Two-loop coefficient functions in deeply virtual Compton scattering: flavor-singlet axial-vector and transversity case

We calculate the two-loop flavor-singlet axial-vector and gluon transversity coefficient functions for deeply virtual Compton scattering in QCD. We observe interesting properties regarding the transcendentality of the transversity coefficient function. Our results complete the calculation of the full next-to-next-to-leading order coefficient function in deeply virtual Compton scattering. Numerically, the two-loop corrections in the axial-vector and transversity channel are comparable to their vector counterpart at moderate skewness parameter {\xi} and hence indispensable for analyzing the upcoming high-precision data from the Electron-Ion Collider.


Introduction
The generalized parton distributions (GPDs) [1][2][3][4] have a rich physical interpretation in terms of transverse spatial probability distribution of partons with a given longitudinal momentum fraction [5], and are essential for studying the decomposition of the proton spin [2] and various inter-and multi-parton correlations inside the hadrons.For the case of nucleon (or more generally spin- 1   2   hardons) targets at leading twist approximation, we can distinguish eight types of GPDs classifiable into three groups: H f , E f (vector), H f , E f (axial-vector), and H f,T , E f,T , H f,T , E f,T (transversity), where f = g, u, d, s, ... runs over parton species.They depend on the averaged parton momentum fraction x, the momenta difference in the forward light-like direction between in and outgoing hadron normalized to their total forward momenta, the so-called skewness ξ, and the square of the momentum transfer ∆ ⊥ projected onto the transverse plane.
The most studied GPD is the vector type H f , whose Fourier transform with respect to ∆ ⊥ at zero skewness gives the impact parameter distribution of an unpolarized parton in an unpolarized nucleon.However, information on other GPDs is necessary to construct a complete picture of the inner structure of the nucleons.In particular, in order to get the impact parameter distribution of a parton of arbitrary polarization in an arbitrarily polarized nucleon, all GPDs are required [6].Another motivation for studying the complete set of GPDs is that their contributions to DVCS can be on the same level for certain observables at leading power, so good control over all channels is required to extract GPDs to a decent accuracy.
The deeply-virtual Compton scattering (DVCS) [2,3], is considered the "golden" channel for experimentally accessing GPDs 1 .All types of leading-twist GPDs enter in DVCS observables at leading-power besides the quark transversity GPDs H q,T , E q,T , H q,T , E q,T , which are forbidden for spin-1/2 targets by selection rules in the hard scattering.This implies that the transversity (also known as maximal helicity/double flip sector) channel provides a unique window into the transversity gluon distribution inside the nucleons without "contaminations" from quarks.The GPDs modulate DVCS observables in a convolution with hard coefficient functions (CFs), which can be calculated in perturbation theory as a series in the strong coupling α s .For DVCS five CFs are required at leading-power, denoted by C q , C g , C q , C g , C g,T .They were first calculated to next-to-leading order (NLO) in [11][12][13][14][15].The next-to-next-to-leading order (NNLO) correction was calculated for the flavor nonsinglet contribution to C q in [16] and to C q in the MS-scheme in [17,18].C g and the flavor singlet contribution to C q were calculated in [19] soon after in MS as well.In this work, we complete the NNLO studies of DVCS CFs by calculating C g , the flavor singlet contribution to C q , and the gluon transversity CF C g,T to the two-loop order in QCD.
This work is organized as follows.In Section 2 we briefly introduce the kinematic definitions for DVCS.In Section 3 we discuss the decomposition of DVCS hadronic tensors into the vector, axialvector and transversity contribution.In Section 4 we describe the non-trivial details of the matching procedure.The non-triviality comes from the axial-vector sector, where a consistent continuation to d = 4 − 2ϵ dimensions must be made without relying on intrinsically four dimensional objects such as γ 5 or the Levi-Civita tensor.For the Levi-Civita tensor a natural generalization is to antisymmetrize the indices that would be contracted with the Levi-Civita tensor in four dimensions.For γ 5 there is a greater deal of arbitrariness.In this case it is consistent to replace γ + γ 5 by an anti-symmetrized combination of three gamma matrices, with one plus and two transverse indices.This corresponds to Larin's scheme [20], which can be implemented into the loop calculations in a straightforward fashion.In Section 5 we discuss how one can convert to another possible scheme for γ 5 , the so-called naive-dimensional regularization (NDR) scheme, where the vector and axial-vector evolution kernels coincide.Then in Section 6 we present the main results, the two-loop CFs for the axial-vector and gluon transversity case.Details of the calculation, regarding the additional subtractions to the partonic amplitudes, can be found in Appendix A. We briefly discuss the weight reduction of tranversity CF in Appendix B.

Preliminaries
We consider the DVCS process where γ * (q) (γ(q ′ )) dentotes a virtual (real) photon with momentum q (q ′ ).N (p, s) is a hadron state carrying momentum p and helicity s.Most commonly N is a proton in experimental studies, but we do not need to require this here, since the CFs do not depend on the target hadron.We introduce the standard momentum variables and Lorentz scalars We use light-cone coordinates with respect to two light-like vectors n, n with n 2 = n2 = 0 with normalization condition n • n = 1.Without loss of generality we can define For a generic vector V we define We will commonly denote a vector V in terms of its light-cone components by (2.5) It is useful to choose a preferred frame of reference to analyze the process.Our choice of frame in this paper is defined by requirement that q ′+ , q ′ ⊥ , P ⊥ = 0 [21] which is most suitable for studying leading-power contributions2 .In particular, q ′ is taken to be proportional to the light-like vector n, whereas the nucleon momenta are proportional to n in the limit of m 2 , t = 0. Explicitly ) Here the transverse components of ∆ ⊥ are defined only up to rotations in the transverse plane with the constraint that In this work we use the following notation for a perturbative, dimensionally regularized quantity (2.8)

Decompositon of the hadronic tensor
The hadronic part of the DVCS amplitude is given by the hadronic tensor where j µ (x) = q e q ψq (x)γ µ ψ q with the sum running over active quark flavors q = u, d, s, ... of electric charge e q .T µν is constrained by the electromagnetic Ward identity This immediately implies that T µ+ = 0 following our convention.Furthermore, the leading twist approximation gives Thus from (3.2), we obtain, It will be convenient to write the projector onto the transverse plane in the following covariant form Using eq.(3.2) gives + higher-power.
A few comments are in order.Firstly, note that the structures proportional to n ν do not contribute in DVCS.Indeed, the hadronic tensor will enter the scattering amplitude contracted with the physical polarization vector ε * ν (q ′ ) of the outgoing photon, which is in the transverse plane, so that ε * ν (q ′ )n ν = 0. Secondly, the projection g νν ′ ⊥ T − ν ′ is of subleading power.To see this, note that the ν ′ sits at a hard scattering vertex, where the leading twist approximation m 2 = t = 0 applies, which implies that all the momenta p, p ′ , q, q ′ have zero transverse component.The leading term in T − ν ′ on the other hand, having only a single Lorentz index, must therefore be proportional to external momenta with zero transverse component, so that the leading contribution to g νν ′ ⊥ T − ν ′ vanishes.Hence we conclude A further decomposition of the tensor product g µµ ′ ⊥ g νν ′ ⊥ can be made in terms of the following Lorentz structures Then we have trivially (3.10) Note that the three basis tensor structures in eq.(3.10) are mutually orthogonal and we have It is easy to see that in d = 4 where and ε µνµ ′ ν ′ is the four-dimensional Levi-Civita tensor with convention ε 0123 = 1, so that ε 12 ⊥ = 1.Thus the decomposition eq.(3.10) reduces to the conventional four-dimensional decomposition used in [15] In four dimensions we can use the decomposition in eq.(3.14) to get where are the vector, axial-vector and transversity Compton form factors (CFFs) respectively.To the leading-power accuracy these contributions can be written in terms of a convolution of CFs with leading-twist GPDs [4,28,29], Here µ UV is the UV renormalization scale that appears in the Lagrangian and enters the scale dependence of α s .µ F is the factorization scale that separates degrees of hard modes of virtuality ∼ Q 2 from the collinear modes of virtuality ∼ t, m 2 .Note that V, A, T µν being given by hadronic matrix elements of conserved currents, are finite and scale independent.In practice, however, these contributions do have scale dependence due to finite α s truncation in pQCD calculations.
The degree of the scale dependence therefore measures the uncertainty of the pQCD predictions.Throughout this work, we will set µ UV = µ F = µ and we will frequently omit the functional dependence on µ for notational brevity.The GPDs can be defined in terms of renormalized hadronic matrix elements of light-ray operators where ψ q is a quark field of flavor q (not to be confused with the virtual photon momentum), G µν is the gluon field strength tensor, and with W n,F (W n,A ) being a collinear Wilson line in the fundamental (adjoint) representation ensuring gauge invariance of the nonlocal operators.
The standard definition of the renormalized GPDs in d = 4 reads For spin-1 2 targets, the GPDs in eq.(3.20) can be further decomposed into the GPDs , mentioned in the introduction, see e.g.[30,31] for details.

Dimensional regularization
To calculate the CFs we use the so-called "matching" procedure, where we evaluate both sides of eq.(3.17) using on-shell external partonic states.There are two types of divergences that appear in the perturbative expansion of the on-shell partonic amplitudes, which are commonly referred to as "bare" CFs.
• UV divergences, which are removed by rewriting the perturbative series in terms of the renormalized coupling.This is sufficient if the external parton propagators are removed according to the Lehmann-Zimmermann-Symanzik formula.In case the renormalized Lagrangian is employed to compute the CFs, the UV divergences would disappear.In our studies, we instead adopt the bare Lagrangian for computational simplicity.
• IR divergences due to having massless and collinear partons.They are removed from the terms corresponding to the renormalization of the GPD.In the graphical treatment, they correspond to the so-called double-counting subtractions.
The divergences are most conveniently regularized using dimensional regularization (dim.reg.) with d = 4 − 2ϵ and subtracted using the MS prescription.It is also most convenient to treat UV and IR divergences using the same ϵ = ϵ UV = ϵ IR .The matching procedure is covered in detail in Appendix A.
In order to use dim.reg., we need to choose a d-dimensional representation for T µν .As usual, this introduces an ambiguity, which must be treated using a consistent scheme.Firstly, we use the decomposition in eq.(3.10).Since everything is written only in terms of g µν ⊥ it can be naturally extended to d dimensions by augmenting the transverse space from 2 to d − 2 dimensions, such that −g µν ⊥ is the unit matrix in that space.This gives where the index f = g, u, d, s, ... denotes the parton species of the external state and the hat indicates that we are dealing with dimensionally regularized partonic quantities.The partonic vector, axial-vector and transversity contributions are then given by The factorization theorem in terms of the partonic quantities reads with the same CFs as in eq.(3.17), since the hard scattering does not depend on the external state.After replacing p, p ′ with the leading twist momenta, the partonic GPDs are defined in the same way as in eq.(3.20), with the exception of the axial-vector sector.We choose which are immediately generalizable to generic d-dimensions advantageous for loop calculations in dim.reg.In four dimensions the definitions in eq. ( 4.4) coincide with the definitions in eq.(3.20) up to a factor of ε µν ⊥ , i.e. they can be recovered by taking µ = 1 and ν = 2.For the gluon case, this is immediately seen by using eq.(3.12).For the quark case, note that This choice to represent γ 5 in d dimensions corresponds to Larin's scheme [20].We mention that recently there has been some work on DVCS [38,39] that calculates the one-loop CF using a finite t regulator.For the bare CF one finds a simple pole 1 t contribution in the vector and axial-vector case, connected to the trace and chiral anomaly respectively.This pole is no contradiction to factorization, since all such IR divergences in the bare CF are removed by double-counting subtractions.Indeed, the authors find that, after applying these subtractions, all singularities as t → 0 are removed, so that after the Taylor expansion one obtains a finite renormalized CF.In particular, the renormalized CF does not depend on the regularization procedure 3 .This is of course perfectly in line with our treatment, since dim.reg.already regulates all IR divergences in the bare CF.For a discussion of the double-counting subtractions for DVCS at two-loops we refer to Appendix A.

Relation to naive-dimensional regularization scheme
Another commonly used choice for γ 5 is the naive-dimensional regularization (NDR) scheme, often simply called "MS-scheme", in which the vector and axial-vector flavor-nonsinglet evolution kernels coincide.This condition comes naturally if a renormalization scheme other than dim.reg. is used.It is clear that adopting the NDR scheme will not lead to any algebraic inconsistencies since γ 5 is absent in the Dirac traces in QCD.
It is possible to convert the CFs between different schemes by enforcing the matrix elements of the so-called evanescent operators to zero.These operators are the differences between operators describing the same physical object but in different schemes.Hence they coincide in d = 4 but are distinct for d ̸ = 4, e.g., the two Dirac structures in eq.(4.5).The condition that the matrix elements of the evanescent operators must vanish implies (finite) renormalization constants allowing one to translate the CFs from one scheme to another.Let us consider the γγ * → π 0 transition form factor as a concrete example 4 .The evanescent operator necessary to relate the nonsinglet (NS) CFs in Larin and MS scheme takes the form [18,40], with γ 5 subscribing to the NDR prescription.The first and second terms in the parentheses correspond to definitions of axial-vector in Larin and MS scheme, respectively.It is clear that in d = 4, the operator O µν NS,E is identically zero due to simple Dirac algebra.However, in d = 4 − 2ϵ dimensions, the operator is nonvanishing and proportional to ϵ. Enforcing the partonic matrix element of the IR-finite UV-renormalized evanescent operator to vanish identically in four dimensions to all orders in perturbation theory [41,42], allows us to determine its renormalization constant Z NS,E which is currently known to the O(α 2 s ) by direct calculation [18].Z NS,E is then used to translate the nonsinglet axial-vector CF from Larin scheme (L) to the MS scheme.At O(α s ), we have, NS,E , with higher-order expressions taking a similar but more complex form.The one-loop renormalization constant Z NS,E for the evanescent operator O µν NS,E is finite in ϵ and * denotes the convolution over partonic momentum fractions.For more technical details, we refer [18,40,43] and references therein.
For our current case, to convert our Larin scheme results for both C q (x/ξ, Q) and C g (x/ξ, Q) to the MS scheme require computing the singlet evanescent contribution to the two-loop order.This is beyond the scope of our current paper and therefore we leave it for future studies.

Results for the coefficient functions
In the following we set d = 4.It will be convenient to introduce the variables for presenting our main results.In this way, at the tree-level the CFs read 2) The one-loop contributions are [11][12][13][14][15]] The calculation procedure is completely analogous to the vector case in [19].For the evaluation of Dirac traces and Lorentz contractions FORM [35] as well as in-house routines were used.The resulting set of scalar integrals was reduced to twelve master integrals making use of the integrationby-parts relations, performed with FIRE [36].No new master integrals appeared compared to the vector nonsinglet case.They were first calculated in [18].For the calculation of the doublecounting/IR subtractions, see Appendix A, the program HyperInt [37] and in-house routines were used for this task.

Two-loop axial-vector CFs
This presents the main results of our paper.The vector contributions C (2) g have been calculated in [19] and we do not repeat the results here.We organize the two-loop axial-vector and transversity contributions in the following way NS (z, L) where "PS" stands for " pure-flavor singlet".The contributions C NS have first been calculated in [17,18], but the results were given only in the NDR scheme.We present our results in Larin's scheme in terms of harmonic polylogarithms (HPLs) [45][46][47] in the following 5 with the argument z for all HPL functions H ⃗ m omitted for brevity.In Figure 1 we have plotted the real and imaginary parts of C q and C g as a function of x/ξ.The higher order corrections look almost identical to the vector case [19].We therefore expect the corrections to the Compton form factors H, E to be of the same size as the corrections to H, E, which were studied in [19] using the Goloskokov-Kroll model [44] for the unpolarized GPDs H q and H g .A more detailed study is left for future work.
We remark that, just as at NLO, the axial-vector CF and vector CF have identical leading terms as z → 0, that is q e 2 q 12z log 3 z.

Two-loop gluon transversity CF
The two-loop gluon transversity CF reads Apparently, this function has transcendental weight reduced by two compared to the vector and axial-vector sectors.From the perspective of loop calculations, such a reduction in complexity, both in transcendentality and the form of the expression, is highly nontrivial considering the fact that it is necessary to compute the same set of Feynman diagrams (with Lorentz projector in (3.9)) for C g,T as for the vector and axial-vector case.The diagrams individually may have nonvanishing contributions to the transversity CF up to transcendental weight four.In other words, the reduction in the weight only occurs when all two-loop diagrams are summed up.At two-loops, this can be seen without any explicit calculation, as a consequence of properties of the two-loop master integrals and the factorization theorem, which guarantees the cancellation of IR singularities in the CF.See Appendix B for a detailled argument.
For a numerical estimation, we have plotted in Figure 1 the real and imaginary parts of C g,T on the real axis.The results are as expected for a first-order correction.

Conclusion
We have presented the two-loop coefficient functions responsible for the flavor singlet axial-vector and gluon transversity contributions in DVCS, which in combination with the previous publications, completes the full NNLO CFs responsible for the leading-power contributions in DVCS.
The calculations are analogous to the vector case, which was considered in [19].The interesting aspects are in the use of dimensional regularization.In terms of the decomposition of the transverse photon indices, the most straightforward continuation given in eq.(3.10) was used.The unavoidable ambiguity concerns the treatment of γ 5 .For the calculations in this paper we have employed the Larin's scheme and explained how the result may be converted to the NDR scheme in Section 5.
We leave the task of making such a conversion for future work.
Our plots of the coefficient functions at varying orders of α s were shown in Figure 1.With regards to the NNLO corrections, no new features compared to the vector case can be observed.We therefore expect the size estimations for the NNLO corrections to the corresponding Compton form factors to be analogous to [19] for particular DVCS channels.A dedicated phenomenological study, including the theory uncertainties in the form of the scale dependence of the Compton form factors, is required to fully assess the NNLO impact in the current (JLAB-12) and future (EIC) experiments.We leave this as a future project.
We observe that, up to two loops, C g,T has transcendental weight reduced by two compared to the vector and axial-vector case.In Appendix B we provide an explanation for the weight reduction up to two-loops.It would be interesting to see if such reductions hold to higher orders.This gives For the axial-vector and transversity case the situation is different, since we have two free Lorentz indices.When choosing external states we will introduce two additional Lorentz indices.Then the resulting factorization formulas in eq. ( 4.3) can be projected with the corresponding rank 4 tensor, χ ⊥ for axial-vector, τ ⊥ for transversity, giving a factorization in terms of scalar quantities.
For axial-vector we take ⟨out| (...) µν |in⟩ → lim for the quark and gluon case respectively.This gives where now there are no free indices compared to eq. (4.4), because they were projected out in the prescriptions in eqs.(A.6) and (A.7).

UV divergences
The partonic tensor T f (dropping the Lorentz indices for simplicity), amputated with the prescriptions eqs.(A.2), (A.4), (A.6), (A.7), (A.10) is formally UV divergent, since we defined the Greens function in terms of bare fields.To make it UV finite we should multiply by inverse wave-function renormalization Z-factors to turn the bare into renormalized fields.We are however only interested in the CFs, which do not care about the external states.After all, multiplying by the wave-function renormalization factors corresponds to multiplying both sides of eq. ( 4.3) by the same constant.In fact, since we do not need to distinguish between ϵ UV and ϵ IR , it is more convenient to not multiply by Z-factors.Any external leg correction is a vanishing scaleless integral and therefore the residues in the poles of the external propagators at / p, / p ′ = 0 are unity.
This leaves us only with the renormalization of the coupling constant Expanding the partonic tensor in terms of both the unrenormalized and renormalized coupling gives f .(A.14) In particular, there is a finite subdivergence subtraction term, which is given by −β 0 T (1,1) f .We remark that although such a subtraction is present also in the gluon case, the two-loop gluon CF turns out to not depend on β 0 .This is because in the matching there is an additional contribution proportional to β 0 , due to the gluon GPD being defined in terms of renormalized gluon fields, which exactly cancels this term.We will come back to this below, see eq. (A.28).

Partonic GPDs
Without renormalization, the GPD with partonic on-shell external states would be given exactly by eqs.(A.3), (A.5), (A.8), (A.9), (A.11) without the O(α s ) terms, since all loop integrals are scaleless.Furthermore, the off-diagonal partonic GPDs would be zero.Formally, this means that the UV and IR singularities "cancel out".But strictly speaking, the factorization formula in eq.(3.17) is formulated in terms of the renormalized GPDs.
After renormalizing the GPDs, only the IR singularities remain, giving purely IR divergent higher order terms to the partonic GPDs.The renormalization Z-factor of the GPDs can be reconstructed from the evolution kernels, which are known to two-loop accuracy [15,48].A complete list of the one-loop evolution kernels in position space can be found in [31], section 4.3.2.
At one-loop we have Z where K f f ′ are the evolution kernels for the mixing of a parton of species f to a parton of species f ′ .Note that Z f f ′ defined here includes a factor of Z −1 A for f ′ = g and Z −1 q for f ′ = q, which comes from the GPD being defined in terms of renormalized fields, such that is given by the tree-level expression, so O R f is entirely determined by Z f f ′ .We list the complete set of one-loop partonic GPDs in terms of position space kernels.For notational simplicity we let • Axial-vector F q/q (x, ξ) = F q/q (x, ξ), F g/q (x, ξ) = −

Double-counting subtractions
To illustrate the matching procedure, consider as an example the quark vector case.The matching for axial-vector and gluon transversity sector are completely analogous.The tree-level factorization formula gives and therefore

B Weight reduction of transversity CF
In this section we discuss the origin of the weight reduction by two, observed for the transversity gluon CF.Let us consider the matching for C g,T .In the following we use the notation introduced in Appendix A. The factorization formula dictates that the UV-renormalized and double-counting subtracted C (2) g,T in d dimensions takes the form6 , C g,T = α s 4π C (1,0) g,T + α s 4π , where n ≥ 3 and i = q, g persist to higher orders in α s .We leave this question for future work.