Operator Product Expansion in QCD Is Not Consistent With Quantum Field Theory For Gluon Distribution Function
aa r X i v : . [ h e p - ph ] M a y Operator Product Expansion in QCD Is Not Consistent WithQuantum Field Theory For Gluon Distribution Function
Gouranga C Nayak ∗ C. N. Yang Institute for Theoretical Physics,Stony Brook University, Stony Brook NY, 11794-3840 USA (Dated: May 29, 2018)
Abstract
Since the operator product expansion (OPE) is applicable at short distance the OPE in QCDdoes not solve the long distance confinement problem involving hadron in QCD where the non-perturbative QCD is applicable. In this paper we show that the gauge invariant definition ofthe non-perturbative gluon distribution function inside the hadron consistent with the operatorproduct expansion (OPE) in QCD at high energy colliders is not consistent with the gauge invariantdefinition of the non-perturbative gluon distribution function in the quantum field theory.
PACS numbers: 12.38.-t; 12.38.Aw; 14.70.Dj; 12.39.St ∗ G. C. Nayak was affiliated with C. N. Yang Institute for Theoretical Physics in 2004-2007.
Typeset by REVTEX 1 . INTRODUCTION
Quantum electrodynamics (QED) is the fundamental theory of the nature describing theinteraction between electrons and photons. Similarly the quantum chromodynamics (QCD)is the fundamental theory of the nature describing the interaction between quarks and gluons.In renormalized QED the coupling decreases at long distance and in renormalized QCD [1]the coupling decreases at short distance due to asymptotic freedom in QCD [2].Since the QCD coupling decreases at short distance the partonic scattering cross sectionat short distance is calculated by using the perturbative quantum chromodynamics (pQCD).However, since the QCD coupling increases at long distance the pQCD can not be applied tostudy hadron formation from partons because the hadron formation from partons involveslong distance physics in QCD. Hence in order to study formation of hadron from partonsthe non-perturbative QCD is required at long distance where confinement happens in QCD.Since the non-perturbative QCD is not solved yet, most of the non-perturbative quantitiesin QCD (such as the parton distribution function (PDF) inside the hadron and the partonto hadron fragmentation function (FF) etc.) are not calculated at present from the firstprinciple in QCD.Because of this reason the non-perturbative PDF and FF are extracted from the exper-iments from the measurement of (physical) hadronic cross section by using the factorizedformula [3–13] dσ ( H H → H + X ) = X i,j,l Z dx Z dx Z dz f i/H ( x , Q ) f j/H ( x , Q ) d ˆ σ ij → kl D H /k ( z, Q ) . (1)In eq. (1) the f i/H ( x , Q ) is the parton distribution function (PDF) of the parton i insidethe hadron H , the d ˆ σ ij → kl is the short distance partonic level scattering differential crosssection calculated by using pQCD, D H /k ( z, Q ) is the fragmentation function (FF) of theparton k to fragment to the outgoing hadron H , the Q is the factorization/renormalizationscale, x ( x ) is the longitudinal momentum fraction of the parton i ( j ) with respect tothe incoming hadron H ( H ), z is the longitudinal momentum fraction of the hadron H with respect to the parton k , the X represents the other (inclusive) outgoing hadrons and i, j, k, l = q, ¯ q, g represent light quark, antiquark, gluon.The non-perturbative parton distribution function (PDF) is an universal quantity, i. e. ,2t does not change from one experiment to another. Hence it is essential to derive the correctdefinition of the parton distribution function inside the hadron from the first principle inQCD. The correct definition of the parton distribution function inside the hadron must beconsistent with the definition of the distribution function in quantum field theory becauseQCD is a quantum field theory.Since the non-perturbative QCD is not solved yet, the operator product expansion (OPE)in QCD is assumed to be valid to study the QCD phenomenology for various processes at highenergy colliders. For example the operator product expansion (OPE) in QCD is assumed tobe valid to define the PDF which is then extracted from the experiments by using eq. (1).The definition of the parton distribution function (PDF) inside the hadron consistent withthe operator product expansion (OPE) in QCD is derived in [3].However, in this paper, we show that the gauge invariant definition of the non-perturbative gluon distribution function inside the hadron consistent with the operatorproduct expansion (OPE) in QCD at high energy colliders is not consistent with the gaugeinvariant definition of the non-perturbative gluon distribution function in quantum field the-ory. Since the operator product expansion is applicable at short distance the OPE in QCDdoes not solve the long distance confinement problem involving hadron in QCD where thenon-perturbative QCD is applicable.The paper is organized as follows. In section II we discuss the definition of the gluondistribution function inside the hadron at high energy colliders using quantum field theory(QCD). In section III we show that the unrenormalized QCD and the renormalized QCDpredict the same hadronic cross section at all orders in coupling constant in QCD. In sectionIV we discuss the operator product expansion and the total cross section in electron-positronannihilation to hadrons. In section V we focus on the operator product expansion in QCDand the confinement in QCD. In section VI we discuss the definition of the gluon distributionfunction inside the hadron at high energy colliders using the operator product expansion(OPE). In section VII we show that the operator product expansion in QCD is not consis-tent with quantum field theory for the gluon distribution function. Section VIII containsconclusions. 3 I. DEFINITION OF THE GLUON DISTRIBUTION FUNCTION INSIDEHADRON USING QUANTUM FIELD THEORY (QCD)
In the factorized formula in eq. (1), the single parton i (a quark or an antiquark or a gluon)from the incoming hadron H interacts with another single parton j (a quark or an antiquarkor a gluon) from the other incoming hadron H in the short distance partonic level crosssection ˆ σ ij → kl to produce the parton k (a quark or an antiquark or a gluon) which fragmentsto the outgoing hadron H . Hence the parton distribution function (PDF) f i/H ( x , Q )represents the probability of finding the single parton i (a quark or an antiquark or a gluon)inside the incoming hadron H in eq. (1). Similarly the parton to hadron fragmentationfunction (FF) D H /k ( z, Q ) represents the probability of a single parton k (a quark or anantiquark or a gluon) fragmenting to the outgoing hadron H in eq. (1).In quantum mechanics the probability of finding a particle can be obtained from thequantum wave function of the particle. Hence in quantum mechanics the probability offinding a particle is proportional to | φ ( x ) | where φ ( x ) is the quantum wave function of theparticle. Therefore in quantum field theory one expects that the distribution function of theparticle be proportional to the correlation function of the type φ † ( x ) ψ (0). In this sectionwe will derive the definition of the quark distribution function inside the hadron and thedefinition of the gluon distribution function inside the hadron by using the quantum fieldtheory.In quantum field theory the free Dirac field containing positive and negative energysolution can be written as [14] X s =1 Z d k (2 π ) √ E k [ a s ( k ) u s ( k ) e − ik · x + b † s ( k ) v s ( k ) e ik · x ] = ψ q ( x ) + ψ † ¯ q ( x ) (2)where a ( k ) is the annihilation operator of the quark and b † ( k ) is the creation operator of theantiquark. The equal-time anti-commutation relations at the initial time, say at t = t in = 0is given by { a r ( p ) , a † s ( p ′ ) } = (2 π ) δ rs δ (3) ( ~p − ~p ′ ) , { a r ( p ) , a s ( p ′ ) } = 0 , { a † r ( p ) , a † s ( p ′ ) } = 0 { b r ( p ) , b † s ( p ′ ) } = (2 π ) δ rs δ (3) ( ~p − ~p ′ ) , { b r ( p ) , b s ( p ′ ) } = 0 , { b † r ( p ) , b † s ( p ′ ) } = 0 . (3)4irac spinors satisfy X s =1 u s ( p )¯ u s ( p ) = p/ + m, X s =1 v s ( p )¯ v s ( p ) = p/ − mu † r ( p ) · u s ( p ) = v † r ( p ) v s ( p ) = δ rs E p , u † r ( ~p ) · v s ( − ~p ) = v † r ( ~p ) · u s ( − ~p ) = 0 . (4)The number operator ˆ n ( k ) of a particle in quantum field theory is given byˆ n ( k ) = a † ( k )( k ) (5)and the distribution function f ( p ) of the particle inside the hadron H in quantum fieldtheory is given by [15] f ( p ) (2 π ) δ (3) ( ~p − ~k ) = < H | a † ( p ) a ( k ) | H > (6)where a † ( k ) is the creation operator of the particle and a ( k ) is the annihilation operator ofthe particle.Let us derive the following equation for the quark case at the initial time t = t in = 0 Z d xe − i~k · ~x < H | ψ † ( x ) ψ (0) | H > = Z d x X r =1 2 X s =1 Z d p (2 π ) q E p Z d p ′ (2 π ) q E p ′ < H | a † r ( p ) a s ( p ′ ) | H > × u † r ( p ) · u s ( p ′ ) e i ( ~p − ~k ) · ~x = X r =1 2 X s =1 Z d p ′ (2 π ) q E p ′ E k < H | a † r ( k ) a s ( p ′ ) | H > u † r ( k ) · u s ( p ′ ) . (7)From eq. (6) we find for the quark case f q/H ( ~p ) (2 π ) δ rs δ (3) ( ~p − ~p ′ ) = < H | a † r ( p ) a s ( p ′ ) | H > (8)where f q/H ( ~p ) is the quark distribution function inside the hadron H .Using eqs. (8) and (4) in (7) we find f q/H ( ~p ) = 12 Z d xe − i~p · ~x < H | ψ † ( ~x ) ψ (0) | H > . (9)In quantum field theory the eq. (9) for the free field theory can be extended to the interact-ing field theory (to the full QCD) by replacing the free fields by the interacting fields. Hencefrom eq. (9) we find that the gauge non-invariant definition of the quark distribution func-tion f q/H ( p ) inside the hadron H which is consistent with the definition of the distributionfunction of the quark in quantum field theory in full (interacting) QCD is given by f q/H ( p ) = 12 Z d x [ e ip · x < H | ψ † i ( x ) ψ i (0) | H > ] t =0 . (10)5here ψ i ( x ) is in the full (interacting) QCD.Similar to quark we find for the antiquark case at the initial time t = t in = 0 Z d xe − i~k · ~x < H | ψ ( x ) ψ † (0) | H > = X r =1 2 X s =1 Z d p ′ (2 π ) q E p ′ E k < H | b † s ( p ′ ) b r ( k ) | H > v † r ( k ) · v s ( p ′ ) . (11)Similar to eq. (8) we find for the antiquark case f ¯ q/H ( ~p ) (2 π ) δ rs δ (3) ( ~p − ~p ′ ) = < H | b † r ( ~p ) b s ( ~p ′ ) | H > (12)where f ¯ q/H ( ~p ) is the antiquark distribution function inside the hadron H .Using eqs. (12) and (4) in (11) we find f ¯ q/H ( ~p ) = 12 Z d xe − i~p · ~x < H | ψ ( ~x ) ψ † (0) | H > . (13)Extending the free field theory equation (13) to interacting field theory one finds thatthe gauge non-invariant definition of the antiquark distribution function f ¯ q/H ( p ) inside thehadron H which is consistent with the definition of the distribution function of the antiquarkin quantum field theory in full (interacting) QCD is given by f ¯ q/H ( p ) = 12 Z d x [ e ip · x < H | ψ i ( x ) ψ † i (0) | H > ] t =0 (14)where ψ i ( x ) is in the full (interacting) QCD.In order to derive the definition of the gluon distribution function inside the hadron H in QCD using quantum field theory let us consider the massless scalar gluon case beforeconsidering the gluon distribution function in QCD. Similar to the non-interacting quarkand antiquark case discussed above one finds that the distribution function f ( ~p ) of masslessscalar gluon inside the hadron in non-interacting quantum field theory is given by f ( ~p ) = 2 E p Z d xe − i~p · ~x < H | φ ( ~x ) φ (0) | H > (15)where φ ( x ) is the massless scalar gluon field in the non-interacting quantum field theory.Extending massless free scalar gluon field equation (15) to massless interacting scalargluon field we find that the definition of the scalar gluon distribution function f ( p ) insidethe hadron H which is consistent with the definition of the distribution function of the scalargluon in interacting quantum field theory is given by f ( ~p ) = 2 E p Z d x [ e ip · x < H | φ ( x ) φ (0) | H > ] t =0 (16)6here φ ( x ) is the massless scalar gluon field in the interacting quantum field theory.Note that the massless scalar gluon field or the scalar gluon distribution function insidethe hadron H does not correspond to any physical situation. We have considered it here forsimplicity to derive the gluon distribution function inside the hadron H in QCD.Hence extending eq. (16) for the scalar gluon distribution function to the gluon distri-bution function f g/H ( p ) inside the hadron in QCD we find that the gauge non-invariantdefinition of the gluon distribution function f g/H ( p ) inside the hadron H in QCD obtainedby using the quantum field theory is given by [8] f g/H ( p ) = 2 E p Z d x [ e ip · x < H | Q cν ( x ) Q νc (0) | H > ] t =0 (17)where Q νc ( x ) is the (quantum) gluon field in the full (interacting) QCD with ν = 0 , , , c = 1 , ..., A aµ ( x )we have proved the factorization of soft and collinear divergences in QCD at all orders incoupling constant in [6–11]. Hence from eq. (10) we find that the gauge invariant definitionof the quark distribution function inside hadron at high energy colliders which is consistentwith the definition of the distribution function in quantum field theory and is consistentwith the factorization of soft and collinear divergences in QCD at all orders in the couplingconstant is given by [7, 13] f q/H ( x ) = 14 π Z dy ′ e − ixp + y ′ < H | ¯ ψ (0 , y ′ , T ) γ + [ P e igT c R y ′ dy ′′ A + c (0 ,y ′′ , ] ψ (0) | H > (18)where A aµ ( x ) is the SU(3) pure gauge background field.Similarly from eq. (17) we find that the gauge invariant definition of the gluon distributionfunction inside hadron at high energy colliders which is consistent with the definition of thedistribution function in quantum field theory and is consistent with the factorization of softand collinear divergences in QCD at all orders in the coupling constant is given by [8, 12, 13] f g/H ( x ) = p + π Z dy ′ e − ixp + y ′ < H | Q bν (0 , y ′ , T ) [ P e igT ( A ) c R y ′ dy ′′ A + c (0 ,y ′′ , ] Q νb (0) | H > . (19)7ince the (quantum) gluon field Q aµ ( x ) in eq. (19) transforms gauge covariantly under thetype I gauge transformation in the background field method of QCD [16] the definition ofthe gluon distribution function in eq. (19) is gauge invariant [8, 12].In eqs. (18) and (19) the Wilson line contains the SU(3) pure gauge background field A aµ ( x ). Since A aµ ( x ) is the SU(3) pure gauge background field it gives vanishing field tensor F aλµ ( x ) = 0 , F bλν ( x ) = ∂ λ A bν ( x ) − ∂ ν A bλ ( x ) + gf bad A aλ ( x ) A dν ( x ) (20)which means the SU(3) pure gauge background field A aµ ( x ) does not contribute to the phys-ical cross section. The only role of the SU(3) pure gauge background field A aµ ( x ) in eqs.(18) and (19) is to maintain the gauge invariance of the quark and gluon distribution func-tions and to prove the factorization of soft and collinear divergences in QCD at all ordersin coupling constant. The color field also plays an important role to study production ofquark-gluon plasma in the laboratory [17–20].It is important to note that since the SU(3) pure gauge background field A aµ ( x ) is theclassical field one finds that the definition of the gauge invariant quark distribution functionin eq. (18) contains quadratic powers of the (quantum) quark field ψ ( x ) consistent with thegauge invariant definition of the distribution function of quark in quantum field theory evenafter the Wilson line is supplied in the gauge invariant definition of the quark distributionfunction in eq. (18).Similarly since the SU(3) pure gauge background field A aµ ( x ) is the classical field one findsthat the definition of the gauge invariant gluon distribution function in eq. (19) containsquadratic powers of the (quantum) gluon field Q aµ ( x ) consistent with the definition of thegauge invariant gluon distribution function in quantum field theory even after the Wilsonline is supplied in the gauge invariant definition of the gluon distribution function in eq.(19).Note that the quark distribution function in eq. (18) in the interacting (full) QCD isinfinite in the unrenormalized QCD. In the renormalized QCD the renormalized quark dis-tribution function is finite. Hence one may argue that since the quark distribution functionin the unrenormalized QCD is divergent it is not necessary to define a quark distributionfunction in the unrenormalized QCD consistent with the definition of the distribution func-tion of the quark in quantum field theory. However, this argument is not correct becausein the next section we will show that the renormalized quark distribution function in the8enormalized QCD and the unrenormalized quark distribution function in the unrenormal-ized QCD predict the same hadronic cross section at all orders in coupling constant in QCD.This is consistent with the fact that the quark is not directly experimentally observed butthe hadron is directly experimentally observed.Similarly the gluon distribution function in eq. (19) in the interacting (full) QCD isinfinite in the unrenormalized QCD. In the renormalized QCD the renormalized gluon dis-tribution function is finite. Hence one may argue that since the gluon distribution functionin the unrenormalized QCD is divergent it is not necessary to define a gluon distributionfunction in the unrenormalized QCD consistent with the definition of the distribution func-tion of the gluon in quantum field theory. However, this argument is not correct becausein the next section we will show that the renormalized gluon distribution function in therenormalized QCD and the unrenormalized gluon distribution function in the unrenormal-ized QCD predict the same hadronic cross section at all orders in coupling constant in QCD.This is consistent with the fact that the gluon is not directly experimentally observed butthe hadron is directly experimentally observed. III. UNRENORMALIZED QCD AND RENORMALIZED QCD PREDICT SAMEHADRONIC CROSS SECTION AT ALL ORDERS IN COUPLING CONSTANT
In eqs. (18) and (19) we have derived the gauge invariant definition of the quark andgluon distribution functions inside the hadron which are consistent with the definition ofthe distribution functions of the quark and gluon in the quantum field theory but thesedistribution functions in eqs. (18) and (19) are divergent in the unrenormalized QCD. Henceone may argue that since the parton distribution function in the unrenormalized QCD isdivergent it is not necessary to define a gluon distribution function in the unrenormalizedQCD consistent with the definition of the distribution function of the gluon in quantumfield theory. However, this argument is not correct because we will show in this section thatthe definition of the gluon distribution function in the unrenormalized QCD can be used topredict the correct hadronic cross section at all orders in coupling constant in QCD (see eq.(21)).Using the path integral formulation of the background field method of QCD in the pres-ence of SU(3) pure gauge background field we have simultaneously proved the renormal-9zation of ultra violet (UV) divergences and the factorization of infrared (IR) and collineardivergences in QCD at all orders in coupling constant in [13]. Hence from eq. (1) and [13]we find dσ ( H H → H + X ) = X i,j,l Z dx Z dx Z dz f i/H ( x , Q ) f j/H ( x , Q ) d ˆ σ ij → kl D H /k ( z, Q )= X i,j,l Z dx Z dx Z dz f UnRenormalized i/H ( x , Q ) f UnRenormalized j/H ( x , Q ) d ˆ σ UnRenormalized ij → kl × D UnRenormalized H /k ( z, Q ) (21)where f ( x, Q ) is the renormalized parton distribution function in the renormalized QCD, f UnRenormalized ( x, Q ) is the unrenormalized parton distribution function in the unrenor-malized QCD, D ( z, Q ) is the renormalized fragmentation function in renormalized QCD, D UnRenormalized ( z, Q ) is the unrenormalized fragmentation function in unrenormalized QCD,ˆ σ is the renormalized partonic level cross section in renormalized QCD and ˆ σ UnRenormalized isthe unrenormalized partonic level cross section in the unrenormalized QCD.Hence one finds from eq. (21) that the correct definition of the parton distributionfunction consistent with the definition of the distribution function from the first principle inthe quantum field theory plays a very important role to prove that the hadronic cross sectionin the renormalized QCD is exactly same as the hadronic cross section in the unrenormalizedQCD at all orders in coupling constant. Note that we are not saying that one does not haveto do renormalization in QCD at fixed orders of coupling constant calculation but whatwe are saying is that at all orders of coupling constant the renormalized QCD and theunrenormalized QCD predict the same hadronic cross section. It is human limitations thatwe can not perform all orders coupling constant calculation in QCD but nature does not workaccording to human limitations. The QCD as a fundamental theory of the nature predictsthat the renormalized QCD and the unrenormalized QCD predict the same hadronic crosssection at all orders in coupling constant. This is consistent with the fact that the quarks andgluons are not directly experimentally observed but the hadrons are directly experimentallyobserved. 10
V. OPERATOR PRODUCT EXPANSION AND TOTAL CROSS SECTION INELECTRON-POSITRON ANNIHILATION TO HADRONS
Since there are no fragmentation functions in the total cross section in electron-positronannihilation to hadrons one may argue that the eq. (21) is not applicable to study the totalcross section in electron-positron annihilation to hadrons. Hence one may argue that therenormalized QCD and the unrenormalized QCD predict the same hadronic cross section atall orders in coupling constant is wrong. However, this argument is not correct which canbe seen as follows.Since one calculates the total cross section in electron-positron annihilation to hadronsone uses in the renormalized QCD X H D Hi = 1 (22)where D Hi is the renormalized fragmentation function for the parton i to fragment to hadron H . Using eq. (60) of [13] in (22) we find in the unrenormalized QCD that X H D Hi UnRenormalized = Z (23)where Z is the (quantum) field divergent renormalization factor of the parton i . Hencethe divergent renormalization factor Z in eq. (23) exactly cancels with the correspondingdivergent factor Z in the partonic level cross section σ e + e − → k ,k ,...,k n in the electron-positronannihilation to partons at all orders of coupling constant in the unrenormalized QCD, similarto eq. (21).Hence one finds that the total cross section in the electron-positron annihilation tohadrons is finite at all orders in coupling constant in the unrenormalized QCD and is exactlythe same total cross section that is obtained in the electron-positron annihilation to hadronsin the renormalized QCD at all orders in coupling constant.Because of eq. (22) there are no fragmentation functions appearing in the total crosssection in the inclusive electron-positron annihilation to hadrons in the renormalized QCD.Hence the current-current correlator that appears in the operator product expansion (OPE)to study the total cross section in the electron-positron annihilation to hadrons becomes thevacuum expectation of the form < | J µ ( z ) j ν ( y ) | > = C ( z − y ) < | O µ...ν ( y ) | > (24)11here C ( z − y ) is the short distance coefficient which is singular as the distance ( z − y ) µ → O µ...ν ( y ) is the local operator.Since the vacuum expectation is used in eq. (24) instead of the hadronic expectation < H | O µ...ν ( y ) | H > one finds that the long distance confinement involving hadron in QCDdoes not play any role to study the total cross section in the electron-positron annihilationto hadrons. This implies that the use of the operator product expansion at short distance[21] is ok to study the total cross section in the electron-positron annihilation to hadronsfor high momentum transfer processes.
V. OPERATOR PRODUCT EXPANSION AND CONFINEMENT IN QCD IN-VOLVING HADRON
For the hadronic expectation < H | O µ...ν ( y ) | H > the long distance confinement involv-ing hadron H in QCD must be included. Since the operator product expansion (OPE) isapplicable at short distance the OPE in QCD does not solve the long distance confinementproblem involving hadron in QCD where the non-perturbative QCD is applicableThe inclusive cross section with identified hadron H in the electron-positron annihilationinvolves the hadronic expectation of the current-current correlator of the form X X < | J µ ( y ) | H + X >< H + X | j ν (0) | > = < | J µ ( y ) a † H a H j ν (0) | >, | H + X > = a † H | X > (25)where a † H is the creation operator of the hadron. In eq. (25) the operator product expansion(OPE) does not apply and no short distance analysis exists for the case of inclusive e + e − → H + X process [22].The first principle issue here is that the operator product expansion (OPE) in QCD isvalid at short distance where pQCD is applicable but the confinement in QCD involvinghadron occurs at long distance where the non-perturbative QCD is applicable. Hence theOPE is applied to short distance pQCD calculation which does not require any informationabout long distance non-perturbative QCD confinement involving hadron, such as the totalcross section in the electron-positron annihilation to hadrons [22]. However, any calculationin pQCD to study physical phenomena which requires long distance non-perturbative QCDconfinement involving hadron may not always be correctly studied from the first principle12n QCD by the operator product expansion (OPE) alone without incorporating the non-perturbative QCD which is not solved yet. VI. DEFINITION OF THE GLUON DISTRIBUTION FUNCTION INSIDEHADRON USING OPERATOR PRODUCT EXPANSION
Let us now discuss the definition of the gluon distribution function consistent with theoperator product expansion (OPE) in QCD which is widely used in the literature at highenergy colliders. The standard procedure in the operator product expansion in the deepinelastic scattering involving lepton and hadron is as follows: The structure functions F and F are related to the product of electromagnetic currents inside the hadron as [23]12 π Z d ye iq · y < H | j µ ( y ) j λ (0) | H > = − g µλ m F ( ν, q ) + p µ p λ mν F ( ν, q ) , ν = p · q (26)where p µ is the hadron momentum and q µ is the momentum transfer to the hadrons. Theproduct of two currents at short distances in the OPE is given by [23] j µ ( y ) j λ (0) ∝ C ( y ) O µ...λ (0) (27)where C ( y ) is the short distance coefficient and O µ...λ is the local operator.Let us consider the gluon sector. Since the quantum gluon field Q aµ ( y ) is not gaugecovariant in QCD the twist two operator that appears in the operator product expansion oftwo currents is given by [3, 23] O µ...νg (0) ≡
12 Tr[ F µδ (0) iD λ [ Q ](0) ...iD β [ Q ](0) F νδ (0)] , D abµ [ Q ] = δ ab ∂ µ + gf acb Q cµ (28)which contains gluon field tensor F aµν ( y ) instead of gluon field Q aµ ( y ) where F aµλ ( y ) = ∂ µ Q aλ ( y ) − ∂ λ Q aµ ( y ) + gf acd Q cµ ( y ) Q dλ ( y ) (29)which transforms gauge covariantly in QCD.In particular the hadronic matrix element of the gauge invariant twist two operator < H |O µ...νg (0) | H > ≡ < H | Tr[ F µδ (0) F νδ (0)] | H > (30)13orresponds to the moment of the gauge invariant gluon distribution function f g/H ( x ) insidethe hadron [3] Z dxf g/H ( x ) ≡ < H |O µ...νg (0) | H > ≡ < H | Tr[ F µδ (0) F νδ (0)] | H > (31)if the gauge invariant definition of the gluon distribution function f g/H ( x ) inside the hadronis given by [3] f g/H ( x ) = 12 πxp + Z dy ′ e − ixp + y ′ < H | F + νb (0 , y ′ , T ) [ P e igT ( A ) c R y ′ dy ′′ Q + c (0 ,y ′′ , ] F + bν (0) | H > . (32)Note that the Wilson line in eq. (32) contains quantum gluon field Q aµ ( y ) whereas the Wilsonline in eq. (19) contains classical SU(3) pure gauge background field A aµ ( y ).It is useful to mention that one can not use the classical SU(3) pure gauge backgroundfield A aµ ( y ) in the Wilson line in eq. (32) because then f g/H ( x ) in eq. (32) will not be gaugeinvariant and will not be consistent with the factorization of soft and collinear divergences atall orders in the coupling constant in QCD. This is because in the background field methodof QCD in the presence of SU(3) pure gauge background field A aµ ( y ) the gauge invariantgluon field is given by [8, 12, 13][ P e − igT ( A ) c R ∞ y ′ dy ′′ A + c (0 ,y ′′ , ] Q bµ (0 , y ′ , T ) (33)where the soft and collinear divergences are factorized into the exponential containing theSU(3) pure gauge background field A aµ ( y ).Hence we find that the gauge invariant definition of the gluon distribution function insidethe hadron consistent with the operator product expansion (OPE) in QCD is given by eq.(32). VII. OPERATOR PRODUCT EXPANSION IN QCD IS NOT CONSISTENTWITH QUANTUM FIELD THEORY FOR GLUON DISTRIBUTION FUNCTION
From eq. (32) one finds that there is no way the gauge invariant definition of the gluondistribution function f g/H ( x ) in eq. (32) can be called as a distribution function in quantumfield theory because it contains infinite powers of quantum gluon field Q aµ ( x ) instead ofquadratic powers of the quantum gluon field Q aµ ( x ). Even without the Wilson line the gauge14on-invariant definition from eq. (32) contains cubic and quartic powers of the quantumgluon field Q aµ ( x ) and hence can not be called as a distribution function in quantum fieldtheory.One can note that the definition of f g/H ( x ) in eq. (32) corresponds to distribution functionin quantum field theory in the light-cone gauge Q + = 0 because in the light-cone gauge itcontains the quadratic powers of the quantum gluon field Q aµ ( x ). However, in any othergauge the gauge invariant definition of f g/H ( x ) in eq. (32) does not correspond to distributionfunction in quantum field theory because in any other gauge (except the light-cone gauge)it contains infinite powers of quantum gluon field Q aµ ( x ) instead of quadratic powers of thequantum gluon field Q aµ ( x ). As mentioned above even without the Wilson line the gaugenon-invariant definition from eq. (32) contains cubic and quartic powers of the quantumgluon field Q aµ ( x ) and hence can not be called as a distribution function in quantum fieldtheory.Since the definition of the gluon distribution function f g/H ( x ) in eq. (32) is gauge invariantit must correspond to the definition of the distribution function in any gauge. Hence wefind that the gauge invariant definition of the gluon distribution function in eq. (32) whichis consistent with the operator product expansion (OPE) in QCD is not consistent withdefinition of the gauge invariant gluon distribution function in quantum field theory.On the other hand the gauge invariant definition of the gluon distribution function ineq. (19) contains quadratic powers of the quantum gluon field Q aµ ( x ) [even after the Wilsonline is supplied] which implies that the gauge invariant definition of the gluon distributionfunction in eq. (19) is consistent with the definition of the gauge invariant gluon distributionfunction in quantum field theory.As mentioned in section III one may argue that since the gluon distribution function in theunrenormalized QCD is divergent it is not necessary to define a gluon distribution functionin the unrenormalized QCD consistent with the definition of the distribution function of thegluon in quantum field theory. However, this argument is not correct because we have shownin section III that the definition of the gluon distribution function in the unrenormalizedQCD can be used to predict the correct hadronic cross section at all orders in couplingconstant in QCD, see eq. (21) and [13].In summary we find that the gauge invariant definition of the non-perturbative gluon dis-tribution function inside the hadron which is consistent with the operator product expansion15OPE) in QCD is not consistent with the gauge invariant definition of the non-perturbativegluon distribution function in the quantum field theory. VIII. CONCLUSIONS
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