Dihadron production in DIS at NLO: the real corrections

By using the formalism of the light-cone wave function along with the colour glass condensate effective theory, we consider next-to-leading order (NLO) corrections to the production of a pair of hadrons in electron-proton, or electron-nucleus, collisions at small Bjorken $x$. To the order of interest, the process involves the fluctuation of a virtual photon into a quark-antiquark pair, followed by the emission of a gluon from either the quark, or the antiquark. For the case of a virtual photon with transverse polarization, we compute the real NLO corrections, where the emitted gluon is present in the final state. We first compute the tree-level cross-section for the production of the quark-antiquark-gluon system and then deduce the real NLO corrections to dihadron production by integrating out the kinematics of the gluon. We verify in detail that, in the limit where the gluon is soft, our calculation reproduces the (real piece of the) B-JIMWLK evolution of the leading-order cross-section for quark-antiquark production. Similarly, in the limit where the gluon is collinear with its emitter, we recover the real terms in the DGLAP evolution of the fragmentation function. The virtual NLO corrections to dihadron production will be presented by one of us in a subsequent publication.


Introduction
One of the main discoveries of the HERA experimental program on deeply inelastic electron-proton scattering is the fact that the gluon density inside a hadron sharply rises as one explores smaller and smaller values of Bjorken x at a fixed virtuality Q 2 [1,2].While conceptually anticipated on the basis of the perturbative evolution in QCD with increasing energy, as encoded in the celebrated BFKL equation [3][4][5], this rapid rise challenges our understanding of QCD scattering in the vicinity of the unitarity limit.The natural solution to this problem, as also emerging from perturbative QCD, is the phenomenon of gluon saturation, which is a consequence of the non-linear dynamics of the highly occupied gluon modes.Predicted already before the advent of the HERA data [6,7], this phenomenon finds its modern, pQCD-based, formulation in the effective theory for the color glass condensate (CGC) [8][9][10][11][12].The saturated gluon matter dubbed as the CGC [13] is believed to be a universal form of hadronic matter which controls the QCD scattering amplitudes for sufficiently high energies.The properties of this matter and the high-energy amplitudes can be reliably computed within pQCD as soon as the characteristic transverse-momentum scale, the saturation momentum squared Q 2 s (a measure of the gluon density per unit transverse area), is sufficiently hard: QCD .This scale rises, roughly, as a power of 1/x and also (for a large nucleus) as a power of the nuclear mass number A: Q 2 s (x, A) ∝ A1/3 (1/x λ ) with λ 0.2.Hence pQCD should be a good tool for studies of gluon saturation at sufficiently small values of x and/or large values of A.
This motivated the use of perturbative techniques for computing the high-energy evolution of the gluon correlation functions -a non-linear generalisation of the BFKL equation known as the B-JIMWLK evolution 1 [13][14][15][16][17][18][19] -as well as the hard impact factors which enter the CGC calculation of hadronic cross-sections for "dilute-dense" collisions, like electron-nucleus (eA) or proton-nucleus (pA).These ingredients have been originally computed to leading order (LO) in the QCD running coupling α s .However, one needs (at least) next-to-leading order (NLO) estimates in order to reach a perturbative accuracy of about 10%, as required for realistic predictions for the phenomenology.And indeed, over the last years one assists at strenuous efforts aiming at promoting the CGC effective theory to NLO accuracy.
Among the processes above listed, the inclusive production of a pair of jets or hadrons is particularly interesting, as this is sensitive to gluon saturation via the azimuthal correlations among the two measured particles [84][85][86][87][88][89][90][91][92][93][94][95][96][97][98].Multiple scattering should lead a broadening in their azimuthal distribution around the back-to-back peak.Such a broadening has been observed in d+Au collisions at RHIC [99,100], although at this level it looks difficult to distinguish the effects of saturation from those of the final state radiation (the "Sudakov factor" [101]).The kinematical conditions for studying this effect are expected to be better at the Electron-Ion Collider (EIC) [102].This explains the interest in having accurate theoretical predictions for this process in ep, or eA collisions.
It is our purpose here to present a calculation of the real NLO corrections to dihadron production at forward rapidities in electron-hadron collisions in the CGC formalism.(The corresponding virtual corrections will be presented by one of us (Y.M.) in a separate paper.)At LO, the respective crosssection involves the production of a quark-antiquark pair, which is initiated by the decay of the virtual equations [19], and applies to gauge-invariant correlators built with products of Wilson lines.In the limit of a large number of colors (Nc 1), the first equation in this hierarchy reduces to a closed, non-linear, equation for the dipole amplitude, known as the Balitsky-Kovchegov equation [19,20].
photon and put on mass-shell by the scattering off the hadronic target [86].The real NLO corrections involve the emission of an additional gluon, by either the quark or the antiquark, which is produced too in the final state, but it is not measured2 .As aforementioned, there are already several independent calculations of these NLO corrections in the literature [72][73][74][76][77][78], with results which appear to agree with each other.So, one may wonder why there is need for yet another calculation.In our opinion, there are in fact several reasons for that.
First, these are complex and tedious calculations, which are technically involved and conceptually subtle.So, it is indeed useful to have several independent calculations, in order to cross check the previous results.Second, the previous calculations generally use different techniques and/or consider different final states: we use the light-cone wave function (LCWF) formalism together with light-cone perturbation theory (LCPT) and look at dihadron final states.The same formalism has been used in Ref. [73], which however considered dijet final states.Ref. [72] uses momentum-space (Feynmanrule) perturbation theory together with dijet final states, while Ref. [74] focuses on dihadrons (like ourselves), but uses helicity techniques when computing Feynman graphs.Moreover, our results and those in Ref. [74] are complementary to each other, in the sense that they refer to different polarisation states for the virtual photon -transverse polarisation in our analysis and, respectively, longitudinal polarisation in [74].Together, our respective papers cover all cases (for real NLO corrections at least).
Despite such formal differences, we have checked that our results for the real NLO corrections fully agree with the corresponding results in the literature, whenever direct comparisons were possible.This is the case, e.g., for the LCWF of the virtual photon to the order of interest and for the tree-level cross-section for producing a system of three partons (a quark, an antiquark and a gluon) -the main ingredients for constructing the real NLO corrections to the production of both dihadrons and dijets.
But even when the results appear to agree with each other, we still believe that our analysis may offer new insights, not only because of the use of different methods, but also because of the different emphasis that we have put on various aspects of the calculation and the associated physical discussions.Notably, we feel that a strong point of our analysis is the particularly transparent discussion of the two special limits, which represent important checks of the NLO results: the soft gluon limit, in which we shall recover the B-JIMWLK evolution of the LO cross-section, and the limit where the gluon becomes collinear with its emitter, where we shall recover the DGLAP evolution of the fragmentation function for the final quark, or antiquark.
As compared to previous studies in the literature, we shall analyse the soft gluon limit diagram by diagram and we shall thus demonstrate that only a particular class of Feynman graphs contribute in this limit -those where the gluon is emitted sufficiently close to the scattering with the nuclear target.There are 16 such graphs and their respective contributions are found to be in an one-to-one correspondence with the (real) terms of the B-JIMWLK equation for the quadrupole [103][104][105].
Furthermore, we shall use an original strategy (that we introduced in the context of proton-nucleus collisions [57]) to verify the emergence of the DGLAP evolution in the collinear limit.While all the other approaches in the literature rely on the transverse momentum representation for that purpose (see e.g.[74] for a discussion in a similar context), we have reformulated the collinear limit directly in term of transverse coordinates (the natural representation for constructing the LCWF in the presence of multiple scattering).While the two representations should be eventually equivalent, we feel that our method is more efficient in practice, as it allows one to easily recognise the diagrams which contribute to the collinear limit and to extract the DGLAP evolution already before the final Fourier transform to momentum space.This paper is structured as follows.In Sect. 2 we schematically describe the results of the LCPT for the virtual photon light-cone wavefunction to the order of interest -that is, for the quark-antiquark (q q) and the quark-antiquark-gluon (q qg) Fock space components.Notice that by "LCWF" we truly mean the outgoing state that would be probed by the detector and which also includes the effects of the collision with the hadronic target.The general but formal results from Sect. 2 are subsequently used to derive fully explicit expressions for the respective wavefunctions and scattering amplitudes.Specifically, in Sect. 3 we deduce the amplitude and the cross-section for the leading order (LO) process, that is, the exclusive production of a q q pair.Then, in Sect. 4 we present the results for the quark-antiquark-gluon Fock component, where the gluon can be emitted from either the quark or antiquark.In Sect. 5 we deduce the real NLO corrections to the dihadron cross-section by integrating out the kinematics of one (anyone) of the three final partons.In particular, we explain the simplifications which occur in the colour structure (i.e. in the partonic S-matrices) due to the fact that one of the partons is not measured.In Sect.6 we show that in the limit where the unmeasured parton is a soft gluon, our NLO results reproduce, as expected, the (real part of the) JIMWLK evolution for the LO dijet cross-section.Finally, in Sect.7, we consider the limit where the unmeasured gluon is produced by a collinear splitting.We show that, in this limit, our NLO results develop collinear singularities, that we isolate to leading logarithmic accuracy and verify that they can be interpreted as one-step in the DGLAP evolution of the fragmentation function.

The virtual photon light-cone wavefunction
We consider high-energy electron-proton or electron-nucleus scattering in a Lorentz frame where the virtual photon γ * is a relativistic right-mover, with 4-momentum q µ = (q + , −Q 2 /2q + , q) in light-cone notations, whereas the nucleus is an ultrarelativistic left-mover, which for the present purposes can be simply treated as a Lorentz-contracted shockwave.
We describe this collision using the light-cone wavefunction (LCWF) formalism, which aims at constructing the wavefunction of the virtual photon as a superposition of Fock states built with "bare quanta" (the eigenstates of the free piece of the Hamiltonian).This superposition is evolving with time, due to the radiation of new quanta, and is constructed via time-dependent perturbation theory.The bare quanta are assumed to be on their mass shall: a bare quark or gluon carries a 4-momentum where k + > 0 and the minus component is the LC energy: A bare single-particle state is specified by the particle 3-momentum (k + , k) and the corresponding quantum numbers, like spin, polarisation, and colour.E.g., a bare quark state is written as |q α λ (k + , k) , where λ = ±1/2 denotes the helicity and α = 1, 2, . . ., N c is a colour index in the fundamental representation.Whenever no confusion is possible, we shall denote bare single-particle states simply as |q , |g , and |γ for a quark, a gluon, and the original photon, respectively.
We would like to construct the final ("outgoing") state at time x + → ∞ for the case where the incoming state at x + → −∞ is a bare, space-like, photon.We work in the projectile light-cone (LC) gauge A + a = 0 and in the interaction representation.As already mentioned, the hadronic target is a shockwave localised near x + = 0, so we can factorise the scattering from the "initial-state" and the "final-state" evolutions -the quantum evolutions occurring before (x + < 0) and respectively after (x + > 0) the collision.We can then write (in compact notations) where |γ is the bare photon state, Ŝ is the scattering operator (or "S-matrix") describing the collision with the shockwave in the eikonal approximation, and the evolution operator is built with the interaction piece H int = H int QCD + H int QED of the full (QCD plus QED) Hamiltonian : (The time-dependence in H int (y + ) is governed by the the free Hamiltonian H 0 = H 0 QCD + H 0 QED , as standard in the interaction representation.)The QED Hamiltonian H int QED generates the splitting of the virtual photon into a quark-antiquark pair in a colour singlet state (a "colour dipole"), whereas the QCD Hamiltonian H int QCD governs the evolution of this q q pair via gluon emissions.The quark and the antiquark can be put on-shell by their scattering off the nuclear shockwave and thus emerge as two "jets" in the final state.Our ultimate goal is to compute the cross-section for dijet production to lowest order in the QED coupling, i.e to O(α em ), and to next-to-leading order in the QCD coupling, i.e. to O(α s ) (as usual, we have α em = e 2 /4π and α s = g 2 /4π).To that aim, it is sufficient to compute single gluon emissions3 , which can be either real -the gluon exists in the final state, albeit it is not measured -, or virtual -the gluon is emitted and reabsorbed at the level of the amplitude, meaning that the final state is just a (bare) q q state.
To the accuracy of interest, one can expand the evolution operators in Eq. (2.1) to linear order in4 H int QED ≡ H QED , for instance with U QCD given by Eq. (2.2) with H int (y + ) → H QCD (y + ).The simpler expression in the second line is sufficient when acting on the photon state |γ .Using this last expression for U (0, −∞) together with the corresponding one for U I (∞, 0), one finds that Eq. (2.1) reduces to The physical interpretation of the above decomposition is quite transparent.The first corrective term, where x + takes negative values, describes a process which starts with the photon decay γ → q q at some time x + < 0, then the q q state evolves according to QCD until it hits the nuclear target at time x + = 0; the scattering between the dressed q q state and the shockwave is quasi-instantaneous (at least, within the eikonal approximation that we employ here) and is described by the S-matrix Ŝ; finally, the partonic state emerging from the scattering evolves up to the time of measurement x + → ∞.
The second corrective term in Eq. (2.4) describes the situation where the photon decays after crossing the shockwave, at some time x + > 0. In this case there is no scattering, but merely the evolution of the q q state from x + up to infinity.It is convenient for what follows to observe that the effect of this second term is to subtract the no-scattering limit ( Ŝ → 1) of the first term; that is, Eq. (2.4) is equivalent to (2.5) Indeed, in the absence of scattering, the virtual photon must eventually return to its initial state, since a space-like photon cannot decay into a system of on-shell partons.This means that |γ out → |γ as Ŝ → 1, or U (∞, −∞)|γ = |γ , which to linear order in H QED implies the result in Eq. (2.5).
The perturbative expansion of the outgoing LCWF to the order of interest can now be obtained by expanding the evolution operators in Eq. (2.5) to second order in H QCD : To evaluate the action of the interaction Hamiltonian, we shall work in the basis of bare multipartonic Fock states, i.e. energy-momentum eigenstates of the free Hamiltonian with fixed numbers of free, on-shell, partons (quarks, antiquarks, and gluons).This not only matches the time-dependence of H QCD (y + ), but is also convenient for computing particle production in the final state: indeed, for asymptotically large times, where the interactions are adiabatically switched off, the bare Fock states are also eigenstates of the full Hamiltonian.If |i and |j are two generic Fock states, then with E i and E j the LC energies of the two states (the sum of the LC energies of the constituent partons).The adiabatic factor e − |y + | was introduced in order to turn off the interactions at large (positive or negative) times.This factor is useful in the intermediate calculations -e.g. it ensures the convergence of time integrations like that in Eq. (2.5) -, but the final, physical, results must have a finite limit when → 0. Similarly, where |q q is a generic quark-antiquark bare state and the sum over all such states, i.e. over the momenta and the quantum numbers of the two bare fermions, is implicitly understood.Of course, this sum is constrained by the conservation properties encoded in the matrix element of H QED ; in particular, the q q state created by the decay of the photon must be a colour-singlet (or "colour dipole").In what follows, we shall systematically use this convention, that repeated indices must be summed over.
Before we turn to a study of the NLO corrections, let us derive the leading-order (LO) result for the q q Fock space component, as obtained by replacing both QCD evolution operators by unity: (2.9) Figure 1: The three possible topologies for the quark-antiquark-gluon outgoing state in the case where the gluon is emitted by the antiquark.We only represent regular graphs, where the intermediate antiquark line represents an on-shell, propagating, fermion.See Fig. 7 for the respective instantaneous graphs.
In the final result, one can neglect the i prescription in the denominator, since the space-like photon cannot be degenerate with the on-shell quark-antiquark pair.

Real gluon emissions
A single gluon emission by either the quark or the antiquark produces a |q qg Fock component in the final state, which contributes to O(g) to the outgoing wavefunction and hence to O(g 2 ) to the dijet cross-section.Such an emission is generated by the term linear in H QCD in the expansion (2.6) of the QCD evolution operator.With reference to Eq. (2.5), it is clear that there are two possible time-orderings: the gluon can be emitted either before, or after, the scattering between the q q pair and the nuclear shockwave.When the emitter is the antiquark, these two possibilities are represented by graphs (a) and (b) in Fig. 1.As a matter of facts, there is also a third graph, illustrated in Fig. 1.c, where the photon decays after crossing the shockwave.But as already explained, this case is implicitly included in Eq. (2.5), via the subtraction of the no-scattering limit from the S-matrix.
Consider the initial-state emission first, cf.Fig. 1.a.The corresponding contribution to Eq. (2.5) reads (cf.Eqs.(2.6) and (2.8)) When inserting intermediate Fock states in the second line, we used the fact that the action of H QCD on the |q q state consists in a gluon emission, leading to a state of the form |q 1 q1 g 1 , whereas the subsequent action of the S-matrix cannot change the partonic content of that state, but only modify the momenta and the discrete quantum numbers (polarisation and colour) of the 3 partons.By energymomentum conservation, the energy denominators in Eq. (2.10) cannot vanish, hence one can safely let → 0 in the final result.Consider similarly the final-state emission, cf.Fig. 1.b.By expanding the final evolution operator in Eq. (2.5) to linear order, one finds the respective contribution as where we have omitted the intermediate steps as well as the i prescription (since unnecessary).
To summarise, the q qg Fock space component of the outgoing LCWF of the virtual photon reads where the first (second) line describes one initial-state (final-state) gluon emission.
It is interesting to notice the structure of the energy denominators in Eq. (2.12).For the initialstate decays, they involve the difference between the LC energies of the intermediate state and of the initial state, respectively; e.g.E q 1 q 1 g 1 −E γ .For the final-state emissions, on the other hand, the energy of the initial state is replaced by that of the final state, e.g.E q 1 q 1 − E q 2 q 2 g 2 .These rules are in fact general and we shall see other examples in what follows.

Virtual corrections
We now turn to the virtual corrections, that is, the one-loop contributions to the amplitude which are generated by the second-order terms in the expansion of the evolution operators in Eq. (2.5).These can be either self-energy graphs, where the gluon is emitted and reabsorbed by the same fermion, or  gluon exchange graphs, where the gluon is emitted by the quark and reabsorbed by the anti-quark, or vice-versa.For both topologies, we encounter three types of time-orderings: initial-state evolution (cf.Fig. 2), final-state evolution (cf.Fig. 3), and mixed evolution, where the gluon is emitted before the scattering with the shockwave and is reabsorbed after that scattering (cf.Fig. 4).
We start with the last case, where the gluon crosses the shockwave and thus can interact with it.We have already shown in Eq. (2.10) the q qg state generated by first emitting a gluon and then scattering with the shockwave.To obtain the "crossing" virtual corrections, one must reabsorb this gluon after the scattering, which can be done by acting with the QCD Hamiltonian (the linear term in the expansion of U QCD (∞, 0)) on the q qg state in Eq. (2.10).Using one deduces (the upper label C stays for "crossing") where the energy denominators cannot vanish.Consider now the one-loop graphs associated with initial-state evolution (cf.Fig. 2).To that aim, one must use the second-order term in the expansion of U QCD (0, −∞), cf.Eq. (2.6), to first emit a gluon by the q q pair and then reabsorb it.The relevant matrix element reads (compare to Eq. (2.10)) After also adding the scattering with the nuclear target, one finds (the upper label B stays for "before") where the i prescription was again ignored, since unimportant.The remaining case, that of the virtual corrections generated via final-state evolution (cf.Fig. 3), turns out to be more subtle.To understand the difficulty, let us first present the formal respective result, as obtained after expanding the evolution operator U QCD (∞, 0) to second order; this reads (the upper label A stays for "after") At a first sight, this contribution has the expected structure; e.g., the energy denominators associated with the QCD transitions are built as differences between the energy of the state prior to the parton branching and that of the final state |q 3 q3 .Consider however the case of a self-energy correction, where the gluon is emitted and reabsorbed by the same fermion -quark or antiquark.In that case, the kinematics of the emitter is clearly the same before and after the loop, hence the intermediate state |q 1 q1 is degenerate with the final state, E q 1 q 1 = E q 3 q 3 , and the respective denominator vanishes.This explains why we have carefully kept the i pieces in the QCD energy denominators in Eq. (2.17).This difficulty is in fact well known: it is related to the proper definition of the final state (the "wavefunction renormalisation").A prescription to circumvent this difficulty has been proposed in [106].For the physical problem at hand, it will be further discussed in the subsequent paper [107], which will be fully devoted to the virtual corrections.From now on, in this paper we shall restrict ourselves to real gluon emissions alone.

Dihadron production in DIS at leading order
As a warm-up, in this section we shall derive the well-known result for the dijet (or dihadron) production in deep inelastic scattering at leading order [86], by following the LCWF formalism outlined in Sect. 2. The corresponding Feynman graphs are shown in Fig. 5.

The leading order photon outgoing state
The outgoing state at leading order (LO) is shown in compact but formal notations in Eq. (2.9).In this section, we shall explicitly compute this state, via the following strategy: First, we shall construct the quark-antiquark Fock state generated by the decay of the virtual photon, by working in momentum space, where the Feynman rules of LCPT are most conveniently formulated; that is, we shall evaluate the graph in Fig. 5.a.Then, we shall perform a Fourier transform to the transverse coordinate representation, which is more convenient for computing the action of the S-matrix in the eikonal approximation; we shall thus deduce the contributions of the graphs in Fig. 5.b and c.
To LO, the q q component of the virtual photon LCWF reads with the following notations: q = (q + , q) is the 3-momentum of the incoming photon with virtuality Q, whereas k ≡ (k + , k) and p ≡ (p + , p) similarly refer to the final quark and anti-quark.Furthermore, λ = T or L denotes the photon polarisation state, α, β are the fermion colour indices and λ 1 , λ 2 are the respective helicity states.The matrix element in Eq. (3.1) involves delta-functions for 3-momentum conservation, which imply p = q − k and p + = q + − k + .The energy denominator is computed as where ϑ is the quark longitudinal momentum fraction, k is the transverse momentum of the quark relative to its parent photon, and Q2 is a measure of the virtuality of the q q pair (see below): We shall consider in more detail the case of a virtual photon with transverse polarisation.The relevant matrix element reads (for a quark flavour with electric charge e f ) where χ λ with λ = ±1/2 are the usual helicity states and in obtaining the second line we have used the following identity (σ i with i = 1, 2, 3 are the usual Pauli matrices) together with the following definition: The coefficient 2ϑ − 1 multiplying δ ij in Eq. (3.6) arises as the difference between the longitudinal momentum fractions, ϑ and 1 − ϑ, of the quark and the antiquark produced by the photon decay.After inserting (3.4) and (3.2) into Eq.(3.1), using 3-momentum conservation and replacing k → k as the integration variable, one finds Its Fourier transform to the transverse coordinate representation is readily obtained as Here x and y are the transverse coordinates of the quark and the antiquark, respectively, R ≡ x − y is their relative separation, R = |R|, c = ϑx + (1 − ϑ)y is their center of energy; we have used the shorthand notation x ≡ d 2 x for the transverse integrations.At this level, it is straightforward to include the effect of the scattering with the nuclear target (the shockwave) in the eikonal approximation: the transverse coordinates, the longitudinal momenta, and the helicity states of the quark and the antiquark remain unchanged, but their colour states get rotated by the interaction with the colour field generated by the left-moving partons from the target: where V (x) and V † (y) are Wilson lines in the fundamental representation of the colour group: We have also introduced here the Wilson line U (x) in the adjoint representation, for later convenience (this describe the colour precession of a gluon).In these equations, A − a is the colour field representing Coulomb exchanges between the quark or the antiquark from the dipole and colour sources (quark and gluons) from the target.In the CGC effective theory, this field is random and must be averaged out at the level of the cross-section (see below).
Thus, finally, the q q outgoing component of the LCWF of a transverse virtual photon reads5 This is a fully explicit version of the outgoing state (2.9), as written in a mixed Fourier representation (longitudinal momentum and transverse coordinates).We recall that the subtraction of unity from the product of Wilson lines inside the square bracket accounts for the process where the photon decays into a q q pair after crossing the shockwave, in which case there is no scattering (see Fig. 5.c).
Similarly, for a virtual photon with longitudinal polarisation we find the following result: (3.12)

The dihadron cross-section at leading order
Given the outgoing LCWF as computed in the previous section, we are now in a position to compute the cross-section for dijet (or dihadron) production at leading order.In this approximation, the final "jets" are simply the quark and the antiquark produced by the decay of the virtual photon and which are put on-shell by their scattering off the hadronic target.The corresponding cross-section is obtained by simply counting the number of q q pairs with a given kinematics in the outgoing state.For a transverse photon, we can write where N q (k 1 ) and N q (k 2 ) are particle number density operators for bare quarks and antiquarks, and the overall factor 1/2 comes from the average over the 2 transverse polarisations.The outgoing photon state in momentum space can be obtained by inverting the Fourier transform in Eq. (3.8).This gives where w and w are the transverse coordinates of the virtual photon in the direct amplitude (DA) and the complex conjugate amplitude (CCA), respectively, and we have also set q = 0 (this entails no loss of generality).The integrations over w and w simply remove the delta-functions like δ (2) (w − c) in Eq. (3.11).To proceed, it is preferable to stick to the transverse coordinate representation, that is, to use Eq.(3.11) for the outgoing state together with the Fourier transform of the number density operators.E.g. for the quarks, we write where the (bare) quark creation and annihilation operators satisfy the anti-commutation relation By using these relations together with the corresponding one for the antiquarks, one finds with R ≡ x − y and ϑ ≡ k + 1 q + .The sum over helicities has been performed as (cf.Eq. (3.6)) In writing Eq. (3.17), we have also performed the average over the random colour fields A − a in the target.This has generated the function W (x, y, y, x), which encodes the effects of the scattering: The 3 non-trivial terms in the r.h.s. are (average) S-matrices describing the forward scattering of colourless systems made with up to four partons: a q q dipole in the DA, a similar q q dipole, S (y, x), in the CCA, and the q qq q quadrupole, where the colour flows connects the q q pair in the DA to that in the CCA.For a virtual photon with longitudinal polarisation one similarly finds (cf.Eq. (3.12)) Eqs. (3.17) and (3.22) are in agreement with previous calculations in the literature [86].
To obtain the corresponding cross-sections for di-hadron production, γA → h 1 h 2 + X, is suffices to convolute the above results for q q production with the fragmentation functions describing the probability to find a hadron within the wavefunction of a quark, or antiquark: The lower script p i = k i /ζ i on the q q cross-section stands for p 4 The tri-parton component of the transverse photon outgoing state In this section, we compute the tri-parton (quark, antiquark, and gluon) Fock-space component γ i T qqg of the outgoing state produced by the evolution and the scattering of an incoming state representing a virtual photon with transverse polarisation.At leading order, this component involves two parton branchings: the decay of the virtual photon into a quark-antiquark pair (γ T → q q), followed by the emission of a gluon from either the quark (q → qg), or the antiquark (q → qg).In practice, it is enough to consider one of these 2 cases -say, gluon emission by the antiquark.The contribution of the other case can then be simply obtained via symmetry operations.Also, as explained in Sect.2, there is no need to explicitly compute the graphs where the photon decays after crossing the shockwave (see Fig. 1.c) -their effects can be accounted for by subtracting the no-scattering limit Ŝ → 1 from the other contributions.So, in practice, it is enough to compute the two graphs in Fig. 1.a and b, together with the corresponding "instantaneous" graphs, where the intermediate antiquark line is replaced with the instantaneous piece of the fermion propagator (see Fig. 7).

Gluon emission by the antiquark
We shall describe in detail the case where the gluon is emitted by the antiquark.We start with the regular graphs in Figs.1.a and b.The respective contributions to the q qg Fock-space component of the outgoing state can be inferred from Eq. (2.12): where the first (second) term within the accolades describes gluon emission prior (after) the shockwave.
The QCD matrix elements ensure that q 1 = q in the first term and, respectively, q 2 = q 1 in the second term.Notice the subscript qqg on the LCWF in the l.h.s.: the antiquark component is shown in boldface to emphasise that this is the fermion which emits the gluon.
As for the leading-order calculation in the previous section, we shall first compute the LCWF in the absence of scattering ( Ŝ → 1) and in transverse momentum space.Then we shall construct its Fourier transform to the transverse coordinate representation.Then it will be easy to insert the effects of the collision, in the form of Wilson lines.Notice that, even after replacing Ŝ → 1, the two terms in Eq. (4.1) are not identical: they differ in the structure of their energy denominators, which in turn reflects the different time-orderings of the gluon emission w.r.t.x + = 0 (the time of scattering).
Our conventions for the momentum-space kinematics are summarised in Fig. 6.a.Let us first work out the relevant energy denominators.They can all be constructed form the LC energy differences at the emission vertices.For the photon decay γ → q q, we have (we recall that Q2 whereas for the gluon emission q → qg, As expected, this energy difference would vanish for a collinear decay, i.e. in the case where the gluon splitting fraction z ≡ ξ/(1 − ϑ) controls not only its longitudinal momentum (p + = z(q + − k + )), but also its transverse momentum (p = z(q − k)).The remaining energy denominator involves the sum of the two energy differences written above: where we introduced the "shifted" momenta which physically express the deviations from collinearity at the emission vertices and which are convenient to use as integration variables when summing over the final states (see below).The matrix element for the virtual photon decay has already been shown in Eq. (3.4).That for the gluon emission from the antiquark reads [57] The structure of this matrix element is consistent with the fact that the antiquark should propagate backwards in time: the colour (δ) and spin (λ 4 ) indices of the daughter antiquark formally appear as the initial quantum numbers in the matrix element in the r.h.s. of Eq. (4.6); similarly, the respective indices β and λ 2 of the emitter appear as final quantum numbers.
For the kinematics shown in Fig. 6, the above spinorial matrix element reads The two arguments of this function are the longitudinal momentum fractions of the daughter partons: the gluon (ξ) and the final antiquark (1−ϑ−ξ).The coefficient 2(1−ϑ)−ξ multiplying δ ij is recognised as the sum of the momentum fractions, 1 − ϑ and 1 − ϑ − ξ, of the antiquark prior and respectively after the gluon emission.Consider now the first term in Eq. (4.1) with Ŝ = 1.This is of course the same as the qqg Fock state at the time of scattering, as shown in Fig. 6.a.Using the energy denominators (4.2) and (4.4) together with the matrix elements in Eqs.(3.4) and (4.6)-(4.7),one finds where the transverse momenta of the final partons can be expressed in terms of the integration variables k and p as follows (cf.Eq. (4.5)) To prepare the Fourier transform to the transverse coordinate representation, let us first introduce the corresponding representation for the 3-parton final state: Also, we shall use the notations w and y for the transverse coordinates of the incoming virtual photon and of the intermediate antiquark, prior to the gluon emission, respectively (see Fig. 1.a).These are not independent coordinates: energy-momentum conservation implies that w must coincide with the center-of-energy of the q q pair produced by the decay of the virtual photon and also with that of the final 3-parton system; similarly, y must be the same as the center-of-energy of the qg pair produced by the decay of the intermediate anti-quark.These conditions imply Using Eq. (4.10) and the above relations for w and y , one can check that the exponent in Eq. (4.11) can be rewritten as where we have introduced the notations R and Y for the transverse separations between the daughter partons after each of the emission vertices: Eq. (4.13) explains why it was more convenient to use the "shifted" transverse momenta k and p as integration variables, instead of the original momenta k and p: the shifted variables are conjugate to the transverse separations between the daughter partons and thus facilitate the calculation of the Fourier transform to the transverse coordinate representation (defined as in Eq. (3.8)).Specifically, by using the following Fourier transform (see e.g.App.A in [61]) one finds x,y,z where c is the center-of-energy of the q qg system, which can be interpreted as the overall transverse size of the q qg partonic fluctuation at the time of scattering (x + = 0).This size is limited by the virtuality of the space-like photon: QD 1. (Indeed, the modified Bessel function exponentially vanishes for large values QD 1 of its argument.)It is interesting to notice that D 2 can be equivalently rewritten as This rewriting shows that the virtuality Q 2 limits both the size R = |x − y | of the q q pair produced by the decay of the virtual photon, namely R 1/ Q, and the size Y = |y − z| of the g q pair produced by the decay of the antiquark (q → qg).In particular, when the gluon is soft, ξ 1, the virtuality constraint on the gluon emission becomes quite loose: Y 2 1/(ξQ 2 ); hence, despite the space-like virtuality, a soft gluon can propagate at relatively large distances from its emitter.
Of course, this whole discussion of the virtuality limits on the transverse size of the partonic fluctuations applies only prior to the scattering (x + ≤ 0).The collision can put the partons on their mass-shell and then they can move from each other arbitrarily far away.
At this point, it is straightforward to add the effects of the scattering in the eikonal approximation and thus deduce our final result for the first term in the r.h.s. of Eq. (4.1): it suffices to insert Wilson lines for the 3 partons which are crossing the shockwave.In practice, this amounts to replacing in the integrand of Eq. (4.16).At this step, it is also convenient to subtract the no-scattering limit ( Ŝ → 1) from the final result 6 and thus directly obtain the first term in the r.h.s. of Eq. (2.12).Our final result for this term therefore reads x,y,z Consider now the second term in Eq. (4.1) which, we recall, corresponds to a gluon emission in the final state (cf.Fig. 1.b).In the absence of scattering ( Ŝ = 1), this term differs from the first term there only in one of the energy denominators: the energy difference E qqg − E γ shown in Eq. (4.4) gets replaced by E qq − E qqg , which is shown (up to a sign) in Eq. (4.3).So, the corresponding contribution (for Ŝ = 1, once again) can be obtained from Eq. (4.9) by replacing in the denominator and changing the overall sign.When computing the Fourier transform to transverse coordinates, the integrals over p and k factorise from each other.The Wilson lines can then be easily added: in this case, they describe the scattering of the intermediate q q pair, with transverse coordinates x and y (see Fig. 1.b).After also subtracting the no-scattering limit, one eventually finds7 x,y,z The above contributions in Eqs.(4.21) and (4.23) can be combined with each other by introducing a convenient notation.Specifically, by using (4.12), (4.18), and the following matrix identity relating the adjoint and the fundamental representations, Figure 7: The two instantaneous graphs contributing to the quark-antiquark-gluon outgoing state in the case where the gluon is emitted by the antiquark.
one sees that the middle line of (4.23) can be obtained from the corresponding line of (4.21) by replacing both y and z by y ; indeed, after this replacement, one has QD → QR.This replacement is merely formal (it should not be applied to the ensemble of Eq. (4.21), but only to its middle line), but it is still useful in that it allows us to introduce a compact notation for the sum of Eqs.(4.21) and (4.23): x,y,z with an "effective vertex" which encompasses the spinor and helicity structure of the whole graph: In our notation in Eq. (4.25), we reintroduced the upper script "reg" to recall that this is the contribution of the "regular" graphs which involve the standard, non-local, piece of the intermediate antiquark propagator.On top on that, we also have the instantaneous contribution.This involves only 2 topologies, since the photon decay and the gluon emission now occur at the same time (see Fig. 7).The respective matrix element reads (see e.g.[61]) A straightforward calculation, similar to that leading to Eq. (4.21), yields x,y,z This has the same Wilson line structure as Eq.(4.21), as expected given the topology of the contributing diagrams in Fig. 7.For what follows, it is convenient to observe that one can combine Eqs.(4.25) and (4.28) in a unique expression by using a generalised version of the effective vertex (4.26), which reads It is understood that the second piece in Eq. (4.29) vanishes after substituting y → y and z → y , so the instantaneous contribution vanishes in this limit, as it should.

Gluon emission by the quark
Our conventions for the case where the gluon is emitted by the quark are summarised in Fig. 6.b in transverse momentum space and in Fig. 8 in transverse coordinate space.(This last figure shows all the possible topologies for the collision with the shock wave and for the case of the "regular" contributions.)There is in fact no need to explicitly compute these graphs: the respective result can be directly inferred from that in the previous section via suitable changes of variables which amount to exchanging the quark and the antiquark, but such that the final state remains unchanged.Specifically, the contribution of the 3 graphs in Fig. 8 to the q qg Fock space component of the LCWF of the transverse photon can be obtained from Eq. (4.25) by (i) exchanging the quark with the antiquark in the final state, (ii) exchanging ϑ ↔ 1 − ϑ − ξ, x ↔ y, λ 1 ↔ λ 2 , and α ↔ β, (iii) taking the complex conjugate of the integrand, and (iv) changing the overall sign (to account for the fact that the colour charge of the quark is minus that of the antiquark) 8 .These transformations have the following consequences on the other transverse coordinates and distances in the problem: (Notice that x is the transverse coordinate of the intermediate quark, prior to the gluon emission; see Fig. 8.a).Furthermore, the spinorial structures change as follows (recall Eq. (3.6)) for the photon decay vertex and, respectively (recall Eq. (4.8)) for the vertex describing the gluon emission.
Remarkably, these transformations do not change the colour structure (including the Wilson lines) for the diagrams where the parton branchings occur either fully before or fully after the collision.That is, the diagram in Fig. 8.a has exactly the same colour structure as that in Fig. 1.a and the same holds for the diagrams in Fig. 8.c and Fig. 1.c, respectively.This is so because the vertex for the photon decay has no incidence on the flow of colour, so the matrix t a describing the gluon emission can be "commuted" through the photon vertex without altering the colour structure.On the other hand, the diagrams where the scattering occurs in the intermediate q q state, that is those in Fig. 8.b and Fig. 1.b respectively, do have different colour structures, since the matrix t a does not commute with the Wilson lines describing the collision.Yet, as in Eq. (4.25), the S-matrix structure of the graph in Fig. 8.b can be obtained from that in Fig. 8.a via an appropriate replacement of variables, which follows as well from the symmetry operations above.
After making all these transformations and also changing the overall sign, one finds (note the subscript qqg on the outgoing state: the quark label is shown in boldface to emphasise that this is the source of the gluon emission): x,y,z where the effective transverse size D is the same as for an emission by the antiquark, cf.Eq. (4.18).(Indeed, this distance is invariant under the symmetry operations exchanging the quark and the antiquark.)When replacing x, z → x , the argument of the Bessel function changes as QD → Q R, where R = | R| = |x − y| and The effective vertex Φ is obtained via the appropriate transformations from Eq. (4.29) and reads Our final result for the q qg Fock-space component of the LCWF of the transverse photon is the sum of the contributions computed above, corresponding to emissions by the antiquark and by the quark, respectively: 5 Next-to-leading order corrections: the real terms Given our previous results for the tri-parton Fock-space component of the outgoing wavefunction of the transverse virtual photon, it is rather straightforward to deduce the "real" next-to-leading order (NLO) corrections to the cross-section for producing a pair of quark-antiquark jets.This involves two steps: first, one computes the (tree-level) cross-section for the production of three partons (quark, antiquark, gluon); then, one integrates out the kinematics of the gluon which is not measured in the final state.

The leading-order trijet cross-section
The leading-order cross-section for q qg production in DIS is computed similarly to Eq. (3.14), that is, as the expectation value of the product of three number-density operators (themselves built with Fock space operators for bare partons) on the photon LCWF in Eq. (4.36): (5.1) It is natural to distinguish between two types of contributions: direct contributions, where the gluon is emitted and reabsorbed by the same fermion (quark or antiquark), and interference terms, where the gluon is absorbed by the quark in the directed amplitude (DA) and reabsorbed by the antiquark in the complex conjugate amplitude (CCA), or vice-versa.

Direct contributions
Two particular diagrams describing direct emissions -one by the quark, the other one by the antiquark -are illustrated in Fig. 9.For each type of emitter, there are nine distinct topologies, corresponding to the product of the three graphs shown in Fig. 1 (for gluon emission by the antiquark) and their complex conjugates.(For simplicity, we explicitly discuss only the case of the regular graphs; the instantaneous contributions are eventually added by modifying the effective vertices as shown in Eqs.(4.29) and (4.35).)Recall however that in our calculation of the LCWF we have grouped the three graphs in Fig. 1 (or in Fig. 8) into two terms: the last graph has been used as a subtraction term for the two previous ones, thus making clear that this particular LCWF component vanishes in the absence of the scattering.Hence, the direct contribution to the 3-jet cross-section will in fact contain only 4 terms.As in the previous section, we first present our result for the emission by the antiquark.A direct calculation using Eq. ( 5.1) together with the LCWF in Eq. (4.25) yields 2) The notations here are similar to those in Eq. (4.25) (see also Fig. 9).Transverse coordinates with a bar refer to the CCA.The longitudinal momentum fractions θ and ξ are the same in the DA and in the CCA, since they are fully fixed by the kinematics of the final state, as follows: 3) The δ-function enforcing longitudinal momentum conservation implies k + 2 = (1 − ϑ − ξ)q + , which in particular requires ϑ + ξ ≤ 1.
The tensorial kernel K imjn 1 is defined as

.4)
As already mentioned, the effective vertices in the numerator also include the contributions from the instantaneous graphs.However, these contributions become irrelevant in some important limits to be later discussed, such as the soft gluon limit ξ → 0. So, it will be useful to have an explicit expression for the product of effective vertices in Eq. ( 5.4) without these instantaneous terms (cf.Eq. (4.26)): where we have also used χ † λ χ λ = 2 and χ † λ 1 σ 3 χ λ 2 = 2λ 1 δ λ 1 λ 2 .The first piece in the last equation, proportional to δ ij δ mn , generates the scalar products of the transverse separations in the DA and, respectively, the CCA, whereas the second piece, proportional to ε ij ε mn , generates their vectorial products, followed by a final scalar product between the 2 vectors previously generated 9 : The structure of the squared vertex in Eq. (5.5) is consistent with that reported in Refs.[78] (see Eq. (2.27) there) and [73] (see Eqs. (9.31-32)).It furthermore agrees with Eq. (B.5) in [72], which refers to the case where the gluon is emitted the final state (i.e. after the collision with the shockwave) in both the DA and the CCA 10 .(For the other cases, the sums over spins and helicities were not explicitly computed in [72], so it is difficult to directly compare the results.) The effects of the collision are encoded in the function W, defined as the following linear combination of partonic S-matrices: The first term in the r.h.s., with the most complex structure, corresponds to the topology depicted in Fig. 9a: all three partons scatter in both the DA and the CCA.So, this is built with a total of six Wilson lines -two in the adjoint representation and four in the fundamental one.Using Fierz identities like (4.24), it is possible to express all the adjoint Wilson lines in terms of fundamental ones.The final result looks particularly simple and suggestive in the large N c limit and will be also exhibited below.Specifically, one has where the approximate equality in the second line holds at large N c : in this limit, the 6-parton S-matrix of interest reduces to the product of two fundamental quadrupoles.Furthermore, the second term S qqg (x, y, z) in Eq. (5.7) corresponds to a diagram where the DA describes initial-state evolution (both parton branchings occur prior to the scattering), whereas the CCA describes final-state evolution (the two parton branchings occur after the scattering).Hence, Wilson lines must be attached only to the three partons from the DA, yielding (5.9) 9 Notice that ε ij = ε ij3 for any i, j = 1, 2 and that a vector product between 2 transverse vectors, like R × R, is a 3-dimensional vector which is oriented along the third axis, i.e. the collision axis. 10As we shall demonstrate in Sect.7, the piece proportional to ε ij ε mn does not contribute to the cross-section in this particular case (gluon emission in the final state), in agreement with the results in [72,73].
Figure 10: A particular interference graph contributing to the leading-order cross-section for q qg production.

Interference terms
Fig. 10 shows a particular interference term, in which the gluon is emitted by the quark in the DA and reabsorbed by the antiquark in the CCA.As for the direct terms, there are 9 possible topologies, but only 4 terms due to our peculiar way of regrouping terms.The other type of diagrams, where the gluon is emitted by the antiquark and absorbed by the quark, are related to those of the first type via complex conjugation.Hence the overall contribution of the interference graphs reads where the new kernel is defined as The product of effective vertices with the instantaneous pieces excluded is evaluated similarly to Eq. (5.5) and reads (5.17) Like in Eq. (5.5), the "squared" vertex is the sum of two contributions, one generating scalar products of transverse vectors, the other one giving rise to vectorial products.The structure in Eq. (5.17) appears to be consistent with the previous results in the literature [72,73,78], whenever direct comparisons are possible (see e.g.Eq. (2.32) in [78] and Eq.(B.7) in [72]).The general S-matrix (colour) structure is the same as for the direct contributions, that is, it is given by the function W shown in Eq. (5.7).When specialised to the "subtraction term" denoted as x, z → x , this function takes the form in Eq. (5.14), whereas for the term y, z → y , it is shown in Eq. (5.12).Finally, for the "double-subtracted term" (x, z → x & y, z → y ), one finds W(x , y, x , x, y , y ) S(y , x)S(x , y) − S(x , y) − S(y , x) + 1. (5.18)

The real NLO corrections to quark-antiquark production in DIS
The real NLO corrections to (forward) dihadron (or dijet) production in DIS at small x are obtained from the leading-order trijet results in the previous subsection, by integrating out the kinematics of the parton that is not measured in the final state.In what follows we shall explicitly consider only the case where the unmeasured parton is the gluon (so the "dijet" is a q q pair, as at leading order).This is indeed the most interesting case, in that it overlaps with both the JIMWLK and the DGLAP evolutions and it corresponds to the virtual corrections to be discussed later on.The results for the two other cases, where one measures either a quark-gluon pair, or an antiquark-gluon one, can be simply obtained via similar manipulations.One therefore has where the subscript "rNLO" stays for real next-to-leading order corrections.The trijet cross-section in the r.h.s. is the sum of direct and interference contributions, as shown in Eqs. ( . By inspection of these expressions, it is clear that the integral over k + 3 can be trivially performed by using the δ-function for longitudinal momentum conservation (which leaves a step-function Θ(q + − k + 1 − k + 2 ) constraining the final momenta), whereas the integral over k 3 yields a factor (2π) 2 δ (2) (z − z), which allows one to identify the coordinates of the unmeasured gluon in the DA and the CCA, respectively.The result of Eq. (5.19) can be succinctly written as where it is understood that the trijet cross-section in the r.h.s. is given by expressions like (5.2), but without the δ-function expressing the conservation of longitudinal momentum.It is understood that the longitudinal momentum fractions of ϑ and ξ must now be expressed in terms of the respective momenta of the two measured particles, that is (cf.Eq. (5.3)) where it is understood that ϑ + ξ ≤ 1.
The fact that the gluon is not measured brings some simplifications in the structure of the S-matrix in Eq. (5.7): when the gluon scatters in both the DA and the CCA, as is the case for the first term in the r.h.s. of Eq. (5.7) (recall Fig. 9a), the associated Wilson lines compensate each other by unitarity, U † (z)U (z) = 1, and then the product of quadrupoles appearing in the second line of Eq. (5.8) reduces to a product of dipoles (we consider large N c , for simplicity): S qqgqqg (x, y, z, x, y, z = z) S(x, x) S(y, y). (5.22) But for the other terms in Eq. (5.7), the identification z = z brings no special simplification, because the gluon is not anymore interacting on both sides of the cut.Altogether, Eq. (5.10) gets replaced by W(x, y, z, x, y, z = z) S(x, x) S(y, y) − S(x, z) S(z, y) − S(z, x) S(y, z) + 1. (5.23) As for the remaining three terms within the square brackets in Eq. (5.2), where the gluon is emitted after the scattering (at least, on one side of the cut), it is clear that there is no special simplification in the respective colour structure.In evaluating these terms, the identification z = z must be performed only at the very end, after making the replacements y, z → y ) or/and (y, z → y .(The two types of identifications do not commute with each other, as one can easily check.)In particular, the only effect of the identification z = z on an expression like that in Eq. (5.13) refers to the "primed" variables y and y , which depend upon z and respectively z, as visible in Eq. ( 4.12).6 The soft gluon limit: recovering the B-JIMWLK evolution In this section, we will show that in the limit where the gluon emission is soft (ξ → 0), our previous results for the NLO corrections reproduce the expected part of the B-JIMWLK evolution of the leadingorder cross-section (3.17).By the "expected part" we mean those terms in the B-JIMWLK equation for the quadrupole Q (x, y, y, x) in which the soft gluon is emitted by the leg at x or at y (i.e. in the DA) and is reabsorbed by the leg at x or at y (in the CCA).All the other pieces of the evolution equation for the quadrupole, as well as the complete evolution equations for the dipole S-matrices S (x, y) and S (y, x) which enter the LO colour structure in Eq. (3.19) will be generated by the virtual NLO corrections to the dijet cross-section, as we shall later check.
For more generality, we shall start with the case where the soft gluon is measured as well, that is, with the LO cross-section for producing a q qg triplet, but such that ξ → 0. In this slightly more general case, we will recognise the action of the "production" version of the JIMWLK Hamiltonian [108][109][110].This is a generalised version of the JIMWLK Hamiltonian which, when acting on the cross-section for particle production in dilute-dense (pA or eA) collisions, leads to the emission of an additional, soft, gluon, which is measured in the final state.The (relevant part of the) standard B-JIMWLK equation for the quadrupole [103][104][105] will be eventually obtained after identifying z = z.
The soft gluon limit is obtained from our previous result by keeping only the leading non-trivial terms in the limit ξ → 0. One then encounters several types of simplifications.
First one can neglect the recoil of the emitter (quark or antiquark) at the respective emission vertex, meaning that its transverse coordinate is not modified by the soft emission.In practice this means that we can neglect the difference between "primed" and "unprimed" coordinates: y y for an emission by the antiquark (recall Fig. 1) and similarly x x for an emission by the quark (cf.Fig. 8).This also implies that the transverse size of the q q is not affected by a soft gluon emission; indeed, when ξ → 0, Eqs.(4.14), (4.18) and (4.30) imply R R |x − y|, whereas D 2 ϑ(1 − ϑ)(x − y) 2 .From now on, in this section we shall use the notation R ≡ |x − y| (and similarly R ≡ |x − y| in the complex conjugate amplitude).One also has Furthermore, one can take the limit ξ → 0 within the emission kernels (5.4) and (5.16), which entails important simplifications -both in the effective vertices and in the energy denominators.In particular, the contributions of the instantaneous graphs to the effective vertices vanish in this limit, cf.Eqs.(4.29) and (4.35), and the same is true for the pieces proportional to ε ij ε mn in the squared vertices (5.5) and (5.17) (indeed, all these contributions are suppressed by additional factors of ξ).If one combines the respective limits of the emission kernels with the explicit factors involving ξ and ϑ in equations like (5.2) and (5.15), one finds that the dependence upon the longitudinal fractions ξ and ϑ becomes the same for all the 3 contributions to the trijet cross-section -the direct contribution of the antiquark (5.2), the similar one due to the quark, and the interference terms (5.15) -, and it is such that the photon decay vertex factorises out from the gluon emission: So, the only difference between these three channels which subsists in the soft gluon limit refers the gluon emission vertex, which involves the appropriate transverse separation, X = x − z or Y = y − z, between the gluon and its emitter.
The "degeneracy" in Eq. ( 6.1) has yet another important consequence: it implies that the four terms contributing to any of the three channels under consideration (i.e. the four terms within the square brackets in Eq. (5.2), or those within the square brackets in Eq. (5.15)) are now multiplied by exactly the same emission kernel.Consider e.g.Eq. (5.2), which we recall represents direct emissions by the antiquark: after the simplifications in Eq. (6.1), an operation like y, z → y ) -which now reduces to z → y, since y = y anyway -has no effect on the kernel, but only on the S-matrix structure exhibited in Eq. (5.7).Accordingly, the colour structures associated with the four terms in Eq. (5.2) are simply added to each other and many of the individual S-matrices (12 over a total of 16) cancel in their sum.The 4 surviving terms correspond to the 4 graphs shown in Fig. 11 for the case of Eq. (5.2).The 12 missing terms, which in fact correspond to only 5 distinct topologies (due to our rearrangement of the three amplitude graphs in Fig. 1), are those where the photon has crossed the shockwave prior to its decay, in either the DA, or the CCA, or both.That is, these are the Figure 12: Diagrams which contribute to the q qg cross-section (5.2) in general, but do not survive in the soft limit ξ 1.
contributions to the cross-section which involve the graph in Fig. 1.c on at least one side of the cut (two such diagrams are shown in Fig. 12).Physically, these simplifications can be understood as follows: the soft gluon with longitudinal momentum p + = ξq + and transverse momentum p has a short formation time τ p = 2ξq + /p 2 , hence in order to affect the scattering (and the ensuing particle production), it must occur relatively close to the shockwave, with a distance ∆x + ∼ τ p away from it.The diagrams shown in Fig. 12 are suppressed in this limit since the photon decay must occur even closer to the shockwave (at least, on one side of the cut), hence the respective phase-space is considerably reduced compared to that available to the diagrams in Fig. 11, where the position of the photon vertex is unconstrained.
This physical interpretation becomes transparent by inspection of the energy denominators corresponding to the three graphs in Fig. 1.For the first 2 graphs, in Fig. 1.a and 1.b respectively, there is only one energy denominator involving the gluon.In the soft limit ξ 1, the respective energy difference is dominated by the gluon light-cone energy p − = p 2 /2ξq + (recall Eqs.(4.3) and (4.4)): which shows that the longitudinal phase-space for the soft gluon emission is of order τ p ∼ ξ, as anticipated.However, the denominator corresponding to the graph in Fig. 1.c involves the product (E qq − E qqg )(E γ − E qqg ) of two large energy differences, hence that terms scales like τ 2 p ∼ ξ 2 and is strongly suppressed as ξ → 0.
After exploiting all the aforementioned simplifications and adding together the direct and interference contributions, the soft gluon limit of the trijet cross-section is obtained as where we have also ignored the soft momentum k + 2 in the δ-function for longitudinal momentum conservation.In the first two lines of this formula, we have factorised the would-be leading-order cross-section (3.17), but without the respective S-matrix structure (3.19).The third line encodes the information about the soft gluon emission: the emission kernels in transverse coordinate space and the colour structures corresponding to all possible topologies.Specifically, one finds The expression in the third line of Eq. ( 6.3) represents indeed the action of the "production" JIMWLK Hamiltonian on the original quadrupole Q(x, y, y, x) from Eq. (3.19).This can be verified e.g. by comparing Eq. (6.4) with Eq. (A.5) in Ref. [110].In making such a comparison, it is useful to recall that at large N c one can approximate 2α s C F α s N c ≡ π ᾱs .Furthermore, the standard version of the B-JIMWLK equation for the quadrupole [103][104][105] is easily recovered after integrating out both the transverse momentum k 3 (which leads to identifying z = z) and the longitudinal momentum k + 3 = ξq + (which introduces the measure factor dk + 3 = q + dξ) of the soft gluon.The result can be written as en evolution equation with ln(1/ξ), namely, whose r.h.s. is easily checked to precisely coincide with all the relevant terms in e.g.Eq. (2.17) from Ref. [104] (the terms which are missing in this comparison will be generated by the virtual NLO corrections to dijet production, as we shall see [107]).
7 The collinear limit: recovering the DGLAP evolution An another interesting limit of our previous results for the "real" NLO corrections to the dihadron production is the limit where the gluon is emitted in the final state and it is nearly collinear with its emitter.In this limit, the cross-section is expected to develop a logarithmic divergence from the integral over the transverse momentum of the gluon relative to its emitter.Physically, this divergence signals the fact that the gluon is emitted longtime after the interaction and should be viewed as a part of the wavefunction of the measured parton.For this interpretation to be correct, the collinear divergence should factorise from the "hard process" -here, the photon decay into a q q pair and the scattering between this pair and the nuclear target -and be recognised as the first step in the DGLAP evolution of the fragmentation function of the parent parton.In this section, we shall verify that this is indeed the case for the gluon emission by the antiquark.Since our results for the cross-sections are expressed in the transverse coordinate representation, it is convenient to also formulate the collinear limit in this representation.For definiteness, les us consider the case where the gluon is emitted by the antiquark.In coordinate space, the collinear limit for this emission is the limit where the transverse separation Y = y − z between the daughter partons (the antiquark and the gluon) is much larger than the separation R = x − y between the q q pair produced by the photon decay: |Y | |R|.To see this, it is convenient to change the transverse momentum variables for the final qg pair from k 2 and k 3 , to K and P , defined as Clearly, K is the total transverse momentum of the qg pair and is conjugated to the transverse position y of its center-of-energy (cf.Eq. (4.12)), whereas P is the relative transverse momentum of that pair and is conjugated to the transverse separation Y .The collinear limit corresponds to P → 0. (Indeed, when P = 0, the gluon takes a transverse momentum fraction equal to its longitudinal splitting fraction ξ/(1 − ϑ), meaning it is collinear with the antiquark; recall the discussion after Eq. ( 4.3).)The structure of the phases in Eq. (7.1) confirms that the collinear limit P → 0 corresponds to large values of Y = |Y |, as anticipated.In turn, this implies that the gluon must be emitted after the scattering with the nuclear shockwave, in order to escape the virtuality barrier.Indeed, as discussed in relation with Eq. (4.19), the amplitude for a gluon emission occurring prior to the collision -as described by the LCFW (4.21) (or, equivalently, by the first term in Eq. (4.25)) -is proportional to K 1 (QD), which implies Y 2 1/(ξQ 2 ).On the other hand, there is no such a restriction for a gluon emitted after the collision, cf.Fig. 1.b -the respective LCWF Eq. (4.21) (or the second term in Eq. (4.25)) is instead proportional to K 1 ( QR), which is independent of Y .This argument also shows that the instantaneous piece of the effective vertex (4.29) does not contribute to the collinear limit.Clearly, a similar argument holds for gluon emissions by the quark.
From the experience with the DGLAP evolution, we expect that the terms expressing interferences between gluon emissions by the quark and the antiquark (cf.Sect.5.1.2) should not contribute in the collinear limit.The respective argument is quite obvious in momentum space, since the gluon cannot be simultaneously collinear to both the quark and the antiquark (except for special configurations of zero measure).It is interesting to see how this argument works in the transverse coordinate representation.Consider a final-state gluon which is emitted by the antiquark in the direct amplitude (DA) and by the quark in the complex-conjugate amplitude (CCA).Since the final momenta are the same on both sides of the cut, one can use the momentum variables K and P defined in Eq. (7.1) (instead of k 2 and k 3 ) in both the DA and the CCA.In both cases, P is conjugated to the transverse separation between the gluon and the final antiquark -that is, to Y = y − z in the DA and, respectively, to Y = y − z in the CCA.The collinear limit for an emission by the antiquark corresponds to P ⊥ = |P | → 0, which implies that Y and Y are simultaneously large.In the DA, where the transverse position y of the intermediate antiquark is different from the final one y, the limit Y → ∞ brings no constraint on the transverse size R = |x − y | of the q q pair prior to the gluon emission 11 .But in the CCA, where the gluon is emitted by the quark, one has y = y, while the transverse coordinate x of the intermediate quark coincides with the center of energy of the final quark-gluon pair (cf.Eq. (4.30)).So, the distance R ≡ |x − y| is typically commensurable with Y and can be arbitrarily large.Accordingly, this interference term is strongly damped by the Bessel function K 1 ( Q R) with R ∼ Y 1/ Q (recall Eq. (4.34)).
To summarise, the dominant contributions to the tri-parton cross-section in the collinear limit come from direct gluon emissions (by either the quark, or the antiquark) in the final state (i.e. after the collision with the SW).E.g., when the emitter is the antiquark, these contributions correspond to the 4 graphs shown in Fig. 13.Their sum is contained in the last term, denoted as y, z → y ) & (y, z → y , in Eq. (5.2).The associated colour structure, shown in Eq. (5.13), is the same as at leading-order, cf.Eq. (3.19), which is a necessary condition for factorisation.
To complete the proof of factorisation in the collinear limit, one must demonstrate that the tensorial kernel K imjn 1 multiplying the fourth term in Eq. (5.2) factorises in this limit.To start with, let us check that the last piece, proportional to ε ij ε mn , in the product (5.5) of the effective vertices does not contribute to this limit.To that aim, it is convenient to change the integration variables in Eq. (5.2) from x, y, z to x, y , Y , so that the Fourier phases take the form in Eq. (7.1) (and similarly in the CCA).This is convenient since, in these new variables and for final-state gluon emissions, the Fourier transforms over Y and Y factorise from the other integrations 12  When this structure is multiplied with the squared vertex (5.5), the last term in the latter, which is antisymmetric in m and n, will clearly yield a vanishing contribution.Using the remaining part of Eq. (5.5), which is symmetric in m and n, together with the fact that QD → QR for final state emissions, one finds where are the DGLAP splitting functions for the processes γ → q q and q → qg, respectively.The factorisation of the collinear gluon emission from the hard process is now manifest: first, the virtual photon fluctuates into a q q pair with transverse coordinates x and y , respectively.Then this pair scatters off the nuclear SW, with the S-matrix shown in Eq. (5.13).Finally, the antiquark emits a gluon with splitting fraction z = ξ/(1 − ϑ).
Putting things together, one finds (after also integrating over the phase-space of the unmeasured gluon, which removes the δ-function for longitudinal momentum conservation and identifies z = z) dσ γA→qq+X where the longitudinal fractions ϑ and ξ are given by Eq. (5.21).At this level, it is convenient to replace the integration variables y and y by y and y , respectively (since the latter are the actual arguments of both the photon decay probability (e.g.R = x − y ) and the S-matrix structure), and also to use ϑ and z 2 as independent longitudinal momentum fractions, instead of ϑ and ξ.Here, z 2 is the splitting fraction of the daughter antiquark at that q → qg vertex: where we have also used P q→g (1 − z 2 ) = P q→q (z 2 ).Via these manipulations, the collinear divergence has been isolated in the last term and also factorised from the leading-order cross-section for the production of a pair of jets (cf.(3.17)): a quark with 3-momentum k 1 = (k + 1 , k 1 ) and an antiquark with 3-momentum p = (p + , p), where p + = k + 2 /z 2 and p = k 2 /z 2 .The collinear divergence refers to the large-|z| limit of the last integral in Eq. (7.7).To exhibit this singularity, it is convenient to make an excursion through momentum space and introduce a lowmomentum cutoff Λ on the transverse momentum of the unmeasured gluon, meaning an upper cutoff where r ≡ |y − y | and it is understood that Λ 1/r (since one is eventually interested in the limit Λ → 0).In the last step, we have split the result between a would-be divergent piece which is independent of r and which will be shortly absorbed into the DGLAP evolution and an r-dependent piece which is a part of the NLO correction to the hard factor.The factorisation scale µ is a priori arbitrary (within the range Λ < µ < 1/r), but ideally it should be chosen of order 1/r ∼ |k 2 |/z 2 , to minimise the NLO corrections.In practice though, it is more convenient to use µ 2 = max(k 2 1 , k 2 2 ), so that the same factorisation scale also applies to a collinear emission by the quark.
The would-be singular contribution to the last line in Eq. (7.7) describes the evolution of the final antiquark state via the emission of a quasi-collinear gluon.This piece must be subtracted from the NLO correction and absorbed into the DGLAP evolution of the (anti)quark fragmentation function into hadrons: where D (0) h/q (ζ) is the (non-perturbative) fragmentation function of a bare quark.Hence, after adding the collinearly-divergent piece of Eq. (7.7) to the LO dihadron cross-section, one finds an approximate version of Eq. (3.23) where the (first step in the) DGLAP evolution is included only on the final antiquark state: In particular, the δ-function for longitudinal momentum conservation inherent in the partonic crosssection, cf.Eq. (3.17), reproduces the normalisation factor in Eq. (7.7), via the following identity 13 : with ϑ = k + 1 /q + and z 2 as defined in Eq. (7.6).Clearly, the case of a collinear emission by the quark can be similarly treated and leads to the analog of Eq. (7.10) in which one step in the DGLAP evolution is included on the final quark state.

Figure 2 :
Figure 2: The 4 possible topologies for virtual corrections occurring in the initial state.

Figure 3 :
Figure 3: The 4 possible topologies for virtual corrections occurring in the final state.

Figure 4 :
Figure 4: The 4 possible topologies for virtual corrections in which the gluon crosses the shockwave.

Figure 5 :
Figure 5: (a) The quark-antiquark fluctuation of the virtual photon.(b,c) The 2 graphs contributing to the leading-order scattering amplitude in coordinate space.

Figure 6 :
Figure 6: Gluon emission by the antiquark (a) or by the quark (b), in the transverse momentum representation.

Figure 8 :
Figure 8: The three possible topologies in the case where the gluon is emitted by the quark.

Figure 9 :
Figure 9: Two direct contributions to the leading-order cross-section for q qg production.The gluon is emitted and reabsorbed by the same fermion: the antiquark in figure (a) and the quark in figure (b).

Figure 11 :
Figure 11:  The four diagrams which give the dominant contribution to the piece of the cross-section in (5.2) (direct emission by the antiquark) in the limit where the emitted gluon is soft (ξ 1).

. 4 )
The diagrammatic interpretation of the various terms in this lengthy expression can be easily traced back by inspection of their respective emission kernel and S-matrix structure.For instance the 4 terms proportional to(Y • Y )/(Y 2 Y2 ) correspond to the 4 graphs in Fig.11.The associated colour structures are recognised as the first terms in Eqs.(5.10), (5.11), (5.12), and (5.13), respectively (with y = y , of course).

Figure 13 :
Figure13: The four graphs surviving in the collinear limit for the gluon emission by the antiquark.These graphs are void of final-state interactions.
and are easily computed asY ,Y e iP •(Y −Y ) Y m Y n Y 2 Y 2 = (2π) 2 P m P n P 4 .(7.2)