Fock space distributions, structure functions, higher twists, and smallx
Larry McLerranTheoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455
Raju VenugopalanNiels Bohr Institute, Blegdamsvej 17, Copenhagen, DK-2100, Denmark
~Received 18 September 1998; published 16 March 1999!
We compute quark structure functions and the intrinsic Fock space distribution of sea quarks in a hadronwave function at smallx. The computation is performed in an effective theory at smallx where the gluon fieldis treated classically. AtQ2 large compared to an intrinsic scale associated with the density of gluonsm2, largecompared to the QCD scaleLQCD
2 , and large compared to the quark mass squaredM2, the Fock spacedistribution of quarks is identical to the distribution function measured in deep inelastic scattering. ForQ2
<M2 but Q2@m2, the quark distribution is computed in terms of the gluon distribution function and explicitexpressions are obtained. ForQ2<m2 but Q2@LQCD
2 we obtain formal expressions for the quark distributionfunctions in terms of the glue. An evaluation of these requires a renormalization group analysis of the gluondistribution function in the regime of high parton density. For light quarks at highQ2, the DGLAP flavorsinglet evolution equations for the parton distributions are recovered. Explicit expressions are given for heavyquark structure functions at smallx. @S0556-2821~99!02305-X#
One of the more interesting problems in perturbatQCD is the behavior of structure functions at small valuesBjorkenx. In deep inelastic scattering~DIS! for instance, fora fixed Q2@LQCD
2 , the operator product expansion~OPE!eventually breaks down at sufficiently smallx @1#. Thereforeat asymptotic energies, the conventional approaches towcomputing observables based on the linear DokshitGribov-Lipatov-Altarelli-Parisi~DGLAP! @2# equations areno longer applicable and novel techniques are required. Eat current collider energies such as those of the DESYepcollider HERA where the conventional wisdom is that tDGLAP equations successfully describe the data, therreason to believe that effects due to large logarithmsaS log(1/x) ~or large parton densities! are not small and wemay be at the threshold of a novel region where non-lincorrections to the evolution equations are large@3,4#. Onereason violations of DGLAP evolution have not been seclearly thus far at HERA is the small phase volume for gluemission@5#. A straightforward way to further probe thiregion at current colider energies would be by using nucbeams at HERA@6# or in electron-proton collisions at thCERN Large Hadron Collider~LHC! where the phase volume for gluon emission is signficantly larger@7#.
In recent years, a non-OPE based effective field theapproach to smallx physics has been developed by Lipatand collaborators@8#. Their initial efforts resulted in an equation known popularly as the Balitskii-Fadin-Kuraev-Lipato~BFKL! equation@9#, which sums the leading logarithms oaS log(1/x) in QCD. In marked contrast to the leading twiAltarelli-Parisi equations for instance, it sums all twist oerators that contain the leading logarithms inx. The solu-tions to the BFKL equation predict a rapidly rising gluodensity and there was much initial euphoria when the H1ZEUS data at HERA showed rapidly rising parton densit
0556-2821/99/59~9!/094002~20!/$15.00 59 0940
f
rdsr-
en
isin
r
n
r
ry
ds
@10#. However, it was shown since:The rapid rise of the structure functions can arguably
accounted for by the next to leading order~NLO! DGLAPequations by appropriate choices of initial parton densi@11,12#.
The next to leading logarithmic corrections to the BFKequation computed in the above mentioned effective fitheory approach arevery large @13–15#. As a consequencethe theoretical situation is wide open and novel approacneed to be explored.
An alternative effective field theory approach to QCDsmallx was put forward in a series of papers@16–20#. In theapproach of Lipatov and collaborators, the fields of thefective theory are composite Reggeons and Pomerons.approach is motivated by the Reggeization of the gluon toccurs in the leading log result. Our approach based on R@16–20# is instead a Wilson renormalization group approawhere the fields are those of the fundamental theory butform of the action at smallx is obtained by integrating oumodes at higher values ofx. Integrating out the higherxmodes results in a set of non-linear renormalization groequations@20#. If the parton densities are not too high, threnormalization group equations can be linearized and hbeen shown to agree identically with the leading log BFKand smallx DGLAP equations@19#. There is much effortunderway to explore and make quantitative predictionsthe non-linear regime beyond@21,22#.
In this paper, we apply the above effective action aproach to study the fermionic degrees of freedom at smalx.At small values ofx, gluon degrees of freedom dominate athe fermionic degrees of freedom present are essentiallysea quarks that are radiatively generated from the glue@23#@and are thereforeO(aS) corrections#. Nevertheless, the sequark distributions are extremely important since theydirectly measured in deep inelastic scattering experimentsthis paper, we will develop a formalism, in the context of t
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
renormalization group approach, which relates structfunctions at smallx to the sea quark distributions, and therfore to the gluon distribution. We derive analytic expressiosumming a particular class of all twist operators whichargue give the dominant contribution at smallx. At leadingtwist, in light cone gauge, these reduce to the well knosimple relation between the structure functionF2 and the seaquark Fock space distribution function@24#
F2~x,Q2!5E0
Q2
dkt2 dNsea
dkt2dx
.
Above, x and Q2 are the usual invariants in deep inelasscattering. We show explicitly that for light quarks andhigh Q2, we reproduce the Altarelli-Parisi evolution equtions for the quark distributions at smallx.
There has been much interest in heavy quark distributimotivated partly by the significant contribution of heaquarks to the structure functions at HERA@25#. Until re-cently, heavy quarks were treated as infinitely massiveQ2 equal to or less then the quark mass squared and masbelow. There now exist approaches which study quarktributions in a unified manner for a range ofQ2 and quarkmasses~for a discussion and further references see Refs.@26,27#!. A nice feature of our formalism is that heavy quaevolution is treated on the same footing as light quarksspecific predictions can be made forF2
charm/bottom/F2 withinour formalism. These can also be related to the diffractcross section at smallx but that issue will not be addressedthis paper. This issue and the relation of our work toabove cited work and other recent works on heavy quproduction at smallx @28,29# will be addressed at a latedate.
The results of our analysis are the following. In our theoof the gluon distribution functions, a dimensionful scale apears which measures the density of gluons per unit are
m251
s
dN
dy, ~1!
where s is the hadronic cross section of interest. Herey5y02 ln(1/x), y0 is an arbitrarily chosen constant andx isBjorken x. When this parameter satisfiesm@LQCD thegluon dynamics, while nonperturbative, is both weakcoupled and semiclassical. We shall always assume thsufficiently smallx this is satisfied.
At small x, the gluon field, being bosonic, has to btreated non-perturbatively. This is analogous to the strfield limit used in Coulomb problems. Fermions, on the othhand, do not develop a large expectation value and matreated perturbatively. To lowest order inaQCD , the gluondistribution function is determined by knowing the fermionpropagator in the classical gluon background field. In geral, this propagator must be determined to all orders inclassical gluon field as the field is strong. This can be ddue to the simple structure of the background field.
At this point it is useful to distinguish between two diferent quantities which are often used interchangeably. Ois the quark structure functions as measured in deep inel
09400
e-s
n
t
s
rlesss-
d
e
ek
-
at
grbe
-ee
etic
scattering and the second is the Fock space distributionquarks. At highQ2 which is usually the case consideredperturbative QCD, these two quantities are essentially idtical. However, for massive quarks whenQ2<M2 even ifQ2@LQCD
2 , the two quantities differ. ForQ2@m2, regard-less of mass, the quark Fock space distribution functionthe quark structure functions may both be simply expresas linear functions of the gluon distribution functions. Thefore, with knowledge of the gluon distribution function, oncan compute the quark distribution function.
For the case of massless quarks, whenm2>Q2@LQCD2 ,
we are still in the weak coupling limit. However, we mukeep all orders in the gluon field. In this region, the integtion over the gluon fields in our effective field theory cannbe directly performed as yet since it requires a full renormization group analysis of the theory. In other words, the msure of integration for the high density regime in the effetive theory has not yet been computed. Nevertheless,obtain an explicit functional dependence on the gluon fiewhich must be integrated over with the right measure.
The power of the technique which we use to analyze tproblem is that it does not rely on a high twist expansionuses only the weak coupling nature of the theory which mbe true at smallx if the gluon density is very high. We artherefore in a position to find non-trivial relations betwethese various parameters in a region where the weakpling analysis is valid but where perturbation theory aleading order operator product expansion methods arevalid.
This paper is organized as follows. In Sec. II we wrdown and review an effective action for the smallx modes inQCD. The action is imaginary and the modes are averaover with a statistical weight
exp†2F@r#‡,
whereF@r# is a functional over the color charge densityr ofthe higherx modes. The functionalF@r# obeys a non-linearrenormalization group equation. Of particular interest in tpaper is the saddle point solution of the effective action sithe sea quark distributions are computed in the classbackground field of this action. This section is a quick rview of known results which are necessary to understandremainder of the paper.
An expression relating the electromagnetic currecurrent correlator to the fermion propagator in the classbackground field is derived in Sec. III. We also discusslight cone Fock space distributions and their relation tostructure functions in this section.
In Sec. IV, we solve the Dirac equation in the classicbackground field and obtain an explicit expression forfermion Green’s function in the classical background field
The Green’s function is used in Sec. V to compute thequark distribution function and the leading twist contributioto the structure functionF2(x,Q2) at small x. The coloraveraging over the functionalF@r# in the distribution func-tion is compactly represented by a functiong(pt), whereptcan be interpreted as the intrinsic transverse momentumthe glue at high parton densities. It is shown explicitly th
2-2
a
onmovncaor
is
uwf
ear-aLth
dat
te
al
f-herto
ceee
inolor
riant
p-o
bengn.
is
ethanu-
n-
the
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
for light quark masses the flavor singlet Altarelli-Parisi equtions at smallx are recovered.
The current-current correlator and the structure functiare computed explicity in Sec. VI. As a check of our coputation, it is shown that the leading twist results are recered in the appropriate limit. The heavy quark structure futions are computed explicitly. The phenomenologicimplications and the connections to the recent literatureheavy quark production will not be addressed in this wobut will be considered at a later date.
Section VII contains a summary of our results and a dcussion of future work.
The first of two Appendixes contains a discussion of onotation and conventions. In the second Appendixpresent an explicit form forg(pt) for the particular case oGaussian color fluctuations.
II. EFFECTIVE FIELD THEORY FOR SMALL xPARTONS IN QCD: REVIEW OF RESULTS
We will discuss below an effective action for the weparton modes in QCD. The action contains an imaginpiece which involves functionalF@r# which satisfies a nonlinear Wilson renormalization group equation. In the wefield limit of this renormalization group equation, the BFKequation is recovered. In the double logarithmic region,evolution equation is also equivalent to DGLAP.1 We nextdiscuss the classical background field which is the sadpoint solution of this action. It is this background field ththe sea quarks couple to at smallx and the properties of thebackground field will be relevant for the discussion in lasections.
A. The effective action and the Wilson renormalization groupat small x
In the infinite momentum frameP1→`, the effectiveaction for the soft modes of the gluon field with longitudinmomentak1!P1 ~or equivalentlyx[k1/P1!1) can bewritten in light cone gaugeA150 as
Se f f52E d4x1
4Gmn
a Gmn,a1i
NcE d2xtdx2ra~xt ,x2!
3Tr„taW2`,`@A2#~x2,xt!…
1 i E d2xtdx2F@ra~xt ,x2!#. ~2!
Above, Gmna is the gluon field strength tensor,ta are the
SU(Nc) matrices in the adjoint representation andW is thepath ordered exponential in thex1 direction in the adjointrepresentation ofSU(Nc),
W2`,`@A2#~x2,xt!5P expF2 igE dx1Aa2~x2,xt!t
aG .~3!
1This was first noticed by Yuri Dokshitzer in his paper in Ref.@2#.
09400
-
s---lnk
-
re
y
k
e
le
r
The above is the most general gauge invariant form@19# ofthe action that was proposed in Ref.@16#.
This is an effective action valid in a limited range oP1!L1 whereL1 is an ultraviolet cutoff in the plus component of the momentum. The degrees of freedom at higvalues ofP1 have been integrated out and their effect isgenerate the second and third terms in the action.
The first term in the above is the usual field strength pieof the QCD action and describes the dynamics of the wpartons at the smallx values of interest. The second termthe above is the coupling of the wee partons to the hard ccharges at higher rapidities, withx values corresponding tovalues ofP1>L1. When expanded to first order inA2 thisterm gives the ordinaryJ•A coupling for classical fields. Thehigher order terms are needed to ensure a gauge invacoupling of the fields to current.
In the infinite momentum frame, only theJ1 componentof the current is large~the other components being supressed by 1/P1). The longer wavelength wee partons dnot resolve the higher rapidity parton sources to within 1/P1
and for all practical purposes, one may write
ra~xt ,x2!→ra~xt!d~x2!. ~4!
The last term in the effective action is imaginary. It canthought of as a statistical weight resulting from integratiout the higher rapidity modes in the original QCD actioExpectation values of gluonic operatorsO(A) are then de-fined as
^O~A!&5*@dr#exp~2F@r#!*@dA#O~A!exp~ iS@r,A# !
*@dr#exp~2F@r#!*@dA#exp~ iS@r,A# !,
~5!
whereS@r,A# corresponds to the first two terms in Eq.~2!.The color averaging procedure for fermionic observablesdiscussed further in Sec. V A.
In Ref. @16# a Gaussian form for the action
E d2xt
1
2m2 rara, ~6!
was proposed, wherem2 was the average color chargsquared per unit area of the sources at higher rapiditiesis appropriate for our effective action, that is. For large nclei A@1 it was shown that
m251
pR2
Nq
2Nc;A1/3/6 fm22. ~7!
This result was independently confirmed in a model costructed in the nuclear rest frame@30#. If we include thecontribution of gluons which have been integrated out byrenormalization group technique, one finds that@31#
m251
pR2 S Nq
2Nc1
NcNg
Nc221D . ~8!
Here Nq is the total number of quarks withx above thecutoff
2-3
an
a-
ta
,
hi
o
tefothurrg
aa
utelar
ixer
ob
a,n
a
alto
coa
inlys
e
ion-
en-
atut
eld
e
n
c-
send
re-
eful
d tod
ed
eld
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
Nq5(iE
x
1
dx8qi~x8! ~9!
where the sum is over different flavors, spins, quarksantiquarks. For gluons, we also have
Ng5Ex
1
dx8g~x8!. ~10!
The value ofpR2 is well defined for a large nucleus. Forsmaller hadron, we must take it to bes, the total cross section for hadronic interactions at an energy correspondingthe cutoff. This quantity will become better defined forhadron in the renormalization group analysis.
The above equation form2 is subtle because, implicitlyon the right hand side, there is a dependence onm throughthe structure functions themselves. This is the scale at wthey must be evaluated. Calculatingm therefore involvessolving an implicit equation. Note that because the gludistribution function rises rapidly at smallx, the value ofmgrows asx decreases.
The Gaussian form of the functionalF@r# is reasonablewhen the color charges at higher rapidity are uncorrelaand are random sources of color charge. This is trueinstance in a very large nucleus. It is also true if we studyFock space distribution functions or deep inelastic structfunctions at a transverse momentum scale which is lathan an intrinsic scale set byaSm. In this equationaS isevaluated at the scalem. At smaller transverse momentscales, one must do a complete renormalization group ansis to determineF@r#. This analysis is not yet complete, bshould be feasible in the context of the weakly coupled fitheory as long as the transverse momentum scale remlarger than thatLQCD . The color averaging procedure fothe case of Gaussian fluctuations is discussed in Append
A final comment about the limit of applicability of thclassical action above concerns limitations in the transvemomentum range. The action above is valid only when pring transverse momenta scalespT<m. This includes theGaussian region sinceaS!1. At higher transverse momentone must use DGLAP evolution, with the structure functioas determined at lower values ofQ2 as boundary conditions@32#. This will be important for the case of heavy quarksthe transverse momentum scale there is very large.
The above comments on the renormalization group ansis show the limitations of our analysis with respectquarks. For transverse momentum scalespt@asm, one canuse a Gaussian source and all relevant quantities can beputed explicitly. At smaller scales, one can derive a formexpression, which, hopefully, will be directly computedthe near future. For heavy quarks, the Gaussian anashould be adequate.
To complete a review of the renormalization group, wbriefly review the procedure used to determineF@r#. It wasshown that a Wilson renormalization group procedure@18#could be applied to derive a non-linear renormalizatgroup equation forF@r#. The procedure, briefly, is as follows. The gauge field is split as
09400
d
o
ch
n
dreeer
ly-
dins
B.
se-
s
s
y-
m-l
is
Ama ~x!5bm
a ~x!1dAma ~x!1am
a ~x!, ~11!
wherebma (x) is the saddle point solution of Eq.~2! and cor-
responds to the hard modes above the longitudinal momtum scaleL. The fluctuation fielddAm
a (x) contains the softmodesL81,k1,L1 andam(x) are soft fields with longi-tudinal momentak1,L81. The cutoffs are chosen such thaS log(L81/L1)!1. Small fluctuations are performed abothe saddle point solutionbm
a (x) to the effective action at thescaleL1, to obtain the effective action for the fieldsam
a (x)at the scaleL81. The new charge density at this scaler8 isgiven byra85ra1dra, wheredra can be expressed as thsum of linear and bi-linear terms in the fluctuation fiedAm
a (x).To leading order inaS , the effective action at the scal
L81 can be expressed in the same form as Eq.~2!, with thefunctional F@r8# satisfying a non-linear renormalizatiogroup equation@19,20#. In terms of the statistical weightZ5exp(2F@r#), it can be expressed as
dZ
d log~1/x!5aSF1
2
d2
drmdrn~Zxmn!2
d
drm~Zsm!G ,
~12!
wheres@r# andx@r# are respectively one and two point funtions obtained by integrating overdA for fixed r. The onepoint function s includes the virtual corrections toF@r#while the two point functionx includes the real contributionto F@r#. Both of these can be computed explicitly from thsmall fluctuations propagator in the classical backgroufield. The propagator was first computed by fixing thesidual gauge freedom to bedA2(x250) @17# but a lessrestrictive gauge choice was later found which may be usfor computings andx @20#.
For weak fields, the free gluon propagator can be useobtain the well known BFKL equation for the unintegrategluon density, which is defined as
dN
dkt2 5E d2xte
2 iktxt^r~xt!r~0!&r . ~13!
Performing the renormalization group procedure definabove to obtain the charge densityr85r1dr, one obtains
^r8r8&r2^rr&r5aS log~1/x!@2^rs&r1^x&r#. ~14!
The one and two point functionss andx respectively can becomputed to linear order in the classical background fiand the results are@19#
sa~kt!52g2Nc
2~2p!3 ra~kt!E d2pt
kt2
pt2~pt2kt!
2 , ~15!
and
x52g2Nc
~2p!3 E d2ptra~pt!r
a~2pt!kt
2
pt2~kt2pt!
2 . ~16!
2-4
ono
lvon
le
tuso
ion
onc
c
cla
ens
ld.inof
wehe
e
lsoionst,ne
atri-thea
bed
canring
heehee
inzed
en-
-
ced
f-
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
The above are respectively the virtual and real contributito the change in the color change density after integratingthe modesL81,k1,L1. Substituting these into Eq.~14!,one obtains the well known BFKL equation
xdN
dkt2dx
5aSNc
2p2 E d2pt
kt2
pt2~pt2kt!
2 F dN
dkt2 22
dN
dpt2G .
~17!
Strenuous efforts are currently underway to computesandx to all orders in the background field and thereby sothe full non-linear Wilson renormalization group equatifor F@r# @21,22#.
B. The classical background field at smallx
The effective action in Eq.~2! has a remarkable saddpoint solution @16,18,30#. It is equivalent to solving theYang-Mills equations
DmGmn5Jndn1, ~18!
in the presence of the sourceJ1,a5ra(xt ,x2). Here we willallow the source to be smeared out inx2 as this is useful inthe renormalization group analysis. It is also useful for initively understanding the nature of the field. One finds alution whereA650 and
Ai521
igV] iV†, ~19!
for i 51,2 is a pure gauge field which satisfies the equat
Di
dAi
dy5gr~y,x'!. ~20!
Here Di is the covariant derivative] i1V] iV† and y5y0
1 log(x2/x02) is the space-time rapidity andy0 is the space-
time rapidity of the hard partons in the fragmentation regiAt small x we will use the space-time and momentum spanotions of rapidity interchangeably@32#. The momentumspace rapidity is defined to bey5y02 ln(1/x) where x isBjorken x. The solution of the above equation is
Ari ~xt!5
1
ig~Peig*
y
y0dy8~1/¹'2
!r~y8,xt!!
3¹ i~Peig*y
y0dy8~1/¹'2
!r~y8,xt!!†. ~21!
To compute the classical nuclear gluon distribution funtion, for instance,
dN
d3k5
1
~2p!3 2uk1u E d3xd3x8eik•~x2x8!
3^Aia~x2,xt!Ai
a~x82,xt8!&r , ~22!
one needs in general to average over the product of thesical fields at two space-time points with the weightF@r# asshown in Eq. ~5! or for the Gaussian measure with thweight in Eq.~6!. In the latter case, exact analytical solutio
09400
sut
e
--
.e
-
s-
are available for correlators in the classical background fieFor the case of interest here, the fermion Green’s functionthe classical background field will depend on correlatorsthe form ^V(xt)V
†(yt)&r where theV’s are SU(Nc) gaugetransformation matrices defined above. In Appendix B,discuss in detail the computation of this correlator for tcase of Gaussian fluctuations.
III. THE CURRENT-CURRENT CORRELATORAT SMALL x
In this section, we will derive a formal expression for thhadron tensorWmn at small x relating it to the fermionGreen’s function in the classical background field. We aderive a relation between the light cone quark distributfunction and the fermion Green’s function. To leading twithe structure functions are simply related to the light coquark distribution function. In general~for example, forheavy quark distributions! this is not true. Nevertheless,simple relationship may be found between the gluon disbution functions and that of the quarks. This is becausequarks distribution functions is given by an integral overpropagator in the classical gluon background field descriabove.
A. Derivation
In deep inelastic electroproduction, the hadron tensorbe expressed in terms of the forward Compton scatteamplitudeTmn by the relation@33#
Wmn~q2,P•q!52Disc Tmn~q2,P•q!
[1
2pIm E d4x exp~ iq•x!
3^PuT„Jm~x!Jn~0!…uP&, ~23!
where ‘‘T’’ denotes time ordered product,Jm5cgmc is thehadron electromagnetic current and ‘‘Disc’’ denotes tdiscontinuity of Tmn along its branch cuts in the variablP•q. Also, q2→` is the momentum transfer squared of tvirtual photon2 andP is the momentum of the target. In thinfinite momentum frame,P1→` is the only large compo-nent of the momentum. The fermion state above is usedthe expectation value for the current operators is normalias ^PuP8&5(2p)3E/md (3)(P2P8) wherem is the mass ofthe target hadron. This definition ofWmn and normalizationof the state is traditional, and we will abide by these convtions in spite of the awkward factors ofm. We will see in theend that all factors ofm cancel from the definition of quantities of physical interest.~The normalization we will use inthis paper for quark and lepton states will have E/m replaby 2P1.)
Let us first describe the computation o^PuT„Jm(x)Jn(0)…uP&. In our computation, we have an ex
2Note that in our metric convention, a space-like photon hasq2
5Q2.0.
2-5
csi
thinone
t
e
ta
e
liveisef
in
enit
thck
rog
ro
pan-
ot
u-veishst.a
hichs, atlerrsest
ehtn
nt
ion
s a
k.
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
ternal source corresponding to the particle whose state veis denoted byuP&. Our source is located at some fixed potion. We must therefore consider the generalization ofWmn
for such a source which has a position dependence. Notefor a given source, we also have a lack of translationalvariance in the transverse direction. Transverse translatiinvariance is restored after integration over the source. This no dependence of our source onx1. Therefore the relevanvariable isx2.
We now argue that the relevant definition ofWmn is
Wmn~q2,P•q!51
2ps
P1
mIm E d4xdX2eiq•x
3^T„Jm~X21x/2!Jn~X22x/2!…&.
~24!
To see this let us first verify that we can writeWmn in thisform for the conventional definition valid for plane wavstatesuP&. Notice that we can defineO&5^PuOuP&/^PuP&whereO is any operator. As mentioned above, the expection value ^PuP&5(2p)3E/md (3)(0)5(2p)3E/mV. Herewe shall take the spatial volumeV to bes times an integralover the longitudinal extent of the state. Using these convtions, we see that we reproduce the above definition ofWmn.
This definition corresponds to treating the variableX2 asa center of mass coordinate andx2 as a relative longitudinaposition. For a translationally invariant state, this would gthe longitudinal dimension of the system. The definitionLorentz covariant as will be shown explicitly in Sec. VI. Thintegration overX2 is required since we must include all othe contributions from quarks at allX2 to the distributionfunction. In our external source language, the variableP1
can be taken to be the longitudinal momentum correspondto the fragmentation region. In the end all of theP1 ~and m!dependence will disappear upon taking the infinite momtum limit. To check this definition later, we shall show thatreproduces the conventional results in the highq2 region.
The expectation value is straightforward to compute inlimit where the gluon field is treated as a classical baground field. If we write
^T„Jm~x!Jn~y!…&5^T„c~x!gmc~x!c~y!gnc~y!…&,~25!
then when the background field is classical, and one ignoquantum corrections arising from either loops of fermionsloops of gluons~a good approximation in the weak couplinlimit of high parton densities!, we obtain
^T„Jm~x!Jn~y!…&5Tr„gmSA~x!…Tr„gnSA~y!…
1Tr„gmSA~x,y!gnSA~y,x!…. ~26!
In this expression,SA(x,y) is the Green’s function for thefermion field in the external fieldA,
SA~x,y!52 i ^c~x!c~y!&A ~27!
for fixed A ~before averaging over A!.
09400
tor-
at-alre
-
n-
g
-
e-
esr
The first term on the right hand side of Eq.~26! is atadpole contribution which does not involve a non-zeimaginary part. It therefore does not contribute toWmn. Wefind then that
Wmn~q2,p•q!51
2ps
P1
mIm E dX2d4xeiq•x
3^Tr„gmSA~X21x/2,X22x/2!
3gnSA~X22x/2,X21x/2!…&. ~28!
B. The light cone Fock distribution functionand structure functions
The expression we derived above forWmn is entirely gen-eral and makes no reference to the operator product exsion. In particular, it is relevant at the smallx values andmoderateq2 where the operator product expansion is nreliable@1#. At sufficiently highq2 though~and for masslessquarks! it should agree with the usual leading twist comptation of the structure functions. The fact that we do not haa valid operator product expansion forces us to distingubetween two quantities which are identical in leading twiThe first is the Fock space distribution of partons withinhadron. The second are the parton structure functions ware measured in deep inelastic scattering. In our analysihigh values ofq2, these expressions are identical. At smalvalues, say those values typical of the intrinsic transvemomentum scaleaSm, they are no longer the same and mube differentiated between.
We will derive below an expression for the light conquark Fock distribution in terms of the propagator in ligcone quantization@34,35#. The quark Fock space distributiois then simply related to the structure functionF2 for q2
@aS2m2. In light cone quantization, only the two compone
spinor projectionc1 is dynamical.~Note: notation and con-ventions are discussed in Appendix A.! The other two spinorcomponentsc2 ~recall thatc5c21c1) are defined via thelight cone constraint relation defined below in Eq.~41!. Thedynamical fermions can then be written in terms of creatand annihilation operators as
c15Ek1.0
d3k
21/4~2p!3 (s561/2
@eik•xw~s!bs~k!
1e2 ik•xw~2s!ds†~k!#. ~29!
Above bs(k) is a quark destruction operator and destroyquark with momentumk while ds
†(k) is an anti-quark cre-ation operator and creates an anti-quark with momentumAlso above the unit spinorsw(s) are defined as
wS 1
2D5S 0100D ; wS 2
1
2D5S 0010D . ~30!
Note that since
2-6
s
eov
tri
s-
s-
io
th
inIn’sie
e
heenot
er-de-as-rce
ure
ndonsstheof
so-lyndll
heting
ich
e
e
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
d†S ktW ,k1;x1;11
2D5bS 2ktW ,2k1;x1;21
2Dd†S ktW ,k1;x1;2
1
2D5bS 2ktW ,2k1;x1;11
2D ,
one can show that
w~s!bs~k!521/4E d3xe2 ik•xc1,s~x!. ~31!
The light cone Fock distribution function is defined in termof the creation and annihilation operators as
dN
d3k5
1
~2p!3 (s
@bs†bs1ds
†ds#52
~2p!3 (s
bs†bs .
~32!
We have assumed above that the sea is symmetric betwquarks and anti-quarks. Combining the two equations abwe get
dN
d3k5
2&
~2p!3 E d3xd3yeik•~x2y!c1† ~x!c1~y!. ~33!
Using the light cone identity
Tr@g1c~x!c~y!#5&c1~x!c1† ~y!, ~34!
we obtain the following expression for the sea quark disbution function
dN
d3k5
2i
~2p!3 E d3xd3ye2 ik•~x2y! Tr@g1SA~x,y!#
~35!
where the fermion propagatorSA(x,y) is the light cone timeordered productSA(x,y)52 i ^T„c(x)c(y)…& in the back-ground fieldAm. In our effective action approach, as dicussed in Secs. II and III A, we can replaceSA(x,y)→SAcl
(x,y) to obtain the sea quark distribution in the clasical background field.
In a nice pedagogical paper~see Ref.@24# and referenceswithin!, Jaffe has shown that the Fock space distributfunction can be simply related to theleading twiststructurefunction F2 by the relation
F2~x,Q2!5E0
Q2
dkt2x
dN
dkt2dx
. ~36!
Actually, Jaffe’s expression is defined as the sum ofquark and anti-quark distributions. At smallx, these areidentical and the resulting factor of 2 is already includedour definition of the light cone quark distribution function.the following section, we will compute the fermion Greenfunction in the classical background field. We will then usein Eq. ~35! and the above equation to show that we do inderecover the standard perturbative result forF2 . For heavyquarks and/or moderateQ2, structure functions should b
09400
ene,
-
n
e
td
computed by inserting the fermion Green’s function in tdefinition for Wmn. We will see that in general the structurfunctions and the Fock space distribution functions arethe same.
IV. THE FERMION GREEN’S FUNCTION IN THECLASSICAL BACKGROUND FIELD
In this section we shall derive an expression for the fmion propagator in the classical gluon background fieldscribed in Sec. II B. The field strength carried by these clsical gluons is highly singular, being peaked about the sou~corresponding to the parton current atx values larger thanthose in the field! localized atx250. Away from the source,the field strengths are zero and the gluon fields are pgauges on both sides ofx250. The fermion wave function isobtained by solving the Dirac equation in the backgroufield on either side of the source and matching the solutiacross the discontinuity atx250. Once the eigenfunctionare known, the fermion propagator can be constructed instandard fashion. We begin this section with a discussionthe notation and conventions, proceed to write down thelution of the Dirac equation and finally, construct explicitthe fermion propagator in the classical gluon backgroufield. This expression is formally exact and is valid to aorders in the source color charge density.
A. The Dirac equation in the classical background field
In order to compute the propagator for a spinor field in tfundamental representation of the gauge group propagain the background gauge field
A1a 50
A2a 50
t•At5u~x2!k t~xt!, ~37!
wherek t(xt), t51,2 is a two dimensional pure gauge,
k t~xt!521
igV~xt!¹ tV
†~xt!, ~38!
we first need to solve the Dirac eigenvalue equation whcan be written as
$aW t•~pW t2gAW t!2&p1a22&p2a11bM %cl~x!5lcl~x!
~39!
for the spinor fieldc and a corresponding equation forc(x).The a’s and b above are defined in Appendix A andpm52 i ]m . For x2,0, the solution is trivial and is just the frespinor plane wave solution. Forx2.0 the solution is lesstrivial and is given by the non-Abelian analogue of th‘‘Baltz’’ ansatz @36#. The full solution of the Dirac equationin the classical background field is
2-7
tioh
s
ionto
und
(-
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
clqa,s~x!5u~2x2!eiq•xus,l
a ~q!1u~x2!1
&
3E d2pt
~2p!2 dzt„V~xt!V†~zt!…
abeipt•xt2 iq2x1
3ezt•~qt2pt! expS 2 i~pt
21M22l!
2p2 x2D3H 11
~a t•pt1bM !
&q2 J a2ulb,s~q!. ~40!
Above, the superscriptsa,b denote the color index in thefundamental representation ands the spinor index. The el-ementary spinors are normalized asus,l(q)us8,l8(q)52Mdss8dll8 and summed over spins (uu)mn5(M2q” )mn .
The interested reader will notice that the above equais not continuous acrossx250. This is because thougc2(x) is continuous acrossx250, c2 is related toc1 viathe light cone constraint equation
eare-
w
itn
Th
09400
n
c151
&q2 Fa t•S 1
i] t2gAtD1bM Gc2 , ~41!
which is discontinuous acrossx250 by the same amount ain the previous equation.
B. Computation of the fermion propagator
Having obtained the eigenfunctions for the Dirac equatin the classical background field we are now in a positioncompute the fermion propagator in the classical backgrofield. This is given by the relation
S~x,y!5E d4q
~2p!4
1
q21M22 i« (pol
cq~x!cq~y!, ~42!
after identifying q15(qt21M22l)/2q2. It is straightfor-
ward to check from the above expression thatq”1M )S(x,y)5(2p)4d (4)(x2y). Substituting the eigenfunctions from Eq.~40! in the above, we have
The translational symmetry of the Green’s function in thex2
direction is of course broken by the presence of the sourcx250. In the absence of the source of color charge, it mbe confirmed that the free fermion propagator is recoveby puttingV5I , where I denotes the unit matrix in the fundamental representation.
The reader will note that the propagator between tpoints on the same side of the source, for eitherx2,y2,0 orx2,y2.0 is the free propagator or a gauge transform ofThe only non-trivial contribution comes from the pieces conecting points on the opposite sides of the source.u(x2)u(2y2) piece can be written more simply as
atyd
o
.-e
S~x,y!52 i E d4zV~xt!S0~x,z!g2d~z2!S0~z,y!V†~zt!.
~45!
An analogous expression holds for theu(2x2)u(y2) pieceof the propagator. A similar simple expression for the scaquark propagator@37# was found recently by Hebecker anWeigert@38# ~see also the recent work of Balitskii@39#!. Theonly difference between the form of the above result and tfor scalar quarks is theg2 matrix present here due to thdifferent spinor structure and a partial derivative]z1 absenthere due to the 1/2q2 factor in Eq.~43!.
If we define
G~xt ,x2!5u~2x2!1u~x2!V~xt! ~46!
which is the gauge transformation matrix which transfor
2-8
ar
he
c-lant
to
r
gthese
a
en
ca-
o
w
incu-nd-inetiont-
inelsst.
tioningg totheag--les,ror-
thet for
r
ar-fi-
Eq
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
the gluon field at hand to a singular field which has onlyplus component,A8m5dm1a(xt), we then see that oupropagator has the form
SA~x,y!5G~x!S0~x2y!G†~y!2 i E d4zG~x!$u~x2!
3u~2y2!„V†~zt!21…2u~2x2!u~y2!
3„V~zt!21…%G†~y!S0~x2z!g2d~z2!S0~z2y!.
~47!
This very simple form of the propagator is useful in tmanipulations below.
In fact the current-current correlation function is expliitly gauge invariant. We may therefore use the singugauge form of the propagator for computing the currecurrent correlation function
SAsing~x,y!5S0~x2y!2 i E d4z$u~x2!u~2y2!„V†~zt!21…
2u~2x2!u~y2!„V~zt!21…%
3S0~x2z!g2d~z2!S0~z2y!. ~48!
A diagrammatic representation of the form of the propagaabove is shown in Fig. 1. In the expressions below forWmn
we will drop the superscriptsingand simply use the singulagauge expression for the propagator.
The Fock space distribution function however is gaudependent. In computing it we must therefore either useform of the propagator with the explicit gauge matricabove, or go back to our original form. In what follows, wshall use the original form of Eq.~43! for computing theFock space distribution function.
Our result for the fermion propagator in the classicbackground field was obtained for ad-function source in thex2 direction. This assumption was motivated by the obsvation that smallx modes with wavelengths greater tha1/P1 perceive a source which is ad-function in x2. Thepropagator above can also be derived for the generalwhere the source has a dependence onx2. The gauge transforms above are transformed fromV(xt)→V(xt ,x2), topath ordered exponentials, whereV(xt ,x2) is given by Eq.~21!. Our result for the propagator is obtained as a smolimit of Dx251/xP1@x2(51/P1). In other words, ourform for the propagator is the correct one as long asinterpret theu-functions andd-functions inx2 to be so onlyfor distances of interest greater than 1/P1, the scale of theclassical source.
FIG. 1. Diagrammatic representation of the propagator in~48!.
09400
r-
r
ee
l
r-
se
th
e
V. THE LEADING TWIST COMPUTATION OF F 2
Now that we have computed the fermion propagatorthe classical background field, we are in a position to callate the sea quark distributions in this background field, ain turn the structure functionF2 and evolution equations using Eq. ~36!. This calculation is accurate to lowest orderaS but to all orders inaSm. Due to the singular nature of thpropagators in the background field, the actual computaof the distribution function is a little subtle and will be oulined below.
Before we go ahead to the computation, we will begwith a discussion of the averaging procedure over the labof color charges at rapidities higher than those of intereWe will obtain a compact expression for it below.
A. Color averaging over the sources of color chargeat higher rapidities
Our expression for the propagator in the previous secmakes no particular assumption about the color averagover the color labels of the external sources correspondinthe valence quarks and/or gluons at higher rapidities thanrapidity of interest. The expression we quote before avering is the quantity which will be useful in loop graph computations. To relate our expression to physical observabas discussed in Sec. II@see Eq.~5!# we need to average oveall the color labels of the external color charge density cresponding to the color charge densityra(xt ,y) at rapiditiesgreater than the rapidity of interest.
Here we have smeared out the source inx2 and are nolonger treating it as a delta function. This means thatsources acquire a rapidity dependence and that the weighthe Gaussian fluctuations over sources is replaced by
E d2xtdy1
2dm2/dyr2~xt ,y! ~49!
which leads us to define
m25Ey
`
dy8dm2
dy8. ~50!
Here the lower limit would be the rapidity of interest foevaluating the structure functions.
If we average the Green’s function in Eq.~43! over allpossible values of the color labels corresponding to the ptons at higher rapidities, we can employ the following denitions for future reference. Defining
1
Nc^Tr„V~xt!V
†~yt!…&r5g~xt2yt!, ~51!
we see that
g~0!51, ~52!
.
2-9
-lu
ns
efs.
niesd
r-
-n
on
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
which follows from the unitarity of the matricesV. Nowdefining the Fourier transform3
g~pt!5E d2xte2 iptxt@g~xt!21#, ~53!
we have the sum rule
E d2pt
~2p!2 g~pt!50. ~54!
The functiong(pt) will appear frequently in our future discussions and as we shall see, can be related to the gdensity at smallx.
For the particular case of a large nucleus@16# the averag-ing procedure has the form
^O&r5E @dr#O~r!expS 2E0
`
dyE d2xt
Tr r2~xt ,y!
dm2~y!/dy D ,
~55!
wherem2 is the average color charge density per unit traverse area per unit rapidity. For an extensive discussion
he
co
e
rre
09400
on
-of
the above averaging procedure we refer the reader to R@16,30,19#. In Appendix B we will explicitly derive an ex-pression forg(pt) for Gaussian fluctuations. A Gaussiaform for the averaging over color charges at higher rapiditis likely valid for very large nuclei or for realistic nuclei anhadrons forx!1 but not at too smallx’s ~or large enoughparton densities! where non-linear corrections to the renomalization group equations are important.
B. Sea quark Fock space distribution function
The relation Eq.~35! can now be combined with our expression in Eq.~43! to compute the sea quark distributiofunction. We shall use below the following identities:
Tr@g1~M2q” !#54q1
1
2q2 Tr@~M2q”2pt”!g2~M2q” !g1#5
2
q2~M21qt21pt•qt!.
~56!
We then obtain the following expression for the distributifunction:
dNf erm
d3k5
2iNc
~2p!3 E d3xd3yE d4q
~2p!4
ei ~q2k!•~x2y!
q21M22 i«
3H 4q1@u~2x2!u~2y2!1u~x2!u~y2!g~xt2yt!#
1E d2pt
~2p!2 d2zt
2
q2 ~M21pt•qt1qt2!Fu~x2!u~2y2!eipt•~xt2zt!
3exp†2 i „@~pt1qt!22qt
2#/2q2
…x2‡g~xt2zt!1u~2x2!u~1y2!e2 ipt•~yt2zt!
3expS i~pt1qt!
22qt2
2q2 y2Dg~zt2yt!G J . ~57!
all
We will now sketch below the procedure used to simplify tabove equation.
~a! First perform the integrations over the transverseordinates. This introduces a common factorpR2 and for theu(2x2)u(2y2) term a factor d (2)(qt2kt) @a factor gwhich was defined in Eq.~53! pops up in the other threpieces#.
~b! Perform the integrations overx2 and y2. For theu(6x2)u(6y2) pieces, we obtain the factors
3We define the Fourier transform in this way because it cosponds to only the connected pieces in the correlator.
-
1
q12k16 i«
1
q12k817 i«, ~58!
respectively. We have introduced above~to ensure smoothconvergence! a slight difference in the momenta (k1 andk81 respectively! multiplying x2 and y2 in the phases. Inthe final step we take the limitk812k1→0.
~c! The simple contour integration overq1 is done next.This introduces the factorsu(6q2).
~d! The final step is to perform the~logarithmic! integra-tion over q2. The ultraviolet cutoffL→` cancels amongthe different terms and we obtain a finite result. Puttingthe terms together and using the identity in Eq.~54!, we-
2-10
nc
caealtir
if-a-
ar
n
nee
ill
-
-
w-
-
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
obtain the general result for the sea quark distribution fution
1
pR2
dNf erm
dk1d2kt
5Nc
2p4
1
k1 E d2pt
~2p!2 g~pt!
3F12~kt
21kt•pt1M2!
pt212kt•pt
logS ~kt1pt!21M2
kt21M2 D G . ~59!
In the region where the logarithm can be expanded, itbe checked analytically that the argument of the abovepression is positive definite. We have checked numericthat the argument remains positive definite in the en(kt ,pt) phase space~as it should be!.
In the next section we will study the above result in dferent limits and relate it to the well known evolution equtions for sea quark distributions.
C. Evolution equations for sea quark distributions at smallx
In this section, we will show that for largeq2, the Fockspace sea quark distribution we derived above in Eq.~59!gives us the Altarelli-Parisi evolution equation for sea quevolution at smallx.
Towards that end, consider the Fock space distributiothe limit of largekt . In the integral in Eq.~59! we approxi-matekt@pt . Then expanding the logarithm and defining
~kt1pt!22kt
2
kt21m2 511k, ~60!
we find that the terms in the square brackets@¯# in Eq. ~59!can be approximated by
@¯#'Fk
22
k2
32
kt•pt
kt21M2 1
~kt•pt!k
kt21M2 2
k2
2
~kt•pt!
kt21M2G .
~61!
Now g(pt) has rotational symmetry in the transverse plaThis helps simplify our expression above since only evterms inkt•pt survive. To leading order inpt
2/kt2 then, Eq.
~59! reduces to
1
pR2 k1dNf erm
dk1d2kt5
Nc
2p4 E d2pt
~2p!2 g~pt!1
2 F pt2
kt21M2
24
3
kt2pt
2
~kt21M2!2 1
kt2pt
2
~kt21M2!2G .
~62!
Since we are interested in the limitkt2@M2, the above ex-
pression can be further simplified to read
1
pR2 k1dNf erm
dk1d2kt5
Nc
4p4
2
3
1
kt2 E d2pt
~2p!2 pt2g~pt!. ~63!
09400
-
nx-lye
k
in
.n
To make contact with the evolution equations we wnow obtain a relation betweeng(pt) and the gluon distribu-tion function at smallx. We begin with the relation we defined in the last section—Eq.~53!:
out the matrixV511 iL(xt)2L2(xt)/21¯ , and doing thesame for the correlator of gauge fieldsAi5(21/ig)V] iV†,we obtain the relation
pt2g~pt!5
g2
2NcE d2xte
ipt•xt^Aia~xt!Ai
a~0!&r . ~65!
The correlator AiaAi
a& can be related to the gluon distribution function by the formula
dN
d3l5
2u l 1u~2p!3 E d3xd3x8e2 i l 1x2
e1 i l 1x82u~x2!u~x82!
3eil t•xt^Aia~xt!Ai
a~0!&r . ~66!
Integrating both sides overl 1, we obtain
E d2xteil t•xt^Ai
a~xt!Aia~0!&
5k1
pR2
~2p!3
2 Ek1
P1 dl1
u l 1udNglue
d2l tdl1 . ~67!
Substituting the right-hand side~RHS! of the above equationin Eq. ~65! and the resulting expression forpt
2g(pt) into Eq.~63!, we obtain
xdNf erm
dxdkt2 5
aS
2p
2
3
1
kt2 E
x
1 dy
y E0
kt2
dpt2x
dNglue
dydpt2 , ~68!
where we definedx5k1/P1 andy5 l 1/P1.Recall that the structure function to leading twist and lo
est order inaS is
F2~x,Q2!5E0
Q2
dkt2x
dNf erm
dxdkt2 5x@q~x,Q2!1q~x,Q2!#
[2xq~x,Q2!, ~69!
and similarly,
xG~x,Q2!5E0
Q2
xdNglue
dxdkt2 , ~70!
wherexq(x,Q2) andxG(x,Q2) are the quark and gluon momentum distributions respectively. At smallx, the sea is
2-11
n
lla
,
e-
ll-in
nc-arksthatich
’s-
we
ov-urturedruc-er-
nstonines
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
symmetric and we takeq(x,Q2)5q(x,Q2). In the limit oflight quark massesQ2@M2 we find then that
d„xq~x,Q2!…
d log~Q2!5
aS
4p
2
3xE
x
1 dy
y2 yG~y,Q2!. ~71!
Now at smallx, yG(y,Q2) is slowly varying. One can forinstance parametrize it by a power law (1/xCaS), where C issome constant. The scale of variation of the structure fution then corresponds to higher orders inaS . We can there-fore takeyG(y,Q2) out of the integrand.~The same is truefor any other slowly varying function@40#.! We then getfinally for the sea quark evolution equation at smallx theresult
d„xq~x,Q2!…
d log~Q2!5
aS
4p
2
3xG~x,Q2!1O~aS!. ~72!
Thus to lowest order inaS , the sea quark evolution at smax is local and simply proportional to the gluon density at thx.
Now consider the Altarelli-Parisi evolution equation
Q2dS
dQ2 5aS~Q2!
2p@S ^ Pqq1G^ 2 f PqG#. ~73!
Above the operation denotes
A^ B[Ex
1 dy
yA~y!BS x
yD ,
and S5( f(q1q), where f is the number of flavors. AlsoPqq and PqG are the well known Altarelli-Parisi splittingfunctions. Since at smallx the quark distribution isaS sup-pressed relative to the glue, takingq5q and f 51 the lead-ing contribution to the sea quark evolution is
d„xq~x,Q2!…
d log~Q2!5
1
2
aS
2p Ex
1 dy
yG~y,Q2!Fx2
y2 1S 12x
yD 2G .~74!
Above we have made use of the relation 2f PqG(z)5 f @z2
1(12z)2#. Let z5y/x. Then the above relation can be rwritten as
d„xq~x,Q2!…
d log~Q2!5
aS
4p E1
1/x dz
z2 zG~zx,Q2!F 2
z2 1122
zG .~75!
Again, as previously, we can argue that since at smaxzG(zx,Q2) is slowly varying, we can take it out of the integral. Doing that and performing the integration, we obtafinally
d„xq~x,Q2!…
d log~Q2!5
aS
4p
2
3xG~x,Q2!1O~aS!. ~76!
09400
c-
t
which is the same as Eq.~72!. This form of the sea quarkevolution equation at smallx was first obtained by Ellis,Kunzt and Levin@41#.
VI. COMPUTATION OF CURRENT-CURRENTCORRELATOR TO ALL TWISTS IN THE CLASSICAL
BACKGROUND FIELD AT SMALL x
In the previous section we used the fermion Green’s fution derived in Sec. IV B to compute the light cone sea qudistributions at smallx and subsequently the leading twiexpression for the structure functions. It was shown tthese structure functions obeyed evolution equations whwere precisely the smallx Altarelli-Parisi evolution equa-tions.
In this section we will again use the fermion Greenfunction in Eq.~43! to derive an explicit result for the hadronic tensorWmn. This result will be valid to all twists atsmall x and for arbitrary quark masses. For light quarks,will compute the structure functionsF1 and F2 and showthat the leading twist result in the previous section is recered as a limit of our general result. We will also use ogeneral result to obtain expressions for heavy quark strucfunctions at smallx. We should note here that Levin ancollaborators have studied screening corrections to the stture functions for light and heavy quarks in the GlaubGribov framework@42#.
A. Analytic result for Wµn at small x
As in the previous sections, we define
Wmn~q,P,X2!5Im E d4zeiq•z
3 K TXJmS X21z
2D JnS X22z
2D CL ,
~77!
where ‘‘Imaginary’’ stands for the discontinuity inq2. Then
FIG. 2. Polarization tensor with arbitrary number of insertiofrom the classical background field. The wavy lines are pholines, the solid circle denotes the fermion look and the dashed lare the insertions from the background field~see Fig. 1!. The imagi-nary part of this diagram givesWmn.
2-12
e
ly,
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
Wmn~q,P!51
2ps
P1
m E dX2Wmn~q,P,X2!
[1
2psP1 Im E dX2E d4zeiq•zTrXSAclS X21
z
2,X22
z
2DgnSAclS X22z
2,X21
z
2DgmC. ~78!
The only terms in the propagator that contribute to the above are theu(6x2)u(7y2) pieces. Using our representation for thpropagator in Eq.~45!,
Wmn~q,P,X!5Im NcE d4zeiq•zE d4pd4l
~2p!8
d4p8d4l 8
~2p!8 d2utd2ut8~2p!d~p22 l 2!
3~2p!d~p822 l 82!eip•~X1z/22ut!e2 i l •~X2z/22ut!el 8•~X2z/22u8!te2 ip8•~X1z/22ut8!
3TrH ~M2p” !g2~M2 l” !gm~M2 l”8!g2~M2p” 8!gn
~p21M22 i«!~ l 21M22 i«!~ l 821M22 i«!~p821M22 i«!J3XuS X21
z
2D uS z
22X2D1uS X22
z
2D uS 2X22z
2D Cg~ut2ut8!. ~79!
We have used above the conditiong(ut ,ut8)5g(ut82ut)5g(ut2ut8), since the correlation functions are translationalrotationally and parity invariant in the transverse plane. We now perform the integration overX2 and use the identity
E dX2ei ~2p11 l 12 l 811p81!X2XuS X21z
2D uS z
22X2D1uS X22
z
2D uS 2X22z
2D C5e~z2!
1
~p12 l 12p811 l 81!2 sinS z2
2~p12 l 11 l 812p81! D . ~80!
We then obtain
Wmn~q,P!5sP1Nc
2pmIm E d4zeiq•ze~z2!E d4pd4l
~2p!8
d4p8d4l 8
~2p!8
3~2p!2d~p22 l 2!d~p822 l 82!~2p!2d~2!~ l t2pt1pt82 l t8!gS l t2pt2pt81 l t8
2 Deizt•~pt1 l t2pt82 l t8!/2eiz1~p22p82!
3TrH ~M2p” !g2~M2 l” !gm~M2 l”8!g2~M2p” 8!gn
~p21M22 i«!~ l 21M22 i«!~ l 821M22 i«!~p821M22 i«!J3
1
~p12 l 12p811 l 81!2 sinS z2
2~p12 l 11 l 812p81! D . ~81!
The subsequent procedure of solving the integrals is as follows:~a! Perform first the integration overl t8 . Then perform the integration overzt andz1. Definingkt5(pt2 l t1pt82 l 8)/2, thissetsl t85pt2kt2qt , l t5pt2kt andqt5pt2pt8 . Also, we getq25p22p82.~b! Next perform the integration overz2. This gives us
E dz2e~z2!eiq1z2~e2 iz2~p12p81!2e2 iz2~ l 12 l 81!!
5 i F l 812 l 12p811p1
~q12p11p811 i«!~q12 l 11 l 811 i«!1~ i«→2 i«!G . ~82!
The numerator above cancels the term 1/(p12 l 12p811 l 81) in Wmn.~c! Lastly, we do the integrations overp81 and l 81. This setsp815p12q1 and l 815 l 12q1. Then we can define inWmn,p85p2q and l 85 l 2q with l 5p2k andk250.
After these considerations, we can writeWmn as
094002-13
ysby the
case,r multiple
theh that0nce canes
r
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
Wmn~q,P!5sP1Nc
2pmIm E d4p
~2p!4
d2kt
~2p!2
dk1
~2p!g~kt!
3TrH ~M2p” !g2~M2 l” !gm~M2 l”8!g2~M2p” 8!gn
~p21M22 i«!~ l 21M22 i«!~ l 821M22 i«!~p821M22 i«!J , ~83!
wherel 8, p8 and l are defined as in step~c! above. Correspondingly, we can writeWmn as the imaginary part of the diagramshown in Fig. 2.
For the case of deep inelastic scattering,q2.0 ~see footnote 1!, and we can cut the above diagram only in the two washown in Fig. 3~the diagram where both insertions from the external field are on the same side of the cut is forbiddenkinematics!.
Also interestingly, the contribution toWmn can be represented solely by the diagram in Fig. 4 and not, as is usually thefrom the sum of this diagram and the standard box diagram. This is because in our representation of the propagatoinsertions from the external field on a quark line can be summarized into a single insertion. See for instance Eq.~48! whichmakes this point clear.
Applying the Landau-Cutkosky rule, and making the shiftp→p1k, Eq. ~83! can be written as
With an appropriate change of variables, the second term is the same as the first except that nowm↔n. We then get
Wmn~q,P!5sP1Nc
2pm E d4p
~2p!4
d2kt
~2p!2
dk1
~2p!g~kt!M
mnu~p11k1!u~2p1!~2p!2d„~p1k!21M2…
3d„~p2q!21M2…
1
p21M2
1
~p1k2q!21M2 , ~85!
where above the trace is represented by4
4Kinematic note: the observant reader will notice we have setq150 here. Since we are working in the infinite momentum frame,hadron has only one large momentum component,P1. The rest are set to zero. For the photon, we choose a left moving frame sucq05uqzu and q150. Then,q25qt
2.0, P•q52P1q2 and xB j52q2/(2P•q)[qt2/(2P1q2). Since in the infinite momentum frame
,xB j,1, this givesq2.0. We are at liberty to choose the above frame since the hadron tensor is clearly Lorentz invariant and hebe expressed purely as a function ofq2 andP•q. The explicit presence ofP1 in Eq. ~79! may give the reader cause for concern. It arisfrom a relativistic normalization of the vacuum states. One can show that, despite appearances, Eq.~79! is Lorentz invariant. Of course oulater results will confirm this fact.
FIG. 3. Cut diagrams corresponding to the imaginary part ofWmn.
094002-14
on
rv
us
of
us
-
q.t
-
eto
nc-e
at
atf
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
Mmn5Tr$~M2p”2k” !g2~M2p” !gm~M2p”1q” !g2~M2p”
2k”1q” !gn1m↔n%. ~86!
Performing thed-function integrations above, our expressifor Wmn can be simplified to
Wmn52sP1Nc
16pm
1
~q2!2 E d2kt
~2p!2
d2pt
~2p!2 g~kt!
3E2`
2M p2q2 /2q2 dp1
2p
Mmn
M p1k2q2 •I ~kt ,pt ,q,p1!,
~87!
with the definitions M p2q2 5(pt2qt)
21M2 and M p1k2q2
5(pt1kt2qt)21M2, and
I ~pt ,kt ,q,p1!51
p12~M p
22M p2q2 !
2q2
31
p12M p2q
2 ~M p1k2 2M p1k2q
2 !
~2q2!M p1k2q2
.
~88!
Equation~87! is our general result for the hadronic tensoWe shall now study the different components of the abotensor and extract from these different limits of interest to
B. Structure functions at small x
The hadronic tensorWmn can be decomposed in termsthe structure functionsF1 andF2 as @33#
mWmn52S gmn2qmqn
q2 DF1
1S Pm2qm~P•q!
q2 D S Pn2qnP•q
q2 D F2
~P•q!,
~89!
FIG. 4. In the singular gauge representation for the propag@see Eq.~48! and Fig. 1#, multiple, higher twist contributions fromthe classical gluon background field to the current-current correl~imaginary part of the LHS! is equivalent to the imaginary part othe RHS.
09400
.e.
where Pm is the four-momentum of the hadron or nucleandP25m2'0 (!q2). In the infinite momentum frame, wehaveP1→` andP2,Pt'0. The above equation can be inverted to obtain expressions forF1 andF2 in terms of com-ponents ofWmn. Since in our kinematicsq150 ~see foot-note 3 for a kinematic note!
F15F2
2x1S q2
~q2!2DW22, ~90!
with
1
2xF252S ~q2!2
q2 DW11. ~91!
It is useful to verify explicitly that our expression forWmn
derived in an external field can be written in the form of E~89!. Recall thatWmn can be written in Lorentz covarianform by using the vectornm5dm1. Usingn•g52g2 in Eq.~83!, we see thatWmn is a Lorentz covariant function of theonly vectors in the problem—qm and nm. Identifying nm
5Pm/P1 in Eq. ~89!, we see that these forms are in complete agreement. We also see that all factors ofm disappearfrom F1 andF2 by the explicit forms of Eqs.~89! and~83!.Henceforth we will takem51 since it disappears from thquantities of interest, and was in fact only introduced duehistorical normalization conventions.
We also see that the structure functions can only be futions ofq2 andn•q by Lorentz invariance. We can therefortakeq150 for the purpose of computingF1 andF2 .
To computeW11 and W22, we need to know the thetracesM 11 and M 22, respectively in Eq.~87!. We cancompute them explicitly and the results are
1
16M 115
1
2~M p
2M p1k2q2 1M p1k
2 M p2q2 !2
1
2qt
2kt2
~92!
and
M 22532~p2!2~p22q2!2. ~93!
From the relations above ofF1 andF2 to W11 andW22,we obtain from Eq.~87! the following general results for thestructure functions for arbitrary values ofQ2, M2 and theintrinsic scalem,5
F25sNc
16p2 E d2pt
~2p!2
d2kt
~2p!2 g~kt!
3M 11
~M p2q2 M p1k
2 2M p1k2q2 M p
2!logS M p1k
2 M p2q2
M p1k2q2 M p
2D ,
~94!
and
5This is implicitly contained in the functiong(kt) in Eq. ~87!.
or
or
2-15
sio
b-
e
ri-ll
ns
g
d af
nal
it
me-
n-om
p-satrk
elurons
y
the
ant.
uonnc-tic
lu-ale
g
ll
ereion
oree
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
F15F2
2x2
q4sNc
2x2p4 E d2ptE d2kt
~2p!2 g~kt!
3F~a,b!
M p2q2 M p1k2q
2 . ~95!
Above, we defined the functionF in terms ofa and b ~inturn functions ofpt , kt , qt andM ) which are defined as
a5S 12M p1k
2
M p1k2q2 D ; b5S 12
M p2
M p2q2 D . ~96!
The general expression forF is
F~a,b!51
ab F1
32
3
2 S 1
a1
1
b D1S 1
a2 11
ab1
1
b2D1
ab
2~b2a! H 1
a2 S 121
a D 2
log~12a!221
b2
3S 121
b D 2
log~12b!2J G . ~97!
Equation ~94! and Eq. ~95! are the central results of thiwork. Since they are the most general possible expressfor arbitrary values ofQ2, M2, andm2, it is inevitable thatthey look complicated. We shall show in the following susection that they simplify considerably in the highq2 limit.
C. Structure functions in the limit q2˜`
We shall now obtain the leading twist limits of Eq.~95!and Eq.~94!. In particular we will show that our structurfunction for M2→0 andq2→` is identical to the structurefunction obtained by integrating the light cone Fock distbution in Eq. ~59!. That this should be the case is a weknown property of the leading twist structure functio@24,43#. Further, we will recover the Callan-Gross result@44#F15F2/2x in this limit.
Consider first our general formula forF2 in Eq. ~94!. Thetrace simplifies considerably when we putM50. For qt@pt ,kt , we obtain
M 11→16qt2~pt
21pt•kt!.
In the logarithm, the ratioM p2q2 /M p1k2q
2 →1. Finally, in thedenominator of the integrals,
~M p2q2 M p1k
2 2M p1k2q2 M p
2!→Mq2~M p1k
2 2M p2!.
Then putting these back into our general expression6 we ob-tain
6The contribution in thept integral in the regionpt;qt is identi-cal to that frompt!qt . This provides a factor of 2 that must btaken into account.
09400
ns
F25sNc
2p4 E d2ptE d2kt
~2p!2 g~kt!
3F12~pt
21kt•pt!
~kt212kt•pt!
logS ~kt1pt!2
pt2 D G . ~98!
A comparison with Eq.~59! immediately reveals that, settinkt↔pt , and integrating the latter overpt up to q2 gives anidentical result to the one above. Thus we have recoverewell known, non-trivial leading twist result as the limit oour general expression forq2→` andM→0.
Even though our general expression for the longitudistructure functionF1 looked terribly complicated, in thelimit considered here it is remarkably simple. In this lima,b→1 and hence Eq.~97! for F→ 1
3 —a constant. Theproduct in the denominator of Eq.~95!, M p2q
2 M p1k2q2 →q4
and cancels theq4 factor outside the integral. From the surule Eq.~54!, we find remarkably that the complicated intgral vanishes and Eq.~95! reduces to
F15F2
2x. ~99!
The above is the well known Callan-Gross relation.We should clarify the result obtained above to avoid co
fusion. The reader may note above that the deviation frthe Callan-Gross relation vanishes as a power law asq2
→`. On the other hand, it is well known in QCD@45,46#that the violations of the Callan-Gross relation only disapear logarithmically asq2→`. The apparent contradiction iresolved by one realizing that the logarithmic violationslarge q2 in QCD come from diagrams where the sea quaemits a gluon~thereby violating Feynman’s parton modhelicity argument!. These diagrams are of higher order in opicture and are therefore not included. In fact, the deviatifrom the Callan-Gross relation of the sort discussed above~atsmall x) should die off faster than logarithmically at verlarge q2 because for sufficiently largeq2, the violations ofthe Callan-Gross relation should come from preciselydiagrams not included here. At moderateq2 however, thecontributions we have discussed above should be import
VII. SUMMARY
In this paper we have used a classical theory of the glfield to derive expressions for Fock space distribution futions of quarks and structure functions for deep inelasscattering. This theory is valid at smallx when the gluondensity is large. In this region, the coupling constant evaated at this density scale is small. With this density scdenoted bym2, we have seen that whenq2@a2m2 and q2
@M2, where M is the mass of the heavy quark beinprobed, all leading twist results are reproduced. Forq2
<a2m2 or q2<M2, we derived an expression valid to aorders in twist but only to leading order inaS . In this kine-matic region, the Callan-Gross relation is not valid, and this no simple relation between the Fock space distributfunction and the structure functions for quarks.
The structure function of heavy quarks deserves m
2-16
nsonio
oou
ed
i-bu
olt
stioe
trior
efolik
shis
u
s.
o
ts
he
s a
k.
ight
il-
s-
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
study. If M2@a2m2, then for q2@a2m2, the heavy quarkdistribution is a linear function of the gluon distributiofunction. The gluon distribution function may neverthelebe computed to all orders in twist. The leading twist relatibetween the structure function and the quark distributfunction is however not valid for largeM2. As we go tosmaller values ofx corresponding to larger values ofm2, thenon-linearities corresponding to higher powers of the gludensity turn on and can be studied systematically inweak coupling formalism.
The situation for light quarks is also amusing and nemore study as well. Ifq2<a2m2, then the non-linearities inthe gluon distribution function become important. In this knematic region, we expect saturation of the gluon distrition function, and our functiong(kt);1/kt
2 up to logarithmsof kt . If this is the case, a look at the definition ofF2 andF1shows that these distributions should be dimensionallyorderq2. In this region a precise analytic estimate is difficusince in the integral representations forF1 andF2 @Eqs.~95!and ~94! respectively#, there is no hierarchy of momentumscales. All momenta are of orderq, and the integrand doenot simplify much. Nevertheless, we see that the saturaof the gluon density is sufficient to imply saturation of thquark density.
Both the study of the heavy quark and light quark disbutions merit more theoretical and phenomenological wwithin the framework described in this paper.
ACKNOWLEDGMENTS
We would both like to thank Leonid Frankfurt, EugenLevin, and Mark Strikman for reading the manuscript andtheir very useful comments and suggestions. We wouldto thank NORDITA ~L.M.! and TPI, Minnesota~R.V.! fortheir hospitality. R.V.’s work was supported by the DaniResearch Council and the Niels Bohr Institute. L.M’s worksupported by DOE grant DE-FG02-87ER40328.
APPENDIX A: NOTATION AND CONVENTIONS
We start by defining our convention and notations. Ometric is the12 metric gmn5(2,1,1,1). The gammamatrices in space-time co-ordinates are denoted by caretthe chiral representation,
g05S 0 I
I 0D ; g i5S 0 s i
2s i 0 D ; g55S I 0
0 2I D ,
and $gm,gn%522gmn. Above,s i , i 51,2,3 are the usual 232 Pauli matrices andI is the 232 identity matrix. In lightcone co-ordinates, g65(g06g3)/& and $gm,gn%522gmn, whereg115g2250, g125g21521 andgt1 ,t251 wheret1 ,t251,2 here stand for the two transverse cordinates. Note for instance that in this conventionA152A2 and At51At. Also, q2522q2q11qt
2 hence a‘‘space-like’’ q2 implying large space-like componenwould correspond toq2.0.
We now define the projection operators
09400
s
n
nr
s
-
f
n
-k
re
r
In
-
a65g0g6
&[
g7g6
2, ~A1!
which project out the two component spinorsc65a6c@34#. Some relevant properties of the projection operatorsa6
are
~a6!25a6; a6a750; a11a251; ~a6!†5a6.~A2!
It follows from the above thatc11c25c. In the follow-ing, we will also use the familiar Dirac conventionsb5g0
anda'5g0g' .The two component spinor is the dynamical spinor in t
light cone QCD HamiltonianPQCD2 and it is defined in terms
of creation and annihilation operators as
c15Ek1.0
d3k
21/4~2p!3 (s561/2
@eik•xw~s!bs~k!
1e2 ik•xw~2s!ds†~k!#. ~A3!
Above bs(k) is a quark destruction operator and destroyquark with momentumk while ds
†(k) is an anti-quark cre-ation operator and creates an anti-quark with momentumAlso above the unit spinorsw(s) are defined as
wS 1
2D5S 0100D ; wS 2
1
2D5S 0010D . ~A4!
The creation and annihilation operators obey the equal lcone time (x1) commutation relations
$bs~kW ,x1!,bs8†
~kW8,x1!%5$ds~kW ,x1!,ds8†
~kW8,x1!%
5~2p!3d~3!~kW2kW8!dss8 . ~A5!
The above definitions ensure that the light cone QCD Hamtonian can be defined asPQCD
2 5P021VQCD , where the non-
interacting piece of the Hamiltonian is defined as
P025E d3k
~2p!3 (s561/2
~kt21M2!
2k1 „bs†~k!bs~k!1ds
†~k!ds~k!….
~A6!
The definition of quark distribution functions is further dicussed in the text of Sec. III B.
The dynamical components of the gauge fieldsAia(x) with
i 51,2 in light cone gaugeA150 are defined as
Aia~x!5E
k1.0
d3k
A2uk1u~2p!3 (l51,2
dl i
3@eik•xala~k!1e2 ik•xal
a†~k!#, ~A7!
2-17
ar
ves-
evi
to
of-
,
fie
t--
ular
en-
-om
e
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
where thel’s here correspond to the two independent polizations andal
a†(ala) creates~destroys! a gluon with mo-
mentumk. They obey the commutation relations
@ala~kW !,al8
b †~kW8!#5~2p!3d~3!~kW2kW8!dabdll8 . ~A8!
The gluon distribution function is then defined as
dN
d3k5
aia†ai
a
~2p!3 . ~A9!
Performing the Fourier transform of the gauge field abowe obtain Eq.~66! in Sec. V C. For a more extensive dicussion of the above formalism see Ref.@47#.
We should mention here that there are several convtions in use. For a discussion of some of these, see the rearticle by Brodsky, Pauli and Pinsky@48#. Our convention islike that of Kogut and Soper@34# but differs from theirs forquark spinor and gauge field normalizations by a facAuk1u/A(2p)3.
APPENDIX B: DERIVATION OF THE FUNCTION g„PT…
FOR GAUSSIAN FLUCTUATIONS
Since g(pt) is defined as the Fourier transform(1/Nc)^Tr„U(xt)U
†(yt)…&r , we need to compute this correlator in co-ordinate space first. Note that the symbol^¯&r
denotes the averaging over with a Gaussian weight. Nowdiscussed by Jalilian-Marianet al., if
U~y,xt!5U`,y~xt!5P expF i Ey
`
dy8L~y8,xt!G , ~B1!
whereU`,y(xt) is the path ordered exponential~in rapidity!which corresponds to the pure gauge potentialAi
52U`,y(xt)¹iU`,y
† (xt)/ ig, thenAi satisfies the Yang-Millsequation
Di
dAi ,a
dy5gra~xt ,y!, ~B2!
if L, the argument of the path ordered exponential, satisthe Laplace equation
¹2La~xt ,y!5ra~xt ,y!. ~B3!
The measure for the functional integral is then
E @dr#expS 2E0
`
dyE d2xt
Tr r2
m2~y! D→E @dL#expS 2E
0
`
dyE d2xt
Tr~L¹4L!
g4m2~y! D .
~B4!
As argued by Jalilian-Marianet al., we can write
09400
-
,
n-ew
r
as
s
U`,y~xt!5:U`,y~xt!:expS 2g4NcG~0!
2 Ey
`
dy8m2~y! D ,
~B5!
where : : denotes normal ordering and
G~xt!51
¹4 [G~0!1xt
2
16plog~xt
2L2!1finite pieces. . . .
~B6!
Above,G(0)}1/L2 whereL corresponds to an infrared cuoff. Writing U`,y(xt) in the above normal ordered form enables us to isolate and exponentiate the infrared singterms coming from disconnected graphs.
Our correlator has then the form
g~xt ,xt ;y,y!5Ncg8~xt ,xt ;y,y!
3expX2 g4NcG~0!
2
3S Ey
`
1Ey
` D dy8m2~y8!C, ~B7!
where
g8~xt ,xt ;y,y!5E @dL#
3expS 2E0
`
dyE d2xt
Tr~L¹4L!
g4m2~y! D3„:U`,y~xt!:…„:U`,y~ xt!…
†. ~B8!
Expanding out first few terms in the path ordered expontials above, we have
g8~0!51,
g8~1!5Nc
221
2NcG~xt2 xt!g
4u~y2 y!Ey
`
dy8m2~y8!,
g8~2!51
2! FNc221
2NcG~xt2 xt!g
4Ey
`
dy8m2~y8!G2
.
~B9!
In the expression forg8(2) above only one of the two possible terms survive on account of the path ordering. Frsimilar considerations it can be argued that in general
g8~n!51
n! FNc221
2NcG~xt2 xt!g
4j~y!G2
, ~B10!
where j(y)5*y`dy8m2(y8). Resumming the terms abov
and including the disconnected pieces, we have
g~xt2 xt ;y,y!5expS g4~Nc221!j
2Nc„G~xt2 xt!2G~0!…D .
~B11!
2-18
s
fare
ion
tiveq.
FOCK SPACE DISTRIBUTIONS, STRUCTURE . . . PHYSICAL REVIEW D 59 094002
The above expression is the complete non-perturbative refor g(xt2 xt ;y,y8). Note that trivially g(0;y,y)51 as wewould expect from the definition ofg.
Taking the Fourier transform ofg,
g~pt!5E d2xteipt•xtg~xt!
[E d2xteipt•xt exp@k„G~xt!2G~0!…#,
~B12!
where k5g4(Nc221)j(y)/2Nc , and expanding outg(xt),
we obtain
g~pt!5Fd~2!~pt!1kE d2xteipt•xtG~xt!2kG~0!d~2!~pt!
1¯ G . ~B13!
-
as
P.
-
t,
09400
ultIf we now recall the definition ofG(xt) from Eq. ~B6!,
G~xt!5E d2kt
~2p!2
eikt•xt
kt4 , ~B14!
and substitute forG(xt) in the above, we find
g~pt!5„12kG~0!…d~2!~pt!1Nc
221
2Nc
~4p!2aS2j
pt4 1¯ .
~B15!
The first term ing(pt) is not relevant for our computation othe sea quark distributions since the relevant momentapt@LQCD . The second term is the perturbative expresscomputed by us previously@37#—up to a factor of 2 whichwas missing in that paper. Note also that the perturbag(pt) above explicitly satisfies the sum rule condition of E~54!.
t,
D
t,
,
J.
@1# A. H. Mueller, Phys. Lett. B396, 251 ~1997!.@2# V. N. Gribov and L. N. Lipatov, Sov. J. Nucl. Phys.15, 78
~1972!; G. Altarelli and G. Parisi, Nucl. Phys.B126, 298~1977!; Yu. L. Dokshitzer, Sov. Phys. JETP73, 1216~1977!.
@3# Eric Laenen and Eugene Levin, Annu. Rev. Nucl. Part. Sci.44,199 ~1994!.
@4# L. Frankfurt, W. Koepf, and M. Strikman, Phys. Rev. D54,3194 ~1996!.
@5# L. Frankfurt and M. Strikman, hep-ph/9806536.@6# M. Arneodo, A. Bialas, M. W. Krasny, T. Sloan, and M. Strik
man, Proceedings ofWorkshop on Future Physics at HERA,Hamburg, Germany, 1995, hep-ph/9610423.
@7# FELIX Collaboration, V. Avatiet al., in Multiparticle Dynam-ics, 1997, Proceedings of 27th International Symposium, Frcati, Italy, 1997, edited by G. Caponet al. @Nucl. Phys. B~Proc. Suppl.! 71, 459 ~1999!#, hep-ex/9801021.
@8# L. N. Lipatov, Phys. Rep.286, 131 ~1997!.@9# E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Sov. Phys. JET
45, 199 ~1977!; Ya. Ya. Balitsky and L. N. Lipatov, Sov. JNucl. Phys.28, 22 ~1978!.
@10# I. Abt et al., Nucl. Phys.B407, 515 ~1993!; M. Derrick et al.,Z. Phys. C72, 399 ~1996!.
@11# M. Gluck, E. Reya, and A. Vogt, Z. Phys. C67, 433 ~1995!;Eur. Phys. J. C5, 461 ~1998!.
@12# H. L. Lai et al., Phys. Rev. D55, 1280~1997!.@13# V. S. Fadin and L. N. Lipatov, Phys. Lett. B429, 127 ~1998!.@14# G. Camici and M. Ciafaloni, Phys. Lett. B412, 396 ~1997!;
417, 390~E! ~1998!.@15# Y. V. Kovchegov and A. H. Mueller, hep-ph/9805208.@16# L. McLerran and R. Venugopalan, Phys. Rev. D49, 2233
~1994!; 49, 3352~1994!.@17# A. Ayala, J. Jalilian-Marian, L. McLerran, and R. Venugo
palan, Phys. Rev. D52, 2935~1995!; 53, 458 ~1996!.@18# J. Jalilian-Marian, A. Kovner, L. McLerran, and H. Weiger
Phys. Rev. D55, 5414~1997!.
-
@19# J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. WeigerNucl. Phys. B504, 415 ~1997!; Phys. Rev. D59, 014014~1998!.
@20# J. Jalilian-Marian, A. Kovner, and H. Weigert, Phys. Rev.59, 014015~1998!.
@21# J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. WeigerPhys. Rev. D59, 034007~1999!.
@22# E. Iancu, A. Leonidov, and L. McLerran~in progress!.@23# A. H. Mueller, Nucl. Phys.B307, 34 ~1988!; B317, 573
~1989!; B335, 115 ~1990!.@24# R. L. Jaffe, Nucl. Phys.B229, 205 ~1983!.@25# C. Adloff et al., Z. Phys. C72, 593 ~1996!; J. Breitweget al.,
Phys. Lett. B407, 402 ~1997!.@26# M. A. G. Aivazis, J. C. Collins, F. I. Olness, and W.-K. Tung
Phys. Rev. D50, 3102~1994!.@27# R. S. Thorne and R. G. Roberts, Phys. Rev. D57, 6871~1998!.@28# M. Buzaet al., Phys. Lett. B411, 211 ~1997!.@29# A. D. Martin, R. G. Roberts, and W. J. Sterling, Eur. Phys.
C 2, 287 ~1998!.@30# Yu. V. Kovchegov, Phys. Rev. D54, 5463 ~1996!; 55, 5445
~1997!.@31# M. Gyulassy and L. McLerran, Phys. Rev. C56, 2219~1997!.@32# L. McLerran and R. Venugopalan, Phys. Lett. B424, 15
~1998!.@33# S. Pokorski,Gauge Field Theories~Cambridge University
Press, Cambridge, England, 1987!.@34# J. Kogut and D. Soper, Phys. Rev. D1, 2901~1970!.@35# J. Bjorken, J. Kogut, and D. Soper, Phys. Rev. D3, 1382
~1971!.@36# A. J. Baltz and L. McLerran, Phys. Rev. C58, 1679~1998!.@37# L. McLerran and R. Venugopalan, Phys. Rev. D50, 2225
~1994!.@38# A. Hebecker and H. Weigert, Phys. Lett. B432, 215 ~1998!.@39# I. Balitskii, Phys. Rev. Lett.81, 2024~1998!.@40# T. Gousset and H. J. Pirner, Phys. Lett. B375, 349 ~1996!.
2-19
s.
ev.
LARRY McLERRAN AND RAJU VENUGOPALAN PHYSICAL REVIEW D59 094002
@41# R. K. Ellis, Z. Kunzt, and E. Levin, Nucl. Phys.B420, 517~1994!.
@42# E. Gotsman, E. Levin, and U. Maor, Eur. Phys. J. C5, 303~1998!; Phys. Lett. B425, 369 ~1998!; E. Gotsmanet al.,hep-ph/9808257.
@43# R. K. Ellis, W. Furmanski, and R. Petronzio, Nucl. PhyB207, 1 ~1982!; B212, 29 ~1983!.
@44# G. G. Callan and D. J. Gross, Phys. Rev. Lett.22, 156~1969!.
09400
@45# A. Zee, F. Wilczek, and S. B. Treiman, Phys. Rev. D10, 2881~1974!.
@46# W. Bardeen, A. J. Buras, D. W. Duke, and T. Muta, Phys. RD 18, 3998~1978!.
@47# R. Venugopalan, nucl-th/9808023.@48# S. Brodsky, H-C. Pauli, and S. S. Pinsky, Phys. Rep.301, 299
