Inclusive prompt photon production in electron-nucleus scattering at small x
Kaushik Roy Raju Venugopalan






kaushik.roy.1@stonybrook.edu
Dept. of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA
Physics Department, Brookhaven National Laboratory, Bldg. 510A, Upton, NY 11973, USA


raju@bnl.gov
Physics Department, Brookhaven National Laboratory, Bldg. 510A, Upton, NY 11973, USA





We compute the differential cross-section for inclusive prompt photon production in deeply inelastic scattering (DIS) of electrons on nuclei at small x in the framework of the Color Glass Condensate (CGC) effective theory. The leading order (LO) computation in this framework resums leading logarithms in x as well as power corrections to all orders in Q_s,A^2/Q^2, where Q_s,A(x) is the nuclear saturation scale. This LO result is proportional to universal dipole and quadrupole Wilson line correlators in the nucleus. In the soft photon limit, the Low-Burnett-Kroll theorem allows us to recover existing results on inclusive DIS dijet production. The k_⊥ and collinearly factorized expressions for prompt photon production in DIS are also recovered in a leading twist approximation to our result. In the latter case, our result corresponds to  the dominant next-to-leading order (NLO) perturbative QCD contribution at small x.  We next discuss the computation of the NLO corrections to inclusive prompt photon production in the CGC framework. In particular, we emphasize the advantages for higher order computations in inclusive photon production, and for fully inclusive DIS, arising from the simple momentum space structure of the dressed quark and gluon “shock wave" propagators in the “wrong" light cone gauge A^-=0 for a nucleus moving with P^+_N→∞. 



Inclusive prompt photon production in electron-nucleus scattering at small x
    Raju Venugopalan
    December 30, 2023
============================================================================






§ INTRODUCTION
  


The measurement of isolated prompt photons in deeply inelastic scattering (DIS) provides a unique precision test of perturbative QCD (pQCD) with two distinct hard scales that are well understood from QED: the exchanged photon virtuality, Q^2 in the initial state and the transverse energy of the emitted prompt photon, E_⊥^γ. Prompt photon cross-sections were measured early on by the H1 and ZEUS experiments <cit.> for the case of photoproduction, where the negative 4-momentum transfer squared, Q^2=-q^2 of the exchanged virtual photon is close to zero. Subsequently, the first measurements of prompt photon production in e+p DIS, isolated and accompanied by jets, were performed by ZEUS and H1 for a wide range of Q^2 <cit.>.

Isolated photons in DIS have proven to be clean and well calibrated probes of QCD dynamics <cit.>. We will explore here their potential for uncovering a novel gluon saturation regime of QCD <cit.> at small Bjorken x. This regime is characterized by the many-body recombination and screening dynamics of gluons that competes with the perturbative bremsstrahlung of increasing numbers of soft gluons at small x. An emergent dynamical scale Q_s,A(x) screens color charges at increasingly smaller distances with decreasing x thereby ensuring that the squared field strengths do not exceed 1/α_S, where α_S is the QCD coupling constant. The dynamics in this regime of QCD is fully nonlinear. Nevertheless, at sufficiently small x, or large enough nuclear size A, where Q_s,A^2(x) ≫Λ_ QCD^2, α_S ≡α_S(Q_s,A) ≪ 1. Many-body weak coupling techniques can therefore be employed to perform systematic computations in this nonlinear QCD regime <cit.>.

The rich dynamics of gluon saturation is captured in an effective field theory (EFT), the Color Glass Condensate (CGC) <cit.>, and we will apply it here to compute the inclusive photon cross-section in DIS off nuclei.  The CGC EFT is formulated in the infinite momentum frame <cit.> of the nucleus with an appropriate choice of gauge. It relies on the separation, in the longitudinal momentum fraction x, of the degrees of freedom into static color sources at large x coupled eikonally to dynamical “wee" gluon fields at small x <cit.>.  Because a large number of large x sources couple to the wee gluons, the color charge of the sources can lie in any one of a number of higher dimensional color representations of the SU(3) algebra <cit.>. This results in a stochastic distribution of classical color sources over a gauge  invariant weight functional W_Y_A[ρ_A] representing the probability density of the color charge density ρ_A(x^-,𝐱_⊥). The subscript Y_A=ln(x_0/x)  denotes the spacetime rapidity separation of small x target gluons relative to those at x≡ x_0 corresponding to the nuclear beam rapidity, Y_beam.   


The expectation value of any physical operator at rapidity Y_A is determined by the average over all possible color charge configurations: 

    ⟨𝒪⟩_Y_A =∫ [𝒟ρ_A ]   W_Y_A [ρ_A]   𝒪 [ρ_A]  ,

where 𝒪 [ρ_A] is the expectation value of the operator for a particular configuration ρ_A of color sources. In the problem of interest, this operator is the  differential cross-section for inclusive photon production for a given ρ_A. 
Requiring that physical observables be independent of the arbitrary scale separation between sources and fields results in the JIMWLK functional renormalization group equation <cit.>

    ∂
    W_Y_A[ρ_A]/∂ Y_A =ℋ[ ρ_A,δ/δρ_A]   W_Y_A [ρ_A]   .

This equation describes the evolution of the distribution of color charges in the nuclear wavefunction from its fragmentation region at large x (or small Y_A) to the small x (or large Y_A) of interest as determined by the kinematics of the process. Here ℋ is the JIMWLK Hamiltonian; explicit expressions and properties of the Hamiltonian are discussed for instance in <cit.>. It is worthwhile to note here that the JIMWLK evolution equation can alternatively be expressed as the Balitsky-JIMWLK hierarchy <cit.> of equations for expectation values of n-point Wilson line correlators. In the limit of large number of colors N_c →∞ and for large nuclei A→∞, the closed form Balitsky-Kovchegov (BK) equation is obtained for the two-point correlator of Wilson lines <cit.>. The BK equation is a good approximation in many practical situations to the full Balitsky-JIMWLK hierarchy <cit.> and is therefore extremely useful in phenomenological studies at collider experiments. 

An attractive feature of the CGC, as in any EFT, is that there is a well defined power counting for systematic higher order computations. As we will show explicitly later for our specific case, this power counting allows one to match results in the CGC framework to those in the collinear factorization and k_⊥ factorization perturbative frameworks in appropriate kinematic limits. For e+A and p+A collisions, the appropriate CGC power counting is in a so-called “dilute-dense" limit <cit.> where  dilute color source density ρ_p in the generic projectile is of order g ≪  1 in the QCD coupling, while the density of color sources ρ_A∼ 1/g in the nuclear target. Strictly speaking, the dilute-dense power counting corresponds to computations that are lowest order in the dimensionless ratios Q_s,p^2/k_p,⊥^2 ≪ 1, and all orders in Q_s,A^2/k_A,⊥^2∼ 1, where k_p,⊥ and k_A,⊥ correspond to the momentum transfer from the projectile and target respectively. The dilute-dense power counting was implemented recently by one of us and collaborators in computing inclusive photon production in p+A collisions to next-to-leading order (NLO) accuracy <cit.> extending earlier leading order (LO) computations in this context <cit.>. 

The power counting in the CGC framework for the e+A DIS case at hand is simpler than the p+A case because we don't have a power counting in ρ_p on account of the lepton probe. In general, at LO in the QCD coupling constant, we obtain the two classes of processes shown in Fig. <ref> that contribute to the inclusive prompt photon cross-section.


  The Class I processes represent the bremsstrahlung of a photon from a valence quark in the wavefunction of the target nucleus. Their contribution to the differential cross-section for inclusive photon production has been calculated <cit.> at small x in the CGC framework albeit in the context of p+A collisions. This result can however be straightforwardly adapted to the present problem. 

The Class II processes correspond to configurations that become important at small x where the virtual photon emitted by the electron 
fluctuates into a long lived quark-antiquark dipole <cit.> and the dipole subsequently has a nearly instantaneous “shock wave" eikonal scattering off the gauge fields in the nuclear target <cit.>. In this dipole picture of e+A DIS, the power counting is dictated by strong color sources ρ_A∼ 1/g in the target, and their energy evolution with respect to the quark-antiquark dipole.  Attaching new sources to the diagram in Fig. <ref> does not change the order in strong coupling constant, because gρ_A∼ 1. Independent powers of g arise only when a vertex is not connected to the source. From this argument, it should be clear that we are performing an all-twist expansion in gρ_A at each order in α_S in this power counting scheme[Note that there is also an LO contribution analogous to that described in <cit.> whereby the real photon is emitted from a quark loop that the virtual photon fluctuates into. Just as in the p+A case, this contribution is highly suppressed relative to the Class II processes we will discuss.]. 

At NLO, typical contributions[For a recent computation of the real gluon emission contribution, see <cit.>.] to Class I processes are shown in Fig. <ref>. For inclusive photon production, these possess the divergence structure inherent in NLO quark production containing i) dominant contributions from the large phase space in transverse momentum available to the emitted gluon (α_Sln(k_⊥)∼ 1), ii) subdominant contribution from logarithms sensitive to x (since valence quark distributions are peaked at x∼ 1) and finally, iii) finite pieces that are not phase space enhanced. By an appropriate choice of scale and scheme for factorization, the finite pieces can be absorbed in the definition of the quark distribution function. These contributions are therefore effectively of order 𝒪(α_e) (where α_e is the QED fine structure constant) if we replace the bare valence quark distribution in the nucleus by one that absorbs the leading logarithmic contributions. These last contributions, to all orders in perturbation theory, are captured by the DGLAP <cit.> renormalization group (RG) evolution of the valence quark distributions.  


At small x, the Class I contributions are strongly suppressed relative to the Class II contribution. The reason for this is that the former are sensitive to the valence quark distribution in the target while the latter are sensitive to the gluon distribution. At small x, as seen in the  e+p HERA DIS data <cit.>, the gluon distribution clearly dominates the valence quark distribution. 
Since our interest is in inclusive photon production at small x, we shall not examine Class I type processes any further and shall focus our attention exclusively on Class II processes. 

 
 
 
In the first part of this paper, we will compute the Class II LO diagrams shown in Fig. <ref>. We will subsequently discuss the structure of the NLO computation of inclusive photon production. We  shall in particular discuss a simple formulation of the dressed gluon propagator in light cone gauge. The NLO computations are important for two reasons. Firstly, as noted in our discussion of Class I NLO contributions, there are logarithmically enhanced contributions of order α_S ln(x_0/x) which can be as large as the LO contributions. Such contributions appear in each order of perturbation theory; they can be isolated and resummed in a so-called leading logarithmic (LLx) RG treatment. There are also pure α_S suppressed NLO contributions. These are important for precision measurements but can also lead to qualitative changes in spectra that impact the discovery potential of such measurements. In particular, they will be important for a quantitative extraction of the saturation scale Q_s,A(x). 

These statements can be understood more clearly by considering the organization of the perturbative series for the argument of ⟨𝒪⟩ in Eq. <ref> as[An extended discussion along these lines for field theories with strong time dependent sources can be found in <cit.>; for the CGC specifically, see <cit.>.]

    𝒪[ρ_A]= ∑_n=0 c_nα_S^n .

Here each coefficient c_n=∑_j=1^∞ξ_nj (gρ_A)^j, where the matrix elements ξ_nj are numbers of order unity, resums the contributions obtained by adding extra sources of magnitude ρ_A∼ 1/g to the allowed graphs.
Before we proceed any further, we must invoke the essential ingredient of the CGC effective field theory – the separation of large x static light cone sources from dynamical small x gauge fields – as represented by the structure of Eq. <ref>. 

The starting point of any CGC computation therefore includes an initial cutoff scale Λ_0^+ (in the `+' longitudinal momentum) or Y_0 (in the rapidity) between sources and fields.  This is shown schematically in  Fig. <ref>. At leading order (LO), the cutoff scale Λ_0^+ (or Y_0=ln(Λ_ beam/Λ_0^+)) distinguishing soft and hard partons is arbitrary and the fast quantum modes with k^+≫Λ^+_0 or Y < Y_0 are represented by the classical color source density ρ_A with the weight functional W_Λ_0^+ (Y_0) [ρ_A].  The quantum evolution of sources and fields is described by the following renormalization group procedure (RG).  One first integrates out quantum fluctuations within the range Λ_1^+< | k^+| < Λ_0^+. Here Λ_1^+ is chosen such that α_Sln(Λ_0^+/Λ_1^+) ≡α_S Δ Y<1, where Δ Y = Y_1-Y_0.  
  

For our computation of inclusive photon production at small x, typical NLO diagrams that generate such logarithms upon change of scale from Λ_0^+ to Λ_1^+ are shown in Fig. <ref>. Their contribution can be absorbed into the evolution of the weight functional describing the distribution of color sources:

    W_Λ_1^+[ρ_A]   =( 1+ln(Λ_0^+/Λ_1^+). ℋ) W_Λ_0^+ [ρ_A]   or equivalently,
    
    W_Y_1[ρ_A]   =( 1+Δ Y . ℋ) W_Y_0 [ρ_A]  .

Here ℋ is the JIMWLK Hamiltonian we alluded to previously. Hence these particular NLO δ𝒪 contributions generate a classical effective theory at this new scale expressed as

    ∫ [𝒟ρ_A ]   W_Λ_0^+ [ρ_A] (𝒪_LO+δ𝒪_NLO)=∫   [𝒟ρ_A ]   W_Λ_1^+[ρ_A] 𝒪 [ρ_A]   ,
 
where the LLx contributions have been absorbed in the JIMWLK evolution of W. To leading order accuracy, the perturbative expansion in Eq. <ref> then has the coefficients:
 
    c_n=∑_j=0^∞  d_nj (g ρ_A)^j ( ln(1/x) )^n,      n=1,2, …  ,

 


 










At NLO however, there are α_S contributions that do not come accompanied with logarithms in x. These would then correspond to coefficients expressed more generally in the expansion of the perturbative series as 

    c_n=∑_i=0^n ∑_j=0^∞ f_nj^i (gρ_A)^j ( ln(1/x) )^i,      n=1,2,…  ,

in particular the coefficients f_nj^n-1. For our process of interest, some of the the corresponding diagrams are shown in Fig. <ref>. These additional contributions are not rapidity ordered relative to the leading order contribution; the phase space integrals therefore do not generate logarithms in x. Thus while the NLO diagrams represented in Fig. <ref> can be absorbed into the LO order computation of photon production accompanied by leading logarithmic JIMWLK evolution of the weight functional W, the NLO graphs represented in Fig. <ref> should be combined with next-to-leading-log (NLLx) JIMWLK evolution to obtain the full NLO result for inclusive photon production in e+A collisions at small x. Fortunately, the NLLx JIMWLK evolution equations are known <cit.> as well as the NLLx BK equations <cit.>. Therefore with the computation of the NLO diagrams represented in Fig. <ref>, all the elements will be in place for quantitative predictions, to NLO accuracy, for inclusive photon production in e+A collisions at small x. This paper represents the first step in this direction. While we will discuss key aspects of the structure of the NLO computation here, the full computation will be presented in future publications in preparation. 

 

The paper is organized as follows. In section <ref>, we begin by discussing the ingredients necessary for the computation of the amplitude for inclusive photon production; a compact expression for this amplitude is given in Eq. <ref>. In section <ref>, we will calculate the cross-section for the production of a direct photon accompanied by a quark-antiquark pair as well as the inclusive differential cross-section for direct photon production at LO. The former provides the rate for inclusive production of a photon with a dijet pair in small x kinematics. Our results for these cross-sections are expressed respectively in Eqs. <ref> and <ref> as a convolution of the lepton tensor L^μν that is familiar from inclusive DIS and a hadron tensor constituted of all-twist lightlike Wilson line correlators. In section <ref>,  which we divide into three subsections, we discuss the important properties of the photon production amplitude at leading order. We first examine the k_⊥ and collinear factorization limits of our computation in the limit of large transverse momentum k_⊥≫ Q^A_S.  In the former case, the cross-section is proportional to the unintegrated gluon distribution within the nucleus while the latter is proportional to the usual leading twist nuclear gluon distribution in perturbative QCD, evaluated at the scale Q^2 of the virtual photon. The corresponding expressions for the hadron tensor in this limit are given in Eqs. <ref> and <ref> respectively. We next demonstrate that the Low-Burnett-Kroll theorem <cit.> is satisfied when k_γ→ 0: we see explicitly that the nonradiative DIS differential cross-section for the inclusive dijet case in DIS matches extant results in the literature for the same <cit.>. In the final subsection, we show that the amplitude derived in Eq. <ref> has a simple and efficient interpretation in terms of a modified fermion propagator in the classical background field of the nucleus. This will prove beneficial for higher order computations.


Section <ref> outlines the machinery for the computation of the amplitude at NLO. This includes a discussion of the motivation behind choosing the “wrong" light cone gauge A^-=0 for the kinematics of our process as the efficient gauge in which to perform our computations. We then derive the corresponding momentum space Feynman rules for the small fluctuation gluon propagator and conclude this section with an analysis of the contributing processes at this order.  We end the paper with a brief summary and an outline of work in progress. 

Appendices <ref>, <ref> and  <ref> supplement the material in the body of the paper. The notations and conventions are clarified in Appendix <ref>. Appendix <ref> includes the proof of gauge invariance by virtue of the Ward identity. It also contains computational detail for the subsection dealing with the soft photon factorization of the photon production amplitude. Appendix <ref> includes a detailed discussion of the techniques used in determining kinematically allowed processes contributing to inclusive photon production at both LO and NLO. 



§ COMPONENTS OF THE LO AMPLITUDE COMPUTATION
 

In this section, we will outline the components needed for the computation of the amplitude for Class II processes at LO in the CGC framework. In the effective theory of CGC, owing to their large occupancy A∼ 1/g, the dynamics of small x gluons to LO is described by the classical Yang-Mills equations[Note that henceforth the factor of g is taken out of the color charge density–this differs from the notation in <cit.> for instance.]

    [D_μ,F^μν](x)=g δ^ν +δ(x^-) ρ_A(𝐱_⊥)    .

The nucleus is assumed to move in the positive z-direction at nearly the speed of light with large light cone longitudinal momentum P^+_N→∞. (See Appendix <ref> for the conventions adopted in this work.) We also assume in this frame that the virtual photon  has a large q^- component of its momentum. 

An essential element in the computation of the LO amplitude is the fermion propagator in the background strong classical color field of the nucleus. In Lorenz gauge ∂_μ A^μ=0, this has the coordinate space representation <cit.>

    S_A(x,y)  =S_0(x,y)+  [θ(x^-)θ(-y^-) ∫d^4z  δ(z^-) (Ũ(𝐳_⊥)-1 )  -  θ(- x^-)θ(y^-)    ∫d^4z  δ(z^-) (Ũ^†(𝐳_⊥)-1 ) ] 
       × S_0(x,z) γ^- S_0(z,y)   ,

where

    S_0(x,y)= ∫d^4p/(2 π)^4 e^-ip.(x-y) S_0(p),    S_0(p)=i(p+m)/p^2-m^2+i ε ,

is the free fermion propagator. 

This effective propagator appears in the momentum space Feynman rules as an effective vertex with the following factors,

    𝒯_ji(q,p)= ± (2 π)δ(p^--q^-) γ^-∫d^2𝐱_⊥  e^i(𝐪_⊥ - 𝐩_⊥).𝐱_⊥[Ũ^(±)(𝐱_⊥) -1 ]_ji  ,

where the plus (minus) sign corresponds to insertions on a quark (antiquark) line respectively and the Wilson line Ũ is written in the fundamental representation of SU(N_c) as

    Ũ(𝐱_⊥)=𝒫_-exp[-ig^2∫_-∞^+ ∞dz^-1/∇^2_⊥ρ_A^a (z^-,𝐱_⊥)t^a] .

In Feynman diagrams, this effective vertex insertion is drawn as depicted in Fig. <ref>. 
These effective vertices do not change the order of the diagrams parametrically by any power of g. In principle therefore there can be diagrams with multiple insertions with the emitted photon sandwiched between any two such vertices. However there is a kinematic constraint that restricts many such possibilities: On the same fermion line(quark or antiquark), there cannot be a photon sandwiched between two effective vertices.  This has the physical meaning that an outgoing fermion (with a definite sign of p^-) can get scattered off of the nucleus and subsequently emit a photon but does not suffer a secondary scattering because the time scale governing the scattering is of order 1/P^+_N, which is nearly instantaneous in the infinite momentum frame[Mathematically, this is manifest in the contour integration over the `+' component of the undetermined momentum with all the poles being on the same side of the real axis. This is discussed in further detail in Appendix <ref>.].



 §.§ Derivation of the amplitude
 
The amplitude for the process

    e(l̃)+A(P) → e(l̃'̃)+Q(k)+Q(p)+γ(k_γ)+X   ,

with X denoting any other particle produced in the collision, can be written as 



    ℳ(𝐥̃, 𝐥̃'̃,𝐪,𝐤,𝐩,𝐤_γ)=e/Q^2u(l̃'̃) γ^μ u(l̃) ℳ_μ(𝐪,𝐤,𝐩,𝐤_γ;λ)  ,

where

    ℳ_μ(𝐪,𝐤,𝐩,𝐤_γ;λ)= ϵ^* α(𝐤_γ,λ) ℳ_μα(𝐪,𝐤,𝐩,𝐤_γ)  ,

represents the amplitude for the hadronic subprocess and is the quantity of interest. Here ϵ^α(𝐤_γ,λ) is the polarization vector for the outgoing photon. The momentum assignments are summarized in Table <ref> and boldface letters stand for the corresponding 3-momenta. Squaring the expression for the amplitude in Eq. <ref>, and performing the necessary averaging and sum over electron spins[We use here the identity

    ∑_λϵ^β(𝐤_γ,λ) ϵ^* α(𝐤_γ,λ)=-g^αβ  ,
 
as the sum over outgoing photon polarizations.], we can write 

    1/2∑_spins, λ|ℳ|^2=L^μν X_μν .

Here 

    L^μν=2e^2/Q^4[(l̃^μl̃'^ν+ l̃^νl̃'^μ)-Q^2/2 g^μν]  ,
 
is the lepton tensor that is identical to that obtained in inclusive DIS. The hadron tensor

    X_μν= - ∑_spinsℳ_μα^* (𝐪,𝐤,𝐩,𝐤_γ) ℳ_ν^α (𝐪,𝐤,𝐩,𝐤_γ)   ,

is of course different because it allows for the emission of a photon from the quark-antiquark pair. After these bare preliminaries, we shall proceed to compute ℳ_μα. 



 §.§ Classifying contributions to the amplitude
 
There are a total of ten contributions, which we will group into diagrams with gluon field insertions on the quark or antiquark line or both. The diagrams for the first group are shown in Fig. <ref>.

The contribution to the amplitude from the diagram labeled (1), representing the emission of the photon by the quark subsequent to its multiple scatterings with the nucleus, is given by

    ℳ^(1)_μα(𝐪,𝐤,𝐩,𝐤_γ)=  - (eq_f)^2 u(𝐤) γ_α S_0(k+k_γ) 𝒯_ij(q-p,k+k_γ)  S_0(q-p) γ_μ v(𝐩)  ,

where q_f is the charge of a quark/antiquark of a given flavor. Employing Eqs. <ref> and <ref>, and choosing a frame where 𝐪_⊥=0 for the virtual photon, we can write the above expression as[The notation here and henceforth closely follows that employed previously in the discussion of photon production in p+A collisions in the CGC 
framework <cit.>.]

    ℳ^(1)_μα(𝐪,𝐤,𝐩,𝐤_γ) =2π (eq_f)^2δ(P^--q^-) ∫_𝐱_⊥e^-i𝐏_⊥.𝐱_⊥ u(𝐤) R^(1)_μα(𝐏_⊥) [Ũ(𝐱_⊥)-1 ]_ijv(𝐩)  ,

where ∫_𝐱_⊥ is a shorthand notation for ∫d^2𝐱_⊥, i (j) represent the color indices of the quark (antiquark) respectively and

    R^(1)_μα(𝐏_⊥)    =γ_α  (k+k_γ+m) /(k+k_γ)^2-m^2+iεγ^-q-p+m/(q-p)^2-m^2+iεγ_μ
       = γ_αk_γ+2k_α/(k+k_γ)^2-m^2+iεγ^-qγ_μ -2 p_μ/(q-p)^2-m^2+iε  .


We use the following relations

    (p-m) γ_μ v(𝐩)    =2p_μ v(𝐩)   , 
    u(𝐤) γ_α (k+m)    = 2k_αu(𝐤)  ,

to obtain the second line of Eq. <ref>. Note that the 𝐏_⊥-dependence appearing in R_μα^(1) includes an implicit dependence on the three transverse momenta constituting 𝐏_⊥=𝐤_⊥+𝐤_γ⊥+𝐩_⊥.

The contributions from diagrams (2) and (3) in Fig. <ref> can be computed similarly; the sum of the contributions from these three diagrams can then be written as 

    ∑_β=1^3ℳ^(β)_μα( 𝐪, 𝐤,𝐩,𝐤_γ) =2π (eq_f)^2δ(P^--q^-) ∫_𝐱_⊥e^-i𝐏_⊥.𝐱_⊥ u(𝐤) T^(q)_μα(𝐏_⊥) [Ũ(𝐱_⊥)-1 ]_ijv(𝐩)  ,

where 

    T^(q)_μα(𝐏_⊥)=∑_β=1^3 R^(β)_μα(𝐏_⊥)   ,

with  

    R^(2)_μα(𝐏_⊥)   = γ^-(q-P)+2k^-/(q-p-k_γ)^2-m^2+iεγ_αqγ_μ-2p_μ/(q-p)^2-m^2+iε  , 
    
    R^(3)_μα(𝐏_⊥)   = - γ^-(q-P)+2k^-/(q-p-k_γ)^2-m^2+iεγ_μk_γγ_α+2p_α/(p+k_γ)^2-m^2+iε  .


Similarly, the contribution from diagrams with insertions on the antiquark line are shown in Fig. <ref>. Their collective contribution can be written as 

    ∑_β=4^6ℳ^(β)_μα( 𝐪, 𝐤,𝐩,𝐤_γ) =2π (eq_f)^2δ(P^--q^-) ∫_𝐲_⊥e^-i𝐏_⊥.𝐲_⊥ u(𝐤) T^(q̅)_μα(𝐏_⊥) [Ũ^†(𝐲_⊥)-1 ]_ijv(𝐩)   ,

where 

    T^(q̅)_μα(𝐏_⊥)=∑_β=4^6 R^(β)_μα(𝐏_⊥)   ,

and

    R^(4)_μα    (𝐏_⊥)= γ_αk_γ+2k_α/(k+k_γ)^2-m^2+iεγ_μ ( q-P)γ^-+2p^-/ (k+k_γ-q)^2-m^2+iε  , 
    
    R^(5)_μα    (𝐏_⊥)= 2k_μ-γ_μq/(q-k)^2-m^2+iεγ^-k_γγ_α+2p_α/(p+k_γ)^2-m^2+iε  , 
    
    R^(6)_μα    (𝐏_⊥)= 2k_μ-γ_μq/(q-k)^2-m^2+iεγ_α(q-P)γ^-+2p^-/(k+k_γ-q)^2-m^2+iε  .


Finally, we  need to consider the diagrams shown in  Fig. <ref> with insertions on both lines. For these cases, we will need to integrate over the nuclear momentum transfer l. 

The contribution from the diagram labeled (7) can be written as

    ℳ^(7)_μα(𝐪,𝐤,𝐩,𝐤_γ)  =  - (eq_f)^2∫d^4l/(2π)^4 u(𝐤) γ_α S_0(k+k_γ)    𝒯_ik(q-p+l,k+k_γ) S_0(q-p+l) 
       ×γ_μS_0(l-p) 𝒯_kj(-p,l-p) v(𝐩)   .

The integration over l^- is trivial because of the δ(l^-) factor from one of the effective vertices. The integration over l^+ can be done using the theorem of residues. The result can be again compactly written as

    ℳ^(7)_μα(𝐪,𝐤,𝐩,𝐤_γ)= 2π (eq_f)^2δ(P^--q^-) ∫_𝐱_⊥∫_𝐲_⊥∫_𝐥_⊥ e^-i𝐏_⊥.𝐱_⊥+i𝐥_⊥.𝐱_⊥    e^-i 𝐥_⊥.𝐲_⊥ u(𝐤) R^(7)_μα(𝐥_⊥,𝐏_⊥) 
       ×[( Ũ(𝐱_⊥)-1) (Ũ^†(𝐲_⊥)-1)   ]_ij v(𝐩)   ,

where 

    R^(7)_μα(𝐥_⊥,𝐏_⊥)=γ_αk_γ+2k_α/(k+k_γ)^2-m^2+iεγ^-q-p+𝐥_⊥+m /N(𝐥_⊥)γ_μ (γ^-𝐥_⊥+2p^-)   .

 The factor
 
    N(𝐥_⊥) = 4p^-(q^--p^-)[ q^+-M^2(𝐥_⊥-𝐩_⊥)/2p^- -M^2(𝐥_⊥-𝐩_⊥)/2(q^--p^-)]   ,

appearing in the denominator is deliberately written in a way to make contact with the expression in Eq. <ref>; this will be exploited in a later part of the calculation. In the above, M^2(𝐩_⊥)=𝐩^2_⊥+m^2 stands for the squared transverse mass and 𝐩_⊥=γ^ip_i. Finally, ∫_𝐥_⊥ is a shorthand notation for ∫d^2𝐥_⊥/(2π)^2.

The remaining three diagrams can be worked out similarly and and the combined result compactly written as

    ∑_β=7^10ℳ^(β)_μα(𝐪,𝐤,𝐩,𝐤_γ)= 2π (eq_f)^2δ(P^--q^-)    ∫_𝐱_⊥∫_𝐲_⊥∫_𝐥_⊥ e^-i𝐏_⊥.𝐱_⊥+i𝐥_⊥.𝐱_⊥ e^-i 𝐥_⊥.𝐲_⊥
       ×u(𝐤) T^(qq̅)_μα(𝐥_⊥,𝐏_⊥)[(Ũ(𝐱_⊥)-1)(Ũ^†(𝐲_⊥)-1)   ]_ij v(𝐩)   ,

where

    T^(qq̅)_μα(𝐥_⊥,𝐏_⊥) =∑_β=7^10 R^β_μα (𝐥_⊥,𝐏_⊥)   .

The various R-factors obtained after the contour integration over l^+ are 

    R^(8)_μα(𝐥_⊥,𝐏_⊥)= -γ^-(q-P+𝐥_⊥)+2k^-/(p+k_γ)^2-m^2+iεγ_μ (p+k_γ-𝐥_⊥-m ) γ^-(k_γγ_α+2p_α)/S(𝐥_⊥)  ,


    R^(9)_μα(𝐥_⊥,𝐏_⊥)= ( γ^-(q-P+𝐥_⊥)+2k^-) γ_α (q-p+l_a+m) γ_μ(γ^-𝐥_⊥+2p^-)/V(𝐥_⊥)  ,

and 

    R^(10)_μα(𝐥_⊥,𝐏_⊥)= ( γ^-(q-P+𝐥_⊥)+2k^-) γ_μ (l_b -p-k_γ+m) γ_α(γ^-𝐥_⊥+2p^-)/W(𝐥_⊥)  .

In the expressions above, the compactly written functions are 

    S(𝐥_⊥)  = 4(p^-+k_γ^-) (q^--p^--k_γ^-) [q^+- M^2(𝐩_⊥+𝐤_γ⊥-𝐥_⊥) q^-/2(p^-+k^-_γ) (q^--k^-_γ-p^-)]   ,


    V(𝐥_⊥)     = 8p^- (q^--p^-) (q^--p^--k^-_γ) [q^+-M^2(𝐥_⊥-𝐩_⊥)q^-/2p^-(q^--p^-)]  
       ×[  q^+-k^+_γ-M^2(𝐥_⊥-𝐩_⊥)/2p^- -  M^2(𝐩_⊥+𝐤_γ⊥-𝐥_⊥)/2(q^--p^--k^-_γ)]    ,

and 

    W(𝐥_⊥)    = 8p^- (p^-+k^-_γ) (q^--p^--k^-_γ) [ q^+-M^2(𝐩_⊥+𝐤_γ⊥-𝐥_⊥)q^-/2(p^-+k^-_γ)(q^--p^-- k^-_γ) ] 
       ×[  q^+-k^+_γ-M^2(𝐥_⊥-𝐩_⊥)/2p^- -  M^2(𝐩_⊥+𝐤_γ⊥-𝐥_⊥)/2(q^--p^-- k^-_γ)]   .

The factors l_a and l_b in Eqs. <ref> and <ref> are obtained by evaluating l  at the enclosed poles depending on the choice of contours:  

    l_a    = γ^-( p^+-M^2(𝐥_⊥-𝐩_⊥)/2p^-) + 𝐥_⊥  , 
    l_b   = γ^-( p^++k^+_γ-q^++M^2(𝐩_⊥-𝐥_⊥+𝐤_γ⊥)/2(q^--p^--k^-_γ)) +𝐥_⊥  .




 §.§ Final expression for the amplitude

The net contribution from all the diagrams is obtained by adding the expressions in Eqs. <ref>, <ref> and <ref>.  Introducing a dummy integration ∫_𝐱_⊥∫_𝐤_⊥ e^-i𝐤_⊥.𝐱_⊥=1, this result can be expressed compactly as 

    ℳ_μα(𝐪, 𝐤,𝐩,𝐤_γ)    = ∑_β=1^10ℳ_μα^β (𝐪,𝐤,𝐩,𝐤_γ)=2π (eq_f)^2δ(P^--q^-)  ∫_𝐱_⊥∫_𝐲_⊥∫_𝐥_⊥ e^-i𝐏_⊥.𝐱_⊥+i𝐥_⊥.𝐱_⊥ e^-i 𝐥_⊥.𝐲_⊥
       ×u(𝐤) [ T^(qq̅)_μα (𝐥_⊥,𝐏_⊥)   [ Ũ(𝐱_⊥) Ũ^† (𝐲_⊥) -1 ] + ( T^(q)_μα (𝐏_⊥) - T^(qq̅)_μα (𝐥_⊥,𝐏_⊥) )  [ Ũ(𝐱_⊥) -1 ] 
        +  ( T^(q̅)_μα (𝐏_⊥) - T^(qq̅)_μα (𝐥_⊥,𝐏_⊥) )  [ Ũ^†(𝐲_⊥) -1 ]   ] v(𝐩)    .

This expression can be further simplified by observing that the following relations holds for the various R-factors given in the previous section,

    R^(1)_μα (𝐏_⊥) - R^(7)_μα (0_⊥,𝐏_⊥)    =0   ,
    
    R^(2)_μα (𝐏_⊥) - R^(9)_μα (0_⊥,𝐏_⊥)    =0    ,
    
    R^(3)_μα (𝐏_⊥) - (R^(8)_μα+ R^(10)_μα ) (0_⊥,𝐏_⊥)     =0   ,

which leads to 

    T^(q)_μα (𝐏_⊥) - T^(q q̅)_μα (0_⊥,𝐏_⊥)=0   .

Since the second term in the sum of the three terms in Eq. <ref> is independent of 𝐲_⊥, that sets 𝐥_⊥=0 for this term. Hence the identity in Eq. <ref> implies that the second term in Eq. <ref> vanishes.

There is an intuitive way of understanding this relation. For this, we identify that the transverse momentum kicks to the antiquark and quark lines are respectively 𝐥_⊥ and 𝐏_⊥-𝐥_⊥. In the limit of 𝐥_⊥ going to zero, diagrams (1) and (7) give identical contributions with the transverse momentum transfer being 𝐏_⊥ for both cases. The same argument holds for processes (2) and (9). For processes (8) and (10), the insertions on the antiquark line are on either side of the photon and hence it is their sum which equals the contribution from process (3) for vanishing 𝐥_⊥. The same argument holds for processes (4), (5) and (6) with the corresponding limit being 𝐥_⊥=𝐏_⊥ so that in this case the transverse momentum kick to the quark line vanishes. One thus gets

    R^(5)_μα (𝐏_⊥) - R^(8)_μα (𝐏_⊥,𝐏_⊥)    =0   , 
    
    R^(6)_μα (𝐏_⊥) - R^(10)_μα (𝐏_⊥,𝐏_⊥)    =0   , 
    
    R^(4)_μα (𝐏_⊥) - (R^(7)_μα+ R^(9)_μα ) (𝐏_⊥,𝐏_⊥)     =0   ,

leading to

    T^(q̅)_μα (𝐏_⊥) - T^(q q̅)_μα (𝐏_⊥,𝐏_⊥)=0   .

For the same reason articulated previously, this identity implies that the third term in Eq. <ref> vanishes as well. 
The relations in Eqs. <ref> and <ref> should be compared to the second line of Eq.  2.46 in <cit.> for the  p+A  NLO calculation. Such qualitative similarities will appear throughout the LO discussion in this paper.   

Since the second and third terms in Eq. <ref> vanish, the LO amplitude therefore reduces to the expression: 

    ℳ_μα(𝐪, 𝐤,𝐩,𝐤_γ) = ∑_β=1^10ℳ_μα^β (𝐪,𝐤,𝐩,𝐤_γ)     =2π (eq_f)^2δ(P^--q^-)  ∫_𝐱_⊥∫_𝐲_⊥∫_𝐥_⊥ e^-i𝐏_⊥.𝐱_⊥+i𝐥_⊥.𝐱_⊥ e^-i 𝐥_⊥.𝐲_⊥
       ×u(𝐤) [ T^(qq̅)_μα (𝐥_⊥,𝐏_⊥)   [ Ũ(𝐱_⊥) Ũ^† (𝐲_⊥) -1 ] ] v(𝐩)   .





§ THE INCLUSIVE PHOTON CROSS-SECTION AT LO
 
In proceeding to write down the expression for the inclusive photon differential cross-section, we note that since Eq. <ref> contains a delta function prefactor, the squared amplitude will naively contain a squared delta function. However this potential problem is resolved by realizing that the photon plane wave hitherto considered should be replaced by a properly normalized wave packet for the incoming virtual photon[A nice discussion of this very question can be found in Sec. IV of <cit.> and in pages 99-107 of <cit.>.].  Following a similar procedure to that described in  <cit.> , and working in a frame where the lepton and the nucleus are moving towards each other at near light speed, we can write the differential probability as

    dσ/dx   dQ^2 = 2 π y^2/64π^3 Q^2d^3𝐤/(2π)^3 2E_kd^3𝐩/(2π)^3 2E_pd^3𝐤_γ/(2π)^3 2E_k_γ1/2q^- ( 1/2∑_spins,
    λ⟨|∼ℳ|^2⟩_Y_A) (2π) δ(P^--q^-)   ,

where ⟨…⟩_Y_A is used to denote the CGC color average, defined in Eq. <ref>, of the modulus squared of the amplitude which is the prescription <cit.> used for inclusive processes. In this expression,  s=(l̃+P_N)^2=Q^2/xy=4EE_N is the e+A squared center-of-mass energy, where E is the incoming electron energy and E_N the energy per nucleon of the incoming nucleus. Further, x=Q^2/2 P_N· q and y=q· P_N/l̃.P_N respectively denote the  familiar DIS Lorentz invariant variables Bjorken x and the inelasticity. Finally, E_k, E_p and E_k_γ are the respective relativistic energies of the quark, antiquark and the outgoing photon. The amplitude squared, averaged over colors, spins and polarizations, can be expressed as

    1/2∑_spins,
    λ⟨|∼ℳ|^2⟩_Y_A=16 π^2α^2 q^4_f N_c   L^μν∼X_μν  ,

where L^μν is given by Eq. <ref> and

    ∼X_μν= ∫_𝐱_⊥,𝐲_⊥,𝐱'_⊥,𝐲'_⊥,𝐥_⊥,𝐥'_⊥e^-i (𝐏_⊥-𝐥_⊥).𝐱_⊥-i𝐥_⊥.𝐲_⊥+i(𝐏_⊥-𝐥'_⊥).𝐱'_⊥+i𝐥'_⊥.𝐲'_⊥ τ_μν^qq̅,qq̅(𝐥_⊥,𝐥'_⊥|𝐏_⊥)  Ξ(𝐱_⊥,𝐲_⊥;𝐱'_⊥,𝐲'_⊥)   .

In the above expression, the Dirac trace is written in the compact notation introduced in <cit.>:

    τ_μν^qq̅,qq̅(𝐥_⊥,𝐥'_⊥|𝐏_⊥) = Tr[ (k+m) T_ν^(qq̅)^α (𝐥_⊥,𝐏_⊥) (m-p) γ̂^0T^(qq̅)_μα^†(𝐥'_⊥,𝐏_⊥) γ̂^0]   .

The nonperturbative information on strongly correlated gluons in the nucleus is entirely contained in the term 

    Ξ(𝐱_⊥,𝐲_⊥;𝐱'_⊥,𝐲'_⊥)= 1-D(𝐱_⊥,𝐲_⊥) -D(𝐲'_⊥,𝐱'_⊥)+Q(𝐱_⊥,𝐲_⊥;𝐲'_⊥,𝐱'_⊥)   ,

where D  and Q are respectively the dipole and quadrupole Wilson line correlators defined as 

    D(𝐱_⊥,𝐲_⊥)    = 1/N_c⟨ Tr(Ũ(𝐱_⊥) Ũ^†(𝐲_⊥))⟩_Y_A  , 
    
    Q(𝐱_⊥,𝐲_⊥;𝐲'_⊥,𝐱'_⊥)   = 1/N_c⟨Tr(Ũ(𝐲'_⊥) Ũ^†(𝐱'_⊥) Ũ(𝐱_⊥) Ũ^†(𝐲_⊥) ) ⟩_Y_A  .

These are universal, gauge invariant quantities that appear in many processes in both p+A and e+A collisions <cit.>.

Be defining these correlators in terms of nuclear unintegrated distributions <cit.>, the expression for ∼X_μν can be given an explicit momentum space `look'. For this, we first introduce a dummy integration over 𝐥_1⊥ and a δ-function representing overall transverse momentum conservation, (2π)^2δ^(2)(𝐏_⊥-𝐥_1⊥). Now, define

    ∫_𝐱_⊥,𝐲_⊥,𝐱'_⊥,𝐲'_⊥e^-i(𝐥_1⊥-𝐥_⊥).𝐱_⊥ -i𝐥_⊥.𝐲_⊥ + i(𝐥_1⊥-𝐥'_⊥).𝐱'_⊥ + i𝐥'_⊥.𝐲'_⊥ D(𝐱_⊥,𝐲_⊥)
     =2α_SN_c/𝐥_1⊥^2ϕ_Y_A^D(𝐥_1⊥-𝐥_⊥,𝐥_⊥)   ,






    ∫_𝐱_⊥,𝐲_⊥,𝐱'_⊥,𝐲'_⊥e^-i(𝐥_1⊥-𝐥_⊥).𝐱_⊥ -i𝐥_⊥.𝐲_⊥ + i(𝐥_1⊥-𝐥'_⊥).𝐱'_⊥ + i𝐥'_⊥.𝐲'_⊥ Q(𝐱_⊥,𝐲_⊥;𝐲'_⊥,𝐱'_⊥)
     = 2α_SN_c/𝐥_1⊥^2ϕ_Y_A^Q(𝐥_1⊥-𝐥_⊥,𝐥_⊥; 𝐥_1⊥-𝐥'_⊥,𝐥'_⊥)   .

where ϕ^D_Y_A and ϕ^Q_Y_A should be contrasted with the correlators appearing in the p+A NLO calculation (see Eqs. 3.11-3.13 of <cit.>) for inclusive photon production. Eq. <ref> therefore becomes

    ∼X_μν=    ∫_𝐥_1⊥∫_𝐥_⊥,𝐥'_⊥ (2π)^2δ^(2)(𝐏_⊥-𝐥_1⊥) τ_μν^qq̅,qq̅(𝐥_⊥,𝐥'_⊥|𝐥_1 ⊥) 
       ×[ϕ_0 -2α_S/N_c𝐥_1⊥^2(ϕ_Y_A^D(𝐥_1⊥-𝐥_⊥,𝐥_⊥)+ ϕ_Y_A^D'(𝐥_1⊥-𝐥'_⊥,𝐥'_⊥)+  ϕ_Y_A^Q(𝐥_1⊥-𝐥_⊥,𝐥_⊥; 𝐥_1⊥-𝐥'_⊥,𝐥'_⊥) )   ]   ,

where

    ϕ_0=(2π)^8δ^(2)(𝐥_1⊥-𝐥_⊥)  δ^(2)(𝐥_⊥)  δ^(2)(𝐥_1⊥-𝐥'_⊥)  δ^(2)(𝐥'_⊥)     .


The form of Eq. <ref> becomes particularly helpful in the next section where we will obtain k_⊥ and collinear factorized limits of these results and compare them to the corresponding leading twist pQCD results.

Following <cit.>, we define the products d^6 K_⊥=d^2𝐩_⊥d^2𝐤_⊥d^2𝐤_γ⊥ and d^3η_K=dη_pdη_kdη_k_γ and write the final form of the triple differential cross-section for inclusive photon production at LO as

    dσ/dx d Q^2d^6 K_⊥d^3η_K= α^2q^4_fy^2N_c/512 π^5 Q^21/(2π)^41/2q^-  L^μν∼X_μν (2π) δ(P^--q^-)   ,

where L^μν and ∼X_μν are given by Eqs. <ref> and <ref> respectively.
In e+A DIS data, the above results can be applied to measurements of isolated photons accompanied by two jets or a quarkonium state, such as the J/Ψ meson. Alternatively, one may integrate over the quark or antiquark to obtain the differential cross-section for direct photon plus jet production. Such measurements were performed for e+p DIS at HERA <cit.> for a wide range of Q^2 and transverse energy of the final state photon E_γ⊥, and are also feasible at future electron-nucleus collider facilities. 

The single inclusive differential cross-section for inclusive prompt photon production. This is obtained by integrating over the quark and antiquark momenta and rapidities. We obtain

    dσ/dx dQ^2d^2𝐤_γ⊥dη_k_γ = α^2q^4_fy^2N_c/512 π^5 Q^21/2q^-∫_0^+ ∞dk^-/k^-∫_0^+ ∞dp^-/p^-∫_𝐤_⊥,𝐩_⊥ L^μν∼X_μν (2π) δ(P^--q^-)   ,

where ∼X_μν is given by Eq. <ref>.






§ PROPERTIES OF THE PHOTON PRODUCTION AMPLITUDE AT LO
 




 §.§ k_⊥-factorization and collinear factorization limits


In this section, we shall consider the large transverse momentum, 𝐥_1 ⊥ limit of the unintegrated distributions defined in Eqs. <ref> - <ref>. This is equivalent to expanding the Wilson line Ũ(𝐱_⊥) defined in Eq. <ref>, to lowest nontrivial order in ρ_A/∇^2_⊥ and using the relations

    ⟨ρ_A^a (x^-,𝐱_⊥) ρ_A^b (y^-,𝐲_⊥) ⟩ = δ^abδ(x^--y^-) δ^(2) (𝐱_⊥-𝐲_⊥) λ_A(x^-)   ,

where

    ∫d x^-λ_A(x^-) =μ^2_A ,

for Gaussian random color sources in a large nucleus  <cit.>. This follows from the arguments outlined in the McLerran-Venugopalan (MV) model describing the nonperturbative gluon distribution at the onset of quantum evolution. In the MV model, μ^2_A=A/2π R^2∼ A^1/3 is the average color charge squared of the valence quarks per color and per unit transverse area of a nucleus having mass number, A. Although there is no transverse coordinate dependence in μ^2_A in the MV model, explicit numerical solutions <cit.> of the Balitsky-JIMWLK hierarchy demonstrate that  a nonlocal μ^2_A→μ^2_A(𝐱_⊥) well-approximates these numerical solutions when   

    ϕ_Y_A(𝐥_1⊥)=2π N_cC_Fg^2/𝐥_1⊥^2∫_𝐱_⊥μ^2_A(Y_A,𝐱_⊥)   ,

satisfies the BK equation.  In the limit of large transverse momentum, ϕ_Y_A(𝐥_1⊥) evolves according to the BFKL equation <cit.>. 

Employing these ingredients, we obtain the following leading twist expression for the hadronic tensor:

    ∼X_μν^LT =2 α_S/N_cϕ_Y_A(𝐏_⊥)/𝐏_⊥^2 Θ_μν(𝐏_⊥)   ,

where

    Θ_μν(𝐏_⊥)= Tr[ (k+m) ( T^(q̅)(𝐏_⊥)-T^(q)(𝐏_⊥))_ν^α (m-p) γ̂^0( T^(q̅)^†(𝐏_⊥)-T^(q)^†(𝐏_⊥))_μαγ̂^0]

is obtained using Eqs. <ref> and <ref> respectively in the Dirac trace defined by Eq. <ref>.

An alternate way to arrive at the above expression is to compute the amplitude perturbatively. This can be done straightforwardly in Lorenz gauge by expanding the Wilson lines appearing in the amplitude expressions corresponding to processes (1)-(10). However the leading twist contribution is of 𝒪(ρ_A); therefore, processes (7)-(10) containing two Wilson line insertions will not be considered here. These leading twist diagrams are shown in Fig. <ref>.

The leading twist amplitude can be written as

    ℳ^LT(𝐪,𝐤,𝐩,𝐤_γ)= ϵ^* α(𝐤_γ,λ) ϵ^μ(𝐪,λ') ℳ^LT_μα (𝐪,𝐤,𝐩,𝐤_γ)    ,

where

    ℳ^LT_μα (𝐪,𝐤,𝐩,𝐤_γ)= A^ν_a,A(P-q) m^a_αμν (𝐪,𝐤,𝐩,𝐤_γ)   .

The Fourier transform of the gluon field is given by

    A^ν_a,A(P-q)= 2π g δ^ν +δ(P^--q^-)  ρ_a,A(𝐏_⊥)/𝐏_⊥^2  ,

and 

    m^a_αμν (𝐪,𝐤,𝐩,𝐤_γ) = ∑_β=1^6 m^β, a_αμν (𝐪,𝐤,𝐩,𝐤_γ)   ,

corresponds to the contributions from the leading twist diagrams (LT1)-(LT6), which can be individually written as

    m^β,a_αμ + (𝐪,𝐤,𝐩,𝐤_γ) =  
    
     -ie^2 g q^2_f u(𝐤) R^(β)_αμ (𝐏_⊥) t^a v(𝐩)   ,   β=1,2,3  
    
     + ie^2 g q^2_f u(𝐤) R^(β)_αμ (𝐏_⊥) t^a v(𝐩)   ,   β=4,5,6   ,

where the various R-factors are given by Eqs. <ref>, <ref> and <ref> respectively. Using these relations <ref> and <ref> in Eq. <ref>, we get 

    ℳ^LT_μα (𝐪,𝐤,𝐩,𝐤_γ)= -ie^2g^2q^2_f(2π) δ(P^--q^-) ρ_a,A(𝐏_⊥)/𝐏_⊥^2 [ u(𝐤) (T^(q̅)(𝐏_⊥)-T^(q)(𝐏_⊥) )_μαt^a v(𝐩)  ]    .


Taking the modulus squared and using 

    ⟨ρ_A^a (𝐏_⊥) ρ_A^b (𝐏_⊥) ⟩_Y_A = δ^ab𝐏_⊥^2/2π N_cC_Fg^2 ϕ_Y_A(𝐏_⊥)   ,

for the correlator of color sources, we obtain

    X^LT_μν   = - ⟨∑_spinsℳ_μα^LT^* (𝐪,𝐤,𝐩,𝐤_γ) ℳ_ν^LT^α (𝐪,𝐤,𝐩,𝐤_γ) ⟩ = 16 π^2α^2 N_c q^4_f ∼X_μν^LT  ,

where is identical to the expression in Eq. <ref>.

Finally, we can take the collinear limit of the leading twist expressions. In our case, this is equivalent to taking the limit 𝐏_⊥→ 0 of Θ_μν(𝐏_⊥) / 𝐏^2_⊥. In analogy to the discussion in the p+A case  <cit.>, we verify that the Ward identity 

    (P-q)^ν m^a_αμν (𝐪,𝐤,𝐩,𝐤_γ) = 0   ,

is indeed satisfied[This follows from the relations

    (P-q)^ν ( m^4,a_αμν )     = - (P-q)^ν ( m^2,a_αμν + m^3,a_αμν )   , 
    
    (P-q)^ν ( m^1,a_αμν )     = - (P-q)^ν ( m^5,a_αμν + m^6,a_αμν )   .
 This Ward identity should not be confused with that for the final state photon; the corresponding proof is given in Appendix  <ref>. ].
Since P^--q^-=0, the Ward identity implies

    m^a_αμ + = P_i m^a_αμ i/P^+-q^+  ,

so that m^a_αμ + vanishes linearly with 𝐏_⊥. Thus the limit

    lim_𝐏_⊥→ 0Θ_μν(𝐏_⊥)/𝐏^2_⊥  ,

is well-defined, leading to the collinear perturbative QCD result for the hadron tensor:

    ∼X_μν^coll. = 2 α_Sπ^2/N_c  (2π)^2δ^(2) (𝐩_⊥+𝐤_⊥+𝐤_γ⊥)   x_A f_g,A(x_A,Q^2) lim_𝐏_⊥→ 0Θ_μν(𝐏_⊥)/𝐏^2_⊥  .

Note that the l.h.s is proportional to the nuclear gluon distribution at small x,

    x_A f_g,A (x_A,Q^2) = 1/π^2∫_𝐥_1⊥ϕ_Y_A(𝐥_1 ⊥)    .



The above result should be compared to inclusive photon production at NLO or 𝒪(α_S) in the power counting of pQCD, first analyzed in detail by Aurenche et. al. <cit.> and later for photon plus jet cross-sections in <cit.>. Excluding photoproduction, the NLO contributions come from two classes of inelastic diagrams; the photon-gluon fusion processes with a prompt photon in the final state and bremsstrahlung processes with a valence quark interacting directly with the virtual photon.  In the power counting of CGC, the latter form a subset of the Class I processes. As clearly shown in Eq. (10) of <cit.>, the cross-section for photon-gluon fusion processes clearly dominate the other inelastic processes at small x by virtue of being proportional to the nucleon/nuclear gluon distribution in e+p/e+A inclusive direct photon production. 

This correspondence between leading twist approximation of CGC  and conventional collinearly factorized pQCD results is useful. Besides corroborating our framework in a well-defined limit, it allows one to extract subleading power corrections to the nuclear gluon distribution.  Further, one recovers higher order results (in the collinear pQCD power counting)  that are the dominant NLO pQCD corrections to inclusive prompt photon production in the limit when both energies and virtualities are large even though x ≪ 1. 



 §.§ Inclusive dijet cross-section in the soft photon limit
 
We will show here that in the infrared limit, k_γ→ 0, the Low-Burnett-Kroll theorem is satisfied by the inclusive photon production amplitude. Towards this end, we shall consider only the hadronic subprocesses for which the amplitude is given by Eqs. <ref> and <ref>. In this limit, only the processes (7) and (8) (see Fig. <ref>) correspond to the photon being radiated after scattering from the nucleus and hence possess the desired structures

    k+k_γ+m/(k+k_γ)^2-m^2  or  p+k_γ-m/(p+k_γ)^2-m^2  ,

where the denominators linearly diverge as k_γ→ 0

    (p+k_γ)^2-m^2→ 2p.k_γ ,      (k+k_γ)^2-m^2→ 2k.k_γ  ,

and the numerators stay finite

    u(𝐤) γ_α( k+k_γ+m)    →  2k_αu(𝐤)  ,  
    
    (p+k_γ-m) γ_α v(𝐩)    → 2p_α v(𝐩)   ,

The contributions from diagrams (9) and (10) are of 𝒪(k_γ^0) in the soft photon limit and are therefore subleading compared to those from diagrams (7) and (8). Combining the two leading contributions in the soft photon limit[As we shall show in the next subsection, and in greater detail in Appendix <ref>, the structures in diagrams (1)–(6) are automatically included in 
diagrams (7)–(10) by a slight modification of the effective vertex. It is therefore sufficient to consider here diagrams (7)–(10) alone.], we get 

    ℳ_μ (𝐪,𝐤, 𝐩, 𝐤_γ)  → - (eq_f) ϵ^* _α (𝐤_γ, λ) ( p^α/p.k_γ -k^α/k.k_γ) ℳ_μ^NR (𝐪,𝐤, 𝐩 )    ,

where ℳ_μ^NR (𝐪,𝐤, 𝐩 ) is the amplitude for inclusive dijet production computed in the CGC framework at small x in <cit.>. The explicit expression is given by Eq. <ref>. It is then straightforward to calculate the differential cross-section for inclusive dijet production and express it in the familiar dipole factorized form. Details of this computation are given in Appendix <ref>. 

The final result can be expressed as 

    dσ^L,T/d^3k d^3p    = α  q_f^2 N_cδ(q^--p^--k^-) ∫d^2𝐱_⊥/(2π)^2d^2𝐱'_⊥/(2π)^2d^2𝐲_⊥/(2π)^2d^2𝐲'_⊥/(2π)^2  e^-i𝐤_⊥.(𝐱_⊥-𝐱'_⊥)  e^-i𝐩_⊥.(𝐲_⊥-𝐲'_⊥)
       ×∑_α, βψ^L,T_αβ(q^-,z,|𝐱_⊥-𝐲_⊥| )  ψ^L,T *_αβ(q^-,z,|𝐱'_⊥-𝐲'_⊥| ) ×Ξ(𝐱_⊥,𝐲_⊥;𝐱'_⊥,𝐲'_⊥)   ,

where Ξ is given by Eq. <ref>. Our result exactly matches that obtained previously in <cit.>  (see Eq. (22) of the same)  for inclusive dijet production at small x. This is a versatile result, because as argued in <cit.>, measurements of this cross-section in e+A collisions provides a direct channel to extract the nuclear Weizsäcker-Williams distribution <cit.>. 



 §.§ The dressed fermion propagator
 
In this subsection, we will outline a convenient modification of the momentum space Feynman rules for the fermion propagator in the background classical color field which enables one to efficiently recover our results thus far. As we shall also shortly discuss, this modification is valid for the dressed gluon propagator as well.

We begin by observing that the factor of unity subtracted from the Wilson line in the effective vertex for the dressed fermion propagator (given in Eq. <ref>) imposes the requirement that the quark (antiquark) has to necessarily scatter off the nucleus. If we now redefine the effective vertex as

    ∼𝒯_ji(q,p)= ± (2 π)δ(p^--q^-) γ^-∫d^2𝐱_⊥ e^i(𝐪_⊥ - 𝐩_⊥).𝐱_⊥Ũ^(±)(𝐱_⊥)_ji  ,

where the plus (minus) signs again correspond respectively to insertions on quark (antiquark) line, this includes the possibility of “no scattering” off the quark or antiquark line.  The LO amplitude expression in Eq. <ref> then has a simple physical interpretation. The first part proportional to Ũ(𝐱_⊥) Ũ^† (𝐲_⊥) represents all possible scatterings of the quark and antiquark off the nucleus, including the case in which they do not scatter at all. We can represent this diagrammatically as the sum of contributions from processes labeled (7)-(10) in Fig. <ref> but with the blobs replaced by crossed dots as shown in Fig. <ref>.

The second part which is proportional to unity gives the “no scattering” contribution and must be subtracted to get the physical result.[This can be shown by using Eq. <ref> and the following relations

    R^(1)_μα(0_⊥) + R^(2)_μα(0_⊥)     = 1/q^+-P^++i ε/2k^-2k_α+γ_αk_γ/(k+k_γ)^2-m^2+iεγ_μ  , 
    
    R^(3)_μα(0_⊥)     =  1/q^+-P^++i ε/2k^-γ_μ-(2p_α+k_γγ_α)/(p+k_γ)^2-m^2+iε  .
 
to decompose the second part as the sum of two processes in which both quark and antiquark do not scatter off the nucleus. ]  
As we will demonstrate in further detail in Appendix <ref>, this seemingly simple modification offers a significant advantage because it is sufficient to compute fewer diagrams in the LO computation than previously. An identical argument holds for the effective vertex for the dressed gluon propagator. These modified quark and gluon vertices will prove very valuable in computations at NLO and higher orders.
 


§ COMPONENTS OF THE NLO COMPUTATION
 



 §.§ Advantages of the “wrong" light cone gauge

Albeit computations of physical observables are gauge independent, an appropriate gauge choice can result in a significant simplification of our path to the final result. In DIS, an attractive gauge choice, in the infinite momentum frame (IMF) where P^+_N→∞,  is the light cone gauge A^+=0 . This follows from the operator definition of parton distribution functions (PDFs), which are defined respectively for quark and gluon fields as <cit.>

    F^q   = 1/2∫dz^-/2 π  e^i xP^+_Nz^-⟨ P_N| q̅(-z/2) γ^+ W ( -z^-/2, z^-/2) q( z/2)  | P_N⟩|_z^+=0, 𝐳_⊥=0  , 
    
    F^g   = 1/P^+_N∫dz^-/2 π  e^i x P^+_N z^-⟨ P_N|   G^+ μ( -z/2) W( -z^-/2, z^-/2) G_μ^+(z/2)  | P_N⟩|_z^+=0, 𝐳_⊥=0  .

In the IMF, the W's are lightlike gauge links which are defined to be 

    W(a,b)= 𝒫_-exp( ig ∫_b^adx^- A^+)   .

These are of course unity in in A^+=0 gauge. In a covariant representation, with off-shell partons and on-shell hadrons, a simple interpretation of the PDFs in terms of Green functions is possible in this gauge <cit.> . Similarly, in a noncovariant representation with on-shell partons, or in other words, light cone quantization <cit.>, a number density interpretation of the quark and gluon PDFs is manifest in A^+=0 gauge.

Though A^+=0 is the right gauge choice for the reasons outlined, it turns out that the “wrong"  gauge choice of A^-=0 is the right one for computations in shock wave background fields. Firstly, we note that in the IMF P^+_N→∞,  the only large component of the current density is J^+ with other components suppressed by 1/P^+_N. The current conservation equation is therefore equivalent to 

    D_+ J^+ =0   .

It ensures that  J^+ is independent of the light cone time x^+, or equivalently, is a static charge on the light cone.  This is implicitly assumed in Eq. <ref> and in the  subsequent analysis. The solution of the classical Yang-Mills equations in A^-=0 gauge gives <cit.> 

    A^+=0,   F^ij=0     with  A^+, A^i   x^+  independent  ,

a solution that is also of course valid in the right light cone gauge A^+=0. 

However unlike A^+=0 gauge, A^-=0 gauge is related to the Lorenz gauge ∂_μ A^μ=0 by a simple gauge transformation. Note that our LO computations thus far were in the latter gauge, where the fermion propagator had a particularly simple structure <cit.>. Most importantly, as noted previously <cit.>, the structure of the small fluctuation propagator in A^-=0 gauge is far simpler than that in A^+=0 gauge. Further, this dressed gluon propagator, when transformed to Lorenz gauge, has a structure that is identical to that of the fermion propagator. 

Leading order computations in shock wave backgrounds, comparing and contrasting results in the Lorenz and A^-=0 light cone gauge have been performed previously in the context of p+A collisions. While it was noted that results for gluon production in A^-=0 gauge <cit.> are less cumbersome relative to those in Lorenz gauge <cit.>, there is no particular virtue of one or the other gauge for quark pair production <cit.> or inclusive photon production <cit.>. Because the shock wave background fields in the two gauges are simply related, intermediate steps may be performed more efficiently in Lorenz gauge before transforming the final result to A^-=0 gauge. 



 §.§ Gluon “shock wave” propagator in A^-=0 gauge


We will begin by rederiving <cit.> the shock wave gluon propagator in A^-=0 gauge.  We can split the gauge field into a classical background field piece (parametrically of order 1/g) and a small fluctuation piece (of order unity) as 

    A^μ(x)=B^μ(x)+b^μ(x)   .

Here B^μ (x) is the classical background field with the components

    B^+(x)=0,   B^-(x)=0,   B^i   =θ(x^-) κ^i(𝐱_⊥)  ,       i=1,2   , 
    where  κ^i (𝐱_⊥)   = i/g  U(𝐱_⊥) ∂^i U^†(𝐱_⊥)   ,

and U is the Wilson line in the adjoint representation of SU(N_c), 

    U(𝐱_⊥)=𝒫_-exp[-ig^2∫_-∞^+ ∞dz^-1/∇^2_⊥ρ_A^a (z^-,𝐱_⊥)T^a]   .

The equations of motion for the small fluctuation field b(x) are

    ( D(B)^2g^μν -D^μ (B) D^ν (B) ) b_ν - 2F^μν b_ν=0   .

Specifically, for the transverse fields, we have a sourceless Klein-Gordon equation

    [2∂^+∂^-- 𝐃_⊥^2(B)]   b^i(x)=0   ,

after fixing the residual gauge freedom with the Gauss' law constraint D^μ (B) b_μ=0 [The covariant derivative, D^μ(B)=∂^μ-igB^μ,aT^a]. Taking μ=- in Eq. <ref>, we get another sourceless equation 

    ∂^-( ∂^-b^+- 𝐃_⊥.𝐛_⊥)=0   ,

which is the Gauss' law constraint imposed on initial field configurations <cit.>. 

Finally, using the above constraint to write

    b^+=1/∂^- 𝐃_⊥.𝐛_⊥  ,

and keeping in mind the discontinuity of the spatial covariant derivative operator at x^-=0, 
we can verify that Eqs. <ref> and <ref> are consistent with the final small fluctuation equation

    [2∂^+∂^- - 𝐃_⊥^2(B) ]   b^+=-2δ(x^-)   κ^i (𝐱_⊥) b_i  .

The transverse field is then assumed to be a simple plane wave for x^-<0 and a linear superposition of plane waves for x^->0. This is done in a manner to ensure continuity of the fields across x^-=0, which again is possible because of the absence of delta function singularities in the above equations at x^-=0. The expressions for the Green's functions for the small fluctuations thus obtained are given in the next subsection.

In contrast, in the right light cone gauge A^+=0, the discontinuity of the fields at the origin and the corresponding singular structure of the electric fields do not allow one to simply  integrate the equations of motion. Hence it is convenient to first obtain the small fluctuation Green functions in A^-=0 gauge and then, if required,  gauge transform the results to A^+=0. The expressions for the propagator in A^+=0 gauge are complicated and not of relevance in our NLO computation.

Following the procedure outlined above and using the identity

    Ũ(𝐱_⊥) t^aŨ^† (𝐱_⊥)=t^b U^ba(𝐱_⊥)   ,

we can express the transverse components of the fluctuation field as <cit.>, 

    b^i(x) =e^ip.xη^i[ θ(-x^-)t^a+θ(x^-) ∫d^2𝐪_⊥/(2π)^2∫d^2𝐳_⊥  e^-i(𝐪_⊥-𝐩_⊥).(𝐱_⊥-𝐳_⊥) e^i(𝐪_⊥^2-𝐩_⊥^2)x^-/2 p^-  t^c(U(𝐱_⊥) U^†(𝐳_⊥) )^ca]   ,

where η^i is a unit vector.
Using the above expression for b^i, we get 

    b^+(x)=1/∂_y^+  [𝐃_⊥,𝐛_⊥]= 1/∂_y^+ [ e^ip.xη^j{θ(-x^-) ip^j t^a +    θ(x^-) ∫d^2𝐪_⊥/(2π)^2∫d^2𝐳_⊥  e^-i(𝐪_⊥-𝐩_⊥).(𝐱_⊥-𝐳_⊥) e^i(𝐪_⊥^2-𝐩_⊥^2)x^-/2 p^-
       × i(p^j+q^j)   t^c(U(𝐱_⊥) U^†(𝐳_⊥) )^ca}]   .

The small fluctuation gluon propagator in coordinate space is defined as

    G^μν;ab(x,y) = ⟨ b^μ,a(x) b^ν,b(y) ⟩  .

The nonzero components of the propagator in this gauge are G^++, G^+i, G^i+ and G^ij. Using the expression for G^ij in <cit.> and Eq. <ref>, it is straightforward to derive the following alternate expressions useful for momentum space calculations:

    G^ij;ab(x,y)    =g(x^-,𝐱_⊥)   G^ij;ab_0(x,y)   g^†(y^-,𝐲_⊥) -  ∫d^4 z  δ(z^-) g(x^-,𝐱_⊥) {θ(x^-) θ(-y^-) [U^†(𝐳_⊥)-1] 
         -θ(-x^-) θ(y^-) [U(𝐳_⊥)-1] }_cd g^†(y^-,𝐲_⊥)   G^ik;ac_0(x,z) (-i2∂_z^+) G^lj;db_0(z,y)  δ_kl  ,

where

    G_0^μν;ab(x,y)=∫d^4p/(2π)^4  e^ip.(x-y)i/p^2+iε( -g^μν+p^μ n^ν+p^νn^μ/n.p) δ^ab ,      n^μ=δ^μ +  ,

is the free gluon propagator in A^-=0 gauge.  Further,  

    g(x^-,𝐱_⊥)=θ(-x^-)+θ(x^-) U(𝐱_⊥)   ,

is a gauge transformation matrix that transforms the background field B^μ(x) to Lorenz gauge in which it is a singular field having only a `plus' component. The same matrix, albeit in the fundamental representation, appears while calculating the fermion propagator <cit.> in the background classical field.

The momentum space expression for the gluon propagator in A^- = 0 gauge with the background field B(x) in Lorenz gauge can be expressed as 

    G^ij;ab(p,p')=(2π)^4δ^(4)(p-p')    G^ij;ab_0(p)+G^ik;ac_0(p)  𝒯_ kl;cd(p,p')   G^lj;db_0(p')   ,

where

    𝒯_ ij;ab(p,p') =2πδ(p^--p'^-)  (2p^-) sign(p^-) ∫d^2𝐳_⊥  e^i(𝐩_⊥-𝐩'_⊥).𝐳_⊥[U^sign(p^-)(𝐳_⊥)-1]_abδ_ij  .

One can similarly derive the following momentum space relations  for the other nonzero components of G^μν:

    G^++;ab(p,p')    =(2π)^4δ^(4)(p-p')    G^++;ab_0(p) + G^+i;ac_0(p)  𝒯_ ij;cd(p,p')   G^j+;db_0(p')   , 
    
    G^i+;ab(p,p')    =(2π)^4δ^(4)(p-p')    G^i+;ab_0(p) + G^ij;ac_0(p)  𝒯_ jk;cd(p,p')   G^k+;db_0(p')   .

The general form for the small fluctuation propagator in momentum space and in A^-=0  gauge can be compactly expressed as 

    G^μν;ab(p,p')=(2π)^4δ^(4)(p-p')    G^μν;ab_0(p)+G^μρ;ac_0(p)  𝒯_ ρσ;cd(p,p')   G^σν;db_0(p')    ,

where the effective vertex of the dressed gluon shock wave is given by

    𝒯_ μν;ab(p,p') =-2πδ(p^--p'^-) (2p^-) g_μν sign(p^-) ∫d^2𝐳_⊥  e^i(𝐩_⊥-𝐩'_⊥).𝐳_⊥[U^sign(p^-)(𝐳_⊥)-1]_ab  ,

a form identical to (with the subsitution of the fundamental Wilson line with its adjoint counterpart) the dressed fermion effective vertex. 
While the gluon shock wave propagator in A^-=0 gauge was first derived in <cit.>, this form for the propagator in momentum space is given in a later paper by Balitsky and Belitsky <cit.> albeit it is also implicit in <cit.>. 

The discussion in Sec. <ref> however suggests that we should modify the effective gluon vertex to include the “no scattering" contribution by omitting the unit matrix in Eq. <ref> above; the modified vertex we will use henceforth in computations is 

    ∼𝒯_ μν;ab(p,p') =-2πδ(p^--p'^-)   (2p^-) g_μν sign(p^-) ∫d^2𝐳_⊥  e^i(𝐩_⊥-𝐩'_⊥).𝐳_⊥U^sign(p^-)_ab (𝐳_⊥)   .









To summarize, higher order all-twist computations at small x can be performed with the conventional Feynman rules of covariant perturbation theory albeit with the dressed 
quark and gluon  effective vertices shown in Fig. <ref>. 




 §.§ Processes contributing to the NLO amplitude
 
Having outlined the necessary components for the computation of the inclusive photon amplitude to NLO, we will discuss here the relevant NLO processes. The general ideas were already presented in the introduction of this paper and we will elaborate further on that discussion. To begin with,  the perturbative expansion in Eq. <ref> of the expectation value of a general operator 𝒪 calculated for a particular source charge density configuration ρ_A can be expanded out to next-to-next-to-leading order (NNLO) as 

    𝒪[ρ_A]   = f^0_0j (gρ_A)^j_LO +f^0_1j(gρ_A)^jα_S +f^1_1j(gρ_A)^jα_S ln (1/x) _NLO
       +f^0_2j(gρ_A)^jα_S^2 +f^1_2j(gρ_A)^jα_S^2 ln(1/x) +f^2_2j(gρ_A)^jα^2_S ln^2 (1/x) _NNLO +…  ,

where the repeated indices are assumed to be summed over. For the differential cross-section for inclusive photon production, at each order in perturbation theory, there are contributions to the small x evolution of the random color sources (encoded in the dipole and quadrupole Wilson line correlators given by Eq. <ref>). These are enhanced by logarithms in x and when α_Sln(1/x)∼ 1, they complicate the naive power counting. In addition, we will have non-log enhanced perturbative corrections to the light cone wavefunction for the virtual photon that fluctuates into a quark-antiquark dipole  (as well as ancillary complications due to the photon in the final state) that should be combined with the ln(1/x) contributions systematically. It is therefore efficient to reexpress the generic expansion in  Eq. <ref> by factorizing the cross-section into a coefficient term and a matrix element of Wilson line correlators between nuclear wavefunctions, at each order in α_S, as 

    dσ [ρ_A]   = 𝒞_LO⊗dσ̂_LO_α_S^0.α_S^0 +𝒞_LO⊗dσ̂_LLx_α_S^0. α_Sln(1/x) + 𝒞_NLO⊗dσ̂_NLO_α_S.α_S^0
        + 𝒞_NLO⊗dσ̂_LLx_α_S. α_Sln(1/x) +𝒞_LO⊗dσ̂_NNLx_α^0_S .α^2_Sln(1/x) +…  .


We computed the first term in the r.h.s of the above equation in the previous sections. The second term is given by contributions from the diagrams of the type depicted in Fig. <ref>. For α_S ln(1/x) ∼ 1, these contributions are as large as the LO contribution and can be absorbed in the LO result by a redefinition of the weight functional W. The resulting LLx RG equation (the JIMWLK equation) efficiently sums such leading contributions to all orders of perturbation theory while preserving the structure of the LO result. 

The third term appearing in Eq. <ref> represents genuine α_S suppressed contributions that do not take into account the small x evolution of the sources. In addition to the real emission diagrams shown in Figs. <ref> and  <ref>, there are contributions from loop graphs depicted by the second and third representative diagrams in Fig. <ref>. Since the cross-section for inclusive gluon production in DIS was computed previously <cit.> at small x, taking the soft-photon limit to the real emission contributions will provide in principle a check of our results. The complete calculation of this term will give the next order correction 𝒞_NLO to the LO coefficient function or “impact factor" 𝒞_LO. It is worth mentioning here that the σ̂_NLO has a different correlator structure than σ̂_LO due to the additional adjoint Wilson lines and hence the distinct nomenclature. The overall contribution of the third term in Eq. <ref> is strongly subleading at small x. 



The two terms in the second line of Eq. <ref> are part of the NNLO contributions to inclusive photon production but are potentially quantitatively important because they are effectively of order α_S when α_Sln(1/x)∼ 1; these form the next-to-leading-log (NLLx) contribution to the cross-section. Under our RG ideology, the two terms can be considered separately. The first term in the second line of  Eq. <ref> is represented by the diagrams in Fig. <ref>. The 
dashed horizontal line represents the rapidity scale Y_0 separating the classical fields that interact with the qq̅γ system from the static color sources RG evolved by the LLx JIMWLK equation from the target. The two diagrams shown are respectively the real and virtual contributions that are characteristic of the terms that generate the JIMWLK kernel. The computation of the NLO impact factor (above the Y_0 cut) using the techniques developed here is in progress and will be reported in a follow-up paper <cit.>. A highly nontrivial check will be  to reproduce the NLO impact factor for fully inclusive DIS <cit.> that should be recovered in the soft photon limit. It will also be important for the consistency of the framework to demonstrate JIMWLK factorization by explicit computation. 

The final term in Eq. <ref> is represented by the diagrams in Fig. <ref>. In this case, there are no radiative corrections above the rapidity cut and the dynamics is described by the LO impact factor. The contributions shown below the cut correspond to NLO contributions to the JIMWLK kernel. These have been computed previously <cit.> (see also <cit.>) and can therefore be used to construct the NLLx result for inclusive photon production in e+A collisions. We note that NLLx corrections have recently been implemented for numerical computations in fully inclusive DIS <cit.>.

An advantage of our approach is that it is fully implemented in momentum space; many shock wave computations use a mixture of momentum and coordinate space variables that is cumbersome. This is especially so in computing running coupling contributions <cit.> where coordinate space prescriptions can be problematic as noted in <cit.>. In our framework, there is no need to switch back and forth; all computations can be realized fully in momentum space with our modified Feynman rules. These issues will be be addressed in future work. 
 






§ SUMMARY AND OUTLOOK


We presented in this paper a first computation of inclusive photon production in e+A DIS at small x within the CGC EFT. At LO, the cross-section is directly proportional to 
universal gauge invariant dipole and quadrupole Wilson line correlators which are ubiquitous in final states that are measured in high energy p+A collisions and potentially in e+A collisions at a future Electron-Ion Collider (EIC). Indeed, since inclusive photon production in DIS has photons in both the initial and final states, it holds promise of being a clean  golden channel for unambiguous discovery of gluon saturation, complementary to other e+A DIS measurements <cit.>. In the soft photon limit, we recover the inclusive DIS dijet cross-section derived previously in  <cit.>. As argued there, this dijet e+A channel may provide direct access to the nuclear Weizsäcker-Williams gluon distribution. 

We next discussed the structure of dominant small x contributions  to the inclusive photon cross-section at next-to-leading order  and next-to-next-to-leading order. The essential ingredients here are the dressed quark and gluon propagators and the corresponding effective vertices in the shock wave classical background field of a nucleus at high energies. These effective vertices are proportional to the respective fundamental and adjoint Wilson lines that carry information about all-twist gluon correlations in the nucleus. The structure of the quark and gluon dressed propagators is remarkably simple in the “wrong" light cone gauge A^-=0 and therefore permits efficient higher order computations. The computations are further simplified by exploiting the natural separation between static sources and dynamical gauge fields in the MV model and the JIMWLK RG treatment thereof. 
In particular, the cross-section can be factorized into impact factor contributions that are convoluted with the RG evolution of products of lightlike Wilson lines. The nontrivial ingredients that need to be computed are the NLO impact factor for inclusive photon production and the NLLx RG evolution of the Balitsky-JIMWLK hierarchy of Wilson line correlators. While the latter is known, the former needs to be determined; these computations are in progress and will be reported on separately <cit.>. 

We note that, motivated by collider experiments, small x computations in the gluon saturation regime are increasingly to next-to-leading order accuracy. Some examples of the studies being performed include, besides inclusive DIS <cit.>, DIS diffractive dijet production <cit.> and exclusive light vector meson production  <cit.>, single inclusive forward hadron production in p+p and p+A collisions <cit.> and more recently inclusive photon production in p+A collisions <cit.>. 
Looking further ahead, we believe that the momentum space methods discussed here can exploited to make progress in these and related computations. 

The forms of the shock wave propagators first derived in <cit.>, and their expressions in terms of effective vertices <cit.>, are identical to the quark-quark-reggeon and gluon-gluon-reggeon propagators <cit.> in Lipatov's reggeon field theory <cit.>. We have shown here that the slightly modified quark and gluon effective vertices in Eqs. <ref> and <ref> significantly simplify the computation of DIS inclusive photon production. 
Because of the form of the shock wave propagators, our results are equally valid for leading twist or all-twist computations. It would therefore be interesting to see if our modified Feynman rules are useful in simplifying multiloop leading twist computations in the Regge limit <cit.> or perhaps, conversely and more interestingly, results derived in those cases applied to advance computations in the saturation regime of high parton densities. 




R.V would like to thank Andrey Tarasov for a useful discussion.
This material is based on work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under Contracts No. DE-SC0012704 and within the framework TMD Theory Topical Collaboration. K. R is supported by an LDRD grant from Brookhaven Science Associates. 





§ APPENDIX



§ NOTATIONS AND CONVENTIONS
 
The metric used is the -2 metric, ĝ=diag(+1,-1,-1,-1), where the `carat' denotes quantities in usual spacetime coordinates. The light cone coordinates are defined as 

    x^+=x̂^0+x̂^3/√(2),    x^-=x̂^0-x̂^3/√(2)  ,

with the transverse coordinates remaining the same and transforming as in Minkowski space. The same definition holds for the gamma matrices γ^+ and γ^- with the Dirac algebra given by

    {γ^μ,γ^ν}=2g^μν  ,

where g^+-=g^-+=1 and g^ij=-δ^ij (i,j=1,2) are the nonzero entries of the metric tensor. In this convention, a.b=a^+b^-+a^-b^+-𝐚_⊥.𝐛_⊥ and a_+=a^-, a_i=-a^i.



§ GAUGE INVARIANCE AND SOFT-PHOTON FACTORIZATION AT LO
 
The first part of this appendix provides a proof of the Ward identity for the final state real photon, thus establishing gauge invariance. In the second part, we will provide an explicit expression  for the nonradiative DIS amplitude recovered using the soft photon limit, k_γ→ 0. We will next derive the differential cross-section for inclusive dijet production in DIS for the case of an incoming longitudinally polarized virtual photon. The case of the transversely polarized virtual photon follows identically and will therefore not be shown.



 §.§ Ward identity
 
The Ward identity for the final state photon requires that

    k^α_γℳ_μα (𝐪, 𝐤,𝐩,𝐤_γ) = 0    .

Towards this end, we make the replacement ϵ^* α(𝐤_γ, λ) → k^α_γ in the amplitude and check the effects of contracting  k^α_γ with T^(q q̅) (𝐥_⊥,𝐏_⊥). Instead of using the final forms for the various R-factors constituting T^(qq̅), we use their original forms in which the contour integrations are not performed. This simplifies the calculation. For example, we have

    u(𝐤) k^α_γ R^(7)_μα (𝐥_⊥, 𝐏_⊥) v(𝐩) = i^3∫dl^+/2π u(𝐤) γ^-q-p+l+m/(q-p+l)^2-m^2+iεγ_μlγ^--2p^-/(l-p)^2-m^2+iε v(𝐩)   .

We can use the second identity in Eq. <ref> and the relation

    u(𝐤) k_γ (k+m)= (2k.k_γ) u(𝐤)   ,

to derive Eq. <ref>. Likewise, we can write

    u(𝐤) k^α_γ R^(9)_μα (𝐥_⊥, 𝐏_⊥) v(𝐩)     = i^3∫dl^+/2π u(𝐤) γ^-(q-p+l+m) k_γ(q-p+l+m) /(q-p+l-k_γ)^2-m^2+iε
       ×γ_μlγ^--2p^-/[(q-p+l)^2-m^2+iε] [(l-p)^2-m^2+iε]  v(𝐩)   .

Adding Eqs. <ref> and <ref>, and using the fact that p^2-m^2= (p+m) (p-m), it is easy to show that one of the terms in square brackets in the denominator cancels out, giving

    u(𝐤) k^α_γ   [   R^(7)_μα (𝐥_⊥, 𝐏_⊥) + R^(9)_μα (𝐥_⊥, 𝐏_⊥) ] v(𝐩)  
        = i^3∫dl^+/2π u(𝐤) γ^-q-p+l -k_γ+m/(q-p+l-k_γ)^2-m^2+iεγ_μlγ^- - 2p^-/(l-p)^2-m^2+iε v(𝐩)   .

Analogously, we can use the first identity in Eq. <ref> and 

    (p - m) k_γ v(𝐩)= (2p.k_γ) v(𝐩) ,

to write 

    u(𝐤) k^α_γ R^(8)_μα (𝐥_⊥, 𝐏_⊥) v(𝐩) = i^3∫dl^+/2π u(𝐤) γ^-q-p+l-k_γ+m/(q-p+l-k_γ)^2-m^2+iεγ_μlγ^--2p^--k_γγ^-/(l-p-k_γ)^2-m^2+iε v(𝐩)   ,

and 

    u(𝐤) k^α_γ R^(10)_μα (𝐥_⊥, 𝐏_⊥) v(𝐩)     = i^3∫dl^+/2π u(𝐤) γ^-q-p+l-k_γ+m/(q-p+l-k_γ)^2-m^2+iεγ_μl-p+m/(l-p)^2-m^2+iε
       ×k_γlγ^--2p^-/(l-p-k_γ)^2-m^2+iε v(𝐩)   .

Adding the expressions in Eqs. <ref> and <ref>, and using a similar reasoning used to simplify the previous sum, we get

    u(𝐤) k^α_γ[   R^(7)_μα (𝐥_⊥, 𝐏_⊥) + R^(9)_μα (𝐥_⊥, 𝐏_⊥) ] v(𝐩) = - u(𝐤) k^α_γ[   R^(8)_μα (𝐥_⊥, 𝐏_⊥) + R^(10)_μα (𝐥_⊥, 𝐏_⊥) ] v(𝐩)    .

This implies that 

    u(𝐤) k^α_γ T^(q q̅)_μα (𝐥_⊥, 𝐏_⊥) v(𝐩) = 0

thereby showing that the Ward identity is indeed satisfied for the outgoing photon. 

It can also be shown that the same relation holds for the case of the exchanged virtual photon if we are considering the hadronic subprocess. This gives us the freedom to consider only the -g^μν part of the photon propagator which is implicitly assumed in writing Eq. <ref>.




 §.§ Soft photon factorization
 
By taking the soft photon limit k_γ→ 0 of our LO amplitude expression in Eq. <ref>, we can recover the nonradiative DIS amplitude given by

    ℳ^NR_μ (𝐪,𝐤, 𝐩 )  = 2π (eq_f) δ(P^--q^-) ∫_𝐱_⊥∫_𝐲_⊥∫_𝐥_⊥ e^-i𝐏_⊥.𝐱_⊥+i𝐥_⊥.𝐱_⊥ e^-i𝐥_⊥.𝐲_⊥
       × - 1/2q^- u(𝐤) γ^-zq^-γ^+- (𝐩_⊥ - 𝐥_⊥) +m /z(1-z) Q^2+M^2(𝐥_⊥-𝐩_⊥)γ_μ (2p^-+γ^-𝐥_⊥) ×( Ũ(𝐱_⊥) Ũ^†(𝐲_⊥) -1 )_ij  v(𝐩)   ,

where P now equals p+k.

We will now show that the above expression can be used to factorize the amplitude squared in terms of products of light cone wavefunctions and a dipole scattering factor. In the soft photon limit, the amplitude for the subprocess 

    γ^*(q) → Q(k) + Q(p) + γ(k_γ)

can be written as

    ℳ(𝐪,𝐤, 𝐩, 𝐤_γ;λ,λ') = ϵ^μ (𝐪,λ') ℳ_μ (𝐪,𝐤, 𝐩, 𝐤_γ)   ,

where ϵ^μ (𝐪,λ') is the polarization vector for the incoming virtual photon and ℳ_μ is given by Eq. <ref>. In order to identify the transverse and longitudinally polarized photon wavefunctions, it is convenient to parametrize the polarization vectors as

    ϵ_T(𝐪,λ'=+1)    = ( 0,0,-1/√(2),-i/√(2))   , 
    ϵ_T(𝐪,λ'=-1)    = ( 0,0,1/√(2),-i/√(2))   , 
    ϵ_L(𝐪,λ'=0)    = ( Q/2q^-,q^-/Q,0,0 )   ,

where T and L stand for transverse and longitudinal respectively. These vectors satisfy the relations

    ϵ^2_T(𝐪,λ') =-1,   ϵ^2_L(𝐪,λ') =1,   ϵ_T(𝐪,λ') ϵ_T^*(𝐪,λ”)= -δ_λ',λ”  , 
        g_μν -q_μq_ν/q^2= - ∑_λ'= ± 1ϵ_μ T(𝐪,λ') ϵ^*_ν T(𝐪,λ') + ϵ_μ L(𝐪,λ'=0) ϵ^*_ν L(𝐪,λ'=0)   .


We will now explicitly compute the cross-section for the longitudinally polarized case.  With the above choice of the λ'=0 polarization vector, Eq. <ref> gives

    ℳ^NR (𝐪,𝐤, 𝐩;λ'=0 ) =ϵ^μ_L(𝐪,λ'=0) ℳ^NR_μ(𝐪,𝐤, 𝐩 ) = 
    2π (eq_f)  δ(P^--q^-) ∫_𝐱_⊥∫_𝐲_⊥∫_𝐥_⊥ e^-i𝐏_⊥.𝐱_⊥+i𝐥_⊥.𝐱_⊥ e^-i𝐥_⊥.𝐲_⊥
       × - 1/2q^- u(𝐤) γ^-zq^-γ^+- (𝐩_⊥ - 𝐥_⊥) +m /z(1-z) Q^2+M^2(𝐥_⊥-𝐩_⊥)(Q/2q^-γ^-+q^-/Qγ^+) (2p^-+γ^-𝐥_⊥) ×( Ũ(𝐱_⊥) Ũ^†(𝐲_⊥) -1 )_ij  v(𝐩)   .

Redefining 𝐥_⊥-𝐩_⊥→𝐥_⊥ and using the property <ref> for gamma matrices along with the Dirac equation, we can simplify the above result to

    ℳ^NR (𝐪,𝐤, 𝐩;λ'=0 )     = 2π (eq_f)  δ(P^--q^-)  ∫_𝐱_⊥∫_𝐲_⊥∫_𝐥_⊥ e^-i𝐤_⊥.𝐱_⊥-i𝐩_⊥.𝐲_⊥ e^i𝐥_⊥.(𝐱_⊥-𝐲_⊥)
       ×( Ũ(𝐱_⊥) Ũ^†(𝐲_⊥) -1 )_ij [ -2z(1-z)Q/𝐥_⊥^2+ϵ_q^2 +1/Q] u(𝐤) γ^- v(𝐩)   ,

where ϵ_q^2=z(1-z)Q^2+m^2. The term proportional to 1/Q vanishes because of the δ^(2)(𝐱_⊥-𝐲_⊥) arising from the integration over 𝐥_⊥ and the identity Ũ(𝐱_⊥) Ũ^† (𝐱_⊥)=1.


Finally using the formula

    ∫d^2𝐥_⊥/(2π)^2e^i𝐥_⊥.(𝐱_⊥-𝐲_⊥)/𝐥_⊥^2+ϵ_q^2 =1/2π K_0(ϵ_q|𝐱_⊥-𝐲_⊥| )   ,

where K_0 is the modified Bessel function of the second kind, we can recast ℳ^NR as

    ℳ^NR(𝐪,𝐤, 𝐩;λ') =  2πδ(P^--q^-) ℳ̃^NR (𝐪,𝐤, 𝐩;λ')   ,

where

    ℳ̃^NR (𝐪,𝐤, 𝐩;λ'=0)    = (eq_f) ∫_𝐱_⊥∫_𝐲_⊥ e^-i𝐤_⊥.𝐱_⊥-i𝐩_⊥.𝐲_⊥ K_0 (ϵ_q|𝐱_⊥-𝐲_⊥| )/2π( Ũ(𝐱_⊥) Ũ^†(𝐲_⊥) -1 )_ij 
       ×(-2z(1-z)Q) u(𝐤) γ^- v(𝐩)   .


The differential cross-section for inclusive dijet production in DIS is given by

    dσ/d^3k d^3p = 1/2q^-1/(2π)^32E_k1/(2π)^32E_p⟨|∼ℳ^NR|^2⟩_Y_A (2π) δ(P^--q^-)  ,

where the δ(0^-) appearing in the amplitude squared is normalized as described earlier in Sec. <ref>. Now using

    Tr[(k+m) γ^- (p-m) γ̂^0γ^+γ̂^0]=(2p^-) (2k^-)   ,

and the form of the λ'=0 photon wavefunction given in <cit.>

    ψ_αβ^L (q^-,z,r)= 2π√(4/q^-) z(1-z)Q K_0(ϵ_qr) δ_αβ  ,

it is a matter of straightforward algebra to show that 

    dσ^L/d^3k d^3p    = N_cα q_f^2δ(q^--p^--k^-) ∫d^2𝐱_⊥/(2π)^2d^2𝐱'_⊥/(2π)^2d^2𝐲_⊥/(2π)^2d^2𝐲'_⊥/(2π)^2  e^-i𝐤_⊥.(𝐱_⊥-𝐱'_⊥)  e^-i𝐩_⊥.(𝐲_⊥-𝐲'_⊥)
       ×∑_α, βψ^L_αβ(q^-,z,|𝐱_⊥-𝐲_⊥| )  ψ^L *_αβ(q^-,z,|𝐱'_⊥-𝐲'_⊥| ) ×[ 1-1/N_c⟨Tr( Ũ(𝐱_⊥) Ũ^†(𝐲_⊥) ) ⟩_Y_A
        -1/N_c⟨Tr( Ũ(𝐲'_⊥) Ũ^†(𝐱'_⊥) ) ⟩_Y_A +1/N_c⟨Tr( Ũ(𝐲'_⊥) Ũ^†(𝐱'_⊥) Ũ(𝐱_⊥) Ũ^†(𝐲_⊥) ) ⟩_Y_A]   .

This result exactly matches Eq. (22) of <cit.> obtained for DIS dijet production at small x. The case of the transversely polarized photon proceeds in a similar fashion.




§ KINEMATICALLY ALLOWED PROCESSES
 
In this appendix, we will explicitly demonstrate the topologies of the LO and NLO Feynman diagrams that are allowed by the kinematics of the process. The techniques described in this section are quite general and can be extended to find kinematically allowed diagrams beyond NLO.

Let us first consider the most general diagram (see Fig. <ref>) for the LO process with all dressed fermion lines and photon emission from the quark line.  We will now show it is possible to find all allowed processes starting from this generic template provided we use the modified Feynman rules discussed in Sec. <ref>. An identical treatment follows for the case of photon emission from the antiquark line.

The amplitude for the hadronic subprocess is given by

    ℳ_μα=-(eq_f)^2∫_l_1,l_2u(𝐤)    𝒯_im(k-l_1,k) S_0(k-l_1) γ_αS_0(k+k_γ-l_1) 𝒯_mn(q+l_2-p,k+k_γ-l_1) 
       × S_0(q+l_2-p) γ_μ S_0(l_2-p)𝒯_nj(-p,l_2-p)v(𝐩)   ,

where ∫_l_i=∫d^4l_i / (2π)^4, i=1,2 and 𝒯 is given by Eq. <ref>.

Integrating out l_1^- and l_2^- using the delta functions embedded in the vertex factors, we are left with the delta function δ(q^--P^-) representing the overall longitudinal momentum conservation and integrations over transverse spatial and momentum coordinates. However the quantity of interest is the integral over l_1^+ and l_2^+ given by

    ℐ=∫dl_1^+∫dl_2^+ N/D  ,
 
where 

    N=u(𝐤) γ^- (k-l_1+m)γ_α (k+k_γ-l_1+m)γ^-(q+l_2-p+m)γ_μ (l_2-p+m) γ^- v(𝐩)   ,

and 

    D    = ( k^+-l_1^+-M^2(𝐤_⊥-𝐥_1⊥)/2k^-+iε/2k^-) ( k^++k^+_γ-l_1^+-M^2(𝐤_⊥+𝐤_γ⊥-𝐥_1⊥)/2(k^-+k_γ^-)+iε/2(k^-+k_γ^-)) 
       ×( q^++l_2^+-p^+-M^2(𝐥_1⊥-𝐩_⊥)/2(k^-+k_γ^-)+iε/2(k^-+k^-_γ)) ( l_2^+-p^++M^2(𝐥_1⊥-𝐩_⊥)/2p^--iε/2p^-)   ,

denote the numerator and denominator respectively. Examining the structure of the poles in the propagator terms of the denominator, there are two l_1^+ poles on the positive side of the real axis and independently, two l_2^+ poles on either side of the real axis. Using the property (γ^-)^2=0, it is easy to see that the numerator does not contain any term proportional to l_1^+ or l_2^+. Therefore the contour for the integration over l_1^+ can be closed below the real axis thereby giving a null result. 

The arguments presented thus far does not invoke the possibility of “no scattering” in our definition of the effective vertices or equivalently the Ũ=1 case. It can be shown easily that for either Ũ_im=δ_im or Ũ_mn=δ_mn in the vertex factors appearing in Eq. <ref>, we will get a nonzero result from the above contour integration. Under these conditions, the two factors appearing in the first line of Eq. <ref> resemble energy denominators that appear in light cone perturbation theory (LCPT) <cit.> and the integration over l_2^+ can be done using the residue theorem. Corresponding to these conditions on the Ũ's, we get the two allowed diagrams with photon emission from the quark line as shown in Fig. <ref>; these are embedded in our generic diagram Fig. <ref>.



One can therefore start with the generic case and eventually deduce the impossibility of a secondary scattering subsequent to emission of the photon by an already scattered fermion. This is a consequence of the eikonal approximation in which the nucleus moving at near light speed interacts instantaneously with the quarks thereby removing the possibility of a second scattering. Once the allowed diagrams are computed, we simply need to deduct the contribution in which both quark and antiquark propagate unscattered. The latter have the same magnitude as the allowed diagrams modulo the Wilson line factors. The net contribution is given by Eq. <ref>.

The same physical principle discussed above for LO also applies to the NLO diagrams which are classified into three broad categories. For the diagrams contributing to the LLx and NLLx JIMWLK evolution, the allowed topologies can be easily extracted from existing literature[We should mention here that our diagrammatic representation is similar in spirit to the “shock wave" approach used in these works.]  <cit.>. Since the upper part of these diagrams has the same structure as LO diagrams, the rules discussed in the previous section apply trivially. In the following, we therefore discuss only the genuine α_S suppressed contributions in Figs. <ref>.

We consider one such representative generic diagram (see  Fig. <ref>) which represents real emission of a gluon from the quark or antiquark in addition to the final state photon. We use the line of reasoning made for the LO case to deduce the allowed processes. The amplitude for this subprocess is given by


    ℳ_μα;a    =-i(eq_f)^2g ( ∏_k=1^4∫_l_k)  u(𝐤) 𝒯_im (k-l_1,k) S_0(k-l_1) γ_α S_0(k+k_γ-l_1) 𝒯_mn(k+k_γ-l_1-l_2,k+k_γ-l_1) 
       × S_0(k+k_γ-l_1-l_2)( t^b)_npγ_βS_0(k+k_γ-l_1-l_2+l_3) 𝒯_pq(q+l_4-p,k+k_γ-l_1-l_2+l_3) 
       ×  S_0(q+l_4-p) γ_μ𝒯_qj(-p,l_4-p) v(𝐩) × G_0^βν;bc(l_3) 𝒯_νρ;ca(l_3,k_g) ϵ^ρ^*(𝐤_g)   ,

where the vertex factors for the fermion and gluon propagators are given respectively by Eqs. <ref> and <ref>.



 By carefully integrating out the l_i^-'s (i=1,…,4) using the δ-functions, the amplitude can be cast in terms of a momentum conserving delta function δ(q^--p^--k^--k^-_γ-k^-_g), integrations over transverse spatial and momentum coordinates and the following integral of interest.

    ℐ_1=( ∏_k=1^4∫dl_k^+)  N_1/D_1


where 

    N_1    = u(𝐤) γ^- [γ^+k^--γ_⊥.(𝐤_⊥-𝐥_1⊥)+m]γ_α[γ^+(k^-+k^-_γ)-γ_⊥.(𝐤_⊥+𝐤_γ⊥-𝐥_1⊥)+m]γ^-
       × [γ^+(k^-+k^-_γ)-γ_⊥.(𝐤_⊥+𝐤_γ⊥-𝐥_1⊥-𝐥_2⊥)+m] γ_β  [γ^+(k^-+k^-_γ+k_g^-)-γ_⊥.(𝐤_⊥+𝐤_γ⊥-𝐥_1⊥-𝐥_2⊥+𝐥_3⊥)+m] 
       ×γ^- [γ^+(k^-+k^-_γ+k_g^-)-γ_⊥.(𝐥_4⊥-𝐩_⊥)+m]γ_μ [γ^+p^-+γ_⊥.(𝐥_4⊥-𝐩_⊥)-m]γ^-( -ϵ^β^*(𝐤_g)+(l_3.ϵ^*(𝐤_g))n^β/k^-_g)   ,

is obtained using n.ϵ^*(𝐤_g)=0 and (γ^-)^2=0 and

    D_1   = ( k^+-l_1^+-M^2(𝐤_⊥-𝐥_1⊥)/2k^-+iε/2k^-) ( k^++k_γ^+-l_1^+-M^2(𝐤_⊥+𝐤_γ⊥-𝐥_1⊥)/2(k^-+k^-_γ)+iε/2(k^-+k^-_γ)) 
       ×( k^++k^+_γ-l_1^+-l^+_2-M^2(𝐤_⊥+𝐤_γ⊥-𝐥_1⊥-𝐥_2⊥)/2(k^-+k^-_γ)+iε/2(k^-+k^-_γ)) (l^+_3-𝐥_3⊥^2/2k^-_g+iε/2k^-_g) 
       ×( k^++k^+_γ-l_1^+-l^+_2+l_3^+ -M^2(𝐤_⊥+𝐤_γ⊥-𝐥_1⊥-𝐥_2⊥+𝐥_3⊥)/2(k^-+k^-_γ+k^-_g)+iε/2(k^-+k^-_γ+k^-_g)) 
       ×( q^++l^+_4-p^+-M^2(𝐥_4⊥-𝐩_⊥)/2(k^-+k^-_γ+k^-_g)+iε/2(k^-+k^-_γ+k^-_g))( l^+_4-p^++M^2(𝐥_4⊥-𝐩_⊥)/2p^--iε/2p^-)   .

 The expressions above clearly demonstrate that the numerator doesn't have any term proportional to l_1^+, l_2^+ or l_3^+ and the poles for all these variables are on the same side of the real axis. Hence the integration contours for any such variable can be deformed in a way so as not to enclose any pole giving a result of zero for general U and Ũ's depicted in Fig. <ref>. However it can be easily shown that for the following three cases
 
 
  * Ũ(𝐱_1)=Ũ(𝐱_3)=1 with rest of the Wilson lines being general (including the identity matrix),
 
  * Ũ(𝐱_2)=Ũ(𝐱_3)=1 with rest of the Wilson lines being general and
 
  * Ũ(𝐱_1)=Ũ(𝐱_2)=1, U(𝐱_5)=1 and remaining two being general, 
  
we get a finite result. Diagrammatically, this corresponds to the processes shown respectively in Fig. <ref> or equivalently the diagrams in Fig. <ref> in the “shock wave" approach. Similar arguments can be applied to find the allowed virtual graphs at NLO.














121
fxundefined [1]
 ifx#1

fnum [1]
 #1firstoftwo
 secondoftwo
 
fx [1]
 #1firstoftwo
 secondoftwo
 





noop [0]secondoftwo

ref[1]@startlink#1@href
href[1]#1@endlink
anitize@url [0]`
12`$12`&12`#12`1̂2`_12`%12
startlink[1]
endlink[0]

rl [1]href #1












@bib@innerbibempty

[Breitweg et al.(2000)Breitweg et al.]Breitweg:1999su
  author author J. Breitweg et al. (collaboration ZEUS), title title Measurement of inclusive prompt photon
  photoproduction at HERA, 10.1016/S0370-2693(99)01450-1
  journal journal Phys. Lett. volume B472, pages 175–188 (year 2000),  http://arxiv.org/abs/hep-ex/9910045 arXiv:hep-ex/9910045 [hep-ex]
  NoStop

[Aktas et al.(2005)Aktas
  et al.]Aktas:2004uv
  author author A. Aktas et al. (collaboration H1), title
  title Measurement of prompt photon cross sections in
  photoproduction at HERA, 10.1140/epjc/s2004-02085-x
  journal journal Eur. Phys. J. volume C38, pages 437–445 (year
  2005), http://arxiv.org/abs/hep-ex/0407018
  arXiv:hep-ex/0407018 [hep-ex] NoStop

[Chekanov et al.(2007)Chekanov et al.]Chekanov:2006un
  author author S. Chekanov et al. (collaboration ZEUS), title title Measurement of prompt photons with
  associated jets in photoproduction at HERA, 10.1140/epjc/s10052-006-0134-1 journal journal
  Eur. Phys. J. volume C49, pages
  511–522 (year 2007), http://arxiv.org/abs/hep-ex/0608028 arXiv:hep-ex/0608028 [hep-ex]
  NoStop

[Chekanov et al.(2004)Chekanov et al.]Chekanov:2004wr
  author author S. Chekanov et al. (collaboration ZEUS), title title Observation of isolated high E(T)
  photons in deep inelastic scattering, 10.1016/j.physletb.2004.05.033 journal journal
  Phys. Lett. volume B595, pages
  86–100 (year 2004), http://arxiv.org/abs/hep-ex/0402019 arXiv:hep-ex/0402019 [hep-ex]
  NoStop

[Aaron et al.(2008)Aaron
  et al.]Aaron:2007aa
  author author F. D. Aaron et al. (collaboration H1), title
  title Measurement of isolated photon production in
  deep-inelastic scattering at HERA, 10.1140/epjc/s10052-008-0541-6 journal journal
  Eur. Phys. J. volume C54, pages
  371–387 (year 2008), http://arxiv.org/abs/0711.4578 arXiv:0711.4578 [hep-ex] NoStop

[Chekanov et al.(2010)Chekanov et al.]Chekanov:2009dq
  author author S. Chekanov et al. (collaboration ZEUS), title title Measurement of isolated photon
  production in deep inelastic ep scattering, 10.1016/j.physletb.2010.02.045 journal journal
  Phys. Lett. volume B687, pages
  16–25 (year 2010), http://arxiv.org/abs/0909.4223
  arXiv:0909.4223 [hep-ex] NoStop

[Abramowicz et al.(2012)Abramowicz et al.]Abramowicz:2012qt
  author author H. Abramowicz et al. (collaboration ZEUS), title title Measurement of isolated photons
  accompanied by jets in deep inelastic ep scattering, 10.1016/j.physletb.2012.07.031 journal journal
  Phys. Lett. volume B715, pages
  88–97 (year 2012), http://arxiv.org/abs/1206.2270
  arXiv:1206.2270 [hep-ex] NoStop

[Gehrmann-De Ridder et al.(2006)Gehrmann-De Ridder, Gehrmann,  and Poulsen]Gehrmann-DeRidder:2006zbx
  author author A. Gehrmann-De Ridder, author T. Gehrmann,  and author E. Poulsen, title title Isolated
  photons in deep inelastic scattering, 10.1103/PhysRevLett.96.132002 journal journal
  Phys. Rev. Lett. volume 96, pages
  132002 (year 2006), http://arxiv.org/abs/hep-ph/0601073 arXiv:hep-ph/0601073 [hep-ph]
  NoStop

[Schmidt et al.(2016)Schmidt, Pumplin, Stump, and  Yuan]Schmidt:2015zda
  author author Carl Schmidt, author Jon Pumplin,
  author Daniel Stump,  and  author C. P. Yuan, title title CT14QED parton distribution functions
  from isolated photon production in deep inelastic scattering, 10.1103/PhysRevD.93.114015 journal journal Phys. Rev. volume D93, pages
  114015 (year 2016), http://arxiv.org/abs/1509.02905 arXiv:1509.02905 [hep-ph] NoStop

[Gribov et al.(1983)Gribov,
  Levin, and Ryskin]Gribov:1984tu
  author author L. V. Gribov, author E. M. Levin,   and author M. G. Ryskin,  title title Semihard Processes in
  QCD, 10.1016/0370-1573(83)90022-4 journal
  journal Phys. Rept. volume 100,  pages 1–150 (year 1983)NoStop

[Mueller and Qiu(1986)]Mueller:1985wy
  author author Alfred H.  Mueller and author Jian-wei  Qiu, title title Gluon
  Recombination and Shadowing at Small Values of x, 10.1016/0550-3213(86)90164-1 journal journal
  Nucl. Phys. volume B268, pages
  427–452 (year 1986)NoStop

[McLerran and Venugopalan(1994a)]McLerran:1993ni
  author author Larry D.  McLerran and author Raju  Venugopalan, title title
  Computing quark and gluon distribution functions for very large nuclei,  10.1103/PhysRevD.49.2233 journal journal Phys. Rev. volume D49, pages
  2233–2241 (year 1994a), http://arxiv.org/abs/hep-ph/9309289 arXiv:hep-ph/9309289 [hep-ph]
  NoStop

[McLerran and Venugopalan(1994b)]McLerran:1993ka
  author author Larry D.  McLerran and author Raju  Venugopalan, title title
  Gluon distribution functions for very large nuclei at small transverse
  momentum, 10.1103/PhysRevD.49.3352 journal journal Phys. Rev. volume
  D49, pages 3352–3355 (year
  1994b), http://arxiv.org/abs/hep-ph/9311205
  arXiv:hep-ph/9311205 [hep-ph] NoStop

[McLerran and Venugopalan(1994c)]McLerran:1994vd
  author author Larry D.  McLerran and author Raju  Venugopalan, title title
  Green's functions in the color field of a large nucleus, 10.1103/PhysRevD.50.2225 journal journal Phys. Rev. volume D50, pages
  2225–2233 (year 1994c), http://arxiv.org/abs/hep-ph/9402335 arXiv:hep-ph/9402335 [hep-ph]
  NoStop

[Iancu and Venugopalan(2003)]Iancu:2003xm
  author author Edmond Iancu and author Raju Venugopalan, title title The Color
  glass condensate and high-energy scattering in QCD, in 10.1142/9789812795533_0005 booktitle In *Hwa, R.C. (ed.)
  et al.: Quark gluon plasma* 249-3363 (year 2003) http://arxiv.org/abs/hep-ph/0303204 arXiv:hep-ph/0303204 [hep-ph]
  NoStop

[Gelis et al.(2010)Gelis,
  Iancu, Jalilian-Marian, and Venugopalan]Gelis:2010nm
  author author Francois  Gelis, author Edmond  Iancu, author Jamal Jalilian-Marian,  and author Raju  Venugopalan, title title
  The Color Glass Condensate, 10.1146/annurev.nucl.010909.083629 journal journal Ann. Rev. Nucl. Part. Sci. volume 60,  pages 463–489 (year 2010), http://arxiv.org/abs/1002.0333 arXiv:1002.0333 [hep-ph] NoStop

[Kovchegov and Levin(2012)]Kovchegov:2012mbw
  author author Yuri V. Kovchegov and author Eugene Levin, http://www.cambridge.org/de/knowledge/isbn/item6803159
  title Quantum chromodynamics at high energy,  Vol. volume 33 (publisher Cambridge University
  Press, year 2012)NoStop

[Blaizot(2017)]Blaizot:2016qgz
  author author Jean-Paul  Blaizot, title title
  High gluon densities in heavy ion collisions, 10.1088/1361-6633/aa5435 journal journal Rept.
  Prog. Phys. volume 80, pages 032301
  (year 2017), http://arxiv.org/abs/1607.04448
  arXiv:1607.04448 [hep-ph] NoStop

[Fubini and Furlan(1965)]Fubini:1964boa
  author author S. Fubini and author G. Furlan, title title Renormalization
  effects for partially conserved currents, @noop  journal journal Physics volume 1,  pages 229–247 (year 1965)NoStop

[Bardakci and Halpern(1968)]Bardakci:1969dv
  author author K. Bardakci and author M. B. Halpern, title title Theories at
  infinite momentum, 10.1103/PhysRev.176.1686 journal journal Phys. Rev. volume
  176, pages 1686–1699 (year 1968)NoStop

[Weinberg(1966)]Weinberg:1966jm
  author author Steven Weinberg, title title Dynamics at
  infinite momentum, 10.1103/PhysRev.150.1313 journal journal Phys. Rev. volume
  150, pages 1313–1318 (year 1966)NoStop

[Susskind(1968)]Susskind:1967rg
  author author Leonard Susskind, title title Model of
  selfinduced strong interactions, 10.1103/PhysRev.165.1535 journal journal Phys.
  Rev. volume 165, pages 1535–1546
  (year 1968)NoStop

[Jeon and Venugopalan(2004)]Jeon:2004rk
  author author Sangyong  Jeon and author Raju  Venugopalan, title title
  Random walks of partons in SU(N(c)) and classical representations of color
  charges in QCD at small x, 10.1103/PhysRevD.70.105012
  journal journal Phys. Rev. volume D70, pages 105012 (year 2004),  http://arxiv.org/abs/hep-ph/0406169 arXiv:hep-ph/0406169 [hep-ph]
  NoStop

[Jalilian-Marian et al.(1997)Jalilian-Marian, Kovner, Leonidov, and  Weigert]JalilianMarian:1997jx
  author author Jamal Jalilian-Marian, author Alex Kovner, author Andrei Leonidov,  and author Heribert Weigert, title title The BFKL
  equation from the Wilson renormalization group, 10.1016/S0550-3213(97)00440-9 journal journal
  Nucl. Phys. volume B504, pages
  415–431 (year 1997), http://arxiv.org/abs/hep-ph/9701284 arXiv:hep-ph/9701284 [hep-ph]
  NoStop

[Jalilian-Marian et al.(1998)Jalilian-Marian, Kovner, and Weigert]JalilianMarian:1997dw
  author author Jamal Jalilian-Marian, author Alex Kovner,  and author Heribert Weigert, title title The Wilson
  renormalization group for low x physics: Gluon evolution at finite parton
  density, 10.1103/PhysRevD.59.014015 journal journal Phys. Rev. volume
  D59, pages 014015 (year 1998), http://arxiv.org/abs/hep-ph/9709432 arXiv:hep-ph/9709432 [hep-ph]
  NoStop

[Iancu et al.(2001a)Iancu, Leonidov,  and McLerran]Iancu:2001ad
  author author Edmond Iancu, author Andrei Leonidov,  and author Larry D. McLerran, title title The
  Renormalization group equation for the color glass condensate, 10.1016/S0370-2693(01)00524-X journal journal Phys. Lett. volume B510, pages 133–144 (year 2001a), http://arxiv.org/abs/hep-ph/0102009 arXiv:hep-ph/0102009 [hep-ph]
  NoStop

[Iancu et al.(2001b)Iancu, Leonidov,  and McLerran]Iancu:2000hn
  author author Edmond Iancu, author Andrei Leonidov,  and author Larry D. McLerran, title title Nonlinear
  gluon evolution in the color glass condensate. 1. 10.1016/S0375-9474(01)00642-X journal journal
  Nucl. Phys. volume A692, pages
  583–645 (year 2001b), http://arxiv.org/abs/hep-ph/0011241 arXiv:hep-ph/0011241 [hep-ph]
  NoStop

[Ferreiro et al.(2002)Ferreiro, Iancu, Leonidov, and  McLerran]Ferreiro:2001qy
  author author Elena Ferreiro, author Edmond Iancu, author Andrei Leonidov,  and author Larry McLerran, title title Nonlinear
  gluon evolution in the color glass condensate. 2. 10.1016/S0375-9474(01)01329-X journal journal
  Nucl. Phys. volume A703, pages
  489–538 (year 2002), http://arxiv.org/abs/hep-ph/0109115 arXiv:hep-ph/0109115 [hep-ph]
  NoStop

[Weigert(2005)]Weigert:2005us
  author author Heribert  Weigert, title title
  Evolution at small x(bj): The Color glass condensate, 10.1016/j.ppnp.2005.01.029 journal journal Prog.
  Part. Nucl. Phys. volume 55, pages
  461–565 (year 2005), http://arxiv.org/abs/hep-ph/0501087 arXiv:hep-ph/0501087 [hep-ph]
  NoStop

[Balitsky(1996)]Balitsky:1995ub
  author author I. Balitsky, title title Operator
  expansion for high-energy scattering, 10.1016/0550-3213(95)00638-9 journal journal
  Nucl. Phys. volume B463, pages
  99–160 (year 1996), http://arxiv.org/abs/hep-ph/9509348 arXiv:hep-ph/9509348 [hep-ph]
  NoStop

[Weigert(2002)]Weigert:2000gi
  author author Heribert  Weigert, title title
  Unitarity at small Bjorken x, 10.1016/S0375-9474(01)01668-2 journal journal
  Nucl. Phys. volume A703, pages
  823–860 (year 2002), http://arxiv.org/abs/hep-ph/0004044 arXiv:hep-ph/0004044 [hep-ph]
  NoStop

[Kovchegov(1999)]Kovchegov:1999yj
  author author Yuri V. Kovchegov, title title Small x
  F(2) structure function of a nucleus including multiple pomeron
  exchanges, 10.1103/PhysRevD.60.034008 journal journal Phys. Rev. volume
  D60, pages 034008 (year 1999), http://arxiv.org/abs/hep-ph/9901281 arXiv:hep-ph/9901281 [hep-ph]
  NoStop

[Rummukainen and Weigert(2004)]Rummukainen:2003ns
  author author Kari Rummukainen and author Heribert  Weigert, title title
  Universal features of JIMWLK and BK evolution at small x, 10.1016/j.nuclphysa.2004.03.219 journal journal Nucl. Phys. volume A739, pages 183–226 (year 2004), http://arxiv.org/abs/hep-ph/0309306 arXiv:hep-ph/0309306 [hep-ph]
  NoStop

[Dumitru et al.(2011)Dumitru, Jalilian-Marian, Lappi,
  Schenke, and Venugopalan]Dumitru:2011vk
  author author Adrian Dumitru, author Jamal Jalilian-Marian, author Tuomas Lappi, author Bjoern Schenke,
   and author Raju Venugopalan,  title title Renormalization group
  evolution of multi-gluon correlators in high energy QCD, 10.1016/j.physletb.2011.11.002 journal journal
  Phys. Lett. volume B706, pages
  219–224 (year 2011), http://arxiv.org/abs/1108.4764 arXiv:1108.4764 [hep-ph] NoStop

[Benic et al.(2017)Benic,
  Fukushima, Garcia-Montero, and  Venugopalan]Benic:2016uku
  author author Sanjin Benic, author Kenji Fukushima, author Oscar Garcia-Montero,  and author Raju  Venugopalan, title title
  Probing gluon saturation with next-to-leading order photon production at
  central rapidities in proton-nucleus collisions, 10.1007/JHEP01(2017)115 journal journal JHEP  volume 01, pages 115 (year
  2017), http://arxiv.org/abs/1609.09424 arXiv:1609.09424
  [hep-ph] NoStop

[Gelis and Jalilian-Marian(2002)]Gelis:2002ki
  author author Francois  Gelis and author Jamal  Jalilian-Marian, title title Photon production in high-energy proton nucleus collisions,  10.1103/PhysRevD.66.014021 journal journal Phys. Rev. volume D66, pages
  014021 (year 2002), http://arxiv.org/abs/hep-ph/0205037 arXiv:hep-ph/0205037 [hep-ph]
  NoStop

[Dominguez et al.(2011)Dominguez, Marquet, Xiao, and  Yuan]Dominguez:2011wm
  author author Fabio Dominguez, author Cyrille Marquet, author Bo-Wen Xiao,
   and author Feng Yuan,  title title Universality of
  Unintegrated Gluon Distributions at small x, 10.1103/PhysRevD.83.105005 journal journal Phys.
  Rev. volume D83, pages 105005
  (year 2011), http://arxiv.org/abs/1101.0715
  arXiv:1101.0715 [hep-ph] NoStop

[Gribov et al.(1966)Gribov,
  Ioffe, and Pomeranchuk]Gribov:1965hf
  author author V. N. Gribov, author B. L. Ioffe,   and author I. Ya. Pomeranchuk, title title What is
  the range of interactions at high-energies, @noop  journal journal Sov. J. Nucl. Phys. volume 2, pages 549 (year 1966), note [Yad. Fiz.2,768(1965)]NoStop

[Ioffe(1969)]Ioffe:1969kf
  author author B. L. Ioffe, title title Space-time
  picture of photon and neutrino scattering and electroproduction cross-section
  asymptotics, 10.1016/0370-2693(69)90415-8 journal journal Phys. Lett. volume
  30B, pages 123–125 (year 1969)NoStop

[Bjorken et al.(1971)Bjorken, Kogut, and Soper]Bjorken:1970ah
  author author J. D. Bjorken, author John B. Kogut,  and author Davison E. Soper, title title Quantum
  Electrodynamics at Infinite Momentum: Scattering from an External Field,  10.1103/PhysRevD.3.1382 journal journal Phys. Rev. volume D3, pages
  1382 (year 1971)NoStop

[Nikolaev and Zakharov(1991)]Nikolaev:1990ja
  author author Nikolai N.  Nikolaev and author B. G.  Zakharov, title title
  Color transparency and scaling properties of nuclear shadowing in deep
  inelastic scattering, 10.1007/BF01483577 journal journal Z. Phys. volume
  C49, pages 607–618 (year 1991)NoStop

[Benic and Fukushima(2017)]Benic:2016yqt
  author author Sanjin Benic and author Kenji Fukushima, title title Photon from
  the annihilation process with CGC in the pA collision, 10.1016/j.nuclphysa.2016.11.003 journal journal
  Nucl. Phys. volume A958, pages 1–24
  (year 2017), http://arxiv.org/abs/1602.01989
  arXiv:1602.01989 [hep-ph] NoStop

[Altinoluk et al.(2018)Altinoluk, Armesto, Kovner, Lublinsky, and Petreska]Altinoluk:2018uax
  author author Tolga Altinoluk, author Néstor Armesto, author Alex Kovner,
  author Michael Lublinsky,  and  author Elena Petreska,  title title Soft photon and two hard
  jets forward production in proton-nucleus collisions, @noop    (year 2018), http://arxiv.org/abs/1802.01398
  arXiv:1802.01398 [hep-ph] NoStop

[Dokshitzer(1977)]Dokshitzer:1977sg
  author author Yuri L. Dokshitzer, title title Calculation
  of the Structure Functions for Deep Inelastic Scattering and e+ e-
  Annihilation by Perturbation Theory in Quantum Chromodynamics.  @noop  journal journal Sov. Phys. JETP  volume 46, pages 641–653 (year 1977), note [Zh. Eksp. Teor.
  Fiz.73,1216(1977)]NoStop

[Gribov and Lipatov(1972)]Gribov:1972ri
  author author V. N. Gribov and author L. N. Lipatov, title title Deep inelastic
  e p scattering in perturbation theory, @noop  journal journal Sov. J. Nucl. Phys. volume 15, pages 438–450 (year 1972),  note [Yad. Fiz.15,781(1972)]NoStop

[Altarelli and Parisi(1977)]Altarelli:1977zs
  author author Guido Altarelli and author G. Parisi, title title Asymptotic
  Freedom in Parton Language, 10.1016/0550-3213(77)90384-4 journal journal
  Nucl. Phys. volume B126, pages
  298–318 (year 1977)NoStop

[Abe et al.(1993)Abe et al.]Abt:1993cb
  author author I. Abe et al. (collaboration H1), title
  title Measurement of the proton structure function F2
  (x, Q^2) in the low x region at HERA, 10.1016/0550-3213(93)90090-C journal journal
  Nucl. Phys. volume B407, pages
  515–538 (year 1993)NoStop

[Ahmed et al.(1995)Ahmed
  et al.]Ahmed:1995fd
  author author T. Ahmed et al. (collaboration H1), title
  title A Measurement of the proton structure function
  f2 (x, Q^2), 10.1016/0550-3213(95)98236-U
  journal journal Nucl. Phys. volume B439, pages 471–502 (year 1995),  http://arxiv.org/abs/hep-ex/9503001 arXiv:hep-ex/9503001 [hep-ex]
  NoStop

[Derrick et al.(1993)Derrick
  et al.]Derrick:1993fta
  author author M. Derrick et al. (collaboration ZEUS), title title Measurement of the proton structure
  function F2 in ep scattering at HERA, 10.1016/0370-2693(93)90347-K journal journal
  Phys. Lett. volume B316, pages
  412–426 (year 1993)NoStop

[Derrick et al.(1995)Derrick
  et al.]Derrick:1994sz
  author author M. Derrick et al. (collaboration ZEUS), title title Measurement of the proton structure
  function F2 from the 1993 HERA data, 10.1007/BF01556128
  journal journal Z. Phys. volume C65, pages 379–398 (year
  1995)NoStop

[Martin et al.(1994)Martin,
  Stirling, and Roberts]Martin:1994kn
  author author Alan D. Martin, author W. James Stirling,  and author R. G. Roberts, title title Parton
  distributions of the proton, 10.1103/PhysRevD.50.6734
  journal journal Phys. Rev. volume D50, pages 6734–6752 (year 1994),  http://arxiv.org/abs/hep-ph/9406315 arXiv:hep-ph/9406315 [hep-ph]
  NoStop

[Lai et al.(1995)Lai,
  Botts, Huston, Morfin,
  Owens, Qiu, Tung, and  Weerts]Lai:1994bb
  author author H. L. Lai, author J. Botts, author J. Huston, author
  J. G. Morfin, author
  J. F. Owens, author
  Jian-wei Qiu, author
  W. K. Tung,  and author
  H. Weerts, title title Global QCD analysis and the CTEQ parton
  distributions, 10.1103/PhysRevD.51.4763 journal journal Phys. Rev. volume
  D51, pages 4763–4782 (year 1995), http://arxiv.org/abs/hep-ph/9410404 arXiv:hep-ph/9410404 [hep-ph]
  NoStop

[Gelis and Venugopalan(2006a)]Gelis:2006yv
  author author Francois  Gelis and author Raju  Venugopalan, title title
  Particle production in field theories coupled to strong external
  sources, 10.1016/j.nuclphysa.2006.07.020 journal journal Nucl. Phys. volume
  A776, pages 135–171 (year
  2006a), http://arxiv.org/abs/hep-ph/0601209
  arXiv:hep-ph/0601209 [hep-ph] NoStop

[Gelis and Venugopalan(2006b)]Gelis:2006cr
  author author Francois  Gelis and author Raju  Venugopalan, title title
  Particle production in field theories coupled to strong external sources.
  II. Generating functions, 10.1016/j.nuclphysa.2006.08.015 journal journal
  Nucl. Phys. volume A779, pages
  177–196 (year 2006b), http://arxiv.org/abs/hep-ph/0605246 arXiv:hep-ph/0605246 [hep-ph]
  NoStop

[Gelis et al.(2007)Gelis,
  Lappi, and Venugopalan]Gelis:2007kn
  author author Francois  Gelis, author Tuomas  Lappi,  and author Raju  Venugopalan, title title
  High energy scattering in Quantum Chromodynamics, booktitle booktitle Hadron physics. Proceedings, 10th
  International Workshop, Florianopolis, Brazil, April 26-31, 2007, 10.1142/S0218301307008331 journal journal Int. J. Mod. Phys. volume E16, pages 2595–2637 (year 2007), http://arxiv.org/abs/0708.0047 arXiv:0708.0047 [hep-ph] NoStop

[Gelis et al.(2008a)Gelis, Lappi,  and Venugopalan]Gelis:2008rw
  author author Francois  Gelis, author Tuomas  Lappi,  and author Raju  Venugopalan, title title
  High energy factorization in nucleus-nucleus collisions, 10.1103/PhysRevD.78.054019 journal journal Phys. Rev. volume D78, pages
  054019 (year 2008a), http://arxiv.org/abs/0804.2630 arXiv:0804.2630 [hep-ph] NoStop

[Gelis et al.(2008b)Gelis, Lappi,  and Venugopalan]Gelis:2008ad
  author author Francois  Gelis, author Tuomas  Lappi,  and author Raju  Venugopalan, title title
  High energy factorization in nucleus-nucleus collisions. II. Multigluon
  correlations, 10.1103/PhysRevD.78.054020 journal journal Phys. Rev. volume
  D78, pages 054020 (year 2008b),  http://arxiv.org/abs/0807.1306 arXiv:0807.1306 [hep-ph]
  NoStop

[Balitsky and Chirilli(2013)]Balitsky:2013fea
  author author Ian Balitsky and author Giovanni A.  Chirilli, title title
  Rapidity evolution of Wilson lines at the next-to-leading order, 10.1103/PhysRevD.88.111501 journal journal Phys. Rev. volume D88, pages
  111501 (year 2013), http://arxiv.org/abs/1309.7644
  arXiv:1309.7644 [hep-ph] NoStop

[Grabovsky(2013)]Grabovsky:2013mba
  author author A. V. Grabovsky, title title Connected
  contribution to the kernel of the evolution equation for 3-quark Wilson loop
  operator, 10.1007/JHEP09(2013)141 journal
  journal JHEP volume 09, pages 141 (year 2013), http://arxiv.org/abs/1307.5414 arXiv:1307.5414 [hep-ph] NoStop

[Kovner et al.(2014)Kovner,
  Lublinsky, and Mulian]Kovner:2013ona
  author author Alex Kovner, author Michael Lublinsky,  and author Yair Mulian, title title
  Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner evolution at
  next to leading order, 10.1103/PhysRevD.89.061704
  journal journal Phys. Rev. volume D89, pages 061704 (year 2014),  http://arxiv.org/abs/1310.0378 arXiv:1310.0378 [hep-ph]
  NoStop

[Kovchegov and Weigert(2007)]Kovchegov:2006vj
  author author Yuri V. Kovchegov and author Heribert Weigert, title title Triumvirate of
  Running Couplings in Small-x Evolution, 10.1016/j.nuclphysa.2006.10.075 journal journal
  Nucl. Phys. volume A784, pages
  188–226 (year 2007), http://arxiv.org/abs/hep-ph/0609090 arXiv:hep-ph/0609090 [hep-ph]
  NoStop

[Balitsky and Chirilli(2008)]Balitsky:2008zza
  author author Ian Balitsky and author Giovanni A.  Chirilli, title title
  Next-to-leading order evolution of color dipoles, 10.1103/PhysRevD.77.014019 journal journal Phys.
  Rev. volume D77, pages 014019
  (year 2008), http://arxiv.org/abs/0710.4330
  arXiv:0710.4330 [hep-ph] NoStop

[Balitsky(2007)]Balitsky:2006wa
  author author Ian Balitsky, title title Quark
  contribution to the small-x evolution of color dipole, 10.1103/PhysRevD.75.014001 journal journal Phys.
  Rev. volume D75, pages 014001
  (year 2007), http://arxiv.org/abs/hep-ph/0609105
  arXiv:hep-ph/0609105 [hep-ph] NoStop

[Low(1958)]Low:1958sn
  author author F. E. Low, title title Bremsstrahlung of
  very low-energy quanta in elementary particle collisions, 10.1103/PhysRev.110.974 journal journal
  Phys. Rev. volume 110, pages
  974–977 (year 1958)NoStop

[Burnett and Kroll(1968)]Burnett:1967km
  author author T. H. Burnett and author Norman M. Kroll, title title Extension of the
  low soft photon theorem, 10.1103/PhysRevLett.20.86
  journal journal Phys. Rev. Lett. volume 20, pages 86 (year
  1968)NoStop

[Bell and Van Royen(1969)]Bell:1969yw
  author author J. S. Bell and author R. Van Royen, title title On the
  low-burnett-kroll theorem for soft-photon emission, 10.1007/BF02823297 journal journal Nuovo Cim.  volume A60, pages 62–68 (year 1969)NoStop

[McLerran and Venugopalan(1999)]McLerran:1998nk
  author author Larry D.  McLerran and author Raju  Venugopalan, title title
  Fock space distributions, structure functions, higher twists and small
  x, 10.1103/PhysRevD.59.094002 journal
  journal Phys. Rev. volume D59,  pages 094002 (year 1999), http://arxiv.org/abs/hep-ph/9809427 arXiv:hep-ph/9809427 [hep-ph]
  NoStop

[Peskin and Schroeder(1995)]Peskin:1995ev
  author author Michael E.  Peskin and author Daniel V.  Schroeder, http://www.slac.stanford.edu/ mpeskin/QFT.html title
  An Introduction to quantum field theory (publisher
  Addison-Wesley, address Reading, USA, year
  1995)NoStop

[Blaizot et al.(2004a)Blaizot, Gelis,  and Venugopalan]Blaizot:2004wu
  author author Jean Paul  Blaizot, author Francois  Gelis,  and author Raju  Venugopalan, title title
  High-energy pA collisions in the color glass condensate approach. 1. Gluon
  production and the Cronin effect, 10.1016/j.nuclphysa.2004.07.005 journal journal
  Nucl. Phys. volume A743, pages
  13–56 (year 2004a), http://arxiv.org/abs/hep-ph/0402256 arXiv:hep-ph/0402256 [hep-ph]
  NoStop

[Jalilian-Marian and Kovchegov(2006)]JalilianMarian:2005jf
  author author Jamal Jalilian-Marian and author Yuri V.  Kovchegov, title title
  Saturation physics and deuteron-Gold collisions at RHIC, 10.1016/j.ppnp.2005.07.002 journal journal Prog. Part. Nucl. Phys. volume 56,  pages 104–231 (year 2006), http://arxiv.org/abs/hep-ph/0505052 arXiv:hep-ph/0505052 [hep-ph]
  NoStop

[Fujii et al.(2006)Fujii,
  Gelis, and Venugopalan]Fujii:2006ab
  author author Hirotsugu  Fujii, author Francois  Gelis,  and author Raju  Venugopalan, title title
  Quark pair production in high energy pA collisions: General features,  10.1016/j.nuclphysa.2006.09.012 journal
  journal Nucl. Phys. volume A780,  pages 146–174 (year 2006), http://arxiv.org/abs/hep-ph/0603099 arXiv:hep-ph/0603099 [hep-ph]
  NoStop

[Kuraev et al.(1977)Kuraev,
  Lipatov, and Fadin]Kuraev:1977fs
  author author E. A. Kuraev, author L. N. Lipatov,
   and author Victor S. Fadin,  title title The Pomeranchuk Singularity
  in Nonabelian Gauge Theories, @noop  journal
  journal Sov. Phys. JETP volume 45,  pages 199–204 (year 1977), note [Zh.
  Eksp. Teor. Fiz.72,377(1977)]NoStop

[Balitsky and Lipatov(1978)]Balitsky:1978ic
  author author I. I. Balitsky and author L. N. Lipatov, title title The
  Pomeranchuk Singularity in Quantum Chromodynamics, @noop 
  journal journal Sov. J. Nucl. Phys. volume 28, pages 822–829 (year
  1978), note [Yad. Fiz.28,1597(1978)]NoStop

[Gelis and Venugopalan(2004)]Gelis:2003vh
  author author Francois  Gelis and author Raju  Venugopalan, title title
  Large mass qq̅ production from the color glass condensate,  10.1103/PhysRevD.69.014019 journal journal Phys. Rev. volume D69, pages
  014019 (year 2004), http://arxiv.org/abs/hep-ph/0310090 arXiv:hep-ph/0310090 [hep-ph]
  NoStop

[Aurenche et al.(1984)Aurenche, Douiri, Baier, Fontannaz, and Schiff]Aurenche:1983hc
  author author P. Aurenche, author A. Douiri,
  author R. Baier, author M. Fontannaz,  and author D. Schiff, title
  title The Deep Compton Scattering: Single Photon
  Spectrum and Photon - Hadron Correlations Beyond Leading Logarithms,  10.1007/BF01410370 journal journal Z. Phys. volume C24, pages
  309 (year 1984)NoStop

[Gehrmann-De Ridder et al.(2000)Gehrmann-De Ridder, Kramer,  and Spiesberger]GehrmannDeRidder:2000ce
  author author A. Gehrmann-De Ridder, author G. Kramer,  and author H. Spiesberger, title title Photon
  plus jet-cross sections in deep inelastic e p collisions at order O(alpha**2
  alpha(s)), 10.1016/S0550-3213(00)00228-5 journal journal Nucl. Phys. volume
  B578, pages 326–350 (year 2000), http://arxiv.org/abs/hep-ph/0003082 arXiv:hep-ph/0003082 [hep-ph]
  NoStop

[Kramer et al.(1998)Kramer,
  Michelsen, and Spiesberger]Kramer:1997nb
  author author G. Kramer, author D. Michelsen,   and author H. Spiesberger,  title title Production of hard photons
  and jets in deep inelastic lepton proton scattering at order O (alpha-s),  10.1007/s100520050272, 10.1007/s100529800836 journal journal Eur. Phys. J. volume
  C5, pages 293–301 (year 1998), http://arxiv.org/abs/hep-ph/9712309 arXiv:hep-ph/9712309 [hep-ph]
  NoStop

[Michelsen(1995)]Michelsen:1995ag
  author author Dirk Michelsen, title Radiative corrections of order
  alpha-s for production of hard photons by HERA, @noop  Ph.D.
  thesis, school Hamburg U. (year
  1995)NoStop

[Gelis and Jalilian-Marian(2003)]Gelis:2002nn
  author author Francois  Gelis and author Jamal  Jalilian-Marian, title title From DIS to proton nucleus collisions in the color glass condensate
  model, 10.1103/PhysRevD.67.074019 journal
  journal Phys. Rev. volume D67,  pages 074019 (year 2003), http://arxiv.org/abs/hep-ph/0211363 arXiv:hep-ph/0211363 [hep-ph]
  NoStop

[Dominguez et al.(2012)Dominguez, Qiu, Xiao, and Yuan]Dominguez:2011br
  author author Fabio Dominguez, author Jian-Wei Qiu, author Bo-Wen Xiao,   and author Feng Yuan,  title title On the linearly polarized
  gluon distributions in the color dipole model, 10.1103/PhysRevD.85.045003 journal journal Phys.
  Rev. volume D85, pages 045003
  (year 2012), http://arxiv.org/abs/1109.6293
  arXiv:1109.6293 [hep-ph] NoStop

[Kovchegov and Mueller(1998)]Kovchegov:1998bi
  author author Yuri V. Kovchegov and author Alfred H. Mueller, title title Gluon
  production in current nucleus and nucleon - nucleus collisions in a
  quasiclassical approximation, 10.1016/S0550-3213(98)00384-8 journal journal
  Nucl. Phys. volume B529, pages
  451–479 (year 1998), http://arxiv.org/abs/hep-ph/9802440 arXiv:hep-ph/9802440 [hep-ph]
  NoStop

[Diehl(2003)]Diehl:2003ny
  author author M. Diehl, title title Generalized
  parton distributions, 10.1016/j.physrep.2003.08.002,
  10.3204/DESY-THESIS-2003-018 journal journal
  Phys. Rept. volume 388, pages
  41–277 (year 2003), http://arxiv.org/abs/hep-ph/0307382 arXiv:hep-ph/0307382 [hep-ph]
  NoStop

[Kogut and Soper(1970)]Kogut:1969xa
  author author John B. Kogut and author Davison E. Soper, title title Quantum
  Electrodynamics in the Infinite Momentum Frame, 10.1103/PhysRevD.1.2901 journal journal Phys.
  Rev. volume D1, pages 2901–2913
  (year 1970)NoStop

[Brodsky and Robertson(1995)]Brodsky:1995rn
  author author Stanley J.  Brodsky and author David G.  Robertson, title title
  Light cone quantization and QCD phenomenology, in http://www-public.slac.stanford.edu/sciDoc/docMeta.aspx?slacPubNumber=SLAC-PUB-7056
  booktitle Confinement physics. Proceedings, 1st ELFE
  Summer School, Cambridge, UK, July 22-28, 1995 (year 1995)  pp. pages 71–110, http://arxiv.org/abs/hep-ph/9511374 arXiv:hep-ph/9511374 [hep-ph]
  NoStop

[Brodsky et al.(1998)Brodsky, Pauli, and Pinsky]Brodsky:1997de
  author author Stanley J.  Brodsky, author Hans-Christian  Pauli,  and author Stephen S.  Pinsky, title title
  Quantum chromodynamics and other field theories on the light cone,  10.1016/S0370-1573(97)00089-6 journal
  journal Phys. Rept. volume 301,  pages 299–486 (year 1998), http://arxiv.org/abs/hep-ph/9705477 arXiv:hep-ph/9705477 [hep-ph]
  NoStop

[Venugopalan(1998)]Venugopalan:1998zd
  author author Raju Venugopalan, title title
  Introduction to light cone field theory and high-energy scattering,  booktitle booktitle Hadrons in dense
  matter and hadrosynthesis. Proceedings, 11th Chris Engelbrecht Summer School,
  Cape Town, South Africa, February 4-13, 1998, 10.1007/BFb0107312   (year 1998), 10.1007/BFb0107312,  note [Lect. Notes Phys.516,89(1999)], http://arxiv.org/abs/nucl-th/9808023 arXiv:nucl-th/9808023 [nucl-th]
  NoStop

[Venugopalan(1999)]Venugopalan:1999wu
  author author Raju Venugopalan, title title Classical
  methods in DIS and nuclear scattering at small x, booktitle booktitle Strong interactions at low and
  high-energies. Proceedings, 39th Cracow School of Theoretical Physics,
  Zakopane, Poland, May 29-June 8, 1999, @noop  journal journal Acta Phys. Polon. volume B30, pages 3731–3761 (year 1999),  http://arxiv.org/abs/hep-ph/9911371 arXiv:hep-ph/9911371 [hep-ph]
  NoStop

[Ayala et al.(1995)Ayala,
  Jalilian-Marian, McLerran, and  Venugopalan]Ayala:1995kg
  author author Alejandro  Ayala, author Jamal Jalilian-Marian, author Larry D. McLerran,  and author Raju Venugopalan, title title The Gluon
  propagator in nonAbelian Weizsäcker-Williams fields, 10.1103/PhysRevD.52.2935 journal journal Phys.
  Rev. volume D52, pages 2935–2943
  (year 1995), http://arxiv.org/abs/hep-ph/9501324
  arXiv:hep-ph/9501324 [hep-ph] NoStop

[Ayala et al.(1996)Ayala,
  Jalilian-Marian, McLerran, and  Venugopalan]Ayala:1995hx
  author author Alejandro  Ayala, author Jamal Jalilian-Marian, author Larry D. McLerran,  and author Raju Venugopalan, title title Quantum
  corrections to the Weizsacker-Williams gluon distribution function at small
  x, 10.1103/PhysRevD.53.458 journal
  journal Phys. Rev. volume D53,  pages 458–475 (year 1996), http://arxiv.org/abs/hep-ph/9508302 arXiv:hep-ph/9508302 [hep-ph]
  NoStop

[Gelis and Mehtar-Tani(2006)]Gelis:2005pt
  author author Francois  Gelis and author Yacine  Mehtar-Tani, title title
  Gluon propagation inside a high-energy nucleus, 10.1103/PhysRevD.73.034019 journal journal Phys.
  Rev. volume D73, pages 034019
  (year 2006), http://arxiv.org/abs/hep-ph/0512079
  arXiv:hep-ph/0512079 [hep-ph] NoStop

[Blaizot et al.(2004b)Blaizot, Gelis,  and Venugopalan]Blaizot:2004wv
  author author Jean Paul  Blaizot, author Francois  Gelis,  and author Raju  Venugopalan, title title
  High-energy pA collisions in the color glass condensate approach. 2. Quark
  production, 10.1016/j.nuclphysa.2004.07.006 journal journal Nucl. Phys. volume
  A743, pages 57–91 (year 2004b),  http://arxiv.org/abs/hep-ph/0402257 arXiv:hep-ph/0402257 [hep-ph]
  NoStop

[Balitsky and Belitsky(2002)]Balitsky:2001mr
  author author I. I. Balitsky and author Andrei V. Belitsky, title title Nonlinear
  evolution in high density QCD, 10.1016/S0550-3213(02)00149-9 journal journal
  Nucl. Phys. volume B629, pages
  290–322 (year 2002), http://arxiv.org/abs/hep-ph/0110158 arXiv:hep-ph/0110158 [hep-ph]
  NoStop

[Hebecker and Weigert(1998)]Hebecker:1998kv
  author author Arthur Hebecker and author H. Weigert, title title Small x parton
  distributions of large hadronic targets, 10.1016/S0370-2693(98)00623-6 journal journal
  Phys. Lett. volume B432, pages
  215–221 (year 1998), http://arxiv.org/abs/hep-ph/9804217 arXiv:hep-ph/9804217 [hep-ph]
  NoStop

[Kovchegov and Tuchin(2002)]Kovchegov:2001sc
  author author Yuri V. Kovchegov and author Kirill Tuchin, title title Inclusive gluon
  production in DIS at high parton density, 10.1103/PhysRevD.65.074026 journal journal Phys.
  Rev. volume D65, pages 074026
  (year 2002), http://arxiv.org/abs/hep-ph/0111362
  arXiv:hep-ph/0111362 [hep-ph] NoStop

[Roy and Venugopalan()]Roy-Venugopalan-2
  author author Kaushik Roy and author Raju Venugopalan, @noop  journal in preparation  NoStop
[Bartels et al.(2001)Bartels, Gieseke, and Qiao]Bartels:2000gt
  journal   author author Jochen Bartels, author S. Gieseke,  and author C. F. Qiao, title title The γ^*→ q q̅ Reggeon vertex in
  next-to-leading order QCD, 10.1103/PhysRevD.63.056014,
  10.1103/PhysRevD.65.079902 journal journal Phys.
  Rev. volume D63, pages 056014
  (year 2001), note [Erratum: Phys.
  Rev.D65,079902(2002)], http://arxiv.org/abs/hep-ph/0009102
  arXiv:hep-ph/0009102 [hep-ph] NoStop

[Bartels et al.(2002a)Bartels, Colferai, Gieseke, and Kyrieleis]Bartels:2002uz
  author author J. Bartels, author D. Colferai,
  author S. Gieseke,  and  author A. Kyrieleis, title title NLO corrections to the photon impact
  factor: Combining real and virtual corrections, 10.1103/PhysRevD.66.094017 journal journal Phys.
  Rev. volume D66, pages 094017
  (year 2002a), http://arxiv.org/abs/hep-ph/0208130 arXiv:hep-ph/0208130 [hep-ph]
  NoStop

[Bartels et al.(2002b)Bartels, Gieseke, and Kyrieleis]Bartels:2001mv
  author author Jochen Bartels, author S. Gieseke,   and author A. Kyrieleis,  title title The Process γ^*L +
  q → q q̅ g + q: Real corrections to the virtual photon impact
  factor, 10.1103/PhysRevD.65.014006 journal journal Phys. Rev. volume
  D65, pages 014006 (year 2002b),  http://arxiv.org/abs/hep-ph/0107152 arXiv:hep-ph/0107152 [hep-ph]
  NoStop

[Balitsky and Chirilli(2011)]Balitsky:2010ze
  author author Ian Balitsky and author Giovanni A.  Chirilli, title title
  Photon impact factor in the next-to-leading order, 10.1103/PhysRevD.83.031502 journal journal Phys.
  Rev. volume D83, pages 031502
  (year 2011), http://arxiv.org/abs/1009.4729
  arXiv:1009.4729 [hep-ph] NoStop

[Beuf(2012)]Beuf:2011xd
  author author Guillaume  Beuf, title title NLO
  corrections for the dipole factorization of DIS structure functions at low
  x, 10.1103/PhysRevD.85.034039 journal
  journal Phys. Rev. volume D85,  pages 034039 (year 2012), http://arxiv.org/abs/1112.4501 arXiv:1112.4501 [hep-ph] NoStop

[Beuf(2016)]Beuf:2016wdz
  author author Guillaume  Beuf, title title
  Dipole factorization for DIS at NLO: Loop correction to the
  γ^*_T,L→ qq light-front wave functions, 10.1103/PhysRevD.94.054016 journal journal Phys. Rev. volume D94, pages
  054016 (year 2016), http://arxiv.org/abs/1606.00777 arXiv:1606.00777 [hep-ph] NoStop

[Beuf(2017)]Beuf:2017bpd
  author author Guillaume  Beuf, title title
  Dipole factorization for DIS at NLO: Combining the qq̅ and
  qq̅g contributions, 10.1103/PhysRevD.96.074033
  journal journal Phys. Rev. volume D96, pages 074033 (year 2017),  http://arxiv.org/abs/1708.06557 arXiv:1708.06557 [hep-ph]
  NoStop

[Hanninen et al.(2017)Hanninen, Lappi, and Paatelainen]Hanninen:2017ddy
  author author H. Hanninen, author T. Lappi,   and author R. Paatelainen,  title title One-loop corrections to
  light cone wave functions: the dipole picture DIS cross section,  @noop    (year 2017), http://arxiv.org/abs/1711.08207 arXiv:1711.08207 [hep-ph] NoStop

[Ducloue et al.(2017a)Ducloue, Hanninen, Lappi, and Zhu]Ducloue:2017ftk
  author author B. Ducloue, author H. Hanninen,
  author T. Lappi,  and author Y. Zhu, title
  title Deep inelastic scattering in the dipole picture
  at next-to-leading order, 10.1103/PhysRevD.96.094017
  journal journal Phys. Rev. volume D96, pages 094017 (year
  2017a), http://arxiv.org/abs/1708.07328
  arXiv:1708.07328 [hep-ph] NoStop

[Kovchegov and Weigert(2008)]Kovchegov:2007vf
  author author Yuri V. Kovchegov and author Heribert Weigert, title title Collinear
  Singularities and Running Coupling Corrections to Gluon Production in
  CGC, 10.1016/j.nuclphysa.2008.04.008 journal journal Nucl. Phys. volume
  A807, pages 158–189 (year 2008), http://arxiv.org/abs/0712.3732 arXiv:0712.3732 [hep-ph] NoStop

[Ducloue et al.(2017b)Ducloue, Iancu,
  Lappi, Mueller, Soyez,
  Triantafyllopoulos, and Zhu]Ducloue:2017dit
  author author B. Ducloue, author E. Iancu,
  author T. Lappi, author A. H. Mueller, author
  G. Soyez, author D. N. Triantafyllopoulos,  and author
  Y. Zhu, title title On the use of a running coupling in the NLO calculation
  of forward hadron production, @noop    (year
  2017b), http://arxiv.org/abs/1712.07480
  arXiv:1712.07480 [hep-ph] NoStop

[Accardi et al.(2016)Accardi
  et al.]Accardi:2012qut
  author author A. Accardi et al., title title
  Electron Ion Collider: The Next QCD Frontier, 10.1140/epja/i2016-16268-9 journal journal Eur.
  Phys. J. volume A52, pages 268
  (year 2016), http://arxiv.org/abs/1212.1701
  arXiv:1212.1701 [nucl-ex] NoStop

[Aschenauer et al.(2017)Aschenauer, Fazio, Lee, Mantysaari, Page, Schenke, Ullrich, Venugopalan, and Zurita]Aschenauer:2017jsk
  author author E. C. Aschenauer, author S. Fazio,
  author J. H. Lee, author H. Mantysaari, author
  B. S. Page, author
  B. Schenke, author T. Ullrich, author R. Venugopalan,  and author P. Zurita, title title
  The Electron-Ion Collider: Assessing the Energy Dependence of Key
  Measurements, @noop    (year 2017), http://arxiv.org/abs/1708.01527 arXiv:1708.01527 [nucl-ex] NoStop

[Boussarie et al.(2014)Boussarie, Grabovsky, Szymanowski, and  Wallon]Boussarie:2014lxa
  author author R. Boussarie, author A. V. Grabovsky, author L. Szymanowski,  and author S. Wallon, title title Impact factor
  for high-energy two and three jets diffractive production, 10.1007/JHEP09(2014)026 journal journal
  JHEP volume 09, pages 026 (year 2014), http://arxiv.org/abs/1405.7676 arXiv:1405.7676
  [hep-ph] NoStop

[Boussarie et al.(2016)Boussarie, Grabovsky, Szymanowski, and  Wallon]Boussarie:2016ogo
  author author R. Boussarie, author A. V. Grabovsky, author L. Szymanowski,  and author S. Wallon, title title On the one loop
  γ^(∗)→ qq impact factor and the
  exclusive diffractive cross sections for the production of two or three
  jets, 10.1007/JHEP11(2016)149 journal
  journal JHEP volume 11, pages 149 (year 2016), http://arxiv.org/abs/1606.00419 arXiv:1606.00419 [hep-ph] NoStop

[Boussarie et al.(2017)Boussarie, Grabovsky, Ivanov, Szymanowski, and Wallon]Boussarie:2016bkq
  author author R. Boussarie, author A. V. Grabovsky, author D. Yu.  Ivanov, author L. Szymanowski,  and author S. Wallon, title title Next-to-Leading
  Order Computation of Exclusive Diffractive Light Vector Meson Production in a
  Saturation Framework, 10.1103/PhysRevLett.119.072002
  journal journal Phys. Rev. Lett. volume 119, pages 072002 (year
  2017), http://arxiv.org/abs/1612.08026 arXiv:1612.08026
  [hep-ph] NoStop

[Chirilli et al.(2012a)Chirilli, Xiao,  and Yuan]Chirilli:2011km
  author author Giovanni A.  Chirilli, author Bo-Wen  Xiao,  and author Feng  Yuan, title title
  One-loop Factorization for Inclusive Hadron Production in pA Collisions
  in the Saturation Formalism, 10.1103/PhysRevLett.108.122301 journal journal
  Phys. Rev. Lett. volume 108, pages
  122301 (year 2012a), http://arxiv.org/abs/1112.1061 arXiv:1112.1061 [hep-ph] NoStop

[Chirilli et al.(2012b)Chirilli, Xiao,  and Yuan]Chirilli:2012jd
  author author Giovanni A.  Chirilli, author Bo-Wen  Xiao,  and author Feng  Yuan, title title
  Inclusive Hadron Productions in pA Collisions, 10.1103/PhysRevD.86.054005 journal journal Phys.
  Rev. volume D86, pages 054005
  (year 2012b), http://arxiv.org/abs/1203.6139 arXiv:1203.6139 [hep-ph] NoStop

[Altinoluk et al.(2016)Altinoluk, Armesto, Beuf, Kovner, and Lublinsky]Altinoluk:2015vax
  author author Tolga Altinoluk, author Néstor Armesto, author Guillaume Beuf, author Alex Kovner,   and author Michael Lublinsky,  title title Heavy quarks in
  proton-nucleus collisions - the hybrid formalism, 10.1103/PhysRevD.93.054049 journal journal Phys.
  Rev. volume D93, pages 054049
  (year 2016), http://arxiv.org/abs/1511.09415
  arXiv:1511.09415 [hep-ph] NoStop

[Iancu et al.(2016)Iancu,
  Mueller, and Triantafyllopoulos]Iancu:2016vyg
  author author E. Iancu, author A. H. Mueller,
   and author D. N. Triantafyllopoulos, title title CGC
  factorization for forward particle production in proton-nucleus collisions at
  next-to-leading order, 10.1007/JHEP12(2016)041
  journal journal JHEP volume 12, pages 041 (year 2016), http://arxiv.org/abs/1608.05293 arXiv:1608.05293 [hep-ph] NoStop

[Caron-Huot(2015)]Caron-Huot:2013fea
  author author Simon Caron-Huot, title title When does
  the gluon reggeize? 10.1007/JHEP05(2015)093 journal journal JHEP volume 05,  pages 093 (year 2015), http://arxiv.org/abs/1309.6521 arXiv:1309.6521 [hep-th] NoStop

[Bondarenko et al.(2017)Bondarenko, Lipatov, Pozdnyakov, and  Prygarin]Bondarenko:2017vfc
  author author S. Bondarenko, author L. Lipatov,
  author S. Pozdnyakov,  and  author A. Prygarin, title title One loop light-cone QCD, effective
  action for reggeized gluons and QCD RFT calculus, 10.1140/epjc/s10052-017-5208-8 journal journal
  Eur. Phys. J. volume C77, pages 630
  (year 2017), http://arxiv.org/abs/1708.05183
  arXiv:1708.05183 [hep-th] NoStop

[Hentschinski(2018)]Hentschinski:2018rrf
  author author Martin Hentschinski, title title The Color
  Glass Condensate formalism, Balitsky-JIMWLK evolution and Lipatov's high
  energy effective action, @noop    (year 2018),  http://arxiv.org/abs/1802.06755 arXiv:1802.06755 [hep-ph]
  NoStop

[Lipatov(1997)]Lipatov:1996ts
  author author L. N. Lipatov, title title Small x
  physics in perturbative QCD, 10.1016/S0370-1573(96)00045-2 journal journal
  Phys. Rept. volume 286, pages
  131–198 (year 1997), http://arxiv.org/abs/hep-ph/9610276 arXiv:hep-ph/9610276 [hep-ph]
  NoStop

[Caron-Huot et al.(2017)Caron-Huot, Gardi, Reichel, and  Vernazza]Caron-Huot:2017zfo
  author author Simon Caron-Huot, author Einan Gardi, author Joscha Reichel,
   and author Leonardo Vernazza, title title Infrared
  singularities of QCD scattering amplitudes in the Regge limit to all
  orders, @noop    (year 2017), http://arxiv.org/abs/1711.04850 arXiv:1711.04850 [hep-ph] NoStop

[Del Duca et al.(2018)Del Duca, Druc, Drummond, Duhr, Dulat, Marzucca, Papathanasiou, and Verbeek]DelDuca:2018hrv
  author author Vittorio  Del Duca, author Stefan  Druc, author James Drummond, author Claude Duhr,
  author Falko Dulat, author Robin Marzucca, author
  Georgios Papathanasiou,  and author Bram Verbeek, title
  title The seven-gluon amplitude in multi-Regge
  kinematics beyond leading logarithmic accuracy, @noop    (year 2018), http://arxiv.org/abs/1801.10605
  arXiv:1801.10605 [hep-th] NoStop




