Parity Violating analogue of GDH
sum rule
Leszek Łukaszuk, Nucl.Phys.A 709 (2002) 289-298)
Krzysztof Kurek & Leszek Łukaszuk,
Phys.Rev.C 70(2004)065204
Frascati, 11 February, 2005
Motivation
The
of p.v.
nucleon-meson
The knowledge
rising interest
in couplings
GDH suminrule
and its
(nucleon-nucleon)
forces
important
Q2 generalizations
has is
started
withfor
the new
understanding
non-leptonic,
weak hadronic
generation of the
precise
spin experiments.
interactions (p.v. couplings are poorly known).
 Polarized photon asymmetry in + photo-production
 near
Newthe
experiments
based
intense
threshold can
be aon
good
candidate to
1. the
measure
pion-nucleon
couling
polarizedp.v.
beams
of photons
giveh
also
 Similar
is expected
lowpart
energy
Compton
opportunity
to testfora the
weak
of photonscattering.
hadron interactions (parity violating, p.v.)
 h1 has been measured in nuclear and atomic
systems; the disagreement between 18F and
133Cs experiments is seen.


Asymptotic states in SM and the
limitations of considerations concerning
the Compton amplitudes
Collision theory and SM:
 Asymptotic states – stable particles (photons,
electrons and neurinos, proton and stable atomic
ions)
 Existence of unstable particles – source of concern
in Quantum Field Theory (Veltman, 1963,
Beenakker et al..,2000)
 Each stable particle should correspond to an
irreducible Poincaré unitary representation –
problem with charged particles, QED infrared
radiation → well established procedure exists in
perturbative calculus only. (Bloch-Nordsic, FadeevKulish, Frohlich, Buchholz et al.. 1991)
Asymptotic states in SM and the
limitations of considerations concerning
the Compton amplitudes



Forward amplitudes – no radiation
Strong interactions: no asymptotic states of quarks and
gluons in QCD (confinement). Physical states are
composite hadrons.
R.Oehme (Int. J. Mod. Phys. A 10 (1995)):
„The analytic properties of physical amplitudes are the
same as those obtained on the basis of an effective
theory involving only the composite, physical fields”
The considerations concerning Compton amplitudes will be
limited to the order  in p.c. part and to the order 2 in
the
p.v. part ( they are infrared safe and at low energies are
GF order contribution; massive Z0 and W or H bosons)
Dispersion relations and low energy
behaviour
Let’s consider forward Compton amplitude:
For Re() >0 we get the physical Compton amplitude;
For Re() <0 the limiting amplitude can be obtained applying complex
conjugation and exploiting invariance with respect to rotation :
Dispersion relations and low energy
behaviour
Coherent amplitudes (related to cross section):
crossing
Normalization (Optics theorem):
We shall not use P, C, T invariance
Dispersion relations and low energy
behaviour
Analyticity, crossing, unitariry  dispersion relation for amplitude f
Dispersion relations and low energy
behaviour
Low Energy Theorem (LET) for any spin of target:
P, K
A.Pais, Nuovo Cimento A53 (1968)433
I.B.Khriplovich et al..,
Sov.Phys.JETP 82(1996) 616
Sum rules for p.v. spin polarizabilities
and superconvergence hypothesis
P.v. analogue of GDH sum rule
Subtraction point is taken at  =0 and - due to LET –
we get the dispersion formula for fh(-)
Unpolarized target
Assuming superconvergence:
fh(-) () → 0 with →

¯¯¯¯¯¯¯
Parity violating analogue of GDH sum rule
GDH (p.c.) sum rule and p.v. analogue
of GDH sum rule
For ½ spin target the above formula is equivalent to:
Nucl.Phys.B 11(1969)2777
Anomalous magnetic moment
Electric dipole moment
(2+ 2)
The photon scattering off elementary
lepton targets
e → Z0e (solid line)
 → We (dotted - multiplied by 0.1)
e →  W (dashed - multiplied by 5)
e →
Z0 e
 →
We
e →
W
P.v. sum rule satisfied for every process
separately, also separately for left- and
right- handed electron target.
First time calculations done (for W boson)
by Altarelli, Cabibo, Maiami ,
Phys.lett.B 40 (1972) 415.
Also discussed by S. Brodsky and I. Schmidt ,
Phys.Lett. B 351 (1995) 344.
(for details see also:
A. Abbasabadi,W.W.Repko
hep-ph/0107166v1 (2001),
D. Seckel, Phys.Rev.Lett.80 (1998) 900).
Proton target
GDH measurement and the saturation:
experimental „point of view”
Saturation hypothesis for p.v. sum rule
Let’s consider sum rule in the form:
And define the F quantity:
Saturation hypothesis for p.v. sum rule
 Requirement that F() does not exceed prescribed small
value at  = sat determines saturation energy.
 The usefulness of such definition of saturation is based
on the assumption that there is no large contribution
to the sum rule integral from photons with energy higher
than sat .
 For the GDH on proton – according to experimental data
sat and F(sat ) can be estimated as follows:
sat  0.5-0.6 GeV and F(sat )  0.1 (10%), respectively.
The pion photoproduction models for
γN → p with weak interactions efects
taken into account

HBχPT
(J-W,Chen, X.Ji, Phys.Rev.Lett.86 (2001)4239;
P.F.Bedaque, M.J.Savage,Phys.Rev.C 62 (2001)018501;
J-W.Chen,T.D.Cohen,C.W.Kao, Phys.Rev.C 64 (2001)055206)

Effective lagrangian approach with one particle exchange
domination and with vertices structure taken into account.
(W-Y.P.Hwang, E.M.Henley, Nucl.Phys.A 356 (1981)365,
S-P.Li, E.M.Henley, W-Y.P.Hwang, Ann.Phys. 143 (1982)372)
Both approaches give similar results close to threshold.
In our paper (KK, LŁ, Phys.Rev.C) the effective lagrangian
approach has been used.
Contribution to the p.v. 0 and +
production amplitude according to
Hwang-Henley pole model
Additional contribution for
charged pion:
a) and b) – nucleon pole,
c) - + pole
a) , b) - nucleon pole ,c) , d) , e) , f) -  pole,
g), h) – vector meson poles
The effective Lagrangians characterizing
the couplings among the hadrons
(Hwang-Henley)
i = 1,2,3 and:
0
Parity violating couplings in
Hwang-Henley model






ρNN – (hρ1, hρ2, hρ3) ; izoscalar, izovector, izotensor
ωNN – (hω0, hω1) ; izoscalar and izovector
NN – h1
N - f , taken 1 (in units 10-7)
γN – μ*, („free” parameter: (-15,15), in units 10-7)
γρ - hE , („free” parameter: (-17,17), in units 10-7)
8 models have been considered
(B. Desplanques, Phys.Rep. 297,(1998)1).
The values of p.v. couplings (in models) are based on the caclulations of
the quark- quark weak interactions with strong interactions corrections,
symetry and exprimental data (hyperon’s decays) taken into account.
Parity violating coupling constants
The p.v. meson-nucleon coupling constants are
calculated from the flavour-conserving part of weak
interactions :
p.v. Hamiltonian
and strong interactions effects from QCD should be
accounted for.
(K label in table presented on next slide, more details in:
B. Desplanques, Phys. Rep. 297 (1998)1. )
Parity violating coupling constants
Ann.Phys.124(80)449


Nucl.Phys.A335(80)147
N.Kaiser,U.G.Meissner,
Nucl.Phys.A 489(88)671,
499(89)699,510(90)759
-7
K=1 - absence of
strong int. corr.
Factorization
approximation
SU(6)W
based on
chiral model
The cross sections and asymmetries
according to Hwang-Henley pole model
Cross sections and asymmetries
(or polarized cross sections) given
by sum of the products of
formfactors and relevant couplings
The unpolarized cross section for pion
photoproduction - good agreement with data.
Having couplings calculated for 8 considered models and the formfactors
taken from Hwang-Henley pole model the differences of the polarized cross
sections are calculated. The saturation hypothesis with saturation energy
sat = 0.55 GeV is assumed and „free” parameters hE and * are selected
to satisfy condition F (sat) < 0.1 .
Results
Results: „non-saturated” models

Models 2 and 3 do not satisfy the quick saturation
hypothesis for any hE and *




additional structure should be seen above 0.55 GeV
to satisfy sum rule;
If saturation energy shifted to 1 GeV then 100 pb
is expected for  in energy of photon between
0.55-1 GeV – quite large.
This might indicate that it is desirable to look for
p.v. effects in this region
Remaining considered models satisfy hypothesis;
additional measurements of asymmetries can help
to distinguish between different models
The asymmetries for different
„saturated” models.
Model 4
(A in 10-7 units, E in GeV)
Model 5
Results: „saturated” models
Combining the measurements of 0 and +
asymmetries together would allow to select
models or group of models.

Let’s define:
A0sat , A+sat , A0th , A+th are 0 and + asymmetries
for saturation and threshold energy region,
respectively.
Then:
A+sat >0 selects models 1 and 8; in addition
A0th > 0 (and/or A0sat < 0) → 1
A0th < 0 (and/or A0sat > 0) → 8

Results: „saturated” models
A+sat <-6*10-7 (large) selects 4 and 5; in addition
A0th  -2*10-7 → 5
A0th  0 → 4
-6*10-7 < A+sat <0 selects 1,4,6,7,8; in addition
A0th < 0(  -1*10-7) → 7
A0th  0 → 1,4,6,8 - then combinnig with A+th and A0sat:
A+th>1 and A0th <0 select (4 and 6) and (1 and 8)
Experimental feasibility
 The intensity and polarization of the electron beam at
JLab allow to produce an intense, circularly polarized
beams of photons from the bremsstrahlung process.
Ch.Sinclair et al.. Letter of intent 00-002, JLab.
B. Wojtsekhowski, W.T.H. van Oers, (DGNP collaboration),PHY01-05,
JLab, AIP Conference proceedings SPIN 2000, 14 –th International Spin
Physiscs Symposium, Osaka, Japan, October 16-21, 2000;
published June 2001, ISBN 0-7354-3.
The 12 GeV upgrade of CEBAF, White Paper prepared for the NSAC
Long Range Planning Exercise, 2000, L.S. Cardman et al..,editors,
Kees de Jager, PHY02-51, JLab.
Experimental feasibility
Taking 60 A current at 12 GeV electron beam and
For
energy
rangetarget
from we
0.137
GeV (threshold)
1mm
Au plate
calculate
the photonto 0.55 GeV
(saturation)
it reads
bremsstrahlung
spectrum as follows:
1.9*109 events/sec.;
0.137 – 0.3 GeV → 7*108 events/sec
0.4 – 0.55 GeV
→ 2.7*108 events/sec
Spectrum
of 9photons
 1/seems
- „bremsstrahlung”
sum
rule
type.
108 -10
events/sec
to be large but
the
same
For 1cm
long
hydrogen
therelevant
number of
9is liquid
rate 10
expected
in LHCtarget
and the
events
/sec. istechniques are feasible
detection
(E.Longo, Nucl.Inst. and Meth.A 486 (2002)7)
Experimental feasibility
To verify quick saturation hypothesis: sum rule ntegral
To
overcome
statisticsup
thetolarge
of events is needed
should
be measured
0.55 number
GeV and:
(signal
fluctuation
of total
 if the higher
resultsthan
comes
40 -110 pb
– theproduction):
hypothesis is not
satisfied - in this case one needs 1013 – 1014 events which
correspond to 6*103 - 6* 104 sec. of beam time;
 much smaller results would indicate the possibility of quick
saturation.
 example: model 5:
low energy contribution (up to 0.3 GeV) is positive: 20-28 pb,
saturation region (0.4-0.55 GeV) is negative: (-10)–(-14) pb,
It demands 4*1013 – 6*1013 and 1.5*1012 – 4.5*1012 events,
respectively. Corresponding beam time:
6*104 – 8.5*104 and 6*103 - 1.7*104 sec.
Concluding remarks



The sum rule has been checked within lowest order
of the electroweak theory for the photon-induced
processes with elementary lepton targets. It would
be interesting to check this sum rule in higher
perturbative orders.
In analogy with observed feature of GDH sum rule
on proton the quick saturation hypothesis has been
formulated.
8 models with different sets of p.v. couplings have
been analyzed in the frame of effective lagrangian
and pole model approach
Concluding remarks



Models with the largest p.v.pion couplings h1 do not
saturate below 0.55 GeV and the contribution from
higher energies cross sections are needed
It is argued that the measurements of the 0 and +
asymmetries at the threshold and close to saturation
point allow to distinguish between „saturated” models
(p.v. couplings)
The verification of our predictions seems to be
experimentally feasible with the beam time of the
order of 105 sec. in the near future experimental
facilities (JLab)
SU(6)W





Bałachandram, Phys.Rev. 153 (1967) 1553
S.Pakwasa, S.P.Rosen, Phys.Rev. 147 (1966)1166
SU(6)W – subgroup of SU(12), all transformations
which leave untouched 0 and 3
Decomposition: SU(3)XSU(2)W
SU(2)W – weak isospin
Generators: ik 5 (SU(2)W)
SU(6)W – symmetry related to fixed direction;
useful in description of two-body decays
Factorization: matrix element factorizes into two parts:
Matrix element of current between vacuum and meson and
Matrix element of another currents between nucleons