Collinear factorization of diphoton photoproduction at next to leading order

We calculate in the framework of collinear factorization the amplitude for the photoproduction of a near forward large mass diphoton at leading twist and next to leading order (NLO) in $\alpha_s$. We demonstrate the validity of factorization at this order, which was never achieved for such a reaction where the coefficient function describes a $2 \to 3$ hard process. While the Born order amplitude was purely imaginary and only probed the $x=\pm \xi$ cross-over line of generalized parton distributions (GPD) domain, the NLO result contains both a real and an imaginary part and probes the whole domain of definition of quark GPDs. The phenomenology of our results for medium (JLab) and higher energy (EIC) experiments will be developed in a future study.


I. INTRODUCTION
In our recent works [1,2] we studied at leading order (LO) in the QCD coupling constant α s the photoproduction of a large mass diphoton on a nucleon target γ(q, ) + N (p 1 , s 1 ) → γ(q 1 , 1 ) + γ(q 2 , 2 ) + N (p 2 , s 2 ) , in the kinematical domain suitable to a factorization of generalized parton distributions (GPDs) and a hard amplitude. This domain is defined by large diphoton invariant mass and center of mass energy and by a small momentum transfer squared t = (p 2 − p 1 ) 2 between the initial and final nucleons. This reaction, which superficially looks like timelike Compton scattering (TCS) [3][4][5][6][7] with the dilepton replaced by a diphoton, has a quite different structure since the hard amplitude describes a 2 → 3 process, contrarily to more familiar reactions -deeply virtual Compton scattering (DVCS), deeply virtual meson production (DVMP) or TCS -which involve a 2 → 2 process. While other 2 → 3 processes have already been studied at leading order [8][9][10][11][12], assuming QCD collinear factorization at leading twist, there is yet no proof of this very basic property, even at the next to leading order (NLO) in α s . It is thus deemed appropriate to calculate at NLO in α s the amplitude of the simplest 2 → 3 process, namely the photoproduction of a diphoton (1). This is the aim of this paper where we demonstrate the factorization at NLO of the amplitude, showing in particular that the divergent parts of the NLO amplitude can be absorbed in the GPDs. We use dimensional regularization as the main tool of our proof. The phenomenological consequences of our NLO description of process (1) will be analyzed in a forthcoming publication.
In Sect. II we present the kinematics of our process, then we briefly recall in Sect. III the structure of a factorization proof at NLO. Section IV discusses in some details the calculation of the vector GPD contribution for our process. In Section V we discuss the corresponding results for the axial amplitude. Section VI contains some remarks and conclusions. Some of the results of this paper have been already presented in [13,14].

II. KINEMATICS
We use the following light-cone four-vectors: arXiv:2110.00048v2 [hep-ph] 5 Oct 2021 the parameter s = 2p · n will be described in more detail later. We parametrize the momenta in the Sudakov basis: Here, M is the mass of the nucleon, ξ is the skewness variable and v u denotes the Euclidean product of transverse vectors. Polarization vectors of the photons are written in the · p = 0 gauge as: The process can be described in terms of the following 4 invariants: For further convenience we relate them to other invariants: At leading twist, the calculation of the hard part uses the approximation ∆ ⊥ = 0, justified by the fact that in the considered kinematics one has |p 2 ⊥ | |∆ 2 ⊥ |. The simplified kinematical relations then read: To reduce the number of considered diagrams, we will consider a process with three incoming photons with momenta (k 1 , k 2 , k 3 ) and polarisation vectors ( 1 , 2 , 3 ), and then sum the amplitudes over (k 1 , 1 ), (k 2 , 2 ), (k 3 , 3 ) corresponding to all permutations of (−q 1 , * (q 1 )), (−q 2 , * (q 2 )), (q, (q)) . We introduce the parameters β i , κ i defined as: It is easy to check that κ i = 2ξβ i . Scalar products of corresponding momenta are: Adding the different diagrams amounts to let the β i and κ i parameters be subject to the following permutations: In the calculation of the hard part it is assumed that p 1 − p 2 ≈ 2ξp, so that 2ξp = q 1 + q 2 − q. Hence, for each polarization vector i : Moreover, k 1 1 = k 2 2 = k 3 3 = 0. In the considered gauge, only the transverse parts of polarizations vectors contribute to their products with momenta. We have:

III. FACTORIZATION
In the collinear factorization regime, the scattering amplitude of our process (1) reads: where F q B and F q B are bare vector and axial quark GPDs; T q B and T q B are bare vector and axial coefficient functions containing infrared divergences at O(α s ), and are written as: where M is the qγ → qγγ hard scattering amplitude with the quark spinors removed.
Only nonsinglet quark GPDs contribute to our process and the O(α s ) relation between the bare and renormalized GPDs in the MS scheme reads: where µ F and µ R are respectively the factorization and renormalization scales, and γ is the Euler constant. The cancellation of divergences -a necessary requirement for a consistent factorization framework -occurs, if: which leads to: where: and the corresponding equation for the axial amplitude.

IV. THE VECTOR AMPLITUDE
Let us first study the vector GPD contribution to the scattering amplitude.

A. Lowest order analysis
At leading (i.e. zeroth) order in α s , the contribution to the vector amplitude coming from the diagram shown in Fig. 1, with a given permutation of photons, reads [1]: where we denote i0 i = i0 · sgn(β i ) and define the trace structure: since it turns out that it appears in all diagrams. The full amplitude of the hard subprocess is obtained by summing over all permutations of photons k i : The trace structure has the important property: perm.
. The only terms proportional to the imaginary part of propagators which survive, are those from diagrams in which the incoming photon is the middle one (so that k 2 → q). Finally: where: This reproduces the result of [1].

B. Self-energy insertions at O(αs)
The contributions of self-energy diagrams shown in Fig. 2 for the left panel diagram, and for the right panel diagram. The left and right vertex corrections, shown in Fig. 3 denoted respectively as M 3.L and M 3.R , read: The middle vertex correction, shown in Fig. 4, has a more complicated structure. Introducing D 1 = (x + ξ)β 1 , D 3 = −(x − ξ)β 3 , we can write it as: where: while the divergent part of the second box diagram reads: The finite contribution resulting from these box diagrams is lengthy and will be included in formulas presented in Sec. IV G. The pentagonal contribution to the amplitude, resulting from the diagram presented on Fig. 6, reads: wherep = (x + ξ)p, p = (x − ξ)p and: Observing that 2(pk) = 1 2ξ and using the following gamma-identities inside the trace: γ µ / pγ µ = (−2 + ε) / p, / k / p/ k = 2(pk)/ k − k 2 / p, allows us to obtain the following identity: and to write the amplitude as the sum of two 4-point integrals: T r iM 5 1,2,3 / p = T r iM 5.L 1,2,3 / p + T r iM 5.R 1,2,3 / p , where The names 5.L and 5.R come from the structure of denominators in integrands, which are the same as in amplitudes 4.L and 4.R. The divergent part of the amplitude reads: The finite contribution resulting from this pentagonal diagram will be included in formulas presented in Sect. IV G.

F. Total divergent part
Summing the various contributions written above in Eqs. (26, 27, 28, 29, 30, 31, 32) and (38) we can write the divergent part of the amplitude corresponding to a given permutation of photons entering the hard part as: which after the summation over permutations gives (see the definition (14)): The resulting expressions for C q 0 and C q coll are, up to a common constant factor, equal to the imaginary part of the corresponding expressions in DVCS (see Eq. (13) in [6]). The non-singlet and singlet kernels K qq are at the α S order equal to each other [15] and real, hence the condition for the cancellation of divergences expressed by Eq. (17) in the case of the considered process is just the imaginary part of the corresponding expression for DVCS.

G. The finite part of C1
The finite parts of the contributions of self energy and vertex diagrams to C 1 are explicitly written in Eqs. (26)-(30). The finite parts emerging from the box and pentagon diagrams are more cumbersome and it is convenient to write them in terms of functions defined by the following equations: It is worth stressing that the integrals in the above equations are finite. Using above definitions, and omitting for brevity the argument of functions F nab and G, we can write the finite part of the box diagram to which this permutation contributes as:

V. THE AXIAL AMPLITUDE
We shall not detail the corresponding analysis for the axial amplitude, which follows the same lines as the vector part, but only quote the main results. The axial LO amplitude corresponding to a given permutation of photons reads [1]: After summation over all permutations of photons one gets: where: ᾱT r / (q) / q/ * (q 2 ) / q 1 / * (q 1 )γ 5 / p + αT r / (q) / q/ * (q 1 ) / q 2 / * (q 2 )γ 5 / p (47) At the first order in α s , an analysis very similar to the one discussed above for the vector part, yields a divergent part equal to: which allows to check factorization by explicit computation, the coefficient function divergences being exactly cancelled by those related to the evolution equations of the axial GPDs.
Since the smallness of the axial contribution has been demonstrated in [1] for the LO computation, we do not write down here the lengthy expressions for the finite part of the NLO axial amplitude. They have an analogous structure as those in the vector part, including the presence of the same functions F nab and G, the only difference being the traces.

VI. CONCLUSION
Our results on the NLO analysis of the scattering amplitude of process (1) is a first step toward a complete proof of QCD collinear factorization for the amplitude of such processes where the coefficient function describes a 2 → 3 hard scattering. We shall soon continue this study by examining the phenomenological consequences of our NLO results, both for reactions at electron machines at medium (JLab) or high energy [17,18] and in ultraperipheral reactions at LHC [19].