Pion structure function within the instanton model

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Pion structure function within the instanton model

A. E. Dorokhov

Instituto de Fı´sica Teo´rica, UNESP, Rua Pamplona, 145, 01405-900, Sa˜o Paulo, Brazil and Bogoliubov Theoretical Laboratory, Joint Institute for Nuclear Research, 141980, Dubna, Russia

Lauro Tomio

Instituto de Fı´sica Teo´rica, UNESP, Rua Pamplona, 145, 01405-900, Sa˜o Paulo, Brazil 共Received 30 April 1999; published 30 May 2000兲

The leading-twist valence-quark distribution function in the pion is obtained at a low normalization scale of an order of the inverse average size of an instanton ␳c. The momentum dependent quark mass and the quark-pion vertex are constructed in the framework of the instanton liquid model, using a gauge invariant approach. The parameters of instanton vacuum, the effective instanton radius and quark mass, are related to the vacuum expectation values of the lowest dimension quark-gluon operators and to the pion low energy observ-ables. An analytic expression for the quark distribution function in the pion for a general vertex function is derived. The results are QCD evolved to higher momentum-transfer values, and reasonable agreement with phenomenological analyses of the data on parton distributions for the pion is found.

PACS number共s兲: 12.38.Aw, 11.10.Hi, 12.38.Lg, 14.40.Aq I. INTRODUCTION

Hadron structure functions, in terms of quark and gluon distributions specifying the fraction x p of the initial hadron momentum p carried by the active parton, play an important role in QCD inclusive processes. Although the evolution of parton distributions at sufficiently large virtuality Q2is con-trolled by the renormalization scale dependence of twist-2 quark and gluon operators within QCD perturbation theory, the derivation of the parton distributions themselves at an initial Q2value from first principles still remains a challenge. Hence, central predictions unknown in QCD are parton dis-tributions at relatively low virtuality determined in a nonper-turbative scheme.

There is some recent progress in the calculation of mo-ments of the pion and ␳ meson parton distributions 关1兴 within lattice QCD 共LQCD兲 using Wilson fermions in the quenched approximation, where internal quark loops are ne-glected. These LQCD predictions for the moments of the pion distribution function confirm the results of previous analyses 关2兴, being also in qualitative agreement with that extracted phenomenologically 关3,4兴 from experiment 关5兴. However, the calculated moments are still of a relatively low accuracy. In addition, only a few lowest-order moments are available, while the reconstruction of the x-dependent distri-butions needs, in principle, the knowledge of all moments. Furthermore, the QCD sum rules calculations of parton dis-tributions in the pion are only moderately successful关6兴, the results being justified in a limited region of the scaling vari-able x. Recently, in Ref.关7兴, the quark distribution function of pion from lowest quark-antiquark Fock state has been ob-tained within the hard scattering approach including trans-verse momentum effects and Sudakov corrections. It feeds phenomenological quark distribution at large x.

The quark distribution function in the pion was consid-ered关8兴 in the framework of the Nambu–Jona-Lasinio 共NJL兲 model 关9兴. These and similar studies are based on the as-sumption that the calculation of the twist-2 matrix elements,

within the QCD inspired effective approaches, gives distri-butions at a low momentum scale ␮0ⱗ1 GeV, where such

effective theories make sense. The distributions obtained are extrapolated to higher experimentally accessible momentum scales using perturbative QCD, so that comparison with ex-perimental data can be made. However, the problem of the NJL model is that it is nonrenormalizable and thus, to avoid this defect, different ad hoc assumptions about momentum cutoff parameters are introduced.

The instanton model of the QCD vacuum 共for recent re-view see, e.g., 关10,11兴兲, which gives the dynamical mecha-nism of chiral symmetry breaking and provides the solution of the UA(1) problem关12兴, describes well the properties of pion关13–15兴 and kaon 关16兴. Moreover, it dynamically gen-erates the momentum-dependent effective quark mass Mq and quark-pion vertex g_␲qq, and, as a consequence, provides inherently a natural ultraviolet cutoff parameter in the quark loop integrals through the effective instanton size ␳_c. On these grounds, one may believe that the instanton vacuum framework represents an important advance with respect to NJL-type models. The first attempt to calculate the pion structure function within the instanton model has been made in关17兴. More recently, important progress has been achieved 关18,19兴 in calculating quark distributions in the nucleon within instanton inspired approaches.

In the present paper, based on the quark-pion dynamics in the framework of the instanton liquid model, we calculate the leading-twist valence-quark distribution in the pion at a low normalization point of the order of the inverse average instanton size␳c. The calculations are performed in a gauge-invariant manner by taking into account P⫺exp factor ex-plicitly in the definition of nonlocal quantities 关20,21兴 and gauging the nonlocal quark-pion interaction 关22,23兴. The momentum dependent quark mass and quark-pion vertex are constructed in terms of nonlocal quark condensate关24兴. The parameters of the instanton vacuum, effective instanton ra-dius and quark mass, are related to the vacuum expectation values共VEV兲 of the lowest dimension quark-gluon operators

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and to the pion low energy observables. We derive the quark distribution in the pion and all its moments for the general form of the effective quark-pion vertex function. The validity of the isospin and total momentum parton sum rules is en-sured by the pion compositeness condition 关26兴, and it is consistent with the gauge invariant approach. As the effec-tive instanton model is valid for values of the quark relaeffec-tive momentum up to p⬃␳_c⫺1⬇0.5– 1 GeV, the parton distribu-tions calculated here are defined at this 共low兲 normalization point␮0⬃␳c⫺1. The results are QCD evolved to higher mo-mentum transfers, and we found reasonable agreement with phenomenological analyses of the data on the pion distribu-tion funcdistribu-tion.

The paper is organized as follows. In Sec. II, we briefly outline the instanton liquid model and introduce the quark-pion vertex. In Secs. III and IV, the parameters of the

instan-ton vacuum model are related to the vacuum expectation values of the lowest dimension quark-gluon operators and to the pion low energy observables. Then, we derive the expres-sions for the moments of the pion distribution共Sec. V兲 and for the x-dependent distribution itself共Sec. VI兲, followed by the QCD evolution to higher values of the momentum trans-fer. In the last section, the results are discussed.

II. THE INSTANTON LIQUID MODEL AND THE QUARK-PION VERTEX

⬘

;y , y

⬘

兲⫽1 2

冕

d

4_{X f}_{共x⫺X兲f 共x}

_⬘

_⫺X兲

⫻ f共y⫺X兲f共y

⬘

⫺X兲. 共2兲

In order to define the form factor f (x) and coupling con-stant G, we consider the dressed quark propagator in the instanton vacuum

S⫺1共p兲⫽S₀⫺1共p兲⫺iMq共p兲,

共3兲

S₀⫺1共p兲⫽pˆ⫺im_c,

where m_cis the current-quark mass. In Eq.共3兲, the momen-tum dependent quark mass Mq( p) is dynamically developed due to the effect of spontaneous breaking of the chiral

sym-metry in the instanton vacuum. We should emphasize that, the momentum dependence of the quark mass is only due to the nonperturbative vacuum interaction and does not contain the strong perturbative corrections at all. In the ladder ap-proximation, the momentum dependent quark mass obeys the well-known gap equation关15兴1

4NcG f˜2共p兲

冕

d4k

共2␲兲4f˜

2_共k兲 Mq共k兲

k2⫹M_q2共k兲⫽Mq共p兲, 共4兲

where f˜(k) is the Fourier transform of f (x). The solution of Eq. 共4兲 has the form

Mq共p兲⫽Mqf˜2共p兲. 共5兲

From another side, the nonperturbative part of the quark propagator has been calculated in the effective single instan-ton model关20,21兴, as

Mq共p兲⫽MqQ˜共p兲, 共6兲

1_{Here and in the following, all Feynman diagrams are calculated}

in the Euclidean space (k2_⫽⫺k

E

2

) where the instanton induced form factor is defined and rapidly decreases, so that no ultraviolet divergences arise. At the very end we simply rotate back to the Minkowski space. One can verify that the numerical dependence of the results on the pion mass and the current quark mass is negligible and can be dispensed with in the following considerations.

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where Mq⫽ 共2␲␳c兲2nc m* , 共7兲 and Q˜共p兲⫽ 1 2共2␲兲2 p2 ␳c 2

冕

d 4_{x exp}_共⫺ip_•x兲Q共x2_兲, 共8兲 Q˜共0兲⫽1 共p⫽兩p兩兲.

In the above equations, ␳_c denotes the effective size of in-stantons, nc is the effective density of instantons, Mq is the effective quark mass, and

Q共x2兲⫽

冓

:q¯共0兲P exp

再

⫺ig␭ a

2

冕

0

x

dz␮A_␮a共z兲

冎

q共x兲:

冔

冒

冑

Q˜共p兲 and G⫽Mq 2

n_c. 共12兲

We have to note that, by using the standard operator

prod-uct expansion共OPE兲, in Refs. 关27兴, the perturbative and non-perturbative contributions to the quark propagator were ob-tained. The nonperturbative part, proportional to

具

q¯ q

典

has the leading term with momentum ␦( p) distribution and the ␣scorrection⬃p⫺4. The first term is gauge invariant and the second one is not. The contribution considered in Eq. 共6兲 corresponds to the first, gauge invariant, leading in␣_sterm, which 共due to nonlocal properties of the instanton vacuum兲 smears the momentum␦ function into a smooth form factor

Q˜ (p). This means that the quarks can scatter in the vacuum

with nonzero virtuality 关compare with ␦( p⫽0) of OPE兴. Formally, it reduces to resummation of the infinite subset of OPE terms. Thus, the nonperturbative mass in this order does not depend on the gauge. In the next-to-leading order, the gauge dependence appears like in 关27兴, but these terms are suppressed by the small coupling constant.共The soft part of the gluon field is effectively integrated out within the effec-tive theory approach.兲

Using the explicit expressions for the instanton field and quark zero mode, the nonlocal quark condensate in the effec-tive single instanton approximation is given by 关28,20,21兴

Q共x2兲⫽8␳c 2 ␲

冕

0 ⬁ drr2

冕

⫺⬁ ⬁ dt cos

冋

r R

冠

arctan

冉

t⫹兩x兩 R

冊

⫺arctan

冉

t R

冊

冡

册

关R2_⫹t2_兴3/2_关R2_{⫹共t⫹兩x兩兲}2_兴3/2 , 共13兲

where R2⫽␳_c2⫹r2,r⫽兩zជ兩,t⫽z4. In the derivation of these equations a reference frame is used, where the instanton is at the

origin and x␮ is parallel to one of the coordinate axes, say ␮⫽4, serving as a ‘‘time’’ direction 共i.e., xជ⫽0,x4⫽兩x兩). The

propagator has the following expansions at small and large Euclidean distances:

Q共x2兲⫽

冦

1⫺1 4 x2 ␳c 2⫹••• as x 2_→0, 2␳c 2 x2⫹••• as x 2_→⬁.

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The gauge-invariant quark propagators in configuration and momentum representation are plotted in Figs. 1 and 2, respectively, along with the propagators derived in neglect-ing P⫺exp factor in Eq. 共9兲 and using the expressions for the quark zero mode in the singular and regular gauges. In the regular gauge, one has

Qreg共x2兲⫽ 2 y2

冉

1⫺ 1

冑

1⫹y2

冊

冏

y⫽x/2␳_c . 共14兲

In the momentum representation, the normalized quark propagators 共without P⫺exp factor兲 are proportional to the square of the quark zero mode in the corresponding regular and singular gauges:

Q˜reg共p兲⫽exp共⫺2␳cp兲, 共15兲 Q˜sing共p兲⫽

再

z d dz关I1共z兲K1共z兲⫺I0共z兲K0共z兲兴兩z⫽␳cp/2

冎

2 . 共16兲 From Fig. 2, one can observe that in the momentum repre-sentation the shape of the propagator is very sensitive to the

P⫺exp factor.2

So, our model is based on the gauge invariant expression for the dressed quark propagator in the instanton vacuum, with the instanton resummation effect given by the P⫺exp factor included. The motivation for this choice is that the physical quantities, like the quark condensate or quark virtu-ality共see below兲, are defined in terms of the gauge invariant objects. Another element of the model is the four-quark ker-nel. In the separable approximation, its form, Eq.共2兲, is com-pletely fixed by the dressed quark propagator through the gap equation. The actual calculations, dominated by the contri-bution of the zero mode quark wave functions in the field of 共anti-兲instanton, can be done in any gauge for the instanton field, including singular one.3

The spin-flavor structure of the action, Eq.共1兲, is invariant under the global axial q(x)→exp(i␥5␶•␪)q(x) and vector

transformations q(x)→exp(i␶•␪)q(x) and it anomalously violates the UA(1) symmetry: q(x)→exp(i␥5␪)q(x). Within

the instanton liquid model关28,14,15兴 it is argued that due to the long range instanton–anti-instanton interaction, configu-rations with large size instantons are strongly suppressed and the instanton density is sharply peaked at some finite average instanton size ␳c in the form n(␳)⫽nc␦(␳⫺␳c). Since the instanton liquid is assumed to be dilute, the mean separation between instantons is much larger than the average instanton size and the effective density nc is a small parameter of the approach. The values of nc and␳c are estimated from the phenomenology of the QCD vacuum and hadron spectros-copy to be nc⬃1 fm⫺4, and ␳c changes within an interval 共1.5–2兲 GeV⫺1_{, where we put less restrictive limits on the}

range of values. The dimensionless parameter, that charac-terizes the diluteness of the instanton liquid vacuum, is ␩ ⫽(␳cMq)2. The smallness of␩ means that the dynamically generated quark mass is not large enough to destroy the

in-2_{To avoid inconsistency with gauge invariance, one cannot use the}

treatment of the quark propagator in the p representation, in factor-izable form, as done in Ref.关20兴.

3_In _{关15兴, the quark propagator is used in the form Q(x}2

) ⫽具:q¯ (0)q(x):典/具:q¯ (0)q(0):典.

FIG. 1. Euclidean configuration space representation of the nor-malized instanton induced nonperturbative part of the gauge-invariant quark propagator, Eq. 共13兲共solid line兲; and the corre-sponding propagators derived without P⫺exp factor in the singular 共short-dashed兲 and regular, Eq. 共14兲, 共long-dashed兲 gauges.

FIG. 2. Normalized momentum space representation for the same propagators given in Fig. 1, corresponding to Eqs.共8兲共solid line兲, 共15兲共long-dashed兲 and Eq. 共16兲共short-dashed兲.

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stanton vacuum. It is important to note that the effective instanton size ␳c, which defines the range of nonlocality, serves as a natural cutoff parameter of the effective low en-ergy model. Moreover, the coupling constants of the model, Eq. 共2兲, are also expressed through the fundamental param-eters describing the QCD instanton vacuum, ncand␳c. The model incorporates all attractive features of the NJL model and, at the same time, is free of arbitrariness in the choice of the ultraviolet cutoff procedure and physically all parameters are well understood. These peculiarities provide important advantages of the instanton model as compared to different versions of the NJL model关9兴.

The instanton induced interaction of quarks is responsible for strong spin-dependent forces in hadron multiplets 共for a review, see 关29兴兲. In particular, this force is attractive for quark-antiquark states with vacuum and pion quantum num-bers, repulsive for the singlet part of␩

⬘

, and absent共in the zero mode approximation兲 in the vectorlike channels ␳,␻, etc. If the attraction is sufficiently large, it can rearrange the vacuum and bind a quark and an antiquark to form a light 共Goldstone兲 meson state.

To study the formation of quark-antiquark bound states in the instanton liquid, it is convenient to rewrite the four-fermion term in the action, Eq. 共1兲, linearizing the bilocals

q

¯ (x)q(y) and q¯(x)␥5␶ជq(y ) by introducing the auxiliary

composite meson4fields ⌽(x) 关30兴共mean field approxima-tion兲 by virtue of the separability of the four-quark kernel. Then, we arrive at the following form of the effective non-local action corresponding to Eq. 共1兲:

S⫽S0⫹Sint, 共17兲

where S0 is the free action for quark and meson fields

S0⫽

冕

d4x

再

¯q共x兲iDˆq共x兲⫹ 1 2关␴共x兲共⌬⫺m␴ 2 兲␴共x兲兴 ⫹1 2关␲ជ共x兲共⌬⫺m␲ 2_{兲␲ជ共x兲兴}

冎

_, _共18兲

and S_intis the quark-meson interaction part

Sint⫽⫺

冕

d4Xd4x1d4x2F共x1,x2;␮0

2_{兲q¯共X⫹x} 1兲

⫻E␥共X⫹x1;X兲关Mq⫹gM q¯ q共⌫•T兲⌽共X兲兴

⫻E␥共X;X⫺x2兲q共X⫺x2兲, 共19兲

with Dirac and isospin matrices for different meson states according to (⌫•T)_␴⫽I•I,(⌫•T)_␲⫽i␥5•␶ជ. In Eq. 共19兲, g_{M q}_{¯ q} is the quark-meson coupling constant and M_q is the effective quark mass fixed in a gauge-invariant manner共see below兲 by the compositeness condition, Eq. 共22兲, and the gap equation 共4兲 in terms of the instanton density nc and the instanton size ␳c. In Eqs. 共18兲,共19兲 we neglect the terms

induced by tensor interaction in Eq. 共1兲 since they do not contribute to the scalar channels. We also neglect in the fol-lowing the current quark mass and restrict ourselves only by the nonstrange quark sector.

To ensure the gauge invariance of the bilocal quark op-erators, which enters in Eq. 共19兲, with respect to external electromagnetic A_␮(z) gauge field, we include into Eq.共19兲, following 关22兴, the path-ordered Schwinger phase factors

E_␥共x;y兲⫽P exp

再

⫺ieQ

冕

x y

dz␮A_␮共z兲

冎

, 共20兲 where the charge matrix is Q⫽(1/3⫹␶_f3)/2, and the partial derivative ⳵_␮ is replaced by the covariant one D_␮⫽⳵_␮ ⫺ieA␮. We adopt here that the integral in the exponent is

evaluated along a straight line with P being the path-ordering operator. The incorporation of a gauge-invariant interaction with gauge fields is of principal importance in order to treat correctly the hadron characteristics probed by external sources such as hadron form factors, structure functions, etc. The Fourier transformed gauge-invariant nonlocal vertex function F˜ (k1,k2;␮0

2

) describes the amplitude of soft transi-tion of a pion with momentum p into a quark and an anti-quark with momenta k1⫽p⫹k/2 and k2⫽p⫺k/2,

respec-tively. This function represents the full interaction vertex with all quark-gluon excitations harder than the scale ␮₀ ⬃1/␳c, strongly 共exponentially兲 suppressed. It is defined through the nonperturbative part of the quark propagator

F共k1,k2;␮0 2_兲⫽

冑

Q˜共k1兲Q˜共k2兲. 共21兲

One of the advantages of using the gauge-invariant for-malism is that the parameters of the model, such as the size of instantons and the effective quark mass, gain physical meaning. As a consequence, all other physical quantities ex-pressed through these parameters become automatically gauge-invariant ones. Moreover, they could be compared with those calculated in the lattice QCD, QCD sum rules or other QCD inspired approaches. In contrast, when one deals with noninvariant-gauge objects there can be chosen any convenient gauge. It is most correct to consider the instanton vacuum field in the singular gauge and to construct the ef-fective action in this specific gauge关10,11兴. In the coordinate space in the singular gauge the instantons fall off rapidly enough to provide small overlaps of neighbor pseudopar-ticles and quasiclassical considerations are justified. But at the end the action has to be independent on the choice of the gauge, otherwise the form of the action and other observ-ables look rather awkward. This explicitly gauge invariant form was assumed by us in Eq. 共19兲 by introducing path-ordered Schwinger P exp factors in Eqs. 共9兲 and 共20兲. The factor E_␥ will effectively take into account the radiation ef-fects of photon field when two quarks become separated. Note also that the quark propagator, Eq. 共9兲, has a direct physical interpretation in the heavy quark effective theory of heavy-light mesons as it describes the propagation of a light quark in the color field of an infinitely heavy quark关28兴.

4_{In this work we do not include explicitly the diquark part of the}

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It is important to emphasize that the meson fields entering the action, Eq.共17兲, are renormalized and the field renormal-ization constants of composite mesons are set equal to zero,

ZM⫽1⫺g_⌽qq¯ 2 ⳵⌸⌽共p 2_兲 ⳵p2

冏

p2_⫽⫺m ⌽ 2 ⫽0, 共22兲

where⌸_⌽( p2) is the meson field polarization operator. This condition关26兴 fixes the couplings of meson fields to quarks,

g_⌽qq¯ 共see Sec. IV兲 and is a consequence of the

composite-ness of hadron states manifesting themselves as poles in the quark-共anti兲quark scattering amplitude. As we will see be-low, it is precisely this condition supplemented by the gauge invariance of the effective action, given by Eq. 共17兲, that leads to the correct parton isospin and momentum sum rules in the model.

III. EXPECTATION VALUES OF QUARK-GLUON OPERATORS

Let us consider the lowest dimensional vacuum expecta-tion values within the instanton model. Given the dynamical mass, Eq.共6兲, the values of the quark condensate,

具

q¯ q

典

⫽⫺4N_c

冕

d 4_k

共2␲兲4

Mq共k兲

k2⫹M_q2共k兲, 共23兲

and the average quark virtuality in the vacuum关24兴,

␭q 2_⬅

具

:q¯ D 2_q:

_典

具

:q¯ q:

典

⫽⫺ 4Nc

具

¯ qq

典

冕

d4k 共2␲兲4k 2 Mq共k兲 k2⫹M_q2共k兲, 共24兲

can be found. The average quark virtuality defines the de-rivative of the quark condensate and thus nonlocal property of it. One of the main suggestions of the QCD sum rule method关31兴 was that the local quark and gluon condensates dominate in the light hadron physics and introduction of higher dimensional corrections or even nonlocal condensates themselves关24兴 have not to change the standard results too much. Thus, at least for consistency of local and nonlocal QCD sum rules, the derivative 共virtuality兲 value has to be relatively small. Phenomenologically, there is rather fine QCD sum rule analysis of this value ␭_q2⬇0.4⫾0.2 GeV2, based on consideration of the light hadrons关32兴 and heavy-light quark meson systems 关33兴. The LQCD calculation yields␭_q2⫽0.55⫾0.05 GeV2 关34兴. Certainly, there is correc-tions from direct instantons to the QCD sum rule result, but they have not to change the result drastically. It would also be urgent if the LQCD estimation could be confirmed by new calculations.

For the moment, it is instructive to consider Eqs. 共4兲,共23兲,共24兲, neglecting the term Mq

2

⬇⫺NcMq⌳NJL

2 _/(4_␲2_{), where} _⌳

NJL is a momentum cutoff for the covariant regularization scheme in the two-flavor model. Since the instanton induced nonlocality is analogous to the covariant regularization, we can estimate⌳NJL

2 _⬇␭

q

2

. The second relation in Eq.共25兲 has recently been obtained in Ref.关21兴, where nonlocal properties of the quark conden-sate are studied within the instanton model. The same result was also obtained in Ref.关36兴 from direct calculations of the

local mixed quark-gluon condensate:

␭q 2 2 ⫽

冓

:q¯

冉

ig␴_␮␯G_␮␯a ␭ a 2

冊

q:

冔

具

:q¯ q:

典

. 共26兲

It is clear, from the expressions for the average quark virtu-ality, that the range of the quark-antiquark interaction is characterized by the effective size ␳c of the instanton fluc-tuations in the QCD vacuum. The natural gauge-invariant definition for the average quark virtuality, Eq.共24兲关and also that in Eq. 共23兲 for the quark condensate兴, with Mq(k) de-fined in Eqs. 共8兲 and 共5兲, is valid only if the zero mode solution in Eq. 共8兲 is written in the gauge-invariant way. If we substituted its expression in the singular gauge 共in ne-glecting P⫺exp factors兲 in Eq. 共24兲, we would obtain ␭_q2 ⫽9/(2␳c

2

), with a coefficient far from the correct one. The reason for such inconsistency, in the calculations of nonlocal quantities, is that the covariance of the derivative in the ma-trix element of Eq. 共24兲 is missing. This implies that one needs to add extra terms in Eq. 共24兲 to take into account effects of gluon field; terms which restore the correct result. However, by using the present invariant approach, with Eq. 共9兲, such effects are taken into account effectively by the P ⫺exp factor.

Inverting the relations, Eq. 共25兲, we express the param-eters of the instanton vacuum model in terms of the funda-mental parameters of QCD vacuum ␳_c2⫽2/␭_q2, Mq ⫽⫺(4␲2_/N

c)(

具

¯ qq

典

/␭q

2_{). If one expected ‘‘standard’’ values}

for the quark condensate,

具

¯ qq

典

⬇⫺(230 MeV)3 _{共see, e.g.,}

关11兴兲, and for the average quark virtuality to take, ␭q

2_⬇0.6

GeV2 关32,34兴, one would obtain ␳_c⫺1⬇0.55 GeV and Mq ⬇0.27 GeV. As we will see in the next section, the joined

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In

冑

Ncnc␭q 2_. _共28兲

These relations have the same parametric dependence as in the estimations obtained in关28,11兴 but with different coeffi-cients. The first one expresses the quark condensate in terms of the effective single instanton contribution times the den-sity of instantons. The reason for the difference in the coef-ficients is that in关28兴, where it looks as

具

¯ qq

典

⫽⫺nc/ Mq, the expressions were obtained from the instanton formula in the gas approximation by ad hoc replacing the current quark mass by the effective quark mass. In contrast, in deriving Eq. 共28兲 this replacing procedure is fixed by the gap equation, Eq. 共4兲, with a definite coefficient mainly defined by the slope of the form factor Mq(k). The second relation repre-sents a self-consistent value of the quark condensate in the instanton vacuum model共cf. 关11兴兲. Further, since the instan-ton contribution to the value of the gluon condensate is given by

具

(␣s/␲)G2

⯝0.012 GeV4. The latest reanalysis 关37兴 of the QCD sum rules for

heavy and light mesons and also recent lattice results 关38兴 provide values which are twice or even larger than the ‘‘stan-dard’’ one.

IV. PION LOW ENERGY OBSERVABLES

Let us now consider the low-energy observables of the pion. The pion-quark coupling constant is determined by the compositeness condition, Eq.共22兲, with the pion mass opera-tor being ⌸␲共p2兲⫽Nc

冕

d4_k 共2␲兲4F˜ 2_{共k,k⫹p;␮} 0 2_兲 ⫻Tr兵␥5S共k⫹p兲␥5S共k兲其, 共30兲

where the normalized nonlocal vertex is given in Eq. 共21兲 and the quark Green function is S(k)⫽关MqQ˜ (k)⫹kˆ兴⫺1with

Q˜ (k) defined in Eq. 共8兲.

From the definition, Eq.共22兲, we derive the expression for the pion-quark coupling constant g_␲qq¯

g_␲qq¯2 ⫽ 2␲ 2 NcIg␲共⫺m_␲

共k兲⫽ d dkQ˜共k兲, D共k兲⫽Mq 2 Q˜2共k兲⫹k2. 共33兲

The expression for g_␲qq¯ given in Eqs. 共31兲 and 共32兲 agrees with that derived in关15兴.

To fix the parameters in the instanton model, we consider the low energy decay constants of the pion. As it has recently been shown in 关23兴, in the presence of nonlocal separable interaction the axial current conserved in the chiral limit can be constructed from the action, Eq. 共1兲, by using a Noether-like method.5The full current is the sum of local,

j_5(loc)␮a 共x兲⫽1

2q¯共x兲␥

␮_␥

5␶aq共x兲, 共34兲

and nonlocal,

5_{One of us} _{共A.E.D.兲 thanks M. C. Birse for discussion of the}

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j_5(nl)␮a 共x兲⫽

冕

dx1dx2dx3dx4K共x1,x2,x3,x4兲 ⫻

再冋

g_␲qq⫽Mq

f_␲ 共36兲

and the␲0→␥␥ decay constant

g_␲␥␥⫽ 1

4␲2f_␲ 共37兲

is consistent with the requirement of axial anomaly. We fix the model parameters to give the pion weak decay constant f_␲, within 1% of accuracy. In Table I, we present the results. For the two model parameters, Mq and␳c, we show the predictions for the quark-pion coupling g_␲qq, the quark condensate

具

¯ qq

典

, the average vacuum quark virtuality ␭q

2

and the instanton density nc.

From Table I it is clear that with growth of the quark mass the values of the quark and gluon condensates decrease. This is an expected effect of suppression of condensates by fer-mion. As shown in Table I, the values of the parameters Mq and ␳c that reproduce the lowest dimension VEV with an accuracy of an order of 30% are in the ‘‘window’’ M_q

⫽(220⫺260) MeV,␳c⫽(1.5⫺2.0) GeV⫺1. In the follow-ing we use the typical set of parameters:

Mq⫽230 MeV, ␳c⫽1.7 GeV⫺1. 共38兲 The diluteness condition ␩Ⰶ1 is satisfied within the whole ‘‘window.’’

The momentum dependence of the vertices in the numera-tors of the integrands共which are defining physical quantities兲 is important because it provides the ultraviolet regulariza-tion. Also, due to momentum dependence of the vertices, the measure in the integrals looks like product of some powers of k2and the function Q˜ (k). This measure has maximum at

k2 of order 1/␳_c2 and, at small momenta, the momentum de-pendent quark mass in denominators can be substituted by an effective constant mass parameter mq⬇Mq(k⬃␳c⫺1). With the form of momentum distribution shown in Fig. 2, it ap-proximately equals to the mass at zero, M_q(0). This mass parameter mqhas to be identified with the standard constitu-ent quark mass. Corresponding to this substitution, we rede-fine the function D(k) given in Eq.共33兲 as D(k)⫽m_q2⫹k2. The choice of the mass parameter,

mq⫽230 MeV, 共39兲

well reproduces the integrals defining the VEV given by Eqs. 共4兲–共24兲. This constant-mass approximation is often used in practice with the quark mass in the region 250–350 MeV 共see, e.g., 关19,39,40兴兲.

The model parameters and predictions for vacuum and pion observables are obtained within a set of approximations. We are working in the chiral limit of zero current quark mass. Further, within the zero mode approximation small contributions coming from vector mesons are neglected. Also only the lowest two-quark Fock intermediate state in the pion is taken into account, which corresponds to the quenched approximation. We regard that all these factors can change a little the numbers in Table I, but the qualitative results discussed are not greatly affected.

V. MOMENTS OF THE QUARK DISTRIBUTION FUNCTION

The standard QCD analysis based on the operator product expansion共OPE兲 relates moments of parton distributions at a given scale to the hadronic matrix elements of local twist-2 operators. This formalism is employed to separate the hard and soft parts of the forward scattering amplitude. Within the OPE, the hard part is calculable within perturbation theory in the form of Wilson coefficients. The soft part is represented by a set of local operators classified by the twist. Their ma-trix elements accumulate information on the nonperturbative structure of the QCD vacuum.

The twist-2 gauge-invariant nonsinglet local quark6 opera-tors with flavor j are defined by

6_{As in Ref.}_{关19兴, we will neglect gluon operators in the covariant}

derivatives, justified by the smallness of the diluteness parameter␩. TABLE I. The values of the low energy vacuum and pion

ob-servables discussed in the text.

Mq ␳c f␲ 兩具q¯ q典兩1/3 ␭q

2

nc 共GeV兲共GeV⫺1₎ _{共MeV兲 g}

␲qq 共MeV兲共GeV2) 共fm⫺4)

0.16 1.0 93 1.7 283 2.2 1.36

0.21 1.5 92 2.3 230 1.0 0.87

0.23 1.7 91 2.5 215 0.83 0.75

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O_␮

1␮2. . .␮N

j _⫽iN_¯_q

j兵␥␮₁D␮₂. . . D␮_N其Sqj, 共40兲 where D_␮⫽⳵_␮⫺igA_␮a␶a is the covariant derivative and the symbol兵•••其Smeans the traceless and symmetric part of the tensor. The matrix elements A_Nj of the local operators O_Nj between pion states兩␲( p)

典

with momentum p, renormalized at the normalization scale␮, are defined by

A_Nj共␮2兲⫽i

N

2

具

␲共p兲兩q¯jnˆ共n␯D

␯_兲N_⫺1_q

j兩␲共p兲

典

兩␮, 共41兲 where n_␯ is a lightlike vector, with n2_{⫽0 and (np)⫽1,}

in-troduced to select the symmetric traceless part of the opera-tor O_Nj , Eq.共40兲. Let us now define the quark distribution for the j th flavor in terms of its moments, viz.

A_Nj共␮2_兲⫽

冕

0

1

dxxN⫺1_q

j共x,␮2兲. 共42兲

The x variable is the fraction of the longitudinal pion mo-mentum carried by a quark in the infinite-momo-mentum frame. The ␮2 dependence of A_Nj is known exactly from the solu-tion of the perturbative QCD evolusolu-tion equasolu-tions, while the nonperturbative dynamical model provides the initial input for this evolution. These initial values of the moments are calculated here in the instanton model, which specifies a low momentum transfer value related to the scale␮₀2⬀1/␳_c2.

The contribution of the lowest Fock quark-antiquark va-lence state to the Nth moment, AN

qq¯ (␮0

2

), in the instanton model, is given by 共see Fig. 3兲

A_Nqq¯共␮₀2兲p_␮ 1p␮2. . . p␮N⫽2Ncg␲qq¯ 2

冕

d 4_k 共2␲兲4

再

F˜ 2_{共k,k⫹p;␮} 0 2_兲Tr关␥ 5S共k⫹p兲兵␥␮₁共k⫹p兲␮₂. . .共k⫹p兲␮_N其SS共k⫹p兲␥5S共k兲兴 ⫺2

冋

⳵F˜ 2_{共k,k⫹p;␮} 0 2_兲 ⳵共k⫹p兲2

册

Tr关␥5S共k⫹p兲兵共k⫹p兲␮1共k⫹p兲␮2. . .共k⫹p兲␮N其S␥5S共k兲兴

冎

. 共43兲

In general the moments are defined in Minkowski space and can be expanded in p_␮and g_␮␯. The terms proportional to g_␮␯are of higher twist nature and have to be ignored. The formal way to do this is to multiply the matrix elements by lightlike vectors n_␮ and use the properties pn⫽1 and n2 ⫽0. The remaining loop integral has to be analytically con-tinued in Euclidean space 关12兴, where the instanton induced vertex is well defined.

In order to obtain the above equation we have considered the Compton scattering amplitude in deep inelastic kinemat-ics at leading order in perturbative QCD, and took its imagi-nary part. During the Compton process, both incoming and outgoing virtual photons have momentum q,q2⬅⫺Q2 and the initial and final pions have momentum p, p2_{⫽0. And the}

Bjorken limit corresponds to large Q2 at fixed x

⫽Q2_{/2( pq). The gauge invariant set of diagrams, induced}

by the nonlocal action 共17兲 and 共19兲 include, in addition to the one-loop box diagrams, the processes ␲␥→␲→␲␥, ␲␥␥→␲. It can be shown共see Appendix A and Ref. 关41兴兲 that only the second type of the process survives in the Bjorken limit of the corresponding amplitudes and leads to the vertex correction 共term with derivative兲 in Eq. 共43兲.

The term with a derivative, which comes from

P exp-factor in Eq. 共20兲, ensures gauge invariance of the

approach关41兴 and enables us to satisfy the isospin and mo-mentum conservation sum rules. Indeed, from Eq. 共43兲 for the first two moments we obtain

A₁qq¯共␮₀2兲⬅

冕

0 1 dxqqq¯共x,␮0 2_兲⫽g ␲qq¯ 2 ⳵⌸␲共p 2_兲 ⳵p2

冏

p2_⫽0 , A₂qq¯共␮₀2兲⬅

冕

0 1 dxxqqq¯共x,␮0 2_兲⫽g␲qq¯ 2 2 ⳵⌸␲共p2兲 ⳵p2

冏

_p2_⫽0 . 共44兲 And, to establish the共normalization兲 isospin and momentum sum rules for the valence-quark distribution function, the compositeness condition, Eq.共22兲, has to be used

FIG. 3. Graphical representation of the operator product expan-sion. The left hand side of this diagram is the imaginary part 共Dis-continuity兲 of the forward scattering amplitude. Within OPE it is represented by the convolution of the Wilson coefficient function CNS(Q2) of a ‘‘hard’’ parton subprocess共upper block of the right

diagram兲 and the ‘‘soft’’ parton distribution function q(x,Q2) 共lower block of the right diagram兲. The constituent quark and pion are depicted by solid and double lines, respectively. The wavy line denotes the virtual photons. ONSis the local operator and the slash

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A₁qq¯共␮₀2兲⫽1 and A₂qq¯共␮₀2兲⫹A¯₂qq¯共␮₀2兲⫽1, 共45兲

where A¯2 is the valence antiquark contribution to the pion

momentum. The fact that, at the low momentum scale ␮0,

the whole momentum in the pion is carried off by the va-lence quarks is due to the quenched approximation used, when only valence quark-antiquark intermediate states are included and all intrinsic quark-gluon sea states are ne-glected.

Thus, in the quenched approximation the dynamical infor-mation contained in the first two moments is strongly re-stricted by the symmetries and kinematics, and as a result, it is model independent. The nontrivial dynamics is contained in the moments with N⬎2. The general structure of the mo-ments of the structure function 共SF兲, from Eq. 共43兲, can be written in the form

A_NSF共␮₀2兲⫽ 1 2N⫺1

兺

i⫽0 [(N⫺1)/2] 1 2i⫹1

冉

N⫺1 2i

冊

Ji SF_共_␮ 0 2_兲, 共46兲 N⫽1,2, . . . ,

with the coefficients J_iSF given by

J_iSF共␮₀2兲⫽ 1 Ig␲共0兲

再

冕

0 ⬁dkk4i⫹3_Q_˜_共k兲2 共k2_⫹m q 2_兲2i⫹3 ⫻关2k2_{⫹共2i⫹3兲m} q 2_兴⫹_•••

冎

, 共47兲

where the vertex terms with derivatives, like that appearing in Eq.共32兲 for Ig␲(0), are denoted by dots. In Eq.共46兲, the square brackets 关•••兴 mean the integer part of the number, and (_ba) are the binomial coefficients.

It is instructive to consider two extreme cases, depending on the physics under consideration. If the QCD vacuum were a very dense medium, ␩Ⰷ1, then J_iSF⫽0 for all i except i ⫽0. As a result, it leads to the set of moments AN ⫽1/2(N⫺1)_{for all n and to a quark distribution which has the}

form of a delta function: q(x)⫽␦(x⫺1/2). This extreme case corresponds to the heavy quark limit, and the coeffi-cients J_iSF represent consequent corrections in inverse pow-ers of the heavy quark mass: ⬃(

具

k2

_典

_/m

q

2₎i_{. In the opposite} extreme case of a very dilute vacuum ␩Ⰶ1 one gets J_iSF ⫽1 for all i and AN⫽1/N for the moments. This extreme case corresponds to the momentum independent quark mass and provides flat quark distribution q(x)⫽1. Moreover, the first term in Eq.共47兲 dominates over the terms indicated by dots, since the latter are small of an order of O(␳cmq) . A realistic situation seems to be somewhere in-between these two extremes. Note that the role of pion mass is negligible, but the interplay of the effective quark mass and the slope of the nonlocality in Q˜ (k) has an important effect.

VI. QUARK DISTRIBUTION FUNCTION AND ITS QCD EVOLUTION

Let us now turn our attention to the quark distribution itself. This distribution for the pion with 4-momentum p is given by 共see a graphical representation in Fig. 3兲

qqq¯共x;␮0 2_兲p_␮_⫽2N cg_␲qq¯ 2

冕

d 4_k 共2␲兲4␦关x⫺1⫺共k•n兲兴 ⫻Tr

再

␥5S共k⫹p兲

冋

F˜ 2_{共k,k⫹p;␮} 0 2_兲␥ ␮ ⫻S共k⫹p兲⫺2

冉

⳵F˜ 2_{共k,k⫹p;␮} 0 2_兲 ⳵共k⫹p兲2

冊

⫻共k⫹p兲␮

册

␥5S共k兲

冎

, 共48兲

where qqq¯(x)⫽u¯(x)val⫽d(x)val for ␲⫺. The subscript qq¯ means that there is taken into account only lowest quark-antiquark component of the pion wave function. In Eq.共48兲, we arrive at the quark distribution defined in a similar man-ner as that used in 关42,43兴. The ␦关x⫺1⫺(k•n)兴 function appearing in Eq. 共48兲 represents the effective vertex related to the composite operator ON

j

given by Eq. 共40兲. The mo-ments of the distribution function 共43兲 are reproduced by making the Mellin transformation of the above Eq. 共48兲, if the light-cone vector n is normalized by ( pn)⫽1. The light-cone vector n serves to project out in Eqs. 共43兲 and 共48兲 symmetric traceless tensors. It can be easily shown that the first moments of qqq¯(x) will reproduce the parton sum rules Eq. 共44兲.

To calculate the k integral in Eq.共48兲, we use␣ represen-tation for the propagators,7

1

k2⫹m2⫽

冕

0 ⬁

d␣exp关⫺␣共k2⫹m2兲兴, 共49兲

and for the vertex␦ function, ␦关x⫺共k•n兲兴⫽₂1_␲

冕

⫺⬁ ⬁

d␣exp关i␣共x⫺k•n兲兴. 共50兲 Then, a direct calculation from Eq. 共48兲 provides the result for the quark distribution, which in the massless case (m_␲2 ⫽⫺p2_{⫽0) is reduced to}

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⫻⌰共x¯␯2⭓x␯1兲⫹共x↔x¯兲

冎

. 共51兲

In the above equation, x¯⫽1⫺x,E₁(z) is the integral expo-nential, and F(␯) is the correlation function related to the vertex function Q˜ (k) by the Laplace transformation. The vertex Q˜ (p), in the essential region of p (0⭐p⭐4/␳_c) is approximated by

Q˜共p兲⫽4.5 exp共⫺1.9␳cp兲⫺3.5 exp共⫺3.6␳cp兲, 共52兲 which leads to the Laplace transform,

F共␯兲⫽ ␳c

冑

␲

冑

␯关8.55 exp共⫺0.9␳c 2_␯_{兲⫺12.6 exp共⫺3.24␳} c 2_␯_兲兴. 共53兲 Let us stress that the expressions Eq.共46兲 for the moments and Eq. 共51兲 for the valence-quark distribution in the pion are universal ones and valid for any shape of the functions

Q˜ (k) and F(␯), which in turn are determined by the con-crete model of the quark-pion dynamics.

The quark distribution qqq¯(x,␮0 2

) and the momentum dis-tribution共structure function兲 xq_qq¯(x,␮₀2) are shown graphi-cally in Fig. 4. We have to note that the shape of the

distri-bution is quite stable with respect to changes of the instanton model parameters, if they are fixed to reproduce the pion low energy properties. Also, we should mention that the ne-glected effects共momentum dependence of the quark mass in the denominators and gluon fields in the covariant deriva-tives兲 are of the order O(␩) and do not change our results qualitatively. The main effect considered in the paper, when calculating the quark distribution in the pion, is related to the nonlocality of quark-pion vertex induced by instanton inter-action. The role of such effect is to modify the leading twist parton model result q(x)⫽1, leading to a smoothed distribu-tion with zeros at the edges of the kinematical region. We remind that we are computing only the leading-twist distri-butions at a low normalization point␮0⬃␳c⫺1rather than the full structure functions which contain also higher-twist cor-rections. The latter may be large at low q2. We have also to note that these results differ strongly from those obtained in calculations with the NJL model 关8兴 which yield distribu-tions that are rather consistent with the strict chiral limit

q(x,␮0 2

)⬇1.

The computed distributions are then used as initial condi-tions for the perturbative evolution to higher values of Q2, where the power corrections are expected to be suppressed,8 so that one can compare them with the available experimen-tal data. Actually, we compare our theoretical predictions with the phenomenological analysis by Sutton et al. 关3兴 of the data taken from Drell-Yan and prompt photon experi-ments performed by the groups NA10 共CERN兲 and E615 共Fermilab兲关5兴. Still some limitations are underlying the analysis, considering the experimental data. The data cover the region of x⭓0.2, and an extrapolation of the proton structure function, for x⭓0.75, has been used as input in the analysis. Also such phenomenological data analysis does not take into account uncertainties induced by theoretical as-sumptions underlying the analysis共e.g., K factor兲.

The form of the evolved distribution qqq¯(x,Q0

2_{) at the}

momentum scale Q₀2⫽4 GeV2 is reconstructed from its mo-ments evolved to this scale in the leading order 共LO兲 and next-to-leading order共NLO兲 perturbative QCD in the modi-fied minimal subtraction (MS) scheme by using the first six Jacobi polynomials. To this goal we use the well-known ex-pressions 关45兴 for the perturbatively calculable coefficient function of the process Ci

N_⫽C 0i N ⫹关␣s(Q2)/4␲兴C1i N and the anomalous dimensions ␥(n) calculated up to LO and NLO.

Thus, the final result for the moments obtained from the factorization procedure is AN共Q2兲⫽

兺

i C_iN共Q2,␮2兲O_iN共␮2兲⫽

冕

0 1 dxxNq共x,Q2兲. 共54兲 In performing the evolution analysis we choose a low mo-mentum scale ␮₀2⫽0.3⫾0.03 GeV2, and a set for the QCD

8_{A careful analysis about the effect of power corrections, in}

par-ticular, the effect of the 1/Q2_{term that imitate the short-string effect}

in QCD as in Ref.关44兴, will be considered elsewhere.

FIG. 4. The valence quark distribution function 共DF兲 q(x;␮0 2 ) 共dashed line兲 and the quark momentum distribution function 共MDF兲 xq(x;␮0

2

) 共solid line兲 for the pion as a function of the longitudinal momentum fraction x at the low momentum scale ␮0

2_{⫽0.3 GeV}2 and density parameter␳cmq⫽0.39.

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scale parameter ⌳_QCD(3) ⫽0.3 GeV in order to be consistent with 关3兴. The resulting distribution qqq¯(x,Q0

2_{) is shown in}

Fig. 5 together with the phenomenological curve derived from the data in 关3兴. The initial distribution function at the low-momentum scale␮₀2 is also shown for comparison.

The values of the first moments of the pion quark distri-bution at Q₀2⫽4 GeV2calculated in LO and NLO are shown in Table II. The error bars quoted in Table II for our calcu-lations are due to accepted uncertainty in the choice of the initial scale of evolution ␮0. These values should be

com-pared with those obtained from the phenomenological analy-sis关3兴 and from LQCD simulations 关1兴. In Table II we also include the moments of quark distribution in the pion ob-tained from the parametrization关4兴.

Let us finally discuss the uncertainties of the QCD evolu-tion from the low momentum scale ␮0. As we see from

Table II, the difference of the LO and NLO results is in the range of 30%. It turns out that the use of a larger initial evolution scale, say␮₀2⭓0.3 GeV2, gives a rather good con-vergence with deviations less than 10%, whereas in the op-posite case, i.e., for scales smaller than about 0.1 GeV2 the deviations increase and perturbative evolution loses any sense. This behavior has also been observed in analyses within the NJL model关8兴.

The comparison shows that our calculations, in particular in NLO, are consistent with the phenomenological analysis of 关3兴 and fairly close to the LQCD results. Both theoretical approaches共LQCD and the instanton model兲 predict moment values systematically larger than the phenomenological one. One of the reasons for this disagreement may be traced to the quenched approximation which does not take into account any sea quark-gluon and higher Fock state contributions at the initial scale, attributing in this way the whole pion mo-mentum to the valence quark-antiquark pair. Indeed, the ori-gin of the A2 moment at the initial scale 共in the quenched

approximation兲 and its subsequent evolution is purely kine-matic and does not depend on the details of the model. In principle, one could match the valence momentum fraction derived in our calculation with that determined in 关3兴 by shifting the initial value ␮₀2 down to 0.01 GeV2 „see, for instance, 关8共a兲兴…. However, to start a perturbative evolution from this very low scale is formally incorrect and technically amounts to a rather unstable procedure.

In our opinion, it is more realistic to expect that by in-cluding in our analysis contributions from the quark-gluon sea and higher Fock states the agreement between the theo-retical predictions and the phenomenological analysis can be considerably improved. The contribution from sea reduces the momentum fraction carried by valence quarks contained in the moment A2. The higher Fock states contribute to both A1 and A2. It means that the overall normalization of the

distribution function and momentum fraction, carried by the valence quark-antiquark component of the wave function, has to be reduced by factors which represent the probability of the missing configurations in the pion wave function. In Fig. 5 we are also presenting the result of our LO calculation, multiplied by a factor 0.65, corresponding to a crude esti-mate of a probability of 35% for higher Fock valence state configurations 共neglecting the contribution of the sea兲. This estimate is found by using the inequality

q共x,Q₀2兲⭓q_qq¯共x,Q₀2兲共55兲 and saturating it as x→1. This inequality considered in Ref.

FIG. 5. The valence quark momentum distribution function xq(x;Q₀2) 共long dashed line兲 for the pion as a function of the vari-able x evolved to the momentum scale Q0

2

⫽4 GeV2_共LO approxi-mation兲, using␳cmq⫽0.39 for the density parameter. The solid line

denotes the phenomenological curve关3兴 on the same scale Q₀2, ex-tracted from the data. The short-dashed line shows the same distri-bution on the low momentum scale␮₀2⫽0.3 GeV2_{. The dash-dotted} line corresponds to the dashed line, multiplied by a factor 0.65, which represents an estimation of the probability 共65%兲 of quenched configuration in the pion wave function.

TABLE II. The values of the first moments at Q0 2

⫽4 GeV2 .

LO NLO LQCD关1兴 Expt. fit关3兴 Expt. fit关4兴

共this calculation兲 A2(Q0 2 ) 0.318⫾0.01 0.275⫾0.017 0.279⫾0.083 0.230⫾0.01 0.193⫾0.01 A3(Q0 2 ) 0.147⫾0.008 0.120⫾0.012 0.107⫾0.035 0.101⫾0.005 0.083⫾0.005 A4(Q0 2 ) 0.081⫾0.006 0.064⫾0.008 0.048⫾0.020 0.057⫾0.005 0.046⫾0.005

(13)

关7兴共and references therein兲 is valid for any partial contribu-tion to the distribucontribu-tion funccontribu-tion. With this assumpcontribu-tion, one can see from Fig. 5 that the lowest quark-antiquark valence contribution to the distribution function is able to saturate the phenomenologically found distribution in the region of x ⭓0.4. The difference in distributions for lower values of x may be attributed to higher Fock states which become domi-nant as x→0. Such a picture is also in agreement with the conclusions made in Ref. 关7兴. In addition to other sources that can change the normalization of our result, the effect of nonperturbative evolution关46兴 from an initial scale␮₀2up to

Q2⬃1 GeV2 may be important.

VII. RESULTS AND DISCUSSIONS

In summary, we have presented theoretical predictions for the valence-quark distribution function, Eq.共51兲, and its mo-ments, Eq.共46兲, for the pion. The calculations are based on the instanton model of the QCD vacuum as a candidate treat-ment of nonperturbative dynamics, expressing the observable hadron properties in terms of fundamental characteristics of the vacuum state. We found that the instanton model de-scribes well the vacuum expectation values of the lowest-dimension quark-gluon operators and the pion low-energy observables. To obtain these results, we have used gauge invariant forms for the dynamically generated quark masses and quark pion vertex, by using path-ordered Schwinger

P exp factors. Such factors enter in the definition of nonlocal

quantities 共like nonlocal quark condensate兲, which effec-tively take into account radiation effects of gluon and photon fields when two quarks are separated. Thus, we are led to express the form of the pion quark distribution function in terms of the effective instanton size␳c, and the quark-mass parameter m_q.

The pion quark distribution function extracted corre-sponds to a low normalization scale, where the effective in-stanton approach is justified. It is shown that the validity of parton sum rules for the isospin and total momentum distri-bution is a consequence of the compositeness condition and the strict implementation of gauge invariance. We have used techniques to derive these results which constitute a comple-mentary approach to lattice simulations and to phenomeno-logical fits to experimental data. Using this distribution func-tion as an input, we obtained the quark distribufunc-tion funcfunc-tion in the pion via standard perturbative evolution to higher mo-mentum values, accessible by experiment. A reasonable agreement with the data was found. In fact, the calculations are performed in the quenched approximation, where the ef-fect of intrinsic quark-gluon sea is neglected. We expect that the effects of the intrinsic quark-gluon component of the pion wave function and the nonperturbative evolution at in-termediate energy scale provide a better agreement between theoretical predictions and phenomenological analysis.

ACKNOWLEDGMENTS

The authors are grateful to I. V. Anikin, M. Birse, S. B. Gerasimov, P. Kroll, A. E. Maximov, S. V. Mikhailov, R. Ruskov, I. L. Solovtsov, N. G. Stefanis, and M. K. Volkov

for fruitful discussions of the results. One of us 共A.E.D.兲 thanks the members of the particle physics groups of the Wuppertal University and Instituto de Fı´sica Teo´rica, UNESP, 共Saõ Paulo兲 for their hospitality and interest in the work. A.E.D. was partially supported by the Russian Foun-dation for Fundamental Research 共RFFR兲 96-02-18096 and 96-02-18097, St. Petersburg center for fundamental research grant 95-0-6.3-20 and the Heisenberg-Landau program. L.T. thanks partial support from Conselho Nacional de Desen-volvimento Cientı´fico e Tecnolo´gico do Brasil共CNPq兲 and, in particular, to the ‘‘Fundaçaõ de Amparo a` Pesquisa do Estado de Saõ Paulo 共FAPESP兲’’ to provide the essential support for this collaboration.

APPENDIX: FEYNMAN RULES FOR NONLOCAL VERTICES

Here we briefly show how to derive Feynman rules for quark-hadron-photon vertices 共for more details, see Ref. 关41兴兲. Rewriting Eq. 共19兲 for the action, we have

Sint⫽⫺

冕

d4Xd4x1d4x2f共x1兲f 共x2兲q¯共X⫹x1兲E共X⫹x1;X兲

⫻关Mq⫹gM q¯ q共⌫•T兲⌽共X兲兴E共X;X⫺x2兲q共X⫺x2兲.

共A1兲 Defining

Q共x,y兲⬅E共x,y兲q共y兲 共A2兲

and performing the Fourier transform in the variables x1and x2, we obtain

Sint⫽⫺

冕

d4X

冉

_Q_¯_共x 1,X兲 ⫻关Mq⫹gM q¯ q共⌫•T兲⌽共X兲兴 ⫻

冋

兺

m f(m)共0兲 m! 共⳵x2 2 _兲m

册

_Q_共X,x 2兲. 共A4兲

This expression can be considered as the generation function for the quark-hadron-photon matrix elements. For example,