ЯДЕРНАЯ ФИЗИКА, 2015, том 78, № 6, с. 519-521



©2015 M. G. Ryskin, E. G. Drukarev*, V. A. Sadovnikova

National Research Center "Kurchatov Institute", B.P. Konstantinov Petersburg Nuclear Physics Institute,

Gatchina, Russia Received November 17, 2014

We suggest a somewhat indirect method for estimation of the nonlocal scalar quark condensate. The approach is based on analysis of the polarization operator of nucleon current in instanton medium.

DOI: 10.7868/S0044002715050104

In this paper we demonstrate how calculation of the polarization operator of the nucleon current in the instanton vacuum enables to find the value of the nonlocal scalar quark condensate.

The nowadays belief is that the qq pairs which compose the vacuum condensate are created by instantons. The average size of instantons is p = = 0.33 fm, and they are separated by the distances of the order R & 1 fm. The theory of the small size instantons (p < R) was developed in [1, 2], where the propagator of a light quark in the instanton medium S(p) was found in an explicit form. The scalar condensate is connected with the quark propagator by the relation

<0|i(0)i(0)|0)=t f ^TrS(p). (1)

The distribution of quarks which compose the qq pair is described by the fermion zero-mode wave function in the instanton field. Thus, one should investigate the x dependence of the expectation value (0|q"(0)q(x)|0). Calculation of the nonlocal condensate is a complicated task. At lx2l < p2 one can employ the Taylor expansion. However, for applications one needs the x dependence at lx2l > p2. Here, we suggest a non-straightforward way for determination of the function

fq (x2) = {0lq(0)q(x)l0). (2)

We shall focus on the case x2 — 1 GeV-2.

Note that the product q(0)q(x) is not gauge invariant. This expression makes sense if we define q(x) as the Taylor expansion near the point x = 0, i.e.

l + x>lDfl + —DflDl/ + ...Uo), (3)

E-mail: drukarev@thd.pnpi.spb.ru

with D^ standing for covariant derivatives.

The key idea is to analyze the proton polarization operator in the instanton vacuum. The polarization operator describes the time—space propagation of a system with the quantum numbers of the proton. It can be written as

n(q2) = ij d4xei(qx)(0lT[j(x)j(0)]l0), (4)

where the current j(x) carries the quantum numbers of the proton, q is the four-momentum of the system. We employ the current

j(x) = (uTa (x)Cjnub(x))j5Y^dc(x)eabc (5)

suggested in [3]. The polarization operator contains the divergent terms originated by the behavior of the integrand at small values of x. They are the polynomials in q2 and can be eliminated by the Borel transform [4]. We analyze the Borel-transformed operator Bn(q2) = 32n4P(M2) at M2 - 1 GeV2. At M2p2 » 1 it can be expanded in powers of 1/M2, but this cannot be done for M2p2 < 1. We demonstrate, however, that the Borel-transformed chirality flipping structure of polarization operator in the instanton vacuum can be approximated by the sum of three terms of the 1/M2 series.

We carry out calculations of the polarization operator under the standard assumption of the zero-mode domination. This means that in the quark propagator G(x, 0) = ^2n (x))($n(0)l we treat the term with n = 0 (zero mode) separately, while the sum of higher states (nonzero mode) is approximated by the propagator of the free massless quark. At the distances of the order x — 1 GeV-1 the quarks which compose the polarization operator can interact with only one instanton. Moreover, two u quarks cannot occupy the same zero mode, while the contribution of u and d quarks, both in zero mode, vanishes in polarization


RYSKIN et al.

operator with current (5) due to its spin structure. Thus, we include the instanton effects only in one propagator, which can be written as

, . p im(p)

babi'P) = H--5—Oab-

p2 p2


Here, the second term on the right-hand side approximates the zero-mode contribution, while the first term is just the propagator of free massless quark. Thus, the u quarks in polarization operator are described by the free propagators, while the d quark is described by the second term on the right-hand side of Eq. (6).

On the other hand, one can present the chirality flipping component of polarization operator as






f (x )e



If the function f (x2) can be approximated as

f (x2) = f (0)(1 + ci x2 + C2x4), (8)

in the region x2 < 1 GeV-2, one can present the coefficients cn in terms of the coefficients of expansion of the Borel-transformed polarization operator. Here x2 is presented in Euclidean metric. Note that the right-hand side of Eq. (8) cannot be treated as the lowest terms of the Taylor series for the function f (x2). The terms x2n with n > 3 of the Taylor series would provide the divergent terms caused by large x behavior of the integrands in the integrals on the right hand side of (7). Such terms are not eliminated by the Borel transform.

In the medium of the small size instantons the u quarks and d quarks of polarization operator are described by the first and second terms of the propagator (6) correspondingly. To obtain results in analytical form we parameterize the dynamical quark

mass as

m(p) =


(p2 + n2)3'







1.2 1.4

M2, GeV2

Consistency of the left-hand side (solid line) and the right-hand side (dashed line) of Eq. (12) for the QCD vacuum filled solely by the small size instantons(ws =1). The values Ci and C2 are given by Eq. (15).

FW) = 2(1 J ^ + -f3) + ^-EM,


£ 3


Em = j dt^,

and F — 1 at M2 — to. Now we define

K (M2) =

a(M 2)


and try to find the function K(M2) as a power series in 1/M2


K (M 2 ) = 1 + J^ Cn/M 2n. (12)


On the other hand, we can write employing Eq. (8)

V(MZ) = 2aM 1 + -i +

8ci 32c2\

M2 ' M4 J '


Thus, if we manage to approximate the function K(M2) by Eq. (12) with N < 2, we can identify



Cl = T; °2 = 32'


The power of denominator ensures the proper be- We shall look for the solution in the interval 0.8 < havior m(p) ~ p-6 at p — to [1, 2]. Parameters A < M2 < 1.4 GeV2 traditional for the sum rules analysis. We find

and n2 are chosen to match the values of m(0) and of <0|g(0)q(0)|0> for p = 0.33 fm and R = 1 fm, employed in [1, 2]. This provides n2 = 1.26 GeV2 .The parameter n2 does not depend on R, dropping with p as p-2. Direct calculation of the polarization operator provides

P (M 2) = 2M 4 a(M2


a(M2) = a • F



a = —(2n)2<0| i(0)g(0)|0>;

Ci = —1.23 GeV2; C2 = 0.54 GeV4. (15)

The accuracy of the solution is illustrated by figure. Employing Eq. (14) we find c1 = —0.16 GeV2 and c2 = 0.017 GeV4. The nonlocality of the quark condensate can be described by the function

K(x2} = f(x2)mf(0) =Cl-X2+C2- X4. (16)



ROEPHAfl OH3HKA tom 78 № 6 2015



Thus, we obtain k = —0.14 for x2 = 1 GeV-2.

Note that much more complicated calculations of the function f (x2) in the framework of the instanton liquid model [5] provided k & —0.1 at x2 = 1 GeV-2.

There were several moves to estimate the parameter m0 defined as

2 _ (о|д<у<?А»/д|о)

m0 =



(0\qq\0) '


with Gt^v the tensor of the gluon field. This parameter may be used to characterize the quark condensate nonlocality since it is connected with second derivative at x2 = 0 via the equation of motion. For the massless quark

(0\d(0)D2d(0)\0) = ^(0\qaß„gß„q\0), (18)

and thus m2 determines the lowest-order term of the Taylor series for the function f (x2), i.e.


The lattice calculations [6] provided m2 = 1.1 GeV2. The value of m0 was evaluated also from the nucleon QCD sum rules analysis, where it was chosen to obtain the best fit between the two sides of the sum rules. The estimation of [7] gave m2 = 0.8 ±

± 0.4 GeV2 [7], while a smaller value m2 = 0.2 GeV2 was proposed in [8]. Note that the value of m0 determines only the lowest-order term of the Taylor expansion of the condensate near the point x2 = 0, while in the sum rule analysis it actually imitates the whole effect of nonlocality of the scalar condensate. Therefore these estimations of the value of m0 are consistent with our results.

We can consider a more general case when only a fraction (01¿7(0)q(0)l0) of the expectation value (0q(0)q(0)l0) (ws < 1) is caused by the small size instantons while the origin of the rest part (1 — — ws)(0l7 (0)q(0)l0) is not clarified. In this case

a(M2) = a

1 - Ws + WsF




For example, in the case ws = 0.65 when the QCD sum rules reproduce the physical value of the nucleon mass [9] we find C1 = —0.80 GeV2 and C2 = 0.35 GeV4. Thus, ci = —0.10 GeV2 and C2 = = 0.011 GeV4. This provides k = —0.09 for x2 = = 1 GeV-2.

Thus, our results are consistent with those obtained in other approaches. They can be used in applications. For example, they can be employed in the analysis of the baryon QCD sum rules in the instanton medium, where the straightforward employing of the operator product expansion is not possible [10].

We acknowledge the support by the grant RSCF no.14-22-00281.


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