aainstitutetext: Deutsches Elektronen-Synchrotron DESY,
Notkestraße 85, 22607, Hamburg, Germany

# Factorisation of soft gluons in multiparton scattering

Markus Diehl a and Riccardo Nagar<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that soft gluons exchanged between the two colliding protons in
multiparton scattering processes can be decoupled, such that their effects are
described by the vacuum expectation values of Wilson lines. Our argument
relies on nonabelian Ward identities and generalises the proof of
factorisation for single Drell-Yan production given by Collins, Soper and
Sterman.

††preprint: DESY 18-204

##  1 Introduction

Significant progress has been made in understanding double parton scattering
(DPS), the process in which two pairs of partons take part in two hard
scattering processes in a single proton-proton collision. Experimental
information comes from an increasing number of DPS studies at the LHC, whilst
on the theory side work is being done both on conceptual aspects and on
phenomenology and modelling. The current status of the field is documented in
the monograph Bartalini:2017jkk  and in the talks at the most recent Workshop
on Multiple Partonic Interactions at the LHC (MPI@LHC
2018111https://indico.cern.ch/event/736470).

The theoretical description of DPS relies on factorisation formulae akin to
those for single parton scattering (SPS). For one of the simplest SPS
processes, namely Drell-Yan production, detailed proofs of factorisation are
available (see the original work Collins:1985ue ; Collins:1988ig ;
Bodwin:1984hc  and the review in Collins:2011zzd ), and it is natural to ask
to which extent they can be generalised to DPS. The work in Diehl:2011yj ;
Diehl:2015bca ; Diehl:2017kgu ; Vladimirov:2017ksc ; Buffing:2017mqm  shows
that indeed factorisation for the double Drell-Yan process can be established
at a comparable level of detail as for single Drell-Yan. An important aspect
left open in that work is however a proof that the effects of soft gluon
exchange between the spectator partons in the two protons can be expressed in
a so-called soft factor, which is defined as the vacuum expectation value of
Wilson line operators. We refer to this as the decoupling of soft gluons.
Given that DPS has a considerably more complicated colour structure than SPS,
a corresponding proof is highly desirable. Indeed, a failure to decouple soft
gluons from the spectator partons would imply that the two colliding protons
are tied together by nonperturbative dynamics, which would seriously curtail
the predictive power of theory.

To close this gap and show that soft gluons can be decoupled in DPS processes
is the aim of the present work. We make use of nonabelian Ward identities,
following closely the derivation of soft gluon decoupling in the work by
Collins, Soper and Sterman Collins:1985ue ; Collins:1988ig  with some
important differences pointed out later. We note that a powerful alternative
method is given by soft-collinear effective theory (SCET), where the
decoupling of soft gluons is achieved by field redefinitions Bauer:2002nz . A
general formulation of DPS factorisation in SCET is given in Manohar:2012jr .
The systematic treatment of the so-called Glauber momentum region in this
framework has however only been formulated recently Rothstein:2016bsq , and a
corresponding analysis for single or double Drell-Yan is still missing. Using
conventional methods, the role of the Glauber region for DPS was analysed in
Diehl:2015bca .

This work is structured as follows. In section 2 we give an overview of the
factorisation proof for double Drell-Yan production, thus putting the analysis
of the present paper into a larger context. In section 3 we introduce our
notation for Wilson and eikonal lines, and we give the expression of the DPS
amplitude after longitudinally polarised collinear gluons have been decoupled
from the hard scattering. Section 4 shows how soft gluons in that expression
can be decoupled from collinear partons, which is the main result of our
paper. In section 5 we show how the momentum-space eikonal lines obtained in
the previous step can be converted into Wilson lines. We briefly summarise our
work in section 6. In an appendix we derive Ward identities for correlation
functions that involve Wilson lines, which is required for the main proof in
section 4.

##  2 Overview of factorisation proof

The decoupling of soft gluons is one step in the factorisation proof for the
double Drell-Yan process. To put this into context, we briefly sketch the
different steps of the overall proof. We largely follow the presentation in
section 2.1 of Diehl:2015bca , which builds on the factorisation proof for
single Drell-Yan production discussed in chapter 14.4 of Collins:2011zzd .

We consider both collinear factorisation, where the transverse momenta of the
Drell-Yan lepton pairs are integrated over, and TMD factorisation, where they
are measured and small compared with the lepton pair invariant masses. The
factorisation proof is readily extended to other channels in which colourless
particles are produced. Examples of hard scattering processes are $q\bar{q}\to
Z$, $q\bar{q}\mskip 1.5mu^{\prime}\to W$, $gg\to H$ and $gg\to\gamma\gamma$.

  1. 1.

Power counting. The approximations leading to hard scattering factorisation
are based on an expansion in powers of $\Lambda/Q$. Here $\Lambda$ represents
all small scales, i.e. hadron masses and the scale of non-perturbative
interactions. In TMD factorisation this includes the measured transverse
momenta $\boldsymbol{q}_{1}$ and $\boldsymbol{q}_{2}$ of the produced bosons.
The large scale $Q$ stands for the invariant masses $Q_{1}$ and $Q_{2}$ of the
two produced systems. In the present work, we will not keep track of the
specific power of $\Lambda/Q$ by which neglected terms are suppressed.

  2. 2.

Leading graphs and momentum regions. Following the analysis of Libby and
Sterman Libby:1978bx ; Libby:1978nr , one identifies the leading graphs
contributing to the process and the leading momentum regions in each leading
graph. For each leading momentum region, the graph can then be decomposed into
subgraphs associated with hard, collinear or soft momenta. Taking a frame in
which the proton with momentum $P$ moves along $z$ and the proton with
momentum $\bar{P}$ along $-z$, these momenta are characterised as follows:

hard $\displaystyle\qquad|\ell^{+}|\sim|\ell^{-}|\sim|\boldsymbol{\ell}|\sim
Q\,,$ right-collinear $\displaystyle\qquad|\ell^{+}|\sim
Q\,,\quad|\ell^{-}|\ll
Q\,,\quad\boldsymbol{\ell}^{2}\sim|\ell^{+}\ell^{-}|\,,$ left-collinear
$\displaystyle\qquad|\ell^{+}|\ll Q\,,\quad|\ell^{-}|\sim
Q\,,\quad\boldsymbol{\ell}^{2}\sim|\ell^{+}\ell^{-}|\,,$ central soft
$\displaystyle\qquad|\ell^{+}|\sim|\ell^{-}|\sim|\boldsymbol{\ell}|\ll Q\,,$
Glauber soft
$\displaystyle\qquad|\ell^{+}\ell^{-}|\ll\boldsymbol{\ell}^{2}\,,\quad|\boldsymbol{\ell}|\ll
Q\,.$ (1)

Here we use light-cone coordinates $v^{\pm}=(v^{0}\pm v^{3})/\sqrt{2}$ for
each four-vector $v^{\mu}$ and write its transverse part as
$\boldsymbol{v}=(v^{1},v^{2})$. Among soft momenta, the Glauber region plays a
special role, which will be recalled shortly. In (2) we do not specify the
scaling of small momentum components in terms of $\Lambda$ and $Q$, referring
to Diehl:2015bca  and to Collins:2011zzd  for detailed discussions. For the
arguments in the present work, the specification in (2) is sufficient. We note
that the scaling of momenta plays an essential role in the analysis of
collinear or TMD factorisation in SCET, see e.g. Bauer:2002nz ; Becher:2010tm
; Chiu:2012ir .

Important features of leading momentum configurations are that soft momenta
may not enter a hard subgraph, and that the only partons that can enter a soft
subgraph are gluons. Exactly one quark and one antiquark enters each hard
subgraph in the amplitude for Drell-Yan production; for other channels like
Higgs production one has two transversely polarised gluons instead. Additional
gluons may connect a collinear with a hard subgraph.

Figure 1:  Structure of the leading graphs in TMD factorisation for the double
Drell-Yan process. $H_{1}$ and $H_{2}$ denote the combination of hard
subgraphs to the left and to the right of the final-state cut, which is
represented by the vertical dashed line. For the sake of clarity, the
electroweak gauge bosons emerging from $H_{1}$ and $H_{2}$ are not shown here
and in the next figures.

The structure of the leading graphs for the double Drell-Yan process in TMD
factorisation is shown in figure 1. We note that the soft subgraph $S$
connecting $J_{A}$ and $J_{B}$ may consist of several connected parts, but
that each of these must connect $J_{A}$ with $J_{B}$. In collinear
factorisation, the leading graphs are more complicated. The final state can
include unobserved jets produced in one of the hard-scattering processes, and
soft gluons can connect the collinear graphs associated with these jets to the
collinear graphs $J_{A}$ and $J_{B}$ of the incoming protons, as shown for
instance in figure 2 of Diehl:2015bca . A series of arguments is needed to
decouple the soft gluons from the unobserved jets and to remove the associated
collinear subgraphs. These arguments, which in particular make use of
unitarity, proceed in the same manner for single and double Drell-Yan
production (and are very similar to those for inclusive $e^{+}e^{-}$
annihilation into hadrons). As our focus is on double parton scattering, we
shall not expand on this issue here. The leading graphs at the end of this
procedure have the same form as in TMD factorisation, except that the hard
scattering subgraphs extend over the final state cut, as shown in figure 3 of
Diehl:2015bca .

The final state of two heavy bosons plus unspecified hadrons can not only be
produced by double parton scattering (DPS) but also by the conventional single
parton scattering (SPS) mechanism. There are Feynman graphs that have leading
momentum regions both of DPS and of SPS type, which leads to a double counting
issue. A scheme to handle this issue is presented in Diehl:2017kgu . Since the
decoupling of soft gluons is not affected by this, we shall not go into
further detail here.

  3. 3.

Kinematic approximations. For each momentum region, a set of kinematic
approximations is applied to the graph. Corrections to these approximations
are power suppressed. Collinear lines entering a hard subgraph are
approximated by

$\ell\to(\ell^{+},0,\boldsymbol{0})$ (2)

for right-collinear momenta and

$\bar{\ell}\to(0,\ell^{-},\boldsymbol{0})$ (3)

for left-collinear ones, where we write the components of a four-vector as
$(v^{+},v^{-},\boldsymbol{v})$. For soft lines entering a collinear subgraph,
we approximate

$\displaystyle\ell$
$\displaystyle\to\tilde{\ell}=(0,\ell^{-},\boldsymbol{\ell}),$
$\displaystyle\bar{\ell}$
$\displaystyle\to\tilde{\bar{\ell}}=(\ell^{+},0,\boldsymbol{\ell}),$ (4)

where the first line refers to the soft gluons entering the right-collinear
graph $J_{A}$ and the second line to the ones entering the left-collinear
graph $J_{B}$. It is important to note that the approximation in (3) is weaker
than the one in the work by Collins, Soper and Sterman Collins:1985ue ;
Collins:1988ig ; Collins:2011zzd , who set the transverse momentum
$\boldsymbol{\ell}$ on the r.h.s. of (3) to zero. The justification for this
stronger approximation relies on a specific routing of loop momenta, for which
the collinear subgraphs are independent of $\boldsymbol{\ell}$ after soft
gluons have been decoupled. Such a routing is possible for SPS but not for
DPS, as shown for a simple example in section 2.3 of Diehl:2015bca . The
calculation in section 5 shows that keeping track of soft transverse momenta
in the collinear factors is essential for getting the correct position
arguments of Wilson lines in the soft factors. One goal of the present work is
to establish that the weaker approximation in (3) is sufficient for showing
the decoupling of soft gluons.

In practice, loop integrations are performed over the full phase space after
the above approximations have been made, in order to avoid the use of explicit
cutoffs. The contribution from the “unwanted” momentum regions is removed by a
recursive subtraction mechanism, as discussed in detail in Collins:2011zzd
and briefly explained in Diehl:2015bca . These subtractions are also relevant
when subgraphs limited to a certain momentum region are expressed in terms of
operator matrix elements, because the latter imply integration over the full
region of all internal momenta. We return to this issue in point 9 below.

  4. 4.

Dirac and Lorentz indices. Fierz transformations are applied to the fermion
lines connecting collinear with hard subgraphs, and approximations are then
made regarding the leading Lorentz components of the Dirac matrices
$\gamma^{\mu}$ arising from these transformations. Corresponding steps are
taken for the Lorentz indices of gluon fields if the hard scattering is
initiated by gluons. Details are given in section 2.2.1 of Diehl:2011yj  for
double parton scattering. In the present context, they are not important.

  5. 5.

Glauber region. To enable step 7 below, the loop integrations over plus or
minus components of soft momenta are deformed in the complex plane so as to
avoid the Glauber region. To establish that such a deformation is possible
requires a sum over all final state cuts of a given graph and is a crucial
step in any factorisation proof. Detailed analyses are given in Collins:1985ue
; Collins:1988ig ; Collins:2011zzd  and Bodwin:1984hc  for the single and in
Diehl:2015bca  for the double Drell-Yan process. Into which region the Glauber
momenta are to be deformed is subtle, as was pointed out in section 4.1 of
Boer:2017hqr . This issue concerns both single and double hard scattering and
merits further investigation, which shall not be attempted here.

The discussion of complex contour deformation in the papers just cited relies
on the use of Feynman gauge for the gluon field, which we therefore use in the
present work as well.

  6. 6.

Collinear decoupling. Collinear gluons are decoupled from the hard subgraphs,
using so-called Grammer-Yennie approximations Grammer:1973db  and nonabelian
Ward identities. This is discussed for single Drell-Yan production in the
appendix of Collins:1985ue . The argument can be immediately transferred to
the double Drell-Yan process since the Ward identity is applied to a hard
scattering graph, which can be done sequentially for the two hard scatters in
DPS. After this procedure, the collinear subgraphs are modified and contain
eikonal lines. The leading graphs for the double Drell-Yan cross section then
have the form shown in figure 2. More detail is given in section 3.2.

Figure 2:  Structure of the TMD cross section after collinear gluons have been
decoupled from the hard subgraphs.

  7. 7.

Soft decoupling. Soft gluons are decoupled from the collinear subgraphs, again
using Grammer-Yennie approximations and nonabelian Ward identities. In this
case, the latter are applied to the collinear graphs and are therefore more
involved for DPS than for SPS processes. After this step, which is the subject
of the present work, the gluons emerging from the soft subgraph couple to
eikonal lines as shown in figure 3. Both collinear and soft subgraphs are now
decoupled from each other.

Let us briefly review the Grammer-Yennie approximation for a soft gluon with
momentum $\ell$ flowing from the soft into the right-collinear subgraph. We
have

$\displaystyle S_{\mu}(\ell)\,J_{A}^{\mu}(\tilde{\ell})$ $\displaystyle\approx
S^{-}(\ell)\,J_{A}^{+}(\tilde{\ell})=S^{-}(\ell)\,\frac{v_{R}^{+}\,\tilde{\ell}^{-}}{v_{R}^{+}\,\ell^{-}+i\epsilon}\,J_{A}^{+}(\tilde{\ell})$
$\displaystyle\approx
S_{\mu}(\ell)\,\frac{v_{R}^{\mu}\,\tilde{\ell}_{\nu}}{v_{R}\cdot\ell+i\epsilon}\,J_{A}^{\mskip
1.5mu\nu}(\tilde{\ell})\,,$ (5)

where $\tilde{\ell}$ is the approximation (3) of the gluon momentum in
collinear graphs, and $v_{R}=(v_{R}^{+},v_{R}^{-},0,0)$ is a spacelike
auxiliary vector satisfying $v_{R}^{+}\gg-v_{R}^{-}>0$. In (7) we have in
particular used that the components of $J_{A}^{\mu}$ scale like a collinear
momentum. The last step can then be justified for
$|\ell^{-}|\sim|\boldsymbol{\ell}|$ but is not generally valid in the Glauber
region, where one can have $|\ell^{-}|\ll|\boldsymbol{\ell}|$. This is the
reason for first making the complex contour deformation mentioned in step 5.
To avoid propagator poles in both $J_{A}$ and $J_{B}$, this deformation is
into the upper half plane for $\ell^{-}$ and the lower half plane for
$\ell^{+}$. The signs of $i\epsilon$ and of $v_{R}^{+}$ and $v_{R}^{-}$ in (7)
are chosen such that one can deform $\ell^{-}$ and $\ell^{+}$ back to the real
axis without obstruction (see step 8).

To simplify the notation in later sections, we implement the Grammer-Yennie
approximation by replacing the metric tensor in the Feynman gauge gluon
propagator as

$\displaystyle g^{\mu\nu}\to Y_{R}^{\mu\nu}(\ell)$
$\displaystyle=\frac{v_{R}^{\mu}\,\tilde{\ell}^{\mskip
1.5mu\nu}}{v_{R}\cdot\ell+i\epsilon}\,,$ $\displaystyle g^{\mu\nu}\to
Y_{L}^{\mu\nu}(\bar{\ell})$
$\displaystyle=\frac{v_{L}^{\mu}\,\tilde{\bar{\ell}}^{\mskip
1.5mu\nu}}{v_{L}\cdot\bar{\ell}\rule{0.0pt}{9.49997pt}+i\epsilon}$ (6)

for right and left collinear gluons, respectively, where
$v_{L}=(v_{L}^{+},v_{L}^{-},0,0)$ is a spacelike auxiliary vector with
$v_{L}^{-}\gg-v_{L}^{+}>0$. In both cases, $\mu$ is to be contracted with the
soft and $\nu$ with the collinear subgraph, and the soft momentum is routed to
flow out of $S$.

Figure 3:  Structure of the TMD cross section after soft gluons have been
decoupled from the collinear subgraphs.

  8. 8.

Back to real momenta. At this point, the loop integrations deformed into the
complex plane in step 5 can be brought back to the real axis. As we will see
in section 4, the application of Grammer-Yennie approximations and Ward
identities in step 7 is intertwined, so that the deformation back to the real
axis cannot be done before.222This was overlooked in section 2.1 of
Diehl:2015bca  but does not affect the remaining arguments in that work.

  9. 9.

Matrix elements. The collinear and soft subgraphs can be expressed in terms of
operator matrix elements. The eikonal lines in these subgraphs (see figure 3)
are generated by Wilson line operators. This step provides a non-perturbative
definition of the soft and collinear factors, which can indeed not be computed
in perturbation theory.

When representing the collinear factors by matrix elements, one needs to
subtract the contribution from regions in which gluons are soft rather than
collinear. This is handled by the subtraction formalism mentioned above. The
same arguments as in step 7 can be used to simplify the subtraction terms and
express them again in terms of operator matrix elements. This was already
sketched in section 2.2 of Diehl:2015bca  and will be briefly reviewed in
section 4.4 here.

  10. 10.

Rapidity dependence. The introduction of eikonal lines in earlier steps
generates so-called rapidity divergences if it is done in a naive way. These
divergences need to be regularised. Following the work in Diehl:2015bca , we
use the regulator of Collins Collins:2011zzd  and take non-lightlike vectors
in the eikonal lines, as already mentioned above. Other rapidity regulators
have been used in the literature, see e.g. Chiu:2011qc ; Chiu:2012ir ;
Echevarria:2015byo ; Echevarria:2016scs  for SPS and Manohar:2012jr ;
Vladimirov:2016qkd ; Vladimirov:2017ksc  for DPS.

By a nontrivial rearrangement, the soft factor can be split into pieces and
absorbed into the collinear factors. The rapidity regulator can then be
removed, but a dependence on a parameter related to rapidities (often called
$\zeta$) remains in the individual factors. This dependence is closely related
with the resummation of Sudakov logarithms and hence has physical
significance. A construction for DPS was worked out in Buffing:2017mqm ,
generalising the formalism of Collins Collins:2011zzd  for single Drell-Yan
production. Interestingly, DPS cross sections contain Sudakov logarithms not
only in TMD but also in collinear factorisation, in contrast to SPS. This was
already pointed out in Mekhfi:1988kj ; Manohar:2012jr  and is due to the more
complicated colour structure of DPS.

  11. 11.

Ultraviolet renormalisation. Up to this point, all subgraphs contain
ultraviolet divergences, which are now renormalised following standard
procedures. This can be done using dimensional regularisation and
$\overline{\text{MS}}$ subtraction. As is well known from SPS, the details of
renormalisation differ between TMD and collinear factorisation. In both cases,
the result is a factorisation formula for the cross section, in which double
parton distributions contain soft and collinear dynamics, whereas hard
scattering cross sections describe the dynamics associated with hard momenta.

This concludes our broad overview of factorisation for double parton
scattering. In the next section, we recall some details about eikonal and
Wilson lines and set up our notation. After that, we show the decoupling of
soft gluons, which is the main result of this work.

##  3 Wilson lines and decoupling of collinear gluons

###  3.1 Wilson and eikonal lines

The decoupling of collinear and of soft gluons using Grammer-Yennie
approximations and Ward identities leads to subgraphs in which collinear or
soft gluons couple to eikonal lines. The Feynman rules for such lines in a
scattering amplitude are given in figure 4, where we follow the conventions
spelled out in Diehl:2011yj ; Buffing:2017mqm : the arrow on the line
indicates colour flow, whereas the open and closed dots at the end specify the
sign of $i\epsilon$. One readily sees the correspondence between these Feynman
rules and the Grammer-Yennie factor in (6).

Figure 4: Feynman rules for eikonal lines in the fundamental representation of
the colour group.

When collinear or soft subgraphs are represented as operator matrix elements,
the eikonal lines turn into Wilson line operators after summing over all
orders of the strong coupling $g$. Wilson lines in the fundamental colour
representation are defined by

$\displaystyle W_{v,\,ij}(x)$
$\displaystyle=\mathcal{P}\exp\left[ig\\!\int_{-\infty}^{0}\\!\mathrm{d}s\,v\cdot
A^{a}(x+sv)\,t^{a}_{ij}\right],$ (7)

where the matrices $t^{a}$ are the generators of the colour group and
$\mathcal{P}$ denotes path-ordering of colour matrices. Wilson lines
$W_{v}^{bc}(x)$ in the adjoint representation are defined in analogy, with
$t^{a}_{ij}$ in (7) being replaced by $-if^{abc}$. The direction of the path
in (7) is appropriate for Drell-Yan production and results from the sign of
$i\epsilon$ in the Grammer-Yennie approximation (6).

To make the correspondence between Wilson and eikonal lines explicit, we
expand the former in the coupling,

$\displaystyle W_{v,\,ij}(x)$
$\displaystyle=\sum_{n=0}^{\infty}\,g^{n}\,W_{v,\,ij}^{(n)}(x)\,,$ (8)

and perform a Fourier transform to momentum space. For a general operator
$\phi(x)$ in position space, we define the momentum-space version

$\displaystyle\phi(k)$ $\displaystyle=\int\mathrm{d}^{D}{z}\;e^{ik\cdot
x}\,\phi(x)\,.$ (9)

The lowest-order term of the Fourier transformed Wilson line is trivial and
reads $W_{v,\,ij}^{(0)}(k)=\delta_{ij}\,(2\pi)^{D}\,\delta^{(D)}(k)$, whilst
for $n>0$ we have

$\displaystyle W_{v,\,ij}^{(n)}(k)$
$\displaystyle=\int\\!\frac{\mathrm{d}^{D}k_{1}}{(2\pi)^{D}}\dots\frac{\mathrm{d}^{D}k_{n}}{(2\pi)^{D}}\,(2\pi)^{D}\,\delta^{(D)}(k-\sum
k_{i})$
$\displaystyle\quad\times\Bigl{[}\mathcal{E}^{(n)}_{v,\,ij}\bigl{(}\\{k_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{n}}_{\mu_{1}\dots\mu_{n}}\,A^{a_{1}\mu_{1}}(k_{1})\dots
A^{a_{n}\mu_{n}}(k_{n})\,,$ (10)

where

$\displaystyle\Bigl{[}{\mathcal{E}^{(n)}_{v,\,ij}}\bigl{(}\\{k_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\cdots
a_{n}}_{\mu_{1}\cdots\mu_{n}}$ $\displaystyle=\frac{-v_{\mu_{n}}}{k_{n}\cdot
v+i\epsilon}\,\frac{-v_{\mu_{n-1}}}{(k_{n}+k_{n-1})\cdot
v+i\epsilon}\cdots\frac{-v_{\mu_{1}}}{(k_{n}+\cdots+k_{1})\cdot v+i\epsilon}$
$\displaystyle\quad\times\bigl{(}t^{a_{n}}\cdots t^{a_{1}}\bigr{)}_{ij}$ (11)

denotes an eikonal line with $n$ gluon insertions. Here and in the following,
curly brackets in function arguments denote a set of momenta, i.e.
$\\{k_{n}\\}$ stands for $k_{1},\ldots,k_{n}$. For conjugate Wilson lines, we
have

$\displaystyle W_{v,\,ji}^{\dagger(n)}(k)$
$\displaystyle=\int\\!\frac{\mathrm{d}^{D}k_{1}}{(2\pi)^{D}}\dots\frac{\mathrm{d}^{D}k_{n}}{(2\pi)^{D}}\,(2\pi)^{D}\,\delta^{(D)}(k-\sum
k_{i})$
$\displaystyle\quad\times\Bigl{[}\bar{\mathcal{E}}^{(n)}_{v,\,ji}\bigl{(}\\{k_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{n}}_{\mu_{1}\dots\mu_{n}}\,A^{a_{1}\mu_{1}}(k_{1})\dots
A^{a_{n}\mu_{n}}(k_{n})$ (12)

with

$\displaystyle\Bigl{[}\bar{\mathcal{E}}^{(n)}_{v,\,ji}\bigl{(}\\{k_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\cdots
a_{n}}_{\mu_{1}\cdots\mu_{n}}$ $\displaystyle=\frac{v_{\mu_{n}}}{k_{n}\cdot
v+i\epsilon}\,\frac{v_{\mu_{n-1}}}{(k_{n}+k_{n-1})\cdot
v+i\epsilon}\cdots\frac{v_{\mu_{1}}}{(k_{n}+\cdots+k_{1})\cdot v+i\epsilon}$
$\displaystyle\quad\times\bigl{(}t^{a_{1}}\cdots t^{a_{n}}\bigr{)}_{ji}\,,$
(13)

where $W^{\dagger(n)}$ is defined from the perturbative expansion of
$W^{\dagger}$ as in (8). Note that $W^{\dagger(n)}(k)$ is the Fourier
transform (9) of $W^{\dagger(n)}(x)$, i.e. the Fourier exponential is still
$e^{ik\cdot x}$. For simplicity, we will refer to $\bar{\mathcal{E}}$ as
“conjugate eikonal line”. A graphical representation with $n=2$ is given in
figure 5. Note that the sign of $i\epsilon$ is the same in the denominators of
(3.1) and (3.1). It is a consequence of the lower limit $-\infty$ for the $s$
integration in the Wilson line (7) and in its complex conjugate.

Figure 5: Graphical representation of eikonal lines
${\mathcal{E}^{(2)}_{ij}}(k_{1},k_{2})$ and
$\bar{\mathcal{E}}^{(2)}_{ji}(k_{1},k_{2})$ with two gluon insertions, defined
by (3.1) and (3.1).

###  3.2 Decoupling of collinear gluons

The decoupling of collinear gluons from the hard scattering subprocesses leads
from the graphs in figure 1 to those in figure 2. In terms of operators, this
corresponds to replacing the quark and antiquark fields in the definition of
the initial collinear factors according to

$\displaystyle q_{i}(x)$ $\displaystyle\to W_{\\!v_{A},\,ij}(x)\,q_{j}(x),$
$\displaystyle\bar{q}_{i}(x)$
$\displaystyle\to\bar{q}_{j}(x)\,W_{\\!v_{A},\,ji}^{\dagger}(x)$ (14)

for right-collinear partons, where
$v_{A}=(v_{A}^{+},v_{A}^{-},\boldsymbol{0})$ is a spacelike auxiliary vector
with either $v_{A}^{-}\sim-v_{A}^{+}>0$ or $v_{A}^{-}\gg-v_{A}^{+}>0$. If only
gluons are exchanged between a hard and a collinear subgraph, then exactly one
of them carries transverse polarisation, whilst all others are longitudinal
and can be decoupled (contributions with more than one transverse gluons are
power suppressed). A transverse gluon in a right-collinear factor is
represented by a gluon field strength operator $G_{\mu\nu}^{a}$. After
decoupling of the longitudinal gluons, this operator is dressed by an adjoint
Wilson line as

$\displaystyle G_{\mu\nu}^{a}(x)$ $\displaystyle\to
W_{\\!v_{A}}^{ab}(x)\,G_{\mu\nu}^{b}(x)\,.$ (15)

A corresponding discussion holds for left-collinear factors, with an auxiliary
vector $v_{B}$ and interchanged roles of plus and minus components. For
definiteness, we restrict our discussion to fundamental Wilson lines in the
following. The extension to adjoint Wilson lines is straightforward.

We will discuss the decoupling of soft gluons at the level of the process
amplitude rather than the cross section. A right-collinear subgraph with a
quark-antiquark pair and $n$ soft gluons in the amplitude is then given
by333We make no difference between colour indices as sub- or superscripts.

$\displaystyle\Bigl{[}J_{AX}^{i_{1}i_{2}}\bigl{(}l_{1},l_{2};\\{\ell_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{n}}_{\mu_{1}\dots\mu_{n}}=\Biggl{[}\mskip
1.5mu\prod_{k=1}^{n}\,(i\ell_{k}^{2})\Biggr{]}\,\Bra{X\rule{0.0pt}{10.50005pt}}TA_{\mu_{1}}^{a_{1}}(-\ell_{1})\dots
A_{\mu_{n}}^{a_{n}}(-\ell_{n})\;$ $\displaystyle\qquad\times\Gamma_{1}\mskip
1.5mu\bigl{[}W_{\\!A}\,q\bigr{]}_{i_{1}}\\!(l_{1})\;\bigl{[}\bar{q}\mskip
1.5muW^{\dagger}_{\\!\mskip-1.5muA}\mskip
1.5mu\bigr{]}_{i_{2}}\\!(l_{2})\,\Gamma_{2}\Ket{\rule{0.0pt}{10.50005pt}P}-\\{\text{subtraction
terms}\\}\,,$ (16)

where the “subtraction terms” remove the contribution from regions of soft
momenta, as was mentioned in point 3 of section 2 and will be expanded on in
the next section. $P$ is the incoming proton and $X$ the set of spectator
partons going into the final state, and $T$ denotes time ordering, as
appropriate for operators in an amplitude. The specific form of the Dirac
matrices $\Gamma_{1,2}$ is not relevant in our context. For brevity we write
$W_{\\!A}$ instead of $W_{v_{A}}$ for a Wilson line along $v_{A}$ and define
$\bigl{[}\mskip 1.5muW_{\\!A}\,q\mskip 1.5mu\bigr{]}(l)$ as the Fourier
transform (9) of $W_{\\!A}(x)\,q(x)$, which is of course equal to the
momentum-space convolution of $W_{\\!A}(k)$ and $q(l-k)$. The same holds for
$[\mskip 1.5mu\bar{q}\,W_{\\!\mskip-1.5muA}^{\dagger}\mskip 1.5mu](l)$. Notice
that in $J_{AX}$ the collinear momenta $l_{1}$ and $l_{2}$ are taken as
outgoing, whilst the soft momenta $\ell_{i}$ are incoming. The product of
$(i\ell_{k}^{2})$ in (3.2) removes the soft gluon propagators, which are
contained in the soft subgraph. The latter is defined by

$\displaystyle\Bigl{[}S_{Z}\bigl{(}\\{\ell_{n}\\},\\{\bar{\ell}_{\bar{n}}\\}\bigr{)}\Bigr{]}_{\mu_{1}\ldots\mu_{n},\,\nu_{1}\ldots\nu_{\bar{n}}}^{a_{1}\ldots
a_{n},\,b_{1}\ldots b_{\bar{n}}}$
$\displaystyle\qquad=\Braket{Z}{TA_{\mu_{1}}^{a_{1}}(\ell_{1})\dots
A_{\mu_{n}}^{a_{n}}(\ell_{n})\,A_{\nu_{1}}^{b_{1}}(\bar{\ell}_{1})\dots
A_{\nu_{\bar{n}}}^{b_{\bar{n}}}(\bar{\ell}_{\bar{n}})}{0}$ (17)

for the process amplitude, where $Z$ denotes the set of partons going across
the final state cut. We note that both $J_{AX}$ and $S_{Z}$ are defined such
that they include an overall delta function for momentum conservation.

The combination of soft and collinear factors to the left of the final state
cut in figure 2 is given by

$\displaystyle\mathcal{L}_{XYZ}^{i_{1}i_{2}\,\bar{\imath}_{1}\bar{\imath}_{2}\,(n,\bar{n})}(q_{1},q_{2})=\frac{1}{n!\,\bar{n}!}\,\int\\!\frac{\mathrm{d}^{D}l_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}l_{2}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{2}}{(2\pi)^{D}}\;\prod_{k=0}^{n}\,\prod_{\bar{k}=0}^{\bar{n}}\int\\!\frac{\mathrm{d}^{D}\ell_{k}}{(2\pi)^{D}}\frac{\mathrm{d}^{D}\bar{\ell}_{\bar{k}}}{(2\pi)^{D}}\,$
$\displaystyle\quad\times(2\pi)^{D}\,\delta^{(D)}\left(q_{1}-l_{1}-\bar{l}_{1}\right)\,(2\pi)^{D}\,\delta^{(D)}\left(q_{2}-l_{2}-\bar{l}_{2}\right)\Bigl{[}J_{AX}^{i_{1}i_{2}}\bigl{(}l_{1},l_{2};\\{\tilde{\ell}_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{n}}_{\mu_{1}\dots\mu_{n}}\;$
$\displaystyle\quad\times\Bigl{[}S_{Z}\bigl{(}\\{\ell_{n}\\},\\{\bar{\ell}_{\bar{n}}\\}\bigr{)}\Bigr{]}^{\mu_{1}\ldots\mu_{n},\,\nu_{1}\ldots\nu_{\bar{n}}}_{a_{1}\ldots
a_{n},\,b_{1}\ldots
b_{\bar{n}}}\;\Bigl{[}J_{BY}^{\bar{\imath}_{1}\bar{\imath}_{2}}\bigl{(}\bar{l}_{1},\bar{l}_{2};\\{\tilde{\bar{\ell}}_{\bar{n}}\\}\bigr{)}\Bigr{]}^{b_{1}\dots
b_{\bar{n}}}_{\nu_{1}\dots\nu_{\bar{n}}}\,,$ (18)

where $q_{1}$, $q_{2}$ are the momenta of the produced gauge bosons. The left-
collinear factor $J_{BY}$ is defined in analogy to (3.2). Soft gluon momenta
entering collinear factors are approximated according to (3). In the term with
$n=\bar{n}=0$ we have a collinear factor $J_{AX}^{i_{1}i_{2}}(l_{1},l_{2})$,
defined as in (3.2) without any external gluon fields, its analogue
$J_{BY}^{j_{1}j_{2}}(\bar{l}_{1},\bar{l}_{2})$, and a trivial factor
$S_{Z}=1$.

It is understood that all gluon momenta $\ell_{k}$ and $\bar{\ell}_{\bar{k}}$
in (3.2) are central soft, because a deformation out of the Glauber region has
already been performed as discussed in section 2. One can therefore insert
Grammer-Yennie factors as specified in (6):

$\displaystyle\Bigl{[}J_{AX}^{i_{1}i_{2}}\bigl{(}l_{1},l_{2};\\{\tilde{\ell}_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{n}}_{\mu_{1}\dots\mu_{n}}\;\Bigl{[}S_{Z}\bigl{(}\\{\ell_{n}\\},\\{\bar{\ell}_{\bar{n}}\\}\bigr{)}\Bigr{]}^{\mu_{1}\ldots\mu_{n},\,\nu_{1}\ldots\nu_{\bar{n}}}_{a_{1}\ldots
a_{n},\,b_{1}\ldots
b_{\bar{n}}}\;\Bigl{[}J_{BY}^{\bar{\imath}_{1}\bar{\imath}_{2}}\bigl{(}\bar{l}_{1},\bar{l}_{2};\\{\tilde{\bar{\ell}}_{\bar{n}}\\}\bigr{)}\Bigr{]}^{b_{1}\dots
b_{\bar{n}}}_{\nu_{1}\dots\nu_{\bar{n}}}$ $\displaystyle\quad\approx
Y_{R}^{\mu_{1}\alpha_{1}}(\tilde{\ell}_{1})\,\dots\,Y_{R}^{\mu_{n}\alpha_{n}}(\tilde{\ell}_{n})\,\Bigl{[}J_{AX}^{i_{1}i_{2}}\bigl{(}l_{1},l_{2};\\{\tilde{\ell}_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{n}}_{\alpha_{1}\dots\alpha_{n}}\;\Bigl{[}S_{Z}\bigl{(}\\{\ell_{n}\\},\\{\bar{\ell}_{\bar{n}}\\}\bigr{)}\Bigr{]}_{\mu_{1}\ldots\mu_{n},\,\nu_{1}\ldots\nu_{\bar{n}}}^{a_{1}\ldots
a_{n},\,b_{1}\ldots b_{\bar{n}}}\;$ $\displaystyle\qquad\times
Y_{L}^{\nu_{1}\beta_{1}}(\tilde{\bar{\ell}}_{1})\,\dots\,Y_{R}^{\nu_{n}\beta_{n}}(\tilde{\bar{\ell}}_{\bar{n}})\,\Bigl{[}J_{BY}^{\bar{\imath}_{1}\bar{\imath}_{2}}\bigl{(}\bar{l}_{1},\bar{l}_{2};\\{\tilde{\bar{\ell}}_{\bar{n}}\\}\bigr{)}\Bigr{]}^{b_{1}\dots
b_{\bar{n}}}_{\beta_{1}\dots\beta_{\bar{n}}}\,.$ (19)

A remark on combinatorics is in order here. The Green functions defined by
operator matrix elements in (3.2) and (3.2) both contain a sum over all
combinations of coupling the gluons with momenta $\ell_{1},\ldots\ell_{n}$ to
the internal lines of the Green function. Contracting them (and integrating
over the gluon momenta) one over-counts by a factor of $n!$ the combinations
of connecting all internal lines of $J_{AX}$ with all internal lines of
$S_{X}$ by $n$ gluons. This factor $n!$ is therefore divided out in (3.2). The
same argument holds of course for the factor $\bar{n}!$.

The amplitude for the double Drell-Yan process is obtained by multiplying
(3.2) with the hard subgraphs. One then needs to sum over all $n$ and
$\bar{n}$, over all relevant combinations of quarks and antiquarks that can be
exchanged between the collinear and hard subgraphs, and over their colour and
polarisation. In the next section, we will show how to decouple the soft
gluons exchanged between subgraphs $J_{A}$ and $S$ in the amplitude. Applying
the same argument to the conjugate amplitude and to the soft gluons exchanged
between $J_{B}$ and $S$, one arrives at the expression represented in figure
3. Further steps towards the final factorisation formula will be discussed in
section 5.

##  4 Soft gluon decoupling

We now show how to decouple soft gluons from collinear factors in double
parton scattering. The steps in this section closely follow the arguments
given for single parton scattering in Collins:1988ig . The main differences
between our treatment and the one in Collins:1988ig  are either due to the
difference in kinematic approximations mentioned below (3) or due to the fact
that for DPS the collinear subgraphs have two external parton lines instead of
one, which in particular requires one to keep track of colour indices. Another
difference between the two treatments is pointed out below.

On the r.h.s. of (3.2), the gluon polarisation indices of $J_{AX}$ are all
contracted with the corresponding approximated gluon momenta, i.e. we have a
factor of the form

$\displaystyle\ell_{1}^{\mu_{1}}\dots\ell_{n}^{\mu_{n}}\;\Bigl{[}J_{AX}^{i_{1}i_{2}}\bigl{(}l_{1},l_{2};\\{\ell_{n}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{n}}_{\mu_{1}\dots\mu_{n}}\,.$ (20)

For better legibility, we omit the tilde for the approximation (3) of soft
momenta in the remainder of this section, i.e. it is understood that

$\displaystyle\ell_{k}^{+}$ $\displaystyle=0$
$\displaystyle\text{~{}for~{}}1\leq k\leq n\,.$ (21)

The form (20) allows us to use the Ward identity

$\displaystyle\ell_{1}^{\mu_{1}}\dots\ell_{n}^{\mu_{n}}\;\Bra{X\rule{0.0pt}{10.50005pt}}TA_{\mu_{1}}^{a_{1}}(-\ell_{1})\dots
A_{\mu_{n}}^{a_{n}}(-\ell_{n})\;\Gamma_{1}\bigl{[}W_{\\!A}\,q\bigr{]}_{i_{1}}(l_{1})\;\bigl{[}\bar{q}\mskip
1.5muW^{\dagger}_{A}\bigr{]}_{i_{2}}(l_{2})\Gamma_{2}\Ket{\rule{0.0pt}{10.50005pt}P}$
$\displaystyle=0\,.$ (22)

This requires, however, a careful discussion because $J_{AX}$ is a _collinear_
subgraph and not a full Green function as appears in (22). This means that an
incoming soft gluon in (20) couples _directly_ to an _internal_ line inside
$J_{AX}$, which by construction carries a collinear momentum. This has two
implications.

  1. 1.

We need the Ward identity not for the full Green function (where internal loop
momenta are integrated over the full phase space) but for the case in which
the internal momenta in the Green function are all fixed. To be more precise,
consider a particular collinear graph with external lines collinear and on
shell, and with all internal momenta collinear and fixed. Connecting $n$
external gluons to this graph in all possible ways and contracting gluon
polarisation indices with the corresponding gluon momenta gives zero. This
form of Ward identities can be proven in a diagrammatic approach
tHooft:1971akt ; tHooft:1972qbu .

An additional complication comes from the presence of the operators
$W_{\\!\mskip-1.5muA}\,q$ and $\bar{q}\mskip
1.5muW^{\dagger}_{\\!\mskip-1.5muA}$ in (22), which are constructed from
nonlocal products of fields in position space and do not correspond to on-
shell external collinear lines. This case was not considered in the work just
quoted. In the appendix we prove (22) for the full Green function using
functional methods. An extension to the case of graphs with fixed internal
momenta is beyond the scope of this work.

  2. 2.

The Green function in (22) contains graphs not included in (20), namely graphs

    1. (a)

in which two or more soft gluons interact with each other before attaching to
an internal collinear line,

    2. (b)

or in which one or more soft gluon couple to the eikonal lines generated by
the operators $W_{\\!\mskip-1.5muA}$ or $W^{\dagger}_{\\!\mskip-1.5muA}$,
either directly or after interacting with each other,

or in which the two previous cases are combined. Applying a Ward identity to
(20) thus schematically gives

$\displaystyle\ell_{1}^{\mu_{1}}\dots\ell_{n}^{\mu_{n}}\;\Bigl{[}J_{AX}\bigl{(}l_{1},l_{2};\\{\ell_{n}\\}\bigr{)}\Bigr{]}_{\mu_{1}\dots\mu_{n}}$
$\displaystyle\quad=-\ell_{1}^{\mu_{1}}\dots\ell_{n}^{\mu_{n}}\;\Bigl{[}\,\text{graphs
(a)}+\text{graphs (b)}+\text{combinations of (a) and (b)}\,\Bigr{]}\,.$ (23)

The treatment of soft gluon interactions is simplified by the following
considerations. Take the collinear graph without external soft gluons
mentioned in point 1 to be of order $g^{c}$ in the strong coupling, and
restrict the corresponding Ward identity for $n$ external gluon attachments to
the smallest possible order, which is $g^{c+n}$. Interactions between the $n$
incoming gluons can result in $m$ soft gluons, each of which then couples to a
collinear line. The latter gives a factor $g^{m}$, so that the interactions
among the soft gluons are of order $g^{n-m}$. It immediately follows that
$m\leq n$. One finds that transitions from $n$ to $m$ gluons of order
$g^{n-m}$ are only realised by tree graphs with sequential transitions from
$2\to 1$ or from $3\to 1$ gluons, as shown in figure 6(a). Any further
interactions, as in figure 6(b) come along with further powers of $g$ and are
hence excluded. The result is that the graphs with soft gluon interactions are
restricted to a disconnected set of transitions from $n_{i}\to 1$ gluons, with
$n_{i}\geq 2$ and $\sum_{i}n_{i}\leq n$. We note that the proof in
Collins:1988ig  did not restrict the order of the coupling and used a
different argument to eliminate soft gluon interactions involving loops.

(a) (b)

Figure 6: Example graphs for $n\to m$ gluon transitions of order $g^{n-m}$ (a)
and of higher order (b).

###  4.1 Leading power approximations

The r.h.s. of (2) admits two types of approximation, valid at leading order in
$\Lambda/Q$, which we now discuss in turn.

#### Soft gluon interactions

Consider a graph with gluon self interactions, as characterised in the
previous subsection. We denote the tree-level graphs for $m\to 1$ gluon
transitions by $V^{(m)}(\\{\ell_{m}\\})$, where the argument specifies the $m$
incoming gluon momenta. The outgoing momentum is of course
$\ell=\sum_{k=1}^{m}\ell_{k}$. The propagators of the incoming gluons are
truncated in $V^{(m)}$, but not the one of the outgoing gluon.

In the graphs on the r.h.s. of (2), a $m\to 1$ vertex is contracted with the
incoming gluon momenta. Due to Lorentz invariance, this contraction can be
written as

$\displaystyle\Bigl{[}\tilde{V}^{(m)}\bigl{(}\\{\ell_{m}\\}\bigr{)}\Bigr{]}_{\nu}=\ell_{1}^{\mu_{1}}\dots\ell_{m}^{\mu_{m}}\,\Bigl{[}V^{(m)}\bigl{(}\\{\ell_{m}\\}\bigr{)}\Bigr{]}_{\mu_{1}\dots\mu_{m}\mskip
1.5mu\nu}$
$\displaystyle=\sum_{k=1}^{m}(\ell_{k})_{\nu}\;F^{(m,k)}\bigl{(}\\{\ell_{m}\\}\bigr{)}\,,$
(24)

where $F^{(m,k)}$ is a set of scalar functions and we drop colour indices for
brevity. The index $\nu$ refers to the outgoing gluon and is to be contracted
with the Lorentz index of a collinear subgraph (for graphs a) or of an eikonal
line along $v_{A}$ (for graphs b). In both cases, one only needs the minus
component of $\tilde{V}^{\nu}$ in the contraction. In the latter case this is
because $\ell_{k}\cdot v_{A}=\ell_{k}^{-}\,v_{A}^{+}$ for momenta satisfying
(21), and in the former case this follows from the same argument as in the
original Grammer-Yennie approximation (7). Using the form (24), one readily
obtains

$\displaystyle\Bigl{[}\tilde{V}^{(m)}\bigl{(}\\{\ell_{m}\\}\bigr{)}\Bigr{]}^{-}$
$\displaystyle=\sum_{k=1}^{m}\ell_{k}^{-}\,F^{(m,k)}\bigl{(}\\{\ell_{m}\\}\bigr{)}\,\frac{v_{R}^{+}\,\ell^{-}}{v_{R}\cdot\ell+i\epsilon}=\Bigl{[}\tilde{V}^{(m)}\bigl{(}\\{\ell_{m}\\}\bigr{)}\Bigr{]}_{\mu}\,Y_{R}^{\mu-}(\ell)\,.$
(25)

This means that, because the incoming gluons in a $m\to 1$ gluon vertex come
with a Grammer-Yennie factor, one may insert the same factor again for the
final gluon. We represent this result graphically in figure 7.

Figure 7:  Graphical representation of the approximation (25). The line at the
bottom can be either a right-collinear parton or an eikonal line along
$v_{A}$. Here and in the following, an arrow on a gluon line represents the
insertion of a Grammer-Yennie factor (6), with the arrow on the side of the
momentum vector. Round blobs denote the tree-level vertices $V^{(m)}$
specified in the text.

We note that the previous argument drastically simplifies if one can use the
soft momentum approximation of Collins et al. Collins:1988ig , described after
(3). The invariant functions $F^{(m,k)}$ in (24) are homogeneous polynomials
in the scalar products of soft momenta, which readily follows from the Feynman
rules for the three- and four-gluon vertices. In the approximation of
Collins:1988ig  one keeps only the minus components of these momenta, so that
the scalar products and hence all $F^{(m,k)}$ are zero. Soft gluon
interactions need then not be considered at all. However, as already noted,
this approximation can be used for SPS but not for DPS graphs.

#### Soft gluon attachments to eikonal lines

We now turn to soft gluon attachments to eikonal lines generated by the
operators $W_{\\!A}$ or $W^{\dagger}_{A}$ in (3.2). It is easy to see that
only those graphs are dominant in which first all soft gluons attach to the
eikonal line and then all collinear ones, as illustrated in figure 8(a). To
see this, we note that for collinear momenta $k_{i}$ one has $v_{A}\cdot
k\approx v_{A}^{-}\,k^{+}$ and for soft momenta $\ell$ approximated as in (21)
one has $v_{A}\cdot\ell=v_{A}^{+}\,\ell^{-}$, so that

$\displaystyle|v_{A}\cdot k\mskip 1.5mu|$
$\displaystyle\gg|v_{A}\cdot\ell|\,.$ (26)

Comparing the eikonal propagators for the two orderings of attaching a soft
and a collinear gluon in figure 8(a) and 8(b), we thus obtain

$\displaystyle\biggl{|}\frac{1}{v_{A}\cdot\ell}\;\frac{1}{v_{A}\cdot(k+\ell)}\biggr{|}\gg\biggl{|}\frac{1}{v_{A}\cdot
k}\;\frac{1}{v_{A}\cdot(k+\ell)}\biggr{|}\,.$ (27)

The argument readily generalises to several collinear and soft momenta.

(a) (b)

Figure 8: The two possibilities of attaching a soft and a collinear gluon to
an eikonal line.

Using (26), one can further split the dominant terms into two eikonal factors.
For one soft and one collinear gluon, one has

$\displaystyle\Bigl{[}\mathcal{E}^{(2)}_{A,ij}(k,\ell)\Bigr{]}_{ab}^{\mu\nu}$
$\displaystyle=\frac{-v_{A}^{\nu}}{v_{A}\cdot\ell+i\epsilon}\,t^{b}_{ih}\;\frac{-v_{A}^{\mu}}{v_{A}\cdot(\ell+k)+i\epsilon}\,t^{a}_{hj}\approx\frac{-v_{A}^{\nu}}{v_{A}\cdot\ell+i\epsilon}\;t^{b}_{ih}\;\frac{-v_{A}^{\mu_{n}}}{v_{A}\cdot
k+i\epsilon}\,t^{a}_{hj}$
$\displaystyle=\Bigl{[}\mathcal{E}^{(1)}_{A,ih}(\ell)\Bigr{]}_{b}^{\nu}\;\Bigl{[}\mathcal{E}^{(1)}_{A,hj}(k)\Bigr{]}_{a}^{\mu}\,.$
(28)

In analogy, one obtains

$\displaystyle\Bigl{[}\mathcal{E}^{(m+n)}_{A,ij}\bigl{(}\\{k_{m}\\},\\{\ell_{n}\\}\bigr{)}\Bigr{]}_{a_{1}\dots
a_{m}b_{1}\dots b_{n}}^{\mu_{1}\dots\mu_{m}\nu_{1}\dots\nu_{n}}$
$\displaystyle\approx\Bigl{[}\mathcal{E}^{(n)}_{A,ih}\bigl{(}\\{\ell_{n}\\})\Bigr{]}_{b_{1}\dots
b_{n}}^{\nu_{1}\dots\nu_{n}}\;\Bigl{[}\mathcal{E}^{(m)}_{A,hj}\bigl{(}\\{k_{m}\\}\bigr{)}\Bigr{]}_{a_{1}\dots
a_{m}}^{\mu_{1}\cdots\mu_{m}}\,,$ (29)

as well as the relation

$\displaystyle\Bigl{[}\bar{\mathcal{E}}^{(m+n)}_{A,ji}\bigl{(}\\{k_{m}\\},\\{\ell_{n}\\}\bigr{)}\Bigr{]}_{a_{1}\dots
a_{m}b_{1}\dots b_{n}}^{\mu_{1}\dots\mu_{m}\nu_{1}\dots\nu_{n}}$
$\displaystyle\approx\Bigl{[}\bar{\mathcal{E}}^{(m)}_{A,jh}\bigl{(}\\{k_{m}\\}\bigr{)}\Bigr{]}_{a_{1}\dots
a_{m}}^{\mu_{1}\cdots\mu_{m}}\;\Bigl{[}\bar{\mathcal{E}}^{(n)}_{A,hi}\bigl{(}\\{\ell_{n}\\}\bigr{)}\Bigr{]}_{b_{1}\dots
b_{n}}^{\nu_{1}\dots\nu_{n}}$ (30)

for conjugate eikonal lines. In both cases one can replace the original
expression with the product of two eikonal lines, one with only soft momenta
and one with only collinear ones.

###  4.2 Decoupling of one soft gluon

Let us now consider the case where a single soft gluon couples to the
collinear subgraph. This is the base case for the inductive proof in section
4.3. We first treat single parton scattering, and then show how it generalises
to DPS. It will become clear that generalisation to $N$ hard scatters is
straightforward.

####  4.2.1 Single parton scattering

The collinear factor for SPS in the amplitude is defined as in (3.2), but with
one parton operator instead of two, which for definiteness we take as
$W_{\\!A}\,q$. Starting point of our consideration is the expression

$Y_{R}^{\mu\nu}(\ell)\,\Bigl{[}J_{AX}^{i}(l;\ell)\Bigr{]}^{b}_{\nu}\,,$ (31)

which includes the Grammer-Yennie factor (6) and is illustrated in the first
line of figure 9. Since $Y_{R}^{\mu\nu}(\ell)\propto\ell^{\nu}$, we can employ
the SPS analogue of Ward identity (2) for $n=1$. The only graph contributing
to the r.h.s. of this identity is shown in the second line of the figure and
has the soft gluon coupling at the end of the eikonal line along $v_{A}$.

Figure 9:  Graphical representation for the decoupling of a single soft gluon
from a collinear graph in the amplitude of an SPS process. In the last row,
eikonal lines point along $v_{A}$ and $v_{R}$ as indicated. The dashed line
between the two eikonal lines is drawn to visualise the contraction of colour
indices.

Let us consider a graph with $m$ collinear gluons coupling to the eikonal line
along $v_{A}$, in addition to the soft gluon. Contracting with the Grammer-
Yennie factor of the latter, we obtain

$\displaystyle
Y_{R}^{\mu\nu}(\ell)\,\Bigl{[}\mathcal{E}^{(m+1)}_{A,ij}\bigl{(}\\{k_{m}\\},\ell\bigr{)}\Bigr{]}^{a_{1}\dots
a_{m}b}_{\mu_{1}\dots\mu_{m}\nu}$
$\displaystyle\approx\frac{v_{R}^{\mu}}{v_{R}\cdot\ell+i\epsilon}\,\frac{-v_{A}\cdot\ell}{v_{A}\cdot\ell+i\epsilon}\;t^{b}_{ih}\,\Bigl{[}\mathcal{E}^{(m)}_{A,hj}\bigl{(}\\{k_{m}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{m}}_{\mu_{1}\dots\mu_{m}}$
$\displaystyle={}-\Bigl{[}\bar{\mathcal{E}}^{(1)}_{R,ih}(\ell)\bigr{)}\Bigr{]}^{\mu}_{b}\;\Bigl{[}\mathcal{E}^{(m)}_{A,hj}\bigl{(}\\{k_{m}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{m}}_{\mu_{1}\dots\mu_{m}}\,.$ (32)

In the first step, we used the relation (29) with $n=1$ and explicitly wrote
out $Y_{R}$ and $\mathcal{E}^{(1)}_{A}$, and in the second step we used the
representation (3.1) of a conjugate eikonal line. The reason for taking
$-\bar{\mathcal{E}}_{R}$ rather than $\mathcal{E}_{R}$ at this point will only
become clear when considering more than one soft gluon attachments. Combining
the minus sign in the Ward identity with the one in (4.2.1), we obtain

$\displaystyle
Y_{R}^{\mu\nu}(\ell)\,\Bigl{[}J_{AX}^{i}(l;\ell)\Bigr{]}^{b}_{\nu}$
$\displaystyle\approx\Bigl{[}\bar{\mathcal{E}}^{(1)}_{R,ih}(\ell)\bigr{)}\Bigr{]}^{\mu}_{b}\;J_{AX}^{h}(l-\ell)\,,$
(33)

where we have summed over all numbers $m=0,1,\dots$ of collinear gluons in
(4.2.1). On the r.h.s. of (33), the eikonal line $\mathcal{E}^{(m)}_{A}$ has
become part of the collinear factor $J_{AX}(l-\ell)$ with no external soft
gluon. The shift of the collinear momentum argument from $l$ to $l-\ell$
follows from momentum conservation, as depicted graphically in figure 9. The
soft gluon is now completely decoupled from the collinear factor, except for
the contraction of colour indices shown in the last line of the figure.

####  4.2.2 Double parton scattering

In the case of double parton scattering we have two eikonal lines along
$v_{A}$ in the collinear subgraph, each belonging to one of the partons
entering a hard scattering in the overall process. To be specific, we consider
the case of a quark and an antiquark here, so to exhibit the difference
between eikonal lines $\mathcal{E}$ and $\bar{\mathcal{E}}$. A graphical
representation of the argument is given in figure 10.

Figure 10: Decoupling of a single soft gluon from a collinear graph in the
amplitude of a DPS process.

Starting point in the DPS case is the expression

$Y_{R}^{\mu\nu}(\ell)\,\Bigl{[}J_{AX}^{i_{1}i_{2}}(l_{1},l_{2};\ell)\Bigr{]}^{b}_{\nu}\,,$
(34)

and the main difference to SPS is that there are now two graphs of type (a)
that appear at the r.h.s. of the Ward identity (2). If the soft gluon couples
to $\mathcal{E}_{\\!A}$, we use again (4.2.1), and if it couples to
$\bar{\mathcal{E}}_{\\!A}$, we use its analogue

$\displaystyle
Y_{R}^{\mu\nu}(\ell)\,\Bigl{[}\bar{\mathcal{E}}^{(m+1)}_{A,ji}\bigl{(}\\{k_{m}\\},\ell\bigr{)}\Bigr{]}^{a_{1}\dots
a_{m}b}_{\mu_{1}\dots\mu_{m}\nu}$
$\displaystyle\approx\Bigl{[}\bar{\mathcal{E}}^{(m)}_{A,jh}\bigl{(}\\{k_{m}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{m}}_{\mu_{1}\dots\mu_{m}}\,\frac{v_{R}^{\mu}}{v_{R}\cdot\ell+i\epsilon}\,\frac{v_{A}\cdot\ell}{v_{A}\cdot\ell+i\epsilon}\;t^{b}_{hi}\,$
$\displaystyle={}-\Bigl{[}\bar{\mathcal{E}}^{(m)}_{A,jh}\bigl{(}\\{k_{m}\\}\bigr{)}\Bigr{]}^{a_{1}\dots
a_{m}}_{\mu_{1}\dots\mu_{m}}\,\Bigl{[}{\mathcal{E}}^{(1)}_{R,hi}(\ell)\bigr{)}\Bigr{]}^{\mu}_{b}\,,$
(35)

which is readily obtained from (30). Summing over the two contributing graphs,
we obtain

$\displaystyle
Y_{R}^{\mu\nu}(\ell)\,\Bigl{[}J_{AX}^{i_{1}i_{2}}(l_{1},l_{2};\ell)\Bigr{]}^{b}_{\nu}$
$\displaystyle\approx\Bigl{[}\bar{\mathcal{E}}^{(1)}_{R,i_{1}h}(\ell)\Bigr{]}^{\mu}_{b}\;J_{AX}^{h\mskip
1.5mui_{2}}(l_{1}-\ell,l_{2})$
$\displaystyle\quad+J_{AX}^{i_{1}h}(l_{1},l_{2}-\ell)\;\Bigl{[}{\mathcal{E}}^{(1)}_{R,h\mskip
1.5mui_{2}}(\ell)\Bigr{]}^{\mu}_{b}\,.$ (36)

On the r.h.s. we have collinear factors with no external soft gluons, with
different shifts of the collinear momentum arguments arising from momentum
conservation. We note in passing that (4.2.2) corresponds to equation (3.22)
in Diehl:2011yj , which was verified for two explicit examples but not
generally proven in that work.

###  4.3 Multiple soft gluon attachments

The formula for the decoupling of $n$ soft gluons from the collinear graph
reads

$\displaystyle
Y_{R}^{\mu_{1}\nu_{1}}(\ell_{1})\,\dots\,Y_{R}^{\mu_{n}\nu_{n}}(\ell_{n})\,\Bigl{[}J_{AX}^{i_{1}i_{2}}\bigl{(}l_{1},l_{2};\\{\ell_{n}\\}\bigr{)}\Bigr{]}^{b_{1}\dots
b_{n}}_{\nu_{1}\dots\nu_{n}}=\sum_{\pi,\sigma_{1},\sigma_{2}\atop|\sigma_{1}|+|\sigma_{2}|=n\rule{0.0pt}{2.58331pt}}\Bigl{[}\bar{\mathcal{E}}^{(|\sigma_{1}|)}_{R,i_{1}h_{1}}\bigl{(}\pi\\{\ell\\}_{\sigma_{1}}\bigr{)}\Bigr{]}^{\pi\\{\mu\\}_{\sigma_{1}}}_{\pi\\{b\\}_{\sigma_{1}}}\;$
$\displaystyle\qquad\quad\times
J_{AX}^{h_{1}h_{2}}(l_{1}-\ell_{\sigma_{1}},l_{2}-\ell_{\sigma_{2}})\;\Bigl{[}{\mathcal{E}}^{(|\sigma_{2}|)}_{R,h_{2}i_{2}}\bigl{(}\pi\\{\ell\\}_{\sigma_{2}}\bigr{)}\Bigr{]}^{\pi\\{\mu\\}_{\sigma_{2}}}_{\pi\\{b\\}_{\sigma_{2}}}\,,$
(37)

where the equality holds up to power corrections. On the r.h.s. the $n$ soft
gluon momenta have been partitioned into two sets $\sigma_{1}$ and
$\sigma_{2}$ with $|\sigma_{1}|$ and $|\sigma_{2}|$ elements, respectively. By
$\pi\\{\ell\\}_{\sigma_{i}}$ we denote a particular permutation $\pi$ of the
momenta in the set, with $\pi\\{\mu\\}_{\sigma_{i}}$ and
$\pi\\{b\\}_{\sigma_{i}}$ being the corresponding permutations of Lorentz and
colour indices. The sum runs over all partitions $\sigma_{1}$, $\sigma_{2}$
and over all permutations of their respective members. The sum of soft momenta
in set $i$ is denoted by $\ell_{\sigma_{i}}$. For $n=1$, the above expression
simplifies to (4.2.2), as it must be.

Figure 11: Decoupling of $n$ soft gluons from a collinear graph in the
amplitude of a DPS process. The sums over different possibilities in the last
three steps are specified in the text.

The l.h.s. of (4.3) is represented in the first row of figure 11, and the
r.h.s. of the equation in the last row of the figure. We give a proof by
induction with $n=1$ as base case. It proceeds as follows.

  1. 1.

Apply the Ward identity (2) to the graph with $n$ external soft gluons.

An example of the graphs contributing to the r.h.s. of (2) is shown in the
second row of figure 11. The soft gluons couple either to internal collinear
lines of the graph or to one of the two eikonal lines along $v_{A}$. The
coupling can be either direct or involve the merging of $m\to 1$ gluons in
tree-level vertices $V^{(m)}$. Only the case in which all $n$ gluons couple
directly to the internal collinear lines is excluded from the sum, since this
appears on the l.h.s. of the Ward identity.

To characterise the different graphs, we introduce a partition of the $n$ soft
gluons into three sets, $\sigma$, $\tau_{1}$ and $\tau_{2}$, such that the
gluons in set $\sigma$ couple to internal collinear lines and those in
$\tau_{i}$ couple to the eikonal line associated with the first or the second
parton, respectively.

  2. 2.

For each of the diagrams obtained, apply the approximations discussed in
section 4.1.

This implies neglecting graphs in which soft gluons couple in the middle of an
eikonal line along $v_{A}$. It also implies factoring the collinear and soft
parts of the eikonal lines for those graphs in which all soft gluons couple at
the end of a line, see (29) and (30). Furthermore, as specified in (25), a
Grammer-Yennie factor $Y_{R}$ is inserted for the outgoing gluon for each of
the vertices $V^{(m)}$. We then have

$\displaystyle
Y_{R}^{\mu_{1}\nu_{1}}(\ell_{1})\,\dots\,Y_{R}^{\mu_{n}\nu_{n}}(\ell_{n})\,\Bigl{[}J_{AX}^{i_{1}i_{2}}\bigl{(}l_{1},l_{2};\\{\ell_{n}\\}\bigr{)}\Bigr{]}^{b_{1}\dots
b_{n}}_{\nu_{1}\dots\nu_{n}}$
$\displaystyle\qquad={}-Y_{R}^{\mu_{1}\nu_{1}}(\ell_{1})\,\dots\,Y_{R}^{\mu_{n}\nu_{n}}(\ell_{n})\,\sum_{\pi,\sigma,\tau_{1},\tau_{2}\atop|\tau_{1}|+|\tau_{2}|>0\rule{0.0pt}{2.58331pt}}\Bigl{[}{\mathcal{E}}^{(|\tau_{1}|)}_{A,i_{1}h_{1}}\bigl{(}\pi\\{\ell\\}_{\tau_{1}}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\tau_{1}}}^{\pi\\{b\\}_{\tau_{1}}}\;$
$\displaystyle\qquad\qquad\times\Bigl{[}J_{AX}^{h_{1}h_{2}}\bigl{(}l_{1}-\ell_{\tau_{1}},l_{2}-\ell_{\tau_{2}};\pi\\{\ell\\}_{\sigma}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\sigma}}^{\pi\\{b\\}_{\sigma}}\,\Bigl{[}{\bar{\mathcal{E}}}^{(|\tau_{2}|)}_{A,h_{2}i_{2}}\bigl{(}\pi\\{\ell\\}_{\tau_{2}}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\tau_{2}}}^{\pi\\{b\\}_{\tau_{2}}}$
$\displaystyle\qquad\quad+\\{\text{terms with vertices }V^{(m)}\\}\,.$ (38)

The terms with vertices $V^{(m)}$ are obtained from those explicitly given by
selecting subsets $\ell_{i}\ldots\ell_{j}$ of gluon momenta in each of the
partitions $\sigma$, $\tau_{1}$ and $\tau_{2}$ and replacing the associated
Grammer-Yennie factors as

$\displaystyle Y_{R}^{\mu_{i}\nu_{i}}\dots Y_{R}^{\mu_{j}\nu_{j}}$
$\displaystyle\to Y_{R}^{\mu_{i}\nu_{i}}\dots
Y_{R}^{\mu_{j}\nu_{j}}\;\Bigl{[}V^{(m)}(\ell_{i}\ldots\ell_{j})\Bigr{]}_{\nu_{i}\ldots\nu_{j}\mu}^{b_{i}\ldots
b_{j}b}\;Y_{R}^{\mu\nu}\,,$ (39)

where $m$ is the number of incoming gluons at the soft vertex, and $\nu$ and
$b$ are the indices for the outgoing gluon. The eikonal factors and $J_{AX}$
then depend on fewer gluon momenta than in the term written out in (2), and
the sum includes the case in which $|\tau_{1}|=|\tau_{2}|=0$.

  3. 3.

Use the decoupling formula (4.3) with $n$ replaced by $n^{\prime}<n$.

Since in all terms on the r.h.s. of (2) the factor $J_{AX}$ has less than $n$
incoming soft gluons, this step removes all external soft gluons from the
collinear factor, and we have

$\displaystyle\eqref{eqn:soft_approximation_multi_gluons}$
$\displaystyle={}-Y_{R}^{\mu_{1}\nu_{1}}(\ell_{1})\,\dots\,Y_{R}^{\mu_{n}\nu_{n}}(\ell_{n})\,\sum_{\pi,\sigma_{1},\sigma_{2},\tau_{1},\tau_{2}\atop|\tau_{1}|+|\tau_{2}|>0\rule{0.0pt}{2.58331pt}}\Bigl{[}{\mathcal{E}}^{(|\tau_{1}|)}_{A,i_{1}h_{1}}\bigl{(}\pi\\{\ell\\}_{\tau_{1}}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\tau_{1}}}^{\pi\\{b\\}_{\tau_{1}}}\;$
$\displaystyle\qquad\times\Bigl{[}\bar{\mathcal{E}}^{(|\sigma_{1}|)}_{R,h_{1}k_{1}}\bigl{(}\pi\\{\ell\\}_{\sigma_{1}}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\sigma_{1}}}^{\pi\\{b\\}_{\sigma_{1}}}\;J_{AX}^{k_{1}k_{2}}(l_{1}-\ell_{\sigma_{1}}-\ell_{\tau_{1}},l_{2}-\ell_{\sigma_{2}}-\ell_{\tau_{2}})$
$\displaystyle\qquad\times\Bigl{[}{\mathcal{E}}^{(|\sigma_{2}|)}_{R,k_{2}h_{2}}\bigl{(}\pi\\{\ell\\}_{\sigma_{2}}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\sigma_{2}}}^{\pi\\{b\\}_{\sigma_{2}}}\;\Bigl{[}\bar{\mathcal{E}}^{(|\tau_{2}|)}_{A,h_{2}i_{2}}\bigl{(}\pi\\{\ell\\}_{\tau_{2}}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\tau_{2}}}^{\pi\\{b\\}_{\tau_{2}}}$
$\displaystyle\quad+\\{\text{terms with vertices }V^{(m)}\\}\,.$ (40)

Here the set $\sigma$ of soft momenta in (2) has been partitioned into sets
$\sigma_{1}$ and $\sigma_{2}$. The terms with vertices $V^{(m)}$ are obtained
in the same way as before, and again the corresponding sum includes the case
$|\tau_{1}|=|\tau_{2}|=0$.

Note that in (3) we put a Grammer-Yennie factor for each external gluon,
including the gluons for which we used the decoupling formula (4.3). This is
possible because $\mathcal{E}_{R}$ and $\bar{\mathcal{E}}_{R}$ remain
unchanged when their Lorentz indices are contracted with $Y_{R}$, as is
readily seen from the definitions (3.1) and (3.1).

  4. 4.

Apply a Ward identity to the combination of eikonal lines.

For given values $l_{1}-\ell_{\sigma_{1}}-\ell_{\tau_{1}}$ and
$l_{2}-\ell_{\sigma_{2}}-\ell_{\tau_{2}}$ of the momentum arguments of
$J_{AX}$, the expression (3) involves a sum over the products
$\mathcal{E}_{A}\,\bar{\mathcal{E}}_{R}$ and
$\mathcal{E}_{R}\,\bar{\mathcal{E}}_{A}$ of eikonal lines, to which the
incoming soft gluons attach either directly or via vertices $V^{(m)}$. To
simplify this sum we can use the Ward identity

$\displaystyle k_{1}^{\mu_{1}}\dots
k_{p}^{\mu_{p}}\;\Bra{0\rule{0.0pt}{10.50005pt}\,}TA_{\mu_{1}}^{a_{1}}(k_{1})\dots
A_{\mu_{p}}^{a_{p}}(k_{p})\;\bigl{[}W_{\\!A,ih}\;W^{\dagger}_{R,hk}\bigl{]}(k)\Ket{\,\rule{0.0pt}{10.50005pt}0}$
$\displaystyle=0\,,$ (41)

where $[W_{\\!A}\,W^{\dagger}_{R}\mskip 1.5mu](k)$ is the Fourier transform
(9) of $W_{\\!A}(x)\,W^{\dagger}_{R}(x)$. We will prove this identity in the
appendix. Expanding the product of Wilson lines to order $g^{\mskip 1.5mup}$,
one gets all tree level graphs in which the gluons couple to the product
$\mathcal{E}_{A}\,\bar{\mathcal{E}}_{R}$ of eikonal lines either directly or
via vertices $V^{(m)}$. A pictorial representation of this result is given in
figure 12. An analogous identity holds for the product
$[W_{R}\;W_{\\!A}^{\dagger}\mskip 1.5mu](k)$ and, truncated to the appropriate
order in $g$, gives a sum over the corresponding gluon attachments to
$\mathcal{E}_{R}\,\bar{\mathcal{E}}_{A}$. It is straightforward to check (41)
at order $g^{\mskip 1.5mup}$ for small values of $p$ by explicit evaluation of
the Feynman graphs.

Figure 12: Pictorial representation of the Ward identity (41) for the product
of two Wilson lines. The sum runs over all possibilities to couple the gluons
to the eikonal lines, either directly or via vertices $V^{(m)}$.

The only graphs missing on the r.h.s. of (3) are those with
$|\tau_{1}|=|\tau_{2}|=0$ and no vertex $V^{(m)}$. In these missing graphs,
all soft gluons couple directly to either $\bar{\mathcal{E}}_{R}$ or
${\mathcal{E}}_{R}$. We thus obtain the r.h.s. of (4.3), which completes the
proof by induction.

###  4.4 Soft gluon subtraction in collinear matrix elements

As announced in section 2, we now return to the subtraction terms in the
definition of the final collinear factor $J_{AX}(l_{1},l_{2})$, whose
unsubtracted form is

$\displaystyle
J_{AX,\,\text{unsub}}^{i_{1}i_{2}}(l_{1},l_{2})=\Bra{X\rule{0.0pt}{10.50005pt}}T\,\Gamma_{1}\bigl{[}W_{\\!A}\,q\bigr{]}_{i_{1}}(l_{1})\;\bigl{[}\bar{q}\mskip
1.5muW^{\dagger}_{\\!\mskip-1.5muA}\mskip
1.5mu\bigr{]}_{i_{2}}(l_{2})\Gamma_{2}\Ket{\rule{0.0pt}{10.50005pt}P}\,.$ (42)

The following discussion builds on the one for single parton scattering in
chapter 10.8.5 of Collins:2011zzd  and expands the argument for DPS in section
2.1 of Diehl:2015bca .

The matrix element (42) receives a leading contribution from momentum regions
in which the gluons that attach to the eikonal lines along $v_{A}$ are not
collinear but soft. For the same reason as in section 4.1, the soft gluons
must couple at the end of an eikonal line. The unsubtracted factor
$J_{AX,\,\text{unsub}}$ can hence be represented as shown in the top row of
figure 13. The soft gluons attached to the eikonal lines can couple among
themselves in the soft subgraph $S_{Z}$, before coupling to the internal lines
of the collinear subgraph $J_{AX}$. Using (29) and (30) one can write the
eikonal factors as products of eikonal lines connected with only soft or only
collinear gluons, as shown in the middle row of the figure. One can
furthermore deform the soft momenta out of the Glauber region, using the same
justification as for the overall DPS cross section Diehl:2015bca . This allows
one to insert Grammer-Yennie factors $Y_{R}$ for the soft gluons entering the
collinear subgraph. One can then decouple the soft gluons using the formula
(4.3) of the previous subsection. The result is shown in the last row of the
figure, where we have the subtracted collinear factor $J_{AX}(l_{1},l_{2})$
and a soft factor with eikonal lines along $v_{A}$ and $v_{R}$.

Figure 13: Decoupling of soft gluons from the unsubtracted collinear matrix
element $J_{AX,\,\text{unsub}}$ defined in (42).

Following similar steps as those shown for the DPS amplitude in the next
section, one obtains the collinear factor as a product of $J_{AX}$ and a
matrix element of Wilson lines along $v_{A}$ and $v_{R}$. Going from the
amplitude to the cross section level, one finds the result given in equation
(2.13) in Diehl:2015bca .

##  5 From eikonal to Wilson lines

Having established the decoupling for an arbitrary number $n$ of soft gluons,
we can now perform a sum over $n$ and convert the eikonal lines obtained in
the previous section into Wilson line operators. We do this at the level of
the amplitude for the DPS cross section.

Using the decoupling formula (4.3) and its analogue for $J_{BY}$ in the
contraction (3.2) of soft and collinear factors and inserting the result into
(3.2), we obtain

$\displaystyle\mathcal{L}_{XYZ}^{i_{1}i_{2}\,\bar{\imath}_{1}\bar{\imath}_{2}\,(n,\bar{n})}=\frac{1}{n!\,\bar{n}!}\,\int\\!\frac{\mathrm{d}^{D}l_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}l_{2}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{2}}{(2\pi)^{D}}\;\prod_{k=0}^{n}\,\prod_{\bar{k}=0}^{\bar{n}}\int\\!\frac{\mathrm{d}^{D}\ell_{k}}{(2\pi)^{D}}\frac{\mathrm{d}^{D}\bar{\ell}_{\bar{k}}}{(2\pi)^{D}}$
$\displaystyle\quad\times(2\pi)^{D}\,\delta^{(D)}\left(q_{1}-l_{1}-\bar{l}_{1}\right)\,(2\pi)^{D}\,\delta^{(D)}\left(q_{2}-l_{2}-\bar{l}_{2}\right)\;$
$\displaystyle\quad\times\sum_{\pi,\sigma_{1},\sigma_{2}\atop|\sigma_{1}|+|\sigma_{2}|=n\rule{0.0pt}{2.58331pt}}\Bigl{[}\bar{\mathcal{E}}^{(|\sigma_{1}|)}_{R,i_{1}h_{1}}\bigl{(}\pi\bigl{\\{}\tilde{\ell}\mskip
1.5mu\bigr{\\}}_{\sigma_{1}}\bigr{)}\Bigr{]}_{\pi\\{\mu\\}_{\sigma_{1}}}\;J_{AX}^{h_{1}h_{2}}\bigl{(}l_{1}-\tilde{\ell}_{\sigma_{1}},l_{2}-\tilde{\ell}_{\sigma_{2}}\bigr{)}\;\Bigl{[}{\mathcal{E}}^{(|\sigma_{2}|)}_{R,h_{2}i_{2}}\bigl{(}\pi\bigl{\\{}\tilde{\ell}\mskip
1.5mu\bigr{\\}}_{\sigma_{2}}\bigr{)}\Bigr{]}_{\pi\\{\mu\\}_{\sigma_{2}}}$
$\displaystyle\quad\times\sum_{\pi,\bar{\sigma}_{1},\bar{\sigma}_{2}\atop|\bar{\sigma}_{1}|+|\bar{\sigma}_{2}|=\bar{n}\rule{0.0pt}{2.58331pt}}\Bigl{[}\bar{\mathcal{E}}^{(|\bar{\sigma}_{2}|)}_{L,\bar{\imath}_{2}\bar{h}_{2}}\bigl{(}\pi\bigl{\\{}\tilde{\bar{\ell}}\mskip
1.5mu\bigr{\\}}_{\bar{\sigma}_{2}}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\bar{\sigma}_{2}}}\;J_{BY}^{\bar{h}_{1}\bar{h}_{2}}\bigl{(}\bar{l}_{1}-\tilde{\bar{\ell}}_{\bar{\sigma}_{1}},\bar{l}_{2}-\tilde{\bar{\ell}}_{\bar{\sigma}_{2}}\bigl{)}\;\Bigl{[}{\mathcal{E}}^{(|\bar{\sigma}_{1}|)}_{L,\bar{h}_{1}\bar{\imath}_{1}}\bigl{(}\pi\bigl{\\{}\tilde{\bar{\ell}}\mskip
1.5mu\bigr{\\}}_{\bar{\sigma}_{1}}\bigr{)}\Bigr{]}_{\pi\\{\nu\\}_{\bar{\sigma}_{1}}}$
$\displaystyle\quad\times\Bigl{[}S_{Z}\bigl{(}\\{\ell_{n}\\},\\{\bar{\ell}_{\bar{n}}\\}\bigr{)}\Bigr{]}^{\mu_{1}\ldots\mu_{n},\,\nu_{1}\ldots\nu_{\bar{n}}}\,.$
(43)

For brevity we have omitted the colour indices of gluons, which are contracted
between eikonal lines and the soft factor in the same way as the gluon
polarisation indices. By contrast, we write again a tilde to indicate the
approximation (3) of soft momenta in the collinear and eikonal factors.

The sum over partitions $\sigma_{1},\sigma_{2}$ of the gluons with momenta
$\ell_{1}\ldots\ell_{n}$ and over the permutations $\pi$ of how these gluons
couple to the eikonal lines can be greatly simplified. This is because the
soft factor $S_{Z}$ is invariant under permutations of its external gluon
momenta, which is readily seen from its definition (3.2). One can hence limit
the sum over $\sigma_{1},\sigma_{2}$ and $\pi$ in (5) to a single partition
and a single ordering. We retain the partition in which set $n_{1}$ includes
the gluons with momenta $\ell_{1}\dots\ell_{|n_{1}|}$ and set $n_{2}$ those
with momenta $\ell_{|n_{1}|+1}\dots\ell_{n}$ and take the permutation in which
the $\ell_{k}$ are ordered sequentially in $k$. This gives a combinatorial
factor

$\binom{n}{|n_{1}|\mskip 1.5mu}\,|n_{1}|\mskip 1.5mu!\;|n_{2}|\mskip
1.5mu!=n!\,,$ (44)

where we used $|n_{1}|+|n_{2}|=n$. The first factor on the l.h.s. is the
number of partitions $\sigma_{1},\sigma_{2}$, whilst the second and third
factor give the number of permutations within each set. Repeating the same
argument for the gluons with momenta
$\bar{\ell}_{1}\dots\bar{\ell}_{\bar{n}}$, we get

$\displaystyle\mathcal{L}_{XYZ}^{i_{1}i_{2}\,\bar{\imath}_{1}\bar{\imath}_{2}\,(n,\bar{n})}=\int\\!\frac{\mathrm{d}^{D}l_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}l_{2}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{2}}{(2\pi)^{D}}\;\prod_{k=0}^{n}\,\prod_{\bar{k}=0}^{\bar{n}}\int\\!\frac{\mathrm{d}^{D}\ell_{k}}{(2\pi)^{D}}\frac{\mathrm{d}^{D}\bar{\ell}_{\bar{k}}}{(2\pi)^{D}}$
$\displaystyle\quad\times(2\pi)^{D}\,\delta^{(D)}\left(q_{1}-l_{1}-\bar{l}_{1}\right)\,(2\pi)^{D}\,\delta^{(D)}\left(q_{2}-l_{2}-\bar{l}_{2}\right)$
$\displaystyle\quad\times\sum_{|n_{1}|+|n_{2}|=n}\Bigl{[}\bar{\mathcal{E}}^{(|n_{1}|)}_{R,i_{1}h_{1}}\bigl{(}\bigl{\\{}\tilde{\ell}\,\bigr{\\}}_{n_{1}}\bigr{)}\Bigr{]}_{\\{\mu\\}_{n_{1}}}\;J_{AX}^{h_{1}h_{2}}\bigl{(}l_{1}-\tilde{\ell}_{n_{1}},l_{2}-\tilde{\ell}_{n_{2}}\bigr{)}\;\Bigl{[}{\mathcal{E}}^{(|n_{2}|)}_{R,h_{2}i_{2}}\bigl{(}\bigl{\\{}\tilde{\ell}\,\bigr{\\}}_{n_{2}}\bigr{)}\Bigr{]}_{\\{\mu\\}_{n_{2}}}$
$\displaystyle\quad\times\sum_{|\bar{n}_{1}|+|\bar{n}_{2}|=\bar{n}}\Bigl{[}\bar{\mathcal{E}}^{(|\bar{n}_{2}|)}_{L,\bar{\imath}_{2}\bar{h}_{2}}\bigl{(}\bigl{\\{}\tilde{\bar{\ell}}\,\bigr{\\}}_{\bar{n}_{2}}\bigr{)}\Bigr{]}_{\\{\nu\\}_{\bar{n}_{2}}}\;J_{BY}^{\bar{h}_{1}\bar{h}_{2}}\bigl{(}\bar{l}_{1}-\tilde{\bar{\ell}}_{\bar{n}_{1}},\bar{l}_{2}-\tilde{\bar{\ell}}_{\bar{n}_{2}}\bigr{)}\;\Bigl{[}{\mathcal{E}}^{(|\bar{n}_{1}|)}_{L,\bar{h}_{1}\bar{\imath}_{1}}\bigl{(}\bigl{\\{}\tilde{\bar{\ell}}\,\bigr{\\}}_{\bar{n}_{1}}\bigr{)}\Bigr{]}_{\\{\nu\\}_{\bar{n}_{1}}}$
$\displaystyle\quad\times\Braket{Z}{TA^{\mu_{1}}(\ell_{1})\dots
A^{\mu_{n}}(\ell_{n})\,A^{\nu_{1}}(\bar{\ell}_{1})\dots
A^{\nu_{\bar{n}}}(\bar{\ell}_{\bar{n}})}{0}\,,$ (45)

where we have explicitly written out the soft factor using (3.2). We now shift
the integration momenta such that $J_{AX}$ and $J_{BY}$ become independent of
soft gluon momenta:

$\displaystyle\mathcal{L}_{XYZ}^{i_{1}i_{2}\,\bar{\imath}_{1}\bar{\imath}_{2}\,(n,\bar{n})}=\int\\!\frac{\mathrm{d}^{D}l_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}l_{2}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{2}}{(2\pi)^{D}}\;\prod_{k=0}^{n}\,\prod_{\bar{k}=0}^{\bar{n}}\int\\!\frac{\mathrm{d}^{D}\ell_{k}}{(2\pi)^{D}}\frac{\mathrm{d}^{D}\bar{\ell}_{\bar{k}}}{(2\pi)^{D}}$
$\displaystyle\quad\times\sum_{|n_{1}|+|n_{2}|=n\atop|\bar{n}_{1}|+|\bar{n}_{2}|=\bar{n}\rule{0.0pt}{2.58331pt}}(2\pi)^{D}\,\delta^{(D)}\bigl{(}q_{1}-l_{1}-\bar{l}_{1}-\tilde{\ell}_{n_{1}}-\tilde{\bar{\ell}}_{\bar{n}_{1}}\bigr{)}\,(2\pi)^{D}\,\delta^{(D)}\bigl{(}q_{2}-l_{2}-\bar{l}_{2}-\tilde{\ell}_{n_{2}}-\tilde{\bar{\ell}}_{\bar{n}_{2}}\bigr{)}$
$\displaystyle\quad\times\Bigl{[}\bar{\mathcal{E}}^{(|n_{1}|)}_{R,i_{1}h_{1}}\bigl{(}\bigl{\\{}\tilde{\ell}\,\bigr{\\}}_{n_{1}}\bigr{)}\Bigr{]}_{\\{\mu\\}_{n_{1}}}\;J_{AX}^{h_{1}h_{2}}(l_{1},l_{2})\;\Bigl{[}{\mathcal{E}}^{(|n_{2}|)}_{R,h_{2}i_{2}}\bigl{(}\bigl{\\{}\tilde{\ell}\,\bigr{\\}}_{n_{2}}\bigr{)}\Bigr{]}_{\\{\mu\\}_{n_{2}}}$
$\displaystyle\quad\times\Bigl{[}\bar{\mathcal{E}}^{(|\bar{n}_{2})|}_{L,\bar{\imath}_{2}\bar{h}_{2}}\bigl{(}\bigl{\\{}\tilde{\bar{\ell}}\,\bigr{\\}}_{\bar{n}_{2}}\bigr{)}\Bigr{]}_{\\{\nu\\}_{\bar{n}_{2}}}\;J_{BY}^{\bar{h}_{1}\bar{h}_{2}}(\bar{l}_{1},\bar{l}_{2})\;\Bigl{[}{\mathcal{E}}^{(|\bar{n}_{1})|}_{L,\bar{h}_{1}\bar{\imath}_{1}}\bigl{(}\bigl{\\{}\tilde{\bar{\ell}}\,\bigr{\\}}_{\bar{n}_{1}}\bigr{)}\Bigr{]}_{\\{\nu\\}_{\bar{n}_{1}}}$
$\displaystyle\quad\times\Braket{Z}{TA^{\mu_{1}}(\ell_{1})\dots
A^{\mu_{n}}(\ell_{n})\,A^{\nu_{1}}(\bar{\ell}_{1})\dots
A^{\nu_{\bar{n}}}(\bar{\ell}_{\bar{n}})}{0}\,.$ (46)

Recalling that $\ell_{n_{1}}$ and $\ell_{n_{2}}$ are defined as sums over soft
momenta, we can insert explicit integrations

$\displaystyle\int\frac{\mathrm{d}^{D}\ell_{n_{1}}}{(2\pi)^{D}}\;(2\pi)^{D}\,\delta^{(D)}\biggl{(}\ell_{n_{1}}-\sum_{k=1}^{n_{1}}\ell_{k}\biggr{)}\;\int\frac{\mathrm{d}^{D}\ell_{n_{2}}}{(2\pi)^{D}}\;(2\pi)^{D}\,\delta^{(D)}\biggl{(}\ell_{n_{2}}-\sum_{k=n_{1}+1}^{n}\ell_{k}\biggr{)}$
(47)

as well as their analogues for $\bar{\ell}_{\bar{n}_{1}}$ and
$\bar{\ell}_{\bar{n}_{2}}$. Combining the delta functions in these expressions
with the integrations over $\ell_{k}$ and $\bar{\ell}_{\bar{k}}$, the gluon
field operators in the soft factor, and the eikonal lines in (5), we recognise
the expressions (3.1) and (3.1) of Wilson lines in momentum space (expanded to
the appropriate order in $g$). We thus obtain

$\displaystyle\mathcal{L}_{XYZ}^{i_{1}i_{2}\,\bar{\imath}_{1}\bar{\imath}_{2}\,(n,\bar{n})}=\int\\!\frac{\mathrm{d}^{D}l_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{1}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}l_{2}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{l}_{2}}{(2\pi)^{D}}\,\sum_{n_{1}+n_{2}=n\atop\bar{n}_{1}+\bar{n}_{2}=\bar{n}\rule{0.0pt}{2.58331pt}}\int\\!\frac{\mathrm{d}^{D}\ell_{n_{1}}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\ell_{n_{2}}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{\ell}_{\bar{n}_{1}}}{(2\pi)^{D}}\,\frac{\mathrm{d}^{D}\bar{\ell}_{\bar{n}_{2}}}{(2\pi)^{D}}$
$\displaystyle\quad\times(2\pi)^{D}\,\delta^{(D)}\bigl{(}q_{1}-l_{1}-\bar{l}_{1}-\tilde{\ell}_{n_{1}}-\tilde{\bar{\ell}}_{\bar{n}_{1}}\bigr{)}\,(2\pi)^{D}\,\delta^{(D)}\bigl{(}q_{2}-l_{2}-\bar{l}_{2}-\tilde{\ell}_{n_{2}}-\tilde{\bar{\ell}}_{\bar{n}_{2}}\bigr{)}$
$\displaystyle\quad\times\int\\!\mathrm{d}^{D}{\zeta_{1}}\,\mathrm{d}^{D}{\zeta_{2}}\,\mathrm{d}^{D}{\bar{\zeta}_{1}}\,\mathrm{d}^{D}{\bar{\zeta}_{2}}\;\exp\bigl{[}i(\ell_{n_{1}}\cdot\zeta_{1}+\ell_{n_{2}}\cdot\zeta_{2}+\bar{\ell}_{\bar{n}_{1}}\cdot\bar{\zeta}_{1}+\bar{\ell}_{\bar{n}_{2}}\cdot\bar{\zeta}_{2})\bigr{]}$
$\displaystyle\quad\times\Braket{Z}{\Bigl{[}W_{R}^{\dagger(n_{1})}(\zeta_{1})\Bigr{]}_{i_{1}h_{1}}\,\Bigl{[}W_{R}^{(n_{2})}(\zeta_{2})\Bigr{]}_{h_{2}i_{2}}\,\Bigl{[}W_{L}^{\dagger(\bar{n}_{1})}(\bar{\zeta}_{2})\Bigr{]}_{\bar{\imath}_{2}\bar{h}_{2}}\,\
\Bigl{[}W_{L}^{(\bar{n}_{2})}(\bar{\zeta}_{1})\Bigr{]}_{\bar{h}_{1}\bar{\imath}_{1}}}{0}$
$\displaystyle\quad\times
J_{AX}^{h_{1}h_{2}}(l_{1},l_{2})\,J_{BY}^{\bar{h}_{1}\bar{h}_{2}}(\bar{l}_{1},\bar{l}_{2})\,,$
(48)

where we have explicitly written out the Fourier transform of the Wilson
lines. Instead of sets of gluon momenta, $n_{1}$ and $n_{2}$ are now integers
(equal to the size of these sets). The delta functions in the second line of
(5) can be simplified by neglecting the small plus or minus components of soft
momenta compared with the large components of $q_{1},q_{2}$ and the collinear
momenta. The transverse components of soft momenta must however be kept.
Representing the corresponding part of the delta functions as Fourier
integrals, we have

$\displaystyle\delta^{(D)}\bigl{(}q-l-\bar{l}-\tilde{\ell}-\tilde{\bar{\ell}}\,\bigr{)}$
$\displaystyle\approx\delta(q^{+}-l^{+})\,\delta(q^{-}-\bar{l}^{-})\int\\!\frac{\mathrm{d}^{D-2}\boldsymbol{\xi}}{(2\pi)^{D-2}}\;e^{-i(\boldsymbol{q}-\boldsymbol{l}-\boldsymbol{\bar{l}}-\boldsymbol{\ell}-\boldsymbol{\bar{\ell}})\cdot\boldsymbol{\xi}}\,,$
(49)

where we omitted indices like $1$ or $n_{1}$ for brevity. Inserting this into
(5), we can perform the integrals over the soft momenta, the positions of the
Wilson line operators, and over the large plus or minus components constrained
by (49). This gives

$\displaystyle\mathcal{L}_{XYZ}^{i_{1}i_{2}\,\bar{\imath}_{1}\bar{\imath}_{2}\,(n,\bar{n})}=\int\\!\mathrm{d}^{D-2}\boldsymbol{\xi}_{1}\,\mathrm{d}^{D-2}\boldsymbol{\xi}_{2}\;e^{-i\boldsymbol{q}_{1}\cdot\boldsymbol{\xi}_{1}-i\boldsymbol{q}_{2}\cdot\boldsymbol{\xi}_{2}}$
$\displaystyle\quad\times\sum_{n_{1}+n_{2}=n\atop\bar{n}_{1}+\bar{n}_{2}=\bar{n}\rule{0.0pt}{2.58331pt}}\Braket{Z}{\Bigl{[}W_{R}^{\dagger(n_{1})}(\boldsymbol{\xi}_{1})\Bigr{]}_{i_{1}h_{1}}\,\Bigl{[}W_{R}^{(n_{2})}(\boldsymbol{\xi}_{2})\Bigr{]}_{h_{2}i_{2}}\,\Bigl{[}W_{L}^{\dagger(\bar{n}_{1})}({\boldsymbol{\xi}}_{2})\Bigr{]}_{\bar{\imath}_{2}\bar{h}_{2}}\,\
\Bigl{[}W_{L}^{(\bar{n}_{2})}({\boldsymbol{\xi}}_{1})\Bigr{]}_{\bar{h}_{1}\bar{\imath}_{1}}}{0}$
$\displaystyle\quad\times\int\\!\frac{\mathrm{d}l_{1}^{-}\,\mathrm{d}^{D-2}\boldsymbol{l}_{1}}{(2\pi)^{D-1}}\,\frac{\mathrm{d}l_{2}^{-}\,\mathrm{d}^{D-2}\boldsymbol{l}_{2}}{(2\pi)^{D-1}}\;e^{i\boldsymbol{l}_{1}\cdot\boldsymbol{\xi}_{1}+i\boldsymbol{l}_{2}\cdot\boldsymbol{\xi}_{2}}\;J_{AX}^{h_{1}h_{2}}(l_{1},l_{2})\,\Big{|}_{l_{1}^{+}=q_{1}^{+},\,l_{2}^{+}=q_{2}^{+}}$
$\displaystyle\quad\times\int\\!\frac{\mathrm{d}\bar{l}_{1}^{-}\,\mathrm{d}^{D-2}\bar{\boldsymbol{l}}_{1}}{(2\pi)^{D-1}}\,\frac{\mathrm{d}\bar{l}_{2}^{-}\,\mathrm{d}^{D-2}\bar{\boldsymbol{l}}_{2}}{(2\pi)^{D-1}}\;e^{i\bar{\boldsymbol{l}}_{1}\cdot\boldsymbol{\xi}_{1}+i\bar{\boldsymbol{l}}_{2}\cdot\boldsymbol{\xi}_{2}}\;J_{BY}^{\bar{h}_{1}\bar{h}_{2}}(\bar{l}_{1},\bar{l}_{2})\,\Big{|}_{\bar{l}_{1}^{-}=q_{1}^{-},\,\bar{l}_{2}^{+}=q_{2}^{-}}\,,$
(50)

where the boldface arguments of Wilson lines indicate that their plus and
minus positions are zero. We note that the collinear factors in the last two
lines are Fourier transformed to the same transverse positions as the Wilson
lines with which they are connected by colour indices. Likewise, the
transverse positions of partons $1$ and $2$ in the right and left moving
proton coincide, in agreement with the physical picture that partons that
enter in a hard subprocess must be at the same position in the transverse
plane.

Summing (5) over all $n$ and $\bar{n}$, one recovers full Wilson lines.
Combining this with the corresponding expression for the complex conjugate
amplitude of the DPS process and summing over the intermediate states $X,Y$
and $Z$, one obtains unsubtracted collinear factors $J_{A}$ and $J_{B}$, as
well as a soft factor $S$ defined as the vacuum expectation value of Wilson
line operators. This is the starting point of the considerations in
Buffing:2017mqm , where the colour structure of these factors is analysed, and
where they are combined into double parton distributions that appear in the
final cross section formula.

We shall not repeat this analysis here, but note only that the colour indices
in (5) are eventually contracted with $\delta_{i_{1}\bar{\imath}_{1}}$ and
$\delta_{i_{2}\mskip 1.5mu\bar{\imath}_{2}}$. For the production of colourless
particles in TMD factorisation, this simply follows from the colour structure
of the hard scattering subgraphs. For collinear factorisation, where hard
unobserved partons can be radiated into the final state, this is the result of
nontrivial simplifications (see section 4.5 in Buffing:2017mqm ).

##  6 Conclusions

In this paper, we have filled a gap in the proof of factorisation for DPS
processes by showing that – after deformation out of the Glauber region – soft
gluons can be decoupled from the right or left collinear subgraphs associated
with the incoming protons. It turns out that the complicated colour structure
of the collinear subgraphs, with two partons emerging in the amplitude and two
in the complex conjugate one, does not pose particular problems but merely
complicates the combinatorics of the proof compared with the case of SPS. On
the technical side, the approximation (3) of soft gluon momenta entering
collinear graphs, which is weaker than in the work of Collins, Soper and
Sterman, can be handled as shown in section 4.1. This requires neglecting
power suppressed terms, which is necessary in other parts of the proof as well
(in particular, for dropping graphs in which soft gluons couple in the middle
of an eikonal line).

Note that the main part of our argument, presented in section 4, does not make
use of the colour structure of the hard scattering subprocesses and is valid
for the production of colourless particles as well as coloured ones. However,
other parts of the factorisation proof, in particular the discussion of the
Glauber region, are more complicated if coloured particles are produced. We
recall that for this reason TMD factorisation has only been established for
the production of colourless particles, even in SPS Rogers:2010dm ;
Boer:2017hqr .

It is straightforward to extend our proof from DPS to the case of $N$ hard
scatters. The result involves a soft factor with more Wilson line operators
than for $N=2$, and the number of colour channels for the collinear and soft
factors is higher. This complicates the analysis but does not pose any
conceptual problems.

There are some technical issues that merit further investigation, in
particular the missing proof of Ward identities for graphs with eikonal lines
and fixed internal momenta, as discussed at the beginning of section 4.
Further open problems are described in section 9 of Diehl:2017wew  and in
section 4.1 of Boer:2017hqr . Nevertheless, we think that the present work
contributes to putting factorisation formulae for DPS processes on
theoretically firm ground.

##  Appendix A Nonabelian Ward identities

The purpose of this appendix is to derive the Ward identities (22) and (41)
for correlation functions involving Wilson lines. Our treatment is based on
the textbook proofs given for instance in Sterman:1994ce ; Itzykson:1980rh ,
which use path integral methods and apply to full amplitudes. As already
discussed, the proof of soft gluon decoupling requires (22) in a more
restricted setting with fixed internal momenta in Feynman graphs. A
corresponding proof should be possible using the diagrammatic approach in
tHooft:1971akt ; tHooft:1972qbu , but this is beyond the scope of this paper.

Consider a theory with generic fields $\phi_{i}$, a Lagrangian density
$\mathcal{L}(\phi_{i})$, and an infinitesimal transformation
$\phi_{i}\to\phi_{i}+\delta\phi_{i}(\\{\phi_{j}\\})$ of the fields. The
corresponding Schwinger-Dyson equation for a generic $n$ point correlation
function is given by

$\displaystyle\delta\bra{0}T\phi_{1}(x_{1})\dots\phi_{n}(x_{n})\ket{0}$
$\displaystyle=\sum_{i=1}^{n}\Braket{0}{T\phi_{1}(x_{1})\dots\delta\phi_{i}\bigl{(}\\{\phi_{j}(x_{j})\\}\bigr{)}\dots\phi_{n}(x_{n})}{0}$
$\displaystyle=-i\int\\!\mathrm{d}^{D}y\,\Braket{0}{T\phi_{1}(x_{1})\dots\phi_{n}(x_{n})\;\delta\mathcal{L}\bigl{(}\\{\phi_{i}(y)\\}\bigr{)}}{0}\,,$
(51)

where in the first line we applied the field transformation to each field of
the $n$ point function. Note that the correlation function is a vacuum
expectation value.

We apply (A) to QCD amplitudes and take $\delta$ to be the infinitesimal BRST
transformation. This leaves the Lagrangian invariant, so that the r.h.s. of
(A) is zero. The transformation acts on the elementary fields in the
Lagrangian as

$\displaystyle\delta\mskip 1.5muq_{i}(x)$
$\displaystyle=ig\,t^{a}_{ij}\,c^{a}(x)\,\eta\,q_{j}(x)\,,$
$\displaystyle\delta\mskip 1.5mu\bar{q}_{i}(x)$
$\displaystyle=-ig\,\bar{q}_{j}(x)\,t^{a}_{ji}\,c^{a}(x)\,\eta\,,$
$\displaystyle\delta\mskip 1.5muA^{a}_{\mu}(x)$
$\displaystyle=\bigl{[}\partial_{\mu}+gf^{abc}A^{c}_{\mu}(x)\bigr{]}\,c^{b}(x)\,\eta\,,$
$\displaystyle\delta\mskip 1.5muc^{a}(x)$ $\displaystyle=-\frac{1}{2}\,g\mskip
1.5muf^{abc}\,c^{b}(x)\,c^{c}(x)\,\eta,$ $\displaystyle\delta\mskip
1.5mu\bar{c}^{a}(x)$ $\displaystyle=\partial_{\mu}A^{a,\mu}(x)\,\eta\,,$ (52)

where $\eta$ is an infinitesimal Grassmann parameter and Feynman gauge is
assumed. As a consequence of (A) the gluon field strength transforms as

$\delta\mskip 1.5muG^{a}_{\mu\nu}(x)=g\mskip
1.5muf^{abc}\,c^{b}(x)\,\eta\,G^{c}_{\mu\nu}(x)\,.$ (53)

The fields $\phi_{i}$ in (A) are not restricted to be elementary fields, but
they can be nonlocal products of elementary fields. Indeed, the correlation
functions we consider involve nonlocal operators of the type $W_{ij}(x)\mskip
1.5muq_{j}(x)$ or $W^{ab}(x)\mskip 1.5muG^{b}(x)$.

As a generalisation of (A), we need identities for operator matrix elements
with an external proton state $\ket{P}$ and partonic final states $\bra{X}$.
These can be rewritten as vacuum expectation values using the LSZ reduction
formula. One finds that the BRST variation of the fields associated with on-
shell external particles does not contribute to the sum over $i$ in (A). For
the proton this is because its interpolating field is gauge invariant and
hence also BRST invariant. For the on-shell parton states in $\bra{X}$ it
follows from the truncation of propagators in the reduction formula (see e.g.
chapter 11.2 of Sterman:1994ce  or chapter 7.4 of Peskin:1995ev ) and from the
restriction to physical gluon polarisation.

The desired Ward identity (22) can be obtained from

$\displaystyle 0$
$\displaystyle=\left(\frac{\mathrm{d}}{\mathrm{d}\eta}\,\delta\right)^{n}\,\Braket{X}{T\,\bar{c}(z_{1})\dots\bar{c}(z_{n})\,\phi_{1}(x_{1})\,\phi_{2}(x_{2})}{P}$
$\displaystyle=\Braket{X}{T\,\partial^{\mu_{1}}\\!{A}_{\mu_{1}}(z_{1})\dots\partial^{\mu_{n}}\\!{A}_{\mu_{n}}(z_{n})\,\phi_{1}(x_{1})\,\phi_{2}(x_{2})}{P}+\\{\text{terms
with $\delta\phi_{1}$ or $\delta\phi_{2}$}\\}\,,$ (54)

where

$\displaystyle\phi_{1}(x)$ $\displaystyle=W_{\\!A}(x)\,q(x)\,,$
$\displaystyle\phi_{2}(x)$ $\displaystyle=\bar{q}(x)\mskip
1.5muW^{\dagger}_{A}(x)\,,$ (55)

and where for brevity we have omitted colour indices. To obtain (A), we
applied (A) $n$ times, removing the Grassmann parameter $\eta$ by
differentiation after each step. Note that applying
$(\mathrm{d}/\mathrm{d}\eta)\,\delta$ more than once to any of the fields on
the l.h.s. of (A) gives zero.

We obtain (22) by Fourier transform to momentum space if we can show that the
terms with $\delta\phi_{1}$ or $\delta\phi_{2}$ on the r.h.s. of (A) are zero.
These terms contain at least one of the BRST variations $\delta\phi_{1}$ and
$\delta\phi_{2}$. To proceed, we make the limit of infinite path length in
$W_{\\!A}$ explicit by writing

$\displaystyle W_{A,ij}(x)$
$\displaystyle=\lim_{t\to-\infty}W_{A,ij}(y_{t},x)$ (56)

where $y_{t}=x+t\mskip 1.5muv_{\mskip-1.5muA}$ is the end of the path in

$\displaystyle W_{A,ij}(y_{t},x)$ $\displaystyle=\mathcal{P}\exp\left[\mskip
1.5muig\\!\int_{t}^{0}\\!\mathrm{d}s\,v_{\mskip-1.5muA}\cdot A^{a}(x+s\mskip
1.5muv_{\mskip-1.5muA})\,t^{a}_{ij}\right]\,.$ (57)

From the BRST transformation

$\delta W(y,x)=ig\left[\mskip
1.5mut^{b}c^{b}(y)\,\eta\,W(y,x)-W(y,x)\,t^{b}c^{b}(x)\,\eta\mskip
1.5mu\right]$ (58)

of a Wilson line one readily obtains

$\displaystyle\delta\mskip 1.5mu\bigl{[}W(y,x)\,q(x)\bigr{]}{}_{i}$
$\displaystyle=ig\,t^{b}_{ij}\,c^{b}(y)\,\eta\,\bigl{[}W(y,x)\,q(x)\bigr{]}{}_{j}\,,$
$\displaystyle\delta\mskip
1.5mu\bigl{[}\bar{q}(x)\,W^{\dagger}(y,x)\bigr{]}{}_{i}$
$\displaystyle=-ig\bigl{[}\bar{q}(x)\,W^{\dagger}(y,x)\bigr{]}{}_{j}\;t^{b}_{ji}\,c^{b}(y)\,\eta\,.$
(59)

The analogous relation for gluon fields reads

$\displaystyle\delta\mskip 1.5mu\bigl{[}W(y,x)\,G(x)\bigr{]}^{a}$
$\displaystyle=gf^{abc}\,c^{b}(y)\,\eta\,\bigl{[}W(y,x)\,G(x)\bigr{]}^{c}\,.$
(60)

The reason why the terms with $\delta\phi_{1}$ or $\delta\phi_{2}$ do not
contribute in (A) is the limit $t\to-\infty$. To see this, we consider the
variation of $W_{\\!A}(y_{t},x)\,q(x)$, isolate the term of order $g$ in the
Wilson line and Fourier transform to momentum space:

$\displaystyle\int\\!\mathrm{d}^{D}x\;e^{il\cdot x}\,\Bra{X}\dots
c(y)\,W_{\\!A}^{(1)}(y_{t},x)\,q(x)\dots\Ket{P}$
$\displaystyle\quad\propto\int\mathrm{d}^{D}x\,\int_{t}^{0}\\!\mathrm{d}s\int\\!\frac{\mathrm{d}^{D}q}{(2\pi)^{D}}\frac{\mathrm{d}^{D}k}{(2\pi)^{D}}\frac{\mathrm{d}^{D}p}{(2\pi)^{D}}\;e^{il\cdot
x}\,e^{-iq\cdot(x+tv_{A})}\,e^{-ik\cdot(x+sv_{A})}\,e^{-ip\cdot x}\,$
$\displaystyle\qquad\times\Bra{X}\dots
c(q)\;v_{A}\\!\cdot\\!{A}(k)\;q(p)\dots\Ket{P}$
$\displaystyle\quad=\int\\!\frac{\mathrm{d}^{D}q}{(2\pi)^{D}}\;e^{-itq\cdot
v_{A}}\int\\!\frac{\mathrm{d}^{D}k}{(2\pi)^{D}}\;\frac{i}{k\cdot
v_{A}+i\epsilon}\,\bigl{(}1-e^{-itk\cdot v_{A}}\bigr{)}\,$
$\displaystyle\qquad\times\Bra{X}\dots
c(q)\;v_{A}\\!\cdot\\!{A}(k)\;q(l-k-q)\dots\Ket{P}\,,$ (61)

where the ellipsis stands for the remaining operators in the correlation
function. For large $t$ the factor $e^{-itq\cdot v_{A}}$ produces strong
oscillations, whereas the remaining integrand has a smooth dependence on $q$.
This gives a vanishing integral over $q$ in the limit $t\to-\infty$. The same
holds if $W^{(1)}_{\\!A}$ in (A) is replaced with $W^{(n)}_{\\!A}$ for
arbitrary $n$, and if one considers the operator $\phi_{2}$ instead of
$\phi_{1}$.

In the same manner, we can derive the Ward identity (41). Instead of (A) we
then consider

$\displaystyle 0$
$\displaystyle=\left(\frac{\mathrm{d}}{\mathrm{d}\eta}\,\delta\right)^{p}\,\Braket{X}{T\,\bar{c}(z_{1})\dots\bar{c}(z_{p})\,\phi(x)}{P}$
$\displaystyle=\Braket{X}{T\,\partial^{\mu_{1}}\\!{A}_{\mu_{1}}(z_{1})\dots\partial^{\mu_{p}}\\!{A}_{\mu_{p}}(z_{p})\,\phi(x)}{P}+\\{\text{terms
with $\delta\phi$}\\}$ (62)

with

$\displaystyle\phi(x)$
$\displaystyle=W_{\\!A}(x)\,W_{R}^{\dagger}(x)=\lim_{t\to-\infty}\,\lim_{u\to-\infty}W_{\\!A}(y_{t},x)\,\bigl{[}W_{R}(z_{u},x)\bigr{]}^{\dagger}\,,$
(63)

where $y_{t}=x+t\mskip 1.5muv_{\mskip-1.5muA}$ and $z_{u}=x+u\mskip
1.5muv_{\mskip-1.5muR}$. Using (58) one finds that the BRST variation of the
r.h.s. gives a ghost field $c(y_{t})$ or $c(z_{u})$. Just as in (A) this
results in vanishing loop integrals when $t$ and $u$ are taken to negative
infinity. Note that the ghost field $c(x)$ in (58) cancels in the
transformation of a product of Wilson lines that meet at the same point $x$.

###### Acknowledgements.

We gratefully acknowledge discussions with Maarten Buffing, Johannes Michel,
Dave Soper and Frank Tackmann. Some calculations useful in the development of
this work were done using FORM Ruijl:2017dtg , and all pictures were produced
using JaxoDraw Binosi:2003yf ; Binosi:2008ig .

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