Impact factor for high-energy two and three jets diffractive production
aa r X i v : . [ h e p - ph ] M a y Preprint typeset in JHEP style - HYPER VERSION
Impact factor for high-energy two and three jetsdiffractive production
R. Boussarie
LPT, Universit´e Paris-Sud, CNRS, 91405, Orsay, FranceEmail:
A. V. Grabovsky
Budker Institute of Nuclear Physics and Novosibirsk State University, 630090Novosibirsk, RussiaEmail:
L. Szymanowski
National Centre for Nuclear Research (NCBJ), Warsaw, PolandEmail:
S. Wallon
LPT, Universit´e Paris-Sud, CNRS, 91405, Orsay, France & UPMC Univ. Paris 06, facult´e de physique, 4 place Jussieu, 75252 Paris Cedex 05,FranceEmail:
Abstract:
We present the calculation of the impact factor for the γ ( ∗ ) → q ¯ qg transitionwithin Balitsky’s high energy operator expansion. We also rederive the impact factor forthe γ ( ∗ ) → q ¯ q transition within the same framework. These results provide the necessarybuilding blocks for further phenomenological studies of inclusive diffractive deep inelas-tic scattering as well as for two and three jets diffractive production which go beyondapproximations discussed in the litterature. ontents
1. Introduction 12. Definitions and necessary intermediate results 33. LO impact factor for γ → q ¯ q transition 64. General expression for γ → q ¯ qg impact factor 85. Diagrams with the gluon crossing the shockwave 96. Diagrams without the gluon crossing the shockwave 127. Impact factor for interaction with color dipole 148. Impact factor in the momentum space and in the linear approximation 179. Conclusions 19 . Introduction Diffraction is one of the key tools to understand the dynamics of strong interaction, andit has been studied since the sixties. In particular, the research program performed atHERA has shown that semi-hard diffractive processes, in which a hard scale allows oneto deal with QCD in its perturbative regime, could provide a quantitative lever-arm tounderstand the internal dynamics of the nucleon in a regime of very high gluon densities .Among the whole set of γ ∗ p → X deep inelastic scattering (DIS) events, almost 10 %reveal a rapidity gap between the proton remnants and the hadrons coming from thefragmentation region of the initial virtual photon, so that the process looks rather like γ ∗ p → X Y [3–10], where Y is the outgoing proton or one of its low-mass excited states.This subset of events is called diffractive deep inelastic scattering (DDIS). DDIS can bestudied at the inclusive level, or further analyzed by considering diffractive jet productionas well as exclusive meson production. One of the main cornerstones of diffraction isthe concept of Pomeron, which carries the quantum numbers of vacuum and which isexchanged at high energy between X and Y . Diffraction can be described according toseveral approaches, important for phenomenological applications. In the perturbative QCDapproach, justified by the existence of a hard scale (like the photon virtuality Q ), one canrely on a QCD factorization theorem [11]. This is the essence of the first approach, whichinvolves a resolved Pomeron contribution: the diffractive structure function is expressedas the convolution of a coefficient function with a diffractive parton distribution, which isanalogous to the usual parton distribution function (PDF), but with the proton replacedby a Pomeron.Besides, at high energies, it is natural to model the diffractive events by a direct
Pomeron contribution involving the coupling of a Pomeron with the diffractive state. Thediffractive states can be modelled in perturbation theory by a q ¯ q pair (for moderate M ,where M is the invariant mass of the diffractively produced state X ) or a q ¯ qg state forlarger values of M . Based on such a model, with a two-gluon exchange picture for thePomeron, a good description of HERA data for diffraction could be achieved [12]. Oneof the important features of this approach is that the q ¯ q component with a longitudinallypolarized photon plays a crucial role in the region of small diffractive mass M , althoughit is a twist-4 contribution. A further analysis was then performed, combining both theresolved and the direct components [13, 14], including a q ¯ q exchange on top of the gluonpair to model the Pomeron.In the direct components considered there, the q ¯ qg diffractive state has been studiedin two particular limits. The first one, valid for very large Q , corresponds to a collinearapproximation in which the transverse momentum of the gluon is assumed to be muchsmaller than the transverse momentum of the emitter [15, 16]. This approximation allowsone to extract the leading logarithm in Q , based on the strong ordering of transversemomenta typical of DGLAP evolution [17–20]. The second one [21, 22], valid for very large For reviews, see refs. [1, 2] , is based on the assumption of a strong ordering of longitudinal momenta, encounteredin BFKL equation [23–26].The main aim of the present article is to compute the γ ∗ → q ¯ qg impact factor andto rederive the γ ∗ → q ¯ q impact factor, both at tree level, with an arbitrary number of t − channel gluons, here described within the Wilson line formalism, also called QCD shock-wave approach [27–30]. In particular, the γ ∗ → q ¯ qg transition is computed without anysoft or collinear approximation for the emitted gluon, in contrast with the above mentionedcalculations. These results provide necessary generalization of buiding blocks for inclusiveDDIS as well as for two- and three-jet diffractive production. Since the results we derivedcan account for an arbitrary number of t − channel gluons, this could allow to include highertwist effects which are suspected to be rather important in DDIS for Q . [31].The QCD shock-wave approach on which we rely is an operator language based on theconcept of factorization of the scattering amplitude in rapidity space and on the extensionto high-energy (Regge limit) of the Operator Product Expansion (OPE) technique, whichwas only known at moderate energy (Bjorken limit) before, as an expansion in terms oflocal operators or in terms of light-ray operators [32]. In DIS off a hadron at high-energy,the matrix elements made of Wilson line operators appearing in the OPE describe the nonperturbative part of the process, and their evolution in rapidity is related to the evolutionof the structure function of the target. The evolution equation can be obtained relyingon background field techniques. The Wilson-line operators in the high-energy OPE evolvewith respect to rapidity according to the Balitsky equation, which reduces to the Balitsky-Kovchegov (BK) equation [27–30,33,34] in the large N c limit. According to the best of ourknowledge, this shock-wave approach was only used for evolution equations and for impactfactors at inclusive level, namely only for the γ ∗ → γ ∗ impact factor at next-to-leadingorder [35, 36]. Its application shows that this method is very powerful [37] when comparedwith usual methods based on summation of contributions of individual Feynman diagramscomputed in momentum space.When describing a diffractive process, the above mentioned approach is natural in orderto implement saturation effects at high energies, since it is formulated in the coordinatespace. Indeed, in the dipole picture [38,39], when probing a nucleon with a virtual photon inthe rest frame of the nucleon, due to the long life-time of the virtual q ¯ q pair produced by the γ ∗ probe with respect to its scattering time, this pair is almost frozen during its interaction.The inclusive cross-section (for DIS) as well as the scattering amplitude (for DDIS) thusnaturally factorizes in the coordinate space in terms of an impact factor involving a dipoleof given transverse size r convoluted with an effective dipole-nucleon cross-section σ ( x, r ),a function of Bjorken x and r . The same picture was extended to the q ¯ qg intermediatestate, at least in the collinear approximation in which case this intermediate state canbe considered as a gluon-gluon dipole [15, 16], the q ¯ q being an effective gluon due to itslocalization in the transverse coordinate space (since the relative transverse momentum ofthis pair is large with respect to the transverse momentum of the emitted gluon). A stepfurther in this spirit was done in the case of vector meson electroproduction at twist 3,including the genuine twist 3 contribution which involves a q ¯ qg intermediate state [40, 41],for which a dipole picture was also obtained [42], based on QCD equations of motion.2his dipole picture provides the natural framework for the formulation of saturation.Indeed, the transverse size r of the dipole is the natural parameter in order to implementboth color transparency (for small r ) and saturation (for large r ). The analysis of low- x sat-uration dynamics of the nucleon target was first introduced in refs. [43,44] by Golec-Biernatand W¨usthoff (GBW) to describe the inclusive and diffractive structure functions of DIS.This is an additionnal reason to rely on the shock-wave analysis, which naturaly provides atool to evaluate the γ ∗ → q ¯ q and γ ∗ → q ¯ qg impact factors in transverse coordinate space.The paper is organized as follows. The section 2 contains the definitions and necessaryintermediate results. Section 3 briefly reproduces the leading order (LO) γ ∗ → q ¯ q impactfactor. In section 4 we give the general expression for the γ → q ¯ qg impact factor, whichis then calculated in sections 5 and 6. Section 7 discusses the linearized impact factor forinteraction with the color dipole. Section 8 is devoted to the impact factor in the momen-tum space. Section 9 summarizes obtained results. Two appendices comprise necessarytechnical details.
2. Definitions and necessary intermediate results
Throughout this paper, we use the following notations. We introduce the light cone vectors n and n n = (1 , , , , n = 12 (1 , , , − , n +1 = n − = n · n = 1 , (2.1)and for any vector p we have p + = p − = p · n = 12 (cid:0) p + p (cid:1) , p + = p − = p · n = p − p , (2.2) p = p + n + p − n + p ⊥ , p = 2 p + p − − ~p , (2.3) p · k = p µ k µ = p + k − + p − k + − ~p · ~k = p + k − + p − k + − ~p · ~k. (2.4)The derivatives and the metric tensor have the form ∂ ± = ∂ ∓ = ∂∂z ± = ∂∂z ∓ , ∂ i = − ∂ i = ∂∂z i = − ∂∂z i , (2.5) g µν = g µν = − − , the indices are + , − , ,
2; (2.6) ǫ + βγ − = − ǫ βγ = − e βγ . (2.7)We denote the initial photon momentum as k, and the outgoing quark, antiquark, andgluon momenta as p q , p ¯ q , and p g . The corresponding longitudinal momentum fractions are p + q k + = x q , p +¯ q k + = x ¯ q , p + g k + = x g . (2.8)3or simplicity, we work with a photon in the forward kinematics ~k = 0 , k µ = k + n µ + k k + n µ , − k = Q > . (2.9)Its longitudinal and transverse polarization vectors read ε αL = 1 √− k (cid:18) k + n α − k k + n α (cid:19) , ε + L = k + Q , ε − L = Q k + , (2.10) ε αT = ε αT ⊥ = 1 √ , , i s, , s = ± . (2.11)Here s is the helicity of the photon. For the outgoing gluon we work in the light cone gauge A · n = 0 . Therefore ε ∗ gν = ( ~ε ∗ g · ~p g ) p + g n ν + ε ∗ g ⊥ ν = (cid:18) g ⊥ να − p g ⊥ α n ν p + g (cid:19) ε ∗ αg , (2.12)and we can use the same transverse polarization vectors as for the photon ε ∗ αg ⊥ = 1 √ , , − i s g , , s g = ± . (2.13)It is convenient to introduce the following vectors ~P ¯ q = ~p g x g − ~p ¯ q x ¯ q , ~P q = ~p g x g − ~p q x q . (2.14)Then ( ~P ¯ q · ~ε ∗ g ) = p ¯ q · ε ∗ g x ¯ q , ( ~P q · ~ε ∗ g ) = p q · ε ∗ g x q . (2.15)To simplify the vector products with the polarization vectors we will use the followingidentities [ ~a × ~ε T ] = i s ( ~a · ~ε T ) , [ ~a × ~ε ∗ g ] = − i s g ( ~a · ~ε ∗ g ) , (2.16)where [ ~a × ~b ] ≡ e γβ a γ b β . The fermion propagator in the shock wave background can beread from ref. [27] and is given by G ( z , z ) = θ ( z +1 z +2 ) G ( z ) − Z d z δ ( z +3 ) G ( z ) γ + G ( z ) × (cid:16) θ ( z +1 ) θ ( − z +2 ) U ~z + θ ( − z +1 ) θ ( z +2 ) U † ~z (cid:17) . (2.17)Here z ij = z i − z j . The free quark propagator reads G ( x ) = 2 i (2 π ) ˆ z ( z − i , G ( p ) = i ˆ pp + i , (2.18)and the Wilson lines U i = U ~z i = U ( ~z i , η ) = P exp (cid:20) ig Z + ∞−∞ b − η ( z + i , ~z i ) dz + i (cid:21) (2.19)4re integrated along the path z − = 0. The operator b − η is the external shock-wave fieldbuilt from only slow gluons which momenta are limited by the longitudinal cut-off definedby the rapidity η b − η = Z d p (2 π ) e − ip · z b − ( p ) θ ( e η − | p + | ) . (2.20)We use the light cone gauge A · n = 0 , (2.21)with A being the sum of the external field b and the quantum field A A = A + b, b µ ( z ) = b − ( z + , ~z ) n µ = δ ( z + ) B ( ~z ) n µ . (2.22)Using the LSZ reduction formulas for the propagators from ref. [27] or summing the dia-grams in this external shockwave field as in ref. [27] one can get the external fermion linesin the shockwave background¯ u ( p, y ) | >y + = Z d z δ ( z + ) e ip · z ¯ u p p p + γ + U ~z G ( z − y ) , (2.23) v ( p, y ) | >y + = − Z d z δ ( z + ) e ip · z G ( y − z ) U † ~z γ + v p p p + . (2.24)In the same way one can get the gluon external line in the shockwave background ǫ ∗ ν ( p, y ) | >y + = − p + ε ∗ α θ ( p + ) Z d z δ ( z + )(2 π ) e ip · z p p + U ~z ∂∂y − g ⊥ αν ( − y + ) − ( z − y ) ⊥ α n ν (( z − y ) − i , (2.25)where we have introduced the notation1 ∂∂y f ( y ) = Z dp π Z du e − ip ( y − u ) − ip f ( u ) , (2.26)In eqs. (2.23, 2.24), the Wilson line is in the adjoint representation. If U → , then¯ u ( p, y ) | >y + → θ ( p + ) ¯ u p p p + e ip · y , v ( p, y ) | >y + → θ ( p + ) v p p p + e ip · y , (2.27) ǫ ∗ ν ( p, y ) | >y + → θ ( p + ) ε ∗ αp p p + (cid:18) g ⊥ αν − p ⊥ α n ν p + (cid:19) e ip · y = θ ( p + ) ε ∗ pν p p + e ip · y , (2.28)and we recover the results without the shockwave.Below, we will need the following integral derived in ref. [36] Z d z ˆ x − ˆ z ( x − z ) (cid:20) γ µ z ν z − γ ν z µ z (cid:21) ˆ z − ˆ y ( z − y ) = − iπ x y ( x − y ) (cid:18) ˆ xγ ν ˆ yx µ − ˆ xγ µ ˆ yx ν x + ˆ xγ ν ˆ yy µ − ˆ xγ µ ˆ yy ν y + ( γ ν γ µ − γ µ γ ν )ˆ y x ( γ µ γ ν − γ ν γ µ )2 + 2 y µ x ν − y ν x µ ( x − y ) [ˆ y − ˆ x ] (cid:19) , (2.29) The factor − i in eq. (2.29) corrects a misprint from ref. [36]. − K (cid:16) √ ab (cid:17) = Z −∞ dzz e i ( a − i z − i b + i z , K ( r ) = − K ′ ( r ) , (2.30)together with the fact that they obey the Bessel equation∆ K ( r ) = (cid:18) ∂ ∂r + 1 r ∂∂r (cid:19) K ( r ) = K ( r ) + 2 πδ ( ~r ) . (2.31)We will also need the following Dirac structures¯ u p q γ + v p ¯ q = λ q ¯ u p q γ + γ v p ¯ q = δ λ q , − λ ¯ q q p + q p +¯ q , λ q = ± , (2.32)¯ u p q γ j v p ¯ q = λ q u p q γ j γ v p ¯ q = δ λ q , − λ ¯ q q p +¯ q p + q p jq p + q + p j ¯ q p +¯ q + iε tj λ q p tq p + q − p t ¯ q p +¯ q !! , (2.33)where λ q is the quark helicity.
3. LO impact factor for γ → q ¯ q transition z z z k p q p ¯ q Figure 1: LO γ → q ¯ q impact factor In this section we will briefly reproduce the known expression for the LO γ ∗ → q ¯ q impact factor, see fig. 1, for completeness of this paper. The LO matrix element for theelectromagnetic current in the shockwave background reads˜ M α = δ nl h | b lp ¯ q ( a p q ) n ψ ( z ) γ α ψ ( z ) e i R L i ( z ) dz | ih | e i R L ( z ) dz | i = Z d~z d~z F ( p q , p ¯ q , z , ~z , ~z ) α tr( U U † ) . (3.1)6ere a and b are the quark and antiquark annihilation operators carrying colour indices n and l , F ( p q , p ¯ q , z , ~z , ~z ) α = Z dz − dz +1 δ ( z +1 ) Z dz − dz +2 δ ( z +2 ) e ip q · z + ip ¯ q · z q p + q p + ¯ q × ¯ u p q γ + G ( z ) γ α G ( z ) γ + v p ¯ q = θ ( p + q ) θ ( p +¯ q ) e − i~p q · ~z − i~p ¯ q · ~z π ) ( z +0 ) q p + q p + ¯ q × ¯ u p q γ + ip + q γ − − γ β ⊥ ∂∂z β ⊥ ! γ α (cid:18) ip +¯ q γ − − γ γ ⊥ ∂∂z γ ⊥ (cid:19) γ + v p ¯ q exp (cid:20) ip + q (cid:18) z − + − z ⊥ + i − z +0 (cid:19)(cid:21) exp (cid:20) ip + ¯ q (cid:18) z − + − z ⊥ + i − z +0 (cid:19)(cid:21) . (3.2)We now introduce M α , built from ˜ M α by substracting the non-interacting term, i.e. M α = Z d~z d~z F ( p q , p ¯ q , z , ~z , ~z ) α (tr( U U † ) − N c ) . (3.3)The Fourier transform of F w.r.t. z is defined as F ( p q , p ¯ q , k, ~z , ~z ) α = Z d z e − ik · z F ( p q , p ¯ q , z , ~z , ~z ) α . (3.4)In the kinematics (2.9) we chose for the photon, we have F ( p q , p ¯ q , k, ~z , ~z ) α = θ ( p + q ) θ ( p +¯ q ) iδ (cid:0) k + − p + q − p +¯ q (cid:1) k + (2 π ) q p + q p + ¯ q e − i~p q · ~z − i~p ¯ q · ~z × ¯ u p q γ + ip + q γ − − γ β ⊥ ∂∂z β ⊥ ! γ α (cid:18) ip +¯ q γ − − γ γ ⊥ ∂∂z γ ⊥ (cid:19) γ + v p ¯ q K (cid:18) Q q x q x ¯ q ~z (cid:19) . (3.5)Calculating the derivatives with α = − we will encounter g γβ ⊥ ∂∂z γ ⊥ ∂∂z β ⊥ K (cid:18) Q q x q x ¯ q ~z (cid:19) = ∆ ~z K (cid:18) Q q x q x ¯ q ~z (cid:19) = Q x q x ¯ q K (cid:18) Q q x q x ¯ q ~z (cid:19) + 2 πδ ( ~z ) , (3.6)where we used (2.31) to derive this expression. However, the term with the δ distributionwill give no contribution to (3.3), therefore we can drop it. That being done, we get F ( p q , p ¯ q , k, ~z , ~z ) α ε Lα = θ ( p + q ) θ ( p +¯ q ) δ (cid:0) k + − p + q − p +¯ q (cid:1) (2 π ) e − i~p q · ~z − i~p ¯ q · ~z × ( − i ) δ λ q , − λ ¯ q x q x ¯ q Q K (cid:18) Q q x q x ¯ q ~z (cid:19) (3.7)7nd F ( p q , p ¯ q , k, ~z , ~z ) j ε T j = θ ( p + q ) θ ( p +¯ q ) δ (cid:0) k + − p + q − p +¯ q (cid:1) (2 π ) e − i~p q · ~z − i~p ¯ q · ~z × δ λ q , − λ ¯ q ( x q − x ¯ q + sλ q ) ~z · ~ε T ~z Q q x q x ¯ q ~z K (cid:18) Q q x q x ¯ q ~z (cid:19) . (3.8)Using the identity F + = F − k + ) Q , (3.9)one can easily check that the electromagnetic gauge invariance F ( p q , p ¯ q , k, ~z , ~z ) α k α = 0 (3.10)is satisfied. The results (3.7) and (3.8) are consistent with the well known result for the γ → q ¯ q wave-function which was derived for example in ref. [45].
4. General expression for γ → q ¯ qg impact factor p q p ¯ q z p g z z z z k z z z z k z z z z k z z z z z k Figure 2:
Impact factor for 3 jet production. The grey ellipse stands for the shockwave at z + = 0.The lines crossing the ellipse are calculated in the shock wave background (2.17), (2.23–2.25).
8e will extract the impact factor from the following matrix element˜ M α = ( t b ) nl h | c bp g b lp ¯ q ( a p q ) n ψ ( z ) γ α ψ ( z ) e i R L i ( z ) dz | ih | e i R L ( z ) dz | i = Z d~z d~z d~z F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α tr( U t a U † t b ) U ba + Z d~z d~z ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α N c − N c tr( U U † ) . (4.1)Here ( t b ) kl is the projector to the color singlet state, and c, a, and b are the gluon, quark andantiquark annihilation operators; F describes the contribution of the first two diagramsand ˜ F stands for diagrams 3 and 4. The space coordinates z , , , , and the momenta p q, ¯ q,g are defined in fig. 2.In this form ˜ M contains contributions from terms without interaction in which alloperators U are reduced to identity. To get the impact factor we have to subtract thoseterms. This amounts in replacing ˜ M by M , which reads M α = Z d~z d~z d~z F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α (cid:20) tr( U t a U † t b ) U ba − N c − (cid:21) + Z d~z d~z ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α N c − N c (cid:16) tr( U U † ) − N c (cid:17) (4.2)= Z d~z d~z d~z F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α (cid:16) tr( U U † ) tr( U U † ) − N c tr( U U † ) (cid:17) + Z d~z d~z F ( p q , p ¯ q , p g , z , ~z , ~z ) α N c − N c (cid:16) tr( U U † ) − N c (cid:17) . (4.3)Here F ( p q , p ¯ q , p g , z , ~z , ~z ) α = ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α + Z d~z F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α , (4.4)and tr( U t a U † t b ) U ba = 12 (cid:16) tr( U U † ) tr( U U † ) − N c tr( U U † ) (cid:17) + N c − N c tr( U U † ) . (4.5)The functions F and ˜ F are the two above mentioned components of the impact factor,which we will calculate in the next two sections.
5. Diagrams with the gluon crossing the shockwave
Using eqs. (2.23–2.25), the sum of the first two diagrams in fig. 2 can be represented in thefollowing way F ( z q , z ¯ q , p g , z , ~z , ~z , ~z ) α = ig (2 π ) Z dz +1 dz − dz +2 dz − dz +3 dz − δ ( z +1 ) δ ( z +2 ) δ ( z +3 ) L jαiβ R iβj , (5.1)9here L jαiβ = Z dz ∂∂z − g ⊥ βν z +34 − z ⊥ β n ν ( z − i × (cid:2) γ + ( G ( z ) γ α G ( z ) γ ν G ( z ) + G ( z ) γ ν G ( z ) γ α G ( z )) γ + (cid:3) ji , (5.2)and R iβj = p + g θ ( p + g ) ε ∗ βg e ip g · z + ip q · z + ip ¯ q · z q p + g p + q p + ¯ q h γ − γ + v p ¯ q ⊗ ¯ u p q γ + γ − i ij . (5.3)Using the integral (2.29) and the fact that z +1 , , = 0 one can write L jαiβ = 2 z +30 (2 π ) ∂∂z − (cid:20) γ + (cid:18) ˆ z γ α ˆ z γ ⊥ β ˆ z z z z z + ˆ z γ ⊥ β ˆ z γ α ˆ z z z z z + 2 z ˆ z γ α ˆ z z z (cid:18) z ⊥ β z − z ⊥ β z (cid:19)(cid:19) γ + (cid:21) ji . (5.4)Integrating w.r.t. z − with the help of eq. (2.26) we have L jαiβ = − π ) Z + ∞ σ dp + p + e − ip + (cid:18) z − − ~z i z +30 (cid:19) (cid:20) γ + (cid:18) ip + ˆ z γ α ( γ − z +30 + ˆ z ⊥ ) γ ⊥ β ˆ z (2 z +30 ) z z z + ip + ˆ z γ ⊥ β ( γ − z +30 + ˆ z ⊥ ) γ α ˆ z z (2 z +30 ) z z + 2 ˆ z γ α ˆ z z z (cid:18) z ⊥ β z − z ⊥ β z (cid:19)(cid:19) γ + (cid:21) ji , (5.5)where σ = e η is the longitudinal cutoff (2.20). As a result, F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α = θ ( p + g − σ ) ig (2 π ) ε ∗ βg e iz − ( p + q + p +¯ q + p + g ) − i~p q · ~z − i~p ¯ q · ~z − i~p g · ~z (2 z +0 ) q p + q p + ¯ q p + g × ¯ u p q γ + (cid:20) z ω ⊥ − ~z (cid:18) ip + q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ α (cid:18) ip + g γ − − γ ρ ⊥ ∂∂z ρ ⊥ (cid:19) γ ⊥ β γ ⊥ ω + ˆ z ⊥ ~z γ ⊥ β (cid:18) ip + g γ − − γ ρ ⊥ ∂∂z ρ ⊥ (cid:19) γ α (cid:18) ip + ¯ q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) + 2 (cid:18) z ⊥ β ~z − z ⊥ β ~z (cid:19) (cid:18) ip + q γ − − γ ρ ⊥ ∂∂z ρ ⊥ (cid:19) γ α (cid:18) ip + ¯ q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19)(cid:21) γ + v p ¯ q × e − i p + g ~z p + q ~z p +¯ q ~z i z +0 . (5.6)Via eq. (2.30) we will calculate the Fourier transform of F F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α = Z d z e − ik · z F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α (5.7)for the photon in our kinematics (2.9). Using our notation (2.8) and denoting Z = q x g x q ~z + x q x ¯ q ~z + x ¯ q x g ~z , (5.8)10e get F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α = − g θ ( p + g − σ )4 πk + δ ( k + − p + g − p + q − p + ¯ q ) ε ∗ gβ e − i~p q · ~z − i~p ¯ q · ~z − i~p g · ~z q p + q p + ¯ q p + g × ¯ u p q γ + (cid:20) z ω ⊥ − ~z (cid:18) ip + q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ α (cid:18) ip + g γ − − γ ρ ⊥ ∂∂z ρ ⊥ (cid:19) γ β ⊥ γ ⊥ ω + ˆ z ⊥ ~z γ β ⊥ (cid:18) ip + g γ − − γ ρ ⊥ ∂∂z ρ ⊥ (cid:19) γ α (cid:18) ip + ¯ q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) + 2 z β ⊥ ~z − z β ⊥ ~z ! (cid:18) ip + q γ − − γ ρ ⊥ ∂∂z ρ ⊥ (cid:19) γ α (cid:18) ip + ¯ q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ + v p ¯ q K ( QZ ) . (5.9)As before for the γ → q ¯ q impact factor, we encounter contributions involving δ distributionsfor α = − . Indeed, e.g. ∂ K ( QZ ) ∂z σ ⊥ ∂z ρ ⊥ = Q ( QZ K ′ ( QZ )) ′ QZ ∂Z ∂z σ ⊥ ∂Z ∂z ρ ⊥ + QZ K ′ ( QZ ) ∂ ln Z ∂z σ ⊥ ∂z ρ ⊥ , (5.10)and from eq. (2.31) we obtain( QZ K ′ ( QZ )) ′ QZ = K ( QZ ) + 4 πδ (cid:0) Q Z (cid:1) δ ( φ ) . (5.11)Again the term with the delta function gives a vanishing contribution to M introduced ineq. (4.2) and we drop it. Then, using eq. (2.31) and the matrix elements (2.32), we get F +1 = F − k + ) Q . (5.12)On one hand this implies the electromagnetic gauge invariance F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α k α = 0 , (5.13)and on the second hand it gives the contribution to the impact factor for longitudinalphoton (2.10) F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε Lα (5.14)= − δ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) Qg e − i~p q · ~z − i~p ¯ q · ~z − i~p g · ~z π q p + g K ( QZ ) × δ λ q , − λ ¯ q ε ∗ βg iλ q ǫ γβ x g (cid:26) x ¯ q z γ ~z + x q z γ ~z (cid:27) + ( (2 x q + x g ) x ¯ q z β ~z − (2 x ¯ q + x g ) x q z β ~z )! . Expressing the vector products e γβ a γ b β = [ ~a × ~b ] through the scalar products usingeq. (2.16) we have F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε Lα = δ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ )2 Qg e − i~p q · ~z − i~p ¯ q · ~z − i~p g · ~z π q p + g K ( QZ ) × δ λ q , − λ ¯ q (cid:26) ( x ¯ q + x g δ − s g λ q ) x q ~z · ~ε ∗ g ~z − ( x q + x g δ − s g λ ¯ q ) x ¯ q ~z · ~ε ∗ g ~z (cid:27) . (5.15)11or transverse photon (2.11) we get the following contribution to the impact factor F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε T α = 2 igQδ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) e − i~p q · ~z − i~p ¯ q · ~z − i~p g · ~z πZ q p + g × δ λ q , − λ ¯ q K ( QZ ) ( − (cid:0) ~z · ~ε ∗ g (cid:1) ( ~z · ~ε T ) ~z x q (cid:0) x q − δ sλ ¯ q (cid:1) (cid:0) x ¯ q + x g δ − s g λ q (cid:1) − (cid:0) ~z · ~ε ∗ g (cid:1) ( ~z · ~ε T ) ~z x q x ¯ q (cid:0) x ¯ q + x g δ − s g λ q − δ sλ q (cid:1)) − ( q ↔ ¯ q ) , (5.16)where ( q ↔ ¯ q ) ≡ ( λ q , x q , ~z , ~p q ↔ λ ¯ q , x ¯ q , ~z , ~p ¯ q ) . (5.17)One can check that the results (5.15) and (5.16) are compatible with the wave functionderived in ref. [46].
6. Diagrams without the gluon crossing the shockwave
Here we will calculate ˜ F (4.3), which gives the contribution from diagrams 3 and 4 infig. 2. It reads˜ F ( z q , z ¯ q , p g , z , ~z , ~z ) α = ig Z dz +1 dz − dz +2 dz − δ (cid:0) z +1 (cid:1) δ (cid:0) z +2 (cid:1) L ′ jαi R ′ ij (6.1)where L ′ jαi = (cid:2) γ + G ( z ) γ α G ( z ) γ + (cid:3) ji , (6.2)and R ′ iβj = Z θ (cid:0) z +4 (cid:1) d z ε ∗ βg e ip g · z q p + g p + q p + ¯ q × h γ − γ + (cid:16) e ip q · z + ip ¯ q · z G ( z ) γ β v p ¯ q ⊗ ¯ u p q + e ip q · z + ip ¯ q · z v p ¯ q ⊗ u p q γ β G ( z ) (cid:17) γ + γ − i ij . (6.3)Integrating eq. (6.3) w.r.t. z , we get R ′ ij = − i ε ∗ βg q p + g p + q p + ¯ q e ip + q z − − i~p q · ~z + ip +¯ q z − − i~p ¯ q · ~z (cid:20) γ − γ + (cid:18) e ip + g z − − i~p g · ~z (ˆ p ¯ q + ˆ p g )( p ¯ q + p g ) γ β v p ¯ q ⊗ ¯ u p q − v p ¯ q ⊗ ¯ u p q γ β (ˆ p q + ˆ p g )( p q + p g ) e ip + g z − − i~p g · ~z (cid:19) γ + γ − (cid:21) ij . (6.4)12s a result,˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α = θ ( p + g − σ ) g π ε ∗ βg θ ( p + g ) q p + g p + q p + ¯ q (cid:0) z +0 (cid:1) e i ( p +¯ q + p + q + p + g ) z − − i~p q · ~z − i~p ¯ q · ~z × (cid:26) − e − i~p g · ~z ¯ u p q γ + (cid:18) ip + q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ α (cid:18) i ( p + g + p + ¯ q ) γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ + (ˆ p ¯ q + ˆ p g )( p ¯ q + p g ) γ β v p ¯ q × e − ik + ( xg + x ¯ q ) ~z xq~z i z +0 + e − i~p g ~z u p q γ β (ˆ p q + ˆ p g )( p q + p g ) γ + (cid:18) i ( p + g + p + q ) γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) × γ α (cid:18) ip + ¯ q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ + v p ¯ q e − ik + x ¯ q~z xg + xq ) ~z i z +0 ) . (6.5)Next we calculate the Fourier transform of ˜ F ˜ F ( p q , p ¯ q , p g , k, ~z , ~z ) α = Z d z e − ik · z ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α (6.6)for the photon in our kinematics (2.9)˜ F ( p q , p ¯ q , p g , k, ~z , ~z ) α = ig k + θ ( p + g − σ ) δ ( k + − p + g − p + q − p + ¯ q ) ε ∗ βg e − i~p q · ~z − i~p ¯ q · ~z q p + g p + q p + ¯ q × (cid:20) − e − i~p g · ~z ¯ u p q γ + (cid:18) ip + q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ α (cid:18) i ( p + g + p + ¯ q ) γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ + × (ˆ p ¯ q + ˆ p g )( p ¯ q + p g ) γ β v p ¯ q K ( QZ ) + e − i~p g · ~z ¯ u p q γ β (ˆ p q + ˆ p g )( p q + p g ) × γ + (cid:18) i ( p + g + p + q ) γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ α (cid:18) ip + ¯ q γ − − γ σ ⊥ ∂∂z σ ⊥ (cid:19) γ + v p ¯ q K ( QZ ) (cid:21) . (6.7)In the arguments of the McDonald functions we encounter the following structures Z = Z | z → z = q x q (cid:0) x g + x ¯ q (cid:1) ~z = q x q (1 − x q ) ~z , (6.8) Z = Z | z → z = q ( x q + x g ) x ¯ q ~z = q (1 − x ¯ q ) x ¯ q ~z . (6.9)Again using the Bessel equation (2.31) as well as the matrix elements (2.32) and (2.33),then dropping the corresponding δ distributions we can check the electromagnetic gaugeinvariance ˜ F ( p q , p ¯ q , p g , k, ~z , ~z ) α k α = 0 . (6.10)Then taking into account Dirac equation and the gauge condition (2.12), we get the con-tribution to the impact factor for longitudinal photon (2.10)˜ F ( p q , p ¯ q , p g , k, ~z , ~z ) α ε Lα = 4 ig Q θ ( p + g − σ ) δ ( k + − p + g − p + q − p + ¯ q ) e − i~p q · ~z − i~p ¯ q · ~z q p + g × δ λ q , − λ ¯ q x q ( x g + x ¯ q ) (cid:0) δ − s g λ q x g + x ¯ q (cid:1) ( p g + p ¯ q ) ( ~P ¯ q · ~ε ∗ g ) e − i~p g · ~z K ( QZ ) − ( q ↔ ¯ q ) , (6.11)13here P q, ¯ q are defined in eq. (2.14). For transverse photons (2.11) we have˜ F ( p q , p ¯ q , p g , k, ~z , ~z ) α ε T α = − g Q θ ( p + g − σ ) δ ( k + − p + g − p + q − p + ¯ q ) e − i~p q · ~z − i~p ¯ q · ~z q p + g δ λ q , − λ ¯ q × ( ~z · ~ε T ) (cid:0) δ λ ¯ q s − x q (cid:1) (cid:0) δ − s g λ q x g + x ¯ q (cid:1) ( ~P ¯ q · ~ε ∗ g ) x q ( x g + x ¯ q )( p g + p ¯ q ) K ( QZ ) Z e − i~p g · ~z − ( q ↔ ¯ q ) . (6.12)
7. Impact factor for interaction with color dipole
In the 2- and 3-gluon approximations (BFKL and BKP) of exchanges in t -channel oneneeds the Green function obeying the linear equation. In the color singlet channel, thesubtracted color dipole is the operator that plays this role U = 1 N c tr (cid:16) U U † (cid:17) − . (7.1)The operator appearing in eq. (4.3) can be rewritten astr( U U † ) tr( U U † ) − N c tr( U U † ) = N c ( U + U − U + U U ) . (7.2)Therefore in the 2- and 3-gluon approximations in which we neglect U U , eq. (4.2)reads M α g = Z d~z d~z U (cid:26) N c −
12 ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α − Z d~z F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α (cid:27) + 12 N c Z d~z d~z U Z d~z F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α + 12 N c Z d~z d~z U Z d~z F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α , (7.3)which we write as M α g = 12 Z d~z d~z U n ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α + (cid:0) N c − (cid:1) ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α o , (7.4)where ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α = Z d~z (cid:2) N c F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α + N c F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α − F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α ] . (7.5)The integrals required in order to calculate expression (7.5) are discussed in appendix A.Using (9.1) and (9.3) for longitudinal photon (2.10) we have Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε Lα = − δ λ q , − λ ¯ q δ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) × Qg q p + g e − i~p q · ~z − i~p ¯ q · ~z (1 − x g ) x g ( x ¯ q + x g δ − s g λ q ) x q e − i~p g · ~z Z dαe α ixq ( ~z · ~pg ) x ¯ q + xq × (cid:18) i ( ~p g · ~ε ∗ g ) Z q ¯ qg K ( Q g ( α ) Z q ¯ qg ) Q g ( α ) + x g x q ( ~z · ~ε ∗ g ) K ( Q g ( α ) Z q ¯ qg ) (cid:19) − ( q ↔ ¯ q ) , (7.6)14nd Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε Lα = δ λ q , − λ ¯ q δ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) 2 Qg q p + g × e − i~p ¯ q · ~z − i~p g · ~z (1 − x q ) x q " ( x ¯ q + x g δ − s g λ q ) x q ~z · ~ε ∗ g ) ~z e − i x ¯ q ( ~pq · ~z xg ( ~pq · ~z xg + x ¯ q Z ¯ qg K ( Z ¯ qg Q q (1)) Q q (1) − ( x q + x g δ − s g λ ¯ q ) x ¯ q e − i~p q · ~z × Z dα e α ix ¯ q ( ~z · ~pq ) xg + x ¯ q (cid:18) i ( ~p q · ~ε ∗ g ) Z ¯ qgq ( α ) Q q ( α ) K ( Q q ( α ) Z ¯ qgq ) − x q x ¯ q ( ~z · ~ε ∗ g ) K ( Q q ( α ) Z ¯ qgq ) (cid:19) . (7.7)Note that Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε Lα = − Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε Lα | q ↔ ¯ q . (7.8)In eqs. (7.6,7.7) we use the notations Z ij = s x i x j x i + x j ~z , Z ijk = s x i x j + (1 − α ) x i x k x j + x i ~z , Q i ( α ) = s α~p i (1 − x i ) x i + Q . (7.9)For forward production ~p q = ~p ¯ q = ~p g = 0 one can simplify these expressions with the helpof eq. (9.6)˜ F ( p q , p ¯ q , p g , k, ~z , ~z ) α ε Lα | ~p q = ~p ¯ q = ~p g =0 = − δ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) × δ λ q , − λ ¯ q gN c q p + g ( ~z · ~ε ∗ g ) ~z (cid:26) − Z q ¯ q K ( QZ q ¯ q )2 N c x g + (cid:0) x ¯ q + x g δ − s g λ q (cid:1) × (cid:18) Z g ¯ q K ( Z ¯ qg Q )(1 − x q ) − Z qg K ( QZ qg ) x ¯ q + Z K ( QZ ) (cid:18) x ¯ q + 1 N c x g (cid:19)(cid:19)(cid:27) − ( q ↔ ¯ q ) . (7.10)For transverse photon (2.11) we can rewrite (5.16) as F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε T α = − igδ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) e − i~p q · ~z − i~p ¯ q · ~z − i~p g · ~z π q p + g ε ∗ jg ε uT δ λ q , − λ ¯ q z j z × (cid:0) ∇ u (cid:0) δ sλ ¯ q (cid:0) x ¯ q + x g δ − s g λ q (cid:1) − x q δ sλ q (cid:1) − x q ∇ u δ sλ q δ − s g λ ¯ q (cid:1) K ( QZ ) − ( q ↔ ¯ q ) . (7.11)15hen using (9.3) and (9.8) we have Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε T α = 2 igδ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) ε ∗ jg ε uT × δ λ q , − λ ¯ q e − i~p q · ~z − i~p ¯ q · ~z q p + g Z dα " e − iαxq ( ~pg · ~z ) x ¯ q + xq − i~p g · ~z n(cid:0) δ sλ ¯ q (cid:0) x ¯ q + x g δ − s g λ q (cid:1) − x q δ sλ q (cid:1) × (cid:18) K ( Q g ( α ) Z q ¯ qg ) x ¯ q + x q (cid:18) ip gj z u Z q ¯ qg ~z x g − αx q ip gu z j ( x ¯ q + x q ) + x q δ ju (cid:19) − x q Q g ( α ) Z q ¯ qg K ( Q g ( α ) Z q ¯ qg ) x ¯ q + x q (cid:18) αp gj p gu Q g ( α ) x g ( x ¯ q + x q ) + z j z u ~z (cid:19)(cid:19) − x q δ sλ q δ − s g λ ¯ q × (cid:18) Z q ¯ qg Q g ( α ) K ( Q g ( α ) Z q ¯ qg )( x ¯ q + x q ) (cid:18) αp gj p gu Q g ( α ) x g + z j z u Z q ¯ qg (1 − α ) x g x q (cid:19) + K ( Q g ( α ) Z q ¯ qg ) (cid:18) αx q z j ip gu x ¯ q + x q + z u ip gj ( α − x q x ¯ q + x q − δ ju (cid:19)(cid:19)(cid:27) − ( q ↔ ¯ q ) (cid:21) , (7.12)which can be also put in the form Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε T α = 2 igδ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) ε ∗ jg ε uT δ λ q , − λ ¯ q e − i~p g · ~z − i~p ¯ q · ~z q p + g × " e − i x ¯ q ( ~pq · ~z ) + xg ( ~pq · ~z ) x ¯ q + xg (cid:26)(cid:0) δ sλ ¯ q (cid:0) x ¯ q + x g δ − s g λ q (cid:1) − x q δ sλ q (cid:1) z j ip qu Z ¯ qg K ( Z ¯ qg Q q (1)) z Q q (1) x q ( x ¯ q + x g )+ δ sλ q δ − s g λ ¯ q x g ( x ¯ q + x g ) z j z (cid:18) z u x ¯ q K ( Z ¯ qg Q q (1)) + ip qu Q q (1) Z ¯ qg K ( Z ¯ qg Q q (1)) (cid:19)(cid:27) − Z dαe i αx ¯ q ( ~pq · ~z ) x ¯ q + xg − i~p q · ~z n(cid:0) x ¯ q δ sλ ¯ q δ − s g λ q + (cid:0) δ sλ q (cid:0) x g δ − s g λ ¯ q + x q (cid:1) − x ¯ q δ sλ ¯ q (cid:1)(cid:1) × (cid:18) x ¯ q Q q ( α ) Z ¯ qgq K ( Q q ( α ) Z ¯ qgq ) x ¯ q + x g (cid:18) αp qj p qu Q q ( α ) x q ( x ¯ q + x g ) + z j z u z (cid:19) + K ( Q q ( α ) Z ¯ qgq ) x ¯ q + x g (cid:18) ip qj z u Z ¯ qgq x q z − z j ip qu αx ¯ q x ¯ q + x g − x ¯ q δ ju (cid:19)(cid:19) + x ¯ q δ sλ ¯ q δ − s g λ q × (cid:18) − Q q ( α ) Z ¯ qgq K ( Q q ( α ) Z ¯ qgq ) x ¯ q + x g (cid:18) αp qj p qu Q q ( α ) x q + z j z u Z qgq (1 − α ) x q x q (cid:19) + K ( Q q ( α ) Z ¯ qgq ) (cid:18) ip qu z j αx ¯ q x ¯ q + x g + δ ju − ip qj z u x ¯ q x ¯ q + x g (1 − α ) (cid:19)(cid:19)(cid:27)(cid:21) . (7.13)Note that Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε T α = − Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε T α | q ↔ ¯ q . (7.14)16or forward production ~p q = ~p ¯ q = ~p g = 0 one can simplify these expressions via eqs. (9.6)and (9.9), thus obtaining Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε T α = 2 igδ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) δ − λ ¯ q λ q x g q p + g × (cid:20) δ − s g s ( ~z · ~ε ∗ g )( ~ε T · ~z ) ~z ( K ( QZ q ¯ q ) + 2 ( x q − K ( QZ )) (cid:0) x q δ sλ q − x ¯ q δ sλ ¯ q (cid:1) − δ ss g (cid:18) K ( QZ ) (cid:0) x q x ¯ q δ sλ q − ( x q − δ sλ ¯ q (cid:1) + x ¯ q δ sλ ¯ q x ¯ q + x q K ( QZ q ¯ q ) (cid:19)(cid:21) − ( q ↔ ¯ q ) , (7.15)which can be also put in the form Z d~z F ( p q , p ¯ q , p g , k, ~z , ~z , ~z ) α ε T α = 2 igδ ( k + − p + g − p + q − p + ¯ q ) θ ( p + g − σ ) δ − λ ¯ q λ q x q q p + g × (cid:20) δ − s g s ( ~z · ~ε ∗ g )( ~ε T · ~z ) ~z (cid:18) x g x q − K ( QZ ¯ qg ) − ( x ¯ q − K ( QZ ) (cid:19) (cid:0) x q δ sλ q − x ¯ q δ sλ ¯ q (cid:1) − δ ss g δ sλ q x g ( x q − K ( QZ ¯ qg ) + (cid:0) x q x ¯ q δ sλ ¯ q − ( x ¯ q − δ sλ q (cid:1) K ( QZ ) ! . (7.16)The expressions (7.10) and (7.16) can be used as a starting point for the description of e.g.meson production within the QCD collinear factorization, see ref. [42].
8. Impact factor in the momentum space and in the linear approximation
Here we will calculate the Fourier transform of the impact factors. We can rewrite thematrix element (4.2) as M α = Z d~p d~p d~p F ( p q , p ¯ q , p g , z , ~p , ~p , ~p ) α (cid:2) N c ( U + U + U U ) − U (cid:3) ( ~p , ~p , ~p )+ Z d~p d~p ˜ F ( p q , p ¯ q , p g , z , ~p , ~p ) α N c − U ( ~p , ~p ) . (8.1)Here the Fourier transforms are defined as F ( p q , p ¯ q , p g , z , ~p , ~p , ~p ) α = Z d~z π d~z π d~z π e i [ ~p · ~z + ~p · ~z + ~p · ~z ] F ( p q , p ¯ q , p g , z , ~z , ~z , ~z ) α , (8.2)˜ F ( p q , p ¯ q , p g , z , ~p , ~p ) α = Z d~z π d~z π e i [ ~p · ~z + ~p · ~z ] ˜ F ( p q , p ¯ q , p g , z , ~z , ~z ) α , (8.3)and (cid:2) N c ( U + U + U U ) − U (cid:3) ( ~p , ~p , ~p ) = Z d~z π d~z π d~z π e − i [ ~p · ~z + ~p · ~z + ~p · ~z ] × (cid:2) N c ( U + U + U U ) − U (cid:3) , (8.4)17 ( ~p , ~p ) = Z d~z π d~z π e − i [ ~p · ~z + ~p · ~z ] U . (8.5)To get the linearized impact factor, one should neglect the term U U and write (cid:2) N c ( U + U + U U ) − U (cid:3) ( ~p , ~p , ~p ) ∼ π (cid:2) N c ( δ ( ~p ) U ( ~p , ~p ) + δ ( ~p ) U ( ~p , ~p )) − δ ( ~p ) U ( ~p , ~p ) (cid:3) . (8.6)Then for the matrix element M α we get M α = 12 Z d~p d~p U ( ~p , ~p ) n ˜ F ( p q , p ¯ q , p g , z , ~p , ~p ) α + (cid:0) N c − (cid:1) ˜ F ( p q , p ¯ q , p g , z , ~p , ~p ) α o , (8.7)˜ F ( p q , p ¯ q , p g , z , ~p , ~p ) α = 2 πN c F ( p q , p ¯ q , p g , z , , ~p , ~p ) α (8.8)+2 πN c F ( p q , p ¯ q , p g , z , ~p , , ~p ) α − πF ( p q , p ¯ q , p g , z , ~p , ~p , α . (8.9)Taking the Fourier transform of eq. (5.15) via eq. (9.10) from appendix B we get for thelongitudinal photon F ( p q , p ¯ q , p g , z , ~p , ~p , ~p ) α ε Lα = δ ( k + − p + g − p + q − p + ¯ q ) δ ( ~p q + ~p q + ~p g ) θ ( p + g − σ ) × δ λ q , − λ ¯ q q p + g iQ g ( x q + x g δ − s g λ ¯ q ) (cid:0) ( ~p q · ~ε ∗ g ) x q + ( ~p q · ~ε ∗ g ) (1 − x ¯ q ) (cid:1) (1 − x ¯ q ) x g x q (cid:16) Q + ~p q x ¯ q (1 − x ¯ q ) (cid:17) (cid:16) Q + ~p q x q + ~p q x ¯ q + ~p g x g (cid:17) − ( q ↔ ¯ q ) , (8.10)where ( q ↔ ¯ q ) ≡ ( λ q , x q , ~p , ~p q ↔ λ ¯ q , x ¯ q , ~p , ~p ¯ q ) . (8.11)One can check that this result is compatible with the wave function derived in ref. [46].Using eq. (9.11), we get˜ F ( p q , p ¯ q , p g , k, ~p , ~p ) α ε Lα = 4 igQ q p + g θ ( p + g − σ ) δ ( k + − p + g − p + q − p + ¯ q ) δ ( ~p q + ~p q − ~p g ) × δ λ q , − λ ¯ q x q ( x g + x ¯ q ) (cid:0) δ − s g λ q x g + x ¯ q (cid:1) ( p g + p ¯ q ) π ( ~P ¯ q · ~ε ∗ g )( Q x q ( x ¯ q + x g ) + ~p q ) − ( q ↔ ¯ q ) . (8.12)18or the transverse photon, using eqs. (9.12) and (9.13) we have F ( p q , p ¯ q , p g , z , ~p , ~p , ~p ) α ε T α = − ig q p + g δ ( k + − p + g − p + q − p + ¯ q ) δ ( ~p q + ~p q + ~p g ) θ ( p + g − σ ) × δ λ q , − λ ¯ q ( ~p q · ~ε ∗ g ) ( x ¯ q + x g ) + ( ~p q · ~ε ∗ g ) x ¯ q ( ~p q ( x ¯ q + x g ) + ~p q x ¯ q ) ( (cid:0) x q − δ sλ ¯ q (cid:1) (cid:0) x ¯ q + x g δ − s g λ q (cid:1) x q × ( ~p q · ~ε T ) ( x g + x q ) + ( ~p q · ~ε T ) x q x g (cid:16) ~p q x ¯ q + ~p g x g + ~p q x q + Q (cid:17) − ( ~p q · ~ε T )( x ¯ q + x g ) (cid:16) ~p q x q ( x ¯ q + x g ) + Q (cid:17) + (cid:0) x ¯ q + x g δ − s g λ q − δ sλ q (cid:1) ( ~p q · ~ε T ) ( x ¯ q + x g ) + ( ~p q · ~ε T ) x ¯ q x g (cid:16) ~p q x ¯ q + ~p g x g + ~p q x q + Q (cid:17) − ( q ↔ ¯ q )= 2 ig q p + g δ ( k + − p + g − p + q − p + ¯ q ) δ ( ~p q + ~p q + ~p g ) θ ( p + g − σ ) δ − λ ¯ q λ q Q (1 − x q ) (cid:16) ~p q x ¯ q + ~p g x g + ~p q x q + Q (cid:17) n δ ss g δ sλ q + 2( ~p q · ~ε T )(( ~p q · ~ε ∗ g )( x ¯ q + x g ) + ( ~p q · ~ε ∗ g ) x ¯ q ) (cid:0) x q − δ sλ ¯ q (cid:1)(cid:0) x g δ − s g λ q + x ¯ q (cid:1) (1 − x q ) x q x ¯ q x g (cid:16) Q + ~p q (1 − x q ) x q (cid:17) − ( q ↔ ¯ q ) . (8.13)Again, one can check that this result is compatible with the wave function derived inref. [46]. Finally, using eq. (9.14) we find˜ F ( p q , p ¯ q , p g , k, ~p , ~p ) α ε T α = − g θ ( p + g − σ ) δ ( k + − p + g − p + q − p + ¯ q ) δ ( ~p q + ~p q − ~p g ) δ λ q , − λ ¯ q q p + g × (cid:0) δ λ ¯ q s − x q (cid:1) (cid:0) δ − s g λ q x g + x ¯ q (cid:1) ( p g + p ¯ q ) πi ( ~P ¯ q · ~ε ∗ g ) ( ~p q · ~ε T ) Q x q ( x ¯ q + x g ) + ~p q − ( q ↔ ¯ q ) . (8.14)The formulas in the momentum space derived in this section constitute a convenient start-ing point for calculations of phenomenologically important observables such as cross sec-tions etc.
9. Conclusions
Based on the QCD shock-wave approach [27, 35, 36], we rederived the γ ∗ → q ¯ q impactfactor. Using the same approach, we computed the general expression for the γ ∗ → q ¯ qg impact factor for the first time. The contribution of the diagrams with the gluons crossingthe shock-wave, calculated using Balitsky’s formalism, are consistent with the results forthe γ ∗ → q ¯ qg wave function obtained in ref. [46], based on old-fashioned perturbationtheory.The results we obtained, in coordinate space, are very suitable for phenomenologicalstudies of diffractive processes since they allow for the implementation of saturation modelswhen considering the color-singlet channel. The measurement of dijet production in DDIS19as recently performed [47], and a precise comparison of dijet versus triple-jet production,which has not been performed yet at HERA [48], would be very useful to get a deeperunderstanding of the QCD mechanism underlying diffraction. Such a ratio would providean observable possibly more independent of any saturation effect. A quantitative firstprinciple analysis of this would require an evaluation of virtual corrections to the γ ∗ → q ¯ q impact factor, which are left for further studies.Our results could also be relevant for photo-production of diffractive jets [49, 50], thehard scale being provided by the invariant mass of the produced state. Indeed, the directcoupling of a Pomeron to the impact factor could be important, in addition to the resolvedPomeron contribution (which is the sum of a direct interaction of the photon with quarksor gluons originating from the pomeron, and a resolved photon-pomeron interaction), inparticular in the region x γ ∼ x γ is the longitudinal momentum fraction carried by thepartons coming from the photon), in view of the collinear factorization breaking which hasbeen the matter of discussions [51, 52]. Since our results are expressed in terms of a shock-wave, they can be used both for inclusive (considering the color octet in the t − channel, bymodifying formula (4.2) and diffractive (in the color-singlet case) jet production, the ratioof amplitudes providing an interesting observable to evaluate gap survival probabilities [51–53]. Furthermore, our results, expressed in terms of a shock-wave, are a natural startingpoint for studies of higher-twist effects, which could be investigated by an appropriateexpansion of U operators in powers of the coupling, in order to study the effect of multigluonexchange in the t − channel.Finally, diffractive open charm production was measured at HERA [54] and studied inthe large M limit based on the direct coupling between a Pomeron and a q ¯ q or a q ¯ qg state,with massive quarks [22]. The extension of our result to the case of massive quark, is leftfor future analysis.Our result is therefore a first step for phenomenological studies of diffraction, whichcould be of relevance in future e − p and e − A colliders like EIC and LHeC, as well as forultraperipheral processes which could be studied at LHC. Acknowledgements
We would like to thank A. Besse, G. Beuf, L. Motyka, Al Mueller, S. Munier and M. Sadzi-kowski for discussions. A. V. G. thanks V. S. Fadin and A. V. Reznichenko for helpfuldiscussions and the LPT Orsay and NCBJ in Warsaw for hospitality while part of this workwas being done. A. V. G. also acknowledges support of president grant MK-525.2013.2 andRFBR grant 13-02-01023. This work was partially supported by the PEPS-PTI PHENO-DIFF, the PRC0731 DIFF-QCD, the Polish Grant NCN No. DEC-2011/01/B/ST2/03915and the Joint Research Activity Study of Strongly Interacting Matter (acronym Hadron-Physics3, Grant Agreement n.283286) under the Seventh Framework Programme of theEuropean Community. 20 ppendix A: Integrals necessary for linearization
We need the following integrals Z d~z e − i~p q · ~z K ( QZ ) | ~z → ~z = 2 πx q (1 − x q ) e − i x ¯ q ( ~pq · ~z xg ( ~pq · ~z xg + x ¯ q Z ¯ qg K ( Z ¯ qg Q q (1)) Q q (1) , (9.1)where we define Z ij = s x i x j x i + x j ~z , Q i ( α ) = s α~p i (1 − x i ) x i + Q . (9.2) Z d~z e − i~p g · ~z ~z ~z K ( QZ ) (9.3)= − πie − i~p g · ~z Z −∞ dte itxgxqQ ~z − i ( x ¯ q + xg )+ i txg t~p g + ~z ~z (cid:16) t~p g + ~z ~z (cid:17) e ixq~z x ¯ q + xq ) t (cid:18) t~p g + ~z ~z (cid:19) − = − πe − i~p g · ~z (1 − x g ) x g Z dαe α ixq ( ~z · ~pg ) x ¯ q + xq (cid:18) i~p g Z q ¯ qg Q g ( α ) K ( Q g ( α ) Z q ¯ qg ) + x g x q ~z K ( Q g ( α ) Z q ¯ qg ) (cid:19) , where Z q ¯ qg = s x q x ¯ q + (1 − α ) x q x g x ¯ q + x q ~z . (9.4)To get the previous result, we used the following parametrization Z d~r ~r~r e i ( a~r + b ( ~r + ~ρ ) ) − i~p · ~r = − π i (2 b~ρ − ~p )(2 b~ρ − ~p ) e ib~ρ (cid:18) e − i (2 b~ρ − ~p )24( a + b ) − (cid:19) = − π a + b ) (2 b~ρ − ~p ) e ib~ρ Z dαe − α i (2 b~ρ − ~p )24( a + b ) . (9.5)In the simpler case ~p q = 0 the integral can be fully reduced to Z d~z ~z ~z K ( QZ ) = − πx g x q Q ~z ~z ( Z q ¯ q K ( QZ q ¯ q ) − Z K ( QZ )) , (9.6)where Z = q x q ( x ¯ q + x g ) ~z . (9.7)Using the integral (9.5), we also get Z d~z e − i~p g · ~z z j ~z ∂∂z l K ( QZ ) = − πe − i~p g · ~z Z dα e α ixq ( ~z · ~pg ) xq + x ¯ q (cid:20)(cid:18) δ jl + ip jg z l x q x ¯ q + x q (cid:19) × K ( Q g ( α ) Z q ¯ qg ) − x q x g x ¯ q + x q z l z j Q g ( α ) Z q ¯ qg K ( Q g ( α ) Z q ¯ qg ) (1 − α ) − x q ( ip lg z j + ip jg z l ) x q + x ¯ q αK ( Q g ( α ) Z q ¯ qg ) + ip lg ip jg x g ( x q + x ¯ q ) α Z q ¯ qg Q g ( α ) K ( Q g ( α ) Z q ¯ qg ) . (9.8)21s before, in the simpler case ~p g = 0 the integral can be fully reduced to Z d~z z j ~z ∂∂z l K ( QZ ) = − πx q x ¯ q x g δ jl − z l z j ~z ! (cid:18) K ( QZ q ¯ q ) QZ q ¯ q − K ( QZ ) QZ (cid:19) +2 π δ jl K ( QZ ) QZ − z l z j ~z K ( QZ ) ! . (9.9)The integrals (9.3) and (9.8) are convergent and can be evaluated numerically. Appendix B: Integrals necessary for Fourier transform
We here provide the set of Fourier transforms of modified Bessel functions we used in ourcalculation. Z d~z π d~z π e i [ ~q · ~z + ~q · ~z ] z j ~z K ( Q q x g x q ~z + x q x ¯ q ~z + x ¯ q x g ~z )= i (cid:0) q j x q + q j (1 − x ¯ q ) (cid:1) (1 − x ¯ q ) x ¯ q x g x q (cid:16) Q + ~q x ¯ q (1 − x ¯ q ) (cid:17) (cid:16) Q + ~q x q + ~q x ¯ q + ( ~q + ~q ) x g (cid:17) . (9.10) Z d~z π e i~q · ~z K ( Q q x q ( x g + x ¯ q ) ~z ) = 1(1 − x q ) x q Q + ~q . (9.11) Z d~z π d~z π e i [ ~q · ~z + ~q · ~z ] z j z β ~z K ( Q q x g x q ~z + x q x ¯ q ~z + x ¯ q x g ~z ) q x g x q ~z + x q x ¯ q ~z + x ¯ q x g ~z (9.12)= x ¯ q (cid:18) δ jβ − ( q j ( x ¯ q + x g )+ q j x ¯ q )( q β ( x ¯ q + x g )+ q β x ¯ q ) ( ~q ( x ¯ q + x g )+ ~q x ¯ q ) (cid:19) Qx q ( ~q ( x ¯ q + x g ) + ~q x ¯ q ) ln q x ¯ q + ( q + q ) x g + q x q + Q q x q (1 − x q ) + Q + q β ( x ¯ q + x g ) + q β x ¯ q Qx q ( ~q ( x ¯ q + x g ) + ~q x ¯ q ) q j ( x g + x q ) + q j x q x g (cid:16) q x ¯ q + ( q + q ) x g + q x q + Q (cid:17) − q j ( x ¯ q + x g ) (cid:16) q x q ( x ¯ q + x g ) + Q (cid:17) . Z d~z π d~z π e i [ ~q · ~z + ~q · ~z ] z j z β ~z K ( Q q x g x q ~z + x q x ¯ q ~z + x ¯ q x g ~z ) q x g x q ~z + x q x ¯ q ~z + x ¯ q x g ~z = ( x ¯ q + x g ) (cid:18) δ jβ − ( q j ( x ¯ q + x g )+ q j x ¯ q )( q β ( x ¯ q + x g )+ q β x ¯ q ) ( ~q ( x ¯ q + x g )+ ~q x ¯ q ) (cid:19) Qx q ( ~q ( x ¯ q + x g ) + ~q x ¯ q ) ln q x ¯ q + ( q + q ) x g + q x q + Q q x q (1 − x q ) + Q + (cid:0) q j ( x ¯ q + x g ) + q j x ¯ q (cid:1) (cid:0) q β ( x ¯ q + x g ) + q β x ¯ q (cid:1) Qx g x q x ¯ q ( ~q ( x ¯ q + x g ) + ~q x ¯ q ) (cid:16) q x ¯ q + ( q + q ) x g + q x q + Q (cid:17) . (9.13) Z d~z π e i~q ~z z j K ( Q p x q ( x g + x ¯ q ) ~z ) p x q ( x g + x ¯ q ) ~z = iq j (1 − x q ) x q Q ((1 − x q ) x q Q + ~q ) . (9.14)22 eferences [1] M. Wusthoff and A. D. Martin, The QCD description of diffractive processes , J.Phys.
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