$\displaystyle\text{for }k=0,1$ (4.29)

in all colour channels except for the mixed octets.

  * •

The terms multiplying $C_{A}\mskip 1.5muC_{F}$ in the square brackets are
colour independent, whereas the terms multiplying $C_{F}^{2}$ have an
additional scaling factor

$\displaystyle\tilde{c}(11)$ $\displaystyle=1\,,$ $\displaystyle\tilde{c}(AA)$
$\displaystyle=\frac{N^{2}}{N^{2}-1}\,,$ $\displaystyle\tilde{c}(SS)$
$\displaystyle=\frac{N^{2}-4}{N^{2}-1}\,,$ $\displaystyle\tilde{c}(27\,27)$
$\displaystyle=0\,.$ (4.30)

  * •

Since $c_{gg,g}(\scalebox{0.95}{$10\,\overline{10}$})=0$, one has
${}^{\scalebox{0.7}{$10\phantom{\overline{1}}$}\mskip
1.5mu\scalebox{0.7}{$\overline{10}$}}V^{(2)}_{gg,q}=0$, and the same holds for
$R_{1}R_{2}=\scalebox{0.95}{$\overline{10}\,10$}$.

In the mixed octet channels, one has

$\displaystyle{}^{AS}V_{gg,q}^{[2,k]}$
$\displaystyle=-\,\,{}^{S\mskip-1.5muA}V_{gg,q}^{[2,k]}=\tilde{c}(AS)\,{}^{AS}\mskip-1.5muR_{gg,q}^{[2,k]}$
$\displaystyle\text{for }k=0,1$ (4.31)

with a global colour factor

$\displaystyle\tilde{c}(AS)$ $\displaystyle=C_{F}\mskip
1.5mu\sqrt{N^{2}-1}\,\sqrt{N^{2}-4}\,.$ (4.32)

#### Pure quark channels.

Using the decomposition (3.1) and the symmetry relations (4.12) and (4.15),
one can deduce results for all pure quark channels from the five kernels

$\displaystyle{}^{RR}V_{qq,q}^{v\mskip 1.5mu(2)}\,,$
$\displaystyle{}^{RR}V_{q\bar{q},q}^{v\mskip 1.5mu(2)}\,,$
$\displaystyle{}^{RR}V_{qq^{\prime}\\!,q}^{(2)}\,,$
$\displaystyle{}^{RR}V_{q\bar{q}^{\prime}\\!,q}^{(2)}\,,$
$\displaystyle{}^{RR}V_{q^{\prime}\mskip-1.5mu\bar{q}^{\prime}\\!,q}^{(2)}\,.$
(4.33)

For $R=1$, the first two have a colour factor $C_{F}\mskip
1.5mu(C_{A}-2C_{F})$, whereas the last three have a colour factor $C_{F}$. One
finds

$\displaystyle{}^{11}V_{qq^{\prime}\\!,q}^{(2)}$
$\displaystyle={}^{11}V_{q\bar{q}^{\prime}\\!,q}^{(2)}\,.$ (4.34)

Because each kernel corresponds to a single two-loop graph, one finds simple
scaling relations between octet and singlet:

$\displaystyle{}^{88}V_{qq,q}^{v\mskip
1.5mu(2)}\big{/}\,{}^{11}V_{qq,q}^{v\mskip 1.5mu(2)}$
$\displaystyle=(N^{2}+1)\,c_{q\bar{q},g}(88)\,,$
$\displaystyle{}^{88}V_{q\bar{q},q}^{v\mskip
1.5mu(2)}\big{/}\,{}^{11}V_{q\bar{q},q}^{v\mskip 1.5mu(2)}$
$\displaystyle=c_{q\bar{q},g}(88)\,,$
$\displaystyle{}^{88}V_{qq^{\prime}\\!,q}^{(2)}\,\big{/}\,{}^{11}V_{qq^{\prime}\\!,q}^{(2)}$
$\displaystyle=2\mskip 1.5muc_{q\bar{q},g}(88)\,,$
$\displaystyle{}^{88}V_{q\bar{q}^{\prime}\\!,q}^{(2)}\,\big{/}\,{}^{11}V_{q\bar{q}^{\prime}\\!,q}^{(2)}$
$\displaystyle=-(N^{2}-2)\,c_{q\bar{q},g}(88)\,,$
$\displaystyle{}^{88}V_{q^{\prime}\mskip-1.5mu\bar{q}^{\prime}\\!,q}^{(2)}\,\big{/}\,{}^{11}V_{q^{\prime}\mskip-1.5mu\bar{q}^{\prime}\\!,q}^{(2)}$
$\displaystyle=c_{q\bar{q},g}(88)\,.$ (4.35)

#### Relations between DPDs at small $y$.

According to (4.25) and (4.28), the two-loop kernels for both $q\to qg$ and
$g\to qg$ follow the same scaling relation between the two octets as the $q\to
qg$ splitting at one loop. We therefore have the relation

$\displaystyle{}^{8A}\mskip-1.5muF_{qg}\mskip 1.5mu\big{/}\mskip
1.5mu{}^{8S}\mskip-1.5muF_{qg}$ $\displaystyle=-N\big{/}\sqrt{N^{2}-4}$ (4.36)

for the DPDs at small $y$. Corrections to this and the following three
relations are of order $\mathcal{O}(a_{s}^{3})$ and
$\mathcal{O}(y^{2}\Lambda^{2})$.

The kernels for the octet combinations $AA$ and $SS$ are opposite to each
other in the $g\to gg$ channel, but they are not for $q\to gg$ because of the
terms with $C_{F}^{2}$ in (4.29). Using the values of the rescaling factors
$c$ and $\tilde{c}$, we obtain the relation

$\displaystyle{}^{AA}\mskip-1.5muF_{gg}+{}^{SS}\mskip-1.5muF_{gg}$
$\displaystyle=-\frac{a_{s}^{2}}{\pi y^{2}}\,\frac{2\mskip
1.5muC_{F}^{2}}{\sqrt{N^{2}-1}}\;\Bigl{[}R_{gg,q}^{F\,[2,0]}+L\mskip
1.5muR_{gg,q}^{F\,[2,1]}\Bigr{]}\,\underset{12}{\otimes}\,\sum_{q}\bigl{(}f_{q}+f_{\bar{q}}\bigr{)}\,.$
(4.37)

We expect the relative difference between
${}^{AA}\mskip-1.5muF_{gg}^{\phantom{[]}}$ and $-\,{}^{SS}\mskip-1.5muF_{gg}$
to grow with $x_{1}+x_{2}$, as the quark singlet distribution then becomes
more important than the gluon distribution in the splitting DPDs. This is
confirmed by the numerical examples in section 5.

The mixed octet distribution is driven by the valence combination
$f_{q}-f_{\bar{q}}$,

$\displaystyle{}^{AS}\mskip-1.5muF_{gg}$
$\displaystyle\underset{\phantom{N=3}}{=}\frac{a_{s}^{2}}{\pi
y^{2}}\,C_{F}\mskip
1.5mu\sqrt{N^{2}-1}\,\sqrt{N^{2}-4}\;\Bigl{[}{}^{AS}\mskip-1.5muR_{gg,q}^{[2,0]}+L\,{}^{AS}\mskip-1.5muR_{gg,q}^{[2,1]}\mskip
1.5mu\Bigr{]}\,\underset{12}{\otimes}\,\sum_{q}\bigl{(}f_{q}-f_{\bar{q}}\bigr{)}\,,$
(4.38)

with the minus sign between quark and antiquark distributions resulting from
the fourth relation in (4). Unlike all other two-gluon distributions,
${}^{AS}F_{gg}(x_{1},x_{2})=-\,{}^{S\mskip-1.5muA}F_{gg}(x_{1},x_{2})$ is odd
under the exchange $x_{1}\leftrightarrow x_{2}$.

For two gluons in the decuplet representation, we find

$\displaystyle{}^{\scalebox{0.7}{$10\phantom{\overline{1}}$}\mskip
1.5mu\scalebox{0.7}{$\overline{10}$}}F_{gg}$
$\displaystyle={}^{\scalebox{0.7}{$\overline{10}$}\,\scalebox{0.7}{$10\phantom{\overline{1}}$}}F_{gg}=0$
(4.39)

at two-loop accuracy. This result holds for all $N$. It is an interesting
question whether these distributions remain small when $y$ is in the non-
perturbative region.

For pure quark channels, we get simple relations between the singlet and octet
distributions if the quark flavors differ, because in this case only the
kernel ${V}_{qq^{\prime}\\!,q}$ or ${V}_{q\bar{q}^{\prime}\\!,q}$ contributes.
As a consequence, one has

$\displaystyle{}^{88}\mskip-1.5muF_{qq^{\prime}}\big{/}\mskip
1.5mu{}^{11}\mskip-1.5muF_{qq^{\prime}}$
$\displaystyle=-2\big{/}\sqrt{N^{2}-1}\,,$
$\displaystyle{}^{88}\mskip-1.5muF_{q\bar{q}^{\prime}}\big{/}\mskip
1.5mu{}^{11}\mskip-1.5muF_{q\bar{q}^{\prime}}$
$\displaystyle=(N^{2}-2)\big{/}\sqrt{N^{2}-1}$ (4.40)

for $q\neq q^{\prime}$. For equal quark flavors, no simple relation is
obtained at two-loop accuracy.

We emphasise that the relations (4.36) to (4.40) should be evaluated for
$\mu\sim\sqrt{x_{1}x_{2}\mskip 1.5mu\zeta_{p}}\sim 1/y$ in order to avoid
logarithmically enhanced contributions from higher orders. An exception is the
relation (4.39), which is stable under evolution.

In the following subsections, we discuss the behaviour of the two-loop kernels
and their convolution with PDFs in various kinematics limits. We closely
follow the presentation for the colour singlet kernels in [64] where
appropriate. In the channels $g\to gg$, $q\to qg$, and $g\to qg$, we will not
give results for kernels for $R_{1}R_{2}=SS$, which are proportional to the
kernels for $R_{1}R_{2}=AA$ according to (4.19), (4.25), and (4.28). For the
mixed octet combinations, we can limit our attention to ${}^{AS}V_{gg,q}$.
Finally, we will no longer discuss the kernels in the decuplet sector, which
are all zero.

###  4.2 Threshold limit: large $x_{1}+x_{2}$

Let us start with the threshold limit $x\to 1$. From (2.13), (4) and (4), we
deduce that the convolution of kernels with PDFs gives333 We note that the
sign of the plus distribution term in equation (149) of [64] is incorrect.

$\displaystyle{}^{R_{1}\mskip-1.5muR_{2}}V^{(2)}\underset{12}{\otimes}f\underset{x\to
1}{=}L\,{}^{R_{1}\mskip-1.5muR_{2}}V^{[2,1]}_{p}(u)\,\log(1-x)\,f(x)$
$\displaystyle\quad{}+\Biggl{[}{}^{R_{1}\mskip-1.5muR_{2}}V^{[2,0]}_{\delta}(u)+L\,{}^{R_{1}\mskip-1.5muR_{2}}V^{[2,1]}_{\delta}(u)+\biggl{(}L\log\frac{\mu^{2}}{x_{1}x_{2}\mskip
1.5mu\zeta_{p}}-\frac{L^{2}}{2}+c_{\scalebox{0.6}{$\overline{\mathrm{MS}}$}}\biggr{)}\frac{{}^{R_{1}}\gamma_{J}^{(0)}}{2}\,{}^{R_{1}\mskip-1.5muR_{2}}V^{(1)}(u)\mskip
1.5mu\Biggr{]}\mskip 1.5muf(x)$
$\displaystyle\quad{}+\mathcal{O}\bigl{[}(1-x)\,\log^{n}(1-x)\bigl{]}\,f(x)\,.$
(4.41)

The power for the subleading contributions in the last line is $n=0$ or $n=1$
and follows from the behaviour of ${V}^{[2,k]}_{\text{reg}}(uz,\bar{u}z)$ for
$z\to 1$.

The contribution enhanced by $\log(1-x)$ is due to plus distribution terms.
Such terms are absent in $V^{(1)}$. They appear in $V^{(2)}$ only for LO
channels, where they are accompanied with a logarithm $L$ and hence vanish for
$\mu=y/b_{0}$. The relevant coefficients read

$\displaystyle{}^{11}V_{gg,g}^{[2,1]}\mskip 1.5mu\bigr{|}_{\mskip 1.5mup}$
$\displaystyle=4C_{A}^{2}\;p_{gg}(u)\,,$
$\displaystyle{}^{27\,27}V_{gg,g}^{[2,1]}\mskip 1.5mu\bigr{|}_{\mskip 1.5mup}$
$\displaystyle={}-c_{gg,g}(27\,27)\;60\,p_{gg}(u)\,,$
$\displaystyle{}^{11}V_{q\bar{q},g}^{[2,1]}\mskip 1.5mu\bigr{|}_{\mskip
1.5mup}$ $\displaystyle={}-(C_{A}-2C_{F})\;p_{qg}(u)\,,$
$\displaystyle{}^{88}V_{q\bar{q},g}^{[2,1]}\mskip 1.5mu\bigr{|}_{\mskip
1.5mup}$ $\displaystyle={}-c_{q\bar{q},g}(88)\;2\mskip
1.5mu(C_{A}-C_{F})\;p_{qg}(u)\,,$ $\displaystyle{}^{11}V_{qg,q}^{[2,1]}\mskip
1.5mu\bigr{|}_{\mskip 1.5mup}$ $\displaystyle=2C_{A}\mskip
1.5muC_{F}\,p_{qq}(u)$ (4.42)

with the functions $p(u)$ given in (4.7). Here we write
$V_{\,\ldots\phantom{p}}^{[2,1]}\mskip 1.5mu|_{\mskip 1.5mup}$ instead of
$V_{\,\ldots,\,p}^{[2,1]}$ for better readability. The plus distribution terms
in the $g\to gg$ and $q\to qg$ channels are zero for $R_{1}R_{2}=AA$ and
$R_{1}R_{2}=SS$.

###  4.3 Small $x_{1}+x_{2}$

To understand how the $x\to 0$ limit is analysed, let us first consider the
ordinary Mellin convolution (2.1) of two functions $P(z)$ and $f(z)$. If both
functions approximately behave like $1/z$ at small $z$, then the convolution
integral approximately goes like $\int\\!dz/z$ over a wide range $x\ll z\ll 1$
in which the arguments of both $P(z)$ and $f(x/z)$ are small. If this range is
wide enough, its contribution dominates the result. The same observation
applies to the convolution (2.13) if $V(uz,\bar{u}z)\sim 1/z^{2}$ for small
$z$.

To be more specific, one finds that for $V(uz,\bar{u}z)=w(u)/z^{2}$

$\displaystyle\Bigl{[}\mskip 1.5muV\underset{12}{\otimes}f\mskip
1.5mu\Bigr{]}(x)=\frac{w(u)}{x}\biggl{[}\mskip 1.5mu\frac{1}{z}\otimes
f\,\biggr{]}(x)$ $\displaystyle=w(u)\,E(x)\,\frac{f(x)}{x}$ (4.43)

with an enhancement factor

$\displaystyle E(x)$ $\displaystyle=\begin{cases}k^{-1}\mskip
1.5mu\log(1/x)&\text{for }\mskip
1.5muxf(x)=c\,\log^{k-1}(1/x)\,,\\\\[3.00003pt]
(1-x^{\alpha})/\alpha&\text{for }\mskip 1.5muxf(x)=c\mskip
1.5mux^{-\alpha}\,,\end{cases}$ (4.44)

where $k$ is a positive integer in the upper line of (4.44). The form of
$f(x)$ in the lower line provides a reasonable first approximation for the
small $x$ behaviour of most PDFs, with $\alpha$ not too far away from zero.

Based on this discussion, we now extract the leading $1/z^{2}$ power behaviour
of the two-loop kernels for the different channels.

#### $\boldsymbol{g\to gg}\mskip 1.5mu:$

$\displaystyle{}^{11}V_{gg,g}^{[2,0]}$
$\displaystyle\approx\frac{2C_{A}^{2}}{z^{2}}\biggl{[}1-6u\bar{u}+\frac{2u^{3}-2u^{2}+4u-1}{u}\log\bar{u}+\frac{2\bar{u}^{3}-2\bar{u}^{2}+4\bar{u}-1}{\bar{u}}\log
u\biggr{]}\,,$ $\displaystyle{}^{AA}V_{gg,g}^{[2,0]}$ $\displaystyle\approx
c_{gg,g}(AA)\,{}^{11}V_{gg,g}^{[2,0]}\,,\phantom{\frac{1}{1}}$
$\displaystyle{}^{27\,27}V_{gg,g}^{[2,0]}$ $\displaystyle\approx
c_{gg,g}(27\,27)\,{}^{11}V_{gg,g}^{[2,0]}\,,\phantom{\frac{1}{1}}$
$\displaystyle{}^{11}V_{gg,g}^{[2,1]}$
$\displaystyle\approx\frac{4C_{A}^{2}}{z^{2}}\,\frac{1+2u\bar{u}-(u\bar{u})^{2}}{u\bar{u}}\,,$
$\displaystyle{}^{AA}V_{gg,g}^{[2,1]}$ $\displaystyle\approx
c_{gg,g}(AA)\,\frac{4C_{A}^{2}}{z^{2}}\,\bigl{(}2-u\bar{u}\bigr{)}\,,$
$\displaystyle{}^{27\,27}V_{gg,g}^{[2,1]}$
$\displaystyle\approx{}-c_{gg,g}(27\,27)\,\frac{12}{z^{2}}\;\frac{5-6u\bar{u}+3(u\bar{u})^{2}}{u\bar{u}}\,.$
(4.45)

#### $\boldsymbol{g\to q\bar{q}}\mskip 1.5mu:$

$\displaystyle{}^{11}V_{q\bar{q},g}^{[2,0]}$
$\displaystyle\approx\frac{C_{A}}{z^{2}}\Bigl{[}2-p_{qg}(u)\bigl{(}\log(u\bar{u})+3\bigr{)}\Bigr{]}\,,$
$\displaystyle{}^{11}V_{q\bar{q},g}^{[2,1]}$
$\displaystyle\approx{}-\frac{C_{A}}{z^{2}}\,p_{qg}(u)\,,$
$\displaystyle{}^{88}V_{q\bar{q},g}^{[2,0]}$ $\displaystyle\approx
c_{q\bar{q},g}(88)\,{}^{11}V_{q\bar{q},g}^{[2,0]}\,,$
$\displaystyle{}^{88}V_{q\bar{q},g}^{[2,1]}$ $\displaystyle\approx
c_{q\bar{q},g}(88)\,{}^{11}V_{q\bar{q},g}^{[2,1]}\,.$ (4.46)

#### $\boldsymbol{q\to gg}\mskip 1.5mu:$

In several (but not all) cases, we observe Casimir scaling by a factor
$C_{F}/C_{A}$ between the $q\to gg$ and $g\to gg$ kernels:

$\displaystyle{}^{R_{1}\mskip-1.5muR_{2}}V_{gg,q}^{[2,0]}$
$\displaystyle\approx\frac{C_{F}}{C_{A}}\;{}^{R_{1}\mskip-1.5muR_{2}}V_{gg,g}^{[2,0]}\qquad\text{for
$R_{1}R_{2}=11,AA,SS,27\,27$}\,,$ $\displaystyle{}^{11}V_{gg,q}^{[2,1]}$
$\displaystyle\approx\frac{4}{z^{2}}\mskip
1.5mu\left[2C_{F}^{2}\,\frac{1}{u\bar{u}}-C_{A}\mskip
1.5muC_{F}\,p_{gg}(u)\right]\,,$ $\displaystyle{}^{AA}V_{gg,q}^{[2,1]}$
$\displaystyle\approx\frac{C_{F}}{C_{A}}\;{}^{AA}V_{gg,g}^{[2,1]}\,,$
$\displaystyle{}^{SS}V_{gg,q}^{[2,1]}$ $\displaystyle\approx
c_{gg,g}(SS)\,\frac{4}{z^{2}}\,\left[\frac{N^{2}-4}{N}\,C_{F}\,\frac{1}{u\bar{u}}-C_{A}\mskip
1.5muC_{F}\,p_{gg}(u)\right]\,,$ $\displaystyle{}^{27\,27}V_{gg,q}^{[2,1]}$
$\displaystyle\approx{}-c_{gg,g}(27\,27)\,\frac{16}{z^{2}}\;p_{gg}(u)\,.\hskip
180.00027pt$ (4.47)

The kernel ${}^{AS}V_{gg,q}^{[2,0]}(uz,\bar{u}z)$ goes like $1/z^{2}$ at small
$z$, but the region $z\ll 1$ does not dominate the small $x$ limit of its
convolution with PDFs in (4.38). This is because the valence combination
$f_{q}(z)-f_{\bar{q}}(z)$ has a small $z$ behaviour far away from the $1/z$
law that underlies the argument at the beginning of this subsection.

#### Remaining channels:

For the splitting $q\to q^{\prime}\bar{q}^{\prime}$, we also find Casimir
scaling:444 The result for
$W^{[2,2]}_{q^{\prime}\mskip-1.5mu\bar{q}^{\prime}\\!,q}$ in equation (161) of
[64] is wrong and should read
$W^{[2,2]}_{q^{\prime}\mskip-1.5mu\bar{q}^{\prime}\\!,q}=(C_{F}/C_{A})\,W^{[2,2]}_{q^{\prime}\mskip-1.5mu\bar{q}^{\prime}\\!,g}\mskip
1.5mu$.

$\displaystyle{}^{RR}V_{q^{\prime}\mskip-1.5mu\bar{q}^{\prime}\\!,q}^{[2,k]}$
$\displaystyle\approx\frac{C_{F}}{C_{A}}\;{}^{RR}V_{q\bar{q},g}^{[2,k]}$
$\displaystyle\text{for all $R$ and }k=0,1\,.$ (4.48)

The kernels for the other pure quark channels, as well as those for $q\to qg$
and $g\to qg$ have no leading $1/z^{2}$ behaviour at small $z$. They behave
like $z^{n}$ with integer $n\geq-1$, and their convolution with a PDF is not
dominated by the region $x\ll z\ll 1$.

We observe that in the channels with a leading $1/z^{2}$ behaviour, the non-
logarithmic part ${}^{R_{1}\mskip-1.5muR_{2}}V^{[2,0]}$ of the kernel scales
with $c(R_{1}R_{2})$ like the LO terms, whereas the logarithmic part
${}^{R_{1}\mskip-1.5muR_{2}}V^{[2,1]}$ in general does not. (An exception to
this rule is the case $R_{1}R_{2}=AS$, which is special as we just explained.)

###  4.4 Small $x_{1}$ or $x_{2}$

We now analyse the limit in which $x_{1}$ or $x_{2}$ becomes small compared
with $x_{1}+x_{2}$, i.e. the limit $u\to 0$ or $u\to 1$. The second limit can
be reduced to the first by swapping the parton indices $a_{1}$ and $a_{2}$,
and we will always take $u\to 0$ for definiteness.

The limit $u\to 0$ is subtle because it gives rise to singularities in the
two-loop kernels that are absent for finite $u$. We isolate these
singularities by writing

$\displaystyle V^{[2,k]}_{\text{reg}}(uz,\bar{u}z)$
$\displaystyle=V^{[2,k]}_{r}(uz,\bar{u}z)+V^{[2,k]}_{s}(uz,\bar{u}z)$
$\displaystyle\text{for }k=0,1\,,$ (4.49)

where the first term is integrable over $z$ in the limit $u\to 0$, whereas the
second one develops a $1/(1-z)$ singularity at $z=1$. This singularity arises
from graphs in which the initial parton $a_{0}$ splits into $a_{2}$ and
another parton. That parton has a momentum fraction $1-z_{2}=1-z(1-u)$, which
appears in the denominator of its propagator.

For $V_{r}$ and for the terms with a plus or $\delta$ distribution in
$V^{(2)}$, one can readily take the limit $u\to 0$. One obtains $1/u$ for the
leading behaviour. For $V_{s}$, one must take the $u\to 0$ limit at the level
of the convolution $V_{s}\otimes f$ and not at the level of the kernel. This
part of the kernel has the general form

$\displaystyle z\mskip 1.5muV_{s}(uz,\bar{u}z)$
$\displaystyle=\sum_{n=2,3,4}\frac{\bar{z}^{n-2}\,h_{n}(\bar{z},u)}{(\bar{z}+u-\bar{z}u)^{n}}+\frac{\log
u-\log(\bar{z}+u-\bar{z}u)}{\bar{z}}\,\frac{h_{1,a}}{\bar{z}+u-\bar{z}u}$
$\displaystyle\quad+\frac{\log\bar{z}-\log(\bar{z}+u-\bar{z}u)}{u}\,\frac{h_{1,b}}{\bar{z}+u-\bar{z}u}+\frac{h_{1,c}(\bar{z},u)}{\bar{z}+u-\bar{z}u}\,,$
(4.50)

where on the r.h.s. we have changed variables from $z$ to $\bar{z}=1-z$. For
ease of writing, we omit the superscript $[2,k]$ on $V_{s}$ and on the
coefficients $h$ henceforth. $h_{1,a}$ and $h_{1,b}$ are numerical constants,
whilst $h_{n}(\bar{z},u)$ and $h_{1,c}(\bar{z},u)$ are functions which in the
limits of small $\bar{z}$ or small $u$ are either finite or have a singularity
of order $\log\bar{z}$ or $\log u$.

Notice that no explicit factor $1/u$ appears when taking the $u\to 0$ limit of
$V_{s}$. We hence erroneously discarded the contribution from $V_{s}$ when
discussing the limit of small $x_{1}$ in our previous work [64]. However, all
terms except for the last one in (4.4) give rise to a $1/u$ behaviour of the
convolution

$\displaystyle V_{s}\underset{12}{\otimes}f$
$\displaystyle=\frac{1}{x}\,\int_{0}^{1-x}\\!d\bar{z}\;V_{s}(uz,\bar{u}z)\,f\left(\frac{x}{1-\bar{z}}\right)\,.$
(4.51)

This is revealed by power counting (with $\log u$ counting as order $1$). In
the region $\bar{z}\sim u$, one finds $V_{s}\sim 1/u^{2}$ and thus obtains a
contribution of order $1/u$ to the integral. By contrast, $V_{s}\sim 1$ for
$\bar{z}\sim 1$, so that this region contributes to the integral only at
subleading level. In the term proportional to $h_{1,b}\mskip 1.5mu$, one must
use $\log\bar{z}-\log(\bar{z}+u+\bar{z}u)\sim u$ for $z\sim 1$ to obtain this
result.

Using the method of regions [101], we can compute the leading behaviour of the
integral by expanding the integrand for small $\bar{z}$ at fixed $\bar{z}/u$.
To implement this, we rescale $u\to\varepsilon u$ and
$\bar{z}\to\varepsilon\bar{z}$, approximate the integrand for $\varepsilon\to
0$, and finally set $\varepsilon=1$. This gives

$\displaystyle
V_{s}\underset{12}{\otimes}f=\frac{1}{x}\,\int_{0}^{1-x}\\!d\bar{z}\;$
$\displaystyle\Biggl{[}\,\sum_{n=2,3,4}\bar{z}^{n-2}\;\frac{h_{n,0}+h_{n,1}\mskip
1.5mu\log\bar{z}}{(\bar{z}+u)^{n}}+\frac{\log
u-\log(\bar{z}+u)}{\bar{z}}\,\frac{h_{1,a}}{\bar{z}+u}$
$\displaystyle\,+\frac{\log\bar{z}-\log(\bar{z}+u)}{u}\,\frac{h_{1,b}}{\bar{z}+u}\,\Biggr{]}\,f(x)+\ldots\,,$
(4.52)

where we used the leading behaviour $h_{n}(\bar{z},u)\sim
h_{n,0}+h_{n,1}\,\log\bar{z}$ in the limit just specified. The ellipsis
denotes subleading terms in the $\varepsilon$ expansion and in particular
contains the full contribution from the term with $h_{1,c}(\bar{z},u)$ in
(4.4). One can readily carry out the integral over $\bar{z}$ for the terms
given in (4.4) and then take the limit $u=0$. The result goes like $1/u$ and
is enhanced by $\log u$ for the terms with nonzero coefficients $h_{n,1}$.

With these preliminaries worked out, we can determine the small $u$ limit of
the two-loop kernels for each channel in turn.

####  $\boldsymbol{g\to gg}\mskip 1.5mu$:

$\displaystyle{}^{11}V_{gg,g}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip 1.5mux}\left(\frac{10}{3}\mskip
1.5muC_{A}\mskip 1.5mu\beta_{0}-\frac{110}{9}\mskip
1.5muC_{A}^{2}\right)f(x)\,,$
$\displaystyle{}^{AA}V_{gg,g}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{gg,g}(AA)}{u\mskip
1.5mux}\left(\frac{10}{3}C_{A}\mskip
1.5mu\beta_{0}-C_{A}^{2}\,\frac{43+3\pi^{2}}{9}\right)f(x)\,,$
$\displaystyle{}^{27\,27}V_{gg,g}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{gg,g}(27\,27)}{u\mskip
1.5mux}\left(10\beta_{0}+\frac{206-24\pi^{2}}{3}\right)f(x)\,,$
$\displaystyle{}^{11}V_{gg,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip 1.5mux}\left[\left(2C_{A}\mskip
1.5mu\beta_{0}-C_{A}^{2}\,\frac{22+12\log
u}{3}\right)f(x)+4C_{A}^{2}\,\left(\mskip
1.5mu\frac{z}{[\bar{z}]_{+}}+\frac{\bar{z}}{z}+z\bar{z}\right)\otimes
f\,\right]\,,$ $\displaystyle{}^{AA}V_{gg,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{gg,g}(AA)}{u\mskip 1.5mux}\left(2C_{A}\mskip
1.5mu\beta_{0}-C_{A}^{2}\,\frac{11+6\log u}{3}\right)f(x)\,,$
$\displaystyle{}^{27\,27}V_{gg,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{gg,g}(27\,27)}{u\mskip
1.5mux}\left[\bigl{(}6\beta_{0}+22+12\log u\bigr{)}\,f(x)-60\left(\mskip
1.5mu\frac{z}{[\bar{z}]_{+}}+\frac{\bar{z}}{z}+z\bar{z}\right)\otimes
f\,\right]\,.$ (4.53)

The terms with an explicit factor $f(x)$ in these expressions are due to
$V^{[2,k]}_{s}$ and $V^{[2,k]}_{\delta}$, whereas the Mellin convolutions

$\displaystyle\varphi(z)\otimes f$
$\displaystyle=\int_{x}^{1}\frac{dz}{z}\,\varphi(z)\,f\biggl{(}\frac{x}{z}\biggr{)}$
(4.54)

originate from $V^{[2,k]}_{r}$ and $V^{[2,1]}_{p}\big{/}\mskip
1.5mu[\bar{z}]_{+}$. For the sake of legibility, we do not indicate the
argument $x$ of these convolutions in (4.4) and the following equations. We
note that the double logarithmic part of the $V^{(2)}$ also contributes to the
small $u$ limit of $V^{(2)}\smash{\underset{12}{\otimes}}\,f$. Its
contribution is readily obtained by extracting the leading term of
${V}^{(1)}(u)$ for $u\to 0$.

####  $\boldsymbol{q\to gq}\mskip 1.5mu$:

$\displaystyle{}^{11}V_{gq,q}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip 1.5mux}\left(\frac{10}{3}\mskip
1.5muC_{F}\beta_{0}-\frac{110}{9}\mskip 1.5muC_{A}\mskip
1.5muC_{F}\right)f(x)\,,$
$\displaystyle{}^{A8}V_{gq,q}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{qg,q}(8A)}{u\mskip
1.5mux}\left[\left(\frac{10}{3}C_{F}\beta_{0}-C_{A}\mskip
1.5muC_{F}\,\frac{43+3\pi^{2}}{9}\right)f(x)+C_{A}\mskip
1.5muC_{F}\,\bar{z}\otimes f\,\right]\,,$
$\displaystyle{}^{11}V_{gq,q}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip
1.5mux}\left[\left(2C_{F}\beta_{0}-C_{A}\mskip 1.5muC_{F}\,\frac{22+12\log
u}{3}\right)f(x)+2C_{A}\mskip 1.5muC_{F}\,\frac{1+z^{2}}{[\bar{z}]_{+}}\otimes
f\,\right]\,,$ $\displaystyle{}^{A8}V_{gq,q}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{qg,q}(8A)}{u\mskip
1.5mux}\left(2C_{F}\beta_{0}-C_{A}\mskip 1.5muC_{F}\,\frac{11+6\log
u}{3}\right)f(x)\,.$ (4.55)

We remark that the only cases for which the $1/u$ behaviour is enhanced by
$\log u$ are the convolutions of ${V}_{gg,g}^{[2,1]}$ and ${V}_{gq,q}^{[2,1]}$
with $f$. The $\log u$ factor is thus always accompanied by a logarithm $L$ of
the scale $\mu$.

####  $\boldsymbol{g\to qg}$ and $\boldsymbol{g\to gq}\mskip 1.5mu$:

$\displaystyle{}^{11}V_{qg,g}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip 1.5mux}\,\frac{10}{9}\mskip
1.5muC_{A}\,f(x)\,,$
$\displaystyle{}^{11}V_{qg,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip 1.5mux}\,\frac{2}{3}\mskip
1.5muC_{A}\,f(x)\,,$
$\displaystyle{}^{8A}V_{qg,g}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{2}\,c_{qg,q}(8A)\;{}^{11}V_{qg,g}^{[2,0]}\,,$
$\displaystyle{}^{8A}V_{qg,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{2}\,c_{qg,q}(8A)\;{}^{11}V_{qg,g}^{[2,1]}\,,$
(4.56) $\displaystyle{}^{11}V_{gq,g}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx-\frac{2}{u\mskip
1.5mux}\,(C_{A}-C_{F})\;(z\bar{z})\otimes f\,,$
$\displaystyle{}^{11}V_{gq,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip
1.5mux}\,C_{A}\,(z^{2}+\bar{z}^{2})\otimes f\,,$
$\displaystyle{}^{A8}V_{gq,g}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx-\frac{c_{qg,q}(8A)}{u\mskip
1.5mux}\,(C_{A}-2C_{F})\;(z\bar{z})\otimes f\,,$
$\displaystyle{}^{A8}V_{gq,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\mathcal{O}(u^{0})\,.$ (4.57)

####  $\boldsymbol{q\to gg}\mskip 1.5mu$:

$\displaystyle{}^{11}V_{gg,q}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{2}{u\mskip 1.5mux}\,C_{F}\mskip
1.5mu(C_{A}-C_{F})\;z\otimes f\,,$
$\displaystyle{}^{11}V_{gg,q}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx-\frac{2}{u\mskip
1.5mux}\,C_{F}(C_{A}-2C_{F})\,\frac{1+\bar{z}^{2}}{z}\otimes f\,,$
$\displaystyle{}^{AA}V_{gg,q}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{gg,g}(AA)}{u\mskip 1.5mux}\,C_{A}\mskip
1.5muC_{F}\,z\otimes f\,,$
$\displaystyle{}^{AA}V_{gg,q}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\mathcal{O}(u^{0})\,,$
$\displaystyle{}^{SS}V_{gg,q}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{gg,g}(SS)}{u\mskip
1.5mux}\,\frac{N^{2}+4}{N}\,C_{F}\,z\otimes f\,,$
$\displaystyle{}^{SS}V_{gg,q}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx 4\mskip
1.5muc_{gg,g}(SS)\;{}^{11}V_{gg,q}^{[2,1]}\underset{12}{\otimes}f\,,$
$\displaystyle{}^{27\,27}V_{gg,q}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{gg,g}(27\,27)}{u\mskip 1.5mux}\;8z\otimes f\,,$
$\displaystyle{}^{27\,27}V_{gg,q}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx-\frac{c_{gg,g}(27\,27)}{u\mskip 1.5mux}\;8\mskip
1.5mu\frac{1+\bar{z}^{2}}{z}\otimes f\,,$
$\displaystyle{}^{AS}V_{gg,q}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{\tilde{c}(AS)}{u\mskip 1.5mux}\,\frac{z}{2}\otimes
f\,,$ $\displaystyle{}^{AS}V_{gg,q}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\mathcal{O}(u^{0})\,,$ (4.58)

where $\tilde{c}(AS)$ is given in (4.32).

#### Remaining channels:

For $q\to q^{\prime}q$ in the colour singlet representation, we have

$\displaystyle{}^{11}V_{q^{\prime}\mskip-1.5muq,q}^{[2,0]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip 1.5mux}\,C_{F}\,\frac{10}{9}\,f(x)\,,$
$\displaystyle{}^{11}V_{q^{\prime}\mskip-1.5muq,q}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{1}{u\mskip 1.5mux}\,C_{F}\,\frac{2}{3}\,f(x)\,.$
(4.59)

Corresponding expressions for ${}^{88}V_{q^{\prime}\mskip-1.5muq,q}$,
${}^{11}V_{\bar{q}^{\prime}\mskip-1.5muq,q}$ and
${}^{88}V_{\bar{q}^{\prime}\mskip-1.5muq,q}$ follow from (4.34) and (4.1)
together with (4.15). The convolutions for $g\to q\bar{q}$, $q\to qg$, and for
the other pure quark channels do not have a leading $1/u$ behaviour.

Notice that in $V_{qg,g}$ and $V_{q^{\prime}\mskip-1.5muq,q}$ we have a $1/u$
enhancement for a slow _quark_ rather than a slow gluon. This enhancement is
due to graphs in which the observed slow quark originates from a slow gluon,
see figures 4(c) and 4(l).

###  4.5 Triple Regge limit

We finally consider the “triple Regge limit” $x_{1}\ll x_{1}+x_{2}\ll 1$ or
$x_{2}\ll x_{1}+x_{2}\ll 1$. The kernels with a leading behaviour turn out the
be symmetric under the interchange of $a_{1}$ and $a_{2}$, so that we can
restrict ourselves to the first of the two cases.

The limit can be approached in two ways. One can either start from the small
$x$ results in section 4.3 and take their limit for $u\to 0$, or one can start
from the small $u$ results in section 4.4 and collect their leading terms for
$x\to 0$. Let us discuss the two procedures in turn.

Taking the limit $u\to 0$ of the kernels expanded for small $z$ in (4.3) to
(4.48) is straightforward and yields the following expressions.

####  $\boldsymbol{g\to gg}\mskip 1.5mu$:

$\displaystyle{}^{11}V_{gg,g}^{[2,0]}$
$\displaystyle\approx\frac{2C_{A}^{2}}{z^{2}}\,\bigl{(}2+3\log u\bigr{)}\,,$
$\displaystyle{}^{11}V_{gg,g}^{[2,1]}$
$\displaystyle\approx\frac{4C_{A}^{2}}{z^{2}}\,\frac{1}{u}\,,$
$\displaystyle{}^{AA}V_{gg,g}^{[2,0]}$ $\displaystyle\approx
c_{gg,g}(AA)\,{}^{11}V_{gg,g}^{[2,0]}\,,$
$\displaystyle{}^{AA}V_{gg,g}^{[2,1]}$ $\displaystyle\approx
c_{gg,g}(AA)\,\frac{8C_{A}^{2}}{z^{2}}\,,$
$\displaystyle{}^{27\,27}V_{gg,g}^{[2,0]}$ $\displaystyle\approx
c_{gg,g}(27\,27)\,{}^{11}V_{gg,g}^{[2,0]}\,,$
$\displaystyle{}^{27\,27}V_{gg,g}^{[2,1]}$ $\displaystyle\approx-
c_{gg,g}(27\,27)\,\frac{60}{z^{2}}\mskip 1.5mu\frac{1}{u}\,.$ (4.60)

####  $\boldsymbol{g\to q\bar{q}}\mskip 1.5mu$:

$\displaystyle{}^{11}V_{q\bar{q},g}^{[2,0]}$
$\displaystyle\approx-\frac{C_{A}}{z^{2}}\,\bigl{(}1+\log u\bigr{)}\,,$
$\displaystyle{}^{11}V_{q\bar{q},g}^{[2,1]}$
$\displaystyle\approx-\frac{C_{A}}{z^{2}}\,,$
$\displaystyle{}^{88}V_{q\bar{q},g}^{[2,0]}$ $\displaystyle\approx
c_{q\bar{q},g}(88)\,{}^{11}V_{q\bar{q},g}^{[2,0]}\,,$
$\displaystyle{}^{88}V_{q\bar{q},g}^{[2,1]}$ $\displaystyle\approx
c_{q\bar{q},g}(88)\,{}^{11}V_{q\bar{q},g}^{[2,1]}\,.$ (4.61)

####  $\boldsymbol{q\to gg}\mskip 1.5mu$:

$\displaystyle{}^{R_{1}\mskip-1.5muR_{2}}V_{gg,q}^{[2,0]}$
$\displaystyle\approx\frac{C_{F}}{C_{A}}\;{}^{R_{1}\mskip-1.5muR_{2}}V_{gg,g}^{[2,0]}\qquad\text{for
$R_{1}R_{2}=11,AA,SS,27\,27$}\,,$ $\displaystyle{}^{11}V_{gg,q}^{[2,1]}$
$\displaystyle\approx-\frac{4}{z^{2}}\mskip 1.5mu\frac{C_{F}\mskip
1.5mu(C_{A}-2C_{F})}{u}\,,$ $\displaystyle{}^{AA}V_{gg,q}^{[2,1]}$
$\displaystyle\approx\frac{C_{F}}{C_{A}}\;{}^{AA}V_{gg,g}^{[2,1]}\,,$
$\displaystyle{}^{SS}V_{gg,q}^{[2,1]}$ $\displaystyle\approx-
c_{gg,g}(SS)\,\frac{16}{z^{2}}\,\frac{C_{F}\mskip 1.5mu(C_{A}-2C_{F})}{u}\,,$
$\displaystyle{}^{27\,27}V_{gg,q}^{[2,1]}$ $\displaystyle\approx-
c_{gg,g}(27\,27)\,\frac{16}{z^{2}}\,\frac{1}{u}\,,\hskip 160.00024pt$ (4.62)

where we omit the case $R_{1}R_{2}=AS$ for the reason given after (4.3).

#### Remaining channels:

$\displaystyle{}^{RR}V_{q^{\prime}\bar{q}^{\prime},q}^{[2,k]}$
$\displaystyle\approx\frac{C_{F}}{C_{A}}\;{}^{RR}V_{q\bar{q},g}^{[2,k]}\,,$
$\displaystyle\text{for all $R$ and }k=0,1\,.$ (4.63)

The kernels for all other channels have no leading $1/z^{2}$ behaviour.

Taking the small $x$ limit of the small $u$ results in section 4.4 means to
select the convolution terms $(1/z)\otimes f$, since this will give a small
$x$ enhancement as explained in section 4.3. Such a behaviour is only found in
very few cases, namely in

$\displaystyle{}^{11}V_{gg,g}^{[2,1]}$
$\displaystyle\approx\frac{4C_{A}^{2}}{z^{2}\mskip 1.5muu}\,,$
$\displaystyle{}^{27\,27}V_{gg,g}^{[2,1]}$
$\displaystyle\approx{}-c_{gg,g}(27\,27)\,\frac{60}{z^{2}\mskip 1.5muu}\,,$
(4.64) and $\displaystyle{}^{11}V_{gg,q}^{[2,1]}$
$\displaystyle\approx{}-\frac{4C_{F}\mskip 1.5mu(C_{A}-2C_{F})}{z^{2}\mskip
1.5muu}\,,$ $\displaystyle{}^{SS}V_{gg,q}^{[2,1]}$ $\displaystyle\approx
4\mskip 1.5muc_{gg,g}(SS)\,{}^{11}V_{gg,q}^{[2,1]}\,,$
$\displaystyle{}^{27\,27}V_{gg,q}^{[2,1]}$
$\displaystyle\approx{}-c_{gg,g}(27\,27)\,\frac{16}{z^{2}\mskip 1.5muu}\,.$
(4.65)

These expressions are in agreement with the ones in (4.5) and (4.5), which
were obtained by taking the opposite order of the two limits. In fact, the
only cases in which the two orders of limits give the same result are those in
which the kernels have a leading behaviour for small $z$ _and_ small $u$.

As an example in which the two limits do not commute, let us take the kernel
${}^{AA}V_{gg,g}^{[2,1]}$. Taking first the small $x$ limit and then $u\to 0$,
we obtain

$\displaystyle{}^{AA}V_{gg,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx
c_{gg,g}(AA)\;\frac{1}{x}\,\biggl{(}\frac{8C_{A}^{2}}{z}\otimes
f\biggr{)}\approx c_{gg,g}(AA)\;8C_{A}^{2}\;E(x)\,\frac{f(x)}{x}\,,$ (4.66)

where in the last step we have performed the convolution for specific choices
of the PDF, as detailed in (4.44). If we first take $u\to 0$ and then the
small $x$ limit, we have instead

$\displaystyle{}^{AA}V_{gg,g}^{[2,1]}\underset{12}{\otimes}f$
$\displaystyle\approx\frac{c_{gg,g}(AA)}{u}\left(2C_{A}\mskip
1.5mu\beta_{0}-C_{A}^{2}\,\frac{11+6\log u}{3}\right)\frac{f(x)}{x}$ (4.67)

with a leading $u^{-1}\,\log u$ behaviour but no small $x$ enhancement.
Clearly, the approach to the triple Regge limit strongly depends on the
direction in the plane of $x_{1}$ and $x_{2}$ in this case.

##  5 Numerical illustration

In this section, we illustrate the impact of NLO corrections on the splitting
DPDs. For easier comparison, we use the same settings as in section 5.4 of our
study [64] for the colour singlet case. We take $y=0.022\operatorname{fm}$ for
the inter-parton distance. This corresponds to a scale

$\displaystyle\mu_{y}$ $\displaystyle=b_{0}/y=10\operatorname{GeV}$ (5.1)

at which the logarithm $L$ in (2.12) is zero.

For the PDFs in the splitting formula, we take the NLO set of CT14 [103],
along with the coupling $\alpha_{s}$ used in that set.555We use the set
CT14nlo obtained from LHAPDF [104] via the ManeParse interface [105]. This
gives

$\displaystyle\alpha_{s}(10\operatorname{GeV})$ $\displaystyle=0.178\,,$
$\displaystyle\alpha_{s}(5\operatorname{GeV})$ $\displaystyle=0.212\,,$
$\displaystyle\alpha_{s}(20\operatorname{GeV})$ $\displaystyle=0.153$ (5.2)

at the scales on which we focus in the following study. We work with $n_{F}=5$
active quark flavours.

The splitting DPD at two-loop accuracy is given by the sum of the LO and NLO
terms $F^{(1)}$ and $F^{(2)}$ defined in (2.102). Note that we evaluate the
one-loop expression $F^{(1)}$ with NLO PDFs here. Thus, the comparison of
$F^{(2)}$ with $F^{(1)}$ directly reveals the impact of the two-loop splitting
kernels. As we saw in [64], the difference between evaluating $F^{(1)}$ with
LO or NLO PDFs is typically quite large, at least in the $gg$ channel. This
simply reflects that PDFs at moderately large scales depend quite strongly on
the order at which they are determined and evolved.

###  5.1 Two-gluon distributions

In figure 8 we show the full splitting DPD $F_{gg}=F^{(1)}_{gg}+F^{(2)}_{gg}$
as a function of $x_{1}$ for different values of $x_{2}$. We take scales
$\mu=\sqrt{x_{1}x_{2}\mskip 1.5mu\zeta_{p}}=\mu_{y}$ and use the standard
definition of the $\overline{\text{MS}}$ scheme, so that according to (2.60)
the scheme parameter in the two-loop kernels (4) is
$c_{\scalebox{0.6}{$\overline{\mathrm{MS}}$}}=-\zeta_{2}/2$. The sign of the
distributions (not visible in the logarithmic plots) is the same as at LO,
i.e. ${}^{11}\mskip-1.5muF_{gg}$ and ${}^{SS}\mskip-1.5muF_{gg}$ are positive
and ${}^{AA}\mskip-1.5muF_{gg}$ and ${}^{27\,27}\mskip-1.5muF_{gg}$ are
negative.

The relative size of ${}^{RR}\mskip-1.5muF_{gg}$ for different $R$ reflects
the scaling factors of the LO kernels given in (4) and (4). In some panels of
figure 8, one can see slight differences in the shape of the distributions,
which are due to the two-loop terms. To see this more clearly, we plot the
ratio ${F}^{(2)}\big{/}{F}^{(1)}$ in figure 9. We find that the $a_{s}^{2}$
corrections exhibit a quite varied structure as a function of the momentum
fractions $x_{1}$ and $x_{2}$. Not exceeding $30\%$, they are of moderate size
in this setting. Note that
${}^{RR}\mskip-1.5muF^{(2)}\big{/}{}^{RR}\mskip-1.5muF^{(1)}$ is nearly equal
for $R=A$ and $R=S$ at small $x_{1}+x_{2}$, whilst for large values of
$x_{1}+x_{2}$ a clear difference between the two representations is observed.
The growing difference with increasing $x_{1}+x_{2}$ is what we anticipated
when discussing our analytic result (4.37).

(a) $x_{2}=x_{1}$

(b) $x_{2}=0.1$

(c) $x_{2}=10^{-3}$

Figure 8:  Absolute value of the splitting DPD
${}^{RR}\mskip-1.5muF_{gg}^{(1)}+{}^{RR}\mskip-1.5muF_{gg}^{(2)}$ evaluated at
two-loop accuracy with central values $\mu=\sqrt{x_{1}x_{2}\mskip
1.5mu\zeta_{p}}=\mu_{y}$ of the scales. The curves for $R=A$ and $R=S$ cannot
be distinguished at the scale of the plots. Unless mentioned otherwise, all
plots in this section are for CT14 NLO PDFs and the standard definition of the
$\overline{\text{MS}}$ scheme (i.e. for
$c_{\scalebox{0.6}{$\overline{\mathrm{MS}}$}}=-\zeta_{2}/2$).

(a) $x_{2}=x_{1}$ (b) $x_{2}=0.1$ (c) $x_{2}=10^{-3}$

Figure 9:  The ratio
${}^{RR}\mskip-1.5muF^{(2)}_{gg}\big{/}{}^{RR}\mskip-1.5muF^{(1)}_{gg}$,
evaluated at central values of the scales as in figure 8.

We also evaluated ${F}^{(2)}\big{/}{F}^{(1)}$ for
$\mu_{y}=5\operatorname{GeV}$ and $\mu_{y}=80\operatorname{GeV}$, always with
central values $\mu=\sqrt{x_{1}x_{2}\mskip 1.5mu\zeta_{p}}=\mu_{y}$ of the
scales. Compared with the choice in (5.1), the overall size of the ratio
changes due to the changed value of $\alpha_{s}(\mu_{y})$. There is also a
mild change in the shape of the curves, due to the scale dependence of the
PDFs in the splitting formula. An example is shown in figure 10. We continue
our study with the value $\mu_{y}=10\operatorname{GeV}$ in (5.1).

Figure 10:  As figure 9(c) but for $\mu_{y}=80\operatorname{GeV}$ and
$\alpha_{s}(80\operatorname{GeV})=0.120$. The corresponding curves for
$\mu_{y}=5\operatorname{GeV}$ are similar in shape but differ in overall size,
with absolute values up to $0.3$ for $R=27$ and up to $0.2$ for the octets and
the singlet.

To gauge the dependence of our results on the PDFs used in the splitting
formula, we produced plots for the NLO sets of ABMP16 [106], NNPDF3.1 [107],
and MSHT20 [108].666The corresponding LHAPDF identifiers are ABMP16_5_nlo,
NNPDF31_nlo_as_0118, and MSHT20nlo_as118. As illustrated in figure 11, the
curves for ABMP16 and MSHT20 are quite similar to those for CT14 up to about
$x_{1}+x_{2}\sim 0.4$, whereas differences become more pronounced for larger
$x_{1}+x_{2}$. The same holds for NNPDF3.1. In this case, the gluon
distribution at $\mu=10\operatorname{GeV}$ has a zero for $x\approx 0.7$,
which leads to a corresponding zero in ${}^{RR}\mskip-1.5muF^{(1)}_{gg}$.

At order $a_{s}^{2}$, the splitting DPDs have a dependence on the
$\overline{\text{MS}}$ scheme definition, which cancels at the level of DPS
cross sections as explained at the end of section 2.6. According to (4), a
change in the scheme parameter $c_{\scalebox{0.6}{$\overline{\mathrm{MS}}$}}$
results in a shift in the ratio
${}^{RR}\mskip-1.5muF^{(2)}\big{/}{}^{RR}\mskip-1.5muF^{(1)}$ that is
proportional to $a_{s}{}^{R}\gamma_{J}^{(0)}$ and independent of $x_{1}$ and
$x_{2}$. Comparing figure 12 with figure 9(c), we see that this shift is
moderate for two-gluon distributions in the octet channels but quite large for
$R=27$.

(a) MSHT20 NLO (b) ABMP16 NLO

Figure 11:  The ratio
${}^{RR}\mskip-1.5muF^{(2)}_{gg}\big{/}{}^{RR}\mskip-1.5muF^{(1)}_{gg}$ for
$x_{2}=10^{-3}$, evaluated at central scales for different PDFs. The
corresponding plot for the CT14 NLO set is in figure 9(c). Figure 12:  As
figure 9(c), but with the $\overline{\text{MS}}$ scheme defined in [68] (i.e.
with $c_{\scalebox{0.6}{$\overline{\mathrm{MS}}$}}=0$).

We now investigate how the importance of $a_{s}^{2}$ contribution changes if
we evaluate the DPDs at different scales. In figure 13, we show the ratio
$F^{(2)}_{gg}\big{/}F^{(1)}_{gg}$ for three different values of $\mu$, taking
$\sqrt{x_{1}x_{2}\mskip 1.5mu\zeta_{p}}=\mu_{y}$ in all three cases. We see
that for both $\mu=\mu_{y}/2$ and $\mu=2\mu_{y}$, the impact of the two-loop
kernels becomes substantial at least in some region of $x_{1}$ and $x_{2}$.
This holds for all colour representations and is in stark contrast to the case
$\mu=\mu_{y}$.

(a) $\mu=\mu_{y}$ (b) $\mu=\mu_{y}/2$ (c) $\mu=2\mu_{y}$

Figure 13:  The ratio
${}^{RR}\mskip-1.5muF^{(2)}_{gg}\big{/}{}^{RR}\mskip-1.5muF^{(1)}_{gg}$ for
$x_{2}=10^{-3}$, evaluated at different values of the scale $\mu$ at
$\sqrt{x_{1}x_{2}\mskip 1.5mu\zeta_{p}}=\mu_{y}$. The plot in panel (a) is the
same as in figure 9(c) and shown again for easier comparison.

A change of the rapidity parameter $\zeta_{p}$ at given $\mu$ affects only the
double logarithmic part of the kernel (4). For the ratio
${}^{RR}\mskip-1.5muF^{(2)}_{gg}\big{/}{}^{RR}\mskip-1.5muF^{(1)}_{gg}$, this
induces a shift proportional to $a_{s}{}^{R}\gamma_{J}^{(0)}$ and independent
of $x_{1}$ and $x_{2}$ (as does the change of $\overline{\text{MS}}$ scheme
definition). We show the ratio for $\sqrt{x_{1}x_{2}\mskip
1.5mu\zeta_{p}}=\mu$ with $\mu=\mu_{y}/2$ and $\mu=2\mu_{y}$ in figure 14 and
see that also in this case the impact of the two-loop terms is in general
large. The large NLO corrections are mainly due to the single logarithmic
terms $L\,V^{[2,1]}$ in the two-loop kernel.

We conclude that, at least for two-gluon DPDs, it is most favourable to use
the fixed-order splitting formula for $\mu$ and $\sqrt{x_{1}x_{2}\mskip
1.5mu\zeta_{p}}$ very close to $\mu_{y}$, so as to avoid large radiative
corrections from higher orders. The DPD at any other scale can then be
obtained by using the evolution equations (2.1) and (2.1).

(a) $\mu=\sqrt{x_{1}x_{2}\mskip 1.5mu\zeta_{p}}=\mu_{y}/2$ (b)
$\mu=\sqrt{x_{1}x_{2}\mskip 1.5mu\zeta_{p}}=2\mu_{y}$

Figure 14:  The ratio $F^{(2)}_{gg}\big{/}F^{(1)}_{gg}$ for $x_{2}=10^{-3}$,
evaluated at different values of $\mu=\sqrt{x_{1}x_{2}\mskip 1.5mu\zeta_{p}}$.

Let us finally take a look at the mixed octet distribution
${}^{AS}\mskip-1.5muF_{gg}$, which starts at order $a_{s}^{2}$ and is driven
by the quark valence distributions. Comparing the plots in figure 15 with
those in figure 8, we see that ${}^{AS}\mskip-1.5muF_{gg}$ is tiny compared
with the two-gluon distributions in the channels that receive a contribution
from $g\to gg$ splitting. The zero of ${}^{AS}\mskip-1.5muF_{gg}$ at
$x_{1}=x_{2}$ is a consequence of the asymmetry of this distribution under the
exchange $x_{1}\leftrightarrow x_{2}$.

(a) $x_{2}=0.1$

(b) $x_{2}=10^{-3}$

Figure 15:  The distribution ${}^{AS}\mskip-1.5muF_{gg}$, evaluated at central
values of the scales as in figure 8.

###  5.2 Quark-gluon distributions

We now turn our attention to the quark-gluon channel. Here, a new phenomenon
occurs compared with the two-gluon case. This is seen in figure 16, where we
show the term $F^{(1)}$ and the separate contributions from the subprocesses
$q\to qg$ and $g\to qg$ to $F^{(2)}$. For $x_{1}\sim x_{2}$ and for
$x_{1}>x_{2}$ , the $a_{s}^{2}$ contributions are small compared with the
$a_{s}$ term, but when $x_{1}$ becomes much smaller than $x_{2}$, the
contribution from $g\to qg$ quickly overtakes the $a_{s}$ term and completely
dominates the sum of terms.

This can be understood from our results for the limit of small
$u=x_{1}/(x_{1}+x_{2})$ in section 4.4. According to (4.4), the two-loop
contribution $F^{(2)}(g\to qg)$ behaves like $1/u$ in that limit, whereas
$F^{(1)}$ and $F^{(2)}(q\to qg)$ do not have a $1/u$ enhancement. Since in
figure 16 we plot contributions to the scaled DPD $x_{1}\mskip
1.5mux_{2}\mskip 1.5muF$ at fixed $x_{2}$, the contribution from $g\to qg$
tends to a constant for small $x_{1}$ and the contributions from $q\to qg$ go
to zero. Closer inspection reveals that $F^{(2)}(q\to qg)$ has a $\log^{2}u$
enhancement compared with $F^{(1)}$ for small $u$, which explains why the
$a_{s}^{2}$ term for $q\to qg$ slowly overtakes the $a_{s}$ term in that
limit.

(a) $R_{1}R_{2}=11$ (b) $R_{1}R_{2}=8A$

Figure 16:  Absolute values of the different contributions to
${}^{R_{1}\mskip-1.5muR_{2}}\mskip-1.5muF_{ug}$ at $x_{2}=0.1$, evaluated at
central scales with the CT14 PDFs. The case $R_{1}R_{2}=8S$ is not shown here,
because according to (4.4) and 4.25 one has
${}^{8S}\mskip-1.5muF_{ug}={}-\sqrt{5/9}\,\,{}^{8A}\mskip-1.5muF_{ug}$ for
each separate contribution.

As explained in section 4.4, the $1/u$ enhancement of $F^{(2)}(g\to qg)$
originates from the graph in figure 4(c), where a gluon with momentum fraction
of order $u$ is emitted from a gluon with much larger momentum fraction. This
is not possible in the $q\to qg$ channel at order $a_{s}^{2}$. As is well
known in small $x$ physics, higher orders will enhance the $1/u$ behaviour by
factors of $\log^{k}u$ but not lead to higher powers like $1/u^{2}$. Hence,
one does not expect a change as drastic as the step from one to two loops in
figure 16 when going to three loops or higher. This situation is reminiscent
of hard-scattering cross sections, where the step from LO to NLO accuracy can
be huge if qualitatively new contributions first appear at NLO.

The graph in figure 4(c) is not only responsible for the $1/u$ behaviour of
$F^{(2)}(g\to qg)$, but it also contributes to the LO splitting $g\ \to gg$
followed by one step of LO evolution with the splitting of a gluon into
$q\bar{q}$. At small $u$, one has $|{F}_{gg}^{(1)}|\gg|{F}_{qg}^{(1)}|$ in all
relevant colour channels, so that $|{F}_{qg}^{(1)}|$ will quickly increase in
size when evolved to higher scales.

##  6 Conclusions

DPDs at small inter-parton distance $y$ are dominated by the splitting of one
parton into two. We computed the kernels that describe this splitting at order
$a_{s}^{2}$ for all colour configurations of the observed partons. All partons
are taken to be unpolarised. The colour dependence of the DPDs is classified
by a pair of irreducible representations $R_{1}$ and $R_{2}$ of the colour
group. For representations different from the singlet, Sudakov double
logarithms in the renormalisation scale $\mu$ and the rapidity parameter
$\zeta_{p}$ appear in the splitting kernels. These logarithms go along with
rapidity divergences in the intermediate stages of the calculation, which need
to be regularised appropriately. We computed the kernels independently with
the Collins regulator, which uses spacelike Wilson lines, and with the
$\delta$ regulator of Echevarria et al. Intermediate results differ in the two
cases, for instance by the order of ultraviolet poles in DPD matrix elements
and the soft factor, but in both cases we find complete cancellation of
ultraviolet and rapidity divergences when all building blocks are assembled.
The resulting splitting kernels are identical for both regulators.

In the context of multi-loop calculations, the Collins regulator is more
involved because eikonal propagators depend not only on plus-momenta. However,
in the limit of large Wilson line rapidities, these propagators take a simple
distributional form. With this form, loop integrals are no more difficult than
those to be computed with the $\delta$ regulator, where eikonal propagators
depend only on plus-momenta. Whilst in our two-loop calculation, there is at
most one eikonal propagator per graph, we see no impediment to using the same
procedure at higher orders.

At order $a_{s}$, the splitting DPDs for a given parton combination are
proportional to each other for the different colour channels $R_{1}R_{2}$.
This no longer holds at order $a_{s}^{2}$, where the colour dependence of the
splitting kernels presents a rather varied picture. Nevertheless, we
identified a number of simple patterns.

  1. 1.

At two-loop accuracy, the splitting DPD for $gg$ in decuplet representations
is zero, and the distributions ${}^{8A}\mskip-1.5muF_{qg}$ and
${}^{8S}\mskip-1.5muF_{qg}$ for the gluon in the antisymmetric and symmetric
octet are proportional to each other.

  2. 2.

At two loops, terms containing Sudakov double logarithms are proportional to
the one-loop splitting result times the one-loop anomalous dimension (2.6) of
the Collins-Soper kernel for the rapidity dependence of the DPD.

  3. 3.

A threshold logarithm $\log(1-x_{1}-x_{2})$ appears in the splitting DPDs at
two loops and is accompanied by a renormalisation group logarithm
$L=2\log(y\mu/b_{0})$. Its coefficient is nonzero for the distributions
${}^{11}\mskip-1.5muF_{gg}$, ${}^{27\,27}\mskip-1.5muF_{gg}$,
${}^{11}\mskip-1.5muF_{q\bar{q}}$, ${}^{88}\mskip-1.5muF_{q\bar{q}}$, and
${}^{11}\mskip-1.5muF_{qg}$, but vanishes for $gg$ and $qg$ distributions in
the octet channels.

  4. 4.

When the sum $z=z_{1}+z_{2}$ of parton momentum fractions in the two-loop
kernel becomes small, a leading $1/z^{2}$ behaviour is found for $g\to gg$,
$q\to gg$, and $g\to q\bar{q}$. This gives rise to a high-energy logarithm
$\log(x_{1}+x_{2})$ when convoluted with PDFs of a suitable shape, as
specified in (4.44). If $\mu$ equals $\mu_{y}=b_{0}/y$, the leading $1/z^{2}$
terms for different channels $R_{1}R_{2}$ scale exactly like the one-loop
kernels, but for other values of $\mu$ they do not.

  5. 5.

There is no simple pattern for the colour dependence of splitting DPDs in the
limit of small $u=x_{1}/(x_{1}+x_{2})$. The strongest enhancement in this
limit is by a factor $u^{-1}\log u$. It appears in the $g\to gg$ and $q\to gq$
channels and goes along with a renormalisation group logarithm $L$.

  6. 6.

In the triple Regge limit $x_{1}\ll x_{1}+x_{2}\ll 1$, a double enhancement
like $u^{-1}\log(x_{1}+x_{2})$ appears in ${}^{11}\mskip-1.5muF_{gg}$,
${}^{SS}\mskip-1.5muF_{gg}$, and ${}^{27\,27}\mskip-1.5muF_{gg}$ for PDFs of a
suitable shape. It originates from a $1/(uz^{2})$ behaviour of the splitting
kernels and is accompanied by a factor $L$. In the case of
${}^{SS}\mskip-1.5muF_{gg}$, the subprocess $q\to gg$ is enhanced in this way,
whereas $g\to gg$ is not. The behaviour of other DPDs depends on the order in
which the limits $x_{1}\ll x_{1}+x_{2}$ and $x_{1}+x_{2}\ll 1$ are taken.

  7. 7.

Because of charge conjugation invariance, the distributions
${}^{AS}\mskip-1.5muF_{gg}$ or ${}^{S\mskip-1.5muA}\mskip-1.5muF_{gg}$ receive
no contribution from $g\to gg$, whilst $q\to gg$ and $\bar{q}\to gg$
contribute with opposite sign. The splitting formula for these channels hence
involves the valence distributions $f_{q}-f_{\bar{q}}$. At two-loop accuracy
we find that
${}^{AS}\mskip-1.5muF_{gg}={}-{}^{S\mskip-1.5muA}\mskip-1.5muF_{gg}$.

Full analytic expressions of the kernels are given in the ancillary files
associated with this paper on arXiv.

A detailed numerical study of two-loop splitting effects in DPDs is beyond the
scope of this work. However, the plots in section 5 show a few examples in
selected kinematics. We find that for $gg$ distributions, the $a_{s}^{2}$
kernels provide corrections not exceeding a few $10\%$ relative to the $a_{s}$
terms if we evaluate the splitting formula at $\mu=\sqrt{x_{1}x_{2}\mskip
1.5mu\zeta_{p}}=\mu_{y}$. However, when we change $\mu$ by a factor of $2$ in
either direction, the $a_{s}^{2}$ corrections become of order $100\%$ and
larger in parts of the $x_{1}$– $x_{2}$ plane. To obtain reliable perturbative
results for scales different from $\mu_{y}$, it is therefore better to
evaluate the splitting formula at $\mu_{y}$ and then use the appropriate
evolution equations. The distribution ${}^{AS}\mskip-1.5muF_{gg}$ is found to
be tiny compared with distributions that receive a contribution from $g\to gg$
splitting.

For $qg$ distributions at $\mu=\sqrt{x_{1}x_{2}\mskip
1.5mu\zeta_{p}}=\mu_{y}$, we find small to moderate $a_{s}^{2}$ corrections
for $x_{1}\sim x_{2}$ and $x_{1}>x_{2}$. For $x_{1}\ll x_{2}$, however, the
$a_{s}^{2}$ contribution from $g\to qg$ splitting completely dominates
${F}_{qg}$, because it has a $1/u$ enhancement that is absent in the $q\to qg$
channel up to that order.

With the results of the present paper, we have added a further ingredient for
an analysis of double parton scattering at NLO. An extension to the case of
polarised DPDs seems straightforward with the methods developed here and in
[64]. Complications due to the use of $\gamma_{5}$ and
$\epsilon^{\mu\nu\rho\sigma}$ in dimensional regularisation can be avoided
using the procedure employed in [78]. We intend to come back to this in future
work.

##  Appendix A Colour space matrices

In this appendix, we list the explicit matrices in colour space that appear in
the relation (2.25) between DPDs for definite colour representations
$R_{1},R_{2}$ and DPDs with open colour indices. We have

$\displaystyle M_{11}^{ii^{\prime}\mskip 1.5mujj^{\prime}}$
$\displaystyle=\delta_{i^{\prime}i}\,\delta_{j^{\prime}\mskip-1.5muj}\,,$
$\displaystyle M_{88}^{ii^{\prime}\mskip 1.5mujj^{\prime}}$
$\displaystyle=\frac{2N}{\sqrt{N^{2}-1}}\,t^{a}_{i^{\prime}i}\,t^{a}_{j^{\prime}\mskip-1.5muj}$
(A.1)

for quark-quark distributions $F_{qq}^{ii^{\prime}\mskip 1.5mujj^{\prime}}$,

$\displaystyle M_{11}^{jj^{\prime}\mskip 1.5muaa^{\prime}}$
$\displaystyle=M_{11}^{aa^{\prime}\mskip
1.5mujj^{\prime}}=\delta_{j^{\prime}\mskip-1.5muj}\,\delta^{aa^{\prime}}\,,$
$\displaystyle M_{8A}^{jj^{\prime}\mskip 1.5muaa^{\prime}}$
$\displaystyle=M_{A8}^{aa^{\prime}\mskip
1.5mujj^{\prime}}=\sqrt{2}\,t^{c}_{j^{\prime}\mskip-1.5muj}\,if^{aa^{\prime}c}\,,$
$\displaystyle M_{8S}^{jj^{\prime}\mskip 1.5muaa^{\prime}}$
$\displaystyle=M_{S8}^{aa^{\prime}\mskip
1.5mujj^{\prime}}=\sqrt{\frac{2N^{2}}{N^{2}-4}}\,t^{c}_{j^{\prime}\mskip-1.5muj}\,d^{aa^{\prime}c}$
(A.2)

for mixed quark-gluon distributions $F_{qg}^{ii^{\prime}\mskip
1.5muaa^{\prime}}$ and $F_{gq}^{aa^{\prime}\mskip 1.5muii^{\prime}}$, and

$\displaystyle M_{11}^{aa^{\prime}\mskip 1.5mubb^{\prime}}$
$\displaystyle=\delta^{aa^{\prime}}\delta^{bb^{\prime}}\,,\phantom{\frac{0}{0}}$
$\displaystyle M_{AA}^{aa^{\prime}\mskip 1.5mubb^{\prime}}$
$\displaystyle=-\frac{\sqrt{N^{2}-1}}{N}\,f^{aa^{\prime}c}f^{bb^{\prime}c}\,,\qquad
M_{SS}^{aa^{\prime}\mskip 1.5mubb^{\prime}}=\frac{N\mskip
1.5mu\sqrt{N^{2}-1}}{N^{2}-4}\,d^{aa^{\prime}c}d^{bb^{\prime}c}\,,$
$\displaystyle M_{AS}^{aa^{\prime}\,bb^{\prime}}$
$\displaystyle=\frac{\sqrt{N^{2}-1}}{\sqrt{N^{2}-4}}\,if^{aa^{\prime}c}d^{bb^{\prime}c}\,,\qquad\;\;M_{S\mskip-1.5muA}^{aa^{\prime}\,bb^{\prime}}=\frac{\sqrt{N^{2}-1}}{\sqrt{N^{2}-4}}\;i\mskip
1.5mud^{aa^{\prime}c}f^{bb^{\prime}c}\,,$ $\displaystyle
M_{\scalebox{0.7}{$10\phantom{\overline{1}}$}\mskip
1.5mu\scalebox{0.7}{$\overline{10}$}}^{aa^{\prime}\mskip 1.5mubb^{\prime}}$
$\displaystyle=\frac{\sqrt{N^{2}-1}}{\sqrt{N^{2}-4}}\,\biggl{[}\frac{1}{2}\bigl{(}\delta^{ab}\delta^{a^{\prime}b^{\prime}}-\delta^{ab^{\prime}}\delta^{a^{\prime}b}\bigr{)}-\frac{1}{N}\mskip
1.5muf^{aa^{\prime}c}f^{bb^{\prime}c}+\frac{i}{2}\bigl{(}d^{abc}f^{a^{\prime}b^{\prime}c}+f^{abc}d^{a^{\prime}b^{\prime}c}\bigr{)}\biggr{]}\,,$
$\displaystyle
M_{\scalebox{0.7}{$\overline{10}$}\,\scalebox{0.7}{$10\phantom{\overline{1}}$}}^{aa^{\prime}\mskip
1.5mubb^{\prime}}$
$\displaystyle=\bigl{[}M_{\scalebox{0.7}{$10\phantom{\overline{1}}$}\mskip
1.5mu\scalebox{0.7}{$\overline{10}$}}^{aa^{\prime}\mskip
1.5mubb^{\prime}}\bigr{]}^{*}\,,\phantom{\frac{0}{0}}$ $\displaystyle
M_{27\,27}^{aa^{\prime}\mskip 1.5mubb^{\prime}}$
$\displaystyle\\!\\!\\!\underset{N=3}{=}\frac{4}{\sqrt{27}}\,\biggl{[}\delta^{ab}\delta^{a^{\prime}b^{\prime}}+\delta^{ab^{\prime}}\delta^{a^{\prime}b}-\frac{1}{4}\mskip
1.5mu\delta^{aa^{\prime}}\delta^{bb^{\prime}}-\frac{6}{5}\mskip
1.5mud^{aa^{\prime}c}d^{bb^{\prime}c}\biggr{]}$ (A.3)

for two-gluon distributions $F_{gg}^{aa^{\prime}\mskip 1.5mubb^{\prime}}$.
These matrices are obtained from equation (I.11) of [78].

The matrices for antiquarks are the same as for quarks. Note that according to
the relation between quark and antiquark operators given below (2.2), the
index $i$ in $\mathcal{O}^{ii^{\prime}}_{q}$ refers to the quark in the
amplitude, whereas in $\mathcal{O}^{ii^{\prime}}_{\bar{q}}$ it refers to the
antiquark in the conjugate amplitude.

#### Charge conjugation.

Let us see how the colour projected products of twist-two operators transform
under charge conjugation. One can use light-cone gauge $A^{+}=0$ for this
purpose, which avoids the need to transform Wilson lines. Denoting the charge
conjugation operator in the space of fields by $U_{c}$, we have

$\displaystyle U_{c}\,\mathcal{O}^{ii^{\prime}}_{q}\,U_{c}^{\dagger}$
$\displaystyle=\mathcal{O}^{i^{\prime}i}_{\bar{q}}\,,$ $\displaystyle
t^{a}_{ij}\,t^{a^{\prime}}_{i^{\prime}\mskip-1.5muj^{\prime}}\,U_{c}\,\mathcal{O}^{aa^{\prime}}_{g}\,U_{c}^{\dagger}$
$\displaystyle=t^{a}_{ji}\,t^{a^{\prime}}_{j^{\prime}\mskip-1.5mui^{\prime}}\;\mathcal{O}^{aa^{\prime}}_{g}\,,$
(A.4)

where the transformation for $\mathcal{O}_{g}$ follows from the transformation
$t^{a}_{ij}\,U_{c}\,A^{\mu a}U_{c}^{\dagger}=-t^{a}_{ji}\,A^{\mu a}$ of the
gluon potential. The space-time arguments of fields do not change under charge
conjugation.

We see that charge conjugation is tantamount to interchanging quark and
antiquark labels and transposing the colour indices in the fundamental
representation, provided that one contracts each adjoint index $a$ of gluonic
operators with a generator $t^{a}$. To make use of this, we rewrite the colour
matrices $M_{R_{1}\mskip-1.5muR_{2}}$ as

$\displaystyle M_{88}^{ii^{\prime}\mskip 1.5mujj^{\prime}}$
$\displaystyle=\frac{N}{\sqrt{N^{2}-1}}\,\biggl{(}\delta_{i^{\prime}j}\,\delta_{j^{\prime}i}-\frac{1}{N}\,\delta_{i^{\prime}i}\,\delta_{j^{\prime}\mskip-1.5muj}\biggr{)}\,,$
$\displaystyle M_{11}^{jj^{\prime}\mskip 1.5muaa^{\prime}}$
$\displaystyle=\delta_{j^{\prime}\mskip-1.5muj}\operatorname{tr}\mskip
1.5mu{}\bigl{\\{}t^{a},t^{a^{\prime}}\bigr{\\}}\,,\qquad\qquad\qquad
M_{8A}^{jj^{\prime}\mskip
1.5muaa^{\prime}}=\sqrt{2}\,\bigl{[}t^{a},t^{a^{\prime}}\bigr{]}{}_{j^{\prime}\mskip-1.5muj}\,,\phantom{\frac{1}{1}}$
$\displaystyle M_{8S}^{jj^{\prime}\mskip 1.5muaa^{\prime}}$
$\displaystyle=\sqrt{\frac{2N^{2}}{N^{2}-4}}\;\biggl{(}\bigl{\\{}t^{a},t^{a^{\prime}}\bigr{\\}}{}_{j^{\prime}\mskip-1.5muj}-\frac{1}{N}\,\delta_{j^{\prime}\mskip-1.5muj}\operatorname{tr}\mskip
1.5mu{}\bigl{\\{}t^{a},t^{a^{\prime}}\bigr{\\}}\biggr{)}$ (A.5)

in the quark and mixed quark-gluon sectors, whereas in the pure gluon sector
we replace all tensors $\delta^{ab}$, $f^{abc}$ and $d^{abc}$ using

$\displaystyle\delta^{aa^{\prime}}$ $\displaystyle=\operatorname{tr}\mskip
1.5mu{}\bigl{\\{}t^{a},t^{a^{\prime}}\bigr{\\}}\,,$ $\displaystyle
f^{aa^{\prime}c}f^{bb^{\prime}c}$ $\displaystyle=-2\operatorname{tr}\mskip
1.5mu{}\bigl{[}t^{a},t^{a^{\prime}}\bigr{]}\bigl{[}t^{b},t^{b^{\prime}}\bigr{]}\,,\qquad\qquad
if^{aa^{\prime}c}d^{bb^{\prime}c}=2\operatorname{tr}\mskip
1.5mu{}\bigl{[}t^{a},t^{a^{\prime}}\bigr{]}\bigl{\\{}t^{b},t^{b^{\prime}}\bigr{\\}}\,,$
$\displaystyle d^{aa^{\prime}c}d^{bb^{\prime}c}$
$\displaystyle=2\operatorname{tr}\mskip
1.5mu{}\bigl{\\{}t^{a},t^{a^{\prime}}\bigr{\\}}\big{\\{}t^{b},t^{b^{\prime}}\bigr{\\}}-2N^{-1}\,\operatorname{tr}\mskip
1.5mu{}\bigl{\\{}t^{a},t^{a^{\prime}}\bigr{\\}}\operatorname{tr}\mskip
1.5mu{}\bigl{\\{}t^{b},t^{b^{\prime}}\bigr{\\}}\,.$ (A.6)

After this, all generators $t^{a}$ in $M_{R_{1}\mskip-1.5muR_{2}}$ are
associated with the adjoint index $a$ of a gluon operator. The charge
conjugation behaviour of the contraction

$\displaystyle{}^{R_{1}\mskip-1.5muR_{2}}\mskip-1.5mu(\mathcal{O}_{a_{1}}\mathcal{O}_{a_{2}})$
$\displaystyle=M_{R_{1}\mskip-1.5muR_{2}}^{r_{1}\mskip-1.5mur_{1}^{\prime}\,r_{2}r_{2}^{\prime}}\,\mathcal{O}^{r_{1}\mskip-1.5mur_{1}^{\prime}}_{a_{1}}\,\mathcal{O}^{r_{2}r_{2}^{\prime}}_{a_{2}}$
(A.7)

is then readily worked out by transposing the fundamental matrices in (A) and
(A). Using $[A^{T},B^{T}]=-[A,B]^{T}$, $\\{A^{T},B^{T}\\}=\\{A,B\\}^{T}$, and
$\operatorname{tr}\mskip 1.5mu{}C^{T}D^{T}=\operatorname{tr}\mskip 1.5mu{}CD$,
we find that

$\displaystyle
U_{c}\;{}^{R_{1}\mskip-1.5muR_{2}}\mskip-1.5mu(\mathcal{O}_{a_{1}}\mathcal{O}_{a_{2}})\,U_{c}^{\dagger}$
$\displaystyle=\eta(R_{1})\mskip
1.5mu\eta(R_{2})\;{}^{\overline{R}_{1}\mskip-1.5mu\overline{R}_{2}}\mskip-1.5mu(\mathcal{O}_{\bar{a}_{1}}\mathcal{O}_{\bar{a}_{2}})\,.$
(A.8)

The sign factor $\eta(R)$ is $-1$ for $R=A$ and $+1$ otherwise, and it
understood that $\bar{g}=g$. The minus sign for each antisymmetric octet
reflects the fact that the tensor $f^{aa^{\prime}c}$ leads to a commutator
$[t^{a},t^{a^{\prime}}]$ in (A) and (A). Furthermore, we see that charge
conjugation interchanges decuplet and antidecuplet.

##  Appendix B Fourier transforms

The Fourier transforms in $2-2\epsilon$ transverse dimensions required in this
work can be performed with the help of the integrals

$\displaystyle\int\frac{d^{2-2\epsilon}\boldsymbol{\Delta}}{(2\pi)^{2-2\epsilon}}\,e^{-i\boldsymbol{\Delta}\boldsymbol{y}}\,\frac{1}{\bigl{(}\boldsymbol{\Delta}^{2}\bigr{)}^{\alpha}}\,$
$\displaystyle=\frac{1}{(\pi\boldsymbol{y}^{2})^{1-\epsilon}}\,\left(\frac{\boldsymbol{y}^{2}}{4}\right)^{\alpha}\,\frac{\Gamma(1-\epsilon-\alpha)}{\Gamma(\alpha)}\,,$
(B.1)
$\displaystyle\int\frac{d^{2-2\epsilon}\boldsymbol{\Delta}}{(2\pi)^{2-2\epsilon}}\,e^{-i\boldsymbol{\Delta}\boldsymbol{y}}\;\frac{\log\boldsymbol{\Delta}^{2}}{\bigl{(}\boldsymbol{\Delta}^{2}\bigr{)}^{\alpha}}\,$
$\displaystyle=\frac{1}{(\pi\boldsymbol{y}^{2})^{1-\epsilon}}\,\left(\frac{\boldsymbol{y}^{2}}{4}\right)^{\alpha}\,\frac{\Gamma(1-\epsilon-\alpha)}{\Gamma(\alpha)}\,$
$\displaystyle\quad\times\biggl{[}\mskip
1.5mu\log\frac{4}{\boldsymbol{y}^{2}}+\psi(\alpha)+\psi(1-\alpha-\epsilon)\mskip
1.5mu\biggr{]}\,,$ (B.2)

where $\psi(x)=\frac{d}{dz}\log\Gamma(z)$ is the digamma function. One can
obtain (B) by differentiation of (B.1) with respect to $\alpha$. A useful
relation is

$\displaystyle\psi(z)$ $\displaystyle=\psi(1+z)-1/z\,,$ (B.3)

which makes the pole at $z=0$ explicit.

##  Appendix C Loop integrals with the Collins regulator

The use of spacelike Wilson lines leads to a non-trivial interplay between
rapidity divergences and loop integrations, because eikonal propagators depend
on both plus- and minus-momenta. After setting gluon momenta on shell, we have
eikonal propagators depending on plus- and transverse momenta. By contrast,
eikonal propagators with the $\delta$ regulator depend only on plus-momenta.
In this appendix, we discuss some implications of the additional transverse-
momentum dependence that appears with the Collins regulator.

Quite remarkably, there are loop integrals that give a $\delta(z_{3})$
contribution for the Collins regulator but not for the $\delta$ regulator. An
example is

$\displaystyle\int
d^{2-2\epsilon}\boldsymbol{k}_{1}\,d^{2-2\epsilon}\boldsymbol{k}_{2}\;\frac{\boldsymbol{k}_{3}^{2}/z_{3}}{\boldsymbol{k}_{2}^{2}\,(\boldsymbol{k}_{1}-\boldsymbol{\Delta})^{2}\,\bigl{(}\boldsymbol{k}_{1}^{2}/z_{1}+\boldsymbol{k}_{2}^{2}/z_{2}+\boldsymbol{k}_{3}^{2}/z_{3}\bigr{)}}\,\frac{z_{3}}{z_{3}^{2}-\boldsymbol{k}_{3}^{2}\mskip
1.5muz_{1}z_{2}\big{/}\rho}\,.$ (C.1)

Using the distributional identity (3.18), one finds that the terms going with
$\delta(z_{3})$ read

$\displaystyle\frac{1}{2}\mskip 1.5mu\delta(z_{3})\int
d^{2-2\epsilon}\boldsymbol{k}_{1}\,d^{2-2\epsilon}\boldsymbol{k}_{2}\;\frac{1}{\boldsymbol{k}_{2}^{2}\,(\boldsymbol{k}_{1}-\boldsymbol{\Delta})^{2}}\,\biggl{[}\log\frac{\rho}{\boldsymbol{\Delta^{2}}}-\log(z_{1}z_{2})-\log\frac{\boldsymbol{k}_{3}^{2}}{\boldsymbol{\Delta^{2}}}\biggr{]}$
$\displaystyle\quad=-\frac{1}{2}\mskip 1.5mu\delta(z_{3})\int
d^{2-2\epsilon}\boldsymbol{k}_{1}\,d^{2-2\epsilon}\boldsymbol{k}_{2}\;\frac{1}{\boldsymbol{k}_{2}^{2}\,(\boldsymbol{k}_{1}-\boldsymbol{\Delta})^{2}}\,\log\frac{\boldsymbol{k}_{3}^{2}}{\boldsymbol{\Delta^{2}}}\,,$
(C.2)

where we recall that
$\boldsymbol{k}_{3}=-\boldsymbol{k}_{1}-\boldsymbol{k}_{2}$. The integrals
going with the first two terms in the square brackets on the l.h.s. are
scaleless and therefore give zero. If one takes the $\delta$ regulator
instead, this applies to all terms going with $\delta(z_{3})$, because the
corresponding identity (3.17) does not involve transverse loop momenta.

This example provides a caveat to our statement in section 3.2 that graphs
which give no rapidity divergence can simply be evaluated with the unregulated
$1/k_{3}^{+}$ eikonal propagator. This is not true for the Collins regulator
when a would-be rapidity divergence is absent because an eikonal propagator is
multiplied by a loop integral that is scaleless in the unregulated case but
not scaleless in the presence of the regulator. In other words, for integrals
of the type (C.1) it matters whether one removes the rapidity regulator before
or after performing the transverse loop integrals. We find, however, that such
integrals do not occur for the graphs in our calculation.

When using the Collins regulator to compute the real graphs, we take the
$\rho\to\infty$ limit (3.18) of the eikonal propagators before evaluating the
loop integrals. One may wonder whether it is advantageous to calculate with
the propagators (3.16) at finite $\rho$ instead, postponing the
$\rho\to\infty$ limit to a later stage.

It is indeed possible to apply integration-by-part reduction to the integrals
with finite $\rho$. The set of master integrals is then augmented, because one
has an additional denominator factor $(z_{3}^{2}-\boldsymbol{k}_{3}^{2}\mskip
1.5muz_{1}z_{2}\big{/}\rho)$. After the reduction, each graph is a sum over
different terms with a $z_{3}$ and $\rho$ dependence of the form

$\displaystyle\frac{z_{3}^{n+1}}{z_{3}^{2}-\boldsymbol{k}_{3}^{2}\mskip
1.5muz_{1}z_{2}\big{/}\rho}$ $\displaystyle\text{ with }n=-2,-1,0,1,2,3$ (C.3)
or $\displaystyle\frac{\rho\mskip
1.5muz_{3}^{n+1}}{z_{3}^{2}-\boldsymbol{k}_{3}^{2}\mskip
1.5muz_{1}z_{2}\big{/}\rho}$ $\displaystyle\text{ with }n=0,1,2,3$ (C.4)

times a function that is finite in the limit $z_{3}\to 0$. Evaluating the
augmented set of master integrals with the differential equation technique
would be a formidable task, so that it seems preferable to take
$\rho\to\infty$ at this point.

The form (C.3) with $n\geq 0$ has a simple $\rho\to\infty$ limit according to
(3.18) and in this sense is unproblematic. For $n<0$, however, one obtains a
non-integrable singularity at $z_{3}=0$ when taking $\rho\to\infty$. Moreover,
terms of the form (C.4) grow linearly with $\rho$ and thus have a stronger
rapidity divergence than the full result, which we know to behave like
$\log\rho$. One thus finds that the individual terms after integration-by-part
reduction are more singular than the graphs one needs to compute. Combining
individual terms so as to obtain less singular expressions would be at odds
with the very idea of a reduction to master integrals. We find this
unattractive and prefer to take the limit $\rho\to\infty$ as early as possible
in the calculation, as described in section 3.2.

##  Appendix D Distributional identities

In this appendix, we derive the distributional identity (3.18) required for
the Collins regulator. Its analogue (3.17) for the $\delta$ regulator can be
established in a very similar manner, as discussed at the end.

To begin with, we cast (3.18) into a simpler form. Replacing $z_{1}$ and
$z_{2}$ in terms of $u$, $\bar{u}$ and $z_{3}$ as specified in (3.14), we
obtain

$\displaystyle\underset{\sigma\to
0}{\lim}\;\operatorname{PV}\frac{t}{t^{2}-\sigma^{2}\,(1-t)^{2}}$
$\displaystyle=\mathcal{L}_{0}(t)-\frac{\log\sigma^{2}}{2}\mskip
1.5mu\delta(t)\,,$ (D.1)

where we write $t=z_{3}$ and
$\sigma^{2}=u\bar{u}\,\boldsymbol{k}_{3}^{2}\big{/}\rho$. It is understood
that the identity (D.1) is to be integrated over $t\in[0,1]$ with a
sufficiently regular test function $\varphi(t)$.

Let us first show that the term $\sigma^{2}\,(1-t^{2})$ on the l.h.s. of (D.1)
can be replaced with $\sigma^{2}$. Integrating over a test function, we have
for the difference of the two versions

$\displaystyle\Delta I$ $\displaystyle=\underset{\sigma\to
0}{\lim}\;\operatorname{PV}\int_{0}^{1}dt\,\varphi(t)\biggl{[}\frac{t}{t^{2}-\sigma^{2}}-\frac{t}{t^{2}-\sigma^{2}\,(1-t)^{2}}\biggr{]}$
$\displaystyle=\underset{\sigma\to
0}{\lim}\;\sigma^{2}\operatorname{PV}\int_{0}^{2\sigma}dt\;\varphi(t)\,\frac{2-t}{t^{2}-\sigma^{2}\,(1-t)^{2}}\;\frac{t^{2}}{t^{2}-\sigma^{2}}$
$\displaystyle\quad+\underset{\sigma\to
0}{\lim}\;\sigma^{2}\int_{2\sigma}^{1}dt\;\varphi(t)\,\frac{2-t}{t^{2}-\sigma^{2}\,(1-t)^{2}}\;\frac{t^{2}}{t^{2}-\sigma^{2}}\,,$
(D.2)

where in the second step we have split the integration into two parts. The PV
prescription applies to both singularities at $t=\sigma$ and
$t=\sigma/(1+\sigma)$. In the first integral on the r.h.s., we can replace
$\varphi(t)$ with $\varphi(0)$, using that the test function is sufficiently
smooth. In the second integral, the factor multiplying $\varphi(t)$ is a
monotonic function if $\sigma$ is sufficiently small, so that we can use the
mean value theorem for integrals and replace $\varphi(t)$ with
$\varphi(t_{\sigma})$ at some point $t_{\sigma}\in[2\sigma,1]$. Rescaling the
integration variable as $t=\sigma\tau$, we obtain

$\displaystyle\Delta I$ $\displaystyle\quad=\underset{\sigma\to
0}{\lim}\;\sigma\varphi(0)\,\operatorname{PV}\int_{0}^{2}d\tau\;\frac{2-\sigma\tau}{\tau^{2}-(1-\sigma\tau)^{2}}\;\frac{\tau^{2}}{\tau^{2}-1}$
$\displaystyle\qquad+\underset{\sigma\to
0}{\lim}\;\sigma\varphi(t_{\sigma})\int_{2}^{1/\sigma}d\tau\;\frac{2-\sigma\tau}{\tau^{2}-(1-\sigma\tau)^{2}}\;\frac{\tau^{2}}{\tau^{2}-1}\,.$
(D.3)

The two integrals over $\tau$ remain finite for $\sigma\to 0$, as can be
checked by explicit calculation, so that $\Delta I=0$ in that limit.

We now show (D.1) without the factor $(1-t)^{2}$. To this end, we decompose
the integral over a test function into three terms

$\displaystyle\underset{\sigma\to
0}{\lim}\;\operatorname{PV}\int_{0}^{1}dt\;\varphi(t)\,\frac{t}{t^{2}-\sigma^{2}}$
$\displaystyle=I_{1}+I_{2}+I_{3}$ (D.4)

with

$\displaystyle I_{1}$
$\displaystyle=\int_{0}^{1}dt\;\frac{\varphi(t)-\varphi(0)}{t}\,,$
$\displaystyle I_{2}$ $\displaystyle=\underset{\sigma\to
0}{\lim}\;\operatorname{PV}\int_{0}^{1}dt\;\frac{\varphi(t)-\varphi(0)}{t}\,\frac{\sigma^{2}}{t^{2}-\sigma^{2}}\,,$
$\displaystyle I_{3}$ $\displaystyle=\varphi(0)\;\underset{\sigma\to
0}{\lim}\;\operatorname{PV}\int_{0}^{1}dt\;\frac{t}{t^{2}-\sigma^{2}}\,.$
(D.5)

One readily sees that $I_{1}=\int_{0}^{1}dt\;\varphi(t)\,\mathcal{L}_{0}(t)$,
and straightforward evaluation of the integral in $I_{3}$ yields

$\displaystyle I_{3}$
$\displaystyle=-\frac{\log\sigma^{2}}{2}\int_{0}^{1}dt\;\varphi(t)\,\delta(t)\,,$
(D.6)

so that (D.1) without the factor $(1-t)^{2}$ is established by showing that
$I_{2}=0$.

We now assume that $\varphi(t)-\varphi(0)$ is continuous and finite in the
interval $t\in[0,1]$. We also assume that it behaves like $ct^{\alpha}$ with
$\alpha>0$ for $t\to 0$. Notice that this is less than requiring $\varphi(t)$
to be differentiable at $t=0$, because $\alpha$ need not be an integer. The
function

$\displaystyle\psi(t)$ $\displaystyle=\frac{\varphi(t)-\varphi(0)}{t^{\beta}}$
$\displaystyle\text{ with }\beta$ $\displaystyle=\min(\alpha,1)$ (D.7)

is then finite for $t\in[0,1]$, with $\psi(0)=c$ for $\alpha=\beta$ and
$\psi(0)=0$ otherwise. Splitting the integration into two terms, we have

$\displaystyle I_{2}$ $\displaystyle=\underset{\sigma\to
0}{\lim}\;\operatorname{PV}\int_{0}^{2\sigma}dt\;\frac{\psi(t)}{t^{1-\beta}}\,\frac{\sigma^{2}}{t^{2}-\sigma^{2}}+\underset{\sigma\to
0}{\lim}\;\int_{2\sigma}^{1}dt\;\frac{\psi(t)}{t^{1-\beta}}\,\frac{\sigma^{2}}{t^{2}-\sigma^{2}}\,.$
(D.8)

For sufficiently small $\sigma$, we can replace $\psi(t)$ with $\psi(0)$ in
the first term on the r.h.s., and for the second term we can use the mean
value theorem for integrals. This gives

$\displaystyle I_{2}$ $\displaystyle=\psi(0)\,\underset{\sigma\to
0}{\lim}\;\operatorname{PV}\int_{0}^{2\sigma}\frac{dt}{t^{1-\beta}}\,\frac{\sigma^{2}}{t^{2}-\sigma^{2}}+\underset{\sigma\to
0}{\lim}\;\psi(t_{\sigma})\int_{2\sigma}^{1}\frac{dt}{t^{1-\beta}}\,\frac{\sigma^{2}}{t^{2}-\sigma^{2}}$
(D.9)

with some value $t_{\sigma}\in[2\sigma,1]$ different from the one that appears
in (D). Rescaling the integration variable as $t=\sigma\tau$, we finally
obtain

$\displaystyle I_{2}$ $\displaystyle=\psi(0)\,\underset{\sigma\to
0}{\lim}\;\sigma^{\beta}\operatorname{PV}\int_{0}^{2}\frac{d\tau}{\tau^{1-\beta}}\;\frac{1}{\tau^{2}-1}+\underset{\sigma\to
0}{\lim}\;\psi(t_{\sigma})\,\sigma^{\beta}\int_{2}^{\infty}\frac{d\tau}{\tau^{1-\beta}}\;\frac{1}{\tau^{2}-1}=0\,,$
(D.10)

which concludes our proof.

The distributional identity (3.17) we use with the $\delta$ regulator has the
same form as (D.1) with $t^{2}+\sigma^{2}\,(1-t)^{2}$ instead of
$t^{2}-\sigma^{2}\,(1-t)^{2}$ on the l.h.s. The PV prescription is not needed
in that case. The proof of this identity proceeds in analogy to what we have
shown, with the simplification that one does not need to split integrals into
two parts as we did in (D) and (D.8).

## Acknowledgements

We gratefully acknowledge discussions with Iain Stewart and Alexey Vladimirov
on rapidity regulators. We also thank Florian Fabry for feedback on the
manuscript.

This work was in part supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Research Unit FOR 2926, grant number 409651613.
The work of JRG is supported by the Royal Society through Grant URF\R1\201500,
and the work of PP in part by the BMBF project 05P2018 (ErUM-FSP T01). MD and
JRG would like to express their thanks to the Mainz Institute for Theoretical
Physics (MITP) of the DFG Cluster of Excellence PRIMSA+ (Project ID 39083149)
for its hospitality and support. The figures in this work were produced with
JaxoDraw [109, 110].

## References

  * [1] P. Landshoff and J. Polkinghorne, _Calorimeter Triggers for Hard Collisions_ , _Phys. Rev. D_ 18 (1978) 3344.
  * [2] R. Kirschner, _Generalized Lipatov-Altarelli-Parisi Equations and Jet Calculus Rules_ , _Phys. Lett._ B84 (1979) 266.
  * [3] H. Politzer, _Power Corrections at Short Distances_ , _Nucl. Phys. B_ 172 (1980) 349.
  * [4] N. Paver and D. Treleani, _Multi-Quark Scattering and Large $p_{T}$ Jet Production in Hadronic Collisions_, _Nuovo Cim._ A70 (1982) 215.
  * [5] V. P. Shelest, A. M. Snigirev and G. M. Zinovev, _The Multiparton Distribution Equations in QCD_ , _Phys. Lett._ B113 (1982) 325.
  * [6] M. Mekhfi, _Multiparton processes: an application to double Drell-Yan_ , _Phys. Rev. D_ 32 (1985) 2371.
  * [7] T. Sjostrand and M. van Zijl, _Multiple Parton-parton Interactions in an Impact Parameter Picture_ , _Phys. Lett. B_ 188 (1987) 149.
  * [8] B. Blok, Y. Dokshitzer, L. Frankfurt and M. Strikman, _The Four jet production at LHC and Tevatron in QCD_ , _Phys. Rev. D_ 83 (2011) 071501 [1009.2714].
  * [9] M. Diehl and A. Schäfer, _Theoretical considerations on multiparton interactions in QCD_ , _Phys. Lett._ B698 (2011) 389 [1102.3081].
  * [10] J. R. Gaunt and W. Stirling, _Double Parton Scattering Singularity in One-Loop Integrals_ , _JHEP_ 06 (2011) 048 [1103.1888].
  * [11] M. G. Ryskin and A. M. Snigirev, _A Fresh look at double parton scattering_ , _Phys. Rev._ D83 (2011) 114047 [1103.3495].
  * [12] B. Blok, Yu. Dokshitser, L. Frankfurt and M. Strikman, _pQCD physics of multiparton interactions_ , _Eur. Phys. J._ C72 (2012) 1963 [1106.5533].
  * [13] M. Diehl, D. Ostermeier and A. Schäfer, _Elements of a theory for multiparton interactions in QCD_ , _JHEP_ 03 (2012) 089 [1111.0910].
  * [14] A. V. Manohar and W. J. Waalewijn, _A QCD Analysis of Double Parton Scattering: Color Correlations, Interference Effects and Evolution_ , _Phys. Rev._ D85 (2012) 114009 [1202.3794].
  * [15] A. V. Manohar and W. J. Waalewijn, _What is Double Parton Scattering?_ , _Phys. Lett. B_ 713 (2012) 196 [1202.5034].
  * [16] M. G. Ryskin and A. M. Snigirev, _Double parton scattering in double logarithm approximation of perturbative QCD_ , _Phys. Rev._ D86 (2012) 014018 [1203.2330].
  * [17] J. R. Gaunt, _Single Perturbative Splitting Diagrams in Double Parton Scattering_ , _JHEP_ 01 (2013) 042 [1207.0480].
  * [18] B. Blok, Y. Dokshitzer, L. Frankfurt and M. Strikman, _Perturbative QCD correlations in multi-parton collisions_ , _Eur. Phys. J. C_ 74 (2014) 2926 [1306.3763].
  * [19] M. Diehl, J. R. Gaunt and K. Schönwald, _Double hard scattering without double counting_ , _JHEP_ 06 (2017) 083 [1702.06486].
  * [20] M. Diehl and R. Nagar, _Factorisation of soft gluons in multiparton scattering_ , _JHEP_ 04 (2019) 124 [1812.09509].
  * [21] Axial Field Spectrometer collaboration, T. Åkesson et al., _Double Parton Scattering in $pp$ Collisions at $\sqrt{s}=63$ GeV_, _Z. Phys. C_ 34 (1987) 163.
  * [22] UA2 collaboration, J. Alitti et al., _A Study of multi-jet events at the CERN anti-p p collider and a search for double parton scattering_ , _Phys. Lett. B_ 268 (1991) 145.
  * [23] CDF collaboration, F. Abe et al., _Double parton scattering in $\bar{p}p$ collisions at $\sqrt{s}=1.8$TeV_, _Phys. Rev. D_ 56 (1997) 3811.
  * [24] D0 collaboration, V. M. Abazov et al., _Study of double parton interactions in diphoton + dijet events in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV_, _Phys. Rev. D_ 93 (2016) 052008 [1512.05291].
  * [25] LHCb collaboration, R. Aaij et al., _Measurement of the J/ $\psi$ pair production cross-section in pp collisions at $\sqrt{s}=13$ TeV_, _JHEP_ 06 (2017) 047 [1612.07451].
  * [26] ATLAS collaboration, M. Aaboud et al., _Study of the hard double-parton scattering contribution to inclusive four-lepton production in $pp$ collisions at $\sqrt{s}=$ 8 TeV with the ATLAS detector_, _Phys. Lett._ B790 (2019) 595 [1811.11094].
  * [27] CMS collaboration, A. M. Sirunyan et al., _Evidence for WW production from double-parton interactions in proton–proton collisions at $\sqrt{s}=13\,\text{TeV}$_, _Eur. Phys. J. C_ 80 (2020) 41 [1909.06265].
  * [28] O. Fedkevych and A. Kulesza, _Double parton scattering in four-jet production in proton-proton collisions at the LHC_ , 2008.08347.
  * [29] P. Bartalini and J. R. Gaunt, eds., _Multiple Parton Interactions at the LHC_ , vol. 29. WSP, 2019, 10.1142/10646.
  * [30] C. Kom, A. Kulesza and W. Stirling, _Pair Production of J/ $\Psi$ as a Probe of Double Parton Scattering at LHCb_, _Phys. Rev. Lett._ 107 (2011) 082002 [1105.4186].
  * [31] C. Kom, A. Kulesza and W. Stirling, _Prospects for observation of double parton scattering with four-muon final states at LHCb_ , _Eur. Phys. J. C_ 71 (2011) 1802 [1109.0309].
  * [32] A. Kulesza and W. Stirling, _Like sign $W$ boson production at the LHC as a probe of double parton scattering_, _Phys. Lett. B_ 475 (2000) 168 [hep-ph/9912232].
  * [33] J. R. Gaunt, C.-H. Kom, A. Kulesza and W. Stirling, _Same-sign W pair production as a probe of double parton scattering at the LHC_ , _Eur. Phys. J. C_ 69 (2010) 53 [1003.3953].
  * [34] F. A. Ceccopieri, M. Rinaldi and S. Scopetta, _Parton correlations in same-sign $W$ pair production via double parton scattering at the LHC_, _Phys. Rev. D_ 95 (2017) 114030 [1702.05363].
  * [35] S. Cotogno, T. Kasemets and M. Myska, _Spin on same-sign $W$-boson pair production_, _Phys. Rev. D_ 100 (2019) 011503 [1809.09024].
  * [36] S. Cotogno, T. Kasemets and M. Myska, _Confronting same-sign W-boson production with parton correlations_ , _JHEP_ 10 (2020) 214 [2003.03347].
  * [37] CMS collaboration, V. Khachatryan et al., _Search for new physics in same-sign dilepton events in proton–proton collisions at $\sqrt{s}=13\,\text{TeV}$_, _Eur. Phys. J. C_ 76 (2016) 439 [1605.03171].
  * [38] CMS collaboration, A. M. Sirunyan et al., _Search for top quark partners with charge 5/3 in the same-sign dilepton and single-lepton final states in proton-proton collisions at $\sqrt{s}=13$ TeV_, _JHEP_ 03 (2019) 082 [1810.03188].
  * [39] H.-M. Chang, A. V. Manohar and W. J. Waalewijn, _Double Parton Correlations in the Bag Model_ , _Phys. Rev. D_ 87 (2013) 034009 [1211.3132].
  * [40] M. Rinaldi, S. Scopetta and V. Vento, _Double parton correlations in constituent quark models_ , _Phys. Rev. D_ 87 (2013) 114021 [1302.6462].
  * [41] W. Broniowski and E. Ruiz Arriola, _Valence double parton distributions of the nucleon in a simple model_ , _Few Body Syst._ 55 (2014) 381 [1310.8419].
  * [42] M. Rinaldi, S. Scopetta, M. Traini and V. Vento, _Double parton correlations and constituent quark models: a Light Front approach to the valence sector_ , _JHEP_ 12 (2014) 028 [1409.1500].
  * [43] W. Broniowski, E. Ruiz Arriola and K. Golec-Biernat, _Generalized Valon Model for Double Parton Distributions_ , _Few Body Syst._ 57 (2016) 405 [1602.00254].
  * [44] T. Kasemets and A. Mukherjee, _Quark-gluon double parton distributions in the light-front dressed quark model_ , _Phys. Rev. D_ 94 (2016) 074029 [1606.05686].
  * [45] M. Rinaldi, S. Scopetta, M. C. Traini and V. Vento, _Correlations in Double Parton Distributions: Perturbative and Non-Perturbative effects_ , _JHEP_ 10 (2016) 063 [1608.02521].
  * [46] M. Rinaldi and F. A. Ceccopieri, _Relativistic effects in model calculations of double parton distribution function_ , _Phys. Rev. D_ 95 (2017) 034040 [1611.04793].
  * [47] M. Rinaldi, S. Scopetta, M. Traini and V. Vento, _A model calculation of double parton distribution functions of the pion_ , _Eur. Phys. J. C_ 78 (2018) 781 [1806.10112].
  * [48] A. Courtoy, S. Noguera and S. Scopetta, _Double parton distributions in the pion in the Nambu–Jona-Lasinio model_ , _JHEP_ 12 (2019) 045 [1909.09530].
  * [49] W. Broniowski and E. Ruiz Arriola, _Double parton distribution of valence quarks in the pion in chiral quark models_ , _Phys. Rev. D_ 101 (2020) 014019 [1910.03707].
  * [50] RQCD collaboration, C. Zimmermann, _Two-current correlations and DPDs for the nucleon on the lattice_ , _PoS_ LATTICE2019 (2019) 040 [1911.05051].
  * [51] G. S. Bali, L. Castagnini, M. Diehl, J. R. Gaunt, B. Gläßle, A. Schäfer et al., _Double parton distributions in the pion from lattice QCD_ , _JHEP_ 02 (2021) 067 [2006.14826].
  * [52] J. R. Gaunt and W. Stirling, _Double Parton Distributions Incorporating Perturbative QCD Evolution and Momentum and Quark Number Sum Rules_ , _JHEP_ 03 (2010) 005 [0910.4347].
  * [53] K. Golec-Biernat and E. Lewandowska, _How to impose initial conditions for QCD evolution of double parton distributions?_ , _Phys. Rev. D_ 90 (2014) 014032 [1402.4079].
  * [54] K. Golec-Biernat, E. Lewandowska, M. Serino, Z. Snyder and A. M. Stasto, _Constraining the double gluon distribution by the single gluon distribution_ , _Phys. Lett. B_ 750 (2015) 559 [1507.08583].
  * [55] M. Diehl, J. Gaunt, D. Lang, P. Plößl and A. Schäfer, _Sum rule improved double parton distributions in position space_ , _Eur. Phys. J. C_ 80 (2020) 468 [2001.10428].
  * [56] B. Blok and P. Gunnellini, _Dynamical approach to MPI four-jet production in Pythia_ , _Eur. Phys. J. C_ 75 (2015) 282 [1503.08246].
  * [57] B. Blok and P. Gunnellini, _Dynamical approach to MPI in W+dijet and Z+dijet production within the PYTHIA event generator_ , _Eur. Phys. J. C_ 76 (2016) 202 [1510.07436].
  * [58] A. Snigirev, N. Snigireva and G. Zinovjev, _Perturbative and nonperturbative correlations in double parton distributions_ , _Phys. Rev. D_ 90 (2014) 014015 [1403.6947].
  * [59] K. Golec-Biernat and E. Lewandowska, _Electroweak boson production in double parton scattering_ , _Phys. Rev. D_ 90 (2014) 094032 [1407.4038].
  * [60] J. R. Gaunt, R. Maciula and A. Szczurek, _Conventional versus single-ladder-splitting contributions to double parton scattering production of two quarkonia, two Higgs bosons and $c\bar{c}c\bar{c}$_, _Phys. Rev. D_ 90 (2014) 054017 [1407.5821].
  * [61] B. Cabouat, J. R. Gaunt and K. Ostrolenk, _A Monte-Carlo Simulation of Double Parton Scattering_ , _JHEP_ 11 (2019) 061 [1906.04669].
  * [62] M. Cacciari, G. P. Salam and S. Sapeta, _On the characterisation of the underlying event_ , _JHEP_ 04 (2010) 065 [0912.4926].
  * [63] M. Mekhfi and X. Artru, _Sudakov Suppression of Color Correlations in Multiparton Scattering_ , _Phys. Rev. D_ 37 (1988) 2618.
  * [64] M. Diehl, J. R. Gaunt, P. Plößl and A. Schäfer, _Two-loop splitting in double parton distributions_ , _SciPost Phys._ 7 (2019) 017 [1902.08019].
  * [65] J. Bartels and M. Ryskin, _Recombination within Multi-Chain Contributions in pp Scattering_ , 1105.1638.
  * [66] T. Becher and M. Neubert, _Drell-Yan Production at Small $q_{T}$, Transverse Parton Distributions and the Collinear Anomaly_, _Eur. Phys. J. C_ 71 (2011) 1665 [1007.4005].
  * [67] T. Becher and G. Bell, _Analytic Regularization in Soft-Collinear Effective Theory_ , _Phys. Lett. B_ 713 (2012) 41 [1112.3907].
  * [68] J. Collins, _Foundations of perturbative QCD_. Cambridge University Press, 2013.
  * [69] M. G. Echevarria, A. Idilbi and I. Scimemi, _Factorization Theorem For Drell-Yan At Low $q_{T}$ And Transverse Momentum Distributions On-The-Light-Cone_, _JHEP_ 07 (2012) 002 [1111.4996].
  * [70] M. G. Echevarría, A. Idilbi and I. Scimemi, _Soft and Collinear Factorization and Transverse Momentum Dependent Parton Distribution Functions_ , _Phys. Lett. B_ 726 (2013) 795 [1211.1947].
  * [71] M. G. Echevarria, I. Scimemi and A. Vladimirov, _Universal transverse momentum dependent soft function at NNLO_ , _Phys. Rev. D_ 93 (2016) 054004 [1511.05590].
  * [72] M. G. Echevarria, I. Scimemi and A. Vladimirov, _Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order_ , _JHEP_ 09 (2016) 004 [1604.07869].
  * [73] J.-y. Chiu, A. Jain, D. Neill and I. Z. Rothstein, _The Rapidity Renormalization Group_ , _Phys. Rev. Lett._ 108 (2012) 151601 [1104.0881].
  * [74] J.-Y. Chiu, A. Jain, D. Neill and I. Z. Rothstein, _A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory_ , _JHEP_ 05 (2012) 084 [1202.0814].
  * [75] Y. Li, D. Neill and H. X. Zhu, _An exponential regulator for rapidity divergences_ , _Nucl. Phys. B_ 960 (2020) 115193 [1604.00392].
  * [76] M. A. Ebert, I. Moult, I. W. Stewart, F. J. Tackmann, G. Vita and H. X. Zhu, _Subleading power rapidity divergences and power corrections for q T_, _JHEP_ 04 (2019) 123 [1812.08189].
  * [77] M. Diehl, J. R. Gaunt, D. Ostermeier, P. Plößl and A. Schäfer, _Cancellation of Glauber gluon exchange in the double Drell-Yan process_ , _JHEP_ 01 (2016) 076 [1510.08696].
  * [78] M. G. A. Buffing, M. Diehl and T. Kasemets, _Transverse momentum in double parton scattering: factorisation, evolution and matching_ , _JHEP_ 01 (2018) 044, Erratum _JHEP_ 07 (2021) 046 [1708.03528].
  * [79] J. C. Collins and D. E. Soper, _Back-To-Back Jets in QCD_ , _Nucl. Phys. B_ 193 (1981) 381.
  * [80] J. C. Collins, D. E. Soper and G. F. Sterman, _Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production_ , _Nucl. Phys. B_ 250 (1985) 199.
  * [81] A. Vladimirov, _Soft factors for double parton scattering at NNLO_ , _JHEP_ 12 (2016) 038 [1608.04920].
  * [82] P. Cvitanović, _Group Theory: Birdtracks, Lie’s, and Exceptional Groups_. Princeton Univ. Press (2008), http://birdtracks.eu.
  * [83] S. Keppeler and M. Sjödahl, _Orthogonal multiplet bases in SU(Nc) color space_ , _JHEP_ 09 (2012) 124 [1207.0609].
  * [84] T. Kasemets and P. J. Mulders, _Constraining double parton correlations and interferences_ , _Phys. Rev. D_ 91 (2015) 014015 [1411.0726].
  * [85] J. C. Collins and T. C. Rogers, _Equality of Two Definitions for Transverse Momentum Dependent Parton Distribution Functions_ , _Phys. Rev. D_ 87 (2013) 034018 [1210.2100].
  * [86] M. A. Ebert, I. W. Stewart and Y. Zhao, _Towards Quasi-Transverse Momentum Dependent PDFs Computable on the Lattice_ , _JHEP_ 09 (2019) 037 [1901.03685].
  * [87] J. R. Gaunt and T. Kasemets, _Transverse momentum dependence in double parton scattering_ , _Adv. High Energy Phys._ 2019 (2019) 3797394 [1812.09099].
  * [88] M. Diehl, P. Plößl and A. Schäfer, _Proof of sum rules for double parton distributions in QCD_ , _Eur. Phys. J. C_ 79 (2019) 253 [1811.00289].
  * [89] C. W. Bauer, D. Pirjol and I. W. Stewart, _Soft collinear factorization in effective field theory_ , _Phys. Rev. D_ 65 (2002) 054022 [hep-ph/0109045].
  * [90] M. Beneke, A. Chapovsky, M. Diehl and T. Feldmann, _Soft collinear effective theory and heavy to light currents beyond leading power_ , _Nucl. Phys. B_ 643 (2002) 431 [hep-ph/0206152].
  * [91] Z. Ligeti, I. W. Stewart and F. J. Tackmann, _Treating the b quark distribution function with reliable uncertainties_ , _Phys. Rev. D_ 78 (2008) 114014 [0807.1926].
  * [92] K. Chetyrkin and F. Tkachov, _Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops_ , _Nucl. Phys. B_ 192 (1981) 159.
  * [93] F. Tkachov, _A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions_ , _Phys. Lett. B_ 100 (1981) 65.
  * [94] R. N. Lee, _LiteRed 1.4: a powerful tool for reduction of multiloop integrals_ , _J. Phys. Conf. Ser._ 523 (2014) 012059 [1310.1145].
  * [95] A. Kotikov, _Differential equations method: New technique for massive Feynman diagrams calculation_ , _Phys. Lett. B_ 254 (1991) 158.
  * [96] Z. Bern, L. J. Dixon and D. A. Kosower, _Dimensionally regulated pentagon integrals_ , _Nucl. Phys. B_ 412 (1994) 751 [hep-ph/9306240].
  * [97] E. Remiddi, _Differential equations for Feynman graph amplitudes_ , _Nuovo Cim. A_ 110 (1997) 1435 [hep-th/9711188].
  * [98] T. Gehrmann and E. Remiddi, _Differential equations for two loop four point functions_ , _Nucl. Phys. B_ 580 (2000) 485 [hep-ph/9912329].
  * [99] J. M. Henn, _Multiloop integrals in dimensional regularization made simple_ , _Phys. Rev. Lett._ 110 (2013) 251601 [1304.1806].
  * [100] O. Gituliar and V. Magerya, _Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form_ , _Comput. Phys. Commun._ 219 (2017) 329 [1701.04269].
  * [101] M. Beneke and V. A. Smirnov, _Asymptotic expansion of Feynman integrals near threshold_ , _Nucl. Phys. B_ 522 (1998) 321 [hep-ph/9711391].
  * [102] J. R. Gaunt, M. Stahlhofen and F. J. Tackmann, _The Quark Beam Function at Two Loops_ , _JHEP_ 04 (2014) 113 [1401.5478].
  * [103] S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky et al., _New parton distribution functions from a global analysis of quantum chromodynamics_ , _Phys. Rev. D_ 93 (2016) 033006 [1506.07443].
  * [104] A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht et al., _LHAPDF6: parton density access in the LHC precision era_ , _Eur. Phys. J. C_ 75 (2015) 132 [1412.7420].
  * [105] D. Clark, E. Godat and F. Olness, _ManeParse : A Mathematica reader for Parton Distribution Functions_ , _Comput. Phys. Commun._ 216 (2017) 126 [1605.08012].
  * [106] S. Alekhin, J. Blümlein, S. Moch and R. Placakyte, _Parton distribution functions, $\alpha_{s}$, and heavy-quark masses for LHC Run II_, _Phys. Rev. D_ 96 (2017) 014011 [1701.05838].
  * [107] NNPDF collaboration, R. D. Ball et al., _Parton distributions from high-precision collider data_ , _Eur. Phys. J. C_ 77 (2017) 663 [1706.00428].
  * [108] S. Bailey, T. Cridge, L. Harland-Lang, A. Martin and R. Thorne, _Parton distributions from LHC, HERA, Tevatron and fixed target data: MSHT20 PDFs_ , 2012.04684.
  * [109] D. Binosi and L. Theussl, _JaxoDraw: A Graphical user interface for drawing Feynman diagrams_ , _Comput. Phys. Commun._ 161 (2004) 76 [hep-ph/0309015].
  * [110] D. Binosi, J. Collins, C. Kaufhold and L. Theussl, _JaxoDraw: A Graphical user interface for drawing Feynman diagrams. Version 2.0 release notes_ , _Comput. Phys. Commun._ 180 (2009) 1709 [0811.4113].