# Chiral and trace anomalies in Deeply Virtual Compton Scattering

Shohini Bhattacharya<EMAIL_ADDRESS>Physics Department, Brookhaven
National Laboratory, Upton, NY 11973, USA  Yoshitaka Hatta<EMAIL_ADDRESS>Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973,
USA  Werner Vogelsang<EMAIL_ADDRESS>Institute for
Theoretical Physics, Tübingen University, Auf der Morgenstelle 14, 72076
Tübingen, Germany

###### Abstract

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 $\Delta\Sigma$ to the
proton spin Kodaira (1980). Soon after the discovery of the ‘spin crisis’ by
the European Muon Collaboration (EMC) in 1988 Ashman et al. (1988), it was
suggested that an anomaly-induced gluon helicity $\Delta G$ contribution to
the ‘intrinsic’ quark helicity $\Delta\tilde{\Sigma}$

$\displaystyle\Delta\Sigma=\Delta\tilde{\Sigma}-\frac{n_{f}\alpha_{s}}{2\pi}\Delta
G\,,$ (1)

could be the key to explaining the unexpectedly small value of $\Delta\Sigma$
Carlitz et al. (1988); Altarelli and Ross (1988). While such a scenario became
popular at the time, subsequent decades-long experiments and global analyses
did not find evidence for a sufficiently large $\Delta G$ to make it
phenomenologically viable de Florian et al. (2014); Nocera et al. (2014); Zhou
et al. (2022). More importantly, the identification of the axial anomaly
contribution as gluon helicity also met with much theoretical objection from
the very start Jaffe and Manohar (1990); Forte (1990); Bodwin and Qiu (1990);
Vogelsang (1991). 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 Jaffe and Manohar (1990), 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
$J_{5}^{\mu}=\sum_{f}\bar{\psi}_{f}\gamma^{\mu}\gamma_{5}\psi_{f}$,

$\displaystyle\langle
p_{2}|J_{5}^{\mu}|p_{1}\rangle=\frac{n_{f}\alpha_{s}}{4\pi}\frac{il^{\mu}}{l^{2}}\langle
p_{2}|F_{a}^{\alpha\beta}\tilde{F}_{\alpha\beta}^{a}|p_{1}\rangle\,,$ (2)

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 $F\tilde{F}$ 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 Tarasov and
Venugopalan (2022, 2020) 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 Jaffe and Manohar (1990) and elaborated
in Tarasov and Venugopalan (2020), at least after the $x$-integration, the
pole should be canceled by another massless pole due to the exchange of the
$\eta_{0}$ 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 Hatsuda (1990); Veneziano (1989)
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 Tarasov
and Venugopalan (2022). 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) $\tilde{H}$ and $\tilde{E}$. 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 $\mu\nu$) part of
the Compton scattering amplitude $T^{\mu\nu}$ 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 Giannotti and Mottola (2009); Armillis
et al. (2009); Donoghue and El-Menoufi (2015) that the off-forward photon
matrix element of the energy momentum tensor $\Theta^{\mu\nu}$ has an anomaly
pole:

$\displaystyle\langle
p_{2}|\Theta^{\mu\nu}|p_{1}\rangle\sim\frac{1}{l^{2}}\langle
p_{2}|F^{\alpha\beta}F_{\alpha\beta}|p_{1}\rangle\,,$ (3)

again from the triangle diagram. The residue is proportional to the matrix
element of the twist-four scalar operator $\langle
F^{\alpha\beta}F_{\alpha\beta}\rangle$ (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

Figure 1: Box diagrams for the Compton amplitude in off-forward kinematics.

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

$\displaystyle T^{\mu\nu}=i\int\frac{d^{4}y}{2\pi}e^{iq\cdot y}\langle
P_{2}|{\rm T}\\{J^{\mu}(y/2)J^{\nu}(-y/2)\\}|P_{1}\rangle=T^{\mu\nu}_{\rm
sym}+iT^{\mu\nu}_{\rm asym}\,,$ (4)

where $J^{\mu}=\sum_{f}e_{f}\bar{\psi}_{f}\gamma^{\mu}\psi_{f}$, with
$f=u,d,s,..$ being a flavor index, is the electromagnetic current and the
subscript ‘sym/asym’ refers to the symmetric/antisymmetric part in the photon
polarization indices $\mu,\nu$. $|P_{1,2}\rangle$ are the proton single-
particle states. The proton mass $P_{1}^{2}=P_{2}^{2}=M^{2}$ is neglected
whenever it is safe to do so. $q=\frac{q_{1}+q_{2}}{2}$ 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 define the following variables

$\displaystyle P=\frac{P_{1}+P_{2}}{2},\qquad q^{2}=-Q^{2},\qquad
x_{B}=\frac{Q^{2}}{2P\cdot q},\qquad\xi=\frac{q_{2}^{2}-q_{1}^{2}}{4P\cdot
q}\approx\frac{-l^{+}}{2P^{+}}\,,$ (5)

where $x_{B}$ is the generalized Bjorken variable and $\xi$ is the skewness
parameter. In forward scattering, $x_{B}$ coincides with the usual Bjorken
variable in DIS. In Deeply Virtual Compton Scattering (DVCS) where
$q_{2}^{2}=0$, $x_{B}\approx\xi$.

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
parameterize as

$p_{1}=p-\dfrac{l}{2}\,,\qquad p_{2}=p+\dfrac{l}{2}\,,\qquad p^{+}\equiv
xP^{+}\,.$ (6)

Note that $p_{2}-p_{1}=P_{2}-P_{1}=l$. We assume the incoming partons to be
massless, $p_{1}^{2}=p_{2}^{2}=0$, which means $p\cdot l=0$ and
$p^{2}=-l^{2}/4$. The kinematical variables at the partonic level are defined
as

$\displaystyle\hat{x}=\frac{Q^{2}}{2p\cdot
q}=\frac{x_{B}}{x},\qquad\hat{\xi}=\frac{q_{2}^{2}-q_{1}^{2}}{4p\cdot
q}=\frac{-q\cdot l}{2p\cdot q}\approx\frac{-l^{+}}{2p^{+}}=\frac{\xi}{x}.$ (7)

The relation $\hat{x}q\cdot l=-\hat{\xi}Q^{2}$ will be used in the calculation
below. One can check that $t=l^{2}$ is spacelike,

$\displaystyle
t=l^{2}\approx-\frac{\vec{l}^{\;2}_{\perp}}{1-\hat{\xi}^{2}}<0\,,$ (8)

and that the photon virtualities can be written as

$\displaystyle
q_{1}^{2}=-Q^{2}\frac{\hat{x}+\hat{\xi}}{\hat{x}}+\frac{l^{2}}{4},\qquad
q_{2}^{2}=Q^{2}\frac{\hat{\xi}-\hat{x}}{\hat{x}}+\frac{l^{2}}{4}\,.$ (9)

Finally, the polarization vectors of the incoming and outgoing gluons,
$\epsilon(p_{1})\equiv\epsilon_{1}$,
$\epsilon^{*}(p_{2})\equiv\epsilon^{*}_{2}$, respectively, satisfy the
physical conditions

$\displaystyle\epsilon_{1}\cdot(p-l/2)=0\quad$
$\displaystyle\to\quad\epsilon_{1}\cdot p=\frac{\epsilon_{1}\cdot l}{2}\,,$
$\displaystyle\epsilon_{2}^{*}\cdot(p+l/2)=0\quad$
$\displaystyle\to\quad\epsilon_{2}^{*}\cdot p=-\frac{\epsilon_{2}^{*}\cdot
l}{2}\,.$ (10)

We note that throughout this paper we use the conventions
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ and
$\epsilon^{0123}=+1$. 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

$\displaystyle{\cal J}^{\alpha}$ $\displaystyle\equiv$
$\displaystyle-\epsilon^{\alpha\beta\mu\nu}P_{\beta}{\rm Im}T^{\rm
asym}_{\mu\nu}\,.$ (11)

In the forward limit $l^{\mu}\to 0$, ${\cal
J}^{\alpha}=g_{1}(x_{B})\bar{u}(P)\gamma^{\alpha}\gamma_{5}u(P)=2g_{1}(x_{B})S^{\alpha}$
is proportional to the $g_{1}$ structure function in polarized DIS. We have
calculated the box diagrams in the near-forward region, using the Mathematica
package ‘Package-X’ Patel (2015). The result is, for massless quarks in the
loop,

$\displaystyle\left.{\cal J}^{\alpha}\right|_{\rm
box}\approx\frac{1}{2}\frac{\alpha_{s}}{2\pi}\left(\sum_{f}e_{f}^{2}\right)\bar{u}(P_{2})\left[\left(\Delta
P_{qg}\ln\frac{Q^{2}}{-l^{2}}+\delta C_{g}^{\rm off}\right)\otimes\Delta
G(x_{B})\gamma^{\alpha}\gamma_{5}+\frac{l^{\alpha}}{l^{2}}\delta C_{g}^{\rm
anom}\otimes\tilde{{\cal F}}(x_{B})\gamma_{5}\right]u(P_{1})\,,$ (12)

where $A\otimes
B(x_{B})\equiv\int_{x_{B}}^{1}\frac{dx}{x}A(\frac{x_{B}}{x})B(x)$. In (12), we
have neglected all terms that vanish in the limit $l\to 0$, including the
$\xi$-dependence of various coefficients. (The dependence on $\xi$ will be
kept in the next section.) $\Delta P_{qg}(\hat{x})=2T_{R}(2\hat{x}-1)$ is the
polarized $g\to q$ splitting function with $T_{R}=\frac{1}{2}$ being the color
factor, and

$\displaystyle\delta C_{g}^{\rm
off}(\hat{x})=2T_{R}(2\hat{x}-1)\left(\ln\frac{1}{\hat{x}(1-\hat{x})}-1\right)$
(13)

is the coefficient function. (In both $\Delta P_{qg}$ and $\delta C_{g}^{\rm
off}$ the explicit factor of 2 accounts for the quark and antiquark
contributions.) $\tilde{{\cal F}}(x)$ is the twist-four pseudoscalar parton
distribution Tarasov and Venugopalan (2020); Hatta (2020); Radyushkin and Zhao
(2022)

$\displaystyle\tilde{{\cal F}}(x,l^{2})$ $\displaystyle\equiv$
$\displaystyle\frac{iP^{+}}{\bar{u}(P_{2})\gamma_{5}u(P_{1})}\int\frac{dz^{-}}{2\pi}e^{ixP^{+}z^{-}}\langle
P_{2}|F_{a}^{\mu\nu}(-z^{-}/2)\tilde{F}^{a}_{\mu\nu}(z^{-}/2)|P_{1}\rangle\,,$
(14)

entering Eq. (12) with

$\displaystyle\delta C_{g}^{\rm anom}(\hat{x})=4T_{R}(1-\hat{x})\,.$ (15)

We note that in the one-loop calculation, $\Delta G(x)$ and $\tilde{{\cal
F}}(x)$ have been identified via the tensor structures
$\frac{-i}{p^{+}}\langle p_{2}|F^{+\mu}\tilde{F}^{+}_{\
\mu}|p_{1}\rangle\propto
i\epsilon^{+\nu\alpha\beta}p_{\nu}\epsilon^{*}_{2\alpha}\epsilon_{1\beta}\equiv
i\epsilon^{+p\epsilon_{2}^{*}\epsilon_{1}}$ and $\langle
p_{2}|F^{\mu\nu}\tilde{F}_{\mu\nu}|p_{1}\rangle\propto\epsilon^{\mu\nu\alpha\beta}l_{\mu}p_{\nu}\epsilon_{1\alpha}\epsilon^{*}_{2\beta}\equiv\epsilon^{lp\epsilon_{1}\epsilon^{*}_{2}}$,
respectively.

The pole term $1/l^{2}$ in (12) agrees with the result derived in Ref. Tarasov
and Venugopalan (2020) using the worldline formalism. In Appendix A, we also
rederive this result via an ordinary Feymnan 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 $\lim_{l\to
0}l^{\alpha}l^{\beta}/l^{2}$ depends on the direction of $l^{\mu}$. In
addition, we find the expected contribution to the $g_{1}$ structure function
where $l^{2}$ plays the role of an infrared cutoff. Both $\Delta P_{qg}$ and
$\delta C_{g}^{\rm off}$ satisfy $\int_{0}^{1}d\hat{x}\Delta
P_{qg}(\hat{x})=\int_{0}^{1}d\hat{x}\delta C_{g}^{\rm off}(\hat{x})=0$. 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 $J_{5}^{\mu}$. Note that in dimensional regularization in
$4-2\epsilon$ dimensions, one has instead

$\displaystyle\Delta P_{qg}\frac{-1}{\epsilon}+\delta C_{g}^{\overline{\rm
MS}},\qquad\delta C_{g}^{\overline{\rm
MS}}(\hat{x})=2T_{R}(2\hat{x}-1)\left(\ln\frac{1-\hat{x}}{\hat{x}}-1\right)+4T_{R}(1-\hat{x})\,,$
(16)

in the coefficient of $\Delta G(x_{B})$. As pointed out in Vogelsang (1991),
the last term $4T_{R}(1-\hat{x})$ is of infrared origin. In off-forward
regularization, this term becomes the pole term.

The cancellation of the pole in (12) and its implications have been discussed
in Tarasov and Venugopalan (2022). 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 Collins and
Freund (1999); Ji and Osborne (1998a), the imaginary part of the Compton
amplitude can be expanded as

$\displaystyle{\cal J}^{\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{f}e_{f}^{2}\bar{u}(P_{2})\left[\gamma^{\alpha}\gamma_{5}(\tilde{H}_{f}(x_{B},\xi,l^{2})+\tilde{H}_{f}(-x_{B},\xi,l^{2}))+\frac{l^{\alpha}\gamma_{5}}{2M}(\tilde{E}^{\rm
bare}_{f}(x_{B},\xi,l^{2})+\tilde{E}^{\rm
bare}_{f}(-x_{B},\xi,l^{2}))\right]u(P_{1})$ (17) $\displaystyle+{\cal
O}(\alpha_{s})+{\cal O}(1/Q^{2})\,,$

where $\tilde{H}_{f}$ and $\tilde{E}_{f}$ 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,
$\tilde{H}_{q}(x_{B})=\Delta q(x_{B})$ and
$\tilde{H}_{q}(-x_{B})=\Delta\bar{q}(x_{B})$, so that ${\cal
J}^{\alpha}=\sum_{f}e_{f}^{2}(\Delta
q_{f}(x_{B})+\Delta\bar{q}_{f}(x_{B}))S^{\alpha}+\cdots\approx
2g_{1}(x_{B})S^{\alpha}$. It is tempting to regard (12) as part of the ${\cal
O}(\alpha_{s})$ corrections in (17). 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 Belitsky and Mueller (1998); Ji and
Osborne (1998a). The reason is that in the GPD-based calculations in the
literature, one sets $l^{\mu}\approx-2\hat{\xi}p^{\mu}$ 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, but such divergences disappear once one makes the replacement

$\displaystyle\epsilon^{*}_{2\alpha}\epsilon_{1\beta}\to\epsilon_{\alpha\beta}^{\perp}$
(18)

($\epsilon^{\perp}_{12}=-\epsilon^{\perp}_{21}=1$ is the antisymmetric tensor
in the transverse plane), in order to project onto the twist-two gluon GPDs
$\tilde{H}_{g}$ and $\tilde{E}_{g}$. Note also that the matrix element of the
twist-four operator $\langle
p_{2}|F^{\mu\nu}\tilde{F}_{\mu\nu}|p_{1}\rangle\propto\epsilon^{\mu\nu\alpha\beta}l_{\mu}p_{\nu}\epsilon_{1\alpha}\epsilon^{*}_{2\beta}$
vanishes if one assumes $l^{\mu}\propto p^{\mu}$.

The problem is that, despite the proportionality to the twist-four
distribution, the pole term cannot be dismissed as a higher twist contribution
because there is no suppression factor $1/Q^{2}$. Instead, $l^{2}=t$ coincides
with the measurable proton momentum transfer squared. Note also that the pole
term has the same Lorentz structure $\propto l^{\alpha}$ as the GPD
$\tilde{E}$ in (17). Clearly, this poses a threat to QCD factorization for the
Compton amplitude in the GPD $\tilde{E}$ sector.111Note that the
experimentally accessible Compton amplitude corresponds to $q_{2}^{2}\geq 0$
or $\xi\gtrsim x_{B}$, whereas we approximated $\xi\approx 0$ in (12).
However, we have checked that the pole $1/l^{2}$ exists also for $\xi>0$.
Besides, the general argument for factorization does not rely on the relative
magnitude of $x_{B}$ and $\xi$.

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 $\tilde{E}$ and let it cancel with a
‘primordial’ pole in $\tilde{E}^{\rm bare}$. Namely, we redefine as

$\displaystyle\tilde{E}_{f}(x_{B},l^{2})+\tilde{E}_{f}(-x_{B},l^{2})=\tilde{E}^{\rm
bare}_{f}(x_{B},l^{2})+\tilde{E}^{\rm
bare}_{f}(-x_{B},l^{2})+\frac{\alpha_{s}}{2\pi}\frac{2M}{l^{2}}\delta
C_{g}^{\rm anom}\otimes\tilde{{\cal F}}(x_{B},l^{2})\,,$ (19)

and postulate that $\tilde{E}^{\rm bare}$ is dominated by a pole

$\displaystyle\tilde{E}^{\rm bare}_{f}(x_{B},l^{2})+\tilde{E}^{\rm
bare}_{f}(-x_{B},l^{2})\approx-\frac{\alpha_{s}}{2\pi}\frac{2M}{l^{2}}\delta
C_{g}^{\rm anom}\otimes\tilde{{\cal F}}(x_{B},l^{2}=0)\,,$ (20)

which exactly cancels the perturbative pole in the last term. We further
postulate that it is this ‘renormalized’ $\tilde{E}$ that integrates to the
pseudoscalar form factor. Namely,

$\displaystyle g_{A}(l^{2})$
$\displaystyle=\sum_{f}\int_{-1}^{1}dx\tilde{H}_{f}(x,\xi,l^{2})=\sum_{f}\int_{0}^{1}dx(\tilde{H}_{f}(x,\xi,l^{2})+\tilde{H}_{f}(-x,\xi,l^{2}))\,,$
(21) $\displaystyle g_{P}(l^{2})$
$\displaystyle=\sum_{f}\int_{-1}^{1}dx\tilde{E}_{f}(x,\xi,l^{2})=\sum_{f}\int_{0}^{1}dx(\tilde{E}_{f}(x,\xi,l^{2})+\tilde{E}_{f}(-x,\xi,l^{2}))\,,$
(22)

where $g_{A}$ and $g_{P}$ are the singlet axial form factors defined by

$\displaystyle\langle
P_{2}|J_{5}^{\alpha}|P_{1}\rangle=\bar{u}(P_{2})\left[\gamma^{\alpha}\gamma_{5}g_{A}(l^{2})+\frac{l^{\alpha}\gamma_{5}}{2M}g_{P}(l^{2})\right]u(P_{1})\,.$
(23)

The prescription (19) is outside the conventional logarithmic renormalization
of twist-two GPDs. In order to better justify it, we show in the Appendix (see
(87)) that exactly the same pole structure arises in the one-loop calculation
of the polarized quark GPD of a gluon222We thank Swagato Mukherjee for
suggesting this calculation.:

$\displaystyle p^{+}\int\frac{dz^{-}}{4\pi}e^{i\hat{x}p^{+}z^{-}}\langle
p_{2}|\bar{\psi}(-z^{-}/2)\gamma^{+}\gamma_{5}\psi(z^{-}/2)|p_{1}\rangle\biggl{|}_{\rm
pole}\sim\frac{\alpha_{s}}{2\pi}T_{R}\frac{2il^{+}}{l^{2}}(1-\hat{x})\otimes\delta(1-\hat{x})\epsilon^{\epsilon_{1}\epsilon_{2}^{*}lp}\,.$
(24)

Moreover, performing the $x$-integral in (22), we obtain

$\displaystyle\frac{g_{P}(l^{2})}{2M}=-\frac{i}{l^{2}}\left(\left.\frac{\langle
P_{2}|\frac{n_{f}\alpha_{s}}{4\pi}F\tilde{F}|P_{1}\rangle}{\bar{u}(P_{2})\gamma_{5}u(P_{1})}\right|_{l^{2}=0}-\frac{\langle
P_{2}|\frac{n_{f}\alpha_{s}}{4\pi}F\tilde{F}|P_{1}\rangle}{\bar{u}(P_{2})\gamma_{5}u(P_{1})}\right)\,.$
(25)

By construction, this is finite at $l^{2}=0$. One can check that (25) is
consistent with the following exact relation obtained by taking the divergence
of (23)

$\displaystyle 2Mg_{A}(l^{2})+\frac{l^{2}g_{P}(l^{2})}{2M}=i\frac{\langle
P_{2}|\frac{n_{f}\alpha_{s}}{4\pi}F\tilde{F}|P_{1}\rangle}{\bar{u}(P_{2})\gamma_{5}u(P_{1})}\,,$
(26)

under the assumption that $g_{A}(l^{2})\approx g_{A}(0)=\Delta\Sigma$ slowly
varies with $l^{2}$. These arguments suggest that the pole should indeed be
regarded as a part of the quark GPD $\tilde{E}$, although a more rigorous,
field-theoretic justification of (19) is needed.

The cancellation of poles within the form factor $g_{P}$ is the central
argument of Tarasov and Venugopalan (2022) following the earlier suggestion in
Jaffe and Manohar (1990). The massless pole in the first term of (25) 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
$\eta_{0}$ meson. This is analogous to the pion pole in the $SU_{A}(2)$
pseudoscalar form factor $g_{P}^{(3)}\sim\frac{1}{l^{2}-m_{\pi}^{2}}$. 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 (25),
turning the massless pole into a massive one at the physical $\eta^{\prime}$
meson mass $m_{\eta}^{2}$. Indeed, the form factor $\langle
P_{2}|F\tilde{F}|P_{1}\rangle$ has a pole at $l^{2}=m_{\eta^{\prime}}^{2}$,
and in the single pole approximation (26) is satisfied by

$\displaystyle\frac{g_{P}(l^{2})}{2M}\approx\frac{-2M\Delta\Sigma}{l^{2}-m_{\eta^{\prime}}^{2}}\,,\qquad
i\frac{\langle
P_{2}|\frac{n_{f}\alpha_{s}}{4\pi}F\tilde{F}|P_{1}\rangle}{\bar{u}(P_{2})\gamma_{5}u(P_{1})}\approx-2M\Delta\Sigma\frac{m_{\eta^{\prime}}^{2}}{l^{2}-m_{\eta^{\prime}}^{2}}\,.$
(27)

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
$\tilde{E}$.333The pion pole is argued to exist also in the GPD $\tilde{E}$ in
the isovector channel
$\tilde{E}_{u}-\tilde{E}_{d}\sim\frac{1}{l^{2}-m_{\pi}^{2}}$ Penttinen et al.
(2000). Moreover, the cancellation of poles in (19) 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
$\tilde{E}$ 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 $\overline{\rm MS}$ scheme.

##  IV Symmetric part

Remarkably, the $1/l^{2}$ pole is also present in the symmetric (in $\mu\nu$)
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 $T_{\rm sym}^{\mu\nu}$ is related to the $F_{1}$
and $F_{2}$ structure functions,

$\displaystyle{\rm Im}T^{\mu\nu}_{\rm
sym}=\left(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}}\right)F_{1}(x_{B})+\left(P^{\mu}-\frac{P\cdot
q}{q^{2}}q^{\mu}\right)\left(P^{\nu}-\frac{P\cdot
q}{q^{2}}q^{\nu}\right)\frac{2x_{B}F_{2}(x_{B})}{Q^{2}}\,.$ (28)

To leading order,

$\displaystyle
F_{2}(x)=\sum_{f}e_{f}^{2}x\big{(}q_{f}(x)+\bar{q}_{f}(x)\big{)}=2xF_{1}(x)\,,$
(29)

where $q_{f}(x)$ ($\bar{q}_{f}(x))$) 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
$\Theta^{\mu\nu}_{f}=\bar{q}_{f}\gamma^{(\mu}iD^{\nu)}q_{f}$:

$\displaystyle\langle P|\Theta^{++}_{f}|P\rangle=2(P^{+})^{2}A_{f}\,,\qquad
A_{f}=\int_{0}^{1}dxx\big{(}q_{f}(x)+\bar{q}_{f}(x)\big{)}\,.$ (30)

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 Patel (2015) and found a more complicated tensor
structure,

$\displaystyle\left.{\rm Im}T^{\mu\nu}_{\rm sym}\right|_{\rm
box}=\left(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}}\right)F^{\rm
off}_{1}(x_{B},l)+\left(P^{\mu}-\frac{P\cdot
q}{q^{2}}q^{\mu}\right)\left(P^{\nu}-\frac{P\cdot
q}{q^{2}}q^{\nu}\right)\frac{2x_{B}F^{\rm off}_{2}(x_{B},l)}{Q^{2}}+\dots,$
(31)

where the ellipses denote terms proportional to tensors like
$l^{(\mu}P^{\nu)}$, $l^{(\mu}q^{\nu)}$ and $l^{\mu}l^{\nu}$ such that they
satisfy the conservation law $q_{\mu}{\rm Im}T^{\mu\nu}_{\rm sym}=0$. This
time $F_{1,2}^{\rm off}$ are related to the unpolarized GPDs instead of the
PDFs. Different from the polarized case, for a reason to become clear shortly,
here we shall keep the full dependence on the skewness parameter $\xi$. For
this purpose, we will make the replacement $l^{\mu}\approx-2\xi
P^{\mu}\approx-2\hat{\xi}p^{\mu}$ in the tensors $l^{(\mu}P^{\nu)}$,
$l^{(\mu}q^{\nu)}$ and $l^{\mu}l^{\nu}$, but not in potential pole terms
$\propto 1/l^{2}$. In doing so, we find that Eq. (31) takes the simple form

$\displaystyle\left.{\rm Im}T^{\mu\nu}_{\rm sym}\right|_{\rm
box}\approx\left(-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}}\right)\bar{F}^{\rm
off}_{1}(x_{B},l)+\left(P^{\mu}-\frac{P\cdot
q}{q^{2}}q^{\mu}\right)\left(P^{\nu}-\frac{P\cdot
q}{q^{2}}q^{\nu}\right)\frac{2x_{B}\bar{F}^{\rm off}_{2}(x_{B},l)}{Q^{2}}\,,$
(32)

where (setting $\xi=0$ in non-pole terms)

$\displaystyle\begin{split}&\bar{F}^{\rm
off}_{1}(x_{B},l)\approx\frac{1}{2}\frac{\alpha_{s}}{2\pi}\left(\sum_{f}e_{f}^{2}\right)\left[\left(P_{qg}\ln\frac{Q^{2}}{-l^{2}}+C^{\rm
off}_{1g}\right)\otimes g(x_{B})+\frac{1}{l^{2}}C^{\rm
anom}\otimes^{\prime}{\cal
F}(x_{B},\xi,l^{2})\frac{\bar{u}(P_{2})u(P_{1})}{2M}\right]\,,\\\
&\bar{F}^{\rm off}_{2}(x_{B},l)\approx
x_{B}\frac{\alpha_{s}}{2\pi}\left(\sum_{f}e_{f}^{2}\right)\left[\left(P_{qg}\ln\frac{Q^{2}}{-l^{2}}+C^{\rm
off}_{2g}\right)\otimes g(x_{B})+\frac{1}{l^{2}}C^{\rm
anom}\otimes^{\prime}{\cal
F}(x_{B},\xi,l^{2})\frac{\bar{u}(P_{2})u(P_{1})}{2M}\right]\,.\end{split}$
(33)

We recognize the expected structure of the one-loop corrections associated
with the unpolarized gluon PDF $g(x)$, with the splitting function
$P_{qg}(\hat{x})=2T_{R}((1-\hat{x})^{2}+\hat{x}^{2})$. The coefficient
functions are given by

$\displaystyle\begin{split}&C_{1g}^{\rm
off}(\hat{x})=2T_{R}((1-\hat{x})^{2}+\hat{x}^{2})\left(\ln\frac{1}{\hat{x}(1-\hat{x})}-1\right)\,,\\\
&C_{2g}^{\rm
off}(\hat{x})=2T_{R}((1-\hat{x})^{2}+\hat{x}^{2})\left(\ln\frac{1}{\hat{x}(1-\hat{x})}-1\right)+8T_{R}\hat{x}(1-\hat{x})\,.\end{split}$
(34)

In addition, we find a pole $1/l^{2}$ in both $\bar{F}_{1}^{\rm off}$ and
$\bar{F}_{2}^{\rm off}$ (but not in the difference $\bar{F}^{\rm
off}_{2}-2x_{B}\bar{F}^{\rm off}_{1}$ relevant to the longitudinal structure
function), with the following convolution formula

$\displaystyle C^{\rm anom}\otimes^{\prime}{\cal
F}(x_{B},\xi,l^{2})\equiv\int_{x_{B}}^{1}\frac{dx}{x}K(\hat{x},\hat{\xi}){\cal
F}(x,\xi,l^{2})-\frac{\theta(\xi-
x_{B})}{2}\int_{-1}^{1}\frac{dx}{x}L(\hat{x},\hat{\xi}){\cal
F}(x,\xi,l^{2})\,,$ (35)

where

$\displaystyle
K(\hat{x},\hat{\xi})=2T_{R}\frac{\hat{x}(1-\hat{x})}{1-\hat{\xi}^{2}}\,,\qquad
L(\hat{x},\hat{\xi})=2T_{R}\frac{\hat{x}(\hat{\xi}-\hat{x})}{1-\hat{\xi}^{2}}\,.$
(36)

The twist-four scalar gluon GPD is defined as

$\displaystyle{\cal F}(x,\xi,l^{2})$ $\displaystyle=$
$\displaystyle-4xP^{+}M\int\frac{dz^{-}}{2\pi}e^{ixP^{+}z^{-}}\frac{\langle
P_{2}|F^{\mu\nu}(-z^{-}/2)F_{\mu\nu}(z^{-}/2)|P_{1}\rangle}{\bar{u}(P_{2})u(P_{1})}\,.$
(37)

Several comments are in order. First, the two terms in (35) come from the
first and third diagrams in Fig. 1, respectively. The imaginary part of the
latter is nonvanishing only in the region $0<x_{B}<\xi$ where the outgoing
virtual photon becomes timelike, $q_{2}^{2}>0$, see (9). The structure of the
convolution (35) has been inferred from the analysis of Compton scattering in
a related context White (2002). Second, concerning the twist-four gluon GPD
(37), the original definition in Hatta and Zhao (2020) was a forward matrix
element $\langle P|...|P\rangle$. The necessity to define ${\cal F}$ in off-
forward kinematics was later emphasized in Radyushkin and Zhao (2021), and
this turns out to be relevant to the present discussion. Accordingly, we have
rescaled the distribution by $\bar{u}(P_{2})u(P_{1})$. We have also introduced
the prefactor $x$ to compensate for the $1/x$ factor in (35). This factor
makes ${\cal F}(x,\xi,l^{2})$ an odd function in $x$, just like $g(x)$.
Finally, in the actual one-loop calculation the pole term
$\frac{1}{l^{2}}{\cal F}(x,\xi,l^{2})$ comes from the structure444Actually the
box diagram generates an additional tensor structure
$\displaystyle\epsilon_{2}^{*\mu}\epsilon_{1}^{\nu}+\epsilon_{2}^{*\nu}\epsilon_{1}^{\mu}-g^{\mu\nu}_{\perp}\epsilon_{2}^{*}\cdot\epsilon_{1}\,,$
(38) and this has been taken into account when determining the coefficient
functions (34). For spin-1 hadrons, (38) induces an additional structure
function in (31) Jaffe and Manohar (1989). For spin-$\frac{1}{2}$ hadrons, the
structure (38) is neglected in inclusive DIS because it vanishes after
averaging over the gluon polarizations (40). In Compton scattering, it becomes
part of the gluon transversity GPD Hoodbhoy and Ji (1998).

$\displaystyle\frac{1}{l^{2}}\langle
p_{2},\epsilon_{2}|F^{\mu\nu}F_{\mu\nu}|p_{1},\epsilon_{1}\rangle\propto\frac{1}{l^{2}}\left(-\epsilon_{1}\cdot
p_{2}\epsilon_{2}^{*}\cdot p_{1}+p_{1}\cdot
p_{2}\epsilon_{1}\cdot\epsilon_{2}^{*}\right)=\frac{1}{l^{2}}\left(\epsilon_{1}\cdot
l\epsilon_{2}^{*}\cdot
l-\frac{l^{2}}{2}\epsilon_{1}\cdot\epsilon_{2}^{*}\right)\,.$ (39)

Similarly, $g(x)$ comes from $\langle-F^{+\mu}F^{+}_{\
\mu}\rangle\propto(p^{+})^{2}\epsilon_{1}\cdot\epsilon^{*}_{2}$. For on-shell
gluon states, the tree-level matrix element $\langle
F^{\mu\nu}F_{\mu\nu}\rangle$ is nonvanishing only in off-forward kinematics
$l^{\mu}\neq 0$. If one averages over the two gluon polarization states by
making the replacement

$\displaystyle\epsilon_{2\alpha}^{*}\epsilon_{1\beta}\to-\dfrac{1}{2}g_{\alpha\beta}^{\perp}\,,$
(40)

where $g^{\perp}_{\alpha\beta}\equiv
g_{\alpha\beta}-n_{\alpha}\bar{n}_{\beta}-\bar{n}_{\alpha}n_{\beta}$ with the
light-like vectors $n=(1,0,0,-1)/\sqrt{2}$ and $\bar{n}=(1,0,0,1)/\sqrt{2}$,
the pole disappears because

$\displaystyle\frac{1}{l^{2}}\left(\epsilon_{1}\cdot l\;\epsilon_{2}^{*}\cdot
l-\frac{l^{2}}{2}\epsilon_{1}\cdot\epsilon_{2}^{*}\right)\to\frac{1}{2l^{2}}(\vec{l}_{\perp}^{\;2}+l^{2})=\dfrac{\hat{\xi}^{2}}{2}\,.$
(41)

Again, this is the reason why the pole has remained unnoticed in the one-loop
GPD calculations in the unpolarized sector Ji and Osborne (1998b); Mankiewicz
et al. (1998); Belitsky and Mueller (1998); Pire et al. (2011) (see also a
recent two-loop calculation in DVCS Braun et al. (2022)) where one projects
onto the twist-two gluon GPDs $H_{g}$ and $E_{g}$ via (40). Without this
projection, the loop integral is power-divergent if one sets
$l^{\mu}=-2\hat{\xi}p^{\mu}$ from the outset.

While the replacement (40) may be justified for the unpolarized DIS structure
functions where the initial and final proton spins are equal and averaged over
as $\sum_{S}\langle PS|...|PS\rangle$, this is not the case for GPDs $\langle
P_{2}S_{2}|...|P_{1}S_{1}\rangle$ 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 $\langle
p_{2}|F^{\mu\nu}F_{\mu\nu}|p_{1}\rangle$ (see (39)), the nonperturbative
proton matrix element $\langle P_{2}|F^{\mu\nu}F_{\mu\nu}|P_{1}\rangle$ does
not vanish in the forward limit, not even after averaging over the proton spin
states. In particular, the zeroth moment of ${\cal F}(x,\xi=0,l^{2}=0)$ is the
so-called gluon condensate:

$\displaystyle\int\frac{dx}{x}{\cal F}(x,0,0)\propto\langle
P|F^{\mu\nu}F_{\mu\nu}|P\rangle\,,$ (42)

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 Giannotti and Mottola (2009); Armillis et
al. (2009); Donoghue and El-Menoufi (2015). 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:

$\displaystyle\langle p_{2}|\Theta_{\rm
QED}^{\mu\nu}|p_{1}\rangle=-\frac{e^{2}}{24\pi^{2}l^{2}}\left(p^{\mu}p^{\nu}+\frac{l^{\mu}l^{\nu}-l^{2}g^{\mu\nu}}{4}\right)\langle
p_{2}|F^{\alpha\beta}F_{\alpha\beta}|p_{1}\rangle+\ldots\,.$ (43)

If one introduces the usual color factor $T_{R}=\frac{1}{2}$ on the right hand
side, essentially the same formula holds for the one-loop gluon matrix element
of the quark energy momentum tensor $\Theta^{\mu\nu}_{f}$ in QCD. Taking the
$++$ component, one finds a relation

$\displaystyle\langle
p_{2}|\Theta_{f}^{++}|p_{1}\rangle\sim-\frac{T_{R}\alpha_{s}\,(p^{+})^{2}}{6\pi
l^{2}}(1+\hat{\xi}^{2})\langle p_{2}|F^{2}|p_{1}\rangle\,.$ (44)

On the other hand, if one takes the trace of (43), the pole disappears.
Applied to the QCD case we find

$\displaystyle\langle p_{2}|(\Theta_{f})^{\mu}_{\mu}|p_{1}\rangle=\langle
p_{2}|\frac{T_{R}\alpha_{s}}{6\pi}F^{2}|p_{1}\rangle\,,$ (45)

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-loop555The
separation of the total trace $\Theta^{\mu}_{\mu}$ into the quark and gluon
parts $(\Theta_{f})^{\mu}_{\mu}$ and $(\Theta_{g})^{\mu}_{\mu}$ is scheme
dependent Hatta et al. (2018). At one loop, the quark part simply accounts for
the $n_{f}$ term in the beta function.

$\displaystyle\Theta^{\mu}_{\mu}=\sum_{f}(\Theta_{f})^{\mu}_{\mu}+(\Theta_{g})^{\mu}_{\mu}=\frac{\beta(g)}{2g}F^{2}=-\frac{\alpha_{s}}{8\pi}\left(\frac{11N_{c}}{3}-\frac{4T_{R}n_{f}}{3}\right)F^{2}+{\cal
O}(\alpha_{s}^{2})\,.$ (46)

These observations suggest that the poles in (33) 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 (33). Treating (4) as a fully off-
forward amplitude, we find from the operator product expansion

$\displaystyle{\rm Im}T_{\rm sym}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{f}\frac{e_{f}^{2}}{4P^{+}}\left[\frac{4x_{B}^{2}}{Q^{2}}\left(P^{\mu}-\frac{P\cdot
q}{q^{2}}q^{\mu}\right)\left(P^{\nu}-\frac{P\cdot
q}{q^{2}}q^{\nu}\right)-g^{\mu\nu}+\frac{q^{\mu}q^{\nu}}{q^{2}}\right]$ (47)
$\displaystyle\times\bar{u}(P_{2})\left[\bigl{(}H^{\rm
bare}_{f}(x_{B},\xi,l^{2})-H^{\rm
bare}_{f}(-x_{B},\xi,l^{2})\bigr{)}\gamma^{+}+\frac{i\sigma^{+\lambda}l_{\lambda}}{2M}\bigl{(}E^{\rm
bare}_{f}(x_{B},\xi,l^{2})-E^{\rm
bare}_{f}(-x_{B},\xi,l^{2})\bigr{)}\right]u(p_{1})$ $\displaystyle+{\cal
O}(\alpha_{s})+{\cal O}(1/Q^{2})\,,$

where, as before, $H_{f}^{\rm bare}$ and $E_{f}^{\rm bare}$ 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 $-H_{f}(-x)=\bar{q}_{f}(x)$ are the
quark and antiquark PDFs. According to QCD factorization theorems Collins and
Freund (1999); Ji and Osborne (1998a), higher order corrections to (47) not
suppressed by $1/Q^{2}$ consist only of the twist-two quark and gluon GPDs to
all orders. However, our result (33) shows that there are diverging
corrections already in the ${\cal O}(\alpha_{s})$ terms which cannot be simply
dismissed as higher-twist, nor be absorbed into the (usual) renormalization of
twist-two GPDs. From the Gordon identity

$\displaystyle\frac{\bar{u}(P_{2})u(P_{1})}{2M}=\frac{1}{2P^{+}}\bar{u}(P_{2})\left[\gamma^{+}-\frac{i\sigma^{+\lambda}l_{\lambda}}{2M}\right]u(P_{1})\,,$
(48)

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, we absorb the pole in (33) into the
definitions of the GPDs (so that $H^{\rm bare}_{f}\to H_{f}$ and $E^{\rm
bare}_{f}\to E_{f}$, cf. (19)) and let it cancel with nonperturbative, pre-
existing poles in $H^{\rm bare}_{f}$ and $E^{\rm bare}_{f}$:

$\displaystyle\begin{split}\sum_{f}e_{f}^{2}x_{B}(H^{\rm
bare}_{f}(x_{B},\xi,l^{2})-H^{\rm
bare}_{f}(-x_{B},\xi,l^{2}))&\approx-\frac{\alpha_{s}}{2\pi}\Bigg{(}\sum_{f}e_{f}^{2}\Bigg{)}\frac{x_{B}}{l^{2}}C^{\rm
anom}\otimes^{\prime}{\cal F}(x_{B},\xi,l^{2}=0)\,,\\\
\sum_{f}e_{f}^{2}x_{B}(E^{\rm bare}_{f}(x_{B},\xi,l^{2})-E^{\rm
bare}_{f}(-x_{B},\xi,l^{2}))&\approx\frac{\alpha_{s}}{2\pi}\Bigg{(}\sum_{f}e_{f}^{2}\Bigg{)}\frac{x_{B}}{l^{2}}C^{\rm
anom}\otimes^{\prime}{\cal F}(x_{B},\xi,l^{2}=0)\,.\end{split}$ (49)

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

$\displaystyle\begin{split}\int_{0}^{1}dx_{B}x_{B}(H_{f}(x_{B},\xi,l^{2})-H_{f}(-x_{B},\xi,l^{2}))&=\int_{-1}^{1}dx_{B}x_{B}H_{f}(x_{B},\xi,l^{2})=A_{f}(l^{2})+\xi^{2}D_{f}(l^{2})\,,\\\
\int_{0}^{1}dx_{B}x_{B}(E_{f}(x_{B},\xi,l^{2})-E_{f}(-x_{B},\xi,l^{2}))&=\int_{-1}^{1}dx_{B}x_{B}E_{f}(x,\xi,l^{2})=B_{f}(l^{2})-\xi^{2}D_{f}(l^{2})\,,\end{split}$
(50)

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 Ji (1997),

$\displaystyle\langle
P_{2}|\Theta^{\mu\nu}_{f}|P_{1}\rangle=\frac{1}{M}\bar{u}(P_{2})\left[P^{\mu}P^{\nu}A_{f}+(A_{f}+B_{f})\frac{P^{(\mu}i\sigma^{\nu)\rho}l_{\rho}}{2}+\frac{D_{f}}{4}(l^{\mu}l^{\nu}-g^{\mu\nu}l^{2})+M^{2}\bar{C}_{f}g^{\mu\nu}\right]u(P_{1})\,.$
(51)

On the other hand, the second moment of the pole term contains the following
integral:

$\displaystyle\int_{0}^{1}dx_{B}x_{B}\left[\int_{x_{B}}^{1}\frac{dx}{x}K\left(\frac{x_{B}}{x},\frac{\xi}{x}\right)-\frac{\theta(\xi-
x_{B})}{2}\int_{-1}^{1}\frac{dx}{x}L\left(\frac{x_{B}}{x},\frac{\xi}{x}\right)\right]{\cal
F}(x,\xi,l^{2})$ $\displaystyle\
=\frac{T_{R}}{6}\int_{0}^{1}dx\frac{x}{1-\frac{\xi^{2}}{x^{2}}}\left(1-\frac{\xi^{4}}{x^{4}}\right){\cal
F}(x,\xi,l^{2})=\frac{T_{R}}{12}\int_{-1}^{1}dxx\left(1+\frac{\xi^{2}}{x^{2}}\right){\cal
F}(x,\xi,l^{2})\,.$ (52)

Note that if it were not for the second term, the result would contain
infinitely many powers of $\xi$ from the expansion

$\displaystyle\frac{1}{1-\frac{\xi^{2}}{x^{2}}}=1+\frac{\xi^{2}}{x^{2}}+\frac{\xi^{4}}{x^{4}}+\ldots\,,$
(53)

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 $\xi^{n}$ with
$n\geq 4$. The final result is thus quadratic in $\xi$, consistent with the
right-hand sides of (50). A cancellation of this sort is known in the GPD
literature, see White (2002). Interestingly, our result (36) coincides with
the ‘simple model’ employed in White (2002).

The last integral in (52) reads

$\displaystyle\int_{-1}^{1}dxx\left(1+\frac{\xi^{2}}{x^{2}}\right){\cal
F}(x,\xi,l^{2})=-\frac{4M}{(P^{+})^{2}}\frac{\langle
P_{2}|F^{\alpha\beta}(i\overleftrightarrow{D}^{+})^{2}F_{\alpha\beta}|P_{1}\rangle}{\bar{u}(P_{2})u(P_{1})}-4\xi^{2}M\frac{\langle
P_{2}|F^{\mu\nu}F_{\mu\nu}|P_{1}\rangle}{\bar{u}(P_{2})u(P_{1})},$ (54)

where
$\overleftrightarrow{D}^{+}\equiv\frac{\overrightarrow{D}^{+}-\overleftarrow{D}^{+}}{2}$.
We thus arrive at

$\displaystyle\begin{split}\sum_{f}e_{f}^{2}\bigl{(}A^{\rm
bare}_{f}(l^{2})+\xi^{2}D^{\rm
bare}_{f}(l^{2})\bigr{)}&\approx\frac{T_{R}\alpha_{s}}{12\pi
l^{2}}\Bigg{(}\sum_{f}e_{f}^{2}\Bigg{)}\left(\frac{\langle
P|F^{\alpha\beta}(i\overleftrightarrow{D}^{+})^{2}F_{\alpha\beta}|P\rangle}{(P^{+})^{2}}+\xi^{2}\langle
P|F^{2}|P\rangle\right)\,,\\\ \sum_{f}e_{f}^{2}\bigl{(}B^{\rm
bare}_{f}(l^{2})-\xi^{2}D^{\rm
bare}_{f}(l^{2})\bigr{)}&\approx-\frac{T_{R}\alpha_{s}}{12\pi
l^{2}}\Bigg{(}\sum_{f}e_{f}^{2}\Bigg{)}\left(\frac{\langle
P|F^{\alpha\beta}(i\overleftrightarrow{D}^{+})^{2}F_{\alpha\beta}|P\rangle}{(P^{+})^{2}}+\xi^{2}\langle
P|F^{2}|P\rangle\right)\,.\end{split}$ (55)

Therefore, $A^{\rm bare}_{f}$, $B^{\rm bare}_{f}$ and $D^{\rm bare}_{f}$ all
develop a pole $1/l^{2}$, but not in the linear combination $A_{f}^{\rm
bare}+B_{f}^{\rm bare}=A_{f}+B_{f}$ relevant to the Ji sum rule Ji (1997). By
construction, these poles cancel against the poles in (33) 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 (33), (55) and the ‘trace
anomaly pole’ (44) known in the literature after taking into account the
factor of 2 from (30) (and also the sign difference between (33) and (55)).
This shows that the pole in the box diagram originates from the QCD trace
anomaly. The ${\cal O}(\xi^{2})$ terms match straightforwardly, but for the
${\cal O}(\xi^{0})$ 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
$\eta_{0}$ 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,

$\displaystyle\langle P_{2}|\Theta^{\mu\nu}|P_{1}\rangle\sim\langle
P_{2}|P_{1}G^{(2)}_{0}\rangle\frac{1}{l^{2}}\langle
G^{(2)}_{0}|\Theta^{\mu\nu}|0\rangle$ $\displaystyle\sim$ $\displaystyle
g^{(2)}_{GNN}f^{(2)}_{G}\frac{\bar{u}(P_{2})P^{(\alpha}\gamma^{\beta)}u(P_{1})}{l^{2}}\sum_{\rm
pol}\epsilon_{\alpha\beta}\epsilon^{\mu\nu}$ (56) $\displaystyle\sim$
$\displaystyle
g^{(2)}_{GNN}f^{(2)}_{G}\frac{1}{l^{2}}\bar{u}(P_{2})P^{(\mu}\gamma^{\nu)}u(P_{1})\,,$

where $g^{(2)}_{GNN}$ and $f^{(2)}_{G}$ are the effective glueball-nucleon
coupling and the glueball decay constant, respectively. $\epsilon^{\mu\nu}$ is
the glueball polarization tensor. Likewise, the exchange of a scalar glueball
gives666A careful analysis shows that both $2^{++}$ and $0^{++}$ glueballs
contribute to the $D$-form factor Fujita et al. (2022), see also Mamo and
Zahed (2020, 2021).

$\displaystyle\langle P_{2}|P_{1}G^{(0)}_{0}\rangle\frac{1}{l^{2}}\langle
G^{(0)}_{0}|\Theta^{\mu\nu}|0\rangle\sim
g^{(0)}_{GNN}f^{(0)}_{G}\frac{l^{\mu}l^{\nu}}{l^{2}}\bar{u}(P_{2})u(P_{1})\,.$
(57)

Comparing with (55), we find

$\displaystyle g^{(2)}_{GNN}f^{(2)}_{G}\sim\frac{\langle
P|\alpha_{s}F^{\alpha\beta}(i\overleftrightarrow{D}^{+})^{2}F_{\alpha\beta}|P\rangle}{(P^{+})^{2}}\,,\qquad
Mg^{(0)}_{GNN}f^{(0)}_{G}\sim\langle P|\alpha_{s}F^{2}|P\rangle\,.$ (58)

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

$\displaystyle\sum_{f}D_{f}(l^{2})\approx M\frac{T_{R}n_{f}\alpha_{s}}{6\pi
l^{2}}\left(\frac{\langle
P_{2}|F^{2}|P_{1}\rangle}{\bar{u}(P_{2})u(P_{1})}\Biggl{|}_{l^{2}=0}-\frac{\langle
P_{2}|F^{2}|P_{1}\rangle}{\bar{u}(P_{2})u(P_{1})}\right)\sim
M\sum_{n}\frac{g_{G_{n}NN}^{(0)}f_{G_{n}}^{(0)}}{l^{2}-m^{2}_{G_{n}}}\,,$ (59)

and similarly for the $A_{f},B_{f}$ form factors. This is in accord with the
recently advocated ‘glueball dominance’ picture Fujita et al. (2022) 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.
Tarasov and Venugopalan (2020, 2022) in the context of polarized DIS. Below we
summarize our main findings:

  * —

Antisymmetric case: Our main result is Eq. (12) which features (i) a pole
$1/l^{2}$ coupled to the gluon fields through the pseudoscalar operator
$F^{\mu\nu}\tilde{F}_{\mu\nu}$, (ii) the usual logarithmic term and the
coefficient function in polarized DIS. We interpret this result in terms of
the GPDs $\tilde{H}$ and $\tilde{E}$ by mapping (12) 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 $\tilde{E}$ (19) in which this perturbative
pole cancels with a nonperturbative pole present a priori in the ‘bare’ GPD
$\tilde{E}^{\rm bare}$. We have further postulated that it is the integral of
this renormalized $\tilde{E}$ that defines the pseudoscalar form factor
$g_{P}$. This requirement has nevertheless led to the relation (25) which we
view as correct and is consistent with the pole cancellation argument in Jaffe
and Manohar (1990); Tarasov and Venugopalan (2022).

  * —

Symmetric case: Our main result is Eq. (33) which features (i) a pole
$1/l^{2}$ coupled to the gluon fields through the scalar operator
$F^{\mu\nu}F_{\mu\nu}$, (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 (33) 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 (12) has a distinct Lorentz structure $\propto
l^{\alpha}$ from the $\Delta G$ term $\propto S^{\alpha}$. The cancellation of
poles occurs entirely within the GPD $\tilde{E}$ sector while $g_{1}$ is
renormalized in the usual way. In the unpolarized case, the pole vanishes
after averaging over the gluon polarizations (41) appropriate to unpolarized
DIS. (See however a caveat above (42).)

In the future, we plan to extend our calculation to the real part of the
Compton amplitude as well as the quark initiated 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.

###### Acknowledgements.

We thank Andrey Tarasov and Raju Venugopalan for all the stimulating
discussions that triggered this work and continued throughout the course of
the work. We also thank Swagato Mukherjee and Jianwei Qiu for valuable
discussions. S. B. thanks Hiren Patel for helpful discussions. S. B. and Y. H.
were supported by the U.S. Department of Energy under Contract No. DE-
SC0012704, and also by Laboratory Directed Research and Development (LDRD)
funds from Brookhaven Science Associates. S. B. has also been supported by the
U.S. Department of Energy, Office of Science, Office of Nuclear Physics and
Office of Advanced Scientific Computing Research within the framework of
Scientific Discovery through Advance Computing (SciDAC) award Computing the
Properties of Matter with Leadership Computing Resources. W. V. has been
supported by Deutsche Forschungsgemeinschaft (DFG) through the Research Unit
FOR 2926 (project 430915220). He is also grateful to Brookhaven National
Laboratory for hospitality during completion of part of this work.

##  Appendix A Analytical derivation of the pole term

In the body of the paper, we have used the Mathematica package ‘Package-X’
Patel (2015) to compute the imaginary part of the relevant Feynman diagrams.
It is instructive to demonstrate analytically how the pole term $1/l^{2}$
arises. This has been done in Tarasov and Venugopalan (2020) in the
‘worldline’ formalism. In this Appendix, we rederive the same result from an
ordinary Feynman diagram calculation.

For definiteness, we consider the antisymmetric case. Our approach is similar
to that in Carlitz et al. (1988) except that we regularize the infrared
singularity by off-forward momentum transfer $l^{\mu}$ instead of the current
quark mass $m_{q}$. The relevant integral is

$\displaystyle\epsilon^{\alpha\beta}_{\ \
\,\mu\nu}p_{\beta}\int\frac{d^{4}k}{(2\pi)^{4}}\delta((q+k)^{2})((k-p)^{2})\frac{{\rm
Tr}[\gamma^{\mu}({\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$q$}})\gamma^{\nu}({\ooalign{\hfil/\hfil\crcr$k$}}-{\ooalign{\hfil/\hfil\crcr$l$}}/2)\gamma^{\rho}({\ooalign{\hfil/\hfil\crcr$k$}}-{\ooalign{\hfil/\hfil\crcr$p$}})\gamma^{\lambda}({\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$l$}}/2)]}{(k-l/2)^{2}(k+l/2)^{2}}\epsilon_{1\rho}\epsilon^{*}_{2\lambda}\,,$
(76)

where we have already taken the imaginary part by putting the final state
partons on-shell. The delta function constraints can be solved as

$\displaystyle\delta((p-k)^{2})\delta((q+k)^{2})\approx\frac{\hat{x}}{2(1-\hat{x})Q^{2}}\Bigl{(}\delta(k^{+}-k_{1}^{+})\delta(k^{-}-k_{1}^{-})+\delta(k^{+}-k_{2}^{+})\delta(k^{-}-k_{2}^{-})\Bigr{)}\,,$
(77)

where

$\displaystyle
k^{+}_{1,2}=\frac{p^{+}}{2}\left(1+\hat{x}\mp(1-\hat{x})\sqrt{1-\frac{16\hat{x}k_{\perp}^{2}}{(1-\hat{x})(4Q^{2}-\hat{x}l^{2})}}\right)\,,$
$\displaystyle
k^{-}_{1,2}=-\frac{1}{16\hat{x}p^{+}}\left(4Q^{2}+\hat{x}l^{2}\mp(4Q^{2}-\hat{x}l^{2})\sqrt{1-\frac{16\hat{x}k_{\perp}^{2}}{(1-\hat{x})(4Q^{2}-\hat{x}l^{2})}}\right)\,.$
(78)

The pole comes from the first set of roots, $k_{1}^{\pm}$, in the
$k_{\perp}\to 0$ region. In this region, we can approximate

$\displaystyle k_{1}^{+}\approx\hat{x}p^{+}\,,\qquad
k_{1}^{-}\approx\frac{-k_{\perp}^{2}}{2(1-\hat{x})p^{+}}-\frac{l^{2}}{8p^{+}}\,.$
(79)

After performing the $k^{\pm}$ integrals using the delta functions, we are
left with an integral of the form

$\displaystyle\int
d^{2}k_{\perp}\frac{f(k_{\perp})}{(k-l/2)^{2}(k+l/2)^{2}}=\int_{0}^{1}da\int
d^{2}k_{\perp}\frac{f(k_{\perp})}{\left(k^{2}+\frac{l^{2}}{4}+(1-2a)k\cdot
l\right)^{2}}\,.$ (80)

Using

$\displaystyle
k^{2}+\frac{l^{2}}{4}\approx\frac{-k_{\perp}^{2}}{1-\hat{x}}+\frac{(1-\hat{x}){l^{2}}}{4}\,,$
$\displaystyle
k^{+}l^{-}+k^{-}l^{+}\approx\frac{(1-\hat{x})l^{2}\hat{\xi}}{4}+\frac{k_{\perp}^{2}\hat{\xi}}{1-\hat{x}}\,,$
(81)

we can evaluate (80) as

$\displaystyle\int_{0}^{1}da\int^{\frac{1-\hat{x}}{4\hat{x}}Q^{2}}d^{2}k_{\perp}\frac{f(k_{\perp})}{\left(\frac{k_{\perp}^{2}}{1-\hat{x}}-\frac{(1-\hat{x})l^{2}}{4}+(1-2a)\left(k_{\perp}\cdot
l_{\perp}-\frac{(1-\hat{x})l^{2}\hat{\xi}}{4}-\frac{k_{\perp}^{2}\hat{\xi}}{1-\hat{x}}\right)\right)^{2}}$
$\displaystyle=\int_{0}^{1}da\int^{\frac{1-\hat{x}}{4\hat{x}}Q^{2}}d^{2}k^{\prime}_{\perp}\frac{f\left(k^{\prime}_{\perp}-\frac{(1-\hat{x})(1-2a)}{2(1-\hat{\xi}(1-2a))}l_{\perp}\right)}{\left(\frac{1-\hat{\xi}(1-2a)}{1-\hat{x}}k^{\prime
2}_{\perp}-\frac{(1-\hat{x})a(1-a)}{1-\hat{\xi}(1-2a)}l^{2}\right)^{2}}$
$\displaystyle\approx\pi\int_{0}^{1}da\frac{f\left(-\frac{1}{2}(1-\hat{x})(1-2a)l_{\perp}\right)}{-a(1-a)l^{2}}\,,$
(82)

where we have neglected $k^{\prime}_{\perp}$ in the numerator since the pole
comes from the infrared. We have also used (8) to obtain the second line. This
relation is crucial to restore Lorentz covariance which was not manifest after
performing the $k^{\pm}$-integrations. The apparent singularities in (82) at
$a=0,1$ are innocuous because the trace in the numerator gives

$\displaystyle f(...)\sim a(1-a)p\cdot q(1-\hat{x})^{2}(\epsilon_{1}\cdot
l\epsilon^{\alpha\epsilon^{*}_{2}lp}-\epsilon^{*}_{2}\cdot
l\epsilon^{\alpha\epsilon_{1}lp})+\ldots\,.$ (83)

This term is quadratic in $l$, so it can be easily missed if one is not
careful in taking the forward limit. With the help of the Schouten identity,
the tensors in (83) can be rewritten as

$\displaystyle\epsilon_{1}\cdot
l\epsilon^{\alpha\epsilon^{*}_{2}lp}-\epsilon_{2}^{*}\cdot
le^{\alpha\epsilon_{1}lp}+l^{2}\epsilon^{\alpha\epsilon_{1}\epsilon^{*}_{2}p}=l^{\alpha}\epsilon^{\epsilon_{1}\epsilon^{*}_{2}lp}\,,$
(84)

where we included a non-pole term proportional to $l^{2}$. This term is beyond
the accuracy of the present schematic derivation, but it must be there due to
gauge invariance. Combining with the prefactor
$\frac{\hat{x}}{(1-\hat{x})Q^{2}}$ from (77), we recover the anomaly pole

$\displaystyle\frac{p\cdot
q}{Q^{2}}\hat{x}(1-\hat{x})\frac{l^{\alpha}}{l^{2}}\epsilon^{\epsilon_{1}\epsilon^{*}_{2}lp}\sim(1-\hat{x})\frac{l^{\alpha}}{l^{2}}\epsilon^{\epsilon_{1}\epsilon^{*}_{2}lp}\,.$
(85)

It is easy to see that the same pole arises in the polarized quark GPD of a
single gluon

$\displaystyle{\cal A}_{g}^{+}\equiv
p^{+}\int\frac{dz^{-}}{4\pi}e^{i\hat{x}p^{+}z^{-}}\langle
p_{2}|\bar{\psi}(-z^{-}/2)\gamma^{+}\gamma_{5}\psi(z^{-}/2)|p_{1}\rangle\,,$
(86)

evaluated to one-loop. For $0<\hat{x}<1$, this can be obtained by replacing
$\epsilon^{+\beta}_{\ \ \
\mu\nu}p_{\beta}\gamma^{\mu}({\ooalign{\hfil/\hfil\crcr$k$}}+{\ooalign{\hfil/\hfil\crcr$q$}})\gamma^{\nu}\to\gamma^{+}\gamma_{5}$
and $\delta((q+k)^{2})\to\delta(k^{+}-\hat{x}p^{+})$ in (76) so that the roots
(79) remain unchanged. The calculation is simpler and here we report on the
complete analytic result in the limit $l^{\mu}\to 0$

$\displaystyle{\cal
A}_{g}^{+}\approx\frac{\alpha_{s}}{2\pi}T_{R}\left[(2\hat{x}-1)\left(\ln\frac{4\Lambda^{2}}{-l^{2}(1-\hat{x})^{2}}-1\right)\otimes\delta(1-\hat{x})i\epsilon^{+p\epsilon_{2}^{*}\epsilon_{1}}+\frac{2il^{+}}{l^{2}}(1-\hat{x})\otimes\delta(1-\hat{x})\epsilon^{\epsilon_{1}\epsilon_{2}^{*}lp}\right]\,.$
(87)

The cutoff $\Lambda$ of the transverse momentum integral has to be chosen such
that the $\hat{x}$-integral of the first term vanishes. This is analogous to
fixing the ambiguity of the triangle diagram for $\langle J_{5}^{\mu}\rangle$
by requiring gauge invariance (vector current conservation). In particular,
with the choice $\Lambda^{2}=\frac{1-\hat{x}}{4\hat{x}}Q^{2}$ (cf., (82)), Eq.
(87) has exactly the same structure as (12) including even the finite terms.
Adding the contribution from the region $-1<\hat{x}<0$ and integrating over
$\hat{x}$, one then recovers (2).

Let us finally comment on the connection to semi-inclusive DIS (SIDIS). Since
the inclusive cross section has a pole, there must be a trace of it in the
SIDIS cross section. This can be seen by undoing the integral over
$k_{\perp}\approx k^{\prime}_{\perp}$ in (82) which is the transverse
component of the final state quark momentum $k_{q}=q+k$. $k_{\perp}$ is
related to the standard variable $\hat{z}=\frac{p\cdot k_{q}}{p\cdot q}$ in
SIDIS as

$\displaystyle\frac{k^{2}_{\perp}}{1-\hat{x}}=\hat{z}(1-\hat{z})\frac{Q^{2}}{\hat{x}}$
(88)

so that $dk^{2}_{\perp}\sim|1-2\hat{z}|d\hat{z}$. We thus see that the pole
comes from the integral

$\displaystyle\frac{\pi}{2}\frac{\hat{x}(1-\hat{x})}{Q^{2}}\int_{0}^{1}d\hat{z}\frac{|1-2\hat{z}|}{(\hat{z}(1-\hat{z})-\hat{x}(1-\hat{x})a(1-a)l^{2}/Q^{2})^{2}}\approx\frac{\pi}{-a(1-a)l^{2}}\,.$
(89)

This shows that the two end points $\hat{z}\to 0,1$ equally contribute to the
pole. We again emphasize that to access this pole in SIDIS, one has to keep
terms quadratic in $l^{\mu}$ in the numerator.

## References

  * Kodaira (1980) J. Kodaira, Nucl. Phys. B 165, 129 (1980).
  * Ashman et al. (1988) J. Ashman et al. (European Muon), Phys. Lett. B 206, 364 (1988).
  * Carlitz et al. (1988) R. D. Carlitz, J. C. Collins, and A. H. Mueller, Phys. Lett. B 214, 229 (1988).
  * Altarelli and Ross (1988) G. Altarelli and G. G. Ross, Phys. Lett. B 212, 391 (1988).
  * de Florian et al. (2014) D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, Phys. Rev. Lett. 113, 012001 (2014), eprint 1404.4293.
  * Nocera et al. (2014) E. R. Nocera, R. D. Ball, S. Forte, G. Ridolfi, and J. Rojo (NNPDF), Nucl. Phys. B 887, 276 (2014), eprint 1406.5539.
  * Zhou et al. (2022) Y. Zhou, N. Sato, and W. Melnitchouk (Jefferson Lab Angular Momentum (JAM)), Phys. Rev. D 105, 074022 (2022), eprint 2201.02075.
  * Jaffe and Manohar (1990) R. L. Jaffe and A. Manohar, Nucl. Phys. B 337, 509 (1990).
  * Forte (1990) S. Forte, Nucl. Phys. B 331, 1 (1990).
  * Bodwin and Qiu (1990) G. T. Bodwin and J.-W. Qiu, Phys. Rev. D 41, 2755 (1990).
  * Vogelsang (1991) W. Vogelsang, Z. Phys. C 50, 275 (1991).
  * Tarasov and Venugopalan (2022) A. Tarasov and R. Venugopalan, Phys. Rev. D 105, 014020 (2022), eprint 2109.10370.
  * Tarasov and Venugopalan (2020) A. Tarasov and R. Venugopalan, Phys. Rev. D 102, 114022 (2020), eprint 2008.08104.
  * Hatsuda (1990) T. Hatsuda, Nucl. Phys. B 329, 376 (1990).
  * Veneziano (1989) G. Veneziano, Mod. Phys. Lett. A 4, 1605 (1989).
  * Giannotti and Mottola (2009) M. Giannotti and E. Mottola, Phys. Rev. D 79, 045014 (2009), eprint 0812.0351.
  * Armillis et al. (2009) R. Armillis, C. Coriano, and L. Delle Rose, Phys. Lett. B 682, 322 (2009), eprint 0909.4522.
  * Donoghue and El-Menoufi (2015) J. F. Donoghue and B. K. El-Menoufi, JHEP 05, 118 (2015), eprint 1503.06099.
  * Patel (2015) H. H. Patel, Comput. Phys. Commun. 197, 276 (2015), eprint 1503.01469.
  * Hatta (2020) Y. Hatta, Phys. Rev. D 102, 094004 (2020), eprint 2009.03657.
  * Radyushkin and Zhao (2022) A. Radyushkin and S. Zhao, JHEP 02, 163 (2022), eprint 2201.02181.
  * Collins and Freund (1999) J. C. Collins and A. Freund, Phys. Rev. D 59, 074009 (1999), eprint hep-ph/9801262.
  * Ji and Osborne (1998a) X.-D. Ji and J. Osborne, Phys. Rev. D 58, 094018 (1998a), eprint hep-ph/9801260.
  * Belitsky and Mueller (1998) A. V. Belitsky and D. Mueller, Phys. Lett. B 417, 129 (1998), eprint hep-ph/9709379.
  * Penttinen et al. (2000) M. Penttinen, M. V. Polyakov, and K. Goeke, Phys. Rev. D 62, 014024 (2000), eprint hep-ph/9909489.
  * White (2002) B. E. White, J. Phys. G 28, 203 (2002), eprint hep-ph/0102121.
  * Hatta and Zhao (2020) Y. Hatta and Y. Zhao, Phys. Rev. D 102, 034004 (2020), eprint 2006.02798.
  * Radyushkin and Zhao (2021) A. Radyushkin and S. Zhao, JHEP 12, 010 (2021), eprint 2111.00887.
  * Jaffe and Manohar (1989) R. L. Jaffe and A. Manohar, Phys. Lett. B 223, 218 (1989).
  * Hoodbhoy and Ji (1998) P. Hoodbhoy and X.-D. Ji, Phys. Rev. D 58, 054006 (1998), eprint hep-ph/9801369.
  * Ji and Osborne (1998b) X.-D. Ji and J. Osborne, Phys. Rev. D 57, 1337 (1998b), eprint hep-ph/9707254.
  * Mankiewicz et al. (1998) L. Mankiewicz, G. Piller, E. Stein, M. Vanttinen, and T. Weigl, Phys. Lett. B 425, 186 (1998), [Erratum: Phys.Lett.B 461, 423–423 (1999)], eprint hep-ph/9712251.
  * Pire et al. (2011) B. Pire, L. Szymanowski, and J. Wagner, Phys. Rev. D 83, 034009 (2011), eprint 1101.0555.
  * Braun et al. (2022) V. M. Braun, Y. Ji, and J. Schoenleber, Phys. Rev. Lett. 129, 172001 (2022), eprint 2207.06818.
  * Hatta et al. (2018) Y. Hatta, A. Rajan, and K. Tanaka, JHEP 12, 008 (2018), eprint 1810.05116.
  * Ji (1997) X.-D. Ji, Phys. Rev. Lett. 78, 610 (1997), eprint hep-ph/9603249.
  * Fujita et al. (2022) M. Fujita, Y. Hatta, S. Sugimoto, and T. Ueda, PTEP 2022, 093 (2022), eprint 2206.06578.
  * Mamo and Zahed (2020) K. A. Mamo and I. Zahed, Phys. Rev. D 101, 086003 (2020), eprint 1910.04707.
  * Mamo and Zahed (2021) K. A. Mamo and I. Zahed, Phys. Rev. D 103, 094010 (2021), eprint 2103.03186.