Nuclear Physics B272 (1986) 457-489
© North-Holland Publishing C o m p a n y
A THEORY OF LIGHT QUARKS
IN THE I N S T A N T O N V A C U U M
D.I. D Y A K O N O V and V.Yu. PETROV
Leningrad Nuclear Physics Institute, Gatchina, Leningrad 188350, USSR
Received 6 May 1985
(Revised 31 July 1985)
A new m e c h a n i s m of spontaneous chiral symmetry breaking in the instanton v a c u u m of Q C D
is proposed. It is based on the delocalization of zero fermionic modes in the pseudoparticles'
medium. This medium exists due to the effective repulsion of pseudoparticles discovered by us
previously. We build a systematic theory of light quarks in such a medium. We find the quark
propagator in the instanton v a c u u m and demonstrate that the quark acquires an effective momentum-dependent mass. It m e a n s that the approximate chiral symmetry of Q C D breaks down
spontaneously, and we find the value of the chiral condensate (t~4,).
Further on, we calculate various current-current correlation functions in the instanton vacuum
and demonstrate explicitly the appearance of a massless pion pole in the pseudoscalar and axial
channels. We calculate the pion constant f~ and its charge radius r~. All observables are expressed
through the average size of pseudoparticles t~ and the average distance between them ~ these
quantities being calculable from a variational principle using the Q C D A parameter. The obtained
numerical values of Meet, (g~b), f~, r~ are in good accordance with the phenomenology.
We also construct the effective chiral langrangian for pions; this contains all powers of
derivatives of the pion field.
I. Introduction
It would not be an exaggeration to say that it is the spontaneous breaking of
chiral symmetry (SBCS) that determines the character of the world of strong
interactions. Indeed, owing to the SBCS, pions are light pseudo-Goldstone bosons
[1], and nucleons are heavy [2] - presumably being fermionic solitons of the 3'5
phases of the chiral condensate (~b) [3]. Therefore, the main particles of which the
matter is built (including nuclei, possibly to be understood as an ensemble of chiral
solitons) are due to the dynamics of the SBCS.
There have been attempts to obtain SBCS in a one-gluon-exchange model [4]
and by summing up leading order terms in the Q C D perturbation theory [5]. It
follows from these references that the SBCS occurs when the coupling as is
sufficiently strong - of the order of unity. In this case, unfortunately, the methods
used in refs. [4, 5] are not justified.
Meanwhile, in Q C D there exist gluon field fluctuations which are of a nonperturbative nature, namely instantons [6]. Taking the instantons into account
457
458
D. L Dyakonov, V. Yu. Petrov / Light quarks
drastically changes the situation as compared to the perturbation theory approach,
in particular with respect to the SBCS. Immediately after the discovery of instantons
it was realized that if instanton-type fluctuations of the gluon field are dominant in
the Q C D vacuum, m a n y of the features of the strong interactions find a natural
explanation. Among those are a clue to the U~ problem [7-9], the appearance of
non-perturbative gluon condensates (F~},
2
(F3), etc. [10], and the possibility of
chiral symmetry violation [9, 11-13]. However, at the end of the 1970s the instanton
calculus was plagued by severe infrared problems resulting from the divergences of
the integrals over the sizes of the instantons. Therefore, the above references compiled
a set of beautiful ideas, rather than providing a quantitative theory.
More recently, Shuryak [ 14] expanded instanton ideas to other strong-interaction
phenomena. Moreover, he has found out the basic characteristic of the "instanton
liquid". To confront the phenomenological needs, he argued, the instanton medium
should be relatively dilute, with the average distance between pseudoparticles being
/~ = (200 MeV) -~ and the average size ~ = (600 MeV) -t. Shuryak demonstrated that
an instanton vacuum with such characteristics can explain the values of both gluon
and quark condensates, the masses of the pseudoscalar nonet, and many other
quantities of interest [14].
Thus, lately much evidence has been accumulated in favour of the instanton
vacuum. However, a quantitative theory of such a vacuum was absent.
In this situation we suggested that the instanton vacuum should be studied by
means of the Feynman variational principle [15]. The idea was to calculate the
Q C D partition function on a trial ansatz, it being a superposition of instantons and
anti-instantons (hereafter denoted by I, I) together with quantum fluctuations around
this ansatz. The partition function was then maximized in the free parameters and
functions of the ansatz. We observed [16] an effective repulsion of I's and I's, which
led to the stabilization of the instanton medium at /~/~ = 3 (for the SU3 cotour).
The infrared problems seem to be removed from the theory; the effective coupling
constant does not grow above the value as/27r-~o. We have found the gluon
condensate ( F ~2 ) , the topological susceptibility (Qt2} and other fundamental quantities as two-loop renormalization-invariant and also scheme-invariant combinations
of the gauge constant and the cut-off - numerically in reasonable agreement with
phenomenology. Furthermore, a variation with respect to the pseudoparticle form
seems to give rise to a mass gap for glueballs [16].
We conclude that the use of the variational principle leads to a reasonable
quantitative theory of the instanton vacuum. Therefore, there is a possibility of
investigating more delicate questions arising when one places light quarks into the
instanton medium. The present paper deals with this problem.
Our logic is as follows. The Yang-Mills sector of the theory generates topological
gluon fields of the I i type (from the variational principle we have learned that such
configurations are necessarily present if only because they lower the vacuum energy
as compared to the perturbative one). The properties of the instanton medium are
D,I. Dyakonov, V. Yu. Petrov / Light quarks
459
basically determined by the gluon sector of the theory. When one switches in light
quarks their high-momentum component renormalizes somewhat the statistical
weight of each pseudoparticle and thus determines, together with the gluon sector,
the statistical ensemble. As to the quark low-momentum component (dominated by
the would-be zero eigenmodes of individual pseudoparticles) it should be considered
in the given statistical ensemble of pseudoparticles. It is the low-momentum quark
field component that is responsible for the chiral symmetry breaking. The feedback
of this component to the medium's properties comes to an overall change of the
chemical potential for pseudoparticles.
For the sake of simplicity we choose to work with a simple ansatz, viz., a sum of
I's and i's. In fact, our results are much more general: actually, we need only that
the vacuum should be populated by singular topological gluon fields which are
characterized by the average density of singularities N / V (4)---/~-4, and by the
average radius fi of the field outside the singularities. We shall demonstrate that in
the medium of such topological singularities the SBCS inevitably occurs, the order
parameter being ( ~ b ) ~ 1//~2~.
The new mechanism that we suggest for the appearance of a non-zero ( ~ ) is
based on the delocalization of zero fermion modes of individual pseudoparticles in
the medium. It is essentially a collective phenomenon and it arises only in the
thermodynamical limit N-> oo, V (4) -> co, ]~r/V-=/~-4 fixed, where N = N+ + N is
the total number of I's and i's. A crude physical description of the mechanism
together with some estimates for (0~b) has been published [17]. In this paper we
proceed in a more systematic way. We start with the QCD partition function in the
instanton medium (sect. 2), and derive the quark propagator in this medium (sect.
3). In sect. 4 we show that the massless free-quark pole l i p 2 cancels out and that
the quark acquires an effective momentum-dependent mass M ( p ) . We calculate the
condensate (~b). In sect. 5 we find the quark determinant in the background field
of the instanton medium.
Since the chiral symmetry is broken, massless Goldstone bosons (here, pions)
should emerge in the theory. The most straightforward and accurate way to see
them is to consider the two-point correlation functions of the mesonic operators
~F0. A systematic way of calculating such correlation functions in the instanton
vacuum is presented in sects. 6, 7. We demonstrate explicitly the appearance of a
massless (or near massless) pion pole in the F = T5 and F = Y~Y5 channels. The
residue of the pole is directly related to the f~ constant, and we calculate it. Further
on, we consider the three-point correlation functions and find the charge radius of
the pion r~. We also check that the ~r°-~ 7T decay occurs in accordance with the
standard current algebra (sects. 8, 9). Our approach paves the way for calculating
the complete chiral langrangian (sect. 10), though this subject remains beyond the
present paper. We also leave for a subsequent publication the calculation of the
singlet 7' mass and a way out of the quenched approximation which, actually, is
used in this paper.
D.L Dyakonov, V. Yu. Petrov / Light quarks
460
2. QCD partition function
Let us consider Ns flavours of light quarks interacting with the gluon field
A . ( x ) - = . ~ . ( x , 3 ` I ) + B . ( x ) , where B.(x) is the quantum field and A.(x, 3`1) is the
classical background field which is a superposition of N÷ I's and N_ I's (3`1 denotes
the set of collective coordinates of the Ith pseudoparticle: its centre z1., the size pl
and the SUN (CO1OUr) orientation matrix UI):
N+
N
A.(x, y)= }~ AS(x , y,)+ ~ A~u(x, 3q),
(1)
I=l
1=1
1
--
2
a~(x, y,)--2ii
Ut(%'r.
- + , (x
p,
-- +--TuT")UI
(X zl)~
ZI) 2 ( X - - Z I ) 2 " q - R 2'
i
1
(2)
2
A.(x, 3q)=~ Ui(r.%
+ - , (x-N),.
+ - "c',r,)Ui
p~
( x - zO 2 (x
-z~) 2 + m2.
(3)
Here r~ are Nc × Nc matrices (N~ is the number of colours) with (r, :vi) matrices
in the left upper corner ('r are the usual 2 x 2 Pauli matrices); the other elements
are zero. Eqs. (2) and (3) are the expressions for the instanton and the anti-instanton,
respectively, in the singular gauge for the arbitrary SU(N¢) colour group.
The partition function in the euclidean formulation is
Z=IDB.
exp [
1
2g2(M)
I d4x Tr F2~(A+ B)I f d~ D~ *
xexp[fd4x~O*(i~(.4+B)+im)~].
(4)
(For a more accurate definition, including gauge fixing, ghosts and the extraction
of the collective coordinates 3', see ref. [16].) This expression should be regularized
and normalized. Following 't Hooft [7] we shall regularize (4) by the Pauli-Villars
method, dividing Z by the same expression with a mass M ~ oo for B, and O fields,
and normalize to the perturbative partition function with fi,, = 0. In the one-loop
approximation we have [9, 16]
Zl-toop
reg .....
1
-N+~N
~v..
×
f N++N
3 I=lll d3'~J(3'i) e -~(o)U~°'(y)
det (iV(3') + ira) det (i0+ im)
det (i~+ ira) det (iV(y) + iM)'
V(3') ~- 3".(0. - iA. (y)).
(5)
Here J(3`i) is the (factorized)jacobian for the collective coordinates of the Ith
pseudoparticle calculated in refs. [7, 18]:
d3`iJ(3'l) =
d4 zl( dpi/ o~) d UI(pIA ) HNj3 CN~[ [J(pl) ]2N'( M / A ) 2NJ3,
CNc = 4.66 exp ( - 1.68 Arc)/7r2(N~ - 1) I (Arc - 2) t,
87r a
fl(P,) =-g2(p,) = (~3'N~ - ~ N ) In (lip,A),
A
=
Apau|i_Villars.
(6)
D.I. Dyakonov, V.Yu. Petrov / Light quarks
461
The factor exp (-flUint(y)) which takes into account both classical and quantum
interactions of instantons and which leads to the stabilization of the pseudoparticles,
was derived and investigated in ref. [16].
Let us rewrite the normalized and regularized fermion determinant entering (5)
as a prcoduct of determinants over low and high eigenfrequencies (in respect to an
as yet free mass parameter Mi):
Det---- Detlow Dethigh
det ( iV + im) det ( iO+ iM 1) det ( iV + iMp) det ( iO+ iM)
--- det (iO+ ira) det (iV + iMO det (iO+ iMp) det (iV + iM)"
(7)
We shall see below that a natural choice of the division between high and low
eigenfrequencies is, roughly speaking, M ~ - 1 / / 5 , t~ being the average size of
pseudoparticles. Then Dethig h c a n be written as a product of determinants calculated
in the background field of individual pseudoparticles, A correction due to nonfactorization is completely under control and can be estimated by the quasiclassical
method of ref. [ 19]. It is small if the packing fraction of the instanton medium/54//~4
is small. We have
Dethigh =
N++N
1]
(Mpl)-2N//3FN/(MlPl)÷O(fi4/R4),
(8)
I=l
where the function F(t) is known in two extreme cases, t >> 1 [20, 19] and t ~ 1 [12]:
F ( t ) = ~ t 2 / 3 ( 1 - ~ ( 1 / t 2 ) + "''),
[1.34t(1 + t2 In t2+ . . . ),
t>>l
t,~l.
(9)
We can now rewrite (5) as
z~-,oop _
--reg .....
- -1
N+ ! N_ 1
I N+~N_d4zt d Ui(dfli/ p5)(piA ) l l
× 3 ( p i ) 2N
Nc/3--2Nf/3 C N c
I= 1
e-~(o)u~nt(V)FN/(Mlpi)Det~ow (y, m, M 0 .
(10)
The problem, thus, is to calculate Dehow in the instanton medium and average
it over the ensemble of pseudoparticles whose grand partition function is given by
eq. (10). Let us note that eq. (10), should not, by construction, depend on the
arbitrary parameter M , . However, we are going to treat Det~ow approximately taking into account only the would-be zero modes (this is justified if M1 is small
enough). Nevertheless, we will check that the dependence of (10) on Ma is extremely
feeble in a broad range of M I : the matching of the quasiclassical treatment of Dethigh
with the diagonalization of zero modes in Detlow is very smooth (see sect. 5 and
also ref. [17]).
The physics behind eq. (10) is quite transparent and was already mentioned in
the introduction. The high-frequency c o m p o n e n t of the fermion field in the instanton
D.L Dyakonou, V. Yu. Petrov / Light quarks
462
background "feels" individual pseudoparticles. Actually, it affects mainly the renormalization of the coupling constant - in accordance with the fermion contribution
to the l-loop G e l l - M a n n - L o w function. Since in the dilute medium Dethigh can be
written approximately in a factorized form (8), it can be said that the high-frequency
component of fermion fields contributes to the statistical weights of individual
pseudoparticles.
As to the low-frequency component represented by Dehow, it is absolutely
necessary to consider it in the background field of all pseudoparticles simultaneously.
Indeed, being factorized, Dehow tends to zero in the chiral limit (m ~ 0) as m NN,
owing to the zero modes in the background field of individual pseudoparticles. To
avoid this disillusioning and, in fact, wrong answer it is necessary first to diagonalize
the zero modes in the medium of pseudoparticles - however dilute this medium
may turn out to be. As a result, the zero modes are delocalized, and their spectral
density v(Z ), instead of being N - Ns6 (3.) as it is in the infinitely dilute limit, becomes
spread over some region of eigenvalues A, whose width is connected with the
averaged overlap integral of zero modes belonging to different pseudoparticles
[17, 21]. The chiral condensate (~4') is then directly related to the averaged spectral
density of the Dirac operator u(Z) at zero eigenvalues A = 0. The Dehow remains
finite in the chiral limit - see sect. 5.
After these general remarks (for more details on the diagonalization of zero modes
see refs. [17, 21]) we would like to proceed now in a more formal way starting from
eq. (10).
Let us introduce the Green function of a quark in the instanton background with
given positions, sizes and orientations of pseudoparticles (i, j stand for both colour
and spinor indices)
So(x, y) ~ (0i(x)0)(y)),
(iVx + im)oSjk(x, y) = --t~ik~(4)(X --y).
(11)
The free Green function is in these notations
o
f d"(x-y) e-~<P'~-Y)(q~i(x)qJ~(Y))o= (\m2+p2/o,
im+fi~
So(P)=
/~=P,,7,,.
(12)
The Dehow can be rewritten identically as
idm'
Dehow(% m, MI) = exp
1
d4xTr[S(x,x,m')-S°(x,x,m')]}
m/
(13)
(ms
is the bare mass of flavour f ) . Eq. (10) can be then rewritten as
1-loop
Zren
norm
'
1
N+!N
!.j ~dylW(yt, Ml)
-- -
x exp
i din'
L f=
1
mI
d"x Tr
[S(x, x, m')- S°(x, x, m')]
,
(14)
D.L Dyakonov, V.Yu. Petrov/ Light quarks
463
where dTitO(Tl, M1) stands for the statistical weight of a given configuration of
pseudoparticles, labelled by collective coordinates 3q and the line denotes averaging
with this weight. Using the inequality
eX >~ex
which holds for any positive weight, we arrive finally at the formula we are going
to work with:
1 I
z reg,oop
. . . . . >_
~ N+! N_~
xexp{~
f= 1
dTitO(T,,
M,)
f M' i d m ' f d 4 x T r [ S ( x , x , m ' ) - S ° ( x , x , m ' ) ] } .
(15)
rnf
Note that the averaging in the exponent means working in the quenched approximation (= no dynamical quark loops); its accuracy is, as usual, of the order of
Ns/No. Though we do not think that the quenched approximation is very good for
the real world with NT --- N~ = 3 and, moreover, it certainly mistreats some important
issues (the solution of the U~ problem being an example), in this paper we confine
ourselves to the averaging in the exponent. The reason is that the chiral symmetry
violation we are mostly interested in here occurs in the instanton medium already
in the quenched approximation. The solution of the U~ problem contained in eq.
(14) (not (15)!) will be a subject of a separate publication.
Let us specify the meaning of the line denoting averaging. It implies that a quantity
in question is calculated in a given configuration of N+ I's and N_ i's and then
averaged over all possible configurations with the weight w(y~, M~) (see eq. (10)).
We have investigated the statistical ensemble of pseudoparticles governed by w(y~)
in a previous work (ref. [16]). We have found that the distribution in the sizes PI
is given by a narrow ~-type function which is the more narrow the larger Nc we
take. In the rest of this paper we shall substitute all pi's by the average size of
instantons f3. Another conclusion which we can draw from ref. [16] is that the
correlation function for two pseudoparticles is small - of the order of the packing
fraction /54//~4< 1. We thus will average over the positions and orientations of
different pseudoparticles without any restrictions. The difference between the number
of I's and I's, N + - N _ , fluctuates as ,J-N, N = N+ + N_. Therefore, we can put
N+=N_ =~N. Finally, we note that at large Nc the density of pseudoparticles
N / V = O(Nc), while f7 = O(1) [16].
Thus, we shall imply the definition
N
A({T'}) -= ~ 2--V f
d4zldUl[I~N-~,
~ 2 v f d4zldUia(Zl, Ul,pi... ) pi=/,
(16)
where the integrations over the positions run over the whole 4-volume V, and S d U
means the integration with the SU(Nc) Haar measure normalized to unity.
464
D.L Dyakonov, V.Yu. Petrov/ Light quarks
3. Quark Green functions in the "pseudopartiele representation"
In this section we derive formulae for the main quantities of the theory, which
we call quark Green functions in the pseudoparticle representation.
Let us recall the definition of the quark Green function in a given background
configuration of pseudoparticles (11) and expand it in the external field ,4, = ~ l A~.
We have
S = - ( i V + ira) -~ = S o + S o , 4 S o + So,4So,4S'o+" • •
(17)
This series can be rearranged so that at first all powers of one pseudoparticle's
field A~ is summed up, then of two pseudoparticles' and so forth. The Green function
S will be thus written as a series in the exact Green function SI in the background
of individual instantons:
S = So~-E ( S l - S o ) Jr E ( S t - S o ) S o I ( S j - S o )
1
I#J
"~ E ( S l - S o ) S o I ( S j -- S o ) S o I ( S K -- SO) q-" " ""
(18)
I#J
J~K
In principle, the exact Green function in the one instanton field S~ is known in
the limit m-->0 [22]: it has a singular term in m coming from zero modes, and a
non-singular part which at the values of the momenta p > 1 / p reduces to the free
Green function. Since we are interested in the SBCS and the latter is governed by
the zero modes, we shall use a model for S~ - So keeping in it only the contribution
of the zero mode. This is a model which interpolates between high momenta where
SI ~ So and very small momenta, where $1 is dominated by the zero mode. In the
intermediate region of momenta p ~ 1/fi the results arising from this simplifying
model should be numerically wrong. Thus, we put
(S,-
So)o(x, y) ~
O,,(x)0~j(y)
-ira
(19)
'
where ~b~(x) is the zero mode for the Ith pseudoparticle: it is a right (left)-handed
Weyl spinor for the (anti)instanton [7]. Explicit expressions for the density matrices
O,(X)O*Jjk(Y) composed of the zero modes of I, J pseudoparticles are given in the
appendix. Inserting (19) in the general eq. (18) we get
S~j(x, y ) ~ S ° ( x , y ) + Y.
l
O , ( x ) O *Jj(y)
-im
÷~ tflli(X)[/, d4tffY~k(t)(_iS_irtl)kt~bjl(t)÷irn3,j ] ~bj*j(y)
÷.....
l.J --Ira
(20)
- lm
The term im,Su has been added here in order to avoid the restriction in the
summation over pseudoparticles (I ~ J) in eq. (18). Indeed, the square bracket is
D.L Dyakonov, V.Yu. Petrov/ Light quarks
465
zero for I = J owing to the chirality of zero modes, and we can sum over pseudoparticles without the limitations I # J, etc. - they are fulfilled automatically. As a result
the series (20) is just a geometrical progression, and we obtain
Sij(x, y) = S°(x, y) + E ~Ol,(x)
l,J
1
x
tkj*j(y),
(21)
where the matrix Tu stands for the overlap integrals of zero modes belonging to
different pseudoparticles:
TIj(Z I - - Z j , flI, flJ, U l , U j ) =
I datO~k(t- Z., p,, Ui)(iO,)kl~bjt(t- Zj, pj, Uj).
(22)
Explicitly, this overlap integral is given in the appendix. At large separations
between pseudoparticles
Tu ~ (z~2 p l zj)
p ~4 (z, - zj),. Tr ( U j r ,~ UI)
,
(23)
(the upper sign corresponds to the case I = anti-instanton, J = instanton, the lower
sign corresponds to the opposite case). We note that the second term in the
denominator of eq. (21) can be safely neglected in the chiral limit (m-+ 0) since it
is small in m as compared to TIj and it is small in the packing fraction of the
instanton medium as compared to the im6u term. Actually, the latter term could be
also neglected in the chiral limit; it is, however, useful to keep it temporarily as an
infrared regulator of the theory. In fact, in the chiral limit one has to find the inverse
matrix ( 1 / T ) u ; for technical reasons it is more convenient to find first ( 1 / T - im)u
and to put m = 0 in the final results.
To obtain the quark Green function in the instanton medium one has to average
(21) over the statistical ensemble of pseudoparticles. To this end it is convenient to
introduce Green functions in the "pseudoparticle representation":
1
( ~Tl / m ) lJ ~
---~tj-DIj(Zl-ZJ'im
UI , Uj),
- - F I J ( Z I -- Zj, U I , U j ) ,
I, J same charge
I,
(24)
J opposite charge.
In eq. (24) the line implies averaging over all pseudoparticles except the end-point
ones, I and J (for the definition of the averaging see the end of the previous section).
A more explicit definition of the D, F functions is
1
- .•
1--~-TIK
.
I~-TKL
.
1--~-TLj--+'''
ll'n K¢I,LCJtm
trn
lm
lm
F I j = I .T I j I - ~.
lm
DIj=
~
K±I,J
1TIK1TKjI+''"
IFH
tm
tm
(odd number of T)
(even number of T) .
(25)
D.L Dyakonov, V. Yu. Petrov / Light quarks
466
'=
I
J
'+
I
+g
J
K.~ I
/
K,t~
I
K,L. ]"
~
"
K
L J
K'~'~'~I K
{''1
L M N
• ~~
//
'-"
J
"\
K
L
K
M
J
K t,.~ "[
K
I..
M
L.
J
K
L
K
L
J
K,L.M,~[
K
L, K
M
K
N
J
Fig. 1
Our nearest goal is to calculate these fundamental quantities of the theory. Let
us introduce a graphical technique. An open circle will represent an instanton; a
dashed one, an anti-instanton. A line connecting two circles will be the overlap
integral T1j (22). A factor I/im should be attributed to each circle. A dashed line
entering a circle will imply averaging over positions and orientations of the corresponding pseudoparticle and summing over the number N± = ½N of such pseudoparticles. If a pseudoparticle enters the series (25) only once we shall denote this
possibility by a closed dashed line. If the same pseudoparticle happens several times
in (25) we shall connect all circles representing this pseudoparticle with a dashed line.
In these notations the series for, say, the FIj Green function given by eq. (25)
takes the form shown in fig. 1. This series can be summed up by a Dyson-type
equation given by fig. 2 where the bold-faced lines represent the exact F and D
functions (to be found). In contrast to fig. 1, all internal circles in fig. 2 carry a
factor im since the F and D functions of fig. 2 have, by definition (25), the factors
1~ira at the end points, and these factors should not be taken twice. For the same
reason a factor 1 is attributed to the first internal circle of fig. 2.
Let us note that all graphs with the intercepting dashed lines (the last term in fig.
2, giving an example) make an Arc times smaller contribution than the planar graphs
with no intercepting. They also are numerically smaller owing to a more complicated
integration over the angles in the relative dispositions of pseudoparticles. We shall
systematically neglect non-planar graphs as giving 1/N~ corrections to our results.
T:.
I
J
K
i
I
K
J
K
[
K
K
f
\
,,
3
\
..°
~"
I
K
K
J
k,b
Fig. 2
K
|
K
K
K
K
~"
D.L Dyakonov, V.Yu. Petrov / Light quarks
467
In principle, these corrections can be taken into account as perturbations. It is
important that they are universally small at all momenta in the Green functions
under consideration. As to the planar graphs they can be easily summed up.
Indeed, let us introduce a notation for the quantity D~j(z~-zj, UI, Uj) at I-=J:
t~ ~ Dti :- Dij(z I - zj, UI, Uj)]zI=Zj, UI=Uj.
(26)
The "mass operator" for the Dyson equation of fig. 2 can be written as a geometrical
progression in ira6, so that the Dyson equations takes the form
1
FIj(ZI--Zj, UI, U j ) : - 7 - T i j ( z l - z j ,
tm
lrn
Nf
d 4 z K d U K 7 TIK(ZI--ZK, UI, UK)
lrn
+-2V
1
X
UI, Uj) L
1 - ira6
DKJ(ZK--ZJ, UK, Uj),
(27)
Nf
D , j ( z , - z j , UI, Uj)=~-~
×- 1-
d4zKdUK--tm TIK(ZI--ZK' UI' UK)
1
ira6
Fro(zK -- zj, UK, Uj).
(28)
Let us note that these equations are highly non-linear since the quantity 8 (26)
is expressed through D I j . TO solve the Dyson equations it is necessary first to find
out the dependence of F, D on the orientation matrices. This can be easily done
from the first iterations of eqs. (27), (28). We pass to the momentum representation
F, D,j(p) = f d 4 ( z l - zj) e'('zJ-z0F, D I j ( Z j - zl) ,
d
and look for the solution in the form
FTI(p)
F i i ( p ) = Tr ( U i p + U * i ) f ( p 2 ) ,
DH, (p)
=
Tr ( UI,~2U*i)d(p2),
=
- T r ( U~p - U~)f ( p2) ,
(29)
D~w(p) = Tr ( Urn2 U~)d(P2).
Here p± = p ~ ' .~, where T~
± and ~2 are 2 x 2 matrices standing in the left-upper corner
of a Nc x Arc matrix with all other elements zero. Integrating with the Haar measure
over the orientation matrices Ut we use the formulae
dU=l,
I
dUUi~IUi~2 utI['U#IJ22=~
N ~1- I
d U U i k U ~ = - ~ t~t6k,
~1,~
+
12]]"
(30)
D.L Dyakonov, V. Yu. Petrov / Light quarks
468
Using the explicit expressions for the T~j matrix from the appendix we obtain a
system of algebraic equations for the scalar functions f(p2) and d(p2):
f(p) = -i,....,
d(p)
q~,2(p) _ i2 VN~ im 1 - im~ ~P'2(P)d(P)'
N
1
1
p2q<2( p )f( p ) ,
i Z VNc im 1 - irn~6
6=2
(31)
Id"
(32)
(-~P)4d(p).
One has first to express f ( p ) , d ( p ) through 6 and then find 6 from the consistency
eq. (32). It is easy to check that the consistency equation has a solution yielding
- 1 -
1
- im6
me,
e =O(m°).
(33)
Keeping this in mind it is convenient to express f, d through e (instead of 6) and
also through the function M ( p ) which, as we shall see in the next section, has the
meaning of the quark effective mass,
' 2,rr2eN/32
eN
2 ,2
M ( P ) = - P,-,~,c
~z~r
~ (p)=
VNc '
72~rze N
VN~4p 6 ,
P << 1/~fi
(34)
p >>1/~.
In terms of these quantities the solution of the Dyson equation (31) is
d(p2 )
VN~ 2i
M2(p)
Ne m 2 M 2 ( p ) + p 2'
f(p2)
VNc 2i
M(p)
Ne m 2 M 2 ( p ) + p 2'
(35)
and the self-consistency eq. (32) reads
f
d4p
MZ(p)
N
=
(1-me)
(2~r) 4 M 2 ( p ) + p 2 4VNc
(36)
which actually is an equation for the quantity e, or for M ( p = 0). From this equation
it follows that e, indeed, has a finite limit when m -->0. Parametrically, e ~ Nlc/2R2/p,
where/~-4 = N~ V. The latter quantity is directly related to the gluon condensate [16]:
N ~ ( F ~ / 3 2 7 r 2)
(200 MeV) 4 .
(37)
The latter value is taken from the phenomenological QCD sum rules [10] though
in principle it is calculable [16] from the QCD A parameter entering the partition
function (10). At the moment we prefer to use the phenomenological value (37).
D.L Dyakonov, V. Yu. Petrov / Light quarks
469
As to the value of t5/R, we have estimated that it does not change significantly with
the inclusion of light quarks and hence take
15//~ ~
(38)
found by us previously in the pure Yang-Mills case [16]. See also ref. [14].
Using the above values of /~ and 15 we find numerically from eq. (36) that at
Arc = 3 and m = O, e = (85 MeV) 1, and hence
M ( p = 0) ~ 345 M e V ,
(39)
which is very close to what one expects for the value of the constituent quark mass.
Note that parametrically both e and M ( 0 ) - - NSl/2R-215 are stable in Arc.
It is very important that M ( p ) has a finite limit at p - 0. The reason can be traced
back to the behaviour of the overlap integral T~j at large separations of pseudoparticles (see (23)), the latter befng determined by the asymptotia of the zero modes at
large distances, which is I / X 4. This circumstance is due to the fact that the instanton
field transforms as the (½,½,0) representation of the SU2(colour)xSU2(left)x
SU2(right) group, and has a unity topological charge. See also the discussion in
ref. [21].
4. Quark propagator in the instanton medium
Using the quark Green functions in the pseudoparticle representation (24) it is
easy now to calculate the quark propagator in the real-space representation (21).
In summing over the end-point pseudoparticles I, J in eq. (21) the following
possibilities should be considered separately: (i) I and J are the same instanton,
(ii) I and J are the same anti-instanton, (iii) I and J are different instantons, (iv) I
and J are different anti-instantons, (v) I is an instanton, J is an anti-instanton, (vi)
I is an anti-instanton, J is an instanton.
In cases (i), (ii) the quantity 6 given by eq. (26) is engaged. It should be taken
into account that the end-point pseudoparticle I can occur many times in the series
(25). Summing over all those possibilities produces a geometrical progression in
ira6, as it has been in the derivation of the Dyson eqs. (27), (28): 6 + im62+ (im6)26 +
. . . . ~/1-ira& In the cases (iii), (iv) and (v), (vi) it should be also taken into
account that the end-point pseudoparticles I, J can occur many times inside (25) see fig. 3; this gives a factor (1 - ira6) -2. Let us note that in iterating the end-point
pseudoparticles I and J they should not be interlocated - otherwise the corresponding
graph with dashed lines would not be planar, which gives an extra factor 1~No.
Fig. 3
D.L Dyakonov, V. Yu. Petrov / Light quarks
470
We, thus, can rewrite (21) in the momentum representation as follows:
So(p)=(_~)
N(1
N 2
+
1
~
+I
P)]
2
\l-ireS] II dUidUi[~li(p)~b}j(p)Fii(p)
\-~]
+ @I,(p)~b~j(p)FIII(P)].
(40)
From the definition (33) we have 1/(1 - imS) 2 = rn2e2, (1/im)+(8/1 - ireS) = -ie,
e = O(m°). Since F, D = O(1/m 2) (see eqs. (35)) we see that the propagator remains
final in the chiral limit. Using the expressions for the density matrices for zero
modes from the appendix, eqs. (29)), (35) for the F, D functions and also eq. (30),
we find the propagator:
S(p)=
_~ iM(p)
-~ p2
iM(p) M2(p)
/~ M2(p)
iM(p)+~ ~(colour)
p2 M2(p)+p2 p~ M2(p)+p2-M2(p)+p2
,
(41)
which is a propagator of a massive fermion with a momentum dependent mass
M(p) (depicted in fig. 4). Note the cancellation of the free-propagator massless
pole at p2= 0, which occurred in eq. (41)! At large momenta the propagator (41)
evidently tends to the free propagator since M(p) decreases rapidly with p.
M(p),MeV
300~
-~=200MeV
200
-- =
t00
0
015
I'0
I.'5
~'.0
"p, GeM
Fig. 4. The momentum dependenceof the quark eflective mass.
D.L Dyakonov, V. Yu. Petrov / Light quarks
471
The appearance of an effective mass M ( p ) for a massless quark clearly signals
the SBCS. Indeed, the chiral condensate is, by definition,
-
(~//l~)Minkowsky = --i(~//~l//)Euclid.
=-4N~ f
~-
i Tr S(x, x) = i I ~d4p Tr S(p)
d4p
M(p)
(2~r) 4 M2(p)+p 2"
(42)
Parametrically (~tp)~ N~/2R-2fi -~ =O(Nc). Using the values (32)-(39) we find
numerically ( ~ ) = - ( 2 5 5 MeV) 3 which should be compared with the
phenomenologically-known value (240-250 MeV) 3. It should be noted that in a
2-loop calculation (which includes not only the background field but also the
quantum gluons) the renorm-invariant combination is gS/q(~b). The above value of
the condensate obtained in a l-loop calculation corresponds to the scale of - 1 / ~
600 MeV.
We conclude that in the instanton vacuum the approximate chiral symmetry of
QCD is spontaneously broken. The order parameter (~b) decreases as 1//~ 2, /~
being the average distance between pseudoparticles. The phenomenon is due to the
delocalization of the would-be zero modes of individual pseudoparticles; it occurs
already in the quenched approximation. The obtained values of the effective mass
M and (0~b) agree well with the phenomenology provided the gluon condensate
2 is considered as being also due to the instanton configurations.
(F~,~)
5. Matching Dehow and Dethig h
The calculation of the quark propagator in the instanton vacuum enables us to
find the averaged spectral density of the Dirac operator u(A) and to calculate the
Detlow given by eq. (13). Using the definition (13) and eq. (21) for the quark
propagator we obtain in the quenched approximation
Detl°w:exp tlf:l~f I mf
M1i din' I
d4x ~ tPli(x) (
1 / ij~ji(x)
t }•
Since S d4XOli(X)tP*Ji(x) ~ (~iJ we get
Detlow = exp
i dm'~ (1~)}
[~f
If= 1fml.rl
I
t.f=~ mj=exp
-N ~I
i dm'N
im'
II
1
-~m'6
dm' e(m') ,
where e(m') is found from the selfconsistency condition (36). Let us neglect M2(p)
in the denominator of eq. (36) (this is consistent with the neglect of the overlap
D.L Dyakonov, V.Yu. Petrov / Light quarks
472
integrals for "same-charged" zero modes, see above) and recall the definition of
the effective mass (34). We obtain the following equation for e(m'):
1-m'e(m')
e2(m ')
N f d'*p
- ~c
~
#2"6.6
K2~(100MeV)2 '
Nc~ _
q,4(p)p2
whence
1 (~/m,2 + 4K2_ m,)"
e ( m ' ) = 2K--~
Integrating this formula over m' we get
[,/4+m~;+~
Detlow=IIy Lj~-~M~-2K2+M, K exp
M2K2-MI K
2\~++M21K2+M,K
~/4+m}K.---~+rnfK/j
}
.
We see that in the chiral limit rnf~ 0 the determinant remains finite. At M1K >>1,
Detlow goes as (MlK) N-N1.
Let us cite without derivation the result for the averaged spectral density of the
Dirac operator:
v(A)-----E6(A-A.)
n
N
"/~'K
•/
A2
1 _
,N~(A)
4 K 2 K~0
This formula demonstrates clearly that the would-be zero modes are spread over
the range 2K =O(/~-2). In the infinitely dilute limit ( / ~ )
the distribution of
eigenvalues tends to the 6-function.
Let us recall now the definition of Dethigh (eqs. (8), (9)). We take the lower
expression in eq. (9) since Mlp will be proved to be less than unity. Then in the
product Dehow × Dethigh the main dependence on M1 cancels out. In the parametrically wide range K < M1 < 1/# the product of the determinants has a very flat
maximum in M~ demonstrating a very smooth matching of high- and low-frequency
contributions to the total quark determinant. We find for the chiral limit
Detlow × Dethigh ~ (0.5
K#)N'lvr.
The answer is to replace the (mp)NNS factor arising in the infinitely dilute medium
of instantons.
Two remarks are in order here. First, the above formula for Dehow has a very
strong dependence on the quark mass ms . For example, taking the strange quark
with ms = 150 MeV - we obtain a three times larger determinant than in the chiral
limit. Second, it should be remembered that we deal here with the quenched
approximation. According to the inequality (15) this gives a lower estimate for the
quark determinant.
D.I. Dyakonov, V. Yu. Petrov / Light quarks
473
6. Two-point correlation functions in the iustanton vacuum
The calculation of the quark propagator performed in previous sections indicated
the SBCS and outlined the physics we are dealing with.* The next step should be
the calculation of the correlation functions of two gauge-invariant currents ~b*F0,
where F is a unity matrix in colour and one of the 5-Fermi matrices in spin. These
quantities are directly related to the spectrum of the theory, that is to hadrons.
The first thing we would like to check is the Goldstone theorem saying that there
should be massless pseudoscalar bosons (= pions) if the chiral symmetry is spontaneously broken. They should manifest themselves as poles a t p2 = 0 in the F = Y5
and F = Y~Y5 channels. O f course, we are not doubting a general theorem, but rather
we are eager to learn how it works: what is the mechanism for a massless boson
excitation in the instanton vacuum?
For the sake of simplicity we shall work in a theory with one quark flavour. In
this case, strictly speaking, there should not be a Goldstone particle owing to the
solution of the U1 problem (for a review see ref. [27]). However, the mass of the
singlet boson arises beyond the quenched approximation we are now dealing with.
Therefore, working with one flavour but neglecting quark loops actually means
dealing with non-singlet quark currents.
Let us consider the correlation function
IIF(p) = f d4x
=
f
e-'(PX)(~b*FO(x),tp*FO(O))
d4x
e -i(p':) Tr
S(x, 0)FS(0, x)F,
F = 1, Ys, %,, Y~"/5, cry,,.
(43)
Here S(x, y) is the quark Green function (21) and the line means averaging over
the statistical ensemble of pseudoparticles. Let us expand eq. (43) in the inverse
powers of the quark mass m. Using the notations of sect. 3 we can present the
correlation function as a sum of graphs of the type shown in fig. 5. The circles
denote pseudoparticles I, J, K , . . . , the lines connecting circles stand for the overlap
integrals T , , T j K , . . . (22); a factor 1~ira is attributed to all circles. Dashed lines
x
Fig. 5. A typical graph for the two-point correlation function.
* Strictly speaking, the quark propagator (in contrast to (rift0)) is not a gauge-invariant object. Eq. (41)
implies the Lorentz gauge for the background field.
D.L Dyakonov, V.Yu. Petrov/ Light quarks
474
connect the same pseudoparticles which are encountered in the expansion of eq.
(43). The crosses represent the external currents; a line entering (leaving) a cross
stands for the zero mode i//I (I//I5-).
It is convenient to divide all graphs for the correlation function (43) into three
groups: (a) graphs with no common pseudoparticles in the upper and lower lines
(fig. 6a); (b) graphs with one common pseudoparticle which, however, can be
repeated many times (fig. 6b); (c) graphs with two or more common pseudoparticles
in the upper and 16wer lines, each of which can be repeated many times (fig. 6c).
We denote the corresponding contributions to Flr(p) as Ho.I,2(P). The bold-faced
lines in fig. 6 stand for the exact Green functions in the pseudoparticle representation
DIj , Flj - see sect. 3.
The graphs of fig. 6a correspond to the factorization of eq. (43) into a product
of two averages. Including the free Green functions So from eq. (21) into IIo(p) we
get for the unconnected part of the correlation function
IIo(p) -= Hu . . . . . (P) = - J d4x e-i(Px) Tr S(x, O)FS(O, x)F
= -I
(dpl dp2) Tr S(p~)FS(p2)F,
(44)
where we have introduced a notation
I
(dp, dp2)---I
d4pl d4p2 t~(4)' "
(2-~)--~
,e,-p2+p)
(45)
and S(p) is the quark propagator in the instanton medium (41).
We next turn to the " B o r n " part HI(p) represented by fig. 6b. First of all, let us
note that the internal part of these graphs which includes a pair of pseudoparticles
"
R
I
@
l p
~)
q" "l
c)
Fig. 6. Three types of planar graphs for Fl(p). (a) unconnected, (b) with one common pseudoparticle,
(c) with two or more common pseudoparticles in the upper and lower lines.
D.L Dyakonov,V.Yu. Petrov/ Lightquarks
:: =
®
',
+
6
',r"
•
+
1
+
,
+
¢'o
475
"'"
+
'
Fig. 7. The definition of the double-dashed line.
I, three I's, four I's, etc., can be easily summed up. The bold-faced lines of fig. 6b
connecting the same pseudoparticles are, by definition, Du-= 8 (see eq. (26)). The
summation over the n u m b e r of the same pseudoparticles in the u p p e r and lower
lines corresponds to a double geometrical progression and gives (see eq. (33)):
im(1-im~)
(l+im~+...)(l+imS+.-.)=
= -e 2.
(46)
This summation is depicted in fig. 7 and its result (46) will be henceforth
represented by a double-dashed line.
Now, to calculate H i ( p ) it is necessary to take into account that both to the left
and to the right from the double-dashed line there can be an arbitrary number of
pseudoparticles which are not c o m m o n for the upper and lower lines. It is natural
to attribute the corresponding graphs to the vertices of the correlation function,
which we denote by YI(P) - see fig. 8. Analytically, the vertex function is
YI(P) =
I
{
'o [I dUkD~k(p2)~i(p2)
(dpi dp2) O~i(p2)-t 2V 1 - i m ~
+ I dUt, FIg(P2)tP~,(P2)]} F,~{$u(Pl)
duLd/Lj(pI)DLI(PO+FdUc
lJ
.IjF
J bcJ(P')FcI(P')']) (47)
+ 2vN 1 -imim~
Since the graphs of fig. 8 end with the same pseudoparticle I, in the m o m e n t u m
representation 7I(P) is an integral over m o m e n t a in a closed loop, which we have
written with a help of the notation (45). The factors ½N in eq. (47) result from the
summation over the intermediate pseudoparticles K(K,) and L(L), the factors 1 / V
come from the averaging over the positions of those pseudoparticles, the factors
im/1-im8 = im2e arise from the summation over the number of times the
pseudoparticles K ( K ) and L([.) are met. The matrix Fe is a unity matrix in colour
and one of the 5 Fermi matrices in spin.
%+(P2)
,", D(~/~K
+
Fig. 8. Graphs for the vertex functions.
D.L Dyakonov, V.Yu. Petrov/ Light quarks
476
The integration over the orientations of K(I() and L(L) can be easily performed
with the help of eq. (30) using the expressions for the D and F functions (29) and
for the density matrices from the appendix. We get
2 VNc f
x/MlM2
1 + Y5
Y'(P) = Ne J (dp~ dp2) (M~+p~)(M~+p~) Tr (/~, + iM,) ~
(fi2+ iM2)F
2 VN~
Ne r,(p)
ML2-= M(p,,2).
(48)
We defined here the " r e d u c e d " vertex F~(p) extracting the factor 2 VNc/Ne. The
factor VNc/Ne emerges here because we prefer to express the vertex through the
effective mass M(p) (34) and not directly through the Fourier transformations of
the zero modes; the coefficient 2 comes from the colour trace of the density matrices
for zero modes.
Eq. (48) gives the left vertex of the correlation function in the case when I is an
instanton. For the right-vertex one should replace p -, - p , and in the case of I being
an anti-instanton one has to replace 75-" - 7 5 . Note that F~(p) does not depend on
the orientation U~ of the pseudoparticle I. This is because we consider colourless
currents.
Summing over the common pseudoparticle I, averaging over its positions and
orientations and recalling the expression for the double-dashed line (46) we get for
the sum of the graphs of fig. 6b:
H~(p)=
N 2VN~
c N [F~(p)F~(-p)+FT(p)F~(-p)].
(49)
Finally, we turn to the graphs of fig. 6c. The vertex functions for H2(p) are,
evidently, the same as for II~(p). However, to calculate II2(p) we have to sum over
the number of common pseudoparticles J, K , . . . that occur in the upper and lower
lines. In the leading order in Nc this summation is presented by ladder graphs of
fig, 9. Indeed, it is easy to check that non-ladder graphs and also graphs with
"renormalized" vertices of the type shown in fig. 10 produce an extra factor 1/Nc
as compared to graphs of fig. 9, and also a small numerical factor arising from the
integration over the angles in the disposition of pseudoparticles involved; we
systematically neglect non-planar graphs.
Let us denote the sum of the ladder graphs of fig. 9 by Sin(p). The bold-faced
lines in fig. 9 are the exact Green functions D, F in the pseudoparticle representation,
and the double-dashed lines stand for the sum of the graphs of fig. 7 and correspond
p ~
I1
+ ~"
'1
':
::
Fig. 9
J(- ~1 'I
II
'11
II
"4-..
D.L Dyakonov, V. Yu. Petrov / Light quarks
477
Fig. 10. Examples of non-planar graphs.
to the factor - e 2 (see (46)). Since the D, F functions contain, by their definition
(25), the end-point factors 1~ira,to avoid double-counting factors (ira) 2 should be
attributed to the both ends of the ladder graphs of fig. 9.
The graphs of fig. 9 can be easily summed up with a help of a Bethe-Salpeter-type
equation (fig. 11). The integral term in this equation is, actually, two terms corresponding to the intermediate pseudoparticle L being an instanton or an antiinstanton. We have
S,K(p, U,, UK)=
m 4 e 4 j (dk,
dk2)DKi(k2, UK, U,)DIK(kl, UI, UK)
-m%2-~
dUL (dkadk2)DKL(k2,UK, UL)
x s,L(p, G , UPDLK(k,, UL, UK)
-m%2-~
dUc
(dk~
dk2)FKc(k2,U~, Ur.)
×S,L(p, U,, U•)F•K(k,, Uc, /dR).
(50)
This equation is written for the case when I, K are instantons. It is easy to see
that Si~: = SIK, SIR = SiK, hence only two quantities out of four are independent. In
the case of opposite-charged pseudoparticles I, K the Bethe-Salpeter equation reads:
SII~= rn4~ 4 ~ (dkl
2V
dk2)Ff(i(k2)FiF~(k,)
dUL
(dk~
dka)Fr~L(k2)SiL(p)FLr~(kO
-mae2~ f dUL f (dk, dk2)D~.~.(k2)S~L(p)DL~(k,).
(51)
In the general case the solution of the system of eqs. (50), (51) is rather cumbersome. However, we have noted previously that with the colourless currents the
K2 ~
K2~
G,,.....4~
I
II
~
r
i.
,(
II
II
II
Fig. 11. Bethe-Salpeter equation for ladder graphs.
478
D.L Dyakonov, V. Yu. Petrov / Light quarks
vertex functions F~a)(p) are independent of the orientation matrices Uia). Therefore,
only the quantities SIL, Sig averaged in the colour matrices of I, K(I() enter the
correlation functions in question. We shall denote these by S~r, S~g. It turns out
that these quantities also satisfy closed Bethe-Salpeter equations obtained from eqs.
(50), (51) by integrating over UI, UK(¢). The integration over the orientation of the
intermediate pseudoparticle L(L) is easily performed with the help of eq. (30), and
we come to a system of algebraic equations for the quantities SIK<~.)(P):
SIK(P)=-
2 Ve 2
N ~(p)+~(p)Sm(p)-~(p)S~g(p),
2 Ve 2
S~(P) =
N
~ ( P ) - ~ ( P ) S m ( P ) - @(P)SIK(P),
(52)
where
4VN~ f
M2~M~
J (dk, dk2) (M2+k2)(Mi+k~),
~(p)-~--~
4VNc f
M I M 2 ( k I , k2)
[ (Ok1 dk2) .-TS-,,
2--TU~,.-T7-, 2 ,
(M~+k~)(M2+k2)
d
~(p) ~-~
M,,2 = M(k,.:).
(53)
The solutions of eqs. (52) can be written as
SIK(P) + '-~II~(P) = VE 2 - f ~ ( P) ~ ~ ( P )
2
N 1-[~(p)~: ~(p)]"
(54)
To obtain II2(p) it is now necessary to add the left- and right-vertex functions
Yla)(P), yK(¢)(-p) and to sum and average over I(I) and K(I() pseudoparticles,
which gives a factor (N/2V) 2. We have
2v/
/
+ ~(P)$Ig(p)F¢(-P) + Fffp)gT¢(p)Fe.(-p)].
(55)
It is convenient to add up the H~ (eq. (49)) and the//2 pieces and thus to obtain
the connected part of the correlation function:
Hoonn(p)=- HI(p) + H2(P)
VNc
1
F
= N~--~-([C~(p) + r d p ) ] R - - ~ [ ~(-P) + rT(-p)]
+ [ r , ( p ) - r~(p)] ~
[Fi(p) - [ ( - p ) ]
,
(56)
where Fia)(+p) are the reduced vertex functions from eq. (48), and R±(p) are,
according to eq. (-54),
4 VSc f
MI M2[ M~MEq: (kl, k2)]
R±(p) = 1 - ~ j (dkl dk2)
2 2
2
(M1 + kl)(M2 + k22)
(57)
D.L Dyakonov,V.Yu. Petrov/ Lightquarks
479
The final answer for the correlation function in the instanton v a c u u m is
H(p) = IIo(p) + II,(p) + HE(p)
= Hu . . . . .
(P) + Hconn(P),
(58)
where Hu . . . . . is given by eq. (44) and H¢onn by eq. (56). Note that both contributions
are finite in the chiral limit m-> 0.
It is r e m a r k a b l e that the connected part (56) has a resonance form: the zeros of
R±(p) determine the positions of the poles in p2, and the reduced vertex functions
F I ( p ) give the residues o f the poles.
7. Goldstone particles in the instanton vacuum
Using the explicit expressions for the vertex functions F~a)(p) (eq. (48)) it is easy
to check the following:
(i) In the vector and tensor channels ( F = y . , tr,~) the vertex functions F~a ) are
zero. This m e a n s that in the a p p r o x i m a t i o n s used the correlation functions for these
channels are d e t e r m i n e d solely by the u n c o n n e c t e d part (44).
(ii) In the scalar channel ( F = 1) FI = F i # 0, so that in the scalar channel there
is a d y n a m i c a l pole connected with the zero of R+(p).
(iii) In the axial and p s e u d o s c a l a r channels ( F = %,3'5, 3'5) F~ = - F i , so that there
is a pole corresponding to the zero of R_(p).
Let us show that at m = 0 the quantity R_(p) given by eq. (57) has a zero at p2 = 0
c o r r e s p o n d i n g to a G o l d s t o n e particle, i.e. to a pion.
To this end let us recall the consistency equation (36) at m = 0 which we shall
n o w rewrite identically as
1=
4VNc f (dkldk2)[ ~ M E ~_1 M 2 ]
N _ _
M2+k 2 2 M 2 2 + k 2 J
"
Putting this expression in place of the unity in eq. (57) we obtain after some simple
algebra
g±(p)=
2VN~ f (dkl dk2) (Mlk2/z :i: M 2 k l / d , ) 2
N .J
(M2+k~)(M2+k~),
M1.2=M(k~.2).
(59)
d4k/(2~r) 4 we have
(
dM(pk) ) 2
M2)k, +½(M~+ M2)p~,] 2 ~ Mp~,-~ dlkl Ikl k.
Using the p a r a m e t r i z a t i o n kl,2, = k,~ :r ½p,, (dkl dk2) =
(MIk2, -
MEk~,) 2 = [ ( M 1 -
R_(p) goes to zero at p -- 0. At small p2 we obtain ( M = M ( k ) ) :
R_(p) =p2 2VN~ f d4k M2-1MIkl(dM/dlkl)+l(Ikl dM/dlkl) 2_
N .I (27r)
--------~
(M2+k2) 2
=tip2.
It is clear that
(60)
The vertex function (48) at small p for the p s e u d o s c a l a r channel ( F = 3's) is
I
Fx(0) -- -FT(0) = - 2
d4k
M(k)
(~)
(27r) 4 M 2 ( k ) + k 2 - 2Nc '
(61)
480
D.L Dyakonov, V. Yu. Petrov / Light quarks
where the expression for the chiral condensate (42) has been exploited. Inserting
(60), (61) into the general formula (56) we see that at small momenta the correlation
function in the pseudoscalar channel is
H5(p) =
c~-\-~-]
tip2,
p~O
(62)
exhibiting a massless pion's pole. We see that the Goldstone theorem works owing
to the same consistency equation (36) that led to the SBCS. This is as it should be
since the existence of a Goldstone excitation cannot depend on the details of the
chiral symmetry breaking, in particular, on the concrete form of the effective quark
mass M(p). Let us emphasize that the massless pole emerges only if consistent
approximations are used in treating the Green functions and the connected part of
the correlation function. In our case this is the planar-graph approximation (justified
at large N~). Corrections in 1/Nc should be taken into account simultaneously for
both quantities - otherwise the Goldstone theorem would be violated.
Let us calculate the pion axial constant f~ defined as
(OIJ~,s[Tr>=
if~p~,,
J~,s= d/%,YsO.
We remind the reader that for the pseudoscalar correlation function there is a
famous Ward identity (see e.g. ref. [23])
lim f d4x
p~0 d
e-i(PX)i(Tt~2imysO(x), (b2imy56(O))= - 4 m ( ~ 0 )
(63)
(we work temporarily in the Minkowsky space). Saturating the 1.h.s. of (63) at low
momenta with the pion pole one gets
1
p~01imf,~m~ m,~2_
p2f~m~=-gm((O0) •
(64)
From this relation one deduces the well-known current algebra result
2
4m(~0)
f~
(65)
Simultaneously, one obtains that at small momenta
I d4x e-i(px)i(T~i,ys~b(x), OiTs~b(O))=
4(5¢) 2
f2p2
•
Returning to the euclidean space according to the formulae
fid4XM=fd4XE,
p2 = _p2,
id/M=O*E,
Y0M= T4E,
"YSM---TSE,
%M = i%E,
(66)
D.L Dyakonov, V. Yu. Petrov / Light quarks
481
we have
_~5(p) :
I
4(~0}2
d4x e-'<'x)(~*V50(x), 0*3'56(0))= f ~ e 2 •
(67)
Comparing this with eq. (62) we obtain
N
f ~ = 4/3-~ = 8No
f d4k
(2~r)4
M 2 - ½ 1 k l M M ' + A k 2 M '=
( M 2 + ke)2
(68)
Let us recall that the effective mass is parametrically small in the packing fraction
of the medium, M ( 0 ) - tS//~ 2. Meanwhile, the scale at which M(k) changes is 1/fi;
dM/dlkl- o when Ikl-, o. Therefore, the integral in (68) is determined by a broad
range of parametrically small momenta 1/R ~ k ~ 1~ft. Calculating (68) with the
logarithmic accuracy we obtain
rr
M(O)~
j52
In T "
(69)
We see that f~ appears to be parametrically small as compared to the characteristic
hadron scale, the latter being determined in the instanton vacuum by the average
size of pseudoparticles, ~ = (600 MeV) -~. Thus, the experimental smallness off~--132 MeV finds a natural explanation in the instanton vacuum. Vice versa, the
smallness of f~ is an argument in favour of a sufficiently dilute instanton vacuum
in QCD.
Numerically, using eqs. (37)-(39) we find from eq. (68) f~ = 138 MeV (f~ =
142 MeV according to the approximate formula (69)) which is very close to the
experimental value of this fundamental quantity. Parametrically, f~---O(Nc), as it
should be.
When one switches in the small quark masses the pion pole shifts to a non-zero
value of p2. To take into account the quark masses (we put mu = rnd= m) it is
necessary to return to our starting formula for the Green function (20) and not to
neglect the masses in the numerator. One has also to take into account the O(m)
correction in the consistency condition (36). We cite without derivation the final
result for the quantity R_(p) at small momenta:
Vm
R_(p) = ---~- (t~O)+/3p2 + O(m 2) + O(p4).
(70)
Hence, the mass of the pion defined as the position of the pole in the HS(p)
correlation function is now
2
m~=
rn(~0) V
/3
N
4m(~)
f~
,
(71)
D.L Dyakonov, V. Yu. Petrov / Light quarks
482
which coincides with the current algebra result (65). Finally, we can find the value
of IIS(p) at p = 0 but m ~ 0 :
. VN~[(~b)'~ 2 N
aS(p=O)=N~-- -N ~ N~) -mV(~tp)
-
(~b)
-m "
(72)
which reproduces the result of the Ward identity (63). Formulae (68), (71), (72)
demonstrate the complete consistency of our understanding of the pion as being a
(pseudo) Goldstone excitation resulting from the SBCS in the instanton vacuum.
8. ~°.--,'7'y decay
We would like to make another consistency check and calculate the ~r° ~ yy decay
amplitude. It is well known that in the chiral limit this decay is completely determined
by the Adler axial anomaly [24]. It is instructive to follow how the triangle anomaly
is formed from the diagonalized zero modes in the instanton vacuum.
Let us consider a three-point correlation function of a pseudoscalar density and
two electromagnetic currents:
TS~,(klk2) = f
d4x d4y e-i(klx)-i(kEY)(~btT5~(O) ' ~btTg~b(X), 0*y~0(y))
= - - f dgx d4y
+Tr
e-i~k'x)--~k2Y~[Try~S(x, 0)y5S(0, y)y~S(y, x)
y~S(y, 0)ysS(0, x)y~S(x, y ) ] .
(73)
Here S(x, y) is the quark propagator (20) and the line means the averaging over
the statistical ensemble of pseudoparticles. To get the amplitude of the ~r°-~ y y
decay we have to calculate (73) and to extract the pole contribution of the pion:
T~(k, k2) p-~O2(~b)f.~ 12 A(k,, k2)e,~k]~k2~,
p = k, + k2.
(74)
The first factor here is the amplitude for the creation of a pion from the vacuum
by the current ~b%/5~b (see eq. (67)). Taking into account the electric charges of
quarks entering the ¢r° we get for the physical amplitude
(y(k~)y(k2)l ~"o) = x/~[(~)
T 2 2 - (~)
~ 2]4~raA(k,,
k2)e~kl~k2t3,
(75)
whence the decay width in the chiral limit is
3
m~ / 4era\
F ( 7r°~ YY)= ~--7--/-~--~/
o~r
\3vz/
2
,
IA(k,,
kgl~,,=k2=o-
(76)
D.L Dyakonov, V. Yu. Petrov / Light quarks
483
From the Adler anomaly it follows [24] (the factor i is due to the euclidean
formulation):
iNc
A(0, 0) = 2rr2f .
(77)
Our goal now is to reproduce eq. (77) from the calculation of (73) in the instanton
vacuum. This is done similarly to the calculation of the two-point correlation
functions. We n o t e that to extract the pion contribution in the T5 channel it is
necessary to consider connected (i.e. ladder in the leading approximation in No)
instanton graphs in this channel. As to the both vector channels, only unconnected
graphs survive there, as has been noted in sect. 7. For this reason the propagator
S(x, y) in (73) should be averaged independently and is, thus, replaced by the exact
propagator in the instanton vacuum (41). The resulting graph for the three-point
correlation function is presented in fig. 12. The instanton ladder of fig. 12 starts
with the vertex function FI(I)(p) (see eq. (48)) and ends with the same vertex
function, but with the two vector vertices and the propagator between them included.
Denoting p = kl+k2, q =½(k2- kl) we have for the graph of fig. 12
VNc F~(p)- r~(p)
N
2
T~,( klk2)lpole = - N o
1 r d4k
2~+M_
x R---~p) J (2rr) 4 [M2+(k+lp)2][M2_+(k-½p)2]
x Tr ( k + ½/~+
iM+)Ts(fc-½#+ iM_)
[
k+~+iM(k+q)
,
fc-~+iM(k-q)
]
x %, M2(k+q)+(k + q)2 T. 1-'y~ M-~-(~_~)-~k_--~) 2 %, ,
M~. = M(k+½p).
(78)
Let us recall (see eqs. (60), (61), (68)) that R_(p)~--flp2=(f2V/4N)p 2, F~(0)=
-F~(0) = (~O)/2Nc. We draw to the attention of the reader that at M = const the
integral in eq. (78) is exactly the expression for the anomalous divergence of the
axial current in a theory with free quarks of the mass M. It is easy to check that
p
s•
II
ii
"'-"
II
II
.-.
~uCO.).fl
Q
'L-'~; ~
M2(Q)+Q z
'~
Fig. 12. A graph for the ~r°-~ TT decay.
484
D.L Dyakonov, V. Yu. Petrov / Light quarks
the account for the momentum dependence of the effective mass in eq. (78) leads
to corrections of the order of M(k = 0)tS'~ 1. This is because the integration in eq.
(78) at p, q ~ 0 goes actually over the range of k ~ M(0). To that accuracy the
diagonalization of the zero modes in the instanton vacuum imitates the free quark
triangle and, therefore, the axial anomaly.
Indeed, at p, q ~ 0 we obtain from eq. (78):
T~,,(k, k2)lpo~
iN~
"]
f~ p2L2---~+O(M~)Je,~,~kl,~k2o.
2 ( t ~ ) 1 [-
(79)
Comparing this calculation with the definition (74) of the ~r°-~ 3'3' amplitude we
reproduce the current algebra's result (77).
9. Pion charge radius
Having a theory of the SBCS in the instanton vacuum and an explicit pion in
this theory we can move further on and investigate properties of the Goldstone
excitations which do not follow from the general PCAC theorems. One such calculation - that of the f~ constant - was performed in sect. 7. In this section we calculate
the pion charge radius, r,.
To this end, let us consider a three-point correlation function of two pseudoscalar
densities ~*Ys~ and of an electromagnetic current. Let t be a pion flavour matrix
and Q the charge matrix, Q - - d i a g (~,-½). We have
T~S(PlP2)= I
d4x d4y ei(ptx)-i(p2Y)(fflt3,5t~(x)' ~bt3,"Q~b(0)' t[tt3,stt~b(Y)>
= - I d4x d4y ei(P~X)-<P2Y)[Tr(Qttt) Tr 3,.S(0,
+Tr
x)3,sS(x, y)3,sS'(y, O)
(Qt*t) Tr 7~S(0, y)3,5S(y, x)3,sS(x, 0)].
(80)
Extracting the pion poles contribution for both 3,5 channels we get the physical
amplitude (Trlj~.ml~r)= (Pl +P2).F((Pl-p2) 2) where F ( q 2) is the pion form factor:
T55(PlP2)lpoles
2(tff~b) l(~ljemlTr ) 1 2(q~tp)
f~ Pl
P2 f•
(81)
The calculation of the three-point correlation function (80) in the instanton
vacuum is quite similar to the previous ones. We leave it as an exercise for a
painstaking reader. The calculation with the D, F functions is rather cumbersome
but the result is amazingly simple: it comes to the calculation of a triangle Feynman
graph with exact quark propagators in the instanton vacuum (41). A factor
2x/MiM2/f~ should be attributed to each 3,5 vertex, where M~,2 = M(kl.2) are the
D.L Dyakonov, V..Yu. Petrov/ Light quarks
485
effective masses depending on the momenta adjacent to a Y5 vertex. (Note that the
amplitude for the 0 ~ YY decay considered in th~ previous section is, in fact, built
according to the same recipe.) We obtain
(Trlj~,ml~') = Nc
I
d 4 k 1"
ttt F 2 ~
f~
(-~)4 ~Tr ( Q ) [ _
2 ~
f~
1
1
1
"]
× T r y " / 3 ~ - i M 1 Ts/~3-iM 3 ")/5 p^2 - t--.
M 2 p3=k-p
l
+ T r ( Q t * t ) [ ]v3=k+e/
, ,
)pl,2= kT~q
q =p2-p,,
v=l(p2+p~).
(82)
Let us expand this expression in the momentum transfer q2. Starting from the
O(q 2) term the integral over the loop momentum k is rapidly convergent, so that
we can neglect the momentum dependence of the effective mass and replace M ( k ) -->
M ( k = O) =- M. The account for the momentum dependence gives rise to corrections
of order of MO, which we neglect. The O(q °) term is then logarithmically divergent
- reflecting the divergency of the f~ constant (see eqs. (68), (69)). It should be cut
at k ~ 1 / 0 since this is the scale where the actual fall-off of M ( k ) comes into play.
Comparing the O(q °) term with the approximate formula for the f~ constant (69)
we are satisfied to see that the form factor at zero transfer is unity (as it should be).
The O(q 2) term gives the charge radius of the pion. Indeed,
( T r l j ~ m l T . r ) = ( p l + p 2 ) ~ , F ( q 2 ) = ( p l + p 2 ) ~ , ( 1 - gr~q
1 2 2-±''').
From eq. (82) we find
2
N~
2rr2f=
(338 MeV) 2,
(83)
which should be compared with the experimental value r~, = (300+ 5 MeV) -~ [25].
10. The chiral lagrangian and baryon solitons
Calculating n-point correlation functions of pseudoscalar densities ~b*ys~Oin the
instanton vacuum, and extracting the pion pole contributions for each channel we
are able to construct the n-point one-particle irreducible pion vertices, i.e. the
effective chiral lagrangian. According to the prescription formulated in the previous
section these vertices can be calculated as given by a free quark loop with the
• a T a
effective mass M. A factor trr
~/~r 2 M / f , ~ should be attributed to every Y5 vertex
in this loop. Consequently, we arrive to a very simple recipe for the effective chiral
486
D.I. Dyakonov, V. Yu. Petrov / Light quarks
lagrangian:*
I d4x~e~(~-) = - I n det [i0+ i M exp (iys~ra(x)r%/2/f~)].
(84)
The O(~r 2) term in the expansion of this determinant is logarithmically divergent reflecting the divergency of the f~ constant (see eqs. (68), (69)). In this particular
case the loop momenta should be cut at 1/~. All other terms in (84) are convergent
in the loop momenta. For this reason we neglect the m o m e n t u m dependence of M
in (84) and use M = M ( p = 0). We remind the reader that earlier in this paper we
have obtained M ( p = O) ~ 345 MeV, f~ ~ 138 MeV.
The effective chiral lagrangian (84) can be expanded in powers of the derivatives
of the pion field using the quasi-classical expansion in the external field. Introducing
a natural variable U = e x p (~rar%/2/f~) we see that the first term is the kinetic
1 2
energy ~f~
Tr 0~ UO~ U*, then come the four-derivatives Wess-Zumino-Witten and
Skyrme terms [3] (the latter acquires a concrete coefficient), and so on. The expansion
goes in the inverse powers of the effective quark mass M; each term is proportional
to No.
The importance of the chiral lagrangian** is due to the idea that its stationary
point with unity baryon charge is a topological soliton that describes baryons [3]
(for a review see ref. [27]). It is evident from eq. (84) that the size of the soliton
(if only a stable solution with unity baryon charge exists) is a numerical constant
times 1 / M . Therefore, all terms in derivatives are equally important in the soliton
and, theoretically speaking, one has no right to cut the expansion after, say, the
four-derivatives Skyrme term.
We would like to suggest that to find the all-terms equation for the soliton it may
not be necessary to calculate exactly the determinant (84), and then vary it in respect
to the pion field. A non-linear equation may exist following directly from the
requirement that the spectrum of the Dirac operator in (84) be stable under small
transformations of the external pion field. Thus, for example, such requirement for
the Schr6dinger equation is known to lead to the Korteweg-de Vries equation for
the external potential. Note that the latter equation possesses a soliton as a non-trivial
solution.
11. Conclusions
We have demonstrated that in the instanton vacuum of Q C D the (approximate)
chiral symmetry is spontaneously broken however dilute the instanton-antiinstanton
medium may turn to be. The mechanism of this important phenomenon is the
delocalization of the would-be zero fermionic modes of individual pseudoparticles
* A similar expressionfor the chiral lagrangian as a functional quark determinant was proposed earlier in
ref. [26].
** Apart from its ability for fixing amplitudes of certain weak and/or electromagnetic decays of
pseudoscalar mesons which results from gauging the chiral lagrangian.
D.L Dyakonov, V. Yu. Petrov / Light quarks
487
in the medium; it is essentially a collective p h e n o m e n o n and it arises only in the
thermodynamical limit: N ~ o~, V ~ ~ , N / V fixed.
At large Nc we have found the quark propagator in the instanton medium. It has
the form of a massive quark propagator with the m o m e n t u m - d e p e n d e n t effective
mass M ( p ) which is finite at p ~ 0 and falls rapidly with p.
Calculating the two-point correlation functions of quark-antiquark currents we
demonstrated explicitly the appearance of the Goldstone bosons associated with
the SBCS. Explicit calculations with the pions of the instanton vacuum proved to
be completely consistent with the general PCAC relations.
Using the phenomenological value of the non-perturbative gluon condensate
(F~J32zr2) = (200 MeV) 4 and attributing its origin solely to the instanton-antiinstanton configurations we obtain the values of the physical quantities ( ~ ) =
- ( 2 5 5 MeV) 3, Mefr= 345 MeV, f~ = 138 MeV, r~ = (338 MeV) -1, which are in good
agreement with the experiment. We consider this matching of quark and gluon
condensates as being a serious argument in favour of the idea that the dominant
gluon field configurations in the euclidean Q C D partition function are of the
relatively dilute IT type. I f so, the gauge constant as always remains small, and the
quasiclassical approach proves to be self-consistent (this idea has been discussed
in detail in refs. [14, 16]).
Evidently, there is no confinement in this scheme. However, we have obtained
the first hadron - the pion - and probably may obtain baryons as solitons of the
constructed chiral lagrangian*. The existence of these most important hadrons seems
to be related to the dynamics of the SBCS, not to confinement. Therefore, the
problem of confinement is to see that there are only Breit-Wigner poles in the
correlation functions of colourless currents and not thresholds corresponding to
quarks with the effective mass M (the latter are what we have at the moment). We
would like to emphasize that after the quarks obtain effective masses (which is due
to the SBCS) so that the correlation functions appear to have exponential, not
power, fall-off at large distances, the problem of confinement becomes not the
problem of small m o m e n t a but that of the minkowskean m o m e n t a of the order of
M. The behaviour of the correlation functions at such m o m e n t a may be well governed
by physics that has little or none impact on the saturation of the euclidean partition
function - that job is presumably done by instantons.
We are grateful to M.I. Eides, L.N. Lipatov, E.V. Shuryak, and N.G. Uraltsev
for constructive discussions.
Appendix
We list formulae for the density matrices of zero fermion modes in the instanton
(anti-instanton) background field, and also the overlap integrals of zero modes.
* This has been done recently in ref. [28].
488
D.I. Dyakonov, V. Yu. Petrov / Light quarks
These formulae are intensively used in the main text. Let ~b~(x- z~, U~, Pz) be the
zero mode in the background field of the pse{ldoparticle I. We have (i,j = 1, 2, 3,
4 are spinor and a, fl = I , . . . , Nc are colour indices):
(i) I, I ' = instantons.
[ ( ~ - 2 i ) y , ~ y . ( y - 2 , , ) ' ,q
2 Jo
qq(X-Zx),~,q,{,(y-zl,)j~=-~q~,(x-z,)~o,,(y-zr)
x [
- + ?
U,~r.
urL~,
PI
~ , ( x ) - ~r(x2),/2(x2 + p~)3/2,
where 2 = x , , y , , r ~ = ( ' r , + i ) , Ul,r are the unitary orientation matrices of the
pseudoparticles I, I' (see eqs. (2), (3)). Passing to the Fourier-transformed zero
modes qJ(k) = S d4x exp ( - i k x ) ¢ ( x ) we have
--
1 q~'(k,)q~'(k2)/,*
,~ 1 ~ )
_ +
,
( UI"I'I~'F v U l , ) o t ~ ,
2~'p
" {k{'
127r
d
~'(k) = 7rp 2 ~z [Io(z)ko(z) - I,(z)k,(z)]z=lklp/2 =
k4p2 ,
kp<l
kp > 1.
( ii) I, I'= anti-instantons.
1 ~p'(k,) ~'(k2) [ t
t 1 + Ys'~
( iii) Instanton-anti-instanton.
t)I(k')'~qJi(k2)j~-Ikd
~j(g,r;g})~i3. (A.3)
Ik~[ (-i)/~lyfl~2
(iv) Anti-instanton- instan ton.
gq(k,),,~t~(k2)j~
~'(k~)
* ~ ~y_2)o(U~r~U,i)~"
[k,i ~p'(k2)
ik21 i ( k,y~k2
(A.4)
For the definition of the overlap integral, see eq. (22). Passing to the Fouriertransformed quantities and using eqs. (A.3) and (A.4) we get
TIt(P, UI, Ui)= f d4(zi--zt)ei(P,q
ZOTli(Zi-zi
= _iq ,2(p) Tr ( UTp + a~),
T~(p, Ui, UI)--~- iq~,2(p)
Tr ( U l p - a ~ ) ,
' UI,
Vi)
p+p.r + •
(A.5)
p-=A,r~.
(A.6)
Let us note that the T matrices are hermitian:
T~(p) = T ~ ( p ) .
(A.7)
D.I. Dyakonov, V. Yu. Petrov / Light quarks
489
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