linear and
quadratic factors does semi-split case. Only semi-split
case satisfies the anomaly factorization condition. In
this case, the gauge group appears Ž E7 = SUŽ2.. 2 =
SO Ž7. at the intersections of two E7 singularities.
Then we get the enhanced gauge symmetry group
Ž E7 = E7 = SO Ž7.. 2 = Ž SUŽ2. = SO Ž7. = SUŽ2.. 4
with extra Ž SUŽ2.. 4 factor due to the intersections of
I0) = III ) .
Let us note that among the various gauge groups
which can appear, we have the possibility of realizing Ž E6 . 3 = Ž E7 = E7 = SO Ž7... For this case, the
discriminant DŽ z,w . vanishes identically since f 1Ž z .
s 0 and g 2 Ž w . s 0. Thus the corresponding vacuum
can not live in the F-theory moduli space where the
couplings remain constant we have studied so far but
live in the full F-theory moduli space in the sense
that the coupling varies. This is also true of the
gauge group Ž E8 = E6 = G 2 . = Ž E7 = E7 = SO Ž7...
We summarize our results in Table 1.
We can compare our findings with those in Table
8 of Ref. w20x. In Table 1, we restricted to only the
cases of intersections between I0) , II ) , III ) and IV )
singularity types. The above five rows correspond to
exactly I0) = I0) , IV ) = I0) , IV ) = IV ) , IV ) =
II ) , III ) = I0) and III ) = III ) respectively when
we intersect the specific two zeroes z s z 1 and w s
w1. Our jŽt Ž z,w .. is related to their J up to constant.
In the remainder of paper, we would like to
consider the SO Ž32. heterotic string. Consider the
following elliptic fibration over Hirzebruch surface
Fn as a base for the CY3 with z the coordinate of
CP 1 fiber of Fn and w the coordinate on the base.
This Weierstrass form may be put into the more
restrictive form w9,22x as follows:
y 2 s x 3 q f Ž z ,w . x q g Ž z ,w .
s Ž x y b Ž z ,w . . Ž x 2 q b Ž z ,w . x q g Ž z ,w . . ,
Ž 11 .
which gives a section along x s b Ž z,w ., y s 0. The
Table 1
Possible enhancements of gauge symmetry for various F-theory orbifolds
model
enhanced gauge group
T rZ 2 = Z 2
T 6 rZ 3 = Z 2
T 6 rZ 3 = Z 3
T 6 rZ 3 = Z 6
T 6 rZ 4 = Z 2
T 6 rZ 4 = Z 4
Ž SO Ž8.. 4 = Ž SO Ž8.. 4
Ž F4 . 3 = Ž G 2 . 4
Ž E6 . 3 = Ž SUŽ3.. 3 = Ž E6 . 3
Ž F4 . 3 = Ž G 2 = SUŽ2. = SUŽ3.. = Ž E8 = E6 = G 2 .
Ž E7 = E7 = SO Ž7.. = Ž SUŽ2.. 2 = Ž SO Ž8.. 4
Ž E7 = E7 = SO Ž7.. = Ž SUŽ2..12 = Ž SO Ž7.. 4 = Ž E7 = E7 = SO Ž7..
6
C. Ahn, S. Nam r Physics Letters B 419 (1998) 132–138
functions b and g can be represented in a sufficiently generic form as follows w9,22x:
b Ž z ,w . s Bz 4 q Cz ,
g Ž z ,w . s Az 8 y 4 BCz 5 y 2C 2 z 2 .
Ž 12 .
Since we constrain above to be a CY space, A is a
polynomial of degree 8 q 4 n in w, B is of degree
4 q 2 n, and C is of degree 4 y n.
We consider the nonzero constant modulus case.
Using the known relation of the modular parameter
in terms of the j function, we get the following
relation in order that jŽt Ž z,w .. should be independent of z and w for nonzero g Ž z,w .,
f3
g
2
s
Žg y b 2 .
Ž bg .
2
3
27
sy
4
.
Ž 13 .
Notice that unlike the case of the compactification to
8 dimensions, where the ratio of f 3 and g 2 was an
arbitrary nonzero constant, here the value of the ratio
gets fixed. For this case actually the discriminant has
to vanish. This means that we must have A s y2 B 2
as we can see from the factorized form of the
discriminant
D Ž z ,w .
3
2
s 4 f Ž z ,w . q 27g Ž z ,w . s z 18 Ž A q 2 B 2 .
= Ž Ž 4 A y B 2 . z 6 y 18 BCz 3 y 9C 2 . .
2
Ž 14 .
For this special case, we get constant couplings.
Now let us be more specific. Suppose A q 2 B 2 is of
order k in w. Then, we can put
k
A q 2 B 2 s Ł Ž w y wi . .
Ž 15 .
is1
In general the loci of the singularities wi can be
different. The zero of the discriminant is of order 2 k
in w for arbitrary value of z. According to Ref. w19x,
it is clear that we have I2 k fiber type. The gauge
group related with I2 k is SUŽ2 k .. We also observe
that this case has a remarkable coincidence with the
case where the point like instantons coalesce w8x,
where the enhanced gauge group is SpŽ k ..
There are other special points when f or g vanishes, rendering the modular parameter of the fiber to
be a constant. First, when f s 0, i.e. b 2 s g , we get
p
jŽt Ž z,w .. s 0, and t s e . Here we must have C s 0
and A s B 2 for generic value of z. Then g s
i
3
137
2 nŽ
yŁ 4q
w y wi . 3 z 12 , and if none of the wi ’s coinis1
cide, we get SO Ž8. type singularity of the discriminant and thus an enhanced gauge symmetry of
Ž G 2 . 4q 2 n for ‘generic’ singularity! Some of these
might be related to the orbifold cases we discussed
in the first part of the paper. In the case of the
orbifold T 6rZ 3 = Z 2 as done in Eq. Ž3. we found
that g Ž z,w . s Ł 3is1Ž z y z i . 4 Ł 4js1Ž w y wj . 3. Merging the zeroes at z i ’s will produce exactly the above
result for n s 0 case. On the other hand, for the case
of bg s 0 where jŽt Ž z,w .. s 13824, the following
relations B s C s 0 or A s C s 0 hold, we get D ;
2 nŽ
Ł 4q
w y wi . 6 z 24 , hence we expect again an enoris1
mous gauge enhancement of Ž G 2 . 4q 2 n. Finally when
we consider the case of A s C s 0, we have f s
2 nŽ
yŁ 4q
w y wi . 2 z 8 which can be realized as the
is1
orbifold T 6rZ 4 = Z 2 with n s 0 by coalescing z i ’s
to vanish in Eq. Ž9..
Recently, the enhanced gauge group in four dimensions by SUŽ n. singular fibers has been studied
w25x in the context of F-theory compactifications on
CY4. Certain classes of gauge symmetry enhancement have been worked out but certainly more works
have to be done especially the cases of colliding
singularities. It would be interesting to extend our
analysis for these cases. One might also consider the
compactification of F-theory on a large class of
CY4’s of the form Ž K 3 = K 3.rZ 2 in four dimensions. The properties of these CY4 are even less
known, but for the elliptically fibered cases with
CY3 as basis will be the starting point of a future
work. For example, the orbifolds of eight torus T 8
by Ž Z 2 . 3 limit of this CY4s can be written as an
elliptic fibration of the form y 2 s x 3 q f Ž z,w,Õ . x q
g Ž z,w,Õ . where Õ,w and z are the coordinates on
Ž CP 1 . 3 and f and g are polynomials of degrees 8,
12 respectively in all their arguments.
Acknowledgements
C.A. thanks Jaemo Park for discussions on the
subject of orientifold. We thank P.S. Aspinwall for
pointing out errors in the original version of the
paper. This work is supported in part by Ministry of
Education ŽBSRI-95-2442., by KOSEF Ž961-0201001-2. and by CTPrSNU through the SRC program
of KOSEF.
138
C. Ahn, S. Nam r Physics Letters B 419 (1998) 132–138
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12 February 1998
Physics Letters B 419 Ž1998. 139–147
Large conformal invariance and duality in self-dual gravity
and ž 2,1/ heterotic string theory
M. Abou Zeid, C.M. Hull
Physics Department, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK
Received 14 October 1997
Editor: P.V. Landshoff
Abstract
A system of gravity coupled to a 2-form gauge field, a dilaton and Yang-Mills fields in 2 n dimensions arises from the
Ž2,1. sigma model or string. The field equations imply that the curvature with torsion and Yang-Mills field strength are
self-dual in four dimensions, or satisfy generalised self-duality equations in 2 n dimensions. The Born-Infeld-type action
describing this system is simplified using an auxiliary metric and shown to be classically Weyl invariant only in four
dimensions. A dual form of the action is found Žno isometries are required.. In four dimensions, the dual geometry is
self-dual gravity without torsion coupled to a scalar field. In D ) 4 dimensions, the dual geometry is hermitian and
determined by a D y 4 form potential K, generalising the Kahler
potential of the four dimensional case, with the
¨
fundamental 2-form given by J˜s i) EE K. The coupling to Yang-Mills is through a term K n tr Ž F n F . and leads to a
Uhlenbeck-Yau field equation J˜i j Fi j s 0. q 1998 Elsevier Science B.V.
1. Introduction
The superstring with Ž2,1. world-sheet supersymmetry provides important insights into M-theory and
superstring theory. The target space of the Ž2,1.
string is 2 q 2 dimensional, with a null reduction
restricting the dynamics to 1 q 1 or 2 q 1 dimensions w1x. The target space dynamics has been shown
in w2–4x to describe critical string worldsheets or
membrane worldvolumes in static gauge and constitutes an explicit realisation of the scenario proposed
in w5x. Furthermore, it was found that all types of
ten-dimensional superstring theories and the elevendimensional supermembrane arise as vacua of the
Ž2,1. heterotic string. More recently, Martinec w7,8x
has argued that the Ž2,1. string may provide the
degrees of freedom needed to define certain com-
pactifications of the matrix model of M-theory proposed in Ref. w9x.
The Ž2,1. heterotic string was shown in w1x to
describe a theory of gravity with torsion coupled to
Yang-Mills gauge fields in 2 q 2 dimensions. The
null reduction mentioned above must be imposed,
and yields a 1 q 1 dimensional space or a 2 q 1
dimensional space depending on the orientation of
the null Killing vector used in the null reduction w1x.
The field equations were found in w13,14x and the
effective action for the gravitational and Yang-Mills
degrees of freedom Žbefore null reduction. was obtained in Refs. w6,12x. The geometry is a generalisation of Kahler
geometry with torsion w10x and a
¨
hypersymplectic structure w12x, and the field equations imply that the curvature with torsion is self-dual
in 2 q 2 dimensions. The Yang-Mills fields are also
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 3 7 - 8
140
M. Abou Zeid, C.M. Hull r Physics Letters B 419 (1998) 139–147
self-dual in 2 q 2 dimensions. In higher dimensions,
the field equations imply that the curvature with
torsion has SUŽ n1 ,n 2 . holonomy, while the YangMills fields satisfy a non-linear form of the Uhlenbeck-Yau equation w12x. The action in 10 q 2 dimensions is the effective space-time theory that is conjectured to give supergravity in 10 q 1 or 9 q 1
dimensions upon null reduction w6x.
Our purpose here is twofold. In Section 3, we will
formulate an equivalent form of the Ž2,1. string
action with an auxiliary metric and will show that it
is Weyl invariant only in four dimensions. In Section
4, we will dualise the vector potential that governs
the geometry to find an equivalent action given in
terms of a D y 4 form potential in D dimensions. In
four dimensions, the dual geometry is Kahler.
¨
2. (2,1) geometry
We begin by recalling the geometric conditions
for Ž2,1. supersymmetry of the Ž1,1. sigma model
with metric g i j and anti-symmetric tensor bi j w10,11x
Žsee also Refs. w14,13x; further discussion of the
geometry, isometry symmetries and gauging of the
Ž2,1. model can be found in Refs. w15–17x.. The
sigma model is invariant under Ž2,1. supersymmetry
w10,13,14x if the target space is even dimensional
Ž D s 2 n. with a complex structure Jji which is
covariantly constant with respect to the connection
with torsion G Žq. and with respect to which the
metric is hermitian, so that Ji j ' g i k Jjk is antisymmetric.
It is useful to introduce complex coordinates z a , zb
in which the line element is ds 2 s 2 g ab dz a dzb and
the exterior derivative decomposes as d s E q E. The
conditions for Ž2,1. supersymmetry imply that H is
given in terms of the fundamental 2-form
J s 12 Ji j d f i n d f j s yig ab dz a n dz
b
Ž 1.
Ž 2.
Then the condition dH s 0 implies
i EE J s 0
Ž 3.
so that locally there is a Ž1,0. form potential k s
k a dz a such that
J s i Ž Ekq Ek . .
gi j s
ž
0
g ab
gab
0
/ ž
, bi j s
0
bab
bab
0
/
Ž 5.
so that the torsion Ž2. is given by
Habg s 12 Ž g ag , b y g bg , a . , Ha bg s 0
Ž 6.
while the constraint Ž3. implies that the metric satisfies
g a w b ,gx d y gd w b ,gx a s 0.
Ž 7.
From Ž4., the geometry is defined locally by
g ab s Ea kb q Eb k a
bab s Ea kb y Eb k a .
Ž 8.
If k a s Ea K for some K, then the torsion vanishes
and the manifold is Kahler
with Kahler
potential K,
¨
¨
but if dk / 0 then the space is a hermitian manifold
of the type introduced in w10x.
The Ž1,1. supersymmetric model will be conformally invariant at one-loop if there is a function F
such that
k
R Žq.
i j y =Ž i=j.F y Hi j=kF s 0
Ž 9.
R Žq.
ij
where
is the Ricci tensor for the connection
with torsion. The curvature and Ricci tensors with
torsion are
k
Žq. k
R Žq.
y E j Gi lŽq. k q Gi mŽq. kGjlŽq. m
l i j s E i Gjl
yGjmŽq. kGi lŽq. m ,
Žq. k
R Žq.
i j sRik j .
Ž 10 .
It will be useful to define the vector
i
Õ s Hjk l J i j J k l
Ž 11 .
together with the UŽ1. part of the curvature
l Žq. k
CiŽq.
j s Jk R l i j
Ž 12 .
and the UŽ1. part of the connection
by
H s i Ž E y E . J.
In a suitable gauge, the metric and torsion potential
are then given by
Ž 4.
a
Gi Žq. s JjkGi kŽq. j s i Ž Gi aŽq. a y Gq
ia . .
Ž 13 .
In a complex coordinate system, Ž12. can be written
as
Žq.
CiŽq.
y E j Gi Žq. .
j s E i Gj
Ž 14 .
If the metric has Euclidean signature, then the holonomy of any metric connection Žincluding G Žq. . is
M. Abou Zeid, C.M. Hull r Physics Letters B 419 (1998) 139–147
141
contained in O Ž2 n., while if it has signature
Ž2 n1 ,2 n 2 . where n1 q n 2 s n, it will be in
O Ž2 n1 ,2 n 2 .. The holonomy H Ž G Žq. . of the connection G Žq. is contained in UŽ n1 ,n 2 .. It will be
contained in SUŽ n1 ,n 2 . if in addition
where g a b is given in terms of k a by Ž8.. It follows
from the form Ž5. of the metric that this action can
be rewritten as
CiŽq.
j s0
which is non-covariant but is invariant under volume-preserving diffeomorphisms.
This can be generalised to include Yang-Mills
fields A i taking values in some group G, in addition
to g i j and bi j . The A i must be a connection for a
holomorphic vector bundle Žso that the field-strength
F is a Ž1,1. form., with Chern-Simons form V Ž A.
satisfying d V s trF 2 , Bott-Chern form F w22x defined by
Ž 15 .
where the UŽ1. part of the curvature is given by Ž12..
As Ci j is a representative of the first Chern class, a
necessary condition for this is the vanishing of the
first Chern class.
It was shown in w13x that geometries for which
Gi Žq. s 0
Ž 16 .
in some suitable choice of coordinate system will
satisfy the one-loop conditions Ž9. provided the dilaton is chosen as
F s y 12 log <det g ab< ,
Ž 17 .
which implies
E iF s Õi .
Ž 18 .
Moreover, the one-loop dilaton field equation is then
satisfied for compact manifolds, or for non-compact
ones in which =F falls off sufficiently fast w12x. This
implies that H Ž G Žq. . 9 SUŽ n1 ,n 2 . and these geometries generalise the Kahler
Ricci-flat or Calabi¨
Yau geometries, and reduce to these in the special
case in which H s 0. These are not the most general
solutions of Ž9. w12x.
The condition that the connection G Žq. has
SUŽ n1 ,n 2 . holonomy can be cast as a generalised
self-duality condition on the curvature R Žq.. Defining the four-form
f i jk l ' y3 J w i j J k l x
Ž 19 .
the condition that H Ž G Žq. . 9 SUŽ n1 ,n 2 . is equivalent to
1
m n p q Žq.
R Žq.
R p qk l .
i jk l s 2 g i m g jn f
i jk l
Ž 20 .
i jk l
For D s 4, f s ye
and this is the usual
anti-self-duality condition, while for D ) 4, this is an
example of the generalised self-duality equations
considered for Riemannian manifolds in Refs. w18–
21x.
The Eq. Ž16. can be viewed as a field equation for
the potential k a . It can be obtained by varying the
action w6,12x
S s d D x <det g ab<
H (
Ž 21 .
S s d D x <det g i j < 1r4 ,
Ž 22 .
H
tr Ž F 2 . s i EEF
Ž 23 .
and a form x defined by
V Ž A . s i Ž E yE. F q d x .
Ž 24 .
Ž
.
The conditions for 2,1 supersymmetry and conformal invariance then imply the existence of a Ž1,0.
form k such that Ž4. is replaced by
J s F q i Ž Ekq Ek .
Ž 25 .
and the metric and torsion potential are given by
g ab s iFab q Ea kb q Eb k a
bab s i xab q Ea kb y Ebk a .
Ž 26 .
The field equations can again be obtained by varying
the action Ž21., but with g a b given by Ž26.. The
Yang-Mills equation is
J i j Fi j s 0.
Ž 27 .
It is sometimes useful to write the metric in terms
of a fixed background metric gˆab Že.g. a flat metric.
which is given in terms of a potential kˆ by gˆa b s
Eb kˆ a q Ea kˆb q iFa b , and a fluctuation given in terms
of a vector field Bi defined by
Ba s i Ž k a y kˆ a . , Ba s yi k a y kˆ a
ž
/
Ž 28 .
with field strength Fi j s 2 E w i Bj x. Then
g ab s gˆab q i Fab .
Ž 29 .
The action Ž21. becomes
(
S s d D x <det Ž gˆab q i Fab . < ,
H
Ž 30 .
which is similar to a Born-Infeld action and is
invariant under the abelian gauge symmetry d B s d l.
M. Abou Zeid, C.M. Hull r Physics Letters B 419 (1998) 139–147
142
3. Weyl-invariant action for (2,1) strings
with the constant T X given by
The Nambu-Goto action for the bosonic string can
be rewritten using an auxiliary world-sheet metric
w23,24x in a way which is useful for many purposes,
such as quantization w25x. Similarly, the action Ž22.
can be written in the classically equivalent alternative form
TqX s
SX s T4X d D x < g < 1r4 g i j g i j y Ž D y 4 . c ,
H
Ž 31 .
where g i j is an auxiliary metric, g s detg i j and c,T X
are Žreal. constants. The field equation for g i j is
gi j s
1
g
c ij
for D / 4, and
4
gi j s k l
g
Žg gk l . i j
Ž 32 .
Ž 33 .
for D s 4. Substituting back in Ž31. one recovers the
action Ž22. with the constant T X given by
D
T4X s 14 c 4
y1
.
Ž 34 .
In complex coordinates, the action Ž31. takes the
form
SX s T X d D x < g < g ab g ab q gab gab y Ž D y 4 . c ,
H (
Ž 35 .
where now g s detga b. The field Eqs. Ž32. or Ž33.
imply that the components g ab and g a b vanish, so
that on-shell the auxiliary metric g i j is hermitian,
JŽji g k . j s 0.
Ž 36 .
It is then consistent to impose the condition Ž36. that
the metric be hermitian off-shell as well, and we
shall do so in what follows.
The action Ž35. is a special case of the general
class of action
S
X
s TqX
Hd
D
x <g <
1r q
ij
g gi j y Ž D y q . c .
Ž 37 .
For D / q, the field equation for the auxiliary tensor
is Ž32., and substituting this back in Ž37. yields
actions of the form
S s d D x <det Ž g i j . < 1r q ,
H
Ž 38 .
1
q
D
y1
c
q
.
Ž 39 .
In the special case in which D s q, the constant
term in the action Ž37. vanishes and there is a
generalised Weyl symmetry under
gi j ™ v Ž x . gi j .
Ž 40 .
The field equation in this case is
q
gi j s k l
g .
Žg gk l . i j
Ž 41 .
For all D,q there is in addition an invariance
under volume preserving diffeomorphisms, i.e. diffeomorphisms of the D-dimensional space-time
which preserve detg i j , so that the vector field j i
generating the diffeomorphism must satisfy =i j i s 0
where =i j i s gy1r2E i Žg 1r2j i . and =i is the usual
covariant derivative for the metric g i j . For D s q,
the symmetry consists of the volume preserving diffeomorphisms, together with the Weyl transformations.
4. Duality
We now discuss the dualisation of the vector
potential k i , starting with the simplest case of four
space-time dimensions and no background metric or
Yang-Mills fields. The discussion will be generalised
below to include the background metric and the
coupling to the Yang-Mills fields.
Consider the action Ž21. and add a Lagrange
multiplier term imposing the constraint Ž8.,
S s d4 x < g <
H
(
y 14 L ab Ž g ab y Ea kb y Eb k a . q c.c.
½
5
,
Ž 42 .
with g ' det g a b Žthe sign of the Lagrange term is
arbitrary.. Eliminating L a b from Ž42., we recover
the action Ž21. subject to the constraint Ž8.. Alternatively, we can first integrate over the vectors k a , k a ,
which are Lagrange multipliers for the constraints
Ea L ab s 0, EbL ab s 0.
Ž 43 .
M. Abou Zeid, C.M. Hull r Physics Letters B 419 (1998) 139–147
In D s 4 dimensions, these can be solved locally in
terms of a scalar K :
L ab s Lab ,
ab
where L
Ž 44 .
is the ‘field strength’ of K given by
Lab ' e agbdEgEdK
Ž 45 .
and e ag b d is the antisymmetric tensor density Žwith
e 1 12 2 s 1.. Then integrating over k,k and solving as
in Ž44., the action takes the form
S s d 4 x < g < y 14 Lab g ab q c.c.
H
(
½
5
.
Ž 46 .
Note that in Ž42. we have chosen L a b to be a tensor
density so that the second term in the action is fully
diffeomorphism invariant, even though the first term
is only invariant under volume preserving diffeomorphisms. This proves to be the most convenient choice,
but equivalent results could have been obtained by
choosing L a b to transform differently, so that the
second term in Ž42. was also only invariant under
volume preserving diffeomorphisms.
Alternatively, one can add a Lagrange multiplier
term imposing the constraint Ž7.,
S s d 4 x < g < y 14 e agbd K Eg Ed g ab q c.c.
H
(
½
5
.
Ž 47 .
Integrating by parts yields
S s d 4 x < g < y 14 e agbdEg Ed Kg ab q c.c.
H
(
½
5
, Ž 48 .
which by definition Ž45. of La b is identical to action
ŽEq. Ž46... Thus it is equivalent to impose either of
the constraints Ž8. or Ž7..
The field equation for g a b which follows from
ŽEq. Ž46.. is
(< g < g
ab
s Lab .
Ž 49 .
Taking determinants in Ž49. yields the constraint
det Lab s y1
Ž 50 .
ab
Ž
.
Ž
for signature 2,2 or det L s 1 for signature 4,0..
Taking the trace gives
Lab g ab s 2 < g < .
(
Ž 51 .
Substituting Ž51. back into the action ŽEq. Ž46.., a
cancellation occurs and the action vanishes,
S s 0.
Ž 52 .
143
The dynamics is contained entirely in the constraint
Ž50.. Consider the Kahler
metric Ga b with potential
¨
K,
Gab ' EaEbK .
Ž 53 .
Then Ž50. implies
detGab s y1
Ž 54 .
for signature Ž2,2., or detGa b s 1 for signature Ž4,0..
Thus the dual metric Ga b is Kahler
and Ricci-flat.
¨
Eq. ŽEq. Ž49.. implies
g ab s V Lab
Ž 55 .
for some scalar field V , and the constraint ŽEq. Ž7..
will be satisfied if ŽEq. Ž54.. and
G abEa Eb V s 0
Ž 56 .
hold. Then V is a harmonic scalar on the dual space.
Writing Ga b s ha b qEa Eb w where ha b is a flat
background metric, Ž54. becomes the following
equation for w :
det
ž
1 q E 1E1 w
E 1E2 w
E 2E1 w
y1 q E 2E2 w
/
s1
Ž 57 .
in the notation of w1x. This equation and ŽEq. Ž56..
can be derived from the action
1
EwEw q wEw n EEw q 'G G abEa VEb V , Ž 58 .
3!
where the first term is the Plebanski action Žalso with
the notation of w1x. and the second term is such that
ŽEq. Ž56.. is the field equation obtained by varying
V and using the constraint ŽEq. Ž54... The action
ŽEq. Ž58.. can be thought of as the dual action.
We have thus established the following remarkable result. We started with the theory of hermitian
gravity with torsion in four dimensions defined by
the action Ž21., the field equations of which implied
that the curvature with torsion was anti-self-dual and
with holonomy SUŽ2. Žfor signature Ž4,0.. or SLŽ2, R .
Žfor signature Ž2,2... We then dualised this to obtain
anti-self-dual Riemannian gravity coupled to a harmonic scalar V , with no torsion and the action ŽEq.
Ž58... Thus in four dimensions, a theory with torsion
is related by a conformal rescaling to a theory without torsion. This is in agreement with the results of
w33,34x. We emphasize that this duality Žunlike the
dualities considered e.g. in w30,11,31,32x do not require any Killing vectors.
H
H
M. Abou Zeid, C.M. Hull r Physics Letters B 419 (1998) 139–147
144
The generalisation to other Ževen. dimensions is
straightforward. Consider the action
S s d D x < g < y 14 L ab Ž g ab y Ea kb y Eb k a .
(
H
½
qc.c. 4 .
Ž 59 .
Eliminating L a b from Ž59., we recover the action
Ž21.. Alternatively, integrating out k a , k a gives the
constraints Ž43.. The solution to Ž43. in D s 2 n
dimensions is
L
ab
ab
sL ,
We now reinstate the fixed background gˆa b,
which will be taken to be of the form gˆa b s Eb kˆ a q
Ea kˆbˆ q iFa b and includes the coupling to Yang-Mills
fields, through Fa b. As a result of Ž23., this background metric satisfies
gˆa w b ,gx d y gˆd w b ,gx a s y4Fa w b Fgx d .
Consider the action
Ss dD x < g <
y 14 L ab Ž g ab y gˆab y Ea kb y Eb k a . q c.c.
½
where
H
(
½
5
,
Ž 62 .
with L given by Ž61.. The field equation for g a b is
(< g < g
ab
s Lab .
Ž 63 .
Eliminating L a b from Ž70., we recover the action
Ž21. Žafter shifting the potentials k ™ k y kˆ .. The
vectors are Lagrange multipliers for the constraints
Ž43., which can be solved locally in terms of a
D y 4 form K as in Ž60.. On integrating out the
vectors, the action takes the form
S s d D x < g < y 14 Lab Ž g ab y gˆab . q c.c.
(
H
½
Ž 64 .
Contracting Ž61. with g a b yields
1
ab
ab ny2
L g ab s n det L
.
It is easily checked that the solution of the field Eq.
Ž63. is of the form
n
Ž 66 .
where m and n are constants given by
ms1 , nsy
1
.
Ž 67 .
ny2
Substituting Ž64. and Ž65. into Ž62. gives the dual
action
1
S s y 12 Ž n y 2 . d D x det Lab
H
ny2
,
where La b is given in Ž61.. Using the field equation
for g a b, taking the determinant and the trace and
substituting back into the action Ž71., we find the
dual action in the form
Ž 65 .
g ab s m det Lab Lab ,
5
Ž 71 .
1
ab ny2
.
Ž 70 .
Taking determinants in Ž63., we find
(< g < s det L
5
Ž 61 .
is the ‘field strength’ of an Ž n y 2,n y 2. form K.
The action then takes the form
S s d D x < g < y 14 Lab g ab q c.c.
(
H
Ž 60 .
Lab ' e ag 1 . . . g ny 1 bdd1 . . . ny 1 Eg 1Ed1 Kg 2 . . . g ny 1dd2 . . . ny 1
Ž 69 .
Ž 68 .
for the field strength L of the D y 4 form potential
K. Again, we have chosen the Lagrange multiplier to
be a tensor density so that the Lagrange multiplier
term in the action is coordinate invariant.
1
Ss
1
4
Hd
D
ab n y2
ab
x L gˆab qc.c.y2 Ž ny2. det L
.
Ž 72 .
Ž n / 2.. In the absence of Yang-Mills fields, Fa b s
0, then the term Li j gˆ i j in Ž72. vanishes after integration by parts, as a result of the form Ž61. of La b and
the fact that the background metric gˆa b satisfies Ž7..
If the Yang-Mills fields do not vanish, then using
the form Ž61. of La b, integrating by parts and using
Ž69., we obtain
1
1
2
D
ab ny2
S s y Ž n y 2 . d x det L
H
y 12 K n tr Ž F n F . ,
H
Ž 73 .
with an interesting coupling of the D y 4 form potential to F n F.
M. Abou Zeid, C.M. Hull r Physics Letters B 419 (1998) 139–147
145
For n s 2, integrating out the metric in Ž71. gives
the constraint
Then the dual geometry is given in terms of the
fundamental two-form
det Lab s 1
J˜s yig˜ab dz a n dz b
Ž 74 .
for signature Ž2,2., or det La b s y1 for signature
Ž4,0., while the action reduces to the term HK n tr Ž F
n F .. The constraint Ž74. can be imposed via a
Lagrange multiplier L, so that the action becomes
S s y 12 K n tr Ž F n F . y 12 d D x L Ž det Lab y 1 . .
H
H
Ž 82 .
by
J˜s i) EEK ,
Ž 83 .
where the Hodge star operation is defined with respect to the metric Ž80.. The dual action Ž73. Žfor
n / 2. can also be expressed in terms of the dual
metric Ž80. and we find
Ž 75 .
Integrating out the scalar L yields the constraint
Ž74., so that one recovers the action HK n Ž F n F .
subject to this constraint. Instead we keep the Lagrange multiplier; using Ž23., Ž61. and integrating by
parts, we find the following field equation for K
EE Ž iF y Ldet Ž Lab . Ly1 . s 0,
Ž 76 .
where Ly1 is the 2-form Ž Ly1 .ab dz a n dz b . This
implies that
L Ž det L . Ly1 s iJ X ,
Ž 77 .
where
X
J X s i Ž Ek q Ek X . q F
Ž 78 .
X
(
Ls"
det L
,
Ž 79 .
which can be substituted back in Ž75..
It is useful to define a dual metric g˜a b Žfor n / 1.
by
1
ab ny1
g˜ab ' det L
Ž Ly1 . ab ,
Ž 80 .
so that
ij
(
D
S s y Ž n y 2 . d x det g˜ab
H
ny2
y 12 ) J˜n F
H
Ž 84 .
using Eqs. Ž23. and Ž83.. The constraint Ž83. defines
a class of hermitian geometries Žwithout torsion. in
which the metric is given in terms of a D y 4 form
potential instead of the scalar potential of Kahler
¨
geometry. The action Ž84. gives a field equation for
such metrics which arises naturally from the dualisation of Ž2,1. geometry.
For n s 2, the Kahler
form JG s i EEK corre¨
sponding to the Kahler
metric Gab defined in Ž53. is
¨
˜ JG s ) J,˜ so that the action becomes
dual to J,
S s y 12 ) J˜n F s y 12 JG n F
H
X
for some Ž1,0. form potential k , so that J is the
X
2-form corresponding to a metric g ab
defining some
Ž
.
dual 2,1 sigma-model. Although Ž77. is not algebraic in K, one can solve for L as a functional of K,
kX and F; taking determinants in Ž77., we find
det Ž g X .
ny 1
1
2
H
Ž 85 .
subject to the constraint Ž83.; this is the Donaldson
action w27x for self-dual Yang-Mills in a self-dual
geometry.
The field equation for the Yang-Mills fields derived from the action Ž84. is
J˜i j Fi j s 0.
Ž 86 .
Thus F satisfies the Uhlenbeck-Yau equation with
respect to the complex structure J˜i j . This can be
derived, for example, by transforming with a complex gauge transformation with parameter h taking
values in the complexification Gc of the gauge group
G w12x,
F s hy1 fh,
Ž 87 .
where
ij
L s det g˜ i j g˜ .
Ž 81 .
f s da q a2 s Ea
Ž 88 .
146
M. Abou Zeid, C.M. Hull r Physics Letters B 419 (1998) 139–147
is the field strength of a holomorphic connection
given by the Ž1,0. form a s Uy1E U. Then using
trF 2 s tr f 2 and varying with respect to the prepotential U gives
K n F n F and gives rise to the Uhlenbeck-Yau-type
field Eq. Ž86..
EEK n f q EK n E f q EK n w a, f x s 0
Ž 89 .
Ž
.
Ž
.
which gives 86 on using 83 and the Bianchi
identity.
The field equations for the dual metric g˜ i j and its
implications for the dual geometry will be discussed
elsewhere. Note that the dualisation procedure carried out in the foregoing can also be applied to the
actions of Section 3 with an auxiliary metric; the
results are equivalent.
Acknowledgements
We would like to thank Emil Martinec for helpful
comments. M.A. would like to thank Jose´ FigueroaO’Farrill for useful discussions. The work of M.A. is
supported in part by the Overseas Research Scheme
and in part by Queen Mary and Westfield College,
London. C.M.H. is supported by an EPSRC Senior
Felowship.
5. Conclusion
References
Summarizing, the Ž2,1. sigma-model or string give
rise to a theory of gravity coupled to a two-form
gauge field, a dilaton and Yang-Mills gauge fields in
D s 2 n dimensions. The field equations imply that
the curvature with torsion is self-dual in four dimensions, or has SUŽ n. holonomy in 2 n dimensions.
The system is described by the Born-Infeld type
action Ž21., where g a b is given in terms of k a by
Ž8.. This action can be simplified using an auxiliary
metric, and the forms Ž35. and Ž31. are classically
equivalent to Ž30. and Ž22. respectively. The four-dimensional action Ž31. is classically invariant under
diffeomorphisms preserving the volume element
constructed from the auxiliary metric and under the
generalised Weyl transformation Ž40.. It would be
interesting to compare this symmetry to the infinite
dimensional current algebra w26x of the Donaldson
action for self-dual Yang-Mills w27,28x.
The action Ž30. can be dualised, with no isometries being required. The dual theory in four dimensions is self-dual gravity without torsion coupled to
a scalar. This recovers the remarkable equivalence
between self-dual hermitian geometries with torsion
and self-dual gravity without torsion. In higher dimensions, dualising gives an aparently new generalisation of Kahler
geometry, in which the metric g˜ i j is
¨
hermitian and is determined by the Ž n y 2,n y 2.
form potential K Žwhich can be thought of as analogous to the scalar potential of Kahler
geometry. via
¨
Ž61. and the dynamics is described by the action
Ž84.. The couplng to Yang-Mills is via the term
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12 February 1998
Physics Letters B 419 Ž1998. 148–156
The rigid symmetries of bosonic D-strings
Friedemann Brandt 1, Joaquim Gomis 2 , Joan Simon
´
3
Departament ECM, Facultat de Fısica,
UniÕersitat de Barcelona and Institut de Fısica
d’Altes Energies, Diagonal 647,
´
´
E-08028 Barcelona, Spain
Received 15 July 1997
Editor: R. Gatto
Abstract
We analyse the classical symmetries of bosonic D-string actions and generalizations thereof. Among others, we show that
the simplest actions of this type have infinitely many nontrivial rigid symmetries which act nontrivially and nonlinearly both
on the target space coordinates and on the UŽ1. gauge field, and form a Kac-Moody
version of the Weyl algebra Žs
ˇ
Poincare´ algebra q dilatations.. q 1998 Elsevier Science B.V.
PACS: 11.25.-w; 11.30.-j
Keywords: D-string; Rigid symmetries; Born-Infeld actions; Kac-Moody algebras
1. Introduction
Much progress has been made lately in constructing k-invariant actions for D-p-branes w1–3x, generalizing earlier work w4x. Typically, the ‘‘bosonic
part’’ of these actions is of the Born–Infeld type,
such as
(
S p s d pq 1s <det Ž Gmn q Fmn . < ,
H
Gmn s hm n Em x m P En x n ,
Fmn s Em an y En am .
Here, the x
1
m
are target space coordinates, hm n is a
E-mail address: [email protected].
E-mail address: [email protected].
3
E-mail address: [email protected].
2
Ž 1.
flat target space metric, and am is an abelian gauge
field living in the world volume.
Important properties of actions are of course their
symmetries. In particular one may ask: What are the
rigid and gauge symmetries of Ž1.? To what degree
is this action determined by symmetries alone? In
this letter we analyse these questions for the case
p s 1. Our results apply not only to the action Ž1. but
also to generalizations thereof which will be given
below. We obtained these results by an analysis of
the BRST cohomology which will be given in w5x.
Although the cohomological analysis parallels
quite closely the one for bosonic strings carried out
in w6x, the results are surprisingly rather different. For
instance, while the usual bosonic string in a flat
target space has only finitely many rigid symmetries
before gauge fixing w6x, we will show that, for p s 1,
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 4 3 - 3
F. Brandt et al.r Physics Letters B 419 (1998) 148–156
the action Ž1. has infinitely many nontrivial rigid
symmetries. Among them there are of course the
obvious Poincare´ symmetries which reflect the
isometries of the target space and coincide with the
rigid symmetries of the bosonic string. However we
find also previously unnoticed rigid symmetries
which are nonlinearly realized and transform both
the x m and the gauge field am . Together with the
familiar Poincare´ symmetries, the new symmetries
form a Kac–Moody
version of the Weyl algebra Žs
ˇ
Poincare´ algebra q dilatations.. We stress that these
symmetries are present already before fixing a gauge.
They should therefore not be confused with the
Kac–Moody
symmetries of sigma models discussed
ˇ
e.g. in w7x as the latter emerge just as residual
symmetries of Weyl and diffeomorphism invariant
actions in suitable gauges.
The fact that the new symmetries transform am
nontrivially has remarkable consequences. In particular there are symmetry transformations which map
solutions of the equations of motion with trivial
gauge field Žzero or pure gauge. to other solutions
with nonvanishing field strength Fmn , and thus usual
bosonic strings to D-strings. As the field strength
contributes to the string tension w8x, the new symmetries therefore also relate strings with different tension and might thus be viewed as ‘‘stringy symmetries’’. It is striking that these properties of the new
symmetries are similar to those of dualities w9,10x
relating bosonic and D-strings. One may speculate
whether the new symmetries reflect part of the symmetry structure of an underlying Ž‘‘M’’. theory. As
they are nonlinearly realized, one might for instance
suspect that they emerge somehow as broken symmetries of that theory.
149
where D s <detŽ Dmn .<. In this formulation, the Dmn are
auxiliary fields Ž D mn denotes the inverse of Dmn ..
Eliminating them, one recovers Ž1..
The action with Lagrangian Ž2. is evidently invariant under world-volume diffeomorphisms and
abelian gauge transformations of am . For p s 1 it is
in addition gauge invariant under Weyl transformations of Dmn and we can decompose the latter according to
'
p s 1: Dmn s gmn q g emn w
Ž 3.
with
e 21 s e 12 s 1
gmn s gnm , g s ydet Ž gmn . ,
where we assumed for definiteness that gmn has
Lorentzian signature. Since g emn behaves as a
covariant 2-tensor field under world-sheet diffeomorphisms and has the same Weyl weight as gmn , w
transforms as a scalar field under diffeomorphisms
and is Weyl invariant. Using Ž3., the Lagrangian Ž2.
for p s 1 reads
'
L1 s
1
2
Ž1 y w 2 .
y1 r2
ž 'g g
mn
Gmn y w e mn Fmn .
/
Ž 4.
We are now looking for generalizations of Ž4.,
guided by its field content and gauge symmetries.
Since the gauge symmetries of Ž4. treat w on an
equal footing with the x m , we can treat w has zeroth
coordinate of an extended target space with coordinates X M ,
X M 4 s w , x m4 , w ' X 0.
2. Actions
To motivate and explain our approach, we note
that Ž1. can be cast in a more convenient form
w11,12x with Lagrangian
Lp s
1
2
'D
D
mn
Ž Gmn q Fmn . y Ž p y 1.
Ž 2.
Furthermore we allow for a set of abelian gauge
fields amI rather than only one such gauge field, and
fix the field content of the models to be studied to
f i 4 s gmn , amI , X M 4 .
Ž 5.
In addition to this field content, we impose gauge
invariance under world-sheet diffeomorphisms, Weyl
transformations of the gmn , and abelian gauge trans-
F. Brandt et al.r Physics Letters B 419 (1998) 148–156
150
formations of the amI . Infinitesimally these gauge
transformations read
We note that Ž7. covers for instance also actions
with Lagrangian
dgmn s ´ rEr gmn q 2gr Ž n Em . ´ r q lgmn ,
L s y det Gˆmn q Fmn y y det Gˆmn
d amI s ´ nEn
which are the counterparts of the Lagrangian considered orignally by Born and Infeld w13x and are
obtained analogously by choosing G m n s
g m nŽ x . w f Ž w . y 1x and the remaining functions as in
Ž8..
amI q anI Em ´ n q Em
(
I
L ,
d X M s ´ mEm X M
Ž 6.
where ´ m, l and L I parametrize world-sheet diffeomorphisms, Weyl transformations and gauge transformations of the amI respectively. Our first result is
that, up to a total derivative, the most general Lagrangian which is Ža. constructible solely of the
fields Ž5., Žb. local Žs polynomial in derivatives of
any order., and Žc. up to a total derivative invariant
under the gauge transformations Ž6., is
Ls
1
2
qe
1
2
ž
/
Ž 10 .
3. Rigid symmetries
Our second result is that the nontrivial rigid symmetries of an action with Lagrangian Ž7. are generated by transformations
BM N Ž X . Em X P En X
N
Dgmn s 0,
qamJ CJI Ž X .
Ž 7.
where the GM N , BM N and DI are arbitrary functions
of the X M. This result is the analogue of a similar
one holding for bosonic strings w6x and will be
proved in w5x. Note that Ž7. covers in particular
D-string actions of a general form, if we choose
(
(
Ž 8.
where f Ž w . is Žalmost. arbitrary Žthis arbitrariness
reflects the freedom of field redefinitions w ™ w˜ Ž w ...
Indeed, upon elimination of gmn and w , Ž8. yields
Born–Infeld actions generalizing Ž1. among others to
curved target spaces:
S1 s d 2s y det Gˆmn q Fmn ,
ž
Ž 11 .
X .,
AMI Ž X .,
BMI Ž X .,
CJI Ž X .
LX GM N s y2 AŽIM E N . DI ,
Ž 12 .
LX BM N s 2 E w N YM x q 2 BwIN E M x DI ,
Ž 13 .
J
CI E M DJ
LX E M DI s yC
E m s g mnEn ,
Bm0 s 0, Bm n s bm n Ž x . f 2 Ž w . y 1 ,
DI s d I Ž x . f 2 Ž w . y 1
with functions X
solving
MŽ
Ž 14 .
for some functions YM Ž X .. Here LX denotes the
standard Lie derivative along X , and we used
GM 0 s 0, Gm n s g m n Ž x . f Ž w . ,
(
(
D amI s y'g emn AMI Ž X . E n X M q BMI Ž X . Em X M
M
q DI Ž X . Em anI
H
/
DX M sX M Ž X . ,
'g g mn GM N Ž X . Em X M P En X N
mn
ž
/
Gˆmn s g m n Ž x . Em x m P En x n ,
Fmn s d I Ž x . Ž Em anI y En amI . q bm n Ž x . Em x m P En x n .
Ž 9.
E M s ErE X M .
Note that Eqs. Ž12. – Ž14. generalize the familiar
Killing vector equations for the target space. The
general form of the latter Žwith nonvanishing BM N .
was discussed in w7,6x and arises from Ž12. – Ž14. for
DI s 0 Žas DI s 0 reproduces the usual bosonic
string, we thus recover for this case the result of w6x
for the rigid symmetries of the bosonic string.. We
will solve these equations explicitly for specific
models in the next section. In w5x we will prove that
the above symmetries exhaust the nontrivial rigid
symmetries of an action with Lagrangian Ž7. 4 . Here
4
The rigid symmetries are obtained from the BRST cohomology at ghost number y1 w14x.
F. Brandt et al.r Physics Letters B 419 (1998) 148–156
we only note that, under transformations Ž11. satisfying Ž12. – Ž14., the Lagrangian Ž7. indeed transforms
into a total derivative as can be easily verified,
YM En X M q anI X ME M DI q DI D anI . .
D L s e mnEm Ž yY
Ž 15 .
The conserved Noether currents j m corresponding to
the symmetries Ž11. are now readily computed,
j m s g g mn GM N X MEn X N
'
qe mn Ž Y˜M En X M y anI X ME M DI .
Ž 16 .
where
Y˜M s YM y BM N X N .
In order to complete the above statements about
the rigid symmetries, we note that the solutions to
the generalized Killing vector Eqs. Ž12. – Ž14. are
determined only up to the redefinitions
AMI ™ AMI q E w I J xE M DJ ,
BMI ™ BMI q E M B I q E Ž I J .E M DJ ,
YM ™ YM q E M Y y B EI M DI
Ž 17 .
where the B I Ž X ., E I J Ž X . and Y Ž X . are arbitrary
functions of the X M. These redefinitions drop out of
Ž12. – Ž14. and affect in Ž11. only the transformations
of the amI according to
D amI ™ D amI q Em B I q EmnI J e nrEr DJ ,
EmnI J s y 1rg gmn E w I J x q emn E Ž I J . .
'
Ž 18 .
These are irrelevant redefinitions of the rigid symmetries, i.e. two rigid symmetries are identified if they
coincide up to such redefinitions. Namely Em B I are
just special gauge transformations, while EmnI J e nrEr DJ
are on-shell trivial symmetries. The latter holds due
EnmJI and e nrEr DJ s d Srd anJ where S
to EmnI J s yE
denotes the action with Lagrangian Ž7..
It is easy to check that the commutator of two
symmetries Ž11. is again a symmetry of this type,
w D1 , D 2 x s D 3 .
Ž 19 .
Namely, using Ž11. and the notation Di X M s Xi M
151
etc. Ži s 1,2,3., a direct computation of w D1 , D2 x
yields
X 3M s X1NE N X 2M y Ž 1 l 2 . ,
A3I M s LX 1 A2I M y C1IJ A2JM y Ž 1 l 2 . ,
B3I M s LX 1 B2I M y C1IJ B2JM y Ž 1 l 2 . ,
C3IJ s LX 1C2I J y Ž 1 l 2 . .
Ž 20 .
Using standard properties of Lie derivatives such as
w LX , LX x s LX , it is easy to verify that the set of
1
2
3
functions Ž20. solves Ž12. – Ž14. with Y 3 M s
L X 1 Y 2 M y L X 2 Y 1 M w henever the sets
Ž Xi M , AiIM , BiIM ,C
Ci IJ , Yi M ., i s 1,2 solve Ž12. – Ž14.
too. In that sense, the commutators of symmetry
transformations Ž11. ‘close’. However, this does not
necessarily imply that the algebra of the rigid symmetries closes off-shell in a particular basis of the
rigid symmetries. Namely suppose that D1 and D2
are two elements of such a basis. Their commutator
Ž19. will in general be a linear combination of
elements of the basis only up to redefinitions Ž18..
Hence, in general the algebra of the elements of the
basis will close only up to gauge transformations and
on-shell trivial transformations of the type occurring
in Ž18..
To summarize, the nontrivial Žinfinitesimal. rigid
symmetries of an action with Lagrangian Ž7. are
exhausted by transformations Ž11. with target space
functions satisfying Ž12. – Ž14., and defined modulo
the redefinitions Ž17.. Furthermore, any solution to
Ž12. – Ž14. which does not vanish modulo redefinitions Ž17. gives rise to a nontrivial rigid symmetry
generated by Ž11.. Hence, one has precisely to solve
Ž12. – Ž14. in order to find all rigid symmetries of an
action with Lagrangian Ž7.. A basis of the rigid
symmetries is obtained from a basis of solutions to
Ž12. – Ž14., i.e. from a complete set of solutions
which are linearly independent up to redefinitions
Ž17.. Needless to say that, on general grounds, rigid
symmetries of Born–Infeld actions Ž9. arise from
those of the corresponding actions with Lagrangians
Ž7., Ž8. by replacing the auxiliary fields gmn and w in
D x m and D amI with a solution to their algebraic
equations of motion, i.e. by substituting for instance
(
f Ž w . ™ det Gˆmn rdet Gˆmn q Fmn ,
gmn ™ Gˆmn .
ž
/
ž
/
Ž 21 .
Note that gmn is actually defined by the equations of
F. Brandt et al.r Physics Letters B 419 (1998) 148–156
152
motion only up to a completely arbitrary function
multiplying Gˆmn due to the Weyl invariance of Ž7..
As this general function drops out of the transformations Ž11., one can indeed choose Ž21. with no loss
of generality.
To illustrate and interpret the general results presented above, we will now solve Ž12. – Ž14. explicitly
for a specific class of models and discuss the corresponding rigid symmetries. The models are characterized by Lagrangians Ž7. involving only one UŽ1.
gauge field am , with
Gm n s f Ž w . hm n ,
B0 m s 0,
Bm n s Bm n Ž w . ,
X Ž p. m s y Ž f Xr2 f . X 0 x m
Ž 24 .
AMŽ p., BMŽ p.
m
Ž p. m
4. Examples and discussion
G 0 M s 0,
AM , BM and YM . The general solution is then the
sum of a particular solution and the general solution
of the homogeneous equations. A particular solution
is given by
and corresponding expressions for
and
YMŽ p. obtained from Ž23. for X ™ X
. The
homogeneous Eqs. Ž12. and Ž13., obtained by setting
X 0 s 0, yield, for Ž M, N . s Ž m,n., Ž M, N . s Ž m,0.
and Ž M, N . s Ž0,0. respectively,
hn k Em X Ž h. k q hm k En X Ž h. k s 0,
hm n Ž X
Ž h. n X
X
AmŽ h. D , 0 s A0Ž h. D ,
. f s yA
En Y˜mŽ h. y Em Y˜nŽ h. s 0,
X
D s DŽ w . .
Ž 22 .
Recall that the special choice Bm n s 0, D s Ž f 2 y
1.1r2 reproduces the action Ž1. for p s 1. We will
first show that, for any choice of f and D / constant,
the general solution of Eqs. Ž12. – Ž14. is, up to
redefinitions Ž17.,
Ž 25 .
X
Ž 26 .
Ž 27 .
X
Ž ỸYmŽ h. . y Em Y0Ž h. y BmŽ h.D
q BmX n X Ž h. n s 0
Ž 28 .
where we used that Bm n depends only on w and
defined
Y˜mŽ h. s YmŽ h. y Bm n X Ž h. n .
X 0sX 0Ž w . ,
Ž25. are just the Killing vector equations for a flat
space with coordinates x m and thus have the general
solution
X m s y Ž f Xr2 f . X 0 x m q a m Ž w . q a m n Ž w . x n ,
X Ž h. m s a m Ž w . q a m n Ž w . x n , a m n s ya n m .
a m n s ya n m ,
X
Am s yhm n Ž frDX . Ž X n . ,
Ž26. can be solved for the AMŽ h. and thus determines
directly these functions. Ž27. implies Y˜mŽ h. s Em Y
for some Y Ž X .. Using this in Ž28., we get BmŽ h. s
Em B q BmX n X Ž h. nrDX with B s Ž Y X y Y0Ž h. .rDX ,
and thus also Y0Ž h. s Y X y BDX . Furthermore, we
have the trivial identity B0Ž h. s BX q E DX with E s
Ž B0Ž h. y BX .rDX . Now, contributions E M Y y BE M D
and E M B q EE M D to YM and BM respectively can
be removed by redefinitions Ž17.. Without loss of
generality, we can thus choose
Bm s Ž 1rDX . BmX n X n q
1
2
A0 s 0,
X
Ž BmX n X 0 . x n
,
B0 s 0,
X
C s y Ž X 0 DX . rDX ,
1
Ym s
2
BmX n X 0 x n q Bm n X n ,
Y0 s 0,
Ž 23 .
where X 0 Ž w ., a m Ž w . and a m n Ž w . are arbitrary
functions of w and we used
X
' ErEw ,
n
x m ' hm n x .
Let us now sketch the derivation of Ž23.. The results
for X 0 and C follow immediately from Ž14. as it
reads in the cases under study DXEm X 0 s 0 for M s
C DX for M s 0. The remaining
m, and Ž X 0 DX .X s yC
Ž
results follow from 12. and Ž13.. To show this, we
regard Ž12. and Ž13., for any fixed function X 0 Ž w .,
as a set of inhomogeneous equations for the X m ,
Ž 29 .
YmŽ h. s Bm n X Ž h. n , Y0Ž h. s 0,
BmŽ h. s BmX n X Ž h. nrDX , B0Ž h. s 0.
Ž 30 .
Altogether this yields Ž23..
Let us now discuss the symmetries Ž11. arising
from Ž23.. As they involve arbitrary functions
X 0 Ž w ., a m Ž w . and a m n Ž w ., we conclude immediately that any model characterized by Ž22. possesses
infinitely many nontriÕial rigid symmetries. To interpret them, we will use the equations of motion, as
rigid symmetries map in general solutions to the
F. Brandt et al.r Physics Letters B 419 (1998) 148–156
equations of motion to other solutions. The equations
of motion for gmn are solved for instance by gmn s Gmn
with Gmn as in Ž1.. The equation of motion for am
yields
e mnEn D Ž w . s 0 ´ w s w 0 s constant
Ž 31 .
i.e. w is on-shell just a constant fixed by initial
conditions. The value of this constant distinguishes
thus partly different solutions. Furthermore it controls among others the coupling of the gauge field to
the string, as the equations of motion for w and x m
yield respectively
e mn Ž Fmn q Bmn . s K Ž w 0 . 'G ,
Ž 32 .
Em Ž 'G G mnEn x m q B˜m n Ž w 0 . e mnEn x n . s 0
Ž 33 .
where we have defined
G s ydet Ž Gmn . ,
Bmn s Bm n Ž w 0 . Em x m P En x n ,
B̃m n Ž w 0 . s Bm n Ž w 0 . D Ž w 0 . rf Ž w 0 . ,
K Ž w 0 . s y2 f X Ž w 0 . rDX Ž w 0 . .
Note that Ž33. are nothing but the equations of
motion for an ordinary bosonic string with constant
B˜m n and that Ž32. relates the abelian gauge field to
this string. Now, D x m reads
D x m s aŽ w . x m q a m Ž w . q a m n Ž w . x n
Ž 34 .
X 0 f Xr2 f. As w is constant for any
where a s yX
solution of the equations of motion, Ž34. generates
on-shell Poincare´ transformations and dilatations 5 of
the target space coordinates. The important property
of the new symmetries is that they transform in
addition the gauge field am nontrivially. In particular, for a transformation Ž34. which generates on-shell
a dilatation of x m , we get in the case Bm n s 0:
Bm n s 0, D x m s a Ž w . x m
´ D am s emn A Ž w . 'G G nr x mEr x m q C Ž w . am
Ž 35 .
X
X
X
X X
X
where A s a frD and C s Ž2 afD rf . rD . Now,
even for am s 0, Ž35. does in general not reduce to a
5
These dilatational symmetries should neither be confused with
the world-sheet Weyl invariance of Ž4. and the world-volume
Weyl invariance of certain formulations of the Žsuper. p-brane,
nor with the linearly realized global target space scale invariance
of the formulation w15x treating the string tension of a p-brane as a
dynamical variable.
153
gauge transformation Žnot even on-shell!.. In particular, it maps thus in general a solution to the equations of motion with am s 0 to another solution with
nonvanishing field strength Fmn . Indeed, Ž32. shows
that solutions with vanishing Fmn correspond in the
case Bm n s 0 to special values of w 0 , namely roots
of the function K,
Bm n s 0: Fmn s 0 l K Ž w 0 . s 0.
As transformations Ž35. are accompanied by transformations Dw s X 0 Ž w . Žrecall that a s
X 0 f Xr2 f ., we conclude: in models with Bm n s 0,
yX
any transformation Dw s X 0 Ž w . which changes the
value of K Ž w 0 . from 0 to a nonvanishing one, is
accompanied by a transformation Fmn s 0 ™ Fmn /
0! Completely analogous considerations apply of
course to models with Bm n / 0.
Let us now discuss the off-shell algebra of the
symmetries arising from Ž23.. As the transformations
Ž34. can be regarded as w-dependent Poincare´ transformations and dilatations of the target space coordinates, their algebra will in any basis be a Kac–Moody
ˇ
version of the Weyl algebra. A basis is obtained by
choosing a suitable basis for the functions X 0 Ž w .,
a m Ž w . and a m n Ž w . occurring in Ž23., adapted to the
properties Že.g. boundary conditions, topology. of
the specific model one wants to study. To give an
explicit example, we consider the case
f Ž w . s exp Ž w .
and functions X 0 Ž w ., a m Ž w . and a m n Ž w . which can
be expanded in integer powers of expŽ w ., i.e.
X 0 Ž w . s c a eya w ,
a m Ž w . s c am eya w , a m n Ž w . s c am n eya w
where the c’s are constant infinitesimal transformation parameters indexed by a g Z, and summation
over a is understood. We now decompose D according to
1
D s c a La q c am Pma q c am n Mma n
2
where La, Pma and Mma n s yMnam are the generators
of rigid symmetries, the algebra of which we want to
compute. Ž11. and Ž23. yield
1
Law s eya w , La x m s y eya w x m ,
2
Pma w s 0, Pma x n s eya wdmn ,
Mma n w s 0,
Mma n x k s eya w Ž dmk x n y dnk x m . .
F. Brandt et al.r Physics Letters B 419 (1998) 148–156
154
Analogously one determines readily the transformations of am . For instance, one gets
differ only by Ym ™ Ym q Em Y . Then solutions to
Ž12. – Ž14. are given by
Laam s Z a Ž w . g emn E n Ž x m x m .
X m s c A Ž w . zAm Ž x . ,
'
Ym s c A Ž w . Ym A Ž x . ,
Zman Ž w . x nEm x m q C a Ž w . am
qZ
Am s y Ž frD . g m n Ž X . ,
X
Bm s Ž hrD . Ž Ym y bm n X
Z a s a e Ž1ya. wr Ž 4 DX . ,
C a s eya w Ž a y DXXrDX . .
It is now very easy to compute the symmetry
algebra on the X M. On the gauge field it is more
involved, but by means of the general arguments
given in the previous section one concludes that the
algebra coincides necessarily on all fields up to
gauge transformations and on-shell trivial symmetries of the type occurring in Ž18.. If the latter are
present, the algebra is open. This turns out to be the
case in general. However, at least for Bm n s 0 the
algebra closes off-shell even on am and reads
w La , Lb x s Ž a y b . Laq b ,
ž
2
y b Pmaqb ,
/
Pma , Pnb s 0,
aqb
Mma n , M pbq s 2hpw m Mnxq
y Ž plq. .
This is what we call a Kac–Moody
version of the
ˇ
Weyl algebra.
Let us briefly point out an immediate generalization of the above results to models characterized by
Gm n s f Ž w . g m n Ž x . ,
D s DŽ w . ,
B0 m s 0,
Ž 36 .
i.e. we allow now of curved target space metrics
g m n Ž x . and x-dependent Bm n . Suppose that
zAm Ž x ., Ym AŽ x .4 is a basis of inequivalent solutions
to
LzA g m n s 0,
B0 s C s 0
Lz A bm n s 2 E w n Ymx A
where c A Ž w . are arbitrary functions. We conclude
that any Killing vector field zAŽ x . of the target space
satisfying Ž37. gives rise to infinitely many rigid
symmetries of the model characterized by Ž36.. The
algebra of these symmetries is a Kac–Moody
version
ˇ
of the Lie algebra of the LzA and generalizes the
Poincare´ Kac–Moody
algebra found above. Whether
ˇ
there is also a generalization of the dilatational symmetries, depends on g m n and bm n and we intend to
elaborate on this point elsewhere.
5. Supersymmetric extensions
'
Mma n , Pkb s hk m Pnaqb y Ž m l n . ,
Bm n s h Ž w . bm n Ž x . ,
.,
One might wonder whether there are supersymmetric extensions of the symmetry structure presented above. We have not investigated this question
in detail by means of a cohomological analysis.
However we have found such extensions in simple
cases. For instance, consider the Lagrangian
1
L s f Ž w . g g mnPmmPn n hm n q D Ž w . e mnEm an
2
Ž 39 .
La , Mmb n s yb Mmaqb
n ,
G 0 M s 0,
A0 s 0,
n X
Ž 38 .
Zman s BmXX n y Ž 1 q a . BmX n eya wr Ž 2 DX . ,
1
Y0 s 0,
n X
X
where
La , Pmb s
X 0 s 0,
Ž 37 .
where two solutions are called equivalent if they
where, as in Ž22., f and D / constant are any
functions of w , and, using the conventions and notation of w2x,
Pmm s Em x m y u G mEm u .
Ž 40 .
The action with Lagrangian Ž39. is invariant under
the following rigid supersymmetry transformations
Qu a s c a B Ž w . ,
Qx m s c G mu B Ž w . ,
Qam s 2 c G mu g emn P mn BX Ž w . f Ž w . rDX Ž w . ,
'
Q w s Qgmn s 0
Ž 41 .
where c a is a constant anticommuting target space
spinor, B Ž w . is an arbitrary function of w , and
P mm s hm n g mnPn n .
F. Brandt et al.r Physics Letters B 419 (1998) 148–156
The commutator of two transformations Ž41. reads
m
w Q1 ,Q2 x s 2 c2 G c1 Pm
Ž 42 .
where Pm generates w-dependent ‘‘translations’’ of
the type found above in the nonsupersymmetric case,
Pm x n s dmn B12 Ž w . ,
X
Pm am s g emn P mn B12
Ž w . f Ž w . rDX Ž w . ,
'
Pm w s Pm gmn s Pm u a s 0
Ž 43 .
with
B12 Ž w . s B1 Ž w . B2 Ž w . .
Together with the analogues of the symmetries arising from Ž11., the supersymmetries Ž41. form a
Kac–Moody
super-Weyl algebra which can be easily
ˇ
constructed explicitly along the lines of the previous
section.
6. Conclusion
Any local action in two dimensions Ž p s 1. with
field content and gauge symmetries given by Ž5. and
Ž6. respectively, has a Lagrangian of the form Ž7..
Specific choices of GM N , BM N and DI provide
actions which turn upon elimination of the auxiliary
fields into D-string actions of the Born–Infeld type
such as Ž1. Žfor p s 1. or, more generally, Ž9. or
Ž10..
The rigid symmetries of an action with Lagrangian Ž7. are determined by the solutions of the
generalized Killing vector Eqs. Ž12. – Ž14.. We have
shown that these equations can have infinitely many
inequivalent solutions which we have spelled out
explicitly for specific models in a flat target space. In
the latter models, we have found a Kac-Moody
ˇ
realization of the Weyl group, the new symmetries
being non-linearly realized. Symmetries of the actions Ž1. Žfor p s 1., Ž9. and Ž10. are obtained from
those of their counterparts Ž7. simply by eliminating
the auxiliary fields. For instance, from Ž35. one
obtains in this way among others a symmetry of the
p s 1-action Ž1. generated by
Dx m sF x m,
D am s emn Ž F 2 y 1 . 'G G nr x mEr x m q 2 F am
Ž 44 .
where, assuming that Gmn has Lorentzian signature,
F s Gy1 r2 e mnEm an ,
G s ydet Ž Gmn . .
155
F is constant on-shell. The value of this constant
characterizes partly a solution to the equations of
motion and contributes to its string tension. Ž44.
generates on-shell a dilatation of the target space
coordinates, but it also transforms the abelian gauge
field nontrivially. In particular, it transforms a solution to the equations of motion with F s 0 to another
one with F / 0, as on-shell one has D F s 2Ž F 2 y 1..
Symmetries such as Ž44. are thus useful, among
others, to connect configurations of the fundamental
string with those of the D-string.
We have also shown that the Kac–Moody
symˇ
metry structure extends analogously to curved target
spaces if the latter possess Killing vector fields
satisfying Ž37.. Furthermore we have given some
examples of supersymmetric extensions where an
infinite number of rigid supersymmetries appears in
addition to the Kac–Moody–Weyl
symmetries.
ˇ
We have obtained our results by an analysis of the
BRST cohomology at ghost numbers 0 Žactions. and
y1 Žsymmetries.. Especially the results on the rigid
symmetries are difficult to guess or to derive by
other means due to highly nonlinear nature of symmetries such as Ž44..
One may speculate whether the infinite number of
symmetries reflects part of the space-time symmetry
structure of an underlying Ž‘‘M’’. theory. This possibility is suggested because D-branes may probe
shorter space-time distances than strings w16x. It
would be interesting to further understand the physical meaning of the Weyl–Kac–Moody
algebra and
ˇ
whether or not the k-invariant formulation of the
D-string has infinitely many rigid symmetries too.
Another interesting point to be investigated will be
to check whether a Weyl–Kac–Moody
algebra apˇ
pears also for other D-p-branes.
Acknowledgements
J. Gomis acknowledges Kiyoshi Kamimura for
discussions on D-brane actions. J. Simon
´ thanks
Fundacio´ Agustı´ Pedro i Pons for financial support.
F. Brandt was supported by the Spanish ministry of
education and science ŽMEC.. This work has been
partially supported by AEN95-0590 ŽCICYT.,
GRQ93-1047 ŽCIRIT. and by the Commission of
European Communities CHRX93-0362 Ž04..
156
F. Brandt et al.r Physics Letters B 419 (1998) 148–156
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Phys. B 485 Ž1997. 85.
12 February 1998
Physics Letters B 419 Ž1998. 157–166
Probing m-term generation mechanism in string models
Yoshiharu Kawamura
a,1,b
, Tatsuo Kobayashi
c,2
, Manabu Watanabe
b,3
a
c
Physik department, Technische UniÕersitat
D-85748 Garching, Germany
¨ of Munchen,
¨
b
Department of Physics, Shinshu UniÕersity, Matsumoto 390, Japan
Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization, Midori-cho, Tanashi, Tokyo 188, Japan
Received 13 May 1997; revised 18 July 1997
Editor: M. Dine
Abstract
We give a generic method to select a realistic m-term generation mechanism based on the radiative electroweak
symmetry breaking scenario and study which type is hopeful within the framework of string theory. We discuss effects of
the moduli F-term condensation and D-term contribution to soft scalar masses. q 1998 Elsevier Science B.V.
1. Introduction
One of important problems in the supersymmetric
standard model ŽSUSY-SM. is how a SUSY Higgs
mass term, the m-term, is derived naturally within
the framework of supergravity ŽSUGRA.. Because
the m-term is not a soft SUSY breaking term and a
natural order of such a mass m is the gravitational
scale M in SUGRA 4 . This problem is called the
m-problem w1x. Some types of natural mechanisms of
m-term generation have been proposed, e.g. the case
with Kahler
potential including a Higgs mixing term
¨
w2x and a superpotential including a suitably suppressed mass term w3,1,4,5x.
In addition to the supersymmetric mass term, soft
SUSY breaking mass terms also contribute to the
Higgs mass terms. Explication of SUSY breaking
mechanism is an important and unsolved problem.
1
E-mail: [email protected].
E-mail: [email protected].
3
E-mail: [email protected].
4
Through this paper, we take M s1.
2
Superstring theory ŽSST. is only the known candidate for unified theory including gravity and is expected to give a natural solution due to some nonperturbative effects. Recently there have been various remarkable developments in studying non-perturbative aspects of SST w6x, but SUSY breaking
mechanism has not been fully understood yet. For
the present, we can derive formulae for soft SUSY
breaking terms assuming the existence of nonperturbative superpotential which induces SUSY breaking
in string models w7–9x. Further these soft terms can
be parametrized simply using several parameters in
most cases. For example, under the assumption that
only the dilaton field S andror the overall moduli
field T trigger SUSY breaking by their F-term condensations, soft scalar masses, gaugino masses and
A-parameters are written model-independently by the
gravitino mass m 3r2 and the goldstino angle u in the
case with the vanishing vacuum energy w10x. We can
extend such a parametrization into cases with several
moduli fields w11,12x. On the other hand, the Bparameter depends on the m-term generation mechanisms w9,10,13,14x. Therefore m and B parameters
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 2 5 6 - 2
158
Y. Kawamura et al.r Physics Letters B 419 (1998) 157–166
are usually treated as arbitrary parameters. The more
arbitrary parameters exist, the harder we analyze and
the less we have a predictivity. It is important to
select m-term generation mechanism in some way.
In this paper, we study several m-term generation
mechanisms based on the radiative breaking scenario
of electroweak symmetry breaking within the framework of string theory. Examining the parameter space
leading to successful electroweak symmetry breaking, we probe a realistic m-term generation mechanism. We take into account D-term contributions to
soft scalar masses w15x in our analysis. 5 The reason
is as follows. String models, in general, have extra
gauge symmetries including an anomalous UŽ1.
symmetry and extra massless matter fields other than
the SUSY-SM ones. We can reduce gauge group and
massless spectrum through symmetry breaking along
flat directions w18x. As discussed in recent papers
w19x, sizable D-term contributions can appear through
such flat direction breaking.
This paper is organized as follows. In the next
section, we give our basic assumptions, formulae of
m-term and B-term, and our strategy. In Section 3,
we study several types of m-term generation mechanisms based on the radiative breaking scenario and
show which type is hopeful. Section 4 is devoted to
conclusions and discussions.
up-type singlet t and the Higgs doublet H2 ,
their modular weights are n k s y1.
4. The top Yukawa coupling f t does not depend on
moduli fields, i.e., E f trE Tm s 0.
5. The modular weight of the Higgs doublet H1 is
y1 Žor y2..
6. The product H1 H2 of Higgs doublets is gauge
invariant under all gauge symmetries which are
broken along a flat direction.
7. Effects of moduli-dependent threshold corrections for gauge couplings and gaugino masses
are negligibly small.
8. The Kac-Moody levels satisfy k a s 1.
9. The gaugino masses Ma , B and m are all real.
10. The dependence of dilaton and moduli fields is
small in the m-parameter, i.e., EmrE S, EmrE Tm
< 1.
The third and fourth assumptions can be justified
from the fact that the coupling f t is strong and
allowed as a renormalized coupling in the untwisted
sector.
Here we discuss the overall-moduli case ŽT s T1
s T2 s T3 . for simplicity. Under the above assumptions, we can obtain the following formulae of soft
Ž0.
scalar masses mŽ0.2
˜3 ,
k , the A-parameter A t among q
Ž0. w
t̃ and H2 , and the gaugino mass M1r2 10x 6 ,
2. Method of analysis
'3 m 3r2 e
Ž0.
M1r2
s '3 m 3r2 eyi a
2.1. Basic assumptions
where we use the following parameterization
2
2
2
mŽ0.2
k s m 3r2 Ž 1 q n k cos u . q d k m 3r2 ,
yi a S
AŽ0.
t sy
We take the SUSY-SM with soft SUSY breaking
parameters expected to be derived as a low-energy
theory from orbifold models through flat direction
breaking as a starting point of our analysis. Let us
first list our basic assumptions.
1. The SUSY is broken by the F-term condensations of dilaton field S andror moduli fields Tm .
2. The vacuum energy V0 vanishes, i.e., V0 s 0, at
the tree level.
3. For the third family of the quark doublet q3 , the
5
Study on the Higgs sector has been done, e.g. within the
framework of string models without D-term contribution w16x and
SUSY grand unified theory ŽSUSY-GUT. with D-term contributions w17x.
² Ž K SS .
²Ž
1r2
S
sin u ,
sin u ,
F S : s '3 m 3r2 e i a S sin u ,
1r2 T
K TT
F : s m 3r2 e i a T cos u .
.
d k m 23r2
Ž 1.
Ž 2.
Ž 3.
Ž 4.
Ž 5.
Here
is a D-term contribution and we get
the relation d H 1 m 23r2 q d H 2 m 23r2 s 0 from the assumption 6 and the fact that its contribution is
proportional to broken diagonal charges of scalar
field. The natural order of d k is expected to be O Ž1.
from some explicit models w19x.
In the limit of the moduli dominant SUSY breaking, we can not neglect moduli-dependent threshold
corrections, but we do not consider such a case for
simplicity in this paper.
6
The same type of formulae were obtained in w8x.
Y. Kawamura et al.r Physics Letters B 419 (1998) 157–166
Table 1
The values of a t and IS . Here we use m t s175 GeV, i.e.,
m t Ž m t . s167.2 GeV and a tŽ0. s a t Ž M X .
tan b
a tŽ0.
atŽ mt .
y2
2
3
4
5
10
9.16=10
8.14=10y2
7.79=10y2
7.62=10y2
7.40=10y2
IS
y1
1.11=10
4.03=10y2
3.20=10y2
2.90=10y2
2.56=10y2
4.179
4.140
4.104
4.083
4.051
159
briefly and we give formulae of the B-parameter for
each m-term generation mechanism.
Ž m-1. The m-term mZ of O Ž m 3r2 . appears after
SUSY breaking in the case where a Kahler
potential
¨
includes a term such as ZH1 H2 w2x. In this case we
have
mZ s m 3r2 ² Z : y ² FT :² Z T : ,
Ž 10 .
BZ s 2 m 3r2 q ² FT : E T log mZ y
ž
The top quark mass is given as m t Ž m t . s
f t Ž m t . Õ <sin b <r '2 7 where tan b s ² h 2 :r² h1 : and
Õ 2 s ² h 2 :2 q ² h1 :2 . Ž h i ’s are neutral components of
Hi ’s.. We use the value m t s 175 GeV as the top
quark mass from the current experiments. Through
our analysis, we take the gauge coupling unification
scale M X s 1.7 = 10 16 GeV as an energy scale where
boundary conditions are imposed. Hence it is supposed that the above formulae ŽEq. Ž1.. – ŽEq. Ž3..
hold at M X .
By the use of renormalization group equations
ŽRGEs. of the SUSY-SM w21,22x, soft scalar masses
of H1 and H2 at the weak scale Žwe use the Z-boson
mass MZ . are given as
m2H 1 s hm23r2 q dm23r2 ,
Ž 6.
m2H 2 s hm23r2 y dm23r2 ,
Ž 7.
where
q
m 3r2
mZ
n H1 q n H 2
²T q T ) :
/
² FT :² Z T : .
Ž 11 .
Hereafter we take Z s 1rŽT q T ) . and then mZ and
BZ are given as
mZ s m 3r2 Ž 1 q e i a T cos u . ,
m 3r2
BZ s
1 q e i a T cos u
Ž 12 .
= 2 y cos u Ž eyi a T Ž 1 q n H 1 q n H 2 . y e i a T .
½
ycos 2u Ž 2 q n H 1 q n H 2 . 4 .
Ž 13 .
Ž m-2. The m-term ml of O Ž m 3r2 . appears after
SUSY breaking in the case where a superpotential W
˜ 1 H2 w5x. Here W˜ is a
includes a term such as lWH
superpotential which induces SUSY breaking. In this
case we have
ml s l m 3r2 ,
2
2
h s 2.56sin u q Ž 1 q n H 1 . cos u ,
Ž 8.
h s 2.56sin 2u y 3 IS sin2u
Ž 9.
Ž 14 .
Bl s m 3r2 2 y eyi a T cos u
= Ž n H 1 q n H 2 y ²T q T ) :² E T log l: . .
5
Ž 15 .
and d ' d H 1 s yd H 2 . Here the second term in RHS
of ŽEq. Ž9.. represents the effect of the top Yukawa
coupling and IS is a function of tan b Žor the top
Yukawa coupling.. The values of a t Ž' f t2r4p . and
IS are given in Table 1.
Ž m-3. The m-term mm can be generated through
some non-perturbative effects such as gaugino condensation w4x and it generally depends on the VEVs
of S and T. In this case we have
2.2. m-parameter and B-parameter
Bm s m 3r2 y1 y '3 eyi a S sin u
Several types of solutions for the m-problem have
been proposed w23x. Here we explain some of them
7
The pole mass of top quark is related with the running mass
as
m tp o l e s m t
Ž m t . 1q
4a 3 Ž m t .
3p
qOŽ
a 32
.
.
mm s mm Ž S,T . ,
Ž 16 .
= Ž 1 y ² S q S ) :² E S log mm : .
yeyi a T cos u Ž 3 q n H 1 q n H 2
y²T q T ) :² E T log mm : . .
5
Ž 17 .
Ž m-4. In the model with a singlet field N which
has a coupling f N NH1 H2 , the m-term appears when
the field N develops its VEV at some lower energy
Y. Kawamura et al.r Physics Letters B 419 (1998) 157–166
160
scale near O Ž m 3r2 . w3x. As a result, the B-parameter
can be generated from the A-term A N f N NH1 H2 .
There can be an admixture of several m-term
generation mechanisms and, in this case, m and B
parameters are given as
m Mix s Ý m p ,
Ž 18 .
p
BMix s Ý m p Bpr Ý m q ,
p
Ž 19 .
q
where the indices p and q run over all m-term
generation mechanisms.
Finally we discuss a multi-moduli case briefly.
Z2 n and Z2 n = ZM orbifold models w24x have U-type
of moduli fields corresponding to complex structures
of orbifolds w25x and a mixing term in the Kahler
¨
potential as
1
Ž
T3 q T3)
. Ž U3 q U3) .
Ž H1 H2 q h.c. . .
Ž 20 .
In this case, the Higgs fields H1 and H2 belong to
the untwisted sector. We assume F-terms of S, Ti
Ž i s 1,2,3. and U3 contribute SUSY breaking and
these are parametrized by m 3r2 , u , Q 3 and Q 3X
following Refs. w10–12x. Then the soft scalar masses,
the m-term and B-term are written at the tree level as
w12,16x
X2
2
2
2
2
mŽ0.2
H 1 s m 3r2 1 y 3cos u Ž Q 3 q Q 3 . q dm 3r2 ,
ž
2.3. Strategy
In this subsection, we give an outline of our
strategy to probe m-term generation mechanism based
on the radiative electroweak symmetry breaking scenario. The neutral fields h1 and h 2 have the following potential w26x:
V Ž h1 ,h 2 . s m12 h12 q m 22 h 22 q Ž m Bh1 h 2 q h.c. .
2
q 18 Ž g 2 q g X 2 . Ž h12 y h22 . ,
m12 ' m2H 1 q m2 ,
2
Ž 27 .
The bounded from below ŽBFB. condition along the
D-flat direction requires
m12 q m 22 ) 2 < m B < .
Ž 28 .
Further the conditions that minimizing the potential
are given as
2mB
sin2 b
,
Ž 29 .
m12 y m22 s ycos2 b Ž MZ2 q m12 q m22 . ,
X2
2
2
2
2
mŽ0.2
H 2 s m 3r2 1 y 3cos u Ž Q 3 q Q 3 . y dm 3r2 ,
/
Ž 22 .
m M s m 3r2 Ž 1 q '3 cos u Ž e i a T 3 Q 3 q e i a T X 3Q 3X . . ,
Ž 23 .
m M BM s 2 m23r2 Ž 1 q '3 cos u e i a T 3 Q 3 .
= Ž 1 q '3 cos u e i a T X 3Q 3X . .
Ž 26 .
m12 m 22 - Ž m B . .
Ž 21 .
ž
m22 ' m2H 2 q m2 ,
where all parameters correspond to the values at MZ .
The condition for the symmetry breaking is given
as
m12 q m 22 s y
/
Ž 25 .
where we use the relation MZ2 s 14 Ž g 2 q g X 2 . Õ 2 .
By the use of stationary conditions ŽEq. Ž29.. and
ŽEq. Ž30.., m and B are expressed by other parameters Ž m 3r2 , cos u , tan b , . . . ., such that
< m<
Ž 24 .
It is easy to show that BM and m M reduce to BZ and
mZ , respectively if we set Q 3 s 1r '3 and Q 3X s 0.
We give formulae of mrm 3r2 and Brm 3r2 in
Table 2. Here we choose eyi a S s 1, eyi a T i s e i a T i s
"1 and n H 1 s y1 and use this choice in our analysis. In general, the Brm 3r2 contains a small number
of free parameters compared with mrm 3r2 and so
the analysis of Brm 3r2 can be more predictable.
Ž 30 .
1
s
m 3r2
'2
ž
hyhq2 d
ycos2 b
yhyhy
MZ
2 1r2
ž //
,
m 3r2
Ž 31 .
< B<
s
m 3r2
sin2 b m 3r2
2
< m<
ž
hyhq2 d
y
ycos2 b
MZ
2
ž //
m 3r2
.
Ž 32 .
By the use of RGEs, we can obtain the m and
Y. Kawamura et al.r Physics Letters B 419 (1998) 157–166
161
Table 2
The formulae of m p rm 3r2 and B p rm 3r2 . The second column shows the dilaton dominant SUSY breaking case and the third one is the
dilaton and overall moduli mixed case. Here we take modular weights n H 1 s n H 2 s y1
Dilaton dominant
Dilaton and Moduli
mZ rm 3r2
mlrm 3r2
m M rm 3r2
1
l
1
1 " cos u
l
X
1 " '3 cos u ŽQ 3 q Q 3 .
BZ rm 3r2
Blrm 3r2
2
2
2
2 " 2cos u
Bmrm 3r2
y1 . '3
y1 y '3 sin u . cos u
BM rm 3r2
2
2
Ž1" 3 cos uQ 3 .Ž1" 3 cos uQ X3 .
'
'
'
X
1" 3 cos u ŽQ 3q Q 3 .
'
B Mix rm 3r2
2 m 3r2 Ž1q l .Ž1"cos u .y mm Ž1q 3 sin u .cos u .
m 3r2 Ž1q l .q mm
m 3r2 Ž1"cos u q l .q mm
B parameters at M X Žthey are denoted by mŽ0.
" and
Ž0.
B"
, respectively.. as follows w22x:
< m<
mŽ0.
"
s "cm
,
Ž 33 .
m 3r2
m 3r2
a 2 Ž tZ .
ž
cm '
a Ž0.
3r2
/ ž
1r22
a 1Ž tZ .
/
a Ž0.
= Ž 1 q 6 a tŽ0. F Ž t Z . .
1r4
Ž 34 .
and
Ž0.
B"
< B<
s.
m 3r2
DB
m 3r2
D B ' 3 AŽ0.
t
,
q
m 3r2
'
2 m 3r2 Ž1q l .y mm Ž1" 3 .
Ž 35 .
a tŽ0. F Ž t Z .
1 q 6 a tŽ0. F Ž t Z .
Here g Ž0. and f tŽ0. are the gauge coupling and top
Yukawa coupling at M X , respectively.
Ž0.
By comparing B "
rm 3r2 and mŽ0.
" rm 3r2 derived
from the stationary conditions of radiative breaking
with Bprm 3r2 and m prm 3r2 Ž p s Z, l, m , M . given
in the last subsection, we can find allowable parameter regions for Ž m 3r2 , cos u , Q 3 , Q 3X , tan b , d, l,
mm . leading to successful electroweak symmetry
breaking and know which type of m-term generation
mechanism is hopeful. Our strategy, ‘bottom-up approach’, is more generic and applicable than the
usual one, ‘top-down approach’, where a realization
of the radiative breaking scenario is examined by
defining the parameters m2k , m and B at M X and
checking if the quantities renormalized at MZ satisfy
stationary conditions ŽEq. Ž29.. – ŽEq. Ž30...
Ž0.
qM1r2
tZ Ž 3 a 2 Ž tZ . q 35 a 1Ž tZ . .
y
3 a tŽ0. Ž t Z F X Ž t Z . y F Ž t Z . .
1 q 6 a tŽ0. F
Ž tZ .
5
3. Which m-term is hopeful?
,
Ž 36 .
where
a Ž0. '
g Ž0.2
4p
F Ž tZ . '
tZ
H0
=
t Z s Ž 4p .
a tŽ0. '
,
ž
ž
y1
/ ž
a Ž0.
a Ž0.
log
M X2
MZ2
4p
16r9
a3Ž t .
a 1Ž t .
f tŽ0.2
,
Ž 37 .
a2 Ž t .
a Ž0.
y3
/
Ž0.
B" p ' B "
y Bp ,
y1 3r99
/
dt ,
.
In this section, we examine which type of m-term
generation is hopeful by taking the cases Ž m-1. – Ž mŽ0.
3. as examples and comparing B "
rm 3r2 and
Ž0.
m " rm 3r2 with Bprm 3r2 and m prm 3r2 determined
by m-term generation mechanism. We also give remarks of several extensions. For our analysis, it is
convenient to define the following functions:
Ž 40 .
Ž 41 .
Ž 38 .
m " p ' mŽ0.
" y mp .
Ž 39 .
The B" p and m " p should be satisfied the conditions B" p s 0 and m " p s 0 by definition if we
assume the radiative breaking of electroweak sym-
162
Y. Kawamura et al.r Physics Letters B 419 (1998) 157–166
Ž0.
Fig. 1. The values of B "
rm 3r2 versus m 3r2 with cos u s d s 0.
metry and the p-th type of m-term generation mechanism.
3.1. Dilaton dominant case
Ž0.
First we give allowed regions for B "
rm 3r2 and
in the limit of dilaton dominant SUSY
breaking in Fig. 1 and Fig. 2, respectively. The
Ž0.
Ž0.
ranges of By
rm 3r2 and Bq
rm 3r2 are y0.74 ;
Ž0.
0.70 and y3.51 ; y1.45. The ranges of mq
rm 3r2
Ž0.
and my rm 3r2 are 3.20 ; 4.98 and y4.98 ; y3.20.
Here we take 2 F tan b F 10 8 , d s 0 and m 3r2 G
50 GeV. The value tan b s 2 Ž10. corresponds to
Ž0.
Ž0.
By
rm 3r2 s 0.70 Žy0.74., Bq
rm 3r2 s y3.51
Ž0.
Žy1.45. and m " rm 3r2 s "4.98 Ž"3.20.. The Fig.
Ž0.
3 shows an allowable region of Ž B "
rm 3r2 , d . in
the limit of m 3r2 4 MZ . In this way, we come to a
mŽ0.
" rm 3r2
8
In the case that we take a larger tan b , it is necessary to
incorporate the contribution of Yukawa couplings other than top
quark neglected here.
conclusion that the m-term generation mechanism
can be realistic if the values of Bprm 3r2 and
m prm 3r2 hit the above ranges.
We study the first case Ž m-1.. It is shown that
there is no region satisfying B" Z s 0 with d s 0
from Fig. 1 and BZrm 3r2 s 2. When tan b s 1.33,
there is a solution of B" Z s 0. However, in this
case, it is not realistic since the top Yukawa coupling
blows up below M X . This result is consistent with
those in Ref. w16x. Further we find no region satisfying m " Z s 0 with d s 0 from in Fig. 2 and
mZrm 3r2 s 1. As discussed in Ref. w19x, D-term
contribution can survive even in the limit of dilaton
dominant SUSY breaking if string model contains an
anomalous UŽ1. symmetry which is cancelled by the
Green-Schwarz mechanism w27x. On the other hand,
D-term contributions related to anomaly-free symmetries vanish at the tree level in the limit of dilaton
dominant SUSY breaking. If we assume the existence of D-term contribution, there appears a region
consistent with the condition B" Z s 0. However it
Y. Kawamura et al.r Physics Letters B 419 (1998) 157–166
is a very narrow region with a relatively large positive value of d. For example, the region with tan b ;
2 and d ; 7 is allowed as given in Fig. 3. On the
other hand, negative D-term contribution is needed
Ž0.
to lower the value of mq
rm 3r2 . Hence it is impossible to realize this m-term generation even with Dterm contribution.
In the second case Ž m-2., we get the same result
for B-parameter as the first one. On m-parameter, we
can estimate the value of l using the condition
m " l s 0. Hence the radiative breaking scenario can
be realized in the models with a relatively large
positive D-term contribution, i.e., d s O Ž10..
In the third case Ž m-3., we have a solution even
in the absence of D-term contribution. The allowable
region of Ž m 3r2 ,tan b . is given in Fig. 4. The favorable value is tan b F 2.8 with d s 0. The introduction of D-term contribution yields tan b ) 2.8. At
present, though we treat m-parameter as a free one as
well as in the second case, we can select a m-term
generation mechanism whose origin is a non-per-
163
turbative one by using the allowed region for
mŽ0.
" rm 3r2 .
3.2. Dilaton and oÕerall moduli case
We consider the effect of overall moduli F-term
condensation. We have the same qualitative result
for cos u / 0 as the dilaton dominant case. That is,
there is no allowed region in the first case, but very
narrow region exists with large positive d in the
second case and there exists an allowed region with
natural values of Ž m 3r2 , tan b , d . in the third case.
The above fact can be understood by the use of the
Eqs. ŽEq. Ž31.. – ŽEq. Ž39.. directly. That is, both
Ž0.
<
<
mŽ0.
" rm 3r2 and B " rm 3r2 are proportional to sin u
in the absence of D-term contribution and the limit
m 23r2 4 MZ2 , and so they decrease as cos 2u inŽ0.
creases. Hence it is impossible to satisfy B"
s 0 in
the first and second case with d s 0. The introduction of d does not improve the situation drastically.
Fig. 2. The values of mŽ0.
" rm 3r2 versus m 3r2 with cos u s d s 0.
164
Y. Kawamura et al.r Physics Letters B 419 (1998) 157–166
Ž0.
Fig. 3. The values of B "
rm 3r2 versus d with cos u s 0 in the limit of m 3r2 4 MZ .
Fig. 4. The values of tan b versus m 3r 2 with cos u s d s 0.
Y. Kawamura et al.r Physics Letters B 419 (1998) 157–166
3.3. Remarks of extension
We discuss a multi-moduli case ŽEq. Ž20.. – ŽEq.
Ž24... For the case with Q 3 s 1r '3 and Q 3X s 0, we
Ž0.
have no allowable regions for B"
s 0 and mŽ0.
" s0
because this case corresponds to the first case Ž m-1..
We can check that there exists an allowable region
with tan b s 2, Q 3 s Q 3X s 0.38 and cos 2u s 0.99.
We can carry out the case with n H 1 s y2. For
tan b s 2 and d s 0, we find the following fact.
Compared with the case with n H 1 s y1, the value of
Ž0.
Ž0.
By
rm 3r2 and Bq
rm 3r2 decreases and increases,
respectively and the absolute value of mŽ0.
" rm 3r2
decreases. The difference between the values in the
case with n H 1 s y1 and n H 1 s y2 increases as
cos 2u increases. Hence the similar conclusion holds
for the reality of the radiative breaking scenario as
the dilaton dominant case with n H 1 s y1.
In the case of an admixture of several m-term
generation mechanisms, we need a dominant contribution of the third mechanism Ž m-3. to get an allowable region with natural values of Ž m 3r2 , cos u , tan u ,
d ..
We discuss the case Ž m-4.. As we have an extra
light singlet field N, the RG flows of m 2H 1 and m2H 2
should be modified owing to the effect of the Yukawa
coupling f N . This case can be applied to a similar
strategy discussed in the last section. The difference
Ž0.
is that B "
receives RGE effects as an A-parameter
above MZ and so we must compare the renormalized
quantity with not B-parameter but A N at M X . It is
not discussed here further because the renormalized
quantity contains an unknown parameter f N .
In w28x, the parameter space is extensively studied
from a non-existence condition of charge and color
breaking minima in the case of dilaton induced
SUSY breaking. Such a condition can also give a
useful information on m-term generation mechanism.
4. Conclusions and discussions
We have given a generic method to select a
realistic m-term generation mechanism based on the
radiative electroweak symmetry breaking scenario
and studied which type is hopeful within the framework of string theory. The m-term generated by
some non-perturbative effects, i.e., Ž m-3., can be
165
hopeful to realize the radiative symmetry breaking
scenario even in dilaton dominant supersymmetry
breaking. We have discussed effects of the moduli
F-term condensation and D-term contribution to soft
scalar masses. In the case of overall moduli, we have
the same qualitative result as in the limit of dilaton
dominant SUSY breaking, that is, the first mechanism is impossible to realize the radiative scenario,
the second one is required to a large D-term contribution of O Ž10 m23r2 . and the third one is hopeful. 9
In the future, we can select a m-term generation
mechanism whose origin is a non-perturbative one
by using the allowed region for mŽ0.
" rm 3r2 .
Our bottom-up approach to select a realistic mterm generation mechanism is so generic and powerful that we can apply it to the models which we
know the soft terms except for m and B-parameters,
the case with an improvement of approximation and
more complex situations. For example, the improvement by the incorporation of 1-loop effective potential w30x, the case with a large tan b , the case with
large moduli-dominant threshold corrections for
gaugino masses, other assignments of modular weight
for matter fields and the modular dominant SUSY
breaking case. 10 In the above situations, extra contributions D h and D h are added to Eqs. ŽEq. Ž8..
and ŽEq. Ž9.., respectively. These studies have to be
considered systematically to select a realistic string
model.
Acknowledgements
The authors are grateful to S. Khalil for useful
discussions.
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12 February 1998
Physics Letters B 419 Ž1998. 167–174
Supersymmetry and gauge theory on Calabi–Yau 3-folds
J.M. Figueroa-O’Farrill
a,1,2
, A. Imaanpur
b,3,4
, J. McCarthy
b,5,6
a
b
Department of Physics, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK
Department of Physics and Mathematical Physics, UniÕersity of Adelaide, Adelaide, SA 5005, Australia
Received 30 September 1997; revised 14 November 1997
Editor: P.V. Landshoff
Abstract
We consider the dimensional reduction of supersymmetric Yang–Mills on a Calabi–Yau 3-fold. We show by
construction how the resulting cohomological theory is related to the balanced field theory of the Kahler
Yang–Mills
¨
equations introduced by Donaldson and Uhlenbeck–Yau. q 1998 Elsevier Science B.V.
1. Introduction
The study of Ricci-flat manifolds is interesting to
both geometers and string theorists for a variety of
reasons. These manifolds provide examples of ‘‘exotic’’ Einstein geometries: in fact, their holonomy
groups have to be SUŽ n., SpŽ n., G 2 or SpinŽ7.,
corresponding to Calabi–Yau n-folds, hyperkahler
¨
manifolds of real dimension 4 n, and exceptional 7and 8-manifolds, respectively. Because they are
Ricci-flat and admit parallel spinors, they are supersymmetric vacua for superstring-related theories. Out
of these parallel spinors one can construct parallel
forms w19,15x which turn out to be calibrations in the
sense of w16x. Indeed, these manifolds have a rich
1
Supported by the EPSRC under contract GRrK57824.
mailto: [email protected].
3
Supported by the Ministry of Culture and Higher Education,
Iran.
4
mailto: [email protected].
5
Supported by the Australian Research Council.
6
mailto: [email protected].
2
geometry of Žcalibrated. minimal submanifolds.
These submanifolds are, in the simplest case, the
supersymmetric cycles w5x around which branes may
wrap to produce BPS states. Yang–Mills theory on
these manifolds is also interesting. The equations of
motion admit instantonic solutions which minimise
the action and are defined by linear equations generalising Žanti.self-duality in four dimensions w11,20x.
This observation forms the basis of the ‘‘Oxford
programme’’ w14x to generalise Donaldson–Floer–
Witten theory to higher dimensional Ricci-flat manifolds.
Perhaps one of the boldest proposals yet to have
emerged out of the ‘‘second superstring revolution’’
is the Matrix Conjecture of Banks et al. w3x. This
conjecture states that the dimensional reduction to
one dimension of 10-dimensional supersymmetric
Yang–Mills in the limit in which the rank of the
gauge group goes to infinity provides an Infinite
Momentum Frame description of M-theory, the 11dimensional theory believed to underlie nonperturbative superstring theory. In this context, it becomes an
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 6 2 - 7
168
J.M. Figueroa-O’Farrill et al.r Physics Letters B 419 (1998) 167–174
important problem to understand the dimensional
reductions on 10-dimensional supersymmetric
Yang–Mills theory. Most research has focused on
toroidal compactifications, since these preserve all of
the sixteen supercharges present in the original theory, and are therefore the most constrained. On the
other hand, reductions on curved Ricci-flat manifolds, also produce manageable theories even though
there is little supersymmetry left. The reason is that
whatever supersymmetry remains becomes BRSTlike, rendering the theory cohomological. This is
explained in more detail below.
The theory we describe in this letter can be
understood as that arising out of euclidean D-branes
wrapping around a Calabi–Yau 3-fold. More prosaically, it is the dimensional reduction of 10-dimensional supersymmetric Yang–Mills theory to such a
manifold. Results in this direction for other manifolds have been obtained in w7,10x, who considered
euclidean D-branes wrapping around calibrated submanifolds. The resulting theories on the D-brane
were seen to be topologically twisted Yang–Mills
theory – the components of the 10-dimensional gauge
field in directions normal to the D-brane being sections of the normal bundle to the calibrated submanifold which need not be trivial. In w1x the dimensional
reductions of supersymmetric Yang–Mills to 7- and
8-manifolds of exceptional holonomy Ž G 2 and
SpinŽ7., respectively. were studied. The theories obtained are cohomological w22x and localise on the
moduli space of generalised instantons and, in the
7-dimensional case, monopoles. The instanton theories agree Žmorally. with the cohomological theories
studied in w4,2x. Similar considerations, in less detail
but in more generality, can be found in w8x.
In this paper we will follow the approach of w1x
and study the theory on a Calabi–Yau 3-fold. We
will recover a cohomological theory which localises
on the moduli space of solutions of the Kahler–
¨
Yang–Mills equations. These equations have been
studied by Donaldson w13x Žfor Kahler
surfaces. and
¨
by Uhlenbeck–Yau w18x Žin complex dimension three
and above., who show that they are in one-to-one
correspondence with stable holomorphic vector bundles. Cohomological theories which localise on this
moduli space have been discussed in w4x as a reduction of the eight-dimensional cohomological theories,
and also briefly in w8x.
We now discribe the general approach of w1x and
at the same time explain the cohomological origin of
this class of theories. We start with 10-dimensional
supersymmetric Yang–Mills theory and reduce it to
6-dimensional euclidean space. The resulting lagrangian can be promoted to any spin 6-manifold M
by simply covariantising the derivatives with respect
to the spin connection; but the supersymmetry transformations will fail to be a symmetry of the action
unless the spinorial parameters are covariantly constant. This requires that M admit parallel spinors,
and that means that the holonomy group must be a
subgroup of SUŽ3.. If we want M to be irreducible
then the holonomy must be SUŽ3.. Covariance of the
supersymmetry algebra under the holonomy group
implies that the commutator of two supersymmetry
transformations with parallel spinors as parameters
will result Žon shell and up to gauge transformations.
in a translation by a parallel vector. Since for the
irreducible manifolds we consider there are no such
vectors, the supersymmetry transformation is a BRST
symmetry. This general argument shows that the
resulting theory is cohomological; so that the partition function localises on a finite-dimensional space
which we will identify with the moduli space of
Kahler–Yang–Mills
instantons.
¨
This paper is organised as follows. In Section 2
we discuss the dimensional reduction of 10-dimensional supersymmetric Yang–Mills theory to 6-dimensional euclidean space. In Section 3 we specialise to the theory defined on a manifold of holonomy SUŽ3.: a Calabi–Yau 3-fold, and show that it is
indeed cohomological. In Section 4 we rewrite the
theory in the form of a balanced cohomological field
theory in the sense of w12x and w9x.
For convenience we briefly summarise our spinor
conventions here. We use the Minkowski signature
Žy1,1,1, . . . ,1.. The unitary charge conjugation matrix for the Clifford algebra generators gm is specified, for given sd , st g "14 , by
Cgm Cy1 s sd gmt
and
C t s st C .
Ž 1.
For the spinor representations of SOŽ3,1. we use
notation along the lines of Wess and Bagger w21x,
s I s Ž1, s i . with indices saIȧ and s I s Žy1, s i .
˙
obeying s Ia˙ a s ye a be a˙ bs bIb˙ . We choose e 12 s 1 and
e a be b c s ydca. The dot distinguishes between the
J.M. Figueroa-O’Farrill et al.r Physics Letters B 419 (1998) 167–174
two spinor representations, ca and c ȧ, for which the
SOŽ3,1. generators are
s I J ' 14 Ž s Is J y s Js I .
s I J ' 14 Ž s Is J y s Js I . .
and
Ž 2.
It is straightforward to see that with Ž ca . † ' cȧ and
Ž c a˙ . † ' c a, we have, e.g., c a˙ s e a˙ b˙c ḃ .
2. Dimensional reduction to six dimensions
Ž 3.
where FM N s E M A N y E N A M q w A M , A N x, and DMC
s E MC q w A M ,C x, and
t
CsC C ,
Ž
sdŽ10. s stŽ10. s y1
..
Ž 4.
This action is hermitian and invariant under the
following supersymmetry transformations,
d A M s i ´GMC and
dC s 12 FM N G M N´ ,
Ž 5.
where ´ is a constant negative chirality Majorana–
Weyl spinor. The supersymmetry algebra only closes
on-shell and up to gauge transformations.
Reducing the theory down to six euclidean dimensions breaks the 10-dimensional Lorentz invariance
down to a subgroup SOŽ3,1. = SOŽ6.. The first step
of the dimensional reduction is then the decomposition of our fields into irreducible representations of
this subgroup. For tensor fields this is the obvious
decomposition M s Ž I, m .; in particular, A M s
Ž f I , Am .. For the spinors we use the representation
G I s g˜ I m g 7 and
G m q4 s |4 m g m ,
The charge conjugation matrix C then decomposes
as C s C˜ m C. For definiteness we choose the representation in which the g m are all antisymmetric
Ž sdŽ6. s ystŽ6. s y1., and the chiral representation
for the 4-dimensional g ’s Ž sdŽ4. s ystŽ4. s 1.; in
terms of Pauli matrices,
g˜ 0 s |2 m Ž i s 2 .
Ž 8.
g˜ i s s i m s 1 for i s 1,2,3
Ž 9.
g˜ 5 s |2 m s 3 ,
Our starting point is 10-dimensional supersymmetric Yang–Mills theory. It can be formulated in
terms of a Lie algebra valued gauge field A M and a
negative chirality Majorana–Weyl adjoint spinor C .
The Lie algebra is assumed to possess an invariant
metric, denoted Žy,y . or sometimes Tr . The lagrangian is then given by
L s y 14 Ž FM N , F M N . q 2i Ž C , G M DMC . ,
169
Ž 6.
m
Ž 10 .
with C˜ s i s 2 m s 3.
Let e a , a s 1,2 denote an orthonormal eigen-basis
of s 3. Then the Weyl condition determines
C s e a m e1 m c L a q e a˙ m e 2 m c Rȧ .
Ž 11 .
The 10-dimensional Majorana condition then reduces
to a reality condition on the 6-dimensional fields,
c L b˙ s yc Rt a˙e a˙ b˙ ,
c Rb s c Lt a e a b
Ž 12 .
c L a s e a b c R) b ,
c Ra˙ s ye a˙ b˙c L)b˙ .
Ž 13 .
Finally then, the lagrangian reduces to
L s y 14 5 Fmn 5 2 y 12 Ž Dm f I , Dm f I .
y 14 Ž w f I , f J x , f I , f J
. q 2i ž cRa ,gm Dm cL a /
˙
q 2i c L a˙ ,gm Dm c Ra˙ q i c Ra , saIb˙ f I , c Rb
ž
/ ž
/,
Ž 14 .
which is invariant under the supersymmetry transformations
dc L a s 12 gmn Fmn e L a q gm Dm c I saIb˙ e Rb˙
q w f I , f J x saI J be L b
Ž 15 .
dc Ra˙ s 12 gmn Fmn e Ra˙ y gm Dm c I s Ia˙ be L b
˙
q w f I , f J x s b˙I J a˙e Rb
Ž 16 .
where g are the generators for the Clifford algebra
C l Ž6,0. and g˜ I for C l Ž3,1.. A straightforward
calculation gives
d Am s i e Ragm c L a q i e L a˙gc Rb˙
Ž 17 .
G 11 s g˜ 5 m g 7 .
df I s yi e L a˙ sI a˙ bc L b q i e Ra sIa b˙ c Rb˙ .
Ž 18 .
Ž 7.
J.M. Figueroa-O’Farrill et al.r Physics Letters B 419 (1998) 167–174
170
3. Reduction to manifolds with SU(3) holonomy
Now that we have a supersymmetric theory defined on a six dimensional euclidean space, it is time
to extend it to a Calabi–Yau 3-fold. The structure
group of the tangent bundle reduces to an SUŽ3.
subgroup of SOŽ6.. Our first task is to decompose
the SOŽ6. fields into irreducible representations of
SUŽ3.. We will actually consider the decomposition
into UŽ3. irreducibles, UŽ3. being the holonomy
group of a 6-dimensional Kahler
manifold. Since
¨
UŽ3. is locally isomorphic to SUŽ3. = UŽ1., we will
be able to read off the SUŽ3. representations easily.
It is, of course, sufficient to work in a local frame.
The embedding SOŽ6. > SUŽ3. = UŽ1. leads to
the branching 4 s Ž1. 3 [ Ž3.y1; thus, under the global
symmetry SUŽ3. = UŽ1. = SOŽ3,1., we have that the
spinors l R and l L transform according to
l R ; Ž 1,3,2 L . [ Ž 3,y 1,2 L .
Ž 19 .
l L ; Ž 1,y 3,2 R . [ Ž 3,1,2 R . .
Ž 20 .
Let u denote the Žcommuting, left-handed. spinor
which is responsible for splitting the 4 above, and let
us normalise it to u †u s 1. Clearly u ) is the righthanded singlet spinor, which splits the 4. We need
the explicit projections onto these representations.
The projector onto the singlet in the 4 is
uu † s 18 Ž 1 y g 7 . y 161 u †gmn u Ž 1 y g 7 . gmn ,
Ž 21 .
as follows from the standard result
We may now introduce the Kahler
form kmn '
¨
i u †gmn u and the 3-form
Vmnl ' u †gmnl u ) .
Ž 26 .
There are no other covariants since
u †g mu ) s 0 ,
Ž 27 .
which follows from sandwiching Žthe complex conjugate of. Eq. Ž25. between u t and u ) .
At this point it is useful to make a special choice
of u which corresponds to the standard choice of
complex coordinates. This reduces the problem to
the usual construction of the spinor representation of
SOŽ6. via linear combinations of the Clifford algebra
generators which obey the algebra of fermionic oscillators. First introduce the combinations Žtaking m s
Ž a , a . in flat Žlocal frame. coordinates.
g a s '12 Ž g a q ig aq3 .
and g a s '12 Ž g a y ig aq3 . .
Ž 28 .
The SU Ž3. generators are then Tba s g agb
y 13 dbag ggg . Requiring that u be an SUŽ3. singlet
Žwith appropriate UŽ1. charge y3. fixes ga u s 0, so
that k ab s k a b s 0 and k a b s i da b . Similarly all
components of V vanish by Eq. Ž27. but for Vabg
' u †gabg u ) and its conjugate. For completeness,
notice that
g m1 . . . m r
s Ž y1 .
1qr Ž ry1 .r2
i
Ž6yr . !
e m 1 . . . m 6g 7gm rq 1 . . . m 6 ,
Ž 22 .
together with a Fierz transformation upon noticing
that by chirality
u †g Ž A.u s 0 , for < A < odd.
Ž 23 .
It follows immediately from
g lgŽ A. gl s Ž y1 .
< A<
Ž 6 y 2 < A < . gŽ A.
Ž 24 .
that
gluu †gl s 34 Ž 1 q g 7 . y 18 u †gmn u Ž 1 q g 7 . gmn , Ž 25 .
which will also be required later.
Vabg Va b Xg X s 8 Ž dbb X dgg X y dbg X dgb X . .
Ž 29 .
Using vielbeins to translate to the coordinate basis,
Ž6d. manithese results apply for an arbitrary Kahler
¨
fold.
The projectors for spinors onto SUŽ3. = UŽ1. covariant fields follow directly by combining Eq. Ž21.
and Žthe complex conjugate of. Eq. Ž25. to get the
appropriate completeness relations; e.g.,
1
2
Ž 1 y g 7 . s uu † q 12 ga u )u tga .
Ž 30 .
For arbitrary symplectic Majorana–Weyl spinors x L
or x R , define
x a s u †x L a and
x a a s u tga x L a .
Ž 31 .
J.M. Figueroa-O’Farrill et al.r Physics Letters B 419 (1998) 167–174
Then the SOŽ3,1. covariant decompositions under
SUŽ3. = UŽ1. are
x L a s ux a q 12 ga u )x a a
x Ra˙ s yu )e a˙ b˙x b˙ q 12 ga ue a˙ b˙x b˙ a ,
and
Ž 32 .
and in terms of these fields, the lagrangian becomes
1
2
1
2
I
Ž Fab , Fa b . y Ž Fa b , Fa b . y Ž Da f , Da fI .
1
y 4 Ž w f I , f J x , f I , f J . q i e a b Ž ca , Da c ba .
Lsy
˙
qi e a˙ b ca˙ , Da c b˙ a y i s
i
2
y
/
Ia˙ b
Ž ca˙ , w fI , c b x .
Ž ca˙ a , fI , c ba .
Vabg e a b Ž ca a , Db c bg .
˙
y 8i Vabg e a˙ b ca˙ a , Db c b˙g .
ž
Ž 33 .
/
This action is invariant with respect to the supersymmetry transformations with the parallel spinors as
parameter. These are obtained from dS just by inserting the SUŽ3. singlet grassmann parameters,
e L a s ue a ,
e Ra˙ s yu )e a˙ b˙e b˙ .
Ž 34 .
In writing the explicit supersymmetries it is convenient to introduce auxiliary fields so that the supersymmetry algebra closes off shell. Because of
SUŽ3. covariance and the fact that there are no
SUŽ3. invariant vectors on a Calabi–Yau 3-fold, the
supersymmetry algebra will be BRST-like, at least
up to gauge transformations. Further, it is convenient
to split the conjugate generators using the complex
structure. Thereto, we introduce supercharges via
a˙
a
dS s i e a˙ Q y i e a Q .
generators Žso QA s B ´ QA† s yB † and QB s A
´ QB † s A†, if A is bosonic..
The algebra of charges on the fields is then easily
found to be
Q a ,Q b 4 s 0 , Q a˙ ,Q b˙ 4 s 0
and
Q a ,Q b˙ 4 s dG Ž y2 i s I b˙ afI . ,
Ž 36 .
Ž
.
where dG u means ‘‘gauge transformation with parameter u ’’. Then dS can be extended to the auxiliary fields
H s yiFaa
Ia˙ b
y s
i
8
ž
Ž 35 .
The sign is such that Q and Q act like canonical
171
and
Ha s 2i Va bg Fbg ,
so that the supersymmetry algebra is maintained off
shell. We have shown the explicit transformations in
Table 1
It is possible now to reduce to a cohomological
theory with a single cohomological symmetry: setting c 2 s c 2˙ s 0, which requires f 1 s f 2 s 0, we
are left with the supersymmetries generated by Q 1
and Q 1̇. Instead we will keep all supersymmetries
and work out a balanced formulation for this cohomological theory.
4. A balanced cohomological field theory
In order to recognise what this theory computes, it
will prove convenient to rewrite it in balanced form
w9,12x; that is, in terms of potentials. Let us first
write the lagrangian in a form linear in Q’s. To this
effect, introduce
L s Q a V a q Q ȧ V ȧ ,
Ž 38 .
7
2
where V a is dimension in the natural units where
the gauge coupling is scaled out, Am and f I have
Table 1
Field
cb
c b˙
c ba
c b˙ a
Aa
Aa
fI
H
Ha
Ha
Q a˙
Qa
0
Hd b˙a˙ q i s b˙I J a˙w f I , f J x
2 iDa f I s Ia˙ ae a b
Ha d b˙a˙
˙
e a˙ bc b˙ a
0
ysI a˙ bc b
i s Ia˙ b w fI , c b x
Hd ba q i s bI J aw f I , f J x
0
˙
4 i e a˙ b Da c b˙ q 2 i s Ia˙ b w f I , c b a x
Ž 37 .
0
Ha d ba
2 iDa f I e a bs bIb˙
0
ye a bc b a
˙
sI b ac b˙
I b˙ a w
is
f I , c b˙ x
˙
4 i e a b Da c b q 2 i s I b aw f I , c b˙ a x
0
J.M. Figueroa-O’Farrill et al.r Physics Letters B 419 (1998) 167–174
172
dimension 1 and c ’s have dimension 32 . Further, it
should be gauge invariant and an SOŽ3,1. doublet.
Taking the most general possible Ansatz and comparing to Eq. Ž33., we find
V a s 4i Ž ca , Fa a . q 18 Ž ca , H .
y 8i saI J b Ž c b , w f I , f J x . y 16i Va bg Ž ca a , Fbg .
˙
q 161 Ž ca a , Ha . y 8i e a˙ bsaIa˙ c b˙ a , Da f I .
ž
/
Ž 39 .
Eliminating the auxiliary fields Žwhich are determined correctly., we find that
L s L q 12 Ž Fab , Fa b . y 12 Ž Fa b , Fa b .
y 21 Ž Faa , Fbb . .
Ž 40 .
1
2
Note that the extra terms can be rewritten as y k n
Tr Ž F n F ., whence their integral only depends on
the Kahler
class and the characteristic class of the
¨
gauge bundle.
We can pursue this a little further, writing L
quadratic in Q’s. The most general form is
—should appear at all is quite interesting, and consistent with the results in w1x for manifolds of G 2
holonomy.
We would like to rewrite this in balanced form
along the lines of w12x Žsee also w9x.. To do that, one
must first choose a global SLŽ2,R. under which the
balanced supercharges will transform as a doublet d.
The lagrangian must then be written Žup to a topological term. in the form
e A B d A d B W for some potential W ,
Ž 48 .
where the critical points of W agree with the fixed
points of the cohomological symmetry.
It is natural, in our case, to take the SOŽ2,1.
subgroup of the global SOŽ3,1. symmetry. Then the
doublet supercharges can be taken as the linear
combinations d A and d˜A , where
˙
ds
ž
Q1 q Q 2
˙
Q 2 q Q1
˙
/ ž
,
d˜s
Q1 y Q 2
˙
Q 2 y Q1
/
,
Ž 49 .
and S can be decomposed. The first term reduces to
L s e a b Q a Q b V q Q asaIa˙ Q ȧ V I q h.c. ,
Ž 41 .
L 1 s y 12 e A B d A d B q e A B d˜A d˜B Ž V q V . ,
ž
and a similar analysis to the above gives
V a s e a b Q b V q saIa˙ Q ȧ V I ,
Ž 42 .
/
while the second term is just
L 2 s 2 Q asa3a˙ Q a˙ V 3 q 2 Q asama˙ Q a˙ Vm .
where
i
32
V s y Va bg C S Ž A . abg y
VIsy
i
16
1
64
Ž fI , Fa a . q sI
b˙ b
1
16
cd
e Ž cc , c d .
žc˙
ba
Ž 43 .
, c ba ,
Ž 44 .
/
with C SŽ A. the holomorphic Chern–Simons 3-form,
C S Ž A . abg s Ž Aa , Fbg . y 13 Ž Aa , Ab , Ag
..
Ž 45 .
Clearly
Ž 51 .
Remarkably, an explicit calculation shows that both
of the terms in L 2 are individually SOŽ3,1. invariant. Since there is a unique such invariant bilinear in
Q a and Q ȧ these two terms in L 2 must be proportional, and we can consider just the first. Thus L 2 is
itself proportional to
2 Q asa3a˙ Q a˙ V 3 s y 12 e A B d A d B y e A B d˜A d˜B V 3 .
ȧ
Q Vs0 ,
Ž 46 .
and V I is real. Thus we can write
˙
s e a b Q a Q b y e a˙ b˙ Q a˙ Q b Ž V q V .
ž
a
saIa˙ Q ȧ V I
.
ž
/
Ž 52 .
This is still not quite in balanced form, since we
have two doublets of supercharges. However, it is
straightforward to check that
L ' L1 q L 2
q2Q
Ž 50 .
/
e A B d A d B Ž V q V . s e A B d˜A d˜B Ž V q V . ,
Ž 47 .
Note that the holomorphic Chern–Simons term cannot be reproduced as a BRST variation, so it isn’t
profitable to continue this process. That such a term
—only invariant under small gauge transformations
Ž 53 .
since by Eq. Ž46. the difference may be written as
anticommutators of supercharges. Moreover, another
explicit calculation shows that
e A B d A d B V 3 s ye A B d˜A d˜B V 3 .
Ž 54 .
J.M. Figueroa-O’Farrill et al.r Physics Letters B 419 (1998) 167–174
Collecting these results we see that L is of
balanced form Eq. Ž48. with potential,
W s 4i Ž w , Fa a . q 32i Va bg C S Ž A . abg
y 32i Vabg C S Ž A . abg y 161 ca˙ a , ca a
ž
˙
q 161 e a b Ž ca , c b . y 161 e a˙ b ca˙ , c b˙ ,
ž
/
/
Ž 55 .
where we have introduced w ' f 3 . Hence the physical supersymmetric Yang–Mills theory differs from
this balanced cohomological field theory by the topological term T in Eq. Ž40.. Note, however, that T of
course depends on the Kahler
class.
¨
Balanced theories localise on the critical points of
W . These points correspond to fermions set to zero,
and bosons obeying Faa s 0 and the Bogomol’nyitype equations
1
4
Vabg Fbg q Da w s 0
and
1
4
Vabg Fbg q Da w s 0 .
Fa a s 0 ,
One direction in which this work may be pursued
is to examine the Ansatz of w7x for the effective
theory of branes wrapped around supersymmetric
cycles in the Calabi-Yau space, here a special lagrangian torus. Considering the embedding of the
torus local coordinates of w17x to see how the topological twisting on the torus arises, we note that
arguments as in w7x Žand w6x. suggest that, for one
UŽ1. case, the resulting path integral on the torus
localises on the moduli space MSL = M Flat , precisely
the local description of the mirror w17x. Details will
appear elsewhere.
Acknowledgements
J.M.F. takes pleasure in thanking Bobby Acharya,
Chris Hull, Chris Kohl
¨ and Bill Spence for conversations on this and other related topics.
Ž 56 .
For a compact Calabi–Yau 3-fold, these equations
reduce to the Kahler–Yang–Mills
equations
¨
Fab s Fa b s 0 and
173
Ž 57 .
together with the trivial Da w s Da w s 0.
5. Conclusions and outlook
We have shown that the dimensional reduction of
supersymmetric Yang–Mills on a compact Calabi–
Yau 3-fold is a cohomological theory which localises
on the moduli space of solutions to the Kahler
¨
Yang–Mills equations or, by the work of Donaldson
and Uhlenbeck–Yau, on the moduli space of stable
holomorphic bundles. Observables in this theory correspond to invariants of this moduli space, which
generalise the Donaldson invariants in four dimensions. Unlike four dimensions, these are not topological invariants of the Calabi–Yau 3-fold, but a priori
only invariants of the SUŽ3. structure. It follows
from the balanced formulation Eq. Ž48. of the theory
that L is invariant under infinitesimal deformations
of the metric which preserve the Calabi–Yau condition. A similar result was shown in w2x for the
cohomological theories on 7- and 8-manifolds of
exceptional holonomy.
References
w1x B.S. Acharya, J.M. Figueroa-O’Farrill, M. O’Loughlin, B.
Spence, Euclidean D-branes and higher dimensional gauge
theory, hep-thr9707118, Nuclear Physics B, in press.
w2x B.S. Acharya, M. O’Loughlin, B. Spence, Nucl. Phys. B 503
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w3x T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys.
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hep-thr9608169.
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Ž1996. 243, hep-thr9606040.
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Ž1986. 257, Correction: Comm. Pure Appl. Math. 42, 703.
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12 February 1998
Physics Letters B 419 Ž1998. 175–178
Characters and modular properties of permutation orbifolds
1
P. Bantay
Institute for Theoretical Physics, Rolland EotÕos
¨ ¨ UniÕersity, Budapest, Hungary
Received 4 September 1997
Editor: L. Alvarez-Gaumé
Abstract
Explicit formulae describing the genus one characters and modular transformation properties of permutation orbifolds of
arbitrary Rational Conformal Field Theories are presented, and their relation to the theory of covering surfaces is
investigated. q 1998 Elsevier Science B.V.
If C denotes a Rational Conformal Field Theory,
its n-th tensor power C mn is straightforward to
describe for any positive integer n, e.g. the primary
fields are just n-tuples of primaries of C , and their
Žgenus one. characters are simply the product of the
corresponding C characters. An interesting feature
of these theories is that any permutation x g Sn of
the n ‘‘replicas’’ is a global symmetry of C mn, so it
is possible to orbifoldize C mn by any permutation
group V - Sn . For reasons to become clear soon, we
shall denote the resulting permutation orbifold by
CX V.
The first systematic investigation of permutation
orbifolds has been performed in w1x Žsee also w2x..
Permutation orbifold techniques have been applied in
w3x to compute the free energy of second quantized
strings. Recently, a detailed analysis of cyclic permutation orbifolds, i.e. the case V s Z n for prime n,
has been presented in w4x, where the explicit form of
the genus one characters and their modular properties can be found for V s Z 2 . The aim of the present
1
Work partially supported by OTKA F019477.
paper is to generalize the above results to arbitrary –
possibly nonabelian – V , and to understand the
underlying geometry. We shall only sketch the main
results, their derivation being left to a future publication.
The basic observation, which lies behind most of
the results to be presented, is that if V 1 X V 2 denotes the wreath product of the permutation groups
V 1 and V 2 Žcf. w5x., then the V 1 X V 2 permutation
orbifold of C is nothing but the V 2 permutation
orbifold of C X V 1 , i.e.
Ž CX V1 . X V2 s CX Ž V1 X V2 . .
Ž 1.
This property, which explains our choice for the
notation, is a straightforward consequence of the
definition of the wreath product. In particular if C is
holomorphic, i.e. it has only one primary with respect to the maximally extended chiral algebra – e.g.
the E8 WZNW model at level 1 – the permutation
orbifold C X V is a holomorphic orbifold model,
whose properties are described by the double D Ž V .
of the group V w6,7x.
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 6 4 - 0
P. Bantayr Physics Letters B 419 (1998) 175–178
176
A most important consequence of Eq. Ž1. is the
following description of the primary field content of
permutation orbifolds: the primary fields of C X V
are in one-to-one correspondence with the pairs
² p, f :, where p is some representative of an orbit
of V acting on the n-tuples ² p 1 , . . . , pn : of primaries pi of C , while f is an irreducible character
of the double D Ž V p . of the stabilizer
For an n-tuple p s ² p 1 , . . . , pn : of primaries of
C let’s introduce the quantity
V p s x g V < xp s p 4
where
mj t q kj
tj s
lj
of the n-tuple p.
If, as usual, 0 denotes the vacuum of C , then the
vacuum of C X V – which we shall also denote by 0
in the sequel – corresponds to the pair ²0, f 0 :,
where w0x is the one point orbit ²0, . . . ,0: whose
stabilizer is obviously V itself, while f 0 Ž x, y . s d x,1
is the trivial character of D Ž V ..
To write down explicitly the characters of the
orbifold theory C X V , we have to introduce some
notation. First, for a primary p of C , we’ll denote
by x p Žt . its genus one character, and by
ž ž
v p s exp 2p ı D p y
c
24
//
x p Ž x , y Nt .
s
½
k j r lj
vy
x pj Ž tj .
pj
Ł
if x , y g V p commute
j g O Ž x, y.
0
otherwise
Ž 3.
Ž 4.
and pj denotes the component of p associated to the
orbit j Žwhich is well defined since x, y g V p ..
Then the character of the primary ² p, f : of C X V
reads
x² p , f :Ž t . s
1
< Vp <
Ý
x p Ž x , y <t . f Ž x , y . .
Ž 5.
x , y ,g V
Note that this formula is meaningful, i.e. it does not
depend on the actual representative p of the orbit,
since for all z g V
its exponentiated conformal weight, so that
x z p Ž x , y <t . s x p Ž x z , y z <t . .
x p Ž t q 1. s v p x p Ž t . .
In the special case V s Z n we recover the results of
w4x for the characters of cyclic permutation orbifolds
of prime order.
The geometry underlying the Eq. Ž3. is clear. If
Et denotes a torus of modular parameter t , then a
commuting pair x, y g V determines an n-sheeted
unramified covering of Et , namely x Žresp. y .
describes how the sheets are permuted when going
around the a-cycle Žresp. b-cycle. of a canonical
homology basis. This covering is usually not connected, its connected components being in one-to-one
correspondence with the orbits j g O Ž x, y .. By the
Riemann–Hurwitz formula, each such connected
component is itself a torus Ej , with modular parameter tj given by Eq. Ž4..
In particular, if ZŽt . denotes the partition function of C , the partition function ZV of C X V reads
For a pair x, y g V of commuting permutations, we
shall denote by O Ž x, y . the set of orbits of the
subgroup generated by x and y. To each ordered
triple ² x, y, j : with j g O Ž x, y ., we associate the
following data:
1. lj Žresp. lj) . is the length of any x orbit Žresp. y
orbit. contained in j
2. mj Žresp. mj) . is the number of the x orbits Žresp.
y orbits.
3. kj Žresp. kj) . denotes the smallest non-negative
)
)
integer for which y mj s x kj Žresp. x mj s y kj .
holds on the points of j .
These quantities are not independent, they are connected by the following relations:
lj mj s lj)mj) s < j <
mj) s gcd Ž lj , kj . and mj s gcd Ž lj) , kj) .
Ž 2.
where < j < is the length of the orbit j , and gcdŽ a,b .
denotes the greatest common divisor of the integers
a and b.
ZV Ž t . s
1
<V <
Ý
Ł
Ž 6.
Z Ž tj . .
Ž 7.
xysyx j g O Ž x , y .
Once we know the explicit expression for the
characters of C X V , we can compute their behavior
P. Bantayr Physics Letters B 419 (1998) 175–178
under modular transformations. Consider for example the S transformation
1
t¨t Xsy .
Ž 8.
t
The modular parameter of the covering torus corresponding the orbit j g O Ž x, y . will change accordingly
mj t X q kj
kj t y mj
tj ¨ tjX s
s
.
Ž 9.
lj
lj t
Besides acting on the modular parameter, S also
transforms the cycles in the canonical homology
basis, thus changing the monodromy of the covering.
In the case at hand, this means that it transforms the
pair ² x, y : into the pair ² y, xy1 :. This last pair
generates the same subgroup as x and y, so that
O Ž x, y . s O Ž y, xy1 ., but the covering torus corresponding to the triple ² y, xy1 , j : has modular parameter
mj)t y kj)
t˜j s
.
Ž 10 .
lj)
Clearly, tjX and t˜j should be related by a modular
transformation:
aj t˜j q bj
tjX s
Ž 11 .
cj t˜j q dj
aj
bj
cj
dj
ž /
for some Sj s
g SLŽ2,Z ., which – by com-
paring Eqs. Ž9. and Ž10. – is easily seen to be
kj
Sj s
mj)
kj kj) y mj mj)
<j <
lj
kj)
mj)
mj
checks that
xp
ž
x, y<y
sÝ
1
t
b
d
ž /
0
.
Ł
LŽpzj q .j Ž Sj . .
Ž 15 .
jg O Ž x , y .
As an example, consider the case x s y s Ž1,2..
Then an orbit j g O Ž x, y . has either length < j < s 1,
in which case Sj s 0 y1 and LŽ Sj . s S, or < j <
ž
1
s 2, in which case Sj s
0
1
2
/
0
1
ž / and
L Ž Sj . s Ty1 r2 Sy1 Ty2 STy1 r2 s T 1r2 ST 2 ST 1r2 ,
Ž 16 .
the latter equality being a consequence of the modular relation TSTSTs S. This is in complete agreement with the results of w4x for the S-matrix of Z 2
permutation orbifolds.
An important characteristic of a primary field is
its S-matrix element with the vacuum of the theory.
From Eq. Ž15. we get
1
S0 ² p , f : s
Ž 17 .
Ý f Ž x ,1 . Ł S0 pj
< V p < xg
jg O Ž x ,1 .
Vp
since for all j g O Ž x,1. we have Sj s S.
Let’s turn to the conformal weights, which can be
determined form the behavior of the characters under
the modular transformation T :t ¨ t q 1. We have
x p Ž x , y <t q 1 . s x p Ž x , xy <t . Ł v pmj j r lj .
Ž 12 .
B ut it is straightforw ard to show that
Ł j g O Ž x, y. v pmj j r lj is independent of y, so finally
one gets
x² p , f :Ž t q 1 . s v² p , f : x² p , f :Ž t . ,
Ž 19 .
Ž 13 .
where
g SLŽ2,Z . with c / 0, then one
/
Ł
x , yg V pl V z q
Ž 18 .
rc
rc
L p q Ž M . s vya
M p q vyd
p
q
a
c
elements of the transformation S in the orbifold
theory C X V :
1
S²² qp ,,cf :: s
Ý f Ž x, y. c Ž y z, x z .
< Vp < < Vq <
zg V
jg O Ž x , y .
If we introduce the notation
for any M s
177
L pj qj Ž Sj . x q Ž y, xy1 <t . ,
v² p , f : s
1
df
Ý
f Ž x, x.
xg V p
q jg O Ž x , y .
which leads to the following formula for the matrix
Ž 20 .
jg O Ž x ,1 .
is the exponentiated conformal weight
nc
v² p , f : s exp 2p ı D² p , f : y
Ž 21 .
24
of the primary ² p, f : of C X V , and df s
Ý x g V pf Ž x,1..
This concludes our presentation of the structure of
ž ž
Ž 14 .
v p1rj < j <
Ł
//
178
P. Bantayr Physics Letters B 419 (1998) 175–178
permutation orbifolds. We have seen that the characters of these theories are completely determined by
the characters of the original theory and the action of
the twist group. The basic characteristics of the
orbifold theory, such as the matrix elements of the
modular transformations T and S have been written
down explicitly. These latter determine through Verlinde’s formula w8x the fusion rules of the permutation orbifold. The connection with the theory of
covering surfaces opens the way to the computation
of arbitrary correlation functions w9–11x. Of course
there’s still much to be done, e.g. one would like to
have explicit expressions like Eqs. Ž15. and Ž20. for
the fusion rules and the Frobenius–Schur indicators
w12x.
Acknowledgements
It is a pleasure to acknowledge discussions with
Zalan
Geoff Mason, Laszlo
´ Horvath,
´
´ ´ Palla and
Christoph Schweigert.
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12 February 1998
Physics Letters B 419 Ž1998. 179–185
Logarithmic conformal field theories with continuous weights
M. Khorrami
a,b,c
, A. Aghamohammadi
a,d
, M.R. Rahimi Tabar
a,e,1
a
Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 5531, Tehran 19395, Iran
b
Department of Physics, Tehran UniÕersity, North Kargar AÕe., Tehran, Iran
c
Institute for AdÕanced Studies in Basic Sciences, P.O. Box 159, GaÕa Zang, Zanjan 45195, Iran
d
Department of Physics, Alzahra UniÕersity, Tehran 19834, Iran
e
Department of Physics, UniÕersity of Science and Technology, Narmak, Tehran 16844, Iran
Received 10 October 1997
Editor: L. Alvarez-Gaumé
Abstract
We study the logarithmic conformal field theories in which conformal weights are continuous subset of real numbers. A
general relation between the correlators consisting of logarithmic fields and those consisting of ordinary conformal fields is
investigated. As an example the correlators of the Coulomb-gas model are explicitly studied. q 1998 Elsevier Science B.V.
1. Introduction
It has been shown by Gurarie w1x, that conformal
field theories ŽCFT. whose correlation functions exhibit logarithmic behaviour, can be consistently defined and if in the OPE of two given local fields
which has at least two fields with the same conformal dimension, one may find some operators with a
special property, known as logarithmic operators. As
discussed in w1x, these operators with the ordinary
operators form the basis of the Jordan cell for the
operators L i .
The logarithmic fields Žoperators. in CFT were
first studied by Gurarie in the c s y2 model w1x.
After Gurarie, thes logarithms have been found in a
1
E-mail: [email protected].
multitude of others models such as the WZNW-model
on GLŽ1,1. w2x, the gravitationally dressed CFT w3x,
c p,1 and non-minimal c p, q models w2,4–6x, critical
disorderd models w7,8x, and the WZNW-models at
level 0 w9,10x. They play a role in the study of
critical polymers and percolation w11,12x, 2D-MHD
turbulence w13–15x, 2D-turbulence w16,17x and quantum Hall states w18–20x. They are also important for
studying the problem of recoil in the string theory
and D-branes w9,21–24x, as well as target space
symmetries in string theory w9x. The representation
theory of the Virasoro algebra for LCFT was developed in w25x. The origin of the LCFT has been
discussed in w26–28x. The modular invariant partition
functions for ceff s 1 and the fusion rules of logarithmic conformal field theories ŽLCFT. are considered
in w4x, see also w29x about consequences for Zamolodchikov’s C-theorem. Structure of the LCFT in D-dimensions has been discussed in w30x.
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 2 6 - 3
M. Khorrami et al.r Physics Letters B 419 (1998) 179–185
180
The basic properties of logarithmic operators are
that, they form a part of the basis of the Jordan cell
for L i ’s and in the correlator of such fields there is a
logarithmic singularity w1x. It has been shown that in
rational minimal models such a situation, i.e. two
fields with the same dimensions, doesn’t occur w14x.
In a previous paper w27x assuming conformal invariance we have explicitly calculated two- and
three-point functions for the case of more than one
logarithmic field in a block, and more than one set of
logarithmic fields for the case where conformal
weights belong to a discrete set. Regarding logarithmic fields formally as derivations of ordinary fields
with respect to their conformal dimension, we have
calculated n-point functions containing logarithmic
fields in terms of those of ordinary fields Žsee also
w31x, about the role of such derivative in the OPE
coefficients of LCFT..
We have done these when conformal weights
belong to a discrete set. In w28x, there is an attempt to
understand the meaning of derivation CFT with respect to conformal weights. Here, we want to consider logarithmic conformal field theories with continuous weights. The simplest example of such theories is the free field theory. The structure of this
article is as follows. In Section 2 we study conformal
theories, in which conformal weights belong to a
continuous subset of real numbers, and calculate the
correlators of these theories. Specifically, we show
that one can calculate the two-point functions of
logarithmic fields in terms of those of ordinary fields
by derivation. This is not possible in the case of
discrete weights. In Section 3 we consider the
Coulomb-gas model as an example.
where D is the conformal weight of the family.
Among the fields F Ž j., the field F Ž0. is primary. It
was shown that one can interpret the fields F Ž j.,
formally, as the j-th derivative of a field with respect
to the conformal weight:
F Ž j. Ž z . s
1
dj
j! d D j
F Ž0. Ž z . ,
Ž 2.
and use this to calculate the correlators containing
F Ž j. in terms of those containing F Ž0. only. The
transformation relation Ž1., and the symmetry of the
theory under the transformations generated by L " 1
and L 0 , were also exploited to obtain two-point
functions for the case where conformal weights belong to a discrete set. There were two features in
two-point functions. First, for two families F 1 and
F 2 , consisting of n1 q 1 and n 2 q 1 members, respectively, it was shown that the correlator
²F 1Ž i.F 2Ž j. : is zero unless i q j G maxŽ n1 ,n 2 .. ŽIt is
understood that the conformal weights of these two
families are equal. Otherwise, the above correlators
are zero.. Another point was that one could not use
the derivation process with respect to the conformal
weights to obtain the two-point functions of these
families from ²F 1Ž0.F 2Ž0. :, since the correlators contain a multiplicative term dD1 , D 2 , which can not be
differentiated with respect to the conformal weight.
Now, suppose that the set of conformal weights of
the theory is a continuous subset of the real numbers.
First, reconsider the arguments resulted to the fact
that ²F 1Ž i.F 2Ž j. : is equal to zero for i q j G
maxŽ n1 ,n 2 .. These came from the symmetry of the
theory under the action of L " 1 and L0 . Symmetry
under the action of Ly1 results in
²F 1Ž i. Ž z . F 2Ž j. Ž w . : s ²F 1Ž i. Ž z y w . F 2Ž j. Ž 0 . :
2. Correlators of a logarithmic CFT with continuous weights
s : Ai j Ž z y w . .
Ž 3.
We also have
In w27x, it was shown that if there are quasi-primary
fields in a conformal field theory, there arises logarithmic terms in the correlators of the theory. By
quasi-primary fields, it is meant a family of operators
satisfying
s Ž z E q D1 q D2 . Ai j Ž z . q Aiy1, j Ž z .
qAi , jy1 Ž z . s 0,
Ž 4.
and
L n ,F Ž j. Ž z .
² L1 ,F 1Ž i. Ž z . F 2Ž j. Ž 0 . :
s z nq 1EzF Ž j. Ž z . q Ž n q 1 . z nDF Ž j. Ž z .
q Ž n q 1 . z nDF Ž jy1. Ž z . ,
² L0 ,F 1Ž i. Ž z . F 2Ž j. Ž 0 . :
Ž 1.
s Ž z 2E q 2 z D1 . Ai j Ž z . q 2 zAiy1, j Ž z . s 0.
Ž 5.
M. Khorrami et al.r Physics Letters B 419 (1998) 179–185
These show that
181
the solution to which is
Ž D1 y D2 . Ai j Ž z . q Aiy1 , j Ž z . y Ai , jy1 Ž z . s 0.
Ž 6.
If D1 / D2 , it is easily seen, through a recursive
calculation, that Ai j ’s are all equal to zero. This
shows that the support of these correlators, as distribution of D1 and D2 , is D1 y D2 s 0. So, one can
use the ansatz
a mi j s
Ž y1.
m
m
Ý Ž sm . a 0iymqs, jys .
m!
Ž 15 .
ss0
From this
iqj
Ai0j
yŽ D 1 q D 2 .
Ž z. sz
Ý Ž ln z .
m
Ž y1.
m
m!
ms0
m
Ai j Ž z . s
Aikj Ž z . d Ž k . Ž D1 y D2 . .
Ý
Ž 7.
=
kG0
y Ž k q 1.
j
Aikq
1
Ž z.
j
q Aiy1,
k
Ž z.
y Aik, jy1
Ž z.
and
k!
d Ž k . Ž D1 y D2 . s 0,
iqjyk
k
1
Aikj Ž z . s
kG0
Ý
Ž y1.
l k
l
Ž . Ý
ls0
Ž ln z .
m
Ž y1.
m
m!
ms0
m
Ž 8.
=
or
Ý Ž sm . a 0iykymqlqs, jylys
zyŽ D1q D 2 . .
ss0
Ž k q 1.
j
Aikq
1
Ž z.
,j
s Aiy1
k
Ž z.
y Aik, jy1
Ž z . , k G 0.
Ž 9.
Ž z. s
1
k!
Ž 17 .
So we have
Ai j Ž z . s zyŽ D1qD 2 .
This equation is readily solved:
Aikj
Ž 16 .
ss0
Inserting this in Ž6., and using x d Ž kq1. Ž x . s yŽ k q
1. d Ž k . Ž x ., it is seen that
Ý
Ý Ž sm . a 0iymqs, jys ,
Ý d Ž k . Ž D1 y D 2 .
kG0
k
k
l
ÝŽ .
, jyl
Aiykql
0
Ž z. ,
Ž 10 .
=
ls0
Ai0j ’s
1
k!
Aikj ’s
where
remain arbitrary. Also note that
with a negative index are zero. We now put Ž7. in
Ž4.. This gives
iqjyk
k
l k
l
Ý Ž y1. Ž . Ý Ž ln z .
ls0
ms0
m
Ž y1.
m
m!
m
=
Ý Ž sm . a 0iykymqlqs, jylys
,
Ž 18 .
ss0
j
Ž z E q D1 q D2 . Aikj Ž z . q Aiy1,
Ž z . q Aik, jy1 Ž z . s 0,
k
Ž 11 .
Using Ž10., it is readily seen that it is sufficient to
write Ž11. only for k s 0. This gives
or
ij
A Ž z. sz
Ý
qq rqs
p!q!r !s!
=a iypyr , jyqys Ž ln z .
rq s
d Ž pqq . Ž D1 y D2 . ,
Ž 19 .
where
iqj
Ai0j Ž z . s zyŽ D1q D 2 .
Ž y1.
p, q , r , sG0
j
Ž z E q D1 q D2 . Ai0j Ž z . q Aiy1,
Ž z . q Ai0, jy1 Ž z . s 0.
0
Ž 12 .
Putting the ansatz
yŽ D 1 q D 2 .
Ý
a mi j Ž ln z .
m
Ž 13 .
a i j :s a 0i j .
Ž 14 .
These constants, defined for nonnegative values of i
and j, are arbitrary and not determined from the
conformal invariance only.
Now differentiate Ž19. formally with respect to
ms0
in Ž12., one arrives at
ij
iy1, j
q a mi , jy1 s 0,
Ž m q 1 . a mq
1 q am
Ž 20 .
M. Khorrami et al.r Physics Letters B 419 (1998) 179–185
182
D1. In this process, a i j ’s are also assumed to be
functions of D1 and D2 . This leads to
and
E Ai j Ž z .
Ai j s
ED1
Ž y1.
yŽ D 1 q D 2 .
sz
Ý
=
Ž ln z .
ED1
qa iypyr , jyqys Ž ln z .
y Ž ln z .
rq sq1
rqs
rq s
d
Ž pqq .
Ž D1 y D 2 .
d Ž pqqq1. Ž D1 y D2 .
d Ž pqq . Ž D1 y D2 .
5,
Ž 21 .
or
E Ai j Ž z .
ED1
Ž y1.
yŽ D 1 q D 2 .
sz
Ý
= Ž ln z .
rq s
d
Ž pqq .
i
²F Ž i. Ž z . : s
iypyr , jyqys
q
EA
,
Ž 27 .
Ž 28 .
Ž 23 .
b ny k
k!
d k Ž D. ,
Ž 29 .
where
b i :s
ij
i q 1 ED1
Ý
ks0
.
ED1
Comparing this with Aiq1, j, it is easily seen that
1
Ž 26 .
and differentiate it with respect to D1 and D2 , to
obtain Ai j. In each differentiation, some new constants appear, which are denoted by a i j ’s but with
higher indices. Note also that the definition is selfconsistent. So that this formal differentiation process
is well-defined.
One can use this two-point functions to calculate
the one-point functions of the theory. We simply put
F 2Ž0. s 1. So, D2 s 0,
Ž D1 y D 2 .
Ž 22 .
Aiq1, j s
A00 .
and
Ea iypyr , jyqys
= Ž pqr . a
i! j! ED1i ED2j
²F Ž0. Ž z . : s b 0d Ž D . ,
qq rqs
p!q!r !s!
p, q , r , s
Ej
A00 Ž z . s zyŽ D1qD 2 .d Ž D1 y D2 . a 00 ,
Ea iypyr , jyqys
½
Ei
These relations mean that one can start from A00 ,
which is simply
qq rqs
p!q!r !s!
p, q , r , s
1
1 d ib 0
i! d D i
.
Ž 30 .
The more than two-point function are calculated
exactly the same as in w27x.
provided
Ea iypyr , jyqys
s Ž i q 1 y p y r . a iq1ypyr , jyqys .
ED1
Ž 24 .
Note, however, that the left hand side of Ž24. is just
a formal differentiation. That is, the functional dependence of a i j ’s on D1 and D2 is not known, and
their derivative is just another constant. Repeating
this procedure for D2 , we finally arrive at
ij
a s
1
Ei
As an explicit example of the general formulation
of the previous section, consider the Coulomb-gas
model characterized by the action w26x
Ss
Ej
i! j! ED1i ED2j
3. The Coulomb-gas model as an example of
LCFT
00
a ,
Ž 25 .
1
4p
2
H d x'g
yg mn Ž EmF . Ž En F . q iQRF ,
Ž 31 .
M. Khorrami et al.r Physics Letters B 419 (1998) 179–185
where F is a real scalar field, Q is the charge of the
theory, R is the scalar curvature of the surface and
the surface itself is of a spherical topology, and is
everywhere flat except at a single point.
Defining the stress tensor as
T mn :s y
4p
dS
'g
d gmn
² f Ž f0 . : s
1
N
s
Ž 32 .
it is readily seen that
T mn s y Ž E mF . Ž E nF . q 12 g mn g a b Ž EaF . Ž Eb F .
yiQ f ; mn y g mn= 2F ,
Ž 33 .
N
f
f
ž
ž
1
d
2 i dQ
1
d
2 i dQ
/
/
Ž N²1: .
Nd Ž Q . .
Ž 42 .
Third, the normal ordering procedure is defined as
following. One can write
f Ž z . s f 0 q fq Ž z . q fy Ž z . ,
Ž 43 .
where ²0 < fy Ž z . s 0, fq Ž z .<0: s 0, and
and
2
T Ž z . :s Tz z Ž z . s y Ž Ef . y iQE 2f ,
Ž 34 .
where in the last relation the equation of motion has
been used to write
F Ž z, z. sf Ž z. qf Ž z. .
Ž 35 .
It is well known that this theory is conformal, with
the central charge
2
c s 1 y 6Q .
Ž 36 .
There are, however, some features which need more
care in our later calculations. First, this theory can
not be normalized so that the expectation value of
the unit operator become unity. In fact, using e S as
the integration measure, it is seen that
²1: A d Ž Q .
Ž 37 .
one can, at most, normalize this so that
²1: s d Ž Q . .
Ž 38 .
Second, f has a z-independent part, which we
denote it by f 0 . The expectation value of f 0 is not
zero. In fact, from the action Ž31.,
²f : s ²f0 : s
1
H df
NŽ Q.
0 f 0 exp Ž 2 iQ f 0 . ,
Ž 39 .
where N is determined from Ž38. and
1
d f 0 exp Ž 2 iQf 0 . .
N
This shows that N Ž0. s p , and
²1: s
More generally
1
,
183
²f0 : s
H
1
2i
w f 0 , f " x s 0.
The normal ordering is so that one puts all ‘-’ parts
at the left of all ‘q’ parts. It is then seen that
² : f w f x :: s ² f Ž f 0 . : .
N Ž 0.
N Ž 0.
d Ž Q. .
Ž 45 .
Here, the dependence of f on f in the left hand side
may be quite complicated; even f can depend on the
values of f at different points. In the right hand
side, however, one simply changes f Ž z . ™ f 0 .
Now consider the two-point function. From the
equation of motion, we have
² f Ž z . f Ž w . : s y 12 ln Ž z y w . ²1: q b;
Ž 46 .
we also have
² : f Ž z . f Ž w . :: s ² f 02 : s y
1
d2
4 N dQ 2
Nd Ž Q . .
Ž 47 .
Note that there is an arbitrary term in Ž46., due to the
ultraviolet divergence of the theory. One can use this
arbitrariness, combined with the arbitrariness in
N Ž Q ., to redefine the theory as
f Ž z . f Ž w . s :y 12 ln Ž z y w . q : f Ž z . f Ž w . :,
Ž 48 .
Ž 40 .
and
X
d X Ž Q. q
Ž 44 .
Ž 41 .
² f Ž f 0 . : :s f
ž
1
d
2 i dQ
/
d Ž Q. ;
Ž 49 .
M. Khorrami et al.r Physics Letters B 419 (1998) 179–185
184
these relations, combined with Ž45. are sufficient to
obtain all of the correlators. One can, in addition, use
Ž34. Žin normal ordered form. to arrive at
From this using Ž45. and Ž48., we have
k
² P js1
WaŽ0.
Ž zj .:
j
ai a j
T Ž z. f Ž w. s
Ew f
iQr2
y
zyw
Ž zyw.
2
q r.t.,
Ž 50 .
2
k
s P1F i- jF k Ž zi y z j .
T Ž z.T Ž w. s
Ew T
2T Ž w .
y
zyw
Ž zyw.
2
q
Ž 1 y 6Q 2 . r2
Ž zyw.
js 1
iQ
2
Ž n q 1. z n .
Ž zyw.
2
Ž 52 .
:e i af Ž w . :q r.t.,
Ž 53 .
which shows that :e i af : is a primary field with
a Ž a q2Q.
4
.
Ž 54 .
To this field, however, there corresponds a quasi
conformal family Žpre-logarithmic operators w26x.,
whose members are obtained by explicit differentiation with respect to a Ž a is not the conformal
weight but is a function of it.:
WaŽ n. : s f n e i af : s Ž yi .
d
n
da n
:e i af :.
Ž 55 .
To calculate the correlators of W ’s, it is sufficient
:.
to calculate ²WaŽ0.
PPP WaŽ0.
1
k
One has, using Wick’s theorem and Ž48.,
k
k
P js1
:e i a j f Ž z j . :
se
1r2
Ý
1F i-jFk
a i a j ln Ž z iyz j .
:e
i
Ý a f Ž z . :.
j
js1
/
.
Obviously, differentiating with respect to any a i ,
leads to logarithmic terms for the correlators consisting of logarithmic fields WaŽ n.. The power of logarithmic terms is equal to the sum of superscripts of
the fields WaŽ n..
zyw
a Ž a q 2 Q . r4
ž
Ý aj
Ž 51 .
Ew :e i af Ž w . :
y
k
d Q q 12
Ž 57 .
This shows that the operators f and 1 are a pair of
logarithmic operators with D s 0 Žin the sense of
Ž1... One can easily show that
T Ž z . :e i af Ž w . : s
2
.
4
Eq. Ž50. can be written in the form
L n , f Ž z . s z nq 1Ef y
d
e1r2 Ý a j d Q d Ž Q .
ai a j
and
Da s
s P1F i- jF k Ž zi y z j .
j
js1
Ž 56 .
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12 February 1998
Physics Letters B 419 Ž1998. 186–194
Effective actions, relative cohomology and Chern-Simons forms
J.A. de Azcarraga
´
a,1
, A.J. Macfarlane
b,2
, J.C. Perez
´ Bueno
a,3
a
b
Departamento de Fısica
Teorica
and IFIC, E-46100 Burjassot (Valencia), Spain
´
´
Department of Applied Mathematics and Theoretical Physics, SilÕer St., Cambridge, CB3 9EW, UK
Received 24 October 1997
Editor: P.V. Landshoff
Abstract
The explicit expression of all the WZW effective actions for a simple group G broken down to a subgroup H is
established in a simple and direct way, and the formal similarity of these actions to the Chern-Simons forms is explained.
Applications are also discussed. q 1998 Elsevier Science B.V.
1. Introduction
Recently it has been shown by D’Hoker and
Weinberg w1,2x that the most general effective action
of Wess–Zumino–Witten ŽWZW. type, with a compact symmetry group G broken down to a subgroup
H, is given by the non-trivial de Rham cocycles on
the homogeneous coset manifold GrH, and a cohomological descent-like procedure has been used in
w2x to obtain explicit expressions for the lower order
examples. Motivated by this work w1,2x and our own
on the properties of symmetric invariant tensors on
simple algebras w3x, we look here at the problem of
finding all the invariant effective actions of WZW
type in terms of the cohomology of the Lie algebra
G relatiÕe to a subalgebra H w4x. By exploiting this
Žequivalent. point of view, we are able to find a
general formula for WZW type actions on GrH for
any compact, connected and simply connected simple Lie group G Žthe case of semisimple G may be
reduced to it. and for arbitrary spacetime dimensions.
The structure of phenomenological Lagrangians
and nonlinear realizations was elucidated thirty years
ago w5x. Their relation to the standard Wigner little
group construction, which is the result of parametrising the coset K ' GrH s gH < g g G4 in terms of
the Goldstone coordinates w a Ž a s 1, . . . ,dim K .,
was emphasised in w6x. Indeed, for the left action of a
global transformation g g G s KH on the coset space
K, g: w a ¨ w X a, we find
gu Ž w . s u Ž w X . h Ž g , w . ,
Ž 1.1 .
1
E-mail: [email protected].
2
E-mail: [email protected].
3
E-mail: [email protected].
where hŽ g, w . g H is simply an element of the ‘little
group’ of the coset reference point. Geometrically,
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 3 4 - 2
J.A. de Azcarraga
et al.r Physics Letters B 419 (1998) 186–194
´
uŽ w . Že.g., exp w a Ta . may be viewed as a section of
the bundle GŽ H, K .; since guŽ w . and uŽ w X . belong
to the same fibre, Eq. Ž1.1. follows 4 .
The key idea of the standard nonlinear realizations in general is that we may write
uy1 du s Ž uy1 du . H q Ž uy1 du . K
' V Ha Ta q V Ka Ta .
Ž 1.2 .
As is well known Žsee w7x, Volume I, p. 103. the
H-valued component of the canonical left invariant
ŽLI. form v on G determines a LI connection on G;
the first term in Ž1.2. is its pull-back to K by the
section uŽ w ., and hence transforms as a connection.
In contrast, V K transforms tensorially through
hŽ g, w . g H under a left transformation of G, its
elements operating linearly for H ; G and nonlinearly for those in G _ H. Hence any H-invariant
expression made out of V K will also be invariant
under the whole G and is a candidate for an invariant
nonlinear Lagrangian. However, as emphasised in
w1x, there are also invariant actions which do not
come from an invariant Lagrangian density. These
are generically referred to as WZW actions w8x, also
discussed in the context of nonlinear sigma models
in w9x. In Witten’s derivation w8x of the simplest
WZW term for G s SUŽ2. Ž H s e . and D s 2
spacetime M ; S 2 , the fields g Ž x . are extended by
means of an interpolating field g Ž x, l. Ž g Ž x,0. s 0,
g Ž x,1. s g Ž x .. and the WZW action is given by
I WZ W s
HBd
2
xd le mn Tr Ž gy1 E grEl WmWn . ,
Ž 1.3 .
where Wm s gy1Em g and E B s M; the construction
uses p 2 Ž G . s 0 and p 3 Ž G . s Z Žwhich hold for any
simple Lie group.. Similar considerations can be
made for higher D, where the existence of a WZW
term requires in particular w10x that there is a nontrivial Ž D q 1.-cocycle Žform on G . for the Chevalley-Eilenberg w4x ŽCE. cohomology.
When H / e, the mappings w a : M ™ GrH are
the Goldstone fields, suitable extended to the analo-
4
Notice that the left action of G on G and on K Ždefined by
the left cosets gH . and the right action of H on G are both
compatible with the bundle projection.
187
gous of B above w1x. The construction of D’Hoker
and Weinberg shows that the WZW actions Ži.e.,
invariant actions associated with non-invariant spacetime Lagrangian densities. on the coset K are given
by non-trivial De Rham cocycles on K, i.e., by
closed non-exact forms on GrH. The result of w1,2x
may be reformulated by stating that the WZW actions are classified by the non-trivial cocycles of the
relative algebra cohomology H0 Ž G , H ;R. Žfor H s e,
H0 Ž G ,e;R. s H0 Ž G ;R.., and this approach will lead
us to a general expression for them. We shall restrict
ourselves here to the ungauged case, and will not
discuss the Žrelated. problem of gauging the WZW
actions w9x.
This paper is organised as follows: in Section 2
we review briefly the forms on coset manifolds and
the relative Lie algebra cohomology. Section 3 is
devoted to finding an explicit general formula for the
non-trivial cocycles on GrH for a simple group, and
in Section 4 we illustrate our result with applications.
The formal connection between the cocycles on GrH
and the expression for the Chern-Simons forms is
exhibited in Section 5, where the relation between
the two is clarified. The indices are as follows:
i, j, . . . refer to G Žor its algebra G ., i s 1, . . . ,dim G;
a , b , . . . to the subgroup H Ž H ., a s 1, . . . ,dim H,
and the indices a,b, . . . parametrise the coset K Žor
H ., a s 1, . . . ,dim K.
the vector space K s GrH
2. Forms on cosets and relative algebra cohomology
From the point of view of physical applications,
Lie algebra G cohomology groups are most conveniently described w4x in terms of forms on the associated simply connected group manifold G. For the
trivial representation r Ž G . s 0, the cohomology
groups H0q Ž G ,R. are characterised by a. closed and
b. Žsay. left invariant ŽLI. q-forms on G Žthe qcocycles. modulo those which are the exterior
derivative d of a LI form Žthe coboundaries.. Let v i
be a basis of LI one-forms on G so that
d v i s y 12 C jki v j n v k
or
d v s yv n v
Ž i , j,k s 1, . . . ,dim G . ,
Ž 2.1 .
J.A. de Azcarraga
et al.r Physics Letters B 419 (1998) 186–194
´
188
Ž v is the G-valued canonical form on G .. Then the
LI q-forms V on G may be written as
Vs
1
q!
V i1 . . . i qv i1 n . . . n v i q .
Ž 2.2 .
Then, those which determine Lie algebra q-cocycles
satisfy the condition
Ž s V . i1 . . . i qq 1 s y
1
1
2 Ž q y 1. !
Cwli 1 i 2 V l i 3 . . . i q x s 0 ,
Ž 2.3 .
where, in the CE formulation, the coboundary operator s may be identified with the exterior derivative
d.
In the language of forms the relatiÕe cohomology
with respect to a subalgebra H ; G is associated
with the notion of projectability of forms on G to the
coset manifold K s GrH. This notion, which plays
an essential role
ˆ in the Chern-Weil theory of characteristic classes, actually means that there is a unique
form V on GrH such that p ) Ž V . s V , where p )
is the pull-back of the canonical projection p :G ™
GrH. A q-form V is projectable if Žsee, e.g. w7x,
Volume II, p. 294.
V Ž X 1 , . . . , X q . s 0 if any X g H ,
Ž 2.4 .
Ž L X V . Ž X1 , . . . , X q .
a
q
sy
cocycles in H0 Ž G , H ;R.. These will define effective
actions of WZW type once they are pulled back to
an enlarged spacetime manifold of the appropriate
dimension.
3. An explicit formula for the cocycles on G r H
As is well known, the non-trivial primitive cocycles Ži.e., that are not the product of other cocycles.
on a simple group G are all of odd order 5 Ž2 m s y 1.,
s s 1, . . . ,l where l is the rank of G . They are
associated with the l primitive invariant symmetric
tensors k i 1 . . . i m of order m s Žand Casimir operators
s
of the same order. which may be constructed on G
w11x, the properties of which have been studied recently w3x. Given such a tensor k i 1 . . . i m , the G-Ž2 m1.-cocycle V Ž2 my1., or simply V , is a form on G
V A k l 1 . . . l my 1 s d v l1 n . . . n d v l my 1 n v s, so that its
coordinates are proportional to
1
V i1 . . . i 2 my 2 s A k l 1 . . . l my 1w s Cil11i 2 . . . Cil2my
my 3 i 2 my 2 x
Ž 3.1 .
Žany non-primitive terms in k do not contribute to
Ž3.1.; see Cor.3.1 in w3x.. Due to the full antisymmetry of the structure constants, any semisimple group
a
is reductive, Cab
s 0, Caba s 0 Ž a , b in H , a, b in
K ., i.e., w H , H x ; H , w H , K x ; K, w K, K x ; G .
The cocycles on GrH are the projectable cocycles
on G. To find general expressions for them for any
simple G consider the Ž2 m-1.-form VŽ p. Žcf. Ž3.1..,
VŽ p. s k a 1 . . . a py 1 i p . . . i my 1 b
Ý V Ž X1 , . . . , w Xa , X s x , . . . , X q . s 0 ,
ss1
1
1
=C aa11a 2 . . . C aa2py
C ai p2 py 1 a 2 p . . . C ai my
py 3 a 2 py 2
2 my 3 a 2 my 2
Ž 2.5 .
i.e., if V is ‘orthogonal’ to H Ž2.4. and it is
invariant under the right action of H Ž2.5.. In Ž2.5.,
L X is the Lie derivative with respect to the LI vector
field X; on LI forms, L X i v j s yC jki v k . As a result,
a q-form V is a non-trivial q-cocycle for the relative Lie algebra cohomology H0q Ž G , H ;R. w4x if a. it
is LI and closed, Eq. Ž2.3. Ži.e., it is a q-cocycle in
Z0q Ž G ;R..; b . it is projectable and c . it is not the
exterior derivative of a LI, projectable form Žin
which case it would be a coboundary.. Our task is
now to find the closed forms on the coset manifold
GrH, parametrised by w a, which are non-trivial
=v a1 n . . . n v a 2 p n v a 2 pq 1 n . . .
nv a 2 my 2 n v b .
Ž 3.2 .
This form clearly satisfies the condition Ž2.4.
Ž i X VŽ p. s 0. since v a Ž Xa . s 0 ; a in K and a in
a
H . The proof that L XaVŽ p. s 0, Eq. Ž2.5. follows
from L Xav a s yCaa b v b and the fact that the constants preceding v a1 n . . . n v a 2 my 2 n v b in Ž3.2.
5
See w3,16x for a detailed list of references on this point.
J.A. de Azcarraga
et al.r Physics Letters B 419 (1998) 186–194
´
may be viewed as products of the invariant polynomials k, C to which we may apply the following:
Lemma 3.1. Let k i 1 . . . i nq 1 and kXj1 . . . j mq 1 be two
invariant tensors on G Žsymmetry is not required
here.. Then,
La Ž k a1 . . . a nb kbX b 1 . . . m v a1 m . . . m v a n m v b 1 m . . .
mv b m . s 0
Ž 3.3 .
Setting i s s a s and jt s bt and using again the reductive property we obtain Ž3.4., q.e.d.
Since the cocycles on the coset manifold are LI
closed forms on K, we look now for a closed form.
Since La VŽ p. s Ž i Xad q di Xa . VŽ p. s i Xad VŽ p. s 0,
when computing d VŽ p. we may ignore the v a
components. A straightforward if somewhat lengthy
calculation, which uses the Jacobi identity and the
fact that the coset is reductive shows that
and
d VŽ p. s y 12 PŽ py1. q
La Ž k a1 . . . a n j kXjb 1 . . . m v a1 m . . . m v a n m v b 1 m . . .
mv b m . s 0.
Ž 3.4 .
Proof: By using the Ž G .-invariance of k we
obtain
189
2myp
2p
PŽ p. ,
Ž 3.5 .
where the 2 m-forms PŽ py1. and PŽ p. are given by
PŽ p. s k a 1 . . . a p i pq 1 . . . i m
1
=C aa11a 2 . . . C aa2ppy 1 a 2 pC ai pq
. . . C ai m2 my 1 a 2 m
2 pq 1 a 2 pq 2
Cai a1 k i a 2 . . . a nb q . . . qCai a nk a1 . . . a ny 1 i b q Cai b k a1 . . . a n i
=v a1 n . . . n v a 2 m .
s0 .
Now, using the fact that the coset is reductive, we
get
Caa a1 k a a 2 . . . a nb q . . . qCaa a nk a1 . . . a ny 1 a b q Cagb k a1 . . . a ng
Ž 3.6 .
For p s 0 we find that that PŽ0. s 0 due to the
G-invariance of k i 1 . . . i m . Let us now define a new
Ž2 m y 1.-form V by
m
s0
Vs
X
and similarly for k . Thus,
Ž Caa a k a a
1
2
a
. . . a nb q . . . qCa a n k a1 . . . a ny 1 a b
sy1
Ý a m Ž s . VŽ s. ,
am Ž s . ' Ł
1
m
a m Ž 1. ' 1 ,
am Ž m. s
q k a1 . . . a nb Ž Caa b 1 kbX a b 2 . . . b m q . . .
qCaa b m kbX b 1 . . . b my 1 a .
Then we find from Ž3.5.
s yCagb k a1 . . . a ng kbX b 1 . . . b m
dVs
y k a1 . . . a nb Cagb kgX b 1 . . . b m
s0 ,
ss1
1
sy
2
from which Ž3.3. follows. Eq. Ž3.4. is deduced from
the fact that k i 1 . . . i n j kXj j1 . . . j m is an invariant tensor on
G so that
žÝ
s
1
sˆ
n
1
m
q Ý Cak j t k i 1 . . . i n j kXj j1 . . . j tˆk . . . j m s 0 .
ts1
sy
2
2s
PŽ s.
/
a m Ž s q 1 . PŽ s.
Ý am Ž s .
1
m
2mys
ss0
m
y
n
ss1
y 12 PŽ sy1. q
Ž 3.7 .
my1
ss1
Ý Cak i k i . . . i k . . . i j kXj j . . . j
ž
,
Ž 2 m y 1. !
.
Ž m y 1 . !m!
m
Ý am Ž s .
r
rs1
ss1
. kbX b . . . b
2myr
2mys
s
PŽ s.
/
my1
ž
Ý
a m Ž s q 1. y a m Ž s .
2mys
ss1
q 12 a m Ž m . PŽ m. s 12 a m Ž m . PŽ m.
s
/
PŽ s.
Ž 3.8 .
J.A. de Azcarraga
et al.r Physics Letters B 419 (1998) 186–194
´
190
Hence d V will be zero if PŽ m. s 0 i.e., if k a 1 . . . a m
s 0. Thus, we have proven the following
Theorem 3.1. Let G be a simple, simply connected group and H a closed subgroup. A primitive
non-trivial Ž2 m y 1.-cocycle in H Ž2 my1. Ž G , H ;R. is
represented by the closed Ž2 m-1.-form V on the
coset GrH given in Ž3.7.. This form is defined
through Ž3.6. by a polynomial of order m on G
which vanishes on H , i.e., when all its indices take
values in H .
4. Applications
As an application of our general formula Ž3.7. let
us find the expression for the three- Ž m s 2. and
five- Ž m s 3. cocycles. Eq. Ž3.7. gives
V Ž3. s VŽ1. q a 2 Ž 2 . VŽ2.
s Ž k i 1 a 3 C ai 11 a 2 q 3k a 1 a 3 C aa11a 2 . v a1 n v a 2 n v a 3 ,
Ž 4.1 .
V Ž5. s VŽ1. q a 3 Ž 2 . VŽ2. q a 3 Ž 3 . VŽ3.
s Ž k i 1 i 2 a 5 C ai 11 a 2 C ai 23 a 4 q 5k a 1 i 2 a 5 C aa11a 2 C ai 23 a 4
q10k a 1a 2 a 5 C aa11a 2 C aa32a 4 . v a1 n . . . n v a 5 .
Ž 4.2 .
These two expressions have also been derived by
D’Hoker by a lengthier cohomological descent-like
procedure w2x, involving the consideration of nontrivial representations r of G . To exhibit the computational convenience of the general formula Ž3.7., we
give one further example, the seven-cocycle,
V Ž7. s VŽ1. q a 4 Ž 2 . VŽ2. q a 4 Ž 3 . VŽ3. q a 4 Ž 4 . VŽ4.
s Ž k i 1 i 2 i 3 a 7 C ai 11 a 2 C ai 23 a 4 C ai 35 a 6
q7k a 1 i 2 i 3 a 7 C aa11a 2 C ai 23 a 4 C ai 35 a 6
q21k a 1a 2 i 3 a 7 C aa11a 2 C aa32a 4 C ai 35 a 6
q35k a 1a 2 a 3 a 7 C aa11a 2 C aa32a 4 C aa53a 6 .
=v a1 n . . . n v a 7 .
Ž 4.3 .
Of course, the existence of these cocycles depends
on the existence of invariant polynomials of the
appropriate degree which are zero on H . These are
all known for all simple algebras Žsee also w3x in this
respect.; in particular all three exist for G s suŽ n.,
n G m.
Let us consider now some specific examples.
a. Ž SUŽ n.rSUŽ m. cosets .
Consider the case K s SUŽ3.rSUŽ2. ; S 5. To
construct a 5-cocycle on S 5 we need a 3rd-order
polynomial vanishing on SUŽ2.. This is provided by
the d i 1 i 2 i 3 polynomial which satisfies da 1a 2 a 3 s 0 ;a
g SUŽ2.. Similarly for the case K s SUŽ4.rSUŽ2.
we have two invariant polynomials of Ž3rd and 4 th
order. vanishing on SUŽ2., which give rise to the 5and 7-cocycles respectively, etc. For the case m ) 2,
we can always construct symmetric invariant polynomials on suŽ n. which are zero on suŽ m. Žsee w3x..
b . Ž Symmetric cosets .
The simplest cases in which formula Ž3.7. gives
rise to a non-trivial result are furnished by symmetric
cosets GrH Žw K, K x ; H .. Such examples, when
they exist, are simple because then all terms in Ž3.7.
have the same structure since C ac b s 0. They require
k a 1 . . . a m s 0 and that the components k a 1 . . . a my 1 a do
not all vanish. For instance, for G s SUŽ n., m s 3
and da 1a 2 a 3 s 0, the five-cocycle becomes proportional to
da 1a 2 a1C aa21a 3 C aa42a 5 v a1 n . . . n v a 5 .
Ž 4.4 .
Symmetric spaces in which the da 1a 2 a 3 vanish and
the da 1a 2 a1 do not, are provided by the families Žsee
w12x, p. 518. SUŽ n.rSO Ž n. and SUŽ2 n.rSpŽ2 n..
The simplest case, one which leads directly to a
Wess-Zumino term in a four dimensional field theory, is that of SUŽ3.rSO Ž3.. To clarify this, we
work with the generators TA s 12 l A of suŽ3., where
the l A are the set of standard Gell-Mann matrices.
For the soŽ3. generators, we take a g 2,5,74 and,
7
since C25
s 12 , w la , lb x s i ea bg lg . Then the coset
indices a g 1,3,4,6,84 . To see that it is correct to
write SUŽ3.rSO Ž3., we note that reduction of the
octet SUŽ3. with respect to SO Ž3. produces j s 1
and j s 2 SO Ž3.-multiplets, and we can argue that
integral j values only arise in the reduction of
triality zero SUŽ3. representations. Explicitly, we
can show that, for c g R
i
i
c Ž l1 y i l3 . ,
'2
'2 c Ž l4 q i l6 . , c l8 ,
i
yi
c Ž l 4 y i l6 . ,
Ž 4.5 .
'2
'2 c Ž l1 q i l3 . ,
J.A. de Azcarraga
et al.r Physics Letters B 419 (1998) 186–194
´
are the standard Racah components Tq , q s Ž2,1,0,y
1,y 2. of a rank 2 tensor operator of SO Ž3.. In this
example, one can see by inspection of the d-tensor
of SUŽ3., that the da 1a 2 a 3 all vanish, but there are
eight non-zero triples for which da 1a 2 a1 / 0. It is in
fact easy to see without explicit calculation that Ž4.4.
is a multiple of v 1 n v 3 n v 4 n v 6 n v 8.
To discuss this and other examples involving
G s SUŽ4., it is most convenient to generalise the
Gell-Mann l-matrices from SUŽ3. to SUŽ4. in a
fashion different from that in w3,13x and to use the d
and f tensors that follow from this new set. Thus, set
si
0
li s
ž
l8 s
(
1
2
0
,
0
/
ž
1
0
0
l iq12 s 0
ž
0
si ,
/
0
,
y1
/
where si are the three Pauli matrices. For i s 4 to 7,
we retain the l i of SUŽ3. so that for i s 4 to 7 and 9
to 12, the l i of w13x are used. In particular, l 3 , l8 , l15
are diagonal, all l’s are hermitian and l i for i g
2,5,7,10,12,144 are antisymmetric. In fact we have
only changed our choices of l8 and l15 , so that very
little further evaluation of d and f tensors is needed.
Consider first GrH s SUŽ4.rSO Ž4. in which
SO Ž4. is generated by the set of six antisymmetric
l’s just mentioned, while the SUŽ4. d-tensors are as
tabulated in w3x. The da 1a 2 a 3 do vanish, since they
correspond to the trace of products of three antisymmetric matrices, while for many triples the da 1a 2 a 3 /
0. Thus a simple five-cocycle is allowed in this
model with nine Goldstone fields. We may contrast
this with the non-symmetric reductive model
SUŽ4.rw SUŽ2. = SUŽ2.x in which the the subgroup
generators are l i for i g 1,2,3,13,14,154 . The relevant da 1a 2 a 3 obviously vanish but, since the C ac b are
not all zero, all the three terms of Ž4.1. survive
giving a much more complicated Wess-Zumino term
for this model with 15 y 6 s 9 Goldstone fields.
Consider next the symmetric coset SUŽ4.rSwUŽ2.
= UŽ2.x, where H is larger than in the previous
example, with the extra generator l8 . The da 1a 2 a 3 do
not all vanish now, and hence there are no fivecocycles of type Ž3.7. w1x. Finally, consider the fivedimensional symmetric coset SUŽ4.rSpŽ4,R.. A presentation of C2 s spŽ4,R. in Cartan-Weyl form with
positive roots r 1 s Ž1,y 1., r 2 s Ž0,2., r 3 s Ž1,1.,
191
r4 s Ž2,0. can be given in terms of the above 4 = 4
l-matrices of SUŽ4.. Writing '2 E " m s Xm " iYm for
m s 1,2,3,4 for the raising and lowering operators
associated with the roots, the realisation is
H1 s l 3 ,
X 4 s l1 ,
H2 s l15 ,
Y4 s l 2 ,
X 2 s l13 ,
Y2 s l14 ,
'2 X1 s l4 y l11 , '2 Y1 s l5 q l12 ,
'2 X 3 s l6 q l9 , '2 Y3 s yl7 q l10 .
Ž 4.6 .
This enables an explicit check that the corresponding
da 1a 2 a 3 are indeed all zero, an easy tabulation of C2
g
structure constants Cab
, Ž1 F a , b ,g F 10., and the
evaluation of the five-cocycle of the model.
There are no seven-dimensional symmetric cosets.
Consider then the case of the 9-cocycle determining
a Wess-Zumino term in D s 8 spacetime. For GrH
symmetric, all the coordinates of V Ž9. s V aŽ9.
v a1
1 . . . a9
a9
n . . . n v become proportional to
k aŽ5.1a 2 a 3a 4 w a1C aa21a 3 C aa42a 5 C aa63a 7 C aa84a 9 x .
Ž 4.7 .
Let GrH s SUŽ n.rSO Ž n. with soŽ n. generated by
the 12 nŽ n y 1. n G 4 imaginary antisymmetric la
matrices, the coset generators l a being real symmetric and traceless. Then, if we take k i 1 . . . i 5 ;
sTrŽ l i 1 . . . l i 5 . it is obvious that sTrŽ la 1 . . . la 5 . is
zero but that sTrŽ la 1 . . . la 4 l a . is not. Hence we will
get a Wess-Zumino term from
V aŽ9.
v a1 n . . . n v a 9 ,
1 . . . a9
V aŽ9.
A k aŽ5.1a 2 a 3a 4 w a1C aa21a 3 C aa42a 5 C aa63a 7 C aa84a 9 x .
1 . . . a9
Ž 4.8 .
In Ž4.8. we may use equivalently d Ž5. for the suŽ n.
polynomial, Žsee w3x..
5. Relative cohomology and Chern-Simons forms
Let us consider VŽ p. in Ž3.2. further. First we
introduce
k a 1 . . . a py 1 i p . . . i my 1 b
1
s
m!
sTr Ž ad Xa 1 . . . ad Xa py 1ad X i p . . .
=ad X i my 1ad X b . ,
Ž 5.1 .
J.A. de Azcarraga
et al.r Physics Letters B 419 (1998) 186–194
´
192
which arises by restricting the indices of the invariant symmetric polynomial of order m on G , given
by a symmetric trace, to the appropriate values.
Next, we make the identifications
which leads to
V Ž2 my1.
2 my 1
s
Ž m y 1. !
W a s y 12 C aab v a n v b ,
my1
p
Ý Ž y1. Ž m y p y 1. ! Ž 2 m y 1. PPP
i
i
U asv < K ,
Ž U n U . ' Ž U 2 . s 12 C ai b v a n v b ,
Ž 5.2 .
and V s v < H , where v s u du. Thus, V determines the LI invariant H-connection and W the
V q V n V Žwhich leads
associated curvature, W s dV
to W a in Ž5.2. using the Maurer-Cartan eqs... Then,
the form in Ž3.2. may be rewritten as
ps0
= Ž 2 m y p . Tr S W p Ž U 2 .
½
y1
2 my 1
py1
m!
sTr W py1 Ž U 2 .
myp
5
U .
/
Now, recalling the expression of the Beta function,
U4 .
1
H0 dt t
ly1
Ž1yt .
sy 1
Ž l y 1. ! Ž s y 1. !
,
Ž l q s y 1. !
s
Ž 5.3 .
If we replace the m! terms in sTr by the sum S
over all possible products Ž‘words’. which contain a
total power Ž p y 1. wŽ m y p .x of the curvature W
wcomponent U x, Eq. Ž5.3. may be written as
my py1
Ž 5.6 .
B Ž l, s . s
VŽ p. s Ž y1 .
ž
Ž 5.7 .
we see that, renaming p ™ Ž m y p y 1., V may be
written in the form
V Ž2 my1.
2 my 1
s
VŽ p. s
2 my 1
Ž m y 1. !
Ž y1.
py1
Ž m y 1. !
Ž p y 1. ! Ž m y p . !
my1
=
½
=Tr S
Ž W py1 Ž U 2 .
my p
.
5
U .
Ý Ž y1. my py1 Ž p . ! Ž 2 m y 1. PPP
ps0
Ž 5.4 .
= Ž m q p q 1 . Tr S W mypy1 Ž U 2 .
As a result, the general expression Ž3.7. for the
Ž2 m y 1.-cocycle on the coset K may be rewritten
as
V Ž2 my1.
2
½
s 2 my1
Ž 2 m y 1. !
Ž m y 1 . ! m!
½
=Tr S
my1
p
U
5
p! m!
Ý Ž y1. my py1 Ž m q p . !
ps0
p
Ž W mypy1 Ž U 2 . .
U
5
my 1
s
Ž m y 1. !
s Ž y1 .
my1
p
Ý Ž y1. Ž p . ! Ž m y p y 1. ! a m Ž p q 1.
ps0
my 1 my1
my1
=
Ý
ps0
=Tr S W p Ž U 2 .
½
ž
my py1
/
5
U ,
Ž 5.5 .
2
1
H0 dt m t
Ž 2 m y 1. !
Ž m y 1 . ! m!
my1
Ž t y 1.
=Tr S W mypy1 Ž U 2 .
½
p
p
5
U .
Ž 5.8 .
J.A. de Azcarraga
et al.r Physics Letters B 419 (1998) 186–194
´
The reader will recognise that the integral in Ž5.8. is
formally identical to that giving the expression of the
Chern-Simons form w14x V Ž2 my1. of the Chern character ch m which are relevant in the theory of nonabelian anomalies Žsee w15x and references therein..
This means that if we know the coefficients which
determine the terms for the Chern-Simons forms of
various orders Žsee, e.g. Žsee w16x, §10.13.., we also
know the cocycles V in Ž5.5., Ž5.6. or Ž5.8. and
viceversa. This explains the similarity between the
two types of Ž2 m-1.-forms. In general, the Ž2 m.-form
Ž2 my1.
d V CS
gives the Chern character ch m ; in contrast, the cocycle form V Ž2 my1. is closed, d V Ž2 my1.
s 0, since PŽ m. in Eq. Ž3.8., Ž3.6. is zero precisely
for the polynomials which vanish on H .
We may now use Ž5.8. to recast from it the
expression of the WZW terms calculated previously.
For m s 2,3, Eq. Ž5.8. gives
The Žglobal. Žy. my 1 2 my1
193
Ž2 m y1 . !
factor in Ž5.8.
Ž m y1 . ! m !
came from the definition of VŽ p. in Ž3.2.. By replacm
ing this factor by Ž 2ip . m1! Ž2p . we may adjust it so
2 my1
that the U
term has the standard factor Žy1. my 1
m
y1
Ž 2ip . ŽŽ2mmyy11..!! Ž2p . Ži.e., 121p , 240yip 2 , 6720
p 3 for the 3,
5, 7 cocycles in Ž5.9., Ž5.10. and Ž5.11...
Acknowledgements
Two of the authors ŽJ.A. de A. and J.C.P.B.. wish
to thank the hospitality of DAMTP, where this work
was started. This paper has been partially supported
by research grants from the MEC, Spain ŽPB96-0756.
and PPARC, UK. J.C.P.B. wishes to thank the Spanish MEC and the CSIC for an FPI grant.
V Ž3. s y3! Tr Ž W U . y 13 Tr Ž U 3 .
s 3k a 1 a 3 C aa11a 2 q k i1 a 3 C ai 11 a 2 v a1 n v a 2 n v a 3 ;
Ž 5.9 .
V Ž5. s 2 2
5!
2! 3!
Tr Ž W 2 U . y 14 Tr Ž W U 3 q U 2 W U .
q 101 Tr Ž U 5 .
s Ž 10k a 1a 2 a 5 C aa11a 2 C aa32a 4 q 5k a 1 i 2 a 5 C aa11a 2 C ai 23 a 4
qk i 1 i 2 a 5 C ai 11 a 2 C ai 23 a 4 . v a1 n . . . n v a 5 ,
Ž 5.10 .
i.e., Eqs. Ž4.1. and Ž4.2. which have been reordered
to show the origin of their terms. Similarly, for
m s 4 we obtain Žcf. Ž4.3..,
7!
V Ž7. s y2 3
Tr Ž W 3 U .
3! 4!
y 15 Tr Ž 2W 2 U 3 q W U 2 W U .
q 151 Tr Ž 3W U 5 . y 351 Tr Ž U 7 .
s Ž 35k a 1a 2 a 3 a 7 C aa11a 2 C aa32a 4 C aa53a 6
q21k a 1a 2 i 3 a 7 C aa11a 2 C aa32a 4 C ai 35 a 6
q7k a 1 i 2 i 3 a 7 C aa11a 2 C ai 23 a 4 C ai 35 a 6
q k i 1 i 2 i 3 a 7 C ai 11 a 2 C ai 23 a 4 C ai 35 a 6 . v a1 n . . . n v a 7 .
Ž 5.11 .
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J.M. Izquierdo, A.J. Macfarlane, Ann.
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Phys. ŽNY. 202 Ž1990. 1.
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w12x S. Helgason, Differential geometry, Lie groups and Symmetric Spaces, Academic Press, 1978.
w13x H. Hayashi, I. Ishiwata, S. Iwao, M. Shako, S. Yakeshita,
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w15x R. Stora, Jaca lectures, in J. Abad et al. eds., New perspectives in quantum field theories, World Scientific, 1986, pp.
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12 February 1998
Physics Letters B 419 Ž1998. 195–198
Macroscopic string-like solutions in massive supergravity
Harvendra Singh
1
2
I.N.F.N. Sezione di PadoÕa, Departimento di Fisica ‘‘Galileo Galilei’’, Via F. Marzolo-8, 35131 PadoÕa, Italy
Received 27 October 1997
Editor: L. Alvarez-Gaumé
Abstract
In this report we obtain explicit string-like solutions of equations of motion of massive heterotic supergravity recently
obtained by Bergshoeff, Roo and Eyras. We also find consistent string source which can be embedded in these backgrounds
when space-time dimension is greater than or equal to six. q 1998 Elsevier Science B.V.
Recently, there has been regenerated interest in
the study of massive supergravity theories w1–4x and
the generalised Scherk and Schwarz dimensional reduction scheme w5,1x. In generalised Scherk-Schwarz
ŽGSS. toroidal reduction some of the fields are given
linear dependence along the coordinates of the torus.
Thus the resultant compactified theory in the lower
dimensions possesses mass like parameters. However, when these parameters are set to zero the
massive theory reduces to the ‘standard’ Žmassless.
supergravity. Another point to be imphasized here is
that the massive supergravities obtained through GSS
reduction, in general, possess smaller duality symmetry groups than their massless counterparts w4,3x.
Fundamental string solution w6x was obtained in
the spacetim es w hich are asym ptotically
Minkowskian. Later on other fundamental solutions
1
2
This work is supported by INFN fellowship.
E-mail: [email protected].
Žwith source terms. as well as solitonic solutions
Žwithout source terms. were obtained for any p-brane
in D spacetime dimensions w7x. For these solutions
masses and their respective charges saturate the Bogomol’nyi-Prasad-Sommerfeld bound and therefore
the supersymmetry in the theory dictates that these
classical solutions are the quantum mechanically exact solutions of the theory. It is now natural to ask
what does happen if the spacetime around a p-brane
is not asymptotically flat or if there is nontrivial
dilatonic potential in the theory. Such examples are
provided when we consider massive supergravities,
e.g., massive IIA supergravity in D s 10 w8x and its
subsequent dimensional reductions w1,4x, various GSS
reductions of massless type II w1,2x and recently of
heterotic strings in ten dimensions w3x. All these
massive theories have some kind of dilatonic potentials either in the NS-NS sector as is the case with
massive heterotic of Bergshoeff, Roo and Eyras or in
the R-R sector as was the case with massive type IIA
of Romans w8,2x or in both the sectors as was antici-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 5 0 6 - 2
H. Singh r Physics Letters B 419 (1998) 195–198
196
pated in w4x in order to obtain maximally symmetric
black hole solutions analogous to w9x.
In the present work, first, we shall obtain explicit
string like solutions of source free string equations of
motion in arbitrary spacetime dimensions with nontrivial dilatonic potential of the form suggessted in
the massive heterotic string w3x. In these solutions
spacetime has explicit O Ž1,1. = O Ž D y 2. symmetry
and does not have asymptotic flatness. Secondly, we
shall show that a string like source could be consistently embedded in these spacetimes when D G 6.
We consider the following effective action in
D-dimensional target space,
motion derived from Ž3. are satisfied for the following choice of the background fields
y Ž Dy4 .
Ž Dy2 .
ds 2 s U
2
qU
H '
2 P 3!
B s 12 Bmn dx m n dx n s y
Hmnl H mnl y 2 m2 ,
Ž 1.
where m is the mass term Žor cosmological constant. as in w3x or an analog of central charge deficit
term, F is the dilaton field and gmn is the s y model
metric. We have taken 2 k 2 s 1 s 2pa X . The antisymmtric field strength is expressed as,
Hmnl s Em Bnl q cyclic permutations.
Ž 2.
Above action for D s 4 can be obtained from the
appropriate truncation of the massive theory described in w3x. Since it would be convenient to work
in the Einstein Žcanonical. frame let us write down
the action in the canonical metric, gmn s e f Gmn
a
2
S s d D x'y G R G y 12 Em fE mf
H
4
ya f
y
2 P 3!
1
U
dx 0 dx 1 ,
Ž 4.
m2
2
Ž t2yx2 . q
Q2
< y y y 0 < Dy 4
,
for D ) 4,
2
1
2
Ž dy12 q PPP qdyDy2
.,
a
f s y lnU,
2
Us1q
1
Ž Dy2 .
provided the potential has the following form:
S s d D x y g ey2F R g q 4 EmFE mF
y
Ž ydt 2 q dx 2 .
e
Hmnl H
mnl
2
y2m e
aŽ Dy2 .
f
,
Ž 3.
8
where a s D y
and rescaled dilaton is F s fa .
2
One can derive equations of motion from the action
Ž3.. We obtain the solutions to these equations in two
cases; when Bmn / 0 and when Bmn s 0. The latter
solutions we describe as the vacuum solution Žpure
dilaton gravity..
Case-I Bmn / 0. We find that the equations of
(
s1q
m2
2
Ž t 2 y x 2 . y Q2 ln < y y y 0 < ,
for D s 4,
Ž 5.
where Q 2 is the charge associated with the 2-from
gauge field Bmn and is given by
Q2 s
HS
eya f ) H.
Dy3
Ž 6.
The symbol ) stands for Hodge dual operation and
y 0 is some point in the transverse y-plane.
Obviously, the background solution obtained in
Ž4. is similar to the spacetime of a fundamental
string solution in w6x. Only difference is of an explicit
m-dependent term in Ž5.. In the limit m ™ 0 background in Ž4. reduces to asymptotically flat spacetime around a fundamental string. There is a curvature singularity at y s y 0 which can be smoothened
by introducing some string-like source at y s y 0 .
Case-II Bmn s 0. In this case we have a pure
dilatonic gravity with the dilatonic potential. The
equations of motion are still satisfied by the same
ansatz as in Ž4. for the remaining fields while the
potential in Ž5. becomes
Us1q
m2
2
Ž t 2 y x 2 . , for D ) 2.
Ž 7.
H. Singh r Physics Letters B 419 (1998) 195–198
Thus we also have solution independent of the antisymmetric field strength. When this background is
substituted into the action it becomes
4 m2
Ss dD x
H
Dy2
,
Ž 8.
2
m
which shows there is finite lagrangian density D4y
2
per unit D-dimensional Minkowski volume.
We now introduce string-like source in the form
of s-model world-sheet action
a
Ss s y
1
2
Hdt d s 'y g g
ij
M
N
E i X E j X GM N e
2
f
q e i jE i X ME j X N BM N ,
Ž 9.
where g i j is the induced metric on the string worldsheet. The f-dependence is chosen in accordance
with w7x so that, when m ™ 0, under the rescaling
4
GM N ™ l
Dy2
B M N ™ l 2 BM N ,
GM N ,
4 Ž Dy4 .
f
e ™l
Ž Dy2 . a
ef ,
g i j ™ l2g i j ,
Ž 10 .
both actions S and Ss scale in the same way
S ™ l2 S,
Ss ™ l2 Ss .
Ž 11 .
When m / 0 such rescaling of the action does not
hold, see w10x. Now, the above string like source can
be embedded in the spacetime Ž4. if we make the
following choice of static gauge for the world-sheet
coordinates,
0
1
X st , X ss ,
r s 1, . . . , D y 2,
X s constant,
a
M
N
g i j s E i X E j X GM N e
2
d2
dt 2
m
X m s 2 G 00m q H00
s 0,
.
Ž 12 .
Ž 13 .
For D - 6 above condition Ž13. is violated near
the end points of the string, i.e., when x ™ "`.
Therefore we have difficulty to find appropriate
Ž 14 .
where Gnsm are the Christoffel connections in the
canonical metric Gmn . The result in Eq. Ž14. suggests
that net transverse force between two such strings
vanishes. Thus it can be argued that a multi-string
solutions could also be obtained in this configuration. For multi-string case the potential U in Ž5.
modifies to
Us1q
m2
2
Ž t2yx2 . q Ý
n
Q2
< y y yn < Dy 4
,
for D ) 4,
m2
2
for D s 4,
f
That is to say, Eqs. Ž4. and Ž12. together satisfy all
the field equations derived from actions S and Ss if
considered together at least for D G 6. The essential
requirement for the embedding of the source Ž9. is
that the string coupling e f vanishes at y s y 0 , i.e.
Uy ™ y 0 ™ `.
source when D - 6. However, if the end points are
excluded, which is possible for temporal evolution
Ž t ) x ., then the source action in Ž9. is a good choice
even for D - 6. Thus in Ž4. we have got a background configuration in which a charged macroscopic string is embedded in a cosmological spacetime. These solutions have explicit O Ž1,1. = O Ž D y
2. symmetry. The symmetry gets automatically enhanced to P2 = O Ž D y 2. when m ™ 0. Here P2
stands for Poincare´ symmetry in two dimensions.
This constitutes our main result.
Next we shall try to find multi-string solution in
asymptotically non-flat spacetime Ž4.. First we calculate the net transverse force a test string is subjected
to when another string is brought close to it. We still
have w6x
s1q
r
197
Ž t 2 y x 2 . y Q2 Ý ln < y y yn < ,
n
Ž 15 .
where n is the number of strings fixed at the positions yn in the y-plane. It is not easy to calculate
ADM energy for these solutions. We do not know
whether their masses and respective charges will
saturate the BPS bound as they do in the case when
m s 0. However, since we can control the value of
the mass parameter m it appears to us that for m ; 0
the solutions obtained above will only be slightly off
extremal.
To analyse briefly the evolution and spacetime
properties of the metric in Ž4. we consider specific
H. Singh r Physics Letters B 419 (1998) 195–198
198
case of D s 6 and of single string source positioned
at y s 0. The metric is
y1
2
ds s U
2
2
2
Ž ydt q dx .
qU Ž dy 12 q PPP qdy42 . ,
Ž 16 .
Acknowledgements
I have been benifited by some useful discussions
with Prof. M. Tonin and specially with Prof. J.
Maharana. The beginning part of this work was
carried out at Institue of Physics, Bhubaneswar.
2
with U s 1 q m2 Ž t 2 y x 2 . q Qy 22 . The spacetime is
non-static and the evolution is dragged by a Lorentz
invariant quantity involving the directions tengential
to the string world-sheet. The evolution could be
light-like if the quantity Ž t 2 y x 2 . is greater than
zero and it could be also space-like if Ž t 2 y x 2 . is
less than zero. The spacetime can be viewed as a
solenoid whose axis is identified with the x-direction. At any given time the metric in Ž16. is not well
defind in whole space since U becomes negative
whenever 1 q m 2 t 2r2 q Q2ry 2 - m 2 x 2r2. However, a light signal travelling in x-direction will not
see any discontinuity and only will detect the geometry around a fundamental string straightened at y s 0.
For a possible m2 - 0 Žanti-de-Sitter. case, situation
is more interesting. There is a critical time t c s < m22 <
such that for t - t c the metric is well defind in the
whole space except at y s 0 which is the position of
the string source. As < m 2 < ™ 0 we can see that
t c ™ `. This implies that for infinitesimally small
values of the cosmological constant, or for a slow
rate of evolution, it will take infinitely longer time
for any irregularities to set in in the space. But m2
cannot become negative if it is the mass term comming from GSS reduction w3x. Certainly there must
be some other source for negative cosmological constant in the NS-NS sector if that has to happen.
(
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12 February 1998
Physics Letters B 419 Ž1998. 199–205
String unification scale and the hyper-charge Kac-Moody level
in the non-supersymmetric standard model
Gi-Chol Cho
b
a,1
, Kaoru Hagiwara
a,b
a
Theory Group, KEK, Tsukuba, Ibaraki 305, Japan
ICEPP, UniÕersity of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan
Received 29 September 1997; revised 19 November 1997
Editor: H. Georgi
Abstract
The string theory predicts the unification of the gauge couplings and gravity. The minimal supersymmetric Standard
Model, however, gives the unification scale ; 2 = 10 16 GeV which is significantly smaller than the string scale ; 5 = 10 17
GeV of the weak coupling heterotic string theory. We study the unification scale of the non-supersymmetric minimal
Standard Model quantitatively at the two-loop level. We find that the unification scale should be at most ; 4 = 10 16 GeV
and the desired Kac-Moody level of the hyper-charge coupling should be 1.33 Q k Y Q 1.35. q 1998 Elsevier Science B.V.
The theory of E8 = E8 heterotic string w1x has
some attractive impacts on the model of low-energy
particle physics. The theory has a potential of explaining the low-energy gauge groups, the quantum
numbers of quarks, leptons and the Higgs bosons,
the number of generations, and the interactions
among these light particles which are not dictated by
the gauge principle. One of the immediate consequences of the string theory is the unification of the
gauge interactions and the gravity. Since, in the
string theory, gravitational and gauge interactions are
naturally related, the strength of the gauge couplings
and the unification scale are both given by the
Newton constant. The unification scale of the heterotic string theory is predicted to be w2,3x
mU < string f 5 = 10 17 GeV,
Ž 1.
in the weak coupling limit where the 1-loop string
1
Research Fellow of the Japan Society for the Promotion of
Science.
effects are taken into account. On the other hand, the
minimal supersymmetric Standard Model ŽMSSM.
predicts the unification scale
mU < MS SM f 2 = 10 16 GeV,
Ž 2.
by using the recent results of precision electroweak
measurements as inputs. The discrepancy between
Ž1. and Ž2. is a few percent of the logarithms of
these scales. However the extrapolation of Ž1. to the
weak scale leads the experimentally unacceptable
values of sin2u W and a s under the hypothesis that
the spectrum below the string scale is that of the
MSSM. Various attempts to modify this naive prediction are reviewed in Ref. w3x. For instance, the
2-loop string effects are not known. On the other
hand, it has been suggested w4x that the strong coupling limit of the E8 = E8 heterotic string theory,
which is considered to be the 11-dimensional M-theory, can give rise to a significantly lower string scale
than the estimation Ž1. in the weak coupling limit.
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 8 8 - 3
G.-C. Cho, K. Hagiwarar Physics Letters B 419 (1998) 199–205
200
Alternatively, the gauge coupling unification scale
can be modified in string theories with non-standard
Kac-Moody levels. The coupling constant g U , which
is related to the Newton constant in the string theory,
is expressed in terms of the SUŽ3. C , SUŽ2.L and
UŽ1. Y gauge couplings and the corresponding KacMoody level k i Ž i s Y,2,3. as w5x
g U2 s k i g i2 ,
Ž 3.
at the unification scale mU . The factor k i should be
positive integer for the non-Abelian gauge group. On
the other hand, for the Abelian group, its value
depends on the structure of four-dimensional string
models. In view of the gauge field theory, k i plays
the role of a normalization factor for g i and, for
example, the set Ž k Y ,k 2 ,k 3 . s Ž5r3,1,1. is taken to
embed the hyper-charge Y in the SUŽ5. GUT group.
It has been known that the SUŽ5. grand unification is not achieved if one extrapolates the observed
three gauge couplings by using the renormalization
group equations ŽRGE. in the minimal Standard
Model ŽSM.. It has been noted w3x, however, that the
trajectories of the SUŽ2.L and the SUŽ3. C couplings
intersect at near the unification scale mU predicted
by the string theory: for example, the leading order
RGE with a certain choice of the weak mixing angle
and the QED coupling in the MS scheme,
sin2u W Ž m Z . MS s 0.2315,
Ž 4a .
1ra Ž m Z . MS s 128,
Ž 4b .
gives the following results,
mU f 1 = 10 17 GeV for a s Ž m Z . s 0.118,
17
f 2 = 10 GeV for a s Ž m Z . s 0.121.
Ž 5a .
Ž 5b .
The above unification scale mU is remarkably close
to the string scale Ž1., which may suggest the string
unification without supersymmetry for the KacMoody level k Y f 1.27 for k 2 s k 3 s 1.
Of course, deserting supersymmetry ŽSUSY. after
compactification into four-dimension means that both
the gauge hierarchy and the fine-tuning problems
have to be solved without SUSY. The existence of a
consistent string theory without the four-dimensional
SUSY has not been demonstrated. It has been argued
that the solution to these problems, if it exists,
should be intimately related to the vanishing of the
cosmological constant; see, e.g., Ref. w3x for a review
of some exploratory investigations. Recently, as an
application of this idea of minimal particle contents,
the mechanism of baryogenesis in non-SUSY, nonGUT string model has been proposed w6x.
In this letter we examine quantitatively at the
next-to-leading-order ŽNLO. level the possibility of
the string unification of the gauge couplings in the
SM without SUSY. Because, in the string theory, the
UŽ1. Y coupling can be rather arbitrarily normalized
by the Kac-Moody level k Y , we define mU as the
scale at which the trajectories of the SUŽ2.L and the
SUŽ3. C running couplings intersect with k 2 s k 3 s
1 2 . Our purposes are to find the scale mU and the
corresponding k Y under the current experimental
and theoretical constraints on the parameters in the
minimal SM. In the NLO level, the scale mU is not
only affected by the uncertainty in the SUŽ3. C coupling but also by threshold corrections due to the SM
particles such as the top-quark and the Higgs boson.
The top-quark Yukawa coupling affects the RGE at
the two-loop level. Therefore, it is interesting to
examine whether the scale mU in the minimal SM
still lie in the string scale ; O Ž10 17 GeV. after the
NLO effects are taken into account.
We first evaluate quantitatively the UŽ1. Y and
SUŽ2.L MS couplings at the weak scale boundary of
the RGE. The magnitudes of the MS couplings are
determined in general by comparing the perturbative
expansions of a certain set of physical observables
with the corresponding experimental data. The correspondence can be made manifest by using the effective charges e 2 Ž q 2 . and s 2 Ž q 2 . of Ref. w7x. The MS
couplings aˆ Ž m . s eˆ 2 Ž m .r4p and aˆ 2 Ž m . s
gˆ 22 Ž m .r4p are related with the effective charges as
1
1
2
s
. aˆ Ž m .
s2 Ž q2 .
1
s
2
aˆ 2 Ž m .
aŽq .
aŽq
QQ
q 4p RePT ,g Ž m ;q 2 . ,
3Q
q 4p RePT ,g Ž m ;q 2 . ,
Ž 6a .
Ž 6b .
where a Ž q 2 . s e 2 Ž q 2 .r4p . The explicit form of
the vacuum polarization functions P TA,VB Ž m ;q 2 . in
the SM can be found in Appendix A of Ref. w7x. The
above expressions are manifestly RG invariant in the
2
No attractive solution is found for k 2 / k 3 .
G.-C. Cho, K. Hagiwarar Physics Letters B 419 (1998) 199–205
one-loop order and give good perturbative expansions at q 2 s m 2Z for m s m Z . We hence need as
inputs a Ž m2Z . and s 2 Ž m2Z .. Recent estimate of the
hadronic contribution to the running of the effective
QED charge finds w8x
1ra Ž m2Z . s 128.75 " 0.09.
Ž 7.
All the other recent estimations w9x find consistent
results. Relation between the running QED charge of
Refs. w8,9x and the effective charge a Ž q 2 . of Ref. w7x
that contain the W-boson contribution is found in
Ref. w10x. The effective charge s 2 Ž m2Z . is measured
directly at LEP1 and SLC from various asymmetries
on the Z-pole w7,10x. In the SM, however, its magnitude can be accurately calculated as a function of m t
and m H through the following formula w7,10x,
s
2
Ž
m2Z
.s
)
y
1
4
1
2
y 4pa Ž m2Z .
ž
4'2 GF m2Z
q
16p
/
where GF and a are the Fermi coupling constant
and the fine structure constant, respectively. Accurate parametrizations of the SM contributions to the
S and T parameters w11x are found in Ref. w10x, as
functions of the scaled mass parameters
m t Ž GeV . y 175 GeV
x H ' log
10 GeV
m H Ž GeV .
,
Ž 9a .
.
Ž 9b .
100 GeV
Finally the MS coupling of the effective 5-quark
QCD has been estimated as w12x
a s Ž m Z . s 0.118 " 0.003.
Ž 10 .
For later convenience, we introduce the following
parametrizations to the observed and calculated values of the three effective charges of the SM:
1
s 128.75 q 0.09 x a ,
Ž 11a.
a Ž m2Z .
s
2
x s ' Ž a s Ž m Z . y 0.118 . r0.003,
Ž 12a.
x a ' Ž 1ra Ž m2Z . y 128.75 . r0.09.
Ž 12b.
The three MS couplings of the SM that enter as the
boundary condition of the 2-loop RGE are then
determined via Eqs. Ž6b. and the corresponding
matching equation of the 5-quark and 6-quark QCD
as follows:
kY
s
aˆ 1 Ž m Z .
m2Z
m 2Z
Ž .
s 29.66 y 0.044 x t q 0.067x H q 0.002 x H2
aŽ .
y0.01 x a ,
a s Ž m Z . s 0.118 q 0.003 x s ,
1 y s 2 Ž m2Z .
aŽ
m2Z
Ž 11b.
Ž 11c.
.
y 0.77 q 0.19log
mt
mZ
,
Ž 13a.
1
s
aˆ 2 Ž m Z .
s 2 Ž m2Z .
aŽ
m2Z
.
aˆ 3 Ž m Z .
y 0.11 q 0.12log
1
s
S
Ž 8.
xt '
where x s and x a are defined as
1
1 q 0.0055 y a T
201
1
q
a s Ž mZ .
3p
log
mt
mZ
.
mt
mZ
,
Ž 13b.
Ž 13c.
We use Ž13a. to Ž13c. as inputs to determine the
unification scale mU , and the relation aˆ 1Ž m . s
k Y aˆ Y Ž m . to fix the desired Kac-Moody level k Y .
The estimates Ž7. and Ž10. give, respectively,
x a s 0 " 1 and x s s 0 " 1. The observed top-quark
mass w13x m t s 175 " 6 GeV gives x t s 0 " 0.6.
The global fit including the electroweak precision
experiments gives w10x m t s 172 " 6 GeV, or x t s
y0.3 " 0.6. The error estimate of Eq. Ž7. is conservative w10x, while that of Eq. Ž10. may be too
optimistic. We will therefore explore the region of
< x a < Q 1, < x t < Q 1 and < x s < Q 2. As for the Higgs boson mass m H , the measurements of s 2 Ž m 2Z . and the
other electroweak observables constrain it indirectly
w10x, while the direct search at LEP gives m H R
70 GeV. In addition, there are theoretical bounds,
both the lower and the upper limits in order for the
minimal SM to be valid up to the unification scale
mU . The lower limit of m H is obtained from the
stability of the SM vacuum. Its recent evaluation
w14,15x finds
m H ) 137.1 q 21 x t q 2.3 x s GeV
for L ; 10 19 GeV.
Ž 14 .
Since the dependence on the cut-off scale L is found
to be small for L ) 10 15 GeV w14x, we can adopt Eq.
Ž14. as the lower limit of m H for L ; mU . On the
other hand, the upper bound is obtained by requiring
202
G.-C. Cho, K. Hagiwarar Physics Letters B 419 (1998) 199–205
the effective Higgs self-coupling to remain finite up
to the cut-off scale L. A recent study finds w16x;
m H - 260 " 10 " 2 GeV for L ; 10 15 GeV,
Ž 15 .
where the first error denotes the uncertainty of theoretical estimation and the second one comes from the
experimental uncertainty in m t . Since the m t-dependence of the upper limit is rather small, and since the
upper limit decreases as L increases, we set the
upper limit of m H to be 270 GeV for L ; mU . In
summary, we consider the following range of the
Higgs boson mass
137.1 q 21 x t q 2.3 x s - m H Ž GeV . - 270,
Ž 16 .
in our analysis. We show in Fig. 1 the allowed
region of the Higgs boson mass which is obtained
from the SM fit to all electroweak precision measurements w10x, where the contours are obtained from
the SM fit to all the electroweak data and the external constraints m t s 175 " 6 GeV w13x, a s s 0.118
" 0.003 w12x and 1ra Ž m 2Z . s 128.75 " 0.09 w8x.
Theoretically allowed region of m H , Eq. Ž16., is also
shown in the figure. It is clearly seen that the
theoretically allowed range of m H with L )
10 16 GeV is in perfect agreement with the constraint
from these precision electroweak experiments.
The 2-loop RGE for the gauge couplings aˆ i Ž m .
in the MS scheme is given as follows:
m
d aˆ i
1
s
dm
2p
bi aˆ i2 q
1
8p
a 2 bi j aˆ j q c i k
2 ˆi
yˆ k2
4p
,
Ž 17 .
where i s 1,2,3 and k s t,b,t . The UŽ1. hypercharge normalization is taken as aˆ 1 s k Y aˆ Y . The
term yˆ k denotes the MS Yukawa coupling. The
coefficients bi ,bi j and c i k are given in the minimal
SM as w17x
bi s
ž
1 41
kY 6
,y
19
6
/
,y 7 ,
Ž 18a.
°1
199
1 9
1 44
k Y2
18
kY 2
kY 3
bi j s
1 3
35
kY 2
6
1 11
9
6
2
¢k
1 17
12
,
y26
Y
y
ci k s
¶
1 5
y
kY 6
Ž 18b.
ß
1 5
y
kY 6
3
kY 2
3
y
y
2
y2
2
y2
1
y
2
0
0
.
Ž 18c.
The MS Yukawa coupling for fermion f is given in
terms of the corresponding pole mass m f as
Fig. 1. Constraint on the Higgs boson mass for the electroweak
precision measurement and the theoretical bounds of the Higgs
potential. The contours are obtained from the SM fit to all
electroweak data with m t s175"6 GeV, a s s 0.118"0.003 and
1r a Ž m2Z . s128.75"0.09. The inner and outer contours correspond to Dx 2 s1 Ž ; 39% CL., and Dx 2 s 4.61 Ž ;90% CL.,
respectively w10x. The upper and lower lines come from the
triviality and vacuum stability bounds for the cut-off scale L ;
10 16 GeV.
yˆ f Ž m . s 2 3r4 GF1r2 m f 1 q d f Ž m . 4 ,
Ž 19 .
where the factor d f Ž m . denotes the QCD and electroweak corrections. Because only the top-quark
Yukawa coupling is found to affect our results significantly, we set yˆ b s yˆt s 0. The explicit form of
d t Ž m . has been given in Ref. w18x. Only the leading
G.-C. Cho, K. Hagiwarar Physics Letters B 419 (1998) 199–205
order m-dependence of yˆt Ž m . is needed in our analysis w17x;
d
yˆt2
m
d m 4p
ž /
ž /
1
s
2p
yˆt2
4p
y8 aˆ 3 q
9
2
1 17
y
k Y 12
ŷt2
ž /
4p
.
aˆ 1 y 94 aˆ 2
203
and 2c. From Fig. 2, it is clearly seen that the 2-loop
RGE gives the unification scale mU which is much
smaller than the 1-loop RGE estimate of Eq. Ž5b..
The scale mU increases for larger a s Ž m Z ., larger
a Ž m2Z ., larger m H , and for smaller m t . We find the
following parametrization:
mU s 2.75 q 0.93 x s q 0.13 x s2 y 0.20 x t q 0.30 x H
Ž 20 .
We can now solve the RGE in the NLO level and
find the unification scale mU and the unified coupling a U as functions of a s Ž m Z ., m t ,m H and a Ž m2Z ..
We show the result of our numerical study in Fig.
2. In order to show the a s Ž m Z .-dependence explicitly, we choose m t s 175 GeV,m H s 100 GeV and
1ra Ž m2Z . s 128.75 in Fig. 2a. In the other figures,
we fixed a s Ž m Z . s 0.118 in Figs. 2b, 2c and 2d,
m t s 175 GeV in Figs. 2c and 2d, m H s 100 GeV in
Figs. 2b and 2d, and 1ra Ž m 2Z . s 128.75 in Figs. 2b
q0.03 x H2 y 0.04 x a
Ž =10 16 GeV. ,
Ž 21a.
ay1
U s 46.15 q 0.16 x s y 0.07x t q 0.12 x H
q0.004 x H2 y 0.02 x a ,
Ž 21b.
for the unification scale mU and the unified coupling a U . It is remarkable that the unification scale
of the minimal SM as determined above is almost
the same as that of the MSSM, Eq. Ž2.. We can find
from Eq. Ž21a. that the largest value of the unification scale is mU ; 4.4 = 10 16 GeV for a s Ž m Z . s
0.121,m t s 165 GeV,m H s 270 GeV and 1ra Ž m2Z .
Fig. 2. Four parameter dependences of the SUŽ2.L and SUŽ3. C running couplings. Each figures correspond to: a. a s Ž m Z . s 0.118 " 0.003,
b. 160 GeV - m t - 190 GeV, c. 100 GeV - m H - 270 GeV, d. 1ra Ž m2Z . s 128.75 " 0.09.
204
G.-C. Cho, K. Hagiwarar Physics Letters B 419 (1998) 199–205
non-standard Kac-Moody level k Y . Taking into account of the threshold corrections at the boundary of
the RGE given by m t and m H , and the uncertainties
in a s Ž m Z . and a Ž m2Z ., we calculated the unification
scale mU in the next-to-leading order. Current theoretical and experimental knowledge then tells us that
the unification scale should satisfy mU Q 4 =
10 16 GeV, which is one order of magnitude smaller
than the naive string scale of 5 = 10 17 GeV w2,3x. If
non-perturbative string effects or perturbative higher
order effects can lower the string scale, then the
hyper-charge Kac-Moody level should be 1.33 Q k Y
Q 1.35.
Fig. 3. Parameter D as a function of the hyper-charge Kac-Moody
level k Y for m t s175 GeV,m H s100 GeV, and 1r a Ž m2Z . s
128.75. The desired k Y is given at D s 0 where the three gauge
couplings are unified.
s 128.66. Even with the extreme choice of a s Ž m Z .
s 0.124 Ž x s s 2., the scale can reach mU ; 5.7 =
10 16 GeV. It is still smaller than the expected string
scale about one order of magnitude.
The above result tells us that the string unification
requires either extra matter particles or non-perturbative effects, as discussed in Ref. w4x, even in the
non-SUSY minimal SM. There may also be a possibility that the 2-loop string effects can lower the
unification scale. The desired Kac-Moody level k Y
is then found by studying the difference
D ' 1raˆ 1 Ž mU . y 1ra U ,
Ž 22 .
where aˆ 1Ž m . s k Y aˆ Y Ž m .. In the absence of the
significant string threshold corrections among the
gauge couplings, the desired range of k Y that gives
the unification of all three gauge couplings is determined by the condition D s 0. We show D as a
function of k Y in Fig. 3 for m t s 175 GeV, m H s 100
GeV and 1ra Ž m2Z . s 128.75. We find that the unification is achieved when 1.33 Q k Y Q 1.35 for a s Ž m Z .
s 0.118 " 0.003. On the other hand, the SUŽ5. case,
k Y s 5r3, gives D s y9.02 " 0.38.
To summarize, we have quantitatively studied the
possibility of the gauge coupling unification of the
minimal non-SUSY SM at the string scale with a
Acknowledgements
We thank H. Aoki and H. Kawai for fruitful
discussions. The work of G.C.C. is supported in part
by Grant-in-Aid for Scientific Research from the
Ministry of Education, Science and Culture of Japan.
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12 February 1998
Physics Letters B 419 Ž1998. 206–210
Modular weights, Už 1/ ’s and mass matrices
G.K. Leontaris
a
a,b
, N.D. Tracas
1
b,c
Theoretical Physics DiÕision, Ioannina UniÕersity, GR-451 10 Ioannina, Greece
b
CERN, Theory DiÕision, GeneÕa, Switzerland
c
Physics Department, National Technical UniÕersity, 157 80 Athens, Greece
Received 2 October 1997
Editor: R. Gatto
Abstract
We derive the scalar mass matrices in effective supergravity models with the standard gauge group augmented by a
UŽ1.F family symmetry. Simple relations between UŽ1.F charges and modular weights of the superfields are derived and
used to express the matrices with a minimum number of parameters. The model predicts a branching ratio for the m ™ eg
process close to the present experimental limits. q 1998 Elsevier Science B.V.
The Minimal Supersymmetric Standard Model
ŽMSSM. emerges as the most natural extension of
the Standard Model ŽSM. in the context of the
unification of all interactions. Although supersymmetric models solve the hierarchy problem, the
plethora of arbitrary parameters requires a further
step beyond the MSSM. The N s 1 supergravity
coupled to matter stands promising w1x. Yet, there are
many essential parameters ŽYukawa couplings, content of the chiral multiplets, etc.. to be chosen by the
model builder. In this scene, string theory appears
the only known candidate theory that can in principle
predict all the required parameters. String theory puts
rather strong constraints on many of the parameters
of the resulting N s 1 effective supergravity, which
appears as its low-energy limit. Thus, the kinetic
1
Research supported in part by TMR contract ERBFMRXCT96-0090, P ENE D-1170r95 and P ENE D-15815r95.
terms must have a certain structure, the Lagrangian
should obey the string duality symmetries, while
several constraints are imposed on the superpotential
and the Yukawa couplings w2x.
The subject of this letter is to reproduce the
observed family hierarchy of the fermion masses and
moreover to predict the corresponding mass matrices
in the scalar supersymmetric sector. This is done in
the context of residual stringy UŽ1. symmetries w3x
left from the large gauge group at a high scale. In
particular, combining modular invariance constraints
and UŽ1. invariance of the superpotential, the scalar
mass matrices are given in terms of powers of an
expansion parameter ² u :rM, where ² u : is the vacuum expectation value of a singlet field and M is a
high Žstring. scale. These powers are written in terms
of modular weight differences. Further, the consequences in the lepton flavour non-conserving reaction m ™ eg are examined. Its branching ratio is
found close to the present experimental limits.
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 1 2 - 3
G.K. Leontaris, N.D. Tracasr Physics Letters B 419 (1998) 206–210
207
We start with a quick review of the N s 1 supergravity, which introduces a real gauge-invariant
Kahler
function with the general form w4x
¨
Q i ’s. non-renormalizable terms can be written in the
form
G Ž z , z . s K Ž z , z . q log N W Ž z . N 2
K ri ijj
Ž 1.
where KŽ z, z . is the Kahler
potential and W Ž z . is
¨
the superpotential. Denoting ŽF ,Q . by z, where F
stands for the dilaton field S and other moduli Ti ,
while Q represents the chiral superfields, the Kahler
¨
potential at tree level can be written as follows
K Ž F ,F ,Q,Q . s ylog Ž S q S . y Sn h n log Ž Tn q Tn .
qZi j Ž F ,F . Q i e 2V Q j q PPP
Ž 2.
The superpotential W Ž z . is a holomorphic function
of the chiral superfields Q i and at the tree level is
given by
W Ž F ,Q . s
1
1
l i jk Ž F . Q i Q j Q k q m i j Ž F . Q i Q j
3
2
q PPP
Ž 3.
In both Ž2. and Ž3., PPP stand for possible non-renormalizable contributions. Terms bilinear in the
fields Q i refer in fact to an effective Higgs mixing
term.
Now under the modular symmetries, the moduli
transform as T ™ Ž aT y ıb .rŽ ıcT q d ., where
a,b,c,d constitute the entries of the SLŽ2,Z . group
elements with a,b,c,d g Z and ad y bc s 1. These
imply the following transformation rules w5x
k
Q i ™ Q i Ł t kn i ,
k
k
k
Zi j ™ Zi j Ł Ž t k . yn i Ž t k .
W ™ Ł t kn W W ,
yn kj
,
k
Ž 4.
k
where we have introduced the notation t k s ıc k Tk q
d k . The exponent n ki is the modular weight of Q i
with respect to the modulus Tk .
Let us now introduce into the Kahler
function
¨
non-renormalizable terms through two fields, u and
u , which are singlets under the low energy standard
gauge group, while they carry charges qu s yqu
under the UŽ1.F family group. The lower order Žin
²u :
ri j
ž /
M1
Q i Q j q K r˜i ijj
²u :
ž /
M2
r̃ i j
Qi Q j .
Ž 5.
These terms should be invariant under the UŽ1.F
symmetry. Assigning UŽ1.F charges qi for the matter fields one gets
qi q q j q qu ri j s 0,
qi q q j q qu r˜i j s 0.
Ž 6.
Similar non-renormalizable terms could also appear
in the superpotential.
After this short review we come to the mass
matrix textures. The SUŽ2.L invariance, together with
the requirement to have symmetric mass matrices,
leads us to assign the same UŽ1.F charge to all quark
members of the same family qi , while the same
should be applied to the leptons of the same family
l i . The full anomaly-free Abelian group involves an
additional family-independent component and with
this freedom we may make UŽ1.F traceless without
any loss of generality. Thus q1 q q2 q q3 s 0 and
l 1 q l 2 q l 3 s 0.
If the light Higgs H2 , responsible for the masses
of the up quarks, and H1 , responsible for the down
quarks and leptons have UŽ1.F charge, so that only
the Ž3,3. renormalizable Yukawa couplings to H2
and H1 are allowed, namely
2 q3 q h 2 s 0,
and 2 l 3 q h1 s 0,
Ž 7.
only the Ž3,3. element of the associated mass matrix
will be non-zero. The remaining entries are generated when the UŽ1.F symmetry is broken. A straightforward consequence of this fact is the equality of
the two Higgs UŽ1.F charges Ž h1 s h 2 ., since H1
provides also the mass to the bottom quark while we
have assumed equal UŽ1.F charges within a family.
A general non-renormalizable relevant term in the
superpotential is of the form
Yi j Q i u cj H2
u
ž /
M
xij
.
Ž 8.
G.K. Leontaris, N.D. Tracasr Physics Letters B 419 (1998) 206–210
208
Owing to UŽ1.F invariance of the superpotential, we
have the constraint
qi q q j q h 2 q x i j qu s 0
Ž 9.
and similarly for the parameter u terms. The allowed
powers of non-renormalizable terms in each entry
are determined by the charges Ž qu s yqu .
xi j s<
q2 y q3
<
qu
2 < a q 1<
< a<
< a q 1<
< a<
2
1
< a q 1<
1
0
0
Ž 10 .
where a s 3q3rŽ q2 y q3 ., and we have used the
condition Ž7.. Suppressing unknown Yukawa couplings Yi j and their phases, which are all expected to
be of order 1, we arrive at the following mass
matrices
e 2 < aq1 <
mU f
e < a<
e < aq1 <
e < a<
e2
e
e < aq1 <
mt
e
1
0
0
Ž 11 .
e˜ 2 < aq1 < e˜ < a< e˜ < aq1 <
mD f
mb
Ž 12 .
e˜ < a<
e˜ 2
e˜
< aq1 <
e˜
e˜
1
² :
² :
where e˜ s Ž Mu 1 . <Ž q 2yq 3 .r qu < , e s Ž Mu 2 . <Ž q 2yq 3 .r qu < Ž M1
and M2 being two high scales.. The charged lepton
mass matrix may similarly be determined. The equality h1 s h 2 , together with Ž7. has also the consequence q3 s l 3 , which implies the successful relation
m b s mt at unification. We then get
e˜ 2 < aqb < e˜ < a< e˜ < aqb <
mL f
Ž 13 .
e˜ < a<
e˜ 2 < b <
e˜ < b < mt
< aqb <
<b<
e˜
e˜
1
where b s Ž l 2 y q3 .rŽ q2 y q3 ..
The powers of the above matrices can be written
in terms of the modular weights as follows. As we
have already discussed in the introduction, the superpotential transforms covariantly under the modular
symmetry. Let us denote by n Q i ,n u i ,n d i ,n h 2 ,nu the
modular weights for the corresponding fields with
respect to a certain modulus. For the non-renormalizable term of the form Ž8., the modular weights obey
the equation n Q i q n u j q n h 2 q x i j nu s n W . Combining this relation with the UŽ1.F invariance and the
fact that Ý3is1 qi s 0, we obtain the general formula
qu
qu
qu
qj s
n Q ji s
n u ji s
Ž 14 .
Ý
Ý
Ýn
3nu i
3nu i
3nu i d ji
0
where n Q ji s n Q j y n Q i and correspondingly for n u ji
and n d ji . The third equality comes from the downquark mass matrix non renormalizable contributions
corresponding to a term like Ž8.. Similar relations
hold for the lepton modular weights. Using the above
relation, we may obtain an elegant form of the
matrix Ž10., which expresses the powers of the allowed non-renormalizable entries only in terms of
modular weight differences w6x. We obtain
xi j s
1
nu
2 n Q 31
n Q 31 q n Q 32
n Q 31
n Q 31 q n Q 32
2 n Q 32
n Q 32
n Q31
n Q32
0
0
Ž 15 .
The positivity of the entries requires the conditions
n Q 31 nu ) 0 and n Q 32 nu ) 0. We can also express the
powers of the matrix Ž12. in terms of modular
weight difference. This is easily done by expressing
the parameter a in the form
n Q q n Q 23
a s 13
.
Ž 16 .
n Q 23
From Ž15. we conclude that the hierarchical fermion
mass spectrum requires all three n Q i’s to be different.
Models with equal n Q i’s, but different qi ’s Žnecessary to create hierarchy., require qu s 0. In this case
the UŽ1.F charges are not related to the modular
weights and the constraint Ž14. does not hold.
We next turn to the lepton fermion mass matrix.
The phenomenological constraint l 3 s q3 imposes
the following relation on the modular weights of the
quark and lepton generations
n L 13 y n Q 13 s n Q 23 y n L 23 ' d .
Ž 17 .
As a result, the UŽ1.F structure permits to express
the powers yi j of the lepton term
Ž y.
L i e j H1
yi j
ž /
u
Ž 18 .
M
by the following matrix
yi j s
1
nu
2 Ž n Q13 q d .
n Q 13 q n Q 23
n Q 13 q d
n Q 13 q n Q 23
2 Ž n Q 23 y d .
n Q 23 y d
n Q 13 q d
n Q 23 y d
0
0
Ž 19 .
whilst the corresponding constraints for the positivity
of the entries are n L 13 nu ) 0 and n L 23 nu ) 0. The
G.K. Leontaris, N.D. Tracasr Physics Letters B 419 (1998) 206–210
powers of the matrix Ž13. can also be expressed in
the same way by writing b in the form
nQ y d
b s 23
.
Ž 20 .
n Q 23
We now turn to the scalar part. At the tree level
the scalar mass matrices receive contributions only
along the diagonal, since terms of the form A Q i Q i)
have zero UŽ1.F charge. Using powers of the fields
u , u scaled by the M, we may fill in the remaining
entries. It can easily be seen that the allowed UŽ1.F
structure of the powers in the scalar mass term is
1
nu
0
n Q 12
n Q 13
n Q 12
0
n Q 23
n Q 13
n Q 23
0
0
Ž 21 .
Thus, the powers of the parameters ² u :,² u : are
simply determined by the differences n Q i j for the
squark matrix and similarly for the sleptons Žremember that since the UŽ1.F charge is the same within a
family, Ž14. tells us that n Q i j s n u i j s n d i j ..
Using again the parameters a and b entered in the
fermion mass matrices, we can express the squark
mass matrix in the form
e N aq6 N e N aq1 N
m23r2
Ž 22 .
e N aq6 N
1
e
N aq1 N
e
e
1
where m 3r2 is the gravitino mass. Similarly, for the
sleptons we obtain
m2Q̃ f
1
0
e˜ N aq6 b N e˜ N aqb N
2
e˜ N aq6 b N
1
e˜ N b N m 3r2 Ž 23 .
e˜ N aqb N
e˜ N b N
1
Obviously in the case of b s 1 the two matrices are
identical since this case corresponds to equal UŽ1.F
charges in the quark and the leptonic sector, l i s qi .
In fact, it can be checked that the phenomenological
analysis of the fermion mass spectrum allows two
values of b, namely b s 1 or 1r2 w7x, while e f
0.053 and e˜ f 0.23.
The above results show that UŽ1.F symmetries
necessarily lead to low energy models where the
Yukawa and its corresponding scalar mass matrices
are not simultaneously diagonalized. As a result,
flavour violation is possible and in general one should
check whether such models can pass also the flavour
m2l˜L , R f
1
209
violation tests. One of the most popular flavour
non-conserving processes is the m ™ eg decay. We
have calculated the branching ratio for this process
in order to compare it with the present experimental
limits. This calculation requires the diagonalization
of the 6 = 6 scalar mass matrix
M̃l˜2 s
ž
m2l˜L
Ž A l ² H1 :q m L mtan b .
A l ² H1 :q m L m tan b
†
m2l˜R
/
Ž 24 .
Here, as usual, A l is the trilinear parameter entering
the scalar potential, m is the Higgs mixing term and
tan b is the Higgs vev ratio. Since lepton mass
matrices are symmetric, left and right diagonalizing
matrices coincide. Further, due to the properties of
the UŽ1.F symmetry of the model, left Ž m2l˜L . and
right Ž m2l˜R . scalar mass matrices are the same. Moreover, here we restrict on the case of small tan b
regime, where the chirality changing diagrams are
suppressed. In the general case, of course, and in the
large tan b scenario, they become important. We
have considered contributions from one loop graphs
involving neutralino-charged slepton or charginosneutrino states in the loop. The diagrams of this
process are shown in Fig. 1. We have diagonalized
the lepton and the slepton mass matrices and found
th e c o rresp o n d in g am p litu d e fo r e a ch
neutralinorchargino graph. Then by diagonalizing
0
Fig. 1. The m ™ eg decay via supersymmetric graphs.
210
G.K. Leontaris, N.D. Tracasr Physics Letters B 419 (1998) 206–210
the Wino mass matrices we evaluated the total amplitude and the branching ratio. For sensible values
of m 3r2 ; m1r2 ; O Ž mW . Žinitial values for the scalar
and gaugino masses respectively. and standard GUT
initial conditions for gauge couplings, the value of
the BRm ™ eg can reach the order of 10y1 2 . Thus, this
rare decay gives the opportunity to test the viability
of the above UŽ1.F-like model in future experiments.
In conclusion, we have considered the scalar mass
matrices in supergravity models with the standard
SUŽ3. = SUŽ2.L = UŽ1. Y gauge group augmented by
a UŽ1.F family symmetry. Using modular invariance
of the Kahler
potential and the superpotential, we
¨
have derived certain relations between UŽ1.F charges
and the modular weights of the fields. As a result,
the scalar mass matrix entries are found to depend
only on certain powers, which are proportional to the
difference of modular weights. We have calculated,
as an example, the process m ™ eg , which is found,
for a wide range of the parameter space Žtan b , m 3r2 ,
m1r2 ., to be very close to the present experimental
limits. This fact makes it possible to test such theories in near future-experiments.
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12 February 1998
Physics Letters B 419 Ž1998. 211–216
Higher order cohomological restrictions on anomalies
and counterterms
G. Barnich
1
Freie UniÕersitat
¨ Berlin, Fachbereich Physik, Institut fur
¨ Theoretische Physik, Arnimallee 14, D-14195 Berlin, Germany
Received 27 October 1997
Editor: P.V. Landshoff
Abstract
Using a regularization with the properties of dimensional regularization, higher order local consistency conditions on one
loop anomalies and divergent counterterms are given. They are derived without any a priori assumption on the form of the
BRST cohomology and can be summarized by the statements that Ži. the antibracket involving the first order divergent
counterterms, respectively the first order anomaly, with any BRST cocycle is BRST exact, Žii. the first order divergent
counterterms can be completed into a local deformation of the solution of the master equation and Žiii. the first order
anomaly can be deformed into a local cocycle of the deformed solution. q 1998 Elsevier Science B.V.
1. Introduction
Cohomological techniques in renormalization theory have attracted a lot of interest since their introduction in the pioneering work by Becchi, Rouet and
Stora w1x in the context of Yang-Mills theories, because they allow to address general problems of
perturbative quantum field theories, like the form of
anomalies or of divergent counterterms by purely
algebraic means Žsee w2x for a recent review.. The
BRST construction and the formulation of Zinn-Justin
thereof in terms of the generating functional for
vertex functions w3x has then been generalized to
theories with general local symmetries by Batalin
and Vilkovisky w4x.
1
Alexander von Humboldt fellow. Starting November 1st 1997:
Charge´ de Recherches du Fonds National Belge de la Recherche
Scientifique at Universite´ Libre de Bruxelles, Faculte´ des Sciences, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium.
In this more general setting, one loop anomalies
are constrained by the cohomology groups in ghost
number 1 of the BRST differential generated by a
solution to the master equation w5,6x. This represents
the generalization of the Wess-Zumino consistency
condition w7x for the case of the gauge anomaly. The
cohomological restrictions on the one loop divergent
counterterms, involving the ghost number 0 group,
have been discussed in w8,9x, where it is stressed that
these techniques are valid in the power counting non
renormalizable case or for the case of higher dimensional operators Žsee also w10x. and hence apply to
effective field theories w11x.
These works only needed the first order consistency conditions following from the quantum action
principle mainly for the following reason. One can
show w12x that for semi-simple Yang-Mills theory or
gravity, the cohomology groups in ghost number 0
and 1 can be described independently of the antifields, so that all higher order constraints to be found
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 6 6 - 1
212
G. Barnichr Physics Letters B 419 (1998) 211–216
below turn out to be trivial. This is however not the
case in general, where the form of the BRST cohomology groups in ghost number 0 and 1 can be more
involved. In this case, higher order considerations
will be relevant.
The purpose of this letter is to give a purely
cohomological description in the space of local functionals of the higher order restrictions on the one
loop anomalies and counterterms in terms of deformation theory. To derive these conditions, we assume that there is a regularization scheme with the
properties of dimensional regularization as used in
w13x, although we expect the cohomological restrictions to be independent of the regularization method.
Notational conventions for the Batalin-Vilkovisky
formalism will be those of the reviews w14,15x. The
analysis applies to local and rigid symmetries if we
understand the master equation to be the generalized
master equation discussed in w16x. In this letter, only
the cocyle condition is considered. More details including a discussion on the coboundary conditions
will appear elsewhere w17x.
A different, but related issue is the problem of
cohomological restrictions on anomalies and counterterms at higher loops. Such considerations have
appeared in the recent literature w18,13,19x from
various points of view, and in particular, the form of
the consistency conditions at higher loops has been
discussed in w20x in the context of non local regularization. These issues will be addressed in the set-up
of this letter in w17x.
Ø the regularized quantum action principle holds
w21x.
Let S˜s St q r )ut , with ut s 21t Ž St ,St ., so that
u 0 s Ž S,S1 ., and r ) a global source in ghost number
y1. On the classical level, we have, using Ž r ) . 2 s
0,
1
2
˜ ˜. s t
Ž S,S
˜
S,
E S˜
ž /
Er )
E S˜
Er )
,
Ž 2.1 .
s 0,
Ž 2.2 .
which translates, according to the quantum action
principle, into the corresponding equations for the
regularized generating functional for 1PI vertex functions:
1
2
Ž G˜ , G˜ . s t
ž
G̃ ,
EG˜
Er )
/
EG˜
Er )
,
Ž 2.3 .
s 0.
Ž 2.4 .
Using Ž r ) . 2 s 0, these equations reduce to
EG˜
1
2
Ž G ,G . st
ž
G,
EG˜
Er )
/
Er )
,
s 0.
Ž 2.5 .
Ž 2.6 .
At one loop, we get
2. Regularization
We will assume that there is a regularization with
the properties of dimensional regularization as explained in Ref. w13x, i.e.,
Ø the regularized action St s Sns0t n Sn is a polynomial or a power series in t , the t independent
part corresponding to the starting point action
S0 s S,
Ø if the renormalization has been carried out up to
n y 1 loops, the divergences of the effective action at n loops are poles in t up to the order n
with residues that are local functionals, and
Ž St , G Ž1. . s tu Ž1. ,
Ž 2.7 .
Ž St ,u Ž1. . q Ž G Ž1. ,ut . s 0,
Ž 2.8 .
where G Ž1. and u Ž1. are respectively the one loop
contributions of G and EG˜rEr ) . By assumption, we
have
G Ž1. s
t nG Ž1. n ,
Ý
Ž 2.9 .
nsy1
u Ž1. s
Ý
t nu Ž1. n ,
Ž 2.10 .
nsy1
where G Ž1.y1 and u Ž1.y1 are local functionals.
G. Barnichr Physics Letters B 419 (1998) 211–216
3. Lowest order cohomological restrictions
and the corresponding equation for the regularized
generating functional G˜j
At 1rt , Eqs. Ž2.7. and Ž2.8. give
Ž S, G Ž1.y1 . s 0,
Ž 3.1 .
Ž S,u Ž1.y1 . q Ž G Ž1.y1 ,u 0 . s 0.
Ž 3.2 .
Using u 0 s Ž S,S1 . and Eq. Ž3.1., Eq. Ž3.2. reduces to
Ž S,u Ž1.y1 y Ž S1 , G Ž1.y1 . . s 0.
Ž 3.3 .
In addition, we get, from the term independent of t
in Eq. Ž2.7.,
Ž S, G
Ž1.0
. su
Ž1.y1
y Ž S1 , G
Ž1.y1
..
Ž 3.4 .
The term linear in t gives
Ž S, G Ž1.1 . s u Ž1.0 y Ž S1 , G Ž1.0 . y Ž S2 , G Ž1.y1 . .
Ž 3.5 .
The one loop renormalized effective action is GR1 s
S q " G Ž1.0 q O Ž " 2 ,t ., where the notation O Ž " 2 ,t .
means that those terms which are not of order at
least two in " are of order at least one in t and
vanish when the regularization is removed Žt ™ 0.,
so that
1
2
Ž GR1 , GR1 . s "A1 q O Ž " 2 ,t . ,
Ž 3.6 .
with, using Eq. Ž3.4.,
A1 s u Ž1.y1 y Ž S1 , G Ž1.y1 . ,
Ž 3.7 .
and the consistency conditions for local functionals
Ž S, G Ž1.y1 . s 0 ´ G Ž1.y1 s c1i Ci q Ž S, J 1 . , Ž 3.8 .
Ž S, A1 . s 0 ´ A1 s a1i A i q Ž S, S 1 . ,
213
Ž 3.9 .
where Ci and A i are respectively a basis of representatives for H 0 Ž s . and H 1 Ž s ..
1
2
EG˜j
ž G˜ , G˜ / s t Er
j
j
)
qOŽ j2 . .
Ž 4.2 .
At one loop, we get for the term independent of r ) ,
Ž Gj Ž1. ,Stj . s tu jŽ1. q O Ž j 2 . .
The term linear in j of order
Ž 4.3 .
1
t
gives
Ž D Ž1.y1 ,S . q Ž D, G Ž1.y1 . s 0,
Ž1.y1
with D
theorem.
s Ž EGj
Ž1.y1
Ž 4.4 .
rE j .< js0 . This gives our first
Theorem 4.1. The antibracket of the diÕergent
one loop part G Ž1.y 1 , which is BRST closed and
local, with any local BRST cocycle is BRST exact in
the space of local functionals.
The theorem can be reformulated by saying that
the antibracket map Žinduced in the local BRST
cohomology groups by the antibracket, see Ref. w22x.
Ž w G Ž1.y1 x , w D x . s w0x
Ž 4.5 .
for all w D x g H g Ž s .. This equation represents a cohomological restriction on the coefficients c1i that
can appear ; it can be calculated classically from the
knowledge of H 0 Ž s . and the antibracket map from
H 0 Ž s . = H g Ž s . to H gq 1 Ž s .. According to the previous section, the theorem holds in particular when
D s G Ž1.y1 or D s A1.
In the same way, the consistency condition is
ž
Gj ,
EG˜j
Er )
/
q O Ž j 2 . s 0,
Ž 4.6 .
4. First order cohomological restrictions
and gives at one loop,
Let D be a BRST cocycle in any ghost number g
and consider S j s S q jD, where the source j is of
ghost number yg. The regularized action is Stj s St
q jDt , with D a polynomial in t starting with D. If
S˜ j s Stj q r )ut j, where ut j s 21t Ž St ,St . q 1t Ž jDt ,St .,
we have
Ž Gj Ž1. ,utj . q Ž Stj ,u jŽ1. . q O Ž j 2 . s 0.
1
2
Ž S˜ ,S˜ . s t
j
j
E S˜ j
Er )
qOŽ j
2
.,
Ž 4.1 .
The term linear in j of order
Ž D Ž1.y1 ,u 0 . y
q Ž D, u
Ž1.y1
ž
Eu 0j
Ej
.y
1
t
gives
, G Ž1.y1
js0
ž
Eu jŽ1.y1
Ej
Ž 4.7 .
/
/
,S s 0.
js0
Ž 4.8 .
G. Barnichr Physics Letters B 419 (1998) 211–216
214
Using u 0 s Ž S,S1 ., Eu 0jrE j < js0 s Ž D 1 ,S . q Ž D,S1 .,
Eqs. Ž3.1., Ž3.7. and Ž4.4., we get
Ž D, A1 . y
ž
Bns
Eu jŽ1.y1
Ej
/
for all w D x g H s . ; it represents a classical cohomological restriction on the coefficients a1i that can
appear.
Let B 0 s S and B 1 s G Ž1.y1. We have the following theorem.
Theorem 5.1. The first order counterterms can be
completed into a local deformation of S, i.e., there
exist local functionals B n such that
Ž S ,S . s 0,
Ý
j B .
Ž 5.2 .
Proof. The theorem is true for j 0 , j 1 and j 2 , if we
take D s G Ž1.y 1 s B 1 in Ž4.4. and B 2 s
1r2Ž EGj Ž1.y1rE j .< js0 . Suppose the theorem true at
order j k i.e., we have
k
k
Ž S˜ j ,S˜ j . s t
k
1
2
ž G˜
Ý
ns1
˜ / st
j k , Gj k
j nB n .
k
k
E S˜ j
k
Er )
q O Ž j kq1 . .
EG˜j k
Er )
q O Ž j kq 1 . .
At one loop, we get, for the part independent of r ) ,
/ s tu
Ž1.
jk q O
Ž j kq 1 . .
At order j k , this equation gives
ž
St ,
E kGj kŽ1.
/ ž
q Bt1 ,
E jk
js0
q . . . q Btk ,Gj kŽ1.
ž
js0
E ky1Gj kŽ1.
E j ky1
/ st
js0
/
k
E ku jŽ1.
.
E jk
js0
At order 1rt , we get, using
E j ny 1
s
js0
Ž1.y1
1
E ny1Gj ny
E j ny1
s nB n ,
js0
for n s 1, . . . ,k y 1 and defining
1. B kq 1 , the relation
1
k
E kG jŽ1.y
js0
E jk
sŽkq
Ž S, Ž k q 1. B kq 1 . q Ž B 1 ,kB k . q . . . q Ž B k , B 1 . s 0,
or equivalently
k
0s
Ý Ž B m , Ž k q 1 y m . B kq1ym .
ms0
k
k
k
The corresponding equation for G˜j k based on the
k
action S˜ j is
Ž S j ,S j . s O Ž j kq1 . ,
S j sSq
k
E ny 1Gj kŽ1.y1
The higher order cohomological restrictions of
such an equation in terms of Lie-Massey brackets is
briefly discussed in w22x. More details will be given
in w17x.
k
k
Ž 5.1 .
n
ns1
1
2
1
2
jk
Ž1.
t , Gj k
j`
j n Btn
Ý
and S˜ j s Stj q r )ut j , with ut j s 21t Ž Stj ,Stj . q
O Ž j kq 1 ., so that
žS
5. Higher orders
S sSq
k
Stj s St q
Ž 4.10 .
gŽ
n
ny 1
k
The theorem can again be reformulated by saying
that the antibracket map
j`
Ž E ny1Gj Ž1.y1rE j ny1 . < js0 .
ns1
Theorem 4.2. The antibracket of the BRST closed
first order anomaly A 1 with any local BRST cocycle
is BRST exact in the space of local functionals.
j`
n
k
Ž 4.9 .
This gives our second result.
Ž w A1 x , w D x . s w 0 x
1
At the regularized level, consider the action
js0
y Ž D 1 , G Ž1.y1 . y Ž D Ž1.y1 ,S1 . ,S s 0.
1
2
and
s
Ž k q 1.
2
kq1
Ý Ž B m , B kq1ym . ,
ms0
Ž 5.3 .
G. Barnichr Physics Letters B 419 (1998) 211–216
which proves the theorem. I
At order 1rt , we get
k
0
Let E s A1 s u
Ž1.y1
y Ž B 1,S1 ..
j`
E s
m
Ž 5.4 .
m
j E .
Ý
Ž 5.5 .
ms0
Proof. The theorem holds for j 0 and j 1 by taking
Ž
in 4.9. D s B 1, and defining
Eu jŽ1.y1
E1 s
Ej
ž
ms0
y Ž D1 , G Ž1.y1 . y Ž D Ž1.y1 ,S1 .
js0
Ž m q 1 . B mq 1 ,
ž
q Bm,
Ý Ž B l , B1kymyl .
ls0
k
E ky mu jŽ1.y1
E j ky m
js0
/
/
s 0.
Ž 5.7 .
Using the Jacobi identity, the first term is given by
k
`
Ž S j , E j . s 0,
kym
Ý
Theorem 5.2. The lowest order contribution to
the anomaly E 0 can be extended to a local cocycle of
`
the deformed solution of the master equation S j ,
i.e., there exist local functionals E m such that
`
215
kym
Ý Ý Ž Ž Ž m q 1. B mq 1 , B kymyl . , B1l .
ms0 ls0
y Ž B l , Ž Ž m q 1 . B mq1 , B1kymyl . . .
m
Changing the sum Ý kms 0 Ý ky
ls0 to the equivalent sum
k
ky l
Ý ls0 Ý ms0 , the first term of this equation vanishes
on account of Ž5.3., while the second term, using the
definition Ž5.6., combines with the second term of
Ž5.7. to give
k
Ý Ž B m , E kym . s 0,
ms0
Eu jŽ1.y1
s
Ej
which proves the theorem. I
y Ž B1 , B11 . y Ž 2 B 2 ,S1 . .
js0
Let us define
m
E mu jŽ1.y1
Em s
m
Ý Ž Ž n q 1. B nq1 , B1myn . .
y
E jm
ns0
js0
Ž 5.6 .
The consistency condition is
ž
Gj k ,
EG˜j k
/
Er )
s O Ž j kq1 . .
At one loop, we have,
žG
Ž1.
jk
j k , ut
jk
Ž1.
t ,u j k
/ qžS
Acknowledgements
/ sOŽ j
kq 1
..
The term of order j k of this equation gives
k
ž
Ý
ms0
q
ž
E mGj kŽ1.
E jm
Btm ,
,
js0
E kymut j
E j kym
k
js0
/
The author wants to thank F. Brandt, M. Henneaux, J. Kottmann, M. Kreuzer, J. Parıs
´ and M.
Tonin for useful discussions and Prof. H. Kleinert
for hospitality in his group while this work has been
completed.
References
k
E ky mu jŽ1.
E j kym
The investigation in this letters is a first step in
order to analyze the cohomological restrictions on
anomalies and counterterms at higher orders in ". To
see this, we note that if we put j s Žy"rt ., the
`
action S Žy" r t . satisfies the Ždeformed. master equa`
`
tion 1r2Ž S Žy" r t . ,S Žy" r t . . s 0, while the corresponding effective action is finite at order ". Its
divergences at order " 2 are poles up to order 2 in t
with residues that are local functionals. A systematic
analysis of the subtraction procedure at higher orders
in " will be presented in w17x.
js0
/
s 0.
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12 February 1998
Physics Letters B 419 Ž1998. 217–222
On radiative CP violation in ž supersymmetric/ two Higgs
doublet models
Otto C.W. Kong
a
a,1
, Feng-Li Lin
b,2
Institute of Field Physics, Department of Physics and Astronomy, UniÕersity of North Carolina, Chapel Hill, NC 27599-3255, USA
b
Department of Physics, and Institute for Particle Physics and Astrophysics, Virginia Polytechnic Institute and State UniÕersity,
Blacksburg, VA 24061-0435, USA
Received 22 October 1997
Editor: H. Georgi
Abstract
We discuss the feasibility of spontaneous CP violation being induced by radiative corrections in 2HDM’s, emphasizing
on a consistent treatment at 1-loop level. Specifically, we analyze the cases of gauginorhiggsino effect on MSSM, and a
new model proposed here with an extra, exotic, quark doublet. The latter model is phenomenologically interesting. One
conclusion is that some fine tuning is in general needed for the scenario to work. The case for the new model requires a tree
level mass mixing between the two Higgses, which fits in the standard SUSY picture. The fine tuning requires is then very
moderate in magnitude and in a way natural. q 1998 Elsevier Science B.V.
1. Introduction
The source of CP violation is one of the most
important unsolved puzzles in particle physics. The
possibility of CP being a spontaneously broken symmetry keeps generating new interest. The most simple setting for achieving the scenario is in a twoHiggs-doublet model Ž2HDM., originally analyzed
by Lee w1x. However, in order to avoid flavor-chang-
1
Present address: Department of Physics and Astronomy, University of Rochester, Rochester NY 14627-0171. E-mail:
[email protected].
2
E-mail: [email protected].
ing-neutral-currents that could result, extra structure
like natural flavor conservation ŽNFC. w2x has to be
imposed on a 2HDM, which then forbids spontaneous CP violation ŽSCPV. w3x. A supersymmetric
version of the standard model ŽSM. is naturally a
2HDM with NFC being imposed automatically by
the holomorphy of the superpotential. Hence, the
bulk of study on the minimal supersymmetric standard model ŽMSSM. concentrates on the CP conserving vacuum. Recently, Maekawa w4x illustrated
that SCPV could actually arise in the model in some
region of the parameter space through radiative correction. In a subsequent paper, Pomarol w5x argued
that mass for the ’’psuedoscalar’’, m A , is roughly
proportional to Ž l5 .1r2 and is at least more than a
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 5 0 - 0
O.C.W. Kong, F.-L. Lin r Physics Letters B 419 (1998) 217–222
218
factor of three too small to be phenomenologically
acceptable. Both analyses, however, are in a way
oversimplified. For instance, the m A mass estimate
is, strictly speaking, obtained from a Higgs mass
matrix of negative determinant, hence indicating the
whole picture is liable to be totally inconsistent. In
this letter, we look into the situation more carefully
and try to address the various issues involved in a
radiative SCPV model of the kind, with or without
supersymmetry ŽSUSY.. The basic idea here is that
in order to have a consistent approximation to the
radiative corrections in whatever interesting region
of the parameter space, loop contributions to all
parameters in the scalar potential at the same order
have to be considered. Our result illustrates that, the
radiative CP violation scenario does not get around
the Georgi-Pais theorem w6x in the way claimed in
the literature. Specifically, fine tuning is in general
needed for it to work.
Despite the Georgi-Pais theorem and the m A mass
limit constraint the radiative CP violation scenario
still generate interest 3. In our opinion, a realistic
model implementing the mechanism, hence realizing
CP violation as a pure quantum mechanical effect, is
of general interest. It provides a natural way of
explaining the smallness of CP violation and a potentially different phenomenology of the latter. Within
the domain of a supersymmetric 2HDM specifically,
it also provides the only possible scenario of SCPV.
Such a model is constructed and presented here. Our
new model, has an extra, exotic, quark doublets. It
could be phenomenologically-viable, in the sense
that l5 resulted could easily be more than an order
of magnitude larger, hence circumventing the likely
small m A obstacle, though the full phenomenological
features of the model still have to be worked out.
Unlike a recent model using right-handed neutrinos
w7x, our model achieves a large l5 without a fine
tuning in the fermion Žquark. masses, and is fully
compatible with SUSY. However, the same type of
fine tuning, though quite moderate numerically, as in
the MSSM case is needed to really have radiative CP
violation. We expect this feature to be generic.
2. The scalar potential and SCPV
The most general two Higgs doublet potential is
given by
V Ž f1 ,f 2 .
s m12 f 1† f 1 q m22 f †2 f 2 y Ž m 23 f 1† f 2 q h.c. .
2
2
ql4 Ž f 1† f 2 .Ž f †2 f 1 . q 12 l5 Ž f 1† f 2 . q h.c.
q 12 f 1† f 2 l6 Ž f 1† f 1 . q l 7 Ž f †2 f 2 . q h.c. 4 .
Ž 1.
Assuming all the parameters in V being real, and
denoting the vacuum expectation values ŽVEV’s. of
the neutral components of the Higgs doublets by
and ² f 20 : s Õ 2 e i d ,
² f 10 : s Õ 1
we have
² V :s m12 Õ12 q m22 Õ 22 q l1Õ14 q l2 Õ 24
q Ž l 3 q l 4 y l5 . Õ 12 Õ 22 q 2 l5 Õ 12 Õ 22 cos 2d
y Ž 2 m 23 y l6 Õ 12 y l7 Õ 22 . Õ 1Õ 2 cos d
s M1Õ12 q M2 Õ 22 q Ž pÕ14 q 2 rÕ12 Õ 22 q qÕ 24 .
q2 l5 Õ 12 Õ 22 Ž cos d y V . y
Vs
2 m 23 y l6 Õ 12 y l7 Õ 22
4 l 5 Õ 1Õ 2
2 l5
;
Ž 2.
,
Ž 3.
and
M1 s m12 q
M2 s m22 q
p s l1 y
q s l2 y
ž
See Refs. w7,8x, for example.
m 43
where
l6 m23
2 l5
l7 m23
2 l5
l26
8 l5
l27
8 l5
,
Ž 4.
,
Ž 5.
,
Ž 6.
,
Ž 7.
r s 12 l 3 q l 4 y l5 y
3
2
ql1 Ž f 1† f 1 . q l2 Ž f †2 f 2 . q l3 Ž f 1† f 1 .Ž f †2 f 2 .
l6 l 7
4 l5
/
.
A nontrivial phase Ž d . then indicates SCPV.
Ž 8.
O.C.W. Kong, F.-L. Lin r Physics Letters B 419 (1998) 217–222
Let us look at the d-dependence of ² V :. The
extremal condition gives
y4l5 Õ 12 Õ 22 Ž cos d y V . sin d s 0 ,
Ž 9.
219
m23 . All these are about the tree level scalar potential.
The interesting point of concern here is whether
radiative corrections can modify the picture.
and the stability condition requires
3. Radiative CP violation
E 2V
Ed
s 4l5 Õ 12 Õ 22
2
2
cos d Ž V y cos d . y sin d ) 0 .
Ž 10 .
cos d s V gives a SCPV solution, provided that l5
) 0 and < V < - 1. Actually, Eq. Ž2. shows that this is
the absolute minimum. Then one can easily obtain
the result that
Õ12 s
Õ 22 s
1 rM2 y qM1
2
pq y r 2
1 rM1 y pM2
2
pq y r 2
,
Ž 11 .
.
Ž 12 .
In order for V to have a lower bound, we have the
extra constraints
p)0 ,
q)0 ,
Ž 13 .
and
pq ) r 2 ,
Ž 14 .
which demand that
rM2 ) qM1 ,
rM1 ) pM2 .
Ž 15 .
For the MSSM case, Maekawa w4x pointed out
that there is a positive contributions to l5 from a
finite 1-loop diagram ŽFig. 1a. involving the gauginos and higgsinos, which could lead to SCPV. The
small value of Dl5 Ž; g 4r16p 2 . resulting, however,
could lead to phenomenological problem w5x. The
lesson here may be that extra, probably vector-like,
fermions with appropriate mixed couplings to both
Higgs doublets could make good candidate models
for the radiative CP violation scenario. The fermions
may be gauginos and higgsinos, or right-handed
neutrinos w7x, or, as illustrated in our new model
below, quarks. These fermions do lead to flavor
changing neutral currents which, however, could be
made to be sufficiently suppressed.
Our new model has an extra pair of vectorlike
quark doublets, Q and Q, with the following couplings
LQ s MQ QQ q lQ tf 1† Q ,
Ž 17 .
as an addition to the SM with two Higgs doublets or
MSSM. Note that f 1† is actually Hd , the Higgs
Note that if p ) 0, q ) 0 and r - 0, then M1 M2 - 0
is required for a consistent solution.
On the other hand, for a CP conserving vacuum,
sin d s 0 and the stability condition reads
"l5 Ž V . 1 . ) 0 ,
Ž 16 .
with the two signs correspond to the cases d s 0 and
p respectively. Only one of them would give a
minimum in whatever region of the parameter space.
For instance, d s p , corresponding to a negative
tan b Žs Õ 2rÕ1 ., is the only minimum for l5 ) 0, V
- y1 and l5 - 0, V ) y1.
Without SUSY, the natural way to impose NFC is
to require that only one of the Higgs, say f 1 , transforms nontrivially under an extra symmetry. This
means that m23 , l6 and l7 , and may be l5 too, all
have to vanish. The same result is obtained from the
superpotential of the MSSM, except that the soft
SUSY breaking B-term gives rise to a nonvanishing
Fig. 1. Gauginorhiggsino-loop diagrams giving rise to modifications to parameters in the potential. ŽEach dot indicates a helicity
flip in the fermion propagator..
220
O.C.W. Kong, F.-L. Lin r Physics Letters B 419 (1998) 217–222
Žsuper.multiplet that gives masses to the down-type
quarks; and f 2 is Hu . So, T3 s y1r2 component of
Q, denoted by T, has the same charge as the top
quark and mixes with it after electroweak ŽEW.
symmetry breaking. The other part of the doublet is a
quark of electric 5r3. The 1-loop diagram, now with
the gauginorhiggsino propagators replaced by that
of the quarks, leads to Dl5 Ž; 3 l2Q l2t r16p 2 . and
could be very substantial for large Yukawa couplings. Note that f 1 and f 2 vertices now have lQ
and l t couplings respectively. If MQ around the
same order as the EW scale, this can easily get
around the above mentioned phenomenological objection without fine tuning of the fermion masses,
unlike the case with right-handed neutrinos. Modification to top quark phenomenology would then be
very interesting. Another interesting point is that the
scalar partner of Q in the SUSY case has exactly the
quantum numbers needed to be a leptoquark that can
produce the high-Q 2 anomaly at HERA w9x. Q-Q
could naturally arise, for example, as the only extra
quarks from some interesting models with a SM-like
chiral fermion spectra embedding the three SM families in a intriguing way w10x.
4. A consistent treatment
Nevertheless, a Dl5 is a necessary but not sufficient condition for radiative CP violation. This is
clearly illustrated in our discussion of the scalar
potential above. For instance, in the MSSM case, if a
Dl5 is taken as the only modification to the tree
level parameters in Eq. Ž2., the constraint given by
Eq. Ž15. is upset and the potential has no lower
bound along cos d s V . An explicit calculation of
the 3 = 3 physical Higgs mass squared matrix m2H i j
actually gives
det Ž m2H . s l5 Ž pq y r 2 . sin2 2 b sin2d ,
the 1-loop effect should of course take into consideration contributions to all the 10 parameters in the
potential V. In Fig. 1, we illustrate all the corresponding 1-loop diagrams involved. Obviously, l5 ,
l6 , l7 as well as Žthe sum of. l3 and l4 get finite
contributions while the other five parameters receive
divergent contributions. Note that in the diagrams,
we do not distinguish Dl3 and Dl 4 ; only their sum
is involved in ² V :. Here we are not going to list all
the tedious analytical expressions of these results, we
plot the numerical results of major interest in Fig. 2
instead. The plots are for the chargino contributions
only, as functions of m s M g̃rm , the gaugino-higgsino mass ratio. Our results here presented give the
1-loop effect before EW symmetry breaking, i.e.
mass mixing between the gaugino and higgsino were
not considered. Further modifications due to the
symmetry breaking are not expected to change the
general features. Here, the neutralino part can simply
be inferred from symmetry. For each charged fermion
loop diagram, there is one from the neutral SUŽ2.L
partner of identical amplitude. Then there is another
one with the SUŽ2.L gaugino and the gauge coupling
replaced by the UŽ1. Y ones.
Now we are in position to discuss the feasibility
of radiative CP violation within a consistent approximation. In the model, Dl6 and Dl7 are identical.
For most region of plot, they are of magnitude
substantially larger than Dl5 . A essential condition
for SCPV is that V , as given by Eq. Ž3., has
magnitude less than one. Here the value of a divergent D m23 is also involved. If we take the finite
result from MS scheme, for example, D m 23 has a
Ž 18 .
which is negative. This inconsistency is also pointed
out by Haba w11x, who then suggested that it would
be fixed when toprstop loop contributions to the
other parameters, mainly l2 , in the potential are
included. In our opinion, it is at least of theoretical
interest to see what the 1-loop gauginorhiggsino
effect alone could do to the vacuum solution of a
supersymmetric 2HDM. A consistent treatment of
Fig. 2. Plots of radiative correction from chargino loop verses
mŽ s M g̃ r m ., with mass mixing from EW symmetry neglected.
Ža. D m23 ŽMS. in 25= g 2m2 r16p 2 ; Žb. Dl5 ; Žc. Dl6 Ž s Dl7 .; both
of the latter curves with values in g 4 r16p 2 .
O.C.W. Kong, F.-L. Lin r Physics Letters B 419 (1998) 217–222
magnitude that increases fast with that of m for
< m < ) 1. Obviously, some fine tuning is needed to get
< V < - 1, though a small window on m always exists
not too far from < m < s 1, for each sign of m , for a
not too small m. With EW-scale m , D mrm of the
admissible regions are of order 10y2 , though the
severe fine tuning can be tamed by having small m.
Apart from the small ’’pseudoscalar’’ mass objection, the admissible V region obviously gives phenomenologically unattractive m and M g̃ values.
Renormalization for m23 here could be tricky, as
there is no definite guideline on what its renormalized value should be, especially in the CP violating
case. In a softly broken SUSY model, such as MSSM,
tree level value for m23 is actually allowed, but then
the B-term involved gives a negative contribution to
Dl5 . Apparently, one can take the renormalized value
of m23 as a free parameter and adjust it to give
< V < - 1 for a fix m, at least when the B-term 1-loop
effects are neglected. In any case, however, the fine
tuning conclusion stands, as suggested by the
Georgi-Pais theorem.
The case of our new model is relatively complicated. First of all, the gauginorhiggsino contributions are there before EW symmetry breaking, as
presented above. In our new model, though the new
quark doublet Q has mass before EW symmetry
breaking, a similar set of 1-loop diagrams, as those
given in Fig. 1, can only be completed when the EW
breaking mass of the top and its mixing with T are
taken into consideration. But then the plausibly large
Yukawa couplings give rise to substantial results.
Otherwise, the qualitative features are similar to the
gauginorhiggsino case, with the exception that here
l3 q l4 also receives logarithmic divergent contributions. In Fig. 3, we presented some numerical results.
The plots use again m as parameter which in this
case denotes MQ rÕ 2 . For simplicity, we assume
lQ s l t . Note that around the CP violating vacuum
of concern here, tan b ; 1 is to be expected in the
SUSY case, as one can easily see by using Eqs. Ž11.
and Ž12., with tree-level values of the various parameters assumed.
Here all the results are independent of the sign of
m. Dl6 and Dl7 are different but always of the
same signs. They are still of substantially larger
magnitude when compared with Dl5 , while the latter is of interesting value for small value of m, say
221
Fig. 3. Plots of radiative correction from t-T loop verses mŽ s
MQ r Õ 2 .. Ža. D m23 ŽMS. in 10=3 l2t Õ 22 r16p 2 ; Žb. Dl5 ; Žc. Dl6 ;
Žd. Dl7 ; all of the latter three curves with values in 3 l4t r16p 2 ;
Ž l t s lQ , tan b s1 assumed..
- 1. However, D m23 ŽMS. always has the opposite
sign to that of Dl6 or Dl7 , making a naive use of
the value to fit in the < V < - 1 condition impossible.
Hence, for SCPV to occur, a tree level m23 value is
needed. The general feature is independent of tan b ;
but making its value really large suppresses Dl5 and
is hence undesirable. Soft SUSY breaking also provides the natural source of a tree level m23 , while the
B-term loop effects would be relatively negligible as
they are suppressed by the small gauge couplings.
While the m23Žtree. then has to be chosen to roughly
match the t-T 1-loop effect, the large Yukawa couplings make this relatively natural, as a value of the
order Õ 2 is all required. Accepting that,
D m23Žtree.rm23Žtree. of the solution region is not quite
small, R .35 for m F 1, representing only a moderate fine tuning. The small m 23 required is natural, as
its zero limit provides the scalar potantial with an
extra Peccei-Quinn type symmetry. We expect a
SUSY version of our model to be phenomenologically interesting. It has extra quarks close to the top
mass scale and can give a feasible radiative CP
violation scenario, though compatibility with topphenomenology awaits detailed investigation.
5. Further discussions
An alternative, and may be more powerful, approach to the problem is to analyse the full 1-loop
effective potential. Actually, we did check explicitly
that we get the same numerical results for the modifications in all the potential parameters, from differ-
222
O.C.W. Kong, F.-L. Lin r Physics Letters B 419 (1998) 217–222
entiating the standard Coleman-Weinberg expression
of the 1-loop potentials for both cases above. With
proper treatment of renormalizations for all the divergent quantities taken, the approach is expected to
give a much better approximation to the quantum
behavior of the potential and the physical Higgs
masses. However, a detailed Coleman-Weinberg
analysis of a 2HDM after symmetry breaking needed
to be done more carefully, for instance, along the
line suggested in Ref. w12x.
Renormalization group runnings and contributions
from other ingredients such as scalars should also be
taken into consideration in the analysis of the detailed phenomenological implications and feasibility
of a full model. The latter is especially important in
the SUSY case. Recall that cancellations between
superpartners is the natural mechanism that eliminates the quadratic divergence, as those arise from
Fig. 1Žg. and Žh., in D m12 and D m22 . For Dl5
specifically, the squarks loop contributions are suppressed by the naturally heavier soft mass scale.
Supersymmetric models with heavy fermion loop
giving the necessary Dl5 , however, would have
problem suppressing the corresponding superparticle
contributions. While the kind of detailed analysis for
the MSSM is abundant w13x, not much attention has
been put into a CP violating scenario. The latter
situation deserves more efforts.
We have presented a consistent 1-loop analysis of
the feasibility of radiatively induced SCPV, for both
MSSM and our proposed new model. The latter is
illustrated to have a good chance to be phenomenologically viable. We will report more on the topic in
a forthcoming publication.
Acknowledgements
O.K. is indebted to P.H. Frampton for discussions
and encouragement, and for suggestions on improving the manuscript. F.L. wants to thank L.N. Chang
and T. Takeuchi for discussions in the early phase of
this work. O.K. was supported by the US Department of Energy under Grant DE-FG05-85ER-40219,
Task B, and a UNC dissertation fellowship. F.L. was
partly supported by the US Department of Energy
under Grant DE-FG05-92ER-40709, Task A.
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12 February 1998
Physics Letters B 419 Ž1998. 223–232
Messenger-matter mixing and lepton flavor violation
S.L. Dubovsky 1, D.S. Gorbunov
2
Physics Department, Moscow State UniÕersity, 117234 Moscow, Russia
Institute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia
Received 6 June 1997; revised 17 September 1997
Editor: P.V. Landshoff
Abstract
We consider the Minimal Gauge Mediated Model ŽMGMM. with messenger-matter mixing and find that existing
experimental limits on m ™ eg decay and m –e-conversion place significant constraints on relevant coupling constants. On
the other hand, the rates of t ™ eg and t ™ mg in MGMM are well below the existing limits. We also point out the
possibility of sizeable slepton oscillations in this model. q 1998 Published by Elsevier Science B.V.
1. Presently, much attention is paid to lepton flavor physics in supersymmetric theories. Lepton flavor
violation naturally emerges in those SUSY models where supersymmetry is broken by supergravity interactions.
The corresponding soft-breaking terms are often assumed to be universal at the Planck scale. This universality
breaks down due to the renormalization group evolution below the scale of Grand Unification. In this way one
gets sizeable mixing in slepton sector at low energies w1x. This mixing leads to lepton flavor violation for
ordinary leptons Ž m ™ eg , etc.. at rates close to existing experimental limits w1x, and also to slepton oscillations
w2x possibly observable at the Next Linear Collider.
Another class of SUSY models assumes gauge mediated supersymmetry breaking w3x. In these models, the
gauge interactions do not lead to lepton flavor violation because the messenger-matter interactions are flavor
blind. Nevertheless there is a way to introduce flavor changing lepton interactions in these theories. This
possibility is based on the observation that some of the messenger fields have the same quantum numbers as
some of the usual fields. So it becomes natural to introduce direct mixing between messenger and matter fields
w4x. In such variation of the gauge mediated models, messengers not only transfer SUSY breaking to usual
matter, but also generate lepton flavor violation.
1
2
E-mail: [email protected].
E-mail: [email protected].
0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 2 5 - 1
S.L. DuboÕsky, D.S. GorbunoÕr Physics Letters B 419 (1998) 223–232
224
The purpose of this paper is to show that in a reasonable range of parameters, messenger-matter mixing in
minimal gauge-mediated models ŽMGMM. w5x gives rise to observable rates of rare lepton flavor violating
processes like m ™ eg and m –e-conversion.
We will see that the tree level mixing between leptons is small due to see-saw type suppression. The tree
level mixing between sleptons is also small in the part of the parameter space that is natural for MGMM.
However, we observe that radiative corrections involving interactions with ordinary Higgs sector induce much
stronger mixing of sleptons. The point is that messengers obtain masses not through interactions with ordinary
Higgs fields, but through interactions with hidden sector. So, the overall matrix of trilinear couplings is not
proportional to the mass matrix Žunlike in the Standard Model.. In the basis of eigenstates of the tree level mass
matrix, the largest trilinear terms involve sleptons as well as messenger and Higgs superfields. It is these terms
that produce slepton mixing at the one loop level.
As a result, at messenger masses of order 10 5 GeV and Yukawa couplings of order 10y2 , the rates of
m ™ eg and m –e-conversion are comparable to their experimental limits.
2. The MGMM contains, in addition to MSSM particles, two messenger multiplets Q s Ž q, cq . and
Q s Ž q, cq . which belong to 5 and 5 representaions of SUŽ5.. In what follows we are interested in lepton sector
so we will consider only colorless components of messenger multiplets. These multiplets couple to a MSSM
singlet Z through the superpotential term
Lms s l ZQQ
Ž 1.
Z obtains non-vanishing vacuum expectation values F and S via hidden-sector interactions,
Z s S q Fuu
Gauginos and scalar particles of MSSM get masses in one and two loops respectively. Their values are w6x
Mli s c i
ai
4p
L f 1Ž x .
Ž 2.
for gauginos and
m
˜ 2 s 2 L 2 C3
a3
ž p/
4
2
q C2
a2
ž p/
4
2
5 Y
q
3
2
a1
ž / ž p/
2
4
2
f2 Ž x .
Ž 3.
for scalars. Here a 1 s arcos 2u W ; c1 s 5r3, c 2 s c 3 s 1; C3 s 4r3 for color triplets Žzero for singlets.,
C2 s 3r4 for weak doublets Žzero for singlets., Y is the weak hypercharge. The two parameters entering Eqs.
Ž2. and Ž3. are:
Ls
F
S
and
xs
lF
l2 S 2
The dependence of masses on x is very mild, as the functions f 1Ž x . and f 2 Ž x . are close to 1. In the absence of
mixing between messengers and leptons Žandror quarks., the predictions of this theory at realistic energies are
determined predominantly by the value of L, while the value of x is almost unimportant. It has been argued in
Ref. w7x that L must be larger than 70 TeV, otherwise the theory would generically be inconsistent with mass
limits from LEP. The characteristic features of the model are large tan b Žan estimate of Ref. w7x is tan b ) 48.
S.L. DuboÕsky, D.S. GorbunoÕr Physics Letters B 419 (1998) 223–232
225
and large squark masses w4,7x. The next lightest supersymmetric particle ŽNLSP. is argued to be a combination
of t˜ R and t˜ L w7x. Bino is slightly heavier, but lighter than m
˜ L, R and e˜L, R .
Unlike the masses of MSSM particles, the masses of messenger fields strongly depend on x. Namely, the
vacuum expectation value of Z mixes scalar components of messenger fields and gives them masses
2
M"
s
L2
x2
Ž1"x .
It is clear that x must be smaller than 1. Masses of fermionic components of messenger superfields are all equal
to Lx .
Components of Q have the same quantum numbers as left leptons Žand d quarks., so one can introduce
messenger-matter mixing w4x
Lmm s HD L i˜Yi˜j Ej
Ž 4.
where
HD s Ž h D , x D .
is one of the Higgs doublet superfields,
°Ž e˜ ˜,e ˜.
/ ~¢Ž q, c .
L i˜s l˜i˜,l i˜ s
ž
Li
Li
q
i˜s 1, . . . ,3
i˜s 4
are left doublet superfields and
Ej s Ž e˜R j ,e R j .
are right lepton singlet superfields. Hereafter i,˜ j˜s 1, . . . ,4 label the three left lepton generations and the
messenger field, i, j s 1, . . . ,3 correspond to the three leptons and Yi˜j is the 4 = 3 matrix of Yukawa couplings,
Yi˜j s
Ye
0
0
Ym
0
0
0
Y41
0
Y42
Yt
Y43
0
In terms of component fields, the tree level scalar potential and Yukawa terms are
V s l2 S 2 q ) q q m2 h D h )D q < l Sq q h D e˜R j Y4 j < 2 q < m hU q Yi˜j l˜i˜e˜R j < 2 q < Yi j h D e˜R j < 2 q < Yi˜j l˜i˜h D < 2
q l Scq cq y l Fqq q Yi˜j h D l i˜e R j q x D l˜i˜e R j q x D l i˜e˜R j q mx D x U q h.c.
ž
ž
/
/
Ž 5.
where m is the usual parameter of MSSM and
HU s Ž hU , x U .
is the second Higgs superfield. Besides these terms, there are soft-breaking terms coming from loops involving
messenger fields. In the absence of messenger-matter mixing Ž4. they have the form Žat the SUSY breaking
scale, which is of order of L.
Lsb s m
˜ 2L i e˜L i e˜L)i q m˜ 2R i e˜R i e˜R)i q Bm hU h D
Ž 6.
S.L. DuboÕsky, D.S. GorbunoÕr Physics Letters B 419 (1998) 223–232
226
where m
˜ 2L j and m˜ 2R j are given by Eq. Ž3.. Messenger-matter mixing modifies Eq. Ž6.; we will consider this
modification later on. We will not discuss sneutrinos in what follows, so we will use the notation e L j for
charged components of lepton doublets.
The main emphasis of this paper is lepton flavor violation induced by the messenger-matter mixing, Eq. Ž4..
However, let us note in passing that another effect of this mixing is the absence of heavy stable charged Žand
colored. particles Žmessengers. in the theory w4x.
3. To see that lepton mixing is small at the tree level, let us write the fermion mass matrix Žthat includes left
and right fermionic messengers. in the following form
Ž 7.
Mf s Uf L Df Uf R
where
°
1
0
0
y
ye y 1)
0
1
0
y
lS
ym y 2)
Uf L s
0
0
ye y 1
¢l S
2
1
ym y 2
2
2
lS
y
2
l2 S 2
yt y 3)
Ž 8.
l2 S 2
yt y 3
2
¶
2
1
ß
y
y 1) y 3
l2 S 2
and
°
1y
y
< y1 < 2
2 l2 S 2
2 l2 S 2
< y2 < 2
1y 2 2
2l S
y 1 y 2)
2 2
2l S
Uf R s
y
¢
y 1) y 2
y
2 l2 S 2
y
2l S
2l S
y1
y2
y3
lS
lS
lS
y
y 2 y 3)
2 2
are mixing matrixes to the leading order in
yi s Y4 i ÕD ,
y
lS
¶
y 1)
lS
y 2) y 3
2 2
2l S
< y3 < 2
1y 2 2
2l S
y 1 y 3)
2 2
y
y
y 2)
lS
y
1y
Ž 9.
y 3)
lS
< y1 < 2 q < y 2 < 2 q < y 3 < 2
2 l2 S 2
ß
. Here ÕU and ÕD are Higgs expectation values,
ye , m ,t s Ye , m ,t ÕD
and
žž
Df s diag ye 1 y
ž
lS 1 q
ye
y1
lS lS
y 1 ye
l2 S 2
2
q
2
/ ž
, ym 1 y
y 2 ym
l2 S 2
2
q
ym
y1
lS lS
y 3 yt
l2 S 2
2
/ ž
, yt 1 y
yt
y1
lS lS
2
/
,
2
//
Ž 10 .
S.L. DuboÕsky, D.S. GorbunoÕr Physics Letters B 419 (1998) 223–232
227
is the matrix of mass eigenvalues. In principle, the off-diagonal terms in Eqs. Ž8. and Ž9. may lead to lepton
flavor violation due to one loop diagrams involving sleptons and gauginos w1x. However, these mixing terms are
negligible due to see-saw type mechanism: in MGMM one definitly has l S ) 10 4 GeV, tan b ) 1 and even at
Yi ; 1 the mixing terms are smaller than 10y4 . The corresponding contributions to lepton flavor violating rates
are too small to be observable.
Mixing in slepton sector is also small at the tree level. Indeed, the mass term of scalars has the form
Vsc s s Msc s†
where
s s e˜L , m
˜ L ,t˜ L ,e˜R) , m˜ )R ,t˜R) ,q,q )
ž
Ž 11 .
/
are the scalar fields, and
° m˜
2
eL
0
m Ye y U
0
0
ye y 1)
0
m
˜ m2 L
0
0
m Ym y U
0
yt y 2)
0
0
0
mt2L
0
0
m Yt y U
yt y 3)
0
m Ye y U
0
0
m
˜ 2e R
y 1) y 2
y 1) y 3
m Y1)y U
l Sy1)
0
m Ym y U
0
y 1 y 2)
m
˜ m2 R
y 2) y 3
m Y2)y U
l Sy 2)
0
0
m Yt y U
y 1 y 3)
y 2 y 3)
m
˜ t2R
m Y3)y U
l Sy 3)
ye y 1
ym y 2
yt y 3
m Y1y U
m Y2 y U
m Y3 y U
l2 S 2
yl F
0
0
0
l Sy1
l Sy 2
l Sy 3
yl F
l2 S 2
0
Msc s
¶
0
¢
˜
ß
After the scalar messengers are integrated out at the tree level, the lepton flavor violating terms in the mass
2 2 2
matrix of right sleptons are of order Yi Yj Ž ÕD2 x 2 q m LÕU2 x .. These terms are smaller than the one loop
contributions Žsee below. at L R 10 TeV, which is certainly the case in MGMM. Flavor violating mixing
between left sleptons is even smaller. The only substantial non-diagonal terms in this matrix are Žyl F . in the
messenger sector and t˜ R y t˜ L mixing. Due to the latter, the NLSP is likely to be t˜ w7x.
4. The most substantial mixing in slepton sector is due to one loop diagrams coming from trilinear terms in
the superpotential, that involve HD . The fact that the one loop mixing terms of sleptons are proportional to the
large parameter L2 is obvious from Eq. Ž5.: say, one of the cubic terms in the scalar potential, w l SY4 j q ) h D e˜R j q
h.c.x, contains Ž l S . s Lx explicitly.
After diagonalizing the messenger mass matrix we obtain the diagrams contributing to slepton mixing to the
order Ž l S . 2 , which are shown in Fig. 1. If supersymmetry were unbroken, their sum would be equal to zero. In
our case of broken supersymmetry the resulting contribution to the mass matrix of sleptons, to the order L 2 is
d m2i j s y
L2
1
16p
2
x
2
½
yln Ž 1 y x 2 . y
x
2
ln
ž
1qx
1yx
/5
Y4)i Y4 j
Ž 12 .
Since sleptons get negative shifts in m
˜ 2R , this equation immediately implies theoretical bounds on Yukawa
S.L. DuboÕsky, D.S. GorbunoÕr Physics Letters B 419 (1998) 223–232
228
Fig. 1. The diagrams giving main contribution to the scalar mixing matrix.
couplings Y4 i which come from the requirement w4x that none of the slepton masses become negative Žsee
below..
5. Let us now consider the effect of slepton mixing on the usual leptons. We begin with m ™ eg . The
dominant contribution to the amplitude of this decay comes from diagrams shown in Fig. 2 where Nn are
neutralino mass eigenstates with masses Mn . In general, their contribution gives
a
G Ž m ™ eg . s mm3 < F < 2
Ž 13 .
4
a
Fs
m d m2 G Ž m
Ž 14 .
˜ 2e R .
4p cos 2u W m 12
where
HB̃2k
4
G Ž m2 . s
Ý
4
ks1 M k
g
m2
ž /
Mk2
,
gŽ r. s
1
6 Ž r y 1.
5
Ž 17 y 9r y 9r 2 q r 3 q 6 Ž 3r q 1. ln r .
Here HB̃ k are coefficients in the decomposition of bino in terms of mass eigenstates. To obtain numerical
estimates we point out that neutralino mixing is small, at least at large mass of bino and large m. Neglecting this
mixing, recalling Eqs. Ž2. and Ž3. and collecting all factors, we find 3
G Ž m ™ eg . s 6.5 P 10 9
mm5
L4
< Y41 Y42 < 2f Ž x .
Ž 15 .
where
fŽ x. s
1
f 18
g
6 f2
2
ln Ž 1 y x 2 . q
ž /
5 f 12
x
4
x
2
ln
1qx
1yx
2
Ž 16 .
and the functions f 1Ž x . and f 2 Ž x . can be found in Refs. w8,9x. This rate strongly depends on x, or, in physical
terms, on the messenger masses. In particular, at small x
f Ž x . s 361 g Ž 56 .
2
x 4 s 4.2 P 10y5 x 4
To see what Eq. Ž15. means, we plot the limit on the product of Yukawa couplings as function of x at L s 100
TeV in Fig. 3, where we make use of the existing experimental limit on the rate of m ™ eg decay w10x. The
case of arbitrary L is straightforward: as follows from Eq. Ž15., the limit on < Y41 Y42 < 1r2 scales like L at fixed x
3
We neglect here the running of slepton masses from the SUSY breaking scale L down to weak scale. In fact, this running is small w7x.
S.L. DuboÕsky, D.S. GorbunoÕr Physics Letters B 419 (1998) 223–232
229
Fig. 2. Diagrams contributing to m ™ eg decay.
and varying L. We conclude that m ™ eg decay is observable in this model in a reasonable part of the
parameter space.
The slepton mixing in this model gives rise also to m –e-conversion. The dominant contribution to
G Ž m ™ eg . is given by penguin-type diagrams, while box diagrams are suppressed by squark masses. So, there
is a simple relation between m –e-conversion and m ™ eg rates w1x:
4
G Ž m ™ e . s 16 a 4 Zeff
Z < F Ž q . < 2G Ž m ™ eg .
Ž 17 .
< Ž .<
w x
For Ti 48
22 with Z s 22, Zeff s 17.6, F q s 0.54 11 one expects
G Ž m™e.
G Ž m ™ eg .
s 2.8 P 10y2
while the ratio of experimental limits w10,12x is
lim
G Ž m ™ e . exp
lim
G Ž m ™ eg . exp
s 1.1 P 10y1
Hence, the existing limit on m –e-conversion gives weaker Žby a factor 1.4. bounds on the product of Yukawa
couplings < Y41 Y42 < 1r2 .
Similar mechanism leads also to flavor changing t-decays, t ™ eg and t ™ mg . Crude estimates of the rates
are straightforward to obtain by neglecting t˜ R y t˜ L mixing. In this approximation, t-decay rates are given by
Eq. Ž15. with obvious substitutions mm ™ mt , Y42 ™ Y43 Žand Y41 ™ Y42 in the case of t ™ mg .. The
corresponding upper bounds derived from experimental limits on G Žt ™ mg . and G Žt ™ eg . w13x are shown in
Fig. 3. Upper limit on Y s < Y41 Y42 < 1r 2 as function of x at L s100 TeV Žsolid line.. Above the dashed line slepton oscillations are
unsuppressed at maximal mixing.
S.L. DuboÕsky, D.S. GorbunoÕr Physics Letters B 419 (1998) 223–232
230
Fig. 4. The upper limits on matrix Yi˜j from flavor changing t - decays. The dashed line is the limit on < Y41Y43 < 1r 2 from t ™ eg decay, the
same line represents also the limit on < Y42 Y43 < 1r 2 from t ™ mg decay. The solid line corresponds to theoretical constraint, which is the same
for < Y41 Y43 < 1r 2 and < Y42 Y43 < 1r 2 .
Fig. 4. In fact, these bounds are weaker than theoretical constraints inherent in this model. Indeed, with loop
corrections Ž12. to slepton mass matrix included, its eigenvalues m
˜ 2i are all positive only if
< Y41 < 2 q < Y42 < 2 q < Y43 < 2 - 103 a 12 f 2 Ž x .
x2
x
2
ln
1yx
1qx
Ž 18 .
2
y ln Ž 1 y x .
Making use of the inequality
< Y41 Y43 < - 12 Ž < Y41 < 2 q < Y42 < 2 q < Y43 < 2 .
1
2
1
2
one obtains from Eq. Ž18. theoretical constraint on < Y41 Y43 < . Precisely the same constraint applies to < Y42 Y43 < .
The result is shown in Fig. 4. We see that self-consistency of the model requires that the rates G Žt ™ eg . and
G Žt ™ mg . are lower than the present experimental limits at least by factor 10y2 .
6. Pair production of right sleptons Žwhich decay into leptons and bino. in eqy ey annihilation at the Next
Linear Collider will result in acoplanar mqy my an eqy ey events with missing energy. The NLSP in the
model under discussion is t˜ , so there will be also four t-leptons produced in each event Žtwo t will come from
bino decays into t and t˜ and two more from subsequent t˜ decays.. In the presence of slepton mixing, the
slepton oscillations leading to lepton flavor violating m "–e . events, are possible w2x.
Oscillations of sleptons are characterized by the mixing angle, which in this model can be found from Eq.
Ž12.
tan2 f s 2
< Y41 Y42 <
< Y41 < 2 y < Y42 < 2
If Y41 ; Y42 then mixing is close to maximal. The cross section of eqey™ e "m .q 4t may, however, be
suppressed even at large mixing if the lifetime of m
˜ R and e˜R is small compared to the period of oscillations. The
condition for the absence of such suppression is w2x:
2 G M e R - < M e2R y Mm2R <
Ž 19 .
S.L. DuboÕsky, D.S. GorbunoÕr Physics Letters B 419 (1998) 223–232
231
where M e R and Mm R denote the true slepton masses. For slepton decay width G we have
Gs
a1
2
ž
m
˜ R 1y
2
M bino
m̃ 2R
2
/
Ž 20 .
By making use of Eqs. Ž2., Ž3. and Ž12., it is straightforward to translate the condition Ž19. into a condition
imposed on Yukawa couplings Y41 and Y42 . For example, at small x one has
Ž < Y41 < 2 q < Y42 < 2 . x 2 ) 3.0 P 10y6
Ž 21 .
In the case of maximal mixing, Y41 s Y42 , the region of validity of Eq. Ž19. is shown in Fig. 3. Therefore, there
is a fairly wide range of parameters in which m ™ eg - decay and slepton oscillations are both allowed. Note,
that unlike m ™ eg , slepton oscillation parameters sin2 f and < M e22 GyMMe m < 2 are independent of L.
R
R
R
7. Messenger-matter mixing is possible also in strongly interacting sector, as some messengers carry quantum
numbers of right down-quarks. It will lead to flavor changing processes involving ordinary quarks w4x, like
b ™ sg or K 0 y K 0 mixing, in addition to contributions considered in Ref. w14x. The Yukawa couplings
involved in that case are not directly related to Yi˜j entering Eq. Ž4., so the limits obtained in this paper cannot be
improved Žin a model-independent way. by considering flavor changing processes in the quark sector. Yet
another possibility emerges in modifications of the model that are obtained by changing the messenger sector.
For example, one can consider messengers belonging to 10 and 10 of SUŽ5.. An interesting feature of this
model is the presence of messengers with quantum numbers of up-quarks. So one can introduce mixing of the
type lQLD and suggest an interpretation of possible HERA leptoquark w15x as a scalar messenger. However, to
have its mass of the order of 200 GeV one has to set x very close to 1, which implies fine tuning
2
Ž lS . y l F
Ž lS .
2
; 10y6
We are not aware of any mechanism that would make such fine tuning natural, but the very possibility of
messenger interpretation of HERA events seems to be interesting.
Acknowledgements
The authors are indebted V.A. Rubakov for stimulating interest and numerous helpful discussions. We thank
M.V. Libanov and S.V. Troitsky for valuable criticism. This work is supported in part by Russian Foundation
for Basic Research grant 96-02-17449a.
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12 February 1998
Physics Letters B 419 Ž1998. 233–242
Sparticle production in electron-photon collisions
V. Barger a , T. Han
a
a,b
, J. Kelly
a
Department of Physics, UniÕersity of Wisconsin, Madison, WI 53706, USA
Department of Physics, UniÕersity of California, DaÕis, CA 95616, USA
b
Received 27 September 1997
Editor: M. Dine
Abstract
We explore the potential of electron-photon colliders to measure fundamental supersymmetry parameters via the
processes eg ™ e˜ x˜ 0 Žselectron-neutralino. and eg ™ nx
˜ ˜y Žsneutrino-chargino.. Given the x˜ 0 and x˜y masses from eq ey
and hadron collider studies, cross section ratios s Žgy.rs Žgq. for opposite photon helicities determine the n˜ L , e˜L and e˜R
masses, independent of the sparticle branching fractions. The difference mn2˜ L y m2e˜L measures MW2 cos2 b in a model-independent way. The e˜L and e˜R masses test the universality of soft supersymmetry breaking scalar masses. The cross section
normalizations provide information about the gaugino mixing parameters. q 1998 Elsevier Science B.V.
A linear eqey collider with 0.5 TeV c.m. energy
Žexpandable to 1.5 TeV. and luminosity 50 fby1 per
year Ž200 fby1 per year at 1.5 TeV. is of special
interest for the study of the properties of supersymmetric particles w1–3x. In many unified models the
0 .
lighter chargino Ž x˜ 1". and neutralinos Ž x˜ 1,2
are
expected to be sufficiently light that they can be pair
produced at the Next Linear Collider ŽNLC.. From
the cross sections for eqey™ x˜q
˜y1 and x˜ 10 x˜ 20 and
1 x
"
the kinematics of the decays x˜ 1 , x˜ 20 ™ x˜ 10 ff X to the
lightest neutralino, the masses of the x˜q
˜ 10 and x˜ 20
1, x
would be determined, along with valuable information regarding their couplings. Additionally, since
one of two contributing Feynman graphs involves
slepton exchange, a determination or upper bound on
the slepton mass may be inferred w2,3x. Experiments
at the LHC may also measure the x˜q
˜ 10 masses
1, x
and deduce coupling information w4x.
In the minimal Supersymmetric Standard Model
ŽMSSM., the two gaugino mass parameters M1 and
M2 , the Higgs mixing m , and the ratio of vacuum
expectation values, tan b s ÕurÕd fully specify the
gaugino masses and mixings. The sfermion masses
Žsuch as m e˜ ,m e˜ , etc.. involve additional soft SUSY
L
R
breaking parameters. In the minimal supergravity
model ŽmSUGRA. w5x with the assumption of universal soft parameters at the GUT scale, the scalar
mass m 0 , gaugino mass m1r2 , the trilinear coupling
A along with tan b and the sign of m are sufficient to
determine all the physical quantities at the electroweak scale. In the minimal gauge-mediated SUSY
breaking ŽGMSB. model w6x, the parameter set is m ,
tan b , the SUSY breaking vacuum expectation value
FX and the messenger mass scale M X ; the effective
SUSY breaking scale is L s FXrM X . Measuring the
x˜q
˜ 10 and x˜ 20 masses may be the first step toward
1, x
deciphering the nature of the SUSY model and especially for testing the MSSM predictions for M1 and
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 3 1 3 - 0
V. Barger et al.r Physics Letters B 419 (1998) 233–242
234
M2 . However, sfermion mass measurements will be
essential to fully understand the SUSY breaking
mechanism.
In this Letter we consider the two processes
eg ™ nx
˜ ˜y
and
eg ™ e˜ x˜ 0
Ž 1.
for use in measuring the sneutrino Ž n˜ . and selectron
Ž e˜. masses and their couplings. We assume that the
masses of the x˜ 1", x˜ 10 and x˜ 20 are known from
experiments at the LHC or NLC. At the LHC the
mass reach for sleptons is only about 250 GeV w7x,
due to the rather small electroweak signal cross
sections and large SM backgrounds. In most SUSY
models, the sfermions are heavier than the lightest
chargino and pair production of the scalar particles
eqey™ e˜qe˜y and nn
˜ ˜ could be inaccessible at an
eq ey collider; the kinematic thresholds for Eq. Ž1.
will then be lower than those for sfermion pair
production. Even when scalar pair production is
kinematically allowed there is a b 3 suppression of
the cross section near threshold, and the event rates
are correspondingly limited. On the other hand, the
cross sections for Eq. Ž1. are proportional to b near
threshold, so high production rates are achievable.
Thus the addition of low energy laser beams to
backscatter from the e " beams, allowing high energy eg collisions w8x, becomes very interesting for
studying sleptons. The luminosity of the backscattered photons is peaked not far below the e " beam
energy, ² Eg : ; 0.83² Ee " : for optimized laser energy, so the c.m. energies Žand luminosity. available
in eg collisions are comparable to those of an eqey
collider, ² s eg : ; 0.9² s e e : w8x. We find that high
degrees of polarization for ey and g beams are
advantageous in the studies of the reactions in Ž1..
Due to approximate decoupling of Higgsinos from
the electron, the cross sections for the processes in
Ž1. are only large when the charginos and neutralinos
are mainly gaugino-like, namely x˜ "; W˜ " and x˜ 10
; B˜ 0 , x˜ 20 ; W˜ 0 . This situation corresponds to < m <
4 M1 , M2 in the MSSM. Fortunately gaugino-like
x˜ 1", x˜ 10 and x˜ 20 are highly favored theoretically for
two reasons: Ži. the radiative electroweak symmetry
breaking in SUSY GUTs theories yields a large < m <
value if tan b is bounded by the infrared fixed point
solutions for the top quark Yukawa coupling w9,10x;
Žii. x 10 ; B˜ is strongly preferred for x 10 to be a
viable cold dark matter candidate w9–11x. In the rest
of our paper, we will thus concentrate on this scenario, although we comment later to what extent a
nx
˜ ˜y signal with a small W˜y component in x˜y can
be detected.
Cross section formulae. The process eyg ™ nx
˜ ˜y
proceeds via s-channel electron and t-channel
chargino exchange; see Fig. 1Ža.. In the x˜y coupling
to n˜ L , the contribution from the higgsino is proportional to m e and thus can be neglected. Consequently
the scattering amplitude is proportional to the wino
(
(
Fig. 1. Feynman graphs for the processes Ža. ey g ™ nx
˜ ˜y and Žb. ey g ™ e˜y x˜ 0 .
V. Barger et al.r Physics Letters B 419 (1998) 233–242
fractions Vj1 of the matrix Vji that diagonalizes the
mass matrix Žthe first index j labels the chargino
mass eigenstate x˜q
˜q2 and the second index i
1 , x
refers to the primordial gaugino and higgsino basis
W˜ ", H˜ ". . Further, the n˜ L state fixes the incoming
electron chirality to be left-handed ‘‘y’’, leaving
just four independent helicity amplitudes. The differential cross sections summed over the chargino helicity are
ds
Ž ey g ™ n˜ L x˜yj .
dcos u y y
pa 2 Vj12
rx̃2
s 2
sin u W s Ž 1 y b 2 .
b
=
Ý Ž 1 q lcos u . Ž 1 q lb .
2
Ž 1 q b cos u . ls"1
=
rx̃
2
2
y Ž 1 q lb .
/
e˜L and four for e˜R . After summing over the neutralino helicities, there are just four independent
helicity cross sections as the helicity of the e˜R Ž e˜L .
matches that of the e R Ž e L .:
ds
Ž ey g ™ e˜yR x˜ i0 .
dcos u q q
ds
s
Ž ey g ™ e˜yL x˜ i0 .
dcos u y y
2 FiŽ2L , R.
r ẽ2
s pa 2
s
Ž1 y b 2 .
b
=
Ý Ž 1 q lcos u .
2
Ž 1 q b cos u . ls"1
= Ž 1 q lb .
ds
,
Ž 2.
dcos u
2
ž
(1 y b
r ẽ
2
/
y Ž 1 q lb . ,
Ž 4.
Ž eyq gy™ e˜yR x˜ i0 .
ds
Ž ey g ™ e˜yL x˜ i0 .
dcos u y q
2 FiŽ2L , R.
r ẽ2
2 b 3 sin2u
s pa 2
s
Ž 1 y b 2 . Ž 1 q b cos u . 2
s
ds
dcos u
s
ž
(1 y b
235
Ž
ey
y gq™ n L
˜ ˜
pa 2 Vj12
sin2u W
=
xy
j
s
.
rx̃2
Ž1 y b 2 .
2 b 3 sin2u Ž 1 y b cos u .
Ž 1 q b cos u .
2
=
.
Ž 3.
The subscripts on e and g refer to the electron and
photon helicities. The angle u specifies the chargino
momentum relative to the incoming electron direction in the c.m. frame, b s prE is the chargino
velocity in the c. m. frame, and rx˜ s mx˜r 's . The
Ž l e , lg . s Žy,y . helicity amplitude is S-wave near
threshold so the cross section of Eq. Ž2. is proportional to b ; the Žy,q . helicity amplitude, which
comes only from the t-channel diagram, is P-wave
near threshold so the cross section of Eq. Ž3. goes
like b 3. For small mn˜ , b ™ Ž1 y rx˜2 .rŽ1 q rx˜2 . and
the helicity amplitudes develop a well-known zero at
cos u s 1 w12x.
Selectron-neutralino associated production eg ™
e˜ x˜ 0 proceeds via s-channel electron and t-channel
selectron exchanges w13–16x; see Fig. 1Žb.. Again,
the contributions from higgsino components Ž H˜10 ,
H˜20 . of x˜ 0 can be neglected and only the neutralino
mixing elements Z j1 and Z j2 enter. There are eight
independent helicity amplitudes to consider: four for
ž
(1 y b
r ẽ
2
/
y Ž 1 y b cos u . .
Ž 5.
The Fi L Ž Fi R . for e˜L Ž e˜R . are effective couplings
given by
1
Zi1
Zi2
yZi1)
Fi L s
q
, Fi R s
.
Ž 6.
2 cos u W
sin u W
cos u W
Here the Z ji are the elements of matrices that diagonalize the neutralino mass matrix Žthe first index j
labels the neutralino mass eigenstate x˜ j0 , j s 1, . . . 4,
and the second index i s 1,2 refers to the primordial
gaugino and higgsino basis Ž B˜ 0 , W˜ 0 , H˜10 , H˜20 ... The
angle u specifies the direction of the selectron with
respect to the direction of the incoming electron in
the c.m. frame, b is the velocity of the selectron,
and r e˜ s m e˜r 's . Again, for small mx˜ , b ™ Ž1 y
r e˜2 .rŽ1 q r e˜2 . and the cross sections develop a zero at
cos u s 1.
Parameters. For our illustrations we choose the
charginorneutralino masses Žin GeV.
mx˜ 10 s 65 ,
mx˜ 20 s 136 ,
mx˜ 1"s 136 ,
mx˜ 2"s 431 ,
Ž 7.
V. Barger et al.r Physics Letters B 419 (1998) 233–242
236
and the mixing matrix elements
V11 s 0.98 ,
V21 s y0.18
Ž 8.
for the charginos and
Z11 s 0.98 ,
Z12 s y0.15 ,
Z21 s y0.18 ,
Z22 s y0.96
Ž 9.
for the neutralinos. These parameters correspond to
the following MSSM parameters at the weak scale
M1 s 62 GeV ,
M2 s 127 GeV ,
m s 427 GeV ,
tanb s 1.8 ,
Ž 10 .
where tan b is at the infrared fixed point value w9x
and the convention for signŽ m . follows Ref. w9x. For
slepton masses, we choose Žin GeV.
m e˜L s 320
m e˜R s 307
mn˜ L s 315,
Ž 11 .
where m e˜L and m e˜R are independent parameters in
MSSM. Our choices for the gaugino and slepton
masses are consistent with renormalization group
evolution w10x to the electroweak scale, with the
following universal mSUGRA parameters
m 0 s 300 GeV ,
y
y
m 1 r 2 s 150 GeV ,
As0 .
Ž 12 .
e g ™ n˜ L x˜ cross section. The total cross sections are shown in Fig. 2Ža. for n˜ L x˜y
1 production
'
and in Fig. 2Žb. for n˜ L x˜y
versus
s . The peak
2
cross section is about 1.5 picobarns for n˜ L x˜y
1 , and is
smaller for n˜ L x˜y
2 partially due to the smaller V21
coupling and partially due to the energy-dependent
factor. We have numerically checked that our calculations agree with Ref. w12x for his particular choice
of mx˜y1 s mn˜ L and couplings.
For realistic predictions we convolute these subprocess cross sections with the appropriate backscattered photon spectrum w8x; these results are shown by
the lower curves in Fig. 2. The effect of the convolution is to decrease the cross sections by about a
factor of two. In our illustrations we choose a prescattering laser beam energy of v 0 s 1.26 eV. For
Ee ) 250 GeV a lower v 0 would be needed to avoid
electron pair production at the backscattering stage.
We assume a polarization PeŽ L, R. s 0.9 of the nonscattered electron beam, a mean helicity l s "0.4
of the pre-scattered electron beam, and a fully polarized Pc s "1 pre-scattered photon beam. We select
l Pc to be negative to have a relatively monochromatic spectrum of s eg , peaked close to 0.9 s e e .
The individual signs of l and Pc are chosen to
illustrate the helicity cross sections.
eyg ™ e˜yx˜ 0 cross section. The total cross sections for e˜yx˜ 0 production versus eg c.m. energy
are illustrated in Fig. 3, where the four panels present results for e˜L and e˜R and for x˜ 10 and x˜ 20 , in all
combinations. Due to the stronger diagonal couplings
Zi i , the cross sections for e˜L x˜ 20 Žmainly e˜LW˜ 0 . and
(
(
Fig. 2. The upper two curves show the total cross section Žin fb. for ey g ™ nx
˜ ˜y versus s eg Žin GeV. for the SUSY and machine
y
Ž
.
Žy,y . and the dashed curves Žy,q .. The lower
parameters given in the text: Ža. nx
;
b
nx
.
The
solid
curves
represent
e,
g
helicities
˜ ˜y
˜
˜
1
2
two curves are corresponding results convoluted with the backscattered photon spectrum versus s e e .
'
'
V. Barger et al.r Physics Letters B 419 (1998) 233–242
237
Fig. 3. The upper two curves show the total cross section Žin fb. for ey g ™ e˜ x˜ 0 versus s eg Žin GeV. for the SUSY and machine
parameters given in the text: Ža. e˜L x˜ 10 ; Žb. e˜L x˜ 20 ; Žc. e˜R x˜ 10 ; Žd. e˜R x˜ 20 . The solid curves represent e, g helicities Žy,y . for Ža., Žb. and
Žq,q . for Žc., Žd.. The dashed curves represent helicities Žy,q . for Ža., Žb. and Žq,y . for Žc., Žd.. The lower two curves are
corresponding results, convoluted with the backscattered photon spectrum, versus s e e .
'
'
e˜R x˜ 10 Žmainly e˜R B˜ 0 . are significantly larger Žsee
Eq. Ž6..; the weaker neutral current couplings and
the more massive scalar propagator in selectron production make the cross sections smaller than for
sneutrino production. The lower curves in Fig. 3
show the effects of convolution with the backscattered laser photon spectrum for the machine parameters detailed previously. Again the effect is to decrease the cross sections by about a factor of two.
We numerically compared our convolution results
Table 1
Sparticle decay modes and branching fractions for the representative parameter choice. Here q generically denotes a quark and l s e or m ;
X
fermion-antifermion pairs ff with net charge y1 Ž0. are denoted by Cy Ž N 0 ..
Decay modes
Branching fraction Ž%.
0 yŽ X .
xy
ff
1 ™ x1 C
0
x 2 ™ x 10 N 0 Ž ff .
e R ™ x 10 ey
X
62 Ž x˜ 10 qq ., 25 Ž x˜ 10 ly n .
˜
˜
˜
˜
˜
˜
66 Ž x˜ 10 qq ., 20 Ž x˜ 10nn ., 9 Ž x˜ 10 lq ly .
97
e˜L ™ x˜y
˜L ™ x˜ 20 ey, e˜L ™ x˜ 10 ey
1 n, e
59, 36, 5
y
n˜ L ™ x˜q
˜ L ™ x˜ 20n , n˜ L ™ x˜ 10n
1 e , n
59, 23, 18
V. Barger et al.r Physics Letters B 419 (1998) 233–242
238
Table 2
Sparticle production and decay in eyg collisions. Branching fractions based on Table 1 are given in the parentheses. Here Eu denotes
missing energy resulting from x˜ 10 and n final state; Cy Ž N 0 . denotes a fermion-antifermion pair of charge y1 Ž0..
Process
y
Final state & Branching fraction
e g ™ n L xy
1™
ey g ™ e R x 10 ™
ey g ™ e L x 20 ™
y
N 0 ey
with Fig. 1 of Ref. w14x and found excellent agreement.
Signal final states and backgrounds. The decays
of the sparticles in these reactions generally give
clean signals: large missing energy Ž Eu ., energetic
charged leptons or jets from light quarks. Table 1
gives predicted branching fractions w17x for the
mSUGRA example discussed earlier. Based on the
predicted cross sections in Figs. 2 and 3, we concentrate on the three leading channels
eyg ™ n˜ L x˜y
1 ,
eyg ™ e˜R x˜ 10
and
eyg ™ e˜L x˜ 20 .
Ž 13 .
n L xy
1
The cross section for ˜ ˜
production is of
O Ž100–1000 fb., while the cross sections for e˜R x˜ 10
and e˜L x˜ 20 are typically of O Ž10–100 fb..
In Table 2, we list observable final states for these
three processes. We have calculated the SM backgrounds, which are presented in Table 3. Wy Ž Z . in
the backgrounds will give the same final state as Cy
Ž N 0 . in the signal Table 3, although the latter will
Table 3
Total cross sections Žin fb. for missing energy plus vector bosons
as Standard Model backgrounds to SUSY signals in eyg collisions at c.m. energies 's eg s 0.5, 1.0 and 1.5 TeV.
eg ™ X
s Žfb.
's eg s 0.5
y
X sW ne
eynn
Wy Zne
Zeynn
Wy Wq eynn
ZZeynn
4.2=10 4
6.5=10 3
210
23
0.62
3 =10y2
Cy N 0 Eu Ž23%., Cy Cq ey Eu Ž59%.
ey Eu Ž100%.
0
Ž
.
Eu 5% , N N 0 ey Eu Ž36%., Cy N 0 Eu Ž59%.
C Eu Ž18%.,
˜ ˜
˜ ˜
˜ ˜
1.0
1.5
4.8=10 4
6.3=10 3
720
79
8.6
0.7
4.9=10 4
6.3=10 3
1.1 =10 3
120
21
2
often be non-resonant ff X pairs. Generally speaking,
the multiple ff X signals for n˜ L x˜y
˜L x˜ 20 have
1 and e
favorable signalrbackground ratios, especially at
's eg ; 0.5 TeV. Given the sizeable signal cross
sections and the distinctive kinematical characteristics from the heavy sparticle decays, such final
states do not have severe SM backgrounds. However, some care needs to be taken for the e˜R x˜ 10
signal, which decays exclusively to ey plus Eu. The
cross section for the SM background ey g ™ ey nn
Žmainly from ey g ™ Wy n . is large, on the order of
a few picobarns. Detailed analysis w15x shows that by
making use of a polarized ey
R beam and kinematical
cuts, the e˜R signal can be separated from the SM
background. In our subsequent discussion we will
simply assume a 30% efficiency associated with
signal branching ratios and acceptance cuts for each
channel in Ž13., and assume no significant backgrounds remain after the appropriate selection cuts
have been implemented.
Before we proceed, a few comments are in order.
First, given the negligibly small background to the
y q y
's eg ; 0.5 TeV, it
signal n˜ L x˜y
1 in C C e Eu at
should be possible to measure even a small signal for
this channel, allowing a probe of a small gaugino
component in V11 . The maximum value of the cross
2
section for n˜ L x˜y
1 is about 600V11 fb, with a realistic
photon spectrum. If a cross section of 20 fb is
needed for a clear signal, a sensitivity down to
V11 ; 0.2 would be feasible. Second, although we
have not discussed the channel eyg ™ e˜L x˜ 10 because it has a smaller cross section for our parameter
choices, there is a cross section complementarity
between e˜L x˜ 10 and e˜L x˜ 20 , depending on which x˜ j0
has a larger W˜ 0 component. Finally, the cross sec-
V. Barger et al.r Physics Letters B 419 (1998) 233–242
tion normalization for e˜R x˜ 10 via the exclusive decay
e˜R ™ x˜ 10 ey will directly measure the neutralino mixing parameter Z11 Žor F1 R .. On the other hand, one
would have to measure the total cross sections
through all the decay channels for n˜ L and e˜L to
determine the other mixing parameters Z ji and Vj1.
The preceding signal and background discussions
are based on the mSUGRA scenario. For gaugemediated SUSY breaking models the signals from
Ž1. may be more spectacular. For instance, if x˜ 10 is
the next-to-LSP ŽNLSP., it will decay to a LSP
gravitino Ž G˜ . plus a photon, resulting in an isolated
photon plus Eu signal if the x˜ 10 decay length is short
w16x. In other GMSB models in which the NLSP is a
right-handed slepton, the signal from x˜ 10 ™ l " l˜R. is
also spectacular. Since signals from GMSB models
should be easily detectable at the eq ey NLC w18x
and the Tevatron w19x, we do not discuss such possibilites further in eg processes.
Slepton mass determination: cross section ratio
239
and energy endpoint measurement. In any of the eg
processes under consideration the ratio of the cross
sections for the two photon helicities provides a
direct measure of the slepton mass if the associated
neutralino or chargino mass is already known from
eqey or pp collider experiments. These ratios are
independent of the cross section normalization factors and the final state branching fractions. Fig. 4
shows the ratios for these three leading channels: Ža.
Ž . ˜R x˜ 10 ; Žc. e˜L x˜ 20 . The three pairs of bands
n˜ L x˜y
1; b e
on each panel correspond to the different energy
choices of s e e s 0.5, 1, and 1.5 TeV at integrated
luminosities of 25, 50, and 100 fby1 , respectively,
convoluted with the backscattered photon spectrum.
Each pair of bands represents the "1 s error bounds
on the ratio. We have included a 30% efficiency
factor for signal identification with rejection of the
SM backgrounds. From the figures, we find that the
"1 s uncertainties determined by the cross section
ratios translate into mass uncertainties of roughly 2
(
.
Ž
. vs mñ ; Žb.
Fig. 4. The ratio of total cross sections versus slepton mass Žin GeV.: Ža. s Ž ey gq™ n˜ L x˜y
˜ L x˜y
1 rs ey gy™ n
1
L
s Ž eq gy™ e˜R x˜ 10 .rs Ž eq gq™ e˜R x˜ 10 . vs m e˜R ; Žc. s Ž ey gq™ e˜L x˜ 20 .rs Ž ey gy™ e˜L x˜ 20 . vs m e˜L. Results for three colliders are presented:
y1
y1
y1
ŽI. s e e s 0.5 TeV, Leg s 25 fb ; ŽII. 1 TeV, 50 fb ; ŽIII. 1.5 TeV, 100 fb . For each collider the upper and lower curves show the
"1 s values for the ratio. The backscattered photon spectrum and the electron polarization are included in the calculations Žsee text..
'
V. Barger et al.r Physics Letters B 419 (1998) 233–242
240
GeV in channel Ža., and 6, 30 and 35 GeV in
channels Žb. and Žc. for the respective energies given
above.
Another way to measure the slepton mass is
through the endpoint of energy distribution in the
two-body decay e˜R ™ x˜ 10 ey w1x. If the maximum
Žminimum. energy of the electron in the lab frame is
Eq Ž Ey ., then the selectron mass can be determined
from
m ẽ R s
(
s Eq Ey
's Ž Eqq Ey . y 2 Eq Ey ,
Ž 14 .
mined in terms of the measured errors D mñ L and
D m ẽ L as
1
D cos2 b
cos2 b
s2
tan4b y 1 D cos2 b
s
(
Ž 15 .
The latter result can be used for a consistency check
with the mx̃ 10 measurement from eqey experiments.
Assuming knowledge of mx̃ 10 , we estimate the error
on m ẽ R measurement as
D m e˜R
m e˜R
s
1 m2e˜R y mx2˜ 10
2
m 2e˜R q mx2˜ 10
D Eq
2
D Ey
ž / ž /
Eq
q
Ey
2
1
2
.
Ž 16 .
For a typical NLC electromagnetic calorimeter, the
single event uncertainty on energy measurement is
D ErE s 12%r 'E q 1%. If there are no large systematical errors in the measurements, the error on
D m e˜Rrm e˜R should be well below 1%.
Deducing tan b . The ratio of vacuum expectation
values, tan b , is of fundamental importance in the
Higgs sector. In the absence of detailed information
about Higgs boson couplings to fermions, tan b is
difficult to measure. Generically, the mass splitting
of the left-handed sleptons satisfies the sum rule w20x
mn2˜ L y m2e˜L s MW2 cos2 b .
4tan2b
tan b
mx2˜ 10 s m2e˜R y 2 m e˜R Eq Ey .
Ž 17 .
/
< mn2˜ y m2e˜ <
L
L
.
Ž 18 .
Hence slepton mass measurements must be more
accurate than the magnitude of the sneutrino-selectron
mass splitting to obtain a significant determination of
cos2 b . The uncertainty on cos2 b translates into an
uncertainty on tan b via the relation
D tan b
and the LSP mass is given by
ž
2
mn2˜ L D mn2˜ L q m2e˜L D m 2e˜L
cos2 b
.
Ž 19 .
Thus a reasonably good sensitivity to tan b is obtained only when tan b ; 1. For the set of parameters
under our consideration at 's s 500 GeV, we find
that the slepton mass uncertainties are D mñ L ; 2.3
GeV and D m ẽ L ; 6 GeV with "1 s cross section
ratio measurements. This corresponds to an indirect
determination of D tan brtan b ; 1 for tan b ; 1.8.
Discriminating between SUSY models. Through
eqey™ x˜q
˜y1 and x˜ 10 x˜ 20 production with x˜y
˜ 20
1 x
1 ,x
subsequent decays, the NLC will most likely determine the MSSM parameters M1 , M2 , tan b and m ,
although it is only possible to obtain lower bounds
on tan b and m when they are large. The information
on F1 R , V11 and F2 L , combined with the corresponding branching fractions associated with the three eg
processes discussed earlier, would provide valuable
additional tests of models, since the Vj1 are functions
of tan b , M2 and m , while FiŽ L, R. depend on tan b ,
M1 , M2 and m.
The measurements of the slepton masses, mñ L,
m e˜R and m e˜L, will probe soft SUSY breaking in the
scalar sector. Although SUŽ2. symmetry predicts the
sum rule between the left-handed doublets in Eq.
Ž17., there is no general relation in the MSSM
between m e˜R and m e˜L. In mSUGRA the left-handed
and right-handed selectron masses are given by
w20,21x
2
which follows from the SUŽ2. structure of the lefthanded scalar partners. Thus the n˜ L and e˜L masses
provide an indirect measure of tan b . From Eq. Ž17.,
the relative error in deducing cos2 b can be deter-
m 2e˜L s Ž m50 . q C2 q 14 C1
q Ž y 12 q sin2u W . MZ2 cos2 b
2
2
2
m2e˜R s Ž m10
0 . q C1 y sin u W MZ cos2 b
Ž 20 .
V. Barger et al.r Physics Letters B 419 (1998) 233–242
where m 50 and m10
0 are the scalar masses in the 5 and
10 representations of SUŽ5., respectively, and
C1 s 112 M12
C2 s 32 M22
a 2 Ž MG .
a 12 Ž MS .
a 2 Ž MG .
a 22 Ž MS .
y1 ,
y1 ,
Ž 21 .
with the GUT scale MG ; 10 16 GeV and the SUSY
mass scale MS . In GMSB models the selectron mass
formulas are the same as above, except that a Ž MG .
and m 0 Ž MG . are replaced by a Ž M X . and m 0 Ž M X .
where M X is the messenger mass scale. Using the
determinations of m e˜L, m e˜R , M1 , M2 and cos2 b ,
Eqs. Ž20. can be solved for m50 and m10
0 to test
.
universality Ž m50 s m10
of
the
soft
supersymmetry0
breaking scalar masses.
Summary. We have shown that the processes eg
™ nx
˜ ˜y and e˜ x˜ 0 offer the opportunity to
Ø measure the selectron and sneutrino masses via
the ratios of cross sections with different initial
electron and photon polarizations,
Ø estimate tan b through the relation in Eqs. Ž17.,
Ž19.,
Ø test the universality of mSUGRA scalar masses at
the GUT scale,
Ø deduce elements of the chargino and neutralino
mass diagonalization matrices from the cross-section normalizations.
This information will be most valuable to the study
of supersymmetric unification models, especially if
the thresholds for e˜qe˜y and nn
˜ ˜ production are
beyond the kinematic reach of the Next Linear eqey
Collider.
V.B. and T.H. would like to thank the Aspen Center
for Physics for warm hospitality during the final
stages of this project. We thank J. Feng and X. Tata
for conversations. This work was supported in part
by the US Department of Energy under Grants No.
DE-FG02-95ER40896 and No. DE-FG0391ER40674. Further support was provided by the
University of Wisconsin Research Committee, with
funds granted by the Wisconsin Alumni Research
Foundation, and by the Davis Institute for High
Energy Physics.
241
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12 February 1998
Physics Letters B 419 Ž1998. 243–252
SUSY-QCD corrections to stop and sbottom decays
into W " and Z 0 bosons
A. Bartl a , H. Eberl b, K. Hidaka c , S. Kraml b, W. Majerotto b, W. Porod a ,
Y. Yamada d
b
a
Institut fur
¨ Theoretische Physik, UniÕersitat
¨ Wien, A-1090 Vienna, Austria
¨
Institut fur
Akademie der Wissenschaften, A-1050 Vienna, Austria
¨ Hochenergiephysik, Osterreichische
c
Department of Physics, Tokyo Gakugei UniÕersity, Koganei, Tokyo 184, Japan
d
Department of Physics, Tohoku UniÕersity, Sendai 980-77, Japan
Received 15 October 1997
Editor: R. Gatto
Abstract
We calculate the supersymmetric O Ž a s . QCD corrections to stop and sbottom decays into vector bosons within the
Minimal Supersymmetric Standard Model. We give analytic formulae and perform a numerical analysis of these decays. We
find that SUSY-QCD corrections to the decay widths are typically y5% to y10% depending on the squark masses, squark
mixing angles, and the gluino mass. q 1998 Elsevier Science B.V.
1. Introduction
In the Minimal Supersymmetric Standard Model
ŽMSSM. w1x every quark has two scalar partners, the
squarks q˜L and q˜R . In general q˜L and q˜R mix to
form mass eigenstates q˜1 and q˜2 Žwith m q˜1 - m q˜2 .,
the size of the mixing being proportional to the mass
of the corresponding quark q w2x. Therefore, the
scalar partners of the top quark Žstops. are expected
to be strongly mixed so that one mass eigenstate t˜1
can be rather light and the other one t˜2 heavy. The
sbottoms b˜ L and b˜ R may also considerably mix for
large tan b s Õ 2rÕ1 Žwhere Õ 1 and Õ 2 are the vacuum expectation values of the two Higgs doublets..
Squark pair production in eqey annihilation including mixing has been studied at tree-level in Ref.
w3x, then including conventional QCD corrections in
w4,5x, and supersymmetric ŽSUSY. QCD corrections
in Ref. w6,7x. The cross section of squark production
at hadron colliders ŽLHC and Tevatron. in next-toleading order of SUSY-QCD was given in Ref. w8x.
While, quite naturally, most studies concentrated
on t˜1 Ž b˜ 1 . production and decays, those of the
heavier t˜2 Ž b˜ 2 . have been discussed much less w9–
11x. These particles could be produced at the LHC or
an eqey Linear Collider. The decay patterns of the
heavier mass eigenstates t˜2 Ž b˜ 2 . can be very complex due to the many possible open decay channels.
There are the decays Ž i, j s 1,2; k s 1 . . . 4.
t̃ i ™ t x˜ k0 , b x˜q
j ,
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 7 5 - 5
b˜ i ™ b x˜ k0 , t x˜y
j ,
Ž 1.
A. Bartl et al.r Physics Letters B 419 (1998) 243–252
244
the renormalization prescription as introduced in w7x,
where it was applied to the case of eq ey™ q˜i q˜ j .
and in case the mass splitting is large enough
t˜i ™ b˜ j Wq,
b˜ i ™ t˜j Wy,
t˜2 ™ t˜1 Z 0 ,
b˜ 2 ™ b˜ 1 Z 0 ,
Ž 2.
2. Tree-level formulae and notation
and
The current eigenstates q˜L and q˜R are related to
their mass eigenstates q˜1 and q˜2 by
t˜i ™ b˜ j Hq, b˜ i ™ t˜j Hy,
t˜2 ™ t˜1 Ž h 0 , H 0 , A0 . ,
b˜ 2 ™ b˜ 1 Ž h0 , H 0 , A0 . .
Ž 3.
Stops and sbottoms can also decay strongly
through t˜i ™ t g˜ and b˜ i ™ b g.
˜ If these decays are
kinematically possible, they are important. The
SUSY-QCD corrections to squark decays into
charginos and neutralinos have been calculated in
w12x. The SUSY-QCD corrections to squark decays
into Higgs bosons have been treated in w13x, and
those into gluino in w14x. The decays to photon and
gluon, q˜2 ™ q˜1 g , q˜1 g, which are absent at tree-level,
are not induced by O Ž a s . SUSY-QCD corrections
either. This is due to SUŽ3.c = UŽ1.em gauge invariance.
In this paper we calculate the O Ž a s . SUSY-QCD
corrections to the squark decays into W " and Z 0
bosons of Ž2.. The stop decays into vector bosons
can be dominant in the parameter region where Ži.
m t Ž A t y m cot b ., appearing in the mass matrix, is
large enough to give the necessary mass splitting, Žii.
M and < m < are relatively large to suppress the decay
modes to chargino, neutralino, and gluino, and Žiii.
the decays into Higgs particles are not so important
Žfor instance, m A is large.. For sbottoms to decay
into vector bosons instead of Ži. it is necessary that
m b Ž A b y m tan b . is large enough to lead to a large
mass splitting of b˜ 1 and b˜ 2 .
In calculating the SUSY-QCD corrections we
work in the on-shell renormalization scheme and use
dimensional reduction w15x, which preserves supersymmetry at least at two-loop level. The gluon loop
corrections are calculated in the Feynman gauge. The
couplings of the squarks to vector bosons depend on
the squark mixing angles for which an appropriate
renormalization procedure is necessary. Here we use
q˜1
q˜L
ž / ž /
s R q̃
q˜2
R q̃ s
ž
q˜R
,
cos uq˜
sin uq˜
ysin uq˜
cos uq˜
/
Ž 0 F uq̃ - p . .
Ž 4.
In the Ž q˜1 , q˜2 . basis the squark interactions with
Z 0 and W " bosons are given by
ig
Lsy
cos u W
ig
y
'2
ig
y
'2
‹
›
Zm Ri1q˜ Rj1q˜ CL q Ri2q˜ Rj2q˜ CR q˜†j E m q˜i
ž
/
‹ ›
˜
Ri1t̃ Rj1b Wmy b˜ †j E m t˜i
‹
›
˜
Ri1b Rj1t˜ Wmq t˜†j E m b˜ i ,
Ž 5.
where CL, R s I3qL, R y e q sin2u W with I3q the third
component of the weak isospin and e q the electric
charge. Here it is assumed that the Ž3,3. element of
the super-CKM matrix K˜33 s 1.
At tree-level the amplitude of a squark decay into
a W " or Z 0 boson has the general form
m
M0 Ž q˜ia ™ q˜ jb V . s yig c i jV Ž k 1 q k 2 . em) Ž k 3 . ,
Ž 6.
with k 1 , k 2 , and k 3 the four-momenta of q˜ia , q˜ jb,
and the vector boson V Ž V s W ",Z 0 ., respectively
ŽFig. 1a.. a and b are flavor indices. In the follow-
A. Bartl et al.r Physics Letters B 419 (1998) 243–252
Fig. 1. Feynman diagrams relevant for the O Ž a s . SUSY-QCD corrections to squark decays into vector bosons.
245
A. Bartl et al.r Physics Letters B 419 (1998) 243–252
246
a
ing we define m i s m q˜ia , m j s m q˜ jb , Ri k s Riq̃k , Rjk
b
s Rjkq̃ for simplicity. Moreover, we shall use primes
to explicitly distinguish between different flavors.
With this notation the q˜ia-q˜ jb-V couplings c i jV are
c i jZ s
1
cos u W
=
ž
Ž Ri1 Rj1CL q Ri2 Rj2 CR . s
y 12 I3qL sin2 uq˜
I3qL sin2uq˜ y e q sin2u W
1
1
s
'2
/
,
ij
X
Ri1 Rj1
ž
cos uq˜ cos uq˜X
ycos uq˜ sin uq˜X
ysin uq˜ cos uq˜X
sin uq˜ sin uq˜X
/
.
The tree-level decay width can thus be written as
w9x
2
G
qia ™ q jb V
Ž˜
˜
.s
g 2 Ž c i jV . k 3 Ž m2i ,m2j ,m2V .
16p m2V m3i
2
3.1. Vertex corrections
The vertex corrections stem from the five diagrams shown in Figs. 1 b–f. The gluon-exchange
graphs of Figs. 1b–d yield
,
d c iŽ ÕjV, g .
2
with k Ž x, y, z . s Ž x q y q z y 2 xy y 2 xz y
2 yz .1r2 .
Note that the decay q˜LŽ R. ™ q˜RŽ L. Z 0 does not
occur at tree-level, and q˜i ™ q˜Xj W " is not allowed
at tree-level if one of the squarks is a pure ‘‘right’’
state.
3. SUSY-QCD corrections
The O Ž a s . loop corrected decay amplitude is
obtained by the shift
sy
,
Ž 10 .
in Ž6.. The superscript Õ denotes the vertex correction ŽFigs. 1 b–f., w the wave function correction
ŽFigs. 1 g–i., and c the counterterm to the couplings.
Thus the O Ž a s . corrected decay width G is given
by
as
3p
c i jV B0 Ž m 2i , l2 ,m2i . q B0 Ž m2j , l2 ,m2j .
y2 Ž m 2i q m 2j y m2V . Ž C0 q C1 q C2 . .
Ž 13 .
B0 and C0,1,2 are the standard two- and three-point
functions w16x for which we follow the conventions
of w17x. Here we use the abbreviation Cm s
CmŽ m2i ,m2V ,m2j ; l2 ,m2i ,m2j .. A gluon mass l is introduced to regularize the infrared divergence.
The gluino-exchange contribution, Fig. 1e, gives
Ž Õ , g̃ .
d c 21
Z
sy
Žw.
Ž c.
c i jV ™ c i jV q d c iŽ Õ.
jV q d c i jV q d c i jV
Ž 12 .
dGreal is the correction due to real gluon emission
ŽFigs. 1j– l . which is included in order to cancel the
infrared divergence. The total correction can be decomposed into a gluon Žexchange and emission.
contribution dG Ž g ., and a gluino-exchange contribution dG Ž g̃ . : G s G 0 q dG Ž g . q dG Ž g̃ .. The contribution from squark loops ŽFigs. 1 f,i. cancels out in our
renormalization scheme.
Ž 9.
2
c i jV Re d c iŽ a.
jV 4
ij
Ž 8.
0
8p m2V m3i
Ž a s Õ,w,c . .
y 12 I3qL sin2 uq˜
'2
g 2 k 3 Ž m2i ,m2j ,m2V .
cos u W
I3qL cos 2uq˜ y e q sin2u W
Ž 11 .
where
dG Ž a. s
1
Ž 7.
c i jW s
G s G 0 q dG Ž Õ. q dG Ž w . q dG Ž c. q dGreal
as
3p cos u W
½I
q
3L
2 m2g˜ C0 q m2q˜2 C1 q m 2q˜1 C2
q Ž m 2g˜ y m2q . Ž C1 q C2 . q B0 sin2 uq˜
q2 m g˜ m q Ž I3qL y 2 e q sin2u W .
= Ž C0 q C1 q C2 . cos2 uq̃ 4 ,
Ž 14 .
A. Bartl et al.r Physics Letters B 419 (1998) 243–252
with Cm s CmŽ m2q˜2 ,m2Z ,m2q˜1;m 2g˜ ,m 2q ,m2q . and B0 s
B0 Ž m2Z ,m 2q ,m2q ., for the decay q˜2 ™ q˜1 Z 0 , and
d c iŽ ÕjW, g˜ . s y
'2
p
3
Ž mq
as
l ™ 0.. The squark self-energy contribution due to
gluon exchange ŽFig. 1g. is
ṠnŽng . Ž m2q˜n . s y
m g˜ Ž C0 q C1 q C2 .
247
2 as
B0 m 2q˜n ,0,m2q˜n
ž
3 p
/
q2 m 2q˜nB˙0 m2q˜n , l2 ,m2q˜n ,
X
X
Ri2 Rj1
q m qX Ri1 Rj2
.
ž
Ž 19 .
/
and that due to gluino exchange ŽFig. 1h. is
y m2i C1 q m2j C2
X
Ri1 Rj1
q m2g˜ Ž 2C0 q C1 q C2 . q B0
X
ym q m qX Ž C1 q C2 . Ri2 Rj2
4
Ž 15 .
2
with Cm s CmŽ m 2i ,mW
,m 2j ;m2g˜ ,m2q ,m 2qX . and B0 s
2
2
2
B0 Ž mW ,m q ,m qX ., for the decay q˜i ™ q˜Xj W ". The
squark loop of Fig. 1f does not contribute because it
is proportional to the four-momentum of the vector
boson. The total vertex correction is thus given by
ŽÕ, g.
Ž Õ , g̃ .
d c iŽ Õ.
jV s d c i jV q d c i jV .
Ž 16 .
ṠnŽng̃ . Ž m2q̃ n .
s
2 as
B0 m 2q˜n ,m2g˜ ,m2q
ž
3 p
/
q m 2q˜n y m2q y m 2g˜ B˙0 m 2q˜n ,m2g˜ ,m2q
ž
/ ž
n
/
y2 m q m g˜ Ž y1 . sin2 uq˜ B˙0 m2q˜n ,m2g˜ ,m2q
ž
/
,
Ž 20 .
Ž g̃ .
S 12
Ž m2q˜n . s S 21Ž g̃ . Ž m2q˜n .
s
4 as
3 p
3.2. WaÕe-function correction
m g˜ m q cos2 uq˜ B0 m2q˜n ,m2g˜ ,m2q .
ž
/
Ž 21 .
The wave-function correction is given by
The four-squark interaction ŽFig. 1i. gives
d c iŽ wjV. s 12 d Z˜i i Ž q˜ia . q d Z˜j j Ž q˜ jb . c i jV
Ž q̃ .
S 12
Ž m2q˜n . s S 21Ž q̃ . Ž m2q˜n .
qd Z˜i k Ž q˜ia . c k jV q d Z˜jl Ž q˜ jb . c i l V
s
as
6p
sin4uq˜ A 0 Ž m2q˜ 2 . y A 0 Ž m2q˜1 . ,
Ž 17 .
Ž 22 .
Z˜n mŽ q˜n . are the squark wave-function renormalization constants. They stem from gluon, gluino, and
squark loops ŽFigs. 1 g–i. and are given by
where A 0 Ž m2 . is the standard one-point function in
the convention of w17x. Note that SnŽnq̃X. Ž m 2q̃1 . s
SnŽnq̃X. Ž m2q̃ 2 ..
Ž i/k, j/l. .
d Z˜nŽ gn , g̃ . Ž q˜n . s yRe S˙ nŽng , g̃ . Ž m2q̃ n . ,
d Z˜nŽ gn˜X, q˜ .
½
5
Re ½ S
Žm .5
,
Ž q˜ . s y
Ž g̃X , q̃ .
nn
n
2
q̃ n
m2q˜n y m q˜nX 2
3.3. On-shell renormalization of squark mixing angles
n / nX ,
Ž 18 .
with S˙ n nŽ m2 . s ESn nŽ p 2 .rE p 2 < p 2sm 2 . ŽThe gluon
loop due to the qqgg
˜˜ interaction gives no contribution because it is proportional to the gluon mass
It is necessary to renormalize the squark mixing
angles uq̃ by adding appropriate counterterms to
obtain ultraviolet finite decay widths:
Ž c.
d c 21
Zsy
1
cos u W
I3qL cos2 uq˜ duq˜ ,
Ž 23 .
A. Bartl et al.r Physics Letters B 419 (1998) 243–252
248
d c iŽ c.
jW
1
s
ž
'2
q
ž
ysin uq˜ cos uq˜X
sin uq˜ sin uq˜X
ycos uq˜ cos uq˜X
cos uq˜ sin uq˜X
ycos uq˜ sin uq˜X
ycos uq˜ cos uq˜X
sin uq˜ sin uq˜X
sin uq˜ cos uq˜X
/
/
grals I, In , and In m have Ž m i ,m j ,m V . as arguments.
Their explicit forms are given in w17x.
We have checked explicitely that the corrected
decay width, Eq. Ž11., is ultraviolet and infrared
finite.
duq̃
duq̃X
.
ij
4. Numerical results and discussion
Ž 24 .
For the definition of the on-shell mixing angles uq̃
we follow the procedure given in Ref. w7x. The
counterterm duq̃ is fixed such that it cancels the
off-diagonal part of the squark wave function corrections to the cross section of eq ey™ q˜1 q˜2 . duq̃ s
duq˜Ž q̃ . q duq˜Ž g̃ . is then given by
duq˜Ž q˜ . s
as
sin4uq̃
6p
m 2q˜1 y m 2q˜2
A 0 Ž m 2q˜2 . y A 0 Ž m2q˜1 . ,
In general, the stop and sbottom sectors are determined by the soft SUSY breaking parameters Ž MQ̃ ,
MU˜ , and MD˜ ., the trilinear couplings Ž A t and A b .,
m , and tan b , which all enter the squark mass matrices. In order to show the importance of the q˜2 ™ q˜1 Z 0
and q˜i ™ q˜XjW " decays, we plot in Fig. 2 the
branching ratios of these modes as a function of m
for MQ˜ s 500 GeV, MU˜ s 444 GeV, MD˜ s 556 GeV,
and A t s A b s 500 GeV. For the SUŽ2. gaugino mass
Ž 25 .
duq˜Ž g˜ . s
as
3p
m g˜ m q
I3qL
2
q1
0
2
q1
B0 m2q˜2 ,m2g˜ ,m2q Õ̃ 11
ž
2
q2
ž m ˜ ym ˜ /
y B ž m ˜ ,m ˜ ,m / Õ˜
2
g
2
q
22
/
,
Ž 26 .
with Õ˜ 11 s 4Ž I3qL cos 2uq˜ y e q sin 2u W . and Õ˜ 22 s
4Ž I3qL sin2uq˜ y e q sin2u W .. Obviously, for the decay
q˜2 ™ q˜1 Z 0 , the counterterm Ž23. completely cancels
the off-diagonal wave function corrections ŽFigs. 1h
Ž i / n. and 1i Ž i / k ... In case of q˜i ™ q˜Xj W " the
contribution of the squark loop, Fig. 1i, is cancelled.
Thus, in both cases the total squark loop contribution
to the correction is zero, dG Ž q̃ . ' 0.
3.4. Real gluon emission
In order
include real
G Ž q˜ia ™ q˜ jb
dGreal s
to cancel the infrared divergence we
Žhard and soft. gluon emission: dGreal s
V g . ŽFigs. 1j, 1k, and 1 l .. The width is
g 2 c i2jV a s
3p 2 m i
½
k2
2 Iy
m2V
I0 q I1 q m2i I00
q m2j I11 q Ž m 2i q m 2j y m2V . I01
5.
Ž 27 .
Again, k s k Ž m2i , m 2j , m2V .. The phase space inte-
Fig. 2. Tree-level branching ratios for stop and sbottom decays
into vector bosons as a function of m , for MQ̃ s 500 GeV,
MU˜ s 444 GeV, MD˜ s 556 GeV, A t s A b s 500 GeV, M s 200
GeV, and m A s 200 GeV. (a) tan b s 2, and (b) tan b s 40.
A. Bartl et al.r Physics Letters B 419 (1998) 243–252
we take M s 200 GeV, and for the mass of the
pseudoscalar Higgs m A s 200 GeV Žfor the UŽ1.
gaugino mass M X and the gluino mass we use the
GUT relations M X s 53 M tan 2u W and m g˜ s
a sra 2 M .. Fig. 2a Ž2b. is for tan b s 2 Ž40.. We see
that the squark decays into vector bosons can have
very large branching ratios under the conditions Ži.,
Žii., and Žiii. given in the introduction.
We now turn to the numerical analysis of the
O Ž a s . SUSY-QCD corrected decay widths. As the
squark couplings to vector bosons depend only on
the squark mixing angles, we use the on-shell squark
masses m q˜1,2 and mixing angles uq˜ Ž0 F uq˜ - p . as
input parameters. Moreover, we take m t s 175 GeV,
m b s 5 GeV, m Z s 91.2 GeV, sin2u W s 0.23, a Ž m Z .
s 1r128.87, and a s Ž m Z . s 0.12. For the running of
a s we use a s Ž Q 2 . s 12prwŽ33 y 2 n f . lnŽ Q 2rL2n f .x
with n f the number of flavors.
We first discuss the decay t˜2 ™ t˜1 Z 0 . Fig. 3 shows
the tree-level and the O Ž a s . SUSY-QCD corrected
widths of this decay as a function of the lighter stop
mass m t˜1, for m t˜2 s 650 GeV, cos u t˜ s y0.6, and
m g̃ s 500 GeV. SUSY-QCD corrections reduce the
tree-level width by y11.7% to y6.8% in the range
of m t̃ 1 s 80 to 508 GeV. It is interesting to note that
the gluonic correction decreases quickly with increasing m t̃ 1 while the correction due to gluino
exchange varies only little with m t̃ 1. In our example,
dG Ž g .rG 0 s y4.5% and dG Ž g̃ .rG 0 s y7.2% at
m t˜1 s 80 GeV, whereas at m t˜1 s 550 GeV dG Ž g .rG 0
, 0% and dG Ž g̃ .rG 0 s y6.8%. The dependence
on the stop mixing angle is shown in Fig. 4. Fig. 4a
Fig. 3. Tree-level Ždashed line. and SUSY-QCD corrected Žsolid
line. widths of the decay t˜2 ™ t˜1 Z 0 as a function of m t˜1 for
m t˜2 s650 GeV, cos u t˜ sy0.6, and m g̃ s 500 GeV.
249
Fig. 4. Ža. Tree-level Ždashed line. and SUSY-QCD corrected
Žsolid line. widths of the decay t˜2 ™ t˜1 Z 0 as a function of cos u ˜t
for m t˜1 s 200 GeV, m t˜2 s 650 GeV, and m g̃ s 500 GeV. Žb.
SUSY-QCD correction relative to the tree-level width of the
decay t˜2 ™ t˜1 Z 0 as a function of cos u ˜t for m t˜1s 200 GeV and
m t̃ 2 s650 GeV. The black Žgray. lines are for m g̃ s 500 Ž1000.
GeV.
shows the tree-level width together with the O Ž a s .
corrected width of t˜2 ™ t˜1 Z 0 as a function of cos u t˜
for m t˜1 s 200 GeV, m t˜2 s 650 GeV, and m g̃ s
500 GeV. Assuming MU˜ - MQ˜ , as suggested by
SUSY-GUT, the stop mixing angle is varied in the
range y '12 - cos u t˜ - '12 . With the t˜1-t˜2-Z 0 coupling proportional to sin2 u t̃ the decay width has
maxima at cos u t˜ s " '12 Žmaximal mixing. and vanishes at cos u t̃ s 0. In Fig. 4b we plot the relative
correction dGrG 0 for m t̃1,24 s 200,6504 GeV and
m g̃ s 500 and 1000 GeV. As the gluonic correction
has the same u t̃ dependence as the tree-level width,
dG Ž g .rG 0 s y3% in our example, the u t̃ dependence in Fig. 4b comes only from the correction due
to gluino exchange. For m g˜ s 500 GeV and cos u t˜ Q
y0.1 Žcos u t̃ R 0.1. the correction is y10% to y12%
Žy3% to y5%.; for m g˜ s 1 TeV and <cos u t˜ < R 0.1
the correction is y2% to y5%. Approaching cos u t̃
s 0, where t˜1Ž2. s t˜RŽ L. , dGrG 0 diverges due to the
250
A. Bartl et al.r Physics Letters B 419 (1998) 243–252
vanishing tree-level coupling c 21 Z while d c 21 Z / 0.
In this case the decay width becomes of O Ž a s2 ..
Note, however, that the appearance of this divergence, as well as the condition cos u t̃ s 0, is renormalization scheme dependent. Taking a closer look
on the gluino mass dependence we find that the
gluino decouples slowly; e.g. for m t̃ 1 s 200 GeV,
m t˜2 s 650 GeV, and cos u t˜ s y0.6 the gluino contribution to the correction dG Ž g̃ .rG 0 s y2.6%,y
0.8%,y 0.4% for m g̃ s 600 GeV, 1 TeV, 1.5 TeV, respectively. On the other hand, the size of the gluino
contribution quickly increases when approaching the
t̃ 2 ™ tg˜ threshold: For m g̃ s 480 GeV dG Ž g̃ .rG 0 s
y17.2%, and for m g̃ s 476 GeV dG Ž g̃ .rG 0 s
y38.8%. For m g˜ - m t˜2 y m t the gluino correction
becomes positive. In our example it reaches the
maximum at m g̃ s 270 GeV with dG Ž g̃ .rG 0 s
Fig. 6. Ža. Tree-level Ždashed line. and SUSY-QCD corrected
Žsolid line. widths of the decay t˜2 ™ b˜ 1Wq as a function of cos u t˜
for m t˜1 s 300 GeV, m t˜2 s650 GeV, m b˜ 1 s 380 GeV, cos u b˜ s
y0.8, m g˜ s 500 GeV, and tan b s 40. m b˜ 2 is a function of the
other parameters and varies with cos u t̃ . Žb. SUSY-QCD corrections relative to the tree-level width of the decay t˜2 ™ b˜ 1Wq as a
function of cos u t˜ for m t˜1s 300 GeV, m t˜2 s650 GeV, m b˜ 1s
380 GeV, cos u b̃ sy0.8, and tan b s 40. The black Žgray. line is
for m g˜ s 500 Ž1000. GeV. m b˜ 2 is a function of the other parameters, as in Fig. 6a.
Fig. 5. Ža. Tree-level Ždashed lines. and SUSY-QCD corrected
Žsolid lines. widths of the decays b˜ 1,2 ™ t˜1Wy as a function of
cos u t˜ for m b˜ 1s 500 GeV, m b˜ 2 s 520 GeV, cos u b˜ sy0.9, m t˜1s
200 GeV, m g˜ s 520 GeV, and tan b s 2. m t˜2 is a function of the
other parameters and varies with cos u t̃ . Žb. SUSY-QCD corrections relative to the tree-level widths of the decays b˜ 1 ™ t˜1Wy
Ždashdotted lines. and b˜ 2 ™ t˜1Wy Žsolid lines. as a function of
cos u t˜ for m b˜ 1s 500 GeV, m b˜ 2 s 520 GeV, cos u b˜ sy0.9, m t˜1s
200 GeV, and tan b s 2. The black Žgray. lines are for m g̃ s 520
Ž1000. GeV. m t̃ is a function of the other parameters, as in Fig.
2
5a.
3.4%. In general, the dependence on the gluino mass
near the threshold is less pronounced for cos u t̃ ) 0.
The decay b˜ 2 ™ b˜ 1 Z 0 can be important for large
tan b Žsee Fig. 2b.. The SUSY-QCD corrections to
this decay behave similarly to those to t˜2 ™ t˜1 Z 0 .
However, the corrections due to gluino exchange are
smaller because of the smaller bottom quark mass
and thus the dependece of dGrG 0 on the sbottom
mixing angle is very weak.
Let us now turn to the squark decays into W "
bosons. Here we discuss two special cases: Ži. b˜ 1
and b˜ 2 decaying into a relatively light t˜1 plus Wy
for small sbottom mixing Žsmall tan b scenario.. In
this case the mass difference of b˜ 1 and b˜ 2 is expected to be rather small and thus the decays b˜ 2 ™
b˜ 1 Ž Z 0 ,h0 , H 0 , A0 . should be kinematically suppressed or even forbidden. Žii. A heavy t˜2 decaying
A. Bartl et al.r Physics Letters B 419 (1998) 243–252
into a relatively light b˜ 1 plus Wq for large sbottom
mixing Žlarge tan b scenario.. Note, however, that in
a combined treatment of both the stop and the sbottom sectors there is a constraint among the parameters m t˜1, m t˜2 , u t˜, m b˜ 1, m b˜ 2 , and u b˜ Žfor a given value
of tan b . due to SUŽ2. L gauge symmetry w10,18x.
Therefore one of these parameters is fixed by the
others. Here one has to take into account that in the
on-shell scheme MQ2˜ Ž t˜. / MQ2˜ Ž b˜ . at O Ž a s ., see Ref.
w18x. The decay widths of b˜ 1,2 ™ t˜1Wy are shown in
Fig. 5a as a function of the stop mixing angle, for
m b˜ 1 s 500 GeV, m b˜ 2 s 520 GeV, cos u b˜ s y0.9, m t˜1
s 200 GeV, m g̃ s 520 GeV, and tan b s 2. The value
of m t̃ 2 is determined by the other parameters as
discussed above. Hence m t̃ 2 varies from 533 GeV to
733 GeV depending on the stop mixing angle in Fig.
5a. ŽHere we have also checked that the squark
parameters do not cause too large A t or A b to avoid
color breaking minima.. Despite the larger phase
space for the b˜ 2 decay, the width of b˜ 2 ™ t˜1Wy is
smaller than that of b˜ 1 ™ t˜1Wy because the W couples only to the ‘‘left’’ components of the squarks
Žnote that b˜ 1 ; b˜ L and b˜ 2 ; b˜ R for cos u b˜ s y0.9..
Fig. 5b shows the relative correction dGrG 0 for the
same squark parameters as in Fig. 5a. For m g̃ s
520 GeV and <cos u t̃ < R 0.1 SUSY-QCD corrections
are y11% to q4%. For m g̃ s 1 TeV the effects are
weaker, i.e. about y5% to y1%. Again, there is a
‘‘singularity’’ at cos u t̃ s 0 where the tree-level t˜1-b˜ iW " coupling vanishes. The dependence of dGrG 0
on the sbottom mixing angle is, in general, much
weaker than that on the stop mixing angle Žapart
from a singularity for the b˜ i being a pure b˜ R .. The
overall dependence on the gluino mass is in general
similar to that of the t˜2 ™ t˜1 Z 0 decay. However, the
enhancement effect of the threshold at m g˜ s m b˜ i y
m b is less pronounced. An example for large tan b is
shown in Fig. 6. In Fig. 6a we plot the tree-level and
the SUSY-QCD corrected widths of t˜2 ™ b˜ 1Wq as a
function of cos u t˜ for m t˜1 s 300 GeV, m t˜2 s 650 GeV,
m b˜ 1 s 380 GeV, cos u b˜ s y0.8, m g̃ s 500 GeV, and
tan b s 40. m b̃ 2 is calculated from the other parameters and thus varies from 615 GeV to 918 GeV. As
expected, the decay width is maximal for t˜2 s t˜L and
vanishes for t˜2 s t˜R . In the example chosen, SUSYQCD corrections are y2.4% to y4.7% for m g̃ s
500 GeV and about y1% to y1.5% for m g̃ s 1 TeV
as can be seen in Fig. 6b, where dGrG 0 is shown
251
as a function of cos u t̃ for the same squark parameters as given above, and m g̃ s 500 GeV and 1 TeV.
Again, there is almost no dependence on cos u b̃ apart
from the singularity at cos u b̃ s 0, i.e. b˜ 1 s b˜ R .
5. Summary
We have calculated the O Ž a s . supersymmetric
QCD corrections to squark decays into vector bosons
in the on-shell renormalization scheme using dimensional reduction. In particular, we have discussed
examples for the decays t˜2 ™ t˜1 Z 0 , t˜2 ™ b˜ 1Wq, and
b˜ 1,2 ™ t˜1Wy. We have found that the correction
dGrG 0 is typically of the order y5% to y10%,
depending on the squark masses, squark mixing angles, and m g̃ . Near the q˜ ™ qg˜ threshold the correction can also exceed y10%. It has also turned out
that the gluino decouples slowly. Moreover, the
gluino-exchange corrections alter the uq̃ dependence
of the tree-level widths. For squark mixing angles
where the decay width vanishes at tree-level, the
gluino corrections do not vanish and may lead to
non-zero widths of O Ž a s2 ..
Acknowledgements
The work of A.B., H.E., S.K., W.M., and W.P.
was supported by the ‘‘Fonds zur Forderung
der
¨
wissenschaftlichen Forschung’’ of Austria, project
no. P10843-PHY.
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12 February 1998
Physics Letters B 419 Ž1998. 253–257
Vortices and flat connections
Sazzad Mahmud Nasir
1
Department of Applied Mathematics and Theoretical Physics, UniÕersity of Cambridge, SilÕer Street, Cambridge CB3 9EW, UK
Received 17 October 1997
Editor: P.V. Landshoff
Abstract
At Bradlow’s limit, the moduli space of Bogomol’nyi vortices on a compact Riemann surface of genus g is determined.
The Kahler
form, and the volume of the moduli space is then computed. These results are compared with the corresponding
¨
results previously obtained for a general vortex moduli space. q 1998 Elsevier Science B.V.
1. The Abelian Higgs model in Ž2 q 1. dimensions is an interesting arena to study vortices. The
coupling constant of the model determines the nature
of interactions among vortices. At the critical coupling, the model admits static and finite energy
Bogomol’nyi vortex solutions w1x. Stability of these
solutions is ensured by topology. We will consider
vortices in a space-time of the form R = M, where
M is a compact two dimensional manifold. The
metric of the space-time is taken to be ds 2 s dx 02 y
V Ž x 1 , x 2 .Ž dx 12 q dx 22 ., where x 1 and x 2 denote local
coordinates on M. Let Am , Ž m s 0,1,2. be a UŽ1.
gauge potential and f be a complex scalar field.
Working in the gauge A 0 s 0, the Lagrangian of the
model at the critical coupling is L s T y V, where
T s 12
˙ ˙/,
HMd x ž A˙ A˙ q Vff
V s 12
HMd
2
i
2
xV
1
2
i
i
Fi j F i j q Di fD f
q 14 Ž < f < 2 y 1 .
2
are respectively, the kinetic and the potential energies. Here, Di s E i y iA i , and F12 s E 1 A 2 y E 2 A1 , is
the magnetic field. The following first order Bogomol’nyi vortex equations are obtained by minimizing
the potential energy
V
Ž 3.
Ž 4.
Ž < f < 2 y 1 . s 0.
2
The above equations admit static multi-vortex solutions. The solutions are parametrized by a 2 N dimensional moduli space, MN , where N is the numF12 q
E-mail address: [email protected].
Ž 1.
Ž 2.
Ž D1 q iD 2 . f s 0
1
i s 1,2
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 6 5 - 2
S.M. Nasirr Physics Letters B 419 (1998) 253–257
254
ber of zeros of the Higgs field counted with multiplicity w2,3x. N y called the vortex number – is
related to the total magnetic flux by N
s 21p HM d 2 xF12 . The potential energy of a configuration of static N-vortices is p N. Topologically, MN
is just the symmetrized N-th power of M. The
moduli space has a natural Riemannian metric induced from the kinetic energy expression Ž1.. This is
given by
1
ds 2 s
d 2 x Ž d A i d A i q Vdfdf . , i s 1,2.
p M
Ž 5.
H
In obtaining Ž5. from Ž1. we have multiplied Ž1. by
2rp to agree with the conventions of w4x. It is to be
noted that there is a Gauss’ law constraint arising
from the equation of motion of A 0 . This constraint
ensures that MN consists of the solutions of Eqs. Ž3.,
Ž4. modulo gauge transformations connected to the
identity. Hereafter, by ‘gauge transformations’ we
will mean those gauge transformations connected to
the identity. Through Manton’s work w5x it is known
that at low energy – when most of the degrees of
freedom remain unexcited – the moduli space can be
used to describe interesting physical phenomena associated with vortices, such as scattering w6x, thermodynamics w4,7,8x, and the phase transition of vortices
at near-critical coupling w9x, etc. In the moduli space
approximation the vortex dynamics can be thought
of as geodesic motion on the moduli space.
For a compact surface, in order for vortex solutions to exist one has to satisfy Bradlow’s bound
w10x. The bound is 4p N F A, where A is the area of
M. This can be obtained by integrating Ž4. over M,
and noticing that the integral of < f < 2 over M is
positive. The bound means that for a given area there
is a limit on the number of vortices one can have on
M. At Bradlow’s limit, A s 4p N, the Higgs field f
must vanish everywhere on M. Then, the
Bogomol’nyi equations reduce to the following single equation
V
F12 s .
Ž 6.
2
This is an equation for a constant magnetic field on
M. The energy of the configuration is still p N.
It might be thought that Ž6. has a unique solution
up to gauge equivalence, which in turn will mean
that the moduli space is just a point. However, if M
has non-contractible loops Žone-cycles., i.e. if the
first homotopy group of M is non-trivial, then the
moduli space of solutions of Ž6. is non-trivial as
well. A solution of Ž6. up to gauge equivalence is
given, in addition to a gauge potential that solves Ž6.,
by specifying holonomies around a basis of onecycles. The holonomies around any two homologous
loops are generally different as the magnetic field is
non-zero on M, but the difference can be completely
determined by using Stokes’ theorem. Now, linearizing Ž6. around a particular solution one can see that
the perturbed gauge potentials satisfy the equations
for flat UŽ1. connections for which the magnetic
field is zero. Flat connections are associated with
large gauge transformations. Flat connections up to
gauge equivalence are also given by specifying their
holonomies around a basis of one-cycles. Hence,
when f s 0 the moduli space of Bogomol’nyi vortices is no longer MN . Locally, the tangent space of
the vortex moduli space at Bradlow’s limit can be
identified with the tangent space of the space of flat
UŽ1. connections on M. With an abuse of notation,
in future the moduli space of solutions of Ž6. and the
space of flat UŽ1. connections will be denoted by
M f . It was demonstrated long ago by Aharonov and
Bohm w11x that flat connections play quite a non-trivial role in quantum physics. It should be noted that
flat connections do not contribute to the moduli of
the Bogomol’nyi equations when the Higgs field is
non-vanishing.
In passing we would like to point out that when
f s 0, one may consider F12 s 0 as solutions of the
static Abelian Higgs model. The energy is then Ar8
with no restriction on the value of A. However,
these solutions cannot be obtained from Bogomol’nyi
equations. Henceforth, we will only consider vortices
at Bradlow’s limit, and M is taken to be of genus
g G 1.
2. Let A s A1 dx 1 q A 2 dx 2 solve Ž6.. Then, A q
d A, where d A s a1 dx 1 q a2 dx 2 is a flat connection,
is also a solution of Ž6.. The equation satisfied by d A
is d d A s 0. The metric on M f is obtained by putting
f s 0 in Ž5.. This is
ds 2 s
1
p
HMd
2
2
2
x Ž a1 . q Ž a 2 . .
Ž 7.
S.M. Nasirr Physics Letters B 419 (1998) 253–257
Notice that the metric of M does not appear in the
above expression. This means that the metric information of M is not carried over to M f . One can see
that M f inherits a complex structure from M. The
map I given by I :a j ™ ye jk a k ,, where e jk is the
antisymmetric tensor with e 12 s 1, leaves invariant
Ž7. and the equation for a flat connection. Moreover,
I 2 s y1 2 . Hence, I defines an almost complex
structure on M f . This almost complex structure can
be used to define the following Ž1,1. form on M f
v Ž d A, d B . s
1
p
HM Ž d A n d B y d B n d A . .
z
z
z
z
Ž 8.
Here, we have used complex coordinates Ž z s x 1 q
ix 2 ., and d A s Ž d A z q c.c... Clearly, v is manifestly real. Using d d A s d d B s 0, it can also be
shown that v is a closed form. Thus, I is a
form.
complex structure and v defines a Kahler
¨
Among many uses of this Kahler
form one can, for
¨
example, compute the volume of M f . Recently, we
obtained an expression for the Kahler
form on MN
¨
and also, we computed the volume of MN w8x. We
will see that when A s 4p N, the Kahler
form on M f
¨
gets mapped to the Kahler
form on MN .
¨
Although a known fact, in order to setup the stage
for the main part of this paper we intend to show that
the space of flat UŽ1. connections on M modulo
gauge transformations, i.e. M f , is parametrized by
˜ of M w12x. J˜ is dual to the
the Picard variety, J,
Jacobian J of M. J is a 2 g-dimensional real torus.
Let n i , Ž i s 1, PPP ,2 g . be a basis of 2 g one-cycles
of M. Let wr , Ž r s 1, PPP , g . be a basis of g holomorphic one-forms on M. Define the period matrix
W s Ž wr i ., where wr i s En i wr , Ž r s 1, PPP , g, i s
1, PPP ,2 g .. The columns Wi , Ž i s 1, PPP ,2 g . of W
can be thought of as spanning a 2 g-dimensional
lattice in C g . J is defined as the torus C grW.
Riemann’s bilinear relations can be used to normalize the period matrix W w13x. One can choose W
such that ŽW t , W t . is a symmetric matrix with
Im ŽW . ) 0. Further, the elements of W can be
restricted to satisfy wr i s dr i for r s 1, PPP , g and
i s 1, PPP , g, the remaining elements being arbitrary
with positive imaginary parts w13x. Let us choose a
basis of 2 g one-cocycles a i such that En i a j s
d i j , Ž i, j s 1, PPP ,2 g .. These one-cocycles are the 2 g
255
generators of the cohomology group H 1 Ž M,Z .. A
canonical basis of n i can be chosen such that the
cocycles also satisfy HM a i a j s d i, jqg , where in the
integration wedge product is implied. The g holomorphic one-forms can be expressed in terms of a i
g
as wr s Ý2is1
wr i a i for Ž r s 1, PPP , g .. Reciprocally,
a i can be expressed in terms of wr as a i s
Ý rgs1Žg i r wr q c.c.., for Ž i s 1, PPP ,2 g . where the
matrix G s Žgr i . is required to satisfy
G tW q G t W s 12 g .
Ž 9.
We note that J˜ is defined as the torus C grG .
Hence, J˜ is dual to J.
Let cr , Ž r s 1, PPP , g . denote complex coordinates on M f Žthat M f is 2 g-dimensional will be
evident below.. Then a real flat connection A f
Žmodulo gauge transformations. can be expressed as
g
A f s 2p
Ý Ž cr wr q c.c. . .
Ž 10 .
rs1
Imposing Gauss’ law on the flat connections one can
see that the above is the most general expansion for
a flat connection. The holonomy, h j , of A f around a
one-cycle n j is given by
g
ž / ž
h j s exp i
En A
f
s exp 2p i
j
Ý Ž cr wr j q c.c. .
rs1
/
.
Ž 11 .
As noted earlier, for a complete specification of A f
one needs to specify all of the holonomies h j . Eq.
Ž11. implies that h j is periodic with the period
matrix being L s Ž lr i ., Ž r s 1, PPP , g , i s
1, PPP ,2 g ., say. Then the 2 g columns of L span a
2 g-dimensional lattice in C g . Thus, M f is
parametrized by a 2 g-dimensional real torus C grL.
Further, the following relation for L is implied by
Ž11.
Lt W q Lt W s 1 2 g .
Ž 12 .
Comparing the above equation with Ž9. one gets
L s G . This identifies M f with J.˜
There are g independent holomorphic one-forms
on M f . Using cr as coordinates of M f , these are
given by dcr , Ž r s 1, PPP , g .. A basis of one-cycles
in M f is given by the 2 g lines t L i , 0 F t F 1, with
L i identified with 0 to produce a closed loop. Then
S.M. Nasirr Physics Letters B 419 (1998) 253–257
256
from the discussion preceeding Ž9., one deduces that
the 2 g generators of H 1 Ž M f ,Z . are given by j˜i s
Ý rgs1Ž wr i dcr q c.c... Now from Ž8., we get the following Kahler
form on M f
¨
g
v˜ s 4p
Ý Ž wr i wr iqg y wr i wr iqg . dcr n dcr .
X
X
X
r , r X ,is1
Ž 13 .
In terms of j˜i , v˜ can be written as
g
v˜ s 4p Ý j˜i j˜iqg .
Ž 14 .
is1
In obtaining the above from Ž13. we have used the
Riemann bilinear relations. The volume of M f is
then
Vol f s
1
g!
HM v˜
g
s Ž 4p .
g
Ž 15 .
f
where use has been made of the fact that
g
HM f Ł is1
j˜i j˜iqg s 1. It is useful to notice that the
volumes of J and J˜ are the same. The computation
of the volume of the space of flat connections on a
compact Riemann surface is not new. For SUŽ2. and
SO Ž3. Yang-Mills theory, Witten w14x computed the
volume of the space of flat connections by a remarkable use of the Verlinde formula w15x in conformal
field theory.
3. The general formula for the Kahler
form on the
¨
vortex moduli space MN , when f is non-vanishing,
is w8x
g
v s Ž A y 4p N . h q 4p Ý j i j iqg
Ž 16 .
mapped in a one-to-one way to the Kahler
form v˜
¨
on J. As J˜ is isomorphic to J it is enough to
establish an isomorphism between H 1 Ž MN ,Z . and
H 1 Ž J,Z .. First, notice that Jacobi’s inversion theorem w13x implies that there is an isomorphism between J and M g where M g is the moduli space of g
vortices, and g is the genus of M. This implies the
isomorphism between H 1 Ž J,Z . and H 1 Ž M g ,Z ..
Next, using the Lefschetz hyperplane section theorem w13x, one sees that there is an isomorphism
between H 1 Ž MN ,Z . and H 1 Ž J,Z . for N G g and
g ) 1. The isomorphism for other values of N and g
can also be easily established by arguments used in
w16x.
It is of interest to see if one can relate Vol f to the
volume of MN near Bradlow’s limit when e s A y
4p N is a small positive quantity. For genus g G 1
and N ) 2 g y 1, MN has a bundle structure, where
the base is J, and the fibre is CPNy g . For N F g,
MN is analytically homeomorphic to a 2 N-dimensional submanifold of the Jacobian. Generically, the
volume of MN is not just a product of the volume of
the base and the volume of the fibre. The volume of
MN as computed in w8x is
Vol N s Ž A y 4p N .
i
g
=
Ý
is0
ž
g
v s 4p Ý j i j iqg .
Vol N s Ž 4p .
g
At this point we should remind the reader that this
Kahler
form is defined on MN , not on J˜ whose
¨
tangent space coincides with the tangent space of the
moduli space of vortices when f s 0. However, it
can be shown that the Kahler
form v on MN is
¨
/
Ž 18 .
e Ny g
Ž Nyg . !
q O Ž e Nygq1 . .
Ž 19 .
Neglecting the higher order corrections the above
can be written as
Ž 17 .
is1
gyi
Ž 4 p . Ž A y 4p N . g !
.
Ž N y i . ! Ž g y i . !i!
In this formula N G g. It is easy to write an analogous formula for N - g. Near Bradlow’s limit the
above volume can be written as
is1
where h is an area form on MN normalized to unity
and j i , Ž i s 1, PPP ,2 g . are the 2 g generators of
H 1 Ž MN ,Z .. When f s 0 is zero, i.e. A s 4p N, the
Kahler
form in Ž16. reduces to
¨
Ny g
e Ny g
Vol N s Vol f =
Ž Nyg . !
Ž 20 .
where the factor e Ny grŽ N y g .! can be thought of
as a contribution coming from the fibre CPNy g .
Indeed, using Ž16. and the cohomolgy class of the
S.M. Nasirr Physics Letters B 419 (1998) 253–257
fibre one can show that the volume of the fibre is
Ž A y 4p N . Ny grŽ N y g .!.
In conclusion, we would like to clarify the following apparent puzzle. In computing the Kahler
form
¨
Ž16., one needs to extract the non-singular parts of
the expressions like Ez log < f < 2 around the zeros of f .
This, however, does not invalidate the derivation of
Ž16. when f s 0 as one may think. From the Bogomol’nyi equations one can always express Ez log < f < 2
in terms of the gauge potentials, which can in principle be used to derive Ž16. regardless of whether f is
zero or not.
Acknowledgements
I would like to thank my supervisor Dr N.S.
Manton for his continuous guidance and very helpful
discussions. Also thanks to P. Irwin for critical comments on this manuscript. This work was supported
by the Overseas Research Council, the Cambridge
Commonwealth Trust and Wolfson college.
257
References
w1x E.B. Bogomol’nyi, Sov. J. Nucl. Phys. 24 Ž1976. 449.
w2x C.H. Taubes, Comm. Math. Phys. 72 Ž1980. 277.
w3x A. Jaffe, C.H. Taubes, Vortices and Monopoles, Birkhauser,
¨
Boston, 1980.
w4x N.S. Manton, Nucl. Phys. B 400 Ž1993. 624.
w5x N.S. Manton, Phys. Lett. B 110 Ž1982. 54.
w6x T.M. Samols, Comm. Math. Phys. 145 Ž1992. 149; Ph.D
Thesis, Cambridge University 1990, unpublished.
w7x P.A. Shah, N.S. Manton, J. Math. Phys. 35 Ž1994. 1171.
w8x N.S. Manton, S.M. Nasir, DAMTP preprint DAMTP-97-90,
1997.
w9x P.A. Shah, Nucl. Phys. B 438 Ž1995. 589.
w10x S. Bradlow, Comm. Math. Phys. 135 Ž1990. 1.
w11x Y. Aharonov, D. Bohm, Phys. Rev. 115 Ž1959. 485.
w12x R.C. Gunning, Lectures on Riemann Surfaces, Princeton
University Press, New Jersey, 1966.
w13x P. Griffiths, J. Harris, Principles of Algebraic Geometry,
John Wiley & Sons, Inc., New York, 1978.
w14x E. Witten, Comm. Math. Phys. 141 Ž1991. 153.
w15x E. Verlinde, Nucl. Phys. B 300 Ž1988. 360.
w16x I.G. MacDonald, Topology 1 Ž1962. 319.
12 February 1998
Physics Letters B 419 Ž1998. 258–262
Abelian Chern-Simons theory and quantum symmetry
G. Grensing
UniÕersitat
¨ Kiel, Fachbereich Physik, D-24118 Kiel, Germany
Received 2 October 1997; revised 13 November 1997
Editor: P.V. Landshoff
Abstract
Quantized abelian Chern-Simons theory on a Riemann surface of arbitrary nonzero genus g carries a quantum symmetry;
the underlying quantum algebra is the deformation Uq Ž sl gq1 ., with q a root of unity. q 1998 Elsevier Science B.V.
Quantum groups originally emerged in the context
of integrable systems w1x. Since then, they have
found their way into various other branches of theoretical physics, such as conformal field theory w2,3x
and Wess-Zumino-Witten models w4x. A further example is provided by the fractional quantum Hall
effect with odd filling fractions n s 1rp, for which
the quantum group Uq Ž su 2 . with deformation parameter q s expŽ2p i n . has recently been proven w5x to
constitute the spectrum generating algebra.
It is the purpose of the present paper, to enlarge
the list of examples by a purely quantum field
theoretical model, abelian Chern-Simons theory w6x
on a Riemann surface of arbitrary nonzero genus.
This theory is completely solvable w7–10x, and we
will show that the finite-dimensional Hilbert space of
solutions gives rise to a cyclic representation of the
quantum algebra Uq Ž sl gq1 ., where g is the genus of
the surface. The deformation parameter is determined by the Chern-Simons coupling constant k s
rrs as q s exp2p irrs; furthermore, the generators
of the quantum algebra may be expressed in terms of
intrinsic quantities, the holonomy operators along the
2g homology cycles of the surface.
To begin with, abelian Chern-Simons theory is
encoded in the Lagrangian
Ls
k
4p
HSd
2
x ´ i j A˙i A j q A 0 Fi j
ž
/
Ž 1.
where the two-dimensional domain of integration S
is chosen to be a closed Riemann surface of nonzero
genus. The kinetic part is linear in time derivatives
and determines the Poisson brackets of the spatial
components A i to be
Ai Ž x . , A j Ž y . 4 s y
2p
k
´i j d Ž x y y . .
Ž 2.
The term proportional to the time-component A 0
plays the role
ˆ of a constraint so that, in particular,
the Hamiltonian is identically zero.
For quantization purposes, it proves to be essential that the theory is topological. Hence, locally a
conformally flat metric g i j Ž x . s r Ž x . d i j with r Ž x .
) 0 exists so that the complex structure is canonical,
and thus the connection 1-form may be written as
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 5 9 - 7
G. Grensingr Physics Letters B 419 (1998) 258–262
A s A z dz q A z dz. Accordingly, we choose the holomorphic polarization with
Aˆz s A z ,
p
Aˆz s
d
Ž 3.
k d Az
where D is the Laplace-Beltrami operator. It is given
in terms of
G0 Ž x , y . s y
ž
HD
A z , A z exp y
k
p
HSd
xA z A z
/
HA v s d
j
i
ij,
HB v s t
j
Ž 4.
i
HS v n v s Im t
j
i
>
Ž 5.
ij
Ž 6.
ij
s G0 Ž x , y . y
1
V
z
Hw v P Ž Im t .
y1
z
Im
Hw v
Ž 8.
,
1
1
V
HS
2
d x g Ž x . < Ž G 0 Ž x , xX .
2 X<
qG 0 Ž x , y . . q
X
1
1
X
V2
HSd
2 X<
X
x gŽ x .<
2
1
=
HSd
2 X<
X
2
y g Ž y . < G 0 Ž x X , yX . .
Ž 9.
The Chern-Simons gauge field is parametrized as
follows Žsee w10x.:
A z Ž x . s Ez x Ž x . q v i Ž z . a i
Ž 10 .
where the complex valued function x is assumed to
be doubly periodic; the additional term with a g C g
is specific for a field theory restricted to a compact
domain. The above relation can be inverted to give x
and a in terms of A z , and we finally obtain
d
where the period matrix t is symmetric with a
positive definite imaginary part. In addition, we need
the doubly periodic propagator on S with the properties 2
D GŽ x , y. s d Ž x , y. y
log < E Ž z ,w . < 2
GŽ x , y.
where the Schrodinger
wave functionals are anti¨
holomorphic, i.e., only depend on the A z-part of the
gauge field.
Specific geometric properties of Riemann surfaces 1 get involved, on parametrizing the gauge
field. Let Ai and Bi with i s 1, . . . ,g be canonical
homology cycles on S with Žsigned. intersection
numbers Ž Ai , Aj . s 0 s Ž Bi , Bj . and Ž Ai , Bj .
s d i j . Then a basis of holomorphic 1- forms v i s
v i Ž z . dz can be chosen with
>
4p
with EŽ z,w . the prime form, and reads Žcf. also w13x.
2
=c 1 A z c 2 A z
>
1
q 12 Im
and the Bargmann inner product reads
²c1 < c2 : s
259
d Az Ž x .
s y d 2 yP Ž x , y .
H
P Ž Im t .
d
dx Ž y .
qv Ž z .
d
y1
Ž 11 .
da
Ž
.
where P x, y s y4Ez G 0 Ž x, y . obeys
Ez P Ž x , y . s d Ž x y y . y v Ž z . P Ž Im t .
y1
vŽ z. .
Ž 12 .
1
HS
2
d x <g Ž x . < GŽ x , y . s 0
2
Ž 7.
This result will be needed in the following paragraph.
Spatial gauge transformations of the wave functional are generated by the operator
Qˆ w a x s yi
1
For relevant background, see, e.g., w11x, and w12–15x.
Notation: V denotes the area of the surface, and < g < the
determinant of the metric tensor.
2
k
p
H d x ž E a Aˆ y E a Aˆ / .
2
z
z
z
z
Ž 13 .
Here, it is crucial that one can not perform a partial
ˆ with Cˆ s
integration to obtain Cˆw a x s Hd 2 x a C,
G. Grensingr Physics Letters B 419 (1998) 258–262
260
i pk Ž Ez Aˆz y Ez Aˆz . the constraint, since the gauge parameter may not be assumed to be periodic: gauge
invariance and the constraint are different issues for
a gauge theory living in a finite domain w5x. A gauge
transformation g Ž x . s expŽyi a Ž x .. must be periodic, and this requirement permits small gauge transformations with periodic parameters a Ž x ., as well
as large gauge transformations with gauge parameters
aŽ n , nX . Ž x . s ip Ž n q t nX . P Ž Im t .
y1
z
Hz v
0
z
y ip
Hz vP Ž Im t .
y1
Ž n q t nX .
Ž 14 .
0
labelled by n,nX g Z g .
On wave functionals, the gauge transformations
are implemented by means of the operator Uˆ w g x s
expŽyiQˆw a x.; in particular, for small gauge transformations, it acts as
Uˆ w g x c A z s exp
ž
k
p
1 k
y
2 p
Hd
Hd
2
xA z Ez a
ž
Hd
2p
2
s exp Ž ik Ž n q t nX . P a
y 12 p k Ž n q t nX . P Ž Im t .
=c a q ip Ž Im t .
y1
y1
Ž n q t nX . .
Ž n q t nX .
Ž 18 .
where we have set x s 0, because only the a-dependence is affected.
The noncommutativity of the operators of large
gauge transformations signalizes a gauge anomaly.
On assuming the Chern-Simons coupling k to be
rational, i.e., k s rrs with r and s coprime integers,
we avoid to come into conflict with the anomaly by
the requirement that only the operators UˆsŽ n, nX . act as
a multiple of the identity up to a phase; we choose
Žw5x, cf. also w10,16,17x.
UˆsŽ0 , n. c w a x s ey2 p i nPnc w a x
/
x Ez aEz a c A z y Ez a
so that the a-dependence remains untouched. The
gauge parameter being time- independent, this is a
Wigner symmetry operation. Hence, it is legitimate
to require the wave functional be left invariant, i.e.,
Uˆ w g xc w A z x s c w A z x. On passing to the complexification UŽ1. C , this requirement fixes the x-dependence to be
k
UˆŽ n , nX . c w a x
UˆsŽ n ,0. c w a x s e 2 p i nPmc w a x ,
2
Ž 15 .
c A z s exp
Furthermore, the wave functionals transform as follows
/
x Ez xEz x c w a x .
Ž 16 .
Turning to large gauge transformations UˆŽ n, nX . s
expŽyiQˆw aŽ n, nX . x., they commute with the operators
of small gauge transformations, but not among themselves:
UˆŽ n , nX .UˆŽ m , mX .
s exp Ž 2p ik Ž n P mX y nX P m . . UˆŽ m , mX .UˆŽ n , nX . . Ž 17 .
Ž 19 .
with m i , n i g w0,1x. These conditions can be solved in
terms of Jacobi theta functions w18x with characteristics a , b g R g
q
a
w<s . s
b Ž
Ý
exp p i Ž n q a . P s Ž n q a .
ngZ g
q2p i Ž n q a . P Ž w q b . 4
Ž 20 .
where w g C g and s is an element of the Siegel
upper half space. The solutions are labelled by m g
Z g with m i s 1, . . . , p and p s rs; they read
ž
cm w a x s Nexp y
ž
= y
i
p
k
2p
1
/
a P Im t a q
/
r Im t a < p t .
p
Ž mqm
n
Ž 21 .
The domain of a is bounded now, since the variable
y pi 1s Imt a can be restricted to be an element of
C grŽZ g q t Z g ., the Jacobian variety. The wave
functionals cm w A z x are proven to be orthonormal
with respect to the Bargmann inner product, where it
G. Grensingr Physics Letters B 419 (1998) 258–262
is crucial that the normalization constant N comes
out to be m-independent.
Finally, the p g-dimensional space of solutions
will be shown to give rise to a quantum group. For
this purpose, observe that there remains a residual
symmetry, large gauge transformations still act as
symmetries of the system; but one can show even
more, namely, we have
ter q g C= has generators K i" 1 and J " i , subject to
the relations
Jqi , Jyj s
1
X
K i y Ky1
i
1yc i j
1 y ci j
k
ks0
X
nPn qnP Ž mq m .yn P n
r
cmqnX w a x
where q s expŽ2p irp .. Strictly speaking, the operators Uˆ Ž n, nX . are large gauge transformations only for
r s 1; to prevent this objection, one could as well
take k s 1rs, without changing the conclusions to
follow. As generators, let us choose Si s Uˆ Ž e i ,0. and
Tiy1 s Uˆ Ž0, e i . , where e i denotes the unit vector in
the i-th direction. They can also be written as Wegner-Wilson operators Žcf. w6x.
1
r
1
r
ž
H Aˆ
s A
i
/
,
ž
Ti s exp y
i
s
HB Aˆ
i
/
1yc i jyk
k
J"
J" j J"
i
i s 0,
qi
Ž 27 .
di
where qi s q ; the q-deformed factorial is defined
as usual by means of w x x q s Ž q x y qyx .rŽ q y qy1 ..
For the sake of brevity, the additional definitions,
which are needed to supply this deformation with a
Hopf algebra structure, are omitted. We restrict ourselves to the first series, Uq Ž sl gq1 ., with c i j s 2 d i, j
y d i, jq1 y d i, jy1 and d i s 1. For < q < s 1, a realization of the quantum group generators turns out to be
1
r
i
Ž 26 .
i/j
Ž 22 .
Si s exp q
di j ,
qi y qy1
i
" ci j
K i J " j Ky1
J" j
i s qi
Ý Ž y1. k
Uˆ 1 Ž n , nX . cm w a x s q 2
261
1
1
y
Jqi s
qyq
Tiy1 ,
i-n
Ž 28 .
1
Ž 23 .
1
y
Jyi s
along the 2g independent homology cycles, and obey
the ‘quantum plane’ relations
q
2 y1
y1
2 S i S iq1 y q S i S iq1
y1
q
2 y1
y1
2 S iy1 S i y q S iy1 S i
y1
qyq
2 y1
K i s Sy1
iy1 S i S iq1 ,
Ti ,
i)1
i s 2, . . . ,n y 1
Ž 29 .
Ž 30 .
and for the remaining cases
Ti Si s q Si Ti .
Ž 24 .
1
1
y
On the basis cm w A z x s ² A z < m:, they act as
Si < m: s q m iqm i < m: ,
Ti < m: s q n i < . . . ,m i y 1, . . . :
Ž 25 .
thus yielding an irreducible, unitary representation.
Hence, the generators of large gauge transformations have geometric significance; beyond this, they
can be used to define a quantum ‘group’ w19x as
follows. Let c i j with i, j s 1, . . . ,n denote the elements of the Cartan matrix of a simple Lie algebra G
over C of rank n, and d i be coprime positive
integers such that d i c i j is symmetric. The quantum
enveloping algebra Uq Ž G . with deformation parame-
Jqn s
q
2 Sn y q
2
Sy1
n
q y qy1
Tny1 ,
1
1
y
Jy1 s
q
2
y1
2 S1 y q S1
y1
qyq
K 1 s S12 Sy1
2 ,
T1
2
K n s Sy1
ny1 S n .
Ž 31 .
Ž 32 .
The proof is a direct verification; only the deformed
Serre relations Ž27. which, for the case at hand, e.g.,
reduce to
2
2
Jqi
Jqiq1 y Ž q q qy1 . Jqi Jqiq1 Jqi q Jqiq1 Jqi
s0
require a somewhat lenghthy computation.
Ž 33 .
G. Grensingr Physics Letters B 419 (1998) 258–262
262
On restricting q to be a root of unity, the representations of Uq Ž sl nq1 ., as inherited from Ž25., take
the form Ž n ) i ) 1.
J " i < m:
s "q . n i m i y m i " 1 q m i y m i " 1 " 12
q
<...,
m i " 1, . . . :
Ž 34 .
K i < m: s qym iy 1q2 m iym iq 1ym iy 1q2 m iym iq 1 < m: .
Let us show that they have quantum dimension zero.
The sum of positive roots can be written in terms of
the simple roots a i as Ý a ) 0 a s Ý i ri a i with ri g N;
on introducing K s Ł i K ir i , the quantum dimension
of a finite-dimensional representation is defined as
the trace d q s tr Ž K .. For the series A n , the ri read
w20x ri s iŽ n y i q 1.; thus, one finds
K s Ł Si2
Ž 35 .
i
and this result entails the assertion.
In general, these representations belong to the
cyclic series Žsee w19x. and have minimal dimension
p n. In w21x Žcf. also w22x. a related approach is given,
but the auxiliary algebra generated by the ‘Verlinde’
operators Si and Ti is larger than ours, since the
index i ranges from 1 to n q 1 there; also the
defining Eqs. Ž28. – Ž32. look different. As a consequence, the resulting cyclic representations are not
irreducible, but decompose into irreducible ones of
minimal dimension, whereas here these representations are irreducible by construction.
To summarize, though having begun with an
abelian gauge theory, at the very end one is faced
with the peculiar fact that the final structure is
noncommutative, being determined by a quantum
group. Furthermore, the above results also demonstrate that Jacobi theta functions are deeply related to
quantum groups at roots of unity.
References
w1x For a recent review, see L.D. Faddeev, Int. J. Mod. Phys. A
10 Ž1995. 1845.
w2x L. Alvarez-Gaume,
´ C. Gomez,
´ G. Sierra, Nucl. Phys. B 330
Ž1990. 347.
w3x C. Gomez,
´ G. Sierra, Nucl. Phys. B 352 Ž1991. 791.
w4x C. Ramirez, H. Ruegg, M. Ruiz-Altaba, Phys. Lett. B 247
Ž1990. 499.
w5x G. Grensing, Fractional quantum Hall effect and quantum
symmetry, Preprint Univ. of Kiel, August 1997.
w6x E. Witten, Commun. Math. Phys. 121 Ž1989. 351.
w7x G. Moore, N. Seiberg, Phys. Lett. B 220 Ž1989. 422.
w8x S. Elitzur, G. Moore, A. Schwimmer, N. Seiberg, Nucl.
Phys. B 326 Ž1989. 108.
w9x G. Dunne, R. Jackiw, C. Trugenberger, Ann. Phys. ŽNY. 194
Ž1989. 197.
w10x M. Bos, V.P. Nair, Int. J. Mod. Phys. A 5 Ž1990. 959.
w11x H.M. Farkas, I. Kra, Riemann Surfaces, Springer, Berlin,
1980.
w12x L. Alvarez-Gaume,
´ G. Moore, C. Vafa, Comm. Math. Phys.
106 Ž1986. 1.
w13x E. Verlinde, H. Verlinde, Nucl. Phys. B 288 Ž1987. 357.
w14x L. Alvarez-Gaume,
´ J.B. Bost, G. Moore, P. Nelson, C. Vafa,
Comm. Math. Phys. 112 Ž1987. 503.
w15x R. Dijkgraaf, E. Verlinde, H. Verlinde, Comm. Math. Phys.
115 Ž1988. 649.
w16x R. Iengo, K. Lechner, Nucl. Phys. B 364 Ž1991. 551; Phys.
Rep. 213 Ž1992. 179.
w17x M. Bergeron, D. Eliezer, G. Semenoff, Phys. Lett. B 311
Ž1993. 137.
w18x D. Mumford, Tata Lectures on Theta, Progress in Math. 28,
Birkhauser,
Boston, 1983.
˝
w19x For a review, see V. Chari, A. Pressley, A Guide to Quantum
Groups, Cambridge University Press, Cambridge, 1994; S.
Majid, Foundations of Quantum Group Theory, Cambridge
University Press, Cambridge, 1995.
w20x N. Bourbaki, Groupes et Algebres
de Lie, Chapitres 4, 5 et 6,
`
Hermann, Paris, 1968.
w21x V. Chari, A.N. Pressley, C.R. Acad. Sci. Paris Ser.
´ I, 313
Ž1991. 429.
w22x D. Arnaudon, A. Chakrabarti, Comm. Math. Phys. 139 Ž1991.
605.
12 February 1998
Physics Letters B 419 Ž1998. 263–271
Constraining differential renormalization
in abelian gauge theories
´
F. del Aguila
a
a,1
, A. Culatti
a,b,2
, R. Munoz
˜ Tapia
a,3
, M. Perez-Victoria
´
a,c,4
Dpto. de Fısica
Teorica
y del Cosmos, UniÕersidad de Granada, 18071 Granada, Spain
´
´
b
Dip. di Fisica, UniÕersita´ di PadoÕa, 35131 PadoÕa, Italy
c
Institut fur
¨ Teoretische Physik, UniÕersitat
¨ Karlsruhe, 76128 Karlsruhe, FRG
Received 18 September 1997
Editor: L. Alvarez-Gaumé
Abstract
We present a procedure of differential renormalization at the one loop level which avoids introducing unnecessary
renormalization constants and automatically preserves abelian gauge invariance. The amplitudes are expressed in terms of a
basis of singular functions. The local terms appearing in the renormalization of these functions are determined by requiring
consistency with the propagator equation. Previous results in abelian theories, with and without supersymmetry, are
discussed in this context. q 1998 Elsevier Science B.V.
Differential regularization and renormalization ŽDR. w1x was introduced as a renormalization method in
coordinate space compatible with gauge and chiral symmetry. In a series of papers this method has been further
developed w2–4x and successfully applied to different theories w5–13x 5. However, it might be considered
unsatisfactory the fact that Ward identities among renormalized Green functions are only satisfied when the
different renormalization scales are conveniently adjusted. Instead, one would like that the gauge symmetry
were automatically preserved, as occurs in dimensional regularization and renormalization w16x.
In this letter we present a procedure to constrain the scales in DR at one loop while preserving abelian gauge
invariance. This is done in two steps. First, each diagram is written in terms of a set of independent functions
Žwith different number of propagators andror different tensor structure.. Second, the singular functions of this
set are renormalized in a way which does not depend on the diagram where they appear. The local terms are
fixed by the requirement that DR be compatible with the equation defining the propagator in the space of
distributions. The propagator equation also allows to treat tadpole diagrams with the usual DR rules. In this
1
E-mail: [email protected].
E-mail: [email protected].
3
E-mail: [email protected].
4
E-mail: [email protected].
5
Different versions of differential renormalization can be found in w14,15x.
2
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 2 7 9 - 3
´
F. del Aguila
et al.r Physics Letters B 419 (1998) 263–271
264
manner one obtains renormalized Green functions which depend on just one arbitrary constant Žthe renormalization group scale. and, as we shall see, fulfil Ward identities in abelian gauge theories. The non-abelian case will
be studied elsewhere.
After describing the method, we discuss the renormalization of the one-loop vacuum polarization in massive
scalar QED, which is the simplest example requiring all the ingredients of the constrained procedure. The
complete one-loop renormalization of this theory will be presented in Ref. w17x. Then we review the one-loop
Ward identities of massive QED w4x and massless QED in an arbitrary gauge w8x, the corresponding ABJ
anomaly w1,8x and the evaluation of Ž g y 2. l in supergravity, where supersymmetry is also preserved w12x.
DR renormalizes diagrams by replacing singular expressions by derivatives of well-behaved distributions
Ž differential reduction.. These derivatives are understood in the sense of distribution theory, i.e., they are
prescribed to act formally by parts on test functions Ž formal integration by parts .. In practice, to carry out this
programme one has to manipulate singular expressions. This gives rise to ambiguities which are usually taken
care of by keeping arbitrary renormalization scales for different diagrams Žor pieces of diagrams.. The scales are
adjusted at the end to enforce the Ward identities Žwhich is equivalent to the addition of finite counterterms.. In
this letter we show that four rules are sufficient to formally manipulate and renormalize the singular expressions,
avoiding the introduction of unnecessary scales. The resulting renormalized amplitudes automatically satisfy the
Ward identities. These rules are summarised as follows:
1. Differential reduction, where we distinguish two cases:
1.1. Functions with singular behaviour worse than xy4 are reduced to derivatives of ‘logarithmically’
singular functions without introducing extra dimensionful constants w18x. For example,
1
x
6
s 18 I
1
x4
.
Ž 1.
1.2. Logarithmically singular functions are written as derivatives of regular functions. At one loop we have
the usual DR identity w1x
1
x4
s y 14 I
log x 2 M 2
x2
,
Ž 2.
which introduces a unique dimensionful constant Žthe renormalization group scale..
2. Formal integration by parts. In particular,
wEF x R sEF R ,
Ž 3.
where F is an arbitrary function and R stands for renormalized.
3. Delta function renormalization rule:
F Ž x , x1 , . . . , x n . d Ž x y y .
R
s F Ž x , x1 , . . . , x n .
R
d Ž xyy. .
Ž 4.
4. The general validity of the propagator equation:
F Ž x , x 1 , . . . , x n . Ž I x y m 2 . Dm Ž x . s F Ž x , x 1 , . . . , x n . Ž yd Ž x . . ,
Ž 5.
where DmŽ x . s 4p1 2 mK 1xŽ mx . and K 1 is a modified Bessel function w19x. This is a valid mathematical identity
between tempered distributions if F is well-behaved enough. This rule formally extends its range of
applicability to an arbitrary function.
The last rule will prove essential in our procedure. For instance, the ‘engineering’ tensor decomposition into
trace and traceless parts is not compatible with it and will be modified by the addition of a finite local term.
To evaluate the diagrams we express them in terms of a set of basic functions using only algebraic
manipulations Žincluding four-dimensional Dirac algebra. and the Leibnitz rule for derivatives. To one loop
´
F. del Aguila
et al.r Physics Letters B 419 (1998) 263–271
265
these functions can be classified according to their number of propagators and their derivative structure. For the
examples we discuss we need functions with one, two and three propagators:
A s DŽ x . d Ž x . ,
Ž 6.
B w O x s D Ž x . O xD Ž x . ,
Ž 7.
T w O x s D Ž x . D Ž y . O xD Ž x y y . ,
Ž 8.
where DŽ x . s 4p1 2 x12 is the massless propagator and O is a differential operator. In massive theories the
following basic functions are also required:
A s DŽ x . d Ž x . ,
Ž 9.
B w O x s D Ž x . O xD Ž x . ,
Ž 10 .
T w O x s D Ž x . D Ž y . O xD Ž x y y . ,
Ž 11 .
where DŽ x . s 14 4p1 2 log x 2 m 2 appears when the massive propagator is expanded in the mass m. Such expansion
allows to properly separate pieces with different degree of singularity. The same type of functions also appear in
massless theories if the photon propagator is written in a general Lorentz gauge. Note that A, A and B functions
are singular, and B and T ŽT. are singular for n G 2 Ž n G 4., where n is the order of the differential operator O .
We renormalize these basic functions using systematically rules 1 to 4. For example,
T w I x s D Ž x . D Ž y . I xD Ž x y y . s y D Ž x .
T R w I x s yB R w 1 x Ž x . d Ž x y y . s
1
1
4 Ž 4p 2 .
2
d Ž x y y . s yB w 1 x Ž x . d Ž x y y . ;
Ix
2
log x 2 M 2
x2
d Ž xyy. .
Ž 12 .
Ž 13 .
Our main observation is that the propagator equation Žrule 4. can be further used to relate the different basic
functions. Thus, by requiring that their renormalization be compatible with these relations, we shall completely
fix the finite local terms Žor scales. which appear in the differentially renormalized functions. Let us illustrate
how to do this with one example, the renormalization of the basic function Tw Em En x. Using rule 4 for a massless
propagator,
F I D Ž x . s yFd Ž x . ,
Ž 14 .
we can write
B Em Ž x . d Ž y . s yEmx I y T w 1 x q I y T Em y 2 EmxEsy T w Es x y Emx T w I x q 2 Esy T Em Es q T Em I .
Ž 15 .
The last basic function on the right-hand side can be easily reduced to
T Em I s 12 Ž Emx y Emy . T w I x ,
Ž 16 .
where rules 3 and 4 have been used. Now, we decompose the basic function Tw Em En x into trace and traceless
parts, adding an arbitrary Žfor the moment. local term to take into account the possible ambiguity introduced by
this operation:
T Em En s 14 dmn T w I x q T Em En y 14 dmn I q
1
64p 2
b d Ž x . d Ž y . dmn .
Ž 17 .
´
F. del Aguila
et al.r Physics Letters B 419 (1998) 263–271
266
Table 1
Renormalized expressions of basic functions
AR
s
0
AR
s
yIBw1x q 2 Es Bw Es x q 1
Ilog x
1
4 Ž4 p 2 . 2
B R w1x
y1
s
Ilog x
1
4 Ž4 p 2 . 2
B R wIx
s
0
B R w Em En x
s
y1
2
M2
12 Ž4 p 2 . 2
B R wIx
s
B R w1x
B R w Em En x
s
y1
ŽI
x2
16 Ž4 p 2 . 2
T R wIx
1
s
1
Ilog x
2
x
2
4 Ž4 p 2 . 2
T R w Em En x
Ž1
s
1
Ilog x
2
x
2
16 Ž4 p 2 . 2
T R wIIx
T R wI Em En x
M2
2
M2
q
x2
4
log x 2 M 2
1
M2
x2
Ž Em En y 1 dmn I.Ilog x
1
2
x2
1
p2
Ž Em En y dmn I. d Ž x .
18 Ž4 p 2 . 2
1
dmn q 2 Em En
.
x2
d Ž x y y.
M2
d Ž x y y. y
1 1
d Ž x . d Ž y .. dmn q Tw Em En y 1 dmn Ix
32 4 p 2
4
T R wIx
T R w Em En x
s
s
The traceless part is finite because of the tensor structure and is not further renormalized. Tw1x and Tw Em x are also
finite. On the other hand, the left-hand side of Eq. Ž15. is
2
B Em Ž x . d Ž y . s D Ž x . EmxD Ž x . d Ž y . s 12 EmxD Ž x . d Ž y . s 12 Emx B w 1 x Ž x . d Ž y . .
Ž 18 .
So, Eq. Ž15. reads for renormalized functions
1 x
2 m
E B R w 1 x Ž x . d Ž y . s yEmx I y T w 1 x q I y T Em y 2 EmxEsy T w Es x y 12 Emx T R w I x q 2 Esy T Em Es y 14 dms I
1
q
1
8 4p 2
b Emy Ž d Ž x . d Ž y . . .
Ž 19 .
Since both members of this equation are finite, we can integrate on x using the integration by parts
prescription 6 :
0 s d x Ž I y T Em q 2 Esy T Em Es y 14 dms I
H
1
1
1
. q 8 4p 2 b Emyd Ž y . s 16p 2 Ž
1
4
q 12 b . Emyd Ž y . .
Ž 20 .
Then this equation fixes
b s y 12 .
Ž 21 .
Note that the engineering trace-traceless decomposition, commonly used in the literature of DR, is not
compatible with the propagator equation in the case of logarithmic singularities. Hence, contraction of indexes
does not commute with renormalization. Generically one must simplify all the tensor and Dirac structure before
identifying the basic functions to be renormalized. The addition of the local term is equivalent to using a
different mass scale M X in the renormalization of the TwIx coming from the trace-traceless decomposition, and
2
then fixing log MMX 2 s b s y 12 . We prefer, however, to use the language of local terms to avoid confusion with
the usual ad hoc adjustment of renormalization scales.
6
Techniques for performing this sort of integrals can be found in Appendix A of Ref. w1x and Appendix C of Ref. w12x.
´
F. del Aguila
et al.r Physics Letters B 419 (1998) 263–271
267
Table 2
Renormalized expressions of massive basic functions, where A m s Dm Ž x . d Ž x . and B m w O x s Dm Ž x . O xDm Ž x .
ARm
s
B mR w1x
s
B mR wIx
s
B mR w Em En x
s
2
p 2 m2 Ž1 y log M . d Ž x .
1
Ž4 p 2 . 2
1
1 ŽI y 4 m
m2
2 .mK 0 Ž mx . K 1Ž mx .
m2 1 ŽI y 4 m 2 .mK 0
1
Ž4 p 2 . 2
m
Ž mx . K 1Ž mx .
2
2
q p 2 Ž2log M y 1. d Ž x .4
x
2
m
2
2
1
Ž4 p
2
q p 2 log M d Ž x .4
x
Ž4 p 2 . 2 2
2 .2
1 Em En wŽI y 4 m2 .ŽmK 0 Ž mx . K 1Ž mx . q 1 m2 Ž K 02 Ž mx . y K 12 Ž mx ... q 2p 2 Žlog M y 1 . d Ž x .x
x
6
1
y dmn wŽI y 4 m
24
2 .Ž
Iy4m
4
2 .mK 0 Ž mx . K 1Ž mx .
x
m
q 2p
2
3
2
2
log M y 2 .I d Ž x . y 4p 2 m2 Ž1 q 3log M . d Ž x .x4
2Ž
m2
3
m2
With the same technique one can determine the renormalization of all the basic functions. In general, besides
the massless propagator equation, Eq. Ž14., one needs
F I D Ž x . s FD Ž x . .
Ž 22 .
Both Eq. Ž14. and Eq. Ž22. are a consequence of the massive propagator equation, Eq. Ž5., and equivalent to the
equation for the photon propagator in a general gauge,
ž
ž
F dmn I y 1 y
1
a
/ /
Em En Dnr Ž x . s yF dm r d Ž x . ,
Ž 23 .
where Dmn Ž x . s 1r16p 2 Ž dmn I q Ž a y 1. Em En .log x 2m2 .
The propagator equation can be further employed to ‘separate’ the tadpole functions into two-point functions,
which can then be treated with the usual DR prescriptions:
A s D Ž x . d Ž x . s yD Ž x . I D Ž x . s yB w I x .
Ž 24 .
In Table 1 we gather the renormalized expressions of the basic functions required in the applications below.
In massive theories it is usually more convenient to work with compact expressions involving modified Bessel
functions w4x. The corresponding DR identities can be obtained by expanding the propagators in the mass
parameter, using Table 1 and resumming the result. In practice, one uses recurrence relations among Bessel
functions Žsee Appendix C of Ref. w12x. and then adds the necessary local terms to agree with Table 1. Table 2
collects the massive renormalization identities used in this paper.
At this point any one-loop diagram can be renormalized: one just has to use the renormalized basic functions
listed in the Tables .
As a simple example which contains all the ingredients of the constrained procedure let us consider in detail
Fig. 1. One-loop diagrams contributing to the vacuum polarization of scalar QED.
´
F. del Aguila
et al.r Physics Letters B 419 (1998) 263–271
268
the vacuum polarization in massive scalar QED. The contributing diagrams are depicted in Fig. 1. Using the
Feynman rules given in Ref. w12x one gets
ll
PmnŽ1. Ž x . s ye 2Dm Ž x . Em En Dm Ž x . ,
Ž 25 .
PmnŽ2. Ž x . s y2 e 2dmn Dm Ž x . d Ž x . ,
Ž 26 .
which expressed in terms of basic functions read
PmnŽ1. Ž x . s ye 2 4B m Em En y Em En Bm w 1 x 4 ,
PmnŽ2.
Ž 27 .
2
Ž x . s y2 e dmn A m .
Ž 28 .
Substituting the renormalized basic functions of Table 2, we obtain for each diagram
PmnŽ1. R Ž x .
e2
sy
½Ž
2 2
Ž 4p .
ž
q 13 p 2 log
PmnŽ2. R Ž x . s
Em En y dmn I .
M2
Ž 4p 2 .
Ž I y 4 m2 .
/
ž
mK 0 Ž mx . K 1 Ž mx .
x
ž
y 43 d Ž x . q dmn 2p 2 m 2 log
m2
e2
1
6
ž
2p 2 m 2 log
2
M2
m2
M2
m2
/
y1 d Ž x.
q m 2 Ž K 02 Ž mx . y K 12 Ž mx . .
5
,
/
Ž 29 .
/
y 1 dmn d Ž x . .
Ž 30 .
The longitudinal terms cancel in the sum, yielding a transverse result:
PmnR Ž x . s y
e2
2 2
Ž 4p .
1
3
q p
2
ž
Ž Em En y dmn I . 16 Ž I y 4 m2 .
log
M2
m2
ž
mK 0 Ž mx . K 1 Ž mx .
x
q m2 Ž K 02 Ž mx . y K 12 Ž mx . .
/
y 43 d Ž x . .
/
Ž 31 .
In the same way, our method recovers previous DR results in abelian gauge theories, without the need to
impose Ward identities a posteriori. Let us consider first the vacuum polarization in massive QED w4x. In terms
of basic functions it reads
Pmn Ž x . s 4 e 2
2
1
mn q 2 dmn I y Em En
½Žm d
. B m w1x q 2B m
Em En y dmn B m w I x
5
Ž 32 .
and the renormalized expression is
PmnR Ž x . s y
4 e2
2 2
Ž 4p .
ž
Ž Em En y dmn I . 16 Ž I y 4 m2 .
q 13 p 2 log
M2
m2
/
q 23 d Ž x . ,
ž
mK 0 Ž mx . K 1 Ž mx .
x
y 12 m2 Ž K 02 Ž mx . y K 12 Ž mx . .
/
Ž 33 .
which is again transverse.
In supersymmetric QED the vacuum polarization is the sum of the spinor QED diagram and twice Žtwo
complex scalars for each Dirac spinor. the scalar QED diagrams. In terms of basic functions this gives directly a
transverse result depending on one basic function only:
Pmn Ž x . s y2 e 2 Ž Em En y dmn I . B m w 1 x .
Ž 34 .
´
F. del Aguila
et al.r Physics Letters B 419 (1998) 263–271
269
In this case one could use an engineering trace-traceless decomposition in renormalizing each diagram. The
gauge-non-invariant terms vanish in the total sum due to supersymmetry cancellations. The complete result
would thus be the same as the one obtained from the renormalized functions of Table 2:
PmnR Ž x . s y
e2
Ž 4p 2 .
2
Ž Em En y dmn I . Ž I y 4 m2 .
mK 0 Ž mx . K 1 Ž mx .
x
q 2p 2 log
M2
m2
d Ž x. .
Ž 35 .
Next we consider the QED vertex Ward identity between the electron self-energy and the electron-electronphoton vertex in an arbitrary Lorentz gauge, which was studied in Ref. w8x. In this case and the next one the
masses play no relevant role, as far as renormalization is concerned, so we consider massless electrons for
simplicity. In order to respect the Ward identity for both the a-dependent and a-independent pieces Žwhere a is
the gauge parameter., the authors of Ref. w8x had to impose two relations among different scales:
MS
log
s 14 ,
Ž 36 .
MV
MS
l ' log X s 3 ,
Ž 37 .
MS
where M V and MS and MSX appear in the vertex and in the two pieces of the electron self-energy, respectively.
With the constrained method we find
½
S R Ž x . s e2
1
1
4 Ž 4p 2 .
Eu I
2
log x 2 M 2
x
2
q Ž a y 1.
1
1
4 Ž 4p 2 .
Eu I
2
log x 2 M 2
x
2
1
q
16p 2
Eud Ž x .
5
Ž 38 .
for the electron self-energy, and
VmR Ž x , y . s ie 3 y2g bgm ga Ž EaxE by T w I x q Eax T w E b I x y E by T w Ea I x . q 4gaT Ea Em
½
y 14 d a m I y
1
1
4 Ž 4p 2 .
g I
2 m
log x 2 M 2
x
2
d Ž xyy. y
1
1
8 4p 2
gm d Ž x . d Ž y .
q Ž a y 1 . gr ga gm g bgs EaxE by T Er Es q Eax T E b Er Es y E by T Ea Er Es
ž
1
y
2
1
2 2
4 Ž 4p .
gm I
log x M
x2
2
d Ž xyy.
5
/
Ž 39 .
for the vertex. These renormalized amplitudes automatically satisfy the Ward identity
Ž Emx q Emy . VmR Ž x , y . s ie S R Ž x y y . Ž d Ž x . y d Ž y . . ,
Ž 40 .
as can be seen by integrating on y.
The chiral triangle anomaly in QED was discussed in Refs. w1,8x. The renormalized triangle diagram
depended on the relation between two renormalization scales. By adequately choosing these scales one could
respect either vector current or axial current conservation, but not both. Imposing conservation of the vector
current, the correct value of the axial anomaly resulted. In contrast, in the constrained DR method everything is
determined and a non-ambiguous result is obtained:
R
Tmnl
Ž x, y.
s ie 3 y2tr Ž g 5gm gl gn ga g bgc . EcxEay T w E b x q 16 Ž elm a b EaxEny y eln a b EmxEay . T w E b x
q16 el bm a Eax T En E b y 14 dn b I y 16 eln b a Eay T Em E b y 14 dm b I
1
q
8p 2
5
emnl a Ž Eax y Eay . Ž d Ž x . d Ž y . . ,
Ž 41 .
´
F. del Aguila
et al.r Physics Letters B 419 (1998) 263–271
270
where the index l corresponds to the axial vertex. Then Žsee appendix B of Ref. w1x.
R
Emx Tmnl
Ž x, y. s0 ,
Ž 42 .
R
Eny Tmnl
Ž x, y. s0 ,
Ž 43 .
R
y Ž Elx q Ely . Tmnl
Ž x, y. s
ie 3
2p 2
emnl r ElxEry Ž d Ž x . d Ž y . . ,
Ž 44 .
so the vector Ward identities are directly preserved while the axial one is broken, giving the known result for the
anomaly.
Finally, let us comment briefly on the calculation of the Ž g y 2. l in unbroken supergravity performed in Ref.
w12x. There, the symmetry to be preserved was supersymmetry, which implies a vanishing anomalous magnetic
moment w20x. This result was obtained thanks to the use of the propagator equation to explicitly relate diagrams
with different topology. Then, only one type of singular basic function, TwIx, appeared. Although engineering
trace-traceless decompositions were performed at intermediate steps, this did not affect the total sum because the
extra local terms cancel, as occurs in the vacuum polarization in supersymmetric QED discussed above. Using
the renormalized basic functions in Tables 1 and 2, Ž g y 2. l also vanishes, although the contribution of each
diagram is different, as is the total graviton contribution.
Summarizing, we have proposed a procedure of differential renormalization to one loop which only
introduces a single renormalization scale. We have verified that the renormalized amplitudes so obtained
automatically satisfy the Ward identities of abelian gauge symmetry in known examples, that the chiral anomaly
is correctly treated, and that supersymmetry is preserved in a relatively complex calculation. In practice, one just
needs to use the renormalized functions of Tables 1 and 2.
In principle, the method could be generalized to higher loops. However this is not straightforward. New more
complicated functions emerge, which could be tackled with the systematic method of Ref. w3x. Still, one should
properly constrain the local terms using rules 1 to 4 or some consistent extension of them.
Acknowledgements
We thank J. Collins and R. Stora for discussions. This work has been supported by CICYT, contract number
AEN96-1672, and by Junta de Andalucıa,
´ FQM101. RMT and MPV thank MEC for financial support.
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12 February 1998
Physics Letters B 419 Ž1998. 272–278
Chiral symmetry breaking in the Nambu–Jona-Lasinio model
in external constant electromagnetic field
A.Yu. Babansky a , E.V. Gorbar
b,1
, G.V. Shchepanyuk
c
a
b
KieÕ State UniÕersity, Ukraine
BogolyuboÕ Institute for Theoretical Physics, Ukraine
c
Institute of Mathematics, Ukraine
Received 19 June 1997; revised 28 October 1997
Editor: P.V. Landshoff
Abstract
Dynamical chiral symmetry breaking ŽDx SB. is studied in the Nambu–Jona-Lasinio model for an arbitrary combination
of external constant electric and magnetic fields. In 3 q 1 dimensions it is shown that the critical coupling constant increases
with increasing of the value of the second invariant of electromagentic field E P B, i.e. the second invariant inhibits Dx SB.
The case of 2 q 1 dimensions is simpler because there is only one Lorentz invariant of electromagnetic field and any
combination of constant fields can be reduced to cases either purely magnetic or purely electric field. q 1998 Elsevier
Science B.V.
PACS: 11.10.Kk; 11.30.Qc
Keywords: Chiral symmetry breaking; External electromagnetic fields
In works w1–3x it was first shown in the so called
ladder approximation that quantum electrodynamics
ŽQED. in the regime of strong coupling has a new
phase with dynamically broken chiral symmetry. The
new phase of QED possesses very interesting properties from the theoretical viewpoint w4–6x. There were
attempts to use them for modelling electroweak symmetry breaking in technicolor-like models w7x. However, at present the new phase of QED has little
relevance to experiment because we need a strong
coupling constant a c f 1 Žrecall that the physical
value of electromagnetic coupling constant is a c
1
s 137
< 1..
1
E-mail: [email protected].
However, as was suggested in w8,9x, the situation
may drastically change in the presence of strong
external electromagnetic fields where dynamical chiral symmetry breaking Ž Dx SB . may occur at the
regime of weak coupling. A breakthrough in this
direction was made in w10,11x, where, in the framework of the Nambu-Jona-Lasinio ŽNJL. model w12x,
it was shown that Dx SB takes place in an external
constant magnetic field at any small attraction between fermions both in 2 q 1 and 3 q 1 dimensions
Žnote that the fact that external magnetic field enhances Dx SB was first noted in the NJL model in
w13x Žsee also w14x...
As shown in w10,11x, in the infrared, the dynamics
of fermions in magnetic field in 2 q 1 and 3 q 1
dimensions resembles the dynamics of fermions in
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 4 5 - 7
A.Yu. Babansky et al.r Physics Letters B 419 (1998) 272–278
0 q 1 and 1 q 1 dimensions, respectively. Therefore,
we have an effective reduction of dimension of
space-time by 2 units and as a result the critical
value of the coupling coustant in external magnetic
field is equal to zero. It was latter shown in w15x that
the same effect takes place in QED in external
magnetic field. Note that although the critical value
of coupling constant is zero extremely strong magnetic fields Ž< B < G 10 13 G . are necessary for experimentally significant consequences because the correction to the physical mass of electron is very tiny
for weak magnetic fields.
The case of constant electric field was considered
in w13x where it was shown that the value of the
critical coupling constant is more in this case than in
the case without electric field. In the present work
we study Dx SB in the NJL model in the case of an
arbitrary combination of constant electric and magnetic fields in 3 q 1 and 2 q 1 dimensions. We first
consider the case of 3 q 1 dimensions. As well
known, electromagnetic field has two Lorentz invariants f 1 s 12 Fm n F mn s B 2 y E 2 and f 2 s
1 mn a b
Fmn Fa b s E P B. Since Dx SB was already
2´
studied in cses of purely electric and magnetic constant external fields when only the first invariant f 1
of electromagnetic field is not equal to zero, in the
present work we consider Dx SB in the case where
f 2 / 0.
The Lagrangian of the NJL model w12x in an
external electromagnetic field reads
By taking integrals over fermion fields, we obtain
the effective action for p and s fields
N
G Ž s ,p . s yi
1
2G
G
q
2
N
Ý
js1
ž CC /
j
2
j
q Cj ig 5Cj
ž
2
/
,
Ž 1.
where Dm is the covariant derivative Dm s Em q ieAm
and j s 1,2, . . . , N flavor index. Lagrangian Ž1. is
invariant with respect to the ULŽ N . = UR Ž N . chiral
group. By using auxiliary fields p and s , we can
rewrite Ž1. in the following form:
N
Ls
1
Ž s 2 q p j2 .
2G j
q p j2 . .
Ž 3.
1
2G
Hd
4
xs 2 .
Ž 4.
By using the method of proper time w16,17x, we
represent the first term in Ž4. as follows:
yiTr Ln Ž iDmg m y s .
i
s y Tr Ln Ž D 2 q s 2 .
2
yi sŽ D 2 q s 2 . <
H 2 s tr ² x < e
x : dsd 4 x
Ž 5.
As well known w17x, vacuum of QED is not stable
in an external electric field and the effective potential has an imaginary part which defines the rate of
birth of fermion-antifermion pairs from vacuum per
unit volume. Since we study the problem of Dx SB,
we can ignore this effect and consider only the real
part of effective potential which is equal to
s2
VŽs . s
N
q
8p
2G
1
`
2
Õ. p.
ys s 2
H1rL ds s e
2
=Mcoth Ž Ms . Lcot Ž Ls . ,
js1
y
2
j
G Ž s . s yiTr Ln ig m Dm y s y
iCjg m DmCj y Cj Ž sj q ig 5p j . Cj
Ý
Hd x Ž s
To obtain the effective potential for s and p
fields, it suffices to consider the case of constant
fields s s const, p s const. Since the effective action is invariant with respect to the ULŽ N . = UR Ž N .
chiral symmetry, the effective potential depends on
p and s fields only through the chirally invariant
combination r 2 s Ý Njs1Ž sj 2 q p j2 .. Therefore, in
what follows it is sufficient to set p k s 0 , s k s 0
for k s 2, . . . , N and consider the effective potential
only for the field s 1 which we simply denote s .
Thus,
s
js1
4
y
i
Ý iCjg m DmCj
Tr Ln ig m Dm y Ž sj q ig 5p j .
Ý
js1
N
Ls
273
Ž 2.
where L2 s e 2
2
f1 s B y E
2
'f
2
2
1 q4 f 2
2
y f1
, M 2 s e2
Ž 6.
'f
2
2
1 q4 f 2
q f1
, and
2
and f 2 s E P B are two invariants of
A.Yu. Babansky et al.r Physics Letters B 419 (1998) 272–278
274
electromagnetic field. In Ž6. we introduced a cut-off
1
L2 and v.p. of the integral in s is present because we
consider only the real part of the effective potential
Žrecall that the imaginary part of the effective potential is given by residues in poles of cotŽ Ls ... The gap
equation d Vrds < ssm s 0 has the form
1
N
y
G
`
4p 2
Õ. p.
ys m
H1rL dse
2
2
Integrating by part, we obtain
`
Õ. p.
ys m 2
Hp dse
2L
`
ym 2 s
sM
Hp e
d Ž ln Ž 2 <sin Ls < . . s yM ln 2 <sin
ž
2L
Mcoth Ž Ms . Lcot Ž Ls .
s 0.
MLcot Ž Ls .
`
qm2 M
ym 2 s
Hp e
Ž 7.
ln Ž 2 <sin Ls < . ds.
p
2
<
/
Ž 10 .
2L
This gap equation was investigated in cases where
only the first invariant of electromagnetic field is not
equal to zero, i.e. for cases of purely electric and
magnetic external fields. In this paper we study how
the presence of nonzero electric field parallel to
magnetic field Ž E P B / 0. affects Dx SB. By using
some inequalities, we first analytically obtain an
estimate from below for the critical coupling constant. We add and subtract 1rs to McothŽ Ms . in the
gap Eq. Ž7.. Then
By using the formula w18x
`
`
yq x
H0 e
ln Ž 2 <sin ax < . dx s yq
1
,
Ý
2
2 2
ks1 k Ž q q 4 k a .
Re q ) 0,
Ž 11 .
p
2L
and the fact that the integral MH1r L 2 lnŽ2 <sin Ls <. ds
is finite, we conclude that the integral m 2 M =
2
H` p eym s lnŽ2 <sin Ls <. ds tends to zero on the critical
2
line Žwhere m2 ™ 0.. Thus, Õ. p. H` p dseysm =
MLcotŽ Ls . s yM ln2. Therefore, in view of Ž9., we
have
2L
`
Õ. p.
ys m 2
H1rL dse
2
s L2 y
p
Lq
2
`
Õ. p.
2L
Mcoth Ž Ms . Lcot Ž Ls .
ys m 2
H1rL dse
2
`
Õ. p.
ys m 2
Hp dse
Ž Mcoth Ž Ms . y 1rs . Lcot Ž Ls .
2L
Ž Mcoth Ž Ms . y 1rs . Lcot Ž Ls . ,
Ž 8.
where we used the result w13x
F yM ln2.
Ž 12 .
It p remains to estimate from below the integral
Ž
Ž .
.
Ž .
H1r
have
L 2 ds Mcoth Ms y 1rs Lcot Ls . We
p
p
H1r L 2 dsŽ McothŽ Ms . y 1rs . LcotŽ Ls . F H1r L 2 Ž McotŽ Ms . y 1rs . dss Žbecause cot x F 1rx for x in the
interval from 0 to p2 .. If M 4 L, then we use the
estimate coth x F 1rx q 1 because cothŽ Ms . is approximately 1 near the upper limit of integration.
Therefore, in this case
2L
2L
`
Õ. p.
ys m 2
H1rL dse
L
cot Ž Ls .
2
s
s L2 y
p
2
L.
Further, we represent the
integral in Ž8. as a sum of
p
`
p
`
Ž
2sH
two integrals H1r
L
1r L 2 q H p note that 2 L is the
first zero of cotŽ Ls ... We now consider the integral
from 2pL to infinity. Since coth x F 1rx q 1 for x ) 0,
we have
2L
2L
`
Õ. p.
ys m
Hp dse
2
p
2L
H1rL ds Ž Mcoth Ž Ms . y 1rs . Lcot Ž Ls .
2
Ž Mcoth Ž Ms . y 1rs . Lcot Ž Ls .
pL2
F M ln
2L
`
F Õ. p.
2L
ys m 2
Hp dse
2L
MLcot Ž Ls . .
Ž 9.
2L
.
Ž 13 .
If M < L, then cothŽ Ms . < 1 in the interval of
integration. By using the estimate coth x F 1 q xr3
x
A.Yu. Babansky et al.r Physics Letters B 419 (1998) 272–278
Ž 1 and xr3 are simply two first terms of the Taylor
x
expansion of coth x ., we obtain
It follows from Ž16. that in this case g cr for f 2 / 0 is
more than w13x
p
H1rL ds Ž Mcoth Ž Ms . y 1rs . Lcot Ž Ls .
2
M
F
3
2
p
ž
1y
y 1rL2 .
/
2L
Ž 14 .
We now analyse the obtained results. We assume in
what follows that < f 1 < 4 < f 2 <. In the magnetic-type
2
case Ž f 1 ) 0., we have L f < e <Ž ff21 .1r2 and M f
1r2
< e < f 1 . By using Ž8., Ž12., and Ž13., we obtain the
following estimate from below for the critical coupling constant in the magnetic-type case:
1
g cr G
1y
Lp
2L
2
L2
M
q
L
ln
2
2L
1
f
M
y
f 11r2
2
L2 f 11r2
1q< e<
ln
4 < ef 2 <
L
L2
1
g cr s
2L
ln2
,
Ž 15 .
where g cr is the dimensionless critical coupling
2
2
constant g cr s 4p NGL . It directly follows from Ž15.
that the presence of electric field parallel to magnetic
field is very important. Indeed, if f 2 / 0, then g cr is
no longer equal to zero Ževen if the magnetic field is
very strong < B < ; L2 . in contrast to the case of
purely magnetic field where g cr s 0. If f 2 increases,
g cr is also increases. If f 2 ™ 0, then g cr ™ 0, i.e. we
recover the result obtained by Gusynin, Miransky,
and Shovkovy w10x. In the electric-type case Ž f 1 - 0.
2
we have L f < e < < f 1 < 1r2 and M f < e <Ž < ff21 < .1r2 . By using
Ž8., Ž12., and Ž14., we obtain
275
p < e < < f 1 < 1r2
2 L2
in the case f 2 s 0. As f 2 goes to zero, our estimate
coincides with the result obtained by Klevansky and
Lemmer w13x in the electric-type case. Thus, we
conclude from the obtained estimates that the second
invariant of electromagnetic field inhibits Dx SB. In
magnetic-type case it looks rather natural Žindeed, if
f 2 / 0, then it means that E / 0 and we know that
electric field inhibits Dx SB .. However, in the electric-type case it appears unlikely. Indeed, let first
E / 0, B s 0 Žtherefore, f 2 s 0.. It is natural to assume that g cr should decrease in the case f 2 / 0
because if f 2 / 0 it means that B / 0 and we know
that magnetic field assists Dx SB. We can understand the cause of growth of g cr with increasing of
E P B as follows. Since we study the dependence on
the second invariant, we keep the first invariant
B 2 y E 2 unchanged. Without loss of generality we
can assume that E I B Žif not, one can perform an
appropriate Lorentz transformation.. If we increase
f 2 , then in order to keep the first invariant unchanged
we have to increase both B and E. Therefore, there
is a competition between increasing of B and increasing of E. It turned out that qualitatively increas-
1
g cr G
1y
Lp
2L
2
M
2
q
3L
2
p
ž
2L
y 1rL2 y
/
1
f
1y
p < e < < f1 <
2 L2
1r2
< e<
y
L2
f 22
ž /
< f1 <
M
L2
.
1r2
ln2
ln2
Ž 16 .
Fig. 1. The dependence of the critical coupling constant on the
second invariant in the magnetic-type case.
276
A.Yu. Babansky et al.r Physics Letters B 419 (1998) 272–278
where SAŽ m. is the fermion propagator in an external
constant electromagnetic field which has the following form in the Fock–Schwinger proper time formalism:
SAŽ m. Ž x , xX . s Ž i Em g m y eAm g m y m .
= Ž yi .
0
X
Hy`dt U Ž x , x ;t . ,
A
Ž 18 .
where e ™ 0 and
Fig. 2. The dependence of the critical coupling constant on the
second invariant in the electric-type case.
UA Ž x , xX ;t .
s- x <
ing of E is more significant for g cr than increasing
of B for any f 1 , therefore, g cr always grows with
increasing of f 2 .
We found a rather rough analytic estimate from
below for the critical coupling constant. To obtain a
more accurate dependence of the critical coupling
constant on f 2 , we numerically calculate the integral
in Ž7.. A typical dependence of g cr on f 2 in the
electric-type case is shown in Fig. 1 Žthis figure
corresponds to Lf14 s y10y4 . and in the magnetictype case in Fig. 2 Žwhere Lf14 s 10y4 ..
We see from these figures that the critical coupling constant increases with increasing of f 2 for
any value of f 1. In the electric-type case g cr is
always more that 1. In the magnetic-type case g cr
abruptly drops to zero as f 2 tends to zero. We also
numerically calculated g cr in the case where the first
invariant is zero f 1 s 0 Ž< E < s < B <. and obtained a
dependence which is similar to the electric-type case,
i.e. g cr increases with increasing of f 2 . Thus, the
numerical analysis of the gap equation confirms that
the second invariant of electromagnetic field inhibits
Dx SB.
We now consider the case of 2 q 1 dimensions.
We use the reducible 4-dimensional representation of
the Dirac algebra for fermion field in order that the
model possess a chiral symmetry Žin fact, there is
two chiral symmetries with g 5 and g 3 matrices, for
more details see w19x. and we do not study parity
breaking. Thus, we have the following gap equation
for parity conserving mass:
m s 2 iGtr Ž SAŽ m. Ž x , x . . ,
Ž 17 .
2
1
e
< xX )
Ž i E y eA . y s
mn
2
Fmn y m
2
p
eyi 4
s
8p 3r2 <t <
i
q
4
x
3r2
Hx d j AŽ j .
exp yie
X
Ž x y xX . eFcoth Ž eFt . Ž x y xX .
1
sinh eFt
y tr ln
2
eFt
ž
/
ie
q
2
s Ft q im2t .
Ž 19 .
By taking trace over the Dirac indices in Ž13., we
obtain
ms
mG
p
3r2
eyi p r4
0
dt
Hy` <t <
3r2
Ž eXt . cot Ž eXt . , Ž 20 .
where X s 12 Fmn F mn s 'B 2 y E 2 Žnote that magnetic field is a pseudoscalar in 2 q 1 dimensions, not
axial-vector..
As well known, in 2 q 1 dimensions there is only
one Lorentz invariant 12 Fmn F mn of electromagnetic
field. Indeed, we can see from Ž20. that the gap
equation depends on electromagnetic fields only
through this Lorentz invariant. Consequently, an arbitrary combination of constant fields can be reduced
to cases either purely magnetic or purely electric
field. The case of constant magnetic field in the
Nambu–Jona–Lasinio model in 2 q 1 dimensions
was considered in w10x. Therefore, we study here
only the case of constant electric field.
(
A.Yu. Babansky et al.r Physics Letters B 419 (1998) 272–278
In the case of external electric field the gap
equation takes the form
ms
mG
p 3r2
eyi p r4
ds
`
H1
s 3r2
eysm
2
Consequently, the gap equation takes the form
p 3r2
2GL
s1q
'p Ž eE . 1r2
2L
L2
mG
q
p
3r2
eyi p r4
yIm z
dt
0
Hy` <t <
e it Ž m
3r2
2
yi e .
Ž 21 .
where we explicitly wrote down the term which
corresponds to the gap equation without external
electric field and E s < E <. Further, by using the fact
that
ž
2
1
e 2t y 1
y
/
t
s
2
m2E
/
// ž /
2
,1 y i
qO
2
m2
L2
.
1
2L
1 y a 2z
1
eE
2
L2
ž /ž /
g cr Ž 0 .
1ya
2
eE
1r2
1r2
,
Ž 27 .
ž /
L2
2
L
where g cr s 2G
p 3r 2 is the dimensionless coupling con-
get
Re IE m2E ,
ž
1
2
/
y Im IE m2E ,
ž
1
2
/
,
Ž 22 .
where
IE m2E ,
ž
1
2
/
`
s
H0 x
ay1 i
e
m2E
2
x
ž
1
1
y
x
e y1
x
/
,
2
m2E s meEE and we also used the equality
`
H
2 0
dx
x
1r2
m2E
2
x y sin
m2E ,a
2
. sy
x
2
ay 1
s 0.
ž /
ay1
G Ž a y 1. e i
m2E
ž
qG Ž a . z a,1 y i
Ž 23 .
m2E
2
/
2
p
.
Ž 24 .
Performing an analytic continuation of Ž24. to the
region 0 - ReŽ a. - 2, we get
IE m 2E ,
1
2
/
ž
s 'p Ž 1 y i . m E q z
2
Thus, the presence of external constant electric
field increases the value of critical coupling constant.
Note that the same is true in 3 q 1 dimensions Žsee
w13x.. It is not difficult to show that in the vicinity of
critical point
2
m2E
ž ž / ž //
cos
stant in 2 q 1 dimension and a 2 s y 'p z Ž 12 . f1.29.
m 2 f C 2 Ž Ecr y E .
IE Ž m2E ,a. is an analytic function of a in the region
0 - ReŽ a. - 2.
For 1 - ReŽ a. - 2, by representing IE Ž m 2E . as a
sum of two integrals, we have
ž
,1 y i
p 3r2
g cr Ž E . s
and performing the change of variable x s , for
dt
0
it Ž m 2 yi e .
Re eyip r4Hy`
Ž eEt . coth Ž eEt . we
<t < 3r2 e
IE Ž
2
m2E
1
From Ž26., we obtain the following value for the
critical coupling constant:
t
2
1
ž
1
ž ž
Re z
Ž 26 .
= Ž eEt . coth Ž eEt . y 1 ,
t cotht y 1 s t 1 q
277
ž
1
2
,1 y i
m2E
2
//
.
Ž 25 .
1r2
,
Ž 28 .
z Ž 1r2 .
Ž 3r2 .
where C s y2 z
f 1.12.
Thus, this phase transition is a phase transition of
the second order.
Since we have only one Lorentz invariant of
electromagnetic field X 2 s 12 Fmn Fmn s B 2 y E 2 in
2 q 1 dimensions, by using an appropriate frame, the
general case of non-zero constant electromagnetic
field can be reduced to the cases of purely electric or
purely magnetic fields.
The authors are grateful to Prof. V.P. Gusynin for
many fruitful discussions and remarks, Prof. V.A.
Miransky for valuable comments, and Dr. I.A.
Shovkovy for the help in drawing figures. The work
of E.V.G. was supported in part through grant INTAS-93-2058-EXT ’’East-West network in constrained dynamical systems’’ and by the Foundation
of Fundamental Research of the Ministry of Science
of the Ukraine through grant No. 2.5.1r003. The
work of A.Yu.B. was supported in part by the Inter-
278
A.Yu. Babansky et al.r Physics Letters B 419 (1998) 272–278
national Soros Science Education Program ŽISSEP.
through grant No. GSU062075.
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12 February 1998
Physics Letters B 419 Ž1998. 279–284
Pulsar velocity with three-neutrino oscillations
in non-adiabatic processes
C.W. Kim
a
a,b,1
, J.D. Kim
a,2
, J. Song
c,3
Department of Physics and Astronomy, The Johns Hopkins UniÕersity, Baltimore, MD 21218, USA
b
School of Physics, Korean Institute for AdÕanced Study, Seoul 130-012, South Korea
c
Center for Theoretical Physics, Seoul National UniÕersity, Seoul 151-742, South Korea
Received 10 October 1997
Editor: M. Dine
Abstract
We have studied the position dependence of neutrino energy on the Kusenko-Segre` mechanism as an explanation of the
proper motion of pulsars. The mechanism is also examined in three-generation mixing of neutrinos and in a non-adiabatic
case. The position dependence of neutrino energy requires the higher value of magnetic field such as B ; 3 = 10 15 Gauss in
order to explain the observed proper motion of pulsars. It is shown that possible non-adiabatic processes decrease the
neutrino momentum asymmetry, whereas an excess of electron neutrino flux over other flavor neutrino fluxes increases the
neutrino momentum asymmetry. It is also shown that a general treatment with all three neutrinos does not modify the result
of the two generation treatment if the standard neutrino mass hierarchy is assumed. q 1998 Elsevier Science B.V.
PACS: 14.60.Pq; 97.60.Bw; 97.60.Gb
In a recent paper, Kusenko and Segre` w1x have
proposed a new explanation for the peculiar velocities Žor kick velocities. of pulsars by considering
magnetically distorted neutrino oscillations. Many
mechanisms w3–6x for the phenomena have been
suggested by some asymmetry during the supernova
collapsing and explosion. However, most of them
have difficulties in explaining the observed average
pulsar velocity 450 " 90 kmrs w2x. On the other
hand, the mechanism suggested in Ref. w1x, hereafter
1
E-mail: [email protected].
E-mail: [email protected].
3
E-mail: [email protected].
2
called the KS mechanism, is successful w7x for a
reasonable value of the magnetic field in pulsars, i.e.,
B ; 10 14 ; 15 Gauss. The KS mechanism assumes
that neutrino oscillations, ne l nt , take place between the electron- and tau-neutrinospheres, where
the matter density is r f 10 11 ; 12 grcm3. Since
neutrinos emitted during the cooling of a protoneutron star have roughly 100 times the momentum of
the proper motion of the pulsar, the neutrino momentum asymmetry of D krk ; 1 % can explain the
observed motion of pulsars. Ref. w1x shows that the
magnetic field of 3 = 10 14 Gauss can produce the
desirable momentum asymmetry. In his comment,
Qian w9x pointed out that the approximate relation
used in Ref. w1x is not a valid assumption and that
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 4 1 - X
C.W. Kim et al.r Physics Letters B 419 (1998) 279–284
280
somewhat higher magnetic field is required in order
to explain the observed kick velocity. The differences between Ref. w1x and Ref. w9x are in the flux
factor 1r3 and the estimate of the matter density
Ž r s 10 11 and 10 12 grcm3 . besides the different
expressions for the ratio of scale heights h NerhT ,
where h Ne ' <dln Nerd r <y1 and hT ' <dlnTrd r <y1 .
Therefore, it is worthwhile to re-examine the KS
mechanism in a more general case for supernovae
and neutrinos. In this paper, first we will consider
the position dependence of neutrino energy inside
the protoneutron star, which was ignored in both
Ref. w1x and w9x. Our expression for the neutrino
momentum asymmetry will be shown to reproduce
the results in Ref. w9x and Ref. w1x, respectively,
under different approximations. We will also study
the treatment of the KS mechanism in a three-neutrino generation scheme and see how the results are
modified for possible non-adiabatic processes. The
possibility of different fluxes for each neutrino flavor
and its effect on the pulsar velocities are also discussed.
First, let us briefly review the KS mechanism in
which only ne l nt conversions are considered. The
difference in the mean free paths of ne and nt
renders the ne neutrinosphere, S e , to be located at
larger radius than the nt neutrinosphere, St . If a
resonant conversion between ne and nt occurs outside St but inside S e , the ne converted from nt is
absorbed or thermalized by the background medium
whereas the nt converted from ne escapes freely. In
effect, the nt ’s are expected to propagate out not
from the St , but from the resonance surface which is
distorted in the presence of magnetic field. The
resonance condition for two neutrino oscillations in
B field is w8x
D02
2kŽ r.
where D02 ' mŽ n 3 . 2 y mŽ n 1 . 2 in the case of ne l nt
conversion, Ne s Ž n eyy n eq ., u is the neutrino mixing angle in vacuum, GF the Fermi constant, B the
magnetic field of pulsars, k , < k < the energy of relativistic neutrinos, f the angle between B and k, and
r the radial coordinate. Around typical neutrinospheres with r f 10 11 ; 12 grcm3 and the electron
fraction per baryon Ye f 0.1, the order of magnitude
of ´ B is estimated to be
2
´ B f 0.002
y
ž /
3
0.1
2
=
ž
y
r
B
3
12
10 grcm
3
/ ž
14
10 Gauss
/
,
Ž 2.
which is indeed small so that a perturbation calculation can be applied. We caution the reader that the
neutrino energy k as well as the electron number
density Ne depend on r. The deeper the neutrinos are
emitted from, the higher their energies are. To the
zeroth order, i.e. in the electron background without
magnetic field, the resonance condition is satisfied at
r 0 as
D02 cos2 u s 2'2 GF Ne Ž r 0 . k Ž r 0 . .
Ž 3.
The presence of B field modifies the resonance
position into
r Ž f . s r 0 q dr cos f ,
Ž 4.
where dr < r 0 because ´ B , the correction due to the
presence of magnetic field, is very smallŽsee Eq.
Ž2... Substituting Eq. Ž4. into Eq. Ž1. and taking into
account the r-dependence, we obtain, to the leading
order in ´ B or drrr 0 ,
cos2 u
1
ž
1
s '2 GF Ne Ž r . q
eGF
'2
ž
3 Ne Ž r .
p4
3
/ˆ
kPB
1
s '2 GF Ne 1 q
Ye
e
2
ž
3
p 4 Ne2
' '2 GF Ne w 1 q ´ B x ,
/
3
Bcos f
Ž 1.
dln Ne
dr
dln k
q
r0
dr
r0
/
dr s
e
2
ž
3
p 4 Ne2
/
3
B,
Ž 5.
where we have used the fact that d Nerd r - 0 and
d krd r - 0. Under the assumption of blackbody radiation for neutrinos, the energy flux of neutrinos
Fn Ž r . is proportional to T 4 so that <dln krd r < s
4 <dlnTrd r <. Defining the scale heights at the resonance point as hT ' <dlnTrd r <y1
and h Ne '
r0
3
2
<dln Nerd r <y1
where m e is
r 0 and using Ne f m e r3p
C.W. Kim et al.r Physics Letters B 419 (1998) 279–284
the chemical potential of the degenerate relativistic
background electrons, Eq. Ž5. reduces to
dr s
3eB
hT h Ne
2 m2e
4 h Ne q hT
.
Ž 6.
The asymmetry of the total neutrino momentum
distribution is, then, given by
Dk
k
p
1 H0 F Ž r q d cos f . cos f sin f d f
n
s
0
r
p
6
H0 F Ž r q d cos f . sin f d f
n
f 0.01
=
ž
ž
0
r
1
1 q Ž hTr4 h Ne .
B
3 = 10 15 Gauss
/ž
/
me
y2
11 MeV
/
,
Ž 7.
where all six species of neutrinos are approximated
to equally contribute to the total energy of a supernova. It is to be emphasized here that unity in the
denominator of the first bracket in Eq. Ž7. is due to
the r-dependence of neutrino energy. Note that we
have not used the relation d NerdT f Ž E NerE T .m e in
Ref. w1x which is, strictly speaking, not valid inside
protoneutron stars as pointed out in Ref. w9x. The
neglect of r-dependent neutrino energy, which
amounts to replacing 1rŽ1 q Ž h T r4 h N e .. by
4 h NerhT , leads to
Dk
f
eB h Ne
2
s
3 m e hT
k
9
3 eB
h Ne
m2e
hT
ž /
2
,
Ž 8.
which is the result in Ref. w9x. Further, if we ignore
the temperature dependence of chemical potential
which justifies d NerdT f Ž E NerE T .m e, the ratio of
the scale heights can be written as h NerhT ;
3 Ner2T 2m e . Then, using the relation Ne ; m 3er3p 2 ,
we can rewrite Eq. Ž8. as
Dk
1
f
k
ž
e
3 2p
B
2
T2
/
,
Ž 9.
which is the result in Ref. w1x Žwith the correct flux
factor 1r3.. If we set B ; 10 15 Gauss, r ;
10 11 grcm3, Ye ; 0.1Ž m e ; 11 MeV. and T ; 3MeV,
we have h NerhT ; 0.7 and therefore D krk ; 0.01.
281
However, numerical supernova calculations w10x
show that the scale heights are h Ne ; 6 km, hr ; 7
km, and hT ; 22 km at the stage of the protoneutron
star in a model of fixed baryon mass, 1.4 M( , without
accretion or convection. Therefore, the ratio of the
scale heights of the model w10x is h NerhT ; 0.3
which is slightly different from the result, 0.7, of the
approximate relation h NerhT ; 3 Ner2T 2m e , increasing the value of B to have D krk ; 0.01. This was
the point stressed in Ref. w9x. If h NerhT turns out to
be large Ž h NerhT ) 1., then the r-dependence of
neutrino energy becomes relatively important and
one has to resort to a general formula in Eq. Ž7.. In
general, the value of h NerhT depends on the model
of the birth of neutron star and on the time evolution
of the collapse-driven supernova explosion. In summary, the incorporation of the r-dependence of the
neutrino energy as manifested in the first parentheses
in Eq. Ž7. requires, in general, higher B values in
order to explain the observed birth velocity of pulsars.
Now, let us discuss the problem in a three-neutrino generation scheme with a mass hierarchy that
can accommodate the solar neutrino data w11x. According to a general theory of three neutrino oscillations, the weak eigenstates of neutrinos ne, m ,t are
superpositions of the mass eigenstates n 1,2,3 via a
3 = 3 unitary matrix U w12x:
ne
n1
nm s U n 2 .
n3
nt
0 0
Ž 10 .
Assuming, for simplicity, that CP is conserved, we
adopt a useful parameterization of U as
U s e i cl 7 e i wl 5 e i v l 2
1
s 0
0
0
Cc
0
Sc
ySc
Cc
Cv
= ySv
0
Sv
Cv
0
0
Cw
0
Sw
0
ySw
1
0
0
Cw
0
0 ,
1
0
0
Ž 11 .
where Cv s cos v , Sv s sin v and the l’s are the
Gell-Mann matrices. In an electron-rich medium, the
C.W. Kim et al.r Physics Letters B 419 (1998) 279–284
282
equations of motion for the weak eigenstates of
neutrinos are
ne
n
i
m
dt
nt
d
0
1
s
2E
U
m12
0
0
0
m 22
0
0
0
m23
0
A
U†q 0
0
ž
0
0
0
ne
= nm ,
nt
0
0
0
0
/
Ž 12 .
where A is the matter-induced mass squared of ne
which is, in the presence of magnetic field,
1
A s 2'2 GF Ne E q '2 eGF
ž
3 Ne Ž r .
p4
/
3
k P B.
Ž 13 .
We will consider the pulsar kick velocity by three
neutrino mixing under two natural conditions: Ž1.
m1 < m 2 < m 3 ; Ž2. the mixing angles in vacuum are
small. It is easy to see then that three neutrino
oscillations in matter separate into two parts, each
being similar to two neutrino oscillations in matter.
Two resonant conversions can take place at well
separated locations. We call R h the resonance surface which occurs at higher density, and R l the one
at lower density. The conditions for each resonance
are w13x
A R h ( m23 y 12 m12 Ž 1 q Cv . q m22 Ž 1 y Cv . 4 C2 w ,
A R l ( Ž m 22 y m12 . C2 v .
Ž 14 .
In a three generation scheme, therefore, the relative
positions of two resonance surfaces to neutrinospheres are essential in order to study the effects
of neutrino oscillations on the birth velocities of
pulsars. First, let us discuss the relative positions of
three neutrinospheres. A neutrinosphere may be considered as an imaginary surface such that inside the
surface neutrinos are trapped due to the interactions
with medium, but outside they escape freely. Therefore, the larger the scattering cross section is, the
outer the neutrinosphere is located. Since the temperature ŽT f 3 MeV. around the neutrinospheres is too
low to thermally create m or t , the nm neutrinosphere is expected to coincide with the nt neutrinosphere, St . As can be seen in Eq. Ž14., the locations of resonance surfaces are dependent on the
masses and the mixing angles of neutrinos in vacuum which have been restricted by various measurements and observations. According to a comprehensive analysis of solar and atmospheric neutrino experiments in a hierarchical three-generation scheme,
m1 < m 2 < m 3 , the Mikheyev-Smirnov-Wolfenstein
ŽMSW. solution to the solar neutrino deficit problem
suggests m 22 y m12 f 10y5 eV 2 w14x and the atmospheric neutrino anomaly leads to m 23 R 10y3 eV 2
w15x. Therefore, the density around neutrinospheres
with r f 10 11 ; 12 grcm3 is too high for the lower
resonance to occur, implying that the lower resonance surface R l lies outside the S e . Within the
current experimental limit of m 3 , the R h Žthe resonant conversion ne l nt 4 . can lie between the S e
and the St w16x.
In the following we assume that the R l for ne l nm
is located outside the S e , but the R h for ne l nt
between the S e and the St . At the R h , nt produced
inside the St will be converted into ne and absorbed,
whereas ne produced inside the St will turn into nt
and will propagate out with the collapsing energy of
the protoneutron star. The nm does not undergo any
resonant conversion at the R h . As neutrinos pass the
lower resonance surface, neutrino flavors are resonantly converted again on the magnetically distorted
surface. However, this resonant conversion outside
the S e does not contribute to the pulsar velocities
because the momentum transfer of neutrinos is independent of whether or not the neutrino flavor is
changed. This is in contrast to the case where the
resonance surface lies between the S e and the St so
that nt converted from ne propagates out but ne
from nt cannot, leading to an asymmetry in the
neutrino momentum distribution. Once neutrinos escape from the S e , therefore, flavor-changing of the
neutrinos does not affect the kick velocity of pulsars.
We summarize that if the value of Ž m 22 y m12 . is
restricted by the MSW solution to the solar neutrino
problem so that the lower resonance occurs outside
4
Of course, this is an approximate interpretation, which is valid
when the neutrino mixing angles in vacuum are very small.
C.W. Kim et al.r Physics Letters B 419 (1998) 279–284
the S e , the consideration of three neutrino oscillations does not affect the result of the KS mechanism
based on the two generation scheme.
Next, we discuss the effects of possible nonadiabatic neutrino oscillations on the birth velocity
of pulsars. It is, of course, obvious that non-adiabatic
processes make ne ™ nt conversion less efficient,
reducing the kick velocity. We wish to study this in
some detail. Let P Ž na ™ nb . be a general probability for na produced deep inside the resonance surface to be observed as nb outside the resonance
surface. In adiabatic cases where instantaneous effective mass eigenstates approximate the energy eigenstates of Hamiltonian, the transition probability
PAŽ na ™ nb . w17x is
PA Ž na ™ nb .
s ² nb <
ž
C f2
S f2
S f2
C f2
1
0
0
1
/ž /ž
Ci2
Si2
Si2
Ci2
/
< na : ,
Ž 15 .
where C f ,i s cos u f ,i and u f Ž u i . is the effective mixing angle of neutrinos in matter after Žbefore. neutrinos go through the resonance surface and
< ne : s 1 , < nt : s 0 .
Ž 16 .
0
1
If the oscillation process is non-adiabatic, a levelcrossing between the effective mass eigenstates takes
place, particularly near the resonance point w12x.
With the level-crossing ŽLandau-Zener. probability
PL Z , we have the following non-adiabatic transition
probabilities
P NA Ž na ™ nb .
ž/
s ² nb <
=
ž
ž
ž/
C f2
S f2
S f2
C f2
Ci2
Si2
Si2
Ci2
/
/ž
1 y PL Z
PL Z
< na : .
PL Z
1 y PL Z
/
not ruled out.. For example, the level-crossing probability is PL Z ; 0.38 for u s 10y4 . In the extremely
non-adiabatic case, flavor changing processes would
not take place, so that the KS mechanism becomes
inoperative.
Let us consider a general non-adiabatic case with
a possibility that the fluxes of ne’s and nt ’s from the
protoneutron star are different. The collapsing supernova core has about 10 57 protons which produce
neutrinos via p q ey™ n q ne , resulting in the formation of a neutron star. Since each ne emitted from
the core carries away an average energy 10 MeV,
about 10 52 ergs are emitted by ne’s during the neutronization processes. This is less than 5% of all the
n ’s radiated w18x. The rest of the neutrinos come
from pair processes such as eqq ey™ n i q n i , where
i s e, m or t . Since nm and nt are produced via
neutral currents but ne via both charged and neutral
currents, there exist some differences between the ne
flux and other neutrino fluxes, defined collectively
by nb . We will introduce the parameter an e such that
neutrino fluxes are given by Fn e s Ž1 q an e . F and
Fnb s F , where F is the common flux factor. The
existence of another enhancing channel for ne production implies an e ) 0. The asymmetry in the neutrino momentum distribution due to non-adiabatic
oscillations is then given by
Dk
ž /
k
f
P NA Ž ne ™ nt . Ž 1 q an e .
Ž 6 q an .
NA
e
p
H0 F Ž r q d cos f . cos f sin f d f
0
=
0
Dk
The level-crossing probability is PL Z s expŽy p4 Q .
where the adiabaticity parameter Q is given by
Q s D02 S22u h NerkC2 u in the two generation case. For
D02 ; 10 4 eV 2 , h Ne ; 6 km, and k ; 10 MeV, the
non-adiabatic effect cannot be ignored when the
vacuum mixing angle u is smaller than 10y4 . ŽRecall that the mixing angle of relevance is ne y nt
mixing angle, therefore, a very small mixing angle is
r
p
H0 F Ž r q d cos f . sin f d f
r
P NA Ž ne ™ nt . Ž 1 q 56 an e . , Ž 18 .
k
where the second approximation is valid for small
value of an e. The term Ž D krk . on the right hand
side of Eq. Ž18. is given by Eq. Ž7.. In the small
vacuum mixing case Ž u f ; 0, u i ; pr2. the transition
probability P NAŽ ne ™ nt . can be further approximated, resulting in
Dk
Dk
f
Ž 1 y PL Z . Ž 1 q 56 an e . .
Ž 19 .
k NA
k
This shows that the non-adiabaticity has a tendency
to decrease the asymmetry in the neutrino momenf
Ž 17 .
283
ž /
ž / ž /
284
C.W. Kim et al.r Physics Letters B 419 (1998) 279–284
tum distribution. Eq. Ž19. shows that extremely nonadiabatic processes Ž PL Z s 1. cause no kick velocity
due to the absence of flavor conversion, as discussed
before. The pulsar velocity increases with an e. This
is also expected because the more ne’s are produced
inside the core, the more they turn to the nt ’s,
leading to a larger birth velocity of pulsar. Therefore,
the effect of non-adiabaticity is opposite to that of
an e ) 0. When we set, for example, the vacuum
mixing angle u s 1.6 = 10y4 and a s 10 %, the
contributions of the non-adiabaticity and the excess
flux of electron neutrinos to the momentum asymmetry are almost canceled out. This cancelation depends
very sensitively on the value of u .
In summary, we have studied the Kusenko-Segrè
mechanism in a more general condition than in Refs.
w1x and w9x. First, the position dependence of neutrino
energy has been taken into account in the calculation
of the neutrino momentum asymmetry to explain the
peculiar pulsar velocity. This aspect becomes relatively important in the case h NerhT ) 1. In addition,
we have clarified the difference between the two
treatments in Refs. w1x and w9x. The KS mechanism is
also studied in a three generation scheme and in a
non-adiabatic case. If the solar neutrino and atmospheric neutrino problems are to be solved with the
standard mass hierarchy, a generalization of the
treatment in Ref. w1x in a three-generation scheme
leads to no new modifications. The non-adiabaticity
has a general tendency to decrease the kick velocity
whereas a larger electron neutrino flux compared to
that of other flavor gives rise to a higher momentum
asymmetry.
Acknowledgements
J.D.K. acknowledges support from the Korean
Science and Engineering Foundation ŽKOSEF. and
J.S. acknowledges support from the Korean Science
and Engineering Foundation ŽKOSEF. through the
Center for Theoretical Physics ŽCTP..
References
w1x A. Kusenko, G. Segre,
` Phys. Rev. Lett. 77 Ž1996. 4872.
w2x A.G. Lyne, D.R. Lorimer, Nature ŽLondon. 369 Ž1994. 127.
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w8x J.C. D’olivo, J.F. Nieves, Phys. Rev. D 40 Ž1989. 3679; S.
Esposito, G. Capone, Z. Phys. C 70 Ž1996. 70.
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B 356 Ž1995. 273.
w12x S.M. Bilenky, A. Pentecorvo, Phys. Rep. 41 Ž1978. 225;
T.K. Kuo, J. Pantaleone, Rev. Mod. Phys. 61 Ž1989. 937;
C.W. Kim, A. Pevsner, Neutrinos in Physics and Astrophysics, Harwood Academic Press, Chur, Switzerland, 1993.
w13x T.K. Kuo, J. Pantaleone, Phys. Rev. Lett. 59 Ž1986. 1805.
w14x GALLEX Collaboration, Phys. Lett. B 285 Ž1992. 390; X.
Shi, D.N. Schramm, J.N. Bahcall, Phys. Rev. Lett. 69 Ž1992.
717; P.I. Krastev, S.T. Petcov, Phys. Lett. B 299 Ž1993. 99;
G.L. Fogli, E. Lisi, Astropart. Phys. 2 Ž1994. 91; N. Hata,
P.G. Langacker, Phys. Rev. 50 Ž1994. 632.
w15x G.L. Fogli, E. Lisi, D. Montanio, Phys. Rev. D 49 Ž1994.
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w16x Particle Data Group, Phys. Rev. D 54 Ž1996..
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Ž1987. 288.
12 February 1998
Physics Letters B 419 Ž1998. 285–290
Gauge invariant formulation and bosonisation
of the Schwinger model
J. Kijowski a , G. Rudolph b, M. Rudolph
a
b
Center for Theor. Phys., Polish Academy of Sciences, al. Lotnikow
´ 32 r 46, 02 - 668 Warsaw, Poland
b
Institut fur
¨ Theoretische Physik, UniÕ. Leipzig, Augustusplatz 10 r 11, 04109 Leipzig, Germany
Received 30 September 1997; revised 17 November 1997
Editor: P.V. Landshoff
Abstract
The functional integral of the massless Schwinger model in Ž1 q 1. dimensions is reduced to an integral in terms of local
gauge invariant quantities. It turns out that this approach leads to a natural bosonisation scheme, yielding, in particular the
famous ‘‘bosonisation rule’’ and giving some deeper insight into the nature of the bosonisation phenomenon. As an
application, the chiral anomaly is calculated within this formulation. q 1998 Elsevier Science B.V.
PACS: 11.15.-q; 11.10.Kk; 11.25.Sq; 11.30.Rd
Keywords: Schwinger model; Gauge invariants; Bosonisation; Chiral anomaly
1. Introduction
In recent years there has been some effort to
understand the classical bosonisation results for
models in Ž1 q 1. dimensions, see w1,2x and further
references therein, in terms of functional integral
techniques, see w4–7x and further references therein.
In this Letter we apply our programme of formulating gauge theories in terms of local gauge invariant
quantities, see w8–10x, to the massless Schwinger
model in 2 dimensions, see w3x. We show that it leads
to a natural explanation of the bosonisation phenomenon, including mass generation. We also
demonstrate that our formulation provides us with
the possibility to calculate the chiral anomaly in a
straightforward, simple way. We stress that this is in
contrast to calculations within the ‘‘ordinary’’ formulation, where one either uses perturbative tech-
niques or topologically motivated regularizations
Žheat kernel, zeta-function., see e.g. w16x.
Our approach is based upon the following ideas:
First one has to analyze the algebra of Grassmann-algebra-valued invariants, which one can build from
the gauge field and the anticommuting matter fields.
For a systematic study of this aspect for the Žmost
complicated. case of QCD we refer to w11x. Typically, there occurs a number of relations between
invariants, which on the level of the algebra cannot
be ‘‘solved’’, because dividing by Grassmann-algebra-valued quantities is in general not well defined.
Next one has to express the Lagrangian, or at least
the Lagrangian multiplied by some nonvanishing
element of the above algebra, in terms of invariants.
The main point of our strategy consists in reducing
the original functional integral measure to a measure
in terms of invariants, for details we refer to w9x and
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 8 6 - X
J. Kijowski et al.r Physics Letters B 419 (1998) 285–290
286
w10x. In particular, under the functional integral we
are able to solve the above-mentioned relations between invariants, leading to theories with the correct
number of effective fields. Along these lines we have
found, both for QED and QCD, a functional integral
formulation in terms of effective bosonic fields.
The paper is organized as follows: In Section 2
we introduce some basic notations and give the
definition of gauge invariants, built from the gauge
potential and the Žanticommuting. spinor fields. Some
algebraic relations among these invariants, necessary
for our purposes, are written down. The main result
of Section 3 is the functional integral completely
rewritten in terms of local Žbosonic. gauge invariants. Finally, in Section 4 we derive the standard
bosonisation rule and calculate the chiral anomaly.
2. Basic notations and gauge invariants
Here Fmn s Em An y En Am denotes the field strength,
Dm c K s Em c K q ie Am c K the covariant derivative
and c L:s c K† Žg 0 .LK s Ž w ) , f ) . .
Now we start to apply our approach to this model.
For this purpose we define the following set of
gauge invariant Grassmann-algebra valued quantities:
L:s w ) w ,
1
2
R:s f ) f ,
H :s f ) w ,
Ž 2.5 .
Bm :s Im H ) Ž f ) Ž Dm w . q w Ž Dm f ) . . .
½
5
Ž 2.6 .
Obviously, L, R and H are Žpointwise. elements of
rank two, obeying the following identity:
L R s y< H < 2 .
Ž 2.7 .
2
)
Moreover, < H < s HH is a nonvanishing element
of Žpointwise. maximal rank. The vector current is
defined by
K
J m :s c K Ž g m . L c L s
ž
RqL
.
RyL
/
Ž 2.8 .
A field configuration of the Schwinger model w3x
is given by a UŽ1.-gauge potential Am and a spinor
field c K . In what follows m , n , . . . s 0,1 denote
space–time indices and K, L, . . . s 1,2 spinor indices. The spinor field is represented by
In 2 dimensions we have the relation J5m s e mn Jn .
Thus, we get for the axial current
f
c Ks w ,
Finally, for Bm we have
c K† s Ž f ) , w ) . ,
ž /
Ž 2.1 .
where the star denotes complex conjugation. The 4
elements Ž c K , c K) . anticommute and Žpointwise.
generate a 16-dimensional Grassmann-algebra.
The Dirac-matrices in Minkowski space with metric h mn s diagŽ1,y 1. are taken as
g 0s
ž
ž
0
1
1
,
0
g1s
/
ž
0
1
y1
,
0
/
Zs
/
i d2 x
H Łx d A d c d c e H
ž
RyL
.
RqL
/
L,
Ž 2.3 .
with the Lagrangian
L s Lgauge q Lmat
Bm s e < H < 2 Am
1
q
4i
w )f ) Em Ž wf . q wf Em Ž f )w ) . 4 .
Ž 2.10 .
In order to be able to calculate expectation values
of physical quantities one usually applies the Faddeev–Popov gauge fixing procedure, see w12x and
w13x, to the functional integral Ž2.3.. Here we use a
completely different procedure, proposed in w9x and
w10x, which consists in reducing Ž2.3. to an integral
in terms of the above defined gauge invariants.
For that purpose we introduce the following auxiliary gauge invariants:
Cm :s f ) Ž Dm w . ,
K
s y 14 Fmn F mn y Im c K Ž g m . L Ž Dm c L . . Ž 2.4 .
½
Ž 2.9 .
3. Functional integral
1
0
g s
.
Ž 2.2 .
0 y1
The completely antisymmetric symbol in 2 dimensions is denoted by emn , with e 01 s 1.
The starting point for quantum theory is the following Žformal. functional integral:
5
J5m s
5
Fm :s f ) Ž Dm f . ,
Em :s w ) Ž Dm w . ,
Ž 3.1 .
J. Kijowski et al.r Physics Letters B 419 (1998) 285–290
which are subject to the relations:
H ) Cm s i Bm q 12 H ) Ž Em H . y 14 R Ž Em L .
q 14 L Ž Em R . ,
Ž 3.2 .
1
2
1
4
REm s yi Bm y H ) Ž Em H . q R Ž Em L .
y 14 L Ž Em R . ,
Ž 3.3 .
LFm s yi Bm q 12 H ) Ž Em H . y 14 R Ž Em L .
q 14 L Ž Em R . q 12 Em Ž LR . .
Ž 3.4 .
One easily proves the following identities:
e < H < 2 Fmn s yi E w m < H < 2 C n x q i Cw)m Cn x
ž
/
yi Ž E w m R . En x ,
Ž 3.5 .
Lmat s y 12 tr Ž g 0g m .
= Ž hmn q emn . Im F n q Ž hmn y emn . Im E n 4
' yIm E0 q F0 y E1 q F1 4 .
Ž 3.6 .
Observe, that Ž3.5. is an identity on the level of
maximal rank in the Grassmann-algebra. Dividing by
a non-vanishing element of maximal rank is a welldefined operation giving a c-number. Thus, we can
express the Lagrangian Ž2.4. completely in terms of
gauge invariants.
With the identities Ž3.2. – Ž3.6. we are now able to
reduce the functional integral Ž2.3. to an integral
over gauge invariants. For that purpose we make use
of the following notion of the d-distribution on
superspace Žsee w17x.
we will call c-number mates. These mates are by
definition gauge invariant: r and l are real scalar
fields, h is a complex scalar field and bm , jm and jm5
are real covector fields. Next we apply the procedure
developed in w9x and w10x: First we use Ž3.5. and
Ž3.6. under the functional integral to express the
Lagrangian in terms of gauge invariants. Next we
eliminate the auxiliary invariants by the help of
Ž2.8., Ž2.9. and Ž3.2. – Ž3.4.. We are left with a
functional integral, where the original gauge dependent field configuration Ž c , Am . occurs only in the
integral measure. It can be integrated out leaving us
with the Žreduced. functional integral in terms of
gauge invariants Žfor details we refer to w9x and w10x..
Finally we stress that, due to Ž2.8., Ž2.9., r, l and jm
can be expressed by jm5 . Thus, for this model we get
a reduction of the functional integral Ž2.3. to an
integral in terms of the gauge invariant set
Ž Õm , j5m, < h <, u ., where h:s < h < e i u and Õm :s e b< hm< 2 :
Zs
m
= e i Hd
2
`
Ž y1.
ns0
n!
Ž 3.7 .
Here u is a c-number variable and U an element of
the Grassmann-algebra built from the matter fields c
and c ) . From this definition we get immediately
1' du d Ž uyU . .
H
Ž 3.8 .
Thus, by inserting identites of the form Ž3.8. under
the functional integral we introduce for each Grassmann-algebra-valued gauge invariant Ž L, R, H, Bm ,
Jm , Jm5 . c-number variables Ž l,r,h,bm , jm , jm5 ., which
5
L w Õm , j5m, u x ,
Ž 3.9 .
K j5m , < h < s
1
2p
q
d2
1
ž
d jm5 d j5m
1
d
4 < h< d < h<
/
y
d2
4 d < h< 2
d Ž jm5 . d Ž < h < 2 . ,
Ž 3.10 .
2
L Õm , jm5 , u s y 14 Ž E w m Õn x . y e e mn Õm jn5
q 12 jm5 Ž E mu . .
n
d Ž n. Ž u . U n .
x
d < h < 2 d u K j5m , < h <
with
H
Ý
m
5
H Łx ½ dÕ d j
d Ž u y U . s d j e 2 p i j Ž uyU .
s
287
Ž 3.11 .
We stress that d u is the Haar-measure on S 1.
The effective Lagrangian Ž3.11. leads to the following set of classical field equations in terms of
gauge invariants:
Ž Em j5m . s 0,
Õ ms
1
2e
e mn Ž En u . ,
Ž Em E w m Õ n x . s e e nm jm5 .
Ž 3.12 .
Ž 3.13 .
Ž 3.14 .
Formula Ž3.12. means that, on the classical level, the
axial current is conserved. Moreover, from Eq. Ž3.14.
J. Kijowski et al.r Physics Letters B 419 (1998) 285–290
288
we get Ž Em j m . s 0, i.e. also the vector current is
conserved. Finally, Eq. Ž3.13. leads to
Ž EmÕ m . s 0.
Ž 3.15 .
We see that, as in the Faddeev–Popov procedure,
a nontrivial singular integral kernel K w j5m, < h <x occurs
in the functional integral Ž3.9.. Obviously, the quantity < h < can be integrated out, because it does not
occur in Ž3.11.. The remaining kernel K w j5m x
d2
s 21p
d Ž j5m . can be treated similarly as in the
ž
d jm5 d j5m
/
Faddeev–Popov procedure: it can be averaged with a
Gaussian measure, or more generally, with a sum of
moments of a Gaussian measure. After this ‘‘regularization’’ a free parameter, say a j , characterizing
the Gaussian measure, enters the theory. In the Faddeev–Popov approach different values of this parameter correspond to different gauge fixings. Here, as
will be seen in what follows, this parameter is fixed,
e.g. by the physical requirement to give the correct
chiral anomaly. After this ‘‘averaging’’ procedure
we get
ZsN
m
5
H Łx dÕ d j
m
d u 4 e i Hd
2
x
L w Õm , j5m, u x ,
Ž 3.16 .
where
2
L Õm , jm5 , u s y 14 Ž E w m Õn x . y e e mn Õm jn5
q 12 jm5 Ž E mu . y
1
2 aj
j5m jm5 .
Ž 3.17 .
Here, N is an infinite normalization constant, which
we omit in what follows. We stress that the same
procedure works also for the standard generating
functional.
where
2
L Õm , u s y 14 Ž E w m Õn x . y
y
aj e
2
aj e2
2
Õm Õ m
e mn Õm Ž En u . q
aj
8
Ž Em u . Ž E mu . .
Ž 4.2 .
After performing the reparameterisation u Ž x . ™
2'p u Ž x . as well as fixing the parameter a j s p1 we
obtain:
Lsy
1
4
Ž E w m Õn x .
2
e2
y
2p
q 12 Ž Em u . Ž E mu . .
Õm Õ m y
e
'p
e mn Õm Ž En u .
Ž 4.3 .
This result has to be compared with other functional
integral approaches explaining the bosonisation phenomenon, see w4–6x. Essentially, our procedure leads
to the same result, but with the following advantages: Within our approach, instead of the electromagnetic potential in a certain gauge, the gauge
invariant field Õm occurs. It becomes – automatically
– massive, with mass m Õ s 'ep . Thus, we see a
dynamical Higgs mechanism characteristic for the
Schwinger model, see w14–16x and references therein.
Moreover, the field u , which in other approaches has
to be rather introduced by hand, shows up as the
‘‘phase’’ of a gauge invariant combination of the
original fermionic fields. This gives – in our opinion
– a deeper insight into the bosonisation phenomenon. Comparing the couplings of Õm in Ž3.17.
and Ž4.3., we read off the famous ‘‘bosonisation
rule’’
4. Standard bosonisation rule and chiral anomaly
Since jm5 does not occur as a dynamical field a
further reduction of Ž3.16. is possible, namely we
can carry out the Gaussian integration over jm5 . This
leads to:
Zs
H Łx dÕ
m
d u 4 e i Hd
2
x
L w Õm , u x ,
Ž 4.1 .
1
m
m
'p Ž E u . s j5 .
Ž 4.4 .
Next, we show that within our approach the chiral
anomaly can be easily calculated, justifying – in
particular – the above made choice a j s p1 . For this
purpose, we treat Õm as an external field. Due to the
J. Kijowski et al.r Physics Letters B 419 (1998) 285–290
bosonisation rule Ž4.4., the chiral anomaly should be
given by the vacuum expectation value
1
²
1
m
Ž E u .:s
'p
1
Hd u 'p
Z w Õm x
Moreover, we have
1
²
m
ŽE u . e
iHd x L˜
2
,
'p
iHd x L˜
2
2
where Z w Õm x s Hdu e
and L˜ s y 2ep Õm Õ m
e
1
mn
m .
Ž
.
Ž
.Ž
y 'p e Õm En u q 2 Em u E u . Since Õm is an
external field, i.e. it obeys the classical field Eq.
Ž3.15., we can write it in the form Õm s emn Ž E nL..
Thus, performing the shift Ž E mu . ™ Ž E mu .
q 'ep e nm Õn ' Ž E mu . y 'ep Ž E mL., which leads to a
trivial change in the integration variable u Ž x . ™
u Ž x . y 'ep LŽ x ., we get:
1
m
:
'p Ž E u .
1
1
Hd u 'p
Zw Õ x
s
m
žŽ
E mu . q
e
'p
e nm Õn
/
=e H
s
Z w Õm x
e
q
p
du
H
1
m
'p
²
m
ŽE u .e H
nm
Õn .
'p
Ž E mu . : ' ²
1
'p
e nm Ž En u . : s
e
p
Ž Em Õ m . ' 0.
Ž 4.8 .
Ž 4.9 .
Due to Ž4.4. and the relation between vector and
axial current, Eq. Ž4.9. shows the conservation of the
vector current jm .
Following the same lines we can discuss Ward
identities. In a forthcoming paper we will show that
in full 4-dimensional QED the vector Ward identity
is automatically fulfilled – in accordance with the
fact that we are dealing with a manifestly gauge
invariant formulation. Here, however, due to j5m s
e mn jn , we encounter the same phenomenon as other
authors, namely, the anomaly can be shifted between
the vector and the axial Ward identity. With Ž4.4. we
have
1
p
e na ² Ž E mu Ž x 1 . .Ž Ea u Ž x 2 . . : .
Using an appropriate regularization procedure and
adding the ‘‘seagull’’ term ag mnP Ž x 1 y x 2 . Žfor a
detailed discussion of regularization schemes in the
Schwinger model we refer to w15x. yields
Ž 4.6 .
ye m a
Taking the partial derivative gives exactly the chiral
anomaly:
1
Õ m.
T5rmne g Ž x 1 y x 2 . s e mn a P Ž x 1 y x 2 .
:
'p Ž E u . s p e
Em²
p
Ž 4.10 .
e nm Õn .
e
Em²
'
i d 2 x 12 Ž Em u .Ž E mu .4
Now observe that the first integral is odd in u and,
consequently, it vanishes identically, leading to
1
e
T5mn Ž x 1 y x 2 . :s ² j5m Ž x 1 . j n Ž x 2 . :
i d 2 x 12 Ž Em u .Ž E mu .4
1
e nm Ž En u . : s
Using again the classical field Eq. Ž3.15. yields
Ž 4.5 .
²
289
1
e
m :
nm
'p Ž Em E u . s p e Ž Em Õn . .
Ž 4.7 .
We see that within our formulation the chiral anomaly
can be calculated in a straightforward way without
using neither perturbative techniques nor regularization techniques of the heat kernel type. We also
stress that this calculation can be – a posteriori –
considered as a deeper justification of the bosonisation rule Ž4.4..
Ea E n
P Ž x 1 y x 2 . , Ž 4.11 .
I
2
where I s Em E m and P Ž x 1 y x 2 .:s p 2eM 2 d Ž x 1 y
x 2 ., M 2 ™ 0 depends on the regularization scheme.
For a ' 1 the vector Ward identity is fulfilled
En T5mn Ž x 1 y x 2 . ' 0, whereas we have an anomalous
axial Ward identity EmT5mn Ž x 1 y x 2 . s e mn Em P Ž x 1 y
x 2 ..
Finally, let us make some remarks. If we consider
the massive Schwinger model, where an additional
term ymcc occurs in the Lagrangian Ž2.4., we get
an additional term y2 mReŽ h. s y2 m < h <cos u in
Ž3.3.. This leads to a field theory of sine–Gordon
type. Comparing with the standard result, see for
instance w1x and w2x, we obtain a slight modification:
290
J. Kijowski et al.r Physics Letters B 419 (1998) 285–290
instead of getting a constant in front of the cosŽ u .term, we obtain < h <. However, since < h < is a nonpropagating field, it can be ‘‘averaged’’ to a constant
– along the same lines as discussed at the end of
Section 3.
Our approach also works for the 2-dimensional
gauged Thirring-model Žsee for instance w7x, where
the Thirring-model and its various generalizations on
curved space–time are analyzed.. Applying our procedure simply leads to a modification of the coefficient of the term quadratic in jm5 occuring in Eq.
Ž3.17..
Acknowledgements
The authors are very much indebted to I. Bialynicki-Birula for helpful discussions and remarks.
References
w1x S. Coleman, Phys. Rev. D 11 Ž1975. 2088.
w2x Frenkel, J. Funct. Anal. 44 Ž1981. 259.
w3x J. Schwinger, Phys. Rev. 128 Ž1962. 2425.
w4x P.H. Damgaard, H.B. Nielsen, Nucl. Phys. B 385 Ž1992. 227.
w5x P.H. Damgaard, H.B. Nielsen, R. Sollacher, CERN-TH6959r93.
w6x P.H. Damgaard, H.B. Nielsen, R. Sollacher, Phys. Lett. B
296 Ž1992. 132.
w7x I. Sachs, A. Wipf, Ann. Phys. ŽNY. 249 Ž1996. 380.
w8x J. Kijowski, G. Rudolph, Lett. Math. Phys. 29 Ž1993. 103.
w9x J. Kijowski, G. Rudolph, M. Rudolph, Lett. Math. Phys. 33
Ž1995. 139.
w10x J. Kijowski, G. Rudolph, M. Rudolph, Effective Bosonic
Degrees of Freedom for One-Flavour Chromodynamics, Ann.
Inst. H. Poincare,
´ in print, preprint-Nr. 22r1996, Univ.
Leipzig.
w11x J. Kijowski, G. Rudolph, M. Rudolph, On the Algebra of
Gauge Invariants for One-Flavour Chromodynamics, Rep.
Math. Phys., in print.
w12x L.D. Faddeev, V.N. Popov, Phys. Lett. B 25 Ž1967. 30.
w13 x L.D. Faddeev, A.A. Slavnov, Gauge Fields, The
BenjaminrCummings Publ. Comp., London, 1980.
w14x E. Abdalla, M.C.B. Abdalla, K.D. Rothe, Non-Perturbative
Methods in 2 dimensional Quantum Field Theory, World
Scientific, Singapore, London 1991.
w15x C. Adam, R.A. Bertlmann, P. Hofer, UWThPh-1993-5.
w16x R.A. Bertlmann, Anomalies in Quantum Field Theory,
Clarendon Press, Oxford, 1996.
w17x F.A. Berezin, The Method of Second Quantization, Academic Press, New York and London, 1966.
12 February 1998
Physics Letters B 419 Ž1998. 291–295
Universal amplitude ratios in the two-dimensional Ising model
G. Delfino
1
2
Laboratoire de Physique Theorique,
UniÕersite´ de Montpellier II, Pl. E. Bataillon, 34095 Montpellier, France
´
Received 17 October 1997
Editor: L. Alvarez-Gaumé
Abstract
We use the results of integrable field theory to determine the universal amplitude ratios in the two-dimensional Ising
model. In particular, the exact values of the ratios involving amplitudes computed at nonzero magnetic field are provided.
q 1998 Elsevier Science B.V.
Universality is one of the most fascinating concepts of statistical mechanics w1x. Briefly stated, it
says that physical systems with different microscopic
structure but having in common some basic internal
symmetry exhibit the same critical behaviour in the
vicinity of a phase transition point. The point is best
illustrated considering the singular behaviour of the
various thermodynamic quantities nearby the critical
point. For a magnetic system exhibiting a second
order phase transition the usual notation is
C , Ž Ara . tya ,
t)0 , hs0 ,
C , Ž AXra X . Ž yt .
ya
ya c
C , Ž A cra c . < h <
b
M , B Ž yt . ,
,
X
,
t-0 , hs0 ,
ts0 , h/0 ,
t - 0 , h s 0q ,
x , G tyg , t ) 0 , h s 0 ,
1
Work supported by the European Union under contract
FMRX-CT96-0012.
2
E-mail: [email protected].
yg
X
x , G X Ž yt .
, t-0 , hs0 ,
yg c
x , Gc < h <
, ts0 , h/0 ,
h , Dc M < M < dy1 ,
ts0 , h/0 ,
j , j 0 tyn , t ) 0 , h s 0 ,
j , j 0X Ž yt .
yn c
j , jc < h <
yn
X
,
t-0 , hs0 ,
ts0 , h/0 ,
where t s a ŽT y Tc . Ž a positive constant., h is the
applied magnetic field and the limit towards the
critical point t s 0, h s 0 is understood. C, M, x
and j denote the specific heat, the magnetisation,
the susceptibility and the correlation length, respectively.
The critical exponents a , b , . . . are the same for
all systems within a given universality class and are
related by the scaling and hyperscaling relations in d
dimensions
,
a s a X , g s g X , n s n X , g s b Ž d y 1. ,
a s 2 y 2 b y g , 2 y a s dn , a c s arbd ,
gc s 1 y 1rd , nc s nrbd .
The critical amplitudes A, B, . . . , on the other hand,
depend on the scale factors used for t and h and are
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 5 7 - 3
G. Delfinor Physics Letters B 419 (1998) 291–295
292
nonuniversal. However, universal ratios of amplitudes can be constructed in which any dependence
on metric factors cancels out. Together with the
critical exponents, these ratios further characterise
the given universality class. The standard amplitude
combinations considered in the literature are w2x
limit of the Ising model at t s 0 and h / 0 is an
integrable theory w5x.
We will regard the scaling limit of the two-dimensional Ising model as described by the euclidean
field theory defined by the action
ArAX ,
S s SCFT y t d 2 x ´ Ž x . y h d 2 x s Ž x . ,
GrG X ,
R C s A GrB 2 ,
Rx s G Dc B
dy1
j 0rj 0X ,
Ž 1.
1r d
Rq
j0 ,
j sA
,
R A s A c DcyŽ1 q a c . By2 r b
Q2 s Ž GrGc . Ž j crj 0 .
gr n
.
Ž 2.
,
Ž 3.
A substantial progress was made over the last years
in the derivation of nonperturbative theoretical results in two-dimensional statistical mechanics and
quantum field theory. The solution of conformal
field theories ŽCFTs. w3,4x provided an almost complete classification of universality classes for second
order phase transitions in d s 2. In particular, it
solved the problem of the exact determination of the
critical exponents. The critical amplitudes, however,
carry information about the scaling region outside
the critical point and are not determined by CFT. In
this respect, the Zamolodchikov’s observation that
specific perturbations of the critical point lead to
integrable off-critical theories w5x is of crucial importance. The integrable theories obtained in this way,
regarded as quantum field theories in 1 q 1 dimensions, are characterisable through the determination
of their exact S-matrix. A number of physical quantities can then be computed using different techniques
w6x. In particular, the results provided by the thermodynamic Bethe ansatz ŽTBA. w7x and the form factor
approach w8–11x enable the determination of amplitude ratios and have been used for this purpose in the
problems of self-avoiding walks w12x and percolation
w13x Žsee also w14x..
It is the purpose of this note to illustrate the
derivation of universal amplitude ratios from Žintegrable. field theory through the very basic example
of the two-dimensional Ising model. Of course, the
purely ‘‘thermal’’ ratios Ž1. and Ž2. are exactly
known for this case since the seventies, when the
correlation functions of the Ising model at h s 0
were first computed on the lattice w15,16x. The possibility to determine the ratios Ž3., on the contrary,
relies on the more recent realisation that the scaling
H
H
Ž 4.
where SCFT stays for the Ising critical point conformal action and s and ´ denote the magnetisation
and energy operators, respectively. All the critical
exponents are determined by the scaling dimensions
xs s 1r8 and x´ s 1 of the operators s and ´ :
a s Ž d y 2 x´ .rŽ d y x´ . s 0, b s xsrŽ d y x´ . s
1r8, g s 7r4, d s 15, n s 1, a c s 0, gc s 14r15,
nc s 8r15. The physical dimensions of the two couplings in Ž4. are t ; m2y x ´ and h ; m2y xs , m being
a mass parameter. What is important for us is that
the theory Ž4. becomes integrable when at least one
of the two couplings is set to zero w5x.
Choosing the scale factors for t and h amounts to
fixing a normalisation for the conjugated operators ´
and s . We will proceed to determine the critical
amplitudes within the standard CFT normalisation
defined by the asymptotic conditions
² s Ž x . s Ž 0 . : s < x <y1 r4 ,
² ´ Ž x . ´ Ž 0 . : s < x <y2 ,
x™0
x™0 .
Ž 5.
Let us begin with the amplitudes for the correlation
length. The latter can be defined in different ways.
For the time being, we will refer to the so called
‘‘true’’ correlation length defined through the large
distance decay of the spin-spin correlation function
² s Ž x . s Ž 0 . :c ;
ey< x < r j
< x < Ž dy1.r2
,
< x<™` ,
Ž 6.
where ² PPP :c denotes the connected correlator.
From the representation of the correlator as a spectral sum over n-particle intermediate states
² s Ž x . s Ž 0 . :c s Ý <²0 < s Ž 0 . < n:< 2 eyE n < x < ,
Ž 7.
n
it is clear that the ‘‘true’’ correlation length is the
inverse of the total mass of the lightest state entering
the decomposition above. For integrable models, the
exact relation between the particle masses and the
G. Delfinor Physics Letters B 419 (1998) 291–295
couplings appearing in the action is provided by the
TBA w17x. The result for the Ising model is 3
m h s C < h < 8r15 ,
p
4sin G Ž 1r5 .
5
Cs
G Ž 2r3 . G Ž 8r15.
mt s 2p <t < ,
2
=
ž
2
G Ž 1r4 . G Ž 3r16.
² s :h s
Ž 9.
The specific heat diverges logarithmically in the
Ising model Ž a s 0. and the specific heat amplitudes
are accordingly defined through C , yAlnt and
analogous relations for AX and A c . Writing the partiS . and the reduced free
tion function as Tr expŽyS
energy as f s y1rV ln Z, the specific heat per unit
volume is given by
2
E f
s d 2 x ² ´ Ž x . ´ Ž 0 . :c .
Ž 10 .
Et 2
It follows from Ž5. that the specific heat amplitudes
in the CFT normalisation are simply
H
A s AX s A c s 2p .
Ž 11 .
The magnetisation per unit volume is given by
Msy
Ef
s²s : .
Ž 12 .
Eh
Vacuum expectation values in the CFT normalisation
are exactly known for integrable models due to the
TBA w17x and some more recent developments w18x.
For the magnetisation operator in the Ising model at
3
/
Ž 13 .
with
Ž 8.
Due to the invariance under spin reversal at h s 0,
the magnetisation operator couples only to the states
with odd Ževen. number of particles for t ) 0 Žt - 0..
When h / 0, s couples to any intermediate state. It
follows
Csy
h1r15
s 1.27758227 . . . h1r15 ,
/
C.
j c s 1rC
2 C2
2p
2p
p
15 sin
q sin
q sin
3
5
15
ž
4r15
s 4.40490857 . . . .
j 0 s 2 j 0X s 1r Ž 2p . ,
t - 0, h s 0 and at t s 0, h ) 0 they are,
respectively 4
² s :t s B Ž yt . 1r8 ,
2
4p G Ž 3r4 . G Ž 13r16.
293
We denote by the subscript t Ž h. the quantities referring to
the theory Ž4. with hs 0 Žt s 0.. m h is the mass of the lightest
among the 8 particles the mass spectrum of the t s 0 theory
consists of.
B s 2 1r12 Ž 2pre .
1r8
A 3r2 s 1.70852190 . . . ; Ž 14 .
we used the Glaisher’s constant
A s 2 7r36py1r6 exp
1
3
q 23
1r2
H0
dx ln G Ž 1 q x .
s 1.28242712 . . . .
Ž 15 .
From Ž13. we deduce
ž
15 sin
Dc s
2p
3
q sin
2C
2p
5
p
q sin
15
15
/
2
s 0.0253610264 . . . .
Ž 16 .
The reduced susceptibility is defined as
xsy
E 2f
Eh
2
s d 2 x ² s Ž x . s Ž 0 . :c .
H
Ž 17 .
The amplitudes G and G X were computed exactly
in w16x integrating the h s 0 spin-spin correlator. In
the CFT normalisation they read 5
G s 0.148001214 . . . ,
G X s 0.00392642280 . . . .
Ž 18 .
When t s 0, s is the operator which perturbs the
critical point. The zeroth moment of the spin-spin
4
In the following we will consider for convenience only positive values of h. Due to the simmetry about hs 0 this involves no
loss of generality.
5
The relations one needs to pass from the lattice normalisation
of w16x to the one used here are s l at s 2 5r 48 e1r8 Ay3r2 s and
wŽT yTc .r Tc x l at s Žp rlnŽ1q'2 .. t .
G. Delfinor Physics Letters B 419 (1998) 291–295
294
correlator is then related to the vacuum expectation
value as
xs
² s :h .
d 2 x ² s Ž x . s Ž 0 . :ch s
Ž 19 .
Ž 2 y xs . h
H
It follows
1
Gc s Dcy1 r d s 0.0851721517 . . . .
Ž 20 .
d
The dependence on the operator normalisations Žthe
only nonuniversal ingredient we had in our computation. cancels when the combinations Ž1., Ž2. and Ž3.
are considered. The exact results we obtain for the
ratios are collected in Table 1. The values for the
combinations of purely thermal amplitudes are well
known. Concerning the ratios involving amplitudes
computed at h / 0, to the best of our knowledge the
only reliable estimates in d s 2 come from the series
analysis of Ref. w19x. The value Rx , 6.78 is quoted
there together with a value for Q2 which uses the
‘‘second moment’’ correlation length Žsee below..
No previous result for R A is known to us.
We conclude this note computing the ratios which
involve the correlation length amplitudes using the
second moment correlation length
2
j 2nd
'
1
2d
Hd
2
x < x < 2² s Ž x . s Ž 0 . :c
.
Hd
2
Ž 21 .
Table 2
Results referring to the second moment correlation length with
Ž j 0 . 2nd and Ž j 0X . 2nd computed in the leading form factor approximation
Ž j 0 r j 0X . 2nd , 3.162
Ž Rq
j . 2nd , 0.3989
Ž Q2 . 2nd , 2.833
elements it contains Žknown as form factors. can be
computed exactly. While the Ising model at h s 0
remains the single example for which the resummation of the spectral series is known, a remarkably fast
convergence of the series emerged in the last years
as a general feature of integrable models w10,12,20x.
This important circumstance makes the form factor
approach extremely effective for obtaining accurate
quantitative results at a relatively little cost.
We compute Ž j 0 . 2nd and Ž j 0X . 2nd in the leading
form factor approximation Ž1-particle contribution
for t ) 0, 2-particle contribution for t - 0.. Using 6
w21x
²0 < s Ž 0 . < u :t s ² s :t ,
u1 yu2
²0 < s Ž 0 . < u 1 , u 2 :t s i ² s :t tanh
,
Ž 22 .
2
one finds
Ž j 0 . 2nd , 1r Ž 2p . , Ž j 0X . 2nd , 1r Ž 2'10 p . .
Ž 23 .
x ² s Ž x . s Ž 0 . :c
The amplitudes Ž j 0 . 2nd Ž j 0X . 2nd could be exactly
computed by numerical integration of the spin-spin
correlation function at h s 0. Here, we will take a
short cut with the purpose of illustrating a point
which is particularly relevant for more general applications. In integrable models the spectral sum Ž7. is
not a purely formal expression because the matrix
Ž j c . 2nd could be similarly estimated using the form
factors for the t s 0 model computed in w20x. However, its exact value is immediately determined reminding that s is the perturbing operator at t s 0.
Then the c-theorem sum rule holds w22x
3
2
cs
2p h Ž 2 y xs .
d 2 x < x < 2² s Ž x . s Ž 0 . :ch ,
4p
Ž 24 .
Table 1
Exact amplitude ratios for the two-dimensional Ising model. The
X
results for j 0 r j 0 , Rq
j and Q 2 refer to the true correlation length
where the central charge c equals 1r2 for the Ising
universality class. Combining with Ž19. one gets
H
X
A r A s1
G r G s 37.6936520 . . .
j 0 r j 0X s 2
R C s 0.318569391 . . .
X
'
Rq
j s1r 2
Rx s6.77828502 . . .
R A s 0.0469985240 . . .
Q2 s 3.23513834 . . .
(
8
D 1r30 s 0.21045990 . . . .
Ž 25 .
45p c
Table 2 contains the results one obtains for the
amplitudes ratios. The value Ž j 0rj 0X . 2nd s 3.16 . . . is
Ž j c . 2nd s
6
On-shell momenta
Ž mt cosh u ,mt sinh u ..
are
parameterised
as
pms
G. Delfinor Physics Letters B 419 (1998) 291–295
quoted in Ref. w23x as the result of the integration of
the exact correlator; Ž Q2 . 2nd s 2.88 " 0.02 is the
series result obtained in Ref. w19x.
References
w1x J.L. Cardy, Scaling and Renormalization in Statistical
Physics, Cambridge U.P., 1996, and references therein.
w2x V. Privman, P.C. Hohenberg, A. Aharony, Universal Critical-Point Amplitude Relations, in: C. Domb, J.L. Lebowitz
ŽEds.., Phase transition and critical phenomena, vol. 14,
Academic Press, 1991.
w3x A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Nucl.
Phys. B 241 Ž1984. 333.
w4x C. Itzykson, H. Saleur, J.B. Zuber ŽEds.., Conformal Invariance and Applications to Statistical Mechanics, World Scientific, Singapore, 1988, and references therein.
w5x A.B. Zamolodchikov, in: Advanced Studies in Pure Mathematics 19 Ž1989. 641; Int. J. Mod. Phys. A 3 Ž1988. 743.
w6x G. Mussardo, Phys. Reports 218 Ž1992. 215, and references
therein.
w7x Al.B. Zamolodchikov, Nucl. Phys. B 342 Ž1990. 695.
w8x M. Karowski, P. Weisz, Nucl. Phys. B 139 Ž1978. 445.
w9x F.A. Smirnov, Form Factors in Completely Integrable Mod-
w10x
w11x
w12x
w13x
w14x
w15x
w16x
w17x
w18x
w19x
w20x
w21x
w22x
w23x
295
els of Quantum Field Theory, World Scientific, 1992, and
references therein.
V.P. Yurov, Al.B. Zamolodchikov, Int. J. Mod. Phys. A 6
Ž1991. 3419.
J.L. Cardy, G. Mussardo, Nucl. Phys. B 340 Ž1990. 387.
J.L. Cardy, G. Mussardo, Nucl. Phys. B 410 wFSx Ž1993. 451.
J.L. Cardy, G. Delfino, in preparation.
A. Smilga, Phys. Rev. D 55 Ž1997. 443.
B.M. Mccoy, T.T. Wu, The two dimensional Ising model,
Harvard U.P., 1982, and references therein.
T.T. Wu, B.M. Mccoy, C.A. Tracy, E. Barouch, Phys. Rev.
B 13 Ž1976. 316.
Al.B. Zamolodchikov, Int. J. Mod. Phys. A 10 Ž1995. 1125;
V.A. Fateev, Phys. Lett. B 324 Ž1994. 45.
S. Lukyanov, A.B. Zamolodchikov, Nucl. Phys. B 493 Ž1997.
571; V.A. Fateev, S. Lukyanov, A.B. Zamolodchikov, Al.B.
Zamolodchikov, Phys. Lett. B 406 Ž1997. 83; hepthr9709034.
H.B. Tarko, M.E. Fisher, Phys. Rev. B 11 Ž1975. 1217.
G. Delfino, G. Mussardo, Nucl. Phys. B 455 Ž1995. 724; G.
Delfino, P. Simonetti, Phys. Lett. B 383 Ž1996. 450.
B. Berg, M. Karowski, P. Weisz, Phys. Rev. D 19 Ž1979.
2477.
A.B. Zamolodchikov, JETP Lett. 43 Ž1986. 730; J.L. Cardy,
Phys. Rev. Lett. 60 Ž1988. 2709.
N.C. Bartelt, T.L. Einstein, L.D. Roelofs, Phys. Rev. B 35
Ž1987. 1776.
12 February 1998
Physics Letters B 419 Ž1998. 296–302
Equivalence of the sine-Gordon and massive Thirring models
at finite temperature
D. Delepine,
R. Gonzalez
´
´ Felipe, J. Weyers
Institut de Physique Theorique,
UniÕersite´ catholique de LouÕain, B-1348 LouÕain-la-NeuÕe, Belgium
´
Received 11 September 1997; revised 5 November 1997
Editor: P.V. Landshoff
Abstract
Using the path-integral approach, the quantum massive Thirring and sine-Gordon models are proven to be equivalent at
finite temperature. This result is an extension of Coleman’s proof of the equivalence between both theories at zero
temperature. The usual identifications among the parameters of these models also remain valid at T / 0. q 1998 Elsevier
Science B.V.
1. Introduction
of the massive Thirring model in Euclidean Ž1 q 1.dimensional space-time is given by
In two dimensional quantum field theories
fermionic degrees of freedom can be expressed as
bosonic ones and vice versa. A remarkable illustration of this property is the equivalence between the
sine-Gordon and Žmassive. Thirring models w1x. The
sine-Gordon model is a Ž1 q 1.-dimensional field
theory of a single scalar field, whose Lagrangian
density in Euclidean space-time is defined classically
by
L SG s 12 Em wEm w y
a0
l2
2
L T s yi cEuc y 12 g 2 Ž cg mc . q imz cc ,
Ž 2.
where z is a cutoff-dependent constant and the gm
matrices are taken in the form:
g0 s
ž
0
1
1
,
0
/
g1 s
ž
0
yi
i
,
0
/
g 5 s ig 0 g 1 .
Ž 3.
Ž cos lw y 1 . ,
Ž 1.
where a 0 plays the role of a squared mass, l is a
coupling constant and the minimum energy is taken
to be zero. On the other hand, the Lagrangian density
The equivalence between the two models was first
derived by Coleman w1x following Klaiber’s work on
the massless Thirring model w2x: he showed that the
perturbation series in the mass parameter m of the
Thirring model is term-by-term identical with a per-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 3 6 - 6
D. Delepine
et al.r Physics Letters B 419 (1998) 296–302
´
turbation series in a 0 for the sine-Gordon model,
provided the following identifications are made w1,3x:
4p
l2
s1q
g2
p
,
Ž 4.
l
cgm c s i
2p
imz cc s y
emn En w ,
a0
l2
cos Ž lw . .
Ž 5.
297
In this letter we give an explicit proof in the path
integral approach w4x of the equivalence between the
massive Thirring and sine-Gordon models at finite
temperature in precisely the terms of Coleman’s
original argument at zero temperature. Namely, we
show that a perturbative expansion in the mass of the
Thirring model is term-by-term identical with a perturbation series in a 0 of the sine-Gordon model,
provided the identifications given in Eqs. Ž4. – Ž6. are
made.
Ž 6.
It is worth noticing that relation Ž4. expresses a
duality symmetry between the Thirring and sineGordon models, i.e. a correspondence that relates the
strong coupling regime in one theory to the weak
coupling one in the other.
Different approaches have been used to prove the
equivalence between the above models. Mandelstam
w3x rederived Coleman’s results by explicitly constructing the operators for creation and annihilation
of quantum sine-Gordon solitons. These operators
satisfy the anticommutation relations and field equations of the fermionic fields in the massive Thirring
model. Alternatively, in the path-integral framework,
Coleman’s proof can be rederived in a very simple
w4x way by making a chiral transformation in the
fermionic path-integral variables.
The results quoted above have been obtained in
the usual context of relativistic quantum field theory,
namely at zero temperature. An obvious question is
whether they remain valid at finite temperature T.
Although a positive answer might be expected from
general arguments Žbosonisation at finite T . contradictory results have been presented in the literature
w5–8x.
The point at issue is the following: on one hand it
is certainly true that finite T calculations are nothing
but a particular combination of zero T matrix elements and hence two theories equivalent at zero T
would trivially appear to remain so at all T ; but on
the other hand, finite T also implies a change in
topology as illustrated by the different boundary
conditions on fermionic or bosonic degrees of freedom and thus the equivalence between a fermionic
and a bosonic theory at finite T is not an obvious
corollary of their equivalence at T s 0.
2. The massive Thirring and sine-Gordon models
at finite temperature
In this section we shall evaluate the partition
functions of the massive Thirring and sine-Gordon
models using the imaginary time formalism and the
path integral approach at finite temperature w9x.
Let us first consider the massive Thirring model.
The Euclidean partition function reads
ZT s N0 Nb Dc Dc ey Hd
H
2
x
LT ,
Ž 7.
Ž
.
where Hd 2 x ' H0b dx 0 Hq`
y` dx 1 b s 1rT and L T is
given by Eq. Ž2.. N0 is an infinite temperature-independent normalization constant, whereas Nb is a
divergent temperature-dependent constant to be determined from the free partition function w9x. The
functional integral is performed over fermionic fields
satisfying antiperiodic boundary conditions in the
time direction,
c Ž x 0 , x 1 . s yc Ž x 0 q b , x 1 . .
Ž 8.
The first step of the evaluation is to introduce an
auxiliary two-component vector field w4x
Am s y
1
g
Ž emn En f y Emh . ,
Ž 9.
where f and h are two scalar fields satisfying the
periodic boundary conditions
f Ž x 0 , x1 . s f Ž x 0 q b , x1 . ,
h Ž x 0 , x1 . s h Ž x 0 q b , x1 . .
Ž 10 .
D. Delepine
et al.r Physics Letters B 419 (1998) 296–302
´
298
With the help of these fields, the quartic interaction
in L T can be eliminated,
exp
ž
g2
2
Hd x ž cg c /
2
m
2
ž
H
H ž
JA s det
1
2
Am2 y g c Auc
LT s yc i Euy gm Ž emn En f y Em h . c q imz cc
1
2g
2
Ž Em f .
2
2
q Ž Em h . .
Ž 11 .
We now perform the chiral transformation
c Ž x. se
,
g2
Ž 17 .
where = 2 s Em Em . Note that this bosonic determinant
is temperature-dependent and hence its contribution
is relevant to the partition function, in contrast to the
zero-temperature case where it plays no role and can
simply be absorbed in the normalization constant.
Finally we have
//,
and we obtain the effective Lagrangian
q
2
y=
/
s D Am exp y d 2 x
The bosonic Jacobian JA on the other hand is easily
evaluated and one finds
ZT s N0 Nb JA Dx Dx Df Dh ey Hd
H
x Ž x. ,
c Ž x. sx Ž x. e
,
Ž 12 .
L T s yx i Eux q imz x e 2g 5 f Ž x .x q
1
with x Ž x . satisfying the boundary condition
q
2
2g
x Ž x 0 , x 1 . s yx Ž x 0 q b , x 1 . ,
x
LT ,
Ž 18 .
with
g 5 f Ž x .qih Ž x .
g 5 f Ž x .yih Ž x .
2
Ž Emh .
1
2k
2
Ž Em f .
2
2
Ž 19 .
Ž 13 .
and
to obtain
LT s yx i Eux q imz x e 2g 5 f Ž x .x
1
q
2g
Ž Em f .
2
2
k s
2
q Ž Em h . .
Ž 14 .
To write the partition function in terms of the new
variables, the Jacobians of the transformations
Dc Dc s JF Dx Dx ,
D Am s JA Df Dh ,
g2
2
1 q g 2rp
.
Ž 20 .
The integration is performed over fields satisfying
the boundary conditions Ž10. and Ž13.. As can be
seen, the h field decouples completely and thus can
be trivially integrated out.
In order to show the equivalence between the
massive Thirring and the sine-Gordon models, let us
expand ZT in the mass parameter zm,
Ž 15 .
have to be properly taken into account. Due to the
anomaly and the fact that we perform a chiral transformation, the first Jacobian JF is not trivial. It has
been computed at finite temperature in Ref. w10x
following Fujikawa’s procedure w11x and the result is
ZT s N0 Nb JA1r2 Dx Dx Df ey Hd
H
`
=
Ž yizm .
Ý
2
x Žy x i Euxq
1
2k 2
Ž Em f . 2 .
n
n!
ns0
n
1
JF s ey 2 p Hd
2
x Ž Em f
.2
.
Ž 16 .
=Ł
js1
Hd
2
x j x Ž x j . e 2g 5 f Ž x j .x Ž x j . .
Ž 21 .
D. Delepine
et al.r Physics Letters B 419 (1998) 296–302
´
Or, equivalently,
ZT s Z FD det
k
`
g
ns0
Ž yizm .
ž /Ý
which, as required, is periodic in x 0 . Note that the
small mass m has been added to avoid infrared
divergences.
Similarly, the fermionic propagator can be defined at finite temperature as
n
n!
n
¦Ł H
=
;
d 2 x j x Ž x j . e 2g 5 f Ž x j .x Ž x j . ,
js1
Ž 22 .
SŽ x yy. s
where ² : denotes the thermal average over the unperturbed ensemble and Z FD is the Fermi-Dirac distribution for massless fermions w9x:
dk
`
ln Z FD s 2
H0
2p
bk
½
2
1
DŽ x y y. s
dk 1
nsy`
`
H0 dse
nsy`
`
H0 ds e
S0 Ž x . s y
,
Ž 24 .
cosh
1
b
cosh
b
cosh
Ý
(
K 0 m Ž x 0 y y 0 y n b . 2q Ž x 1 y y1 . 2 .
Ž
. Ž 25.
Q1 Ž x . s ysinh
When m ™ 0, we get from the last equation
DŽ x y y .
1
sy
2p
1
sy
2p
x 0p
sin
b
2 x 0p
,
Ž 28 .
.
Ž 29 .
ž / ž /
b
x 1p
y cos
b
x 0p
ž / ž /
b
2 x 1p
cos
b
2 x 0p
ž / ž /
b
y cos
b
SŽ x y y . is thus antiperiodic in x 0 with a period
equal to b , as it should be.
The scalar and fermion propagators can be rewritten in a more familiar way, using the following
dimensionless ‘‘generalized coordinates’’ Q '
Ž Q0 ,Q1 .,
x 1p
x 0p
ž / ž /
ž / ž /
Q 0 Ž x . s ycosh
nsy`
x 1p
ž / ž /
b
2 x 1p
sinh
1
yi kŽ xyy.
q`
2p
Ž 27 .
with k 0 s Ž2 n q 1.prb . The functions S0 Ž x . and
S1Ž x . can be expressed as:
DŽ x y y .
s
Ž k 0g 0 q k1g 1 .
Ž 23 .
with k 2 s k 02 q k 12 and k 0 s 2p nrb - the Matsubara
frequencies. Eq. Ž24. can be expressed in terms of
the modified Bessel function K 0 Ž z .,
1
yi kŽ xyy.
ys k 2
=
S1 Ž x . s y
ys Ž k 2 q m 2 .
=
dk 1
H 2p e
Ý
b
H 2p e
Ý
b
q`
1
' ig 0 S0 Ž x y y . q ig 1 S1 Ž x y y . ,
5
q ln Ž 1 q eyb k . .
At this stage, it is worth noticing that Eq. Ž22.
reproduces in the limit m s 0 the partition function
for the massless Thirring model at finite temperature
w12,13x.
To evaluate the thermal averages in Eq. Ž22. we
need the boson and fermion free propagators at finite
temperature. In the imaginary time formalism and
using the Schwinger representation w14x, the propagator for a scalar with mass m can be defined as
q`
299
b
x 1p
b
sin
cos
b
x 0p
b
,
.
Ž 30 .
and
q`
Ý
ž
(
ln m Ž x 0 y y 0 y n b . 2q Ž x 1 y y1 . 2
Ž
nsy`
(
ln mb
cosh
ž
2 x 1p
b
/ ž
ycos
2 x 0p
b
//
,
.
Q 2 Ž x . ' Q02 Ž x . q Q12 Ž x .
Ž 26 .
s cosh
2 x 1p
2 x 0p
ž / ž /
b
y cos
b
.
Ž 31 .
D. Delepine
et al.r Physics Letters B 419 (1998) 296–302
´
300
In terms of these generalized coordinates, DŽ x . and
SŽ x . are now given by
DŽ x . s y
SŽ x . s
1
ln Ž mb < Q Ž x . < . ,
2p
i Qu Ž x .
Ž 32 .
.
b Q2 Ž x .
Ž 33 .
Ž y1.
Introducing the composite operators
sqs x
1 q g5
2
sys x
x,
1 y g5
2
where r is a renormalization scale, QŽ x i . are the
generalized coordinates defined in Eqs. Ž30. and k is
given by Eq. Ž20.. The presence of the latter factor is
due to the kinetic term of the scalar Lagrangian Žcf.
Eq. Ž19...
The fermionic average is also easily evaluated, if
we recall the identity w15x
x,
nq 1
det
i-j
s
Ž 35 .
Q0 s
=² e
Ž yizm .
ž /Ý
g
n!2
ns0
y2 Ž f Ž x j .y f Ž y j .. q
2n
n
i
2
Ž sinhw y sinhw . ,
Ł Hd 2 x j d 2 y j
Q1 s y 12 Ž sinhw q sinhw . ,
y
Q 2 s sinhwsinhw,
js1
s Ž x j . s Ž yj . :.
Ž 36 .
To compute the bosonic thermal average, we use
Wick’s theorem and the well-known identity
Ž 40 .
with w s Ž x 1 q ix 0 . prb , and thus
¦
n
;
Ł sq Ž x j . sy Ž yj .
js1
T eyiHd
ž
2
x jŽ x . f Ž x .
s :ey iHd
2 x jŽ x . f Ž x .
n
/
Ł ž b 2 QŽ xi y x j .
1
:ey 2 Hd
2
x d 2 y jŽ x . D Ž xyy . jŽ y .
,
n
j
js1
Łž r b
s
Q Ž x i y x j . Q Ž yi y y j .
i)j
2k 2
n
Ł ž rb Q Ž x i y yj .
i, j
/
Ž 41 .
Substituting Eqs. Ž38. and Ž41. into ZT , we obtain
finally
/
`
ž /Ý
g
ns0
Ž zm .
n!2
2n
n
ž ŁH
d 2 x j d 2 yj
js1
2
Łžr b
,
=
2
Q Ž x i y x j . Q Ž yi y y j .
i)j
2k 2
n
p
Ł ž rb Q Ž x i y yj .
i, j
Ž 38 .
/
2k 2
n
p
y
.
/
i, j
y
2
2
/
2
Ł ž b Q Ž x i y yj .
2k 2
n
2
n
ZT s Z FD det
renorm .
Q Ž yi y y j .
n i)j
k
;
Ł ey2 Ž f Ž x .y f Ž y ..
j
s Ž y1 .
Ž 37 .
where T is the x 0-ordering chronological product
and : : denotes the normal product; DŽ x y y . is the
propagator of the f field and jŽ x ., any space-time
function. We have then
¦
Ž 39 .
which holds only for the analytic functions f Ž w . s w
and f Ž w . s sinhŽ a w .. In our case
Eq. Ž22. now reads
ZT s Z FD det
,
Ł f Ž wi y wjX .
i, j
x e 2g 5 fx s e 2 fsqq ey2 fsy,
`
f Ž wi y wjX .
Ł f Ž wi y wj . f Ž wiX y wjX .
Ž 34 .
and using the relation
k
1
2y
/
2y
/
p
.
p
Ž 42 .
D. Delepine
et al.r Physics Letters B 419 (1998) 296–302
´
To compare ZT with the sine-Gordon model, we
have to expand in a 0 the sine-Gordon partition
function
ZSG s N0 NbX Dw ey Hd
H
2
x
L SG
,
with M an arbitrary scale and z is an ultravioletcutoff-dependent coefficient.
Comparing ZT and ZSG , we see that the two
perturbation series are indeed identical provided the
relations
Ž 43 .
4p
where LSG is given in Eq. Ž1. and the integration
runs over scalar fields periodic in the time direction:
l2
za 0
w Ž x 0, x 1 . s w Ž x 0 q b , x 1 . .
Ž 44 .
`
Ý
ns0
n!
2n
a0
1
2
žl /
2
n
¦Ł H
=
;
Ž 45 .
5
Ž 46 .
e i lw Ž x j . eyi lw Ž y j . d 2 x j d 2 yj ,
js1
where
`
ln Z BE s y
H0
dk
2p
½
bk
2
q ln Ž 1 y eyb k .
l2
s1q
s zm,
Msr
This perturbative expansion in a 0 yields
ZSG s Z BE
301
is the Bose-Einstein distribution for massless bosons.
Comparing Eqs. Ž23. and Ž46. it is straightforward to
check that Z BE s CZ FD , where C is an irrelevant
Žinfinite. constant related to the zero-point energies.
Evaluating the bosonic thermal average with the
help of relation Ž37., we obtain
g2
p
,
Ž 48 .
Ž 49 .
Ž 50 .
are satisfied. The first relation is independent of the
renormalization scheme, while the last two equations
depend on it and hence have only a convention-dependent meaning. We also recover Coleman’s relations Ž5. and Ž6. between the two theories.
In conclusion, we have shown using the path
integral method that the compactification of the time
variable into a circle of radius b s 1rT preserves
the equivalence between the sine-Gordon and massive Thirring models in Coleman’s sense: at fixed
radius b Žor at fixed temperature., the perturbation
series in the mass parameter of the Thirring model is
term-by-term identical with a perturbation series in
a 0 for the sine-Gordon model, provided the identifications given in Eqs. Ž4. – Ž6. are made.
References
`
ZSG s Z BE
Ý
ns0
za 0
1
n!
2
2n
n
ž / ž ŁH
l2
d 2 x j d 2 yj
js1
/
l2
n
Ł ž b 2M 2 QŽ xi y x j .
=
Q Ž yi y y j .
i)j
l2
n
Ł ž M b Q Ž x i y yj .
i, j
/
/
2p
,
2p
Ž 47 .
w1x S. Coleman, Phys. Rev. D 11 Ž1975. 2088.
w2x B. Klaiber, in: A. Barut, W. Brittin ŽEds.., Lectures in
Theoretical Physics, Boulder, 1967, Gordon and Breach,
New York, 1968.
w3x S. Mandelstam, Phys. Rev. D 11 Ž1975. 3026.
w4x C.M. Naon,
´ Phys. Rev. D 31 Ž1985. 2035.
w5x D. Wolf, J. Zittartz, Z. Phys. B 51 Ž1983. 65.
w6x T.R. Klassen, E. Melzer, Int. J. Mod. Phys. A 8 Ž1993. 4131.
w7x H. Itoyama, P. Moxhay, Phys. Rev. Lett. 65 Ž1990. 2102.
w8x T. Fujita, H. Takahashi, Finite size corrections in massive
Thirring model, hep-thr9706061.
w9x C.W. Bernard, Phys. Rev. D 9 Ž1974. 3312.
302
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´
w10x M. Reuter, W. Dittrich, Phys. Rev. D 32 Ž1985. 513.
w11x K. Fujikawa, Phys. Rev. Lett. 42 Ž1979. 1195; Phys. Rev. D
21 Ž1980. 2848.
w12x F. Ruiz Ruiz, R.F. Alvarez-Estrada, Phys. Rev. D 35 Ž1987.
3161.
w13x M.V. Manıas,
C.M. Naon,
´
´ M.L. Trobo, Path-integral
fermion-boson decoupling at finite temperature, hepthr9701160.
w14x J. Schwinger, Phys. Rev. 82 Ž1951. 664.
w15x J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1996.
12 February 1998
Physics Letters B 419 Ž1998. 303–310
Antiferromagnetic Ož N / models in four dimensions
a
H.G. Ballesteros a , J.M. Carmona b, L.A. Fernandez
, A. Tarancon
´
´
a
b
1
Departamento de Fısica
Teorica
I, Facultad de CC. Fısicas
, UniÕersidad Complutense de Madrid, 28040 Madrid, Spain
´
´
´
b
Departamento de Fısica
Teorica,
Facultad de Ciencias 2 , UniÕersidad de Zaragoza, 50009 Zaragoza, Spain
´
´
Received 8 April 1997
Editor: R. Gatto
Abstract
We study the antiferromagnetic OŽ N . model in the F4 lattice. Monte Carlo simulations are applied for investigating the
behavior of the transition for N s 2,3. The numerical results show a first order nature but with a large correlation length. The
N ™ ` limit is also considered with analytical methods. q 1998 Elsevier Science B.V.
PACS: 05.70.Jk; 64.60.Fr; 75.40.Mg; 75.50.Ee
Keywords: Lattice; Monte Carlo; Antiferromagnetic; Phase transitions; OŽ N . models
1. Introduction
The antiferromagnetic formulations of field theories in four dimensions have been recently paid
considerable attention w1–5x. The hope is to give
some insight into the well known triviality problem
in field theory w6x. Also there are other interesting
phenomena as the apparition of new particles w5x.
A spin model, in a simple cubic lattice with first
neighbor interactions, becomes antiferromagnetic if
the coupling is negative. However, with some exceptions w4,7x, a simple staggered transformation maps
the antiferromagnetic phase into the usual ferromagnetic one.
1
2
e-mail addresses: hector, [email protected]
e-mail addresses: carmona, [email protected]
To obtain a non-equivalent antiferromagnetic
phase one has to include further couplings or modify
the lattice geometry Žsee for instance ref w8x..
Perhaps the simplest method to obtain non-trivial
antiferromagnetism in four dimensions is to work in
an F4 lattice. It is defined by taking out the odd sites
Žthe sum of the coordinates is odd. of a simple
hypercubic lattice.
Four dimensional antiferromagnetic OŽ N . models
have been already studied by Monte CarloŽMC.
means in this lattice. The OŽ1. model ŽIsing model.
was considered in Ref. w2x: a weak first order transition was found. A study of the OŽ4. model appears
in Ref. w3x: in the range of the lattice sizes simulated,
the behavior pointed to a second order transition.
In this letter we consider the intermediate cases:
OŽ2. and OŽ3., to know if the order of the transition
changes with N. We will give evidence that the
transitions are in both cases first order, but the
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 2 9 2 - 6
H.G. Ballesteros et al.r Physics Letters B 419 (1998) 303–310
304
numerical difficulties grow with N. We also present
an analytical study in the N ™ ` limit.
tary cells, m i , defined as the normalized sum of the
magnetization for all 2 4 cells for each of the 8 sites.
m I s m X Y ZT s
2. The model
We label the coordinates of an F4 lattice as a set
of integers x, y, z,t 4 such that x q y q z q t is even.
We consider the action
S s b H s yb
Ý
Fi P Fj ,
Ž 1.
²i , j:
1
V
MAx F H s
1
...
¦( ;
1
8
Ms
Ý m 2I
,
xs
Is1
V
8
8
¦ ;
Ý m 2I
.
Ž 4.
Is1
The Binder Cumulants are defined in a such way
that VM bs0 s 0 and VM bs` s 1:
VMOŽ N . s
Nq2
2
¦
8
;
¦Ý ;
Ý Ž m 2I .
8N
2
Is1
1y
2
8
m 2I
Ž N q 2.
Is1
0
.
Ž 5.
For the connected susceptibility we use the definition
8
xc s V
ž
1
8
Ý ² m 2I :y M 2
Is1
/
.
Ž 6.
Ý
3. Critical behavior
Fx y z t Ž y1 . ,
For an operator O that diverges as Ž b y bc .yx O ,
its mean value at a coupling b in a size L lattice can
be written, in the critical region, assuming the
finite-size scaling ansatz as w9x
x , y , z ,t
1
V
The quantities Ž2. can be expressed as linear
combinations of these magnetizations.
The mean magnetization and the susceptibility are
defined respectively as
x
...
MAxyy
FP s
Ž 3.
x , y , z ,t
xyX , . . . even
Fx y z t ,
Ý
x , y , z ,t
V
Fx y z t .
Ý
V
8
where the sum runs over 24 pairs of nearest neighbors and the field, F, is a normalized N component
real vector. We work in a hypercubic lattice of size
V s L4r2 with periodic boundary conditions.
In the ferromagnetic region Ž b ) 0. this model is
expected to belong to the same universality class of
the simple cubic model: it presents a second order
transition with mean field critical exponents.
In the antiferromagnetic sector Ž b - 0. the system also presents an ordering phase transition, but
the structure of the ordered phase is much more
complex. In fact, it can be easily checked that the
ground state presents frustration. Moreover, the ordered vacuum is not isotropic.
The independent order parameters we can construct with periodicity 2 are
MF s
8
Ý
Fx y z t Ž y1 .
xq y
,
O Ž L, b . s Lx O r n Ž FO Ž j Ž L, b . rL . q O Ž Lyv . . ,
x , y , z ,t
Ž 7.
Ž 2.
where the sums are extended to all the F4 sites. The
dots stand for the other 3 combinations of hyperplanes and 2 of planes.
We label the site I s 1, . . . ,8 inside the 2 4 elementary cell with its Cartesian coordinates X,Y,Z,T
s 0,1. In practice we measure the 8 different magnetizations associated to a given position in the elemen-
where FO is a smooth scaling function and v is the
universal leading corrections-to-scaling exponent. In
order to eliminate the unknown FO function we can
measure at the coupling where FO presents a maximum as for xc or for the specific heat C V . Another
method w7x is to study the behavior of the quantities
Q O s O Ž sL, b . rO Ž L, b . .
Ž 8.
H.G. Ballesteros et al.r Physics Letters B 419 (1998) 303–310
We use VM L as scaling variable w7x. It is direct to
obtain
305
We use that xx s g and xEb log Ž M . s 1 to obtain the
critical exponents.
The expected FSS behavior of a first-order transition w10x corresponds to apparent exponents: n s
1rd, a s 1, g s 1.
In order to extrapolate to the neighborhood of the
critical point, we used the usual reweighting method
w11x. We also simulated at the maxima of the specific
heat because the region where the extrapolation was
reliable was not large enough. These simulations
were about one fourth of the total CPU time.
The errors were computed with a jack-knife
method, performing 50 blocks in order to achieve
statistical error bars within 10% of precision.
4. The simulation
5. The vacuum
We will consider in this letter the cases N s 2,3.
We have used a Metropolis algorithm followed by
No overrelaxation steps as update method, No depending on the model and lattice size. We have
worked in lattice sizes up to 48. We used for the
computation the dedicated machine RTNN, consisting in 16 Dual Pentium Pro units. For the largest
lattices we parallelized, using shared memory, in
each dual motherboard. Every f sweeps we measured the energy, the specific heat and the 8 periodtwo magnetizations m I . In Table 1 we report the
parameters of the simulation at the critical region.
For large < b <, all the magnetizations m I go to
unitary vectors, confirming the assumption of a period 2 vacuum, so that we can restrict our analysis
only to a unit cell. We also find that MA F P is non
zero, while M F and MA F H vanish.
When M F s MA F H s 0, it follows that m X Y ZT s
m1y X ,1yY ,1yZ,1yT . We checked that the cosine between m X Y ZT and m1yX ,1yY ,1yZ,1yT goes to 1, with
the expected Ly4 behavior in the broken phase
corresponds to AFP order.
This ordering is also found near the transition. So
we can restrict ourselves to study only 4 independent
spins.
At T s 0 the frustrated ground state is described
in w3x; it consists in two couples of antiparallel spins,
but the angle between couples is free. The different
choices for the couples determine the plane for AFP
symmetry breaking.
At T ) 0 we do not know whether there is a
privileged angle or not. It is necessary to determine
the pattern of the symmetry breaking.
A very important point is to determine if the two
couples are aligned or not. To clarify this point we
construct the following tensor:
QO
Q V M L ss s s
x O rn
q O Ž Ly v . .
Ž 9.
Table 1
Description of the simulation in the critical region for OŽ2. Župper
part. and OŽ3. Žlower part.. We report the lattice size, the frequency of measures, number of over-relaxation steps for each
Metropolis one, autocorrelation time Žt . for x , iterations in t
units, and the coupling.
L
4
6
8
12
16
24
32
48
f
No
tŽ x.
20
20
20
20
20
25
24
32
3
3
3
3
3
4
7
7
0.775Ž9.
1.144Ž14.
1.63Ž4.
3.16Ž6.
6.5Ž5.
15.8Ž7.
32Ž1.
85Ž7.
a of t
100000
87000
55000
60000
11400
5000
1700
312
b
y0.352
y0.352
y0.352
y0.352
y0.352
y0.3516
y0.3513
y0.35125
4
T a b s 14
Ý
m Ka = m Kb ,
Ž 10 .
Ks1
4
6
8
12
16
24
32
48
20
20
20
20
20
24
28
28
3
3
3
3
3
5
6
6
0.871Ž11.
1.19Ž2.
1.52Ž3.
2.07Ž4.
3.15Ž6.
4.99Ž14.
8.7Ž4.
21Ž1.
98000
82000
98700
63000
48000
29000
5000
2000
y0.53
y0.53
y0.53
y0.53
y0.53
y0.5287
y0.5287
y0.5286
where the superindices run for the components of the
N vectors and the sum over the four independent
magnetizations in the elementary cell. It is clear that
if the two couples are aligned, the four tensors as
well as the sum tensor can be simultaneously diagonalized, and so we expect in the broken phase a
non-zero value for the largest eigenvalue and zero
H.G. Ballesteros et al.r Physics Letters B 419 (1998) 303–310
306
Table 2
Eigenvalues for the tensor Ž10. for the OŽ2. model Župper part. at
b sy0.37 and for the OŽ3. model Žlower part. at b sy0.57.
Eigenvalue
x 2 rd.o.f.
Value
lm a x
lm i n
0.16
0.13
0.20346Ž15.
y0.0Ž2.2.=10y6
lm a x
lm e d
lm i n
0.11
0.89
0.05
0.18986Ž7.
2.3Ž2.7.=10y7
y1.7Ž1.2.=10y6
values Žup to Ly4 effects. for the rest Ž N y 1. of
them.
If this holds true, we also expect in the critical
region a Ly2 b r n behavior for the biggest one and a
Ly4 for the others. This will be checked in the next
section.
In Table 2 we show the eigenvalues for both
models in the broken phase. We note that the largest
one goes to a non-zero value while the rest go to
zero. The fit parameters are obtained with a linear
extrapolation in Ly2 for the former case and in Ly4
for the latter.
6. Critical exponents
A determination of the critical exponents is obtained by studying the height of the peaks of C V and
xc . For a first order transition both quantities should
diverge as the volume Ž arn s 4,grn s 4.. For small
lattices it is usual to find the apparent critical exponents of a weak first order transition: arn s 1,grn
s 1 Žsee Ref. w12x., which are precursors of a first
order transition.
To analyze the divergence of C V a bilogarithmic
plot is not adequate due to the presence of a nonnegligible constant term. In order to compare with
the first order behavior it is better to plot C V and xc
as a function of several powers of L. This is done in
Fig. 1. Specific heat and connected susceptibility for OŽ2. and OŽ3. as a function of several powers of the lattice sizes.
H.G. Ballesteros et al.r Physics Letters B 419 (1998) 303–310
Fig. 1. We remark that in the OŽ2. case for L in the
interval w8,24x Žw12,32x for OŽ3.. there is an excellent
linear fit for n s 1 which is the value predicted in a
weak first order transition. This frequently produces
a misunderstanding of the order of the transition.
However, it is clear from Fig. 1 that this is a
transient effect: for the larger lattices the divergences
are faster than linear, and presumably they would
reach the first order behavior for very large lattices.
The susceptibility also shows a fast divergence.
Although we are not able to observe the asymptotic
first order behavior the trend seems rather clear.
A more accurate measure of the critical exponents
can be obtained from Eq. Ž9.. We have always used
the ratio s s 2. In Fig. 2 we plot several determinations of exponents using different operators. We
remark that there is a systematic error in the arn
determination because of the analytic term in C V .
307
We observe no asymptotic behavior in all cases
although the values in the larger lattices are hardly
compatible with a second order transition.
In the upper right part of Fig. 2 we plot the
exponents related with each of the eigenvalues of the
matrix Ž10.. While the maximum eigenvalues should
behave as Ly2 b r n Žwith brn s 0 for a first order
transition., the others should go to zero as Lyd . The
latter can be used as a control of when the asymptotic regime is reached. We observe that we are far
from this regime but the first order limit seems rather
clear.
A comparison between the curves for OŽ2. and
OŽ3. shows a roughly similar shape, differing in a
horizontal shift that corresponds to multiplying the
lattice size by a factor near 2. This fact can be
understood as a correlation length at the critical point
which is twice larger for OŽ3. than for OŽ2..
Fig. 2. Critical exponents for OŽ2. Žsolid lines. and OŽ3. Ždashed ones. measured through the relation Ž9.. The filled circles Ždiamond for the
non-maximum eigenvalues. mark the first order limit.
H.G. Ballesteros et al.r Physics Letters B 419 (1998) 303–310
308
7. Critical point and energy histograms
To calculate an estimation for the critical coupling
we study the bL values where VM Ž2 L, bL . s
VM Ž L, bL .. In both models, these bL for the largest
lattices are compatible within the error bars. We can
fit to the functional form bcL y bc Ž`. A Lyx in order
to estimate the error bars. We perform these fits with
the full covariance matrix. In both cases we obtain
fitting for L G 6 a wide valid range for x and very
good x 2 Ž x 2 rd.o.f.s 1.9r2 for O Ž2. and
x 2rd.o.f.s 0.8r2 for OŽ3... The results are compatible with the values for the largest lattices. We get
bc Ž ` .
O Ž2 .
bc Ž ` .
O Ž3 .
s y0.351216 Ž 10 .
s y0.52857 Ž 2 . .
Ž 11 .
point. In Fig. 3 we show the energy histograms for
both models at bc . In the OŽ2. case we note that the
width of the energy distribution is nearly constant for
the larger lattices, being an indication of the existence of a two peak distribution that cannot be
resolved. In the OŽ3. case, up to L s 48 there is not
a similar behavior.
8. The N ™ ` limit
The partition function of the model can be written
as
Z s Z0
Ž 12 .
Finally let us comment on the energy distribution
of the configurations. A direct check of the first
order character of a transition is the observation of a
latent heat. Unfortunately, a sharp double peak structure can be observed only when the lattice size is
much larger than the correlation length at the critical
s Z0
N
H Łj d F d ŽF
j
N daj
N
H Łj d F
2
j y1
j
2p
X
. eyN b H
2
e N Ý Ž i a jq l j .Ž 1y F j .y b
j
X
H4
,
Ž 13 .
where b X s brN, and Z0 is a normalization factor
such that Z ™ 1 when b ™ 0. We have introduced
Fig. 3. Normalized energy histogram at bc Ž`. for both models.
H.G. Ballesteros et al.r Physics Letters B 419 (1998) 303–310
the conjugate parameters a j , l j to give an integral
representation of the constraint F j2 s 1 w13x.
Writing the quadratic form in the exponent of Eq.
Ž13. as
y1
Ý fn Q n m fm ,
2
Ž 14 .
n, m
the integration over F yields
Table 3
Critical couplings divided by N for different OŽ N . models. We
include also the obtained in Refs. w2x and w3x.
N
1
2
3
4
bc
y0.17459Ž15.
y0.175608Ž5.
y0.17619Ž1.
y0.1766Ž1.
1r2 Ž Ny2 .V
2p
ž /
Z s Z0
N
H Łj d a
=
j
e N r2 Ý 2Ž l jqi a j .yTrln Q 4 .
j
Ž 15 .
In the limit N ™ `, a variational equation with
respect to 2Ž l k q i a k . gives
1 s Ž Qy1 . i i .
Ž 16 .
In order to study the disorder-AF transition in the
F4 lattice Ž b - 0., we perform a change of variables
which transforms the plane-AF vacuum Žsuppose
x y y . into a ferromagnetic one defining
FxX y z t s
Ž y1.
xq y
Fx y z t .
Ž 17 .
Q matrix changes and then the propagator, Qy1 ,
reads
GŽ p. s
1
1
X
y2
Ž yb . j q 4 Ž 2 q g Ž p . .
,
Ž 18 .
where
g Ž p.
s cos p x cos p y q cos pz cos pt y cos p x cos pz
ycos p x cos pt y cos p y cos pz y cos p y cos pt ,
Ž 19 .
and j is defined from Žtranslationally invariant.
auxiliary fields as
2 Ž l i q i a i . ' Ž yb X . Ž jy2 q 8 . .
Ž 20 .
From the variational Eq. Ž16. we obtain bcX imposing
j s `:
bcX s
309
d4 p
H Ž 2p .
1
4
s y0.178972.
y8 y 4 g Ž p .
when N ™ ` points to the absence of an abrupt
change of the critical properties as a function of N.
Exactly at N s ` the order of the transition is not
clear, because the divergence in the correlation length
can simply be caused by the Goldstone bosons of the
symmetry breaking.
9. Conclusions
In this letter we present a MC study of the four
dimensional antiferromagnetic OŽ2. and OŽ3. models
in the F4 lattice. We study the critical behavior of
these models with FSS techniques. There is an apparent asymptotic behavior which gives false critical
exponents for not large enough lattice sizes. This
transitory effect can be understood as caused by a
large correlation length whose presence can be
demonstrated for some observables Žas the eigenvalues of the sum tensor of the period-two magnetizations.. This must be very carefully controlled, because as we see in our case, the behavior changes
drastically when larger lattice sizes are considered,
revealing the true first order nature of the OŽ2., OŽ3.
transitions. We also see that this effect is bigger as N
grows, so that for larger values of N, it is very
difficult to study numerically the critical properties
of the system. However, the great accuracy in the
determination of the critical point obtained by the
analytical calculation at N s ` points to a similar
qualitative behavior for all values of N.
Ž 21 .
In Table 3 we compare Ž21. with MC results for
N s 1,2,3,4. The good agreement between the simulations for these values of N and the analytical limit
10. Acknowledgments
We thank to the CICyT Žcontracts AEN94-0218,
AEN96-1634. for partial financial support. We have
310
H.G. Ballesteros et al.r Physics Letters B 419 (1998) 303–310
employed for the simulations dedicated Pentium Pro
machines ŽRTNN project.. J.M. Carmona is a Spanish MEC fellow.
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w11x A.M. Ferrenberg, R.H. Swendsen, Phys. Rev. Lett. 61 Ž1988.
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12 February 1998
Physics Letters B 419 Ž1998. 311–316
Non-perturbative renormalization constants on the lattice
from flavour non-singlet Ward identities
G.M. de Divitiis a , R. Petronzio
a
b,1
Department of Physics and Astronomy, UniÕersity of Southampton, Southampton SO17 1BJ, UK
b
CERN, Theory DiÕision, CH-1211 GeneÕe
` 23, Switzerland
Received 23 October 1997
Editor: R. Gatto
Abstract
By imposing axial and vector Ward identities for flavour-non-singlet currents, we estimate in the quenched approximation the non-perturbative values of combinations of improvement coefficients, which appear in the expansion around the
massless case of the renormalization constants of axial, pseudoscalar, vector, scalar non-singlet currents and of the
renormalized mass. These coefficients are relevant for the completion of the improvement programme to O Ž a. of such
operators. The simulations are performed with a clover Wilson action non-perturbatively improved. q 1998 Elsevier Science
B.V.
1. Introduction
The programme of the improvement w1x of the
Wilson action has been actively developed at the
non-perturbative level over the last years w2–5x. At
first, in the framework of the Schrodinger
functional
¨
it was possible to determine non perturbatively the
dependence upon the bare coupling constant of the
coefficient c SW w3x of the clover term in the improved action w6x.
Besides improving the action, the programme includes the improvement of the operators appearing in
the correlation functions related to phenomenological
interesting quantities such as pseudoscalar meson
1
Also at: Dipartimento di Fisica, Universita` di Roma Tor
Vergata and INFN, Sezione di Roma II, Via della Ricerca Scientifica 1, 00133 Rome, Italy.
decay constants and the matrix element of the fourfermion operators of the weak effective Hamiltonian.
In general the operator improvement consists of
two parts: the mixing with higher-dimensional operators with the same quantum numbers Žin the literature the mixing coefficients are called c . and the
multiplication by a suitable renormalization constant.
The ultraviolet-finite renormalization constants can
be expanded around the massless case:
Z O Ž m / 0 . s Z O P Ž 1 q bO ma q ... . ,
Ž 1.
where we have omitted corrections due to lattice
artefacts of order a 2 and higher.
Some of these quantities have been calculated at
the perturbative level for axial Ž A., vector Ž V .,
pseudoscalar Ž P . and scalarŽ S . currents as well as
for the renormalized mass Ž m. w7–9x: non perturbative estimates are available for ZA , Z V , b V w5x and
for c A w3x and c V w10x.
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 4 4 - 5
G.M. de DiÕitiis, R. Petronzior Physics Letters B 419 (1998) 311–316
312
In this letter, we present a non-perturbative determination of the quantities:
bA y bP ,
b V y bS ,
bm
Ž 2.
m jk s m Rjk
and of the ratios:
Zm ZPrZA ,
The quantities m jk and D m jk are related respectively to the average and the difference of the renormalized mass, according to
Zm ZSrZ V
Ž 3.
Z P 1 q bP m q a
ZA 1 q bA m q a
ZS 1 q bS m q a
from a set of axial and vector Ward identities.
D m jk s D m Rjk
2. The method
m Rjk s
The extraction of the b coefficients and of the Z
ratios is based on the following Ward identities:
where m q a is the average of the bare masses j and
k:
Em² AmI Ž x . V † Ž 0 . : s 2 m jk² P Ž x . V † Ž 0 . : q O Ž a2 .
m q as
Em² VmI Ž x . V † Ž 0 . : s D m jk² S Ž x . V † Ž 0 . : q O Ž a2 . ,
Ž 4.
am q j s
which, after x integration, become:
E t² A 0I Ž t . V † Ž 0 . : s 2 m jk² P Ž t . V † Ž 0 . : q O Ž a2 .
E t² V0I Ž t . V † Ž 0 . : s D m jk² S Ž t . V † Ž 0 . : q O Ž a2 . .
Ž 5.
The suffix I for the axial current indicates that the
current is improved by the appropriate mixing with
the pseudoscalar density multiplied by the coefficient
c A . The value for c A is taken from its non-perturbative determination in Ref. w3x. For the vector current,
the contribution of the mixing with the tensor current
Žsee Ref. w10x for non-perturbative determination of
the mixing coefficient c V . vanishes because of the
antisymmetry of tensor indices.
The indices j,k refer to the flavour content of the
bilinear operators, which can be written as
O Ž t . s Ý O Ž x . s Ý c j Ž x . GO c k Ž x . .
x
1
Ž m Rj q m kR . ,
2
1
2
ž
Eqs. Ž4. hold for any operator V at any time
different from t, reflecting the fact that the W.I. are
identities among operators. For the axial W.I. we use
V s P Ž0. and V s A 0 Ž0. while for the vector W.I.
we use V s SŽ0. and V s V0 Ž0.; in both cases the
latter operator leads to much noisier results.
D m Rjk s m Rj y m kR ,
Ž m q j a q m qka.
1
2kj
1
y
/
2 kc
Ž 8.
with k c the critical value of the Wilson hopping
parameter. The parameter k c is determined from the
chiral extrapolation of the mass defined through the
axial Ward identities themselves.
The renormalization constants can be determined
by replacing the ‘‘unrenormalized current masses’’
m jk and D m jk with the renormalized ones through
the above equation, and then the renormalized masses
in terms of the bare masses:
m R s Zm m q Ž 1 q bm m q a . .
Ž 9.
Indeed, by including the lattice artefacts up to O Ž a.:
m jk a s
ZP Zm
ZA
ž m ay Ž b y b . Žm a.
q
qbm Ž m q a .
Ž 6.
x
Ž 7.
ZV 1 q b Vm q a
D m jk a s
ZS Z m
ZV
A
2
P
2
q
/
Ž m q j a y m qka.
= Ž 1 q Ž 2 bm y Ž b V y bS . . m q a .
Ž 10 .
From a fit of the bare mass dependence of these
results we can determine non-perturbatively the combinations in Eqs. Ž2,3..
G.M. de DiÕitiis, R. Petronzior Physics Letters B 419 (1998) 311–316
Within the approximation used in the above formulae, we can derive an expression for Ž bA y bP .:
m jk a y
s
Ž m j j a q m k k a.
2
ZP Zm 1
ZA
4
2
Ž bA y bP . Ž m q i y m q j . a2
Ž 11 .
which can be used to determine a value for the
combination independently from the knowledge of
the critical value of k . We have compared such a
determination of bA y bP with the one coming from
a global fit to expression Eq. Ž10. and used it as a
sensitive check of our estimate of k c .
The presence of flavour-non-diagonal currents is
important for the fit of the axial W.I. and essential
for the vector ones. We want to point out that m jk a
2
depends either upon Žm q a. 2 or upon Ž m q a . , which
are different variables if the flavours are not degenerate, making then possible to disentangle the coefficient bm from the combination bA y bP .
Our fits are to the dependence upon the quark
mass, and order a2 corrections linear in the quark
mass can in general fake the extraction of the coefficients. The simple lattice discretization of the time
derivative 21a Ž f Ž t q a. y f Ž t y a.. has an error of the
order of f Ž3. a 2 , which becomes f Ž1. M 2 a 2 when a
single state of mass M dominates the correlation
function f. In the pseudoscalar channel, chiral symmetry makes this term proportional to the quark
mass. In other channels, the mass M acquires a
non-zero value when the quark masses vanish. The
quantity Ž Ma. 2 still contains a term that is linear in
the quark mass but not in the lattice spacing. In order
to minimize such effects we have used the lattice
discretization of the time derivative aE t correct up to
term f Ž5. a 5. While for the pseudoscalar case with
spontaneous symmetry breaking the improvement to
the fifth order of the derivative removes these fake
linear terms in the quark mass, for the vector current
a linear term survives at any finite order n, although
suppressed by a coefficient of order 1rŽ n y 1.!. We
have checked that improving the derivative to the
next order does not change our results beyond their
accuracy.
We cannot exclude the presence of other lattice
artefacts in the ratio of matrix elements formally of
order a 2 but linear in the quark mass. We have
313
checked in the case of the axial current that the
results are stable with respect to the choice of the
operators and we interpret this as a sign for the
absence of large extra terms of order m L a 2 .
This method of computing the combinations in
Eqs. Ž2,3. is valid in the quenched approximation.
The presence of dynamical quark loops would introduce an additional sea quark mass dependence, which
would involve flavours different from those in the
currents.
3. The numerical results
We have performed several simulations at different values of b , in order to derive the coupling
constant dependence of the quantities in Eqs. Ž2,3..
The values of b used in the simulations and the
corresponding volumes are collected in Table 1, with
a list of the values of k and of our best estimate of
the critical k obtained from the W.I. themselves.
The values of k c are well compatible within errors
with those of Ref. w3x. The variation of our results
under a change of k c within the quoted error is
smaller than the accuracy by which we can extract
the non-perturbative quantities from our fit.
Simulations are performed with an updating sequence made by a standard heat-bath followed by 3
over-relaxation steps. The improved fermion propagator is calculated every 1000 gauge update using a
Table 1
The values of k used in our simulations at various b
L3 T
b
a confs
16 3 48
6.2
50
16 3 32
6.8
50
16 3 32
8.0
80
16 3 32
12.0
80
k
0.124
0.1275
0.1295
0.132
0.13326
0.13362
0.134
0.1345
0.135
0.13535
0.124967
0.127517
0.128831
0.131198
0.132589
0.132942
0.133296
0.133796
0.134371
0.134660
0.129382
0.130055
0.130736
0.131078
0.131423
0.131700
0.131908
0.132222
0.132467
0.132749
0.126299
0.126941
0.127589
0.127915
0.128243
0.128507
0.128705
0.129004
0.129237
0.129505
kc
0.13578Ž2.
0.13511Ž1.
0.13318Ž1.
0.129915Ž8.
G.M. de DiÕitiis, R. Petronzior Physics Letters B 419 (1998) 311–316
314
stabilized biconjugate algorithm. For our runs we
have used the 25 Gflops machine of the APE series,
made of 512 nodes working in SIMD ŽSingle Instruction Multiple Data. mode.
Each propagator was summed over the space
volume distributed to the single node Ž3 = 3 = 2.
and stored on disk. The use of these propagators
leads to correlation functions that contain the correct
local-gauge-invariant terms and other non-local,
gauge-non-invariant terms that go to zero after summing over the gauge configurations. We have explicitly checked that with our statistics the residual noise
due to imperfect cancellation of the gauge-non-invariant terms is much below the statistical fluctuations. Storing fermion propagators allows for an
off-line calculation of all flavour-non-singlet correlations.
The W.I. are satisfied separately at each time:
after some initial time, up to which higher-order
lattice artefacts still dominate, m jk and D m jk show a
plateau. At b s 6.2 we run two temporal extensions
Ž32 and 48. in order to monitor the stability of the
plateau. We have used two methods of analysis:
either we first average the result over the time
interval of the plateau and then perform a fit, or we
first perform a fit at each time value inside the
plateau and then average the results of the fit at
different times. The two procedures give very similar
results.
The choice of the spectator operator V affects the
statistical error of the final results. We have found
that the pseudoscalar and the scalar densities give the
best results for the axial and the vector case respectively.
We perform a fit of 2 m jk with the function Žhere
all masses are in lattice spacing units.:
a1 Ž m q i q m q j . q a 2 m2q i q m2q j q a3 Ž m q i q m q j .
ž
/
2
q a 4 m3q i q m3q j q a5 m q i m q jŽ m q i q m q j .
ž
/
The first three coefficients of the fit are related to the
renormalization constants as follows: a1 s Zm ZPrZA ;
a 2ra1 s bm ; a3ra1 s yŽ bA y bP .r2. The last two
coefficients in the fit can be introduced to parametrize
order-a2 corrections compatible with the flavour exchange symmetry of the Ward identity.
For the quantity D m jk we perform the fit with:
Õ 1 Ž m q i y m q j . q Õ 2 m 2q i y m2q j q Õ 3 m3q i y m3q j
ž
/
ž
/
q Õ4 m q i m q jŽ m q i y m q j .
where Õ 1 s Zm ZSrZ V ; Õ 2rÕ1 s bm y Ž b V y bP .r2.
As before, the extra coefficients Õ 3 and Õ4
parametrize the possible order-a2 corrections.
Our results normally refer to the fit with three
parameters for the axial case and two for the vector
case. Increasing the number of parameters in general
does not improve the value of x 2 much while it
considerably increases the error. The results are compatible with the lower parameter fit, with the exception of the determination of b V , which comes systematically higher with the four-parameter fit. We
have included this effect in the corresponding error.
Table 2 contains the main results, the values for
the various renormalization parameters at different b
and volumes. The b s 6.2 results on the smaller
temporal extension are fully compatible.
With the values of the fit we can check if the
renormalized W.I. depend only upon the sum Žfor the
axial. or the difference Žfor the vector. of the renormalized masses.
The renormalized masses and currents manage to
bring on the same straight line points that appear
misaligned and on a curved line for the bare quantities. For large values of the masses, higher-order
terms enter the game and produce again a misalignment of the corresponding points.
The fits in general do not include all k values;
we exclude the heavier masses until the stability of
the results is reached.
Our results, when compared with those available
from perturbation theory, show that higher-order
Table 2
The results of our calculations
L3 T
b
a confs
16 3 48
6.2
50
16 3 32
6.8
50
16 3 32
8.0
80
16 3 32
12.0
80
Zm ZP rZA
1.09Ž1.
1.08Ž1.
1.08Ž1. 1.060Ž6.
Zm ZS rZ V
1.24Ž2.
1.21Ž1. 1.142Ž4. 1.080Ž5.
bA y bP
0.15Ž2.
0.10Ž2.
0.06Ž2.
0.04Ž2.
bm y0.62Ž3. y0.58Ž3. y0.57Ž3. y0.53Ž2.
bm yŽ b V y bS .r2 y0.69Ž4. y0.63Ž4. y0.54Ž3. y0.52Ž2.
G.M. de DiÕitiis, R. Petronzior Physics Letters B 419 (1998) 311–316
Fig. 1. The non-perturbative result for bA y bP . The perturbative
result of O Ž g 2 . is negligible on this scale.
315
Fig. 3. The non-perturbative result for b V is compared, after using
Luscher’s
relation, with the one of Ref. w6x Ždotted curve. and with
¨
the perturbative result Ždashed curve..
terms seem to dominate for the differences bA y bP ,
which is very small at one-loop order Žsee Fig. 1.,
while for bm the presence of sizeable terms of order
g 2 makes the effect of g 4 terms less prominent.
Indeed, our results are not far from lowest-order
perturbation theory for this case Žsee Fig. 2..
For b V y bS q 2 bm , there is an argument due to
w11x relating bm with bS and ZS with
Martin Luscher
¨
Zm in the quenched approximation: 2 bm y bS s 0
and ZS Zm s 1, which implies that from the W.I. for
the vector current we actually obtain b V and Z V .
Fig. 2. The non-perturbative estimate of bm is compared with the
perturbative result.
Fig. 4. The non-perturbative result for Z V is compared, after using
Luscher’s
relation, with the one of Ref. w6x.
¨
316
G.M. de DiÕitiis, R. Petronzior Physics Letters B 419 (1998) 311–316
The comparison with those of Ref. w5x is shown in
Figs. 3 and 4: while for Z V there is a perfect
agreement, for b V we are generally closer to the
perturbative result. Our large errors are mainly systematic and reflect the instability of a four-parameter
fit. Running at lower quark masses could reduce the
discrepancy which might also be due to residual
order-a2 lattice artefacts.
The use of axial and vector Ward identities with
flavour-non-singlet currents allows the determination
in the quenched approximation of various non-perturbative renormalization constants. The calculation
that we have presented could be refined by using the
Schrodinger
functional method which would allow a
¨
safe investigation of the very low quark mass region.
Acknowledgements
We thank M. Masetti for his collaboration to the
first part of this work. The program for the inversion
of the fermion propagator with the biconjugate gradient algorithm was written for the APE machine in
Tor Vergata by A. Cucchieri and T. Mendes. We
thank M. Luscher
for various interesting remarks and
¨
for stimulating discussions.
References
w1x K. Symanzik, Nucl. Phys. B 226 Ž1983. 187, 205
w2x M. Luscher,
S. Sint, R. Sommer, P. Weisz, Nucl. Phys. B
¨
478 Ž1996. 365
w3x M. Luscher,
S. Sint, R. Sommer, P. Weisz, U. Wolff, Nucl.
¨
Phys. B 491 Ž1997. 323.
w4x K. Jansen, C. Liu, M. Luscher,
H. Simma, S. Sint, R.
¨
Sommer, P. Weisz, U. Wolff, Phys. Lett. B 372 Ž1996. 275.
w5x M. Luscher,
S. Sint, R. Sommer, H. Wittig, Nucl. Phys. B
¨
491 Ž1997. 344.
w6x B. Sheikholeslami, R. Wohlert, Nucl. Phys. B 259 Ž1985.
572.
w7x M. Luscher,
P. Weisz, Nucl. Phys. B 479 Ž1996. 429.
¨
w8x S. Sint, P. Weisz, Further results on O Ž a. improved lattice
QCD the one-loop order,hep-latr9704001.
w9x S. Sint, Proceedings of the International Symposium on
Lattice Field Theory, 21–27 July 1997, Edinburgh, U.K.
w10x M. Guagnelli, Proceedings of the International Symposium
on Lattice Field Theory, 21–27 July 1997, Edimburgh, U.K.
w11x M. Luscher,
private communication
¨
12 February 1998
Physics Letters B 419 Ž1998. 317–321
Confinement and scaling of the vortex vacuum
of SUž 2/ lattice gauge theory 1
Kurt Langfeld, Hugo Reinhardt, Oliver Tennert
Institut fur
D–72076 Tubingen,
Germany
¨ Theoretische Physik, UniÕersitat
¨ Tubingen,
¨
¨
Received 24 October 1997
Editor: P.V. Landshoff
Abstract
The magnetic vortices which arise in SUŽ2. lattice gauge theory in center projection are visualized for a given time slice.
We establish that the number of vortices piercing a given 2-dimensional sheet is a renormalization group invariant and
therefore physical quantity. We find that roughly 2 vortices pierce an area of 1 fm2. q 1998 Elsevier Science B.V.
1. Introduction
The conjecture w1x that confinement is realized as
a dual Meissner effect by a condensate of magnetic
monopoles recently received strong support by lattice calculations which show that in certain gauges
the magnetic monopole configurations account for
about 90% of the string tension w2–4x. The dominance of magnetic monopoles is most pronounced in
the so-called Maximum Abelian gauge, where the
influence of the charged components of the gauge
field is minimized and which is a precursor of the
Abelian projection, where the charged components
of the gauge fields are neglected. The existence of
magnetic monopoles is not restricted to the Maximum Abelian gauge. Monopoles occur, if the gauge
fixing procedure leaves an UŽ1. degree of freedom
1
Supported in part by DFG under contract Re 856r1–3.
unconstrained w5,6x. The monopoles carry magnetic
charge with respect to this residual UŽ1. gauge freedom.
If confinement is realized as a dual Meissner
effect, i.e. by a condensation of the magnetic
monopoles, the residual UŽ1. degree of freedom
should be broken in the confining phase by the
Higgs mechanism. This suggests that the relevant
infrared degrees of freedom may be more easily
identified in a gauge where the residual gauge is
fixed. The UŽ1. gauge symmetry is explicitly broken
in the Maximum Center gauge of the lattice theory,
which has been implemented on top of the Maximum Abelian gauge w7x. The Maximum Center gauge
preserves a residual Z2 gauge symmetry. Analogously to the Abelian projection, one has studied the
center projection of SUŽ2. lattice gauge theory, i.e.
the projection of SUŽ2. link variables Žin Maximum
Center gauge. onto Z2 center elements. The important finding in w7x Žby numerical studies. has been a
significant center dominance, i.e. the center pro-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 3 5 - 4
K. Langfeld et al.r Physics Letters B 419 (1998) 317–321
318
jected links carry most of the information about the
string tension of the full theory w7x. This naturally
leads to the conjecture that the field configurations
which are eliminated by the center projection are not
relevant for confinement. Furthermore, the numerical
calculations also reveal that the signal of the scaling
behavior of the string tension is much clearer in
center projection than in Abelian projection.
The center projection gives rise to vortices which
are defined by a string of plaquettes. Plaquettes are
part of the string, if the product of the corresponding
center–projected links is Žy1. Žfor details see w7x..
The results in w7x indicate that these Z2-vortices are
the ’’confiners’’, i.e. the field configurations relevant
for the infra-red behavior of the theory. This result
supports a previous picture by ’t Hooft w8x and Mack
w9x in which the random fluctuations in the numbers
of such vortices linked to a Wilson loop explain the
area law.
In this letter, we will further investigate the properties of the vortices introduced by center projection
on top of Abelian projection as considered by Del
Debbio et al. in w7x. We will present a visualization
of these vortices in coordinate space at a given time
slice. The splitting of the vortices into branches is
briefly addressed. By calculating the vortex distribution at several values of the inverse coupling b , we
will establish that the string density is renormalization group invariant and therefore a physical quantity. On the average, we will find two vortices
piercing an area of 1 fm2 .
In order to quantify the error done by this projection,
we introduce an angle u A by
tan u
A
¦(a
s
¦(a
2
2
1 q a2
;,
;
2
2
0 q a3
Ž 3.
which measures the strength of the charged components relative to the strength of the neutral ones. The
brackets in Ž3. indicate that the lattice average of the
desired quantity is taken.
Center projection is then defined by assigning to
each ŽAbelian. link variable a value "1 according
the rule AŽ x . ™ sign Žcos u Ž x ... The error which is
induced by the center projection can be measured by
u C , i.e.
(
cos u C s ²cos 2u : .
Ž 4.
Our numerical result for the angles u A and u C is
shown in Fig. 1 as a function of the inverse coupling
b . In the scaling window, i.e. b g w2,3x, the u-angle
is of the order 208 in either case. The relative error of
a measured quantity on the lattice induced first by
Abelian and subsequently by center projection is
generically given by sin u A q sin u C f 75%. Hence,
the charged components of the link variable, i.e. a1
and a 2 , are not small compared with the neutral
components, i.e. a 0 and a3 .
In Fig. 2, we compare the numerical result for the
Creutz ratios of full SUŽ2. lattice gauge theory with
the result of Abelian projected and center projected
theory, respectively. As observed by many people
before w3,7x, the string tension is almost unaffected
2. The vortex vacuum in SU(2) lattice gauge theory
Center projection was defined in w7x on top of
Abelian projection. The Maximum Abelian gauge w1x
makes a link variable UmŽ x . as diagonal as possible,
and Abelian projection replaces a link variable
Um Ž x . s a 0 Ž x . q i a Ž x . t ,
a 02 q a 2 s 1
Ž 1.
by the Abelian link variable
As
a0 Ž x . q i a3Ž x . t 3
(a
2
2
0 q a3
s cos u Ž x . q isin u Ž x . t 3 .
Ž 2.
Fig. 1. The angle u is a measure for the error induced by Abelian
and center projection, respectively.
K. Langfeld et al.r Physics Letters B 419 (1998) 317–321
319
Fig. 2. The Creutz ratios of the full SUŽ2. gauge theory compared with the values obtained in Abelian projected Žleft picture. and in center
projected Žright picture. theory.
by the projection. There is no signal of an error of
the order of 70% in the string tension as one would
naively expect from Fig. 1. This shows that the
degrees of freedom which are relevant for the string
tension are largely untouched by the projection
mechanism.
Center projection of lattice gauge theory induces
magnetic vortices. We follow the definition of Del
Debbio et al. w7x. A plaquette is defined to be part of
the vortex, if the product of the center projected links
which span the plaquette is y1. As explained in w7x,
a Žcenter projected. field configuration contributes
Žy1. n to the Wilson loop, where n is the number of
vortices piercing the loop area. In particular, it was
shown in w7x that the string tension vanishes, if the
configurations with n ) 0 are discarded.
Stimulated by these results, we reduce the lattice
ground state to a medium of vortices with the help of
the center projection in order to construct a pure
vortex vacuum. We then calculate the Creutz ratios
for several values of b employing the vortex model.
In particular, we measure the probability of finding
n vortices which pierce the minimal area of the
Wilson loop. The Wilson loop W is then obtained in
the vortex vacuum model by
n
W s Ý Ž y1 . P Ž n . r
n
Ý P Ž n. .
index indicating the direction of the link is not
shown. and Q x the product of center projected links
of a particular plaquette at position x Žindices suppressed.. If A denotes the minimal area of Wilson
loop and E A its boundary, one finds Žsee Fig. 3.
Ł
xg E A
Sx s
Ł Q x s Ž y1.
n
,
Ž 6.
xg A
where n is the number of vortices piercing the area.
Here the first equality follows since center projected
links S x which do not belong to the boundary of the
Ž 5.
n
The validity of this formula is easily checked:
Define S x g "14 to be the center projected link Žthe
Fig. 3. Schematic plot of the contributions of a center projected
configuration on the lattice to the Wilson loop.
K. Langfeld et al.r Physics Letters B 419 (1998) 317–321
320
Fig. 4. The distribution of the magnetic vortices in the 10 3 hypercube of the 10 4 lattice for a given time slice for b s 2.5 Žleft. and b s 3
Žright..
loop appear twice and hence give no contribution
since S x2 s Ž"1. 2 s 1.
The results for the Creutz ratios calculated from
Ž5. as function of b is identical to the results
obtained by taking the product of center projected
links along the boundary of the Wilson loop Žsmall
symbols in the right picture of Fig. 2.. Hence the
vortex vacuum model reproduces almost the full
string tension as well as the correct scaling behavior,
if the continuum limit is approached.
3. Vortices scale
In order to get a more explicit picture of the
vortex structure of the center projected gauge theory,
we visualize the vortex distribution in coordinate
space for a given time slice. In Fig. 4, we provide
two generic configurations of such vortex filled state
for b s 2.5 and b s 3. The crucial observation is
that the gas of vortices is more dilute in the case of
b s 3 than in the case of b s 2.5. This behavior is
anticipated, if we assume that the vortex gas is a
physical object and therefore renormalization group
invariant. From the Creutz ratios Žsee Fig. 2., one
extracts the value k a 2 , where k is the string tension
Žwe use k f Ž440 MeV. 2 . and a denotes the lattice
spacing, as function of b . In practice, we extracted
k a2 from the Creutz ratios x Ž3,3. using the center
projected configurations, since there is a clear signal
of the perturbative renormalization group flow in the
center projected theory Žsee Fig. 2 and also w7x..
From k a 2 the actual value of the lattice spacing a
Žin units of the string tension. is extracted for a given
value of b . Larger values of b imply a smaller
value of the lattice spacing due to asymptotic freedom. This implies that we zoom into the medium of
the vortices, when we go to larger values of b , if the
vortices are physical objects Žlike the string tension..
In order to establish the physical nature, i.e. the
renormalization group invariance of the vortex
medium, we investigate the average number N of
vortices which pierce through an area of 10 2 a 2 as
function of b . We then relate the lattice spacing a to
the physical scale given by the string tension, and
extract the density r of vortices which pierce an area
Table 1
Results of calculations
b
N
k a2
L wfmx
r w1r fm2 x
n
2.0
2.1
2.2
2.3
2.4
2.5
22.0 " 0.3
18.9 " 0.4
15.1 " 0.6
11.1 " 0.4
8.0 " 0.7
5.8 " 0.9
0.35
0.46
0.40
0.26
0.18
0.11
2.7
3.1
2.9
2.3
1.9
1.5
3.0
2.0
1.8
2.0
2.1
2.5
3.5
3.2
3.0
3.0
2.9
2.6
K. Langfeld et al.r Physics Letters B 419 (1998) 317–321
of 1 fm2 . We find that r is almost independent of b
in the scaling window b g w2.1,2.4x. This indicates
that r is a renormalization group invariant and
therefore physical quantity. Our results are summarized in Table 1. L denotes the spatial extension of
our 10 4 lattice in one direction in physical units.
For b F 2, the lattice configurations are far off
the continuum limit a ™ 0, whereas the numerical
uncertainties in k a 2 grow for b ) 2.5. From the
numerical results, we estimate
r s Ž 1.9 " 0.2 .
1
fm2
.
Ž 7.
Finally, we extract the number n of nearest neighbors which a particular point of the vortex has. It is a
measure of the branching of the vortices. If the
vortices form closed loops without branches, this
number would exactly be two. If the vortices are
open strings without branches, this number would
slightly be smaller than two. Table 1 shows the
numerical value of branching value n for several
values of b . In the scaling window, a value n f 3 is
consistent with the data, while a significant deviation
of n from 3 is observed for b G 2.5. This deviation
is likely due to finite size effects.
4. Conclusion
In this letter, we have studied the vortices arising
in center projected lattice gauge theory, considered
previously in Ref. w7x. We have evaluated the Creutz
ratios in a pure vortex vacuum defined via center
projection, and have obtained almost the full string
tension as well as the right scaling behavior towards
the continuum limit. We have visualized the vortex
distribution in coordinate space for a given time
321
slice. In particular, we have obtained the density r
of vortices piercing the minimal Wilson area as a
function of b . We have observed an approximate
scaling, which suggests that the vortices are not
lattice artifacts but physical objects. On the average,
we find r f Ž1.9 " 0.2.rfm2 .
Our investigations support the observations of
Ref. w7x that the vortices play the role of ’’confiners’’
in SUŽ2. Yang-Mills theory.
Acknowledgements
We thank M. Engelhardt for helpful discussions
and useful comments on the manuscript.
References
w1x G.’t Hooft, High energy physics, Bologna, 1976; S. Mandelstam, Phys. Rep. C 23 Ž1976. 245; G.’t Hooft, Nucl. Phys. B
190 Ž1981. 455.
w2x For a review see e.g. T. Suzuki, Lattice 92, Amsterdam, Nucl.
Phys. B ŽProc. Suppl.. 30 Ž1993. 176,
w3x M.N. Chernodub, M.I. Polikarpov, A.I. Veselov, Talk given at
International Workshop on Nonperturbative Approaches to
QCD, Trento, Italy, 10–29 July 1995; published in Trento
QCD Workshop, 1995 pp. 81–91; M.I. Polikarpov, Nucl.
Phys. B ŽProc. Suppl.. 53 Ž1997. 134.
w4x G.S. Bali, V. Bornyakov, M. Mueller-Preussker, K. Schilling,
Phys. Rev. D 54 Ž1996. 2863.
w5x A.S. Kronfeld, G. Schierholz, U.-J. Wiese, Nucl. Phys. B 293
Ž1987. 461.
w6x K. Langfeld, H. Reinhardt, M. Quandt, Monopoles and strings
in Yang-Mills theories, hep-thr9610213.
ˇ Olejnık,
w7x L. Del Debbio, M. Faber, J. Greensite, S.
´ Nucl. Phys.
B ŽProc. Suppl.. 53 Ž1997. 141; L. Del Debbio, M. Faber, J.
ˇ Olejnık
Greensite, S.
´ Phys. Rev. D 53 Ž1996. 5891.
w8x G.’t Hooft, Nucl. Phys. B 153 Ž1979. 141.
w9x G. Mack, in: G.’t Hooft et al. ŽEds.., Recent developments in
gauge theories, Plenum, New York, 1980.
12 February 1998
Physics Letters B 419 Ž1998. 322–325
An exact QED 3q1 effective action
G. Dunne, T.M. Hall
Physics Department, UniÕersity of Connecticut, Storrs, CT 06269, USA
Received 10 October 1997
Editor: M. Dine
Abstract
We compute the exact QED 3q1 effective action for fermions in the presence of a family of static but spatially
inhomogeneous magnetic field profiles. An asymptotic expansion of this exact effective action yields an all-orders derivative
expansion, the first terms of which agree with independent derivative expansion computations. These results generalize
analogous earlier results by Cangemi et al. in QED 2q1. q 1998 Elsevier Science B.V.
The effective action plays a central role in quantum field theory. Here we consider the effective
action in quantum electrodynamics ŽQED. for
fermions in the presence of a background electromagnetic field. Using the proper-time technique w1x,
Schwinger showed that the QED effective action can
be computed exactly for a constant Žand for a plane
wave. electromagnetic field. For general electromagnetic fields the effective action cannot be computed
exactly, so one must resort to some sort of perturbative expansion. A common perturbative approach is
known as the derivative expansion w2–4x in which
one expands formally about the constant field case,
assuming that the background is ‘‘slowly varying’’.
However, even first-order derivative expansion calculations of the effective action are cumbersome, and
somewhat difficult to interpret physically. A complementary approach is to seek other Ži.e., inhomogeneous. background fields for which the effective
action can be computed exactly, with the hope that
this will lead to a better nonperturbative understanding of the derivative expansion. There are two technical impediments to such an exact computation of
the effective action. First, the background field must
be such that the associated Dirac operator has a
spectrum that is known exactly. Second, this spectrum will Žin general. contain both discrete and
continuum states, and so an efficient method is
needed to trace over the entire spectrum. ŽNote that
in the constant field case the spectrum is purely
discrete so this trace is a simple sum.. Cangemi et al.
w5x used a resolvent technique to obtain an exact
answer for the effective action in 2 q 1-dimensional
QED for massive fermions in the presence of static
but spatially inhomogeneous magnetic fields of the
form B Ž x, y . s Bsech2 Ž lx .. In this Letter, we extend
this result to 3 q 1-dimensional QED.
Consider the QED 3q 1 effective action
i
S s yilndet Ž iDu y m . s y lndet Ž Du 2 q m 2 .
2
Ž 1.
where Du s g n Ž En q ieAn ., and An is a fixed classical gauge potential with field strength tensor Fmn s
Em An y En Am . We work in Minkowski space, and the
Dirac gamma matrices g n satisfy the anticommutation relations g n ,g s 4 s 2 diagŽ1,y 1,y 1,y 1..
Schwinger’s proper-time formalism w1x involves rep-
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 2 9 - 9
G. Dunne, T.M. Hall r Physics Letters B 419 (1998) 322–325
resenting lndet as Tr ln and using an integral representation for the logarithm:
Ss
i
`
H
2 0
ds
s
Tr exp yis Ž Du 2 q m 2 .
Ž 2.
In the constant field case one can compute exactly
the proper time propagator Tr expwyisŽ Du 2 q m 2 .x,
leading to an exact integral representation w1x for the
effective action Žsee below - Eq. Ž11... Furthermore,
for non-constant but ‘‘slowly varying’’ fields, the
first order derivative expansion contribution has been
computed in w3,4x.
To make contact with the exact QED 2q 1 result of
Cangemi et al. w5x, we restrict our attention to static
background magnetic fields that point in a fixed
direction Žsay, the x 3 direction. in space, and whose
magnitude only depends on one of the spatial coordinates Žsay x 1 .. This type of configuration can be
represented by a gauge field Am s Ž0,0, A 2 Ž x 1 .,0.,
where A 2 is only a function of x 1. Then, it is
straightforward to show that the operator Du 2 q m2
diagonalizes as
Du 2 q m 2 s m 2 q p 32 y p 02 1
q diag Ž Dq , Dy , Dy , Dq .
323
so this trick is not directly applicable. Instead, we
note that an alternative way to view this generalization from 2 q 1 to 3 q 1 is simply to take the 2 q 1
expression and replace m2 by m2 q p 32 , and then
integrate over x 3 and p 3 , as is clear from Ž3.. Thus,
if we take the final answer from w5x and perform this
operation, we obtain the exact effective action for
QED 3q 1 in the background of a family of static
magnetic fields of the form
ž
B s 0,0, Bsech2
x1
ž //
Ž 5.
l
where B is a constant setting the scale of the magnetic field strength, and l is a length scale describing the ‘‘width’’ of the inhomogeneous profile of
< B < in the x 1 direction.
From w5x, the exact parity-even QED 2q 1 effective
action in the family of magnetic backgrounds with
profile B Ž x, y . s Bsech2 Ž xrl. is
S2q 1 s y
L
4pl
dt
`
2
H0
e
2p t
ž
= Ž eBl2 y it .
Ž 3.
y1
Ž l2 m 2 q Õ 2 .
Õ
ln
l m y iÕ
l m q iÕ
Ž 6.
qc.c. .
where
2
2
D"s p 12 q Ž p 2 y eA 2 Ž x 1 . . " eB Ž x 1 .
Ž 4.
and B Ž x 1 . is the magnitude of the background magnetic field. These operators D" are precisely the
ones that appear in the computation of the parity-even
QED 2q 1 effective action Žwith static magnetic background depending on just one of the spatial coordinates. – as was done in w5x. Thus, the only difference
between the QED 3q 1 case described above and the
computation in w5x is the appearance of an extra trace
over x 3 and p 3 , the momentum corresponding to the
free motion in the x 3 direction Žplus an extra overall
factor of 2 from the Dirac trace..
In terms of the proper-time representation Ž2. this
is a straightforward generalization – one performs
the p 3 integral, which Žup to overall factors. simply
changes the power of s in the proper time integral
from sy1 to sy3 r2 . This much is obvious and wellknown. However, the computation of the exact
QED 2q 1 effective action in w5x was done using the
resolvent method rather than the proper time method,
where c.c. denotes the complex conjugate, and Õ '
t 2 q 2 iteBl2 . Here L is the length scale of the y
direction, and we suppress the overall time scale as
all fields are static. An asymptotic expansion of this
integral for large Bl2 corresponds physically to
expanding about the uniform B field case wrecall that
Bl2 ™ ` is the limit of uniform backgroundx, and
yields the following all-orders derivative expansion
w5,6x:
S2q 1 s y
Lm3l
8p
`
Ý
js0
`
=
=
1
ž
1
j! 2 eBl2
j
/
Ž 2 k q j y 1 . ! B2 kq2 j
Ý
1
3
ks1 Ž 2 k . ! Ž 2 k q j y 2 .Ž 2 k q j y 2 .
2 eB
ž /
m2
2 kqj
Ž 7.
where Bk is the k t h Bernoulli number w7x.
To generalize these results Ž6., Ž7. to 3 q 1 dimensions, we need to replace m2 by m2 q p 32 , and
G. Dunne, T.M. Hall r Physics Letters B 419 (1998) 322–325
324
integrate over p 3 . We first do this for the all-orders
derivative expansion expression Ž7., and return later
to the generalization of the exact integral representation Ž6. of the effective action.
The j s 0 and k s 1 term in Ž7. must be treated
separately as it is logarithmically divergent:
L Ž0. s y
e2B2
L2
24p
m2
dp 3
2
L
y
2
H
2p
Ž m2 q p 32 .
y1 r2
;
1
2p
L
ln
8p
2
m2
,
For the remaining terms Ži.e., excluding the j s 0
and k s 1 term.
Hy` 2p Ž m q p .
2 2y 2 kyj
s
2'p
coth Ž p t . s
S3q 1 s y
L ld B
18p
2
L2l m4
y
8p 3r2
`
=
Ý
L
ln
`
Ý
js0
2 eB
ž /
m2
pt
srŽ e B .
ž
coth s y
1
s
y
s
/
3
Ž 11 .
1
Ý
p
Ž 12 .
2
k qt2
ks1
L2
24p
m2
ln
2
8p 2
2 eB
B2 k
Ý
ks2
2k
ž /
m2
2 k Ž 2 k y 1. Ž 2 k y 2.
Ž 13 .
m2
1
2
`
2t
q
`
=
2
ž
1
j! 2 eBl2
Substituting Bsech2 Ž x 1rl. for B and integrating,
we obtain the zeroth order Žin the derivative expansion. contribution to the effective action:
j
/
G Ž 2 k q j . G Ž 2 k q j y 2 . B2 kq2 j
ks1
=
2
eym
m4
y
We therefore obtain an all-orders derivative expansion of the QED 3q 1 effective action for the inhomogeneous magnetic background Ž5.:
2
1
e2B2
Ž 9.
G Ž 2 k q j y 3r2 .
2
s
2
Using the expansion w7x
L Ž0. s y
G Ž 2 k q j y 2.
Žm .
ds
`
H0
it is straightforward to develop the expansion
2 3r2y2 kyj
3
2
2
L™`
Ž 8.
dp 3
ln
2
e2B2
y
L
`
to obtain the contribution to the action. From w1x the
constant field effective Langrangian is
G Ž 2 k q 1 . G Ž 2 k q j q 12 .
S Ž0. s y
L2l e 2 B 2
18p 2
`
2 kqj
Ž 10 .
Here it is understood that the double sum excludes
the j s 0 and k s 1 term.
Each power in B1l 2 in Ž7., and therefore also in
Ž10., corresponds to a fixed order in the derivative
expansion w5,6x. We now compare the j s 0 and
j s 1 terms in Ž10. with independent QED 3q 1
derivative expansion calculations w1,3,4x. To compute the zeroth Ži.e., leading. order contribution to
the derivative expansion of the effective action, we
take the constant field result for the effective Lagrangian Žnot action!. and substitute the magnetic
field Ž5., and then integrate over space-time in order
=
Ý
ks2
L2
ln
m2
L2l m 4
y
8p 3r2
1 B2 k G Ž 2 k y 2 .
G Ž 2 k q 12 .
2k
2 eB
2k
ž /
Ž 14 .
m2
This result agrees exactly with the j s 0 term from
Ž10. including the form and magnitude of the logarithmic divergence. As shown in w1x, the logarithmically divergent piece corresponds to a charge renormalization.
The first-order Žin the derivative expansion. contribution to the QED 3q 1 effective Lagrangian has
been computed in w3,4x:
L Ž1. s ye
E 1 BE 1 B
64p
2
`
H
B 0
ds
s
eym
2
srŽ e B .
Ž scoth s .
XXX
Ž 15 .
G. Dunne, T.M. Hall r Physics Letters B 419 (1998) 322–325
Once again, using Ž12. this can be expanded as
L Ž1. s ye 2
`
E 1 BE 1 B
B2 kq2
Ý
4p 2 m 2
ks1
2ky1
2 eB
2 ky2
ž /
Ž 16 .
m2
Substituting B Ž x 1 . s Bsech2 Ž x 1rl. and integrating,
we obtain the first order Žin the derivative expansion.
contribution to the effective action:
S Ž1. s y
L2 m2
`
Ý
8 lp 3r2
B2 kq2 G Ž 2 k y 1 .
G Ž 2 k q 32 .
ks1
2 eB
2k
ž /
m2
Ž 17 .
This agrees exactly with the j s 1 term from Ž10..
Having understood how Ž7. generalizes to an
all-orders derivative expansion Ž10. of the QED 3q 1
effective action, we conclude by presenting the exact
integral representation for the QED 3q 1 effective action. This is obtained from the corresponding exact
expression Ž6. in 2 q 1 dimensions by substituting
m2 with m2 q p 32 and tracing over p 3 , as before. We
find
S3q 1 s y
2 L2
2 3
3p l
ž
dt
`
H0
2
e
2p t
= Ž eBl y it .
qc.c.
/
325
answer, and terms quadratic in B as they may be
absorbed by renormalization.
The expression Ž18. is the exact QED 3q 1 effective action for fermions in the family of inhomogeneous magnetic backgrounds Ž5.. It is interesting to
note that it is not so much more complicated than
Schwinger’s answer Ž11. for the exact effective action in the constant field case. It is a straightforward
exercise to check that an asymptotic expansion of
this exact result Ž18. for large Bl2 yields the allorders derivative expansion Ž10.. We regard these
results as further evidence that the formal derivative
expansion should be understood as an asymptotic
series expansion.
Acknowledgements
This work has been supported by the DOE grant
DEFG02- 92ER40716.00, and by the University of
Connecticut Research Foundation. We thank Daniel
Cangemi for helpful correspondence.
y1
Ž l2 m 2 q Õ 2 .
Õ
3r2
arcsin
iÕ
ž /
lm
Ž 18 .
where, as before, c.c. denotes the complex conjugate,
Õ 2 ' t 2 q 2 iteBl2 , and we have neglected terms independent of B as they cancel against the zero-field
References
w1x
w2x
w3x
w4x
w5x
J. Schwinger, Phys. Rev. 82 Ž1951. 664.
I. Aitchison, C. Fraser, Phys. Rev. D 31 Ž1985. 2605.
H.W. Lee, P.Y. Pac, H.K. Shin, Phys. Rev. D 40 Ž1989. 4202.
V.P. Gusynin, I.A. Shovkovy, Can. J. Phys. 74 Ž1996. 282.
D. Cangemi, E. D’Hoker, G. Dunne, Phys. Rev. D 52 Ž1995.
3163.
w6x G. Dunne, Int. J. Mod. Phys. A 12 Ž1997. 1143.
w7x I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and
Products, Academic Press, New York, 1979.
12 February 1998
Physics Letters B 419 Ž1998. 326–332
h y h X mixing from Už 3/ L m Už 3/ R chiral perturbation theory
P. Herrera-Siklody
´ 1, J.I. Latorre 2 , P. Pascual 3, J. Taron
4
Departament d’Estructura i Constituents de la Materia
UniÕersitat de Barcelona, Diagonal 647,
` Facultat de Fısica,
´
E-08028 Barcelona, Spain
and I. F. A. E., UniÕersitat Autonoma
de Barcelona, E-08193 Bellaterra, Spain
`
Received 17 October 1997
Editor: L. Alvarez-Gaumé
Abstract
We obtain explicit expressions for the h and hX masses and decay constants using ULŽ3. m UR Ž3. chiral perturbation
theory at next-to-leading order in a combined expansion in p 2 and 1rNc . A numerical fit of the parameters appearing to this
order of the expansion is also discussed. q 1998 Elsevier Science B.V.
The masses of the I s 0 pseudoscalar h , hX particles as well as the value of their mixing angle have
long been the subject of discussion and theoretical
interest, also in recent years. Phenomenologically,
the situation of the h y hX mixing remains not completely settled, the reason being the high sensitivity
of the mixing angle u on the deviation Ž D . of the
h 8 5 mass from the Gell-Mann–Okubo Ž D s 0. relation w1x
mh2 8 s 13 Ž 4 m2K y mp2 . Ž 1 q D . .
Ž 1.
Indeed, if D s 0 the singlet-octet mixing yields u ;
y108, whereas data on JrC ™ h Žh X .g and
h ŽhX ,p 0 . ™ gg favour a higher mixing u ; y208
1
E-mail: [email protected].
E-mail: [email protected].
3
E-mail: [email protected].
4
E-mail: [email protected].
5
h 0 , h 8 are the eigenstates of SUŽ3.RqL . Although in the text
X
we are occasionally loose when referring to h , h instead of h 0 ,
h 8 , the distinction is carefully made in the calculation.
2
w2x, a factor of two bigger, that can be accommodated from Ž1. with just a small D ; 0.16, Žsee also
Ref. w3x, which advocates for a smaller mixing..
The value of the mixing angle is linked to the
value of the hX mass, which is heavier than the octet
of light pseudoscalars, MhX s 958 MeV ; 2 MK , but
still lighter than the next I s 0 candidate h Ž1279.,
which is so distant from the h mass that it might not
significantly mix with it. Therefore, we only consider h y hX mixing.
Within the SUŽ3.L m SUŽ3.R Chiral Lagrangian
w4x analysis, the shift downwards of the h mass due
to this mixing is an O Ž p 4 . effect of SUŽ3.Rq L
breaking, proportional to Ž m s y m u q2 m d . 2 and multiplied by the constant Lw7SU x, which contains information of the hX state that has been integrated out.
In this article we re-analyse these issues on the
basis of large-Nc Chiral Perturbation Theory Ž x PT..
The hX is regarded as the ninth Goldstone boson of a
nonet, together with the octet of light pseudoscalars,
and the corrections are treated in a double expansion,
both in powers of quark masses m quark and 1rNc .
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 5 0 7 - 4
P. Herrera-Siklody
´ et al.r Physics Letters B 419 (1998) 326–332
Such a formulation for this problem is plausible
2
is an effect induced by the
because Mh2X y Moctet
UAŽ1. anomaly, which vanishes in the large-Nc limit
as 1rNc . Furthermore, the fact that phenomenologically it is found that MhX y Moctet ; Moctet , together
with the usual bookkeeping of quark masses as
2 .
m quark ; O Ž Moctet
, suggests that the relative magnitude of the double expansion may be regarded as
2
m q ; 1rNc ; Moctet
; p2 ; O Ž d . ;
Ž 2.
this is the approach that we adopt and henceforth we
shall refer to it as the combined expansion. It has
already been used in the literature w5x, in particular
Leutwyler has forcefully pursued the analysis of the
masses of h and hX in order to discard the possibility
that m u s 0. In this note we reproduce Leutwyler’s
results and proceed further to obtain a fit for the
masses and decay constants of the Goldstone boson
nonet w6x.
Let us start from the ULŽ3. m UR Ž3. chiral lagrangian which has been recently proposed to O Ž p 4 .
w7x, from which we share the notation and conventions, and which we refer to for references therein.
The nonet of fields are gathered in a unitary 3 = 3
matrix U g U Ž3., whose determinant detU s
expŽ i'6 h 0rf . differs from unity by the presence of
the h 0 field. The chiral lagrangian LŽ p 0 . q LŽ p 2 .
q . . . reads
LŽ p 0 . s yW0 Ž X . ,
LŽ p 2 .
s W1 Ž X . ² Dm U †Dm U : q W2 Ž X . ²U †x q x †U :
q iW3 Ž X . ²U †x y x †U :
q W5 Ž X . ²U † Ž Dm U . :² iam : .
Ž 3.
We have only kept the axial source because we take
the divergence of the axial current as the interpolating fields for the nonet. The axial source am appears
in the last term as well as through the covariant
derivatives of the fields:
i
DmU s Em U y Ž am U q Uam . .
2
The quark masses appear in
x s 2 B diag Ž m u ,m d ,m s . .
Ž 4.
Ž
.
As a consequence of the UA 1 anomaly, the lagrangian has coefficient functions that depend on
X s logŽdetU . s i'6 h 0rf, which are not fixed by
327
symmetry arguments. From the large-Nc perspective
however, only the lowest powers of X survive, for
each new power of X is suppressed by a factor of
1rNc . Of interest to us, the only terms that prevail
are
W0 Ž X . s Constantq
W1 Ž X . s
f2
4
W3 Ž X . s yi
W5 Ž X . s
f2
4
q... ,
f2
4
f2
4
Õ 02 X 2 q . . . ,
W2 Ž X . s
f2
4
q... ,
Õ 31 X q . . . ,
Õ50 q . . . ,
Ž 5.
with Õ 02 , Õ 31 , Õ50 ; O Ž1rNc .. We shall expand the
chiral lagrangian and keep terms up to O Ž d 2 ., according to the combined power counting Ž2.. Recall
that f 2 ; O Ž Nc ., B ; O Ž1.. Within this approximation the chiral logarithms are suppressed Žthey appear at O Ž d 3 .., although some terms of the same
order O Ž p 4 . have to be included at tree level. More
precisely, the contributions
LŽ p 4 . s L5 ² Dm U † D m U Ž U †x q x †U . :
q L8 ² x †Ux †U q U †x U †x :
Ž 6.
6
should be included, since L5 and L8 are O Ž Nc ..
Notice that a term L 7 ²U †x y x †U :2 appears at
O Ž d 4 . - L 7 is O Ž1. in 1rNc Žsee w8x.. The rest of
terms all remain subleading.
The masses and the decay constants can be obtained from the two-point function of axial currents,
which is most easily done by taking functional
derivatives with respect to am . We shall thus keep
only terms quadratic in am . The part of the effective
action quadratic in the nonet fields finally reads
S s 12
4
H d x žŽE f
m
a
y bma . Aa b Ž Em f b y bmb .
yf a Ba b f b q f a Ca b Em bmb ,
/
6
Ž 7.
In the UŽ3.LmUŽ3.R chiral lagrangian, instead of constants
the L i ’s are functions of the h 0 field. When we write L5 or L8
throughout the article we mean the value of those functions with
the argument set to zero, L5 Ž0., L8 Ž0.. They are not to be
confused with the constants in w4x, for which we reserve a
superscript of w SU x Lwi SU x.
P. Herrera-Siklody
´ et al.r Physics Letters B 419 (1998) 326–332
328
where bm s 2 fam , f a Ž a s 0,1, . . . ,8. are the fields
with SUŽ3.Rq L quantum numbers and
AsIqD A ,
B s 2 mB Ž D q D D . ,
C s DC .
Ž 8.
The mass matrix D is non-diagonal already at leading order:
D 11 s D 22 s D 33 s 1 ,
D44 s D55 s D66 s D 77 s 1 q
D 88 s 1 q 23 x ,
D 08 s y
D 00 s 1 q 13 x y
3 Õ 02
2 mB
'2
3
,
2
x,
,
Ž 9.
with m s Ž m u q m d .r2, x s Ž m s y m.rm. On the
other hand, the matrix D A brings the next-to-leading
order corrections to the kinetic term,
D A11 s D A 22 s D A 33 s 16
f2
L5 ,
D A 88 s 16
D A 08 s y
D A 00 s 16
mB
f
2
mB
f2
ž
L5 1 q
mB
f2
2
/
f2
DC comes entirely from the UAŽ1. anomaly:
D D 88 s 32
mB
f2
Ž 12 .
From the correlator of two axial currents we can
read off the physical masses and the decay constants,
the former from the location of the poles, the latter
from their residues. Following the steps of Ref. w4x,
the correlator is the second derivative of Ž7. with
respect to am when evaluated for am that minimises
the effective action; it boils down to a simultaneous
diagonalisation of A and B for the masses,
AsF† I F ,
BsF† M 2 F ,
Ž 13 .
whereas the decay constants read Ž P is a diagonal
index, a is non-diagonal.,
y1
ž
Aq
C
2
//
ž
L5 x ,
.
Ž 14 .
Pa
/
only the 0 and 8 components do mix,
L5 Ž 1 q 13 x . ,
Ž 10 .
2 mB F0 DF0† s M 2 ,
Ž 16 .
with
mB
f2
L8 ,
D D44 s D D55 s D D66 s D D 77
f2
L8 Ž 1 q 23 x q 13 x 2 . y 2 Õ 31 Ž 3 q x . .
Consistently, we perform a perturbative diagonalization, order by order in the expansion. First, we
diagonalize to leading order and only D has a nondiagonal part; it is already diagonal in p and K Žwe
shall neglect the isospin breaking.,
x
2 mB s Mp2 , 2 mB 1 q
s MK2 ,
Ž 15 .
2
,
D D 11 s D D 22 s D D 33 s 32
mB
f2
L8 x Ž 2 q x . q '2 Õ 31 x ,
Ž 11 .
and D D corrects the mass matrix,
s 32
mB
ž
L5 Ž 1 q 23 x . ,
16'2 mB
3
x
f2
3
f Pa s f Ž F † .
D A 44 s D A 55 s D A 66 s D A 77
s 16
D D 00 s 32
32'2 mB
DC00 s 32 Õ50 .
x
mB
D D 08 s y
L8 Ž 1 q x q 14 x 2 . ,
L8 Ž 1 q 43 x q 23 x 2 . ,
F0 s
ž
cos u 0
sin u 0
ysin u 0
.
cos u 0
/
Ž 17 .
From Ž15. we can fix mB and x unambiguously.
Upon diagonalization of Ž16. we fix Õ 02 so as to
describe correctly Mh2X . This implies
u 0 , y208,
Õ 02 , y0.22 GeV 2 .
Ž 18 .
In this way, we get a prediction for Mh2 ,
Ž495.5 MeV. 2 which is only a 10% off. This is a
known feature of the leading 1rNc approximation: it
P. Herrera-Siklody
´ et al.r Physics Letters B 419 (1998) 326–332
was proven in Ref. w9x that the ratio Mh2rMh2X always
comes out too small; the next-to-leading corrections
are thus needed to reconcile the 1rNc expansion with
the observed masses w10x.
Let us emphasize that we had two masses, Mh2
and Mh2X , to fit and just one parameter, Õ 02 , to tune.
In principle, we had the choice to fit either one to its
experimental value. We could have, instead, fixed
the value of Õ 02 in order to describe correctly Mh2 ,
obtaining a prediction for Mh2X . This choice would
give, though, a poor prediction for Mh2X since it is the
hX that gets the bulk of the contribution from Õ 02
being, therefore, more sensitive to it.
The next-order expressions for the masses can
also be found in w5x. The D A corrections are absorbed into a wave-function renormalization and the
p and K are still diagonal
Mp2 s 2 mB
ž
1 q 16
MK2 s 2 mB 1 q
ž
ž
= 1 q 16
x
mB
f2
Fs
ž
2
m288 s
m208 s
3
ž
/
y
3
/
MK2 q
(9 q 2 y q y
y
y
3
ž
Ž 2 L 8 y L5 . 1 q
x
2
//
,
Ž 19 .
ž
ž
q 1y
3(9 q 2 y q y
(
2
/Ž
MK2 y Mp2 . DM
2
Ž 3 Ž 2 MK2 y 3 Mp2 .
/
Ž 20 .
where
2
.
Mp2
9qy
3 9q2 yqy
. Ž 1 q DM y DN . ,
q 23 Ž MK2 y Mp2 . DM y 3Õ 02 ,
2
0
Ž MK2 y Mp2 .
q
Ž
1q
2
D A 00
q y 13 Ž 2 MK2 q Mp2 .
q 43 Ž MK2 y Mp2 . DM ,
MK2 y Mp2
2
D A 08
D A 08
2
3
m200 s 13 Ž 2 MK2 q Mp2 . Ž 1 y 2 DN .
DM s
/
1
y2'2
3
ysin u
cos u
D A 88
where we have explicitly separated the rotation that
diagonalizes the kinetic term. Note that u also gets
corrections with respect to u 0 . For the physical
masses we find
whereas the 0 and 8 components remain to be diagonalized and can be written as
4 MK2 y Mp2
cos u
sin u
1q
Ž 22 .
/
2
mB
f
dence, on L5 only, due to the presence of Õ 02 , i.e.,
the UAŽ1. anomaly.
Let us now proceed to diagonalise to next-to-leading order. In order to preserve the form of the above
partial result, we write the transformation matrix as
Mh2 s 1 y
Ž 2 L 8 y L5 . ,
329
qy Ž y2 MK2 y Mp2 . . DN ,
Ž 23 .
Õ 02
2
MK y Mp2
Ž 24 .
where
8
f2
Ž MK2 y Mp2 . Ž 2 L8 y L5 . ,
DN s 3Õ 31 y
12
f2
Õ 02 L5 .
y'
Ž 21 .
Note the explicit violation of the Kaplan–Manohar
symmetry w11x in SUŽ3.L m SUŽ3.R - usually stated as
the fact that in the expressions for the masses L8 and
L5 always come through the combination Ž2 L8 y L5 .
- in UŽ3. chiral perturbation theory which is apparent
in DN : this correction brings about a different depen-
9
2
and
Mh2 q Mh2X
s y 23 Ž y3 q y . MK2 q 23 yMp2
q2 Ž MK2 y Mp2 . DM y 32 Ž 2 MK2 q Mp2 . DN ;
Ž 25 .
P. Herrera-Siklody
´ et al.r Physics Letters B 419 (1998) 326–332
330
the angle u being
2'2
tan2 u s
1qy
1
y
ž
1q
y
1qy
bation theory. The violation of the original formula
is related to DM ,
DM y DN
2 MK2 q Mp2
1 q y MK2 y Mp2
4 MK2 y Mp2 y 3 Mh2 y 3sin2u Ž Mh2X y Mh2 .
s y4 Ž MK2 y Mp2 . DM .
/
DN .
Ž 26 .
Let us now turn to the numerical exploitation of
these results. The global counting of parameters
versus data goes as follows: we are left with y
Žwhich is proportional to Õ 02 ., DM and DN to fix
Mh2 and Mh2X . Thus, no prediction can be made. Yet,
if we fix a rotation angle, then there is a fit for all
three parameters. The results for angles which are
close to the zeroth order u 0 are displayed in the
following table:
The best fit, if UŽ3. chiral perturbation theory is
to make sense, is expected to be near the zeroth
order result. One could play with different convergence criteria to argue in favor of a given angle. A
nice feature of our table is that DM and DN move in
different directions as we change the mixing angle.
This leads to a minimization of corrections around
u ; y208. Moreover, one may select the optimal u
where the masses of h and hX receive the smallest
correction in the sense that corrections proportional
to DM and DN tend to cancel. Interestingly enough,
we find it to be very close to y208 as well.
Of course none of these arguments is a substitute
for a global fit to data. However, our results agree in
hinting at a mixing angle close to y208 G u G y218,
which correspond to values of DM and DN compatible with those given in w5x Ž DM , 0.18, DN , 0.24..
We stress that since from this point of view the angle
u is the only variable, the values of DM and DN are
correlated.
At this point let us comment on the Gell-Mann–
Okubo formula, with corrections from chiral pertur-
The leading order equation, thus, corresponds to
setting the r.h.s. to zero. At this order, u ; y108 in
order to fulfill the equation. This is not the procedure
we chose in Eq. Ž18., as we took Mh2X to fix u ; y208
at leading order. Eq. Ž27. thus provides a different
way of fitting the mixing, which is strongly corrected
by DM . Of course, at next-order, the best fit at e.g.
DM ; 0.156 is u ; y208 as well.
We may also use our analytical results to get the
form of the decay constants for the whole nonet. The
diagonal elements of relevance are given by the
combination
f physical s f
ž
=
=
cos u
sin u
ysin u
cos u
ž
cos u
ysin u
y188
y208
y228
y248
y Õf022
0.113
0.156
0.203
0.254
0.270
0.220
0.178
0.143
30.8
29.3
27.9
26.5
1q
2
2
D A 00
2
q
D E00
2
sin u
.
cos u
/
0
Ž 28 .
This expression yields
ž
ž
fp s f 1 q 4
L5
fK s f 1 q 4
ž
fh s f 1 q 4
Table 1
DN
D A 08
2
D A 08
f2
L5
f2
L5
f
2
/
/
Mp2 ,
MK2 ,
ž
ž
DM
/
D A 88
1q
q 38 Õ50 1 q
u
Ž 27 .
ž
fhX s f 1 q 4
ž
L5
f
2
ž
q 38 Õ50 1 y
MK2 y
MK2 y Mp2
(9 q 2 y q y
3
1qy
(9 q 2 y q y
MK2 q
9qy
2
//
9qy
(9 q 2 y q y
3
1qy
2
//
/
,
MK2 y Mp2
(9 q 2 y q y
2
.
2
/
Ž 29 .
P. Herrera-Siklody
´ et al.r Physics Letters B 419 (1998) 326–332
Note that in the exact SUŽ3.Rq L limit of quarks
degenerate in mass there is no mixing and one finds
ž
ž
fh ™ f 1 q 4
fhX ™ f 1 q 4
L5 4 MK2 y Mp2
f2
3
L5 2 MK2 q Mp2
f2
3
/
,
/
q 34 Õ50 ,
Ž 30 .
which is the y ™ y` limit of Ž29., as expected.
Let us compare these results with data. We can
take fp and f K to fix f and L5 , Ž fp s 92.4 MeV,
f K s 1.223 fp .
f s 90.8 MeV ,
L5 s 2.0 10y3 .
Ž 31 .
Lw5SU x r
This is the same value as obtained for
in w4x
for it is extracted from the same source. In our
approximation the issue of the running of the L5 , L8
cannot be addressed because the loop corrections are
dropped altogether. As a matter of principle one
expects L5 ; Lw5SU x because integrating out the hX
does not give any contribution to Lw5SU x, Lw8SU x at tree
level, except for Lw7SU x.
We are left with fh and fh X to be described by y
Žagain, proportional to Õ 02 . and Õ50 . Assuming that
convergence criteria fix a rotation angle around u ;
y208, this leaves room for a fit of Õ50 and a
prediction. The values of h and hX decay constants
are, though, poorly determined. Indeed, the experimental values of fh and fh X w6x have large errors:
fh
fh X
0.943 F F 1.091 , 0.912 F
F 1.015 . Ž 32 .
fp
fp
Let us present the output of our formulae as a
function of the mixing angle u . At each given u , the
experimental values of ffph allow for a range of values
of the parameter Õ50 and of ffhp . We again display the
results for angles which are close to the zeroth order
u0.
A word of caution should be added as regards the
use of the fhX value, as made in this paper. As
pointed out by Shore and Veneziano w12x, from the
X
Table 2
u
y Õ50 minrmax
y208
y228
y248
2.293r0.601
1.775r0.365
1.394r0.197
fhX
fp
minrmax
y0.218r0.902
0.171r1.080
0.457r1.205
331
decay rate of hX ™ gg one cannot obtain the value of
fhX because the singlet axial current AmŽ5. is not
conserved in the chiral limit, due to the UAŽ1.
anomaly, which makes fhX from ²0 < AmŽ5. <hX Ž k .: s
ikm fh X depend on a subtraction point and thus not be
observable. Of course this dependence is 1rNc suppressed because in the large-Nc limit the anomaly is
a subleading effect. The relation between our fhX and
what is measured in hX ™ gg and given in w6,13x can
be worked out and amounts to include one further
term in the effective action coupled to an external
electromagnetic source w7x. This would still bring in
a new unknown constant and we would face the
problem of having one more constant to fit than
measured constants available. Nevertheless, we overcome this shortcoming by using the criterion of
minimum sensitiÕity that minimizes the size of the
corrections to a given order and conclude that the
combined expansion Ž2., within the framework of
ULŽ3. m UR Ž3. chiral perturbation theory, is able to
accommodate very naturally the observed values of
masses and decay constants fp , f K . The big uncertainty in fh makes the method inconclusive for fh X
Žwhich, strictly speaking, is not experimentally
known., as can be seen from the last table. If data
from hX ™ gg were more precise, we could with our
method pursue to determine the new constant that is
involved in the process, and how well it would
accommodate in the framework of the combined
expansion.
We finish by quoting the values of all the parameters in one batch, as functions of the mixing angle in
the range 208 - u - 248:
0.15 F DM F 0.26,
2 mB
0.980 F
F 0.988,
Mp2
18.3 F x F 20.9,
y4.7 F y F y4.2,
26 F yÕ 02 rf 2 F 29,
0.14 F DN F 0.22,
1.35 10y3 F L8 F 1.57 10y3 ,
y0.164 F Õ 31 F y0.161.
Ž 33 .
We should like to add that after this work was
finished there appeared the document of Ref. w14x in
which the author reports some results, as yet unpublished, on the same issue that concerns our study.
332
P. Herrera-Siklody
´ et al.r Physics Letters B 419 (1998) 326–332
Acknowledgements
Financial support from CICYT, contract AEN950590, and from CIRIT, contract GRQ93-1047 are
acknowledged. J.I.L. acknowledges the Benasque
Center for Physics where part of this work was
completed. P.H.-S. acknowledges a Grant from the
Generalitat de Catalunya. J.T. acknowledges the
Theory Group at CERN for the hospitality extended
to him.
References
w1x J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the
Standard Model, Cambridge Univ. Press, 1992.
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w5x H. Leutwyler, Phys. Lett. B 374 Ž1996. 163.
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Suppl.. 39B,C Ž1995. 266.
w11x D.B. Kaplan, A.V. Manohar, Phys. Rev. Lett. 56 Ž1986.
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w12x G.M. Shore, G. Veneziano, Nucl. Phys. B 381 Ž1992. 3.
w13x J. Bijnens, Int. J. Mod. Phys. A 18 Ž1993. 3045.
w14x H. Leutwyler, On the 1rN-expansion in chiral perturbation,
talk given at QCD 97, Montpellier, July 1997, hepphr9709408.
12 February 1998
Physics Letters B 419 Ž1998. 333–339
Higgs-mass dependence of two-loop corrections to D r
Stefan Bauberger
a
a,1
, Georg Weiglein
b
Institut fur
Am Hubland, D-97074 Wurzburg,
Germany
¨ Theoretische Physik, UniÕersitat
¨ Wurzburg,
¨
¨
b
Institut fur
¨ Theoretische Physik, UniÕersitat
¨ Karlsruhe, D-76128 Karlsruhe, Germany
Received 5 August 1997; revised 13 November 1997
Editor: P.V. Landshoff
Abstract
The Higgs-mass dependence of the Standard Model contributions to the correlation between the gauge-boson masses is
studied at the two-loop level. Exact results are given for the Higgs-dependent two-loop corrections associated with the
fermions, i.e. no expansion in the top-quark and the Higgs-boson mass is made. The results for the top quark are compared
with results of an expansion up to next-to-leading order in the top-quark mass. Agreement is found within 30% of the
two-loop result. The remaining theoretical uncertainties in the Higgs-mass dependence of D r are discussed. q 1998 Elsevier
Science B.V.
The remarkable accuracy of the electroweak precision data allows to thoroughly test the predictions
of the electroweak Standard Model ŽSM. at its quantum level, where all the parameters of the model
enter the theoretical predictions. In this way it has
been possible to predict the value of the top-quark
mass, m t , within the SM prior to its actual experimental discovery w1x, and the predicted value turned
out to be in impressive agreement with the experimental result.
After the discovery of the top quark the Higgs
boson remains the only missing ingredient of the
minimal SM. At the moment, the mass of the Higgs
1
Work supported by the German Federal Ministry for Research
and Technology ŽBMBF. under contract number 05 7WZ91P Ž0..
boson, M H , can still only rather mildly be constrained by confronting the SM with precision data
w2x. An important goal for the future is therefore a
further reduction of the experimental and theoretical
errors, not only in order to obtain stronger bounds
for the Higgs-boson mass but also to achieve improved sensitivity to effects of physics beyond the
SM.
Concerning the reduction of the theoretical error
due to missing higher-order corrections, in particular
a precise prediction for the basic relation between
the masses M W , MZ of the vector bosons, the Fermi
constant Gm , and the fine structure constant a is of
interest. This relation is commonly expressed in
terms of the quantity D r w3x derived from muon
decay. With the prospect of the improving accuracy
of the measurement of the W-boson mass at LEP2
0370-2693r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 7 0 - 2 6 9 3 Ž 9 7 . 0 1 4 5 8 - 5
334
S. Bauberger, G. Weigleinr Physics Letters B 419 (1998) 333–339
and the Tevatron, the importance of D r for testing
the electroweak theory becomes even more pronounced.
At the one-loop level the largest contributions to
D r in the SM are the QED induced shift in the fine
structure constant, Da , and the contribution of the
toprbottom weak isospin doublet, which gives rise
to a term that grows as m 2t . The SM one-loop result
for D r w3x has been supplemented by resummations
of certain one-loop contributions w4,5x. While QCD
corrections at O Ž aa s . w6,7x and O Ž aa s2 . w8x are
available and the remaining theoretical uncertainty of
the QCD corrections has been estimated to be small
w9,10x, the electroweak results at the two-loop level
have so far been restricted to expansions in either m t
or M H . The leading top-quark and Higgs-boson contributions have been evaluated in Refs. w11,12x. The
full Higgs-boson dependence of the leading Gm2 m4t
contribution was calculated in Ref. w13x, and recently
also the next-to-leading top-quark contributions of
O Ž Gm2 m2t MZ2 . were derived w14x.
In the global SM fits to all available data w2x,
where the O Ž Gm2 m2t MZ2 . correction obtained in Ref.
w14x is not yet included, the error due to missing
higher-order corrections has a strong effect on the
resulting value of M H , shifting the upper bound for
M H at 95% C.L. by ; q100 GeV. For the different
precision observables this error is comparable to the
error caused by the parametric uncertainty related to
the experimental error of the hadronic contribution to
a Ž MZ2 . w15x. In Refs. w10,16x it is argued that inclusion of the O Ž Gm2 m 2t MZ2 . contribution will lead to a
significant reduction of the error from the missing
higher-order corrections.
Both the Higgs-mass dependence of the leading
m4t contribution and the inclusion of the next-to-leading term in the m t expansion turned out to yield
important corrections. In order to further settle the
issue of theoretical uncertainty due to missing
higher-order corrections therefore a more complete
calculation would be desirable, where no expansion
in m t or M H is made.
In this paper the Higgs-mass dependence of the
two-loop contributions to D r in the SM is studied.
The corrections associated with the fermions are
evaluated exactly, i.e. without an expansion in the
masses. The results are compared with the expansion
up to next-to-leading order in the top-quark mass.
The relation between the vector-boson masses in
terms of the Fermi constant reads w3x
ž
2
MW
1y
2
MW
MZ2
/
pa
s
'2 Gm Ž 1 q D r . ,
Ž 1.
where the radiative corrections are contained in the
quantity D r. In the context of this paper we treat D r
without resummations, i.e. as being fully expanded
up to two-loop order,
D r s D rŽ1. q D rŽ2. q O Ž a 3 . .
Ž 2.
The theoretical predictions for D r are obtained by
calculating radiative corrections to muon decay.
From a technical point of view the calculation of
top-quark and Higgs-boson contributions to D r and
other processes with light external fermions at low
energies requires in particular the evaluation of twoloop self-energies on-shell, i.e. at non-zero external
momentum, while vertex and box contributions can
mostly be reduced to vacuum integrals. The problems encountered in such a calculation are due to the
large number of contributing Feynman diagrams,
their complicated tensor structure, the fact that scalar
two-loop integrals are in general not expressible in
terms of polylogarithmic functions w17x, and due to
the need for a two-loop renormalization, which has
not yet been worked out in full detail.
The methods that we use for carrying out such a
calculation have been outlined in Ref. w18x. The
generation of the diagrams and counterterm contributions is done with the help of the computer-algebra
program FeynArts w19x. Making use of two-loop
tensor-integral decompositions, the generated amplitudes are reduced to a minimal set of standard scalar
integrals with the program TwoCalc w20x. The renormalization is performed within the complete on-shell
scheme w21x, i.e. physical parameters are used
throughout. The two-loop scalar integrals are evaluated numerically with one-dimensional integral representations w22x. These allow a very fast calculation
of the integrals with high precision without any
approximation in the masses. As a check of our
calculations, all results presented in this paper have
been derived using a general Rj gauge, which is
specified by one gauge parameter j i , i s g ,Z,W, for
each vector boson, and we have explicitly verified
that the gauge parameters actually drop out. As input
S. Bauberger, G. Weigleinr Physics Letters B 419 (1998) 333–339
parameters for our numerical analysis we use
unless otherwise stated the following values:
a Ž M Z2 .y 1 ' Ž1 y Da .r a Ž0. s 128.896, Gm s
1.16639 10y5 GeVy2 , MZ s 91.1863 GeV, m t s
175.6 GeV, m b s 4.7 GeV.
In order to study the Higgs-mass dependence of
the two-loop contributions to D r we consider the
subtracted quantity
D rŽ2.,subtr Ž M H . s D rŽ2. Ž M H .
yD rŽ2. Ž M H s 65 GeV . ,
335
contributions to the transverse part of the W-boson
self-energy S W Ž p 2 . and the counterterm d Z vert to
top
Ž M H . reads
the Wy ene vertex, the quantity D rŽ2.,subtr
top
D rŽ2.,subtr
Ž MH .
s
W
SŽ2.
Ž 0 . y Re SŽ2.W Ž M W2 .
2
MW
vert
q 2 d ZŽ2.
W
W
q2 Ž SŽ1.,t
Ž 0 . y Re SŽ1.,t
Ž M W2 . .
Ž 3.
where D rŽ2.Ž M H . denotes the two-loop contribution
to D r.
Potentially large M H -dependent contributions are
the corrections associated with the top quark, since
the Yukawa coupling of the Higgs to the top quark is
proportional to m t , and the contributions which are
proportional to Da . We first consider the Higgs-mass
dependence of the two-loop top-quark contributions
top
Ž M H . which deand calculate the quantity D rŽ2.,subtr
notes the contribution of the toprbottom doublet to
D rŽ2.,subtr Ž M H ..
From the one-particle irreducible diagrams obvitop
Ž M H . that
ously those graphs contribute to D rŽ2.,subtr
contain both the top andror bottom quark and the
Higgs boson. It is easy to see that only two-point
functions enter in this case, since all graphs where
the Higgs boson couples to the muon or the electron
may safely be neglected. Although no two-loop
three-point function enters, there is nevertheless a
contribution from the two-loop and one-loop vertex
counterterms. If the field renormalization constants
of the W boson are included Žwhich cancel in the
complete result., the vertex counterterms are separately finite.
The technically most complicated contributions
arise from the mass and mixing-angle renormalization. Since it is performed in the on-shell scheme,
the evaluation of the W- and Z-boson self-energies
are required at non-zero momentum transfer. 2
Expressed in terms of the one-loop and two-loop
2
It should be noted at this point that in the context of this paper
the question is immaterial whether the mass definition of unstable
particles at the two-loop level should be based on the real part of
the complex pole of the S matrix or on the real pole. For the
contributions investigated here both definitions are equivalent.
=
W
W
Ž 0 . y Re SŽ1.,H
Ž M W2 . .
Ž SŽ1.,H
q2
q2
4
MW
W
W
vert
Ž 0 . y Re SŽ1.,t
Ž M W2 . . d ZŽ1.,H
Ž SŽ1.,t
2
MW
W
W
vert
Ž 0 . y Re SŽ1.,H
Ž M W2 . . d ZŽ1.,t
Ž SŽ1.,H
2
MW
vert
vert
q2 d ZŽ1.,t
d ZŽ1.,H
subtr
,
Ž 4.
where it is understood that the two-loop contributions to the self-energies contain the subloop renormalization. The two-loop terms denote those graphs
that contain both the top quark and the Higgs boson,
while for the one-loop terms the top-quark and the
Higgs-boson contributions are indicated by a subscript. The two-loop vertex counterterm is expressible in terms of the charge counterterm d Z e and the
mixing-angle counterterm d s W rs W ,
vert
d ZŽ2.
s d Z e ,Ž2. y
d s W ,Ž2.
sW
y d Z e ,Ž1.,t
q2
d s W ,Ž1.,H
sW
d s W ,Ž1.,t d s W ,Ž1.,H
sW
,
sW
Ž 5.
and similarly the one-loop vertex counterterm is
given by
vert
d ZŽ1.
s d Z e ,Ž1. y
d s W ,Ž1.
sW
,
Ž 6.
2
2
2
with s W
' 1 y cW
s 1 y MW
rMZ2 . For the Higgsdependent fermionic contributions the charge counterterm is related to the photon vacuum polarization
according to
AA
d Z e ,Ž2. s y 12 d ZA A ,Ž2. s 12 PŽ2.
Ž 0. ,
Ž 7.
S. Bauberger, G. Weigleinr Physics Letters B 419 (1998) 333–339
336
which is familiar from QED. The validity of Ž7. can
also be understood by observing that the contributions considered here are precisely the same as the
ones obtained within the framework of the background-field method w23x.
The mixing angle counterterm d s W,Ž2.rs W is expressible in terms of the on-shell two-loop W-boson
and Z-boson self-energies and additional one-loop
contributions,
d s W ,Ž2.
Fig. 1. Two-loop top-quark contribution to D r subtracted at
M H s65 GeV.
sW
sy
y
2
1 cW
2
2 sW
ž
W
Re SŽ2.
Ž M W2 .
2
MW
MZ2
top
Ž M H . amounts to about 10% of the one-loop
D rŽ2.,subtr
contribution, D rŽ1.,subtr Ž M H ., which is defined in
analogy to Ž3..
The other M H -dependent two-loop correction that
is expected to be sizable is the contribution of the
terms proportional to Da . It reads
/
MZ2
ZZ
d s W ,Ž1.,H Re SŽ1.,t
Ž MZ2 .
Da
D rŽ2.,subtr
Ž MH .
MZ2
sW
y
ZZ
Re SŽ2.
Ž MZ2 .
ZZ
d s W ,Ž1.,t Re SŽ1.,H
Ž MZ2 .
sW
y
y
d s W ,Ž1.,t d s W ,Ž1.,H
sW
sW
s 2 Da
,
Ž 8.
y2
W
W
SŽ1.,H
Ž 0 . y Re SŽ1.,H
Ž M W2 .
2
MW
d s W ,Ž1.,H
sW
where
s2 Da D rŽ1.,subtr Ž M H . ,
subtr
Ž 10 .
d s W ,Ž1.
sW
sy
2
1 cW
2
2 sW
ž
W
Re SŽ1.
Ž M W2 .
2
MW
y
ZZ
Re SŽ1.
Ž MZ2 .
MZ2
/
.
Ž 9.
In Ž4. – Ž6. the field renormalization constants of the
W boson have been omitted. In our calculation of
top
Ž M H . we have explicitly kept the field
D rŽ2.,subtr
renormalization constants of all internal fields and
have checked that they actually cancel in the final
result.
top
Ž M H . is shown in Fig. 1
The result for D rŽ2.,subtr
for various values of m t . The Higgs-boson mass is
varied in the interval 65 GeV F M H F 1 TeV. The
top
Ž M H . over this interval is about
change in D rŽ2.,subtr
0.001, which corresponds to a shift in M W of about
20 MeV. It is interesting to note that the absolute
value of the correction is maximal just in the region
of m t s 175 GeV, i.e. for the physical value of the
top-quark mass. For m t ; 175 GeV the correction
and can easily be obtained by a proper resummation
of one-loop terms w5x.
lf
,
The remaining fermionic contribution, D rŽ2.,subtr
is the one of the light fermions, i.e. of the leptons
and of the quark doublets of the first and second
generation, which is not contained in Da . Its structure is analogous to Ž4., but due to the negligible
coupling of the light fermions to the Higgs boson
much less diagrams contribute. The scalar two-loop
integrals needed for the light-fermion contribution
can be solved analytically in terms of polylogarithmic functions. They can be found in Ref. w24x.
The total result for the one-loop and fermionic
two-loop contributions to D r, subtracted at M H s
65 GeV, reads
top
Da
D rsubtr ' D rŽ1.,subtr q D rŽ2.,subtr
q D rŽ2.,subtr
lf
q D rŽ2.,subtr
.
Ž 11 .
It is shown in Fig. 2, where separately also the
one-loop contribution D rŽ1.,subtr , as well as D rŽ1.,subtr
top
top
Da
q D rŽ2.,subtr
, and D rŽ1.,subtr q D rŽ2.,subtr
q D rŽ2.,subtr
are
S. Bauberger, G. Weigleinr Physics Letters B 419 (1998) 333–339
Fig. 2. One-loop and two-loop contributions to D r subtracted at
M H s65 GeV. D rsubtr is the result for the full one-loop and
fermionic two-loop contributions to D r, as defined in the text.
shown for m t s 175.6 GeV. In Table 1 numerical
values for the different contributions are given
for several values of M H . It can be seen that the
top
Ž M H . and
higher-order contributions D rŽ2.,subtr
Da
D rŽ2.,subtr Ž M H . are of about the same size and to a
large extent cancel each other. The light-fermion
contributions which are not contained in Da add a
relatively small correction. Over the full range of the
Higgs-boson mass it amounts to about 4 MeV. In
total, the inclusion of the higher-order contributions
discussed here leads to a slight increase in the sensitivity to the Higgs-boson mass compared to the pure
one-loop result.
Regarding the remaining Higgs-mass dependence
of D r at the two-loop level, there are only purely
bosonic corrections left, which contain no specific
source of enhancement. They can be expected to
yield a contribution to D rŽ2.,subtr Ž M H . of about the
bos Ž
bos
same size as Ž D rŽ1.
M H .. 2 < subtr , where D rŽ1.
denotes the bosonic contribution to D r at the one-loop
bos Ž
level. The contribution of Ž D rŽ1.
M H .. 2 < subtr amounts
337
top
Ž M H . corresponding
to only about 10% of D rŽ2.,subtr
to a shift of about 2 MeV in the W-boson mass. This
estimate agrees well with the values obtained for the
Higgs-mass dependence from the formula in the
second paper of Ref. w7x for the leading term proportional to M H2 in an asymptotic expansion for large
Higgs-boson mass. The Higgs-mass dependence of
the term proportional to M H2 amounts to less than
top
Ž M H . for reasonable values of M H .
15% of D rŽ2.,subtr
top, Da
top
The result for D rsubtr
' D rŽ1.,subtr q D rŽ2.,subtr
q
Da
D rŽ2.,subtr can be compared to the result obtained via
an expansion in m t up to next-to-leading order, i.e.
O Ž Gm2 m2t MZ2 . w14,16x. The results for M W as a function of M H according to this expansion Žwithout
QCD corrections; m t s 175.6 GeV. are given in
Table 2 w25x. Extracting from Table 2 the corresponding values of D r and subtracting at M H s
top, Da ,expa Ž
65 GeV yields the values D rsubtr
M H . as results of the expansion in m t . These are compared to
top, Da Ž
the exact result D rsubtr
M H . in Table 3. In the last
column of Table 3 the approximate shift in M W is
given which corresponds to the difference between
exact result and expansion. The results agree within
top
Ž M H ., which amounts to a
about 30% of D rŽ2.,subtr
difference in M W of up to about 4 MeV.
In Table 4 the shift in M W corresponding to
D rsubtr Ž M H ., i.e. the change in the theoretical prediction for M W when varying the Higgs-boson mass
from 65 GeV to 1 TeV, is shown for three values of
the top-quark mass, m t s 170,175,180 GeV. The dependence on the precise value of m t is rather mild,
which is expected from the fact that m t enters here
top
Ž MH .
only at the two-loop level and that D rŽ2.,subtr
has a local maximum in the region of m t s 175 GeV
Žsee Fig. 1.. From the shift in M W given in Table 4
Table 1
The dependence of one-loop and two-loop contributions to D r on the Higgs-boson mass for m t s 175.6 GeV Žsee text.
M H r GeV
D rŽ1.,subtrr10y3
top
D rŽ2.,subtr
r10y3
Da
D rŽ2.,subtr
r10y3
D rsubtrr10y3
65
100
200
300
400
500
600
1000
0
1.42
4.12
5.90
7.24
8.31
9.20
11.8
0
y0.101
y0.282
y0.397
y0.474
y0.565
y0.663
y1.04
0
0.148
0.431
0.620
0.762
0.876
0.971
1.25
0
1.49
4.34
6.23
7.66
8.78
9.69
12.2
S. Bauberger, G. Weigleinr Physics Letters B 419 (1998) 333–339
338
Table 2
The results for M W as a function of M H Žwithout QCD corrections; m t s175.6 GeV. obtained via an expansion up to next-toleading order in m t w25x
M H r GeV
M W r GeV
65
100
300
600
1000
80.4819
80.4584
80.3837
80.3294
80.2901
the theoretical prediction for the absolute value of
M W can be obtained using as input one value of M W
for a given M H . From Ref. w26x we infer for the
subtraction point M H s 65 GeV the corresponding
values M W s 80.374,80.404,80.435 GeV for m t s
170,175,180 GeV, respectively. The accuracy of this
subtraction point, being taken from the expansion up
to O Ž Gm2 m2t MZ2 ., is of course lower than the accuracy of the shift in M W as given in Table 4.
In summary, we have discussed the Higgs-mass
dependence of the two-loop corrections to D r by
considering the subtracted quantity D rsubtr Ž M H . s
D r Ž M H . y D r Ž M H s 65 GeV.. Exact results have
been presented for the Higgs-dependent fermionic
contributions, i.e. no expansion in the top-quark and
the Higgs-boson mass has been made. The contribution associated with the top quark has been compared with the result of an expansion up to next-toleading order in m t . Agreement within about 30% of
the two-loop top-quark correction has been found,
which corresponds to a difference in M W of about
4 MeV in the range 65 GeV F M H F 1 TeV of the
Higgs-boson mass. The Higgs-dependence of the
Table 3
top, Da Ž
Comparison between the exact result, D rsubtr
M H ., and the
result of an expansion up to next-to-leading order in m t ,
top, Da ,expa Ž
D rsubtr
M H .. In the last column the approximate shift in
M W is displayed which corresponds to the difference between the
two results
top, Da
top, Da ,expa
M H r GeV D rsubtr
r10y3 D rsubtr
r10y3 d M W r MeV
65
100
300
600
1000
0
1.48
6.16
9.56
12.0
0
1.52
6.32
9.79
12.3
0
0.6
2.5
3.6
4.1
Table 4
The shift in the theoretical prediction for M W caused by varying
the Higgs-boson mass in the interval 65 GeV F M H F1 TeV for
three values of m t
M H r GeV
65
100
200
300
400
500
600
1000
D M W Ž M H .r MeV
m t s170 GeV
m t s175 GeV
m t s180 GeV
0
y22.6
y65.8
y94.5
y116
y133
y147
y185
0
y22.8
y66.3
y95.2
y117
y134
y148
y187
0
y23.0
y66.8
y96.0
y118
y135
y149
y188
light-fermion contributions leads to a shift of M W of
up to 4 MeV. The only missing part in the Higgs-dependence of D r at the two-loop level are the purely
bosonic contributions, which have been estimated to
yield a relatively small correction of up to about
2 MeV in the W-boson mass. Considering the envisaged experimental error of M W from the measurements at LEP2 and the Tevatron of ; 20 MeV, we
conclude that the theoretical uncertainties due to
unknown higher-order corrections in the Higgs-mass
dependence of D r are now under control.
We thank M. Bohm,
P. Gambino, W. Hollik and A.
¨
Stremplat for useful discussions. We also thank P.
Gambino for sending us the results displayed in
Table 2.
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Physics Letters B 419 Ž1998. 340–347
Two Higgs doublet models and CP violating Higgs exchange
in eqey™ ttZ
S. Bar-Shalom a , D. Atwood b, A. Soni
c
a
b
Physics Dept., UniÕersity of California, RiÕerside, CA 92521, USA
Theory Group, Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA
c
Physics Dept., BrookhaÕen Nat. Lab., Upton, NY 11973, USA
Received 9 July 1997
Editor: M. Dine
Abstract
Appreciable CP asymmetries Ž; 10%. can arise in the reaction eqey™ ttZ already at tree-leÕel in models with two
Higgs doublets. For a neutral Higgs particle, h, with a mass in the range 50 GeV Q m h Q 400 GeV, it may be possible to
detect a 2–3 sigma CP-odd effect in eqey™ ttZ in ; 1–2 years of running of a future high energy eqey collider with c.m.
energies of ; 1–2 TeV and an integrated luminosity of 200–500 inverse fb. q 1998 Published by Elsevier Science B.V.
A future high energy eqey collider running at
c.m. energies of 0.5–2 TeV, often referred to as the
Next Linear Collider ŽNLC., will no doubt serve as a
very useful laboratory for a detailed study of the
properties of the Higgs particleŽs. and that of the top
quark w1x. In particular, it may unveil new phenomena, beyond the Standard Model ŽSM. associated
with the top Yukawa couplings to scalar particleŽs..
Evidence of such new ttH couplings, if detected at
the NLC, can give us important clues about the
nature of the scalar potential and of the properties of
the scalar particleŽs..
In the SM, the scalar potential is economically
composed of only one scalar doublet. Even a mild
exten