# Probing the polarized photon content of the proton in $ep$ collisions at the
EIC

Daniel Rein<EMAIL_ADDRESS>Marc Schlegel
<EMAIL_ADDRESS>Werner Vogelsang  werner.vogelsang@uni-
tuebingen.de Institute for Theoretical Physics, University of Tübingen, Auf
der Morgenstelle 14, D-72076 Tübingen, Germany

###### Abstract

We study the single-inclusive production of prompt photons in electron proton
collisions, $ep\to\gamma X$, for kinematics relevant at the Electron-Ion
Collider (EIC). We perform a perturbative calculation of the differential
cross section to next-to-leading order in QCD and to lowest order in QED. We
consider unpolarized collisions as well as scattering of longitudinally
polarized incident electrons and protons. We show that the cross sections are
sensitive to the parton distribution functions of photons inside the proton,
which we find to generate the dominant contributions in certain kinematical
regions at the EIC. We also investigate the effects of photon isolation on the
unpolarized and polarized cross sections.

††preprint: arXiv:XXXXX.XXXXX [hep-ph]

##  I Introduction

Production of photons with large transverse momentum in hadronic scattering
serves as an important probe of the partonic structure of hadrons, especially
of their gluon distributions. Compared to hadronic final states, photons offer
the advantage that a substantial – and often strongly dominant – part of the
production mechanism comes from “point-like” contributions for which the
produced photon couples directly to a partonic hard-scattering process. In the
present paper, we study photon production in a different and somewhat simpler
setting, in electron proton collisions. Previous studies in this area have
considered the process $ep\to e\gamma X$, with detected final-state electron.
Here we will address its truly single-inclusive counterpart, $ep\to\gamma X$,
for which the scattered electron is not explicitly observed. Our motivation
for investigating $ep\to\gamma X$ is twofold. First, as we will show, the
process may offer access to the proton’s photon parton distribution function
(PDF). Second, the process is also interesting from a theoretical point of
view, especially when the proton is transversely polarized. Although we do not
consider this particular case here, our calculation provides an important
basis for future dedicated studies. We will now describe our two motivations
in a little more detail.

There is a long history of interest in the proton’s photon PDF, starting from
discussions of its possible relevance for new-physics searches at hadron
colliders Drees and Zeppenfeld (1989); Ohnemus et al. (1994a); Drees et al.
(1994); Ohnemus et al. (1994b); Gluck et al. (1995). As part of the QED
corrections to proton structure, the distribution soon became part of the
global analysis of PDFs Martin et al. (2005); Ball et al. (2013); Schmidt et
al. (2014). Important progress was made by Refs. Manohar et al. (2016, 2017)
which showed that the photon PDF may in fact be computed from first
principles, although the result relies on elastic contributions as well as on
understanding of the proton structure functions down to low scales. The
formalism of Manohar et al. (2016, 2017) has become the paradigm for modern
studies of the photon content of the proton when incorporated in global
analyses of proton structure Bertone et al. (2018); Harland-Lang et al.
(2019); Cridge et al. (2022).

The usefulness of electron-proton scattering in obtaining information on the
proton’s photon PDF has been established in numerous studies, primarily in the
context of the HERA collider Bawa and Stirling (1988); Bawa et al. (1991);
Aurenche et al. (1992); Courau and Kessler (1992); Blumlein et al. (1993);
Gordon and Storrow (1994); Gordon and Vogelsang (1995); De Rujula and
Vogelsang (1999); Hoyer et al. (2000); Fontannaz et al. (2001); Gluck et al.
(2002, 2003); Mukherjee and Pisano (2003); Mukherjee and Pisano (2004a, b);
Lendermann et al. (2003); Aktas et al. (2005); Chekanov et al. (2010);
Gehrmann-De Ridder et al. (2006a, b). The photon PDF is also central to
studies of QED radiative corrections to the DIS and semi-inclusive DIS cross
sections Liu et al. (2021a); Akushevich et al. (2023). However, to our
knowledge, single-inclusive production of photons has not been considered so
far in the literature, and it is useful to explore its potential ability to
put further constraints on the photon PDF, especially so in view of the future
Electron-Ion Collider (EIC) now under construction Accardi et al. (2016);
Abdul Khalek et al. (2022). Here our focus will also be on the polarized
(helicity) photon PDF, whose presence in $ep\to e\gamma X$ has been discussed
in Refs. Gluck et al. (2002, 2003); Mukherjee and Pisano (2003); Mukherjee and
Pisano (2004a, b). Most of the previous studies did not take into account the
contributions arising from production of photons in jet fragmentation, which
we will develop here to full next-to-leading order (NLO).

Single-inclusive processes have also been of more general interest in the
context of achieving a better understanding of single-transverse spin
asymmetries (TSSAs) in hadronic scattering. The theoretical description of
TSSAs in reactions of the type $p^{\uparrow}p\to hX$ in terms of collinear
factorization has turned out to be remarkably complex Qiu and Sterman (1991,
1992, 1999); Kouvaris et al. (2006); Kanazawa and Koike (2010); Kang et al.
(2011a); Kanazawa et al. (2014); Benić et al. (2024), even at the lowest order
(LO) of QCD perturbation theory. The main reason for this is that the
asymmetry is power-suppressed in the hard scale of the reaction, the
transverse momentum of the produced hadron, and hence involves twist-3
hadronic matrix elements and correspondingly hard-scattering diagrams with
three partons connecting to these matrix elements. As the leading-power
unpolarized cross section shows, NLO QCD corrections are vital for a
successful phenomenology of $pp\to hX$ Jager et al. (2003). For the TSSA, the
computation of NLO corrections is extremely complicated and, although
essential for phenomenology, is currently not on the horizon. As a first step
in this direction, it is useful to consider reactions with similar kinematics,
but less complexity. One possibility is to go from purely hadronic scattering
to $ep$ reactions, essentially replacing one initial proton by the simpler
electron. References Kang et al. (2011b); Gamberg et al. (2014); Kanazawa et
al. (2016); Anselmino et al. (2000, 2010); D’Alesio et al. (2017); Fitzgibbons
et al. (2024) proposed the process $ep\to\pi X$ in this context, which – as it
turns out – by itself could offer promising opportunities to gain new
information on twist-3 correlation functions in the proton at the EIC. NLO
corrections to the spin-averaged cross section for $ep\to\pi X$ were presented
in Hinderer et al. (2015).

The process $ep\to\gamma X$ which we consider in the present paper, is an
equally valuable (and in some ways simpler) testing ground for QCD
calculations. One may consider it a less complex version of the analogous
single-inclusive photon production in proton collisions, $pp\to\gamma X$.
Historically, the latter reaction with transverse polarization,
$p^{\uparrow}p\to\gamma X$, played an important role in finding explanations
for transverse spin asymmetries Qiu and Sterman (1991, 1992). We stress again
that the process we investigate differs from the semi-inclusive production of
photons in electron-proton collisions, $ep\to e\gamma X$ , for which the
final-state electron is detected. The TSSA for the latter has been proposed in
Ref. Albaltan et al. (2020) as an observable well-suited to obtain insights
into higher-twist quark-gluon-quark correlation functions.

In order to set the stage for future work on the TSSA for single-inclusive
photon production $ep\to\gamma X$, we present in this paper a full NLO
calculation of the unpolarized and the longitudinally polarized (helicity
dependent) cross sections for this reaction, which are both of leading twist.
Focusing on inelastic contributions, we carefully discuss the various
production channels. We present phenomenological studies for the EIC,
identifying kinematical regions that are favorable with respect to obtaining
information on the photonic distributions and fragmentation functions. We hope
that establishing the basic NLO leading-twist theoretical framework for
$ep\to\gamma X$ will prove useful for future NLO studies of the TSSA for the
process.

Our paper is organized as follows. In section II we first discuss the various
LO and NLO contributions to the differential unpolarized cross section for
$ep\to\gamma X$ and introduce the contributing PDFs and fragmentation
functions, especially the photonic ones. We then present the analytic
calculation of the full unpolarized cross section. Section III addresses the
corresponding spin-dependent cross section $\vec{e}\vec{p}\to\gamma X$ . Based
on these results, in section IV we present our numerical predictions for the
cross sections and longitudinal double spin asymmetries at the EIC. We
conclude in section V.

##  II The unpolarized cross section

###  II.1 LO and NLO contributions

The process we consider is the single-inclusive production of prompt photons
in electron-proton collisions, $e(l)+p(P)\to\gamma(P_{\gamma})+X$ , where we
have denoted the relevant four-momenta. In contrast to the reactions usually
considered in electron-proton scattering, $X$ denotes an unobserved multi-
particle final state that consists not only of the hadronic remnants of the
initial proton , but also of the final-state electron.

At large c.m. energies $\sqrt{s}$ and reasonably large transverse momenta of
the detected photons we may assume that collinear factorization may be used to
write the cross section in terms of short-distance partonic hard-scattering
cross sections and long-distance hadronic matrix elements.

We will work to the lowest order in the QED coupling $\alpha_{\mathrm{em}}$,
but will derive the cross section for the process to NLO in QCD. The counting
of perturbative orders in the strong coupling $\alpha_{s}$ for the process is
somewhat special, but familiar from other hadronic processes with identified
photons. At LO, the process can proceed in two ways: (i) a photon plays the
role of a parton inside the proton and participates in a QED Compton reaction;
(ii) a QED electron-quark scattering process is followed by fragmentation of
the quark to a photon. The two types of contributions are depicted in Fig. 1.
Both the photon PDF of the proton and the quark-to-photon fragmentation
functions are of perturbative order ${\cal
O}(\alpha_{\mathrm{em}}/\alpha_{s})$, owing to an explicit logarithm arising
in the calculation of higher-order diagrams. Thus, when combined with the
overall factor $\alpha_{\mathrm{em}}^{2}$ arising from the Compton or the $eq$
scattering process, respectively, the lowest order for the process
$e(l)+p(P)\to\gamma(P_{\gamma})+X$ becomes
$\alpha^{3}_{\mathrm{em}}/\alpha_{s}$.

We note that the photon PDF also includes an elastic part for which the proton
remains intact and that gives rise to an exclusive final state Kniehl (1991);
De Rujula and Vogelsang (1999); Mukherjee and Pisano (2003); Manohar et al.
(2016, 2017). Such elastic contributions are thus included in our analysis.

Figure 1:  LO contributions to $e(l)+p(P)\to\gamma(P_{\gamma})+X$ . We show
sketches of factorized amplitudes where photons play the role of a parton in
the proton (upper panel), and where photons are produced in a quark
fragmentation process (lower panel). Figure 2:  NLO contributions to
$e(l)+p(P)\to\gamma(P_{\gamma})+X$ . The hard scattering amplitude consists of
four diagrams where the detected photon is radiated off the electron line or
off the quark line.

NLO contributions arise at order
$\alpha_{s}\times\alpha^{3}_{\mathrm{em}}/\alpha_{s}=\alpha^{3}_{\mathrm{em}}$.
They again come in two classes. The first class consists of all $2\to 3$
diagrams for $eq\to\gamma eq$ , as shown in Fig. 2.

In the context of semi-inclusive photon production, $ep\to e\gamma X$,
contributions associated with these diagrams are often referred to as “virtual
Compton scattering” contributions in the literature Mukherjee and Pisano
(2003); Mukherjee and Pisano (2004a, b). For the single-inclusive process we
consider, the outgoing electron and quark are not observed, so that their
momenta need to be integrated over their full phase space. The corresponding
calculations will be presented below. Treating quarks as massless, the
diagrams will exhibit collinear singularities when the incoming quark splits
into a quark+photon pair, or when the outgoing quark radiates the observed
photon collinearly. These singularities are absorbed into the photon PDF of
the proton and the quark-to-photon fragmentation function, respectively,
giving rise to the “inhomogeneous” part of the evolution equations for these
functions in the LO contribution.

An interesting issue concerns the treatment of the electron . Clearly, at the
energies we consider, the electron mass $m_{e}$ is expected to be negligible.
However, for vanishing $m_{e}$ one encounters additional singularities in the
$2\to 3$ diagrams for $eq\to\gamma eq$ , arising when the initial or final
electron is accompanied by collinear emission. One option is to keep a finite
electron mass throughout the full calculation, neglecting it wherever
possible. The collinear singularities mentioned above would then be replaced
by finite (but, potentially large) logarithms of the form $\log(m_{e}/Q)$ ,
with $Q$ a hard scale in the problem. Using this approach, the phase-space
integrations with non-zero electron mass become slightly more involved.
Alternatively – and this is the approach we will show here – one may introduce
“Weizsäcker-Williams (WW)” type distributions that correspond to a “photon-in-
electron” PDF or an “electron-to-photon” fragmentation function, respectively,
as shown in Fig. 3. The calculation may then be performed with $m_{e}=0$,
regularizing collinear divergences associated with the electron by dimensional
regularization and absorbing them into the electron distributions. As
explained in Ref. Hinderer et al. (2015), both approaches, i.e., with or
without a finite electron mass, will yield equivalent results.

Indeed, in addition to the WW approach, we have carried out the non-zero
electron mass calculation and found complete agreement between the two
methods. We also refer the reader to Refs. Liu et al. (2021b, a) for a
discussion of radiative corrections in lepton scattering using similar
concepts.

Figure 3:  Contributions with a photon-in-electron PDF (upper panel) and an
electron-to-photon fragmentation function (lower panel).

The second class of NLO contributions involves the hard-scattering functions
for the processes $eq\to eqg$ and $eg\to eq\bar{q}$, followed by fragmentation
of a final-state parton to the photon. Using again that the photon
fragmentation functions are of order $\alpha_{\mathrm{em}}/\alpha_{s}$, the
net contribution by these processes is of order
$\alpha_{\mathrm{em}}^{2}\alpha_{s}\times\alpha_{\mathrm{em}}/\alpha_{s}=\alpha_{\mathrm{em}}^{3}$,
making them indeed NLO. The corresponding calculations for $eq\to eqg$ and
$eg\to eq\bar{q}$ where already carried out in Refs. Hinderer et al. (2015,
2017), where inclusive-hadron production in $ep$ scattering was considered.
The collinear singularities encountered in the phase space integration for
these reactions give rise to the homogeneous part of the evolution of the
photon fragmentation functions. Technically, this contribution can be obtained
simply by replacing the parton-to-hadron fragmentation functions in the
calculation of Refs. Hinderer et al. (2015, 2017) with the corresponding
parton-to-photon ones. In the following we will therefore not further discuss
this contribution which will, however, be included in our numerical results
below.

We close this discussion with a note of caution. As we have described, we
count the photon-in-proton PDF and the quark-to-photon fragmentation functions
as LO, that is, ${\cal O}(\alpha_{\mathrm{em}}/\alpha_{s})$. In contrast to
this we regard the corresponding photon-in-electron PDF and electron-to-photon
FF as ${\cal O}(\alpha_{\mathrm{em}})$, despite the fact that they enter with
an additional large logarithm $\log(m_{e}/Q)$ . Although this way of counting
is arguably sensible from the point of view of QCD, it is clear that one
should expect the contributions coming with the leptonic PDF and FF to be
numerically as important as the ones entering with the photon-in-proton PDF
and the quark-to-photon fragmentation functions. Furthermore, the evolution of
the latter resums large logarithms in the hard scale, while the logarithm
$\log(m_{e}/Q)$ in the leptonic functions is only kept at fixed order. It
would be possible to resum these logarithms as well to all orders in
$\alpha_{\mathrm{em}}$, but this is beyond the scope of the present work.

###  II.2  Parton Distribution Functions (PDFs) and Fragmentation Functions
(FFs)

Before proceeding to the actual calculation of the cross section to order
$\mathcal{O}(\alpha_{\mathrm{em}}^{3})$, we collect all of the relevant
hadronic/leptonic matrix elements and their parameterizations in terms of PDFs
and FFs that we need for a factorized description of the cross section.

The most common matrix elements that we encounter are those for the
distribution functions for unpolarized or longitudinally polarized quarks in
protons, $f_{1}^{q/p}(x,\mu)$ and $g_{1}^{q/p}(x,\mu)$ , respectively. It is
well-known that the two PDFs parameterize the following bilocal matrix
element,

$\displaystyle\int_{-\infty}^{\infty}\tfrac{\mathrm{d}\lambda}{2\pi}\,\mathrm{e}^{i\lambda
x}\langle p(P,S)|\,\bar{q}_{j}(0)\,\mathcal{W}[0;\lambda n]\,q_{i}(\lambda
n)\,|p(P,S)\rangle$ (1) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\hbox
to0.0pt{\hbox to7.80904pt{\hfil/\hfil}\hss}P\,f_{1}^{q/p}(x,\mu)-S_{L}\,\hbox
to0.0pt{\hbox
to7.80904pt{\hfil/\hfil}\hss}P\gamma_{5}\,g_{1}^{q/p}(x,\mu)\right)_{ij}+...\,.$

Note that this matrix element (1) incorporates a light-cone vector $n^{\mu}$
that satisfies $n^{2}=0$ and $P\cdot n=1$, but is otherwise arbitrary. In
particular the Wilson line $\mathcal{W}[0;\lambda n]$ that renders the bilocal
quark operator color-gauge invariant is along a straight line in the light-
cone direction $n^{\mu}$. We explicitly keep chiral-even leading twist
structures in the second line of (1), but neglect chiral-odd and/or subleading
twist structures (indicated by the dots). We note that the apparent dependence
on the explicit choice of the light-cone vector $n^{\mu}$ cancels out for the
leading twist observables that we are interested in in this paper111The
situation is more complicated for subleading-twist observables Kanazawa et al.
(2016).. We also see that the first term in (1) is independent of the proton
spin vector $S^{\mu}$ (with $S^{2}=-1$ and $P\cdot S=0$) while the second is
proportional to the longitudinal projection of $S^{\mu}$ on $n^{\mu}$, i.e.,
$S_{L}=M\,(n\cdot S)$. In the approximation of a massless proton , $S_{L}$ may
be considered as the proton ’s helicity. Lastly, we note that the matrix
element (1) is to be considered a subtracted matrix element in which ultra-
violet divergences are $\overline{\mathrm{MS}}$-subtracted. Then, the PDFs are
to be read as $f_{1}^{q/p,\overline{\mathrm{MS}}}(x,\mu)$ and
$g_{1}^{q/p,\overline{\mathrm{MS}}}(x,\mu)$ , functions of the renormalization
scale $\mu$.

As discussed in the previous subsection, the process $e+p\to\gamma+X$ is also
sensitive to the photon-in-proton distribution function $f_{1}^{\gamma/p}$
(and its polarized companion $g_{1}^{\gamma/p}$ ), which enters the factorized
description of the cross section already at LO. It is in fact the appearance
of these lesser-known distribution functions that makes the process especially
interesting. The definition of the photon-in-proton distribution functions
shares some similarities with that for the proton ’s gluon distribution
function $f_{1}^{g/p}(x,\mu)$ . To be specific, the unpolarized
($f_{1}^{\gamma/p}$ ) and polarized ($g_{1}^{\gamma/p}$ ) photon PDFs
parameterize the following matrix element,

$\displaystyle\int_{-\infty}^{\infty}\tfrac{\mathrm{d}\lambda}{2\pi}\,\mathrm{e}^{i\lambda
x}\langle p(P,S)|F^{n\nu}(0)\,F^{n\mu}(\lambda\,n)|p(P,S)\rangle$ (2)
$\displaystyle=$
$\displaystyle\frac{x}{2}\left[-\frac{g_{\perp}^{\mu\nu}}{1-\varepsilon}\,f_{1}^{\gamma/p}(x,\mu)+S_{L}\,i\epsilon^{Pn\nu\mu}\,g_{1}^{\gamma/p}(x,\mu)\right]\,.$

This matrix element (2) incorporates the photonic gauge-invariant field-
strength tensor $F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$, with
short-hand notations of the form $F^{n\nu}=n_{\mu}F^{\mu\nu}$ etc. The photon
field itself is represented by $A^{\mu}(x)$. The transverse projector
$g_{\perp}^{\mu\nu}$ is given by
$g_{\perp}^{\mu\nu}=g^{\mu\nu}-P^{\mu}n^{\nu}-P^{\nu}n^{\mu}$, and the
antisymmetric tensor $\epsilon^{\mu\nu\rho\sigma}$ follows the convention
$\epsilon^{0123}=+1$. We have anticipated the later use of dimensional
regularization with $d=4-2\varepsilon$ space-time dimensions.

Furthermore, at order $\mathcal{O}(\alpha_{\mathrm{em}}^{3}/\alpha_{s})$, the
photon can also be produced within a fragmentation process of a quark in the
final state. This process is described by a quark-to-photon fragmentation
matrix element,

$\displaystyle\frac{1}{N_{c}}\sum_{X}\int\int_{-\infty}^{\infty}\tfrac{\mathrm{d}\lambda}{2\pi}\,\mathrm{e}^{i\frac{\lambda}{z}}\langle
0|\,\mathcal{W}[\infty n;\lambda n]q_{i}(\lambda
n)\,|\gamma(P_{\gamma});X\rangle\times$ (3) $\displaystyle\hskip
85.35826pt\langle\gamma(P_{\gamma});X|\,\bar{q}_{j}(0)\,\mathcal{W}[0;\infty
n]\,|0\rangle$ $\displaystyle=$ $\displaystyle z^{-1+2\varepsilon}(\hbox
to0.0pt{\hbox
to7.80904pt{\hfil/\hfil}\hss}P_{\gamma})_{ij}\,D_{1}^{\gamma/q}(z,\mu)\,.$

The quark-to-photon fragmentation functions $D_{1}^{\gamma/q}$, like the
photonic PDF counterparts, are only poorly known, despite their relevance for
direct-photon production at hadron colliders. Hence, measurements of
$e+p\to\gamma+X$ cross sections could potentially further constrain these
fragmentation functions.

Like the quark PDFs (1) and any collinear matrix element, also the photon
PDFs/FFs (2) and (3) contain ultraviolet divergences that need to be
regularized and subtracted, thereby introducing scale dependence. This is a
standard procedure, and a convenient way is to consider perturbative
corrections to the collinear functions and to introduce “renormalized”
$\overline{\mathrm{MS}}$-PDFs/FFs. To the order we consider, we have

$\displaystyle f_{1}^{\gamma/p}(x,\mu)$ $\displaystyle=$ $\displaystyle
f_{1}^{\gamma/p,\overline{\mathrm{MS}}}(x,\mu)+\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{S_{\varepsilon}}{\varepsilon}\times$
(4) $\displaystyle\int_{x}^{1}\tfrac{\mathrm{d}w}{w}\,P_{\gamma
q}(w)\sum_{q}e_{q}^{2}\,f_{1}^{(q+\bar{q})/p,\overline{\mathrm{MS}}}(x/w,\mu),$
$\displaystyle g_{1}^{\gamma/p}(x,\mu)$ $\displaystyle=$ $\displaystyle
g_{1}^{\gamma/p,\overline{\mathrm{MS}}}(x,\mu)+\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{S_{\varepsilon}}{\varepsilon}\times$
(5) $\displaystyle\int_{x}^{1}\tfrac{\mathrm{d}w}{w}\Delta P_{\gamma
q}(w)\sum_{q}e_{q}^{2}\,g_{1}^{(q+\bar{q})/p,\overline{\mathrm{MS}}}(x/w,\mu),$
$\displaystyle D_{1}^{\gamma/q}(z,\mu)$ $\displaystyle=$ $\displaystyle
D_{1}^{\gamma/q,\overline{\mathrm{MS}}}(z,\mu)+\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{S_{\varepsilon}}{\varepsilon}e_{q}^{2}\,P_{\gamma
q}(z)\,.$ (6)

The kernels in these equations are the LO QED splitting functions $P_{\gamma
q}(w)=\frac{1+(1-w)^{2}}{w}$ and $\Delta P_{\gamma q}(w)=2-w$. The customary
factor $S_{\varepsilon}=(4\pi)^{\varepsilon}/\Gamma(1-\varepsilon)$
facilitates the correct $\overline{\mathrm{MS}}$ subtraction at this order. In
(4),(5) the sum is over quark flavors, $e_{q}$ are the fractional quark
charges, and $f^{(q+\bar{q})/p}$ indicates a sum of quark and anti-quark
distributions. Finally, the last term in (6) indicates the perturbative
radiation of a collinear photon that is part of the quark-to-photon
fragmentation.

As described earlier, it turns out to be convenient to similarly introduce
photon PDFs for an electron rather than a proton and, likewise, electron-to-
photon FFs. To be specific, we may define photon-in-electron distributions
$f_{1}^{\gamma/e}(x,\mu)$ , $g_{1}^{\gamma/e}(x,\mu)$ in the same way as the
photon-in-proton distributions by replacing the proton state $p(P,S)$ by an
electron state $e(l,s)$ . The main difference is that the
$f_{1}^{\gamma/e}(x,\mu)$ , $g_{1}^{\gamma/e}(x,\mu)$ (sometimes referred to
as ”Weiszäcker-Williams” (WW) distributions von Weizsacker (1934); Williams
(1934)) can be calculated perturbatively for a non-zero electron mass $m_{e}$
, and that they do not couple to the quark distributions. The corresponding
calculations to order $\mathcal{O}(\alpha_{\mathrm{em}})$ have been performed
in Refs. Hinderer et al. (2015, 2017), and we simply quote the results:

$\displaystyle f_{1}^{\gamma/e}(x,\mu)$ $\displaystyle=$ $\displaystyle
f_{1}^{\gamma/e,\overline{\mathrm{MS}}}(x,\mu)+\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{S_{\varepsilon}}{\varepsilon}\,P_{\gamma
q}(x)+\mathcal{O}(\alpha_{\mathrm{em}}^{2})\,,$ (7) $\displaystyle
g_{1}^{\gamma/e}(x,\mu)$ $\displaystyle=$ $\displaystyle
g_{1}^{\gamma/e,\overline{\mathrm{MS}}}(x,\mu)+\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{S_{\varepsilon}}{\varepsilon}\,\Delta
P_{\gamma q}(x)+\mathcal{O}(\alpha_{\mathrm{em}}^{2})\,,$ (8)

with

$\displaystyle f_{1}^{\gamma/e,\overline{\mathrm{MS}}}(x,\mu)$
$\displaystyle=$ $\displaystyle\frac{\alpha_{\mathrm{em}}}{2\pi}\,P_{\gamma
q}(x)\,\left[\ln\left(\frac{\mu^{2}}{x^{2}\,m_{e}^{2}}\right)-1\right]\,,$ (9)
$\displaystyle g_{1}^{\gamma/e,\overline{\mathrm{MS}}}(x,\mu)$
$\displaystyle=$ $\displaystyle\frac{\alpha_{\mathrm{em}}}{2\pi}\,\Delta
P_{\gamma q}(x)\,\ln\left(\frac{\mu^{2}}{x^{2}\,m_{e}^{2}}\right)\,.$ (10)

In similar fashion, we also introduce an electron-to-photon fragmentation
function $D_{1}^{\gamma/e}(z)$ by replacing the quark fields in (3) by
electron fields. As for the electron PDFs, we can calculate all involved
quantities in QED perturbation theory, and obtain

$\displaystyle
D_{1}^{\gamma/e}(z,\mu)=D_{1}^{\gamma/e,\overline{\mathrm{MS}}}(z,\mu)+\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{S_{\varepsilon}}{\varepsilon}\,\,P_{\gamma
q}(z)+\mathcal{O}(\alpha_{\mathrm{em}}^{2})\,,$ (11)

with

$\displaystyle D_{1}^{\gamma/e,\overline{\mathrm{MS}}}(z,\mu)$
$\displaystyle=$ $\displaystyle\frac{\alpha_{\mathrm{em}}}{2\pi}\,P_{\gamma
q}(z)\,\left[\ln\left(\frac{\mu^{2}}{z^{2}\,m_{e}^{2}}\right)-1\right]\,.$
(12)

We note that leptonic PDFs have also been introduced in more general terms in
the context of factorized expressions for QED radiative corrections to the DIS
and semi-inclusive DIS cross sections Liu et al. (2021a).

###  II.3 LO partonic cross sections

We introduce the Mandelstam variables $s=(P+l)^{2}>0$,
$t=(P-P_{\gamma})^{2}<0$ and $u=(l-P_{\gamma})^{2}<0$, which satisfy
$s+t+u\geq 0$. We further define

$\displaystyle x_{0}=\frac{-u}{s+t}\,,\quad\quad v=\frac{s+t}{s}<1\,.$ (13)

We start by analyzing the contribution to the cross section associated with
the photon PDF. As indicated in the upper panel of Fig. 1, we factorize this
contribution into $f_{1}^{\gamma/p}$ and the hard-scattering function for QED
Compton scattering $\gamma e\to\gamma e$, which may computed from the diagrams
shown in the figure. Although the result is finite at this order, we compute
it in dimensional regularization with $d=4-2\varepsilon$ space-time
dimensions. The reason is that we need the $d$-dimensional cross section later
in the factorization of the $2\to 3$ scattering contributions. A
straightforward calculation gives

$\displaystyle
E_{\gamma}\frac{\mathrm{d}^{d-1}\sigma^{\gamma\mathrm{PDF}}}{\mathrm{d}^{d-1}\bm{P}_{\gamma}}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{2}}{s(-u)}\,\hat{\sigma}^{\gamma\mathrm{PDF}}(v,\varepsilon)\,f_{1}^{\gamma/p}(x_{0},\mu)\,,$
(14)

with

$\displaystyle\hat{\sigma}^{\gamma\mathrm{PDF}}(v,\varepsilon)$
$\displaystyle=$ $\displaystyle
2\frac{1+v^{2}}{v}-2\varepsilon\frac{(1-v)^{2}}{v}\,.$ (15)

Here $\bm{P}_{\gamma}$ denotes the three-momentum of the produced photon. If
(14) were the only contribution to $eN\to\gamma X$, the process would be
ideally suited to constrain the photon-in-proton PDF $f_{1}^{\gamma/p}$.

However, as already discussed in subsection II.1, there are also LO
contributions associated with quark-to-photon fragmentation. The corresponding
factorization is sketched in the lower part of Fig. 1. We find

$\displaystyle
E_{\gamma}\frac{\mathrm{d}^{d-1}\sigma^{\gamma\mathrm{FF}}}{\mathrm{d}^{d-1}\bm{P}_{\gamma}}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{2}}{s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\hat{\sigma}^{\gamma\mathrm{FF}}(v,w,\varepsilon)\times$
(16)
$\displaystyle\sum_{q}e_{q}^{2}\,f_{1}^{(q+\bar{q})/p}\left(\tfrac{x_{0}}{w},\mu\right)\,D_{1}^{\gamma/q}(1-v+vw,\mu)\,,$

where

$\displaystyle\hat{\sigma}^{\gamma\mathrm{FF}}(v,w,\varepsilon)$
$\displaystyle=$
$\displaystyle\frac{2vw(1-v+vw)^{2\varepsilon}}{(1-v)^{2}(1-v+vw)}\times$ (17)
$\displaystyle\left[1+v^{2}-2v(1-w)(1+vw)-\varepsilon(1-v)^{2}\right].$

As indicated in (16), quarks and antiquarks have identical fragmentation
functions to photons.

###  II.4 Contributions at $\mathcal{O}(\alpha_{\mathrm{em}}^{3})$

In this subsection we discuss the contributions sketched in Figs. 2 and 3. We
first address the $2\to 3$ diagrams shown in Fig. 2, for which the observed
photon can be radiated either from the electron or from the quark line.
Radiation purely off the electron is known as the Bethe-Heitler (BH)
contribution Brodsky et al. (1972); Albaltan et al. (2020), while the diagrams
with radiation off quarks may be regarded as a quark-photon Compton (C)
contribution. Finally, there is also an interference (I) contribution of the
two types of radiation.

The three types of contributions, BH,C,I, can be distinguished by the
weightings with different quark charges. Furthermore, the Bethe-Heitler and
Compton contributions come with the sums of quark and anti-quark
distributions, while the interference contribution is generated by the valence
quark distributions. To be specific, we encounter the following generic
combinations of quark PDFs Albaltan et al. (2020):

$\displaystyle f^{{\textrm{BH}}}(x,\mu)$ $\displaystyle=$
$\displaystyle\sum_{q}e_{q}^{2}\big{(}f^{q/p}(x,\mu)+f^{\bar{q}/p}(x,\mu)\big{)}\,,$
(18) $\displaystyle f^{{\textrm{C}}}(x,\mu)$ $\displaystyle=$
$\displaystyle\sum_{q}e_{q}^{4}\big{(}f^{q/p}(x,\mu)+f^{\bar{q}/p}(x,\mu)\big{)}\,,$
(19) $\displaystyle f^{{\textrm{I}}}(x,\mu)$ $\displaystyle=$
$\displaystyle\sum_{q}e_{q}^{3}\big{(}f^{q/p}(x,\mu)-f^{\bar{q}/p}(x,\mu)\big{)}\,.$
(20)

In order to obtain the hard-scattering cross section for $qe\to qe\gamma$ we
need to integrate its squared amplitude over the phase space of the momenta of
the undetected electron and quark ($r_{1}$ and $r_{2}$ in Fig. 2), which is
done in $d=4-2\varepsilon$ dimensions. As mentioned earlier, we use a
vanishing electron mass $m_{e}=0$ to simplify the calculation. Although the
phase space integrations become quite standard then, we found the recent paper
Lyubovitskij et al. (2021) particularly helpful in this context. After
performing the phase-space integrations we encounter various types of
$1/\varepsilon$-poles associated with the Bethe-Heitler and Compton
contributions, arising when the radiated photon becomes either collinear to
the quark or the electron momentum. In contrast to the Bethe-Heitler and
Compton contributions, the interference contributions are finite.

The collinear divergences in the Bethe-Heitler and Compton contributions can
be systematically absorbed into the hadronic and leptonic PDFs and FFs. For
initial-state radiation off the quark, the collinear pole is canceled by that
in the perturbative photon PDF in (4), inserted into the LO cross section in
(14). Likewise, the final-state collinear singularity cancels against the
quark-to-photon fragmentation contribution (6), which appears via the LO term
(17). In both cases, we perform the subtraction in the
$\overline{\mathrm{MS}}$ scheme.

For collinear radiation off the electron line, the situation is slightly less
standard. As discussed in subsection II.1, our approach is to set $m_{e}=0$,
in which case collinear initial-state and final-state divergences will occur.
We subsequently add perturbative “Weizsäcker-Williams” contributions
associated with a photon-in-electron PDF and an electron-to-photon FF, which
cancel the divergences and “reinstate” the explicit logarithms $\log(m_{e}/Q)$
that would arise in a calculation that keeps a finite electron mass.
Expressions for the relevant PDF and FF were given in Eqs. (7) and (11),
respectively. They need to be convoluted with the appropriate Born processes
which are $\gamma q\to\gamma q$ in the PDF case and $eq\to eq$ in the FF one.
We obtain:

$\displaystyle
E_{\gamma}\frac{\mathrm{d}^{3}\sigma^{\gamma\mathrm{PDF}}_{\mathrm{WW}}}{\mathrm{d}^{d-1}\bm{P}_{\gamma}}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{2}}{s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\hat{\sigma}^{\gamma\mathrm{PDF}}_{\mathrm{WW}}(v,w,\varepsilon)\times\,$
(21) $\displaystyle
f_{1}^{\gamma/e}\left(\tfrac{1-v}{1-vw},\mu\right)\,\sum_{q}e_{q}^{4}\,f_{1}^{(q+\bar{q})/p}\left(\tfrac{x_{0}}{w},\mu\right)\,,$

where

$\displaystyle\hat{\sigma}^{\gamma\mathrm{PDF}}_{\mathrm{WW}}(v,w,\varepsilon)$
$\displaystyle=$
$\displaystyle\frac{2vw(2-vw(2-vw(1-\varepsilon)))}{(1-v)(1-vw)}\,,$ (22)

and

$\displaystyle
E_{\gamma}\frac{\mathrm{d}^{3}\sigma^{\gamma\mathrm{FF}}_{\mathrm{WW}}}{\mathrm{d}^{d-1}\bm{P}_{\gamma}}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{2}}{s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\hat{\sigma}^{\gamma\mathrm{FF}}_{\mathrm{WW}}(v,w,\varepsilon)\times$
(23) $\displaystyle
D_{1}^{\gamma/e}(1-v+vw,\mu)\sum_{q}e_{q}^{2}\,f_{1}^{(q+\bar{q})/p}\left(\tfrac{x_{0}}{w},\mu\right)\,,$

with

$\displaystyle\hat{\sigma}^{\gamma\mathrm{FF}}_{\mathrm{WW}}(v,w,\varepsilon)$
$\displaystyle=$
$\displaystyle\frac{2(1-v+vw)^{2\varepsilon}}{vw(1-v+vw)}\times$ (24)
$\displaystyle\left[2(1-v)(1-v+vw)+(1-\varepsilon)\,v^{2}w^{2}\right].$

As mentioned before, resummation of the logarithm $\log(m_{e}/Q)$ may
eventually be necessary.

###  II.5 Final result for the unpolarized cross section

We separate the invariant cross section for $e(l)+p(P)\to\gamma(P_{\gamma})+X$
into five pieces: the contributions associated with (1) the photon-in-proton
PDF and (2) the quark-to-photon FF (which both start at
$\mathcal{O}(\alpha_{\mathrm{em}}^{2}/\alpha_{s})$), and the ones by the (3)
Bethe-Heitler, (4) Compton, (5) interference channels, which all arise only at
$\mathcal{O}(\alpha_{\mathrm{em}}^{3})$ and are hence NLO:

$\displaystyle E_{\gamma}\frac{\mathrm{d}^{3}\sigma^{ep\to\gamma
X}}{\mathrm{d}^{3}\bm{P}_{\gamma}}$ $\displaystyle\equiv$
$\displaystyle\sigma^{\gamma\mathrm{PDF}}+\sigma^{\gamma\mathrm{FF}}+\sigma^{\mathrm{BH}}+\sigma^{\mathrm{C}}+\sigma^{\mathrm{I}}\,.$
(25)

We emphasize again that the first contribution in (25) is associated with the
photon-in-proton PDF $f_{1}^{\gamma/p}$ while the second term is generated by
the quark-to-photon fragmentation functions $D_{1}^{\gamma/q}$. We find

$\displaystyle\sigma^{\gamma\mathrm{PDF}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha_{\mathrm{em}}^{2}}{s(-u)}\,\hat{\sigma}^{\gamma\mathrm{PDF}}(v)\,f_{1}^{\gamma/p,\overline{\mathrm{MS}}}(x_{0},\mu)\,,$
(26) $\displaystyle\sigma^{\gamma\mathrm{FF}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha_{\mathrm{em}}^{2}}{s(-u)}\,\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\hat{\sigma}^{\gamma\mathrm{FF}}(v,w)\times$
(27)
$\displaystyle\sum_{q}e_{q}^{2}\,f_{1}^{(q+\bar{q})/p,\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,D_{1}^{\gamma/q,\overline{\mathrm{MS}}}(1-v+vw,\mu)\,,$
$\displaystyle\sigma^{\mathrm{BH}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{3}}{\pi\,s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\hat{\sigma}^{\mathrm{BH}}\left(v,w,\tfrac{-u}{m_{e}^{2}},\tfrac{-u}{\mu^{2}}\right)\times$
(28) $\displaystyle\hskip
71.13188ptf_{1}^{\mathrm{BH},\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,,$
$\displaystyle\sigma^{\mathrm{C}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{3}}{\pi\,s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\hat{\sigma}^{\mathrm{C}}\left(v,w,\tfrac{-u}{m_{e}^{2}},\tfrac{-u}{\mu^{2}}\right)\times$
(29) $\displaystyle\hskip
71.13188ptf_{1}^{\mathrm{C},\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,,$
$\displaystyle\sigma^{\mathrm{I}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{3}}{\pi\,s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\hat{\sigma}^{\mathrm{I}}(v,w)\,f_{1}^{\mathrm{I},\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,.$
(30)

The various partonic hard-scattering functions $\hat{\sigma}$ appearing in
these equations are collected in Appendix A.

###  II.6  Photon isolation

In collider experiments, in order to cope with the large photon background
arising from neutral-pion decay, one usually introduces an isolation of the
photon, for which one imposes a limit on the hadronic activity in the vicinity
of the photon. Typically one defines a cone in azimuthal-pseudorapidity space
(as defined in the electron-proton c.m. frame) around the photon candidate,
i.e.

$\mathrm{cone}_{\gamma}(R)=\left\\{(\eta,\phi)|\sqrt{(\eta-\eta_{\gamma})^{2}+(\phi-\phi_{\gamma})^{2}}\leq
R\right\\}\,.$ (31)

Inside this cone one then requires that the energy of any hadrons accompanying
the photon is smaller than a fixed fraction of the photon energy, i.e.

$E_{\textrm{had}}\leq\xi E_{\gamma}\,.$ (32)

where $\xi\sim 0.1$ is a typical value. Anticipating that isolation would also
be imposed in photon measurements at the EIC, we will extend our calculation
to the isolated case. Actually, a benefit of isolation is that it also reduces
the contribution of photons that were generated in fragmentation of quarks or
gluons. The reason is that for the fragmentation component photons will indeed
always be accompanied by hadronic energy, so that the isolation condition (32)
puts restrictions on the size of the contributions, although it obviously does
not eliminate it completely. To be more specific, one can see that criterion
(32) translates to a condition on the light-cone fraction $z=1-v+vw$ that
appears in the fragmentation contribution (27) Berger and Qiu (1990):

$\displaystyle E_{\textrm{had}}\leq\xi E_{\gamma}$
$\displaystyle\Leftrightarrow$ $\displaystyle z\geq\frac{1}{1+\xi}$ (33)
$\displaystyle\Leftrightarrow$ $\displaystyle
w\geq\mathrm{max}\left(x_{0},1-\frac{\xi}{v(1+\xi)}\right)\equiv w_{0}\,.$

Thus the integration domain over the variable $w$ in (27) is reduced, and the
smaller the fraction $\xi$ the stronger is the suppression of the parton-to-
photon fragmentation.

Beyond LO, partons can also be radiated into the photon isolation cone, rather
than just enter it via fragmentation. This effect must be taken into account
in the ${\cal O}(\alpha_{\mathrm{em}}^{3})$ contribution. In order to do this,
we adopt the small cone approximation (SCA), see Ref. Gordon and Vogelsang
(1994). The basic idea is to use the results obtained for the fully inclusive
(i.e., non-isolated) prompt photon cross section, Eqs. (28)–(30), and subtract
the NLO contribution where partons have entered the isolation cone with energy
in excess of what is allowed by the condition (32). For small cone radii $R$
the subtraction contribution defined in this way may be computed analytically.
More specifically, for the case of interest here it can be shown to have the
form

$\sigma_{\mathrm{sub}}=\ln(R)\,A+B+\mathcal{O}(R^{2})\,.$ (34)

The two leading terms associated with $A$ and $B$ can only arise from
configurations where the photon is radiated by an outgoing quark or antiquark.
For nearly collinear emission, the quark propagator before emission goes
nearly on-shell, so that phase space integration leads to the structure of
logarithm plus constant shown in (34). The coefficients $A$ and $B$ contain a
Born cross section which in this case is the one for the Compton process
$eq\to eq$. The explicit calculation of the full subtraction term in the SCA
is straightforward, and for details we refer the reader to Ref. Gordon and
Vogelsang (1994). In the end we obtain

$\displaystyle\sigma^{\mathrm{C}}_{\mathrm{sub}}$ $\displaystyle=$
$\displaystyle\theta(w_{0}-x_{0})\frac{\alpha_{\mathrm{em}}^{3}}{\pi\,s(-u)}\int_{x_{0}}^{w_{0}}\tfrac{\mathrm{d}w}{w}\,\times$
(35)
$\displaystyle\hat{\sigma}^{\mathrm{C}}_{\mathrm{sub}}\left(v,w,x_{0},R,\tfrac{-u}{\mu^{2}}\right)\,f_{1}^{\mathrm{C},\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,,$

where

$\displaystyle\hat{\sigma}^{\mathrm{C}}_{\mathrm{sub}}\left(v,w,x_{0},R,\tfrac{-u}{\mu^{2}}\right)$
$\displaystyle=$ $\displaystyle-\frac{vw}{(1-v)^{2}(1-v+vw)^{2}}\times$ (36)
$\displaystyle(1+v^{2}-2v(1-w)(1+vw))\,\times$
$\displaystyle\left[(1-v+vw)^{2}+(1+v^{2}(1-w)^{2})\times\right.$
$\displaystyle\left.\ln\left(-\frac{R^{2}\,u\,v(1-v)(1-w)(1-v+vw)^{2}x_{0}}{\mu^{2}\,w(1-v+vx_{0})^{2}}\right)\right]\,.$

The term in (35) is to be subtracted from the Compton contribution in (25).
Empirically, the SCA is known to work well for values of $R$ up to $R\sim 0.7$
Gordon and Vogelsang (1994).

As we discussed earlier, there is a second class of NLO contributions to
$e(l)+p(P)\to\gamma(P_{\gamma})+X$ which is generated by the hard-scattering
processes $eq\to eqg$ and $eg\to eq\bar{q}$, followed by fragmentation of a
final-state parton to the photon. In principle, we would need to introduce
appropriate subtraction terms also for this contribution. However, in our
numerical studies to be presented below we found that this contribution is
strongly reduced already by the condition (33), so that it becomes irrelevant.
We therefore refrain from deriving the subtraction piece for this part.

We finally note that the Bethe-Heitler and interference contributions only
contribute to the terms of order $R^{2}$ in (34). The reason for this is that
we only impose isolation in terms of hadronic energy. In principle, one could
introduce also an “electromagnetic isolation”, restricting the amount of
energy of additional electromagnetic particles around the photon. Such an
isolation would affect especially the size of the Bethe-Heitler contribution,
where it would then also generate logarithms of the cone size. Whether or not
such a constraint would be desirable (or even feasible experimentally) remains
to be studied.

##  III The polarized cross section

In this section we extend our calculation to photon production in collisions
of longitudinally polarized electrons and protons. To be specific, we consider
the following spin dependent cross section:

$\displaystyle\Delta\sigma^{ep\to\gamma X}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4}\left[E_{\gamma}\frac{\mathrm{d}^{3}\sigma^{e^{+}p^{+}\to\gamma
X}}{\mathrm{d}^{3}\bm{P}_{\gamma}}-E_{\gamma}\frac{\mathrm{d}^{3}\sigma^{e^{-}p^{+}\to\gamma
X}}{\mathrm{d}^{3}\bm{P}_{\gamma}}\right.$ (37)
$\displaystyle\left.-E_{\gamma}\frac{\mathrm{d}^{3}\sigma^{e^{+}p^{-}\to\gamma
X}}{\mathrm{d}^{3}\bm{P}_{\gamma}}+E_{\gamma}\frac{\mathrm{d}^{3}\sigma^{e^{-}p^{-}\to\gamma
X}}{\mathrm{d}^{3}\bm{P}_{\gamma}}\right]\,,$

where the superscripts refer to the electron and proton helicities. Since
$\\--$ like the unpolarized cross section $\\--$ also $\Delta\sigma$ is a
leading-power observable, the discussions of subsections II.2 \- II.4 will
hold for $\Delta\sigma$ as well, and the calculations can be performed in the
same fashion. A slight complication compared to the unpolarized cross section
is the treatment of the Dirac matrix $\gamma_{5}$ in dimensional
regularization which arises from projecting on longitudinal polarization of
the electron and the proton. In this work we choose to work within the ’t
Hooft-Veltman-Breitenloher-Maison scheme for $\gamma_{5}$ ’t Hooft and Veltman
(1972); Breitenlohner and Maison (1977).

As before, we split the spin dependent cross section into five pieces (cf. Eq.
(25)),

$\displaystyle\Delta\sigma^{ep\to\gamma X}$ $\displaystyle=$
$\displaystyle\Delta\sigma^{\gamma\mathrm{PDF}}+\Delta\sigma^{\gamma\mathrm{FF}}+\Delta\sigma^{\mathrm{BH}}+\Delta\sigma^{\mathrm{C}}+\Delta\sigma^{\mathrm{I}}.$
(38)

The individual contributions are as follows:

$\displaystyle\Delta\sigma^{\gamma\mathrm{PDF}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha_{\mathrm{em}}^{2}}{s(-u)}\,\Delta\hat{\sigma}^{\gamma\mathrm{PDF}}(v)\,g_{1}^{\gamma/p,\overline{\mathrm{MS}}}(x_{0},\mu)\,,$
(39) $\displaystyle\Delta\sigma^{\gamma\mathrm{FF}}$ $\displaystyle=$
$\displaystyle\frac{2\alpha_{\mathrm{em}}^{2}}{s(-u)}\,\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\Delta\hat{\sigma}^{\gamma\mathrm{FF}}(v,w)\times$
(40)
$\displaystyle\sum_{q}e_{q}^{2}\,g_{1}^{(q+\bar{q})/p,\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,D_{1}^{\gamma/q,\overline{\mathrm{MS}}}(1-v+vw,\mu)\,,$
$\displaystyle\Delta\sigma^{\mathrm{BH}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{3}}{\pi\,s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\Delta\hat{\sigma}^{\mathrm{BH}}\left(v,w,\tfrac{-u}{m_{e}^{2}},\tfrac{-u}{\mu^{2}}\right)\times$
(41) $\displaystyle\hskip
71.13188ptg_{1}^{\mathrm{BH},\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,,$
$\displaystyle\Delta\sigma^{\mathrm{C}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{3}}{\pi\,s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\Delta\hat{\sigma}^{\mathrm{C}}\left(v,w,\tfrac{-u}{m_{e}^{2}},\tfrac{-u}{\mu^{2}}\right)\times$
(42) $\displaystyle\hskip
71.13188ptg_{1}^{\mathrm{C},\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,,$
$\displaystyle\Delta\sigma^{\mathrm{I}}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{\mathrm{em}}^{3}}{\pi\,s(-u)}\int_{x_{0}}^{1}\tfrac{\mathrm{d}w}{w}\,\Delta\hat{\sigma}^{\mathrm{I}}(v,w)\,g_{1}^{\mathrm{I},\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,.$
(43)

We observe that the helicity distributions $g_{1}^{q/p}$, $g_{1}^{\gamma/p}$,
$g_{1}^{\gamma/e}$ (see Eqs. (1), (2)) appear in these expressions. The
explicit analytic forms of the polarized hard-scattering functions
$\Delta\hat{\sigma}$ in Eqs. (39)–(43) are presented in Appendix B.

In order to construct the spin-dependent cross section for isolated-photon
production we repeat the steps described in the previous section and again
compute the relevant subtraction term for the polarized Compton contribution
(42):

$\displaystyle\Delta\sigma^{\mathrm{C}}_{\mathrm{sub}}$ $\displaystyle=$
$\displaystyle\theta(w_{0}-x_{0})\frac{\alpha_{\mathrm{em}}^{3}}{\pi\,s(-u)}\int_{x_{0}}^{w_{0}}\tfrac{\mathrm{d}w}{w}\,\times$
(44)
$\displaystyle\Delta\hat{\sigma}^{\mathrm{C}}_{\mathrm{sub}}\left(v,w,x_{0},R,\tfrac{-u}{\mu^{2}}\right)\,g_{1}^{\mathrm{C},\overline{\mathrm{MS}}}\left(\tfrac{x_{0}}{w},\mu\right)\,,$

with

$\displaystyle\Delta\hat{\sigma}^{\mathrm{C}}_{\mathrm{sub}}\left(v,w,x_{0},R,\tfrac{-u}{\mu^{2}}\right)$
$\displaystyle=$ $\displaystyle-\frac{vw(1-v+2vw)}{(1-v)(1-v+vw)^{2}}\times$
(45) $\displaystyle\left[(1-v+vw)^{2}+(1+v^{2}(1-w)^{2})\times\right.$
$\displaystyle\left.\ln\left(-\frac{R^{2}\,u\,v(1-v)(1-w)(1-v+vw)^{2}x_{0}}{\mu^{2}\,w(1-v+vx_{0})^{2}}\right)\right]\,.$

##  IV Numerical Predictions

In this section we estimate the size of the various contributions to the
unpolarized and polarized cross sections at the EIC. Our main goal is to
investigate in how far the future EIC experiments could help to probe the
proton’s photon PDF.

Throughout our calculations we use the NLO unpolarized parton distributions of
Ref. Martin et al. (2009), referred to as MSTW2008. For the helicity parton
distributions we use the latest NLO set of Ref. de Florian et al. (2014)
(DSSV). For the parton-to-photon fragmentation functions we adopt the GRV NLO
set Gluck et al. (1993). The photon-in-proton PDF is implemented using a
recent NLO fit performed by the LuxQED collaboration Bertone et al. (2018) and
included in the NNPDF3.1 parameterizations.

Figure 4:  The unpolarized cross section at the EIC for c.m.-energy
$\sqrt{s}=100\,\mathrm{GeV}$, plotted as function of the photon’s
pseudorapidity $\eta_{\gamma}$ for fixed transverse momenta. Left:
$p_{T}^{\gamma}=5\,\mathrm{GeV}$; center: $p_{T}^{\gamma}=15\,\mathrm{GeV}$;
right: $p_{T}^{\gamma}=30\,\mathrm{GeV}$. We show the five contributions to
the cross section as defined in Eq. (25), as well as their sum. We show
results both for the non-isolated and for the isolated case. Figure 5:  Same
as Fig. 4, but for the dependence on $x_{T}^{\gamma}$ for fixed
pseudorapidities $\eta_{\gamma}=-1.5$ (left), $\eta_{\gamma}=0$ (center),
$\eta_{\gamma}=1.5$ (right).

To the best of our knowledge no parameterization is available for the
polarized photon-in-proton PDF. In order to obtain an estimate for the
longitudinal double-spin asymmetry we adopt a simple model that relates the
photon PDF to the gluon one:

$g_{1}^{\gamma/p}(x,\mu)=\alpha_{\mathrm{em}}\,g_{1}^{g/p}(x,\mu)\,,$ (46)

where $g_{1}^{g/p}$ denotes the proton’s gluon helicity PDF. Of course, this
can at best be a crude model. Future data may very well shed further light on
$g_{1}^{\gamma/p}$ and falsify this ansatz.

We anticipate a center-of-mass energy of $\sqrt{s}=100\,\mathrm{GeV}$ for
electron-proton collisions at the EIC. We translate the invariant cross
section to

$\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}\eta_{\gamma}\,\mathrm{d}x_{T}^{\gamma}}=\frac{\pi}{2}\,x_{T}^{\gamma}\,s\,E_{\gamma}\frac{\mathrm{d}^{3}\sigma}{\mathrm{d}^{3}\bm{P}_{\gamma}}\,,$
(47)

where $\eta_{\gamma}$ is the photon’s pseudorapidity in the c.m.-frame, where
we count positive rapidity in the forward proton direction. Furthermore,
$x_{T}^{\gamma}=2p_{T}^{\gamma}/\sqrt{s}$ with the photon’s transverse
momentum $p_{T}^{\gamma}$. Our plots will show results always for
$\mathrm{d}^{2}\sigma/\mathrm{d}\eta_{\gamma}\,\mathrm{d}x_{T}^{\gamma}$.

###  IV.1 Unpolarized cross section

The dependence of the differential unpolarized cross section (47) on the
photon’s pseudorapidity $\eta_{\gamma}$ is shown in Fig. 4 along with the
various individual contributions of the channels defined in Eq. (25). We
observe that the contributions $\sigma^{\gamma\mathrm{PDF}}$ and
$\sigma^{\mathrm{BH}}$ tend to dominate in the backward region (electron
direction). Hence, this region offers the best opportunity to experimentally
learn about the photon-in-proton PDF $f_{1}^{\gamma/p}(x)$ and to further
constrain this function. On the other hand, in the forward region (proton
direction) $\sigma^{\gamma\mathrm{FF}}$ dominates, along with the Compton
channel, possibly allowing access to the quark-to-photon fragmentation
functions $D_{1}^{\gamma/q}(z)$. This appears to be the case especially at
lower $p_{T}^{\gamma}$. We further observe that the interference contribution
$\sigma^{\mathrm{I}}$ is overall small.

We also study in Fig. 4 the effect of photon isolation on the cross section at
the EIC. Here we assume a cone radius of $R=0.7$ and an isolation parameter
$\xi=0.1$. It is interesting to see that photon isolation indeed strongly
suppresses the photon fragmentation channel, which improves the overall
sensitivity to the photon PDF $f_{1}^{\gamma/p}$.

The dependence of the unpolarized cross section on the transverse photon
momentum is shown in Fig. 5 in the backward, central, and forward regions.
Again we observe a clear hierarchy of the various contributions, in particular
in the backward and forward regions. The photon PDF contribution
$\sigma^{\gamma\mathrm{PDF}}$ is dominant in the backward region over a large
range of transverse photon momenta, in particular for the isolated case.
Hence, the backward region may be the preferred region for further
constraining $f_{1}^{\gamma/p}$, and the transverse photon momentum dependence
appears well-suited for that purpose.

By contrast, the Compton channel and photon fragmentation contributions
dominate the transverse photon momentum dependence in the forward region. In
the mid-rapidity region the hierarchy of channels is not that strict, and the
various channels are approximately of similar size for larger transverse
photon momenta. However, for smaller transverse momenta of about 2.5 to 10
GeV, the Compton channel and the photon fragmentation contribution dominate as
well.

###  IV.2 Polarized Cross Section

Next we discuss our numerical estimates for the polarized cross section (38)
at the EIC and the sizes of the various contributions. We focus in particular
on the sensitivity to the polarized photon-in-proton PDF $g_{1}^{\gamma/p}$
via the channel $\Delta\sigma^{\gamma\mathrm{PDF}}$. We remind the reader that
our results are based on our ad-hoc model ansatz (46). For the polarized case
we will only consider the isolated cross section, in order to suppress the
photon fragmentation contribution. Clearly, for the purpose of studying quark-
to-photon fragmentation the unpolarized cross section is the preferred
observable.

Figure 6:  Same as Fig. 4, but for the polarized case. We only consider the
isolated cross section here. Figure 7:  Same as Fig. 5, but for the polarized
case. We only consider the isolated cross section here. Figure 8:  The
longitudinal double-spin asymmetry $A_{{\textrm{LL}}}$ at the EIC for
c.m.-energy of $\sqrt{s}=100\,\mathrm{GeV}$, plotted as function of the
photon’s pseudorapidity $\eta_{\gamma}$ for fixed transverse momenta (left)
and as function of $x_{T}^{\gamma}$ for fixed values of pseudorapidity
(right).

Figure 6 shows the dependence of the polarized cross section on the photon’s
pseudorapidity $\eta_{\gamma}$, again for three different transverse momenta.
The largest contributions of the polarized photon-in-proton PDF to the
polarized cross section can be found in the backward region for small to
medium transverse photon momenta. The large transverse momentum region is not
suitable to constrain $g_{1}^{\gamma/p}$. In Fig. 7 we also show the
$p_{T}^{\gamma}$ distributions for the various individual channels. Again, we
see that the backward region is the preferred region for obtaining information
on $g_{1}^{\gamma/p}$.

In experiment, one typically does not directly access the polarized cross
section, but rather measures the longitudinal double-spin asymmetry
$A_{{\textrm{LL}}}$, defined by

$A_{{\textrm{LL}}}=\frac{\Delta\sigma}{\sigma}\,.$ (48)

Figure 8 shows the dependence of the double-spin asymmetry $A_{{\textrm{LL}}}$
on the photon’s pseudorapidity $\eta_{\gamma}$ as well as on the photon’s
transverse momentum. Overall, large asymmetries of about 20% to 40% could be
possible. However, in the main region of interest, i.e., the backward region
at moderate transverse momenta, which shows the best sensitivity to the
polarized photon-in-proton PDF $g_{1}^{\gamma/p}$, the size of the asymmetry
$A_{{\textrm{LL}}}$ is only about a few percent. However, this might still be
sufficient to obtain valuable information on $g_{1}^{\gamma/p}$.

##  V Conclusions

We have analyzed the single-inclusive production of prompt photons in
electron-proton scattering at the EIC, computing the spin-averaged as well as
the longitudinally polarized cross section of this reaction to NLO in QCD. At
NLO, there are several production channels that contribute to both the cross
sections. We have presented analytical results for the partonic cross sections
for each of the channels. We have also discussed the implementation of a
photon isolation cut and its effect on the cross sections.

The purpose of our computation was to investigate the accessibility of the
proton’s photonic parton distributions at the EIC, the unpolarized photon-in-
proton PDF $f_{1}^{\gamma/p}$ and its polarized counterpart
$g_{1}^{\gamma/p}$. In particular, $g_{1}^{\gamma/p}$ is only poorly known,
but could potentially provide (albeit likely small) contributions to the
proton spin.

Our numerical studies for the Electron-Ion Collider with a center-of-mass
energy of $\sqrt{s}=100\,\mathrm{GeV}$ suggest that the future data should
very well help to constrain the photonic functions mentioned above, and
possibly even the quark-to-photon fragmentation functions, provided a careful
selection of kinematical regions.

Future extensions of our work will address the transverse single-spin
asymmetry for the process $ep\to\gamma X$, which could be of great interest
for understanding power-suppressed phenomena in QCD and for accessing the
proton’s quark and gluon structure at twist-three.

###### Acknowledgements.

We are grateful to Gunar Schnell for helpful discussions. This work has been
supported by Deutsche Forschungsgemeinschaft (DFG) through the Research Unit
FOR 2926 (project 409651613).

##  Appendix A Unpolarized partonic cross sections

Here we list the unpolarized partonic hard-scattering functions that appear in
Eqs. (26)–(30). For the LO terms we have

$\displaystyle\hat{\sigma}^{\gamma\mathrm{PDF}}(v)$ $\displaystyle=$
$\displaystyle\frac{1+v^{2}}{v}\,,$ (49)
$\displaystyle\hat{\sigma}^{\gamma\mathrm{FF}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{vw\left[1+v^{2}-2v(1-w)(1+vw)\right]}{(1-v)^{2}(1-v+vw)}\,.$
(50)

We note (see subsection II.1) that the photon fragmentation channel receives
${\cal O}(\alpha_{s})$ corrections that are known from Refs. Hinderer et al.
(2015, 2017) and will not be recalled here. They are, however, included in our
numerical results.

For the Bethe-Heitler contribution we find:

$\displaystyle\hat{\sigma}^{\mathrm{BH}}\left(v,w,\tfrac{-u}{m_{e}^{2}},\tfrac{-u}{\mu^{2}}\right)$
$\displaystyle=$ $\displaystyle
C_{1}^{\textrm{BH}}(v,w)\,\ln\left(\tfrac{-u}{\mu^{2}}\right)$ (51)
$\displaystyle+C_{2}^{\textrm{BH}}(v,w)\,\ln\left(\tfrac{-u}{m_{e}^{2}}\right)+C_{3}^{\textrm{BH}}(v,w)\,\ln(1-v+vw)$
$\displaystyle+C_{4}^{\textrm{BH}}(v,w)\,\ln(1-w)+C_{5}^{\textrm{BH}}(v,w)\,\ln(w)$
$\displaystyle+C_{6}^{\textrm{BH}}(v,w)\,,$

with the coefficient functions

$\displaystyle C_{1}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{1+v^{2}}{v}\frac{1+(1-w)^{2}}{w}\,,$ $\displaystyle
C_{2}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{1+v^{2}(1-w)^{2}}{vw(1-v+vw)^{2}}$
$\displaystyle\times(2+2v^{2}-v(2-w)(2+vw))\,,$ $\displaystyle
C_{3}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle-\frac{2}{vw}(2+v^{2}(1+(1-w)^{2}))\,,$ $\displaystyle
C_{4}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{4(1+v^{2})-w(2-w)(1+2v^{2})}{vw}$
$\displaystyle+\frac{2v^{2}(1-w)-2}{(1-v+vw)^{2}}\,,$ $\displaystyle
C_{5}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{2(1-v^{2}(1-w))}{(1-v+vw)^{2}}+\frac{2-w}{v}\,,$
$\displaystyle C_{6}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{2(1-v)(1+2v(1-w))}{(1-v+vw)^{2}}-4\frac{1-w}{w}$ (52)
$\displaystyle+\frac{1-v}{v}(1-w)+vw\,,$

In the Compton channel, we have

$\displaystyle\hat{\sigma}^{\mathrm{C}}\left(v,w,\tfrac{-u}{m_{e}^{2}},\tfrac{-u}{\mu^{2}}\right)$
$\displaystyle=$ $\displaystyle
C_{1}^{\textrm{C}}(v,w)\,\ln\left(\tfrac{-u}{\mu^{2}}\right)$ (53)
$\displaystyle+C_{2}^{\textrm{C}}(v,w)\,\ln\left(\tfrac{-u}{m_{e}^{2}}\right)+C_{3}^{\textrm{C}}(v,w)\,\ln\left(\tfrac{1-v}{1-vw}\right)$
$\displaystyle\,+C_{4}^{\textrm{C}}(v,w)\,\ln(1-v+vw)$
$\displaystyle+C_{5}^{\textrm{C}}(v,w)\,\ln(\tfrac{1-w}{w})+C_{6}^{C}(v,w),$

with

$\displaystyle C_{1}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{vw\left(1+v^{2}(1-w)^{2}\right)}{(1-v)^{2}(1-v+vw)^{2}}\times$
$\displaystyle(1+v^{2}-2v(1-w)(1+vw))\,,$ $\displaystyle
C_{2}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{vw(2-vw(2-vw))}{(1-v)^{2}(1-vw)^{2}}\times$
$\displaystyle(1+v^{2}-2vw(1+v-vw))\,,$ $\displaystyle
C_{3}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{2vw(1+v-2vw)}{(1-v)(1-vw)^{2}}\,,$ $\displaystyle
C_{4}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle-\frac{4vw(1-v(1-w)(1-vw))}{(1-v)(1-v+vw)^{2}}\,,$
$\displaystyle C_{5}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{4vw(2-vw-v^{2}w+v^{2}w^{2})}{(1-v)^{2}}$
$\displaystyle-\frac{2(2-v+v^{2}w)}{1-v}+\frac{8-2v}{1-v+vw}$
$\displaystyle-\frac{3-v}{(1-v)(1-vw)}-\frac{2(1-v)}{(1-v+vw)^{2}}$
$\displaystyle+\frac{1}{(1-vw)^{2}}\,,$ $\displaystyle
C_{6}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{-vw(2+3v-v^{2}-4vw)}{(1-v)^{2}}-\frac{v}{1-v}$ (54)
$\displaystyle-\frac{2(4-v)}{1-v+vw}+\frac{3(2-v)}{(1-v)(1-vw)}$
$\displaystyle+\frac{4(1-v)}{(1-v+vw)^{2}}-\frac{2}{(1-vw)^{2}}\,,$

Finally, our results for the interference contribution are

$\displaystyle\hat{\sigma}^{\textrm{I}}(v,w)$ $\displaystyle=$ $\displaystyle
C_{1}^{\textrm{I}}(v,w)\,\ln(1-v)+C_{2}^{\textrm{I}}(v,w)\,\ln(v)$ (55)
$\displaystyle+C_{3}^{\textrm{I}}(v,w)\,\ln(1-w)+C_{4}^{\textrm{I}}(v,w)\,\ln(w)$
$\displaystyle+C_{5}^{\textrm{I}}(v,w)\,\ln((1-vw)(1-v+vw)^{2})$
$\displaystyle+C_{6}^{\textrm{I}}(v,w)\,,$

where

$\displaystyle C_{1}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{8}{1-v+vw}-4\,,$ $\displaystyle C_{2}^{\textrm{I}}(v,w)$
$\displaystyle=$
$\displaystyle-\frac{4v^{2}(w(3w-4)+2)-8vw+8}{(1-v)(1-v+vw)},$ $\displaystyle
C_{3}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle-\frac{2v^{2}(w(5w-8)+5)-4vw+6}{(1-v)(1-v+vw)},$ $\displaystyle
C_{4}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{4vw(1+v-vw)}{(1-v)(1-v+vw)}\,,$ $\displaystyle
C_{5}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle-\frac{2(1+v-vw)}{1-v}\,,$ $\displaystyle
C_{6}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle-\frac{2w(1+v-vw)}{(1-v+vw)^{2}(1-vw)}\times$ (56)
$\displaystyle\left(1+v^{2}-v\left(4-2w-vw^{2}\right)\right)\,.$

##  Appendix B Longitudinally polarized partonic cross sections

Here we list the polarized partonic hard cross sections that appear in Eqs.
(39)–(43). To LO,

$\displaystyle\Delta\hat{\sigma}^{\gamma\mathrm{PDF}}(v)$ $\displaystyle=$
$\displaystyle\frac{1-v^{2}}{v}\,,$ (57)
$\displaystyle\Delta\hat{\sigma}^{\gamma\mathrm{FF}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{vw(1-v+2vw)}{(1-v)(1-v+vw)}\,,$ (58)

again with NLO corrections in the fragmentation channel that are known
Hinderer et al. (2015, 2017).

For the Bethe-Heitler process,

$\displaystyle\Delta\hat{\sigma}^{\mathrm{BH}}\left(v,w,\tfrac{-u}{m_{e}^{2}},\tfrac{-u}{\mu^{2}}\right)$
$\displaystyle=$ $\displaystyle
D_{1}^{\textrm{BH}}(v,w)\,\ln\left(\tfrac{-u}{\mu^{2}}\right)$ (59)
$\displaystyle+D_{2}^{\textrm{BH}}(v,w)\,\ln\left(\tfrac{-u}{m_{e}^{2}}\right)+D_{3}^{\textrm{BH}}(v,w)\,\ln(1-v+vw)$
$\displaystyle+D_{4}^{\textrm{BH}}(v,w)\,\ln(1-w)+D_{5}^{\textrm{BH}}(v,w)\,\ln(w)$
$\displaystyle+D_{6}^{\textrm{BH}}(v,w)\,,$

with the coefficient functions

$\displaystyle D_{1}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{1-v^{2}}{v}(2-w)\,,$ $\displaystyle
D_{2}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{(2-2v+vw)\left(1+v^{2}(1-w)^{2}\right)}{(1-v+vw)^{2}}\,,$
$\displaystyle D_{3}^{\textrm{BH}}(v,w)$ $\displaystyle=$ $\displaystyle
2v(2-w)\,,$ $\displaystyle D_{4}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{2(1-v)}{(1-v+vw)^{2}}-2v(2-w)$
$\displaystyle+\frac{2v}{1-v+vw}+\frac{2-w}{v}\,,$ $\displaystyle
D_{5}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle-\frac{2-2v^{2}(1-w)}{(1-v+vw)^{2}}-\frac{2-w}{v}\,,$
$\displaystyle D_{6}^{\textrm{BH}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{4(1-v)}{1-v+vw}+1-w-\frac{3-4w}{v}$ (60)
$\displaystyle-\frac{6(1-v)}{(1-v+vw)^{2}}+2v(1-w)\,,$

The results for the Compton channel are

$\displaystyle\Delta\hat{\sigma}^{\mathrm{C}}\left(v,w,\tfrac{-u}{m_{e}^{2}},\tfrac{-u}{\mu^{2}}\right)$
$\displaystyle=$ $\displaystyle
D_{1}^{\textrm{C}}(v,w)\,\ln\left(\tfrac{-u}{\mu^{2}}\right)$ (61)
$\displaystyle+D_{2}^{\textrm{C}}(v,w)\,\ln\left(\tfrac{-u}{m_{e}^{2}}\right)+D_{3}^{\textrm{C}}(v,w)\,\ln(\tfrac{1-v}{1-vw})$
$\displaystyle+D_{4}^{\textrm{C}}(v,w)\,\ln(1-v+vw)$
$\displaystyle+D_{5}^{\textrm{C}}(v,w)\,\ln(\tfrac{1-w}{w})+D_{6}^{C}(v,w),$

with

$\displaystyle D_{1}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{vw\left(1+v^{2}(1-w)^{2}\right)(1-v+2vw)}{(1-v)(1-v+vw)^{2}}\,,$
$\displaystyle D_{2}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{v^{2}w^{2}(2-vw)(1+v-2vw)}{(1-v)(1-vw)^{2}}\,,$
$\displaystyle D_{3}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle-\frac{2vw(1+v-2vw)}{(1-v)(1-vw)^{2}}\,,$ $\displaystyle
D_{4}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{4vw(1-v(1-w)(1-vw))}{(1-v)(1-v+vw)^{2}}\,,$ $\displaystyle
D_{5}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{vw}{(1-v)(1-v+vw)^{2}(1-vw)^{2}}\times$
$\displaystyle\left[1-v+2vw+v^{2}(1-2w-3w^{2})\right.$
$\displaystyle\left.-2v^{4}w(1-w)(2w-1)\times\right.$
$\displaystyle\left.(2-w-vw(1-w))\right.$
$\displaystyle\left.+v^{3}\left((7-4w)w^{2}-1\right)\right]\,,$ $\displaystyle
D_{6}^{\textrm{C}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{8-2v}{1-v+vw}+\frac{v(1+2w+vw)}{1-v}$ (62)
$\displaystyle+\frac{5v-8}{(1-v)(1-vw)}+\frac{4(v-1)}{(1-v+vw)^{2}}$
$\displaystyle+\frac{4}{(1-vw)^{2}}\,,$

The interference contribution reads

$\displaystyle\Delta\hat{\sigma}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle
D_{1}^{\textrm{I}}(v,w)\,\ln(1-v)+D_{2}^{\textrm{I}}(v,w)\,\ln(v)$ (63)
$\displaystyle+D_{3}^{\textrm{I}}(v,w)\,\ln(1-w)+D_{4}^{\textrm{I}}(v,w)\,\ln(w)$
$\displaystyle+D_{5}^{\textrm{I}}(v,w)\,\ln((1-vw)(1-v+vw)^{2})$
$\displaystyle+D_{6}^{\textrm{I}}(v,w)\,,$

with the coefficients

$\displaystyle D_{1}^{\textrm{I}}(v,w)$ $\displaystyle=$ $\displaystyle
4-\frac{8}{1-v+vw}\,,$ $\displaystyle D_{2}^{\textrm{I}}(v,w)$
$\displaystyle=$ $\displaystyle-\frac{4vw(2-vw)}{(1-v)(1-v+vw)}\,,$
$\displaystyle D_{3}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{-2-4vw+2v^{2}\left(1+w^{2}\right)}{(1-v)(1-v+vw)}\,,$
$\displaystyle D_{4}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle-\frac{4vw(1+v-vw)}{(1-v)(1-v+vw)}\,,$ $\displaystyle
D_{5}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{2+2v(1-w)}{1-v}\,,$ $\displaystyle
D_{6}^{\textrm{I}}(v,w)$ $\displaystyle=$
$\displaystyle\frac{2w(1+v-vw)}{(1-v+vw)^{2}(1-vw)}$ (64)
$\displaystyle\left[1-4v+v^{2}+vw\left(2+vw\right)\right]\,.$

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