From conservation of transverse momentum, equation (124c), we have
$k_{\Pgluon\perp}^{2}=(\vec{q}_{\perp}-\vec{}k_{\perp})^{2}$, so

$x_{g}^{\text{NLO}}p_{\textmd{\HepGenParticle{A}{}{}\xspace}}^{-}=k_{\perp}e^{-Y}+\frac{(\vec{q}_{\perp}-\vec{}k_{\perp})^{2}}{k_{\perp}}\frac{\xi}{1-\xi}e^{-Y}$
(137)

and finally, setting
$p_{\textmd{\HepGenParticle{A}{}{}\xspace}}^{-}=\sqrt{s}$, we can solve for
$x_{g}^{\text{NLO}}$ as

$x_{g}^{\text{NLO}}=\frac{k_{\perp}}{\sqrt{s}}e^{-Y}+\frac{(\vec{}q_{\perp}-\vec{}k_{\perp})^{2}}{k_{\perp}\sqrt{s}}\frac{\xi}{1-\xi}e^{-Y}=x_{g}^{\text{LO}}\biggl{(}1+\frac{(\vec{}q_{\perp}-\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}\frac{\xi}{1-\xi}\biggr{)}$
(138)

In practice, it is sometimes beneficial to assume the gluon transverse
momentum $\vec{}q_{\perp}$ is negligible or zero. This is approximately valid
because the gluon distribution $\mathcal{F}_{x_{g}}$ falls off exponentially
outside $q_{\perp}\lesssim Q_{s}$, so the contribution of the region
$q_{\perp}\gtrsim Q_{s}$ to the cross section can be neglected. When $Q_{s}\ll
k_{\perp}$ that means we drop terms subleading in $q_{\perp}/k_{\perp}$,
resulting in

$x_{g}^{\text{NLO}}=\frac{k_{\perp}}{\sqrt{s}}e^{-Y}+\frac{k_{\perp}}{\sqrt{s}}\frac{\xi}{1-\xi}e^{-Y}=\frac{k_{\perp}}{\sqrt{s}}\frac{e^{-Y}}{1-\xi}$
(139)

As a momentum fraction, $x_{g}^{\text{NLO}}$ has to satisfy
$x_{g}^{\text{NLO}}<1$. Using (138), that means the gluon momentum fraction
$\xi$ is kinematically limited to

$\xi<\frac{1-x_{g}^{\text{LO}}}{1-x_{g}^{\text{LO}}+x_{g}^{\text{LO}}(\vec{}q_{\perp}-\vec{}k_{\perp})^{2}/k_{\perp}^{2}}$
(140)

This excludes $\xi=1$ unless $\vec{}k_{\perp}=\vec{}q_{\perp}$; physically, it
corresponds to the fact that the gluon can only carry all the longitudinal
momentum if the final state quark (prior to fragmentation) carries all the
transverse momentum. For simplicity, we ignore this limit for the basic
computation described in chapter Computational Results and Analysis and
reintroduce it in chapter Extensions Beyond NLO.

### NLO Cross Section Expressions

As previously mentioned, several papers [94, 82, 95, 96, 97] spanning the
decade from 2002 to 2012 calculated progressively more precise formulas for
the $\mathrm{p}A\to hX$ cross section. However, reference [98] was the first
to incorporate the complete NLO corrections, the formulas implemented by SOLO.
In this section I’ll review the derivation of selected terms to demonstrate
the process.

Figure 25: A representative Feynman diagram for AhX at next-to-leading order
in the quark-quark channel

#### Leading order

The leading order contributions, shown in figure 26, can be calculated in much
the same way as in chapter Drell-Yan Correlations. We can compute the
amplitude for interaction between a quark and the gluon field of the nucleus,
AX, as the Wilson line (48) in the fundamental representation (or its Fourier
transform, to get it as a function of momentum):

$\mathcal{A}=\int\frac{\uddc\vec{}x_{\perp}}{2\pi}e^{-i\vec{}k_{\perp}\cdot\vec{}x_{\perp}}U(\vec{}x_{\perp})$
(141)

To get the cross section, we multiply this by its complex conjugate to get

$\abs{\mathcal{A}}^{2}=\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}x_{\perp}}e^{i\vec{}k_{\perp}\cdot\vec{}y_{\perp}}\trace
U(\vec{}x_{\perp})\herm{U}(\vec{}y_{\perp})$ (142)

We then average over the unknown color charge distribution of the nucleus
(just as we would average over initial spins in a QED calculation). The
averaging operation is represented by
$\frac{1}{N_{c}}\braket{\cdots}_{x_{g}}$, and the resulting parton-level cross
section is the momentum space dipole gluon distribution $\mathcal{F}_{x_{g}}$
from equation (32).

$\displaystyle\frac{\udddc\sigma_{\text{LO}}^{qA\to qA}}{\udc
Y\uddc\vec{}k_{\perp}}$
$\displaystyle=\frac{1}{N_{c}}\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{2}}e^{-i\vec{}k_{\perp}\cdot(\vec{}x_{\perp}-\vec{}y_{\perp})}\Braket{\trace
U(\vec{}x_{\perp})\herm{U}(\vec{}y_{\perp})}$
$\displaystyle=\frac{1}{N_{c}}\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{2}}e^{-i\vec{}k_{\perp}\cdot(\vec{}x_{\perp}-\vec{}y_{\perp})}S_{x_{g}}^{(2)}(\vec{}x_{\perp},\vec{}y_{\perp})$
(143) $\displaystyle=\mathcal{F}_{x_{g}}(\vec{}k_{\perp})$ (144)

The subscript $x_{g}$ indicates the longitudinal momentum fraction of the
gluons involved in the interaction. This gives the cross section for one
flavor of quark interacting with the gluon field of the nucleus.

We also have to include the contribution for a gluon interacting with the
nucleus, AX. This is the same but using the adjoint Wilson line (50).
Accordingly, the square of the amplitude is101010Here all repeated indices, in
particular $a$ and $b$, are summed over. This is a variant of the Einstein
summation convention.

$\abs{\mathcal{A}}^{2}=\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}x_{\perp}}e^{i\vec{}k_{\perp}\cdot\vec{}y_{\perp}}\trace
W^{ab}(\vec{}x_{\perp})\herm{W}^{ab}(\vec{}y_{\perp})$ (145)

Using identity (609) the product becomes

$4\trace\Bigl{\\{}\trace\bigl{[}T^{a}U(\vec{}x_{\perp})T^{b}\herm
U(\vec{}x_{\perp})\bigr{]}\trace\bigl{[}\herm
U(\vec{}y_{\perp})T^{b}U(\vec{}y_{\perp})T^{a}\bigr{]}\Bigr{\\}}$ (146)

With the matrix indices explicit, this is

$4\trace\Bigl{\\{}T_{ij}^{a}U_{jk}(\vec{}x_{\perp})T_{kl}^{b}\herm
U_{li}(\vec{}x_{\perp})\herm
U_{pq}(\vec{}y_{\perp})T_{qr}^{b}U_{rs}(\vec{}y_{\perp})T_{sp}^{a}\Bigr{\\}}$
(147)

Using the identity
$T_{ij}^{a}T_{kl}^{b}=\frac{1}{2}\delta_{il}\delta_{jk}-\frac{1}{2N_{c}}\delta_{ij}\delta_{kl}$,
after a slightly tedious calculation and much relabeling of indices this can
be reduced to

$\trace\Bigl{[}U_{jk}(\vec{}x_{\perp})\herm
U_{kj}(\vec{}y_{\perp})U_{il}(\vec{}y_{\perp})\herm
U_{li}(\vec{}x_{\perp})-\frac{1}{N_{c}}U_{jk}(\vec{}x_{\perp})\herm
U_{ki}(\vec{}x_{\perp})U_{il}(\vec{}y_{\perp})\herm U_{lj}(\vec{}y_{\perp})\\\
-\frac{1}{N_{c}}U_{il}(\vec{}x_{\perp})\herm
U_{lj}(\vec{}y_{\perp})U_{jk}(\vec{}y_{\perp})\herm
U_{ki}(\vec{}x_{\perp})+\frac{1}{N_{c}^{2}}\underbrace{U_{ik}(\vec{}x_{\perp})\herm
U_{ki}(\vec{}x_{\perp})}_{N_{c}}\underbrace{U_{jl}(\vec{}y_{\perp})\herm
U_{lj}(\vec{}y_{\perp})}_{N_{c}}\Bigr{]}$ (150)

In the large-$N_{c}$ limit, the middle two terms vanish, and we’re left with

$\trace\bigl{[}U(\vec{}x_{\perp})\herm
U(\vec{}y_{\perp})\bigr{]}\trace\bigl{[}U(\vec{}y_{\perp})\herm
U(\vec{}x_{\perp})\bigr{]}-1$ (151)

which, after color-averaging with
$\frac{1}{N_{c}^{2}-1}\braket{\cdots}_{x_{g}}$, leaves us with

$\displaystyle\tilde{\mathcal{F}}_{x_{g}}(k_{\perp})$
$\displaystyle\coloneqq\frac{\udddc\sigma_{\text{LO}}^{gA\to gA}}{\udc
Y\uddc\vec{}k_{\perp}}$ (152)
$\displaystyle=\frac{1}{N_{c}^{2}-1}\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{2}}e^{-i\vec{}k_{\perp}\cdot(\vec{}x_{\perp}-\vec{}y_{\perp})}\Braket{\trace\bigl{[}U(\vec{}x_{\perp})\herm
U(\vec{}y_{\perp})\bigr{]}\trace\bigl{[}U(\vec{}y_{\perp})\herm
U(\vec{}x_{\perp})\bigr{]}-1}_{x_{g}}$
$\displaystyle=\frac{1}{N_{c}^{2}-1}\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{2}}e^{-i\vec{}k_{\perp}\cdot(\vec{}x_{\perp}-\vec{}y_{\perp})}\bigl{(}S_{x_{g}}^{(2)}(\vec{}x_{\perp},\vec{}y_{\perp})S_{x_{g}}^{(2)}(\vec{}y_{\perp},\vec{}x_{\perp})-1\bigr{)}$
(153)

In the last line I’ve used the mean-field limit to assume we can consider the
two contracted Wilson line pairs to be uncorrelated. If we separate the two
terms inside parentheses, the $-1$ leads to a delta function
$\delta^{(2)}(\vec{}k_{\perp})$ after integration, but in practice we always
evaluate the gluon distribution at finite momentum, so we can safely drop that
term.

To construct the final cross section for AhX, we multiply the parton-level
quark cross section by the quark distribution function $x_{p}q_{i}(x_{p})$ and
the fragmentation function $D_{\HepGenParticle{h}{}{}\xspace/i}(z)/z^{2}$, and
similarly for the gluon cross section, and integrate over the longitudinal
momentum space parametrized by $z$, $x_{p}$, or $x_{g}$ as described in
section Kinematics:

$\frac{\udddc\sigma_{\text{LO}}^{\mathrm{p}A\to hX}}{\udc
Y\uddc\vec{p}_{\perp}}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}\Biggl{[}\sum_{\text{flavors
}q}x_{p}q(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}(z)\mathcal{F}_{x_{g}}(\vec{}k_{\perp})+x_{p}g(x_{p})D_{\HepGenParticle{h}{}{}\xspace/g}(z)\tilde{\mathcal{F}}_{x_{g}}(x_{p},k_{\perp})\Biggr{]}$
(154)

where $\tau=\frac{p_{\perp}}{\sqrt{s}}e^{Y}$.

Figure 26: Diagrams contributing to AhX at leading order

#### Next-to-Leading Order Example

Figure 27: Diagrams contributing to the quark-quark channel at next-to-leading
order. The left column shows real diagrams, in which the emitted gluon is an
unmeasured final-state particle, and the right column shows virtual diagrams,
in which the gluon is emitted and reabsorbed.

The next-to-leading order contributions to the cross section arise from
Feynman diagrams in which a gluon is emitted and/or absorbed by the parton
coming from the proton. Naturally, this leads to a greatly increased number of
Feynman diagrams; for example, figure 27 shows only those diagrams in which
the initial- and final-state partons are both a quark, the $qq$ (or $q\to qg$
where the hadron comes from the final-state quark) channel. In addition to
those, there are diagrams in which the initial- and final-state partons are
both a gluon, the $gg$ channel (which comprises both $g\to gg$, and $g\to gq$
where the produced hadron comes from the gluon rather than the quark). There
are also off-diagonal channels in which the initial- and final-state partons
are different: the $qg$ channel ($q\to qg$ where the hadron comes from the
final-state gluon) and $gq$ channel ($g\to gq$ where the hadron comes from the
final-state quark).

The formulas for all these contributions are derived in references [98], which
uses earlier results from [92, 38]. Let’s look at the quark-quark channel, for
which the diagrams are shown in figure 27, as an example.

The derivation proceeds in much the same way as that of the Drell-Yan process
of chapter Drell-Yan Correlations. Using the general framework of light-cone
perturbation theory and the CGC rules described in chapter Hadron Structure
From QCD, it’s straightforward to see that the real diagrams, on the left side
of figure 27, correspond to the expression

$\frac{\udc^{6}\sigma}{\udddc\vec{k}_{\textmd{\Pquark}}\udddc\vec{k}_{\textmd{\Pgluon}}}=\alpha_{\text{s}}C_{F}\delta(x_{p}p_{\textmd{\Pproton}}^{+}-k_{\textmd{\Pquark}}^{+}-k_{\textmd{\Pgluon}}^{+})\int\frac{\uddc\vec{}x_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}x_{\perp}^{\prime}}{(2\pi)^{2}}\frac{\uddc\vec{}b_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}b_{\perp}^{\prime}}{(2\pi)^{2}}\\\
\times
e^{-i\vec{k}_{\textmd{\Pgluon}\perp}\cdot(\vec{}x_{\perp}-\vec{}x_{\perp}^{\prime})}e^{-i\vec{k}_{\textmd{\Pquark}\perp}\cdot(\vec{}b_{\perp}-\vec{}b_{\perp}^{\prime})}\sum_{\lambda\alpha\beta}\conj{(\psi_{\alpha\beta}^{\lambda})}(\vec{}u_{\perp}^{\prime})\psi_{\alpha\beta}^{\lambda}(\vec{}u_{\perp})\\\
\times\bigl{[}S_{x_{g}}^{(6)}(\vec{}b_{\perp},\vec{}x_{\perp},\vec{}b_{\perp}^{\prime},\vec{}x_{\perp}^{\prime})+S_{x_{g}}^{(2)}(\vec{}v_{\perp},\vec{}v_{\perp}^{\prime})-S_{x_{g}}^{(3)}(\vec{}b_{\perp},\vec{}x_{\perp},\vec{}v_{\perp}^{\prime})-S_{x_{g}}^{(3)}(\vec{}v_{\perp},\vec{}x_{\perp}^{\prime},\vec{}b_{\perp}^{\prime})\bigr{]}$
(158)

where

$\displaystyle\vec{}u_{\perp}$ $\displaystyle=\vec{}x_{\perp}-\vec{}b_{\perp}$
$\displaystyle\vec{}v_{\perp}$
$\displaystyle=(1-\xi)\vec{}x_{\perp}+\xi\vec{}b_{\perp}$ (159)

and similarly for the primed quantities, and

$\displaystyle
S_{x_{g}}^{(3)}(\vec{}x_{\perp},\vec{}b_{\perp},\vec{}y_{\perp})$
$\displaystyle=\frac{1}{C_{F}N_{c}}\Braket{\trace\bigl{[}U(\vec{}x_{\perp})T^{d}\herm{U}(\vec{}y_{\perp})T^{c}\bigr{]}W^{cd}(\vec{}b_{\perp})}_{x_{g}}$
(160) $\displaystyle
S_{x_{g}}^{(6)}(\vec{}x_{\perp},\vec{}y_{\perp},\vec{}x_{\perp}^{\prime},\vec{}y_{\perp}^{\prime})$
$\displaystyle=\frac{1}{C_{F}N_{c}}\Braket{\trace\bigl{[}U(\vec{}x_{\perp})\herm{U}(\vec{}x_{\perp}^{\prime})T^{d}T^{c}\bigr{]}\bigl{[}W(\vec{}y_{\perp})\herm{W}(\vec{}y_{\perp}^{\prime})\bigr{]}^{cd}}_{x_{g}}$
(161)

As explained in chapter Hadron Structure From QCD, each product of the form
$UW$ arises from the interaction of the gluon field of the nucleus with the
quark and the emitted gluon, whereas “isolated” fundamental Wilson lines (a
$U$ without an associated factor of $W$) comes from interaction of the quark
with the nucleus prior to emission of the gluon. Accordingly, we can tell that
the $S_{x_{g}}^{(6)}$ term arises from the top diagram on the left in figure
27, the $S_{x_{g}}^{(2)}$ term comes from the second diagram down, and each of
the two $S_{x_{g}}^{(3)}$ terms comes from one of the bottom diagrams on the
left side of the figure.

Since we are computing an inclusive cross section, we’ll integrate over the
phase space of the unmeasured particle, here the emitted gluon. After
integrating over $\vec{k}_{\textmd{\Pgluon}}$, we get a factor of
$\delta^{(2)}(\vec{}x_{\perp}-\vec{}x_{\perp}^{\prime})$, which simplifies
$S_{x_{g}}^{(6)}(\vec{}b_{\perp},\vec{}x_{\perp},\vec{}b_{\perp}^{\prime},\vec{}x_{\perp}^{\prime})\to
S_{x_{g}}^{(2)}(\vec{}b_{\perp},\vec{}b_{\perp}^{\prime})$. The expression
(158) becomes

$\frac{\udc^{3}\sigma}{\udddc\vec{k}_{\textmd{\Pquark}}}=\alpha_{\text{s}}C_{F}\int\frac{\uddc\vec{}x_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}b_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}b_{\perp}^{\prime}}{(2\pi)^{2}}e^{-i\vec{k}_{\textmd{\Pquark}\perp}\cdot(\vec{}b_{\perp}-\vec{}b_{\perp}^{\prime})}\sum_{\lambda\alpha\beta}\conj{(\psi_{\alpha\beta}^{\lambda})}(\vec{}u_{\perp}^{\prime})\psi_{\alpha\beta}^{\lambda}(\vec{}u_{\perp})\\\
\times\bigl{[}S_{x_{g}}^{(2)}(\vec{}b_{\perp},\vec{}b_{\perp}^{\prime})+S_{x_{g}}^{(2)}(\vec{}v_{\perp},\vec{}v_{\perp}^{\prime})-S_{x_{g}}^{(3)}(\vec{}b_{\perp},\vec{}x_{\perp},\vec{}v_{\perp}^{\prime})-S_{x_{g}}^{(3)}(\vec{}v_{\perp},\vec{}x_{\perp},\vec{}b_{\perp}^{\prime})\bigr{]}$
(164)

where now

$\displaystyle\vec{}u_{\perp}^{\prime}$
$\displaystyle=\vec{}x_{\perp}-\vec{}b_{\perp}^{\prime}$
$\displaystyle\vec{}v_{\perp}^{\prime}$
$\displaystyle=(1-\xi)\vec{}x_{\perp}+\xi\vec{}b_{\perp}^{\prime}$ (165)

For the virtual diagrams, in the right column of figure 27, the same procedure
results in the expression

$\frac{\udddc\sigma}{\udddc\vec{k}_{\textmd{\Pquark}}}=-2\alpha_{\text{s}}C_{F}\int\frac{\uddc\vec{}v_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}v_{\perp}^{\prime}}{(2\pi)^{2}}\frac{\uddc\vec{}u_{\perp}}{(2\pi)^{2}}e^{-i\vec{k}_{\textmd{\Pquark}\perp}\cdot(\vec{}v_{\perp}-\vec{}v_{\perp}^{\prime})}\\\
\times\sum_{\lambda\alpha\beta}\conj{(\psi_{\alpha\beta}^{\lambda})}(\vec{}u_{\perp})\psi_{\alpha\beta}^{\lambda}(\vec{}u_{\perp})\bigl{[}S_{x_{g}}^{(2)}(\vec{}v_{\perp},\vec{}v_{\perp}^{\prime})-S_{x_{g}}^{(3)}(\vec{}b_{\perp},\vec{}x_{\perp},\vec{}v_{\perp}^{\prime})\bigr{]}$
(168)

In this expression we only have one momentum, the quark momentum, because only
the quark line passes through the cut, and this removes the need for the delta
function because the initial quark momentum doesn’t occur anywhere in the
expression. The integral over the loop momentum is already incorporated into
the definition of the Wilson line. And the factor of $2$ arises because the
top two diagrams in the right column of figure 27 give the same contribution
(specifically, the $S_{x_{g}}^{(3)}$ term), and likewise for the bottom two
diagrams (the $S_{x_{g}}^{(2)}$ term).

Putting together both the real (164) and virtual (168) contributions gives the
complete expression for the quark-quark channel.

$\frac{\udc^{3}\sigma}{\udddc\vec{k}_{\textmd{\Pquark}}}=\alpha_{\text{s}}C_{F}\int\frac{\uddc\vec{}u_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}v_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}v_{\perp}^{\prime}}{(2\pi)^{2}}e^{-i\vec{k}_{\textmd{\Pquark}\perp}\cdot(\vec{}b_{\perp}-\vec{}b_{\perp}^{\prime})}\sum_{\lambda\alpha\beta}\conj{(\psi_{\alpha\beta}^{\lambda})}(\vec{}u_{\perp}^{\prime})\psi_{\alpha\beta}^{\lambda}(\vec{}u_{\perp})\\\
\times\bigl{[}S_{x_{g}}^{(2)}(\vec{}b_{\perp},\vec{}b_{\perp}^{\prime})-S_{x_{g}}^{(2)}(\vec{}v_{\perp},\vec{}v_{\perp}^{\prime})+S_{x_{g}}^{(3)}(\vec{}b_{\perp},\vec{}x_{\perp},\vec{}v_{\perp}^{\prime})-S_{x_{g}}^{(3)}(\vec{}v_{\perp},\vec{}x_{\perp},\vec{}b_{\perp}^{\prime})\bigr{]}$
(171)

We can substitute in the definition of the splitting wavefunction,

$\psi_{\alpha\beta}^{\lambda}(x_{p}p_{\textmd{\Pproton}}^{+},k_{\textmd{\Pquark}}^{+},\vec{}r_{\perp})=2\pi
i\sqrt{\frac{2}{k_{\textmd{\Pquark}}^{+}}}\begin{cases}\frac{\vec{}r_{\perp}\cdot\epsilon_{\perp}^{(1)}}{r_{\perp}^{2}}(\delta_{\alpha-}\delta_{\beta-}+\xi\delta_{\alpha+}\delta_{\beta+}),&\lambda=1\\\
\frac{\vec{}r_{\perp}\cdot\epsilon_{\perp}^{(2)}}{r_{\perp}^{2}}(\delta_{\alpha+}\delta_{\beta+}+\xi\delta_{\alpha-}\delta_{\beta-}),&\lambda=2\end{cases}$
(172)

and simplify the gluon distributions using the identity

$S_{x_{g}}^{(3)}(\vec{}x_{\perp},\vec{}b_{\perp},\vec{}y_{\perp})=\frac{N_{c}}{2C_{F}}\biggl{[}S_{x_{g}}^{(4)}(\vec{}x_{\perp},\vec{}b_{\perp},\vec{}y_{\perp})-\frac{1}{N_{c}^{2}}S_{x_{g}}^{(2)}(\vec{}x_{\perp},\vec{}y_{\perp})\biggr{]}$
(173)

to get

$\frac{\udc^{3}\sigma}{\udddc\vec{k}_{\textmd{\Pquark}}}=\alpha_{\text{s}}\int\frac{\uddc\vec{}u_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}v_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}v_{\perp}^{\prime}}{(2\pi)^{2}}e^{-i\vec{k}_{\textmd{\Pquark}\perp}\cdot(\vec{}b_{\perp}-\vec{}b_{\perp}^{\prime})}\sum_{\lambda\alpha\beta}\conj{(\psi_{\alpha\beta}^{\lambda})}(\vec{}u_{\perp}^{\prime})\psi_{\alpha\beta}^{\lambda}(\vec{}u_{\perp})\\\
\times\Biggl{(}C_{F}S_{x_{g}}^{(2)}(\vec{}b_{\perp},\vec{}b_{\perp}^{\prime})-C_{F}S_{x_{g}}^{(2)}(\vec{}v_{\perp},\vec{}v_{\perp}^{\prime})-\frac{1}{2N_{c}}S_{x_{g}}^{(2)}(\vec{}b_{\perp},\vec{}v_{\perp}^{\prime})+\frac{1}{2N_{c}}S_{x_{g}}^{(2)}(\vec{}v_{\perp},\vec{}b_{\perp}^{\prime})\\\
+\frac{N_{c}}{2}S_{x_{g}}^{(4)}(\vec{}b_{\perp},\vec{}x_{\perp},\vec{}v_{\perp}^{\prime})-\frac{N_{c}}{2}S_{x_{g}}^{(4)}(\vec{}v_{\perp},\vec{}x_{\perp},\vec{}b_{\perp}^{\prime})\Biggr{)}$
(177)

which is in terms of only two gluon distributions, $S_{x_{g}}^{(2)}$ and
$S_{x_{g}}^{(4)}$. If we further use the mean field approximation (83), that
reduces to only $S_{x_{g}}^{(2)}$; we do this in the SOLO code, but in this
document I’ll keep $S_{x_{g}}^{(4)}$ explicit.

The last step in finding the cross section for AhX in the quark-quark channel
is to convolve with the parton distributions of the proton and the
fragmentation functions.

$\frac{\udddc\sigma}{\udddc\vec{p}}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}\int_{\tau/z}^{1}\udc\xi
x_{p}q(x_{p})D_{h/j}(z)\frac{\udddc\sigma}{\udddc\vec{k}_{\textmd{\Pquark}}}$
(178)

#### Removal of Divergences

In reference [98], the quark-quark channel cross section is given as the
following two expressions:

$\frac{\alpha_{\text{s}}}{2\pi^{2}}\int_{\tau}^{1}\int_{\frac{\tau}{z}}^{1}\frac{\udc
z\udc\xi}{z^{2}}x_{p}q(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}(z)\frac{1+\xi^{2}}{1-\xi}\biggl{[}C_{F}\int\uddc\vec{q}_{\perp}\mathcal{I}(\vec{k}_{\textmd{\Pquark}\perp},\vec{q}_{\perp})\\\
+\frac{N_{c}}{2}\int\uddc\vec{q}_{\perp}\uddc\vec{}k_{\Pgluon
1\perp}\mathcal{J}(\vec{k}_{\textmd{\Pquark}\perp},\vec{q}_{\perp},\vec{}k_{\Pgluon
1\perp})\biggr{]}$ (181)

from the real diagrams and

$-\frac{\alpha_{\text{s}}}{2\pi^{2}}\int_{\tau}^{1}\int_{0}^{1}\frac{\udc
z\udc\xi}{z^{2}}x_{p}q(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}(z)\frac{1+\xi^{2}}{1-\xi}\biggl{[}C_{F}\int\uddc\vec{q}_{\perp}\mathcal{I}(\vec{q}_{\perp},\vec{k}_{\textmd{\Pquark}\perp})\\\
+\frac{N_{c}}{2}\int\uddc\vec{q}_{\perp}\uddc\vec{}k_{\Pgluon
1\perp}\mathcal{J}(\vec{q}_{\perp},\vec{k}_{\textmd{\Pquark}\perp},\vec{}k_{\Pgluon
1\perp})\biggr{]}$ (184)

from the virtual diagrams, where

$\displaystyle\mathcal{I}(\vec{k}_{\textmd{\Pquark}\perp},\vec{q}_{\perp})$
$\displaystyle=\mathcal{F}_{x_{g}}(\vec{q}_{\perp})\biggl{[}\frac{\vec{k}_{\textmd{\Pquark}\perp}-\vec{q}_{\perp}}{(\vec{k}_{\textmd{\Pquark}\perp}-\vec{q}_{\perp})^{2}}-\frac{\vec{k}_{\textmd{\Pquark}\perp}-\xi\vec{q}_{\perp}}{(\vec{k}_{\textmd{\Pquark}\perp}-\xi\vec{q}_{\perp})^{2}}\biggr{]}^{2}$
(185)
$\displaystyle\mathcal{J}(\vec{k}_{\textmd{\Pquark}\perp},\vec{q}_{\perp},\vec{}k_{\Pgluon
1\perp})$
$\displaystyle=\begin{multlined}\biggl{[}\mathcal{F}_{x_{g}}(\vec{q}_{\perp})\delta^{(2)}(\vec{q}_{\perp}-\vec{}k_{\Pgluon
1\perp})-\mathcal{G}_{x_{g}}(\vec{q}_{\perp},\vec{}k_{\Pgluon
1\perp})\biggr{]}\\\
\times\frac{2(\vec{k}_{\textmd{\Pquark}\perp}-\xi\vec{q}_{\perp})\cdot(\vec{k}_{\textmd{\Pquark}\perp}-\vec{}k_{\Pgluon
1\perp})}{(\vec{k}_{\textmd{\Pquark}\perp}-\xi\vec{q}_{\perp})^{2}(\vec{k}_{\textmd{\Pquark}\perp}-\vec{}k_{\Pgluon
1\perp})^{2}}\end{multlined}\biggl{[}\mathcal{F}_{x_{g}}(\vec{q}_{\perp})\delta^{(2)}(\vec{q}_{\perp}-\vec{}k_{\Pgluon
1\perp})-\mathcal{G}_{x_{g}}(\vec{q}_{\perp},\vec{}k_{\Pgluon
1\perp})\biggr{]}\\\
\times\frac{2(\vec{k}_{\textmd{\Pquark}\perp}-\xi\vec{q}_{\perp})\cdot(\vec{k}_{\textmd{\Pquark}\perp}-\vec{}k_{\Pgluon
1\perp})}{(\vec{k}_{\textmd{\Pquark}\perp}-\xi\vec{q}_{\perp})^{2}(\vec{k}_{\textmd{\Pquark}\perp}-\vec{}k_{\Pgluon
1\perp})^{2}}$ (188)

It’s clear that these formulas are divergent in two ways: the rapidity
divergence, where the integrand becomes undefined at $\xi\to 1$, and the
collinear divergence when $\vec{k}_{\textmd{\Pquark}\perp}=\vec{q}_{\perp}$.
Regulating these divergences constitutes the largest obstacle in computing the
full NLO corrections to the cross section.

Broadly speaking, here is how the process works. A simplified version of the
quark-quark channel result looks like

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}x_{p}q^{(0)}(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)\mathcal{F}_{x_{g}}^{(0)}(\vec{q}_{\perp})\\\
+\frac{\alpha_{\text{s}}}{2\pi}\int_{\tau}^{1}\frac{\udc
z}{z^{2}}\int_{0}^{1}\udc\xi\,x_{p}q^{(0)}(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)\frac{A\mathcal{F}_{x_{g}}^{(0)}(\vec{q}_{\perp})+B\mathcal{G}_{x_{g}}^{(0)}(\vec{q}_{\perp},\vec{k}_{\textmd{\Pgluon}\perp})}{1-\xi}$
(191)

where $A$ and $B$ are nonsingular coefficients. The leading order term is
finite on its own, but the next-to-leading order term is divergent at $\xi\to
1$. (In this simplified example, I’ll omit the collinear divergence until
later.)

We can rewrite this as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}x_{p}q^{(0)}(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)\\\
\times\Biggl{[}\mathcal{F}_{x_{g}}^{(0)}(\vec{}k_{\perp})+\frac{\alpha_{\text{s}}}{2\pi}\int_{0}^{1}\udc\xi\frac{A\mathcal{F}_{x_{g}}^{(0)}(\vec{q}_{\perp})+B\mathcal{G}_{x_{g}}^{(0)}(\vec{q}_{\perp},\vec{k}_{\textmd{\Pgluon}\perp})}{1-\xi}\Biggr{]}$
(194)

$\mathcal{F}_{x_{g}}^{(0)}(\vec{q}_{\perp})$ is not actually a physical
quantity, so there’s no reason for it to be finite. We can then assume that
$\mathcal{F}_{x_{g}}^{(0)}(\vec{q}_{\perp})$ is divergent in a way that
cancels out the rapidity divergence, so the total of the terms in square
brackets is finite.

At this point we could define the renormalized gluon distribution
$\mathcal{F}_{x_{g}}(\vec{q}_{\perp})$ to be exactly the term in brackets, so
that the quark-quark cross section would just be

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}x_{p}q^{(0)}(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)\mathcal{F}_{x_{g}}(\vec{q}_{\perp})$
(195)

but that would make $\mathcal{F}_{x_{g}}$ dependent on the exact coefficients
$A$ and $B$. If we do that, we can’t generalize $\mathcal{F}_{x_{g}}$ to any
other calculation. Instead, we should define the renormalized distribution as
the sum of $\mathcal{F}_{x_{g}}^{(0)}$ and some other divergent term that
cancels out the singularity in the remaining part of the expression. That will
leave us with

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}x_{p}q^{(0)}(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)\bigl{[}\mathcal{F}_{x_{g}}(\vec{q}_{\perp})+C(z,\xi,\vec{q}_{\perp})\bigr{]}$
(196)

$C$ is called the coefficient function. The procedure for how we choose to
split the term in brackets between $\mathcal{F}_{x_{g}}$ and $C$ is a
renormalization scheme.111111For a detailed accessible description of
subtractions, see sections 4.3 and 7.1 of reference [19].

When there are multiple divergences, namely the rapidity divergence and
collinear divergence, we can generalize the procedure to incorporate these
divergences into the parton distribution and fragmentation function as well.
In doing so, it’s worth considering the physical nature of the divergences
[98]:

  * •

The rapidity divergence occurs when the quark momentum fraction $\xi$ goes to
$1$, so that the emitted gluon’s momentum becomes negligible in the rest frame
of the target. These extremely low-momentum gluon emissions are physically
indistinguishable from the target’s own gluon field. When we renormalize the
rapidity divergence using the target gluon distribution, we are effectively
absorbing the low-momentum gluon emissions into the structure of the target.

  * •

The collinear divergence, on the other hand, occurs when the quark and the
emitted gluon are traveling in the same direction with nearly equal momenta.
However, we know that a propagating quark is dressed with a cloud of virtual
gluons (and other particles). An emitted gluon with nearly the same momentum
as the quark will effectively be part of that cloud, so it makes sense to
absorb the collinear divergence into the function that represents the
nonperturbative dynamics of the propagating physical quark. This is either the
fragmentation function, if the gluon is emitted after the interaction, or the
integrated parton distribution, if it’s emitted before the interaction.

With this in mind, let’s look at a slightly less simplified version of the
quark-quark channel cross section, just like (191) except that this contains
the collinear divergence as well,

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}x_{p}q^{(0)}(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)\mathcal{F}_{x_{g}}^{(0)}(\vec{q}_{\perp})\\\
+\frac{\alpha_{\text{s}}}{2\pi}\int_{\tau}^{1}\frac{\udc
z}{z^{2}}\int_{\tau/z}^{1}\udc\xi\,x_{p}q^{(0)}(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)\frac{A\mathcal{F}_{x_{g}}^{(0)}(\vec{q}_{\perp})+B\mathcal{G}_{x_{g}}^{(0)}(\vec{q}_{\perp},\vec{k}_{\textmd{\Pgluon}\perp})}{(1-\xi)(\vec{q}_{\perp}-\vec{k}_{\textmd{\Pgluon}\perp})^{2}}$
(199)

With all three of the functions $x_{p}q^{(0)}(x_{p})$,
$D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)$, and
$\mathcal{F}_{x_{g}}^{(0)}(\vec{q}_{\perp})$ playing a role in removing the
divergences, we should wind up with an expression of the form

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}\Biggl{[}x_{p}q^{(0)}(x_{p})-\frac{\alpha_{\text{s}}}{2\pi}\int_{x_{p}}^{1}\udc\xi^{\prime}\frac{C^{\prime}x_{p}q(x_{p}/\xi^{\prime})}{(\vec{q}_{\perp}-\vec{k}_{\textmd{\Pgluon}\perp})^{2}}\Biggr{]}\\\
\times\Biggl{[}D_{\HepGenParticle{h}{}{}\xspace/q}^{(0)}(z)-\frac{\alpha_{\text{s}}}{2\pi}\int_{z}^{1}\udc\xi^{\prime\prime}\frac{C^{\prime\prime}D_{\HepGenParticle{h}{}{}\xspace/q}(z/\xi^{\prime\prime})}{(\vec{q}_{\perp}-\vec{k}_{\textmd{\Pgluon}\perp})^{2}}\Biggr{]}\\\
\times\Biggl{[}\mathcal{F}_{x_{g}}^{(0)}(\vec{}k_{\perp})+\frac{\alpha_{\text{s}}}{2\pi}\int_{0}^{1}\udc\xi\frac{A\mathcal{F}_{x_{g}}^{(0)}(\vec{}k_{\perp})+B\mathcal{G}_{x_{g}}^{(0)}(\vec{}k_{\perp},\vec{q}_{\perp})}{1-\xi}\Biggr{]}$
(203)

where $C^{\prime}$ and $C^{\prime\prime}$ are nonsingular coefficients (like
$A$ and $B$). For simplicity, let me express this as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}(q_{0}-q_{1})(D_{0}-D_{1})(F_{0}+F_{1})$ (204)

where $q_{0}$ and $q_{1}$ stand for the bare parton distribution and the
subtraction, respectively, and similarly for $D$ and $F$. Distributing we find

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}\bigl{[}q_{0}D_{0}F_{0}+q_{0}D_{0}F_{1}-q_{1}D_{0}F_{0}-q_{1}D_{0}F_{1}\\\
-q_{0}D_{1}F_{0}-q_{0}D_{1}F_{1}+q_{1}D_{1}F_{0}+q_{1}D_{1}F_{1}\bigr{]}$
(207)

Note that each term is at an order in $\alpha_{\text{s}}$ corresponding to the
sum of its subscripts (e.g. the last term is
$\orderof{\alpha_{\text{s}}^{3}}$).

Now, if we go back to (199), which is the bare result,

$\displaystyle\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{qq}$ $\displaystyle=\int_{\tau}^{1}\frac{\udc
z}{z^{2}}\bigl{[}q_{0}D_{0}F_{0}$ (208) and perform the rapidity subtraction
$\displaystyle+q_{0}D_{0}F_{1}$ (209) and then incorporate the parton
distribution and fragmentation function $\displaystyle-
q_{1}D_{0}F_{0}-q_{0}D_{1}F_{0}$ (210) and the rapidity subtractions that have
to be incorporated into those subtractions $\displaystyle-
q_{1}D_{0}F_{1}-q_{0}D_{1}F_{1}\bigr{]}$ (211)

then we reproduce the distributed result (207) up to terms which are finite
and subleading in $\alpha_{\text{s}}$ (specifically
$\orderof{\alpha_{\text{s}}^{2}}$). This is the basic procedure by which
subtracting divergences gives a finite result.

Performing the full subtractions for the real and virtual contributions
involves a fairly tedious calculation which is the subject of reference [98].
That work uses the $\overline{\text{MS}}$ renormalization scheme, which
entails regularizing integrals by dimensional regularization and bringing a
term proportional to $-\frac{1}{\epsilon}+\gamma_{E}$ into the gluon
distribution; the details of $\overline{\text{MS}}$ renormalization are
described in many standard references. The result works out to

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{nqq}^{(1)}=\frac{\alpha_{\text{s}}}{2\pi}\int_{\tau}^{1}\int_{\frac{\tau}{z}}^{1}\frac{\udc
z\udc\xi}{z^{2}}x_{p}q(x_{p})D_{\HepGenParticle{h}{}{}\xspace/q}(z)\biggl{[}\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\mathcal{H}^{(1)}_{2qq}\\\
+\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}b_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})\mathcal{H}^{(1)}_{4qq}\biggr{]}$
(214)

where the hard factors $\mathcal{H}^{(1)}_{2qq}$ and $\mathcal{H}^{(1)}_{4qq}$
are given in equations (306) and (325).

### Adapting Formulas for Numerical Implementation

As they are given in reference [98], the formulas are integrals over various
sets of position coordinates. A computational implementation needs code to
evaluate the integrands and also to perform the integration. The integrands
involve various singular terms proportional to $\delta(1-\xi)$ and the plus
prescription $\frac{1}{(1-\xi)_{+}}$, which need to be rewritten in terms of
well-behaved (nonsingular) functions in order to be numerically evaluated.
There are also Fourier factors like
$e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}$ in the integrals, which are
numerically difficult to evaluate, so in the interest of accuracy it will be
useful to integrate out those factors symbolically where possible. This
section describes the details of that process.

#### Eliminating Singularities in $\xi$

Each term in the cross section can be written as a sum of three parts: one
proportional to a plus prescription-regularized fraction, one proportional to
a delta function, and one “normal” contribution which has no singularities
that will prevent its numerical evaluation.

$\int_{\tau}^{1}\udc
z\int_{\frac{\tau}{z}}^{1}\udc\xi\biggl{[}\frac{F_{s}(z,\xi)}{(1-\xi)_{+}}+F_{d}(z,\xi)\delta(1-\xi)+F_{n}(z,\xi)\biggr{]}$
(215)

To make the $F_{s}$ term numerically tractable, we need to simplify it using
the definition of the plus prescription,

$f(x)_{+}\udc x\coloneqq f(x)\udc x-\delta(1-x)\udc x\int_{0}^{1}f(y)\udc y$
(216)

When this is integrated from some $a\in[0,1]$ to $1$, that integral can be
simplified as follows:

$\displaystyle\int_{a}^{1}\frac{g(x)}{(1-x)_{+}}\udc x$
$\displaystyle=\int_{a}^{1}\frac{g(x)}{1-x}\udc
x-g(1)\int_{0}^{1}\frac{1}{1-y}\udc y$ (217)
$\displaystyle=\int_{a}^{1}\frac{g(x)}{1-x}\udc
x-g(1)\int_{a}^{1}\frac{1}{1-x}\udc x-g(1)\int_{0}^{a}\frac{1}{1-x}\udc x$
(218) $\displaystyle=\int_{a}^{1}\frac{g(x)-g(1)}{1-x}\udc x+g(1)\ln(1-a)$
(219)

Applying that to the $F_{s}$ term from (215), we get

$\int_{\tau}^{1}\udc
z\int_{\frac{\tau}{z}}^{1}\udc\xi\frac{F_{s}(z,\xi)}{(1-\xi)_{+}}=\int_{\tau}^{1}\udc
z\int_{\frac{\tau}{z}}^{1}\udc\xi\,\frac{F_{s}(z,\xi)-F_{s}(z,1)}{1-\xi}+\int_{\tau}^{1}\udc
z\,F_{s}(z,1)\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}$ (220)

The $F_{d}$ term is easier, as it only requires us to do the integration over
$\xi$:

$\int_{\tau}^{1}\udc z\int_{\frac{\tau}{z}}^{1}\udc\xi
F_{d}(z,\xi)\delta(1-\xi)=\int_{\tau}^{1}\udc zF_{d}(z,1)$ (221)

And of course, the $F_{n}$ term of (215) requires no symbolic integration.

Putting the three terms together, we get the following expression, which is
equivalent to (215) but is suitable for numerical evaluation:

$\int_{\tau}^{1}\udc
z\int_{\frac{\tau}{z}}^{1}\udc\xi\biggl{[}\frac{F_{s}(z,\xi)-F_{s}(z,1)}{1-\xi}+F_{n}(z,\xi)\biggr{]}+\int_{\tau}^{1}\udc
z\biggl{[}F_{s}(z,1)\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F_{d}(z,1)\biggr{]}$
(222)

It is easy to split each hard factor up into the parts $F_{s}$, $F_{n}$, and
$F_{d}$ and then the integration can be done automatically with an
implementation of this formula.

#### Reducing Additional Degrees of Freedom

In general, each term of the cross section depends on several other variables
beyond $z$ and $\xi$. Which variables those are varies from one term to
another, though, and even depends on how a given term is expressed.

Broadly speaking there are three different ways that a term might be
expressed:

Cartesian position space

the term will be expressed as an integral over $z$, $\xi$, and transverse
displacements

$\displaystyle\vec{}r_{\perp}$ $\displaystyle=\vec{}x_{\perp}-\vec{}y_{\perp}$
$\displaystyle\vec{}s_{\perp}$ $\displaystyle=\vec{}x_{\perp}-\vec{}b_{\perp}$
$\displaystyle\vec{}t_{\perp}$ $\displaystyle=\vec{}y_{\perp}-\vec{}b_{\perp}$
(223)

This is closest to the original form given in reference [98], but is not
particularly useful for computation because it includes Fourier factors like
$e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}$, which are numerically unstable.

Radial position space

the term will be expressed as an integral over $z$, $\xi$, and the radial part
of transverse positions $r_{\perp}$, $s_{\perp}$, and $t_{\perp}$. In order to
express the term in this way, we’ll assume angular independence of the gluon
distribution $S_{x_{g}}^{(2)}$.

Momentum space

the term will be expressed as an integral over $z$, $\xi$, and some number of
momentum variables $k_{\Pgluon\perp}$, $k_{\Pgluon 1\perp}$, etc. This is the
most accurate form for numerical computation, but it requires us to do some
symbolic Fourier integrals, which are not possible for every term. Also, the
details of how to convert a given term into momentum space vary significantly
from one term to another, even when it is possible.

##### Dipole-Type Terms

Dipole-type terms involve integrals over only two transverse position
coordinates, $\vec{}x_{\perp}$ and $\vec{}y_{\perp}$. A generic term of this
type can be written as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2ji}^{(n)}=\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}f_{i}(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/j}(z,\mu)\\\
\times\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{2}}\mathcal{S}(\vec{}x_{\perp},\vec{}y_{\perp})e^{-iC_{0}\vec{}k_{\perp}\cdot(\vec{}x_{\perp}-\vec{}y_{\perp})}\hat{}\mathcal{H}^{(n)}_{2ji}(\vec{}x_{\perp},\vec{}y_{\perp})$
(226)

where $n=0$ for leading order or $1$ for next-to-leading order, and $i$ and
$j$ represent parton flavors (gluon, up, down, strange, and heavy quarks).
$C_{0}$ is some factor independent of the position and momentum coordinates,
and $\mathcal{S}$ represents one or more factors of the dipole gluon
distribution $S_{x_{g}}^{(2)}$. The hard factor $\mathcal{H}^{(n)}_{2ji}$ from
reference [98] corresponds to the combination

$\mathcal{H}^{(n)}_{2ji}=e^{-iC_{0}\vec{}k_{\perp}\cdot(\vec{}x_{\perp}-\vec{}y_{\perp})}\hat{}\mathcal{H}^{(n)}_{2ji}$
(227)

To manipulate these terms, we first simplify the expression by changing
coordinates to

$\displaystyle\vec{R}_{\perp}$ $\displaystyle=\vec{}x_{\perp}+\vec{}y_{\perp}$
$\displaystyle\vec{}r_{\perp}$ $\displaystyle=\vec{}x_{\perp}-\vec{}y_{\perp}$
(228)

In this work, we assume that $S_{x_{g}}^{(2)}$, and thus $\mathcal{S}$, is
independent121212$\vec{R}_{\perp}$ is the impact parameter, conventionally
denoted $\vec{b}$. Incorporating impact parameter dependence can have a
noticeable effect on cross sections, as described in refs. [51, 52, 54], but
that will not be explored here. of $\vec{R}_{\perp}$ and the angle of
$\vec{}r_{\perp}$, so we can write
$\mathcal{S}(\vec{}x_{\perp},\vec{}y_{\perp})$ as $\mathcal{S}(r_{\perp})$.
It’s also easy to verify, by inspection of the expressions in ref. [98], that
all the hard factors $\mathcal{H}^{(n)}_{2ji}$ are independent of
$\vec{R}_{\perp}$. Accordingly, we can change variables

$\uddc\vec{}x_{\perp}\uddc\vec{}y_{\perp}=\frac{1}{4}\uddc\vec{R}_{\perp}\uddc\vec{}r_{\perp}$
(229)

and integrate out the $\vec{R}_{\perp}$ dependence, defining

$S_{\perp}\coloneqq\int\frac{\uddc\vec{R}_{\perp}}{4}$ (230)

$S_{\perp}$ can be interpreted as the cross-sectional area of the target.
After this integration, the term (226) becomes

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2ji}^{(n)}=S_{\perp}\biggl{(}\frac{\alpha_{\text{s}}}{2\pi}\biggr{)}^{n}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}f_{i}(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/j}(z,\mu)\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\mathcal{S}(r_{\perp})e^{-iC_{0}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\hat{}\mathcal{H}^{(n)}_{2ji}(\vec{}r_{\perp})$
(231)

Recall from section Eliminating Singularities in $\xi$ that a term can be
expressed as a combination of a plus-regularized contribution, a delta-
function contribution, and a normal contribution, as in equation (215). We can
obtain the three functions $F^{(n)}_{s2ji}$, $F^{(n)}_{d2ji}$, and
$F^{(n)}_{n2ji}$ for each term by breaking down the term into these three
contributions and pulling out all the integrals — not only those over $z$ and
$\xi$, but also over position and momentum variables. Different terms (and
different representations of those terms) will be integrated over different
variables, so the third and additional arguments of $F^{(n)}_{s2ji}$ etc. will
be used to indicate what is being integrated over. When the third argument is
a 2D position vector, that indicates the Cartesian representation; when it is
the magnitude of a 2D position vector, that indicates the radial
representation; when it is a 2D momentum vector, that indicates the momentum-
space representation.

###### Cartesian position space

It’s straightforward to break down equation (231) into the three functions
$F_{s}$, $F_{n}$, and $F_{d}$ from section Eliminating Singularities in $\xi$
as follows:

$\displaystyle\frac{F^{(n)}_{s2ji}(z,\xi,\vec{}r_{\perp})}{(1-\xi)_{+}}$
$\displaystyle=S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp})}{(2\pi)^{2}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}e^{-iC_{0}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\bigl{(}\text{plus-
regulated part of }\hat{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (232a)
$\displaystyle F^{(n)}_{n2ji}(z,\xi,\vec{}r_{\perp})$
$\displaystyle=S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp})}{(2\pi)^{2}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}e^{-iC_{0}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\bigl{(}\text{normal
part of }\hat{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (232b) $\displaystyle
F^{(n)}_{d2ji}(z,\xi,\vec{}r_{\perp})\delta(1-\xi)$
$\displaystyle=S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp})}{(2\pi)^{2}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}e^{-iC_{0}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\bigl{(}\text{delta
function part of }\hat{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (232c)

The entire term (231) can then be written as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2ji}^{(n)}=\int\udc
z\udc\xi\int\uddc\vec{}r_{\perp}\biggl{[}\frac{F^{(n)}_{s2ji}(z,\xi,\vec{}r_{\perp})}{(1-\xi)_{+}}+F^{(n)}_{n2ji}(z,\xi,\vec{}r_{\perp})+F^{(n)}_{d2ji}(z,\xi,\vec{}r_{\perp})\delta(1-\xi)\biggr{]}$
(233)

or, using equation (222), as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2ji}^{(n)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}r_{\perp}\biggl{[}\frac{F^{(n)}_{s2ji}(z,\xi,\vec{}r_{\perp})-F^{(n)}_{s2ji}(z,1,\vec{}r_{\perp})}{1-\xi}+F^{(n)}_{n2ji}(z,\xi,\vec{}r_{\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}\biggl{[}F^{(n)}_{s2ji}(z,1,\vec{}r_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(n)}_{d2ji}(z,1,\vec{}r_{\perp})\biggr{]}$
(236)

This is the integral actually performed by SOLO when this term is evaluated in
position space. The functions $F^{(n)}_{s2ji}$, $F^{(n)}_{n2ji}$, and
$F^{(n)}_{d2ji}$ can be picked out from the corresponding expression for
$\mathcal{H}^{(n)}_{2ji}$ in reference [98] using (232), transcribed directly
into the program, and then equation (236) is implemented by a generic
integration routine.

###### Radial position space

It turns out that the non-Fourier part of each dipole-type term,
$\hat{}\mathcal{H}^{(n)}_{2ji}$, is independent of the angle of
$\vec{}r_{\perp}$. This means we can integrate over the angle in equation
(231),

$\iint\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\mathcal{S}(r_{\perp})e^{-iC_{0}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\hat{}\mathcal{H}^{(n)}_{2ji}(r_{\perp})=\frac{1}{2\pi}\int_{0}^{\infty}\udc
r_{\perp}r_{\perp}\mathcal{S}(r_{\perp})J_{0}(C_{0}k_{\perp}r_{\perp})\hat{}\mathcal{H}^{(n)}_{2ji}(r_{\perp})$
(237)

and again, pick out the pieces of the term as

$\displaystyle\frac{F^{(n)}_{s2ji}(z,\xi,r_{\perp})}{(1-\xi)_{+}}$
$\displaystyle=\frac{S_{\perp}}{z^{2}}x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}\frac{S_{x_{g}}^{(2)}(\vec{}r_{\perp})}{2\pi}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}r_{\perp}J_{0}(k_{\perp}r_{\perp})\bigl{(}\text{plus-
regulated part of }\hat{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (238a)
$\displaystyle F^{(n)}_{n2ji}(z,\xi,r_{\perp})$
$\displaystyle=\frac{S_{\perp}}{z^{2}}x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}\frac{S_{x_{g}}^{(2)}(\vec{}r_{\perp})}{2\pi}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}r_{\perp}J_{0}(k_{\perp}r_{\perp})\bigl{(}\text{normal
part of }\hat{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (238b) $\displaystyle
F^{(n)}_{d2ji}(z,\xi,r_{\perp})\delta(1-\xi)$
$\displaystyle=\frac{S_{\perp}}{z^{2}}x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}\frac{S_{x_{g}}^{(2)}(\vec{}r_{\perp})}{2\pi}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}r_{\perp}J_{0}(k_{\perp}r_{\perp})\bigl{(}\text{delta
function part of }\hat{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (238c)

The entire term, analogous to equation (236), will be

$\biggl{(}\frac{\udddc\sigma}{\udc Y\uddc
p_{\perp}}\biggr{)}_{\text{dip}}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int_{0}^{\infty}\udc
r_{\perp}\biggl{[}\frac{F^{(n)}_{s2ji}(z,\xi,r_{\perp})-F^{(n)}_{s2ji}(z,1,r_{\perp})}{1-\xi}+F^{(n)}_{n2ji}(z,\xi,r_{\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc z\int_{0}^{\infty}\udc
r_{\perp}\biggl{[}F^{(n)}_{s2ji}(z,1,r_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(n)}_{d2ji}(z,1,r_{\perp})\biggr{]}$
(241)

Essentially, the change from Cartesian to radial coordinates, equation (237),
amounts to making the replacement

$e^{-iC_{0}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\to 2\pi
r_{\perp}J_{0}(C_{0}k_{\perp}r_{\perp})$ (242)

and adjusting the variable of integration accordingly. In practice, this is
the easiest way to determine the radial formulas: start with the Cartesian
position-space expression from [98], replace the Fourier factors with Bessel
functions à la (242), and then break the formula down into $F^{(n)}_{s2ji}$,
$F^{(n)}_{n2ji}$, and $F^{(n)}_{d2ji}$.

This formulation improves numerical accuracy compared to the Cartesian
implementation, by virtue of requiring one fewer oscillatory integral.

###### Momentum space

In some cases, we can express a contribution to the cross section in terms of
the momentum space gluon distribution as defined in equation (32). The
mathematical procedure for doing this is fairly specific to each individual
term, but it will work out to some convolution of $\mathcal{F}_{x_{g}}$ and
the Fourier transform of the rest of the term,
$\hat{}\mathcal{H}^{(n)}_{2ji}$:

$\tilde{}\mathcal{H}^{(n)}_{2ji}(k_{\perp})=\iint\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}e^{i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\hat{}\mathcal{H}^{(n)}_{2ji}(\vec{}r_{\perp})$
(243)

For example, suppose $\mathcal{S}=S_{x_{g}}^{(2)}$ in equation (237). We can
express $S_{x_{g}}^{(2)}$ as

$S_{x_{g}}^{(2)}(\vec{}r_{\perp})=\frac{1}{S_{\perp}}\iint\uddc\vec{}q_{\perp}e^{i\vec{}q_{\perp}\cdot\vec{}r_{\perp}}\mathcal{F}_{x_{g}}(\vec{}q_{\perp})$
(244)

and plug it in to the position integral of equation (231) to get

$\iint\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-iC_{0}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\hat{}\mathcal{H}^{(n)}_{2ji}(r_{\perp})=\frac{1}{S_{\perp}}\iint\uddc\vec{}q_{\perp}\mathcal{F}_{x_{g}}(\vec{}q_{\perp})\tilde{}\mathcal{H}^{(n)}_{2ji}(\vec{}q_{\perp}-C_{0}\vec{}k_{\perp})$
(245)

Or if $\mathcal{S}=\bigl{[}S_{x_{g}}^{(2)}\bigl{]}^{2}$, we get

$\iint\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}e^{-iC_{0}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\hat{}\mathcal{H}^{(n)}_{2ji}(r_{\perp})\\\
=\frac{1}{S_{\perp}^{2}}\iint\uddc\vec{}q_{1\perp}\mathcal{F}_{x_{g}}(\vec{}q_{1\perp})\iint\uddc\vec{}q_{2\perp}\mathcal{F}_{x_{g}}(\vec{}q_{2\perp})\tilde{}\mathcal{H}^{(n)}_{2ji}(C_{0}\vec{}k_{\perp}-\vec{}q_{1\perp}-\vec{}q_{2\perp})$
(248)

or so on. In general, there will be as many factors of $\mathcal{F}_{x_{g}}$
as there were of $S_{x_{g}}^{(2)}$, and an extra 2D integration for every
$\mathcal{F}_{x_{g}}$ beyond the first.

In the end, we’ll get an expression for the contribution to the cross section
of the form:

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{}p_{\perp}}\biggr{)}_{\text{dip}}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\iint\uddc\vec{}q_{\perp}\biggl{[}\frac{F^{(n)}_{s2ji}(z,\xi,\vec{}q_{\perp})-F^{(n)}_{s2ji}(z,1,\vec{}q_{\perp})}{1-\xi}+F^{(n)}_{n2ji}(z,\xi,\vec{}q_{\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\iint\uddc\vec{}q_{\perp}\biggl{[}F^{(n)}_{s2ji}(z,1,\vec{}q_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(n)}_{d2ji}(z,1,\vec{}q_{\perp})\biggr{]}$
(251)

where131313While the functions $F^{(n)}_{s2ji}$, $F^{(n)}_{n2ji}$, and
$F^{(n)}_{d2ji}$ do depend on $\vec{}k_{\perp}$, recall that
$\vec{}k_{\perp}=\frac{\vec{}p_{\perp}}{z}$, so the dependence on
$\vec{}k_{\perp}$ does not add any additional degrees of freedom beyond those
encapsulated in $z$. This is why the dependence on $\vec{}k_{\perp}$ is not
explicit in equations (252).

$\displaystyle\frac{F^{(n)}_{s2ji}(z,\xi,\vec{}q_{\perp})}{(1-\xi)_{+}}$
$\displaystyle=\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\mathcal{F}_{x_{g}}(\vec{}q_{\perp})\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\bigl{(}\text{plus-
regulated part of }\tilde{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (252a)
$\displaystyle F^{(n)}_{n2ji}(z,\xi,\vec{}q_{\perp})$
$\displaystyle=\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\mathcal{F}_{x_{g}}(\vec{}q_{\perp})\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\bigl{(}\text{normal
part of }\tilde{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (252b) $\displaystyle
F^{(n)}_{d2ji}(z,\xi,\vec{}q_{\perp})\delta(1-\xi)$
$\displaystyle=\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\mathcal{F}_{x_{g}}(\vec{}q_{\perp})\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\bigl{(}\text{delta
function part of }\tilde{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (252c)

As mentioned, the number of momentum variables involved depends on how many
factors of $S_{x_{g}}^{(2)}$ there are.

The advantage of this formulation is that, in momentum space,
$F^{(n)}_{s2ji}$, $F^{(n)}_{n2ji}$, and $F^{(n)}_{d2ji}$ don’t oscillate.
General-purpose integration algorithms can be wildly inaccurate when applied
to oscillating functions. Although specialized algorithms [102, 103, 104] do
exist for performing Fourier-type integrals, it’s still easier and more
accurate to use equation (251) so we can work with a relatively smooth, non-
oscillatory integrand.

However, it’s not always possible to express a given term in momentum space.
Unless the Fourier transform of $\hat{}\mathcal{H}^{(n)}_{2ji}$ (243) can be
computed symbolically, we won’t have a momentum-space expression to use. One
could compute that Fourier transform numerically, but that’s no different from
what we do in the radial position-space representation.

Furthermore, in deriving equation (252) we haven’t really _eliminated_ the
Fourier factor; we’ve just “shifted” it into the Fourier transform of the
dipole gluon distribution, equation (244). We still need to compute
$\mathcal{F}_{x_{g}}$ from $S_{x_{g}}^{(2)}$ before calculating the cross
section with equation (251). Still, the Fourier transform of $S_{x_{g}}^{(2)}$
tends to be “better behaved” than the hard factor $\mathcal{H}^{(n)}_{2ji}$.
$S_{x_{g}}^{(2)}$ is a relatively smooth function which generally takes a
roughly Gaussian form, and in fact under the assumption of angular
independence, we can perform one integration symbolically:

$\mathcal{F}_{x_{g}}(\vec{}k_{\perp})=S_{\perp}\iint\uddc\vec{}r_{\perp}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}S_{x_{g}}^{(2)}(\vec{}r_{\perp})=2\pi
S_{\perp}\int_{0}^{\infty}\udc
r_{\perp}r_{\perp}J_{0}(k_{\perp}r_{\perp})S_{x_{g}}^{(2)}(\vec{}r_{\perp})$
(253)

This is still an oscillatory integral, but a one-dimensional one, so standard
methods are much better able to handle it than they would a more complicated
multidimensional integral.

In fact, it’s possible to compute the solution to the BK equation in momentum
space directly. Theoretically we should be able to use that in equations
(252), but as described in section Numerical Analysis of BK Evolution, there
are some difficulties ensuring that the solutions in momentum space and
position space are consistent.

##### Quadrupole-Type Terms

The other terms in [98] are quadrupole-type, which involve integrals over
three transverse position coordinates, $\vec{}x_{\perp}$, $\vec{}b_{\perp}$,
and $\vec{}y_{\perp}$. These terms can be written

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4ji}^{(n)}=\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}f_{i}(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/j}(z,\mu)\int\frac{\uddc\vec{}x_{\perp}\uddc\vec{}b_{\perp}\uddc\vec{}y_{\perp}}{(2\pi)^{4}}\\\
\times\mathcal{S}(\vec{}x_{\perp},\vec{}b_{\perp},\vec{}y_{\perp})e^{-iC_{1}\vec{}k_{\perp}\cdot(\vec{}x_{\perp}-\vec{}b_{\perp})}e^{-iC_{2}\vec{}k_{\perp}\cdot(\vec{}y_{\perp}-\vec{}b_{\perp})}\hat{}\mathcal{H}^{(n)}_{4ji}(\vec{}x_{\perp},\vec{}b_{\perp},\vec{}y_{\perp})$
(256)

where $C_{1}$ and $C_{2}$ are, like $C_{0}$, coefficients independent of
position and momentum. For this type of term, the hard factor
$\mathcal{H}^{(n)}_{4ji}$ as given in [98] corresponds to the combination

$\mathcal{H}^{(n)}_{4ji}=e^{-iC_{1}\vec{}k_{\perp}\cdot(\vec{}x_{\perp}-\vec{}b_{\perp})}e^{-iC_{2}\vec{}k_{\perp}\cdot(\vec{}y_{\perp}-\vec{}b_{\perp})}\hat{}\mathcal{H}^{(n)}_{4ji}$
(257)

Handling these terms proceeds along the same general lines as the dipole-type
terms, with a few differences to account for the additional coordinates. First
we change variables to

$\displaystyle\vec{R}_{\perp}$ $\displaystyle=\vec{}x_{\perp}+\vec{}y_{\perp}$
$\displaystyle\vec{}s_{\perp}$ $\displaystyle=\vec{}x_{\perp}-\vec{}b_{\perp}$
$\displaystyle\vec{}t_{\perp}$ $\displaystyle=\vec{}y_{\perp}-\vec{}b_{\perp}$
(258)

and $\vec{}r_{\perp}=\vec{}s_{\perp}-\vec{}t_{\perp}$. As in the dipole case,
$\hat{}\mathcal{H}^{(n)}_{4ji}$ is independent of $\vec{R}_{\perp}$ so we can
integrate over that degree of freedom.

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4ji}^{(n)}=S_{\perp}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}f_{i}(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/j}(z,\mu)\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}\\\
\times\mathcal{S}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-iC_{1}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-iC_{2}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\hat{}\mathcal{H}^{(n)}_{4ji}(\vec{}s_{\perp},\vec{}t_{\perp})$
(261)

Now, $\mathcal{S}\hat{}\mathcal{H}^{(n)}_{4ji}$ is a scalar, so it depends on
$\vec{}s_{\perp}$ and $\vec{}t_{\perp}$ through scalar functions of those
variables: $s_{\perp}^{2}$, $t_{\perp}^{2}$, and
$\vec{}s_{\perp}\cdot\vec{}t_{\perp}=\frac{1}{2}(s_{\perp}^{2}+t_{\perp}^{2}-r_{\perp}^{2})$.
This means we can’t integrate over the angular variables of both
$\vec{}s_{\perp}$ and $\vec{}t_{\perp}$, for the following reason. In
Cartesian position space, $\mathcal{S}\hat{}\mathcal{H}^{(n)}_{4ji}$ is
written as a function of four variables, $s_{x\perp}$, $s_{y\perp}$,
$t_{x\perp}$, and $t_{y\perp}$. We can express it as a function of three
variables, $r_{\perp}$, $s_{\perp}$, and $t_{\perp}$, but not two. So we can
only hope to integrate over one angle. In principle, we could do that
integration, but we’re not going to have a simple substitution rule to do so,
as we did with the dipole-type terms (equation (242)).

In practice, all the quadrupole-type terms in [98] can be expressed in
momentum space, so it’s not worth the effort to find the radial representation
at all.141414If one were inclined to do so, it could serve as a potentially
useful check on the momentum-space implementation.

###### Cartesian position space

As with the dipole terms, we can practically copy the Cartesian position space
formula from reference [98].

$\displaystyle\frac{F^{(n)}_{s4ji}(z,\xi,\vec{}r_{\perp})}{(1-\xi)_{+}}$
$\displaystyle=\begin{multlined}S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp},s_{\perp},t_{\perp})}{(2\pi)^{4}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\\\
\times
e^{-iC_{1}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-iC_{2}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\bigl{(}\text{plus-
regulated part of
}\hat{}\mathcal{H}^{(n)}_{4ji}\bigr{)}\end{multlined}S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp},s_{\perp},t_{\perp})}{(2\pi)^{4}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\\\
\times
e^{-iC_{1}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-iC_{2}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\bigl{(}\text{plus-
regulated part of }\hat{}\mathcal{H}^{(n)}_{4ji}\bigr{)}$ (262c)
$\displaystyle F^{(n)}_{n4ji}(z,\xi,\vec{}r_{\perp})$
$\displaystyle=\begin{multlined}S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp},s_{\perp},t_{\perp})}{(2\pi)^{4}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\\\
\times
e^{-iC_{1}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-iC_{2}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\bigl{(}\text{normal
part of
}\hat{}\mathcal{H}^{(n)}_{4ji}\bigr{)}\end{multlined}S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp},s_{\perp},t_{\perp})}{(2\pi)^{4}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\\\
\times
e^{-iC_{1}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-iC_{2}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\bigl{(}\text{normal
part of }\hat{}\mathcal{H}^{(n)}_{4ji}\bigr{)}$ (262f) $\displaystyle
F^{(n)}_{d4ji}(z,\xi,\vec{}r_{\perp})\delta(1-\xi)$
$\displaystyle=\begin{multlined}S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp},s_{\perp},t_{\perp})}{(2\pi)^{4}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\\\
\times
e^{-iC_{1}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-iC_{2}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\bigl{(}\text{delta
function part of
}\hat{}\mathcal{H}^{(n)}_{4ji}\bigr{)}\end{multlined}S_{\perp}\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\frac{\mathcal{S}(r_{\perp},s_{\perp},t_{\perp})}{(2\pi)^{4}}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\\\
\times
e^{-iC_{1}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-iC_{2}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\bigl{(}\text{delta
function part of }\hat{}\mathcal{H}^{(n)}_{4ji}\bigr{)}$ (262i)

and by analogy with equation (236), the term becomes

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4ji}^{(n)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}\\\
\times\biggl{[}\frac{F^{(n)}_{s4ji}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})-F^{(n)}_{s4ji}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})}{1-\xi}+F^{(n)}_{n4ji}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}\biggl{[}F^{(n)}_{s4ji}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(n)}_{d4ji}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})\biggr{]}$
(266)

###### Momentum space

Also by analogy with the dipole terms, we can Fourier transform quadrupole-
type terms into momentum space. The resulting expression includes the momentum
space hard factor:

$\tilde{}\mathcal{H}^{(n)}_{4ji}(k_{\perp})=\iint\frac{\uddc\vec{}s_{\perp}}{(2\pi)^{2}}\iint\frac{\uddc\vec{}t_{\perp}}{(2\pi)^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\hat{}\mathcal{H}^{(n)}_{4ji}(\vec{}s_{\perp},\vec{}t_{\perp})$
(267)

and either $\mathcal{G}_{x_{g}}$ or factors of $\mathcal{F}_{x_{g}}$,
depending on the gluon distribution in each individual term. As a
straightforward example, the structure

$\int\frac{\uddc\vec{}s_{\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}t_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-iC_{1}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{-iC_{2}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\hat{}\mathcal{H}^{(n)}_{4ji}(\vec{}s_{\perp},\vec{}t_{\perp})$
(268)

becomes

$\int\frac{\uddc\vec{}q_{1\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}q_{2\perp}}{(2\pi)^{2}}\mathcal{G}_{x_{g}}(\vec{}q_{1\perp},\vec{}q_{2\perp})\tilde{}\mathcal{H}^{(n)}_{4ji}(\vec{}q_{1\perp}-C_{1}\vec{}k_{\perp},\vec{}q_{2\perp}-C_{2}\vec{}k_{\perp})$
(269)

Of course the exact nature of the expression varies significantly from term to
term.

In the end the contribution of a quadrupole-type term takes the form

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{}p_{\perp}}\biggr{)}_{\text{quad}}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\\\
\times\biggl{[}\frac{F^{(n)}_{s4ji}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp})-F^{(n)}_{s4ji}(z,1,\vec{}q_{1\perp},\vec{}q_{2\perp})}{1-\xi}+F^{(n)}_{n4ji}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\biggl{[}F^{(n)}_{s4ji}(z,1,\vec{}q_{1\perp},\vec{}q_{2\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(n)}_{d4ji}(z,1,\vec{}q_{1\perp},\vec{}q_{2\perp})\biggr{]}$
(273)

with

$\displaystyle\frac{F^{(n)}_{s4ji}(z,\xi,\vec{}q_{\perp})}{(1-\xi)_{+}}$
$\displaystyle=\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\mathcal{G}_{x_{g}}(\vec{}q_{\perp})\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\bigl{(}\text{plus-
regulated part of }\tilde{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (274a)
$\displaystyle F^{(n)}_{n4ji}(z,\xi,\vec{}q_{\perp})$
$\displaystyle=\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\mathcal{G}_{x_{g}}(\vec{}q_{\perp})\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\bigl{(}\text{normal
part of }\tilde{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (274b) $\displaystyle
F^{(n)}_{d4ji}(z,\xi,\vec{}q_{\perp})\delta(1-\xi)$
$\displaystyle=\frac{x_{p}f_{i}D_{\HepGenParticle{h}{}{}\xspace/j}}{z^{2}}\mathcal{G}_{x_{g}}(\vec{}q_{\perp})\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}\bigl{(}\text{delta
function part of }\tilde{}\mathcal{H}^{(n)}_{2ji}\bigr{)}$ (274c)

As mentioned, the number of momentum variables involved depends on how many
factors of $S_{x_{g}}^{(2)}$ there are.

### Summary of Terms

The formulas in reference [98] generically take the form

$\frac{\udddc\sigma}{\udc
Y\uddc\vec{}p_{\perp}}=x_{p}f(x_{p},\mu)\otimes\mathcal{S}\otimes\mathcal{H}\otimes
D(z,\mu^{2})$ (275)

where $\mathcal{S}$ is some combination of dipole or quadrupole gluon
distributions, and $\mathcal{H}$ is a hard factor, representing the
perturbative part of the calculation. It’s straightforward to extract the
Cartesian position space formulas from the expressions given in [98], but an
accurate numerical implementation requires the radial or momentum space
formulas, which need to be derived. The purpose of this section is to show
those derivations.

Table 1 summarizes which equations give the contribution corresponding to each
hard factor in each implementation, for easy reference.

Term |  Cartesian |  Radial |  Momentum
---|---|---|---
$\mathcal{H}^{(0)}_{2qq}$ |  (291) with (292) |  |  (296) with (295)
$\mathcal{H}^{(1)}_{2qq}$ |  (316) with (309), (313) |  (321) with (317), (318) |
$\mathcal{H}^{(1)}_{4qq}$ |  (355)
with (328), (350), (351) |  |  (378)
with (366), (374)
$\mathcal{H}^{(0)}_{2gg}$ |  (298) with (299) |  (300) with (301) |  (302) with (303)
$\mathcal{H}^{(1)}_{2gg}$ |  (396)
with (385), (388), (393) |  (408)
with (399), (402), (405) |
$\mathcal{H}^{(1)}_{2q\bar{q}}$ |  (421) with (424) |  |  (432) with (431)
$\mathcal{H}^{(1)}_{6gg}$ |  (455)
with (439), (450), (451) |  |  (482)
with (468), (477)
$\mathcal{H}^{(1,1)}_{2gq}$ |  (487) with (486) |  (489) with (488) |
$\mathcal{H}^{(1,2)}_{2gq}$ |  (494) with (493) |  (496) with (493) |
$\mathcal{H}^{(1)}_{4gq}$ |  (503) with (502) |  |  (509) with (512)
$\mathcal{H}^{(1,1)}_{2qg}$ |  (519) with (518) |  (523) with (522) |
$\mathcal{H}^{(1,2)}_{2qg}$ |  (530) with (529) |  (534) with (533) |
$\mathcal{H}^{(1)}_{4qg}$ |  (541) with (540) |  |  (545) with (544)
Table 1: Summary of formulas for each hard factor

#### $\mathcal{H}^{(0)}_{2qq}$

In reference [98], the term from the leading order $qq$ channel is given in
equations (41) and (42) as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qq}^{(0)}=S_{\perp}\int\frac{\udc
z\udc\xi}{z^{2}}\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\delta(1-\xi)$
(290)

This can be expressed as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qq}^{(0)}=\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}F^{(0)}_{d2qq}(z,1,\vec{}r_{\perp})$ (291)

where

$F^{(0)}_{d2qq}(z,\xi,\vec{}r_{\perp})=\frac{S_{\perp}}{(2\pi)^{2}}\frac{1}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}$
(292)

We can translate this into the radial formulation easily using equation (242),
obtaining

$F^{(0)}_{d2qq}(z,\xi,r_{\perp})=\frac{S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\frac{S_{x_{g}}^{(2)}(\vec{}r_{\perp})}{2\pi}\biggl{(}\frac{\alpha_{s}}{2\pi}\biggr{)}^{n}r_{\perp}J_{0}(k_{\perp}r_{\perp})$
(293)

which gets integrated as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qq}^{(0)}=\int_{\tau}^{1}\udc
z\int_{0}^{\infty}\udc r_{\perp}F^{(0)}_{d2qq}(z,1,r_{\perp})$ (294)

And for the momentum space implementation, equation (245) gives us
$\tilde{}\mathcal{H}^{(0)}_{2qq}(\vec{}k_{\perp})=\delta(1-\xi)\delta^{(2)}(\vec{}k_{\perp})$,
and then (252) we get

$F^{(0)}_{d2qq}(z,\xi,\vec{}k_{\perp})=\frac{1}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\mathcal{F}_{x_{g}}(\vec{}k_{\perp})$
(295)

This case is particularly simple because no momentum integration at all is
needed:

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qq}^{(0)}=\int_{\tau}^{1}\udc
zF^{(0)}_{d2qq}(z,1,\vec{}k_{\perp})$ (296)

In each case $F_{s}$ and $F_{n}$ are, of course, zero.

#### $\mathcal{H}^{(0)}_{2gg}$

The expression for the leading-order gluon contribution comes from equations
(74) and (75) of [98]:

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gg}^{(0)}=S_{\perp}\int\frac{\udc
z\udc\xi}{z^{2}}\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\delta(1-\xi)$
(297)

The Cartesian position space implementation is straightforward,

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gg}^{(0)}=\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}F^{(0)}_{d2gg}(z,1,\vec{}r_{\perp})$ (298)

where

$F^{(0)}_{d2gg}(z,\xi,\vec{}r_{\perp})=\frac{S_{\perp}}{(2\pi)^{2}}\frac{1}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)[S_{x_{g}}^{(2)}(\vec{}r_{\perp})]^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}$
(299)

and again using (242) for the radial implementation, we find

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gg}^{(0)}=\int_{\tau}^{1}\udc z\int\udc
r_{\perp}r_{\perp}F^{(0)}_{d2gg}(z,1,r_{\perp})$ (300)

where

$F^{(0)}_{d2gg}(z,\xi,r_{\perp})=\frac{S_{\perp}}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\frac{[S_{x_{g}}^{(2)}(\vec{}r_{\perp})]^{2}}{2\pi}r_{\perp}J_{0}(k_{\perp}r_{\perp})$
(301)

For the momentum space implementation, we get an extra integration, as in
(248). We wind up with

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gg}^{(0)}=\int_{\tau}^{1}\udc
z\iint\uddc\vec{}q_{\perp}F^{(0)}_{d2gg}(z,1,\vec{}q_{\perp})$ (302)

where

$F^{(0)}_{d2gg}(z,\xi,\vec{}q_{\perp})=\frac{1}{S_{\perp}z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\mathcal{F}_{x_{g}}(\vec{}q_{\perp})\mathcal{F}_{x_{g}}(\vec{}k_{\perp}-\vec{}q_{\perp})$
(303)

It will be typical for most terms that the momentum-space expressions have to
be derived on a case-by-case basis. There are not as many patterns that can be
exploited, as there are with the Cartesian expressions.

#### $\mathcal{H}^{(1)}_{2qq}$

From equations (41) and (43) of [98], the $\mathcal{H}^{(1)}_{2qq}$
contribution to the cross section in the large-$N_{c}$ limit is given as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qq}^{(1)}=\int\frac{\udc
z\udc\xi}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}C_{F}\biggl{[}\frac{1+\xi^{2}}{(1-\xi)_{+}}+\frac{3}{2}\delta(1-\xi)\biggr{]}\\\
\times\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\frac{\vec{}k_{\perp}}{\xi}\cdot\vec{}r_{\perp}}\biggr{)}-3C_{F}\delta(1-\xi)e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}k_{\perp}^{2}}$
(306)

This expression consists of individual terms which follow the pattern of
equation (226), so we can just substitute into the right side of (232) to get

$F^{(1)}_{s2qq}(z,\xi,\vec{}r_{\perp})=\frac{1}{4\pi^{2}}\frac{\alpha_{s}}{2\pi}\frac{C_{F}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})(1+\xi^{2})\\\
\times\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\frac{\vec{}k_{\perp}}{\xi}\cdot\vec{}r_{\perp}}\biggr{)}$
(309)

and

$F^{(1)}_{d2qq}(z,\xi,\vec{}r_{\perp})=\frac{3}{4\pi^{2}}\frac{\alpha_{s}}{2\pi}\frac{C_{F}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})\\\
\times\biggl{[}\frac{1}{2}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\frac{\vec{}k_{\perp}}{\xi}\cdot\vec{}r_{\perp}}\biggr{)}-e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}k_{\perp}^{2}}\biggr{]}$
(312)

As a delta contribution, this last equation will only ever be evaluated at
$\xi=1$, so we can plug that in immediately and get a simpler expression:

$F^{(1)}_{d2qq}(z,1,\vec{}r_{\perp})=\frac{3}{4\pi^{2}}\frac{\alpha_{s}}{2\pi}\frac{C_{F}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\ln\frac{k_{\perp}^{2}}{\mu^{2}}$
(313)

Implementing this expression instead of the general one (312) reduces the
chances of mistakes in the implementation and reduces any numerical errors in
the evaluation.

Expressions (309) and (313) go into (236)

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qq}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}r_{\perp}\frac{F^{(1)}_{s2qq}(z,\xi,\vec{}r_{\perp})-F^{(1)}_{s2qq}(z,1,\vec{}r_{\perp})}{1-\xi}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}\biggl{[}F^{(1)}_{s2qq}(z,1,\vec{}r_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(1)}_{d2qq}(z,1,\vec{}r_{\perp})\biggr{]}$
(316)

to produce the final contribution to the cross section.

To transform into the radial representation, we use equation (242) to get

$\displaystyle F^{(1)}_{s2qq}(z,\xi,r_{\perp})$
$\displaystyle=\frac{1}{2\pi}\frac{\alpha_{s}}{2\pi}\frac{C_{F}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})$
$\displaystyle\quad\times(1+\xi^{2})r_{\perp}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{[}J_{0}(k_{\perp}r_{\perp})+\frac{1}{\xi^{2}}J_{0}\biggl{(}\frac{k_{\perp}r_{\perp}}{\xi}\biggr{)}\biggr{]}$
(317) $\displaystyle F^{(1)}_{d2qq}(z,1,r_{\perp})$
$\displaystyle=\frac{3}{2\pi}\frac{\alpha_{s}}{2\pi}\frac{C_{F}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})r_{\perp}J_{0}(k_{\perp}r_{\perp})\ln\frac{k_{\perp}^{2}}{\mu^{2}}$
(318)

and the contribution to the cross section will be

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qq}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}r_{\perp}\frac{F^{(1)}_{s2qq}(z,\xi,\vec{}r_{\perp})-F^{(1)}_{s2qq}(z,1,\vec{}r_{\perp})}{1-\xi}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}\biggl{[}F^{(1)}_{s2qq}(z,1,\vec{}r_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(1)}_{d2qq}(z,1,\vec{}r_{\perp})\biggr{]}$
(321)

The logarithmic factors $\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$ can’t be
Fourier transformed symbolically, so there is no momentum-space expression for
this term.

#### $\mathcal{H}^{(1)}_{4qq}$

From equations (41) and (45) of [98], this contribution to the cross section
is

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4qq}^{(1)}=-4\pi
N_{c}S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\int\frac{\udc
z\udc\xi}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\\\
\times\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\Biggl{\\{}e^{-i\frac{1-\xi}{\xi}\vec{}k_{\perp}\cdot\vec{}s_{\perp}}\frac{1+\xi^{2}}{(1-\xi)_{+}}\frac{1}{\xi}\frac{\vec{}s_{\perp}}{s_{\perp}^{2}}\cdot\frac{\vec{}t_{\perp}}{t_{\perp}^{2}}\\\
-\delta(1-\xi)\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{(1-\xi^{\prime})_{+}}\Biggl{[}\frac{e^{-i(1-\xi^{\prime})\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}-\delta^{(2)}(\vec{}t_{\perp})\int\uddc\vec{}r_{\perp}^{\prime}\frac{e^{i\vec{}k_{\perp}\cdot\vec{r_{\perp}^{\prime}}}}{r_{\perp}^{\prime
2}}\Biggr{]}\Biggr{\\}}$ (325)

where $\vec{}s_{\perp}=\vec{}x_{\perp}-\vec{}b_{\perp}$ and
$\vec{}t_{\perp}=\vec{}y_{\perp}-\vec{}b_{\perp}$. The first term (first and
second lines) is easy: it directly translates into a plus-regulated
contribution,

$F^{(1)}_{s4qq}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})=-\frac{1}{4\pi^{3}}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})\\\
\times
e^{-i\vec{}k_{\perp}\cdot(\vec{}s_{\perp}/\xi-\vec{}t_{\perp})}\frac{1+\xi^{2}}{\xi}\frac{\vec{}s_{\perp}}{s_{\perp}^{2}}\cdot\frac{\vec{}t_{\perp}}{t_{\perp}^{2}}$
(328)

where I’ve used
$\vec{}r_{\perp}+\vec{}s_{\perp}\frac{1-\xi}{\xi}=\vec{}r_{\perp}+\frac{\vec{}s_{\perp}}{\xi}-\vec{}s_{\perp}=\frac{\vec{}s_{\perp}}{\xi}-\vec{}t_{\perp}$
to construct the Fourier exponent.

The delta contribution is somewhat more complicated, though, because it has an
additional delta function $\delta^{(2)}(\vec{}t_{\perp})$ which we need to
integrate symbolically first. We’ll have to calculate this integral:

$\iint\uddc\vec{}s_{\perp}\iint\uddc\vec{}t_{\perp}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})\\\
\times\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{(1-\xi^{\prime})_{+}}\Biggl{[}\frac{e^{-i(1-\xi^{\prime})\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}-\delta^{(2)}(\vec{}t_{\perp})\int\uddc\vec{}r_{\perp}^{\prime}\frac{e^{i\vec{}k_{\perp}\cdot\vec{}r_{\perp}^{\prime}}}{r_{\perp}^{\prime
2}}\Biggr{]}$ (331)

Each of the two terms in square brackets is individually divergent, so we need
to regulate them, but we can’t do the entire integral symbolically because of
the unknown gluon distribution $S_{x_{g}}^{(4)}$.

The trick is to distribute the $\vec{}t_{\perp}$ integration and impose a
lower cutoff $\epsilon$:

$\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{(1-\xi^{\prime})_{+}}\Biggl{[}\iint\uddc\vec{}s_{\perp}\iint_{t_{\perp}>\epsilon}\uddc\vec{}t_{\perp}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot(\vec{}s_{\perp}-\vec{}t_{\perp})}\frac{e^{-i(1-\xi^{\prime})\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}\\\
-\iint\uddc\vec{}s_{\perp}\iint_{r_{\perp}^{\prime}>\epsilon}\uddc\vec{}r_{\perp}^{\prime}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{0})e^{-i\vec{}k_{\perp}\cdot\vec{}s_{\perp}}\frac{e^{i\vec{}k_{\perp}\cdot\vec{}r_{\perp}^{\prime}}}{r_{\perp}^{\prime
2}}\Biggr{]}$ (334)

Note that integrating with $\delta^{(2)}(\vec{}t_{\perp})$ sets
$\vec{}r_{\perp}\to\vec{}s_{\perp}$ in the second integral. Then we change
variables $\vec{}r_{\perp}^{\prime}\to\xi^{\prime}\vec{}t_{\perp}$.

$\iint\uddc\vec{}s_{\perp}\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{(1-\xi^{\prime})_{+}}\Biggl{[}\iint_{t_{\perp}>\epsilon}\uddc\vec{}t_{\perp}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}s_{\perp}}\frac{e^{i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}\\\
-S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{0})\iint_{t_{\perp}>\epsilon/\xi^{\prime}}\uddc\vec{}t_{\perp}e^{-i\vec{}k_{\perp}\cdot\vec{}s_{\perp}}\frac{e^{i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}\Biggr{]}$
(337)

Now we split the second integral using

$\int_{t_{\perp}>\epsilon/\xi^{\prime}}=\int_{t_{\perp}>\epsilon}-\int_{t_{\perp}>\epsilon}^{t_{\perp}<\epsilon/\xi^{\prime}}$
(338)

The first term of (338) combines with the first term of (337) to give

$\iint\uddc\vec{}s_{\perp}\iint_{t_{\perp}>\epsilon}\uddc\vec{}t_{\perp}\bigl{[}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})-S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{0})\bigr{]}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{(1-\xi^{\prime})_{+}}\frac{e^{-i(1-\xi^{\prime})\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}$
(339)

and the second term becomes

$-\iint\uddc\vec{}s_{\perp}\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{(1-\xi^{\prime})_{+}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{0})\iint_{t_{\perp}>\epsilon}^{t_{\perp}<\epsilon/\xi^{\prime}}\uddc\vec{}t_{\perp}\frac{e^{i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}\\\
=-\iint\uddc\vec{}s_{\perp}\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{(1-\xi^{\prime})_{+}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{0})2\pi\ln\frac{1}{\xi^{\prime}}$
(342)

where, in performing the $\vec{}t_{\perp}$ integral, I’ve used the fact that
$t_{\perp}$ is small so the exponential factor can be approximated as $1$.

Finally, the integrals over $\xi^{\prime}$ in (339) and (342) can be done
symbolically, giving

$\iint\uddc\vec{}s_{\perp}\iint_{t_{\perp}>\epsilon}\uddc\vec{}t_{\perp}\bigl{[}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})-S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{0})\bigr{]}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\frac{\mathcal{I}_{1}(\vec{}k_{\perp}\cdot\vec{}t_{\perp})}{t_{\perp}^{2}}\\\
-\iint\uddc\vec{}s_{\perp}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{0})2\pi\biggl{(}-\frac{5}{4}+\frac{\pi^{2}}{3}\biggr{)}$
(345)

where

$\mathcal{I}_{1}(x)=-2\gamma_{E}+\frac{i(2-e^{-ix})}{x}+\frac{e^{-ix}-1}{x^{2}}+2\operatorname{Ci}(x)-2i\operatorname{Si}(x)-2\ln
x$ (346)

with $\operatorname{Ci}$ and $\operatorname{Si}$ defined as

$\displaystyle\operatorname{Ci}(z)$
$\displaystyle=-\int_{z}^{\infty}\frac{\cos t}{t}\udc t$
$\displaystyle\operatorname{Si}(z)$ $\displaystyle=\int_{0}^{z}\frac{\sin
t}{t}\udc t$ (347)

Note that the imaginary parts of $\operatorname{Ci}(x)$ and $\ln x$ for $x<0$
cancel out on the principal branch of each function.

So in Cartesian position space, $\mathcal{H}^{(1)}_{4qq}$ actually contributes
both a quadrupole term,

$F^{(1)}_{d4qq}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})=\frac{1}{4\pi^{3}}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\\\
\times\Bigl{[}S_{Y}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})-S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{0})\Bigr{]}\frac{1}{t_{\perp}^{2}}e^{-i\vec{k}_{\perp}\cdot\vec{r}_{\perp}}\mathcal{I}_{1}(\vec{k}_{\perp}\cdot\vec{t}_{\perp})$
(350)

_and_ a dipole term

$F^{(1)}_{d2^{\prime}qq}(z,\xi,\vec{}r_{\perp})=\frac{1}{2\pi^{2}}\biggl{(}-\frac{5}{4}+\frac{\pi^{2}}{3}\biggr{)}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{Y}^{(4)}(\vec{}r_{\perp},\vec{0})e^{-i\vec{k}_{\perp}\cdot\vec{r}_{\perp}}$
(351)

and the complete contribution to the cross section is

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4qq}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}\biggl{[}\frac{F^{(1)}_{s4qq}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})-F^{(1)}_{s4qq}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})}{1-\xi}\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}\biggl{[}F^{(1)}_{s4qq}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(1)}_{d4qq}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}F^{(1)}_{d2^{\prime}qq}(z,1,\vec{}r_{\perp})$ (355)

To get the momentum space formulation of the plus-regulated part, it will be
easiest to start with the Cartesian space result from equations (328) as
plugged into (355). For simplicity, I’ll extract only the parts that depend on
$\vec{}s_{\perp}$ and $\vec{}t_{\perp}$:

$\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot(\vec{}s_{\perp}/\xi-\vec{}t_{\perp})}\frac{\vec{}s_{\perp}}{s_{\perp}^{2}}\cdot\frac{\vec{}t_{\perp}}{t_{\perp}^{2}}$
(356)

Although it is possible to convert this into momentum space by plugging in the
inverse Fourier transform of $S_{x_{g}}^{(4)}$, equation (59), it’s easier to
judiciously insert some delta functions.

$\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot(\vec{}s_{\perp}/\xi-\vec{}t_{\perp})}\int\uddc\vec{s_{\perp}^{\prime}}\uddc\vec{t_{\perp}^{\prime}}\delta^{(2)}(\vec{}s_{\perp}-\vec{s_{\perp}^{\prime}})\delta^{(2)}(\vec{}t_{\perp}-\vec{t_{\perp}^{\prime}})\frac{\vec{s_{\perp}^{\prime}}}{s_{\perp}^{\prime
2}}\cdot\frac{\vec{t_{\perp}^{\prime}}}{t_{\perp}^{\prime 2}}$ (357)

Note that this expression is exactly equivalent to the previous one. We can
then replace the delta functions with their Fourier transforms according to
equation (645)

$\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot(\vec{}s_{\perp}/\xi-\vec{}t_{\perp})}\\\
\times\int\uddc\vec{s_{\perp}^{\prime}}\uddc\vec{t_{\perp}^{\prime}}\int\frac{\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}}{(2\pi)^{4}}e^{-i\vec{}q_{1\perp}\cdot(\vec{}s_{\perp}-\vec{s_{\perp}^{\prime}})}e^{-i\vec{}q_{2\perp}\cdot(\vec{}t_{\perp}-\vec{t_{\perp}^{\prime}})}\frac{\vec{s_{\perp}^{\prime}}}{s_{\perp}^{\prime
2}}\cdot\frac{\vec{t_{\perp}^{\prime}}}{t_{\perp}^{\prime 2}}$ (360)

which can be rearranged to separate the two groups of position integrals:

$\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i(\vec{}k_{\perp}/\xi+q_{1\perp})\cdot\vec{}s_{\perp}}e^{i(\vec{}k_{\perp}-\vec{}q_{2\perp})\cdot\vec{}t_{\perp}}\\\
\times\int\uddc\vec{s_{\perp}^{\prime}}\uddc\vec{t_{\perp}^{\prime}}e^{i\vec{}q_{1\perp}\cdot\vec{s_{\perp}^{\prime}}}e^{i\vec{}q_{2\perp}\cdot\vec{t_{\perp}^{\prime}}}\frac{\vec{s_{\perp}^{\prime}}}{s_{\perp}^{\prime
2}}\cdot\frac{\vec{t_{\perp}^{\prime}}}{t_{\perp}^{\prime 2}}$ (363)

The second line works out to
$-(2\pi)^{2}\frac{\vec{}q_{1\perp}\cdot\vec{}q_{2\perp}}{q_{1\perp}^{2}q_{2\perp}^{2}}$,
from equation (630), and the integral over $\vec{}s_{\perp}$ and
$\vec{}t_{\perp}$ on the first line is equal to
$\frac{1}{S_{\perp}}\mathcal{G}_{x_{g}}\bigl{(}\frac{\vec{}k_{\perp}}{\xi}+\vec{}q_{1\perp},\vec{}k_{\perp}-\vec{}q_{2\perp}\bigr{)}$
according to equation (53). Putting this result together with the extra
factors that were dropped to get equation (356), we get

$F^{(1)}_{s4qq}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp})=\frac{1}{\pi}\frac{\alpha_{s}}{2\pi}\frac{N_{c}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\\\
\times\frac{1+\xi^{2}}{\xi}\mathcal{G}_{x_{g}}\biggl{(}\frac{\vec{}k_{\perp}}{\xi}+\vec{}q_{1\perp},\vec{}k_{\perp}-\vec{}q_{2\perp}\biggr{)}\frac{\vec{}q_{1\perp}\cdot\vec{}q_{2\perp}}{q_{1\perp}^{2}q_{2\perp}^{2}}$
(366)

which will be integrated over $\vec{}q_{1\perp}$ and $\vec{}q_{2\perp}$ in
equation (378).

For the delta function term, we can use the identity in equation (34) of [98],
with $\bar{\vec{r}}_{\perp}=\vec{}t_{\perp}$. Multiplying both sides of that
equation by $-\frac{1}{4\pi}e^{i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}$ and
taking the complex conjugate gives

$-\frac{1}{4\pi}\iint\uddc\vec{}q_{1\perp}e^{i(\vec{}q_{1\perp}-\vec{}k_{\perp})\cdot\vec{}t_{\perp}}\ln\frac{(\vec{}q_{1\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}\\\
=\biggl{[}\frac{e^{-i(1-\xi^{\prime})\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}-\delta^{(2)}(\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\iint\frac{\uddc\vec{}r_{\perp}^{\prime}}{r_{\perp}^{\prime
2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}^{\prime}}\biggr{]}$ (369)

Since this is going to be integrated over $\vec{}t_{\perp}$, the
$e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}$ factor in the second term will be
equal to 1 anyway (because of the delta function), and taking that into
account, the right side of this equation is exactly the last factor of the
delta-function term in equation (325). Restoring the factors and the integral
from the second line of (325) gives

$-\frac{1}{4\pi}\iint\uddc\vec{}q_{1\perp}\underbrace{\iint\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}s_{\perp}}e^{i\vec{}q_{1\perp}\cdot\vec{}t_{\perp}}}_{\mathcal{G}_{x_{g}}(\vec{}k_{\perp},-\vec{}q_{1\perp})/S_{\perp}}\ln\frac{(\vec{}q_{1\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}$
(370)

Restoring the factors from the first line of (325) gives the delta
contribution. The plus prescription factor, $\frac{1}{(1-\xi^{\prime})_{+}}$,
can be handled symbolically:

$\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{(1-\xi^{\prime})_{+}}\ln\frac{(\vec{}q_{1\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}=\int_{0}^{1}\udc\xi^{\prime}\frac{1+\xi^{\prime
2}}{1-\xi^{\prime}}\ln\frac{(\vec{}q_{1\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}-\frac{2}{1-\xi^{\prime}}\ln\frac{(\vec{}q_{1\perp}-\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}$
(371)

making the final expression for the delta contribution

$F^{(1)}_{d4qq}=-N_{c}\frac{\alpha_{\text{s}}}{2\pi}\frac{1}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\mathcal{G}_{x_{g}}(\vec{}k_{\perp},-\vec{}q_{1\perp})\\\
\times\biggl{[}\frac{1+\xi^{\prime
2}}{1-\xi^{\prime}}\ln\frac{(\vec{}q_{1\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}-\frac{2}{1-\xi^{\prime}}\ln\frac{(\vec{}q_{1\perp}-\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}\biggr{]}$
(374)

The complete contribution from $\mathcal{H}^{(1)}_{4qq}$ in momentum space is
then

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4qq}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\biggl{[}\frac{F^{(1)}_{s4qq}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp})-F^{(1)}_{s4qq}(z,1,\vec{}q_{1\perp},\vec{}q_{2\perp})}{1-\xi}\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}F^{(1)}_{s4qq}(z,1,\vec{}q_{1\perp},\vec{}q_{2\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}q_{1\perp}\int_{0}^{1}\udc\xi^{\prime}F^{(1)}_{d4qq}(z,1,\vec{}q_{1\perp},\xi^{\prime})$
(378)

with the functions $F^{(1)}_{s4qq}$ and $F^{(1)}_{d4qq}$ from equations (366)
and (374) respectively.

#### $\mathcal{H}^{(1)}_{2gg}$

From equations (74) and (76) in [98], the contribution of the
$\mathcal{H}^{(1)}_{2gg}$ term to the cross section is

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gg}^{(1)}=N_{c}S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\\\
\times\Biggl{[}\biggl{\\{}2\biggl{[}\frac{\xi}{(1-\xi)_{+}}+\frac{1-\xi}{\xi}+\xi(1-\xi)\biggr{]}+\biggl{(}\frac{11}{6}-\frac{2N_{f}T_{R}}{3N_{c}}\biggr{)}\delta(1-\xi)\biggr{\\}}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\biggr{)}\\\
-\biggl{(}\frac{11}{3}-\frac{4N_{f}T_{R}}{3N_{c}}\biggr{)}N_{c}\delta(1-\xi)e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}k_{\perp}^{2}}\Biggr{]}$
(382)

As with $\mathcal{H}^{(1)}_{2qq}$, this consists of terms that follow the
pattern of equation (215), so it’s quite straightforward to split it into the
three contributions:

$\displaystyle F^{(1)}_{s2gg}(z,\xi,\vec{}r_{\perp})$
$\displaystyle=\begin{multlined}\frac{2N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\xi\\\
\times\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\biggr{)}\end{multlined}\frac{2N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\xi\\\
\times\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\biggr{)}$
(385) $\displaystyle F^{(1)}_{n2gg}(z,\xi,\vec{}r_{\perp})$
$\displaystyle=\begin{multlined}\frac{2N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\biggl{(}\frac{1-\xi}{\xi}+\xi(1-\xi)\biggr{)}\\\
\times\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\biggr{)}\end{multlined}\frac{2N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\biggl{(}\frac{1-\xi}{\xi}+\xi(1-\xi)\biggr{)}\\\
\times\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\biggr{)}$
(388) $\displaystyle F^{(1)}_{d2gg}(z,\xi,\vec{}r_{\perp})$
$\displaystyle=\begin{multlined}\frac{N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\\\
\times\biggl{[}\biggl{(}\frac{11}{6}-\frac{2N_{f}T_{R}}{3N_{c}}\biggr{)}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\biggr{)}\\\
-\biggl{(}\frac{11}{3}-\frac{4N_{f}T_{R}}{3N_{c}}\biggr{)}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}k_{\perp}^{2}}\biggr{]}\end{multlined}\frac{N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\\\
\times\biggl{[}\biggl{(}\frac{11}{6}-\frac{2N_{f}T_{R}}{3N_{c}}\biggr{)}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{(}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}+\frac{1}{\xi^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\biggr{)}\\\
-\biggl{(}\frac{11}{3}-\frac{4N_{f}T_{R}}{3N_{c}}\biggr{)}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}k_{\perp}^{2}}\biggr{]}$
(392)

though since $F^{(1)}_{d2gg}$ will only be evaluated at $\xi=1$, we can make
that substitution (as with equation (313)) and get a simpler result:

$F^{(1)}_{d2gg}(z,1,\vec{}r_{\perp})=\frac{N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\biggl{(}\frac{11}{3}-\frac{4N_{f}T_{R}}{3N_{c}}\biggr{)}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\ln\frac{k_{\perp}^{2}}{\mu^{2}}$
(393)

where the contribution to the cross section is given by (236) as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gg}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}r_{\perp}\biggl{[}\frac{F^{(1)}_{s2gg}(z,\xi,\vec{}r_{\perp})-F^{(1)}_{s2gg}(z,1,\vec{}r_{\perp})}{1-\xi}+F^{(1)}_{n2gg}(z,\xi,\vec{}r_{\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}\biggl{[}F^{(1)}_{s2gg}(z,1,\vec{}r_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(1)}_{d2gg}(z,1,\vec{}r_{\perp})\biggr{]}$
(396)

Now using equation (242) we can obtain the radial position space formulation:

$\displaystyle F^{(1)}_{s2gg}(z,\xi,r_{\perp})$
$\displaystyle=\begin{multlined}\frac{2N_{c}S_{\perp}}{2\pi}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\xi\\\
\times
r_{\perp}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{[}J_{0}(k_{\perp}r_{\perp})+\frac{1}{\xi^{2}}J_{0}\biggl{(}\frac{k_{\perp}r_{\perp}}{\xi}\biggr{)}\biggr{]}\end{multlined}\frac{2N_{c}S_{\perp}}{2\pi}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\xi\\\
\times
r_{\perp}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{[}J_{0}(k_{\perp}r_{\perp})+\frac{1}{\xi^{2}}J_{0}\biggl{(}\frac{k_{\perp}r_{\perp}}{\xi}\biggr{)}\biggr{]}$
(399) $\displaystyle F^{(1)}_{n2gg}(z,\xi,r_{\perp})$
$\displaystyle=\begin{multlined}\frac{2N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\biggl{(}\frac{1-\xi}{\xi}+\xi(1-\xi)\biggr{)}\\\
\times
r_{\perp}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{[}J_{0}(k_{\perp}r_{\perp})+\frac{1}{\xi^{2}}J_{0}\biggl{(}\frac{k_{\perp}r_{\perp}}{\xi}\biggr{)}\biggr{]}\end{multlined}\frac{2N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\biggl{(}\frac{1-\xi}{\xi}+\xi(1-\xi)\biggr{)}\\\
\times
r_{\perp}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}\biggl{[}J_{0}(k_{\perp}r_{\perp})+\frac{1}{\xi^{2}}J_{0}\biggl{(}\frac{k_{\perp}r_{\perp}}{\xi}\biggr{)}\biggr{]}$
(402) $\displaystyle F^{(1)}_{d2gg}(z,1,r_{\perp})$
$\displaystyle=\begin{multlined}\frac{N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\biggl{(}\frac{11}{3}-\frac{4N_{f}T_{R}}{3N_{c}}\biggr{)}\\\
\times
r_{\perp}J_{0}(k_{\perp}r_{\perp})\ln\frac{k_{\perp}^{2}}{\mu^{2}}\end{multlined}\frac{N_{c}S_{\perp}}{(2\pi)^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}\biggl{(}\frac{11}{3}-\frac{4N_{f}T_{R}}{3N_{c}}\biggr{)}\\\
\times r_{\perp}J_{0}(k_{\perp}r_{\perp})\ln\frac{k_{\perp}^{2}}{\mu^{2}}$
(405)

and

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gg}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int_{0}^{\infty}\udc
r_{\perp}\biggl{[}\frac{F^{(1)}_{s2gg}(z,\xi,r_{\perp})-F^{(1)}_{s2gg}(z,1,r_{\perp})}{1-\xi}+F^{(1)}_{n2gg}(z,\xi,r_{\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc z\int_{0}^{\infty}\udc
r_{\perp}\biggl{[}F^{(1)}_{s2gg}(z,1,r_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(1)}_{d2gg}(z,1,r_{\perp})\biggr{]}$
(408)

#### $\mathcal{H}^{(1)}_{2q\bar{q}}$

From equations (74) and (77) of [98], this contribution to the cross section
is given as

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2q\bar{q}}^{(1)}=S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})\\\
\times 8\pi
N_{f}T_{R}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\delta(1-\xi)\int_{0}^{1}\udc\xi^{\prime}\bigl{[}\xi^{\prime
2}+(1-\xi^{\prime})^{2}\bigr{]}\biggl{[}\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}}{r_{\perp}^{2}}-\delta^{(2)}(\vec{}r_{\perp})\int\uddc\vec{}r_{\perp}^{\prime}\frac{e^{i\vec{}k_{\perp}\cdot\vec{}r_{\perp}^{\prime}}}{r_{\perp}^{\prime
2}}\biggr{]}$ (411)

Despite the notation $2q{\bar{q}}$, this is actually a quadrupole-type term.
It’s proportional to $\delta(1-\xi)$, so only $F_{d}$ will be nonzero.

To put $F^{(1)}_{d2q\bar{q}}$ in a a form suitable for numerical integration,
we’ll have to regularize the terms in brackets. The procedure is basically
similar to the one used with $\mathcal{H}^{(1)}_{4qq}$, with some slight
differences because the formula is different. We start by extracting the
position-dependent factors,

$\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\biggl{[}\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}}{r_{\perp}^{2}}-\delta^{(2)}(\vec{}r_{\perp})\int\uddc\vec{}r_{\perp}^{\prime}\frac{e^{i\vec{}k_{\perp}\cdot\vec{}r_{\perp}^{\prime}}}{r_{\perp}^{\prime
2}}\biggr{]}$ (412)

After changing variables $\vec{}s_{\perp}\to\vec{}r_{\perp}$ (with Jacobian
$1$) and excising a region of radius $\epsilon$ around the poles in
$\vec{}r_{\perp}$ and $\vec{}r_{\perp}^{\prime}$, then changing variables
$\vec{}r_{\perp}^{\prime}\to-\xi^{\prime}\vec{}r_{\perp}$ in the second term,
we get

$\int\frac{\uddc\vec{}t_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(2)}(\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\int_{r_{\perp}>\epsilon}\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(2)}(\vec{}t_{\perp}-\vec{}r_{\perp})\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}}{r_{\perp}^{2}}\\\
-\int\frac{\uddc\vec{}t_{\perp}}{(2\pi)^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}t_{\perp})\bigr{]}^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\int_{r_{\perp}>\epsilon/\xi^{\prime}}\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}}{r_{\perp}^{2}}$
(415)

We split the second integral into two pieces,
$\int_{r_{\perp}>\epsilon/\xi^{\prime}}=\int_{r_{\perp}>\epsilon}-\int_{r_{\perp}>\epsilon}^{r_{\perp}<\epsilon/\xi^{\prime}}$,
so the total integral turns into

$\int\frac{\uddc\vec{}t_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(2)}(\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\int_{r_{\perp}>\epsilon}\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}t_{\perp}-\vec{}r_{\perp})-S_{x_{g}}^{(2)}(\vec{}t_{\perp})\bigr{]}\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}}{r_{\perp}^{2}}\\\
+\int\frac{\uddc\vec{}t_{\perp}}{(2\pi)^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}t_{\perp})\bigr{]}^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\underbrace{\int_{r_{\perp}>\epsilon}^{r_{\perp}<\epsilon/\xi^{\prime}}\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}r_{\perp}}}{r_{\perp}^{2}}}_{\ln(1/\xi^{\prime})/2\pi}$
(418)

This gets multiplied by $\xi^{\prime 2}+(1-\xi^{\prime})^{2}$ and integrated,
so we will need the integrals

$\displaystyle\int_{0}^{1}\udc\xi^{\prime}[\xi^{\prime
2}+(1-\xi^{\prime})^{2}]e^{-i\xi^{\prime}x}$
$\displaystyle=\biggl{(}\frac{4i}{x^{3}}-\frac{i}{x}\biggr{)}(1-e^{-ix})+\frac{2}{x^{2}}(1+e^{-ix})\coloneqq\mathcal{I}_{2}(x)$
(419) $\displaystyle\int_{0}^{1}\udc\xi^{\prime}[\xi^{\prime
2}+(1-\xi^{\prime})^{2}]\ln\frac{1}{\xi^{\prime}}$
$\displaystyle=\frac{13}{18}$ (420)

Putting everything together, the final Cartesian position space formula is

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2q\bar{q}}^{(1)}=\int_{\tau}^{1}\udc
z\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}F^{(1)}_{d2q\bar{q}}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})+\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}F^{(1)}_{d2^{\prime}q\bar{q}}(z,1,\vec{}r_{\perp})$
(421)

with

$F^{(1)}_{d2q\bar{q}}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})=\\\
\frac{1}{2\pi^{3}}\frac{\alpha_{\text{s}}}{2\pi}\frac{N_{f}T_{R}S_{\perp}}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\bigl{[}S_{x_{g}}^{(2)}(\vec{}s_{\perp})-S_{x_{g}}^{(2)}(\vec{}t_{\perp})\bigr{]}S_{x_{g}}^{(2)}(\vec{}t_{\perp})\frac{1}{r_{\perp}^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\mathcal{I}_{2}(\vec{}k_{\perp}\cdot\vec{}r_{\perp})$
(424)

and

$F^{(1)}_{d2^{\prime}q\bar{q}}(z,\xi,\vec{}r_{\perp})=\frac{13}{18\pi^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{N_{f}T_{R}S_{\perp}}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}$
(425)

To get the momentum space expression, we use equation (649) to rewrite (411)
as

$-S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})\\\
\times 8\pi
N_{f}T_{R}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\delta(1-\xi)\int_{0}^{1}\udc\xi^{\prime}\bigl{[}\xi^{\prime
2}+(1-\xi^{\prime})^{2}\bigr{]}\frac{1}{4\pi}\int\uddc\vec{}q_{1\perp}e^{-i\vec{}q_{1\perp}\cdot\vec{}r_{\perp}}\ln\frac{(\vec{}q_{1\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}$
(428)

We can do the integrals over $\vec{}s_{\perp}$ and $\vec{}t_{\perp}$,
resulting in
$\frac{1}{S_{\perp}^{2}}\mathcal{F}_{x_{g}}(\vec{}q_{1\perp})\mathcal{F}_{x_{g}}(\vec{}k_{\perp}-\vec{}q_{1\perp})$.
After that it’s straightforward to extract the function

$F^{(1)}_{d2q\bar{q}}(z,\xi,\vec{}q_{1\perp},\xi^{\prime})=-\frac{1}{S_{\perp}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\\\
\times 2N_{f}T_{R}\bigl{[}\xi^{\prime
2}+(1-\xi^{\prime})^{2}\bigr{]}\mathcal{F}_{x_{g}}(\vec{}q_{1\perp})\mathcal{F}_{x_{g}}(\vec{}k_{\perp}-\vec{}q_{1\perp})\ln\frac{(\vec{}q_{1\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}$
(431)

which goes into (273) in the form of

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2q\bar{q}}^{(1)}=\int_{\tau}^{1}\udc
z\int\uddc\vec{}q_{1\perp}\int_{0}^{1}\udc\xi^{\prime}F^{(1)}_{d2q\bar{q}}(z,1,\vec{}q_{1\perp},\xi^{\prime})$
(432)

#### $\mathcal{H}^{(1)}_{6gg}$

From equations (74) and (78) of [98] we get this term to be

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{6gg}^{(1)}=-S_{\perp}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(x_{p},\mu)\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})S_{x_{g}}^{(2)}(\vec{}r_{\perp})\frac{\alpha_{\text{s}}}{2\pi}\\\
\times 16\pi
N_{c}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\Biggl{\\{}-e^{-i\frac{\vec{}k_{\perp}}{\xi}\cdot\vec{}t_{\perp}}\frac{[1-\xi(1-\xi)]^{2}}{(1-\xi)_{+}}\frac{1}{\xi^{2}}\frac{\vec{}r_{\perp}}{r_{\perp}^{2}}\cdot\frac{\vec{}t_{\perp}}{t_{\perp}^{2}}\\\
-\delta(1-\xi)\int_{0}^{1}\udc\xi^{\prime}\biggl{[}\frac{\xi^{\prime}}{(1-\xi^{\prime})_{+}}+\frac{1}{2}\xi^{\prime}(1-\xi^{\prime})\biggr{]}\Biggl{[}\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}-\delta^{(2)}(\vec{}t_{\perp})\int\uddc\vec{}r_{\perp}^{\prime}\frac{e^{i\vec{}k_{\perp}\cdot\vec{}r_{\perp}^{\prime}}}{r_{\perp}^{\prime
2}}\Biggr{]}\Biggr{\\}}$ (436)

This is a quadrupole-type term with
$\mathcal{S}(\vec{}s_{\perp},\vec{}t_{\perp})=S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})S_{x_{g}}^{(2)}(\vec{}r_{\perp})$.
(Recall that $\vec{}r_{\perp}=\vec{}s_{\perp}-\vec{}t_{\perp}$.)

The first and second lines lead to a plus-regulated contribution which is
straightforward to extract in Cartesian position space:

$F^{(1)}_{s6gg}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})=\frac{1}{\pi^{3}}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})S_{x_{g}}^{(2)}(\vec{}r_{\perp})\\\
\times
e^{-i\vec{}k_{\perp}\cdot(\vec{}r_{\perp}+\vec{}t_{\perp}/\xi)}\frac{[1-\xi(1-\xi)]^{2}}{\xi^{2}}\frac{\vec{}r_{\perp}}{r_{\perp}^{2}}\cdot\frac{\vec{}t_{\perp}}{t_{\perp}^{2}}$
(439)

For the delta contribution, which comes from the first and third lines, we
again need to use cutoff regularization. The position-dependent factors are

$\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\Biggl{[}\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}-\delta^{(2)}(\vec{}t_{\perp})\int\uddc\vec{}r_{\perp}^{\prime}\frac{e^{i\vec{}k_{\perp}\cdot\vec{}r_{\perp}^{\prime}}}{r_{\perp}^{\prime
2}}\Biggr{]}$ (440)

We’ll first need to change variables $\vec{}s_{\perp}\to\vec{}r_{\perp}$ so
that the Fourier factor $e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}$ doesn’t
change when we apply the delta function of $\vec{}t_{\perp}$, but otherwise,
the procedure is the same as in section $\mathcal{H}^{(1)}_{4qq}$. It gives
two terms,

$\int\frac{\uddc\vec{}r_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigl{[}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})-S_{x_{g}}^{(2)}(\vec{}r_{\perp})S_{x_{g}}^{(2)}(\vec{0})\bigr{]}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\frac{e^{-i\xi^{\prime}\vec{}k_{\perp}\cdot\vec{}t_{\perp}}}{t_{\perp}^{2}}\\\
-\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}S_{x_{g}}^{(2)}(\vec{0})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\frac{1}{2\pi}\ln\frac{1}{\xi^{\prime}}$
(443)

Then we need the integrals

$\int_{0}^{1}\udc\xi^{\prime}\biggl{[}\frac{\xi^{\prime}}{(1-\xi^{\prime})_{+}}+\frac{1}{2}\xi^{\prime}(1-\xi^{\prime})\biggr{]}e^{-i\xi^{\prime}x}\\\
=i(e^{-ix}-1)\biggl{(}\frac{1}{x^{3}}-\frac{1}{x}\biggr{)}-\frac{e^{-ix}+1}{2x^{2}}+e^{-ix}[-\gamma_{E}+\operatorname{Ci}(x)-\ln(x)+i\operatorname{Si}(x)]\coloneqq\mathcal{I}_{3}(x)$
(446)

and

$\int_{0}^{1}\udc\xi^{\prime}\biggl{[}\frac{\xi^{\prime}}{(1-\xi^{\prime})_{+}}+\frac{1}{2}\xi^{\prime}(1-\xi^{\prime})\biggr{]}\ln\frac{1}{\xi^{\prime}}=-\frac{67}{72}+\frac{\pi^{2}}{6}$
(447)

Putting everything together, we wind up with the quadrupole-type contribution

$F^{(1)}_{d6gg}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})=\frac{1}{\pi^{3}}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\\\
\times
S_{x_{g}}^{(2)}(\vec{}r_{\perp})\Bigl{[}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})-S_{x_{g}}^{(2)}(\vec{}r_{\perp})S_{x_{g}}^{(2)}(\vec{0})\Bigr{]}\frac{1}{t_{\perp}^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\mathcal{I}_{3}(\vec{}k_{\perp}\cdot\vec{}t_{\perp})$
(450)

and the dipole-type contribution

$F^{(1)}_{d2^{\prime\prime}gg}(z,1,\vec{}r_{\perp})=\biggl{(}-\frac{67}{36\pi^{2}}+\frac{1}{3}\biggr{)}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}S_{x_{g}}^{(2)}(\vec{0})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}$
(451)

which both go into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{6gg}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}\frac{F^{(1)}_{s6gg}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})-F^{(1)}_{s6gg}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})}{1-\xi}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}\biggl{[}F^{(1)}_{s6gg}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}+F^{(1)}_{d6gg}(z,1,\vec{}s_{\perp},\vec{}t_{\perp})\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}r_{\perp}F^{(1)}_{d2^{\prime\prime}gg}(z,1,\vec{}r_{\perp})$
(455)

To get the plus-regulated contribution in momentum space, we start with the
expression from the first two lines of (436), which can be rewritten

$16\pi N_{c}S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\int\frac{\udc
z\udc\xi}{z^{2}}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\frac{[1-\xi(1-\xi)]^{2}}{(1-\xi)_{+}}\frac{1}{\xi^{2}}\\\
\times\int\frac{\uddc\vec{}r_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}/\xi}\frac{\vec{}r_{\perp}\cdot\vec{}t_{\perp}}{r_{\perp}^{2}t_{\perp}^{2}}$
(458)

We take the expression in the second line and insert delta functions as
follows:

$\int\frac{\uddc\vec{}r_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}\int\uddc\vec{}s_{\perp}\int\uddc\vec{}r_{\perp}^{\prime}\uddc\vec{}t_{\perp}^{\prime}\delta^{(2)}(\vec{}r_{\perp}-\vec{}r_{\perp}^{\prime})\delta^{(2)}(\vec{}t_{\perp}-\vec{}t_{\perp}^{\prime})\delta^{(2)}(\vec{}s_{\perp}-\vec{}r_{\perp}-\vec{}t_{\perp})\\\
\times
S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}/\xi}\frac{\vec{}r_{\perp}^{\prime}\cdot\vec{}t_{\perp}^{\prime}}{r_{\perp}^{\prime
2}t_{\perp}^{\prime 2}}$ (461)

This “elevates” $\vec{}s_{\perp}$ to an independent integration variable,
rather than simply a shorthand for $\vec{}r_{\perp}+\vec{}t_{\perp}$. Next, we
replace each of these delta functions with its Fourier transform according to
equation (645), using $\vec{}q_{1\perp}$, $\vec{}q_{2\perp}$, and
$\vec{}q_{3\perp}$ as the momentum variables, respectively. The resulting
expression can be rearranged into

$(2\pi)^{2}\int\frac{\uddc\vec{}q_{1\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}q_{2\perp}}{(2\pi)^{2}}\frac{\uddc\vec{}q_{3\perp}}{(2\pi)^{2}}\Biggl{(}\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i(\vec{}k_{\perp}+\vec{}q_{3\perp}-\vec{}q_{1\perp})\cdot\vec{}r_{\perp}}\Biggr{)}\Biggl{(}\int\frac{\uddc\vec{}s_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(2)}(\vec{}s_{\perp})e^{i\vec{}q_{3\perp}\cdot\vec{}s_{\perp}}\Biggr{)}\\\
\times\Biggl{(}\int\frac{\uddc\vec{}t_{\perp}}{(2\pi)^{2}}S_{x_{g}}^{(2)}(\vec{}t_{\perp})e^{-i(\vec{}k_{\perp}/\xi+\vec{}q_{3\perp}-\vec{}q_{2\perp})\cdot\vec{}t_{\perp}}\Biggr{)}\Biggl{(}\int\uddc\vec{}r_{\perp}^{\prime}\uddc\vec{}t_{\perp}^{\prime}e^{-i\vec{}q_{1\perp}\cdot\vec{}r_{\perp}^{\prime}}e^{-i\vec{}q_{2\perp}\cdot\vec{}t_{\perp}^{\prime}}\frac{\vec{}r_{\perp}^{\prime}\cdot\vec{}t_{\perp}^{\prime}}{r_{\perp}^{\prime
2}t_{\perp}^{\prime 2}}\Biggr{)}$ (464)

The first three integrals in parentheses give factors of
$\mathcal{F}_{x_{g}}/S_{\perp}$ according to (32), and the last is an
application of (630), so we wind up with

$-\frac{1}{(2\pi)^{2}}\frac{1}{S_{\perp}^{3}}\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\uddc\vec{}q_{3\perp}\mathcal{F}_{x_{g}}(\vec{}k_{\perp}+\vec{}q_{3\perp}-\vec{}q_{1\perp})\mathcal{F}_{x_{g}}(\vec{}q_{3\perp})\mathcal{F}_{x_{g}}\biggl{(}\frac{\vec{}k_{\perp}}{\xi}+\vec{}q_{3\perp}-\vec{}q_{2\perp}\biggr{)}\frac{\vec{}q_{1\perp}\cdot\vec{}q_{2\perp}}{q_{1\perp}^{2}q_{2\perp}^{2}}$
(465)

Restoring the prefactors gives

$F^{(1)}_{s6gg}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp},\vec{}q_{3\perp})=-\frac{4}{\pi}\frac{N_{c}}{S_{\perp}^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\frac{[1-\xi(1-\xi)]^{2}}{\xi^{2}}\\\
\times\mathcal{F}_{x_{g}}(\vec{}k_{\perp}+\vec{}q_{3\perp}-\vec{}q_{1\perp})\mathcal{F}_{x_{g}}(\vec{}q_{3\perp})\mathcal{F}_{x_{g}}\biggl{(}\frac{\vec{}k_{\perp}}{\xi}+\vec{}q_{3\perp}-\vec{}q_{2\perp}\biggr{)}\frac{\vec{}q_{1\perp}\cdot\vec{}q_{2\perp}}{q_{1\perp}^{2}q_{2\perp}^{2}}$
(468)

For the rest of the term, we use equation (649) to turn the third line of
(436) into

$\delta(1-\xi)\int_{0}^{1}\udc\xi^{\prime}\biggl{[}\frac{\xi^{\prime}}{(1-\xi^{\prime})_{+}}+\frac{1}{2}\xi^{\prime}(1-\xi^{\prime})\biggr{]}\frac{1}{4\pi}\int\uddc\vec{}q_{2\perp}e^{-i\vec{}q_{2\perp}\cdot\vec{}t_{\perp}}\ln\frac{(\vec{}q_{2\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}\\\
=\frac{\delta(1-\xi)}{4\pi}\int_{0}^{1}\udc\xi^{\prime}\int\uddc\vec{}q_{2\perp}e^{-i\vec{}q_{2\perp}\cdot\vec{}t_{\perp}}\biggl{\\{}\biggl{[}\frac{\xi^{\prime}}{1-\xi^{\prime}}+\frac{1}{2}\xi^{\prime}(1-\xi^{\prime})\biggr{]}\ln\frac{(\vec{}q_{2\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}\\\
-\frac{1}{1-\xi^{\prime}}\ln\frac{(\vec{}q_{2\perp}-\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}\biggr{\\}}$
(472)

This basically takes the form
$\iint\uddc\vec{}q_{2\perp}e^{-i\vec{}q_{2\perp}\cdot\vec{}t_{\perp}}(\cdots)$.
If we combine that with the other position-dependent factors from (436), we
get

$\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(2)}(\vec{}s_{\perp})S_{x_{g}}^{(2)}(\vec{}t_{\perp})S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\iint\uddc\vec{}q_{2\perp}e^{-i\vec{}q_{2\perp}\cdot\vec{}t_{\perp}}(\cdots)$
(473)

Inserting
$\iint\uddc\vec{}r_{\perp}\delta^{(2)}(\vec{}r_{\perp}-\vec{}s_{\perp}+\vec{}t_{\perp})$
and expanding it with (645), with $\vec{}q_{1\perp}$ as the momentum variable,
turns this into

$\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\mathcal{F}_{x_{g}}(\vec{}q_{1\perp})\mathcal{F}_{x_{g}}(\vec{}k_{\perp}-\vec{}q_{2\perp})\mathcal{F}_{x_{g}}(\vec{}q_{1\perp}+\vec{}q_{2\perp}-\vec{}k_{\perp})(\cdots)$
(474)

All in all, the delta contribution is

$F^{(1)}_{d6gg}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp},\xi^{\prime})=-\frac{4N_{c}}{S_{\perp}^{2}}\frac{\alpha_{\text{s}}}{2\pi}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(x_{p},\mu)}{z^{2}}\mathcal{F}_{x_{g}}(\vec{}q_{1\perp})\mathcal{F}_{x_{g}}(\vec{}k_{\perp}-\vec{}q_{2\perp})\\\
\times\mathcal{F}_{x_{g}}(\vec{}q_{1\perp}+\vec{}q_{2\perp}-\vec{}k_{\perp})\biggl{[}\biggl{(}\frac{\xi^{\prime}}{1-\xi^{\prime}}+\frac{1}{2}\xi^{\prime}(1-\xi^{\prime})\biggr{)}\ln\frac{(\vec{}q_{2\perp}-\xi^{\prime}\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}-\frac{1}{1-\xi^{\prime}}\ln\frac{(\vec{}q_{2\perp}-\vec{}k_{\perp})^{2}}{k_{\perp}^{2}}\biggr{]}$
(477)

This and equation (468) go into (273) in the form of

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{6gg}^{(1)}=\\\ \int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\uddc\vec{}q_{3\perp}\biggl{[}\frac{F^{(1)}_{s6gg}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp},\vec{}q_{3\perp})-F^{(1)}_{s6gg}(z,1,\vec{}q_{1\perp},\vec{}q_{2\perp},\vec{}q_{3\perp})}{1-\xi}\biggr{]}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\uddc\vec{}q_{3\perp}F^{(1)}_{s6gg}(z,1,\vec{}q_{1\perp},\vec{}q_{2\perp},\vec{}q_{3\perp})\ln\biggl{(}1-\frac{\tau}{z}\biggr{)}\\\
+\int_{\tau}^{1}\udc
z\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}\int_{0}^{1}\udc\xi^{\prime}F^{(1)}_{d4qq}(z,1,\vec{}q_{1\perp},\vec{}q_{2\perp},\xi^{\prime})$
(482)

#### $\mathcal{H}^{(1,1)}_{2gq}$

From equations (82) and (83) in [98], we get this term to be

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gq}^{(1,1)}=\frac{\alpha_{\text{s}}}{2\pi}\frac{S_{\perp}N_{c}}{2}\int\frac{\udc
z\udc\xi}{z^{2}}\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})\\\
\times\frac{1}{\xi^{2}}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\frac{1}{\xi}\bigl{[}1+(1-\xi)^{2}\bigr{]}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(485)

This has no plus-regulated or delta-function contribution, so the Cartesian
position space representation is fairly trivial to find: we just compare the
form of the term to equation (232) and find

$F^{(1,1)}_{n2gq}(z,\xi,\vec{}r_{\perp})=\frac{1}{8\pi^{2}}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\frac{1+(1-\xi)^{2}}{\xi^{3}}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(486)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{12g}^{(1)}q=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}r_{\perp}F^{(1,1)}_{n2gq}(z,\xi,\vec{}r_{\perp})$
(487)

to produce the contribution to the cross section.

Because of the logarithmic factor, this can’t be transformed into momentum
space, but we can get the radial representation using equation (242):

$F^{(1,1)}_{n2gq}(z,\xi,r_{\perp})=\frac{1}{4\pi}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})J_{0}\biggl{(}\frac{k_{\perp}r_{\perp}}{\xi}\biggr{)}\frac{1+(1-\xi)^{2}}{\xi^{3}}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(488)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{12g}^{(1)}q=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\udc r_{\perp}F^{(1,1)}_{n2gq}(z,\xi,r_{\perp})$
(489)

#### $\mathcal{H}^{(1,2)}_{2gq}$

Again from equations (82) and (83) in [98]

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2gq}^{(1,2)}=\frac{\alpha_{\text{s}}}{2\pi}\frac{S_{\perp}N_{c}}{2}\int\frac{\udc
z\udc\xi}{z^{2}}\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\\\
\times\frac{1}{\xi}\bigl{[}1+(1-\xi)^{2}\bigr{]}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(492)

And like $\mathcal{H}^{(1,1)}_{2gq}$, this has only a normal contribution (no
plus-regulated or delta-function terms), so again we can compare it to (232)
to find

$F^{((1,2))}_{n2gq}(z,\xi,\vec{}r_{\perp})=\frac{1}{8\pi^{2}}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\frac{1+(1-\xi)^{2}}{\xi}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(493)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{22g}^{(1)}q=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}r_{\perp}F^{(1,2)}_{n2gq}(z,\xi,\vec{}r_{\perp})$
(494)

The radial representation is easily obtained using (242):

$F^{((1,2))}_{n2qg}(z,\xi,\vec{}r_{\perp})=\frac{1}{4\pi}\frac{\alpha_{s}}{2\pi}\frac{N_{c}S_{\perp}}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})J_{0}(k_{\perp}r_{\perp})\frac{1+(1-\xi)^{2}}{\xi}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(495)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{22g}^{(1)}q=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\udc r_{\perp}F^{(1,2)}_{n2gq}(z,\xi,r_{\perp})$
(496)

Again, because of the logarithmic factor, there is no momentum-space
representation.

#### $\mathcal{H}^{(1)}_{4gq}$

Equations (82) and (83) in [98] give us

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4gq}^{(1)}=4\pi
N_{c}S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\int\frac{\udc
z\udc\xi}{z^{2}}\sum_{q}x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)\\\
\times\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\frac{1+(1-\xi)^{2}}{\xi^{2}}\frac{\vec{}r_{\perp}\cdot\vec{}t_{\perp}}{r_{\perp}^{2}t_{\perp}^{2}}$
(499)

Again there is only a normal contribution, no plus-regulated or delta-function
term, so the Cartesian position space representation of this term can be
directly extracted from the last expression:

$F^{(1)}_{n4gq}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})=\frac{1}{4\pi^{3}}N_{c}S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\sum_{q}\frac{x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\\\
\times
S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\frac{1+(1-\xi)^{2}}{\xi^{2}}\frac{\vec{}r_{\perp}\cdot\vec{}t_{\perp}}{r_{\perp}^{2}t_{\perp}^{2}}$
(502)

and that goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4gq}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}F^{(1)}_{n4gq}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})$
(503)

To convert this into momentum space, we rewrite the position-dependent parts
of (499) as

$\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\frac{\vec{}r_{\perp}\cdot\vec{}t_{\perp}}{r_{\perp}^{2}t_{\perp}^{2}}$
(504)

and insert delta functions, in the form of (645), to get

$\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}}\\\
\times\int\uddc\vec{}r_{\perp}^{\prime}\uddc\vec{}t_{\perp}^{\prime}\int\frac{\uddc\vec{}q_{1\perp}}{(2\pi)^{2}}e^{-i\vec{}q_{1\perp}\cdot(\vec{}s_{\perp}-\vec{}t_{\perp}-\vec{}r_{\perp}^{\prime})}\int\frac{\uddc\vec{}q_{2\perp}}{(2\pi)^{2}}e^{-i\vec{}q_{2\perp}\cdot(\vec{}t_{\perp}-\vec{}t_{\perp}^{\prime})}\frac{\vec{}r_{\perp}^{\prime}\cdot\vec{}t_{\perp}^{\prime}}{r_{\perp}^{\prime
2}t_{\perp}^{\prime 2}}$ (507)

which simplifies to

$-\frac{1}{S_{\perp}}\int\frac{\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}}{(2\pi)^{2}}\mathcal{G}_{x_{g}}\biggl{(}\frac{\vec{}k_{\perp}}{\xi}+\vec{}q_{1\perp},\frac{\vec{}k_{\perp}(\xi-1)}{\xi}-\vec{}q_{1\perp}+\vec{}q_{2\perp}\biggr{)}\frac{\vec{}q_{1\perp}\cdot\vec{}q_{2\perp}}{q_{1\perp}^{2}q_{2\perp}^{2}}$
(508)

The complete contribution is then

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4gq}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}F^{(1)}_{n4gq}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp})$
(509)

with

$F^{(1)}_{n4gq}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp})=-\frac{1}{\pi}N_{c}\frac{\alpha_{\text{s}}}{2\pi}\sum_{q}\frac{x_{p}q(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/g}(z,\mu)}{z^{2}}\frac{1+(1-\xi)^{2}}{\xi^{2}}\\\
\times\mathcal{G}_{x_{g}}\biggl{(}\frac{\vec{}k_{\perp}}{\xi}+\vec{}q_{1\perp},\frac{\vec{}k_{\perp}(\xi-1)}{\xi}-\vec{}q_{1\perp}+\vec{}q_{2\perp}\biggr{)}\frac{\vec{}q_{1\perp}\cdot\vec{}q_{2\perp}}{q_{1\perp}^{2}q_{2\perp}^{2}}$
(512)

#### $\mathcal{H}^{(1,1)}_{2qg}$

The $qg$ channel is very similar to the $gq$ channel. From equations (87) and
(88) of [98] we get

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qg}^{(1,1)}=\frac{\alpha_{\text{s}}}{2\pi}\frac{S_{\perp}}{2}\int\frac{\udc
z\udc\xi}{z^{2}}\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\sum_{q}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\\\
\times\bigl{[}(1-\xi)^{2}+\xi^{2}\bigr{]}\biggl{(}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}-1\biggr{)}$
(515)

Again we compare it to (232) and find only a normal term:

$F^{((1,1))}_{n2qg}(z,\xi,\vec{}r_{\perp})=\frac{1}{8\pi^{2}}\frac{\alpha_{s}}{2\pi}\frac{S_{\perp}}{z^{2}}\sum_{i}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\\\
\times
S_{x_{g}}^{(2)}(\vec{}r_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}\bigl{[}(1-\xi)^{2}+\xi^{2}\bigr{]}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(518)

in the Cartesian representation, which gets plugged into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qg}^{(1,1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}r_{\perp}F^{(1,1)}_{n2qg}(z,\xi,\vec{}r_{\perp})$
(519)

or

$F^{((1,1))}_{n2qg}(z,\xi,r_{\perp})=\frac{1}{4\pi}\frac{\alpha_{s}}{2\pi}\frac{S_{\perp}}{z^{2}}\sum_{i}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)S_{x_{g}}^{(2)}(\vec{}r_{\perp})\\\
\times
J_{0}(k_{\perp}r_{\perp})\bigl{[}(1-\xi)^{2}+\xi^{2}\bigr{]}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(522)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qg}^{(1,1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\udc r_{\perp}F^{(1,1)}_{n2qg}(z,\xi,r_{\perp})$
(523)

in the radial representation. Again, the logarithmic factor precludes a
momentum-space formula.

#### $\mathcal{H}^{(1,2)}_{2qg}$

From equations (87) and (88) of [98] we get

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{2qg}^{(1,2)}=\frac{\alpha_{\text{s}}}{2\pi}\frac{S_{\perp}}{2}\int\frac{\udc
z\udc\xi}{z^{2}}\int\frac{\uddc\vec{}r_{\perp}}{(2\pi)^{2}}\sum_{q}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\\\
\times\bigl{[}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\bigr{]}^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\frac{\bigl{[}(1-\xi)^{2}+\xi^{2}\bigr{]}}{\xi^{2}}\biggl{(}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}-1\biggr{)}$
(526)

which again, has no plus-regulated or delta contribution. Comparing to (232)
we find

$F^{((1,2))}_{n2qg}(z,\xi,\vec{}r_{\perp})=\frac{1}{8\pi^{2}}\frac{\alpha_{s}}{2\pi}\frac{S_{\perp}}{z^{2}}\sum_{i}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\\\
\times\Bigl{(}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\Bigr{)}^{2}e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}/\xi}\biggl{(}\frac{(1-\xi)^{2}}{\xi^{2}}+1\biggr{)}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(529)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{22q}^{(1)}g=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\udc r_{\perp}F^{(1,2)}_{n2qg}(z,\xi,r_{\perp})$
(530)

for Cartesian position space and

$F^{((1,2))}_{n2qg}(z,\xi,r_{\perp})=\frac{1}{4\pi}\frac{\alpha_{s}}{2\pi}\frac{S_{\perp}}{z^{2}}\sum_{i}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\\\
\times\Bigl{(}S_{x_{g}}^{(2)}(\vec{}r_{\perp})\Bigr{)}^{2}J_{0}\biggl{(}\frac{k_{\perp}r_{\perp}}{\xi}\biggr{)}\biggl{(}\frac{(1-\xi)^{2}}{\xi^{2}}+1\biggr{)}\ln\frac{c_{0}^{2}}{r_{\perp}^{2}\mu^{2}}$
(533)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{22q}^{(1)}g=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\udc r_{\perp}F^{(1,2)}_{n2qg}(z,\xi,r_{\perp})$
(534)

for radial position space.

#### $\mathcal{H}^{(1)}_{4qg}$

Equations (87) and (88) of [98] give us

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4qg}^{(1)}=4\pi
S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\int\frac{\udc
z\udc\xi}{z^{2}}\sum_{q}x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)\\\
\times\int\frac{\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}}{(2\pi)^{4}}S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}/\xi}\frac{(1-\xi)^{2}+\xi^{2}}{\xi}\frac{\vec{}r_{\perp}\cdot\vec{}t_{\perp}}{r_{\perp}^{2}t_{\perp}^{2}}$
(537)

This is almost identical to $\mathcal{H}^{(1)}_{4gq}$, so the same procedure
used in section $\mathcal{H}^{(1)}_{4gq}$ is applicable here. In Cartesian
space, we get

$F^{(1)}_{n4qg}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})=\frac{1}{4\pi^{3}}S_{\perp}\frac{\alpha_{\text{s}}}{2\pi}\sum_{q}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)}{z^{2}}\\\
\times
S_{x_{g}}^{(4)}(\vec{}s_{\perp},\vec{}t_{\perp})e^{-i\vec{}k_{\perp}\cdot\vec{}r_{\perp}}e^{-i\vec{}k_{\perp}\cdot\vec{}t_{\perp}/\xi}\frac{(1-\xi)^{2}+\xi^{2}}{\xi}\frac{\vec{}r_{\perp}\cdot\vec{}t_{\perp}}{r_{\perp}^{2}t_{\perp}^{2}}$
(540)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4qg}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}s_{\perp}\uddc\vec{}t_{\perp}F^{(1)}_{n4qg}(z,\xi,\vec{}s_{\perp},\vec{}t_{\perp})$
(541)

In momentum space, we get

$F^{(1)}_{n4qg}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp})=-\frac{1}{\pi}\frac{\alpha_{\text{s}}}{2\pi}\sum_{q}\frac{x_{p}g(x_{p},\mu)D_{\HepGenParticle{h}{}{}\xspace/q}(z,\mu)}{z^{2}}\frac{(1-\xi)^{2}+\xi^{2}}{\xi}\\\
\times\mathcal{G}_{x_{g}}\biggl{(}\vec{}k_{\perp}+\vec{}q_{1\perp},\frac{\vec{}k_{\perp}(1-\xi)}{\xi}-\vec{}q_{1\perp}+\vec{}q_{2\perp}\biggr{)}\frac{\vec{}q_{1\perp}\cdot\vec{}q_{2\perp}}{q_{1\perp}^{2}q_{2\perp}^{2}}$
(544)

which goes into

$\biggl{(}\frac{\udddc\sigma}{\udc
Y\uddc\vec{p}_{\perp}}\biggr{)}_{4qg}^{(1)}=\int_{\tau}^{1}\udc
z\int_{\tau/z}^{1}\udc\xi\int\uddc\vec{}q_{1\perp}\uddc\vec{}q_{2\perp}F^{(1)}_{n4qg}(z,\xi,\vec{}q_{1\perp},\vec{}q_{2\perp})$
(545)

### Integration Methods

The analytic formulas derived in the previous section, indexed in table 1, are
numerically well-behaved and thus straightforward to implement in a computer
program. In a full calculation, SOLO integrates over anywhere from one