Chiral and trace anomalies in Deeply Virtual Compton Scattering

Inspired by recent work by Tarasov and Venugopalan, we calculate the one-loop quark box diagrams relevant to polarized and unpolarized Deep Inelastic Scattering (DIS) by introducing off-forward momentum $l^\mu$ as an infrared regulator. In the polarized case, we rederive the pole $1/l^2$ related to the axial (chiral) anomaly. In addition, we obtain the usual logarithmic term and the DIS coefficient function. We interpret the result in terms of the generalized parton distributions (GPDs) $\tilde{H}$ and $\tilde{E}$ and discuss the possible violation of QCD factorization for the Compton scattering amplitude. Remarkably, we also find poles in the unpolarized case which are remnants of the trace anomaly. We argue that these poles are cancelled by the would-be massless glueball poles in the GPDs $H$ and $E$ as well as in their moments, the nucleon gravitational form factors $A,B$ and $D$. This mechanism sheds light on the connection between the gravitational form factors and the gluon condensate operator $F^{\mu\nu}F_{\mu\nu}$.


I. INTRODUCTION
The role of the U A (1) axial (chiral) anomaly in polarized Deep Inelastic Scattering (DIS) has a long and winding history.Originally in the late 1970s it was used to constrain the one-loop gluonic correction to the first moment of the singlet g 1 (x) structure function, as well as the two-loop anomalous dimension of the quark helicity contribution ∆Σ to the proton spin [1].Soon after the discovery of the 'spin crisis' by the European Muon Collaboration (EMC) in 1988 [2], it was suggested that an anomaly-induced gluon helicity ∆G contribution to the 'intrinsic' quark helicity ∆ Σ could be the key to explaining the unexpectedly small value of ∆Σ [3,4].While such a scenario became popular at the time, subsequent decades-long experiments and global analyses did not find evidence for a sufficiently large ∆G to make it phenomenologically viable [5][6][7].More importantly, the identification of the axial anomaly contribution as gluon helicity also met with much theoretical objection from the very start [8][9][10][11].The gluonic contribution in (1) comes from the infrared region of the triangle diagram which contains the Adler-Bell-Jackiw anomaly.As pointed out by Jaffe and Manohar [8], a proper way to regulate the infrared singularity of this diagram is to calculate it in off-forward kinematics.The anomaly then manifests itself as a pole in momentum transfer l = p 2 − p 1 in the matrix element of the singlet axial current J µ 5 = f ψf γ µ γ 5 ψ f , where the incoming and outgoing gluons are on shell and the n f quarks in the loop are massless.The appearance of the pole 1/l 2 and the twist-four pseudoscalar operator F F seem alarming, as they signal some underlying nonperturbative physics that does not fit into the standard perturbative QCD framework.Yet, in ordinary perturbative calculations done in forward kinematics, this problem is superficially avoided by certain choices of infrared regularization such as dimensional regularization.Decades after the initial controversy, infrared sensitivity and the subtleties of taking the forward limit seemed to have been largely forgotten.Nowadays, forward kinematics is routinely used in the higher-order computations of polarized cross sections and asymmetries.However, recently the issue of the anomaly pole has been rekindled by Tarasov and Venugopalan [12,13] who pursued and crystallized the original suggestion by Jaffe and Manohar.They have demonstrated, within the worldline formalism, that the box diagram (see the left diagram in Fig. 1) contains a pole 1/l 2 if it is calculated in off-forward kinematics.This may be viewed as a nonlocal generalization of the local relation (2) unintegrated in the Bjorken variable x.As envisaged in [8] and elaborated in [13], at least after the xintegration, the pole should be canceled by another massless pole due to the exchange of the η 0 meson, the would-be Nambu-Goldstone boson of U A (1) symmetry breaking.This requirement leads to an independent derivation of the U A (1) Goldberger-Treiman relation [14][15][16] between the pseudoscalar and pseudovector form factors.
Motivated by these developments, in this paper we further explore the physics of anomaly poles in two different directions.First, we calculate the box diagram in off-forward kinematics in the standard perturbation theory.This is a useful cross-check of the result obtained in the worldline formalism [12].In addition to reproducing the pole term, we obtain the 'usual' perturbative corrections to the g 1 (x) structure function which features the DGLAP splitting function and a coefficient function.We then interpret the result in terms of the generalized parton distributions (GPDs) H and Ẽ.The emergence of the pole is potentially problematic for the QCD factorization of the Compton amplitude.We discuss how factorization may still be justified following the possibility of cancellation of poles.
Second, we point out that entirely analogous poles can arise in unpolarized DIS, or more precisely, in the symmetric (in Lorentz indices µν) part of the Compton scattering amplitude T µν in off-forward kinematics.Just as the pole in the polarized sector is related to the axial (chiral) anomaly, that in the unpolarized sector is related to the trace anomaly.Indeed, it is known in QED and other gauge theories [17][18][19] that the off-forward photon matrix element of the energy momentum tensor Θ µν has an anomaly pole: again from the triangle diagram.The residue is proportional to the matrix element of the twist-four scalar operator F αβ F αβ (or the 'gluon condensate' in QCD) which characterizes the trace anomaly.We shall derive the unintegrated (in x) version of (3) by evaluating the quark box diagrams and interpret the result in terms of the unpolarized GPDs H and E. We then make a connection to the gravitational form factors of the proton and discuss the possibility of cancellation of poles.

II. PRELIMINARIES
In this section, we set up our notations for the kinematical variables that enter the calculation of the quark box diagrams in DIS.More precisely, since we generalize the calculation to off-forward kinematics as explained in the Introduction, we consider the Compton scattering amplitude where . being a flavor index, is the electromagnetic current and the subscript 'sym/asym' refers to the symmetric/antisymmetric part in the photon polarization indices µ, ν. |P 1,2 are the proton single-particle states.q = q1+q2 2 is the average of the incoming and outgoing virtual photon momenta, and the momentum transfer is denoted by P 2 − P 1 = q 1 − q 2 = l.We assume −q 2 ≡ Q 2 > 0 is large and neglect 'higher-twist' terms of order O(M 2 /Q 2 ) where M 2 = P 2 1 = P 2 2 is the proton mass squared.Their inclusion is understood in the literature [20].We also assume that t ≡ l 2 < 0 is much smaller than the hard scale |t| Q 2 and neglect terms of order l 2 /Q 2 .We define the following variables where x B is the generalized Bjorken variable and ξ is the skewness parameter.In forward scattering, x B coincides with the usual Bjorken variable in DIS.In Deeply Virtual Compton Scattering (DVCS) where q 2 2 = 0, x B ≈ ξ.The quark box diagrams of interest are part of the perturbative expansion of (4) at one-loop.There are three topologies, see Fig. 1.The diagrams consist of two insertions of photon fields with momenta (q 1 , q 2 ) and two insertions of gluon fields with partonic momenta (p 1 , p 2 ) which we parametrize as Note that p 2 − p 1 = P 2 − P 1 = l, so the momentum transfer t = l 2 < 0 is the same in both the hadronic and partonic processes.We assume the incoming partons to be massless, p 2 1 = p 2 2 = 0, which means p • l = 0 and p 2 = −l 2 /4.The kinematical variables at the partonic level are defined as The relation xq • l = − ξQ 2 will be used in the calculation below.The photon virtualities can be written as Finally, the polarization vectors of the incoming and outgoing gluons, (p 1 ) ≡ 1 , * (p 2 ) ≡ * 2 , respectively, satisfy the physical conditions We note that throughout this paper we use the conventions γ 5 = iγ 0 γ 1 γ 2 γ 3 and 0123 = +1.This becomes relevant for the antisymmetric part of the Compton amplitude (4), to which we turn first.

III. ANTISYMMETRIC PART
In the antisymmetric case we define In the forward limit l µ → 0, J α = g 1 (x B )ū(P )γ α γ 5 u(P ) = 2g 1 (x B )S α is proportional to the g 1 structure function in polarized DIS.We have calculated the box diagrams in the near-forward region |l µ | Q, using the Mathematica package 'Package-X' [21].The result is, for massless quarks in the loop, where A⊗B(x B ) ≡ x )B(x).In (11), we have neglected all terms that vanish in the limit l → 0, including the ξ-dependence of various coefficients.(The dependence on ξ will be kept in the next section.)∆P qg (x) = 2T R (2x − 1) is the polarized g → q splitting function with T R =1 2 being the color factor, and is the coefficient function.(In both ∆P qg and δC off g the explicit factor of 2 accounts for the quark and antiquark contributions.)F(x) is the twist-four pseudoscalar parton distribution [13,22,23] F(x, l2 ) ≡ iP + ū(P 2 )γ 5 u(P 1 ) entering Eq. ( 11) with We note that in the one-loop calculation, ∆G(x) and F(x) have been identified via the tensor structures The pole term 1/l 2 in (11) agrees with the result derived in Ref. [13] using the worldline formalism.In the Appendix A, we also rederive this result via an ordinary Feynman diagram calculation.Although not manifest, the numerator actually contains an extra factor of l, so this term is singular in the sense that the limit lim l→0 l α l β /l 2 depends on the direction of l µ .In addition, we find the expected contribution to the g 1 structure function where l 2 plays the role of an infrared cutoff.Both ∆P qg and δC off g satisfy They thus do not contribute to the first moment of g 1 .This is a necessary condition in order to be consistent with the perturbative result (2) for the local operator J µ 5 .Note that in dimensional regularization in 4 − 2 dimensions, one has instead in the coefficient of ∆G(x B ).As pointed out in [11], the last term 4T R (1 − x) is of infrared origin.In off-forward regularization, this term becomes the pole term.
The cancellation of the pole in (11) and its implications have been discussed in [12].Here we point out that the description of the pole is most naturally phrased in terms of the generalized parton distributions (GPDs).For this purpose we return to (4) and fully treat it as an off-forward amplitude.In the GPD factorization framework [24,25], the imaginary part of the Compton amplitude can be expanded as where Hf and Ẽf are the polarized quark GPDs for flavor f .The meaning of the superscript 'bare' will be explained shortly.In the forward limit, the polarized quark and antiquark PDFs are recovered, Hq (x B ) = ∆q(x B ) and It is tempting to regard (11) as part of the O(α s ) corrections in (16).However, clearly the pole term 1/l 2 does not fit into the general structure of factorization.This term remained unnoticed in the calculation of the antisymmetric part of the one-loop corrections to the Compton amplitude [25,26].The reason is that in the GPD-based calculations in the literature, one sets l µ ≈ −2 ξp µ before the loop integration.Since l 2 = 0 in this approximation, the pole 1/l 2 cannot arise.Without this regulator, the box diagrams are power divergent in the infrared unless some cancellation mechanism is at work.In the present case, the matrix element of the twist-four operator The problem is that, despite the proportionality to the twist-four distribution, the pole term in (11) cannot be dismissed as a higher twist contribution because there is no suppression factor 1/Q2 .In fact, it is parametrically of the same order as the twist-two ∆G(x) term in (11) since l α F F /l 2 ∼ l α l β /l 2 ∼ O(1) as mentioned above.Note also that the pole term has the same Lorentz structure ∝ l α as the GPD Ẽ in (16).Clearly, this poses a threat to QCD factorization for the Compton amplitude in the GPD Ẽ sector. 2  On the other hand, the Compton amplitude must have a well-defined forward limit.Thus, there must be a mechanism to cancel the pole at l 2 = 0 which could ultimately 'rescue' the factorization theorem.Noticing that the pole term comes entirely from the infrared region of the box diagram, we absorb it into the definition of the GPD Ẽ and let it cancel with a 'primordial' pole in Ẽbare .Namely, we redefine as and postulate that Ẽbare is dominated by a pole Ẽbare which exactly cancels the perturbative pole in the last term.We further postulate that it is this 'renormalized' Ẽ that integrates to the pseudoscalar form factor. Namely, where g A and g P are the singlet axial form factors defined by The prescription (17) is outside the conventional logarithmic renormalization of twist-two GPDs.In order to better justify it, we show in the Appendix (see (A13)) that exactly the same pole structure arises in the one-loop calculation of the polarized quark GPD of a gluon3 : Moreover, performing the x-integral in (20), we obtain By construction, this is finite at l 2 = 0.One can check that ( 23) is consistent with the following exact relation obtained by taking the divergence of ( 21): under the assumption that g A (l 2 ) ≈ g A (0) = ∆Σ slowly varies with l 2 .These arguments suggest that the pole should indeed be regarded as a part of the quark GPD Ẽ, although a more rigorous, field-theoretic justification of ( 17) is needed.
The cancellation of poles within the form factor g P is the central argument of [12] following the earlier suggestion in [8].The massless pole in the first term of ( 23) can be nonperturbatively generated by the t-channel exchange of the would-be Nambu-Goldstone boson of spontaneous U A (1) symmetry breaking, the primordial η 0 meson.This is analogous to the pion pole in the SU A (2) pseudoscalar form factor g . The difference, however, is that the latter is physical while the former is not.In reality, the U A (1) symmetry is explicitly broken by the anomaly.In the present context, this effect is represented by the second term in (23), turning the massless pole into a massive one at the physical η meson mass m 2 η .Indeed, the form factor P 2 |F F |P 1 has a pole at l 2 = m 2 η , and in the single-pole approximation (24) is satisfied by Returning to GPDs, we add that, in order to guarantee the finiteness of the Compton amplitude, there must be a massless pole already in the GPD Ẽ. 4 Moreover, the cancellation of poles in (17) must somehow occur for all values of x B and Q 2 (the latter enters as the renormalization scale), or else the factorization of the Compton amplitude is in danger.On the other hand, since the cancellation of poles occurs within the GPD Ẽ sector alone, we do not see any issues with the usual factorization of the g 1 structure function in polarized DIS.In this sense, the above off-forward calculation may be regarded as an alternative to other regularization schemes such as the MS scheme.

IV. SYMMETRIC PART
Remarkably, the 1/l 2 pole is also present in the symmetric (in µν) part of the Compton amplitude (4).As mentioned in the Introduction, and will be studied in detail below, in this case the pole is related to the QCD trace anomaly.In this section we obtain the fully analytic expression of the pole terms and discuss their implications for the Compton amplitude and the gravitational form factors.
Let us first recall that in the forward limit, the imaginary part of the symmetric Compton amplitude T µν sym is related to the F 1 and F 2 structure functions, To leading order, where q f (x) (q f (x))) is the unpolarized quark (antiquark) PDF.The first moment of F 2 is related to the matrix element of the quark part of the QCD energy momentum tensor Θ µν f = qf γ (µ iD ν) q f : Physically, A f is the momentum fraction of the proton carried by f -quarks and antiquarks.
Turning to the off-forward case, we have calculated the contribution from the box diagrams using Package-X [21] and found a more complicated tensor structure, where the ellipses denote terms proportional to tensors like l (µ P ν) , l (µ q ν) and l µ l ν such that they satisfy the conservation law q µ ImT µν sym = 0.This time F off 1,2 are related to the unpolarized GPDs instead of the PDFs.Different from the 4 The pion pole is argued to exist also in the GPD Ẽ in the isovector channel Ẽu − Ẽd ∼ polarized case, for a reason to become clear shortly, here we shall keep the full dependence on the skewness parameter ξ.For this purpose, we will make the replacement l µ ≈ −2ξP µ ≈ −2 ξp µ in the tensors l (µ P ν) , l (µ q ν) and l µ l ν , but not in potential pole terms ∝ 1/l 2 .In doing so, we find that Eq. ( 29) takes the simple form ImT µν sym box ≈ −g µν + q µ q ν q 2 F off 1 (x B , l) where (setting ξ = 0 in non-pole terms) We recognize the expected structure of the one-loop corrections associated with the unpolarized gluon PDF g(x), with the splitting function ).The coefficient functions are given by In addition, we find a pole 1/l 2 in both F off 1 and F off 2 relevant to the longitudinal structure function), with the following convolution formula where The twist-four scalar gluon GPD is defined as Several comments are in order.First, the two terms in (33) come from the first and third diagrams in Fig. 1, respectively.The imaginary part of the latter is nonvanishing only in the region 0 < x B < ξ where the outgoing virtual photon becomes timelike, q 2 2 > 0, see (8).The structure of the convolution (33) has been inferred from the analysis of Compton scattering in a related context [28].Second, concerning the twist-four gluon GPD (35), the original definition in [29] was a forward matrix element P |...|P .The necessity to define F in off-forward kinematics was later emphasized in [30], and this turns out to be relevant to the present discussion.Accordingly, we have rescaled the distribution by ū(P 2 )u(P 1 ).We have also introduced the prefactor x to compensate for the 1/x factor in (33).This factor makes F(x, ξ, l 2 ) an odd function in x, just like g(x).Finally, in the actual one-loop calculation the gluon PDF can be identified as g 5 Actually the box diagram generates an additional tensor structure * µ 2 and this has been taken into account when determining the coefficient functions (32).For spin-1 hadrons, (36) induces an additional structure function in (29) [31].For spin-1 2 hadrons, the structure ( 36) is neglected in inclusive DIS because it vanishes after averaging over the gluon polarizations (38).In Compton scattering, it becomes part of the gluon transversity GPD [32].This is again order unity l α l β /l 2 ∼ O(1), the same order as the twist-two g(x) term in (31).Equation (37) shows that, for on-shell gluon states, the tree-level matrix element F µν F µν is nonvanishing only in off-forward kinematics l µ = 0.If one averages over the two gluon polarization states by making the replacement * where g ⊥ αβ ≡ g αβ − n α nβ − nα n β with the light-like vectors n = (1, 0, 0, −1)/ √ 2 and n = (1, 0, 0, 1)/ √ 2, the pole disappears because Again, this is the reason why the pole has remained unnoticed in the one-loop GPD calculations in the unpolarized sector [26,[33][34][35] (see also a recent two-loop calculation in DVCS [36]) where one projects onto the twist-two gluon GPDs H g and E g via (38).Without this projection, the loop integral is power divergent if one sets l µ = −2 ξp µ from the outset.While the replacement ( 38) may be justified for the unpolarized DIS structure functions where the initial and final proton spins are equal and averaged over as S P S|...|P S , this is not the case for GPDs P 2 S 2 |...|P 1 S 1 in Compton scattering where the initial and final spin states S 1 and S 2 are typically unconstrained.Moreover, in contrast to the partonic matrix element p 2 |F µν F µν |p 1 (see (37)), the nonperturbative proton matrix element P 2 |F µν F µν |P 1 does not vanish in the forward limit, not even after averaging over the proton spin states.In particular, the zeroth moment of F(x, ξ = 0, l 2 = 0) is the so-called gluon condensate: responsible for the generation of the proton mass via the QCD trace anomaly.As a matter of fact, the connection between trace anomalies and massless poles 1/l 2 is known in the literature [17][18][19].An explicit calculation of the triangle diagram shows that the matrix element of the QED energy momentum tensor between on-shell photon states has a pole: If one introduces the usual color factor T R = 1 2 on the right-hand side, essentially the same formula holds for the one-loop gluon matrix element of the quark energy momentum tensor Θ µν f in QCD.Taking the ++ component, one finds a relation On the other hand, if one takes the trace of (41), the pole disappears.Applied to the QCD case we find and the right-hand side is nothing but the contribution to the QCD trace anomaly from the single-flavor quark energy momentum tensor to one-loop6 These observations suggest that the poles in (31) are physically there, at least in Compton scattering, and may carry profound implications possibly touching the issue of nonperturbative mass generation in QCD.
Let us now explore the consequences of (31).Treating (4) as a fully off-forward amplitude, we find from the operator product expansion where, as before, H bare f and E bare f are the 'bare' versions of the standard unpolarized quark GPDs H f and E f .In the forward limit, H f (x) = q f (x) and −H f (−x) = qf (x) are the quark and antiquark PDFs.According to QCD factorization theorems [24,25], higher-order corrections to (45) not suppressed by 1/Q 2 consist only of the twist-two quark and gluon GPDs to all orders.However, our result (31) shows that there are diverging corrections already in the O(α s ) terms which cannot be simply dismissed as higher twist, nor be absorbed into the (usual) renormalization of twist-two GPDs.From the Gordon identity we see that the pole affects both H f and E f , with opposite signs.This raises the concern that factorization is violated also in the symmetric part of the Compton amplitude.Nevertheless, the Compton amplitude must have a smooth forward limit.Similar to the antisymmetric case discussed above (cf.( 17)), we absorb the pole in (31) into the definitions of the GPDs and let it cancel with nonperturbative, pre-existing poles in H bare Again, this cancellation must occur for all values of x B and Q 2 in order to save the QCD factorization of the Compton amplitude.The second moments of the 'renormalized' GPDs thus defined are where A f , B f , D f are the gravitational form factors defined by the off-forward matrix element of the (quark part of the) QCD energy momentum tensor [38], On the other hand, the second moment of the pole term contains the following integral: Note that if it were not for the second term, the result would contain infinitely many powers of ξ from the expansion which would violate the polynomiality of GPD moments.In fact, the only role of the second term is to subtract all the higher-order terms ξ n with n ≥ 4. The final result is thus quadratic in ξ, consistent with the right-hand sides of (50).A cancellation of this sort is known in the GPD literature, see [28].Interestingly, our result (34) coincides with the 'simple model' employed in [28].
The last integral in (52) reads where . We thus arrive at = A f + B f relevant to the Ji sum rule [38].By construction, these poles cancel against the poles in (31) from the box diagrams.The total form factors A f , B f and D f are free of a massless pole, as they should.
We now see a clear connection between our results (31), (55) and the 'trace anomaly pole' (42) known in the literature after taking into account the factor of 2 from (28) (and also the sign difference between (31) and ( 55)).This shows that the pole in the box diagram originates from the QCD trace anomaly.The O(ξ 2 ) terms match straightforwardly, but for the O(ξ 0 ) terms there is an additional complication due to the convolution integral in x.
Let us finally speculate on the origin of the nonperturbative poles introduced in (49) in an ad hoc way.Just like the massless pole in g P is induced by the would-be Nambu-Goldstone boson of unbroken U A (1) symmetry, the η 0 meson, those in the gravitational form factors might be induced by the would-be massless tensor (2 ++ ) and scalar (0 ++ ) glueballs G (2) 0 and G (0) 0 in conformally symmetric QCD.Indeed, the exchange of a massless tensor glueball gives, schematically, where g GN N and f G are the effective glueball-nucleon coupling and the glueball decay constant, respectively.µν is the glueball polarization tensor.Likewise, the exchange of a scalar glueball gives7 Comparing with (55), we find g GN N f (2) Of course, in QCD conformal symmetry is explicitly broken by the trace anomaly and glueballs acquire nonzero masses.After the cancellation, poles in the gravitational form factors are shifted to the physical glueball masses and similarly for the A f , B f form factors.This is in accord with the recently advocated 'glueball dominance' picture [39] of the gravitational form factors (similar to the vector meson dominance of the electromagnetic form factors).Further investigations and discussions of these ideas are certainly needed.
V. SUMMARY We have calculated the imaginary part of the one-loop quark box diagrams for the antisymmetric and symmetric parts of the Compton amplitude relevant to DVCS.The results in the forward limit correspond to the DIS structure functions.Our calculations are distinct from all previous work in the GPD literature in the sense that we have kept t = l 2 finite.This is mandatory in order to correctly isolate the infrared 1/l 2 poles related to the chiral anomaly (in the antisymmetric channel) and the trace anomaly (in the symmetric channel).The need to keep t = l 2 finite was pointed out in Refs.[12,13] in the context of polarized DIS.Below we summarize our main findings: -Antisymmetric case: Our main result is Eq. ( 11) which features (i) a pole 1/l 2 coupled to the gluon fields through the pseudoscalar operator F µν Fµν , (ii) the usual logarithmic term and the coefficient function in polarized DIS.
We interpret this result in terms of the GPDs H and Ẽ by mapping (11) to the QCD factorization formula for the Compton amplitude.The appearance of a genuine pole 1/l 2 is a serious concern in this context since such a term does not fit in this framework.However, in order to still justify factorization and to ensure a well-defined forward limit for the Compton amplitude, we have introduced a renormalized definition of the GPD Ẽ (17) in which this perturbative pole cancels with a nonperturbative pole present a priori in the 'bare' GPD Ẽbare .We have further postulated that it is the integral of this renormalized Ẽ that defines the pseudoscalar form factor g P .This requirement has nevertheless led to the relation (23) which we view as correct and is consistent with the pole cancellation argument in [8,12].
-Symmetric case: Our main result is Eq. ( 31) which features (i) a pole 1/l 2 coupled to the gluon fields through the scalar operator F µν F µν , (ii) the usual logarithmic term and the coefficient functions in unpolarized DIS.We again interpret this result in terms of the GPDs H and E by mapping (31) to the QCD factorization formula.To the best of our knowledge, the observation of such a genuine pole in the symmetric sector is new, but once again this raises concerns for the validity of QCD factorization.Following the same line of reasoning as in the antisymmetric case, we have introduced renormalized definitions of the GPDs H and E which undergo cancellation of the perturbative poles with the nonperturbative, pre-existing poles from massless glueball exchanges.The same cancellation occurs in the gravitational form factors, shifting massless poles to massive glueball poles (59).
We emphasize that, while the above arguments are to some extent speculative, without such arguments the QCD factorization of the Compton amplitude is violated or at least requires revisions in both the symmetric and antisymmetric parts.On the other hand, from these one-loop calculations we do not foresee problems with the QCD factorization in inclusive DIS.In the polarized case, the pole term (11) has a distinct Lorentz structure ∝ l α from the ∆G term ∝ S α .The cancellation of poles occurs entirely within the GPD Ẽ sector while g 1 is renormalized in the usual way.In the unpolarized case, the pole vanishes after averaging over the gluon polarizations (39) appropriate to unpolarized DIS.(See however a caveat above (40).) In the future, we plan to extend our calculation to the real part of the Compton amplitude as well as the quarkinitiated diagrams in DVCS and DIS.The implications of our result for the gravitational form factors will also be further investigated.We hope that our work will motivate further studies that shed additional light on the role of anomalous contributions to DVCS and their potential impact on factorization.so that dk 2 ⊥ ∼ |1 − 2ẑ|dẑ.We thus see that the pole comes from the integral π 2 x(1 − x) (A15) This shows that the two end points ẑ → 0, 1 equally contribute to the pole.We again emphasize that to access this pole in SIDIS, one has to keep terms quadratic in l µ in the numerator.

Therefore, A bare f , B bare f and D bare f all develop a pole 1 /l 2 ,
but not in the linear combination A bare f + B bare f