One-loop evolution of twist-2 generalized parton distributions

We revisit the evolution of generalised parton distributions (GPDs) at the leading order in the strong coupling constant α s for all of the twist-2 quark and gluon operators. We rederive the relevant one-loop evolution kernels, expressing them in a form suitable for implementation

GPDs provide a 1+2 dimensional picture of the partonic structure of hadrons related to both the longitudinal-momentum and the transverse-spatial distribution of partons [18].Furthermore, the second Mellin moments of unpolarised GPDs give access to the so-called Ji spin sum rule [2] and the D-term, which encode information on the mechanical properties of hadrons [19][20][21][22][23].In general, GPDs contain a wealth of information on hadron structure that has led the upcoming Electron-Ion Collider (EIC) [24] and the JLab22 upgrade [25] to make the study of GPDs a cornerstone of their research programs.
However, the phenomenological study of GPDs presents us with many challenges.One of the primary hurdles in extracting meaningful information from experimental data lies in the intricate structure of the factorisation theorems, which renders the analysis of GPDs considerably more difficult than that of PDFs.A pivotal aspect in the exploration of GPDs concerns the ability to disentangle their dependence on both the partonic longitudinal-momentum fraction x and the skewness ξ.This task proved exceptionally challenging in that the convolution of GPDs with the partonic cross sections involved in the computation of CFFs intertwines these variables, preventing a straightforward separation.The longstanding belief that evolution effects may help achieve this separation was finally disproven in Ref. [26], where the concept of "shadow" GPDs, i.e.GPDs having an arbitrary small imprint on CCFs, was introduced.However, more recently Ref. [27] revived this debate.Nonetheless, it is broadly accepted that a solid extraction of GPDs should not only rely on DVCS data but rather on a simultaneous analysis of different processes, as routinely done for PDFs.
This underscores the rationale behind the substantial efforts invested in recent years to derive the perturbative structure of both the CFFs and the evolution kernels governing the evolution of GPDs [13-17, 28, 29, 29-38].This pursuit has driven extensive investigations into the computation of higher-order results, primarily employing conformal-space techniques.This approach, not only enhances computational efficiency, but also offers an independent alternative to the more traditional Feynman-diagram-based methods.
On the phenomenological side, comparatively much less effort has been devoted to developing GPD evolution codes.The first leading-order (LO) momentum-space evolution code was developed by Vinnikov and presented in Ref. [39].However, the only surviving public version of this code is available only through PARTONS [40].Freund and McDermott [41] implemented a version of GPD evolution specifically tailored for DVCS at next-to-LO (NLO) accuracy.However, the code is not fully open-source and cannot be easily obtained.On the other hand, a public, open-source implementation of GPD evolution in conformal space at NLO also exists [28,42,43].
All codes mentioned above are typically able to evolve only specific models or class of parameterisations, and can thus hardly be used out-of-the-box with arbitrary input GPDs.This is a significant shortcoming in view of a possible extraction of GPDs from experimental data based on flexible parameterisations.Another disadvantage of these codes is that none of them includes heavy-flavour threshold crossing.This is a limitation because much of the current experimental data lies significantly above the charm threshold and, all the more so, the future EIC will deliver data that will require charm and bottom to be treated as active flavours.It is therefore clear that the lack of open-source public codes to perform GPD evolution in momentum space without any assumption on the initial-scale model is now becoming a bottleneck.
With this work, we aim to provide a fully open-source implementation for all of the twist-2 GPD evolution equations in momentum space with no a-priori assumptions on the input models and allowing for heavy-flavour threshold crossing.We extend the work of Ref. [44], devoted to the one-loop evolution of unpolarised GPDs, re-computing and implementing the evolution kernels for longitudinally polarised quarks and gluons, and for transversely polarised quarks and circularly polarised gluons.These quantities had already been computed (see e.g.Refs.[7,45]), but a numerical implementation only exists for longitudinally polarised partons [39] (with the limitations discussed above) and was absent so far for transversely polarised quarks and circularly polarised gluons.The implementation of the full set of twist-2 evolution equations is made public through the code APFEL++ [46,47] and available within the numerical framework PARTONS [40].
The paper is organised as follows.In Sect.2, we give a brief overview of definitions and conventions on both the structure of GPDs and their evolution equations.The explicit form of the leading-order kernels is given in Sect. 3 and presented in a form that is well-suited for numerical implementation.In the same section, we also discuss some relevant properties of the kernels.In Sect.4, we present some numerical results of our implementation, and finally in Sect. 5 we draw some conclusions.

Definitions
Let us first start with a summary of our notation.We will denote scalar products as a µ b µ ≡ (ab).We introduce two light-cone vectors, n and n, such that n 2 = n2 = 0 and (nn) = 1.We parametrise the transverse space to n and n in terms of two vectors R and L that satisfy the normalisation conditions: In addition, we will use the short-hand notation: The bare quark and gluon GPD correlators associated with the hadron H are defined in terms of off-forward matrix elements of the collinear operators as follows (see e.g.Ref. [8] for * An explicit parametrisation of all these vectors is: basic definitions): where ξ = −∆ + /(2P + ), t = ∆ 2 , and W is the Wilson line defined as: where P denotes the path ordering.The representation of the color-group generators T c is the fundamental one in the quark case and the adjoint one in the gluon case.In the hadronic states, Λ and Λ ′ denote the helicity states of the incoming and outgoing hadron H, respectively.The relation between the spin vector S and the helicity Λ for a state of momentum p and mass M is given by: At twist-2, three relevant projections are to be considered for both quarks and gluons: unpolarised, longitudinally polarised, and transversely (quark)/circularly (gluon) polarised.For the quark operator, the three cases are projected out as follows: with: Γ ∈ {/ n, / nγ 5 , in β σ αβ γ 5 } .
In the gluon case, the projection is instead defined as: where the tensor Γ µν is to be selected amongst the following structures: where we use the convention ε 12 T ≡ nα n β ε αβ12 = +1.The evolution kernels do not depend on the specific external states and therefore they are independent of how the correlators Ff←H are parametrised in terms of the single scalar GPDs.In principle, one could study the evolution equations in position space [48], where the independence of the external states is made transparent.This observation also implies that evolution equations for transition GPDs [9,49] are identical to those for standard (flavour diagonal) GPDs.
The goal of this work is the evaluation of the one-loop (leading-order (LO)) evolution kernels in momentum space for all of the three twist-2 GPD correlators introduced above.Although these quantities are already known in the literature, † we aim to achieve an efficient numerical implementation and to lay the foundations for a systematic Feynman-graph approach to the computation.For this reason, we choose to work in light-cone gauge, which allows us to consider a significantly smaller number of diagrams.The light-cone gauge is obtained by enforcing the following condition on the gluon field: It is well known that in light-cone gauges the condition above is not enough to completely fix the gauge [50].Indeed, the transverse components of the gauge field at light-cone infinity are left unconstrained by Eq. (10).However, in the context of GPDs, the specific boundary condition on these transverse components is irrelevant.Indeed, the gauge link (Wilson line) runs along the light-cone direction and all operators are compact, thus shielding the GPDs from being sensitive to the boundary conditions at light-cone infinity.
A complication of working in light-cone gauge is that the gluon propagator has a more convoluted structure that reads: The subscript "reg."indicates that the linear propagator (nk) −1 , which gives rise to the so-called rapidity divergences, has to be regularised.These spurious divergences, which are present in single diagrams, cancel out when summing over all diagrams, so that the regulator can eventually be safely removed.However, at one-loop the cancellation of the rapidity divergences is apparent and we find it unnecessary to specify a particular regularisation procedure.
The GPD correlators defined in Eq. ( 3) require UV renormalisation.Using dimensional regularisation in d = 4−2ε dimensions and the MS renormalisation scheme, GPDs are renormalised in a multiplicative fashion by means of a set of renormalisation constants Z as follows: where U , L, and T stand for unpolarised, longitudinally polarised and transversely/circularly polarised projections, respectively.‡ In addition, the sums over f ′ and/or f ′′ run over active partons.The corresponding evolution equations for the renormalised GPDs take the general form [44]: The evolution kernel P are related to the renormalisation constants as: The MS renormalisation constants Z can depend on the renormalisation scale µ only through the strong coupling α s .Therefore, defining a s = α s /(4π), we have that: Expanding the evolution kernel in powers of a s : at LO we immediately obtain: Here and in the following we refer to transversely/circularly polarised GPDs with the index T , implicitly understanding transversely polarised quark GPDs and circularly polarised gluon GPDs.
Since the renormalisation constants are universal and independent of the external states, we can compute them perturbatively using the parton-in-parton GPDs, which are defined from the hadronic GPDs by replacing the external hadronic states with on-shell free partonic states [44].Expanding the bare and renormalised parton-in-parton GPDs in powers of a s : we can derive the renormalisation constants by plugging these expansions into Eq.( 12) and requiring that the renormalised GPDs be finite order by order in a s .This eventually produces the iterative set of equations: The coefficients are obtained by matching the UV divergences produced by the dia- f ←f ′ are all finite.

Analytic results
As discussed above, the evolution kernels derive from the renormalisation of the parton-inparton GPDs and only depend on the operator involved in their definition.As a consequence, the unpolarised GPDs H and E share the same evolution kernels, and so do the longitudinally polarised GPDs H and E, and the full set of transversely/circularly polarised GPDs H T , E T , H T , and E T .The general form of the one-loop evolution kernels for each of these three classes of GPDs can be presented as follows: where the θ function is normalised such that θ(0) = 1.The constants K i and C i are the same for all polarisations Γ and read: with C q = C F and C g = C A .Conversely, the functions p Γ i/k are different for each polarisation.Those associated with the unpolarised GPDs H and E have already been presented in Ref. [44], but we report them here for completeness: In the longitudinally polarised case, instead, they read: while in the transversely/circularly polarised case, they read: It is interesting to observe that in the transversely/circularly polarised case, due to the fact that the off-diagonal functions p T q/g and p T g/q are identically zero, transversely polarised quark GPDs and circularly polarised gluon GPD do not couple under evolution.More details on the computation of the functions above can be found in Appendix A.
Defining appropriate GPD combinations allows for a partial diagonalisation of the splitting matrix that is best suited for numerical implementations.At LO, these combinations are the non-singlet: and the singlet: where n f is the number of active quark flavours and i←H = H T,i←H , E T,i←H , H T,i←H , E T,i←H with i = q, g.In addition, we have defined anti-quark GPDs using the charge-conjugation symmetry relations: where the upper sign applies to the unpolarised and transversely/circularly polarised cases (Γ = U, T ), while the lower sign applies to the longitudinally polarised case (Γ = L).Non-singlet and singlet GPD combinations obey their own decoupled evolution equations that at one loop read: with κ = ξ/x.The evolution kernels P can be decomposed as: with the non-singlet combinations given by: and the singlet combinations given by: where again the upper sign refers to unpolarised and transversely/circularly polarised distributions (Γ = U, T ), while the lower sign refers to the longitudinally polarised ones (Γ = L).
Explicit expressions for the one-loop non-polarised splitting kernels in this notation can be found in Ref. [44], but we report them here for completeness: The longitudinally polarised ones read: (y, κ) , while the (non-zero) transversely polarised ones read: The +-distribution (with round brackets) in the expressions above is defined as: while the ++-distribution instead is defined as:

DGLAP limit
One of the fundamental requirements of the GPD evolution kernels is that, in the limit of vanishing ξ, the well-known DGLAP splitting functions have to be recovered.This limit amounts to taking κ → 0 and, given the decomposition in Eq. (37), it is such that P drops, leaving only the term proportional to P . The presence of θ(1−y) in this term reduces the integral in the r.h.s. of Eq. (36) to a "standard" Mellin convolution which is precisely what enters the DGLAP evolution equations.What is left to verify is that P [Γ],±,[0] 1 in the limit κ → 0 tend to the known one-loop DGLAP splitting functions.This has already been verified in Ref. [44] in the unpolarised case.Using Eq. ( 41), for the longitudinally polarised evolution kernels we find: that indeed coincide with the corresponding DGLAP splitting functions [51].For the transversely polarised evolution kernels we take the limit for κ → 0 of Eq. ( 42) and find: These expressions coincide with those from Refs.[52][53][54].§

ERBL limit
The evolution equations in Eq. ( 36) can be alternatively written in a form that resembles the ERBL equation [44,55]: with: (48) § Note that in Eq. ( 13) of Ref. [54] the term proportional to the δ-function is correct for the ∆T P qq , while it should be equal to Kg in the case of ∆LP (0) gg (see Refs. [52,53]).
Taking the ξ → 1 limit and performing the changes of variable x = 2v − 1 and y = 2u − 1, the GPD evolution equations turn into ERBL evolution equations [55] for distribution amplitudes (DAs): where the DAs Φ [Γ],± are related to the GPDs through the following identity: and the corresponding evolution kernels are defined as: Their explicit expressions in the unpolarised case are given by: In Ref. [44], it was verified that, at least in the non-singlet case, these expressions coincide with those present in the literature.The longitudinally polarised expressions instead read: while the transversely/circularly polarised ones are: The +-prescription in the expressions above, this time with square brackets, is defined differently from Eq. ( 43) and reads: To the best of our knowledge, the one-loop ERBL kernels for the full set of twist-2 distributions for both singlet and non-singlet combinations have not been presented anywhere.

Continuity at x = ξ and spurious divergences
As it can be verified explicitly, all of the P 2 functions in Eqs. ( 40)-( 42) are such that: This property ensures that the r.h.s. of Eq. ( 36) is continuous at κ = 1, i.e. at the crossover point x = ξ.This is essential to ensure the continuity of GPDs at the crossover point.
Of course, GPD continuity also requires that the integral in the r.h.s. of Eq. ( 36) converges for all values of κ.However, as it can be seen from Eqs. ( 40)-( 42), all of the single expressions for P 1 and P 2 are affected by a spurious pole at y = 1/κ.For κ ≤ 1, P 2 does not contribute to the evolution, while the pole of P 1 , that is to be integrated only up to y = 1, falls outside the integration region.As a consequence, the integral in this region converges.For κ > 1, both P 1 and P 2 contribute.As shown in Ref. [44] in the unpolarised case, it so happens that the coefficients of the poles at y = 1/κ of P 1 and P 2 are equal in absolute value but opposite in sign, such that the singularity cancels out leaving a finite result also for κ > 1.
In the following, we will prove that the same cancellation takes place also for the longitudinally and transversely/circularly polarised evolution kernels.Having ascertained that the cancellation needs to happen only for κ > 1, we concentrate on this region.For each evolution kernel we compute the following quantities: and verify that they are finite.In the longitudinally polarised case we find: while in the transversely/circularly polarised case, we have: that are indeed all finite for κ > 1, thus guaranteeing that the evolution at one loop leaves GPDs continuous at the crossover point x = ξ.

Sum rules
In this section, we discuss the sum rules.Specifically, it can be shown that polynomiality of GPDs implies some integral constraints of the evolution kernels.Ref. [44] discusses these constraints in the unpolarised case.In the longitudinally polarised case the conservation of the first moment leads to: where the independence of ξ also implies the independence of y.Indeed, we find: It is interesting to observe that, in the cases ij = qq, qg, the integrals above evaluate to zero.This implies that, at least at one-loop accuracy, the first moment the longitudinally polarised quark GPDs is independent of the scale µ and can thus be identified with a physical observable.This is indeed related to the anti-symmetric part of the energy-momentum tensor.It is known that the anti-symmetric form factor is related to the axial form factor that at one loop does not need renormalisation (see e.g.Refs.[21,56,57]).The same does not hold for the gluon part because at the level of the energy-momentum tensor no anti-symmetric and gauge invariant operator exists.Therefore, the third and fourth integrals in Eq. ( 61) are not forced to vanish.The conservation of the longitudinally polarised second moment instead implies: and indeed we find: In the transversely/circularly polarised case, and accounting for the fact that the qg and gq splitting kernels are identically zero, the sum rules imply that: and indeed we find: The fulfilment of the sum rules provides a strong check of the correctness of the evolution kernels derived here.

Conservation of polynomiality
In this section, we prove analytically that GPD polynomiality is conserved by the evolution.The proof presented below is limited to the unpolarised non-singlet quark GPD, but a similar demonstration can be given also for the other cases.The polynomiality property reads: Using Eq. ( 47), one finds: In order for this equality to be fulfilled, the following identity has to hold: where V [U ],−,[0] n is a constant to be evaluated.The integral in the l.h.s. of the equation above can be computed using the results in Appendix C of Ref. [44], thus proving that the equality in Eq. ( 68) is indeed true with: Interestingly, this allows us to derive evolution equations for the coefficients A (n) k that read: and that admit the solution: 4 Numerical results Having presented the expression for the evolution kernels in Sect.3, we are now in a position to implement them in the numerical code APFEL++ [46,47].
To showcase the effect of the evolution, we have used as a set of initial-scale GPDs the realistic model of Refs.[58][59][60], referred to as Goloskokov-Kroll (GK) model, as implemented in PARTONS [40].For the unpolarised evolution we selected the GPD H, in the longitudinally polarised case we instead used H, while in the transversely/circularly polarised case we used H T .Since the GK model does not provide a circularly polarised gluon GPD, we used the unpolarised H g as a proxy to test the evolution.Since the evolution of the circularly polarised gluon GPD is completely decoupled, no spurious effects are introduced in the evolution of the transversely polarised quark distributions that also evolve independently.GPDs are evolved from µ 0 = 2 GeV to µ = 10 GeV in the variable-flavour-number scheme (VFNS), i.e. allowing for heavy-flavour threshold crossing, with charm and bottom thresholds set to m c = 2.1 GeV and m b = 4.75 GeV, respectively.‖ The strong coupling is consistently evolved at LO in the VFNS using α s (M Z ) = 0.118 as a boundary condition.We set the value of the momentum transfer squared, that does not directly participate in the evolution, to t = −0.1 GeV 2 throughout.Figs. 1, 2, and 3 show the effect on GPDs of unpolarised, longitudinally polarised, and transversely/circularly polarised evolutions, respectively.GPDs are displayed as functions of x ‖ The unusually large value of the charm threshold is due to the fact that the lowest available scale accessible to the GK model is µ0 = 2 GeV [58][59][60].However, at this scale, no distribution associated with the charm quark is provided.Therefore, we assumed n f = 3 active flavours at µ0 which required setting mc > µ0. for four different values of ξ, including the DGLAP (ξ = 0) and ERBL (ξ = 1) limits.The upper panels display the absolute distributions at the final scale µ = 10 GeV multiplied by a factor of x for the quark GPDs and x 2 for the gluon GPDs for a better visualisation.The lower panels display their ratio to the corresponding distributions evolved using the DGLAP equations.We first note that for all three cases, setting ξ = 0 exactly reproduces the DGLAP evolution, as expected.For increasing values of ξ, the evolution gradually deviates from DGLAP for all considered distributions.The deviations are particularly pronounced for x ≲ ξ where GPD evolution causes a strong slowdown of the evolution as compared to DGLAP.We also observe that GPDs at the crossover point x = ξ are continuous, as expected from the discussion in Sect.3.3.However, they develop a cusp (discontinuity of the derivative in x) that is a consequence of the fact that the evolution kernels are continuous but not smooth at x = ξ.
As discussed in Sect.3.5, a crucial property of GPDs, that must be preserved by the evolution, is polynomiality.Figs. 4, 5, and 6 show the behaviour as functions of ξ of the first three even (left plots) and odd (right plots) moments of the up-quark distributions H, H, and H T , respectively.The bullets correspond to the values obtained by integrating numerically the evolved GPDs for different values of ξ, while the dashed lines show the fits using the expected polynomial laws in ξ.It is clear that in all cases, the expected behaviour is accurately reproduced.It is also interesting to observe that both even and odd first moments (n = 0) for all polarisations are constant in ξ.In fact, this is the expected behaviour in all cases, except for the unpolarised even moment that would in principle admit a quadratic term in ξ.However, this contribution, often referred to as D-term, evolves independently from the rest of the GPD.Since the GK model does not include any D-term, the evolution does not generate it and it is thus absent at all scales, finally producing a constant first even moment also for the unpolarised GPD H.
Finally, in Figs.7 and 8, we present a comparative analysis of the evolution of unpolarised and longitudinally polarised distributions between the code developed in Ref. [39], which we refer to as Vinnikov's code, and our implementation in APFEL++.* * To match the capabilities of Vinnikov's code, the comparison is performed without heavy-flavour threshold crossing.All other settings are consistent with those applied in the numerical results presented above.This comparison was already presented in Ref. [44] in the unpolarised case, where it was observed that, for small enough values of ξ (ξ ≲ 0.6), a generally good agreement between the two codes was achieved.However, for larger values of ξ (ξ ≳ 0.6) a significant deterioration in the agreement was noted.Subsequently, we conducted a deeper investigation of this issue † † that revealed that Vinnikov's code was indeed affected by a bug in the region ξ > 2/3.This issue is due to the way the x-space interpolation grid is constructed (see Eq. ( 6) of Ref. [39]).Specifically, the definition of the grid parameter γ in Eq. ( 6) of Ref. [39] guarantees that the constraints in Eqs. ( 7) and ( 9) are fulfilled only when ξ ≤ 2/3, while they are broken for ξ > 2/3.Following the resolution of this issue, ‡ ‡ as illustrated in Figs.7 and 8, a better agreement between APFEL++ and Vinnikov's code is found across a broad range of ξ, extending beyond ξ ≃ 0.6, for both unpolarised and longitudinally polarised evolutions.
In the unpolarised case (Fig. 7), the agreement generally remains at or below the percent level, with the only exception of the up-quark singlet distribution H + u at ξ = 1, which displays a larger departure of the order of 10-20% for small values of x.As far as the longitudinally polarised evolution is concerned (Fig. 8), the agreement between the two codes is excellent for H + u , while it tends to deteriorate at small x and large ξ for H − u .The disagreement is more pronounced for H g where differences tend to increase with growing ξ, reaching the 40% level at a solid implementation in the numerical code APFEL++ [46,47].The computation is performed following a Feynman-graph approach in momentum space using the light-cone gauge (see Appendix A), paving the way for the extraction of the two-loop evolution kernels along the same lines of Ref. [61].This will eventually allow us to achieve GPD evolution at next-to-leading order accuracy that, with the advent of the Electron-Ion Collider [24], will soon become a necessary ingredient for accurate phenomenology.
We have performed a number of analytic checks of the expressions obtained for the evolution kernels.Specifically, we have verified the correctness of the DGLAP (ξ → 0) and ERBL (ξ → 1) limits, ensured that the evolution does not cause any discontinuity of the GPDs at x = ξ, and proven that the kernels preserve polynomiality.Finally, we have thoroughly checked our numerical implementation of the evolution using as a set of initial-scale GPDs the realistic GK model [58][59][60] as provided by PARTONS [40].We showed that the DGLAP limit is accurately recovered for all polarisations and that polynomiality is conserved upon evolution.Our implementation was also compared against an independent code (Vinnikov's code [39]), revealing a generally good agreement.where Λ ss ′ is a projector introduced for convenience to select the desired polarisation of the external quark state.The three relevant polarisations (unpolarised, longitudinally polarised, and transversely polarised) produce the following outcome when summed over the spin states s and s ′ : Λ ss ′ ūs ′ (P ) ∈ { / P , / P γ 5 , iσ σρ P ρ γ 5 }, respectively.The trace in Eq. ( 73) is easily computed yielding: As expected, in the limit ξ → 0, the leading-order quark-in-quark PDFs are recovered.However, it is also interesting to observe that in the limit ξ → 1 the leading-order GPD vanishes.This is indeed the correct behavior because the limit ξ → 1 corresponds to a distribution amplitude that encodes the creation of meson bound state out of a pair of incoming quarkantiquark.Of course, this cannot happen without any interaction between the quark and the anti-quark.Therefore, the leading-order graph has to vanish.Now, we can use the same procedure to compute the one-loop correction to the quark-inquark GPDs whose relevant diagram is displayed in Fig. 10.The integral to compute is: with This integral will give different results depending on the relative position of x and ξ.In particular, the region x < ξ corresponds the ERBL region, while x > ξ corresponds to the DGLAP region.The functions p Γ q/q can then be obtained by extracting the MS UV pole part of this integral in these regions as shown in the Appendix of Ref. [44].

A.2 Gluon-in-gluon GPD
We now consider the gluon-in-gluon GPD whose operator definition in light-cone gauge reduces to: (77) The leading-order contribution is obtained using free fields and plugging in the Lorentz structures Γ µν given in Eq. ( 9).The result is: The next-to-leading order correction is obtained considering the diagrams in Fig. 11. where: V µνρ (q, l, r) = − [g µν (q − l) ρ + g νρ (l − r) µ + g ρµ (r − q) ν ] .
A.3 Quark-in-gluon and gluon-in-quark GPDs We now consider the off-diagonal quark-in-gluon and gluon-in-quark GPDs whose operator definitions are: (87) These GPDs have no tree-level contribution and the first non-vanishing contribution appears at O(α s ).The corresponding diagrams are shown in Fig. 12.  (91) Once again, the UV pole part of these integrals allows us to obtain the functions p Γ q/g and p Γ g/q .

Figure 4 :Figure 5 :
Figure4: Effect of the evolution on the Mellin moments of the unpolarised up-quark GPD H.The bullets display the value of the moments computed numerically as integrals of the distributions, whereas the dashed lines show the fits to the bullets using the expected polynomial law.The left panel (4a) displays the first three even moments related to the non-singlet combination, while the right panel (4b) displays the first three odd moments that are instead related to the singlet combination.

Figure 10 :
Figure 10: One-loop diagram contributing to the quark-in-quark GPD.