#  One-Loop Corrections to Dihadron Production in DIS at Small $x$

Filip Bergabo<EMAIL_ADDRESS>Department of Natural Sciences,
Baruch College, CUNY, 17 Lexington Avenue, New York, NY 10010, USA  City
University of New York Graduate Center, 365 Fifth Avenue, New York, NY 10016,
USA  Jamal Jalilian-Marian<EMAIL_ADDRESS>Department
of Natural Sciences, Baruch College, CUNY, 17 Lexington Avenue, New York, NY
10010, USA  City University of New York Graduate Center, 365 Fifth Avenue, New
York, NY 10016, USA

###### Abstract

We calculate the one-loop corrections to dihadron production in Deep Inelastic
Scattering (DIS) at small $x$ using the Color Glass Condensate formalism. We
show that all UV and soft singularities cancel while the collinear divergences
are absorbed into quark and anti quark-hadron fragmentation functions.
Rapidity divergences lead to JIMWLK evolution of dipoles and quadrupoles
describing multiple-scatterings of the quark anti-quark dipole on the target
proton/nucleus. The resulting cross section is finite and can be used for
phenomenological studies of dihadron angular correlations at small $x$ in a
future Electron-Ion Collider (EIC).

##  I Introduction

Since the experimental observation of the fast rise of parton (especially
gluon) distribution functions at HERA Aaron et al. (2009), the phenomenon of
gluon saturation Gribov et al. (1983); Mueller and Qiu (1986) at small $x$ has
been an active field of study. The Color Glass Condensate formalism Iancu and
Venugopalan (2003); Jalilian-Marian and Kovchegov (2006); Weigert (2005);
Morreale and Salazar (2021) is an effective theory of QCD at small $x$ which
provides a robust platform that can be used to investigate saturation
dynamics. Nevertheless and despite the intense ongoing theoretical and
experimental efforts, clear and unambiguous evidence for gluon saturation
remains elusive. There is hope that the proposed Electron-Ion Collider (EIC)
Accardi et al. (2016) will be able to unambiguously discover gluon saturation
and to establish the kinematic region in which it is applicable. Perhaps the
most promising process in which to discover gluon saturation is two-particle
angular correlations which has been extensively studied using Leading Order
(LO) expressions obtained in the Color Glass Condensate formalism Kovner and
Wiedemann (2001); Jalilian-Marian and Kovchegov (2004); Jalilian-Marian
(2006); Marquet (2007); Albacete and Marquet (2010); Stasto et al. (2012);
Lappi and Mantysaari (2013); Jalilian-Marian and Rezaeian (2012a, b); Zheng et
al. (2014); Stasto et al. (2018); Albacete et al. (2019); Mäntysaari et al.
(2020); Hatta et al. (2021a); Jia et al. (2020); Dominguez et al. (2011); Metz
and Zhou (2011); Dominguez et al. (2012); Iancu and Laidet (2013); Iancu et
al. (2022); Altinoluk et al. (2016); Hatta et al. (2016); Dumitru et al.
(2015); Kotko et al. (2015); Marquet et al. (2016); van Hameren et al. (2016);
Marquet et al. (2018); Dumitru et al. (2019); Mäntysaari et al. (2019);
Salazar and Schenke (2019); Boussarie et al. (2021); Kotko et al. (2017);
Hagiwara et al. (2017); Klein and Mäntysaari (2019); Hatta et al. (2021b);
Iancu and Mulian (2021); Caucal et al. (2021); Kolbé et al. (2021); Gelis and
Jalilian-Marian (2002); Altinoluk et al. (2019); Boussarie et al. (2019, 2016,
2014); Dumitru and Jalilian-Marian (2010). There are already strong hints for
the presence of gluon saturation dynamics in the observed disappearance of
away side hadrons in inclusive dihadron production in deuteron (proton)-gold
collisions at RHIC Aschenauer et al. (2016); Braidot (2011); Adare et al.
(2011). Nevertheless the EIC will measure dihadrons with much more accuracy
and with much better control over the range of target $x$ contributing to the
process. Therefore it is essential to go beyond a LO calculation in order to
achieve a reasonable quantitative accuracy. Toward this end we calculate the
one-loop corrections to inclusive dihadron production in DIS here. As expected
there are several divergences that appear when going beyond a tree level
calculation. We show that all such divergences either cancel or can be
absorbed into evolution (scale dependence) of physical quantities. Our
conclusion on factorization/cancellation of all singularities are in agreement
with recent studies of one-loop corrections to dijet production in DIS Caucal
et al. (2021); Taels et al. (2022), nevertheless our treatment of collinear
divergences and their absorption into evolution of hadron fragmentation
functions is new. Our final results are completely finite and can be used to
investigate gluon saturation dynamics in inclusive dihadron production and
angular correlations in DIS at small $x$.

In the small $x$ limit scattering of a virtual photon on a target hadron or
nucleus can be understood as a two-step process; first the virtual photon
splits into a quark anti-quark pair (a dipole), which then scatters from the
target hadron or nucleus. The total virtual photon-target scattering
probability is expressed as a convolution of the probability for a photon to
split into a quark at transverse position $\mathbf{x}_{1}$ and an anti-quark
at position $\mathbf{x}_{2}$, with the probability for this dipole to scatter
from the target. The leading order double inclusive production cross section
can be written as

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to
q\bar{q}X}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}$ $\displaystyle=$
$\displaystyle\frac{e^{2}Q^{2}(z_{1}z_{2})^{2}N_{c}}{(2\pi)^{7}}\delta(1-z_{1}-z_{2})\int\differential^{8}\mathbf{x}\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]$
(1) $\displaystyle
e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\bigg{[}4z_{1}z_{2}K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})+$
$\displaystyle(z_{1}^{2}+z_{2}^{2})\,\frac{\mathbf{x}_{12}\cdot\mathbf{x}_{1^{\prime}2^{\prime}}}{|\mathbf{x}_{12}||\mathbf{x}_{1^{\prime}2^{\prime}}|}\,K_{1}(|\mathbf{x}_{12}|Q_{1})K_{1}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\bigg{]}.$

where the first and second terms above correspond to the contribution of the
longitudinal and transverse polarizations of the virtual photon. Here
$l^{\mu}$ is the momentum of the virtual photon with $l^{2}=-Q^{2}$. We set
the transverse momentum of the photon to zero without any loss of generality.
Also $p^{\mu}$ ($q^{\mu}$) is the momentum of the outgoing quark (anti-quark)
and $z_{1}$ ($z_{2}$) is its longitudinal momentum fraction relative to the
photon. $\mathbf{x}_{1}$ ($\mathbf{x}_{2}$) is the transverse coordinate of
the quark (anti-quark), and primed coordinates are used in the conjugate
amplitude. Note that we write the differential variables on the left side
using quark and anti-quark rapidities $y_{1}$ and $y_{2}$ which are related to
their momentum fractions $z_{1}$ and $z_{2}$ via $\differential
y_{i}=\differential z_{i}/z_{i}$. The electric charge of the quark is written
as $e$, and in principle one needs to sum over quark flavors. We also use the
following shorthand notations:

$\displaystyle
Q_{i}=Q\sqrt{z_{i}(1-z_{i})},\,\,\,\,\,\,\mathbf{x}_{ij}=\mathbf{x}_{i}-\mathbf{x}_{j},\,\,\,\,\,\,\differential^{8}\mathbf{x}=\differential^{2}\mathbf{x}_{1}\,\differential^{2}\mathbf{x}_{2}\,\differential^{2}\mathbf{x}_{1^{\prime}}\,\differential^{2}\mathbf{x}_{2^{\prime}}.$
(2)

All the dynamics of the strong interactions and gluon saturation are contained
in the dipoles $S_{ij}$ and quadrupoles $S_{ijkl}$, normalized correlation
functions of two and four Wilson lines

$\displaystyle S_{ij}=\frac{1}{N_{c}}{\rm tr}\left\langle
V_{i}V_{j}^{\dagger}\right\rangle,\,\,\,\,\,\,\,\,\,S_{ijkl}=\frac{1}{N_{c}}{\rm
tr}\left\langle V_{i}V_{j}^{\dagger}V_{k}V_{l}^{\dagger}\right\rangle,$ (3)

where the index $i$ refers to the transverse coordinate $\mathbf{x}_{i}$ and
the following notation is used for Wilson lines,

$\displaystyle V_{i}$ $\displaystyle=\hat{P}\exp\left(ig\int\differential
x^{+}A^{-}(x^{+},\mathbf{x}_{i})\right).$ (4)

The Wilson lines efficiently resum the multiple scatterings of the quark and
anti-quark from the target hadron or nucleus. The angle brackets in Eq. 3
signify _color averaging_. It is important to keep in mind that as this is a
classical result the cross section has no non-trivial $x$ (or rapidity/energy)
dependence. It is also easy to check that if one integrates over the phase
space of the quark and anti-quark one recovers the standard expressions for
the virtual photon-target total cross section at small $x$.

##  II One-loop corrections

One-loop corrections to the LO cross section above involve radiation of a
gluon from either the quark or the anti-quark. The radiated gluon can either
be real (radiated in the amplitude and absorbed in the complex conjugate
amplitude) or virtual (radiated and absorbed in either the amplitude or the
complex conjugate amplitude). The real contributions were already computed in
Ayala et al. (2016, 2017), here we will repeat the calculation and reproduce
the previous results. The virtual corrections have also been computed, but for
a different process, namely inclusive dijets Caucal et al. (2021); Taels et
al. (2022) rather than dihadrons. Furthermore we use spinor helicity methods
to evaluate the Dirac Algebra which leads to a tremendous simplification of
the calculations. The real corrections are shown in Fig. (1) and are given by
111We define $i\mathcal{M}_{i}$ via
$i\mathcal{A}_{i}=2\pi\delta(l^{+}-p^{+}-q^{+})i\mathcal{M}_{i}$. for quark
initiated radiation,

Figure 1: The real corrections
$i\mathcal{A}_{1}^{a},...,i\mathcal{A}_{4}^{a}$. The arrows on Fermion lines
indicate Fermion number flow, all momenta flow to the right. The thick solid
line indicates interaction with the target.

$\displaystyle i\mathcal{M}_{1}^{a}$ $\displaystyle=$ $\displaystyle-
ieg\int\frac{\differential^{3}k_{1}}{(2\pi)^{3}}\differential^{4}\mathbf{x}\frac{16(l^{+})^{2}N_{1}\,t^{a}V_{1}V_{2}^{\dagger}}{k_{1}^{2}(k_{1}-l)^{2}}e^{i\mathbf{k}_{1}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}e^{-i(\mathbf{p}+\mathbf{k})\cdot\mathbf{x}_{1}}e^{-i\mathbf{q}\cdot\mathbf{x}_{2}},\,\,\,\,\,k_{1}^{+}=p^{+}+k^{+},$
$\displaystyle i\mathcal{M}_{3}^{a}$ $\displaystyle=$ $\displaystyle-
eg\int\frac{\differential^{3}k_{1}}{(2\pi)^{3}}\frac{\differential^{3}k_{2}}{(2\pi)^{3}}\differential^{6}\mathbf{x}\frac{16(l^{+})^{2}N_{3}\,(2k^{+})V_{1}t^{b}V_{2}^{\dagger}U_{3}^{ab}e^{i\mathbf{k}_{1}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}e^{i\mathbf{k}_{2}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}e^{-i(\mathbf{p}\cdot\mathbf{x}_{1}+\mathbf{q}\cdot\mathbf{x}_{2}+\mathbf{k}\cdot\mathbf{x}_{3})}}{k_{1}^{2}k_{2}^{2}(k_{1}-l)^{2}(k_{1}-k_{2})^{2}},$
$\displaystyle k_{1}^{+}$ $\displaystyle=$ $\displaystyle
p^{+}+k^{+},\,\,\,\,\,\,k_{2}^{+}=k^{+}.$

Here we use $\frac{\differential^{3}k}{(2\pi)^{3}}=\frac{\differential
k^{-}}{(2\pi)}\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}$ and the Dirac
numerators $N_{i}$ are defined as

$\displaystyle N_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{16(l^{+})^{2}}\frac{\bar{u}(p)\not{\epsilon}(k)^{*}(\not{p}+\not{k})\not{n}\not{k}_{1}\not{\epsilon}(l)(\not{k}_{1}-\not{l})\not{n}v(q)}{(p+k)^{2}}.$
$\displaystyle N_{3}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{16(l^{+})^{2}}\bar{u}(p)\not{n}(\not{k}_{1}-\not{k}_{2})\gamma^{\mu}\not{k}_{1}\not{\epsilon}(l)(\not{k}_{1}-\not{l})\not{n}v(q)d_{\mu\nu}(k_{2})\epsilon^{\nu}(k)^{*}.$
(6)

We define the gauge vector $n^{\mu}$ in $(+,-,\perp)$ notation as
$(0,1,\mathbf{0})$. The anti-quark diagrams $i\mathcal{M}_{2}$ and
$i\mathcal{M}_{4}$ are related to the quark diagrams via swapping quark and
anti-quark momenta (i.e.: $\mathbf{p}\leftrightarrow\mathbf{q}$ and
$z_{1}\leftrightarrow z_{2}$). Note that when a gluon line multiply scatters
from the target one gets a Wilson line in the adjoint representation
$U_{j}^{ab}$.

$\displaystyle U_{j}^{ab}=$ $\displaystyle\hat{P}\exp\left(ig\int\differential
x^{+}A_{c}^{-}(x^{+},\mathbf{x}_{j})T^{c}\right)_{ab},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{where}\,\,\,\,\,\,\,\,T^{c}_{ab}=if^{abc}.$
(7)

Virtual diagrams are shown in figure 2, and the amplitudes can be written as
follows,

$\displaystyle i\mathcal{M}_{5}$
$\displaystyle=ieg^{2}\int\frac{\differential^{3}k_{1}}{(2\pi)^{3}}\frac{\differential^{3}k_{2}}{(2\pi)^{3}}\frac{\differential^{4}k_{3}}{(2\pi)^{4}}\differential^{6}\mathbf{x}\frac{N_{5}e^{i\mathbf{k}_{1}\cdot(\mathbf{x}_{3}-\mathbf{x}_{2})}e^{i\mathbf{k}_{2}\cdot(\mathbf{x}_{1}-\mathbf{x}_{3})}e^{i\mathbf{k}_{3}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}e^{-i\mathbf{p}\cdot\mathbf{x}_{3}}e^{-i\mathbf{q}\cdot\mathbf{x}_{2}}}{k_{1}^{2}k_{2}^{2}k_{3}^{2}(k_{1}-l)^{2}(k_{2}-k_{1})^{2}(k_{3}-p)^{2}}2(k_{3}^{+}-p^{+})$
$\displaystyle
t^{a}\left[\theta(k_{3}^{+})V_{1}-\theta(-k_{3}^{+})V^{\dagger}_{1}\right]t^{b}V^{\dagger}_{2}\left[\theta(k_{3}^{+}-p^{+})U^{ba}_{3}-\theta(p^{+}-k_{3}^{+})U_{3}^{{\dagger}ba}\right],\,\,\,\,\,k_{2}^{+}=k_{3}^{+},\,\,\,\,\,\,k_{1}^{+}=p^{+}.$
(8) $\displaystyle i\mathcal{M}_{7}$
$\displaystyle=ieg^{2}\int\frac{\differential^{3}k_{1}}{(2\pi)^{3}}\frac{\differential^{3}k_{2}}{(2\pi)^{3}}\frac{\differential^{4}k_{3}}{(2\pi)^{4}}\differential^{6}\mathbf{x}\frac{N_{7}e^{i\mathbf{k}_{3}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}e^{i\mathbf{k}_{2}\cdot(\mathbf{x}_{2}-\mathbf{x}_{3})}e^{i\mathbf{k}_{1}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}e^{-i\mathbf{p}\cdot\mathbf{x}_{3}}e^{-i\mathbf{q}\cdot\mathbf{x}_{2}}}{k_{1}^{2}k_{2}^{2}k_{3}^{2}(l-k_{1})^{2}(k_{1}-k_{2})^{2}(k_{3}-p)^{2}}2(k_{3}^{+}-p^{+})$
$\displaystyle
t^{a}\Big{[}\theta(k_{3}^{+})V_{1}-\theta(-k_{3}^{+})V_{1}^{\dagger}\Big{]}t^{b}V_{2}^{\dagger}\Big{[}\theta(k_{3}^{+}-p^{+})U_{3}^{ba}-\theta(p^{+}-k_{3}^{+})U_{3}^{{\dagger}ba}\Big{]},\,\,\,\,\,\,k_{2}^{+}=q^{+},\,\,\,\,k_{1}^{+}=l^{+}-k_{3}^{+}.$
(9) $\displaystyle i\mathcal{M}_{9}$
$\displaystyle=\frac{-eg^{2}}{p^{2}}\int\frac{\differential^{3}k_{1}}{(2\pi)^{3}}\frac{\differential^{4}k_{2}}{(2\pi)^{4}}\differential^{4}\mathbf{x}\frac{N_{9}\,t^{a}t^{a}V_{1}V_{2}^{\dagger}\,e^{i\mathbf{k}_{1}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}e^{-i(\mathbf{p}\cdot\mathbf{x}_{1}+\mathbf{q}\cdot\mathbf{x}_{2})}}{k_{1}^{2}k_{2}^{2}(k_{1}-l)^{2}(k_{2}-p)^{2}},\,\,\,\,\,\,\,k_{1}^{+}=p^{+}.$
(10) $\displaystyle i\mathcal{M}_{11}$
$\displaystyle=-eg^{2}\int\frac{\differential^{3}k_{1}}{(2\pi)^{3}}\frac{\differential^{4}k_{2}}{(2\pi)^{4}}\differential^{4}\mathbf{x}\frac{N_{11}\,V_{1}t^{a}t^{a}V_{2}^{\dagger}\,e^{i\mathbf{k}_{1}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}e^{-i(\mathbf{p}\cdot\mathbf{x}_{1}+\mathbf{q}\cdot\mathbf{x}_{2})}}{k_{1}^{4}k_{2}^{2}(k_{1}-l)^{2}(k_{1}-k_{2})^{2}},\,\,\,\,\,\,k_{1}^{+}=p^{+}.$
(11) $\displaystyle i\mathcal{M}_{13}$
$\displaystyle=eg^{2}\int\frac{\differential^{3}k_{1}}{(2\pi)^{3}}\frac{\differential^{4}k_{2}}{(2\pi)^{4}}\differential^{4}\mathbf{x}\frac{N_{13}e^{i\mathbf{k}_{1}\cdot\mathbf{x}_{12}}e^{i\mathbf{k}_{2}\cdot\mathbf{x}_{21}}e^{-i(\mathbf{p}\cdot\mathbf{x}_{1}+\mathbf{q}\cdot\mathbf{x}_{2})}}{k_{1}^{2}k_{2}^{2}(k_{1}-l)^{2}(k_{2}+p)^{2}(k_{2}-q)^{2}}$
$\displaystyle\times
t^{a}\Big{[}\theta(k_{2}^{+}+p^{+})V_{1}-\theta(-k_{2}^{+}-p^{+})V_{1}^{\dagger}\Big{]}\Big{[}\theta(k_{2}^{+}-q^{+})V_{2}-\theta(q^{+}-k_{2}^{+})V_{2}^{\dagger}\Big{]}t^{a},\,\,\,\,\,\,k_{1}^{+}=k_{2}^{+}+p^{+}.$
(12) $\displaystyle i\mathcal{M}_{14}$
$\displaystyle=-eg^{2}\int\frac{\differential^{4}k_{1}}{(2\pi)^{4}}\frac{\differential^{3}k_{2}}{(2\pi)^{3}}\differential^{4}\mathbf{x}\frac{N_{14}\,V_{1}t^{a}t^{a}V_{2}^{\dagger}e^{i\mathbf{k}_{2}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}e^{-i(\mathbf{p}\cdot\mathbf{x}_{1}+\mathbf{q}\cdot\mathbf{x}_{2})}}{k_{1}^{2}k_{2}^{2}(k_{1}-l)^{2}(k_{2}-l)^{2}(k_{1}-k_{2})^{2}},\,\,\,\,\,k_{2}^{+}=p^{+}.$
(13)

with the virtual Dirac numerators defined as

$\displaystyle N_{5}$
$\displaystyle\equiv\bar{u}(p)\gamma^{\mu}\not{k}_{3}\not{n}\not{k}_{2}\gamma^{\nu}\not{k}_{1}\not{\epsilon}(l)(\not{k}_{1}-\not{l})\not{n}v(q)d_{\nu\sigma}(k_{2}-k_{1})d_{\mu}^{\,\sigma}(k_{3}-p).$
(14) $\displaystyle N_{7}$
$\displaystyle\equiv\bar{u}(p)\gamma^{\mu}\not{k}_{3}\not{n}(\not{l}-\not{k}_{1})\not{\epsilon}(l)\not{k}_{1}\gamma^{\nu}\not{k}_{2}\not{n}v(q)d_{\nu\sigma}(k_{2}-k_{1})d_{\mu}^{\sigma}(k_{3}-p).$
(15) $\displaystyle N_{9}$
$\displaystyle\equiv\bar{u}(p)\gamma^{\mu}(\not{p}-\not{k}_{2})\gamma^{\nu}\not{p}\not{n}\not{k}_{1}\not{\epsilon}(l)(\not{k}_{1}-\not{l})\not{n}v(q)d_{\mu\nu}(k_{2}).$
(16) $\displaystyle N_{11}$
$\displaystyle\equiv\bar{u}(p)\not{n}\not{k}_{1}\gamma^{\mu}\not{k}_{2}\gamma^{\nu}\not{k}_{1}\not{\epsilon}(l)(\not{k}_{1}-\not{l})\not{n}v(q)d_{\mu\nu}(k_{1}-k_{2}).$
(17) $\displaystyle N_{13}$
$\displaystyle\equiv\bar{u}(p)\gamma^{\mu}(\not{k}_{2}+\not{p})\not{n}\not{k}_{1}\not{\epsilon}(l)(\not{k}_{1}-\not{l})\not{n}(\not{k}_{2}-\not{q})\gamma^{\nu}v(q)d_{\mu\nu}(k_{2}).$
(18) $\displaystyle N_{14}$
$\displaystyle\equiv\bar{u}(p)\not{n}\not{k}_{2}\gamma^{\mu}\not{k}_{1}\not{\epsilon}(l)(\not{k}_{1}-\not{l})\gamma^{\nu}(\not{k}_{2}-\not{l})\not{n}v(q)d_{\mu\nu}(k_{1}-k_{2}).$
(19)

Here again the anti-quark diagrams
($i\mathcal{M}_{6},i\mathcal{M}_{8},i\mathcal{M}_{10},i\mathcal{M}_{12}$) are
related to the corresponding quark diagrams via swapping quark and anti-quark
momenta. The tensor $d_{\mu\nu}$ is defined as follows.

$\displaystyle
d_{\mu\nu}(p)=-g_{\mu\nu}+\frac{p_{\mu}n_{\nu}+p_{\nu}n_{\mu}}{n\cdot p}.$
(20)

In this study, we focus on the contribution from longitudinally polarized
photons and we compute the numerators using the spinor helicity formalism. The
necessary real numerators are in table 1 and the virtual numerators are in Eq.
21 \- 26.

Figure 2: The ten virtual NLO diagrams $i\mathcal{A}_{5},...,i\mathcal{A}_{14}$. The arrows on fermion lines indicate fermion number flow, all momenta flow to the right, except for gluon momenta. The thick solid line indicates interaction with the target. Numerator | $\lambda_{\gamma};\lambda_{q},\lambda_{g}$ | $N_{i}^{\lambda_{\gamma};\lambda_{q},\lambda_{g}}$
---|---|---
$N_{1}$ | $L;+,+$ | $-Q(z_{1}z_{2})^{3/2}(1-z_{2})\frac{[(z_{1}\mathbf{k}-z_{3}\mathbf{p})\cdot\bm{\epsilon}]}{(z_{1}\mathbf{k}-z_{3}\mathbf{p})^{2}}$
$L;+,-$ | $-Q(z_{2})^{3/2}\sqrt{z_{1}}(1-z_{2})^{2}\frac{[(z_{1}\mathbf{k}-z_{3}\mathbf{p})\cdot\bm{\epsilon}]}{(z_{1}\mathbf{k}-z_{3}\mathbf{p})^{2}}$
$N_{2}$ | $L;+,+$ | $Q(z_{1})^{3/2}\sqrt{z_{2}}(1-z_{1})^{2}\frac{[(z_{2}\mathbf{k}-z_{3}\mathbf{q})\cdot\bm{\epsilon}]}{(z_{2}\mathbf{k}-z_{3}\mathbf{q})^{2}}$
$L;+,-$ | $Q(z_{1}z_{2})^{3/2}(1-z_{1})\frac{[(z_{2}\mathbf{k}-z_{3}\mathbf{q})\cdot\bm{\epsilon}^{*}]}{(z_{2}\mathbf{k}-z_{3}\mathbf{q})^{2}}$
$N_{3}$ | $L;+,+$ | $Q(z_{1}z_{2})^{3/2}(1-z_{2})\left(\frac{\mathbf{k}_{2}\cdot\bm{\epsilon}}{z_{3}}-\frac{\mathbf{k}_{1}\cdot\bm{\epsilon}}{1-z_{2}}\right)$
$L;+,-$ | $Q(z_{2})^{3/2}\sqrt{z_{1}}(1-z_{2})^{2}\left(\frac{\mathbf{k}_{2}\cdot\bm{\epsilon}^{*}}{z_{3}}-\frac{\mathbf{k}_{1}\cdot\bm{\epsilon}^{*}}{1-z_{2}}\right)$
$N_{4}$ | $L;+,+$ | $-Q(z_{1})^{3/2}\sqrt{z_{2}}(1-z_{1})^{2}\left(\frac{\mathbf{k}_{2}\cdot\bm{\epsilon}}{z_{3}}-\frac{\mathbf{k}_{1}\cdot\bm{\epsilon}}{1-z_{1}}\right)$
$L;+,-$ | $-Q(z_{1}z_{2})^{3/2}(1-z_{1})\left(\frac{\mathbf{k}_{2}\cdot\bm{\epsilon}^{*}}{z_{3}}-\frac{\mathbf{k}_{1}\cdot\bm{\epsilon}^{*}}{1-z_{1}}\right)$
Table 1: The minimal set of real numerators $N_{1}$ to $N_{4}$ in momentum
fraction notation. Complex conjugation results in the numerator with all
helicities flipped (while leaving longitudinal helicities unchanged), and any
numerator where $\lambda_{q}=\lambda_{\bar{q}}$ is zero. We have also defined
$z_{3}$ to be the gluon’s momentum fraction relative to the photon
($z_{3}=k^{+}/l^{+}$).

We also need the Dirac numerators for the virtual diagrams, they are

$\displaystyle N_{5}^{L;+}=$
$\displaystyle\frac{2^{5}(l^{+})^{2}Q(z_{1}z_{2})^{3/2}}{(z_{1}-z_{3})^{2}}\Bigg{\\{}z_{1}^{2}\left[\left(\mathbf{k}_{3}-\frac{z_{3}}{z_{1}}\mathbf{p}\right)\cdot\bm{\epsilon}\right]\left[\left(\mathbf{k}_{2}-\frac{z_{3}}{z_{1}}\mathbf{k}_{1}\right)\cdot\bm{\epsilon}^{*}\right]+z_{3}^{2}\left[\left(\mathbf{k}_{3}-\frac{z_{3}}{z_{1}}\mathbf{p}\right)\cdot\bm{\epsilon}^{*}\right]\left[\left(\mathbf{k}_{2}-\frac{z_{3}}{z_{1}}\mathbf{k}_{1}\right)\cdot\bm{\epsilon}\right]\Bigg{\\}}.$
(21) $\displaystyle N_{7}^{L;+}=$
$\displaystyle\frac{2^{5}(l^{+})^{2}Qz_{3}\sqrt{z_{1}z_{2}}}{(z_{1}-z_{3})^{2}}\Bigg{\\{}z_{1}z_{2}\left[\left(\mathbf{k}_{3}-\frac{z_{3}}{z_{1}}\mathbf{p}\right)\cdot\bm{\epsilon}\right]\Big{[}\Big{(}z_{2}\mathbf{k}_{1}-(1-z_{3})\mathbf{k}_{2}\Big{)}\cdot\bm{\epsilon}^{*}\Big{]}$
$\displaystyle+z_{3}(1-z_{3})\left[\left(\mathbf{k}_{3}-\frac{z_{3}}{z_{1}}\mathbf{p}\right)\cdot\bm{\epsilon}^{*}\right]\Big{[}\Big{(}z_{2}\mathbf{k}_{1}-(1-z_{3})\mathbf{k}_{2}\Big{)}\cdot\bm{\epsilon}\Big{]}\Bigg{]}.$
(22) $\displaystyle N_{9}^{L;+}=$ $\displaystyle
2^{4}Q(l^{+})^{2}(z_{1}z_{2})^{3/2}\left[k_{2}^{2}+\frac{(2z_{1}-z)}{z}(k_{2}-p)^{2}-\frac{[z_{1}^{2}+(z_{1}-z)^{2}]}{z_{1}z}p^{2}\right].$
(23) $\displaystyle N_{11}^{L;+}=$ $\displaystyle
2^{4}(l^{+})^{2}Q(z_{1}z_{2})^{3/2}\left[\frac{\left[(k_{2}^{+})^{2}+(p^{+})^{2}\right]}{p^{+}(k_{2}^{+}-p^{+})}k_{1}^{2}+\frac{(p^{+}+k_{2}^{+})}{(p^{+}-k_{2}^{+})}k_{2}^{2}+(k_{1}-k_{2})^{2}\right].$
(24) $\displaystyle N_{13}^{L;+}=$ $\displaystyle
2^{5}Q(l^{+})^{2}(z_{1}+z_{3})(z_{2}-z_{3})\sqrt{z_{1}z_{2}}\Bigg{[}z_{1}z_{3}\left(\frac{\mathbf{p}\cdot\bm{\epsilon}}{z_{1}}-\frac{\mathbf{q}\cdot\bm{\epsilon}}{z_{2}}\right)\left(\frac{\mathbf{p}\cdot\bm{\epsilon}^{*}}{z_{1}}-\frac{\mathbf{k}_{2}\cdot\bm{\epsilon}^{*}}{z_{3}}\right)+z_{3}^{2}\left(\frac{\mathbf{k}_{2}\cdot\bm{\epsilon}}{z_{3}}-\frac{\mathbf{q}\cdot\bm{\epsilon}}{z_{2}}\right)\left(\frac{\mathbf{p}\cdot\bm{\epsilon}^{*}}{z_{1}}-\frac{\mathbf{k}_{2}\cdot\bm{\epsilon}^{*}}{z_{3}}\right)$
$\displaystyle-
z_{2}z_{3}\left(\frac{\mathbf{k}_{2}\cdot\bm{\epsilon}}{z_{3}}-\frac{\mathbf{q}\cdot\bm{\epsilon}}{z_{2}}\right)\left(\frac{\mathbf{p}\cdot\bm{\epsilon}^{*}}{z_{1}}-\frac{\mathbf{q}\cdot\bm{\epsilon}^{*}}{z_{2}}\right)-p\cdot
q-\frac{(z_{1}+z_{3})}{2z_{3}}(k_{2}-q)^{2}+\frac{(z_{2}-z_{3})}{2z_{3}}(k_{2}+p)^{2}\Bigg{]}.$
(25) $\displaystyle N_{14}^{L;+}=$
$\displaystyle\frac{2^{5}Q(l^{+})^{2}(z_{1}z_{2})^{3/2}z_{3}}{(z_{3}-z_{1})}\Bigg{[}2l^{+}(1-z_{3})(k_{1}^{-}-k_{2}^{-})+\frac{(z_{3}-z_{2})}{z_{3}}(\mathbf{k}_{1}\cdot\bm{\epsilon})(\mathbf{k}_{1}\cdot\bm{\epsilon}^{*})+\frac{(z_{1}-z_{3})}{z_{2}}(\mathbf{k}_{2}\cdot\bm{\epsilon})(\mathbf{k}_{1}\cdot\bm{\epsilon}^{*})$
$\displaystyle+\frac{(1-z_{3})(z_{1}-z_{3})}{z_{1}z_{3}}(\mathbf{k}_{1}\cdot\bm{\epsilon})(\mathbf{k}_{2}\cdot\bm{\epsilon}^{*})+\frac{(z_{2}-z_{3})(1-z_{3})}{z_{1}z_{2}}(\mathbf{k}_{2}\cdot\bm{\epsilon})(\mathbf{k}_{2}\cdot\bm{\epsilon}^{*})\Bigg{]}.$
(26)

In all these virtual numerators, $z_{3}$ is the momentum fraction for the
remaining internal $+$ momentum (see Eq. 8 \- 13). We have also defined the
transverse vector $\bm{\epsilon}$ as

$\displaystyle\bm{\epsilon}=\frac{1}{\sqrt{2}}(1,i).$ (27)

##  III Results

To calculate the $\mathcal{O}(\alpha_{s})$ corrections to the production cross
section we need to multiply the helicity amplitudes with the corresponding
conjugate amplitudes (real amplitudes need to be multiplied with each other,
virtual amplitudes need to be multiplied by the leading order amplitude
$i\mathcal{M}$), include the phase space factors $\differential\Phi^{n}$, and
divide by the incident flux $\mathcal{F}$ which we’ll take to be $2l^{+}$. We
also relabel and shift (where necessary) on the remaining $z_{3}$ integral in
each expression so that the gluon’s momentum fraction is always labeled $z$.
We’ll write the real corrections as $\sigma_{i\times j}$ for $i,j=1,...,4$ and
the virtual corrections as $\sigma_{i}$ for $i=5,...,14$.

$\displaystyle\differential\sigma^{L}_{\text{NLO}}=\sum_{i,j=1}^{4}\differential\sigma^{L}_{i\times
j}+2\,\text{Re}\sum_{i=5}^{14}\differential\sigma_{i}^{L}.$ (28)

$\displaystyle\differential\sigma^{L}_{i\times
j}=\frac{1}{\mathcal{F}}\int_{z,\mathbf{k}}\left[(i\mathcal{M}_{i}^{a})(i\mathcal{M}_{j}^{a})^{*L}\right]\differential\Phi^{(3)},\,\,\,\,\,\,\differential\sigma^{L}_{i}=\frac{1}{\mathcal{F}}\left[(i\mathcal{M}_{i})(i\mathcal{M})^{*,L}\right]\differential\Phi^{(2)}.$
(29)

$\displaystyle\differential\Phi^{(3)}=2l^{+}\frac{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential^{2}\mathbf{k}\,\differential
y_{1}\,\differential y_{2}\,\differential
z}{(2\pi)^{8}(4l^{+})^{2}z}\delta(1-z_{1}-z_{2}-z),\,\,\,\,\,\,\,\differential\Phi^{(2)}=2l^{+}\frac{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}{2(2\pi)^{5}(2l^{+})^{2}}\delta(1-z_{1}-z_{2}).$
(30)

Here the $L$ signifies that we are including contributions only from
longitudinally polarized photons, and imply that we have summed over all
outgoing polarizations. We note that all real corrections come with a
$\delta(1-z_{1}-z_{2}-z)$ whereas all virtual corrections come with a
$\delta(1-z_{1}-z_{2})$. We omit these delta functions here for brevity and
restore them at the end. In many cases, it is easiest to write the results in
coordinate space with the radiation kernel $\Delta^{(3)}_{ij}$ defined as
follows.

$\displaystyle\Delta^{(3)}_{ij}=\frac{\mathbf{x}_{3i}\cdot\mathbf{x}_{3j}}{\mathbf{x}_{3i}^{2}\mathbf{x}_{3j}^{2}}.$
(31)

The real corrections are:

$\displaystyle\frac{\differential\sigma^{L}_{1\times
1}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{2}^{3}(1-z_{2})^{2}(z_{1}^{2}+(1-z_{2})^{2})}{(2\pi)^{10}z_{1}}\int\frac{\differential
z}{z}\int\differential^{10}x\,K_{0}(|\mathbf{x}_{12}|Q_{2})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{2})\Delta^{(3)}_{11^{\prime}}$
$\displaystyle\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{i\frac{z}{z_{1}}\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}.$
(32) $\displaystyle\frac{\differential\sigma^{L}_{2\times
2}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}^{3}(1-z_{1})^{2}(z_{2}^{2}+(1-z_{1})^{2})}{(2\pi)^{10}z_{2}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\Delta^{(3)}_{22^{\prime}}$
$\displaystyle[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\frac{z}{z_{2}}\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}.$
(33) $\displaystyle\frac{\differential\sigma^{L}_{1\times
2}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}z_{2}(1-z_{1})(1-z_{2})(z_{1}(1-z_{1})+z_{2}(1-z_{2}))}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{2})$
$\displaystyle
K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\Delta^{(3)}_{12^{\prime}}[S_{12}S_{1^{\prime}2^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{i\frac{z}{z_{1}}\mathbf{p}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}e^{i\frac{z}{z_{2}}\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{3})}.$
(34) $\displaystyle\frac{\differential\sigma^{L}_{3\times
3}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}z_{2}^{3}(z_{1}^{2}+(1-z_{2})^{2})}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(QX)K_{0}(QX^{\prime})\Delta^{(3)}_{11^{\prime}}$
$\displaystyle[S_{11^{\prime}}S_{22^{\prime}}-S_{13}S_{23}-S_{1^{\prime}3}S_{2^{\prime}3}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}.$
(35) $\displaystyle\frac{\differential\sigma^{L}_{4\times
4}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}^{3}z_{2}(z_{2}^{2}+(1-z_{1})^{2})}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(QX)K_{0}(QX^{\prime})\Delta^{(3)}_{22^{\prime}}$
$\displaystyle[S_{11^{\prime}}S_{22^{\prime}}-S_{13}S_{23}-S_{1^{\prime}3}S_{2^{\prime}3}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}.$
(36) $\displaystyle\frac{\differential\sigma^{L}_{3\times
4}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}^{2}z_{2}^{2}(z_{1}(1-z_{1})+z_{2}(1-z_{2}))}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(QX)K_{0}(QX^{\prime})\Delta^{(3)}_{12^{\prime}}$
$\displaystyle[S_{11^{\prime}}S_{22^{\prime}}-S_{13}S_{23}-S_{1^{\prime}3}S_{2^{\prime}3}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}.$
(37) $\displaystyle\frac{\differential\sigma^{L}_{1\times
3}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}z_{2}^{3}(1-z_{2})(z_{1}^{2}+(1-z_{2})^{2})}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}x\,K_{0}(|\mathbf{x}_{12}|Q_{2})K_{0}(QX^{\prime})\Delta^{(3)}_{11^{\prime}}$
$\displaystyle[S_{122^{\prime}3}S_{1^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{i\frac{z}{z_{1}}\mathbf{p}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}.$
(38) $\displaystyle\frac{\differential\sigma^{L}_{1\times
4}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}z_{2}^{2}(1-z_{2})(z_{1}(1-z_{1})+z_{2}(1-z_{2}))}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}xK_{0}(|\mathbf{x}_{12}|Q_{2})K_{0}(QX^{\prime})\Delta^{(3)}_{12^{\prime}}$
$\displaystyle[S_{122^{\prime}3}S_{1^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{i\frac{z}{z_{1}}\mathbf{p}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}.$
(39) $\displaystyle\frac{\differential\sigma^{L}_{2\times
3}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}^{2}z_{2}(1-z_{1})(z_{1}(1-z_{1})+z_{2}(1-z_{2}))}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}xK_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(QX^{\prime})\Delta^{(3)}_{21^{\prime}}$
$\displaystyle[S_{1231^{\prime}}S_{2^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{i\frac{z}{z_{2}}\mathbf{q}\cdot(\mathbf{x}_{3}-\mathbf{x}_{2})}.$
(40) $\displaystyle\frac{\differential\sigma^{L}_{2\times
4}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}^{3}(1-z_{1})(z_{2}^{2}+(1-z_{1})^{2})}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}xK_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(QX^{\prime})\Delta^{(3)}_{22^{\prime}}$
$\displaystyle[S_{1231^{\prime}}S_{2^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{i\frac{z}{z_{2}}\mathbf{q}\cdot(\mathbf{x}_{3}-\mathbf{x}_{2})}.$
(41)

This concludes the real corrections, next we have the virtual corrections:

$\displaystyle\frac{\differential\sigma^{L}_{5}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{2}^{3}z_{1}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,[S_{322^{\prime}1^{\prime}}S_{13}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1](z_{1}^{2}+(z_{1}-z)^{2})$
$\displaystyle\frac{K_{0}(QX_{5})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})}{\mathbf{x}_{31}^{2}}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{-i\frac{z}{z_{1}}\mathbf{p}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}.$
(42)
$\displaystyle\frac{\differential\sigma^{L}_{6}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}^{3}z_{2}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}[S_{132^{\prime}1^{\prime}}S_{23}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1](z_{2}^{2}+(z_{2}-z)^{2})$
$\displaystyle\frac{K_{0}(QX_{6})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})}{\mathbf{x}_{32}^{2}}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{-i\frac{z}{z_{2}}\mathbf{q}\cdot(\mathbf{x}_{3}-\mathbf{x}_{2})}.$
(43)
$\displaystyle\frac{\differential\sigma^{L}_{7}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}z_{2}^{2}z_{1}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z\,(z_{1}-z)}{z}\differential^{10}\mathbf{x}\,[S_{322^{\prime}1^{\prime}}S_{13}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1](z_{1}z_{2}+(z_{1}-z)(z_{2}+z))$
$\displaystyle
K_{0}(QX_{5})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\Delta^{(3)}_{12}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{-i\frac{z}{z_{1}}\mathbf{p}\cdot(\mathbf{x}_{3}-\mathbf{x}_{1})}.$
(44)
$\displaystyle\frac{\differential\sigma^{L}_{8}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}z_{1}^{2}z_{2}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z\,(z_{2}-z)}{z}\differential^{10}\mathbf{x}\,[S_{132^{\prime}1^{\prime}}S_{23}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1](z_{1}z_{2}+(z_{2}-z)(z_{1}+z))$
$\displaystyle
K_{0}(QX_{6})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\Delta^{(3)}_{12}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{-i\frac{z}{z_{2}}\mathbf{q}\cdot(\mathbf{x}_{3}-\mathbf{x}_{2})}.$
(45)
$\displaystyle\frac{\differential\sigma_{9}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{8}}\int\differential^{8}\mathbf{x}\big{[}S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\big{]}K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}$
$\displaystyle\times\int_{0}^{z_{1}}\frac{\differential
z}{z}\frac{[z_{1}^{2}+(z_{1}-z)^{2}]}{z_{1}^{2}}\int\frac{\differential^{2}\mathbf{k}_{2}}{(2\pi)^{2}}\frac{1}{\left(\mathbf{k}_{2}-\frac{z}{z_{1}}\mathbf{p}\right)^{2}}.$
(46)
$\displaystyle\frac{\differential\sigma_{10}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{8}}\int\differential^{8}\mathbf{x}\big{[}S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\big{]}K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}$
$\displaystyle\times\int_{0}^{z_{2}}\frac{\differential
z}{z}\frac{[z_{2}^{2}+(z_{2}-z)^{2}]}{z_{2}^{2}}\int\frac{\differential^{2}\mathbf{k}_{2}}{(2\pi)^{2}}\frac{1}{\left(\mathbf{k}_{2}-\frac{z}{z_{2}}\mathbf{q}\right)^{2}}.$
(47)
$\displaystyle\frac{\differential\sigma_{11}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{7}}\int\differential^{8}\mathbf{x}\big{[}S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\big{]}K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}$
$\displaystyle\int_{0}^{z_{1}}\frac{\differential
z}{z}\frac{[z_{1}^{2}+(z_{1}-z)^{2}]}{z_{1}^{2}}\int\frac{\differential^{2}\mathbf{k}_{2}}{(2\pi)^{2}}\int\frac{\differential^{2}\mathbf{k}_{1}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}_{1}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}}{\big{[}\mathbf{k}_{1}^{2}+Q_{1}^{2}\big{]}\left[\left(\mathbf{k}_{2}-\frac{z}{z_{1}}\mathbf{k}_{1}\right)^{2}+\frac{z(z_{1}-z)}{z_{1}^{2}z_{2}}\mathbf{k}_{1}^{2}+\frac{z}{z_{1}}(z_{1}-z)Q^{2}\right]}.$
(48)
$\displaystyle\frac{\differential\sigma_{12}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{7}}\int\differential^{8}\mathbf{x}\big{[}S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\big{]}K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}$
$\displaystyle\int_{0}^{z_{2}}\frac{\differential
z}{z}\frac{[z_{2}^{2}+(z_{2}-z)^{2}]}{z_{2}^{2}}\int\frac{\differential^{2}\mathbf{k}_{2}}{(2\pi)^{2}}\int\frac{\differential^{2}\mathbf{k}_{1}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}_{1}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}}{\big{[}\mathbf{k}_{1}^{2}+Q_{1}^{2}\big{]}\left[\left(\mathbf{k}_{2}-\frac{z}{z_{2}}\mathbf{k}_{1}\right)^{2}+\frac{z(z_{2}-z)}{z_{2}^{2}z_{1}}\mathbf{k}_{1}^{2}+\frac{z}{z_{2}}(z_{2}-z)Q^{2}\right]}.$
(49)

$\displaystyle\frac{\differential\sigma_{13(1)}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential
y_{2}}=\frac{-e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{2}}{(2\pi)^{8}}\int_{0}^{z_{2}}\frac{\differential
z}{z}(z_{1}+z)(z_{2}-z)\int\differential^{8}\mathbf{x}\,K_{0}\left(|\mathbf{x}_{12}|Q\sqrt{(z_{1}+z)(z_{2}-z)}\right)K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})$
$\displaystyle
e^{i\mathbf{p}\cdot(\mathbf{x}_{1^{\prime}}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2^{\prime}}-\mathbf{x}_{2})}[S_{12}S_{1^{\prime}2^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\Bigg{[}\frac{-\frac{z(z_{2}-z)[z_{1}z_{2}+(z_{1}+z)(z_{2}-z)]}{z_{1}^{2}z_{2}^{3}}(z_{2}\mathbf{p}-z_{1}\mathbf{q})^{2}}{\left[(\mathbf{k}+(z_{2}-z)\mathbf{p}-(z_{1}+z)\mathbf{q})^{2}-\frac{(z_{1}+z)(z_{2}-z)}{z_{1}z_{2}}(z_{2}\mathbf{p}-z_{1}\mathbf{q})^{2}-i\epsilon\right]\left(\mathbf{k}-\frac{z}{z_{2}}\mathbf{q}\right)^{2}}$
$\displaystyle+\frac{\frac{z[z_{1}(z_{1}+z)+z_{2}(z_{2}-z)]}{z_{1}z_{2}}}{\left[(\mathbf{k}+(z_{2}-z)\mathbf{p}-(z_{1}+z)\mathbf{q})^{2}-\frac{(z_{1}+z)(z_{2}-z)}{z_{1}z_{2}}(z_{2}\mathbf{p}-z_{1}\mathbf{q})^{2}-i\epsilon\right]}-\frac{\frac{[z_{1}z_{2}+(z_{1}+z)(z_{2}-z)]}{z_{1}z_{2}}}{\left(\mathbf{k}-\frac{z}{z_{2}}\mathbf{q}\right)^{2}}\Bigg{]}e^{i\mathbf{k}\cdot(\mathbf{x}_{2}-\mathbf{x}_{1})}.$
(50)
$\displaystyle\frac{\differential\sigma_{13(2)}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential
y_{2}}=\frac{-e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{2}}{(2\pi)^{8}}\int_{0}^{z_{1}}\frac{\differential
z}{z}(z_{2}+z)(z_{1}-z)\int\differential^{8}\mathbf{x}\,K_{0}\left(|\mathbf{x}_{12}|Q\sqrt{(z_{2}+z)(z_{1}-z)}\right)K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})$
$\displaystyle
e^{i\mathbf{p}\cdot(\mathbf{x}_{1^{\prime}}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2^{\prime}}-\mathbf{x}_{2})}[S_{12}S_{1^{\prime}2^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\Bigg{[}\frac{-\frac{z(z_{1}-z)[z_{1}z_{2}+(z_{2}+z)(z_{1}-z)]}{z_{2}^{2}z_{1}^{3}}(z_{2}\mathbf{p}-z_{1}\mathbf{q})^{2}}{\left[(\mathbf{k}+(z_{1}-z)\mathbf{q}-(z_{2}+z)\mathbf{p})^{2}-\frac{(z_{2}+z)(z_{1}-z)}{z_{1}z_{2}}(z_{2}\mathbf{p}-z_{1}\mathbf{q})^{2}+i\epsilon\right]\left(\mathbf{k}-\frac{z}{z_{1}}\mathbf{p}\right)^{2}}$
$\displaystyle+\frac{\frac{z[z_{2}(z_{2}+z)+z_{1}(z_{1}-z)]}{z_{1}z_{2}}}{\left[(\mathbf{k}+(z_{1}-z)\mathbf{q}-(z_{2}+z)\mathbf{p})^{2}-\frac{(z_{2}+z)(z_{1}-z)}{z_{1}z_{2}}(z_{2}\mathbf{p}-z_{1}\mathbf{q})^{2}+i\epsilon\right]}-\frac{\frac{[z_{1}z_{2}+(z_{2}+z)(z_{1}-z)]}{z_{1}z_{2}}}{\left(\mathbf{k}-\frac{z}{z_{1}}\mathbf{p}\right)^{2}}\Bigg{]}e^{i\mathbf{k}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}.$
(51)

Here we have kept the $i\epsilon$ in the long denominators, since it proves
useful for evaluating the $\mathbf{k}$ integral.

$\displaystyle\frac{\differential\sigma_{14(1)}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{2}}{(2\pi)^{7}}\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}$
$\displaystyle\int_{0}^{z_{1}}\frac{\differential
z}{z}\frac{\differential^{2}\mathbf{k}_{1}}{(2\pi)^{2}}\frac{\differential^{2}\mathbf{k}_{2}}{(2\pi)^{2}}e^{i\mathbf{k}_{2}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}\Bigg{[}\frac{(z_{1}z_{2}+(z_{1}-z)(z_{2}+z))}{\Big{[}\mathbf{k}_{2}^{2}+Q_{1}^{2}\Big{]}\left[\left(\mathbf{k}_{1}-\frac{z_{1}-z}{z_{1}}\mathbf{k}_{2}\right)^{2}+\frac{z(z_{1}-z)}{z_{2}z_{1}^{2}}\mathbf{k}_{2}^{2}+\frac{z}{z_{1}}(z_{1}-z)Q^{2}\right]}$
$\displaystyle+\frac{\frac{(z_{1}-z)(z_{2}+z)}{z_{1}z_{2}}(z_{1}z_{2}+(z_{1}-z)(z_{2}+z))}{\Big{[}\mathbf{k}_{1}^{2}+(z_{1}-z)(z_{2}+z)Q^{2}\Big{]}\left[\left(\mathbf{k}_{1}-\frac{z_{1}-z}{z_{1}}\mathbf{k}_{2}\right)^{2}+\frac{z(z_{1}-z)}{z_{2}z_{1}^{2}}\mathbf{k}_{2}^{2}+\frac{z}{z_{1}}(z_{1}-z)Q^{2}\right]}$
$\displaystyle-\frac{\frac{z(z_{1}-z)}{z_{1}}(z_{1}z_{2}+(z_{1}-z)(z_{2}+z))Q^{2}}{\Big{[}\mathbf{k}_{1}^{2}+(z_{1}-z)(z_{2}+z)Q^{2}\Big{]}\Big{[}\mathbf{k}_{2}^{2}+Q_{1}^{2}\Big{]}\left[\left(\mathbf{k}_{1}-\frac{z_{1}-z}{z_{1}}\mathbf{k}_{2}\right)^{2}+\frac{z(z_{1}-z)}{z_{2}z_{1}^{2}}\mathbf{k}_{2}^{2}+\frac{z}{z_{1}}(z_{1}-z)Q^{2}\right]}$
$\displaystyle-\frac{z(z_{1}(z_{1}-z)+z_{2}(z_{2}+z))}{\Big{[}\mathbf{k}_{1}^{2}+(z_{1}-z)(z_{2}+z)Q^{2}\Big{]}\Big{[}\mathbf{k}_{2}^{2}+Q_{1}^{2}\Big{]}}\Bigg{]}.$
(52)
$\displaystyle\frac{\differential\sigma_{14(2)}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{2}}{(2\pi)^{7}}\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}$
$\displaystyle\int_{0}^{z_{2}}\frac{\differential
z}{z}\frac{\differential^{2}\mathbf{k}_{1}}{(2\pi)^{2}}\frac{\differential^{2}\mathbf{k}_{2}}{(2\pi)^{2}}e^{i\mathbf{k}_{2}\cdot(\mathbf{x}_{1}-\mathbf{x}_{2})}\Bigg{[}\frac{(z_{1}z_{2}+(z_{1}+z)(z_{2}-z))}{\Big{[}\mathbf{k}_{2}^{2}+Q_{1}^{2}\Big{]}\left[\left(\mathbf{k}_{1}-\frac{z_{2}-z}{z_{2}}\mathbf{k}_{2}\right)^{2}+\frac{z(z_{2}-z)}{z_{1}z_{2}^{2}}\mathbf{k}_{2}^{2}+\frac{z}{z_{2}}(z_{2}-z)Q^{2}\right]}$
$\displaystyle+\frac{\frac{(z_{1}+z)(z_{2}-z)}{z_{1}z_{2}}(z_{1}z_{2}+(z_{1}+z)(z_{2}-z))}{\Big{[}\mathbf{k}_{1}^{2}+(z_{1}+z)(z_{2}-z)Q^{2}\Big{]}\left[\left(\mathbf{k}_{1}-\frac{z_{2}-z}{z_{2}}\mathbf{k}_{2}\right)^{2}+\frac{z(z_{2}-z)}{z_{1}z_{2}^{2}}\mathbf{k}_{2}^{2}+\frac{z}{z_{2}}(z_{2}-z)Q^{2}\right]}$
$\displaystyle-\frac{\frac{z(z_{2}-z)}{z_{2}}(z_{1}z_{2}+(z_{1}+z)(z_{2}-z))Q^{2}}{\Big{[}\mathbf{k}_{1}^{2}+(z_{1}+z)(z_{2}-z)Q^{2}\Big{]}\Big{[}\mathbf{k}_{2}^{2}+Q_{1}^{2}\Big{]}\left[\left(\mathbf{k}_{1}-\frac{z_{2}-z}{z_{2}}\mathbf{k}_{2}\right)^{2}+\frac{z(z_{2}-z)}{z_{1}z_{2}^{2}}\mathbf{k}_{2}^{2}+\frac{z}{z_{2}}(z_{2}-z)Q^{2}\right]}$
$\displaystyle-\frac{z(z_{2}(z_{2}-z)+z_{1}(z_{1}+z))}{\Big{[}\mathbf{k}_{1}^{2}+(z_{1}+z)(z_{2}-z)Q^{2}\Big{]}\Big{[}\mathbf{k}_{2}^{2}+Q_{1}^{2}\Big{]}}\Bigg{]}.$
(53)

These expressions constitute the full result for the one-loop corrections to
inclusive quark anti-quark production cross section. We have written these
results all in terms of the dipole and quadrupole functions defined in Eq. (3)
in the large $N_{c}$ limit and ignored all subleading $N_{c}$ terms. We have
also used the following notation for the coordinate dependence of some of the
Bessel functions:

$\displaystyle X$
$\displaystyle=\sqrt{z_{1}z_{2}\mathbf{x}_{12}^{2}+z_{1}z\mathbf{x}_{13}^{2}+z_{2}z\mathbf{x}_{23}^{2}},$
$\displaystyle X_{5}$
$\displaystyle=\sqrt{z_{2}(z_{1}-z)\mathbf{x}_{12}^{2}+z(z_{1}-z)\mathbf{x}_{13}^{2}+z_{2}z\,\mathbf{x}_{23}^{2}},$
$\displaystyle X_{6}$
$\displaystyle=\sqrt{z_{1}(z_{2}-z)\mathbf{x}_{12}^{2}+z_{1}z\,\mathbf{x}_{13}^{2}+z(z_{2}-z)\mathbf{x}_{23}^{2}}.$
(54)

Note that when $z\to 0$ these all become $|\mathbf{x}_{12}|\sqrt{z_{1}z_{2}}$.
The primed version $X^{\prime}$ that appears in some real corrections is the
same as $X$ above but with
$\mathbf{x}_{1},\mathbf{x}_{2}\to\mathbf{x}_{1}^{\prime},\mathbf{x}_{2}^{\prime}$.

##  IV Divergences

The expressions for the one-loop corrections to quark anti-quark inclusive
production cross section given above are formal, i.e. they contain divergences
that need to be regulated and/or eliminated before one obtains a meaningful
result. There are $4$ categories of divergences:

$\bullet$ Ultraviolet (UV) divergences when loop momentum
$\mathbf{k}\rightarrow\infty$ or equivalently in coordinate space, when the
transverse coordinate of the radiated gluon approaches the transverse
coordinate $\mathbf{x}_{i}$ of either quark or anti-quark when integrated,
i.e. $\mathbf{x}_{3}\rightarrow\mathbf{x}_{i}$ such that
$|\mathbf{x}_{3}-\mathbf{x}_{i}|\rightarrow 0$. We will show that the real
corrections are UV finite and that all UV divergences in the virtual
corrections cancel against each other so that the virtual corrections are also
UV finite by themselves.

$\bullet$ Soft divergences when $k^{\mu}\rightarrow 0$, which in this context
corresponds to both transverse momentum in the loop $\mathbf{k}$ and the
radiated gluon momentum fraction $z$ go to zero simultaneously,
$\mathbf{k},z\rightarrow 0$. Both the real and virtual corrections contain
soft divergences, however all soft divergences cancel between real and virtual
corrections.

$\bullet$ Collinear divergences when the radiated gluon momentum becomes
parallel to either quark or anti-quark momentum, i.e. when the angle $\theta$
between the radiated gluon $3$-momentum and its parent quark (anti-quark)
$3$-momentum goes to zero at finite $\mathbf{k}$ and $z$ (note that finite in
this context means that $\mathbf{k}$ is neither zero nor infinite). These
appear, for example as
${1\over(p+k)^{2}}\sim{1\over(z_{1}\mathbf{k}-z\mathbf{p})^{2}}\sim\frac{1}{1-\cos\theta}$
for radiation from a quark and similarly for radiation from a anti-quark
($z_{1},\mathbf{p}\rightarrow z_{2},\mathbf{q}$). These collinear divergences
are absorbed into quark-hadron (anti quark-hadron) fragmentation functions
which makes the fragmentation functions scale dependent, i.e. they evolve with
the renormalization scale $\mu^{2}$.

$\bullet$ Rapidity divergences when the momentum fraction $z$ of the gluon
goes to zero while the transverse momentum $\mathbf{k}$ of the gluon remains
finite. These rapidity divergences are absorbed into ($x$) renormalization of
the dipoles and quadrupoles making them energy/rapidity dependent and evolving
according to the BK and JIMWLK evolution equations Balitsky (1996); Kovchegov
(2000); Jalilian-Marian et al. (1997, 1998a, 1998b); Kovner et al. (2000);
Iancu et al. (2001); Ferreiro et al. (2002).

Here we give more details of the divergences and their cancellation/absorption
into renormalized quantities. The easiest way to see cancellations of
divergences is via power counting which is what we will use in this section.
We will also provide results using dimensional regularization in the
$\overline{\text{MS}}$ scheme which is the more common method. Using
dimensional regularization raises issues when using spinor helicity methods as
one changes the number of spacetime dimensions from $4$ (where spinor helicity
methods are formulated) to $4-2\epsilon$. There are various ways of dealing
with this issue, here we will use the FDH (Four Dimensional Helicity) scheme
which treats both internal and external states in $4$-dimensions. As expected
the finite terms after regulating divergences will depend on the
regularization scheme.

###  IV.1 UV divergences

For the real corrections (Eq. 32-41) all momentum integrations have been
performed and results are given in terms of a two-dimensional coordinate space
integration $\differential^{2}\mathbf{x}_{3}$ and an integration over momentum
fraction $z$ (relative to the LO result in Eq. 1). In this context UV
divergences manifest themselves as poles when
$\mathbf{x}_{3}\rightarrow\mathbf{x}_{1}$ or
$\mathbf{x}_{3}\rightarrow\mathbf{x}_{2}$, i.e. when the transverse coordinate
of the radiated gluon $\mathbf{x}_{3}$ approaches that of the quark
$\mathbf{x}_{1}$ or the anti-quark $\mathbf{x}_{2}$. We focus on the quark
case and consider the limit $\mathbf{x}_{3}\rightarrow\mathbf{x}_{1}$. The
case of radiation from the anti-quark is identical.

We note that the only possible UV singularity in our expressions is in the
gluon radiation kernel $\Delta_{ij}^{(3)}$ which appears in all real
corrections. Writing it as

$\displaystyle\Delta^{(3)}_{ij}=\frac{\mathbf{x}_{3i}\cdot\mathbf{x}_{3j}}{\mathbf{x}_{3i}^{2}\mathbf{x}_{3j}^{2}}=\frac{1}{2}\left[\frac{1}{\mathbf{x}_{3i}^{2}}+\frac{1}{\mathbf{x}_{3j}^{2}}-\frac{{\mathbf{x}_{ij}^{2}}}{\mathbf{x}_{3i}^{2}\mathbf{x}_{3j}^{2}}\right],$
(55)

we see that the integral of this kernel over $\mathbf{x}_{3}$ is finite as
$\mathbf{x}_{3}\rightarrow\mathbf{x}_{i}$ due to cancellations between the
first and third terms. Similarly, as $\mathbf{x}_{3}\rightarrow\mathbf{x}_{j}$
there is a cancellation between the second and third terms. The other factors
appearing in the real corrections are all non-singular in the UV limit. We
therefore conclude that there are no UV divergences in the real corrections
and that all real corrections are UV finite.

For the virtual corrections, several of the diagrams are UV divergent. However
when combined they all cancel each other in the UV limit, specifically

$\displaystyle\left[\differential\sigma_{5}+\differential\sigma_{11}\right]_{\text{UV}}=0,$
$\displaystyle\left[\differential\sigma_{6}+\differential\sigma_{12}\right]_{\text{UV}}=0,$
$\displaystyle\left[\differential\sigma_{9}+\differential\sigma_{10}+\differential\sigma_{14(1)}+\differential\sigma_{14(2)}\right]_{\text{UV}}=0.$
(56)

The remaining virtual diagrams (7,8,13) are all UV finite. Therefore, all UV
divergences are canceled when adding up all the relevant pieces of the
differential cross section. The UV-finiteness of the production cross section
holds for any value of $z$. We provide more details in Appendix A.

###  IV.2 Soft Divergences

Soft divergences appear when all components of a loop momentum go to zero,
i.e. $k^{\mu}\rightarrow 0$ which in this context means $\mathbf{k}$ and $z$
both go to zero (in coordinate space this becomes $\mathbf{x}_{3}\to\infty$
and $z\to 0$). Soft divergences in the real corrections generically appear
either as

$\int{\differential^{2}\mathbf{k}\over\mathbf{k}^{2}}\,e^{i\mathbf{k}\cdot(\mathbf{x}_{i}-\mathbf{x}_{j})}\,\,\,\,\,\,\,\,\text{or}\,\,\,\,\,\,\,\,\int{\differential^{2}\mathbf{k}\over\mathbf{k}^{2}}$
(57)

in the virtual corrections (after possibly shifting the transverse momentum).
In coordinate space soft divergences are contained in

$\int\differential^{2}\mathbf{x}_{3}\,\Delta^{(3)}_{ij}={1\over
2}\,\int\differential^{2}\mathbf{x}_{3}\,\left[{1\over(\mathbf{x}_{3}-\mathbf{x}_{i})^{2}}+{1\over(\mathbf{x}_{3}-\mathbf{x}_{j})^{2}}-{(\mathbf{x}_{i}-\mathbf{x}_{j})^{2}\over(\mathbf{x}_{3}-\mathbf{x}_{i})^{2}\,(\mathbf{x}_{3}-\mathbf{x}_{j})^{2}}\right]$
(58)

from which it is clear that the first two terms in $\Delta^{(3)}_{ij}$ are
divergent when $\mathbf{x}_{3}\rightarrow\infty$. Adding all the singular
parts of the production cross section we get the following cancellations
between the various terms,

$\displaystyle\left[\differential\sigma_{1\times
1}+2\,\differential\sigma_{9}\right]_{\text{soft}}=0,$
$\displaystyle\left[\differential\sigma_{2\times
2}+2\,\differential\sigma_{10}\right]_{\text{soft}}=0,$
$\displaystyle\left[\differential\sigma_{1\times
2}+\differential\sigma_{13(1)}+\differential\sigma_{13(2)}\right]_{\text{soft}}=0,$
$\displaystyle\left[\differential\sigma_{3\times
3}+\differential\sigma_{4\times 4}+2\,\differential\sigma_{3\times
4}\right]_{\text{soft}}=0,$ $\displaystyle\left[\differential\sigma_{1\times
3}+\differential\sigma_{1\times 4}\right]_{\text{soft}}=0,$
$\displaystyle\left[\differential\sigma_{2\times
3}+\differential\sigma_{2\times 4}\right]_{\text{soft}}=0,$
$\displaystyle\left[\differential\sigma_{5}+\differential\sigma_{7}\right]_{\text{soft}}=0,$
$\displaystyle\left[\differential\sigma_{6}+\differential\sigma_{8}\right]_{\text{soft}}=0,$
$\displaystyle\left[\differential\sigma_{11}+\differential\sigma_{14(1)}\right]_{\text{soft}}=0,$
$\displaystyle\left[\differential\sigma_{12}+\differential\sigma_{14(2)}\right]_{\text{soft}}=0.$
(59)

The necessary relative factors of $2$ in these expressions can be seen from
Eq. 28. These cancellations are easiest to see when the cross section is
written in coordinate space in the soft $z$ limit (Appendix B). It is worth
noting that there are singular terms $\sim{1\over z}\,\log({1\over z})$ which
can be seen in dimensional regularization of the remaining integrals which
cancel between $\differential\sigma_{11,\text{soft}}$ and
$\differential\sigma_{14(1),\text{soft}}$ and similarly between
$\differential\sigma_{12,\text{soft}}$ and
$\differential\sigma_{14(2),\text{soft}}$ (this is shown in Appendix C). Thus
there are no soft singularities (both $\mathbf{k},z\rightarrow 0$) left after
adding all the real and virtual terms.

###  IV.3 Rapidity Divergences

From eqs. (32 \- 53) it is clear that the production cross section contains
terms which are singular in the $\int{\differential z\over z}$ integral as
$z\rightarrow 0$. This happens at finite $\mathbf{k}$ (or $\mathbf{x}_{3}$) so
that rapidity divergences are different from the soft divergences that have
been shown to cancel among various terms. Following Caucal et al. (2021) we
introduce a longitudinal momentum fraction factorization scale $z_{f}$ and
divide the $z$ integration into two regions: $z>z_{f}$ and $z<z_{f}$ (here we
write the upper limit as $1$, in our expressions the upper limit is always
either $z_{1}$ or $z_{2}$).

$\displaystyle\int_{0}^{1}\frac{\differential
z}{z}f(z)=\left\\{\int_{0}^{z_{f}}\frac{\differential
z}{z}+\int_{z_{f}}^{1}\frac{\differential z}{z}\,\right\\}\,f(z).$ (60)

In the region where $z>z_{f}$ there are formally no divergences left anymore
(there are still collinear divergences in this region but they will be
absorbed into fragmentation functions - this is the topic of the next section)
so the cross section is completely finite. On the other hand in the second
region where $z<z_{f}$ we encounter the familiar rapidity divergence
$y=\log(1/z)$ which we expect will be absorbed into JIMWLK evolution of the
quadrupoles and dipoles. We now focus on the $z\rightarrow 0$ limit of our
expressions while $\mathbf{k}$ (or equivalently $\mathbf{x}_{3}$) is kept
finite (see Appendix B).

$\displaystyle\frac{\differential\sigma_{\text{NLO}}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential
y_{2}}=\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{f}}\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot\mathbf{x}_{1^{\prime}1}}e^{i\mathbf{q}\cdot\mathbf{x}_{2^{\prime}2}}\delta(1-z_{1}-z_{2})$
$\displaystyle\Bigg{\\{}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]\left(\frac{3}{\mathbf{x}_{31}^{2}}+\frac{3}{\mathbf{x}_{32}^{2}}+\frac{1}{\mathbf{x}_{31^{\prime}}^{2}}+\frac{1}{\mathbf{x}_{32^{\prime}}^{2}}-\frac{8}{\mathbf{x}_{3}^{2}}-\tilde{\Delta}_{11^{\prime}}-\tilde{\Delta}_{22^{\prime}}-2\tilde{\Delta}_{12}\right)$
$\displaystyle+[S_{12}S_{1^{\prime}2^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]\left(\frac{2}{\mathbf{x}_{32}^{2}}-\frac{2}{\mathbf{x}_{32^{\prime}}^{2}}+2\tilde{\Delta}_{12^{\prime}}-2\tilde{\Delta}_{12}\right)$
$\displaystyle+[S_{11^{\prime}}S_{22^{\prime}}-S_{13}S_{23}-S_{1^{\prime}3}S_{2^{\prime}3}+1]\left(-\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{31^{\prime}}^{2}}+\frac{1}{\mathbf{x}_{32}^{2}}-\frac{1}{\mathbf{x}_{32^{\prime}}^{2}}-\tilde{\Delta}_{11^{\prime}}-\tilde{\Delta}_{22^{\prime}}+2\tilde{\Delta}_{12^{\prime}}\right)$
$\displaystyle+[S_{122^{\prime}3}S_{1^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]\left(\frac{2}{\mathbf{x}_{32^{\prime}}^{2}}-\frac{2}{\mathbf{x}_{31^{\prime}}^{2}}+2\tilde{\Delta}_{11^{\prime}}-2\tilde{\Delta}_{12^{\prime}}\right)$
$\displaystyle+[S_{1231^{\prime}}S_{2^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]\left(\frac{2}{\mathbf{x}_{31^{\prime}}^{2}}-\frac{2}{\mathbf{x}_{32^{\prime}}^{2}}+2\tilde{\Delta}_{22^{\prime}}-2\tilde{\Delta}_{21^{\prime}}\right)$
$\displaystyle+[S_{322^{\prime}1^{\prime}}S_{13}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1]\left(\frac{2}{\mathbf{x}_{31}^{2}}-\frac{2}{\mathbf{x}_{32}^{2}}+2\tilde{\Delta}_{12}\right)$
$\displaystyle+[S_{132^{\prime}1^{\prime}}S_{23}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1]\left(\frac{2}{\mathbf{x}_{32}^{2}}-\frac{2}{\mathbf{x}_{31}^{2}}+2\tilde{\Delta}_{12}\right)\Bigg{\\}}.$
(61)

Here we have defined

$\displaystyle\tilde{\Delta}_{ij}=\frac{\mathbf{x}_{ij}^{2}}{\mathbf{x}_{3i}^{2}\mathbf{x}_{3j}^{2}}.$
(62)

Inside the curly brackets, the first line comes from adding $\sigma_{1\times
1}+\sigma_{2\times
2}+2\sigma_{9}+2\sigma_{10}+2\sigma_{11}+2\sigma_{12}+2\sigma_{14(1)}+2\sigma_{14(2)}$.
The second line is $2\sigma_{1\times 2}+2\sigma_{13(1)}+2\sigma_{13(2)}$. The
third line is $\sigma_{3\times 3}+\sigma_{4\times 4}+2\sigma_{3\times 4}$.
Fourth line is $2\sigma_{1\times 3}+2\sigma_{1\times 4}$. Fifth line is
$2\sigma_{2\times 3}+2\sigma_{2\times 4}$. Sixth line is
$2\sigma_{5}+2\sigma_{7}$, and the last line is $2\sigma_{6}+2\sigma_{8}$.
Notice that cross terms have been doubled since we need to add their complex
conjugate. To simplify the expression we use a combination of shifting
$\mathbf{x}_{3}$ where possible and using symmetry between primed and unprimed
coordinates. We find that everything can be simplified down to the following
expression.

$\displaystyle\frac{\differential\sigma_{\text{NLO}}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential
y_{2}}=\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\,\delta(1-z_{1}-z_{2})\int_{0}^{z_{f}}\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})$
$\displaystyle
e^{i\mathbf{p}\cdot\mathbf{x}_{1^{\prime}1}}e^{i\mathbf{q}\cdot\mathbf{x}_{2^{\prime}2}}\Bigg{\\{}\left(\tilde{\Delta}_{12}+\tilde{\Delta}_{22^{\prime}}-\tilde{\Delta}_{12^{\prime}}\right)S_{132^{\prime}1^{\prime}}S_{23}+\left(\tilde{\Delta}_{1^{\prime}2^{\prime}}+\tilde{\Delta}_{22^{\prime}}-\tilde{\Delta}_{21^{\prime}}\right)S_{1^{\prime}321}S_{2^{\prime}3}$
$\displaystyle+\left(\tilde{\Delta}_{12}+\tilde{\Delta}_{11^{\prime}}-\tilde{\Delta}_{21^{\prime}}\right)S_{322^{\prime}1^{\prime}}S_{13}+\left(\tilde{\Delta}_{1^{\prime}2^{\prime}}+\tilde{\Delta}_{11^{\prime}}-\tilde{\Delta}_{12^{\prime}}\right)S_{32^{\prime}21}S_{1^{\prime}3}$
$\displaystyle-\left(\tilde{\Delta}_{11^{\prime}}+\tilde{\Delta}_{22^{\prime}}+\tilde{\Delta}_{12}+\tilde{\Delta}_{1^{\prime}2^{\prime}}\right)S_{122^{\prime}1^{\prime}}-\left(\tilde{\Delta}_{12}+\tilde{\Delta}_{1^{\prime}2^{\prime}}-\tilde{\Delta}_{12^{\prime}}-\tilde{\Delta}_{21^{\prime}}\right)S_{12}S_{1^{\prime}2^{\prime}}$
$\displaystyle-\left(\tilde{\Delta}_{11^{\prime}}+\tilde{\Delta}_{22^{\prime}}-\tilde{\Delta}_{12^{\prime}}-\tilde{\Delta}_{21^{\prime}}\
\right)S_{11^{\prime}}S_{22^{\prime}}-2\tilde{\Delta}_{12}\left(S_{13}S_{23}-S_{12}\right)-2\tilde{\Delta}_{1^{\prime}2^{\prime}}\left(S_{1^{\prime}3}S_{2^{\prime}3}-S_{1^{\prime}2^{\prime}}\right)\Bigg{\\}}.$
(63)

Here the first seven terms inside the curly bracket give the JIMWLK evolution
of the quadrupole $S_{122^{\prime}1^{\prime}}$ with the factorization scale
$z_{f}$ (See Eq. $5$ in Jalilian-Marian (2012)). The last two terms correspond
to the rapidity evolution of the dipoles $S_{12}$ and
$S_{1^{\prime}2^{\prime}}$ as given by the BK equation Balitsky (1996);
Kovchegov (2000). This demonstrates that the rapidity divergences can be
absorbed into rapidity evolution of the LO cross section. The contribution of
the $z$-integration above the factorization scale $z_{f}$ is finite and
constitutes part of the NLO corrections to the Leading Order cross section.

###  IV.4 Collinear divergences

####  IV.4.1 Leading Order dihadron production

We now focus on the collinear divergences present in some of the one-loop
corrections in (Eq. 32 \- 53) and show that they can absorbed into evolution
of quark-hadron and anti quark-hadron fragmentation functions. As this is
completely new we will provide the full details of the calculation. We start
by writing the inclusive dihadron production cross section in terms of the
partonic cross section as

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to
h_{1}h_{2}X}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\,\frac{1}{z_{h_{1}}^{2}z_{h_{2}}^{2}}\frac{\differential\sigma^{\gamma^{*}A\to
q\bar{q}X}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential
y_{2}}D_{h_{1}/q}(z_{h_{1}})D_{h_{2}/\bar{q}}(z_{h_{2}}).$ (64)

where $\mathbf{p},\mathbf{q}$ are the quark and anti-quark transverse momenta
while $\mathbf{p}_{h},\mathbf{q}_{h}$ are the transverse momenta of the two
produced hadrons. As partons and hadrons are assumed to be massless their
rapidities are the same so that $y_{1}\equiv y_{p}=y_{h}$ (and similarly for
the anti-quark). Hadronization is assumed to be describable in terms of a non-
perturbative quark (or anti-quark)-hadron fragmentation function
$D_{h/q}(z_{h_{1}})$ which is then perturbatively renormalized (becomes
$\mu^{2}$ dependent) after absorption of the collinear divergences. The
momentum fractions $z_{h_{1}},z_{h_{2}}$ are defined via
$p_{h_{1}}^{\mu}=z_{h_{1}}p^{\mu}$ and $q_{h_{2}}^{\mu}=z_{h_{2}}q^{\mu}$ so
that $0<z_{h_{i}}<1$. Using our Leading Order result from Eq. (1) (only the
longitudinal part) we can write this as

$\displaystyle\frac{\differential\sigma_{LO}^{\gamma^{*}A\to
h_{1}h_{2}X}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=$ $\displaystyle\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\,e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}\,e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}$
$\displaystyle
D_{h_{1}/q}(z_{h_{1}})D_{h_{2}/\bar{q}}(z_{h_{2}})\,\delta(1-z_{1}-z_{2}),$
(65)

where $z_{1}$ and $z_{2}$ are related to hadronic momentum fractions by

$\displaystyle
z_{1}=\frac{p_{h}^{+}}{z_{h_{1}}l^{+}},\,\,\,\,\,\,z_{2}=\frac{q_{h}^{+}}{z_{h_{2}}l^{+}}.$
(66)

Anticipating factorization of the one-loop corrections and to make our
expression more compact we define the common factor
$H(\mathbf{p},\mathbf{q},\mathbf{x},z_{1})$ as

$\displaystyle
H(\mathbf{p},\mathbf{q},\mathbf{x},z_{1})\equiv\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]K_{0}\left(|\mathbf{x}_{12}|Q_{1}\right)\,K_{0}\left(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1}\right)\,e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})},$
(67)

with $Q_{1}^{2}\equiv z_{1}(1-z_{1})\,Q^{2}$ so that the common factor
implicitly depends on $Q^{2}$. The Leading Order cross section can be then
expressed in the following compact form

$\displaystyle\frac{\differential\sigma_{LO}^{\gamma^{*}A\to
h_{1}h_{2}X}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=$ $\displaystyle\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H\left(\mathbf{p},\mathbf{q},\mathbf{x},z_{2}\right)$
$\displaystyle\delta\left(1-z_{1}-z_{2}\right)D_{h_{1}/q}(z_{h_{1}})D_{h_{2}/\bar{q}}(z_{h_{2}}).$
(68)

####  IV.4.2 The virtual correction

First we’ll focus on the fragmentation of the quark into a hadron $h_{1}$.
Treatment of anti quark fragmentation will be identical. Schematically, the
fragmentation function receives corrections from the following diagrams:

Figure 3: The full fragmentation function $D_{h_{1}/q}(z_{h_{1}},\mu^{2})$
(solid black circle) can be written as the bare fragmentation function (faint
gray circle) plus the $\mathcal{O}(\alpha_{s})$ corrections shown. These need
to be added at the cross section level, and we’ll keep terms only up to
$\mathcal{O}(\alpha_{s})$.

Here in Fig. 3 the first diagram after the equals sign corresponds to the
leading order hadronic cross section computed in the previous subsection. The
two $\mathcal{O}(\alpha_{s})$ corrections correspond to our virtual correction
$\sigma_{9}$ and the real correction $\sigma_{1\times 1}$. So, let’s now
compute the hadronic cross section contribution from the virtual correction
$\sigma_{9}$ (combining Eq. 46 with Eq. 64).

$\displaystyle\frac{\differential\sigma_{9}^{\gamma^{*}A\to
h_{1}h_{2}X}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=-\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{1}/q}(z_{h_{1}})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})$
$\displaystyle\times\alpha_{s}C_{F}\int_{0}^{z_{1}}\frac{\differential
z}{z}\left[\frac{z_{1}^{2}+(z_{1}-z)^{2}}{z_{1}^{2}}\right]\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\left(\mathbf{k}-\frac{z}{z_{1}}\mathbf{p}\right)^{2}}\delta(1-z_{1}-z_{2}).$
(69)

Here we relaxed the large $N_{c}$ approximation taking one factor of $N_{c}\to
2C_{F}$, and have also used $g^{2}\to 4\pi\alpha_{s}$. Let’s define a new
variable for the $z$ integration

$\displaystyle\xi=\frac{z_{1}-z}{z_{1}}.$ (70)

This defines $\xi$ to be the fraction of longitudinal momentum that the quark
carries after radiating the gluon. In terms of this, we get

$\displaystyle\frac{\differential\sigma_{9}^{\gamma^{*}A\to
h_{1}h_{2}X}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=-\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})D^{0}_{h_{1}/q}(z_{h_{1}})$
$\displaystyle\times\alpha_{s}\,C_{F}\,\int^{1}_{0}\differential\xi\,\frac{(1+\xi^{2})}{(1-\xi)}\,\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\left(\mathbf{k}-(1-\xi)\mathbf{p}\right)^{2}}\delta(1-z_{1}-z_{2}).$
(71)

This can now be added to the leading order hadronic cross section. Note that
we actually need to add $2\sigma_{9}$ because it appears twice in the formula
since it’s a cross term (Eq. 28).

####  IV.4.3 The real correction

Looking at the relevant diagrams in Fig. 3, the last one we need for the
$\mathcal{O}(\alpha_{s})$ corrections to the quark fragmentation function is
the real correction $\sigma_{1\times 1}$.

$\displaystyle\frac{\differential\sigma^{L}_{1\times
1}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential
y_{2}}=\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{2}^{3}(1-z_{2})^{2}(z_{1}^{2}+(1-z_{2})^{2})}{(2\pi)^{8}z_{1}}\int\differential^{8}x\,K_{0}(|\mathbf{x}_{12}|Q_{2})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{2})$
$\displaystyle\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int\frac{\differential
z}{z}\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{z}{z_{1}}\mathbf{p}\right)^{2}}\delta(1-z_{1}-z_{2}-z).$
(72)

We have rewritten the integral over the radiation kernel (in Eq. 32) in
momentum space using

$\displaystyle
e^{i\frac{z}{z_{1}}\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}\int\differential^{2}\mathbf{x}_{3}\Delta^{(3)}_{11^{\prime}}=\int\differential^{2}\mathbf{k}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{z}{z_{1}}\mathbf{p}\right)^{2}}.$
(73)

Looking at Eq. 64 we see that this correction contributes the following
expression to the hadronic cross section.

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to h_{1}h_{2}X}_{1\times
1}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=$ $\displaystyle\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{2}^{3}(1-z_{2})^{2}(z_{1}^{2}+(1-z_{2})^{2})}{(2\pi)^{8}z_{1}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}x\,K_{0}(|\mathbf{x}_{12}|Q_{2})$
$\displaystyle
K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{2})\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}$
$\displaystyle\int\frac{\differential
z}{z}\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{z}{z_{1}}\mathbf{p}\right)^{2}}\delta(1-z_{1}-z_{2}-z)D^{0}_{h_{1}/q}(z_{h_{1}})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}}).$
(74)

Now we’ll write this using $H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})$ defined
in Eq. 67 and again relax the large $N_{c}$ approximation taking one factor of
$N_{c}$ to $2C_{F}$, and also write $g^{2}=4\pi\alpha_{s}$.

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to h_{1}h_{2}X}_{1\times
1}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{8e^{2}Q^{2}N_{c}(z_{1}z_{2})^{3}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{1}/q}(z_{h_{1}})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})$
$\displaystyle\times\alpha_{s}C_{F}\int\frac{\differential
z}{z}\frac{(z_{1}+z)^{2}(z_{1}^{2}+(z_{1}+z)^{2})}{z_{1}^{4}}\delta(1-z_{1}-z_{2}-z)\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{z}{z_{1}}\mathbf{p}\right)^{2}}.$
(75)

We factored out $(z_{1}z_{2})^{3}$ to match the leading order expression and
used the delta function to write $1-z_{2}$ as $z_{1}+z$. We then perform a
change of variables in the $z$ integral by defining

$\displaystyle\xi=\frac{z_{1}}{z_{1}+z}.$ (76)

This $\xi_{1}$ is then the fraction of the outgoing quark’s momentum relative
to the parent quark that emitted the gluon (i.e.: the same definition as we
used for the virtual correction).

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to h_{1}h_{2}X}_{1\times
1}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{8e^{2}Q^{2}N_{c}(z_{1}z_{2})^{3}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})$
$\displaystyle\times\alpha_{s}C_{F}\int\frac{\differential\xi}{\xi^{5}}\frac{(1+\xi^{2})}{(1-\xi)}\delta(1-z_{2}-z_{1}/\xi)D^{0}_{h_{1}/q}(z_{h_{1}})\,\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{(1-\xi)}{\xi}\mathbf{p}\right)^{2}}.$
(77)

Now, $z_{1}$ is no longer an external variable, as remarked earlier it can be
written in terms of $z_{h_{1}}$ using Eq. 66. So one can then write $z_{1}$ in
terms of $z_{h_{1}}$,

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to h_{1}h_{2}X}_{1\times
1}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{8e^{2}Q^{2}N_{c}(p_{h}^{+})^{3}z_{2}^{3}}{(2\pi)^{7}(l^{+})^{3}z_{h_{1}}^{5}z_{h_{2}}^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})$
$\displaystyle\times\alpha_{s}C_{F}\int\frac{\differential\xi}{\xi^{5}}\frac{(1+\xi^{2})}{(1-\xi)}\delta\left(1-z_{2}-\frac{p_{h}^{+}}{z_{h_{1}}l^{+}\xi}\right)D^{0}_{h_{1}/q}(z_{h_{1}})\,\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{(1-\xi)}{\xi}\mathbf{p}\right)^{2}}.$
(78)

Now that the $z_{h_{1}}$ dependence is all explicit we make a substitution in
the $z_{h_{1}}$ integral, defining the new variable

$\displaystyle z_{h_{1}}^{\prime}=z_{h_{1}}\xi,\,\,\,\,\,\,\differential
z_{h_{1}}^{\prime}=\xi\differential z_{h_{1}}.$ (79)

In order to complete this substitution, we need to take
$z_{h_{1}}\to\frac{z_{h_{1}}^{\prime}}{\xi}$ everywhere. This also means that
the bounds on the $z_{h_{1}}^{\prime}$ integral go from 0 to $\xi$. We’ll
write this using a step function $\theta(\xi-z_{h_{1}}^{\prime})$ so we can
keep the explicit bounds from $0$ to $1$.

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to h_{1}h_{2}X}_{1\times
1}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=\int_{0}^{1}\differential
z_{h_{1}}^{\prime}\int_{0}^{1}\differential
z_{h_{2}}\frac{8e^{2}Q^{2}N_{c}(p_{h}^{+})^{3}z_{2}^{3}}{(2\pi)^{7}(l^{+})^{3}z_{h_{1}}^{\prime
5}z_{h_{2}}^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})$
$\displaystyle\times\alpha_{s}C_{F}\int\frac{\differential\xi}{\xi}\frac{(1+\xi^{2})}{(1-\xi)}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}^{\prime}}{\xi}\right)\theta(\xi-
z_{h_{1}}^{\prime})\delta\left(1-z_{2}-\frac{p_{h}^{+}}{z_{h_{1}}^{\prime}l^{+}}\right)\,\,\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{(1-\xi)}{\xi}\mathbf{p}\right)^{2}}.$
(80)

We can now write things back in terms of $z_{1}$ to get rid of $p_{h}^{+}$ and
$l^{+}$ to match the leading order and virtual correction. We can also use the
step function to set the bounds on the $\xi$ integral. Finally, we remove the
primes from $z_{h_{1}}^{\prime}$ since it’s an integration variable.

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to h_{1}h_{2}X}_{1\times
1}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{8e^{2}Q^{2}N_{c}(z_{1}z_{2})^{3}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})$
$\displaystyle\times\delta\left(1-z_{1}-z_{2}\right)\alpha_{s}C_{F}\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}\frac{(1+\xi^{2})}{(1-\xi)}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)\,\,\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{(1-\xi)}{\xi}\mathbf{p}\right)^{2}}.$
(81)

####  IV.4.4 Evolution of the fragmentation function

Now we would like to add up the three terms in Fig. 3. These terms are Eq. 68,
71, and 81. Note that we also need to double Eq. 71 since it’s a cross term.
Adding up these three gives the following,

$\displaystyle\int_{0}^{1}\differential z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})\delta(1-z_{1}-z_{2})$
$\displaystyle\times\Bigg{[}D^{0}_{h_{1}/q}(z_{h_{1}})+2\alpha_{s}C_{F}\Bigg{\\{}\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}\frac{(1+\xi^{2})}{(1-\xi)}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\frac{\mathbf{k}}{\xi}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-(1-\xi)\mathbf{p}\right)^{2}}$
$\displaystyle-D^{0}_{h_{1}/q}(z_{h_{1}})\int_{0}^{1}\differential\xi\frac{(1+\xi^{2})}{(1-\xi)}\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{(\mathbf{k}-(1-\xi)\mathbf{p})^{2}}\Bigg{\\}}\Bigg{]}.$
(82)

Now we turn our attention to the two transverse momentum integrals. The one
with the exponential comes from the real correction $\sigma_{1\times 1}$, and
contains both a soft divergence, $\mathbf{k}\to 0$ and $z\to 0$ (equivalently
$\xi\to 1$) at the same time, and a collinear divergence as
$\mathbf{k}\to(1-\xi)\mathbf{p}$. The integral without an exponential (from
the virtual correction $\sigma_{9}$) contains a soft divergence, a collinear
divergence, and a UV divergence as $|\mathbf{k}|\to\infty$. We showed in
section IV.2 that the soft divergences cancel between these two terms. In
section IV.1 we also showed that the UV divergence in the virtual correction
is canceled with other terms in the cross section. Therefore, the only
divergence left to consider here is the collinear divergence in both
integrals.

To see how regulating these divergences leads to renormalization of the quark-
hadron fragmentation function it is perhaps easiest to demonstrate this using
an explicit cutoff where the lower limit on $|\mathbf{k}|$ is $\Lambda$ and
the upper limit is $\mu$.

$\displaystyle\int_{0}^{1}\differential z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})\delta(1-z_{1}-z_{2})$
$\displaystyle\times\Bigg{[}D^{0}_{h_{1}/q}(z_{h_{1}})+\frac{\alpha_{s}C_{F}}{2\pi}\Bigg{\\{}\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}\frac{(1+\xi^{2})}{(1-\xi)}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)-\int_{0}^{1}\differential\xi\frac{(1+\xi^{2})}{(1-\xi)}D^{0}_{h_{1}/q}(z_{h_{1}})\Bigg{\\}}\log\left(\frac{\mu^{2}}{\Lambda^{2}}\right)\Bigg{]}.$
(83)

We would like to combine the terms inside the curly brackets, to do so we’ll
define the quark-quark splitting function $P_{qq}(\xi)$:

$\displaystyle P_{qq}(\xi)\equiv
C_{F}\left[\frac{(1+\xi^{2})}{(1-\xi)_{+}}+\frac{3}{2}\delta(1-\xi)\right],$
(84)

where the $+$ distribution is defined inside an integral via

$\displaystyle\int_{z}^{1}\differential\xi\frac{f(\xi)}{(1-\xi)_{+}}\equiv\int_{z}^{1}\frac{\differential\xi}{(1-\xi)}(f(\xi)-f(1))+f(1)\log(1-z).$
(85)

This allows us to write Eq. 83 as

$\displaystyle\int_{0}^{1}\differential z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})\delta(1-z_{1}-z_{2})$
$\displaystyle\times\Bigg{[}D^{0}_{h_{1}/q}(z_{h_{1}})+\frac{\alpha_{s}}{2\pi}\Bigg{\\{}\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}P_{qq}(\xi)D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)\Bigg{\\}}\log\left(\frac{\mu^{2}}{\Lambda^{2}}\right)\Bigg{]}.$
(86)

Here we have used the following result:

$\displaystyle\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}P_{qq}(\xi)D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)=\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}\frac{(1+\xi^{2})}{(1-\xi)}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)-\int_{0}^{1}\differential\xi\frac{(1+\xi^{2})}{(1-\xi)}D^{0}_{h_{1}/q}(z_{h_{1}}).$
(87)

To prove this, one first expands the $P_{qq}$ function using Eq. 84, then
further expands the term with the plus distribution according to Eq. 85. One
of the terms is then exactly the first term on the right side of Eq. 87. In
the remaining term with a $\xi$ integral one separates into two terms using
$\int_{z_{h_{1}}}^{1}=\int_{0}^{1}-\int_{0}^{z_{h_{1}}}$. One can then use the
fact that

$\displaystyle\frac{2}{1-\xi}=\frac{(1+\xi^{2})}{(1-\xi)}+1+\xi.$ (88)

The first term here paired with the $\int_{0}^{1}$ integrals then gives
exactly the second term on the right side of Eq. 87. In all the other terms,
the integral over $\xi$ can be performed and these terms all cancel in the
full expression.

Next, we can also combine with the leading order term (first term in the
square brackets) by including a $\xi$ integral that evaluates to $1$:

$\displaystyle\int_{0}^{1}\differential z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})\delta(1-z_{1}-z_{2})$
$\displaystyle\times\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)\Bigg{[}\delta(1-\xi)+\frac{\alpha_{s}}{2\pi}P_{qq}(\xi)\log\left(\frac{\mu^{2}}{\Lambda^{2}}\right)\Bigg{]}.$
(89)

Then, we define the DGLAP evolved quark-hadron fragmentation function
$D_{h_{1}/q}(z_{h_{1}},\mu^{2})$ as

$\displaystyle
D_{h_{1}/q}(z_{h_{1}},\mu^{2})=\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)\Bigg{[}\delta(1-\xi)+\frac{\alpha_{s}}{2\pi}P_{qq}(\xi)\log\left(\frac{\mu^{2}}{\Lambda^{2}}\right)\Bigg{]},$
(90)

in terms of which our expression becomes

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to
h_{1}h_{2}X}_{LO+9+1\times
1}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=$ $\displaystyle\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})$
$\displaystyle
D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})D_{h_{1}/q}(z_{h_{1}},\mu^{2})\delta(1-z_{1}-z_{2}).$
(91)

It must be noted that here we used an explicit cutoff $\mu$ to regulate these
collinear divergences. However the most common way to handle these divergences
is via dimensional regularization (dim reg). This is done in the next
subsection.

####  IV.4.5 Evolution of the Fragmentation Function in Dimensional
Regularization

As most parametrizations of the fragmentation functions are defined in the
$\overline{\text{MS}}$ scheme it is important to relate our result with that
given in the $\overline{\text{MS}}$ scheme and using dimensional
regularization as this would affect the finite pieces. The real integral can
be regulated without complication via dimensional regularization.

$\displaystyle\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{(1-\xi)}{\xi}\mathbf{p}\right)^{2}}\to\mu^{2-d}\int\frac{\differential^{d}\mathbf{k}}{(2\pi)^{d}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\left(\mathbf{k}-\frac{(1-\xi)}{\xi}\mathbf{p}\right)^{2}}$
$\displaystyle=\mu^{2-d}e^{i\frac{(1-\xi)}{\xi}\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}\int\frac{\differential^{d}\mathbf{k}}{(2\pi)^{d}}\frac{e^{i\mathbf{k}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{\mathbf{k}^{2}}=\frac{\mu^{2-d}e^{i\frac{(1-\xi)}{\xi}\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{(2\pi)^{d}}\int\differential
k\,k^{d-3}\int\differential\Omega_{d}e^{ik|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|\cos\theta}.$
(92)

The limit $d\to 2$ is safe in the angular integral.

$\displaystyle\frac{\mu^{2-d}e^{i\frac{(1-\xi)}{\xi}\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{(2\pi)^{d-1}}\int\differential
k\,k^{d-3}J_{0}\left(k|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|\right)$
$\displaystyle=\frac{\mu^{2-d}e^{i\frac{(1-\xi)}{\xi}\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{(2\pi)^{d-1}}2^{d-3}|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|^{2-d}\frac{\Gamma[d/2-1]}{\Gamma[2-d/2]},\,\,\,\,\,\,2<d<7/2,$
$\displaystyle=\frac{e^{i\frac{(1-\xi)}{\xi}\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}}{2\pi}\left[\frac{1}{\epsilon}-\log\left(\pi
e^{\gamma_{E}}\mu|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|\right)\right]+\mathcal{O}(\epsilon),\,\,\,\,\,\epsilon=d-2>0.$
(93)

This completes our result for regulating the real integral via dimensional
regularization. Next we look at the virtual integral which, after a shift,
becomes

$\displaystyle\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\mathbf{k}^{2}}.$
(94)

This divergent integral is zero in dimensional regularization Leibbrandt
(1975, 1987). This can be understood as the result of the collinear (here
looks soft due to the shift) divergence that exactly cancels the ultraviolet
divergence in dimensional regularization. One can see this by separating the
integration into two terms with a cutoff and doing each separatelyChirilli et
al. (2012a, b); Caucal et al. (2021). Now, it’s important to note that in our
case we have already used the UV divergence to cancel other UV divergences in
the virtual corrections (see section IV.1). Thus we need to be careful with
this integral. Following Caucal et al. (2021) let’s separate into two regions,

$\displaystyle\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\mathbf{k}^{2}}\to\mu^{2-d}\int\frac{\differential^{d}\mathbf{k}}{(2\pi)^{d}}\frac{1}{\mathbf{k}^{2}}=\frac{\mu^{2-d}2\pi^{d/2}}{(2\pi)^{d}\Gamma[d/2]}\left[\int_{0}^{\Lambda}\differential
k\,k^{d-3}+\int_{\Lambda}^{\infty}\differential k\,k^{d-3}\right].$ (95)

The first term contains the infrared (collinear) divergence, and the second
term contains the UV divergence. These integrals can both be done, which
yields

$\displaystyle\frac{\mu^{2-d}2\pi^{d/2}}{(2\pi)^{d}\Gamma[d/2]}\left[\frac{\Lambda^{-\epsilon_{UV}}}{\epsilon_{UV}}-\frac{\Lambda^{-\epsilon_{IR}}}{\epsilon_{IR}}\right].$
(96)

Here $\epsilon=2-d$ with $\epsilon_{UV}>0$ and $\epsilon_{IR}<0$. If one
formally sets $\epsilon_{UV}=\epsilon_{IR}$, then one finds that the
expression is zero. For our purposes, let’s write both expressions separately
and expand each for small $\epsilon$:

$\displaystyle\left[\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\mathbf{k}^{2}}\right]_{IR}=\frac{1}{2\pi}\left[-\frac{1}{\epsilon_{IR}}-\frac{1}{2}\log\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{\Lambda^{2}}\right)\right]+\mathcal{O}(\epsilon).$
(97)

$\displaystyle\left[\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\mathbf{k}^{2}}\right]_{UV}=\frac{1}{2\pi}\left[\frac{1}{\epsilon_{UV}}+\frac{1}{2}\log\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{\Lambda^{2}}\right)\right]+\mathcal{O}(\epsilon).$
(98)

Notice that the finite terms cancel if one adds both of these together to form
the full integral (which gives Eq. 5.45 in Caucal et al. (2021)). We would
like for the finite term to look similar to the finite term in the result for
the real integral (Eq. 93), so let’s add a finite term to the IR piece and
also subtract it from the UV piece:

$\displaystyle\left[\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\mathbf{k}^{2}}\right]_{IR}=\frac{1}{2\pi}\left[-\frac{1}{\epsilon_{IR}}-\frac{1}{2}\log\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{\Lambda^{2}}\right)+\log\left(\frac{2e^{-\frac{3}{2}\gamma_{E}}}{\pi^{1/2}\Lambda|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|}\right)\right]+\mathcal{O}(\epsilon).$
(99)

$\displaystyle\left[\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\mathbf{k}^{2}}\right]_{UV}=\frac{1}{2\pi}\left[\frac{1}{\epsilon_{UV}}+\frac{1}{2}\log\left(\frac{4\pi
e^{-\gamma_{E}}\mu^{2}}{\Lambda^{2}}\right)-\log\left(\frac{2e^{-\frac{3}{2}\gamma_{E}}}{\pi^{1/2}\Lambda|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|}\right)\right]+\mathcal{O}(\epsilon).$
(100)

We have added and subtracted the same quantity from the full expression.
Combining the logarithms in both terms gives

$\displaystyle\left[\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\mathbf{k}^{2}}\right]_{IR}=\frac{1}{2\pi}\left[-\frac{1}{\epsilon_{IR}}-\log\left(e^{\gamma_{E}}\pi\mu|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|\right)\right]+\mathcal{O}(\epsilon).$
(101)

$\displaystyle\left[\int\frac{\differential^{2}\mathbf{k}}{(2\pi)^{2}}\frac{1}{\mathbf{k}^{2}}\right]_{UV}=\frac{1}{2\pi}\left[\frac{1}{\epsilon_{UV}}+\log\left(e^{\gamma_{E}}\pi\mu|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|\right)\right]+\mathcal{O}(\epsilon).$
(102)

Then, since $\epsilon_{IR}$ is negative, we can label it as $-\epsilon$ with
$\epsilon>0$ to match the real case. This addition and subtraction of a finite
piece can also be thought of as choosing a particular value for the cutoff
$\Lambda$. So, Eq. 101 will contribute to the fragmentation function, while
Eq. 102 will contribute to the rest of the NLO corrections. The divergence in
Eq. 101 is collinear and will be absorbed into the bare quark hadron
fragmentation function to make it scale dependent. The divergence in Eq. 102
is canceled with the other UV divergences in virtual diagrams as before.
Finally we note that the real integral (Eq. 93) has an overall exponential
which we set to $1$. This can be motivated by noticing that it does not affect
the nature of the collinear singularity and that splitting function favors
$\xi\rightarrow 1$ (soft radiation).

Using the results of our transverse integrals in the expression for the
hadronic cross section we get

$\displaystyle\int_{0}^{1}\differential z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})\delta(1-z_{1}-z_{2})$
$\displaystyle\times\Bigg{[}D^{0}_{h_{1}/q}(z_{h_{1}})+\frac{\alpha_{s}C_{F}}{\pi}\Bigg{\\{}\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}\frac{(1+\xi^{2})}{(1-\xi)}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)$
$\displaystyle-D^{0}_{h_{1}/q}(z_{h_{1}})\int_{0}^{1}\differential\xi\frac{(1+\xi^{2})}{(1-\xi)}\Bigg{\\}}\left(\frac{1}{\epsilon}-\log\left(\pi
e^{\gamma_{E}}\mu|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|\right)\right)\Bigg{]}.$
(103)

Next we again introduce the quark-quark splitting function as in the previous
subsection (Eq. 84). In terms of this, the full expression becomes

$\displaystyle\int_{0}^{1}\differential z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})\delta(1-z_{1}-z_{2})$
$\displaystyle\times\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)\Bigg{[}\delta(1-\xi)+\frac{\alpha_{s}}{\pi}P_{qq}(\xi)\left(\frac{1}{\epsilon}-\log\left(\pi
e^{\gamma_{E}}\mu|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|\right)\right)\Bigg{]}.$
(104)

Then, we define the DGLAP evolved quark-hadron fragmentation function
$D_{h_{1}/q}(z_{h_{1}},\mu^{2})$ in dimensional regularization as

$\displaystyle
D_{h_{1}/q}(z_{h_{1}},\mu^{2})=\int_{z_{h_{1}}}^{1}\frac{\differential\xi}{\xi}D^{0}_{h_{1}/q}\left(\frac{z_{h_{1}}}{\xi}\right)\Bigg{[}\delta(1-\xi)+\frac{\alpha_{s}}{\pi}P_{qq}(\xi)\left(\frac{1}{\epsilon}-\log\left(\pi
e^{\gamma_{E}}\mu|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|\right)\right)\Bigg{]},$
(105)

in terms of which our expression becomes

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to
h_{1}h_{2}X}_{LO+9+1\times
1}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=$ $\displaystyle\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{2})$
$\displaystyle
D^{0}_{h_{2}/\bar{q}}(z_{h_{2}})D_{h_{1}/q}(z_{h_{1}},\mu^{2})\delta(1-z_{1}-z_{2}),$
(106)

where we recall that

$\displaystyle
z_{1}=\frac{p_{h}^{+}}{z_{h_{1}}l^{+}},\,\,\,\,\,\,z_{2}=\frac{q_{h}^{+}}{z_{h_{2}}l^{+}}.$
(107)

Finally, if one repeats the exact same procedure for the anti-quark line
(i.e.: adding up $LO+10+2\times 2$), one gets

$\displaystyle\frac{\differential\sigma^{\gamma^{*}A\to
h_{1}h_{2}X}_{LO+10+2\times
2}}{\differential^{2}\mathbf{p}_{h}\,\differential^{2}\mathbf{q}_{h}\,\differential
y_{1}\,\differential y_{2}}=$ $\displaystyle\int_{0}^{1}\differential
z_{h_{1}}\int_{0}^{1}\differential
z_{h_{2}}\frac{4e^{2}Q^{2}(z_{1}z_{2})^{3}N_{c}}{(2\pi)^{7}(z_{h_{1}}z_{h_{2}})^{2}}\int\differential^{8}\mathbf{x}\,H(\mathbf{p},\mathbf{q},\mathbf{x},z_{1})$
$\displaystyle
D^{0}_{h_{1}/q}(z_{h_{1}})D_{h_{2}/\bar{q}}(z_{h_{2}},\mu^{2})\delta(1-z_{1}-z_{2}).$
(108)

This shows that absorption of collinear divergences in
$\sigma_{9},\sigma_{1\times 1},\sigma_{10}$, and $\sigma_{2\times 2}$ into
bare quark and anti quark-hadron fragmentation functions lead to evolution,
i.e. scale dependence of these fragmentation functions.

Therefore our final result for the regulated dihadron production cross section
can be written as the sum of several terms (Eq. 109). The first term contains
the $z$ integration region below $z_{f}$ where the leading order cross section
is evolved with the BK/JIMWLK evolution equations. The second term includes
the integration region $z>z_{f}$ where the leading order cross section
multiplies the DGLAP evolved fragmentation functions for both quark and anti-
quark. Finally the last term constitutes all the remaining contributions to
the NLO cross section which is finite.

$\displaystyle\differential\sigma^{\gamma^{*}A\to
h_{1}h_{2}X}=\differential\sigma_{LO}\otimes\text{JIMWLK}+\differential\sigma_{LO}\otimes
D_{h_{1}/q}(z_{h_{1}},\mu^{2})D_{h_{2}/\bar{q}}(z_{h_{2}},\mu^{2})+\differential\sigma_{NLO}^{\text{finite}}.$
(109)

Here we imply the presence of the bare fragmentation functions in the first
and last terms.

In summary, we have calculated the one-loop corrections to inclusive quark
anti-quark production in DIS at small $x$ for longitudinal photons. We have
shown that all divergences that appear at the one-loop level are either
canceled or absorbed into JIMWLK evolution of dipoles and quadrupoles, and
into DGLAP evolution of parton-hadron fragmentation functions. These results
are well suited for further phenomenological studies of angular correlations
of the dihadrons produced in DIS at small $x$ Bergabo and Jalilian-Marian
(2022a). A particularly interesting limit is the so-called back to back limit
where a suppression of the away side peak is observed experimentally. Here one
will be sensitive to Sudakov radiation which will be significant Mueller et
al. (2013); Taels et al. (2022); Zheng et al. (2014) and must be included in a
full phenomenological study. One can also use our results here and integrate
over the phase space of one of the outgoing particles, thus obtaining the
single inclusive hadron production cross section in DIS at small $x$ at Next-
to-Leading Order. This is under investigation and will be reported elsewhere
Bergabo and Jalilian-Marian (2022b).

##### Acknowledgements:

We gratefully acknowledge support from the DOE Office of Nuclear Physics
through Grant No. DE-SC0002307 and by PSC-CUNY through grant No. 63158-0051.
We would like to thank T. Altinoluk, G. Beuf, R. Boussarie, P. Caucal, L.
Dixon, Y. Kovchegov, C. Marquet, F. Salazar, M. Tevio, R. Venugopalan, W.
Vogelsang and B. Xiao for helpful discussions.

##  Appendix A Cancellation of UV Divergences

UV divergences appear in the corrections $\differential\sigma_{5}$,
$\differential\sigma_{6}$, $\differential\sigma_{9}$,
$\differential\sigma_{10}$, $\differential\sigma_{11}$,
$\differential\sigma_{12}$, $\differential\sigma_{14(1)}$, and
$\differential\sigma_{14(2)}$. In this appendix, we will show that these
divergences all cancel between these corrections. First, we note that the UV
divergence in $\differential\sigma_{5}$ appears in the limit
$\mathbf{x}_{3}\to\mathbf{x}_{1}$. If we take this limit in our result (Eq.
42) everywhere except where it causes the divergence, we get the following.

$\displaystyle\frac{\differential\sigma_{5,UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\left[\frac{z_{1}^{2}+(z_{1}-z)^{2}}{z_{1}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{\mathbf{x}_{3}\to\mathbf{x}_{1}}\frac{\differential^{2}\mathbf{x}_{3}}{\mathbf{x}_{31}^{2}}.$
(110)

Here we have used the notation $\int_{\mathbf{x}_{3}\to\mathbf{x}_{1}}$ to
signify that the limit has been taken inside the $\mathbf{x}_{3}$ integral.
The divergence as $\mathbf{x}_{3}\to\mathbf{x}_{1}$ can equivalently be
written as a divergent momentum integral as the transverse momentum goes to
infinity:
$\int_{\mathbf{k}\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}$
. Making this substitution in $\differential\sigma_{5}$, and similarly for the
UV divergence as $\mathbf{x}_{3}\to\mathbf{x}_{2}$ in
$\differential\sigma_{6}$, and writing the UV divergent parts of all the other
UV divergent diagrams, we have

$\displaystyle\frac{\differential\sigma_{5,UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\left[\frac{z_{1}^{2}+(z_{1}-z)^{2}}{z_{1}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(111)
$\displaystyle\frac{\differential\sigma_{6,UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\left[\frac{z_{2}^{2}+(z_{2}-z)^{2}}{z_{2}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(112)
$\displaystyle\frac{\differential\sigma_{9,UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\left[\frac{z_{1}^{2}+(z_{1}-z)^{2}}{z_{1}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(113)
$\displaystyle\frac{\differential\sigma_{10,UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\left[\frac{z_{2}^{2}+(z_{2}-z)^{2}}{z_{2}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(114)
$\displaystyle\frac{\differential\sigma_{11,UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\left[\frac{z_{1}^{2}+(z_{1}-z)^{2}}{z_{1}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(115)
$\displaystyle\frac{\differential\sigma_{12,UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\left[\frac{z_{2}^{2}+(z_{2}-z)^{2}}{z_{2}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(116)
$\displaystyle\frac{\differential\sigma_{14(1),UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\left[1+\frac{(z_{1}-z)(z_{2}+z)}{z_{1}z_{2}}-z\frac{z_{1}(z_{1}-z)+z_{2}(z_{2}+z)}{z_{1}z_{2}}\right]$
$\displaystyle\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(117)
$\displaystyle\frac{\differential\sigma_{14(2),UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\left[1+\frac{(z_{1}+z)(z_{2}-z)}{z_{1}z_{2}}-z\frac{z_{1}(z_{1}+z)+z_{2}(z_{2}-z)}{z_{1}z_{2}}\right]$
$\displaystyle\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(118)

One can immediately verify from the above expressions that the UV divergent
part of $\sigma_{5}$ is canceled by the UV divergent part of $\sigma_{11}$.
Similarly $\sigma_{6,UV}$ cancels with $\sigma_{12,UV}$. The remaining terms
can be written (after some simplification of the $z$ factors) as follows:

$\displaystyle\frac{\differential\sigma_{9+14(1),UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\differential
z\,\left[\frac{2(z_{1}-z)}{z_{1}}-\frac{z}{z_{1}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(119)

$\displaystyle\frac{\differential\sigma_{10+14(2),UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\differential
z\,\left[\frac{2(z_{2}-z)}{z_{2}}-\frac{z}{z_{2}^{2}}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(120)

The integrals over $z$ in these two expressions can now be done, we find the
following:

$\displaystyle\frac{\differential\sigma_{9+14(1),UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\left[z_{1}-\frac{1}{2}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(121)

$\displaystyle\frac{\differential\sigma_{10+14(2),UV}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\left[z_{2}-\frac{1}{2}\right]\int\differential^{8}\mathbf{x}[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]$
$\displaystyle
K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int_{|\mathbf{k}|\to\infty}\frac{\differential^{2}\mathbf{k}}{\mathbf{k}^{2}}.$
(122)

Now, if we add up $\sigma_{9},\sigma_{10},\sigma_{14(1)}$ and $\sigma_{14(2)}$
we see that the result is proportional to $z_{1}+z_{2}-1$ which is zero by
virtue of the delta function (not shown here). So, what we have found is
precisely Eq. 56. Therefore, all UV divergences are canceled when adding up
all the relevant pieces of the differential cross section.

##  Appendix B Coordinate Space Results at Soft $z$

Here we have taken the limit $z\to 0$ in all our expressions everywhere except
where it causes a divergence. The result for each expression has also been
written in coordinate space where we have recalled the definition of the
radiation kernel (Eq. 57). The expressions in this appendix are useful both
for showing that soft divergences are canceled and that rapidity divergences
can be combined to show evolution of the dipoles and quadrupoles according to
the BK and JIMWLK equations.

$\displaystyle\frac{\differential\sigma^{L}_{1\times
1}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}x\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{31^{\prime}}^{2}}-\frac{\mathbf{x}_{11^{\prime}}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{31^{\prime}}^{2}}\right]$
$\displaystyle\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(123) $\displaystyle\frac{\differential\sigma^{L}_{2\times
2}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{32}^{2}}+\frac{1}{\mathbf{x}_{32^{\prime}}^{2}}-\frac{\mathbf{x}_{22^{\prime}}^{2}}{\mathbf{x}_{32}^{2}\mathbf{x}_{32^{\prime}}^{2}}\right]$
$\displaystyle[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}\delta(1-z_{1}-z_{2}).$
(124) $\displaystyle\frac{\differential\sigma^{L}_{1\times
2}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32^{\prime}}^{2}}-\frac{\mathbf{x}_{12^{\prime}}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32^{\prime}}^{2}}\right]$
$\displaystyle[S_{12}S_{1^{\prime}2^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(125) $\displaystyle\frac{\differential\sigma^{L}_{3\times
3}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{31^{\prime}}^{2}}-\frac{\mathbf{x}_{11^{\prime}}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{31^{\prime}}^{2}}\right]$
$\displaystyle[S_{11^{\prime}}S_{22^{\prime}}-S_{13}S_{23}-S_{1^{\prime}3}S_{2^{\prime}3}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(126) $\displaystyle\frac{\differential\sigma^{L}_{4\times
4}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{32}^{2}}+\frac{1}{\mathbf{x}_{32^{\prime}}^{2}}-\frac{\mathbf{x}_{22^{\prime}}^{2}}{\mathbf{x}_{32}^{2}\mathbf{x}_{32^{\prime}}^{2}}\right]$
$\displaystyle[S_{11^{\prime}}S_{22^{\prime}}-S_{13}S_{23}-S_{1^{\prime}3}S_{2^{\prime}3}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(127) $\displaystyle\frac{\differential\sigma^{L}_{3\times
4}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32^{\prime}}^{2}}-\frac{\mathbf{x}_{12^{\prime}}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32^{\prime}}^{2}}\right]$
$\displaystyle[S_{11^{\prime}}S_{22^{\prime}}-S_{13}S_{23}-S_{1^{\prime}3}S_{2^{\prime}3}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(128) $\displaystyle\frac{\differential\sigma^{L}_{1\times
3}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}x\,K_{0}(|\mathbf{x}_{12}|Q_{2})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{31^{\prime}}^{2}}-\frac{\mathbf{x}_{11^{\prime}}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{31^{\prime}}^{2}}\right]$
$\displaystyle[S_{122^{\prime}3}S_{1^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(129) $\displaystyle\frac{\differential\sigma^{L}_{1\times
4}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}xK_{0}(|\mathbf{x}_{12}|Q_{2})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32^{\prime}}^{2}}-\frac{\mathbf{x}_{12^{\prime}}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32^{\prime}}^{2}}\right]$
$\displaystyle[S_{122^{\prime}3}S_{1^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(130) $\displaystyle\frac{\differential\sigma^{L}_{2\times
3}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}xK_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{32}^{2}}+\frac{1}{\mathbf{x}_{31^{\prime}}^{2}}-\frac{\mathbf{x}_{21^{\prime}}^{2}}{\mathbf{x}_{32}^{2}\mathbf{x}_{31^{\prime}}^{2}}\right]$
$\displaystyle[S_{1231^{\prime}}S_{2^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(131) $\displaystyle\frac{\differential\sigma^{L}_{2\times
4}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int\frac{\differential
z}{z}\int\differential^{10}xK_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{32}^{2}}+\frac{1}{\mathbf{x}_{32^{\prime}}^{2}}-\frac{\mathbf{x}_{22^{\prime}}^{2}}{\mathbf{x}_{32}^{2}\mathbf{x}_{32^{\prime}}^{2}}\right]$
$\displaystyle[S_{1231^{\prime}}S_{2^{\prime}3}-S_{1^{\prime}3}S_{2^{\prime}3}-S_{12}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(132)

$\displaystyle\frac{\differential\sigma^{L}_{5}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{4e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\frac{1}{\mathbf{x}_{31}^{2}}$
$\displaystyle[S_{322^{\prime}1^{\prime}}S_{13}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(133)
$\displaystyle\frac{\differential\sigma^{L}_{6}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{4e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\frac{1}{\mathbf{x}_{32}^{2}}$
$\displaystyle[S_{132^{\prime}1^{\prime}}S_{23}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(134)
$\displaystyle\frac{\differential\sigma^{L}_{7}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32}^{2}}-\frac{\mathbf{x}_{12}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32}^{2}}\right]$
$\displaystyle[S_{322^{\prime}1^{\prime}}S_{13}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(135)
$\displaystyle\frac{\differential\sigma^{L}_{8}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32}^{2}}-\frac{\mathbf{x}_{12}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32}^{2}}\right]$
$\displaystyle[S_{132^{\prime}1^{\prime}}S_{23}-S_{13}S_{23}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(136)

$\displaystyle\frac{\differential\sigma_{9}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\frac{1}{\mathbf{x}_{3}^{2}}$
$\displaystyle\big{[}S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\big{]}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(137)
$\displaystyle\frac{\differential\sigma_{10}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-2e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\frac{1}{\mathbf{x}_{3}^{2}}$
$\displaystyle\big{[}S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\big{]}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(138)
$\displaystyle\frac{\differential\sigma_{11}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-4e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\frac{1}{\mathbf{x}_{3}^{2}}$
$\displaystyle\big{[}S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\big{]}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(139)
$\displaystyle\frac{\differential\sigma_{12}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{-4e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\frac{1}{\mathbf{x}_{3}^{2}}$
$\displaystyle\big{[}S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\big{]}e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(140)

$\displaystyle\frac{\differential\sigma_{13(1)}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32}^{2}}-\frac{\mathbf{x}_{12}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32}^{2}}\right]$
$\displaystyle[S_{12}S_{1^{\prime}2^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(141)
$\displaystyle\frac{\differential\sigma_{13(2)}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\int\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32}^{2}}-\frac{\mathbf{x}_{12}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32}^{2}}\right]$
$\displaystyle[S_{12}S_{1^{\prime}2^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(142)

$\displaystyle\frac{\differential\sigma_{14(1)}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{1}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{1}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{2}{\mathbf{x}_{3}^{2}}+\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32}^{2}}-\frac{\mathbf{x}_{12}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32}^{2}}\right]$
$\displaystyle[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(143)
$\displaystyle\frac{\differential\sigma_{14(2)}^{L}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{1}}=$
$\displaystyle\frac{e^{2}g^{2}Q^{2}N_{c}^{2}(z_{1}z_{2})^{3}}{(2\pi)^{10}}\int_{0}^{z_{2}}\frac{\differential
z}{z}\differential^{10}\mathbf{x}\,K_{0}(|\mathbf{x}_{12}|Q_{1})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{1})\left[\frac{2}{\mathbf{x}_{3}^{2}}+\frac{1}{\mathbf{x}_{31}^{2}}+\frac{1}{\mathbf{x}_{32}^{2}}-\frac{\mathbf{x}_{12}^{2}}{\mathbf{x}_{31}^{2}\mathbf{x}_{32}^{2}}\right]$
$\displaystyle[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1]e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\delta(1-z_{1}-z_{2}).$
(144)

##  Appendix C Dimensional Regularization

The results in section III include divergent integrals over the transverse
momentum $\mathbf{k}$ or in coordinate space as divergent integrals over
$\mathbf{x}_{3}$. Ideally, one would like to regulate these divergences using
dimensional regularization. In many of the results, we have not found a way to
evaluate the regularized integral, but in several it is possible. Here we show
the ones that can be done.

$\displaystyle\frac{\differential\sigma^{L}_{1\times
1}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{2}^{3}(1-z_{2})^{2}(z_{1}^{2}+(1-z_{2})^{2})}{(2\pi)^{9}z_{1}}\int\differential^{8}x\,K_{0}(|\mathbf{x}_{12}|Q_{2})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{2})\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]$
$\displaystyle
e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int\frac{\differential
z}{z}e^{i\frac{z}{z_{1}}\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}\left[\frac{1}{\epsilon}-\log(e^{\gamma_{E}}\pi\mu|\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1}|)\right].$
(145) $\displaystyle\frac{\differential\sigma^{L}_{2\times
2}}{\differential^{2}\mathbf{p}\,\differential^{2}\mathbf{q}\,\differential
y_{1}\,\differential y_{2}}=$
$\displaystyle\frac{2e^{2}g^{2}Q^{2}N_{c}^{2}z_{2}^{3}(1-z_{1})^{2}(z_{2}^{2}+(1-z_{1})^{2})}{(2\pi)^{9}z_{2}}\int\differential^{8}x\,K_{0}(|\mathbf{x}_{12}|Q_{2})K_{0}(|\mathbf{x}_{1^{\prime}2^{\prime}}|Q_{2})\left[S_{122^{\prime}1^{\prime}}-S_{12}-S_{1^{\prime}2^{\prime}}+1\right]$
$\displaystyle
e^{i\mathbf{p}\cdot(\mathbf{x}_{1}^{\prime}-\mathbf{x}_{1})}e^{i\mathbf{q}\cdot(\mathbf{x}_{2}^{\prime}-\mathbf{x}_{2})}\int\frac{\differential