aainstitutetext: Deutsches Elektronen-Synchroton DESY, Notkestraße 85, 22607
Hamburg, Germanybbinstitutetext: Nikhef Theory Group and VU University
Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

# Transverse momentum in double parton scattering:
factorisation, evolution and matching

Maarten G. A. Buffing a Markus Diehl b and Tomas Kasemets
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We give a description of double parton scattering with measured transverse
momenta in the final state, extending the formalism for factorisation and
resummation developed by Collins, Soper and Sterman for the production of
colourless particles. After a detailed analysis of their colour structure, we
derive and solve evolution equations in rapidity and renormalisation scale for
the relevant soft factors and double parton distributions. We show how in the
perturbative regime, transverse momentum dependent double parton distributions
can be expressed in terms of simpler nonperturbative quantities and compute
several of the corresponding perturbative kernels at one-loop accuracy. We
then show how the coherent sum of single and double parton scattering can be
simplified for perturbatively large transverse momenta, and we discuss to
which order resummation can be performed with presently available results. As
an auxiliary result, we derive a simple form for the square root factor in the
Collins construction of transverse momentum dependent parton distributions.

††preprint:  NIKHEF 2016-028, DESY 17-014

##  1 Introduction

Proton-proton collisions at high energies are sensitive to regions of phase
space where partons have small momentum fractions. This implies high parton
densities and thus increases the importance of events in which two or more
partons in each proton take part in a hard interaction. The most frequent and
best studied case of such multiple hard interactions is double parton
scattering (DPS). This mechanism can be especially prominent in cross sections
depending on transverse momenta in the final state. Its theoretical
description involves double parton distributions (DPDs), which quantify the
joint distribution of two partons inside a proton and contain a wealth of
information on correlations between the proton constituents.

Experimental measurements of double parton scattering contributions to
different final states dates back to experiments at the ISR Akesson:1986iv
and SPS Alitti:1991rd . A wide range of DPS processes has been investigated at
the Tevatron Abe:1993rv ; Abe:1997bp ; Abe:1997xk ; Abazov:2009gc ;
Abazov:2011rd ; Abazov:2014fha ; Abazov:2014qba ; Abazov:2015fbl ;
Abazov:2015nnn  and in run I of the LHC Aaij:2011yc ; Aaij:2012dz ;
Aaij:2015wpa ; Aad:2013bjm ; Aad:2014rua ; Aad:2014kba ; Aaboud:2016dea ;
Aaboud:2016fzt ; Chatrchyan:2013xxa ; Khachatryan:2016ydm . An overview of
most of these measurements can be found in figure 14 of Aaboud:2016fzt . The
importance of such processes will be even more pronounced at the full LHC
energy (first results from run II are reported in CMS:2017jwx ) and at future
hadron colliders.

Theoretical and phenomenological analyses of DPS have a long history; an
overview of early work can be found in Sjostrand:2004pf . Following the
seminal papers Paver:1982yp  and Mekhfi:1983az , significant effort has been
invested in recent years to achieve a systematic theory description of DPS
Diehl:2011tt ; Diehl:2011yj ; Manohar:2012jr ; Diehl:2015bca , aiming at the
same level of rigour as has been achieved for the familiar single parton
scattering (SPS) mechanism. A formalism for combining the single and double
parton scattering contributions to the physical cross section without double
counting was recently presented in Diehl:2017kgu  – a different scheme was
proposed earlier Manohar:2012pe . However, there remain several gaps in our
understanding, and the present paper aims at closing some of them.

A crucial aspect for understanding DPS are correlations between the two
partons that are probed in the reaction. Correlations in spin and in colour
have been classified systematically Mekhfi:1985dv ; Diehl:2011yj ;
Manohar:2012jr  and will play an important role in the present work. Their
size is poorly known, but can be limited by positivity bounds Diehl:2013mla ;
Kasemets:2014yna , which have similar theoretical status as positivity
constraints on single parton distribution functions (PDFs). Quark model
calculations Chang:2012nw ; Rinaldi:2013vpa ; Rinaldi:2014ddl ;
Rinaldi:2016jvu ; Rinaldi:2016mlk ; Broniowski:2013xba ; Broniowski:2016trx ;
Kasemets:2016nio  typically yield strong correlations in the valence region,
but for the region of small momentum fractions $x$ there is little guidance
from models so far. The decrease of spin correlations under evolution to
higher scales has been studied in Diehl:2014vaa , and their influence on
final-state distributions has been investigated for several processes
Manohar:2012jr ; Kasemets:2012pr ; Echevarria:2015ufa . The generation of
parton correlations by the splitting of a single parton into two has been
investigated by several groups Diehl:2011yj ; Gaunt:2012dd ; Blok:2011bu ;
Blok:2013bpa ; Ryskin:2011kk ; Ryskin:2012qx ; a simplified implementation
into the event generator Pythia is described in Blok:2015rka ; Blok:2015afa .

At the level of integrated cross sections, double parton scattering is
suppressed by $\Lambda^{2}/Q^{2}$ compared to single parton scattering, where
$Q$ denotes the scale of the hard scattering and $\Lambda$ the scale of
nonperturbative interactions Diehl:2011tt ; Blok:2010ge . However, there are
situations where DPS can nevertheless compete with SPS, for instance when the
latter is suppressed by higher powers in coupling constants. A prominent
example is same sign $W$ pair production Kulesza:1999zh ; Gaunt:2010pi ;
Ceccopieri:2017oqe , which is an important search channel for physics beyond
the Standard Model. A generic mechanism enhancing DPS over SPS in processes
involving small parton momentum fractions $x$ is the fact that with decreasing
$x$ the density of two partons increases roughly like the square of the single
parton density.

If the transverse momenta $q_{T}$ of the particles produced by a hard
scattering are small compared with $Q$, then DPS has the same power behaviour
in $1/Q$ as SPS Diehl:2011tt ; Blok:2010ge . If $q_{T}$ is of order $\Lambda$,
then one needs information about the nonperturbative “intrinsic” transverse
momentum of partons. However, in the region $\Lambda\ll q_{T}\ll Q$ one can
reliably compute the $q_{T}$ spectrum in perturbation theory, provided that
one resums the large logarithms of $Q/q_{T}$ that arise in this regime. This
resummation is intricately related with evolution in the rapidity of emitted
gluons. For SPS processes producing colourless particles, like Drell-Yan
lepton pairs or a Higgs boson, a powerful theoretical formalism has long been
established Collins:1984kg ; Catani:2000vq  and been pushed to high
perturbative accuracy, see e.g. Catani:2013tia  and references therein.
Formulations using soft-collinear effective theory (SCET) have been given in
GarciaEchevarria:2011rb ; Becher:2012yn ; Neill:2015roa . A precise theory for
DPS at small measured $q_{T}$ is of obvious interest. In Blok:2015rka ;
Blok:2015afa  the DDT formalism Dokshitzer:1978hw  has been extended to DPS
processes. The analysis in Diehl:2011tt ; Diehl:2011yj  was based on
transverse-momentum dependent (TMD) factorisation in the original formulation
of Collins and Soper Collins:1981uk , which also underlies the CSS resummation
formalism Collins:1984kg . Since then, an improved version of TMD
factorisation for SPS processes has been formulated by Collins Collins:2011zzd
; a brief review is given in Aybat:2011zv ; Collins:2011ca  and the
differences between the old and new versions are described in Collins:2017oxh
.

The aim of the present paper is twofold. Firstly, we complete the formulation
of DPS in Diehl:2011tt ; Diehl:2011yj  and adapt it to the new factorisation
formalism of Collins:2011zzd , providing a systematic analysis of soft-gluon
effects, rapidity evolution and colour correlations. Secondly, we show how the
theory simplifies for intermediate transverse momenta $\Lambda\ll q_{T}\ll Q$,
where transverse-momentum dependent DPDs can be matched on transverse-momentum
integrated distributions, with the transverse-momentum dependence being
computed in perturbation theory. This significantly increases the predictive
power of the theoretical framework. Some of our main results have been
reported in Buffing:2016qql ; Buffing:2017sxz .

TMD factorisation in proton-proton collisions can be established to all orders
in perturbation theory for the production of colourless particles such as a
Higgs boson or electroweak gauge bosons Collins:2011zzd . Because of serious
complications from soft gluon exchange, it is not known if and how the
formalism could be extended to hard-scattering processes with coloured
particles in the final state, such as jets or heavy quarks Rogers:2010dm . We
will therefore limit our discussion of TMD factorisation in DPS to colourless
final states as well. Important channels are the production of two electroweak
gauge bosons (often called the double Drell-Yan process), of a Higgs boson and
an electroweak gauge boson, or of a Higgs boson pair. Instead of a heavy
boson, one may also consider a photon pair of large invariant mass.

This paper is organised as follows. In section 2 we recall some of the
concepts and results for TMD factorisation in single parton scattering. In
section 3 we discuss properties of collinear matrix elements and of the soft
factor, which are the ingredients in the definition of transverse-momentum
dependent and transverse-momentum integrated DPDs, which we will call DTMDs
and DPDFs, respectively. Our definition generalises the combination of
collinear and soft factors in Collins:2011zzd  to double parton distributions,
and it provides an alternative form of this construction for single parton
TMDs. The colour structure of DPS is significantly more complicated than the
one of SPS, and we show in section 4 how this structure can be handled in a
general and efficient way. We find significant simplifications for transverse-
momentum integrated quantities. In section 5 we present the general
factorisation formula for DPS at low $q_{T}$, its combination with SPS, as
well as the evolution equations for DTMDs and DPDFs and their general
solution. Section 6 is devoted to the region $\Lambda\ll q_{T}\ll Q$. We
establish the matching of DTMDs onto different types of transverse-momentum
integrated distributions. The multi-scale nature of the problem leads to
different matching regimes, which we combine in a consistent way using a
subtraction formalism. In section 7 we give one-loop expressions for
perturbative quantities that appear in the DPS cross section, extending
previous work in the literature and discussing several technical aspects of
the computation. We summarise our main results in section 8. A variety of
technical details and results are given in the appendices.

##  2 Reminder: single TMD factorisation

To begin with, let us recall a few results from TMD factorisation, as laid out
in Collins:2011zzd . The cross section depends on TMDs that describe the
distribution of partons inside the proton in both longitudinal and transverse
momentum. Throughout this work, we consider unpolarised protons. The
factorisation formula for Drell-Yan production then reads

$\displaystyle\frac{d\sigma}{dx\,d\bar{x}\,d^{2}\boldsymbol{q}}=\sum_{q}\hat{\sigma}_{q\bar{q}}(Q,\mu)\int\frac{d^{2}\boldsymbol{z}}{(2\pi)^{2}}\,e^{-i\boldsymbol{q}\boldsymbol{z}}\,W_{q\bar{q}}(x,\bar{x},\boldsymbol{z};\mu)+\\{q\leftrightarrow\bar{q}\\}\,,$
(1)

where $\hat{\sigma}_{q\bar{q}}$ is the cross section for $q\bar{q}$
annihilation into an electroweak gauge boson and

$\displaystyle
W_{q\bar{q}}(x,\bar{x},\boldsymbol{z};\mu)=f_{q}(x,\boldsymbol{z};\mu,\zeta)\,f_{\bar{q}}(\bar{x},\boldsymbol{z};\mu,\bar{\zeta})$
(2)

is the product of a quark and an antiquark TMD. The invariant mass of the
boson is $Q$ and its transverse momentum is $\boldsymbol{q}$. The TMDs depend
on longitudinal momentum fractions ($x$ or $\bar{x}$) that are fixed by final-
state kinematics, and on a transverse distance $\boldsymbol{z}$ that is
Fourier conjugate to the transverse parton momentum (and often denoted by
$\boldsymbol{b}$ in the literature). They also depend on an ultraviolet
renormalisation scale $\mu$ and on a rapidity parameter ($\zeta$ or
$\bar{\zeta}$) as we will review later. Notice that the rapidity parameter
dependence cancels in $W_{q\bar{q}}$. The parton-level cross section
$\hat{\sigma}_{q\bar{q}}$ and the overall cross section may be taken
differential in additional variables if more than one particle is produced in
the hard scattering. An example is the angular distribution of the leptons
into which the electroweak gauge boson decays in Drell-Yan production. In this
case one must include the TMDs for transverse quark and antiquark polarisation
in (2).

The dependence of a TMD on the renormalisation scale is given by
Collins:2011zzd ; Aybat:2011zv

$\displaystyle\frac{\partial}{\partial\log\mu}\,f_{a}(x,\boldsymbol{z};\mu,\zeta)$
$\displaystyle=\gamma_{F,a}(\mu,\zeta)\,f_{a}(x,\boldsymbol{z};\mu,\zeta)\,,$
(3)

where $a=q,\bar{q},g$ labels the parton type. The rapidity dependence of the
anomalous dimension $\gamma_{F,a}$ is given by

$\displaystyle\frac{\partial}{\partial\log\zeta}\,\gamma_{F,a}(\mu,\zeta)$
$\displaystyle={}-\frac{1}{2}\mskip 1.5mu\gamma_{K,a}(\mu)\,,$ (4)

where $\gamma_{K,a}$ is called the cusp anomalous dimension. It depends on
$\mu$ via $\alpha_{s}(\mu)$, i.e.
$\gamma_{K,a}(\mu)=\gamma_{K,a}\bigl{(}\alpha_{s}(\mu)\bigr{)}$. From (4) one
readily finds

$\displaystyle\gamma_{F,a}(\mu,\zeta)$
$\displaystyle=\gamma_{a}(\mu)-\gamma_{K,a}(\mu)\,\log\frac{\sqrt{\zeta}}{\mu}$
(5)

with

$\displaystyle\gamma_{a}(\mu)$ $\displaystyle=\gamma_{F,a}(\mu,\mu^{2})\,.$
(6)

The evolution of TMDs with the rapidity scale $\zeta$ is governed by the
Collins-Soper equation111We generally follow the notation of Collins:2011zzd ;
Collins:2017oxh  in the present paper. We do however not use a tilde to denote
quantities in transverse position space, thus writing $f$ and $K$ instead of
$\tilde{f}$ and $\tilde{K}$.

$\displaystyle\frac{\partial}{\partial\log\zeta}\,f_{a}(x,\boldsymbol{z};\mu,\zeta)$
$\displaystyle=\frac{1}{2}\mskip
1.5muK_{a}(\boldsymbol{z};\mu)\,f_{a}(x,\boldsymbol{z};\mu,\zeta)\,,$ (7)

whose kernel satisfies

$\displaystyle\frac{\partial}{\partial\log\mu}\,{K_{a}(\boldsymbol{z};\mu)}={}-\gamma_{K,a}(\mu)\,.$
(8)

If the transverse boson momentum $\boldsymbol{q}$ is large compared with the
scale $\Lambda$ of non-perturbative interactions, one can use a short-distance
expansion that connects a TMD $f_{a}(x,\boldsymbol{z})$ at small
$\boldsymbol{z}$ with a conventional collinear PDF $f_{b}(x)$ according to

$\displaystyle f_{a}(x,\boldsymbol{z};\mu,\zeta)$
$\displaystyle=\sum_{b}{C}_{ab}(x^{\prime},\boldsymbol{z};\mu,\zeta)\underset{x}{\otimes}f_{b}(x^{\prime};\mu)\,,$
(9)

with the convolution product defined by

$\displaystyle C(x^{\prime})\underset{x}{\otimes}f(x^{\prime})$
$\displaystyle=\int_{x}^{1}\frac{dx^{\prime}}{x^{\prime}}\,C(x^{\prime})\,f\biggl{(}\frac{x}{x^{\prime}}\biggr{)}\,.$
(10)

The expansion (9) has power corrections in the parameter
$\Lambda|\boldsymbol{z}|$. Combining it with the solution of the evolution
equations in $\mu$ and $\zeta$, one obtains

$\displaystyle f_{a}(x,\boldsymbol{z};\mu,\zeta)$
$\displaystyle=\exp\,\biggl{\\{}\int_{\mu_{0}}^{\mu}\frac{d\mu^{\prime}}{\mu^{\prime}}\,\biggl{[}\gamma_{a}(\mu^{\prime})-\gamma_{K,a}(\mu^{\prime})\,\log\frac{\sqrt{\zeta}}{\mu^{\prime}}\biggr{]}+K_{a}(\boldsymbol{z};\mu_{0})\log\frac{\sqrt{\zeta}}{\mu_{0}}\mskip
1.5mu\biggr{\\}}$
$\displaystyle\quad\times\sum_{b}{C}_{ab}(x^{\prime},\boldsymbol{z};\mu_{0},\mu_{0}^{2})\underset{x}{\otimes}{f_{b}(x^{\prime};\mu_{0})}\,.$
(11)

The short-distance coefficient $C$ and the Collins-Soper kernel $K$ should be
evaluated with a scale choice that avoids large logarithms, so that they can
be reliably calculated in fixed-order perturbation theory. In the non-
perturbative region of $\boldsymbol{z}$ one needs a model ansatz for
$f_{a}(x,\boldsymbol{z})$. The so-called $b^{*}$ prescription Collins:1981va ;
Collins:1984kg  can be used to smoothly interpolate between such an ansatz and
the perturbative result (2).

In the following sections, we will show how these results can be extended to
the case of double parton scattering. There are several aspects that make this
extension far from trivial. One is the larger number of coloured particles
involved in the process, which leads to a non-trivial colour structure of
DTMDs and DPDFs. As a consequence, even DPDFs depend on the rapidity parameter
$\zeta$, unlike PDFs for a single parton Diehl:2011yj ; Manohar:2012jr . A
second aspect is that DPS involves several transverse distances, which makes
the analogue of the short-distance expansion (9) more complicated.

##  3 Defining double parton distributions

As reviewed in Diehl:2015bca , factorisation of DPS processes involves
separating the leading graphs for the cross sections into subgraphs that are
hard, soft, or collinear to one of the two incoming protons. The treatment of
the soft subgraph is intimately related with the rapidity parameter $\zeta$
mentioned in the previous subsection.

There are in fact different alternatives for such a treatment. The analysis of
the double Drell-Yan process in Diehl:2011yj  followed the original procedure
for TMD factorisation by Collins and Soper Collins:1981uk  and did not work
out all relevant aspects of the problem. In the present paper, we perform a
systematic analysis of DPS in the factorisation framework of Collins
Collins:2011zzd . We explicitly show how the soft factors relevant for the
cross section can be entirely absorbed into DTMDs or DPDFs, and we derive the
resulting evolution equations in $\zeta$, as well as the ones in $\mu$. Other
schemes to handle soft factors and rapidity dependence will briefly be
discussed in section 8.

The starting point of our discussion is an intermediate expression of the DPS
cross section, given in section 2.1 of Diehl:2015bca . The cross section for
the production of two sets of colourless particles involves a term

$\displaystyle
H_{1,q\bar{q}}\,H_{2,q\bar{q}}\;F^{T}_{\text{us},\bar{q}\bar{q}}(v_{R})\,S_{qq}^{-1}(v_{L},v_{R})\,S_{qq}(v_{L},v_{R})\,S_{qq}^{-1}(v_{L},v_{R})\,F_{\text{us},qq}(v_{L})$
(12)

for the annihilation of two quarks in one proton with two antiquarks in the
other proton, and corresponding terms for the other parton combinations. For
definiteness we have only written down the DPDs for unpolarised partons;
polarised terms have the same soft factor. $H_{i,q\bar{q}}$ denotes the
squared hard-scattering amplitudes, with appropriate spin projections (see
section 2.2 in Diehl:2011yj ) but with the colour structure removed as
specified in (116) below. $F_{\text{us}}$ denotes unsubtracted collinear
matrix elements, and $S$ is a soft factor. The inverse of this factor removes
contributions of soft gluons from the unsubtracted collinear matrix elements,
so that $S^{-1}\mskip 1.5muF_{\text{us}}$ receives only contributions from
collinear gluons. For brevity we have omitted momentum fraction and position
space arguments in (12), as well as renormalisation and factorisation scales
and colour indices. $F_{\text{us}}$ is a row vector in colour space (with one
index for each of the four parton legs), and $S$ is a matrix with two times
four indices. The spacelike four-vectors $v_{L}$ and $v_{R}$ denote the
directions of Wilson lines and will be specified later. As discussed in
Collins:2011zzd ; Diehl:2015bca , $v_{L}$ and $v_{R}$ have to be chosen such
that the effects of so-called Glauber gluon exchange on the cross section can
be subsumed into the soft and collinear factors in (12).

The aim of the following sections is to combine soft factors and collinear
matrix elements in such a way that the product (12) takes on a simple form.
After introducing the necessary notation, we derive a number of symmetry
properties of soft factors in section 3.2 and then discuss our central
hypothesis for their rapidity dependence, given in (35) and (36). This leads
us to the definition (3.3) of DPDs and to their rapidity evolution equation
(45) in section 3.3. We discuss renormalisation in section 3.4 and derive the
basic evolution equations (3.4) and (55) relevant for DTMDs. Applying our
construction to TMDs for a single parton in section 3.5, we are led to the
definition (60) and see that it is equivalent to the definition of Collins
Collins:2011zzd  by virtue of the relation (59).

###  3.1 Collinear matrix elements

To begin with, let us recall the definitions of unsubtracted DTMDs and DPDFs
in terms of proton matrix elements. These will later be combined with soft
factors in order to define the double parton distributions that appear in the
cross section formula.

For two partons $a_{1}$ and $a_{2}$, the unsubtracted DTMDs are defined in
terms of matrix elements as Diehl:2011yj ; Diehl:2011tt

$\displaystyle
F_{\text{us},a_{1}a_{2}}(x_{1},x_{2},\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y})$
$\displaystyle=2p^{+}(x_{1}\mskip 1.5mup^{+})^{-n_{1}}\,(x_{2}\mskip
1.5mup^{+})^{-n_{2}}\int\frac{dz^{-}_{1}}{2\pi}\,\frac{dz^{-}_{2}}{2\pi}\,dy^{-}\,e^{i\mskip
1.5mu(x_{1}z_{1}^{-}+x_{2}z_{2}^{-})\mskip 1.5mup^{+}}$
$\displaystyle\quad\times\langle\mskip 1.5mup\mskip
1.5mu|\,\mathcal{O}_{a_{1}}(y,z_{1})\,\mathcal{O}_{a_{2}}(0,z_{2})\,|\mskip
1.5mup\mskip 1.5mu\rangle\,,$ (13)

where $n_{i}=1$ if parton number $i$ is a gluon and $n_{i}=0$ otherwise. We
use light-cone coordinates $w^{\pm}=(w^{0}\pm w^{3})/\sqrt{2}$ and the
transverse component $\boldsymbol{w}=(w^{1},w^{2})$ for any four-vector $w$.
The definition (3.1) is natural for a proton moving to the right, i.e. for
$p^{3}>0$. For a left moving proton, i.e. for $p^{3}<0$, one would interchange
the roles of plus and minus coordinates. It is understood that
$\boldsymbol{p}=\boldsymbol{0}$ in both cases, and that the proton
polarisation is averaged over. Setting
$\boldsymbol{z}_{1}=\boldsymbol{z}_{2}=\boldsymbol{0}$ in (3.1), one obtains
DPDFs, which are relevant for collinear factorisation. As we will discuss
later, this changes the ultraviolet behaviour of the operators.

The operators for quarks in a right moving proton read

$\displaystyle\mathcal{O}_{a}(y,z)$
$\displaystyle=\bar{q}\bigl{(}y-{\textstyle\frac{1}{2}}z\bigr{)}\,W^{\dagger}\bigl{(}y-{\textstyle\frac{1}{2}}z,v_{L}\bigr{)}\,\Gamma_{a}\,W\bigl{(}y+{\textstyle\frac{1}{2}}z,v_{L}\bigr{)}\,q\bigl{(}y+{\textstyle\frac{1}{2}}z\bigr{)}\Big{|}_{z^{+}=y^{+}=0}$
(14)

with spin projections

$\displaystyle\Gamma_{q}$ $\displaystyle={\textstyle\frac{1}{2}}\gamma^{+}\,,$
$\displaystyle\Gamma_{\Delta q}$
$\displaystyle={\textstyle\frac{1}{2}}\gamma^{+}\gamma_{5}\,,$
$\displaystyle\Gamma_{\delta
q}^{j}={\textstyle\frac{1}{2}}i\sigma^{j+}\gamma_{5}\quad(j=1,2)$ (15)

onto unpolarised quarks ($q$), longitudinally polarised quarks ($\Delta q$)
and transversely polarised quarks ($\delta q$). We do not explicitly display
the transverse index $j$ of the operator $\mathcal{O}_{\delta q}$ and of the
corresponding DPDs, unless it is needed. The field with argument
$y+{\textstyle\frac{1}{2}}z$ in $\mathcal{O}_{q}(y,z)$ is associated with a
quark in the amplitude of a double scattering process and the field with
argument $y-{\textstyle\frac{1}{2}}z$ with a quark in the complex conjugate
amplitude. The Wilson lines are defined as

$\displaystyle W(\xi,v)$ $\displaystyle=\operatorname{P}\exp\biggl{[}\mskip
1.5muig\mskip 1.5mut^{a}\int_{-\infty}^{0}\\!ds\,vA^{a}(\xi+sv)\mskip
1.5mu\biggr{]}\,,$ (16)

where $\operatorname{P}$ denotes path ordering, such that fields
$t^{a}A^{a}(\xi+sv)$ with smaller $s$ stand further to the left in the
expanded exponential. Our convention for the strong coupling $g$ is specified
in appendix D. Throughout this work, we only consider the case
$\boldsymbol{v}=\boldsymbol{0}$. In the matrix element (3.1) for a right-
moving proton, one takes a direction $v_{L}$ with $v_{L}^{-}\gg-v_{L}^{+}>0$,
and in its analogue for a left-moving proton one has a direction $v_{R}$ with
$v_{R}^{+}\gg-v_{R}^{-}>0$. In both cases, the Wilson lines are past-pointing.
Analogous operators are defined for antiquarks, with some sign changes as
specified in section 2.2 of Diehl:2011yj . For gluons, one has

$\displaystyle\mathcal{O}_{a}(y,z)$
$\displaystyle=\Pi_{a}^{jj^{\prime}}\,G^{+j^{\prime}}\bigl{(}y-{\textstyle\frac{1}{2}}z\bigr{)}\,W^{\dagger}\bigl{(}y-{\textstyle\frac{1}{2}}z,v_{L}\bigr{)}\,W\bigl{(}y+{\textstyle\frac{1}{2}}z,v_{L}\bigr{)}\,G^{+j}\bigl{(}y+{\textstyle\frac{1}{2}}z\bigr{)}\Big{|}_{z^{+}=y^{+}=0}$
(17)

with spin projections

$\displaystyle\Pi_{g}^{jj^{\prime}}$ $\displaystyle=\delta^{jj^{\prime}}\,,$
$\displaystyle\Pi_{\Delta g}^{jj^{\prime}}$
$\displaystyle=i\epsilon^{jj^{\prime}}\,,$ $\displaystyle\bigl{[}\Pi_{\delta
g}^{kk^{\prime}}\bigr{]}{}^{jj^{\prime}}$
$\displaystyle=\tau^{jj^{\prime}\\!,kk^{\prime}}$ (18)

onto unpolarised gluons ($g$), longitudinally polarised gluons ($\Delta g$)
and linearly polarised gluons ($\delta g$). The indices
$j,j^{\prime},k,k^{\prime}=1,2$ run over transverse components,
$\epsilon^{jj^{\prime}}$ is the antisymmetric tensor with $\epsilon^{12}=1$,
and $\tau^{jj^{\prime}\\!,kk^{\prime}}$ is defined as

$\displaystyle\tau^{jj^{\prime}\\!,kk^{\prime}}$
$\displaystyle=\frac{1}{2}\,\Bigl{(}\delta^{jk}\delta^{j^{\prime}k^{\prime}}+\delta^{jk^{\prime}}\delta^{j^{\prime}k}-\delta^{jj^{\prime}}\delta^{kk^{\prime}}\Bigr{)}\,.$
(19)

The Wilson lines in (17) are in the adjoint representation rather than in the
fundamental one (see section 3.2). Making the colour indices of the operators
explicit, we have

$\displaystyle\mathcal{O}_{q,\,jj^{\prime}}$
$\displaystyle=\bar{q}_{k^{\prime}}\,(W^{\dagger})_{k^{\prime}j^{\prime}}\,\Gamma\,W_{jk}\,q_{k}\,,$
$\displaystyle\mathcal{O}_{g,\,aa^{\prime}}$
$\displaystyle=\Pi\,G_{b^{\prime}}\,(W^{\dagger})_{b^{\prime}a^{\prime}}\,W_{ab}\,G_{b}$
(20)

for quarks and gluons, respectively.

For the discussion of ultraviolet renormalisation and of the short-distance
expansion, it is useful to introduce operators that correspond to a definite
light-cone momentum fraction $x$ of a parton:

$\displaystyle O_{a}(x,\boldsymbol{y},\boldsymbol{z})$
$\displaystyle=2p^{+}\mskip
1.5mu(xp^{+})^{-n}\int\frac{dz^{-}}{2\pi}\,dy^{-}\;e^{ixz^{-}p^{+}}\,\mathcal{O}_{a}(y,z)\,.$
(21)

Using translation invariance, one readily finds that unsubtracted single and
double parton TMDs are then given by matrix elements

$\displaystyle
2\pi\delta(p^{+}-p^{\prime+})\,2p^{+}f_{\text{us},a}(x,\boldsymbol{z})$
$\displaystyle=\langle
p^{\prime}|\,O_{a}(x,\boldsymbol{0},\boldsymbol{z})\,|p\rangle\,,$
$\displaystyle
2\pi\delta(p^{+}-p^{\prime+})\,2p^{+}F_{\text{us},a_{1}a_{2}}(x_{1},x_{2},\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y})$
$\displaystyle=\langle
p^{\prime}|\,O_{a_{1}}(x_{1},\boldsymbol{y},\boldsymbol{z}_{1})\,O_{a_{2}}(x_{2},\boldsymbol{0},\boldsymbol{z}_{2})\,|p\rangle\,,$
(22)

where $\boldsymbol{p}^{\prime}=\boldsymbol{p}=\boldsymbol{0}$. The
corresponding collinear distributions are obtained by setting
$\boldsymbol{z}$, $\boldsymbol{z}_{1}$ and $\boldsymbol{z}_{2}$ to zero.

We note that in the DPS cross section, there are also distributions describing
the interference between different parton types in the process amplitude and
its complex conjugate, i.e. between quarks and antiquarks, between quarks and
gluons, or between quarks of different flavour Diehl:2011yj ; Manohar:2012jr ;
Diehl:2013mla ; Kasemets:2014yna . At low values of $x_{1}$ and $x_{2}$, such
interference DPDs are expected to be negligible, because they have no dynamic
cross talk with gluon distributions, which grow most strongly with decreasing
$x$. Although we do not consider interference DPDs in this work, they can be
treated with the methods presented in the following.

Let us remark that an analysis of transverse-momentum dependent DPDs in the
small-$x$ limit has recently been given in Golec-Biernat:2016vbt . The
quantities considered in that work are Fourier transformed w.r.t. our variable
$\boldsymbol{z}_{i}$ and – more importantly – integrated over
$\boldsymbol{y}$. To make contact with the DPS cross section, one would need
to restore the dependence of these distributions on $\boldsymbol{y}$, or on
the Fourier conjugated momentum.

###  3.2 Soft factors

Before constructing the final DPDs, we must take a closer look at the soft
factor, which is defined as the vacuum expectation value of Wilson lines. For
reasons that will become clear later, we define an “extended soft factor” with
open colour indices of all Wilson lines as

$\displaystyle\bigl{[}S_{qq}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{L},v_{R})\bigr{]}^{i_{1}i_{1}^{\prime}i_{2}i_{2}^{\prime},l_{1}l_{1}^{\prime}l_{2}l_{2}^{\prime}}_{j_{1}j_{1}^{\prime}j_{2}j_{2}^{\prime},k_{1}k_{1}^{\prime}k_{2}k_{2}^{\prime}}$
$\displaystyle\qquad=\bigl{\langle}\,0\,\big{|}\mskip
1.5mu\bigl{[}O_{S,q}(\boldsymbol{y},\boldsymbol{z}_{1};v_{L},v_{R})\bigr{]}^{i_{1}i_{1}^{\prime},l_{1}l_{1}^{\prime}}_{j_{1}j_{1}^{\prime},k_{1}k_{1}^{\prime}}\,\bigr{[}O_{S,q}(\boldsymbol{0},\boldsymbol{z}_{2};v_{L},v_{R})\bigr{]}^{i_{2}i_{2}^{\prime},l_{2}l_{2}^{\prime}}_{j_{2}j_{2}^{\prime},k_{2}k_{2}^{\prime}}\,\big{|}\,0\,\bigr{\rangle}\,.$
(23)

This factor appears in the cross section with two $q\bar{q}$ annihilation
subprocesses, and the subscripts $q$ refer to the right-moving parton in each
subprocess. The operator

$\displaystyle\bigl{[}O_{S,q}(\boldsymbol{y},\boldsymbol{z};v_{L},v_{R})\bigr{]}^{ii^{\prime},ll^{\prime}}_{jj^{\prime},kk^{\prime}}$
$\displaystyle\qquad=W_{ij}(\boldsymbol{y}+{\textstyle\frac{1}{2}}\boldsymbol{z},v_{L})\,W_{kl}^{\dagger}(\boldsymbol{y}+{\textstyle\frac{1}{2}}\boldsymbol{z},v_{R})\,W_{l^{\prime}k^{\prime}}(\boldsymbol{y}-{\textstyle\frac{1}{2}}\boldsymbol{z},v_{R})\,W_{j^{\prime}i^{\prime}}^{\dagger}(\boldsymbol{y}-{\textstyle\frac{1}{2}}\boldsymbol{z},v_{L})$
(24)

is a product of four Wilson lines, which are defined as in (16), but with
$\xi^{+}=\xi^{-}=0$, so that their position arguments are only in the
transverse plane and hence written in boldface. Primed and unprimed indices
$j$ and $k$ in the Wilson lines of (3.2) correspond to gluon fields $A(\xi)$
at light-cone zero ($\xi^{+}=\xi^{-}=0$), whereas primed and unprimed indices
$i$ and $l$ correspond to gluon fields at light-cone infinity, $A(\xi-\infty
v)$. A pictorial representation of (3.2) is given in figure 1.

Figure 1:  Wilson lines and colour indices of the extended soft factor defined
by (3.2) and (3.2). Wilson lines with subscripts $1$ are grouped into the
operator $O_{S,q}(\boldsymbol{y},\boldsymbol{z}_{1})$ and those with
subscripts $2$ into the operator $O_{S,q}(\boldsymbol{0},\boldsymbol{z}_{2})$.

For right-moving antiquarks instead of quarks, one replaces $W_{ij}$ with
$W^{*}_{ij}=W^{\dagger}_{ji}$ and vice versa, which corresponds to replacing
$it^{a}_{ij}$ with $(it^{a}_{ij})^{*}=-it^{a}_{ji}$ in the exponential (16).
For gluons, one takes adjoint Wilson lines $W_{bc}$, obtained by replacing
$it^{a}_{ij}$ in (16) with $f^{abc}$. This reflects the fact that the
generators $T^{a}$ of the colour group in the adjoint representation are given
by $(T^{a})_{bc}=-if^{abc}$. We therefore have

$\displaystyle S_{qq}$ $\displaystyle=S_{\bar{q}\bar{q}}^{*}\,,$
$\displaystyle S_{q\bar{q}}$ $\displaystyle=S_{\bar{q}q}^{*}\,,$
$\displaystyle S_{qg}$ $\displaystyle=S_{\bar{q}g}^{*}\,,$ $\displaystyle
S_{gg}$ $\displaystyle=S_{gg}^{*}\,,$ (25)

where we have omitted colour indices for brevity. Note that, by construction,
the Wilson line operators for all representations are unitary, i.e.
$W(\xi,v)\mskip 1.5muW^{\dagger}(\xi,v)=\mathbb{1}$, where $\mathbb{1}$ is the
unit matrix in the relevant colour space.

In processes producing colourless particles, one needs (3.2) with all index
pairs $j$, $k$ contracted, i.e.

$\displaystyle\bigl{[}S_{qq}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{L},v_{R})\bigr{]}^{i_{1}i_{1}^{\prime}i_{2}i_{2}^{\prime},\,l_{1}l_{1}^{\prime}l_{2}l_{2}^{\prime}}$
$\displaystyle\qquad=\bigl{\langle}\,0\,\big{|}\mskip
1.5mu\bigl{[}O_{S,q}(\boldsymbol{y},\boldsymbol{z}_{1};v_{L},v_{R})\bigr{]}^{i_{1}i_{1}^{\prime},l_{1}l_{1}^{\prime}}\,\bigr{[}O_{S,q}(\boldsymbol{0},\boldsymbol{z}_{2};v_{L},v_{R})\bigr{]}^{i_{2}i_{2}^{\prime},l_{2}l_{2}^{\prime}}\,\big{|}\,0\,\bigr{\rangle}$
(26)

with

$\displaystyle\bigl{[}O_{S,q}(\boldsymbol{y},\boldsymbol{z};v_{L},v_{R})\bigr{]}^{ii^{\prime},ll^{\prime}}$
$\displaystyle\qquad=\bigl{[}W(\boldsymbol{y}+{\textstyle\frac{1}{2}}\boldsymbol{z},v_{L})\,W^{\dagger}(\boldsymbol{y}+{\textstyle\frac{1}{2}}\boldsymbol{z},v_{R})\bigr{]}_{il}\,\bigl{[}W(\boldsymbol{y}-{\textstyle\frac{1}{2}}\boldsymbol{z},v_{R})\,W^{\dagger}(\boldsymbol{y}-{\textstyle\frac{1}{2}}\boldsymbol{z},v_{L})\bigr{]}_{l^{\prime}i^{\prime}}\,.$
(27)

We simply call this the “soft factor” (without the specification “extended”).
Regarding this as a matrix in the index pairs
$(i_{1}i_{1}^{\prime}i_{2}i_{2}^{\prime})$ and
$(l_{1}l_{1}^{\prime}l_{2}l_{2}^{\prime})$, we define the transposed matrix as

$\displaystyle\bigl{[}S_{qq}^{T}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{L},v_{R})\bigr{]}^{i_{1}i_{1}^{\prime}i_{2}i_{2}^{\prime},l_{1}l_{1}^{\prime}l_{2}l_{2}^{\prime}}$
$\displaystyle=\bigl{[}S_{qq}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{L},v_{R})\bigr{]}^{l_{1}l_{1}^{\prime}l_{2}l_{2}^{\prime},i_{1}i_{1}^{\prime}i_{2}i_{2}^{\prime}}$
(28)

and the Hermitian conjugate as
$S_{qq}^{\smash{\dagger}\phantom{T}}=\bigl{[}S_{qq}^{T}\bigr{]}^{*}$.
Corresponding definitions hold for $S_{gg}$, $S_{gq}$, $S_{\bar{q}\bar{q}}$
etc. There are a number of symmetry constraints on the soft factor:

  1. 1.

With the restrictions on the directions of Wilson lines in $S(v_{L},v_{R})$
specified below (16), one can always perform a longitudinal boost such that
$v_{L}=(\alpha,\beta,\boldsymbol{0})$ and
$v_{R}=(\beta,\alpha,\boldsymbol{0})$. A parity transformation interchanges
plus- and minus components and thus, combined with the boost, exchanges
$v_{L}\leftrightarrow v_{R}$. Parity and boost invariance therefore give

$\displaystyle
S_{a_{1}a_{2}}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{L},v_{R})$
$\displaystyle=S_{a_{1}a_{2}}(-\boldsymbol{z}_{1},-\boldsymbol{z}_{2},-\boldsymbol{y};v_{R},v_{L})$
$\displaystyle=S_{a_{1}a_{2}}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{R},v_{L})\,,$
(29)

where $a_{1},a_{2}$ denote the parton types, $q,\bar{q},g$. In the final step
we used that the soft factor depends on position arguments only via scalar
products between $\boldsymbol{z}_{1},\boldsymbol{z}_{2}$ and $\boldsymbol{y}$.
This follows from rotational and parity invariance, combined with the fact
that the Wilson line directions $v_{L}$ and $v_{R}$ are purely longitudinal.

  2. 2.

A combined parity and time reversal transformation reverses the vectors
$v_{L}$ and $v_{R}$, as well as the position arguments. This gives

$\displaystyle
S_{a_{1}a_{2}}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{L},v_{R})$
$\displaystyle=S_{a_{1}a_{2}}(-\boldsymbol{z}_{1},-\boldsymbol{z}_{2},-\boldsymbol{y};-v_{L},-v_{R})$
$\displaystyle=S_{a_{1}a_{2}}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};-v_{L},-v_{R})\,,$
(30)

where in the last step we used the same symmetry argument as in (1).

  3. 3.

Since the soft factor is constructed from products
$\bigl{[}W(z,v_{L})\,W^{\dagger}(z,v_{R})\bigr{]}_{il}$ for quarks,
$\bigl{[}W(z,v_{R})\,W^{\dagger}(z,v_{L})\bigr{]}_{li}$ for antiquarks, and
$\bigl{[}W(z,v_{L})\,W^{\dagger}(z,v_{R})\bigr{]}_{ad}$ for gluons, we have

$\displaystyle
S_{a_{1}a_{2}}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{L},v_{R})$
$\displaystyle=S_{a_{1}a_{2}}^{\dagger}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{R},v_{L})$
$\displaystyle=S_{a_{1}a_{2}}^{\dagger}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y};v_{L},v_{R})\,,$
(31)

where in the second step we have used (1). The soft matrix is thus Hermitian
in the groups of indices for the Wilson lines along the different directions
$v_{L}$ and $v_{R}$. Combining this result with the relations (25), we find

$\displaystyle S_{qq}$ $\displaystyle=S_{\bar{q}\bar{q}}^{T}\,,$
$\displaystyle S_{q\bar{q}}$ $\displaystyle=S_{\bar{q}q}^{T}\,,$
$\displaystyle S_{qg}$ $\displaystyle=S_{\bar{q}g}^{T}\,,$ $\displaystyle
S_{gg}$ $\displaystyle=S_{gg}^{T}\,,$ (32)

where the matrices on the left and the right hand side have identical
arguments. We can thus identify the product
$F^{T}_{\text{us},\bar{q}\bar{q}}\,S_{qq}^{-1}=(S_{\bar{q}\bar{q}}^{-1}\,F_{\text{us},\bar{q}\bar{q}})^{T}$
in the expression (12) as the analogue of the product $S_{qq}^{-1}\mskip
1.5muF_{\text{us},qq}$.

From now on we restrict ourselves to spacelike Wilson line directions in
$S(v_{L},v_{R})$ that satisfy

$\displaystyle v_{L}^{-}$ $\displaystyle>0\,,$ $\displaystyle v_{R}^{+}$
$\displaystyle>0\,,$ $\displaystyle Y_{L}\ll Y_{R}\,,$ (33)

where the rapidity $Y_{v}$ of the spacelike vector $v$ is defined as

$\displaystyle Y_{v}=\frac{1}{2}\log\biggl{|}\frac{v^{+}}{v^{-}}\biggr{|}\,.$
(34)

Cases other than (33), such as $v_{L}^{-}<0,\,v_{R}^{+}<0$, can be realised
using the symmetry relations (1) and (2). Owing to boost invariance and the
fact that the Wilson lines $W(\xi,v)$ are invariant under rescaling
$v\to\lambda v$, the soft factor depends on $v_{L}$ and $v_{R}$ only via the
difference of the Wilson line rapidities. We can hence write
$S(v_{L},v_{R})=S(Y_{R}-Y_{L})$.

At this point we make an _assumption_ on the properties of the soft factor,
which we cannot fully prove (we specify below what can be proven at present).

For the soft factor $S(Y_{R}-Y_{L})$ with $Y_{R}-Y_{L}\gg 1$ (for brevity we
omit the indices $a_{1}$, $a_{2}$ and arguments
$\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y}$), we assume that there
exists a matrix function $s(Y)$ satisfying the following three properties:

Property 1a:

The rapidity dependence is given by

$\displaystyle\frac{\partial}{\partial Y}\,s(Y)$ $\displaystyle=s(Y)\mskip
1.5muK$ (35)

with $K=K^{\dagger}$ independent of $Y$.

Property 1b:

$s(Y)$ is nonsingular. It is sufficient to establish this property at an
arbitrary value $Y_{1}$; its validity at any other $Y$ then follows from the
solution of (35).

Property 1c:

One has

$\displaystyle S(Y)$ $\displaystyle=s(Y-Y_{0})\,s^{\dagger}(Y_{0})$
$\displaystyle\text{for $Y\gg 1$ and arbitrary $Y_{0}$}\,.$ (36)

It is easy to see that $s(Y)$ is not unique, since properties 1a – 1c remain
true if one redefines $s(Y)\to s(Y)\,U$ and $K\to U^{\dagger}KU$ with a $Y$
independent unitary matrix $U$. Conversely, if properties 1a – 1c hold for two
matrix functions $s(Y)$ and $s^{\prime}(Y)$, one can show that
$s^{\prime}(Y)=s(Y)\,U(Y)$ with a unitary matrix $U$ (which is not necessarily
$Y$ independent).

In appendix A we will show that properties 1a – 1c are equivalent to the
following two properties:

Property 2a:

$S$ satisfies the Collins-Soper equation

$\displaystyle\frac{\partial}{\partial Y}\,S(Y)$
$\displaystyle=\widehat{K}\mskip 1.5muS(Y)$ $\displaystyle\text{for $Y\gg
1$}\,.$ (37)

$\widehat{K}$ is not necessarily Hermitian, and its relation with $K$ is
specified in (332). Both (36) and (37) are meant to be approximations for
$Y\gg 1$. We can define a matrix $\widehat{S}(Y)$ that approximates $S(Y)$ for
$Y\gg 1$ whilst being an exact solution of the differential equation (37) at
all $Y$. This equation is then solved by

$\displaystyle\widehat{S}(Y)$
$\displaystyle=e^{Y\widehat{K}}\,\widehat{S}(0).$ (38)

Property 2b:

There is a value $Y_{1}$ for which $\widehat{S}(Y_{1})$ is positive definite.

We show in appendix A that $\widehat{S}(Y)$ is then in fact positive definite
for all $Y$. This guarantees that the matrix $S(Y)$ at large $Y$ has an
inverse, which according to (12) is needed in the cross section formula.

Evidence for the properties just discussed comes from perturbation theory,
which can be used to compute the soft factor when the distances
$\boldsymbol{z}_{1}$, $\boldsymbol{z}_{2}$ and $\boldsymbol{y}$ are all small.

  * •

Properties 2a and 2b are easily checked at one-loop order from our explicit
calculations in all colour channels, which are reported in section 7. In this
case one finds $\widehat{S}(0)=\mathbb{1}$ and, taking $U=\mathbb{1}$ one has
$K=\widehat{K}$ according to (332).

  * •

Given property 2a, one can motivate property 2b using the perturbative
expansions $S(Y)=\mathbb{1}+\mathcal{O}(\alpha_{s})$ and
$\widehat{K}=\mathcal{O}(\alpha_{s})$. One may worry that the expansion of
$S(Y)$ has poor convergence, because higher orders in $\alpha_{s}$ can come
with higher powers of the large rapidity $Y$. However, according to (38) we
have $\widehat{S}(0)=\exp(-Y\widehat{K})\,S(Y)$ for sufficiently large $Y$. If
the perturbative expansions of $S(Y)$ and $\exp(-Y\widehat{K})$ are valid at
least in a formal sense, we get an expansion
$\widehat{S}(0)=\mathbb{1}+\mathcal{O}(\alpha_{s})$ that is free of any large
parameter. The eigenvalues of $\widehat{S}(0)$ then have a perturbative
expansion around $1$, which supports property 2b with $Y_{1}=0$, at least for
sufficiently small $\alpha_{s}$.

  * •

$S_{qq}$ and $S_{q\bar{q}}$ have been calculated at two-loop order in
Vladimirov:2016qkd . The validity of properties 1a and 1c can be explicitly
verified from the results in sections 4.2 and 4.4 of that work. This requires
a translation between the rapidity regulator used there and the one used here,
which we discuss in appendix B. We note that Vladimirov:2016qkd  defines
$S=s^{\dagger}s$, whilst we choose the order $S=ss^{\dagger}$ in (36).

Property 1b can then be motivated by the perturbative expansion of $s(Y)$,
arguing along the same lines as in the previous point.

  * •

An all-order derivation of the rapidity evolution equation of the DPS soft
factor has recently been given in Vladimirov:2017ksc . Our equation (37) can
be obtained from equation (5.17) in the arXiv version 2 of Vladimirov:2017ksc
using the relation (336) between the regulator variables
$\delta^{+},\delta^{-}$ and $Y_{L},Y_{R}$. However, more work is needed to
establish whether the derivation in Vladimirov:2017ksc  holds if one uses
either of the associated rapidity regulators.

A few more comments are in order.

  * •

While $S(0)=\mathbb{1}$ by construction (the Wilson line pairs at equal
positions give unit matrices, $W^{\dagger}W=\mathbb{1}$), one generally has
$\widehat{S}(0)\neq\mathbb{1}$. This implies that the evolution equation (37)
holds for $Y\gg 1$ but not when $Y$ becomes small. We see this already at one
loop: as follows from sections 3.3.1 and 3.2.2 of Diehl:2011yj , one has
$S-\mathbb{1}\propto Y\tanh(Y)$ for the one-loop soft factor, whilst (37)
truncated to $\mathcal{O}(\alpha_{s})$ gives $S-\mathbb{1}\propto Y$.

  * •

For the soft factor needed in collinear factorisation, i.e. for
$\boldsymbol{z}_{1}=\boldsymbol{z}_{2}=\boldsymbol{0}$, we will see in section
4.4 that $S(Y)$ is diagonal in the basis of colour representations for all
$Y$. As a consequence, $\widehat{K}$ is diagonal as well. In appendix A we
show that one can then also choose $s(Y)$ and $K$ to be diagonal. For the soft
factor in collinear factorisation, all matrix multiplications in colour space
thus become trivial.

  * •

The matrix $s$ replaces the square roots of soft factors in the construction
of single parton TMDs by Collins Collins:2011zzd , as we will show in section
3.5.

###  3.3 Definition of DPDs

According to (12), the DPS cross section involves a product of soft matrices,
which using (36) can be rewritten as

$\displaystyle S^{-1}(v_{L},v_{R})\,S(v_{L},v_{R})\,S^{-1}(v_{L},v_{R})$
$\displaystyle=s^{\dagger-1}(Y_{R}-Y_{C})\,s^{-1}(Y_{C}-Y_{L})\,,$ (39)

where $Y_{C}$ is a central rapidity, $Y_{L}\ll Y_{C}\ll Y_{R}$. Restoring
parton labels, we then define DPDs by

$\displaystyle F_{a_{1}a_{2}}(Y_{C})$
$\displaystyle=\lim_{Y_{L}\to-\infty}s_{a_{1}a_{2}}^{-1}(Y_{C}-Y_{L})\,F_{\text{us},a_{1}a_{2}}(Y_{L})\,,$
$\displaystyle F_{b_{1}b_{2}}(Y_{C})$
$\displaystyle=\lim_{Y_{R}\to\infty\phantom{-}}s_{b_{1}b_{2}}^{-1}(Y_{R}-Y_{C})\,F_{\text{us},b_{1}b_{2}}(Y_{R})$
(40)

for the distributions in a right-moving and a left-moving proton,
respectively. An analogous definition in the $\delta$ regulator scheme was put
forward in Vladimirov:2016qkd .

From the construction in appendix A it follows that the symmetry relations
(25) for $S_{a_{1}a_{2}}$ imply corresponding relations

$\displaystyle s_{qq}=$ $\displaystyle s_{\bar{q}\bar{q}}^{*}\,,$
$\displaystyle s_{q\bar{q}}$ $\displaystyle=s_{\bar{q}q}^{*}\,,$
$\displaystyle s_{qg}$ $\displaystyle=s_{\bar{q}g}^{*}\,,$ $\displaystyle
s_{gg}$ $\displaystyle=s_{gg}^{*}$ (41)

if the matrices $U_{a_{1}a_{2}}$ in (331) are chosen such that they satisfy
the relations (334). In the expression (12) for double Drell-Yan production we
can thus rewrite

$\displaystyle
F^{T}_{\text{us},\bar{q}\bar{q}}(Y_{R})\,s_{qq}^{\dagger-1}(Y_{R}-Y_{C})\,s_{qq}^{-1}(Y_{C}-Y_{L})\,F_{\text{us},qq}(Y_{L})$
$\displaystyle=F^{T}_{\bar{q}\bar{q}}(Y_{C})\,F_{qq}(Y_{C})\,,$ (42)

where we have omitted the hard-scattering factors $H_{i,q\bar{q}}$ for
brevity. A corresponding argument leads to the combination
$F^{T}_{\bar{q}q}(Y_{C})\,F_{q\bar{q}}(Y_{C})$ in the same process. The
production of one Higgs and one electroweak gauge boson involves a term

$\displaystyle
F^{T}_{\text{us},\bar{q}g}(Y_{R})\,s_{qg}^{\dagger-1}(Y_{R}-Y_{C})\,s_{qg}^{-1}(Y_{C}-Y_{L})\,F_{\text{us},qg}(Y_{L})$
$\displaystyle=F^{T}_{\bar{q}g}(Y_{C})\,F_{qg}(Y_{C})$ (43)

with mixed quark-gluon and antiquark-gluon DPDs, and likewise one obtains the
product $F^{T}_{gg}(Y_{C})\,F_{gg}(Y_{C})$ in double Higgs boson production.

In (3.3) we can take the limit of infinite Wilson line rapidities $Y_{L}$ and
$Y_{R}$ in the unsubtracted matrix elements. This means that the Wilson lines
in the parton operators $\mathcal{O}_{a}$ are along lightlike paths, which
leads to important simplifications as discussed in chapter 10.11 of
Collins:2011zzd . Let us see that this limit is well behaved in (3.3). The
rapidities $Y_{L}$ and $Y_{R}$ in the cross section formula originate from
Grammer-Yennie approximations (see e.g. Diehl:2015bca ), and one finds that
their precise values in (42) and (43) do not matter as long as
$|Y_{L}|,|Y_{R}|\gg 1$. One can thus take the limit of infinite $Y_{L}$ and
$Y_{R}$ in these expressions, and thus also in the individual factors in
(3.3). By contrast, taking infinite Wilson line rapidities in individual
unsubtracted DPDs or in the soft factor would lead to rapidity divergences.

The dependence of the DPDs on the central rapidity $Y_{C}$ is easily obtained
from (35), which gives

$\displaystyle\frac{\partial}{\partial Y}\,s^{-1}(Y)$ $\displaystyle=-K\mskip
1.5mus^{-1}(Y)$ (44)

and thus

$\displaystyle\frac{\partial}{\partial Y_{C}}\mskip 1.5muF(Y_{C})$
$\displaystyle=-KF(Y_{C})$ (45)

for DPDs in a right-moving proton, whilst for a left-moving proton one has the
opposite sign on the r.h.s. of (45).

The preceding construction can be performed for DTMDs and DPDFs alike.
Important differences between the two types of distributions arise at the
level of ultraviolet renormalisation. In the next subsection, we discuss this
for the case of DTMDs, postponing the case of DPDFs to section 5.2.

###  3.4 Renormalisation of DTMDs

The ultraviolet (UV) renormalisation of TMDs arises from vertex and self
energy corrections associated with composite operators that contain fields at
the same transverse position. Interactions between fields at different
transverse positions do not give rise to UV divergences: the finite spacelike
distance acts as an effective UV cutoff in the corresponding graphs. Operators
that require renormalisation are therefore $[W(\xi)\,q(\xi)]_{i}$,
$[W(\xi)\,G^{+i}(\xi)]_{a}$ and their Hermitian conjugates in the collinear
matrix elements, as well as $[W(\xi,v_{L})\,W^{\dagger}(\xi,v_{R})]_{ij}$,
$[W(\xi,v_{L})\,W^{\dagger}(\xi,v_{R})]_{ab}$ and their Hermitian conjugates
in the soft factor. Due to colour SU(3) invariance, the corresponding
renormalisation factors are all colour independent. We could in principle
choose a different renormalisation scale for each of the four parton operators
in $F_{\text{us}}$ and for each of the four corresponding Wilson line products
in the soft factor. We choose a slightly simpler variant, taking common
renormalisation scales $\mu_{1}$ and $\mu_{2}$ for the operators associated
with partons carrying momentum fraction $x_{1}$ and $x_{2}$, respectively.
Denoting bare quantities with a subscript $B$, we then have
$O_{a}(x,\boldsymbol{y},\boldsymbol{z})=Z_{\text{us},a}\,O_{B,a}(x,\boldsymbol{y},\boldsymbol{z})$,
where $Z_{\text{us},a}$ is the product of renormalisation factors for the
composite operators at transverse positions
$\boldsymbol{y}-{\textstyle\frac{1}{2}}\boldsymbol{z}$ and
$\boldsymbol{y}+{\textstyle\frac{1}{2}}\boldsymbol{z}$. Likewise, for the
operator defined in (3.2) we have
$O_{S,a}(\boldsymbol{y},\boldsymbol{z})=Z_{S,a}\,O_{BS,a}(\boldsymbol{y},\boldsymbol{z})$,
where $Z_{S,a}$ is the product of renormalisation factors for the Wilson line
pairs at equal transverse positions.222Our convention for renormalisation
factors $Z$ of composite operators corresponds to the one in Collins:1984xc ;
Collins:2011zzd . Other authors, such as the ones of Peskin:1995ev , use
$Z^{-1}$ instead. Both $Z_{\text{us},a}$ and $Z_{S,a}$ are independent of spin
and colour, but they differ for quarks and gluons (being equal for quarks and
antiquarks due to charge conjugation invariance). They depend on a scale $\mu$
and on Wilson line rapidities, and $Z_{\text{us},a}$ also depends on the plus-
momentum $xp^{+}$ of the relevant parton. We thus have

$\displaystyle F_{\text{us},a_{1}a_{2}}(x_{i};\mu_{i},Y_{L})$
$\displaystyle=Z_{\text{us},a_{1}}(\mu_{1},Y_{L},x_{1}\mskip
1.5mup^{+})\,Z_{\text{us},a_{2}}(\mu_{2},Y_{L},x_{2}\mskip
1.5mup^{+})\,F_{B,\text{us},a_{1}a_{2}}(x_{i};Y_{L})\,,$ $\displaystyle
S_{a_{1}a_{2}}(\mu_{i},Y)$
$\displaystyle=Z_{S,a_{1}}(\mu_{1},Y)\,Z_{S,a_{2}}(\mu_{2},Y)\,S_{B,a_{1}a_{2}}(Y)$
(46)

with $Y=Y_{R}-Y_{L}$. Throughout this work we use the convention that a
function with arguments $x_{i}$ ($\mu_{i}$) depends on both $x_{1}$ and
$x_{2}$ ($\mu_{1}$ and $\mu_{2}$). In this and the next subsection, we
consider a right-moving proton for definiteness, and we omit transverse
position arguments for brevity.

We now derive the renormalisation properties of the matrix
$s_{a_{1}a_{2}}(Y)$. Assuming that the $Y$ dependence of $S_{B}(Y)$ is given
by (37) with a bare kernel $\widehat{K}_{B}$ and defining

$\displaystyle\Lambda_{a}(Y)$ $\displaystyle=\frac{\partial}{\partial Y}\,\log
Z_{S,a}(Y)\,,$ (47)

we readily get

$\displaystyle\frac{\partial}{\partial Y}\,S_{a_{1}a_{2}}(Y)$
$\displaystyle=\widehat{K}_{a_{1}a_{2}}(Y)\,S_{a_{1}a_{2}}(Y)$ (48)

with

$\displaystyle\widehat{K}_{a_{1}a_{2}}(Y)$
$\displaystyle=\Lambda_{a_{1}}(Y)\,\mathbb{1}+\Lambda_{a_{2}}(Y)\,\mathbb{1}+\widehat{K}_{B,a_{1}a_{2}}\,,$
(49)

where we have dropped the dependence on the renormalisation scales for
brevity. The renormalised soft factor hence satisfies (37) if $\Lambda_{a}$ is
independent of $Y$. This is easily shown for the MS scheme. Expanding $Z_{S}$
in the renormalised coupling, we have
$Z_{S}=1+\sum_{n=1}^{\infty}\alpha_{s}^{n}Z_{S}^{(n)}(\epsilon)$ with
coefficients $Z_{S}^{(n)}$ being a finite sum of poles in $\epsilon$ for each
$n$. The corresponding coefficients in
$\Lambda=\sum_{n=1}^{\infty}\alpha_{s}^{n}\Lambda^{(n)}(\epsilon)$ are hence
pure pole terms as well. Since $S$ is finite for $\epsilon=0$, the same holds
for $\widehat{K}$ according to (48), so that the $\epsilon$ poles of $\Lambda$
must cancel those of $\widehat{K}_{B}$ in (49). Since the latter are $Y$
independent, the same holds for $\Lambda$. In the $\overline{\text{MS}}$
scheme one can repeat the previous argument after rescaling $\alpha_{s}$ in
the expansion of $Z_{S}$ by a factor $S_{\epsilon}$ (see section 7.3.3 for
further explanation).

With $\Lambda_{a}$ being $Y$ independent, the solution of (47) reads
$Z_{S,a}(Y)=Z_{S,a}(0)\,e^{Y\Lambda_{a}}$. Using (331) and (332), one finds
that

$\displaystyle K_{a_{1}a_{2}}(\mu_{i})$
$\displaystyle=\Lambda_{a_{1}}(\mu_{1})\,\mathbb{1}+\Lambda_{a_{2}}(\mu_{2})\,\mathbb{1}+K_{B,a_{1}a_{2}}\,,$
$\displaystyle s_{a_{1}a_{2}}(\mu_{i},Y)$
$\displaystyle=Z_{s,a_{1}}(\mu_{1},Y)\,Z_{s,a_{2}}(\mu_{2},Y)\,s_{B,a_{1}a_{2}}(Y)$
(50)

with $Z_{s,a}(Y)=\sqrt{Z_{S,a}(0)}\,e^{Y\Lambda_{a}}$, where we have restored
the $\mu$ dependence of the factors. We thus find that the renormalisation of
$s(Y)$ is multiplicative and independent of the colour channel, just as the
one of $S(Y)$. With bare and renormalised DTMDs defined as in (3.3), we then
have

$\displaystyle F_{a_{1}a_{2}}(x_{i};\mu_{i},Y_{C})$
$\displaystyle=Z_{F,a_{1}}(\mu_{1},Y_{C},x_{1}\mskip
1.5mup^{+})\,Z_{F,a_{2}}(\mu_{2},Y_{C},x_{2}\mskip
1.5mup^{+})\,F_{B,a_{1}a_{2}}(x_{i};Y_{C})$ (51) with $\displaystyle
Z_{F,a}(\mu,Y_{C},xp^{+})$
$\displaystyle=\lim_{Y_{L}\to-\infty}Z^{-1}_{s,a}(\mu,Y_{C}-Y_{L})\,Z_{\text{us},a}(\mu,Y_{L},xp^{+})$
(52)

for a right-moving proton, and correspondingly for a left-moving one. Note
that, as explained in chapter 10.11 of Collins:2011zzd , one should first take
the limit of infinite $Y_{L}$ in the product (52) and in $F_{B}(Y_{C})$, and
then let $\epsilon$ go to zero on the r.h.s. of (51).

We can now derive the $\mu$ dependence of the DTMDs and of their Collins-Soper
kernel. Defining

$\displaystyle\gamma_{F,a}(\mu,Y_{C},xp^{+})$
$\displaystyle=\frac{\partial}{\partial\log\mu}\,\log
Z_{F,a}(\mu,Y_{C},xp^{+})\,,$ $\displaystyle\gamma_{K,a}(\mu)$
$\displaystyle=-\frac{\partial}{\partial\log\mu}\,\Lambda_{a}(\mu)\,,$ (53)

we obtain for a right-moving proton

$\displaystyle\frac{\partial}{\partial\log\mu_{1}}\,F_{a_{1}a_{2}}(x_{i};\mu_{i},Y_{C})$
$\displaystyle=\gamma_{F,a_{1}}(\mu_{1},Y_{C},x_{1}\mskip
1.5mup^{+})\,F_{a_{1}a_{2}}(x_{i};\mu_{i},Y_{C},)\,,$
$\displaystyle\frac{\partial}{\partial\log\mu_{1}}\,K_{a_{1}a_{2}}(\mu_{i})$
$\displaystyle={}-\gamma_{K,a_{1}}(\mu_{1})\,\mathbb{1}\,,$ (54)

and analogous relations for the $\mu_{2}$ dependence, as well as the
additional condition

$\displaystyle\frac{\partial}{\partial Y_{C}}\,\gamma_{F,a}(\mu,Y_{C},xp^{+})$
$\displaystyle=\gamma_{K,a}(\mu)\,.$ (55)

We note that $Z_{F,a}$ and hence $\gamma_{F,a}$ can depend on $Y_{C}$ and
$xp^{+}$ only via the boost invariant combination $xp^{+}e^{-Y_{C}}$. This
will be used to introduce the rapidity parameter $\zeta$ later on (see (61)
and (125)).

It is easy to repeat the above arguments with four different renormalisation
scales for the four parton legs of $F$. One then obtains analogues of (3.4)
for each scale, with $\gamma_{K,a}$ replaced by $\gamma_{K,a}/2$ and
$\gamma_{F,a}$ by $\gamma_{E,a}$ or $(\gamma_{E,a})^{*}$, where
$\gamma_{E,a}+(\gamma_{E,a})^{*}=\gamma_{F,a}$. The anomalous dimension
$\gamma_{E,a}$ has an imaginary part and is for a parton momentum leaving $F$,
whereas $(\gamma_{E,a})^{*}$ is for a parton momentum entering $F$. This can
be seen in the study of the quark form factor Collins:2017oxh  (see section
10.12.2 of Collins:2011zzd  for an explicit calculation at leading order).

###  3.5 Definition of single parton TMDs

It is instructive to revisit the definition of single parton TMDs in the
framework we have just laid out. The colour structure is considerably
simplified in this case. The operator $O_{a}(x,\boldsymbol{0},\boldsymbol{z})$
in the matrix element for an unsubtracted TMD in (3.1) must carry colour
singlet quantum numbers, which is achieved by contracting the operators in
(20) with $\delta_{jj^{\prime}}$ or $\delta_{aa^{\prime}}$. Likewise, the soft
factor for single hard scattering involves the colour singlet projection of
the operator $O_{S,a}(\boldsymbol{0},\boldsymbol{z})$ defined in (3.2). One
can easily adapt the arguments in the previous subsections to this case: the
matrices $S$, $\widehat{S}$, $s$, $K$ and $\widehat{K}$ then become single
real valued functions. One finds $K=\widehat{K}$, whilst the definition (331)
simplifies to

$\displaystyle s(Y)$ $\displaystyle=e^{YK}\,\widehat{S}^{\,1/2}(0)\,,$ (56)

which is positive and satisfies (35). This gives

$\displaystyle
S(Y)=\widehat{S}(Y)=e^{YK}\,\widehat{S}(0)=s(Y-Y_{0})\,s(Y_{0})$ (57)

for $Y\gg 1$ and arbitrary $Y_{0}$. The ambiguity in the choice of $s$, due to
unitary transformations in the matrix case, is no longer present. Combining
the two previous equations, we deduce that

$\displaystyle s(Y)=\sqrt{S(2Y)}\,.$ (58)

The square-root factor in the construction of single parton TMDs by Collins,
given in equation (13.106) of Collins:2011zzd , can be rewritten as

$\displaystyle\sqrt{\frac{S(Y_{R}-Y_{C})}{S(Y_{R}-Y_{L})\,S(Y_{C}-Y_{L})}}$
$\displaystyle=s^{-1}(Y_{C}-Y_{L})$ (59)

using (57) and (58). The final TMD is then given by

$\displaystyle f_{a}(Y_{C})$
$\displaystyle=\lim_{Y_{L}\to-\infty}s_{a}^{-1}(Y_{C}-Y_{L})\,f_{\text{us},a}(Y_{L})\,,$
(60)

where we have restored the parton label $a$ to denote quarks, antiquarks and
gluons. This form closely resembles the definition in Echevarria:2015byo ;
GarciaEchevarria:2011rb ; we will expand on this further in appendix B. A
related discussion has been given in Collins:2012uy , but the separation into
different factors made there differs from ours.

The renormalisation of single parton TMDs is done with the same factors
$Z_{\text{us},a}$, $Z_{S,a}$ and $Z_{F,a}$ as in section 3.4, and hence
involves the same anomalous dimensions. Coming back to our remark after (55)
and defining

$\displaystyle\zeta$ $\displaystyle=2\mskip 1.5mu(xp^{+})^{2}\,e^{-2Y_{C}}$
(61)

for a right-moving proton, we recover the familiar evolution equations given
in (3) to (8).

##  4 Colour

An essential element in the description of DPS is the colour structure, which
is much more involved than in single hard scattering. To deal efficiently with
matrices in colour space, we introduce projection operators and make use of
the fact that certain colour indices must couple to an overall colour singlet.
We derive the combination (85) of colour projected quantities that enters the
DPS cross section in section 4.2 and establish a number of symmetry relations
in section 4.3. In sections 4.4 and 4.5 we consider the case of collinear
factorisation. Projector identities such as the one in figure 2 lead to
dramatic simplifications in this case, with soft matrices that are diagonal
and depend on only three independent functions as specified in (108) to (110).
These simplifications also allow us to derive the structure (4.5) of the DPS
cross section for produced particles carrying colour. A reader who is not
interested in technical details may skip the following derivations and will
find the principal results in the equations just mentioned.

###  4.1 Projection operators

For the fundamental representation of the group SU($N$), we introduce the
operators

$\displaystyle P_{1}^{ii^{\prime}\,jj^{\prime}}$
$\displaystyle=\frac{1}{N}\mskip
1.5mu\delta_{ii^{\prime}}\delta_{jj^{\prime}}\,,$ $\displaystyle
P_{8}^{ii^{\prime}\,jj^{\prime}}$ $\displaystyle=2\mskip
1.5mut^{a}_{ii^{\prime}}t^{a}_{jj^{\prime}}\,,$ (62)

which project the index pairs $ii^{\prime}$ and $jj^{\prime}$ onto a colour
singlet or a colour octet, respectively.333We use the term “octet” to denote
the adjoint representation, which of course has dimension 8 only for SU(3). In
these and all following projection operators, the four indices are coupled to
an overall colour singlet. We take $T_{F}=1/2$ throughout this paper and do
not distinguish between upper and lower colour indices. We will make repeated
use of the colour Fierz identity

$\displaystyle 2\mskip 1.5mut^{a}_{ii^{\prime}}t^{a}_{jj^{\prime}}$
$\displaystyle=\delta_{ij^{\prime}}\delta_{i^{\prime}j}-\frac{1}{N}\mskip
1.5mu\delta_{ii^{\prime}}\delta_{jj^{\prime}}\,.$ (63)

For the adjoint representation we use Bartels:1993ih ; Mekhfi:1985dv ;
Keppeler:2012ih

$\displaystyle P_{1}^{aa^{\prime}\,bb^{\prime}}$
$\displaystyle=\frac{1}{N^{2}-1}\mskip
1.5mu\delta^{aa^{\prime}}\delta^{bb^{\prime}}\,,$ $\displaystyle
P_{A}^{aa^{\prime}\,bb^{\prime}}$ $\displaystyle=\frac{1}{N}\mskip
1.5muf^{aa^{\prime}c}f^{bb^{\prime}c}\,,$ $\displaystyle
P_{S}^{aa^{\prime}\,bb^{\prime}}$ $\displaystyle=\frac{N}{N^{2}-4}\mskip
1.5mud^{aa^{\prime}c}d^{bb^{\prime}c}\,,$ $\displaystyle
P_{D}^{aa^{\prime}\,bb^{\prime}}$ $\displaystyle=\frac{1}{2}\mskip
1.5mu\bigl{(}\delta^{ab}\delta^{a^{\prime}b^{\prime}}-\delta^{ab^{\prime}}\delta^{a^{\prime}b}\bigr{)}-P_{A}^{aa^{\prime}\,bb^{\prime}}\,,$
$\displaystyle P_{27}^{aa^{\prime}\,bb^{\prime}}$
$\displaystyle=\frac{1}{2}\mskip
1.5mu\bigl{(}\delta^{ab}\delta^{a^{\prime}b^{\prime}}+\delta^{ab^{\prime}}\delta^{a^{\prime}b}\bigr{)}-P_{S}^{aa^{\prime}\,bb^{\prime}}-P_{1}^{aa^{\prime}\,bb^{\prime}}\,.$
(64)

The subscript on $P$ always denotes the colour representation onto which index
pairs $aa^{\prime}$ and $bb^{\prime}$ are projected, with $A$ denoting the
antisymmetric and $S$ the symmetric octet.444For $N>3$ a further
representation appears when decomposing the product of two octets (see table
9.4 in Cvitanovic:2008zz ). This representation is sometimes labelled as
$R=0$, since it has dimension $0$ for $N=3$. The projector $P_{27}$ in (4.1)
corresponds to $P_{27}+P_{0}$ in the case of general $N$ Keppeler:2012ih .
Useful relations for calculating with the $f$ and $d$ tensors are given in
MacFarlane:1968vc .

The operator $P_{D}=P_{10}+P_{\;\smash{\overline{10}}}$ projects on the sum of
the decuplet and antidecuplet representations. For any tensor
$M^{aa^{\prime}bb^{\prime}}$ that transforms as an overall colour singlet, the
projections on decuplet and antidecuplet are equal (and can hence be added
without loss of information). Following appendix A of Kasemets:2014yna , one
can show this by first decomposing $M^{aa^{\prime}bb^{\prime}}$ on
$P_{R}^{ab\,a^{\prime}b^{\prime}}$ with $R=1,A,S,10,\overline{10},27$ and then
projecting on $P_{R}^{aa^{\prime}\,bb^{\prime}}$. This result can be traced
back to the fact that $(P_{10})^{\mskip
1.5muaa^{\prime}\,bb^{\prime}}-(P_{\;\smash{\overline{10}}})^{\mskip
1.5muaa^{\prime}\,bb^{\prime}}$ is odd but all
$P_{R}^{ab\,a^{\prime}b^{\prime}}$ are even under the simultaneous exchange
$(a\leftrightarrow b,a^{\prime}\leftrightarrow b^{\prime})$.

We will further need projectors for mixed fundamental and adjoint indices:

$\displaystyle P_{1}^{ii^{\prime}\,aa^{\prime}}$
$\displaystyle=P_{1}^{aa^{\prime}\,ii^{\prime}}=\frac{1}{\sqrt{N(N^{2}-1)}}\,\delta_{ii^{\prime}}\delta^{aa^{\prime}}\,,$
$\displaystyle P_{A}^{ii^{\prime}\,aa^{\prime}}$
$\displaystyle=P_{A}^{aa^{\prime}\,ii^{\prime}}=\sqrt{\frac{2}{N}}\,t^{c}_{ii^{\prime}}\mskip
1.5muf^{aa^{\prime}c}\,,$ $\displaystyle P_{S}^{ii^{\prime}\,aa^{\prime}}$
$\displaystyle=P_{S}^{aa^{\prime}\,ii^{\prime}}=\sqrt{\frac{2N}{N^{2}-4}}\,t^{c}_{ii^{\prime}}\mskip
1.5mud^{aa^{\prime}c}\,.$ (65)

The normalisation factors are chosen such as to yield the simple symmetry and
projection properties to be discussed next.

To ease our notation, we introduce double indices
$\underline{i}=(ii^{\prime})$, $\underline{a}=(aa^{\prime})$. Here and in the
following, $\underline{i},\underline{j},\underline{k}$ are in the fundamental
representation, $\underline{a},\underline{b},\underline{c}$ are in the
adjoint, whereas
$\underline{r},\underline{s},\underline{t},\underline{u},\underline{v},\underline{w}$
can belong to either representation. Repeated double indices are summed over,
so that for adjoint indices we have

$\displaystyle
M_{1}^{\underline{r}\,\underline{a}}\,M_{2}^{\underline{a}\,\underline{s}}$
$\displaystyle=M_{1}^{\underline{r}\,aa^{\prime}}M_{2}^{aa^{\prime}\,\underline{s}}\,.$
(66)

Some care is required for indices of the fundamental representation. In
fundamental Wilson lines $W$ and in projection operators $P_{R}$, the first
index always transforms as a triplet and the second index as an antitriplet.
For such cases, we define matrix multiplication with a transposition

$\displaystyle
M_{1}^{\underline{r}\,\underline{i}}\,M_{2}^{\underline{i}\,\underline{s}}$
$\displaystyle=M_{1}^{\underline{r}\,ii^{\prime}}M_{2}^{i^{\prime}\\!i\,\underline{s}}\,.$
(67)

In soft matrices $S$ and in the matrix elements giving DPDs, the order of
triplet and antitriplet indices depends on the parton channel ($q$ or
$\bar{q}$) and in $S$ also on the direction of the Wilson lines (along $v_{L}$
or $v_{R}$). Contraction of double indices should always be done such that a
triplet index is contracted with an antitriplet one. This ensures proper
behaviour of the result under gauge transformations.

With these definitions, the projection operators have the symmetry

$\displaystyle
P_{R}^{\underline{r}\,\underline{s}}=P_{R}^{\underline{s}\,\underline{r}}$
(68)

and the projection property

$\displaystyle
P_{R}^{\underline{r}\,\underline{s}}\,P_{R^{\prime}}^{\underline{s}\,\underline{t}}$
$\displaystyle=\delta_{RR^{\prime}}\,P_{R}^{\underline{r}\,\underline{t}}$
(69)

for any pair of projectors $P_{R},P_{R^{\prime}}$ with a common double index
$\underline{s}$ in either the fundamental or the adjoint representation. In
the octet sector, it is understood that
$\delta_{8A}=\delta_{A8}=\delta_{8S}=\delta_{S8}=1$ but
$\delta_{AS}=\delta_{SA}=0$ in (69). The normalisation is given by

$\displaystyle
P_{R}^{\underline{r}\,\underline{s}}\,P_{R^{\prime}}^{\underline{r}\,\underline{s}}$
$\displaystyle=\delta_{RR^{\prime}}\,m(R)\,,$ (70)

where

$\displaystyle m(R)=P_{R}^{\underline{r}\,\underline{r}}$ (71)

is the multiplicity of the representation, i.e. $m(1)=1$, $m(8)=m(A)=m(S)=8$,
$m(D)=20$ and $m(27)=27$ for SU(3). Since the projectors above form a basis of
the space of rank-four tensors $M$ that transform as an SU(3) singlet, any
such tensor $M$ can be decomposed as

$\displaystyle M^{\underline{r}\,\underline{s}}$
$\displaystyle=\sum_{R}\frac{1}{m(R)}\,P_{R}^{\underline{r}\,\underline{s}}\,\bigl{(}P_{R}^{\underline{t}\,\underline{u}}M^{\underline{t}\,\underline{u}}_{\phantom{1}}\bigr{)}\,,$
(72)

where the sum is over all representations $R$ given in (4.1), (4.1) and (4.1),
as applicable. The contraction of two colour singlet tensors is then given by

$\displaystyle
M_{1}^{\underline{r}\,\underline{s}}\,M_{2}^{\underline{r}\,\underline{s}}$
$\displaystyle=\sum_{R}\frac{1}{m(R)}\,\bigl{(}P_{R}^{\underline{r}\,\underline{s}}M_{1}^{\underline{r}\,\underline{s}}\bigr{)}\,\bigr{(}P_{R}^{\underline{t}\,\underline{u}}M_{2}^{\underline{t}\,\underline{u}}\bigr{)}\,.$
(73)

For representation labels $R$, we will _not_ use the summation convention,
i.e. summation over $R$ will always be indicated explicitly.

###  4.2 Colour structure of the DPS cross section

The projection operators just introduced allow us to rewrite the DPS cross
section in a compact way. We start by defining DPDs that are projected on a
definite colour representation $R$. Using the general decomposition in (72),
we can write

$\displaystyle F_{a_{1}a_{2}}^{\underline{r}_{1}\underline{r}_{2}}$
$\displaystyle=\sum_{R}\frac{1}{\varepsilon_{a_{1}}(R)\,\varepsilon_{a_{2}}(R)}\,\frac{1}{\mathcal{N}_{a_{1}}\mathcal{N}_{a_{2}}}\,\frac{1}{\sqrt{m(R)}}\;{}^{R}\mskip-1.5muF_{a_{1}a_{2}}\,P_{R}^{\underline{r}_{1}\underline{r}_{2}}$
(74)

with

$\displaystyle{}^{R}\mskip-1.5muF_{a_{1}a_{2}}$
$\displaystyle=\varepsilon_{a_{1}}(R)\,\varepsilon_{a_{2}}(R)\;\mathcal{N}_{a_{1}}\mathcal{N}_{a_{2}}\,\frac{1}{\sqrt{m(R)}}\;P_{R}^{\underline{s}_{1}\underline{s}_{2}}F_{a_{1}a_{2}\phantom{R}}^{\underline{s}_{1}\underline{s}_{2}}\\!\\!.$
(75)

Note that the lower indices $a_{1},a_{2}$ denote the parton species and
polarisation, not colour. The factors

$\displaystyle\varepsilon_{a}(R)$ $\displaystyle=\begin{cases}\mskip
1.5mui&\text{if $a$ is a gluon and $R=A$}\\\ \mskip
1.5mu1&\text{otherwise}\end{cases}$ (76)

ensure that the collinear distributions
${}^{A}\mskip-1.5muF_{qg}(x_{i},\boldsymbol{y})$ and
${}^{A}\mskip-1.5muF_{\bar{q}g}(x_{i},\boldsymbol{y})$ and their polarised
counterparts are real valued (rather than imaginary), as are the collinear
distributions in all other channels. This is shown in the next subsection. The
prefactors $\mathcal{N}_{a}$ are given by

$\displaystyle\mathcal{N}_{q}=\mathcal{N}_{\bar{q}}$
$\displaystyle=\sqrt{N}\,,$ $\displaystyle\mathcal{N}_{g}$
$\displaystyle=\sqrt{N^{2}-1}$ (77)

and likewise for polarised partons. They ensure that colour singlet
distributions ${}^{1}F_{a_{1}a_{2}}$ involve a _sum_ over the colours of the
two partons. The definitions here coincide with the ones in Diehl:2011yj ;
Kasemets:2014yna , with the notational change

$\displaystyle{}^{D}\mskip-1.5muF_{gg}\mskip 1.5mu\big{|}_{\text{here}}$
$\displaystyle={}^{10+\overline{10}}\mskip-1.5muF_{gg}\mskip
1.5mu\big{|}_{\text{Ref.\leavevmode\nobreak\ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Kasemets:2014yna}{\@@citephrase{(}}{\@@citephrase{)}}}}}=\frac{1}{\sqrt{2}}\mskip
1.5mu\bigl{(}{}^{10}\mskip-1.5muF_{gg}+{}^{\overline{10}}\mskip-1.5muF_{gg}\bigr{)}\mskip
1.5mu\big{|}_{\text{Ref.\leavevmode\nobreak\ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Diehl:2011yj}{\@@citephrase{(}}{\@@citephrase{)}}}}}$ (78)

and its analogues for polarised gluons. For the discussion of renormalisation
and the short-distance expansion, it is useful to project the partonic
operators (21) on definite colour representations as well. Introducing

$\displaystyle{}^{R}O^{\mskip 1.5mu\underline{r}}_{a\rule{0.0pt}{3.46594pt}}$
$\displaystyle=\varepsilon_{a}(R)\;\mathcal{N}_{a}\,P_{R}^{\underline{r}\underline{s}}\,O^{\mskip
1.5mu\underline{s}}_{a\rule{0.0pt}{3.46594pt}}$ (79)

and defining unsubtracted DPDs ${}^{R}\mskip-1.5muF_{\text{us},a_{1}a_{2}}$ in
analogy to (75), we obtain

$\displaystyle
2\pi\delta(p^{+}-p^{\prime+})\,2p^{+}\,{}^{R}\mskip-1.5muF_{\text{us},a_{1}a_{2}}$
$\displaystyle=\frac{1}{\sqrt{m(R)}}\,\big{\langle}p^{\prime}\big{|}\,{}^{R}O^{\mskip
1.5mu\underline{r}}_{a_{1}}\;{}^{R}O^{\mskip
1.5mu\underline{r}}_{a_{2}}\,\big{|}p\big{\rangle}$ (80)

from (3.1) and the projection property (69).

The soft factor for DPS producing colour singlet particles in the hard
interactions carries two times four indices. Its projection on different
representations is defined by

$\displaystyle{}^{RR^{\prime}}\\!S_{a_{1}a_{2}}$
$\displaystyle=\frac{\varepsilon_{a_{1}}(R)\,\varepsilon_{a_{2}}(R)}{\varepsilon_{a_{1}}(R^{\prime})\,\varepsilon_{a_{2}}(R^{\prime})}\frac{1}{\sqrt{m(R)\,m(R^{\prime})}}\,P_{R}^{\underline{r}_{1}\underline{r}_{2}}\,S_{a_{1}a_{2}\phantom{R}}^{\underline{r}_{1}\underline{r}_{2},\underline{s}_{1}\underline{s}_{2}}\,P_{R^{\prime}}^{\underline{s}_{1}\underline{s}_{2}}\,.$
(81)

$R$ specifies the colour representation for the Wilson lines with positive
rapidity, and $R^{\prime}$ the one for the Wilson lines with negative
rapidity. Notice that the $\varepsilon$ factors cancel for $R=R^{\prime}$. In
analogy to (81) we define projections ${}^{RR^{\prime}}\\!s$ and
${}^{RR^{\prime}}\\!K$ of the matrix $s$ and of the Collins-Soper kernel $K$,
as well as projections ${}^{RR^{\prime}}(S^{-1})$ and
${}^{RR^{\prime}}(s^{-1})$ of the inverse matrices $S^{-1}$ and $s^{-1}$ in
colour space. Using the same argument as in (73), one can show that

$\displaystyle\sum_{R^{\prime}}{}^{RR^{\prime}}\mskip-1.5mu(S_{a_{1}a_{2}}^{-1})\;{}^{R^{\prime}R^{\prime\prime}}\\!S_{a_{1}a_{2}}=\sum_{R^{\prime}}{}^{RR^{\prime}}\mskip-1.5mu(s_{a_{1}a_{2}}^{-1})\;{}^{R^{\prime}R^{\prime\prime}}\\!s_{a_{1}a_{2}}$
$\displaystyle=\delta_{RR^{\prime\prime}}\,.$ (82)

We see that matrix multiplication in the space of four colour indices
$(\underline{r}_{1}\underline{r}_{2})=(r_{1}r_{1}^{\prime}r_{2}r_{2}^{\prime})$
turns into matrix multiplication in the space of colour representations $R$.
The matrix decomposition (36) can thus be rewritten as

$\displaystyle{}^{RR^{\prime\prime}}\mskip-1.5muS_{a_{1}a_{2}}(Y)$
$\displaystyle=\sum_{R^{\prime}}{}^{RR^{\prime}}\mskip-1.5mus_{a_{1}a_{2}}(Y-Y_{0})\;{}^{R^{\prime}R^{\prime\prime}}\\!\bigl{(}s_{a_{1}a_{2}}^{\dagger}(Y_{0})\bigr{)}\,,$
(83)

and the definition (3.3) of DPDs in a right-moving proton as

$\displaystyle{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(Y_{C})$
$\displaystyle=\lim_{Y_{L}\to-\infty}\sum_{R^{\prime}}{}^{RR^{\prime}}\mskip-1.5mu\bigl{(}s_{a_{1}a_{2}}^{-1}(Y_{C}-Y_{L})\bigr{)}\;{}^{R^{\prime}}\\!F_{\text{us},a_{1}a_{2}}(Y_{L}).$
(84)

According to (12) and (42), the cross section for the production of colour
neutral particles involves the product

$\displaystyle X$
$\displaystyle=H_{a_{1}b_{1}}\,H_{a_{2}b_{2}}\,F_{b_{1}b_{2}}^{\underline{r}_{1}\underline{r}_{2}}\,F_{a_{1}a_{2}\phantom{b}}^{\underline{r}_{1}\underline{r}_{2}}=\frac{H_{a_{1}b_{1}}}{\mathcal{N}_{a_{1}}\mskip
1.5mu\mathcal{N}_{b_{1}}}\,\frac{H_{a_{2}b_{2}}}{\mathcal{N}_{a_{2}}\mskip
1.5mu\mathcal{N}_{b_{2}}}\,\sum_{R}\eta_{a_{1}a_{2}}(R)\;{}^{R}\mskip-1.5muF_{b_{1}b_{2}}{}^{R}\mskip-1.5muF_{a_{1}a_{2}}\,.$
(85)

Here we have combined the factors
$\varepsilon_{a_{1}}(R)\,\varepsilon_{b_{1}}(R)\,\varepsilon_{a_{2}}(R)\,\varepsilon_{b_{2}}(R)$
into

$\displaystyle\eta_{a_{1}a_{2}}(R)$
$\displaystyle=\frac{1}{\varepsilon^{2}_{a_{1}}(R)\,\varepsilon^{2}_{a_{2}}(R)}\,,$
(86)

using that the only channels that produce colourless particles are $q\bar{q}$
and $gg$ annihilation. Multiplied with a flux factor, the combinations
$H_{ab}/(\mathcal{N}_{a}\mskip 1.5mu\mathcal{N}_{b})$ turn into the hard-
scattering cross sections in the final factorisation formula. The factor
$\mathcal{N}_{a}\mskip 1.5mu\mathcal{N}_{b}$ is $N$ for $q\bar{q}$
annihilation and $N^{2}-1$ for $gg$ annihilation and ensures that one takes an
_average_ over the colour states of the two colliding partons, as is
appropriate for a parton-level cross section.

###  4.3 Symmetry properties

As already remarked, multiplication of matrices and vectors, $S$ and $F$, in
the space of colour indices $(\underline{r}_{1}\underline{r}_{2})$ turns into
multiplication of matrices and vectors in the space of representations $R$.
For brevity, we refer to the former as “colour space” and to the latter as
“representation space”. Note that our representation space only corresponds to
the part of colour space in which the indices
$(\underline{r}_{1}\underline{r}_{2})$ are coupled to an overall colour
singlet. We now derive some important properties of $S$ and of the related
matrices $s$ and $K$ in representation space.

In (3) we have seen that the matrix $S_{a_{1}a_{2}}$ is Hermitian in colour
space,

$\displaystyle
S_{a_{1}a_{2}}^{\underline{r}_{1}\underline{r}_{2},\underline{s}_{1}\underline{s}_{2}}$
$\displaystyle=\bigl{(}S_{a_{1}a_{2}}^{\underline{s}_{1}\underline{s}_{2},\underline{r}_{1}\underline{r}_{2}}\bigr{)}^{*}\,.$
(87)

Under Hermitian conjugation, an index pair $\underline{i}=(i,i^{\prime})$ in
$S$ changes its transformation properties: if $i$ transforms as an SU(3)
triplet in $S$, it transforms as an antitriplet in $S^{\dagger}$ and vice
versa. The colour projectors satisfy
$P_{R}^{ii^{\prime}\,jj^{\prime}}=(P_{R}^{i^{\prime}\\!i\,j^{\prime}\\!j})^{*}$,
$P_{R}^{ii^{\prime}\,aa^{\prime}}=(P_{R}^{i^{\prime}\\!i\,aa^{\prime}})^{*}$
and $P_{R}^{aa^{\prime}\,bb^{\prime}}=(P_{R}^{aa^{\prime}\,bb^{\prime}})^{*}$,
so that the rule for contraction with the colour projectors stated below (67)
is satisfied if $S$ is contracted with $P_{R}$ and $S^{\dagger}$ with
$P_{R}^{*}$. We thus find that

$\displaystyle{}^{RR^{\prime}}\\!S_{a_{1}a_{2}}$
$\displaystyle=\left({}^{R^{\prime}\\!R}\mskip-1.5muS_{a_{1}a_{2}}\right)^{*}\,,$
(88)

i.e. $S$ is also Hermitian in representation space. In analogy, the fact that
the Collins-Soper kernel $K_{a_{1}a_{2}}$ is Hermitian in colour space implies
that it is Hermitian in representation space as well.

We now turn to the relations between soft factors for quarks and antiquarks.
The relations (25) imply that

$\displaystyle{}^{RR^{\prime}}\\!S_{a_{1}a_{2}}$
$\displaystyle=\frac{\eta_{a_{1}a_{2}}(R^{\prime})}{\eta_{a_{1}a_{2}}(R)}\,\left({}^{RR^{\prime}}\\!S_{\bar{a}_{1}\bar{a}_{2}}\right)^{*}=\frac{\eta_{a_{1}a_{2}}(R^{\prime})}{\eta_{a_{1}a_{2}}(R)}\;{}^{R^{\prime}\\!R}\mskip-1.5muS_{\bar{a}_{1}\bar{a}_{2}}\,,$
(89)

where $\bar{a}$ denotes the antiparton of $a$ (with the convention
$\bar{g}=g$). The complex conjugation of projection operators implicit in the
first step corresponds to the fact that an index $i$ transforming as a triplet
in $S_{a_{1}a_{2}}$ transforms as an antitriplet in
$S_{\bar{a}_{1}\bar{a}_{2}}$ and vice versa. Complex conjugation of the
$\varepsilon$ factors in the definition (81) gives the factors $\eta$ in (89).
In the second step we have simply used the property (88). We now recall that
the relations (25) for $S$ translate into analogous relations (41) for $s$. By
taking the rapidity derivative, we find corresponding relations for $K$ as
well. In analogy to (89) we can thus derive

$\displaystyle{}^{RR^{\prime}}\\!K_{a_{1}a_{2}}$
$\displaystyle=\frac{\eta_{a_{1}a_{2}}(R^{\prime})}{\eta_{a_{1}a_{2}}(R)}\;{}^{R^{\prime}\\!R}\mskip-1.5muK_{\bar{a}_{1}\bar{a}_{2}}\,.$
(90)

We will use this in section 5.1 when making rapidity evolution explicit in the
DPS cross section. A different way to connect soft factors for quarks and
antiquarks is to note that the operator $O_{S,q}$ in (3.2) is related with its
analogue $O_{S,\bar{q}}$ for antiquarks by

$\displaystyle\bigl{[}O_{S,q}(\boldsymbol{y},\boldsymbol{z};v_{L},v_{R})\bigr{]}^{ii^{\prime},ll^{\prime}}$
$\displaystyle=\bigl{[}O_{S,\bar{q}}(\boldsymbol{y},-\boldsymbol{z};v_{L},v_{R})\bigr{]}^{i^{\prime}\\!i,\,l^{\prime}\mskip-1.5mul}\,.$
(91)

Multiplying with colour projectors according to the rule stated below (67),
one obtains

$\displaystyle{}^{RR^{\prime}}\\!S_{qq}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y})$
$\displaystyle={}^{RR^{\prime}}\\!S_{q\bar{q}}(\boldsymbol{z}_{1},-\boldsymbol{z}_{2},\boldsymbol{y})$
$\displaystyle={}^{RR^{\prime}}\\!S_{\bar{q}q}(-\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y})={}^{RR^{\prime}}\\!S_{\bar{q}\bar{q}}(-\boldsymbol{z}_{1},-\boldsymbol{z}_{2},\boldsymbol{y})\,,$
(92)

where the arguments $v_{L},v_{R}$ are the same in all four expressions.
Relations analogous to (4.3) hold for ${}^{RR^{\prime}}\\!K_{a_{1}a_{2}}$,
provided that ${}^{RR^{\prime}}\mskip-1.5muU_{a_{1}a_{2}}$ satisfies the
requirements (A).

Let us finally investigate the consequences of charge conjugation invariance.
Charge conjugation transforms $t^{a}A^{\mu a}(\xi)\to-(t^{a})^{*}A^{\mu
a}(\xi)$, which can be derived from the familiar transformation properties of
quark fields and charge conjugation invariance of the interaction term
$-g\mskip 1.5mu\bar{q}\mskip 1.5mut^{a}\mskip-1.5muA^{\mu a}\gamma_{\mu}\mskip
1.5muq$ in the QCD Lagrangian. As a result, a fundamental Wilson line
transforms as

$\displaystyle W_{ij}\to W_{ij}^{*}\,.$ (93)

For soft factors ${}^{RR^{\prime}}\\!S$ involving gluons, we rewrite all
adjoint Wilson lines as Bomhof:2006dp

$\displaystyle W^{ab}=2\operatorname{tr}(t^{a}Wt^{b}W^{\dagger})\,,$ (94)

where the Wilson lines on the r.h.s. are in the fundamental representation,
express $f^{abc}$ and $d^{abc}$ as traces of products of matrices $t^{a}$,
$t^{b}$, $t^{c}$ and then eliminate all these matrices by repeated use of the
colour Fierz identity (63). After these steps, we have for all parton
combinations $a_{1}$ and $a_{2}$ a representation of
${}^{RR^{\prime}}\\!S_{a_{1}a_{2}}$ in terms of traces of fundamental Wilson
lines $W$ and $W^{\dagger}$, where thanks to the $\epsilon$ factors in the
definition (81) no explicit factors of $i$ appear. The transformation (93)
then readily implies that ${}^{RR^{\prime}}\\!S_{a_{1}a_{2}}$ is real valued,
and with (88) we furthermore obtain that it is symmetric in representation
space, i.e. we have ${}^{RR^{\prime}}\\!S={}^{R^{\prime}\\!R}\mskip-1.5muS$,
where for brevity we omit the subscripts $a_{1}a_{2}$.

To show that ${}^{RR^{\prime}}\\!s$ and ${}^{RR^{\prime}}\\!K$ are also real
valued, we translate the derivation starting in (327) into representation
space, projecting all relevant matrices as specified in (81). It is then easy
to see that ${}^{RR^{\prime}}\\!\widehat{S}$ and
${}^{RR^{\prime}}\\!\widehat{K}$ are real valued. To establish that
${}^{RR^{\prime}}\\!(\widehat{S}^{\,1/2})$ is real valued, we note that
${}^{RR^{\prime}}\\!\widehat{S}$ has positive eigenvalues, which is readily
seen by projecting the eigenvalue equation $\widehat{S}v=\lambda v$ from
colour space to representation space and recalling that in colour space
$\widehat{S}$ is a positive matrix. According to the discussion below (330),
the square root of a positive real matrix is real. Choosing the matrix $U$ in
(331) and (332) such that ${}^{RR^{\prime}}\mskip-1.5muU$ is real, we find
that both ${}^{RR^{\prime}}\\!s$ and ${}^{RR^{\prime}}\\!K$ are real. Since
${}^{RR^{\prime}}\\!K$ is Hermitian, it is also symmetric:
${}^{RR^{\prime}}\\!K={}^{R^{\prime}\\!R}\mskip-1.5muK$.

Combining the symmetry relations (89) and (4.3) with the fact that
${}^{RR^{\prime}}\\!S$ is symmetric in $R$ and $R^{\prime}$, we obtain

$\displaystyle{}^{RR^{\prime}}\\!S_{qq}(\boldsymbol{z}_{1},\boldsymbol{z}_{2},\boldsymbol{y})$
$\displaystyle={}^{RR^{\prime}}\\!S_{qq}(-\boldsymbol{z}_{1},-\boldsymbol{z}_{2},\boldsymbol{y})\,.$
(95)

An analogous relation holds for ${}^{RR^{\prime}}\\!K_{qq}$.

The quark and gluon operators in (14) and (17) satisfy
$\bigl{[}\mathcal{O}_{a}^{rr^{\prime}}\bigr{]}^{\dagger}(y,z)=\mathcal{O}_{a}^{r^{\prime}\\!r}(y,-z)$,
where the colour index $r$ pertains to the fields at
$y+{\textstyle\frac{1}{2}}z$ and $r^{\prime}$ to the fields at
$y-{\textstyle\frac{1}{2}}z$. For colour projected unsubtracted DPDFs this
yields the relation

$\displaystyle\bigl{[}{}^{R}\mskip-1.5muF_{\text{us},a_{1}a_{2}}(x_{i},\boldsymbol{y})\bigr{]}^{*}$
$\displaystyle={}^{R}\mskip-1.5muF_{\text{us},a_{1}a_{2}}(x_{i},\boldsymbol{y})\,.$
(96)

Let us explicitly show this for ${}^{A}\mskip-1.5muF_{\text{us},qg}$. Using
that $\varepsilon_{q}(A)\mskip 1.5mu\varepsilon_{g}(A)=i$ and
$(P_{A}^{jj^{\prime}\,aa^{\prime}})^{*}=-P_{A}^{j^{\prime}\\!j\;a^{\prime}\\!a}$,
we have

$\displaystyle\bigl{[}{}^{A}\mskip-1.5muF_{\text{us},qg}(x_{i},\boldsymbol{y})\bigr{]}^{*}$
$\displaystyle\qquad=c\int\\!dz^{-}_{1}\,dz^{-}_{2}\,dy^{-}\,\Bigl{[}\mskip
1.5mui\mskip 1.5muP_{A}^{jj^{\prime}\,aa^{\prime}}e^{i\mskip
1.5mu(x_{1}z_{1}^{-}+x_{2}z_{2}^{-})\mskip 1.5mup^{+}}\,\langle
p|\,\mathcal{O}^{jj^{\prime}}_{q}(y,z_{1})\,\mathcal{O}^{aa^{\prime}}_{g}(0,z_{2})\,|p\rangle\Bigr{]}^{*}$
$\displaystyle\qquad=c\int\\!dz^{-}_{1}\,dz^{-}_{2}\,dy^{-}\,i\mskip
1.5muP_{A}^{j^{\prime}\\!j\;a^{\prime}\\!a}\,e^{-i\mskip
1.5mu(x_{1}z_{1}^{-}+x_{2}z_{2}^{-})\mskip 1.5mup^{+}}\,\langle
p|\,\mathcal{O}^{j^{\prime}\\!j}_{q}(y,-z_{1})\,\mathcal{O}^{a^{\prime}\\!a}_{g}(0,-z_{2})\,|p\rangle$
$\displaystyle\qquad={}^{A}\mskip-1.5muF_{\text{us},qg}(x_{i},\boldsymbol{y})\,,\phantom{\int}$
(97)

where in the last step we used that
$\boldsymbol{z}_{1}=\boldsymbol{z}_{2}=\boldsymbol{0}$. Here $c=c^{*}$ is a
product of kinematic and numerical factors not essential for the argument. We
see that the factor $\varepsilon_{g}(A)=i$ compensates the behaviour of
$P_{A}^{jj^{\prime}\,aa^{\prime}}$ under complex conjugation. One easily
checks that $\bigl{[}\varepsilon_{a_{1}}(R)\mskip
1.5mu\varepsilon_{a_{2}}(R)\,P_{R}^{rr^{\prime}\,ss^{\prime}}\bigr{]}^{*}=\varepsilon_{a_{1}}(R)\mskip
1.5mu\varepsilon_{a_{2}}(R)\,P_{R}^{r^{\prime}\mskip-1.5mur\;s^{\prime}\\!s}$
for all projectors, which ensures the general validity of (96). Given that
${}^{RR^{\prime}}\\!s$ is real valued, we finally find that

$\displaystyle\Bigl{[}\mskip
1.5mu{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{y})\mskip
1.5mu\Bigr{]}^{*}$
$\displaystyle={}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{y})\,.$ (98)

In all colour channels and for all parton combinations, DPDFs are thus real
valued.

###  4.4 Simplification of soft factors in collinear factorisation

In the soft factors relevant for collinear factorisation, one should set
$\boldsymbol{z}_{1}=\boldsymbol{z}_{2}=\boldsymbol{0}$ in the definitions
(3.2) and (3.2). Corresponding Wilson lines in the scattering amplitude and
its complex conjugate are then at the same transverse position. This
simplifies the colour structure significantly, as we will now demonstrate. In
the remainder of this subsection, it is understood that all soft factors are
taken at $\boldsymbol{z}_{1}=\boldsymbol{z}_{2}=\boldsymbol{0}$.

Figure 2:  Graphical representation of the relation (99). The Wilson lines are
understood to be interacting, i.e. any number of gluons may be attached to
them in a Feynman graph.

The projection operators on Wilson lines obey the relation

$\displaystyle
W_{ij}\,P_{R}^{jj^{\prime},k^{\prime}k}\,W^{\dagger}_{j^{\prime}i^{\prime}}$
$\displaystyle=W_{jk}\,P_{R}^{ii^{\prime},j^{\prime}j}\,W^{\dagger}_{k^{\prime}j^{\prime}}$
(99)

for $R=1,8$, where it is understood that the Wilson lines $W$ and
$W^{\dagger}$ are in the same direction and at the same position. For $R=1$
this is trivial, and for $R=8$ it readily follows from the colour Fierz
identity (63). A graphical representation of (99) is shown in figure 2.
Remarkably, an analogous relation holds also for adjoint Wilson lines

$\displaystyle
W_{ab}\,P_{R}^{bb^{\prime},cc^{\prime}}\,W^{\dagger}_{b^{\prime}a^{\prime}}$
$\displaystyle=W_{bc}\,P_{R}^{aa^{\prime},bb^{\prime}}\,W^{\dagger}_{c^{\prime}b^{\prime}}$
(100)

and for the mixed case

$\displaystyle
W_{ab}\,P_{R}^{bb^{\prime},k^{\prime}k}\,W^{\dagger}_{b^{\prime}a^{\prime}}$
$\displaystyle=W_{jk}\,P_{R}^{aa^{\prime},j^{\prime}j}\,W^{\dagger}_{k^{\prime}j^{\prime}}$
(101)

in all relevant representations $R$. This can be shown by following the steps
in and after (94). For the extended soft factor, the relations (99), (100) and
(101) imply

$\displaystyle
P_{R}^{\underline{s}\,\underline{t}}\;(S_{a_{1}a_{2}})^{\cdots\,\underline{r}\,\cdots}_{\cdots\,\underline{s}\,\cdots}$
$\displaystyle=P_{R}^{\underline{r}\,\underline{s}}\;(S_{a_{1}a_{2}})^{\cdots\,\underline{s}\,\cdots}_{\cdots\,\underline{t}\,\cdots}\,,$
(102)

where we have displayed only one of the four double index pairs
$(\underline{r}_{i},\underline{s}_{i})$ in
$(S_{a_{1}a_{2}})^{\underline{r}_{1}\underline{r}_{2},\underline{r}_{3}\underline{r}_{4}}_{\underline{s}_{1}\underline{s}_{2},\underline{s}_{3}\underline{s}_{4}}$
for better readability, the other three pairs remaining untouched. Instead of
making a colour projection for the fields at $\xi^{+}=\xi^{-}=0$, one can thus
make the same projection for the fields at infinity. This yields an important
relation between a specific projection of the extended soft factor in (3.2)
and the basic one in (3.2). With (102) and the projection property (69) we
have

$\displaystyle
P_{R_{1}}^{\underline{s}_{1}\underline{t}_{1}}\,P_{R_{2}}^{\underline{s}_{2}\underline{t}_{2}}\,(S_{a_{1}a_{2}})^{\underline{r}_{1}\underline{r}_{2},\underline{u}_{1}\underline{u}_{2}}_{\underline{s}_{1}\underline{s}_{2},\underline{t}_{1}\underline{t}_{2}}$
$\displaystyle=P_{R_{1}}^{\underline{s}_{1}\underline{v}_{1}}\,P_{R_{1}}^{\underline{t}_{1}\underline{v}_{1}}\,P_{R_{2}}^{\underline{s}_{2}\underline{v}_{2}}\,P_{R_{2}}^{\underline{t}_{2}\underline{v}_{2}}\,(S_{a_{1}a_{2}})^{\underline{r}_{1}\underline{r}_{2},\underline{u}_{1}\underline{u}_{2}}_{\underline{s}_{1}\underline{s}_{2},\underline{t}_{1}\underline{t}_{2}}$
$\displaystyle=P_{R_{1}}^{\underline{r}_{1}\underline{s}_{1}}\,P_{R_{1}}^{\underline{u}_{1}\underline{t}_{1}}P_{R_{2}}^{\underline{r}_{2}\underline{s}_{2}}\,P_{R_{2}}^{\underline{u}_{2}\underline{t}_{2}}(S_{a_{1}a_{2}})^{\underline{s}_{1}\underline{s}_{2},\underline{t}_{1}\underline{t}_{2}}_{\underline{v}_{1}\underline{v}_{2},\underline{v}_{1}\underline{v}_{2}}\,,$
(103)

where $\underline{r},\underline{s},\ldots,\underline{v}$ can each be in the
fundamental or adjoint representation. Projecting this on the remaining open
indices, we obtain

$\displaystyle
P_{R}^{\underline{r}_{1}\underline{r}_{2}}\,P_{R^{\prime}}^{\underline{u}_{1}\underline{u}_{2}}\,P_{R_{1}}^{\underline{s}_{1}\underline{t}_{1}}\,P_{R_{2}}^{\underline{s}_{2}\underline{t}_{2}}\,(S_{a_{1}a_{2}})^{\underline{r}_{1}\underline{r}_{2},\underline{u}_{1}\underline{u}_{2}}_{\underline{s}_{1}\underline{s}_{2},\underline{t}_{1}\underline{t}_{2}}$
$\displaystyle=\delta_{RR_{1}}\,\delta_{RR_{2}}\,\delta_{RR^{\prime}}\,P_{R}^{\underline{s}_{1}\underline{s}_{2}}\,P_{R}^{\underline{t}_{1}\underline{t}_{2}}\,(S_{a_{1}a_{2}})^{\underline{s}_{1}\underline{s}_{2},\underline{t}_{1}\underline{t}_{2}}_{\underline{v}_{1}\underline{v}_{2},\underline{v}_{1}\underline{v}_{2}}$
$\displaystyle=\delta_{RR_{1}}\,\delta_{RR_{2}}\,\delta_{RR^{\prime}}\,m(R)\,{}^{RR}\mskip-1.5muS_{a_{1}a_{2}}\,,$
(104)

where in the last step we have used the definition (81). We furthermore have

$\displaystyle{}^{RR^{\prime}}\\!S_{a_{1}a_{2}}\,\frac{\varepsilon_{a_{1}}(R^{\prime})\,\varepsilon_{a_{2}}(R^{\prime})}{\varepsilon_{a_{1}}(R)\,\varepsilon_{a_{2}}(R)}\,\sqrt{m(R)m(R^{\prime})}$
$\displaystyle=P_{R}^{\underline{r}_{1}\underline{r}_{2}}\,P_{R^{\prime}}^{\underline{t}_{1}\underline{t}_{2}}\,(S_{a_{1}a_{2}})^{\underline{r}_{1}\underline{r}_{2},\underline{t}_{1}\underline{t}_{2}}_{\underline{s}_{1}\underline{s}_{2},\underline{s}_{1}\underline{s}_{2}}$
$\displaystyle=P_{R}^{\underline{r}_{2}\underline{s}_{2}}\,P_{R^{\prime}}^{\underline{t}_{2}\underline{s}_{2}}\,(S_{a_{1}a_{2}})^{\underline{r}_{1}\underline{r}_{1},\underline{t}_{1}\underline{t}_{1}}_{\underline{s}_{1}\underline{r}_{2},\underline{s}_{1}\underline{t}_{2}}$
$\displaystyle=\delta_{RR^{\prime}}\,P_{R}^{\underline{r}_{2}\underline{t}_{2}}\,(S_{a_{1}a_{2}})^{\underline{r}_{1}\underline{r}_{1},\underline{t}_{1}\underline{t}_{1}}_{\underline{s}_{1}\underline{r}_{2},\underline{s}_{1}\underline{t}_{2}}\,,$
(105)

which shows that at $\boldsymbol{z}_{1}=\boldsymbol{z}_{2}=\boldsymbol{0}$ the
soft matrix is diagonal in the colour representations $R$ and $R^{\prime}$.
Note that this is not the case for nonzero
$\boldsymbol{z}_{1},\boldsymbol{z}_{2}$, as follows immediately from the
explicit one-loop expressions in section 7.2.

The soft factors discussed so far correspond to the same parton species
entering a hard scattering process in the amplitude and its conjugate. As
already mentioned at the end of section 3.1, there is also interference
between $q$ and $\bar{q}$, between $q$ and $g$, and between $\bar{q}$ and $g$.
The corresponding Wilson line products in the soft factor are then
$W_{ij}W_{i^{\prime}j^{\prime}}$, $W_{ij}W_{a^{\prime}b^{\prime}}$, etc. The
relevant projection operators for coupling colour indices in these cases are
given in equations (9) and (11) of Mekhfi:1985dv . Relations analogous to (99)
can be easily derived for these cases, where again the colour projections for
the gluon fields at $\xi^{+}=\xi^{-}=0$ in the Wilson lines can be traded for
the colour projections of the fields at infinity. From this, relations
analogous to (4.4) and (4.4) can be derived for the soft factors associated
with interference between parton species in the hard scattering.

We now relate soft factors in the adjoint representation with those in the
fundamental one. For the mixed quark-gluon soft factor we have

$\displaystyle m(R)\,{}^{RR}\\!S_{qg}$
$\displaystyle=P_{R}^{\underline{i}_{1}\underline{a}_{2}}\,P_{R}^{\underline{k}_{1}\underline{c}_{2}}\,(S_{qg})^{\underline{i}_{1}\underline{a}_{2},\underline{k}_{1}\underline{c}_{2}}_{\underline{j}_{1}\underline{b}_{2},\underline{j}_{1}\underline{b}_{2}}=P_{R}^{\underline{i}_{2}\underline{b}_{2}}\,P_{R}^{\underline{k}_{2}\underline{b}_{2}}\,(S_{qq})^{\underline{i}_{1}\underline{i}_{1},\underline{k}_{1}\underline{k}_{1}}_{\underline{j}_{1}\underline{i}_{2},\underline{j}_{1}\underline{k}_{2}}$
$\displaystyle=P_{R}^{\underline{i}_{2}\underline{j}_{2}}\,P_{R}^{\underline{k}_{2}\underline{j}_{2}}\,(S_{qq})^{\underline{i}_{1}\underline{i}_{1},\underline{k}_{1}\underline{k}_{1}}_{\underline{j}_{1}\underline{i}_{2},\underline{j}_{1}\underline{k}_{2}}=P_{R}^{\underline{i}_{1}\underline{i}_{2}}\,P_{R}^{\underline{k}_{1}\underline{k}_{2}}\,(S_{qq})^{\underline{i}_{1}\underline{i}_{2},\underline{k}_{1}\underline{k}_{2}}_{\underline{j}_{1}\underline{j}_{2},\underline{j}_{1}\underline{j}_{2}}$
$\displaystyle=m(R)\,{}^{RR}\\!S_{qq}$ (106)

if $R$ is in the singlet or octet channels, where in the second and the forth
step we have used (102) for the indices associated with parton $a_{2}$ in
$S_{a_{1}a_{2}}$. Writing

$\displaystyle m(R)\,{}^{RR}\\!S_{gg}\ $
$\displaystyle=P_{R}^{\underline{a}_{1}\underline{a}_{2}}\,P_{R}^{\underline{c}_{1}\underline{c}_{2}}\,(S_{gg})^{\underline{a}_{1}\underline{a}_{2},\underline{c}_{1}\underline{c}_{2}}_{\underline{b}_{1}\underline{b}_{2},\underline{b}_{1}\underline{b}_{2}}$
$\displaystyle=P_{R}^{\underline{i}_{1}\underline{a}_{1}}\,P_{R}^{\underline{i}_{1}\underline{a}_{2}}\,P_{R}^{\underline{k}_{1}\underline{c}_{1}}\,P_{R}^{\underline{k}_{1}\underline{c}_{2}}\,(S_{gg})^{\underline{a}_{1}\underline{a}_{2},\underline{c}_{1}\underline{c}_{2}}_{\underline{b}_{1}\underline{b}_{2},\underline{b}_{1}\underline{b}_{2}}$
(107)

for $R=1,A,S$ and using (102) for all index pairs, one can reduce $S_{gg}$ to
$S_{qq}$ in the colour singlet and octet channels as well. The only
independent soft matrix elements for collinear DPS factorisation (in channels
with the same parton species in the amplitude and its conjugate) are therefore

$\displaystyle{}^{11}\mskip-1.5muS$
$\displaystyle={}^{11}\mskip-1.5muS_{qq}={}^{11}\mskip-1.5muS_{qg}={}^{11}\mskip-1.5muS_{gq}={}^{11}\mskip-1.5muS_{gg}=1$
(108) and $\displaystyle{}^{88}\mskip-1.5muS$
$\displaystyle={}^{88}\mskip-1.5muS_{qq}={}^{AA}\mskip-1.5muS_{qg}={}^{SS}\mskip-1.5muS_{qg}={}^{AA}\mskip-1.5muS_{gq}={}^{SS}\mskip-1.5muS_{gq}={}^{AA}\mskip-1.5muS_{gg}={}^{SS}\mskip-1.5muS_{gg}\,,$
$\displaystyle{}^{DD}\mskip-1.5muS$
$\displaystyle={}^{DD}\mskip-1.5muS_{gg}\,,$
$\displaystyle{}^{27\,27}\mskip-1.5muS$
$\displaystyle={}^{27\,27}\mskip-1.5muS_{gg}\,,$ (109)

where one can also replace one or both labels $q$ with $\bar{q}$ in the soft
factors involving quarks. We recall that each factor in (4.4) depends on
$\boldsymbol{y}$, on a rapidity difference, and on two renormalisation scales
$\mu_{1}$ and $\mu_{2}$. In the colour singlet channel the soft matrix
elements are unity because all Wilson lines are contracted pairwise to
$WW^{\dagger}=W^{\dagger}W=\mathbb{1}$.

As noted in Vladimirov:2016qkd , there is a remarkable identity

$\displaystyle{}^{88}\mskip-1.5muS(\boldsymbol{y})$
$\displaystyle=S_{g}(\boldsymbol{y})\,,$ (110)

where $S_{g}$ is the soft factor for single TMD factorisation with a gluon
initiated hard scattering such as $gg\to H$ Echevarria:2015uaa . Its
definition reads

$\displaystyle S_{g}(\boldsymbol{z})$ $\displaystyle=\bigl{\langle}0\mskip
1.5mu\big{|}\mskip
1.5muP_{1}^{\underline{a}\,\underline{b}}\,\bigl{[}O_{S,g}(\boldsymbol{0},\boldsymbol{z})\bigr{]}^{\underline{a},\underline{b}}\mskip
1.5mu\big{|}\mskip 1.5mu0\bigr{\rangle}\,,$ (111)

where $O_{S,g}$ is obtained from (3.2) by replacing the fundamental Wilson
lines with adjoint ones. One can easily prove (110) by applying the colour
Fierz identity (63) to ${}^{88}\mskip-1.5muS$ in the quark representation and
to $S_{g}$ after converting adjoint Wilson lines into fundamental ones using
(94). It is understood that the two renormalisation scales $\mu_{1},\mu_{2}$
in ${}^{88}\mskip-1.5muS(\boldsymbol{y})$ must be taken equal in (110).

###  4.5 Interlude: collinear factorisation for coloured particle production

Figure 3:  Graphical representation of colour index contractions in the
factorised double dijet cross section. The indices in this figure correspond
to the ones given in (4.5). The corresponding primed indices in the complex
conjugate part of the graph are not shown.

In collinear factorisation, we may consider the production of coloured
particles, a prominent example for DPS being double dijet production. As
pointed out in section 2.1 of Diehl:2015bca , the steps for showing
factorisation in this case lead to the extended soft factor (3.2). Instead of
the product in (12), the cross section then involves products of the form

$\displaystyle
X=H_{a_{1}b_{1}}^{\underline{s}_{1}\underline{t}_{1}}\,H_{a_{2}b_{2}}^{\underline{s}_{2}\underline{t}_{2}}\,$
$\displaystyle\bigl{[}F^{T}_{\text{us},b_{1}b_{2}}(Y_{R})\,S^{-1}(Y_{R}-Y_{L})\bigr{]}^{\underline{r}_{1}\underline{r}_{2}}\,$
$\displaystyle\times\bigl{[}S(Y_{R}-Y_{L})\bigr{]}^{\underline{r}_{1}\underline{r}_{2},\underline{u}_{1}\underline{u}_{2}}_{\underline{s}_{1}\underline{s}_{2},\underline{t}_{1}\underline{t}_{2}}\,\bigr{[}S^{-1}(Y_{R}-Y_{L})\,F_{\text{us},a_{1}a_{2}}(Y_{L})\bigr{]}^{\underline{u}_{1}\underline{u}_{2}}\,.$
(112)

A pictorial representation is given in figure 3. Colour indices $s,t$ in the
hard scattering factor $H^{ss^{\prime}tt^{\prime}}$ are for the incoming
partons in the amplitude, whilst $s^{\prime},t^{\prime}$ are for the incoming
partons in the conjugate amplitude. These indices are contracted with colour
indices of the extended soft factor. The inverse soft factors $S^{-1}$ in
(4.5) remove soft gluons from the unsubtracted collinear factors
$F_{\text{us}}$ as usual; they are hence the soft factors with left and right
moving Wilson lines colour contracted as in (3.2). We have omitted their
parton labels, given that after colour projection they are independent of the
parton species as shown in the previous subsection. In the extended soft
factor, the representation of each Wilson line must match the parton entering
in the corresponding hard-scattering subprocess. This includes combinations
such as
$W_{ab}(\boldsymbol{\xi},v_{L})\,W^{\dagger}_{kl}(\boldsymbol{\xi},v_{R})$,
which corresponds to a right-moving quark scattering on a left-moving gluon.
It is not difficult to generalise the relation (4.4) between the extended and
usual soft factors to such cases, so that we can omit parton labels in the
extended soft factor as well. Inserting projection operators as in (73), we
can thus rewrite (4.5) as

$\displaystyle X$
$\displaystyle=\sum_{R,R^{\prime},R_{1},R_{2}}\frac{P_{R_{1}}^{\underline{s}_{1}\underline{t}_{1}}\,H_{a_{1}b_{1}}^{\underline{s}_{1}\underline{t}_{1}}\,P_{R_{2}}^{\underline{s}_{2}\underline{t}_{2}}\,H_{a_{2}b_{2}}^{\underline{s}_{2}\underline{t}_{2}}}{m(R)\mskip
1.5mum(R^{\prime})\mskip 1.5mum(R_{1})\mskip
1.5mum(R_{2})}\;P_{R}^{\underline{r}_{1}\underline{r}_{2}}\,\bigl{[}F^{T}_{\text{us},b_{1}b_{2}}(Y_{R})\,S^{-1}(Y_{R}-Y_{L})\bigr{]}^{\underline{r}_{1}\underline{r}_{2}}$
$\displaystyle\qquad\qquad\times\delta_{RR_{1}}\,\delta_{RR_{2}}\,\delta_{RR^{\prime}}\,m(R)\,{}^{RR}\mskip-1.5muS(Y_{R}-Y_{L})\;P_{R^{\prime}}^{\underline{u}_{1}\underline{u}_{2}}\,\bigr{[}S^{-1}(Y_{R}-Y_{L})\,F_{\text{us},a_{1}a_{2}}(Y_{L})\bigr{]}^{\underline{u}_{1}\underline{u}_{2}}$
$\displaystyle=\sum_{R}\frac{P_{R}^{\underline{s}_{1}\underline{t}_{1}}\,H_{a_{1}b_{1}}^{\underline{s}_{1}\underline{t}_{1}}\,P_{R}^{\underline{s}_{2}\underline{t}_{2}}\,H_{a_{2}b_{2}}^{\underline{s}_{2}\underline{t}_{2}}}{m^{3}(R)}\;P_{R}^{\underline{r}_{1}\underline{r}_{2}}\,F^{\underline{r}_{1}\underline{r}_{2}}_{\text{us},b_{1}b_{2}}(Y_{R})\;{}^{RR}\mskip-1.5muS^{-1}(Y_{R}-Y_{L})\,P_{R}^{\underline{u}_{1}\underline{u}_{2}}\,F^{\underline{u}_{1}\underline{u}_{2}}_{\text{us},a_{1}a_{2}\rule{0.0pt}{3.91806pt}}(Y_{L})\,,$
(113)

where in the last step we used that ${}^{RR^{\prime}}\\!S^{-1}$ is diagonal in
$R$ and $R^{\prime}$. Splitting ${}^{RR}\mskip-1.5muS$ as in (83) and using
that ${}^{RR}\mskip-1.5mus$ is real valued, we obtain colour projected DTMDs
from (84) and the analogue of (75) for $F_{\text{us}}$. This yields the main
result of this subsection:

$\displaystyle X$
$\displaystyle=\sum_{R}\frac{1}{\varepsilon_{a_{1}}(R)\,\varepsilon_{b_{1}}(R)\,\varepsilon_{a_{2}}(R)\,\varepsilon_{b_{2}}(R)}\,\frac{1}{\mathcal{N}_{a_{1}}\mskip
1.5mu\mathcal{N}_{b_{1}}\mskip 1.5mu\mathcal{N}_{a_{2}}\mskip
1.5mu\mathcal{N}_{b_{2}}}\frac{P_{R}^{\underline{s}_{1}\underline{t}_{1}}\,H_{a_{1}b_{1}}^{\underline{s}_{1}\underline{t}_{1}}\,P_{R}^{\underline{s}_{2}\underline{t}_{2}}\,H_{a_{2}b_{2}}^{\underline{s}_{2}\underline{t}_{2}}}{m^{2}(R)}$
$\displaystyle\qquad\quad\times{}^{R}\mskip-1.5muF_{\text{us},b_{1}b_{2}}(Y_{R})\,{}^{RR}\mskip-1.5mus^{-1}(Y_{R}-Y_{C})\,{}^{RR}\mskip-1.5mus^{-1}(Y_{C}-Y_{L})\,{}^{R}\mskip-1.5muF_{\text{us},a_{1}a_{2}}(Y_{L})$
$\displaystyle=\sum_{R}{}^{R}\mskip-1.5muH_{a_{1}b_{1}}\,{}^{R}\mskip-1.5muH_{a_{2}b_{2}}\,{}^{R}\mskip-1.5muF_{b_{1}b_{2}}\,{}^{R}\mskip-1.5muF_{a_{1}a_{2}}\,.$
(114)

The colour structure of the collinear cross section is thus surprisingly
simple: we have the same DPDs in (4.5) as in the formula (85) for colour
singlet production, and the only new ingredients are the colour projected
hard-scattering factors 555The factors ${}^{R}\mskip-1.5muH$ in (115) differ
from the ones in equation (2.11) of Diehl:2015bca . Here we project index
pairs in the $t$-channel ($s,s^{\prime}$ etc.) onto a definite colour
representation $R$, whereas in Diehl:2015bca  this was done for index pairs in
the $s$-channel ($s,t$ etc.).

$\displaystyle{}^{R}\mskip-1.5muH_{ab}$
$\displaystyle=\frac{1}{\varepsilon_{a}(R)\,\varepsilon_{b}(R)}\,\frac{1}{\mathcal{N}_{a}\,\mathcal{N}_{b}}\,\frac{P_{R}^{\underline{s}\,\underline{t}}\,H_{ab}^{\underline{s}\,\underline{t}}}{m(R)}\,.$
(115)

For the production of a colour singlet system, the hard scattering subprocess
has the structure

$\displaystyle H_{ab}^{\underline{s}\,\underline{t}}$
$\displaystyle=\delta_{st}\,\delta_{s^{\prime}t^{\prime}}\,H_{ab}\,.$ (116)

Using (71) we thus find

$\displaystyle
P_{R}^{\underline{s}\,\underline{t}}\,H_{ab}^{\underline{s}\,\underline{t}}$
$\displaystyle=m(R)\,H_{ab}\,,$ (117)

so that ${}^{R}\mskip-1.5muH_{ab}$ reproduces $H_{ab}/(\mathcal{N}_{a}\mskip
1.5mu\mathcal{N}_{b})$ divided by $\varepsilon_{a}(R)\,\varepsilon_{b}(R)$,
which is absorbed into the sign factor $\eta_{a_{1}a_{2}}(R)$ in (85).

##  5 Factorisation at small transverse momenta

We are now in a situation to give the TMD factorisation formula for DPS
processes. Let us consider the production of two particles or groups of
particles with transverse momenta $\boldsymbol{q}_{1},\boldsymbol{q}_{2}$ and
invariant masses $Q_{1},Q_{2}$. Both $Q_{1}$ and $Q_{2}$ are required to be
large, and we denote their generic size by $Q$ for the purpose of power
counting. Both transverse momenta, whose generic size we denote by $q_{T}$,
are required to be small compared with $Q$. Up to power corrections in
$\Lambda/Q$ and $|\boldsymbol{q}_{i}|/Q$, the DPS cross section with measured
transverse momenta reads

$\displaystyle\frac{d\sigma_{\text{DPS}}}{dx_{1}\,dx_{2}\,d\bar{x}_{1}\,d\bar{x}_{2}\,d^{2}\boldsymbol{q}_{1}\,d^{2}\boldsymbol{q}_{2}}=\frac{1}{C}\,\sum_{a_{1},a_{2},b_{1},b_{2}}\\!\\!\\!\hat{\sigma}_{a_{1}b_{1}}(Q_{1},\mu_{1})\,\hat{\sigma}_{a_{2}b_{2}}(Q_{2},\mu_{2})\,$
$\displaystyle\qquad\times\int\frac{d^{2}\boldsymbol{z}_{1}}{(2\pi)^{2}}\,\frac{d^{2}\boldsymbol{z}_{2}}{(2\pi)^{2}}\,d^{2}\boldsymbol{y}\;e^{-i\boldsymbol{q}_{1}\boldsymbol{z}_{1}-i\boldsymbol{q}_{2}\boldsymbol{z}_{2}}\,W_{a_{1}a_{2}b_{1}b_{2}}(\bar{x}_{i},x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\nu)\,,$
(118)

with

$\displaystyle
W_{a_{1}a_{2}b_{1}b_{2}}(\bar{x}_{i},x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\nu)=\Phi(\nu\boldsymbol{y}_{+})\,\Phi(\nu\boldsymbol{y}_{-})$
$\displaystyle\qquad\times\sum_{R}\eta_{a_{1}a_{2}}(R)\;{}^{R}\mskip-1.5muF_{b_{1}b_{2}}(\bar{x}_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\bar{\zeta})\,{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)$
(119)

and

$\displaystyle\boldsymbol{y}_{\pm}=\boldsymbol{y}\pm{\textstyle\frac{1}{2}}(\boldsymbol{z}_{1}-\boldsymbol{z}_{2})\,.$
(120)

We recall that we write arguments $x_{i}$ if a function depends on both
$x_{1},x_{2}$, and likewise for all other variables. The combinatorial factor
$C$ is equal to $2$ if the systems produced by the two hard scatters are
identical and equal to $1$ otherwise. The factor $\eta_{a_{1}a_{2}}(R)$ was
defined in (86); it is equal to $-1$ if $R=A$ and exactly one of $a_{1}$ and
$a_{2}$ is a gluon and equal to $1$ otherwise. The subprocess cross sections
are given by

$\displaystyle\hat{\sigma}_{a_{1}b_{1}}(Q_{1}^{2},\mu_{1}^{2})$
$\displaystyle=\frac{1}{2Q_{1}^{2}}\,\frac{H_{a_{1}b_{1}}(Q_{1}^{2},\mu_{1}^{2})}{\mathcal{N}_{a_{1}}\mskip
1.5mu\mathcal{N}_{b_{1}}}\,$ (121)

and likewise for index $2$. Here, $1/(2Q_{1}^{2})$ is the standard flux
factor, and $1/(\mathcal{N}_{a_{1}}\mskip 1.5mu\mathcal{N}_{b_{1}})$
implements colour averaging over the initial state as noted after (85). The
sum over $a_{1},a_{2},b_{1},b_{2}$ in (5) runs over both parton species and
polarisations, following the notation specified in section 3.1. Hard-
scattering cross sections and DPDs carry transverse Lorentz indices for
transverse quark and linear gluon polarisation, which must be contracted
appropriately. For later discussion, it is useful to rewrite the integration
measure in (5) as

$\displaystyle
d^{2}\boldsymbol{y}\,d^{2}\boldsymbol{z}_{1}\,d^{2}\boldsymbol{z}_{2}\;e^{-i\boldsymbol{q}_{1}\boldsymbol{z}_{1}-i\boldsymbol{q}_{2}\boldsymbol{z}_{2}}$
$\displaystyle=d^{2}\boldsymbol{Z}\,d^{2}\boldsymbol{y}_{+}\,d^{2}\boldsymbol{y}_{-}\;e^{-i\mskip
1.5mu(\boldsymbol{q}_{1}+\boldsymbol{q}_{2})\mskip 1.5mu\boldsymbol{Z}-i\mskip
1.5mu(\boldsymbol{q}_{1}-\boldsymbol{q}_{2})(\boldsymbol{y}_{+}-\boldsymbol{y}_{-})/2}$
(122)

with

$\displaystyle\boldsymbol{Z}$
$\displaystyle={\textstyle\frac{1}{2}}(\boldsymbol{z}_{1}+\boldsymbol{z}_{2})\,.$
(123)

The function $\Phi(\nu\boldsymbol{y}_{\pm})$ in (5) regulates the ultraviolet
region. It is intimately related with the cross talk between double and single
hard scattering and was introduced in the recent work Diehl:2017kgu . This is
discussed in section 5.3, where we show how to combine the contributions from
DPS, SPS and their interference to obtain the physical cross section. Apart
from this function, the factorisation formula (5) is identical to the form
that emerged from the analysis of lowest-order graphs in section 2 of
Diehl:2011yj . This is because our DPD definition in section 3.3 absorbs all
relevant effects of soft gluon exchange in the cross section, leading to the
crucial relations (42) and (43).

Of course, the scale and rapidity parameter dependence of the DTMDs in (5)
does not arise from lowest-order graphs but from the full analysis in section
3.4. In the next section, we give the final forms (124) and (128) of the
corresponding evolution equations, as well as their explicit solutions (5.1)
and (5.1) at the level of DPDs and of the cross section, respectively. In
section 5.2 we derive the evolution equations (139) to (142) of DPDFs and give
the explicit form (5.2) of their rapidity dependence.

When combining SPS and DPS in the physical cross section, one must take care
of interference effects and of double counting. A systematic framework for
this has been proposed in Diehl:2017kgu , and in section 5.3 of the present
paper we characterise the individual terms in the master formula (5.3) for the
cross section with measured transverse momenta.

###  5.1 DTMD evolution: renormalisation scale and rapidity

Let us rewrite the evolution equations for the DTMDs using the projection
operators introduced in section 4. The renormalisation group equation (3.4)
then reads

$\displaystyle\frac{\partial}{\partial\log\mu_{1}}\,{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)$
$\displaystyle=\gamma_{F,a_{1}}(\mu_{1},x_{1}\zeta/x_{2})\,{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)$
(124)

for the scale $\mu_{1}$, and in analogy for $\mu_{2}$. Here we have introduced
the rapidity parameter

$\displaystyle\zeta$ $\displaystyle=2x_{1}x_{2}\,(p^{+})^{2}\,e^{-2Y_{C}}\,,$
(125)

where $p$ is the proton momentum.666A different definition (going back to
Ji:2004wu ) was used in Diehl:2011yj , namely $x_{1}x_{2}\mskip
1.5mu\zeta^{2}|_{\text{Ref.\leavevmode\nobreak\ \cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Diehl:2011yj}{\@@citephrase{(}}{\@@citephrase{)}}}}}=\zeta|_{\text{here}\,}$.
This extends the definition (61) for SPS to the case of DPS in a way that is
symmetric in the two momentum fractions $x_{1},x_{2}$. As discussed after
(55), the correct rapidity argument of $\gamma_{F,a}$ in (124) is
$x_{1}\zeta/x_{2}=2(x_{1}\mskip 1.5mup^{+})^{2}e^{-2Y_{C}}$, since the
corresponding UV divergent subgraphs depend on the momentum of parton $1$
(whose plus-component is $x_{1}\mskip 1.5mup^{+}$), but not on the momentum of
parton $2$. For the DPD of the left moving proton with momentum $\bar{p}$ we
define accordingly

$\displaystyle\bar{\zeta}$
$\displaystyle=2\bar{x}_{1}\bar{x}_{2}\,(\bar{p}^{-})^{2}\,e^{2Y_{C}}\,.$
(126)

Note that

$\displaystyle\zeta\,\bar{\zeta}$ $\displaystyle=x_{1}\bar{x}_{1}x_{2}\mskip
1.5mu\bar{x}_{2}\,(2p^{+}\bar{p}^{\,-})^{2}=Q_{1}^{2}\,Q_{2}^{2}\,.$ (127)

Rewriting the rapidity derivative as $\partial/\partial Y_{C}=-2\mskip
1.5mu\partial/\partial\log\zeta=2\mskip
1.5mu\partial/\partial\log\bar{\zeta}$, one obtains from (45) the Collins-
Soper equation

$\displaystyle\frac{\partial}{\partial\log\zeta}{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)$
$\displaystyle=\frac{1}{2}\sum_{R^{\prime}}{}^{RR^{\prime}}\\!K_{a_{1}a_{2}}(\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i})\,{}^{R^{\prime}}\\!F_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)$
(128)

for colour projected DTMDs in a right moving proton. The Collins-Soper
equation for a left moving proton has the same form, with $\zeta$ replaced by
$\bar{\zeta}$. The scale dependence of the Collins-Soper kernel in (3.4) now
reads

$\displaystyle\frac{\partial}{\partial\log\mu_{1}}\,{}^{RR^{\prime}}\\!K_{a_{1}a_{2}}(\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i})$
$\displaystyle=-\gamma_{K,a_{1}}(\mu_{1})\,\delta_{RR^{\prime}}$ (129)

and correspondingly for $\mu_{2}$. It is thus only the diagonal elements of
${}^{RR^{\prime}}\\!K$ that have UV divergences and depend on the
renormalisation scales $\mu_{1}$ and $\mu_{2}$. Let us recall that the kernel
$K_{a_{1}a_{2}}$ and the anomalous dimensions $\gamma_{F,a}$ and
$\gamma_{K,a}$ depend on the colour representation of the parton (quarks or
antiquarks vs. gluons) but not on their flavour or polarisation.

It is easy to solve the rapidity and renormalisation scale evolution with
general starting scales $\mu_{01}$, $\mu_{02}$ for $\mu_{1}$, $\mu_{2}$ and
$\zeta_{0}$ for $\zeta$. Renormalisation scale evolution gives

$\displaystyle{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)={}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{0i},\zeta)$
$\displaystyle\qquad\qquad\times\exp\biggl{[}\int_{\mu_{01}}^{\mu_{1}}\frac{d\mu}{\mu}\,\gamma_{F,a_{1}}(\mu,x_{1}\zeta/x_{2})+\int_{\mu_{02}}^{\mu_{2}}\frac{d\mu}{\mu}\,\gamma_{F,a_{2}}(\mu,x_{2}\zeta/x_{1})\biggr{]}\,.$
(130)

Using the rapidity evolution in (128) we write

$\displaystyle{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)$
$\displaystyle=\sum_{R^{\prime}}{}^{RR^{\prime}}\\!\exp\biggl{[}K_{a_{1}a_{2}}(\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i})\log\frac{\sqrt{\zeta}}{\sqrt{\zeta_{0}}}\,\biggr{]}{}^{R^{\prime}}\\!F_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta_{0})\,,$
(131)

where ${}^{RR^{\prime}}\\!\exp$ is to be understood as a matrix exponential,
i.e.

$\displaystyle{}^{RR^{\prime}}\\!\exp(M)$
$\displaystyle=\delta_{RR^{\prime}}+{}^{RR^{\prime}}\\!M+\sum_{n=2}^{\infty}\,\sum_{R_{2},\ldots,R_{n}}\,\frac{{}^{RR_{2}}\mskip-1.5muM\,\cdots\,{}^{R_{n}R^{\prime}}\\!M}{n!}\,.$
(132)

We now split

$\displaystyle{}^{RR^{\prime}}\\!K_{a_{1}a_{2}}(\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i})$
$\displaystyle=\delta_{RR^{\prime}}\,\bigl{[}{}^{1}\mskip-1.5muK_{a_{1}}(\boldsymbol{z}_{1};\mu_{1})+{}^{1}\mskip-1.5muK_{a_{2}}(\boldsymbol{z}_{2};\mu_{2})\bigr{]}+{}^{RR^{\prime}}\\!M_{a_{1}a_{2}}(\boldsymbol{z}_{i},\boldsymbol{y})\,,$
(133)

where ${}^{1}\mskip-1.5muK_{a}(\boldsymbol{z};\mu)$ is the kernel in the
Collins-Soper equation (7) for a single parton TMD. The colour singlet label
$1$ on $K$ is given for consistency with our later notation in section 6.2.2.
By virtue of (8) and (129), the matrix ${}^{RR^{\prime}}\\!M$ is independent
of $\mu_{1}$ and $\mu_{2}$. Combining (5.1) and (131) with (133), we obtain
the relation between the DTMD at initial and final scales in the form

$\displaystyle{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)$
$\displaystyle\quad=\exp\,\biggl{\\{}\int_{\mu_{01}}^{\mu_{1}}\frac{d\mu}{\mu}\,\biggl{[}\gamma_{a_{1}}(\mu)-\gamma_{K,a_{1}}(\mu)\,\log\frac{\sqrt{x_{1}\zeta/x_{2}}}{\mu}\biggr{]}+{}^{1}\mskip-1.5muK_{a_{1}}(\boldsymbol{z}_{1};\mu_{01})\log\frac{\sqrt{\zeta}}{\sqrt{\zeta_{0}}}$
$\displaystyle\qquad\hskip
15.00002pt+\int_{\mu_{02}}^{\mu_{2}}\frac{d\mu}{\mu}\,\biggl{[}\gamma_{a_{2}}(\mu)-\gamma_{K,a_{2}}(\mu)\,\log\frac{\sqrt{x_{2}\mskip
1.5mu\zeta/x_{1}}}{\mu}\biggr{]}+{}^{1}\mskip-1.5muK_{a_{2}}(\boldsymbol{z}_{2};\mu_{02})\log\frac{\sqrt{\zeta}}{\sqrt{\zeta_{0}}}\,\biggr{\\}}$
$\displaystyle\qquad\times\sum_{R^{\prime}}{}^{RR^{\prime}}\\!\exp\,\biggl{[}M_{a_{1}a_{2}}(\boldsymbol{z}_{i},\boldsymbol{y})\log\frac{\sqrt{\zeta}}{\sqrt{\zeta_{0}}}\,\biggr{]}\,{}^{R^{\prime}}\\!F_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{01},\mu_{02},\zeta_{0})\,,$
(134)

where we have used the explicit form (5) of $\gamma_{F,a}$. The exponential in
the second and third lines is just the generalisation to two partons of the
evolution factor for a single parton TMD, as is readily seen by solving the
system of equations in (3) to (8). It resums both double and single
logarithms. The last line in (5.1) describes the mixing between different
colour representations $R$ under rapidity evolution and involves a single
logarithm. The double logarithms in the evolution of
${}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{i},\zeta)$
are thus the same as those for a product of two single TMDs with appropriate
arguments.

The symmetry relation (90) for ${}^{RR^{\prime}}\\!K$ translates into an
analogous relation for ${}^{RR^{\prime}}\\!M$. Using that $b_{i}$ is the
antiparton of $a_{i}$ in the cross section formula (5) and that
$M_{a_{1}a_{2}}$ does not depend on parton spins, we can thus rewrite

$\displaystyle\eta_{a_{1}a_{2}}(R)\;{}^{RR^{\prime}}\\!\exp\,\Biggl{[}M_{b_{1}b_{2}}\log\frac{\sqrt{\bar{\zeta}\rule{0.0pt}{9.04166pt}}}{\sqrt{\zeta_{0}}}\,\Biggr{]}\,{}^{R^{\prime}}\\!F_{b_{1}b_{2}}$
$\displaystyle=\eta_{a_{1}a_{2}}(R^{\prime})\;{}^{R^{\prime}}\\!F_{b_{1}b_{2}}\mskip
1.5mu{}^{R^{\prime}\\!R}\mskip-1.5mu\exp\,\Biggl{[}M_{a_{1}a_{2}}\log\frac{\sqrt{\bar{\zeta}\rule{0.0pt}{9.04166pt}}}{\sqrt{\zeta_{0}}}\,\Biggr{]}$
(135)

when inserting the evolved form of ${}^{R}\mskip-1.5muF_{b_{1}b_{1}}$ into the
expression (5) of $W$. This allows us to combine the matrix exponentials
associated with the two DPDs. Further using that one has equal anomalous
dimensions $\gamma_{a_{i}}=\gamma_{b_{i}}$ and
$\gamma_{K,a_{i}}=\gamma_{K,b_{i}}$ and equal Collins-Soper kernels
${}^{1}\mskip-1.5muK_{a_{i}}={}^{1}\mskip-1.5muK_{b_{i}}$, we obtain

$\displaystyle W_{a_{1}a_{2}b_{1}b_{2}}$
$\displaystyle=\exp\,\biggl{\\{}\int_{\mu_{01}}^{\mu_{1}}\frac{d\mu}{\mu}\,\biggl{[}\gamma_{a_{1}}(\mu)-\gamma_{K,a_{1}}(\mu)\,\log\frac{Q_{1}^{2}}{\mu^{2}}\biggr{]}+{}^{1}\mskip-1.5muK_{a_{1}}(\boldsymbol{z}_{1};\mu_{01})\log\frac{Q_{1}Q_{2}}{\zeta_{0}}$
$\displaystyle\quad\hskip
15.00002pt+\int_{\mu_{02}}^{\mu_{2}}\frac{d\mu}{\mu}\,\biggl{[}\gamma_{a_{2}}(\mu)-\gamma_{K,a_{2}}(\mu)\,\log\frac{Q_{2}^{2}}{\mu^{2}}\biggr{]}+{}^{1}\mskip-1.5muK_{a_{2}}(\boldsymbol{z}_{2};\mu_{02})\log\frac{Q_{1}Q_{2}}{\zeta_{0}}\,\biggr{\\}}$
$\displaystyle\quad\times\Phi(\nu\boldsymbol{y}_{+})\,\Phi(\nu\boldsymbol{y}_{-})\sum_{RR^{\prime}}\eta_{a_{1}a_{2}}(R)\;{}^{R}\mskip-1.5muF_{b_{1}b_{2}}(\bar{x}_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{01},\mu_{02},\zeta_{0})\,$
$\displaystyle\quad\hskip
15.00002pt\times{}^{RR^{\prime}}\\!\exp\,\biggl{[}M_{a_{1}a_{2}}(\boldsymbol{z}_{i},\boldsymbol{y})\log\frac{Q_{1}Q_{2}}{\zeta_{0}}\,\biggr{]}\,{}^{R^{\prime}}\\!F_{a_{1}a_{2}}(x_{i},\boldsymbol{z}_{i},\boldsymbol{y};\mu_{01},\mu_{02},\zeta_{0})\,.$
(136)

We have used the relation (127) for combining the arguments of the logarithms
and expressing them in terms of the physical scales $Q_{1}$ and $Q_{2}$. The
variables $\zeta$ and $\bar{\zeta}$ have thus completely disappeared from the
physical cross section. This is not the case for the factorisation scales
$\mu_{i}$ in (5.1) and in the hard scattering cross sections $\hat{\sigma}$.
As is well known, the dependence on these scales only cancels up to un-
calculated higher orders in $\alpha_{s}$.

###  5.2 DPDF evolution

We now turn to DPDFs, which appear not only in the factorisation formula for
DPS with integrated transverse momenta, but also in the short-distance
expansion of DTMDs, as we will see in section 6.2. We have derived in section
4.4 that the soft factor ${}^{RR^{\prime}}\\!S_{a_{1}a_{2}}$ at
$\boldsymbol{z}_{1}=\boldsymbol{z}_{2}=\boldsymbol{0}$ is diagonal in $R$ and
$R^{\prime}$ and independent of the parton species. Following the construction
in appendix A, one finds that the matrix $s$ in (36) is also diagonal in $R$
and $R^{\prime}$ and satisfies the relation

$\displaystyle{}^{RR}\mskip-1.5mus(Y)$
$\displaystyle=\sqrt{{}^{RR}\mskip-1.5muS(2Y)}\,,$ (137)

which is the analogue of (58) for SPS. The definition of DPDFs from
unsubtracted collinear matrix elements in (3.3) thus turns into

$\displaystyle{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(Y_{C})$
$\displaystyle=\lim_{Y_{L}\rightarrow-\infty}{}^{RR}\mskip-1.5mus^{-1}(Y_{C}-Y_{L})\,{}^{R}\mskip-1.5muF_{\text{us},a_{1}a_{2}}(Y_{L})$
(138)

for a right moving proton, and correspondingly for a left moving one. The
kernel $K=\widehat{K}$ controlling the $Y$ dependence of $S(Y)$ and $s(Y)$ is
of course also diagonal in $R$ and $R^{\prime}$. We denote it by ${}^{R}\\!J$
in order to avoid confusion with the evolution kernel for TMDs. In terms of
the variable $\zeta$, the rapidity evolution of DPDFs is then given by

$\displaystyle\frac{\partial}{\partial\log\zeta}\,{}^{R}\mskip-1.5muF(x_{i},\boldsymbol{y};\mu_{i},\zeta)$
$\displaystyle=\frac{1}{2}\,{}^{R}\\!J(\boldsymbol{y};\mu_{i})\,{}^{R}\mskip-1.5muF(x_{i},\boldsymbol{y};\mu_{i},\zeta)\,,$
(139)

where we have explicitly given all arguments of the functions. Since in the
colour singlet channel ${}^{11}\mskip-1.5muS=1$, one has
${}^{1}\mskip-1.5muJ=0$ for the corresponding evolution kernel, i.e. the
collinear colour singlet DPDFs are independent of the rapidity parameter
$\zeta$.

Let us now discuss renormalisation. Setting
$\boldsymbol{z}_{1}=\boldsymbol{z}_{2}=\boldsymbol{0}$ in the soft factor $S$
and in the hadronic matrix elements $F_{\text{us}}$ induces additional UV
divergences compared to the case where these distances are finite. The graphs
requiring renormalisation are now not only vertex corrections and self
energies, but also involve the exchange of hard gluons between partons or
Wilson lines associated with the scattering amplitude and its complex
conjugate. This is of course well known from the renormalisation of the twist-
two operators that define ordinary PDFs, which is not purely multiplicative
but involves a convolution in the momentum fraction $x$ and mixing between
quark and gluon operators. In the renormalisation group equations for PDFs,
splitting kernels $P_{ab}$ take the role of the anomalous dimensions
$\gamma_{F,a}$ in the TMD case.

In the case of DPDFs we must pay particular attention to colour. Taking the
twist-two operators (21) at $\boldsymbol{z}=\boldsymbol{0}$ and projecting
their colour indices on a definite representation $R$ as in (79), we obtain
the colour projected matrix element ${}^{R}\mskip-1.5muF_{\text{us}}$
according to (80). Due to gauge invariance, operators ${}^{R}O_{a}$ belonging
to different colour representations do not mix under renormalisation. To
obtain the soft factor ${}^{RR}\mskip-1.5muS$, we evaluate the Wilson line
products (3.2) at $\boldsymbol{z}=\boldsymbol{0}$ and project them on definite
colour channels $R$ as
$P_{R}^{\underline{r}\,\underline{s}}\,O_{S,a}^{\underline{r},\underline{s}}$.
Again, operators with different $R$ do not mix under renormalisation. The
renormalisation of the soft factor remains multiplicative, because it is
identical for quarks and gluons and because it does not depend on $x$.

With this in mind, one can repeat the arguments we developed in section 3.4
for the renormalisation of DTMDs and of their Collins-Soper kernels. One then
obtains

$\displaystyle\frac{\partial}{\partial\log\mu_{1}}\,{}^{R}\\!J(\boldsymbol{y};\mu_{i})$
$\displaystyle=-{}\,{}^{R}\gamma_{J}(\mu_{1})$ (140)

for the Collins-Soper kernel and

$\displaystyle\frac{\partial}{\partial\log\mu_{1}}\,{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{y};\mu_{i},\zeta)$
$\displaystyle=2\sum_{b_{1}}{}^{R}\mskip-1.5muP_{a_{1}b_{1}}(x_{1}^{\prime};\mu_{1},x_{1}\zeta/x_{2})\underset{x_{1}}{\otimes}{}^{R}\mskip-1.5muF_{b_{1}a_{2}}(x_{1}^{\prime},x_{2},\boldsymbol{y};\mu_{i},\zeta)$
(141)

for DPDFs, as well as analogous equations for the $\mu_{2}$
dependence.777Regarding factors of 2, our convention for splitting kernels
follows Collins:2011zzd , so that $2P$ takes the place of the anomalous
dimension $\gamma_{F,a}$ in TMD evolution. The kernel ${}^{R}\mskip-1.5muP$ in
(141) should not be confused with the colour projector $P_{R}$. Let us
emphasise that the evolution equation (141) for DPDFs depending on
$\boldsymbol{y}$ is homogeneous. As discussed in Diehl:2011yj ; Diehl:2017kgu
, an additional term for the splitting of one parton into two arises on the
r.h.s. if one integrates the distributions over $\boldsymbol{y}$ or Fourier
transforms them w.r.t. that variable. The resulting inhomogeneous equation has
been extensively studied in the literature Kirschner:1979im ; Shelest:1982dg ;
Snigirev:2003cq ; Gaunt:2009re ; Ceccopieri:2010kg .

By construction, the kernels ${}^{1}\mskip-1.5muP_{a_{1}b_{1}}$ for the colour
singlet sector are the ordinary DGLAP kernels for the evolution of PDFs. They
are hence independent of $\zeta$, which is not the case in the other colour
channels. Indeed, combining (139), (140), (141) and requiring equality of the
derivatives
$\partial/(\partial\log\mu_{i})\,\partial/(\partial\log\zeta)\,{}^{R}\mskip-1.5muF$
and
$\partial/(\partial\log\zeta)\,\partial/(\partial\log\mu_{i})\,{}^{R}\mskip-1.5muF$,
we find

$\displaystyle\frac{\partial}{\partial\log\zeta}\,{}^{R}\mskip-1.5muP_{ab}(x;\mu,\zeta)$
$\displaystyle=-\frac{1}{4}\,\delta_{ab}\,\delta(1-x)\,{}^{R}\gamma_{J}(\mu)$
(142)

and thus

$\displaystyle{}^{R}\mskip-1.5muP_{ab}(x;\mu,\zeta)$
$\displaystyle=-\frac{1}{2}\,\delta_{ab}\,\delta(1-x)\,{}^{R}\gamma_{J}(\mu)\log\frac{\sqrt{\zeta}}{\mu}+{}^{R}\mskip-1.5muP_{ab}(x;\mu,\mu^{2})\,.$
(143)

These are the analogues of the relations (4) and (5) for the anomalous
dimension $\gamma_{F,a}(\mu,\zeta)$ in TMD evolution.

We can make the $\zeta$ dependence of the DPDFs fully explicit in the form

$\displaystyle{}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{y};\mu_{i},\zeta)$
$\displaystyle=\exp\,\biggl{[}-\int_{\mu_{0}}^{\mu_{1}}\frac{d\mu}{\mu}\,{}^{R}\gamma_{J}(\mu)\,\log\frac{\sqrt{x_{1}\zeta/x_{2}}}{\mu}-\int_{\mu_{0}}^{\mu_{2}}\frac{d\mu}{\mu}\,{}^{R}\gamma_{J}(\mu)\,\log\frac{\sqrt{x_{2}\mskip
1.5mu\zeta/x_{1}}}{\mu}$
$\displaystyle\qquad\quad\;+{}^{R}\\!J(\boldsymbol{y};\mu_{0},\mu_{0})\log\frac{\sqrt{\zeta}}{\sqrt{\zeta_{0}}}\mskip
1.5mu\biggr{]}\;{}^{R}\mskip-1.5mu\widehat{F}_{a_{1}a_{2},\,\mu_{0},\zeta_{0}}(x_{i},\boldsymbol{y};\mu_{i})\,,$
(144)

where $\widehat{F}$ is defined by the differential equation

$\displaystyle\frac{\partial}{\partial\log\mu_{1}}\,{}^{R}\mskip-1.5mu\widehat{F}_{a_{1}a_{2},\,\mu_{0},\zeta_{0}}(x_{i},\boldsymbol{y};\mu_{1},\mu_{2})$
$\displaystyle\qquad\qquad=2\sum_{b_{1}}{}^{R}\mskip-1.5muP_{a_{1}b_{1}}(x_{1}^{\prime};\mu_{1},\mu_{1}^{2})\underset{x_{1}}{\otimes}{}^{R}\mskip-1.5mu\widehat{F}_{b_{1}a_{2},\,\mu_{0},\zeta_{0}}(x_{1}^{\prime},x_{2},\boldsymbol{y};\mu_{1},\mu_{2})$
(145)

and its analogue for $\mu_{2}$, with the initial condition

$\displaystyle{}^{R}\mskip-1.5mu\widehat{F}_{a_{1}a_{2},\,\mu_{0},\zeta_{0}}(x_{i},\boldsymbol{y};\mu_{0},\mu_{0})$
$\displaystyle={}^{R}\mskip-1.5muF_{a_{1}a_{2}}(x_{i},\boldsymbol{y};\mu_{0},\mu_{0},\zeta_{0})\,.$
(146)

Unlike $F$, the distribution $\widehat{F}$ evolves with $\zeta$ independent
kernels. The correct scale dependence of (5.2) can be verified by inserting
(142) into (141), and its $\zeta$ dependence can be verified by using (139)
and (140).

The integrals over $\mu$ in (5.2) give rise to double logarithms, which have
been exponentiated to a Sudakov factor in the solution of the evolution
equation. Since the anomalous dimensions ${}^{R}\gamma_{J}$ are positive for
all $R\neq 1$ (see section 7.2.1), the contribution of all colour non-singlet
DPDs to the cross section is Sudakov suppressed for sufficiently large hard
scales. This was recognised long ago in Mekhfi:1988kj  and confirmed in
Manohar:2012jr , where a simple numerical estimate was also given. In
Manohar:2012jr , scale and rapidity evolution was discussed for collinear DPS
factorisation, keeping unsubtracted DPDs and an explicit soft factor in the
cross section. The individual evolution equations hence differ between their
approach and ours, but must of course give the same effects at the cross
section level.

###  5.3 Combining DPS with SPS

The DPS cross section given in (5) is not a physical observable, because the
same final state can also be produced by SPS. Dimensional analysis readily
shows that DPS and SPS contribute to the TMD cross section at the same power
in $1/Q$. The distinction between the two mechanisms is in fact nontrivial,
since there are Feynman graphs that contribute to both of them. These are
graphs in which one parton splits into two, as shown in figure 5a. The
kinematic region where the $q\bar{q}$ pairs are nearly collinear with their
parent gluon is naturally described as DPS, with the splitting being included
in the DPDs. By contrast, the region where the quarks and antiquarks have
large transverse momenta (and hence large virtualities) is adequately
described as a loop correction to SPS, with gluon TMDs describing the initial
state of the hard scattering.

The interference between SPS and DPS has the same power behaviour in $1/Q$ as
SPS and DPS and hence should also be included in the cross section. Again,
there is a double counting problem, since graphs like the one in figure 5b
contribute to either the interference term or to DPS, depending on whether the
splitting takes place in the collinear or in the hard momentum region. Note