Nikhef 2023-027

Exponentiation of soft quark effects from the replica trick
Melissa van Beekvelda,b, Leonardo Vernazzac and Chris D. Whited

a Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Parks
Road, University of Oxford, Oxford OX1 3PU, UK
bNikhef, Science Park 105, 1098 XG Amsterdam, NL
c INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy
d Centre for Theoretical Physics, Department of Physics and Astronomy, Queen
Mary University of London, 327 Mile End Road, London E1 4NS, UK

###### Abstract

In this paper, we show that multiple maximally soft (anti-)quark and gluon
emissions exponentiate at the level of either the amplitude or cross-section.
We first show that such emissions can be captured by introducing new _soft
emission operators_ , which serve to generalise the well-known Wilson lines
describing emissions of maximally soft gluons. Next, we prove that vacuum
expectation values of these operators exponentiate using the replica trick, a
statistical-physics argument that has previously been used to demonstrate
soft-gluon exponentiation properties in QCD. The obtained results are general,
i.e. not tied to a particular scattering process. We illustrate our arguments
by demonstrating the exponentiation of certain real and virtual corrections
affecting subleading partonic channels in deep-inelastic scattering.

##  1 Introduction

The understanding of contemporary collider physics experiments continues to
rely on high-precision predictions in perturbative Quantum Chromodynamics
(QCD). Predictions in QCD are typically done through a small-coupling
expansion, resulting in a perturbative series. Next-to-leading order (NLO)
accuracy has been achieved for all processes, whereas for particular processes
we can reach as high as N3LO [1, 2, 3, 4, 5]. The direct calculation of
higher-order contributions is difficult in general, and often requires the
development of new techniques to make obtaining results computationally
feasible. This is further complicated by the fact that the coefficients of the
perturbation expansion can diverge in certain kinematic regions. A well-known
case of this is production of heavy (or off-shell) particles near threshold,
for which perturbative cross-sections become enhanced due to the emission of
soft and/or collinear radiation. In such cases, one may define a dimensionless
threshold variable $\xi$, involving a ratio of kinematic invariants, and such
that $\xi\rightarrow 0$ at threshold. The differential partonic cross-section
turns out to involve terms of the form

$\frac{{\rm d}\hat{\sigma}}{{\rm
d}\xi}\sim\sum_{n=0}\alpha_{s}^{n}\sum_{m=0}^{2n-1}\left[c_{nm}^{(0)}\left(\frac{\log^{m}\xi}{\xi}\right)_{+}+c_{nm}^{(1)}\log^{m}\xi+\ldots\right].$
(1.1)

That is, at each power of the coupling, one finds a series of logarithmic
terms in $\xi$, where the first set (governed by the coefficients
$c_{nm}^{(0)}$) involves the well-known plus distribution, thus ensuring that
the differential cross-section is integrable, as required. These are known as
leading-power (LP) threshold contributions, as they correspond to keeping the
leading terms in a systematic expansion of the differential cross-section in
powers of the threshold variable $\xi$, whose physical origin is the emission
of radiation which is strictly soft and/or collinear.111We have neglected to
show terms in Eq. (1.1) which involve a pure delta function $\delta(\xi)$, and
which contribute at leading power. We will not be considering such
contributions in this paper. According to this nomenclature, the second set of
terms in Eq. (1.1) comprises next-to-leading power (NLP) contributions, which
are suppressed by a single power of $\xi$, and which potentially arise from
next-to-soft and/or collinear emissions. Finally, the ellipsis in Eq. (1.1)
denotes terms which are suppressed by further powers of $\xi$, and which are
not formally singular as $\xi\rightarrow 0$.

The well-understood origin of LP terms allows us to sum them up to all orders
in perturbation theory for some (classes of) observables, a procedure known as
resummation. Different procedures exist, such as Feynman diagrammatic
approaches [6, 7, 8, 9, 10, 11, 12], renormalisation group arguments [13], use
of Wilson lines [14, 15], and soft-collinear effective theory (SCET) [16, 17,
18, 19]. All of these have at heart that soft and collinear radiation
factorises in a process-independent manner, making precise the quantum
mechanical notion that radiation with a low (transverse) momentum has a large
wavelength, and is thus insensitive to short-distance physics. One may then
write down certain universal functions describing the threshold radiation,
whose calculation at fixed orders in $\alpha_{s}$ leads to the resummation of
entire towers of logarithms in perturbation theory. The most divergent logs at
each order ($\sim\xi^{-1}\log^{2n-1}\xi$) are referred to as leading
logarithmic (LL) terms, followed by next-to-leading logarithmic (NLL) and so
on. The development of new resummation techniques forms a crucial part of the
theory programme for current and forthcoming experiments.

Until relatively recently, much less has been known about NLP threshold
contributions, but there are by now well-established motivations for trying to
include them in cross-section predictions. They may, for example, be
numerically significant in certain scattering processes, such that they are
comparable with missing higher-order contributions at either LP or fixed-order
accuracy [20, 21, 22, 23, 24, 25, 26, 27]. There is then a lengthy programme
to be carried out of classifying the various NLP contributions, and either:
(i) resumming them alongside LP contributions; (ii) using results for NLP
terms at fixed-order to estimate missing higher-order corrections or improving
fixed-order slicing schemes [28, 29, 30, 31, 32, 33, 34]. However, it should
also be emphasised that the characterisation of new types of contribution in
QCD – especially those that are potential windows to all-order structures in
perturbation theory – is a highly interesting field theoretical question in
its own right. Recent years have seen a number of developments in the more
formal theoretical physics literature, such as the relation of next-to-soft
physics with asymptotic symmetries at past or future null infinity [35, 36],
which be in turn be interpreted in terms of a conformal field theory living on
the celestial sphere [37, 38]. There has also been examination of the role
that (next-to)-soft physics can play in (quantum) gravity in the high-energy
limit [39, 40, 41, 42, 43, 44, 45, 46, 47, 48] (see Ref. [49] for earlier
work), which is indeed the exact limit probed by gravitational wave
experiments such as LIGO. It is interesting that these very formal topics are
directly relatable to cutting-edge phenomenology (see e.g. Refs. [50, 51, 52,
53, 54] for papers and reviews that make these links explicit), and the next
few years are sure to see an interesting interplay between very different
topics and ideas.

Aside from the pioneering early works of Refs. [55, 56, 57], studies of NLP
effects in collider physics have utilised a number of techniques [58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79,
80, 81, 28, 29, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 27, 92, 93, 94, 95,
96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
112, 113, 114, 115, 116, 117, 118, 119], mirroring the situation at leading
power. As these works make clear, the resummation of NLP contributions is
starting to become possible, thus opening up a new chapter in the history of
QCD perturbation theory. What makes this significantly more difficult than the
study of LP contributions, however, is the fact that there are many different
sources of NLP effect, each of which must be analysed in detail. Indeed, not
all NLP terms in Eq. (1.1) arise from the emission of next-to-soft radiation.
This has been emphasised in recent studies [117, 120, 121, 122, 123, 124, 79],
which have examined the emission of soft (anti-)quarks in a range of collider
processes including Higgs boson production, Drell-Yan (DY) production of a
heavy vector boson, and deep inelastic scattering (DIS) (see also Ref. [113]
for closely related work). Soft fermion emission is kinematically suppressed
relative to gluon emission, such that the presence of soft (anti-)quarks in
the final state is possible only from NLP onwards in the threshold expansion.
However, there is a well-defined sense in which this soft fermion emission is
simpler than other types of NLP contribution, such as those arising from next-
to-soft radiation, next-to-soft operators for collinear gluon emission, or
phase space effects. From a diagrammatic point of view, it is only the
presence of fermion emission vertices – evaluated in the leading soft
approximation – that makes such contributions differ from their soft gluonic
counterparts. All other simplifications that are associated with soft gluon
emission still apply, including factorisation of the phase space for all
emitted soft partons. This suggests that it should be particularly simple to
resum soft quark effects, at least at LL order, and it is the aim of this
paper to present a general argument for this resummation.

The main ideas are as follows. We will first build on the results of Refs.
[79], which defined emission factors describing the radiation of arbitrary
partons from a given hard external leg of an amplitude.222Ref. [79] used the
term emission operator for the emission factors alluded to above. We here
change this terminology to avoid confusion with the generalised soft-emission
line operators to be discussed in what follows. We will show that the emission
of arbitrary numbers of soft partons can be expressed by combining these
emission factors into an operator that generalises the well-known Wilson line
governing the emission of soft gluons alone. Note that this is a different
operator to the generalised Wilson line that has previously been considered in
the study of next-to-soft physics[125, 41, 53]: the latter describes the
emission of gluon radiation that is not strictly soft. By contrast, our
generalised soft-emission line operator here encompasses purely soft partons,
which may be gluons, or (anti-)quarks. It is thus matrix-valued in colour,
spin and flavour space, which we will define shortly.

Similarly to the radiation of soft gluons, emission of multiple soft partons
can be expressed in terms of vacuum expectation values of our generalised
soft-emission line operators, which may in turn be written as a certain field
theoretic path integral. We may then apply a novel argument known as the
replica trick, which first arose in statistical physics (see e.g. Ref. [126]),
to show that soft-parton emissions of any type exponentiate at LL order, at
either amplitude or cross-section level. Our use of this argument is very
similar to an ongoing programme of work in QCD, aimed at calculating higher-
loop soft gluon effects in multiparton scattering [127, 128, 129, 130, 131,
132, 133, 134, 135] (see Refs. [136, 137] for reviews, and Refs. [138, 139,
140] for related work). To make the similarities clear we will review this
below. A strength of the replica trick argument is that our results will be
general, i.e. not tied to any particular scattering process. Indeed, our
analysis reveals that a large class of purely soft multiparton emissions
exponentiates entirely. Given that fermion emissions are kinematically
suppressed, this statement goes beyond NLP in the soft expansion, although the
practical utility of this may be limited given that the associated terms in
perturbation theory may overlap with those which have a different (next-
to)soft origin.

A convenient corollary of our results is that recent related conjectures
emerge as a special case. In particular, Ref. [117] considered Higgs-boson
induced DIS, and focused on the kinematically subleading partonic channel in
which a quark or anti-quark scatters in the initial state, rather than a
gluon. This is the analogue of the gluon-initiated channel in conventional
photon DIS, and the fact that it commences only at NLP is due to the emission
of a fermion.333Similar calculations to those of Ref. [117] exist for the
cases of conventional DIS, and also for DY production [141]. Ref. [117] showed
that it is possible to resum LL NLP terms in this channel, provided one
assumes that the leading singular terms in the virtual corrections to this
process exponentiate, in the limit in which the emitted fermion is soft. A
similar conjecture was made in the earlier Ref. [113], which considered the
process $e^{+}e^{-}\rightarrow q\bar{q}g$, where the (anti-)quarks are
collinear. If the quark and anti-quark carry momentum fractions $z$ and
$\bar{z}=1-z$ of the total fermion momentum respectively, the authors noted
that care is needed in examining the soft quark limit $z\rightarrow 0$: when
including the virtual corrections, one must keep track of non-analytic
dependence in $z$ as $z\rightarrow 0$, as this will affect the LL NLP terms
after integration over the final state phase space. Both Refs. [113, 117]
referred to terms in the soft quark limit of the fermion emission processes as
endpoint contributions, and their conjectures amount to the statement that the
one-loop virtual corrections to these endpoint contributions must
exponentiate. As discussed in Refs. [117, 122], the interpretation of endpoint
contributions in SCET is delicate, and involves a refactorisation of SCET
operators such as to manifestly extract the correct dependence on the soft
momentum fraction. In the present paper, however, such contributions will
emerge as a special case of our generalised soft-emission line operators.
Exponentiation of endpoint contributions is then part of a more general
exponentiation, involving arbitrary numbers of soft parton emissions. We hope
that our results thus provide a useful complementary viewpoint to those of
Refs. [113, 117], whilst also providing an interesting further application of
the replica trick.

The structure of our paper is as follows. In Section 2, we review the replica
trick argument for the exponentiation of soft gluons, both at amplitude and
cross-section level. In Section 3, we introduce our generalised soft-emission
lines for the emission of soft (anti-)quarks and/or gluons, and explain how
the replica trick argument generalises in this case. We then show two examples
of how to apply this argument: in Section 4, we show how the replica trick
reproduces the exponentiation of real emission corrections in deep-inelastic
scattering (DIS), that was conjectured in Ref. [142] and proven using much
more cumbersome methods in Ref. [123]. Then, in Section 5, we show how a
slightly modified version of the replica trick argument of Section 3 can be
used to confirm the expectations of Refs. [113, 117], namely that the leading
virtual corrections to the kinematically subleading partonic channels at NLO
in DIS exponentiate. We discuss our results and conclude in Section 6, and
certain technical details are collected in the appendix.

##  2 The replica trick at leading power

In this section, we examine the replica trick argument for soft gluon
exponentiation, first presented in Refs. [125, 127]. Let us consider a
scattering amplitude ${\cal A}_{n}$ with $n$ external partons and momenta
labelled as shown in Fig. 1.

Figure 1: An $n$-point scattering amplitude, where all external momenta have
been taken to be outgoing.

As is well known, virtual QCD corrections to the amplitude lead to infrared
(IR) singularities, associated with those kinematic regions in which the
exchanged parton is soft (i.e. it has vanishing 4-momentum), or collinear to
any of the hard external legs. The correction has a universal form, meaning
that for a general amplitude it may be factorised according to the following
schematic formula [143]

${\cal A}_{n}(\\{p_{i}\\})=\left[{\cal
H}_{n}(\\{p_{i}\\},\\{n_{i}\\})\otimes\,{\cal
S}(\\{\beta_{i}\\})\right]\frac{\prod_{i=1}^{n}J(p_{i},n_{i})}{\prod_{i=1}^{n}{\cal
J}(\beta_{i},n_{i})}\,.$ (2.1)

Here ${\cal H}_{n}(\\{p_{i}\\},\\{n_{i}\\})$ is a process-dependent hard
function, free of IR singularities. The soft function ${\cal
S}(\\{\beta_{i}\\})$ depends only on the 4-velocities of the incoming and
outgoing particles, and collects all singularities arising from soft
radiation. This object is colour-connected to the hard function, as indicated
by the symbol $\otimes$. The jet function $J(p_{i},n_{i})$ collects collinear
singularities associated with emissions off the external leg with momentum
$p_{i}$, and has a universal definition involving an auxiliary 4-vector
$n_{i}$. Including both the soft and jet functions leads to a double-counting
of radiation that is both soft and collinear, such that one has to remove each
double-counted contribution. This is done though dividing out an eikonal jet
function ${\cal J}(\beta_{i},n_{i})$ associated with leg $i$. The soft and
(eikonal) jet functions all have universal definitions involving the
appropriate partonic fields and Wilson line operators. Note that, given that
the auxiliary vectors $\\{n_{i}\\}$ are arbitrary (apart from the condition
$n_{i}\cdot p_{i}\neq 0$), dependence on them must cancel between the
(eikonal) jet and hard functions, which in practice amounts to fixing the
definition of the hard function in terms of the jet functions.

Eq. (2.1) is known as the soft-collinear factorisation formula, and simplifies
considerably if one only cares about the leading IR singularities of a given
amplitude. In that case, all radiation must be soft (and collinear) so that
one may remove the (eikonal) jet functions. Furthermore, higher-order
contributions to the hard function can be neglected, such that this may be
replaced by its first-order approximation. Note that this approximation does
not have to correspond to the leading-order (LO) of a given inclusive
scattering process, depending on whether or not we are considering a
subleading partonic channel that only turns on at next-to-leading order (NLO)
or higher. Thus, Eq. (2.1) reduces to

${\cal A}_{n}(\\{p_{i}\\})={\cal H}^{(0)}_{n}(\\{p_{i}\\})\,\otimes\,{\cal
S}(\\{\beta_{i}\\})\,,$ (2.2)

for some suitably defined first-order hard function ${\cal
H}^{(0)}_{n}(\\{p_{i}\\})$ for an $n$-legged hard-scattering processes.
Classifying the leading IR singularities then amounts to calculating the soft
function, which has the following operator definition as a vacuum expectation
value [143]:

${\cal S}\equiv{\cal S}(\\{\beta_{i}\\})=\left\langle
0\left|\prod_{i=1}^{n}\Phi_{i}(A)\right|0\right\rangle$ (2.3)

where

$\Phi_{i}(A)={\cal P}\exp\left[ig_{s}\beta_{i}^{\mu}{\bf T}_{i}^{a}\int{\rm
d}t_{i}A^{a}_{\mu}(t_{i}\beta_{i})\right],$ (2.4)

is a Wilson line operator along the classical straight-line contour
$x_{i}^{\mu}=t_{i}\beta_{i}^{\mu}$ of external parton $i$, with
$\beta_{i}^{\mu}$ the relevant 4-velocity. Furthermore, ${\bf T}^{a}_{i}$ is a
colour generator in the appropriate representation of hard line $i$, $g_{s}$
is the strong coupling, and ${\cal P}$ denotes path-ordering of these colour
generators along the integration contour. We can formally write the vacuum
expectation value in Eq. (2.3) as a path integral over the soft gauge field
$A^{a}_{\mu}$ that couples to each external line:

${\cal S}=\int[{\cal D}A]\,\prod_{i=1}^{n}{\cal
P}\exp\left(ig_{s}\beta_{i}^{\mu}{\bf T}_{i}^{a}\int{\rm
d}t_{i}A_{\mu}^{a}(t_{i}\beta_{i})\right)\,e^{iS[A]}\,,$ (2.5)

where $S[A]$ is the action for the soft gauge field (note that this involves
additional fields, which we have left implicit). Contracting the gluon fields
with the action and carrying out the path integral (order-by-order) generates
all possible Feynman diagrams in which the Wilson lines are connected by soft
gluon graphs.

It is well-known that one may write the soft function in an exponential form,
where the logarithm contains certain sets of diagrams known as webs. The
concept of a web was first introduced in QCD in the case of processes
involving two coloured particles only [10, 11, 12], but has more recently been
generalised to arbitrary multiparton scattering processes [127] (see Refs.
[136, 137] for reviews). This property may be derived using the replica trick.
To this end, we start with ${\cal S}$ defined as in Eq. (2.5), and consider
its counterpart in a distinct, duplicated theory that contains $N$ identical
copies, or replicas, of the original soft gauge field. Let us label these
different fields by $\\{A_{\mu}^{{\color[rgb]{0,0,0}{(i)}}a}\\}$, where
${\color[rgb]{0,0,0}{(i)}}$ is the replica index. The soft gauge fields of
different replicas are required to be non-interacting, so that the action for
the replicated theory can simply be expressed as the sum of actions
$S[A_{\mu}^{{\color[rgb]{0,0,0}{(i)}}}]$ for each gauge field.444Note that in
diagrams where gluons couple via other fields off the Wilson lines (e.g. to
fermion bubbles), we must also replicate these other fields. We will not need
to consider this complication explicitly here. The soft function in the
replicated theory, denoted by $\color[rgb]{0,0,0}{{\cal S}_{N}}$, then takes
the form

$\displaystyle\color[rgb]{0,0,0}{{\cal S}_{N}}$
$\displaystyle={\color[rgb]{0,0,0}{\prod_{i=1}^{N}}}\int[{\cal
D}A^{{\color[rgb]{0,0,0}{(i)}}}]\,\prod_{j}^{n}\Phi_{j}(A^{{\color[rgb]{0,0,0}{(i)}}})\,e^{iS[A^{{\color[rgb]{0,0,0}{(i)}}}]}$
(2.6) $\displaystyle={\cal S}^{N}.$

In the first line, we now have a product of Wilson lines on each parton leg:
one for each replica number. In the second line, we have recognised that the
soft function in the replicated theory amounts to the original soft function
raised to the power $N$, and the non-interacting nature of the replicas
(leading to additivity of their actions) is crucial in this regard. A simple
mathematical identity allows us to write

${\cal S}^{N}=1+N\log{\cal S}+{\cal O}(N^{2})\,,$ (2.7)

from which we can immediately conclude that the logarithm of the soft function
contains only those diagrams which in the replicated theory are ${\cal O}(N)$.
This gives the following recipe for determining $\log{\cal S}$ (and thus the
exponentiated soft function):

  1. 1.

Draw all possible Feynman diagrams in the replicated theory. These consist of
Feynman diagrams that also occur in the original theory, but where each gluon
emitted from a Wilson line now carries a replica index.

  2. 2.

Find the dependence of each diagram on the number of replicas $N$.

  3. 3.

Take the ${\cal O}(N)$ part of each diagram, which gives the contribution of
each diagram to the logarithm of the soft function.

By taking the exponent of the logarithm of the soft function one directly
resums soft (and possibly collinear) eikonal emissions.

The simplest case to consider is that of QED. There are then no non-commuting
colour matrices to worry about, and the soft function for the replicated
theory reduces to

$\displaystyle{\color[rgb]{0,0,0}{{\cal
S}_{N}}}=\left[{\color[rgb]{0,0,0}{\prod_{i=1}^{N}}}\int{\cal
D}A^{{\color[rgb]{0,0,0}{(i)}}}\right]\left[\prod_{k=1}^{n}\exp\left(ie\,{\color[rgb]{0,0,0}{\sum_{j=1}^{N}}}\int{\rm
d}x_{k}^{\mu}A^{{\color[rgb]{0,0,0}{(j)}}}_{\mu}(x_{k})\right)\right]\exp\left(i{\color[rgb]{0,0,0}{\sum_{l=1}^{N}}}S[A^{{\color[rgb]{0,0,0}{(l)}}}]\right).$
(2.8)

This generates Feynman diagrams where each external line can emit photons,
which may be joined by other fields off the Wilson lines. If we consider a
diagram such as that shown in Fig. 2, which has 2 connected soft photon
subdiagrams, there are $N$ choices for the first replica index $i$, and $N$
choices for the second replica index $j$. The diagram is thus ${\cal
O}(N^{2})$ and so does not contribute to the logarithm of the soft function.
Likewise, a diagram with $m$ connected soft photon subdiagrams will be ${\cal
O}(N^{m})$, such that only diagrams with a single connected piece $m=1$
contribute to the logarithm of the soft function. This is the well-known
exponentiation of connected soft photon diagrams that was first derived a long
time ago using combinatoric methods [144].

Figure 2: A diagram occurring in the replicated soft photon theory of Eq.
(2.8), for the case of two outgoing Wilson lines exchanging soft photons.
There are two connected soft photon subdiagrams once the hard external lines
are removed, consisting of individual soft photons, where $i$ and $j$ denote
their replica indices.

Returning to a non-abelian context, we must face the fact that colour matrices
associated with different replica fields do not commute with each other. To
deal with this complication, Refs. [127, 125] noted that one may rewrite the
replicated soft function of Eq. (2.6) in a form that instead makes clear the
product of Wilson line operators associated with each individual hard leg $k$:

$\displaystyle{\color[rgb]{0,0,0}{{\cal S}_{N}}}$ $\displaystyle=\int[{\cal
D}A^{{\color[rgb]{0,0,0}{(1)}}}]\ldots[{\cal
D}A^{{\color[rgb]{0,0,0}{(N)}}}]\,e^{i{\color[rgb]{0,0,0}{\sum_{j=1}^{N}}}S[A^{{\color[rgb]{0,0,0}{(j)}}}]}\,\prod_{k=1}^{n}\left[\Phi_{k}(A^{{\color[rgb]{0,0,0}{(1)}}})\ldots\Phi_{k}(A^{{\color[rgb]{0,0,0}{(N)}}})\right].$
(2.9)

Here we have used the commuting nature of Wilson-line operators associated
with different external legs, as they act on different partonic colour
indices. Note that we are not allowed to reorder Wilson-line operators
associated with the same leg, as they are non-commuting. There are then $N$
Wilson-line operators on each leg $k$, one for each replica gauge field, and
such that the replica index increases as we go along the leg. Following Ref.
[127, 125], we can implement this constraint in a convenient way by
introducing a replica-ordering operator ${\cal R}$, such that the product of
Wilson line operators on a given leg can be written

$\prod_{{\color[rgb]{0,0,0}{j=1}}}^{\color[rgb]{0,0,0}{N}}{\cal
P}\exp\left[ig_{s}\int{\rm
d}x_{k}^{\mu}A_{\mu}^{{\color[rgb]{0,0,0}{(j)}}}\right]={\cal R}{\cal
P}\exp\left[ig_{s}{\color[rgb]{0,0,0}{\sum_{j=1}^{N}}}\int{\rm
d}x_{k}^{\mu}A_{\mu}^{{\color[rgb]{0,0,0}{(j)}}}(x_{k})\right].$ (2.10)

On the right-hand side, we now have a single path-ordered exponential, but
where, in performing the Taylor expansion, ${\cal R}$ acts to reorder colour
matrices where necessary, such that the replica index is increasing along the
line. To clarify this, the explicit action of ${\cal R}$ on two colour
generators ${\bf T}_{k}^{{\color[rgb]{0,0,0}{(i)}}}$ and ${\bf
T}_{k}^{{\color[rgb]{0,0,0}{(j)}}}$ associated with replica indices $i$ and
$j$ is defined as

${\cal R}\left[{\bf T}_{k}^{{\color[rgb]{0,0,0}{(i)}}}{\bf
T}_{k}^{{\color[rgb]{0,0,0}{(j)}}}\right]=\begin{cases}{\bf
T}_{k}^{{\color[rgb]{0,0,0}{(i)}}}{\bf
T}_{k}^{{\color[rgb]{0,0,0}{(j)}}},\quad i\leq j\\\ {\bf
T}_{k}^{{\color[rgb]{0,0,0}{(j)}}}{\bf
T}_{k}^{{\color[rgb]{0,0,0}{(i)}}},\quad i>j,\end{cases}$ (2.11)

so that ${\cal R}$ reorders the matrices only if the replica numbers are not
increasing. This definition is straightforward to generalise to higher numbers
of colour matrices. Armed with the replica-ordering operator ${\cal R}$, our
replicated soft function of Eq. (2.9) now assumes the form

$\displaystyle{\color[rgb]{0,0,0}{{\cal S}_{N}}}$ $\displaystyle=\int[{\cal
D}A^{{\color[rgb]{0,0,0}{(1)}}}]\ldots[{\cal
D}A^{{\color[rgb]{0,0,0}{(N)}}}]\,e^{i{\color[rgb]{0,0,0}{\sum_{j=1}^{N}}}S[A^{{\color[rgb]{0,0,0}{(j)}}}]}\,\left[\prod_{k=1}^{n}{\cal
R}\,{\cal P}\,e^{ig_{s}\,{\color[rgb]{0,0,0}{\sum_{j=1}^{N}}}\int{\rm
d}x_{k}^{\mu}A_{\mu}^{{\color[rgb]{0,0,0}{(j)}}}(x_{k})}\right].$ (2.12)

This provides a systematic prescription for ascertaining which diagrams
contribute to the logarithm of the soft function. Contracting the fields
within each replicated field theory leads to a set of Feynman diagrams. The
colour factor of each such diagram will not be the usual colour factor of QCD
perturbation theory, but will instead have colour matrices reordered by the
operator ${\cal R}$. The contribution of each diagram in the replicated theory
will depend upon the number of replicas $N$ in general. One may then take the
${\cal O}(N)$ part of each diagram as before.

Replica-ordering hierarchy $h$ | ${\cal R}[C(a)|h]$ | ${\cal R}[C(b)|h]$ | Multiplicity
---|---|---|---
$i=j$ | $C(a)$ | $C(b)$ | $N$
$i<j$ | $C(a)$ | $C(a)$ | $N(N-1)/2$
$i>j$ | $C(a)$ | $C(a)$ | $N(N-1)/2$
Table 1: Replica trick analysis of the diagrams in Fig. 3. For each diagram
$D$ we give the various hierarchies $h$ of the ordering of the replica
indices, together with the colour factor ${\cal R}[C(D)|h]$ one obtains by
applying the replica-ordering operator ${\cal R}$. Finally, we give the number
of ways of choosing replica numbers consistent with each hierarchy.  Figure 3:
Two diagrams that potentially contribute to the logarithm of the QCD soft
function, involving two outgoing Wilson lines. Replica indices $i$ and $j$ are
labelled, as are the numbers of the Wilson lines.

The simplest non-trivial example of this procedure is the case of two separate
gluon emissions between two Wilson lines, for which the two Feynman diagrams
in the replicated theory are shown in Fig. 3. To calculate the colour factors,
we must take into account all possible hierarchies of replica numbers, given
that ${\cal R}$ potentially acts differently in each case. For example, the
colour factor of diagram (a) is

${\cal R}\left[{\bf T}_{1}^{{\color[rgb]{0,0,0}{(i)}}}{\bf
T}_{1}^{{\color[rgb]{0,0,0}{(j)}}}{\bf T}_{2}^{{\color[rgb]{0,0,0}{(i)}}}{\bf
T}_{2}^{{\color[rgb]{0,0,0}{(j)}}}\right],$ (2.13)

with ${\bf T}_{k}$ a general colour operator acting on the external line with
label $k=1,2$. If $i=j$ or $i<j$, the replica ordering operator will leave the
colour factor intact. If $i>j$, however, it will reorder the colour matrices
on each line, so that one has

${\cal R}\left[{\bf T}_{1}^{{\color[rgb]{0,0,0}{(i)}}}{\bf
T}_{1}^{{\color[rgb]{0,0,0}{(j)}}}{\bf T}_{2}^{{\color[rgb]{0,0,0}{(i)}}}{\bf
T}_{2}^{{\color[rgb]{0,0,0}{(j)}}}\right]=\left[{\bf
T}_{1}^{{\color[rgb]{0,0,0}{(j)}}}{\bf T}_{1}^{{\color[rgb]{0,0,0}{(i)}}}{\bf
T}_{2}^{{\color[rgb]{0,0,0}{(j)}}}{\bf
T}_{2}^{{\color[rgb]{0,0,0}{(i)}}}\right],\quad i>j\,.$ (2.14)

This has the same form as the original colour factor with the indices $i$ and
$j$ interchanged, and so for all hierarchies of replica number, we obtain the
colour factor of diagram (a). For a given hierarchy of replica orderings $h$
and diagram $D$, we denote the replica-ordered colour factor by ${\cal
R}[C(D)|h]$. We may then summarise the above analysis as in table 1, which
gives the different hierarchies of the ordering of the two replica indices $i$
and $j$, together with the colour factor obtained by applying the replica
ordering operator in each case. The table also details the findings for
diagram (b). For this diagram, for some hierarchies, the action of ${\cal R}$
is to disentangle the crossed gluon pair, thus creating the colour factor of
diagram (a). Finally, we give the multiplicity factor corresponding to the
number of ways of choosing replica numbers consistent with the given
hierarchy. The total contribution of a given diagram $D$ in the replicated
theory will be given by its kinematic factor ${\cal K}(D)$, multiplied by the
various colour factors weighted by the appropriate multiplicities. For diagram
(a), for example, we have

$\displaystyle{\cal K}(a)C_{N}(a)\,,\quad
C_{N}(a)=NC(a)+2C(a)\left(\frac{N(N-1)}{2}\right)=N^{2}C(a)\,,$ (2.15)

where $C_{N}(D)$ denotes the effective colour factor for diagram $D$.
According to the replica trick, we must take the ${\cal O}(N)$ part of this as
the contribution to the logarithm of the soft function, and following Ref.
[127] we may write this as

${\cal K}(D)\tilde{C}(D),\quad\tilde{C}(D)\equiv C_{N}(D)\Big{|}_{{\cal
O}(N)}.$ (2.16)

For the case of diagram (a), we thus find $\tilde{C}(a)=0$. In the case of
diagram (b), we instead find

$\displaystyle\tilde{C}(b)=C(b)-C(a)\,,$ (2.17)

such that this diagram indeed contributes to the logarithm of the soft
function. It does so, however, with a modified colour weight, and this is a
special case of the previously known fact that in two-parton processes, only
those diagrams which are two-particle irreducible (i.e. webs) exponentiate
[10, 11, 12], with appropriately modified colour factors. In general, we find
then the exponentiated soft function can be written as

$\displaystyle\mathcal{S}={\rm
exp}\left[\sum_{D}\mathcal{K}(D)\,\tilde{C}(D)\right],$ (2.18)

where the argument of the exponent are the so-called web diagrams.

The above methods generalise to arbitrary multiparton scattering processes in
QCD, where the simple criterion of two-particle irreducibility no longer
applies. It is nevertheless possible to write a general structure for the
logarithm of the soft function. To do so, note that the ${\cal R}$ operator
reorders gluon attachments on all external legs, so that the exponentiated
colour factor of a given diagram potentially depends on all other diagrams
related by gluon permutations. There is a closed set of such permutations (as
exemplified by the diagrams of Fig. 3), and Ref. [127] defined a multiparton
web to consist of such a closed set of diagrams. This interpretation survives
after the effects of renormalisation are taken into account [129], and a
further refinement of this notion has been introduced in Refs. [138, 139]. In
general, the exponentiated colour factor of a diagram $D$ in each web $W$ will
be a superposition of all colour factors in the web

$\tilde{C}(D)=\sum_{D^{\prime}\in W}R_{DD^{\prime}}\,C(D^{\prime})\,.$ (2.19)

Here the quantity $R_{DD^{\prime}}$ is known as a web mixing matrix. It has a
purely combinatorial definition [130], of interdisciplinary interest [145,
146], and which has been further explored in Refs. [131, 138, 139]. Combining
the exponentiated colour factor of each diagram with its kinematic part, the
contribution of a given web to the logarithm of the soft function is

$w=\sum_{D,D^{\prime}\in W}{\cal K}(D)\,R_{DD^{\prime}}\,C(D^{\prime})\,.$
(2.20)

So far we have only considered virtual emissions. Comments regarding the
generalisation to real emissions were made in Ref. [83], which postulates a
soft function at the level of the squared amplitude/cross-section. That is,
one may write

${\cal S}(\bar{v})=\sum_{m}{\rm
Tr}\left[\Big{\langle}0\Big{|}\Phi_{1}\ldots\Phi_{n}\Big{|}m\Big{\rangle}\Big{\langle}m\Big{|}\Phi_{n}^{\dagger}\ldots\Phi_{1}^{\dagger}\Big{|}0\Big{\rangle}\right]\delta(v(m)-\bar{v})\,.$
(2.21)

Here the trace is over the colour indices of the outgoing parton legs, and the
two vacuum expectation values (VEVs) are taken between a vacuum state, and a
final state $|m\rangle$ containing $m$ soft gluons together with a measurement
function $\delta(v(m)-\bar{v})$ corresponding to the (dimensionless)
observable $\bar{v}$ and the $m$-particle definition of that observable. We
use a compact notation for the sum over final states, which implicitly
contains the integral over the combined phase space of the hard partons and
soft gluons for each $m$.555For particular scattering processes in the
literature, the precise definition of the soft function may vary, depending on
whether it is defined for the squared amplitude or a normalised differential
cross-section. It is possible to write a formal generating functional for soft
functions such as those in Eq. (2.21) which allows to apply the replica trick
directly, although this was not proven in Ref. [83]. We thus return to this
point in what follows.

In this section, we have reviewed how the replica trick can be used to show
how soft gluon emissions exponentiate. It is perhaps more correct to say that
the replica trick constitutes a systematic procedure for determining the
logarithm of the soft function. Scrutiny of the above arguments, however,
reveals that they do not crucially depend on the nature of the Wilson lines in
Eq. (2.5). We are free to replace the latter with a generalised soft-emission
operator, and indeed Ref. [125] did precisely this in arguing that certain
classes of next-to-soft contributions formally exponentiate. Here, we will
seek a different generalised soft-emission line, which includes the effects of
soft quark emission. This is the subject of the following section.

##  3 The replica trick for generalised soft-parton emissions

The replica trick may be extended to show that not only soft-gluon, but also
soft-quark emissions can be exponentiated. Indeed, in applying the replica
trick to prove the exponentiation of soft gluons, the precise definition of
the Wilson line (and all its gauge-transformation properties) never entered.
All that matters is that we are able to write down an operator definition that
generates the soft emissions in QCD up to all orders in the strong coupling.
When considering also the emission of soft quarks, we must apply a similar
argument, but the difficulty lies in finding a suitable replacement for the
operator that describes the emission of soft (anti-)quarks in addition to
gluons. We will successfully seek such a generalised soft-emission operator
below. Our strategy to prove that soft quark emissions exponentiates consists
of three key ingredients:

  1. 1.

We will consider a hard external leg of an amplitude, and show that the
emission of a single soft (anti-)quark or gluon can be written in terms of
universal emission factors (Section 3.1). The latter have been considered in
Ref. [79], and we build on those results here.

  2. 2.

These emission factors are used to build a generalised soft-emission operator
for the emission of arbitrary numbers of soft partons from a hard leg (Section
3.2).

  3. 3.

The generalised soft-emission operator is used to define a generalised soft
function, which captures all soft parton emissions. This generalised soft
function will have a path integral representation, allowing the replica trick
to be used in an analogous fashion to the case of soft gluon emission (Section
3.3).

In carrying out this programme, we will also provide a more rigorous treatment
of the replica trick for real emissions than has previously appeared in the
literature.

###  3.1 Emission factors for soft partons

Figure 4: Emission of various soft partons from incoming hard legs, where
$A_{X}$ denotes the amplitude with an incoming parton $X$, but where the
external wavefunction for the particle $X$ is removed. Here $\\{i,j,\dots\\}$
($\\{a,b,c,\dots\\})$ represent the fundamental (adjoint) colour indices,
while Lorentz and spinor indices are suppressed, though explicitly written in
the corresponding Eqs. (3.1) – (3.7).

Consider the emission of a soft gluon from an incoming quark leg, as shown in
Fig. 4(a). Scaling the gluon momentum with $k^{\mu}\to\lambda k^{\mu}$ and
taking $\lambda\to 0$ leads to the well-known eikonal Feynman rule, such that
the matrix element for diagram (a) is written as

$[\mathcal{M}_{q}^{\rm in}(p)]^{c}_{\mu}=\frac{g_{s}}{p\cdot
k}\,[\mathcal{A}_{q}(p)]_{\hat{b}}^{j}\,t_{ji}^{c}\left(p_{\mu}\delta_{\hat{b}\hat{a}}\right)u^{i}_{\hat{a}}(p)\,.$
(3.1)

Here $\mathcal{A}_{q}(p)$ denotes the amplitude for an incoming quark leg with
the external spinor removed. We denote spinor indices by hatted lower-case
Latin letters $\\{\hat{a},\hat{b},\dots\\}$ (raised or lowered spinor indices
carry no meaning). The fundamental (adjoint) colour indices are denoted with
$\\{i,j,\dots\\}$ ($\\{a,b,c,\dots\\})$. The colour generator of the ${\rm
SU}(3)$ gauge group in the fundamental representation is denoted by
$t^{a}_{ij}$. The external wavefunction of the quark carries a label $i$,
which refers to the colour of the incoming quark, whereas the amplitude
carries the label $j$, referring to the colour of the internal quark entering
the hard scattering. We have not included an external particle wavefunction
for the emitted soft gluon, leaving open the possibility of this gluon being
either real or virtual.

Similarly, if the emitting particle were an anti-quark instead of a quark
(diagram (b)) we would have

$[\mathcal{M}_{\bar{q}}^{\rm in}(p)]^{c}_{\mu}=\frac{g_{s}}{p\cdot
k}\,\bar{v}^{i}_{\hat{a}}(p)\,(-t_{ij}^{c})\,\left(p_{\mu}\delta_{\hat{a}\hat{b}}\right)\,[\mathcal{A}_{\bar{q}}(p)]^{j}_{\hat{b}}\,.$
(3.2)

Performing the same exercise for diagram (c) leads to

$[\mathcal{M}_{g}^{\rm in}(p)]^{c}_{\mu}=\frac{g_{s}}{p\cdot
k}\,if^{bca}\left(p_{\mu}\eta^{\alpha\beta}\right)[\mathcal{A}_{g}(p)]_{\beta}^{b}\,\epsilon_{\alpha}^{a}(p)\,,$
(3.3)

with $f^{abc}$ the colour generator of the ${\rm SU}(3)$ gauge group in the
adjoint representation.

To obtain the Feynman rules for the emission of a soft-quark (soft anti-quark)
from an incoming quark (anti-quark), we may, in analogy with the soft-gluon
case, scale the (anti-)quark momentum with $k^{\mu}\to\lambda k^{\mu}$ and
take $\lambda\to 0$ [79]. The soft (anti-)quark emission contributions from a
quark (anti-quark) are shown respectively in diagrams (d) and (e) of Fig. 4.
We obtain

$\displaystyle[\mathcal{M}_{q}^{\rm in}(p)]^{j}_{\hat{c}}$
$\displaystyle=\frac{g_{s}}{p\cdot
k}\,t^{b}_{ji}\left(-\frac{1}{2}\gamma^{\beta}_{\hat{c}\hat{a}}\right)u^{i}_{\hat{a}}(p)\,[\mathcal{A}_{g}(p)]_{\beta}^{b}\,,$
(3.4) $\displaystyle[\mathcal{M}_{\bar{q}}^{\rm in}(p)]^{j}_{\hat{c}}$
$\displaystyle=\frac{g_{s}}{p\cdot
k}\,\bar{v}^{i}_{\hat{a}}(p)\,t^{b}_{ij}\left(-\frac{1}{2}\gamma^{\beta}_{\hat{a}\hat{c}}\right)[\mathcal{A}_{g}(p)]_{\beta}^{b}\,.$
(3.5)

Finally, the case of a soft quark (soft anti-quark) emission from an incoming
gluon, shown respectively in diagrams (f) and (g) of Fig. 4, reads

$\displaystyle[\mathcal{M}_{g}^{\rm in}(p)]^{i}_{\hat{c}}$
$\displaystyle=\frac{g_{s}}{p\cdot
k}\,(-t^{a}_{ij})\,\frac{1}{2}\big{(}\gamma^{\alpha}\not{p}\big{)}_{\hat{c}\hat{b}}\,[\mathcal{A}_{\bar{q}}(p)]^{j}_{\hat{b}}\,\epsilon^{a}_{\alpha}(p)\,,$
(3.6) $\displaystyle[\mathcal{M}_{g}^{\rm in}(p)]^{i}_{\hat{c}}$
$\displaystyle=\frac{g_{s}}{p\cdot
k}\,[\mathcal{A}_{q}(p)]_{\hat{b}}^{j}\,t^{a}_{ji}\,\frac{1}{2}\big{(}\not{p}\gamma^{\alpha}\big{)}_{\hat{b}\hat{c}}\,\epsilon^{a}_{\alpha}(p)\,.$
(3.7)

Figure 5: Emission of a parton with (spinor/vector) index $C$ from two partons
with indices $B$ and $A$. The barred letters denote colour indices.

In each case, the additional factors in the amplitude take the form of an
eikonal denominator depending on the soft momentum $k$, with everything else
depending only on the hard momentum $p$. The additional factors are sandwiched
between the non-radiative amplitude on the one-hand, and the external
wavefunction for the hard leg on the other, meaning we can write a general
form that covers all cases. Let us first introduce an array of fields

$\left[\Theta_{I}\right]^{\bar{A}}_{A}\equiv(\psi^{i}_{\hat{a}},\bar{\psi}^{i}_{\hat{a}},A^{a}_{\alpha})\,,$
(3.8)

where $\psi^{i}$ ($\bar{\psi}^{i}$) denotes the (anti-)quark field, and $i$,
$a$ are colour labels. On the left-hand side, we have introduced a flavour
index $I\in\\{q,\bar{q},g\\}$, and an index $A$ in Minkowski/spinor space,
labeling the correct index for the relevant fields (i.e.
$A\in\\{\hat{a},\hat{a},\alpha\\}$). Furthermore, the index $\bar{A}$ labels
the colour index associated with the field whose species index is $A$:
$\bar{A}\in\\{i,i,a\\}$. We stress that this is purely a book-keeping device
that is useful for our purposes – we do not wish to imply that these fields
form a multiplet that is acted upon by any sort of symmetry transformation (as
would be the case in e.g. a supersymmetric theory). Next, using the same
basis, let us define an array of external initial-state particle wavefunctions
that go with the fields of Eq. (3.8)

$\left[\xi_{I}\right]^{\bar{A}}_{A}=(u^{i}_{\hat{a}},\bar{v}^{i}_{\hat{a}},\epsilon^{a}_{\alpha})\,.$
(3.9)

We will also need an array corresponding to the various wave-function stripped
amplitudes occurring in Fig. 4

$\left[\mathcal{A}_{I}\right]^{\bar{A}}_{A}=\left([\mathcal{A}_{q}(p)]^{i}_{\hat{a}},[\mathcal{A}_{\bar{q}}(p)]^{i}_{\hat{a}},[\mathcal{A}_{g}(p)]^{a}_{\alpha}\right).$
(3.10)

In this notation, we may summarise Eqs. (3.1)–(3.7) in a compact notation by
introducing a transition matrix for soft emissions off incoming hard partons,
$[{\cal T}^{\rm in}_{IJ}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}$, such that

$[\mathcal{M}_{I}^{\rm in}(p)]_{C}^{\bar{C}}=\frac{g_{s}}{p\cdot
k}\,[\xi_{I}(p)]^{\bar{A}}_{A}\,[{\cal T}^{\rm
in}_{IJ}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}\,[\mathcal{A}_{J}(p)]^{\bar{B}}_{B}\,,$
(3.11)

with666Note that the indices $I$,$J$ in Eq. (3.12) are not summed, i.e. there
is no matrix multiplication in the indices $I$,$J$. These simply labels the
flavour of the transitions, both on the l.h.s and the r.h.s of Eq. (3.12).

$[{\cal T}^{\rm in}_{IJ}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}=[{\bf
T}_{IJ}]^{\bar{C}}_{\bar{A}\bar{B}}\,[Q^{\rm in}_{IJ}(p)]_{C,AB}.$ (3.12)

The labeling is defined such that it follows the momentum flow of the hard
line, which goes from $A$ to $B$. One sees that this object consists of a
kinematic part $[Q^{\rm in}]_{C,AB}$ and a colour factor ${\bf
T}^{\bar{C}}_{\bar{A}\bar{B}}$. In the basis Eq. (3.10) we may write

$[{\cal T}^{\rm
in}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}=\left(\begin{array}[]{ccc}[{\cal
T}_{qq}^{\rm in}(p)]^{c,ij}_{\gamma,\hat{a}\hat{b}}&0&[{\cal T}_{qg}^{\rm
in}(p)]^{k,ib}_{\hat{c},\hat{a}\beta}\\\ 0&[{\cal T}_{\bar{q}\bar{q}}^{\rm
in}(p)]^{c,ij}_{\gamma,\hat{a}\hat{b}}&[{\cal T}_{\bar{q}g}^{\rm
in}(p)]^{k,ib}_{\hat{c},\hat{a}\beta}\\\ ({\cal T}_{gq}^{\rm
in}(p))^{k,aj}_{\hat{c},\alpha\hat{b}}&[{\cal T}_{g\bar{q}}^{\rm
in}(p)]^{k,aj}_{\hat{c},\alpha\hat{b}}&[{\cal T}_{gg}^{\rm
in}(p)]^{c,ab}_{\gamma,\alpha\beta}\end{array}\right),$ (3.13)

where we will refer to the entries of this matrix as _emission factors_ , and
where we have explicitly indicated where no valid non-zero entry is possible.
The assignment of the colour factor $[{\bf
T}_{IJ}]^{\bar{C}}_{\bar{A}\bar{B}}$ follows the usual eikonal Feynman rules.
In our notation, we have used the colour labels $a,b$ ($c$) for colour indices
of hard (soft) gluons, and $i,j$ ($k$) for that of hard (soft) quarks and
anti-quarks. The assignment of the kinematic factor $[Q^{\rm in}]_{C,AB}$ can
be inferred by direct comparison with Eqs. (3.1)–(3.7), and is summarised in
Appendix A. Combining the colour and kinematic factors we find for the
transition matrix

$[{\cal T}^{\rm
in}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}=\left(\begin{array}[]{ccc}t_{ji}^{c}\left(p_{\gamma}\delta_{\hat{b}\hat{a}}\right)&0&t^{b}_{ki}\left(-\frac{1}{2}\gamma^{\beta}_{\hat{c}\hat{a}}\right)\\\
0&(-t_{ij}^{c})\,\left(p_{\gamma}\delta_{\hat{a}\hat{b}}\right)&t^{b}_{ik}\left(-\frac{1}{2}\gamma^{\beta}_{\hat{a}\hat{c}}\right)\\\
t^{a}_{jk}\,\frac{1}{2}\big{(}\not{p}\gamma^{\alpha}\big{)}_{\hat{b}\hat{c}}&(-t^{a}_{kj})\,\frac{1}{2}\big{(}\gamma^{\alpha}\not{p}\big{)}_{\hat{c}\hat{b}}&if^{bca}\left(p_{\gamma}\eta^{\beta\alpha}\right)\end{array}\right).$
(3.14)

To summarise, in Eq. (3.11) we introduced a compact notation for the emission
of a single soft parton with Lorentz/spinor and colour indices $C,\bar{C}$ off
a hard incoming line with flavour $I$, space-time index $A$, and colour index
$\bar{A}$. This hard parton then continues with internal flavour index $J$,
space-time index $B$ and colour index $\bar{B}$. The notation in Eq. (3.11) is
also illustrated in Fig. 5. We do not include an external wave-function for
the emitted soft parton, leaving open the possibility of this parton being
either real or virtual. Multiple soft emissions off a single hard parton can
easily be described through repeated insertion of the transition matrix, where
care of course has to be taken that the eikonal pre-factor $1/p\cdot k$ is
updated accordingly. The final product of transition matrices may in the end
then be sandwiched between a hard matrix element on one end, and the external
wave function for the hard parton on the other. This will be demonstrated in
Section 3.2.

One may perform the same exercise for a final-state emitter with hard momentum
$p^{\mu}$. We may summarise the results as

$[\mathcal{M}_{I}^{\rm out}(p)]_{C}^{\bar{C}}=\frac{g_{s}}{p\cdot
k}\,[\bar{\xi}_{I}(p)]^{\bar{A}}_{A}\,[{\cal T}^{\rm
out}_{IJ}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}\,[\mathcal{A}_{J}(p)]^{\bar{B}}_{B}\,,$
(3.15)

The outgoing wave function can be any of

$\left[\bar{\xi}_{I}\right]^{\bar{A}}_{A}=(\bar{u}^{j}_{\hat{a}},v^{j}_{\dot{a}},\epsilon^{\dagger
a}_{\alpha})\,.$ (3.16)

The transition matrix now describes soft emissions off hard outgoing partons,
and can be written as

$[{\cal T}^{\rm
out}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}=\left(\begin{array}[]{ccc}[{\cal
T}_{qq}^{\rm out}(p)]^{c,ij}_{\gamma,\hat{a}\hat{b}}&0&[{\cal T}_{qg}^{\rm
out}(p)]^{k,ib}_{\hat{c},\hat{a}\beta}\\\ 0&[{\cal T}_{\bar{q}\bar{q}}^{\rm
out}(p)]^{c,ij}_{\gamma,\hat{a}\hat{b}}&[{\cal T}_{\bar{q}g}^{\rm
out}(p)]^{k,ib}_{\hat{c},\hat{a}\beta}\\\ ({\cal T}_{gq}^{\rm
out}(p))^{k,aj}_{\hat{c},\alpha\hat{b}}&[{\cal T}_{g\bar{q}}^{\rm
out}(p)]^{k,aj}_{\hat{c},\alpha\hat{b}}&[{\cal T}_{gg}^{\rm
out}(p)]^{c,ab}_{\gamma,\alpha\beta}\end{array}\right),$ (3.17)

Again, the colour factor follows the rules for eikonal emissions listed above.
Together with the kinematic factor $\left[Q^{\rm out}\right]_{C,AB}$ (written
out explicitly in Appendix A for all cases) we then obtain

$[{\cal T}^{\rm
out}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}=\left(\begin{array}[]{ccc}(-t_{ij}^{c})\left(p_{\gamma}\delta_{\hat{b}\hat{a}}\right)&0&t^{b}_{ik}\left(\frac{1}{2}\gamma^{\beta}_{\hat{a}\hat{c}}\right)\\\
0&t_{ji}^{c}\,\left(p_{\gamma}\delta_{\hat{b}\hat{a}}\right)&t^{b}_{ki}\left(\frac{1}{2}\gamma^{\beta}_{\hat{c}\hat{a}}\right)\\\
(-t^{a}_{kj})\,\frac{1}{2}\big{(}\gamma^{\alpha}\not{p}\big{)}_{\hat{c}\hat{b}}&(t^{a}_{jk})\,\frac{1}{2}\big{(}\not{p}\gamma^{\alpha}\big{)}_{\hat{b}\hat{c}}&if^{acb}\left(p_{\gamma}\eta^{\alpha\beta}\right)\end{array}\right).$
(3.18)

It is easy to check that $[{\cal T}^{\rm
out}(p)]^{\bar{C},\bar{A}\bar{B}}_{C,AB}=[\overline{{\cal T}}^{\rm
in}(p)]^{\bar{C},\bar{B}\bar{A}}_{C,BA}$, as required. We may now work towards
a generalisation of the conventional Wilson lines, that includes emission of
any soft partons. Let us start from the case of emission from an incoming
line. In momentum space, the Feynman rule describing this process is given in
Eq. (3.11), i.e.

$\frac{g_{s}}{p\cdot k}\,\left[{\cal T}^{\rm
in}\right]^{\bar{C},\bar{A}\bar{B}}_{C,AB},$ (3.19)

which is sandwiched between the external incoming wave function
$\xi^{\bar{A}}_{A}$ and the stripped hard-scattering amplitude ${\cal
A}^{\bar{B}}_{B}$. In position space this Feynman rule reads

$\displaystyle ig_{s}\int_{-\infty}^{0}\,{\rm d}t\,\left[{\cal T}^{\rm
in}\right]^{\bar{C},\bar{A}\bar{B}}_{C,AB}\,\Theta^{\bar{C}}_{C}(tp)\,e^{\varepsilon
t}\,.$ (3.20)

The integral is defined over the incoming particle contour (parameterised by
$x^{\mu}=tp^{\mu}$), and the factor $e^{\varepsilon t}$ makes sure the
integral converges. To see the equivalence between the two Feynman rules, we
may write the field in momentum space via the Fourier transform

$\Theta^{\bar{C}}_{C}(x)=\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\,\tilde{\Theta}^{\bar{C}}_{C}(k)\,e^{ik\cdot
x}=\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\,\tilde{\Theta}^{\bar{C}}_{C}(k)\,e^{it(k\cdot p)}.$
(3.21)

Inserting this into Eq. (3.20) yields

$\displaystyle ig_{s}\int_{-\infty}^{0}{\rm d}t\,\left[{\cal T}^{\rm
in}\right]^{\bar{C},\bar{A}\bar{B}}_{C,AB}\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\tilde{\Theta}^{\bar{C}}_{C}(k)\left[\frac{e^{it(k\cdot
p)+t\varepsilon}}{ik\cdot p+\varepsilon}\right]^{0}_{-\infty}=\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\tilde{\Theta}^{\bar{C}}_{C}(k)\left[\frac{g_{s}}{p\cdot
k-i\varepsilon}\left[{\cal T}^{\rm
in}\right]^{\bar{C},\bar{A}\bar{B}}_{C,AB}\right],$ (3.22)

where the lower limit of the $t$ integral vanishes because of the
$\varepsilon$ regulator. The contents of the square bracket is the required
momentum-space Feynman rule, which indeed agrees with Eq. (3.19). We may
define Feynman rules for outgoing emissions analogously, sending
$\varepsilon\to-\varepsilon$ to make the integral well-defined. To further
clarify Eq. (3.20), note that for the diagonal elements of $\left[{\cal
T}^{\rm in}\right]^{\bar{C},\bar{A}\bar{B}}_{C,AB}$ (i.e. soft gluon
emission), we may simplify it to

$\displaystyle ig_{s}{\bf T}^{c}\int_{-\infty}^{0}\,{\rm
d}t\,p^{\mu}\,A^{c}_{\mu}(tp)\,,$ (3.23)

which is the usual exponent of the Wilson-line operator (see Eq. (2.4)),
omitting the explicit identity matrices $\delta^{\hat{a}\hat{b}}$ and
$\eta^{\alpha\beta}$ in spinor/Lorentz space. This is as it should be: the
effect of a single gluon emission should be given by a Wilson line. However,
this also suggests an interpretation of Eq. (3.20): it acts as a generalised
Wilson line exponent that describes the emission of soft (anti-)quarks in
addition to gluons. That we can indeed interpret the object in Eq. (3.20) in
this way is discussed in the following section.

###  3.2 Multiple emissions from a generalised soft-emission operator

The effect of multiple emissions off a single (possibly flavour-changing) hard
line with momentum $p^{\mu}$ may be generated by the following path-ordered
object:

${\cal F}^{\bar{A}\bar{B}}_{AB}={\cal P}\exp\left[ig_{s}\int_{-\infty}^{0}{\rm
d}t\,\left[{\cal
T}\right]^{\bar{C}}_{C}\,\Theta^{\bar{C}}_{C}(tp)\right]^{\bar{A}\bar{B}}_{AB}.$
(3.24)

We will call this a _generalised soft-emission operator_. It carries open
indices and needs to be sandwiched in-between the matrix element that the hard
line originated from, and its external state. The notation of Eq. (3.24) is
appropriate for an incoming hard line, and the regularisation is implicitly
understood. The soft-emission operator for an outgoing hard line is easily
obtained by switching the integration boundaries to $(0,\infty)$, and adapting
the regularisation accordingly. To see that the object defined in Eq. (3.24)
indeed defines multiple insertions of the transition matrix, we may expand it
and examine the $\mathcal{O}(g_{s}^{l})$ contribution, which is given by

${\cal
F}^{\bar{A}\bar{B}}_{AB}\Big{|}_{\mathcal{O}(g_{s})^{l}}=(ig_{s})^{l}\int_{-\infty}^{0}{\rm
d}t_{1}\,\left[{\cal
T}\right]^{\bar{C}_{1},\bar{A}\bar{A}_{1}}_{C_{1},AA_{1}}\Theta^{\bar{C}_{1}}_{C_{1}}(t_{1}p)\,\,\dots\,\,\int_{t_{l-1}}^{0}{\rm
d}t_{l}\,\left[{\cal
T}\right]^{\bar{C}_{l},\bar{A}_{l-1}\bar{B}}_{C_{l},A_{l-1}B}\Theta^{\bar{C}_{l}}_{C_{l}}(t_{l}p)\,.$
(3.25)

Writing again the field in momentum space through Eq. (3.21), we end up with a
string of integrals of the form

$\int_{-\infty}^{0}{\rm d}t_{1}\int_{t_{1}}^{0}{\rm
d}t_{2}\,\,\dots\,\,\int_{t_{l-1}}^{0}{\rm
d}t_{l}\,\exp\left[i\sum_{j=1}^{l}(k_{j}\cdot
p)t_{j}\right]=\frac{(-i)^{l}}{k_{1}\cdot p\,(k_{1}+k_{2})\cdot
p\,\dots\,(k_{1}+\dots+k_{l})\cdot p}\,,$ (3.26)

where $k_{j}$ is the four-momentum of each of the soft fields. With this, the
full soft-emission operator at $\mathcal{O}(g_{s}^{l})$ becomes

${\cal
F}^{\bar{A}\bar{B}}_{AB}\Big{|}_{\mathcal{O}(g_{s})^{l}}=\prod_{j=1}^{l}\left[\int\frac{{\rm
d}^{d}k_{j}}{(2\pi)^{d}}\widetilde{\Theta}^{\bar{C}_{l}}_{C}(k_{j})\right]\frac{g_{s}^{l}}{k_{1}\cdot
p\,\dots\,(k_{1}+\dots k_{l})\cdot p}\left[{\cal
T}\right]^{\bar{C}_{1},\bar{A}\bar{A}_{1}}_{C_{1},AA_{1}}\,\dots\,\left[{\cal
T}\right]^{\bar{C}_{l},\bar{A}_{l-1}\bar{B}}_{C_{l},A_{l-1}B}\,.$ (3.27)

We may now see how one may use this object to generate soft emissions. For
instance, we can consider a configuration involving an incoming quark with
colour label $i_{1}$ and spinor label $\hat{a}_{1}$, transitioning into a
gluon of colour index $b$, and space-time index $\beta$, entering into the
hard scattering ${\cal A}_{g}$, at third order in the strong coupling $g_{s}$.
The expansion of the generalised Wilson line up to third order automatically
generates all contributions given in Figs. 6 and 7, given that the transitions
operators ${\cal T}_{IJ}$ are matrices in the flavour space $IJ$, and matrix
multiplication generates all possible transitions.

Figure 6: Diagrams corresponding to the first three contributions in Eq.
(3.28), originating from expanding the generalised Wilson line in Eq. (3.24)
to third order in $g_{s}$, and selecting the components giving rise to a
transition between an initial quark and a gluon incoming into the amplitude
${\cal A}_{g}$.

We obtain

$\displaystyle\frac{g_{s}^{3}}{p\cdot
k_{1}\,p\cdot(k_{1}+k_{2})\,p\cdot(k_{1}+k_{2}+k_{3})}u^{i_{1}}_{\hat{a}_{1}}(p)\Big{\\{}\left[{\cal
T}^{{\rm
in}}_{qq}(p)\right]^{c_{1},i_{1}i_{2}}_{\gamma_{1},\hat{a}_{1}\hat{a}_{2}}\,\left[{\cal
T}^{\rm
in}_{qq}(p)\right]^{c_{2},i_{2}i_{3}}_{\hat{c}_{2},\hat{a}_{2}\hat{a}_{3}}\,\left[{\cal
T}^{\rm in}_{qg}(p)\right]^{j_{3},i_{3}b}_{\hat{c}_{3},\hat{a}_{3}\beta}$
(3.28) $\displaystyle+\,\left[{\cal T}^{{\rm
in}}_{qq}(p)\right]^{c_{1},i_{1}i_{2}}_{\gamma_{1},\hat{a}_{1}\hat{a}_{2}}\,\left[{\cal
T}^{\rm
in}_{qg}(p)\right]^{j_{2},i_{2}a_{3}}_{\hat{c}_{2},\hat{a}_{2}\alpha_{3}}\,\left[{\cal
T}^{\rm in}_{gg}(p)\right]^{c_{3},a_{3}b}_{\gamma_{3},\alpha_{3}\beta}$
$\displaystyle+\,\left[{\cal T}^{{\rm
in}}_{qg}(p)\right]^{j_{1},i_{1}a_{2}}_{\hat{c}_{1},\hat{a}_{1}\alpha_{2}}\,\left[{\cal
T}^{\rm
in}_{gg}(p)\right]^{c_{2},a_{2}a_{3}}_{\gamma_{2},\alpha_{2}\alpha_{3}}\,\left[{\cal
T}^{\rm in}_{gg}(p)\right]^{c_{3},a_{3}b}_{\gamma_{3},\alpha_{3}\beta}$
$\displaystyle+\,\left[{\cal T}^{{\rm
in}}_{qg}(p)\right]^{j_{1},i_{1}a_{2}}_{\hat{c}_{1},\hat{a}_{1}\alpha_{2}}\,\left[{\cal
T}^{\rm
in}_{gq}(p)\right]^{j_{2},a_{2}i_{3}}_{\hat{c}_{2},\alpha_{2}\hat{a}_{3}}\,\left[{\cal
T}^{\rm in}_{qg}(p)\right]^{j_{3},i_{3}b}_{\hat{c}_{3},\hat{a}_{3}\beta}$
$\displaystyle+\,\left[{\cal T}^{{\rm
in}}_{qg}(p)\right]^{j_{1},i_{1}a_{2}}_{\hat{c}_{1},\hat{a}_{1}\alpha_{2}}\,\left[{\cal
T}^{\rm
in}_{g\bar{q}}(p)\right]^{j_{2},a_{2}i_{3}}_{\hat{c}_{2},\alpha_{2}\hat{a}_{3}}\,\left[{\cal
T}^{\rm
in}_{\bar{q}g}(p)\right]^{j_{3},i_{3}b}_{\hat{c}_{3},\hat{a}_{3}\beta}\Big{\\}}\left[{\cal
A}_{g}(p)\right]^{b}_{\beta}\,.$

Filling in the transition-matrix elements we get

$\displaystyle\frac{g_{s}^{3}}{p\cdot
k_{1}\,p\cdot(k_{1}+k_{2})\,p\cdot(k_{1}+k_{2}+k_{3})}\Big{\\{}-\,\frac{1}{2}[t^{b}t^{c_{2}}t^{c_{1}}]_{j_{3}i_{1}}p_{\gamma_{1}}p_{\gamma_{2}}[\gamma^{\beta}u(p)]^{i_{1}}_{\hat{c}_{3}}$
$\displaystyle-\,\frac{i}{2}f^{bc_{3}a_{3}}[t^{a_{3}}t^{c_{1}}]_{j_{2}i_{1}}p_{\gamma_{1}}p_{\gamma_{3}}[\gamma^{\beta}u(p)]^{i_{1}}_{\hat{c}_{2}}$
$\displaystyle+\,\frac{1}{2}f^{bc_{3}a_{3}}f^{a_{3}c_{2}a_{2}}[t^{a_{2}}]_{j_{1}i_{1}}p_{\gamma_{2}}p_{\gamma_{3}}[\gamma^{\beta}u(p)]^{i_{1}}_{\hat{c}_{1}}$
$\displaystyle+\,\frac{1}{4}[t^{a_{2}}]_{j_{1}i_{1}}[t^{a_{2}}t^{b}]_{j_{3}j_{2}}[\gamma^{\beta}\not{p}\gamma^{\alpha_{2}}]_{\hat{c}_{3}\hat{c}_{2}}[\gamma_{\alpha_{2}}u(p)]^{i_{1}}_{\hat{c}_{1}}$
$\displaystyle-\,\frac{1}{4}[t^{a_{2}}]_{j_{1}i_{1}}[t^{a_{2}}t^{b}]_{j_{2}j_{3}}[\gamma^{\alpha_{2}}\not{p}\gamma^{\beta}]_{\hat{c}_{2}\hat{c}_{3}}[\gamma_{\alpha_{2}}u(p)]^{i_{1}}_{\hat{c}_{1}}\Big{\\}}\left[{\cal
A}_{g}(p)\right]^{b}_{\beta}\,.$ (3.29)

Each line correctly reproduces the corresponding diagrams in Figs. 6 and 7, at
the lowest order at which each diagram contributes within the soft expansion.
In this respect, it is important to highlight that the diagrams in Figs. 6 and
7 contribute at different orders in the soft expansion, although this is not
apparent from the expressions in Eq. (3.2). The correct power counting emerges
when inserting the expressions above in a full squared matrix element. In the
virtual case, the soft gluons/quarks are contracted to produce a soft
propagator, whereas in the real-emission case, the sum over physical
polarisation/spinor degrees-of-freedom produces additional terms. In both
cases, the contribution of a soft quark produces an additional factor of the
soft momentum $k_{i}$ w.r.t. the emission of a soft gluon, i.e., it is power
suppressed. Thus, in the example at hand the three diagrams in Fig. 6 involve
the emission of a soft quark, so contribute at NLP compared to diagrams where
only soft gluons appear. The remaining two diagrams in Fig. 7 involve the
emission of three soft quarks, so contribute at N3LP in a squared matrix
element. This observation can be encoded directly in the matrices of Eqs.
(3.14) and (3.18), by noticing that the off-diagonal elements give rise to
power-suppressed contributions. In this way one has a definite power counting,
that can be used to determine at which order in the power expansion a given
term in Eq. (3.27) will contribute.

As a last comment, note that, by taking only the diagonal elements of ${\cal
T}$, we directly see that the generalised soft-emission operator defined in
Eq. (3.24) is equal to the standard Wilson line. However, one must bear in
mind that although the normal Wilson line is an object that transforms
covariantly under the gauge group, our soft-emission operator does not
straightforwardly share this property. Whilst it may be possible to make
rigorous the gauge-transformation properties of the generalised soft emission
operator (e.g. by exploiting the property that it lives in a reducible
representation of the gauge group), this is unnecessary for our purposes.
Instead, we can simply consider our operator as a convenient book-keeping
device, casting soft-quark emissions in the same framework as soft-gluon
emissions, thereby allowing us to use the replica trick.

Figure 7: Diagrams corresponding to the last two contributions in Eq. (3.28),
originating from expanding the generalised Wilson line in Eq. (3.24) to third
order in $g_{s}$, and selecting the components giving rise to a transition
between an initial quark and a gluon incoming into the amplitude ${\cal
A}_{g}$ These two diagrams are suppressed by a power of $\lambda^{2}$ w.r.t.
the first three diagrams in 6.

###  3.3 The replica trick for emissions of soft partons

Figure 8: Amplitude with $n$ external parton legs, each of which carries an
abstract index $A_{i}$ in species space.

Let us consider a scattering process with $n$ external parton legs, carrying
hard momenta $\\{p_{i}\\}$. Each external line $i$ will carry an index $A_{i}$
in the flavour/spin space that has been introduced above. Following Fig. 8
(c.f. Fig 5), we label the amplitude with external wavefunctions removed as
${\cal A}^{\bar{A}_{1}\ldots\bar{A}_{n}}_{A_{1}\ldots A_{n}}$, where barred
indices are colour indices as before. The full hard-scattering amplitude then
takes the form

${\cal A}(\\{p_{i}\\})={\cal A}^{\bar{A}_{1}\ldots\bar{A}_{n}}_{A_{1}\ldots
A_{n}}(\\{p_{i}\\})\prod_{i=1}^{n}\xi^{\bar{A}_{i}}_{A_{i}}(p_{i})\,,$ (3.30)

where we again use $\xi^{\bar{A}}_{A}$ to denote the external wave function
for a particle with species index $A$ and colour index $\bar{A}$. We may then
describe the emission of any number of virtual soft partons from the external
lines by dressing each line of the stripped amplitude with the generalised
soft-emission operator of Eq. (3.24), such that this is sandwiched between the
amplitude and the external wavefunctions. That is, we may define a generalised
(and flavour-changing) soft function

${\cal
A}(\\{p_{i}\\})=\left(\prod_{i=1}^{n}\xi^{\bar{A}_{i}}_{A_{i}}(p_{i})\right){\cal
S}^{\bar{A}_{1}\ldots\bar{A}_{n},\bar{B}_{1}\ldots\bar{B}_{n}}_{A_{1}\ldots
A_{n},B_{1}\ldots B_{n}}\,{\cal A}^{\bar{B}_{1}\ldots\bar{B}_{n}}_{B_{1}\ldots
B_{n}}\,,$ (3.31)

where now ${\cal A}$ is the full amplitude dressed by soft parton emission,
and the generalised soft function is given by

${\cal
S}^{\bar{A}_{1}\ldots\bar{A}_{n},\bar{B}_{1}\ldots\bar{B}_{n}}_{A_{1}\ldots
A_{n},B_{1}\ldots B_{n}}=\Big{\langle}0\Big{|}\prod_{i=1}^{n}{\cal
F}^{\bar{A}_{i}\bar{B}_{i}}_{A_{i}B_{i}}(p_{i})\Big{|}0\Big{\rangle}\,.$
(3.32)

That is, the soft function is defined similarly to the pure gluon emission
case, by a VEV of generalised soft-emission operators acting along each
incoming particle contour. We may note that it is matrix-valued in the tensor
product space of the flavour/spin and colour indices associated with each hard
line. Note that this is directly analogous to how the usual soft function is
matrix-valued in the tensor product space of colour indices, so the definition
in Eq. (3.32) presents no significant additional complication from the
conceptual point of view. As in Eq. (2.5), we may write a path-integral
formula for the definition of Eq. (3.32). Let us first introduce the compact
notation

${\cal D}\Theta^{\bar{C}}_{C}\equiv{\cal D}\psi^{i}\,{\cal
D}\bar{\psi}^{j}\,{\cal D}A_{\mu}^{a}\,,$ (3.33)

i.e. the path-integral over the individual parton fields appearing in Eq.
(3.8) can be simply written as a single measure involving the array of fields
$\Theta_{C}^{\bar{C}}$. Then Eq. (3.32) may be written as

${\cal S}_{A_{1}\ldots A_{n},B_{1}\ldots
B_{n}}^{\bar{A}_{1}\ldots\bar{A}_{n},\bar{B}_{1}\ldots\bar{B}_{n}}=\int{\cal
D}\Theta^{\bar{C}}_{C}\,\left(\prod_{i=1}^{n}{\cal
F}^{\bar{A}_{i}\bar{B}_{i}}_{A_{i}B_{i}}(p_{i})\right)e^{iS[\Theta^{\bar{C}}_{C}]}\,,$
(3.34)

which is similar in form to the pure gluon case of Eq. (2.5), and

$\displaystyle S[\Theta^{\bar{C}}_{C}]=\int{\rm d}^{d}x\,\mathcal{L}_{\rm
QCD}\,$ (3.35)

is the QCD action. Consequently, the replica trick argument for exponentiation
of arbitrary soft parton corrections proceeds by direct analogy to the soft
gluon case. Carrying out the path integral over the compound soft parton field
$\Theta^{\bar{C}}_{C}$ generates all possible Feynman diagrams in which the
external hard partons are connected by soft partons. We may then introduce $N$
identical replicas of the original soft parton field, such that Eq. (2.9) is
replaced by

$\displaystyle\left[{\cal S}_{N}\right]_{A_{1}\ldots A_{n},B_{1}\ldots
B_{n}}^{\bar{A}_{1}\ldots\bar{A}_{n},\bar{B}_{1}\ldots\bar{B}_{n}}$
$\displaystyle=\int{\cal D}[\Theta^{(1)}]^{\bar{C}}_{C}\ldots{\cal
D}[\Theta^{(n)}]^{\bar{C}}_{C}\,\exp\left\\{i\sum_{j=1}^{N}S\Big{[}[\Theta^{(j)}]^{\bar{C}}_{C}\Big{]}\right\\}$
(3.36) $\displaystyle\quad\times\,\prod_{k=1}^{n}{\cal R}\left\\{{\cal
P}\exp\left(ig_{s}{\bf T}^{\bar{C}}\,\sum_{j=1}^{N}\int{\rm d}t_{k}\,{\bf
Q}^{C}[\Theta^{(j)}]^{\bar{C}}_{C}(t_{k}p_{k})\right)\right\\}^{\bar{A}_{i}\bar{B}_{i}}_{A_{i}B_{i}}.$

There is now a product of generalised soft-emission operators on each external
leg, associated with the different soft parton replicas that may be emitted.
As before, there is a replica-ordering operator, which in any given diagram
acts to reorder parton emissions as necessary. We must bear in mind, however,
that the action of the replica-ordering operator ${\cal R}$ in Eq. (3.36) is
more complicated than for pure soft gluon emission. In the present case, the
replica-ordering operator acts on the full transition matrix $[{\cal
T}_{IJ}]_{C,AB}^{\bar{C},\bar{A}\bar{B}}$, instead of only on the colour
matrices as was the case for soft-gluon emissions. This leads us to introduce
$[{\cal T}_{N,IJ}(D)]_{C,AB}^{\bar{C},\bar{A}\bar{B}}$, which is the
transition matrix of a replicated theory. This matrix will determine both the
colour factor and the kinematic numerator of a given diagram, whereas the
denominator is untouched. To see this, let us decompose the kinematic part of
a given soft diagram $D$ in momentum space as

${\cal K}(D)=\frac{{\cal N}(D)}{{\cal D}(D)}\,,$ (3.37)

where ${\cal N}(D)$ is a kinematic numerator factor, which results from
combining the emission factors $\\{Q^{C,AB}\\}$ of the transition functions
for each emitted soft parton, as well as coupling factors, external
wavefunctions etc. The denominator ${\cal D}$ arises from performing the
integrals over the distance parameters $\\{t_{k}\\}$, which results in a
product of nested momentum factors as exemplified in Eq. (3.26). While the
numerator depends on the spinor structure of the diagrams, the denominator
only depends on the kinematic assignment. If only soft gluons are emitted, the
kinematic numerator for a diagram $D$ with $n_{j}$ soft emissions on a given
line $j$ takes the simple form

${\cal N}(D)\propto\prod_{j}p^{\mu_{1}}p^{\mu_{2}}\ldots p^{\mu_{j}}\,.$

That is, there is a simple factor of the hard momentum for each emission, such
that the emission factors for individual emissions commute with each other,
and the numerator for any diagram with the same number of emissions on each
line is the same. This indeed follows straightforwardly from the conventional
Wilson line definition of Eq. (2.4), where the only non-commuting quantity in
the exponent is the colour generator ${\bf T}_{i}$, and it is for this reason
that the replica ordering operator ${\cal R}$ acts to modify the colour
factors of graphs $C(D)\rightarrow\tilde{C}(D)$, but not their kinematics.
This ceases to be the case for the generalised soft emission operator of Eq.
(3.24). In this case the emission factors are not mutually commuting, and so
the action of ${\cal R}$ is to reorder them, in addition to the colour
generators. In the replicated theory, we denote by ${\cal N}_{N}(D)$ the
kinematic numerator of diagram $D$, such that its full contribution is

$C_{N}(D)\,{\cal K}_{N}(D)=\frac{{\cal N}_{N}(D)\,C_{N}(D)}{{\cal D}(D)}\,,$
(3.38)

where the replicated colour factor $C_{N}(D)$ is defined as in the case of
pure gluon emission. Despite this additional complication, the replica trick
argument proceeds as before, where the logarithm of the soft amplitude can be
directly obtained by taking the ${\cal O}(N)$ part of each soft diagram.
Defining exponentiated numerators and colour factors via

$\tilde{{\cal N}}(D)={\cal N}_{N}(D)\Big{|}_{{\cal
O}(N)}\,,\quad\tilde{C}(D)=C_{N}(D)\Big{|}_{{\cal O}(N)}\,,$ (3.39)

we may write the contribution of a given soft diagram to the logarithm of the
amplitude as

$\tilde{E}(D)=\frac{\tilde{\cal N}(D)\,\tilde{C}(D)}{{\cal D}(D)}\,.$ (3.40)

An exponentiated form of the soft function is then simply found as

$\displaystyle\mathcal{S}={\rm exp}\left[\sum_{D}\tilde{E}(D)\right],$ (3.41)

where $\tilde{E}(D)$ are the generalised webs.

Figure 9: Diagrams (a) and (b) constitute a generalization at NLP of the web
in Fig. 3.

As in the pure gluon emission case, the replica trick provides a completely
systematic way of calculating the logarithm of the soft amplitude directly,
rather than the amplitude itself. However, the abstract nature of the
generalised soft function of Eq. (3.36), and the exponentiated numerators
appearing in Eq. (3.40), will doubtless look rather baffling to the reader. To
explain our notation, let us consider an example. We take an amplitude with
external partons $q(p_{1})$ and $\bar{q}(p_{2})$, which define two hard
outgoing legs. At one-loop level, these fermions can either remain the same
flavour (LP), or transition into gluons (NLP). When allowing for two loops, we
may either consider two gluon exchanges (LP), one gluon and one fermion
exchange (NLP), or two fermionic exchanges (NNLP). For this example we focus
on the NLP contribution to the two-loop diagram, meaning that the external
particles will have to be gluons. We thus consider two insertions of the
replicated transition matrix on each leg, as shown in Fig. 9. For diagram (a)
we denote the ordering of transition matrices as

$\displaystyle\left[{\cal
T}^{(i)}(p_{1})\right]^{\bar{C}_{1},\bar{A}_{1}\bar{a}_{1}}_{C_{1},A_{1}a_{1}}\left[{\cal
T}^{(j)}(p_{1})\right]^{\bar{C}_{2},\bar{a}_{1}\bar{B}_{1}}_{C_{2},a_{1}B_{1}}\left[{\cal
T}^{(i)}(p_{2})\right]^{\bar{C}_{1},\bar{A}_{2}\bar{a}_{2}}_{C_{1},A_{2}a_{2}}\left[{\cal
T}^{(j)}(p_{2})\right]^{\bar{C}_{2},\bar{a}_{2}\bar{B}_{2}}_{C_{2},a_{2}B_{2}}\,.$
(3.42)

This object is sandwiched in between two external wave functions
$\epsilon^{\dagger}_{B_{1}}(p_{1})$ and $\epsilon^{\dagger}_{B_{2}}(p_{2})$,
and a hard-scattering matrix
$\mathcal{A}(p_{1},p_{2})_{A_{1}A_{2}}^{\bar{A}_{1}\bar{A}_{2}}$. In Eq.
(3.42), lower-case letters correspond to intermediate particles on the hard
lines. The replica-reordering operator leaves the order the same for all
choices of $i,j$. This is not true for diagram (b), for which the ordering of
transition matrices is

$\displaystyle{\cal T}^{(i)}(p_{1}){\cal T}^{(j)}(p_{1}){\cal
T}^{(j)}(p_{2}){\cal T}^{(i)}(p_{2})\,.$ (3.43)

Here we leave the indices implicit for ease of notation. The reordering acts
as follows:

$\displaystyle{\cal R}\left[{\cal T}^{(i)}(p_{1}){\cal T}^{(j)}(p_{1}){\cal
T}^{(j)}(p_{2}){\cal T}^{(i)}(p_{2})\right]=\begin{cases}{\cal
T}^{(i)}(p_{1}){\cal T}^{(j)}(p_{1}){\cal T}^{(i)}(p_{2}){\cal
T}^{(j)}(p_{2})&\,i<j\quad\left(\frac{N(N-1)}{2}\right),\\\ {\cal
T}^{(j)}(p_{1}){\cal T}^{(i)}(p_{1}){\cal T}^{(j)}(p_{2}){\cal
T}^{(i)}(p_{2})&\,j<i\quad\left(\frac{N(N-1)}{2}\right),\\\ {\cal
T}^{(j)}(p_{1}){\cal T}^{(j)}(p_{1}){\cal T}^{(j)}(p_{2}){\cal
T}^{(i)}(p_{2})&\,i=j\quad(N)\,,\\\ \end{cases}$ (3.44)

where we have also denoted the multiplicity of each replica assignment in the
round brackets. In direct analogy with the soft-gluon case, we see that the
first two expressions correspond to the ordering for diagram (a), whereas the
last entry is the ordering for diagram (b). When combined with the external
wave functions, we can write the contribution of diagram (b) to the logarithm
of the generalised soft function as

$\displaystyle\widetilde{E}(b)=\frac{{\cal N}(b)C(b)-{\cal N}(a)C(a)}{{\cal
D}(b)}\,.$ (3.45)

The projection onto external wave functions and the hard-scattering amplitude
will turn the matrix notation into a sum expression, and will contract the
open indices of the transition matrices.

So far we have discussed the use of the replica trick for virtual corrections
only. Let us now generalise this to include real emissions.

###  3.4 The replica trick for real emissions

The starting point for using the replica trick in the case of virtual
exchanges is to consider a generating functional for virtual soft parton
diagrams (Eq. (3.34) in the general case). In the replicated theory, the
corresponding functional is simply given by the original one raised to a
power, as in Eq. (2.6). It is this fact that allows one to conclude that the
${\cal O}(N)$ contribution to diagrams in the replicated theory formally
exponentiates. To include real emissions, we must therefore identify a
suitable generating functional, such that the appropriate analogue of Eq.
(2.6) can be applied. To this end, we simplify our arguments for the moment by
considering a toy model of a single scalar field $\phi$, and a generating
functional of the form

$Z=\int{\cal D}\phi\,\Psi[\phi]e^{iS[\phi]}\,,$ (3.46)

where $\Psi[\phi]$ is some function of the field that is the analogue of the
product of generalised soft emission operators in Eq. (3.34). As is well-
known, functionals such as those of Eqs. (3.34, 3.46) generate all possible
Green’s functions, or vacuum expectation values of (time-ordered) products of
fields. Usually in scalar field theory, one introduces a current $J$ conjugate
to $\phi$, such that differentiation with respect to this current can be used
to pick out Green’s functions of arbitrary multiplicity. Here, we will instead
use the alternative approach of Ref. [147]. First, we may separate the action
into its free-field part and an interaction term

$S=S_{0}+S_{I}=\int{\rm d}^{d}x\,\left(\frac{1}{2}\phi K\phi+S_{I}\right)\,.$
(3.47)

Here $K$ is the operator appearing in the quadratic term in the Lagrangian. To
carry out the path integral, one may use the integral identity

$\int{\rm
d}u\,e^{-uMu/2}f(u+v)\,\sim\,e^{\partial_{v}M^{-1}\partial_{v}/2}f(v)\,,$
(3.48)

where we have neglected an overall factor. By applying the appropriate
functional generalisation of this around $v=0$, the path integral in Eq.
(3.46) can be carried out (after substituting Eq. (3.47)) to give

$Z=\left.\exp\left[-\int{\rm d}^{d}x\,{\rm
d}^{d}x^{\prime}\,\frac{1}{2}\frac{\delta}{\delta\phi(x)}\Delta(x-x^{\prime})\frac{\delta}{\delta\phi(x^{\prime})}\right]e^{iS_{I}[\phi]}\Psi[\phi]\right|_{\phi=0}\,.$
(3.49)

Here $\Delta(x-x^{\prime})$ is the Feynman propagator, corresponding (up to a
numerical factor) with the inverse of the operator $K$. This is a less
standard form of the generating functional, but nevertheless generates the
usual Feynman diagram expansion. To see this, note that Taylor expanding the
factors on the right-hand side yields a sum of products of the field $\phi$,
where these come manifestly from interaction vertices of the theory. The
latter may be contained in the interaction Lagrangian appearing in $S_{I}$, or
the function $\Psi[\phi]$. In the case of soft parton emission, this
corresponds to vertices off or on the hard particle lines respectively. The
prefactor in Eq. (3.49) takes pairs of fields appearing in each term, and
strips them off in favour of a propagator. After setting $\phi\rightarrow 0$,
one is left with a sum of contributions in which interaction vertices are
joined by propagators. These are the usual Feynman diagrams, and the
exponential dependence in the prefactor turns out to be precisely such as to
yield the required combinatorial factors associated with the usual Feynman
rules. Examples can be found in Ref. [147] for conventional scalar field
theory, although we have here modified the argument slightly so that it
provides a scalar analogue of Eq. (3.34). All Feynman diagrams will then
correspond to hard particle lines which are connected by conventional
propagators and interaction vertices. That is, Eq. (3.49) generates virtual
diagrams only, as expected from the fact that the original generating function
of Eq. (3.49) is a vacuum-to-vacuum transition amplitude.

So much for virtual corrections. Next, we can include the effect of real
corrections, and before doing so we note that the exponential prefactor
appearing in Eq. (3.49) can be written in momentum space as [147]

$\exp\left[-\int{\rm d}^{d}x\,{\rm
d}^{d}x^{\prime}\,\frac{1}{2}\frac{\delta}{\delta\phi(x)}\Delta(x-x^{\prime})\frac{\delta}{\delta\phi(x^{\prime})}\right]=\exp\left[\int\frac{{\rm
d}^{d}p}{(2\pi)^{d}}\frac{1}{2}\frac{\delta}{\delta\phi(p)}\Delta(p)\frac{\delta}{\delta\phi(-p)}\right],$
(3.50)

where we have used the same symbol for the field and propagator in position or
momentum space, but where the argument resolves any ambiguity. To include real
corrections, we may consider the cut propagator

$\Delta_{+}(p)=2\pi\Theta(p^{0})\delta\left[(p^{2}+m^{2})\right].$ (3.51)

Then, a generating functional for the squared amplitude is

$Z_{\rm sq}=Z\exp\left[-\int\frac{{\rm
d}^{d}p}{(2\pi)^{d}}\frac{1}{2}\frac{\overleftarrow{\delta}}{\delta\phi(p)}\Delta_{+}(p)\frac{\overrightarrow{\delta}}{\phi(-p)}\right]Z^{\dagger}.$
(3.52)

Here the arrows on the derivatives indicate that they act to the left or
right, and we have introduced the generating functionals for virtual diagrams
$Z$ and $Z^{\dagger}$ on either side of the final-state cut. By direct analogy
with Eqs. (3.49, 3.50), the factor in the middle acts to strip off pairs of
fields, and replace them with a propagator. Now, however, this is a cut
propagator, and connects fields on different sides of the cut, as is required
for a real particle. We have here considered the simplified case of a scalar
field to simplify our arguments. However, it is straightforward to generalise
this to the soft parton case. Let ${\cal S}$ be the (virtual) soft function of
Eq. (3.34) with colour and species indices suppressed. A suitable generating
functional for the squared amplitude is

$Z_{{\rm sq}}={\cal S}\exp\left[-\int\frac{{\rm
d}^{d}p}{(2\pi)^{d}}\frac{1}{2}\frac{\overleftarrow{\delta}}{\delta\Theta^{\bar{C}}_{C}(p)}\Delta_{+\phantom{,}CD}^{\phantom{+,}\bar{C}\bar{D}}(p)\frac{\overrightarrow{\delta}}{\delta\bar{\Theta}^{\bar{D}}_{D}(-p)}\right]{\cal
S}^{\dagger},$ (3.53)

where we have been careful to note that the functional differentiation acting
to the right involves the conjugate field to that acting to the left.777 The
Keldysh formalism [148, 149] provides provide an alternative approach to
arrive at an equivalent form of Eq. (3.53) (also see Appendix C of Ref. [150]
for an application to the Drell-Yan Soft function).

We can now apply the replica trick. Upon replicating the soft parton theory,
Eq. (3.53) is replaced by

$Z_{{\rm sq},N}={\cal
S}_{N}\left\\{\prod_{i=1}^{N}\exp\left[-\int\frac{d^{d}p}{(2\pi)^{d}}\frac{1}{2}\frac{\overleftarrow{\delta}}{\delta\Theta^{(i)\bar{C}}_{C}(p)}\Delta_{+\phantom{,}CD}^{\phantom{+,}\bar{C}\bar{D}}(p)\frac{\overrightarrow{\delta}}{\delta\bar{\Theta}^{(i)\bar{D}}_{D}(-p)}\right]\right\\}{\cal
S}^{\dagger}_{N},$ (3.54)

where $\Theta_{C}^{(i)\bar{C}}$ is the compound soft parton field associated
with replica $i$. Furthermore, we have introduced the replicated generating
functional for virtual soft-parton exchange diagrams of Eq. (3.36), again with
colour and flavour indices suppressed. Given that this functional satisfies
${\cal S}_{N}={\cal S}^{N}$, we obtain

$Z_{{\rm sq},N}=(Z_{\rm sq})^{N}=1+N\log(Z_{\rm sq})+{\cal O}(N^{2})\,.$
(3.55)

Thus, the usual replica trick arguments apply,888In extending this statement
to a given observable, one must make sure that the corresponding measurement
function does not differentiate between the exchanged soft partons. This will
indeed be the case for infrared-safe observables. and the ${\cal O}(N)$ piece
of an arbitrary diagram contributes to the logarithm of the squared soft
parton amplitude. This succeeds in proving the exponentiation of either real
or virtual soft parton corrections, as well as providing a systematic method
for deriving how the colour and/or kinematic dependence of Feynman diagrams
becomes modified upon their entering the logarithm of the soft function. In
the following sections, we illustrate the use of the abstract arguments of
this and the preceding sections, by applying them to processes of
phenomenological interest.

##  4 Real emission corrections in DIS

Figure 10: Squared Feynman diagram for the DIS process at LO.

As a testing ground for applying the replica trick to the generalised soft
function, we will consider photon-initiated deep-inelastic scattering (DIS)
process. This serves two purposes: we will reproduce a known exponentiation
property of soft quark and gluon emissions, which was recently used in Ref.
[123] (see also Ref. [117]) to resum NLP logarithms in the off-diagonal DGLAP
splitting functions. Secondly, the discussion in this section paves the way
for the proof (in the following section) of a conjecture made in Refs. [117,
113], namely that the leading virtual corrections in the quark channel should
formally exponentiate.

First we introduce some notation. At LO, the incoming virtual photon couples
to a valence quark, leading to the process

$q(p)+\gamma^{*}(q)=q(p_{2})\,,$ (4.1)

The LO squared Feynman diagram is shown in Fig. 10, and it is conventional to
define

$Q^{2}=-q^{2}>0\,,\qquad\qquad x=\frac{Q^{2}}{2\,p\cdot q}\,,$ (4.2)

where the latter is the so-called Björken $x$ variable. In terms of these
parameters, one may define the structure function

$F_{2}(x,Q^{2})=\int{\rm d}\Phi_{1}\,T_{2}^{\alpha\beta}\,\overline{|{\cal
A}|^{2}_{\alpha\beta}}\,,$ (4.3)

with ${\rm d}\Phi_{1}$ the one-particle phase-space. The integrand contains
the squared amplitude summed (averaged) over final (initial) colours and spins
respectively, with photon Lorentz indices as in Fig. 10. We have also
introduced the projector

$T_{2}^{\alpha\beta}=-\frac{1}{4\pi}\frac{1}{1-\epsilon}\left(\eta^{\alpha\beta}+(3-2\epsilon)\frac{q^{2}}{(p\cdot
q)^{2}}p^{\alpha}p^{\beta}\right),$ (4.4)

where we work in $d=4-2\epsilon$ dimensions. The LO amplitude and complex
conjugate amplitude read

$\displaystyle\big{[}\mathcal{A}^{(0)}\big{]}^{\alpha}$
$\displaystyle=\bar{u}_{\hat{b}}^{j}(p_{2})\,ie\gamma_{\hat{b}\hat{a}}^{\alpha}\delta^{ji}\,u_{\hat{a}}^{i}(p_{2}),$
(4.5a) $\displaystyle\big{[}\mathcal{A}^{{\dagger}(0)}\big{]}^{\beta}$
$\displaystyle=\bar{u}_{\hat{a}^{\prime}}^{i^{\prime}}(p)\,(-i)e\gamma_{\hat{a}^{\prime}\hat{b}^{\prime}}^{\beta}\delta^{i^{\prime}j^{\prime}}\,u_{\hat{b}^{\prime}}^{j^{\prime}}(p_{2}),$
(4.5b)

leading to

$\overline{|{\cal A}^{(0)}|^{2}_{\alpha\beta}}=\frac{e^{2}}{2}{\rm
Tr}\big{[}\not{p}\gamma_{\beta}\not{p}_{2}\gamma_{\alpha}\big{]}.$ (4.6)

The matrix element squared is then inserted in Eq. (4.3), where at LO one has

$\int{\rm d}\Phi_{1}=\int\frac{{\rm
d}^{d}p_{2}}{(2\pi)^{d-1}}\,(2\pi)^{d}\delta^{(d)}(p+q-p_{2})\,\delta(p_{2}^{2})\theta(p_{2}^{0})=\frac{2\pi}{Q^{2}}\delta(1-x)\theta(Q).$
(4.7)

With the normalization given by the projector in Eq. (4.4), one easily finds

$F^{(0)}_{2}(x,Q^{2})=\delta(1-x),$ (4.8)

where the superscript (0) indicates that this is the leading order (Born)
result.

At higher orders the emission of soft particles can be expressed in terms of
the generalised soft emission operators introduced in Section 3.2. To this
end, let us define the vector of amplitudes with stripped-off
spinors/polarisation vectors:

$\big{[}\mathcal{A}^{(0)}_{\alpha}\big{]}^{\bar{B}\bar{A}}_{BA}=\left([\mathcal{A}^{(0)}_{q,\alpha}]^{ji}_{\hat{b}\hat{a}},0,0\right),\qquad\qquad\big{[}\mathcal{A}^{{\dagger}(0)}_{\beta}\big{]}^{\bar{A}^{\prime}\bar{B}^{\prime}}_{A^{\prime}B^{\prime}}=\left([\mathcal{A}^{{\dagger}(0)}_{q,\beta}]^{i^{\prime}j^{\prime}}_{\hat{a}^{\prime}\hat{b}^{\prime}},0,0\right).$
(4.9)

The stripped-off amplitudes can be obtained by direct comparison with (4.5),
and we obtain

$\big{[}\mathcal{A}_{q}^{(0)\alpha}\big{]}^{ji}_{\hat{b}\hat{a}}=ie\gamma_{\hat{b}\hat{a}}^{\alpha}\delta^{ji},\qquad\qquad\big{[}\mathcal{A}_{q}^{{\dagger}(0)\beta}\big{]}^{i^{\prime}j^{\prime}}_{\hat{a}^{\prime}\hat{b}^{\prime}}=-ie\gamma_{\hat{a}^{\prime}\hat{b}^{\prime}}^{\beta}\delta^{i^{\prime}j^{\prime}}.$
(4.10)

In this notation, the tree level matrix element squared reads

$\displaystyle\overline{|{\cal A}^{(0)}|^{2}_{\alpha\beta}}$ $\displaystyle=$
$\displaystyle\frac{1}{2N_{c}}\bar{\xi}^{\bar{A}^{\prime}}_{A^{\prime}}(p)\big{[}\mathcal{A}^{{\dagger}(0)\beta}\big{]}^{\bar{A}^{\prime}\bar{B}^{\prime}}_{A^{\prime}B^{\prime}}\xi^{\bar{B}^{\prime}}_{B^{\prime}}(p_{2})\,\bar{\xi}^{\bar{B}}_{B}(p_{2})\big{[}\mathcal{A}^{(0)\alpha}\big{]}^{\bar{B}\bar{A}}_{BA}\xi^{\bar{A}}_{A}(p)\,,$

where the vector of spinors/polarisation vectors in the first line has been
defined in Eqs. (3.9), (3.16), and we use the summation convention for
repeated species indices. Inserting the expressions from (4.10) in the second
line above one readily reproduces Eq. (4.6).

The hard-scattering amplitude may now be dressed with emissions of soft
particles through adding generalised soft emission operators:

$\displaystyle\overline{|{\cal A}|^{2}_{\alpha\beta}}$ $\displaystyle=$
$\displaystyle\sum_{m}\frac{1}{{\cal N}_{s}{\cal N}_{c}}\left\langle
0\left|\Big{(}\bar{\xi}^{\bar{A}_{1}^{\prime}}_{A_{1}^{\prime}}(p)\,\big{[}{\cal
F}_{1,\rm
out}^{{\dagger}}\big{]}^{\bar{A}^{\prime}_{1}\bar{A}^{\prime}_{2}}_{A^{\prime}_{1}A^{\prime}_{2}}\,\big{[}\mathcal{A}^{{\dagger}(0)\beta}\big{]}^{\bar{A}_{2}^{\prime}\bar{B}_{2}^{\prime}}_{A_{2}^{\prime}B_{2}^{\prime}}\,\big{[}{\cal
F}_{2,\rm
in}\big{]}^{\bar{B}^{\prime}_{2}\bar{B}^{\prime}_{1}}_{B^{\prime}_{2}B^{\prime}_{1}}\,\xi^{\bar{B}^{\prime}_{1}}_{B^{\prime}_{1}}(p_{2})\Big{)}\right|m\right\rangle$
(4.11) $\displaystyle\hskip 42.67912pt\times\,\left\langle
m\left|\Big{(}\bar{\xi}^{\bar{B}_{1}}_{B_{1}}(p_{2})\,\big{[}{\cal F}_{2,\rm
out}^{{\dagger}}\big{]}^{\bar{B}_{1}\bar{B}_{2}}_{B_{1}B_{2}}\,\big{[}\mathcal{A}^{(0)\alpha}\big{]}^{\bar{B}_{2}\bar{A}_{2}}_{B_{2}A_{2}}\,\big{[}{\cal
F}_{1,\rm
in}\big{]}^{\bar{A}_{2}\bar{A}_{1}}_{A_{2}A_{1}}\,\xi^{\bar{A}_{1}}_{A_{1}}(p)\Big{)}\right|0\right\rangle,$

where the factors ${\cal N}_{s}$, ${\cal N}_{c}$ represent the spin and color
average over the initial-state particle. At tree level the initial state
particle is a quark, thus ${\cal N}_{s_{q}}=2$, ${\cal N}_{c_{q}}=N_{c}$, as
in Eq. (4). At higher orders it is possible to have gluons in the initial
state, leading to ${\cal N}_{s_{g}}=(d-2)$, ${\cal N}_{c_{g}}=N_{c}^{2}-1$.
Introducing a soft function with $m$ real gluon emissions:

$\big{(}{\cal
S}_{m}\big{)}^{\bar{A}^{\prime}_{1}\bar{A}^{\prime}_{2}\,\bar{B}^{\prime}_{2}\bar{B}^{\prime}_{1}\,\bar{B}_{1}\bar{B}_{2}\,\bar{A}_{2}\bar{A}_{1}}_{A^{\prime}_{1}A^{\prime}_{2}\,B^{\prime}_{2}B^{\prime}_{1}\,B_{1}B_{2}\,A_{2}A_{1}}=\left\langle
0\left|\big{[}{\cal F}_{1,\rm
out}^{{\dagger}}\big{]}^{\bar{A}^{\prime}_{1}\bar{A}^{\prime}_{2}}_{A^{\prime}_{1}A^{\prime}_{2}}\,\big{[}{\cal
F}_{2,\rm
in}\big{]}^{\bar{B}^{\prime}_{2}\bar{B}^{\prime}_{1}}_{B^{\prime}_{2}B^{\prime}_{1}}\right|m\right\rangle\left\langle
m\left|\big{[}{\cal F}_{2,\rm
out}^{{\dagger}}\big{]}^{\bar{B}_{1}\bar{B}_{2}}_{B_{1}B_{2}}\,\big{[}{\cal
F}_{1,\rm
in}\big{]}^{\bar{A}_{2}\bar{A}_{1}}_{A_{2}A_{1}}\right|0\right\rangle,$ (4.12)

we can write Eq. (4.11) as

$\displaystyle\overline{|{\cal A}|^{2}_{\alpha\beta}}$ $\displaystyle=$
$\displaystyle\frac{1}{{\cal N}_{s}{\cal
N}_{c}}\sum_{m}\,\Big{(}\bar{\xi}^{\bar{A}_{1}^{\prime}}_{A_{1}^{\prime}}(p)\,\big{[}\mathcal{A}^{{\dagger}(0)\beta}\big{]}^{\bar{A}_{2}^{\prime}\bar{B}_{2}^{\prime}}_{A_{2}^{\prime}B_{2}^{\prime}}\,\xi^{\bar{B}^{\prime}_{1}}_{B^{\prime}_{1}}(p_{2})\Big{)}$
(4.13) $\displaystyle\hskip 56.9055pt\times\,\big{(}{\cal
S}_{m}\big{)}^{\bar{A}^{\prime}_{1}\bar{A}^{\prime}_{2}\,\bar{B}^{\prime}_{2}\bar{B}^{\prime}_{1}\,\bar{B}_{1}\bar{B}_{2}\,\bar{A}_{2}\bar{A}_{1}}_{A^{\prime}_{1}A^{\prime}_{2}\,B^{\prime}_{2}B^{\prime}_{1}\,B_{1}B_{2}\,A_{2}A_{1}}\,\Big{(}\bar{\xi}^{\bar{B}_{2}}_{B_{2}}(p_{2})\,\big{[}\mathcal{A}^{(0)\alpha}\big{]}^{\bar{B_{2}}\bar{A}_{2}}_{B_{2}A_{2}}\,\xi^{\bar{A}_{1}}_{A_{1}}(p)\Big{)}.$

Upon inserting this into Eq. (4.3) and performing the relevant phase-space
integrals, one obtains the LL result for the structure function at any power.
Let us stress that the above amplitude only should reproduces the maximally
soft momentum configuration, i.e. the LL contribution. Corrections beyond LL
could in principle be taken into account by modifying the diagonal terms
${\cal T}_{ii}^{\rm in}(p)$ of the transition matrix in Eq. (3.13), with
$i=q$, $\bar{q}$, $g$, in such a way to include NLP corrections to these
transitions, but are not considered here.999They have been obtained in [125,
151], see in particular Eq. 5.7 of [151], which provide the NLP contribution
to the diagonal terms of the generalised soft emission operator ${\cal F}$. In
Appendix B we show explicitly how to calculate $F_{2}(x,Q^{2})$ at NLO, for
which one needs to consider the diagrams in Figs. 11 and 12.

Figure 11: Diagrams corresponding to the LP NLO soft function involving the
exchange of a virtual (diagram a) or real (diagrams b, c, d) soft gluon. The
corresponding soft function is given in Eq. (B.3), Eq. (B.4) and Eq. (B.3).
The complex conjugate of diagrams (a) and (b) are not shown in this figure,
but are taken into account in Eqs. (B.3) and (B.4). Figure 12: Diagrams
corresponding to the NLP NLO soft function arising from the exchange of a soft
(anti-)quark. The corresponding soft functions ${\cal S}_{\bar{q}}^{(1)}$ and
${\cal S}_{q}^{(1)}$ are given in Eq. (B.13) and (B.14) respectively. Notice
that the label $\bar{q}$, $q$ refers to the particle that crosses the cut, not
to the particle in the hard scattering amplitude ${\cal A}_{i}$, where the
label is actually opposite compared to the label in the soft function. There
are additional diagrams where the soft (anti-)quark is emitted from the line
with momentum $p_{2}$. These however contribute beyond LL, as explained in the
main text, and therefore are not shown here.

We now turn our attention to the exponentiation of the real emission diagrams
by means of the replica trick. First of all, let us notice that the structure
function $F_{2}$ is gauge invariant. We can exploit this by choosing a gauge
where the calculation of higher-order contributions simplifies. To this end it
proves useful [152, 153, 154] to consider physical gluon polarisation vectors
defined via the requirements

$k\cdot\epsilon(k)=c\cdot\epsilon(k)=0\,,$ (4.14)

where $c$ is a null vector ($c^{2}=0$). This means that the sum over physical
gluon polarisation states take the form

${\cal
P}_{\mu\nu}(k)=-\eta_{\mu\nu}+\frac{k_{\mu}c_{\nu}+k_{\nu}c_{\mu}}{c\cdot
k}\,.$ (4.15)

In what follows we consider

$c=p_{2},$ (4.16)

a choice which is particularly convenient, as it eliminates the need to
consider ${\cal F}_{2}$ in the case of soft gluon emissions. For soft quark
emissions, at NLP, ${\cal F}_{2}$ will contribute to at most NLL, which is
also beyond what we are considering here. That is, we only need to take into
account emissions from the generalised Wilson line ${\cal F}_{1,\rm in}$,
${\cal F}_{1,\rm out}^{{\dagger}}$. Indeed, it is easy to check that by
setting $c=p_{2}$, only diagram (c) in Fig. 11 contributes to the real
emission at LP, while diagrams (b) and (d) are zero.

Figure 13: (a) Single soft topology corresponding to each of the partonic
assignments in Figs. 11 (c) and 12 (a); (b) and (c) soft topologies generated
at NNLO in the squared amplitude, with replica indices indicated.

With this, we note that the diagrams of Fig. 11(c) and Fig. 12(a), (b) are all
given by the single topology in Fig. 13(a). At higher orders, using the gauge
of Eq. (4.14), the LL of the real emission is given in terms of
(crossed-)ladder diagrams, where emission of any partonic species is expressed
by the transition matrix ${\cal T}_{IJ}$. In the case of DIS, at NNLO in the
gauge defined by Eqs. (4.15) (4.16) one needs to take into account the two
real topologies (b) and (c) in Fig. 13, namely an uncrossed and a crossed
ladder, with replica index $i$ and $j$. One immediately sees that diagrams (b)
and (c) of Fig. 13 are the analog of the diagrams in Fig. 9. Therefore,
following the discussion in Section 3.3, the contributions of diagrams (b) and
(c) to the logarithm of the generalised soft function in Eq. (3.41) read (Cf.
Eq. (3.45))

$\displaystyle\widetilde{E}(b)=0,\qquad\qquad\widetilde{E}(c)=\frac{{\cal
N}(c)C(c)-{\cal N}(b)C(b)}{{\cal D}(c)}\,.$ (4.17)

This result is consistent with exponentiation of the generalised soft-emission
operator in the generalised soft function. To see this, let us explicitly
construct the $\mathcal{O}(\alpha_{s}^{2})$ expansion of the soft function
using the reduced diagrams of Eq. (4.17). With only one emission present,
there is only one choice of replica index, and thus all graphs with one soft
emission are ${\cal O}(N)$. They thus contribute to the logarithm of the soft
function unmodified. When two partons are emitted, the only contribution that
needs to be added according to Eq. (4.17) is $\widetilde{E}(c)$, the rest is
already generated by the one-emission diagram. Therefore the logarithm of the
soft function becomes

$\ln{\cal S}=E(a)+\tilde{E}(c)\,,$ (4.18)

where the first term corresponds to diagram (a) in Fig. 13. Focusing on the
NNLO contribution, obtained after exponentiating the above result and then
expanding $\cal S$ up to $\mathcal{O}(\alpha_{s}^{2})$, we obtain

$\exp\left[\ln{\cal
S}\right]\Big{|}_{\mathcal{O}(\alpha_{s}^{2})}=\frac{1}{2}\Big{[}E(a)\Big{]}^{2}+\frac{{\cal
N}(c)C(c)-{\cal N}(b)C(b)}{{\cal D}(c)}\,.$ (4.19)

In the first term, squaring the one-emission contribution will generate
kinematic numerators, which are products of two distinct emissions on the
left-and right-hand side of the cut. Selecting the correct partonic assignment
will yield the kinematic numerator and colour factor of diagram (b). However,
the denominator arising from squaring the one-emission graphs will have
uncoupled momenta

$\displaystyle\Big{[}D(a)\Big{]}^{2}=(p\cdot k_{1})^{2}(p\cdot k_{2})^{2}\,,$
(4.20)

rather than the actual denominator of diagram (b), which is

$\displaystyle{\cal D}(b)=(p\cdot k_{1})^{2}[p\cdot(k_{1}+k_{2})]^{2}\,.$
(4.21)

The denominator for diagram (c) is

$\displaystyle{\cal D}(c)=(p\cdot k_{1})(p\cdot
k_{2})[p\cdot(k_{1}+k_{2})]^{2}\,.$ (4.22)

Using the eikonal identity

$\frac{1}{(p\cdot k_{1})[p\cdot(k_{1}+k_{2})]}+\frac{1}{(p\cdot
k_{2})[p\cdot(k_{1}+k_{2})]}=\frac{1}{(p\cdot k_{1})}\frac{1}{(p\cdot
k_{2})}\,,$ (4.23)

we can write

$\frac{1}{(p\cdot k_{1})^{2}(p\cdot k_{2})^{2}}=2\left[\frac{1}{{\cal
D}(b)}+\frac{1}{{\cal D}(c)}\right],$ (4.24)

where we have used the freedom to relabel $k_{1}\leftrightarrow k_{2}$ to
identify $2{\cal D}(b)$, which is allowed given that all emission momenta are
integrated over. We thus find

$\frac{1}{2}\Big{[}E(a)\Big{]}^{2}={\cal N}(b)C(b)\left[\frac{1}{{\cal
D}(b)}+\frac{1}{{\cal D}(c)}\right].$ (4.25)

Combining this with the second term in Eq. (4.19) gives

$\frac{1}{2}\Big{[}E(a)\Big{]}^{2}+\tilde{E}(c)=\frac{{\cal N}(b)C(b)}{{\cal
D}(b)}+\frac{{\cal N}(c)C(c)}{{\cal D}(c)}\,.$ (4.26)

That is, exponentiating the logarithm of the generalised soft function and
extracting a given partonic assignment does indeed reproduce the conventional
diagrams in the amplitude, with their correct kinematic numerators and colour
factors. Similar arguments can be made at higher orders.

We now use the replica trick to obtain the LL contribution of the emission of
a soft parton, in which case we need only consider the first term of $\ln\cal
S$, namely the one consisting of a sum over all possible one real-emission
contributions. In the gauge defined by Eq. (4.16), we may therefore simply
exponentiate the NLO soft function $S^{(1)}$. The corresponding matrix element
squared becomes

$\displaystyle\overline{|{\cal A}|^{2}_{\alpha\beta}}\big{|}_{\rm real}$
$\displaystyle=$ $\displaystyle\frac{1}{{\cal N}_{s}{\cal
N}_{c}}\,\Big{(}\bar{\xi}^{\bar{A}_{1}^{\prime}}_{A_{1}^{\prime}}(p)\,\big{[}\mathcal{A}^{{\dagger}(0)\beta}\big{]}^{\bar{A}_{2}^{\prime}\bar{B}_{2}^{\prime}}_{A_{2}^{\prime}B_{2}^{\prime}}\,\xi^{\bar{B}^{\prime}_{1}}_{B^{\prime}_{1}}(p_{2})\Big{)}$
(4.27) $\displaystyle\hskip 14.22636pt\times\,\exp\Big{[}{\cal
S}^{(1)}\Big{]}^{\bar{A}^{\prime}_{1}\bar{A}^{\prime}_{2}\,\bar{B}^{\prime}_{2}\bar{B}^{\prime}_{1}\,\bar{B}_{1}\bar{B}_{2}\,\bar{A}_{2}\bar{A}_{1}}_{A^{\prime}_{1}A^{\prime}_{2}\,B^{\prime}_{2}B^{\prime}_{1}\,B_{1}B_{2}\,A_{2}A_{1}}\,\Big{(}\bar{\xi}^{\bar{B}_{2}}_{B_{2}}(p_{2})\,\big{[}\mathcal{A}^{(0)\alpha}\big{]}^{\bar{B_{2}}\bar{A}_{2}}_{B_{2}A_{2}}\,\xi^{\bar{A}_{1}}_{A_{1}}(p)\Big{)},$

Using this expression, we make contact with the known results at NLP for the
quark-gluon channel in photon-induced DIS [123]. To this end, let us recall
that expansion of the exponential in Eq. (4.27) generates contributions at all
subleading power, but we can select the NLP term by associating a book-keeping
parameter $\lambda\ll 1$ to the quark-gluon transitions in the matrix Eq.
(3.13). The parameter $\lambda$ is chosen such that the soft momentum scales
as $k\sim\lambda^{2}$, with $\lambda^{2}\sim(1-x)$ for $x\to 1$. Then it is
easy to see that the LP and NLP soft functions scale respectively as
$\lambda^{-4n}$ and $\lambda^{-4n+2}$ at order $\alpha_{s}^{n}$. To select the
gluon-initiated DIS channel we need at least the exchange of a single soft
fermion, which is suppressed by a power of $\lambda^{2}$ compared to the
exchanged a soft gluon. Therefore, at ${\cal O}(\alpha_{s}^{n})$ the NLP
contributions is obtained by allowing only one insertion of the soft function
${\cal S}_{\bar{g}q}$, $m$ insertions of ${\cal S}_{gg}$, and $n-1-m$
insertions of ${\cal S}_{qq}$. Given its length, the explicit expression is
given in Eq. (B.17). Substituting the results for the partonic emission
factors, contracting these with the LO amplitudes, and applying the eikonal
identity as discussed e.g. in Sec 2.2.1 of [123] one finds for the squared NLP
amplitude

$\displaystyle\overline{|{\cal A}|^{2}_{\alpha\beta}}\big{|}_{\rm
qg,\,real,\,LL}^{(n)}$
$\displaystyle=\frac{g_{s}^{2n}\,e^{2}\,n_{f}\,T_{R}}{n!(d-2)}\left[\prod_{i=1}^{n-1}\frac{p^{\mu_{i}}p^{\nu_{i}}}{(p\cdot
k_{i})^{2}}{\cal
P}_{\mu_{i}\nu_{i}}(k_{i})\right]\sum_{m=0}^{n-1}C_{A}^{m}\,C_{F}^{n-1-m}$
$\displaystyle\quad\times\,\bigg{\\{}{\rm
Tr}\big{[}\not{k}_{q}\gamma^{\alpha_{1}^{\prime}}\not{p}\gamma_{\beta}\not{p}_{2}\gamma_{\alpha}\not{p}\gamma^{\alpha_{1}}\big{]}+{\rm
Tr}\big{[}\not{k}_{q}\gamma^{\alpha_{1}}\not{p}\gamma_{\alpha}\not{p}_{2}\gamma_{\beta}\not{p}\gamma^{\alpha^{\prime}_{1}}\big{]}\bigg{\\}}\frac{{\cal
P}_{\alpha_{1}\alpha_{1}^{\prime}}(p)}{(2p\cdot k_{q})^{2}},$ (4.28)

This indeed agrees with the known form of the squared amplitude with one soft
quark emission and $n-1$ soft gluon emissions, as conjectured in Ref. [142]
and proven in Ref. [123].

As discussed above, in Eqs. (B.17), (4) we have only considered those graphs
which are non zero in the gauge defined by Eqs. (4.15) and (4.16). One could
of course choose a different gauge, in which case the remaining diagrams would
have to be considered. These will exponentiate by the same arguments as above,
leading the to same final (gauge-invariant) result.

Whilst the arguments of this section may take some getting used to, they
greatly streamline the derivation of the all-order structure of the leading
real emission corrections at arbitrary order in perturbation theory. That is,
exponentiating the one-emission contribution to the generalised soft function
leads immediately to the fact that at NLP, the soft quark emission process is
dressed by $n-1$ soft gluons at ${\cal O}(\alpha_{s}^{n})$, where the emission
momenta are uncoupled, and where there is a combinatorial factor of $1/n!$.
The correct colour factor is also obtained, namely that associated with a sum
over all possible uncrossed ladder graphs, in which gluons are emitted either
before or after the soft quark emission.

To summarise, in this section we have used the structure of real emission
corrections in DIS to illustrate the use of the replica trick in determining
the leading singular behaviour of scattering amplitudes in the soft limit. As
well as recovering previously proven results, however, we may also use the
replica trick to prove a recent conjecture regarding virtual corrections in
the same process. This is the subject of the following section.

##  5 Endpoint contributions in deep inelastic scattering

Ref. [117] considered the so-called off-diagonal splitting functions in deep-
inelastic scattering, arising from kinematically subleading partonic channels
that open up for the first time at NLO. The authors then derived the all-order
structure of leading logarithmic terms, agreeing with the previous conjectures
of Ref. [142]. However, in doing so, it was assumed that the leading IR
divergent virtual corrections to each subleading partonic channel
exponentiate. A related conjecture was made in Ref. [113], namely that leading
virtual corrections to a certain form factor involving soft quark emission
also exponentiate, and indeed this is the form factor that would replace the
usual quark form form factor as applied to DIS, in generating subleading
partonic emissions at NLO.

The aim of this section is to show how the above results naturally arise from
using the replica trick arguments of this paper. To this end, we must again
consider the generating functional for soft parton diagrams at the level of
the squared amplitude, as constructed in Eq. (3.53). As explained after Eq.
(3.54), this generating functional implies that the ${\cal O}(N)$ part of an
arbitrary diagram enters the logarithm of the squared soft parton amplitude.
However, to confirm the above conjecture, we must consider only those diagrams
that have a single soft quark crossing the final state cut. This is
straightforward using the generating functional of Eq. (3.54), but
unfortunately does not lead directly to a proof (or otherwise) of the desired
conjecture. To see why, it is sufficient to note that diagrams containing a
single soft real quark will arise at any order in perturbation theory, such
that some of them will indeed exponentiate. An example is shown in Fig. 14,
and we can clearly add more soft gluons to this as desired. Thus, a naïve
application of the replica trick results in a complicated form of the soft
function for the squared amplitude, in which single soft quark corrections
enter the logarithm at arbitrary orders. What the above conjecture then states
is that it must somehow be possible to reorder the diagrammatic expansion, so
that the soft quark emission factorises from the remaining soft gluon
emissions, leading to a simple exponential form for the latter. Put more
simply, the conjecture states that the additional soft gluon emissions
dressing a single soft quark emission exponentiate. What is desirable is to
isolate the single quark emission from the outset, such that the replica trick
can be applied to the remaining soft gluon emissions alone.

Figure 14: Example higher-order diagram with a single soft quark crossing the
final state cut.

It is indeed possible to achieve this by appealing again to Eq. (3.53). The
exponential operator in the middle of the diagram contracts pairs of
(conjugate) fields on either side of the final state cut. If we are to have a
single soft quark as the only real emission, then the exponential may be
expanded to first-order only, retaining only the term involving the
(anti-)quark fields. We may also simplify the generating functional ${\cal S}$
that appears on the left-hand side. This contains a generalised soft emission
operator associated with each hard particle. However, only one soft quark
emission needs to be included, from the incoming parton leg. On the latter,
the operator of Eq. (3.24) takes the form

${\cal P}\exp\left[ig_{s}\int{\rm d}t\left({\bf Q}_{\Psi}{\bf
T}_{\Psi}\Psi(tp_{1})+{\bf Q}_{\bar{\Psi}}{\bf
T}_{\bar{\Psi}}\bar{\Psi}(tp_{1})+{\bf
T}^{a}p^{\mu}\sum_{i=1}^{N}A_{\mu}^{(i),a}(tp_{1})\right)\right],$ (5.1)

where ${\bf Q}_{\Psi}$ (${\bf Q}_{\bar{\Psi}}$) and ${\bf T}_{\Psi}$ (${\bf
T}_{\bar{\Psi}}$) are the kinematic and colour factors associated with soft
(anti-)quark emission respectively. To NLP order, inclusion of a single soft
quark emission amounts to expanding Eq. (5.1) to first order in ${\bf
Q}_{\Psi}$, such that Eq. (5.1) becomes

$\int_{-\infty}^{0}{\rm d}t_{q}\,\Phi_{1}(-\infty,t_{q})\left[{\bf
Q}_{\Psi}{\bf T}_{\Psi}\Psi(t_{q}p)\right]\Phi_{1}(t_{q},0),$ (5.2)

where $\Phi_{1}(a,b)$ is the conventional Wilson line operator on parton leg
1, between distance parameters $a$ and $b$. We have also introduced $t_{q}$ as
the distance parameter at which the soft quark is emitted. All remaining soft
emission operators in the squared amplitude can then be replaced with
conventional Wilson lines, such that the generating functional for diagrams
containing a single real soft quark emission becomes

$\displaystyle Z_{sq}\Big{|}_{q}$ $\displaystyle={\cal
S}_{q}\left[-\int\frac{{\rm
d}^{d}k_{q}}{(2\pi)^{d}}\frac{1}{2}\frac{\overleftarrow{\delta}}{\delta\Psi(k_{q})}\Delta^{(q)}_{+}(k_{q})\frac{\overrightarrow{\delta}}{\delta\bar{\Psi}(-k_{q})}\right]{\cal
S}^{\dagger}_{q},$ (5.3)

where $\Delta^{(q)}_{+}$ is the appropriate cut propagator for a quark of
momentum $k_{q}$, and we have introduced the generating functional for
diagrams containing a single soft real quark:

$\displaystyle{\cal S}_{q}$ $\displaystyle=\int_{-\infty}^{0}{\rm
d}t_{q}\int{\cal D}\Psi\int{\cal D}\bar{\Psi}\,{\bf Q}_{\Psi}{\bf
T}_{\Psi}\Psi(t_{q}p_{1}){\cal S}_{A};$ $\displaystyle{\cal S}_{A}$
$\displaystyle=\left[\int{\cal
D}A_{\mu}\Phi_{1}(-\infty,t_{q})\Phi_{1}(t_{q},0)\Phi_{2}(0,\infty)e^{iS[A_{\mu},\Psi,\bar{\Psi}]}\right].$
(5.4)

Here we have isolated the path integral over the soft gluon field, together
with the Wilson lines that source the soft gluons.101010Although it appears
that we may combine the two Wilson lines associated with the incoming parton
leg with momentum $p_{1}$, it must be remembered that they contain colour
matrices in different representations, due to acting on either side of the
soft quark emission. This path integral acts to dress the amplitude for the
emission of a single soft quark with additional gluon emissions, thus
generating all possible virtual corrections to the single soft quark emission
process. The soft gluons may interact either with the hard parton legs
(through the Wilson lines), or with the emitted soft quark (via the
interaction Lagrangian in the soft parton action). That these additional
contributions exponentiate can now be surmised by replicating the soft gluon
field $A_{\mu}$, but not the soft (anti-)quark fields. Then, in the replicated
theory with $N$ replicas $A_{\mu}^{(i)}$, Eq. (5.4) becomes

$\displaystyle{\cal S}_{q,N}$ $\displaystyle=\int_{-\infty}^{0}{\rm
d}t_{q}\int{\cal D}\Psi\int{\cal D}\bar{\Psi}\,{\bf Q}_{\Psi}{\bf
T}_{\Psi}\Psi(t_{q}p_{1}){\cal S}_{A}^{N}.$ (5.5)

Following similar arguments to our previous cases, the ${\cal O}(N)$ piece of
the soft gluon dressings exponentiates, precisely confirming the desired
conjecture that virtual corrections to the soft quark emission process
exponentiate.

For completeness, we mention a subtlety in the above argument. At higher
orders, there will be diagrams in which fermion bubbles occur off the hard
parton lines. These are generated by pulling down multiple interaction
vertices from the action $S[A_{\mu},\Psi,\bar{\Psi}]$, and a potential problem
occurs given that we have only replicated the soft gluons: gluons of different
replica number may interact with each other through a fermion bubble, thus
contradicting the assumptions that go into the replica trick. That is, one
would no longer be able to conclude that the factor ${\cal S}_{A}$ gets
replaced, in the replicated theory, by ${\cal S}_{A}^{N}$. One way around this
would be to introduce an additional set of soft (anti-)quark fields, that
enter the action, but cannot be sourced by the hard parton legs. In any case,
this issue is irrelevant for proving the conjecture of Refs. [117, 113],
namely that leading virtual corrections to the quark channel in DIS
exponentiate: the relevant diagrams contain only soft gluons in addition to
the single radiated soft (anti-)quark.

In this section, we have shown that the additional soft gluon corrections to
the squared amplitude with a single soft quark emission exponentiate. Given
the rather combinatorial nature of the above analysis, however, it is useful
to make things more concrete. In particular, we may explicitly confirm that
our generalised soft-emission lines do indeed match the results of a complete
calculation of virtual corrections, where the soft expansion is performed
after the integral over the virtual gluon momentum has been carried out.

###  5.1 Calculation of endpoint contributions

Figure 15: Diagrams contributing to the one-real, one-virtual contributions to
DIS, in which a soft quark is emitted in the final state.

The various diagrams contributing to the leading virtual corrections to the
quark channel of DIS are shown in Fig. 15. Dressing the appropriate NLO
squared amplitude with additional factors from the generalised soft-emission
line, the kinematic part of diagram (a) in Fig. 15 is given (in the Feynman
gauge) by

$\displaystyle{\cal K}(a)$
$\displaystyle=T_{2\alpha\beta}\,P_{\mu\nu}(p)\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{1}{k^{2}}\left(\frac{p^{\sigma}}{p\cdot(k-k_{q})}\right)\left(\frac{p_{2\sigma}}{p_{2}\cdot
k}\right){\rm
Tr}\left[\not\\!{k}_{q}\left(\frac{\gamma^{\mu}\not\\!{p}}{2p\cdot
k_{q}}\right)\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}\left(\frac{\not\\!{p}\gamma^{\nu}}{2p\cdot
k_{q}}\right)\right]$ $\displaystyle=4p\cdot
p_{2}\,T_{1}(p,p_{3};k_{q})T_{2\alpha\beta}\,P_{\mu\nu}(p){\rm
Tr}\left[\not\\!{k}_{q}\left(\frac{\gamma^{\mu}\not\\!{p}}{2p\cdot
k_{q}}\right)\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}\left(\frac{\not\\!{p}\gamma^{\nu}}{2p\cdot
k_{q}}\right)\right],$ (5.6)

where we have also included the projector for the structure function
$F_{2}(x,Q^{2})$, as given in Eq. (4.4). Furthermore, in the second line of
Eq. (5.6) we have introduced the soft triangle integral

$T_{1}(p_{1},p_{2};k_{2})=\int\frac{{\rm
d}^{d}k_{1}}{(2\pi)^{d}}\frac{1}{k_{1}^{2}\,2p\cdot(k_{1}-k_{2})\,2p_{2}\cdot
k_{1}}.$ (5.7)

We calculate this integral in Appendix C, and the result is given in Eq.
(C.14). To express the so-called endpoint contribution at which all emitted
partons are soft, we can define the Mandelstam invariants

$s=(k_{q}+p_{2})^{2},\quad t=(p-k_{q})^{2},\quad u=(p-p_{2})^{2}$ (5.8)

satisfying

$s+t+u=q^{2}\equiv-Q^{2}.$ (5.9)

We may then parameterise these as

$s=\frac{Q^{2}(1-x)}{x},\quad t=-\frac{z}{x}{Q^{2}},\quad
u=-\frac{(1-z)Q^{2}}{x}.$ (5.10)

The first of these follows from the definition of the Björken variable in Eq.
(4.2). The second corresponds to introducing an additional parameter $z$, such
that Eq. (5.9) is respected, and such that the threshold limit in which the
final-state quark (rather than anti-quark) is soft corresponds to

$x\rightarrow 1,\quad z\rightarrow 0.$ (5.11)

Substituting the results of Eqs. (5.8, 5.10, C.14) into Eq. (5.6) and
expanding about $d=4-2\epsilon$ dimensions in the limit of Eq. (5.11), one
finds

${\cal
K}(a)=-\frac{i}{4\pi^{3}}\frac{1}{z}\frac{1}{\epsilon^{2}}\left[\left(\frac{\mu^{2}}{Q^{2}}\right)^{\epsilon}-\left(\frac{\mu^{2}}{zQ^{2}}\right)^{\epsilon}\right]+{\cal
O}(\epsilon^{-1})+{\cal O}[(1-x)]+{\cal O}(z^{0}).$ (5.12)

We see that this diagram has a leading $\epsilon$ pole, so will indeed
contribute to the leading logarithm in $(1-x)$ after integrating over the
final state phase space. Furthermore, we may note the non-analytic dependence
on $z$ in the second term, which is important to keep track of, given that
this will affect the leading logarithmic threshold behaviour after integration
over the final state phase space.

We may apply similar methods to the other diagrams in Fig. 15. First, we may
make things easier by noting that diagram (f) will have at most a bubble
integral, which cannot produce a leading $\epsilon$ pole: it is ${\cal
O}(\epsilon^{-1})$ rather than ${\cal O}(\epsilon^{-2})$. Furthermore, diagram
(b) also fails to contribute a leading pole. To see this, we may first
evaluate its kinematic part, which turns out to yield

$\displaystyle{\cal K}(b)$ $\displaystyle=\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{p_{\sigma}}{[-p\cdot(k-k_{q})]}\frac{P_{\mu\nu}(p)\,T_{2\alpha\beta}}{k^{2}}$
$\displaystyle\quad\times{\rm
Tr}\left[\not\\!{k}_{q}\left(\frac{\gamma^{\sigma}(\not\\!{k}_{q}-\not\\!{k})}{(k_{q}-k)^{2}}\right)\left(\frac{\gamma^{\mu}\not\\!{p}}{(2p\cdot
k_{q})}\right)\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}\left(\frac{\gamma^{\beta}\not\\!{p}}{p\cdot
k_{q}}\right)\right],$ (5.13)

which involves the virtual integral

$\int\frac{{\rm
d}^{d}k_{1}}{(2\pi)^{d}}\frac{1}{k^{2}\,[2p\cdot(k-k_{q})](k-k_{q})^{2}}.$
(5.14)

However, the Dirac trace (after contraction with polarisation and projection
tensors) evaluates to

$-\frac{64(p\cdot k_{q})(p\cdot p_{2})[p\cdot(k-k_{q})]}{\pi}+{\cal
O}(\epsilon),$ (5.15)

such that the $k$-dependent factor in the numerator cancels the similar factor
in the denominator of Eq. (5.14), leaving a bubble integral which is
necessarily ${\cal O}(\epsilon^{-1})$.

Carrying out a similar analysis for the remaining diagrams yields

$\displaystyle{\cal K}(c)$ $\displaystyle=\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{2p^{\sigma}}{(2p\cdot
k_{q})^{2}}\left\\{T_{2}(p,p_{2};k_{q}){\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\sigma}\not\\!{k_{q}}\gamma^{\mu}\not\\!{p}\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}\not\\!{p}\gamma^{\nu}\right]\right.$
$\displaystyle\left.\hskip 56.9055pt-\,T_{2}^{\delta}(p,p_{2};k_{q}){\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\sigma}\gamma_{\delta}\gamma^{\mu}\not\\!{p}\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}\not\\!{p}\gamma^{\nu}\right]\right\\}P_{\mu\nu}(p);$
(5.16a) $\displaystyle{\cal K}(d)$ $\displaystyle=\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{p_{2\sigma}}{p\cdot k_{q}}{\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\sigma}\gamma_{\delta}\gamma^{\mu}\not\\!{p}\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}\not\\!{p}\gamma^{\nu}\right]\left[B_{2}(p,p_{2};k_{q})k_{q}^{\delta}-B_{2}^{\delta}(p,p_{2};k_{q})\right]P_{\mu\nu}(p);$
(5.16b) $\displaystyle{\cal K}(e)$ $\displaystyle=\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{p\cdot p_{2}}{p\cdot k_{q}}B_{1}(p,p_{2};-k_{q}){\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\mu}\not\\!{p}\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}\not\\!{p}\gamma^{\nu}\right]P_{\mu\nu}(p),$
(5.16c)

where we have introduced various integrals, which are defined and calculated
in Appendix C. Substituting those results into Eq. (5.16) before expanding in
the limit of Eq. (5.11) gives

$\displaystyle{\cal K}(c)$
$\displaystyle=\frac{i}{4\pi^{3}}\frac{1}{z}\frac{1}{\epsilon^{2}}\left(\frac{\mu^{2}}{zQ^{2}}\right)^{\epsilon};$
(5.17a) $\displaystyle{\cal K}(d)$
$\displaystyle=-\frac{i}{4\pi^{3}}\frac{1}{z}\frac{1}{\epsilon^{2}}\left(-\frac{\mu^{2}}{Q^{2}z(1-x)}\right)^{\epsilon};$
(5.17b) $\displaystyle{\cal K}(e)$
$\displaystyle=-\frac{i}{4\pi^{3}}\frac{1}{z}\frac{1}{\epsilon^{2}}\left(\frac{\mu^{2}}{zQ^{2}}\right)^{\epsilon}.$
(5.17c)

To verify our results, we may compare Eqs. (5.12, 5.17) with the result of
carrying out an exact calculation of each diagram in Fig. 15, before expanding
in the threshold limit after the integration over the virtual gluon momentum
has already been carried out. First, we may note results for the kinematic
part of each diagram, using the same conventions as above. That is, we ignore
overall coupling and spin averaging factors, but nevertheless contract with
polarisation and projection tensors. We then find kinematic parts:

$\displaystyle(a):$ $\displaystyle\quad\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{{\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\mu}(\not\\!{k}_{q}-\not\\!{p})\gamma^{\sigma}(\not\\!{k}_{q}-\not\\!{k}-\not\\!{p})\gamma^{\alpha}(-\not\\!{p}_{2}-\not\\!{k})\gamma_{\sigma}\not\\!{p}_{2}\gamma^{\beta}(\not\\!{k}_{q}-\not\\!{p})\gamma^{\nu}\right]}{k^{2}(k_{q}-p)^{2}(k_{q}-k-p)^{2}(k+p_{2})^{2}(k_{q}-p)^{2}};$
(5.18a) $\displaystyle(b):$ $\displaystyle\quad\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{{\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\sigma}(\not\\!{k}_{q}-\not\\!{k})\gamma^{\mu}(\not\\!{k}_{q}-\not\\!{k}-\not\\!{p})\gamma_{\sigma}(\not\\!{k}_{q}-\not\\!{p})\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}(\not\\!{k}_{q}-\not\\!{p})\gamma^{\nu}\right]}{k^{2}(k_{q}-k-p)^{2}(k_{q}-k)^{2}(k_{q}-p)^{4}};$
(5.18b) $\displaystyle(c):$ $\displaystyle\quad\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{{\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\sigma}(\not\\!{k}_{q}-\not\\!{k})\gamma^{\tau}(\not\\!{p}-\not\\!{k}_{q})\gamma^{\alpha}\not\\!{p}_{2}\gamma^{\beta}(\not\\!{p}-\not\\!{k}_{q})\gamma^{\nu}\right]}{k^{2}(k_{q}-k)^{2}(p-k)^{2}(p-k_{q})^{4}}V^{\tau\sigma\mu}(p-k,k,-p]);$
(5.18c) $\displaystyle(d):$ $\displaystyle\quad\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{{\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\sigma}(\not\\!{k}_{q}-\not\\!{k})\gamma^{\mu}(\not\\!{k}_{q}-\not\\!{k}-\not\\!{p})\gamma^{\alpha}\not\\!{p}_{2}\gamma_{\sigma}\not\\!{p}_{2}\gamma^{\beta}(\not\\!{p}-\not\\!{k}_{q})\gamma^{\nu}\right]}{k^{2}(k_{q}-k)^{2}(p+k-k_{q})^{2}(p_{2}+k)^{2}(p-k_{q})^{2}};$
(5.18d) $\displaystyle(e):$ $\displaystyle\quad\int\frac{{\rm
d}^{d}k}{(2\pi)^{d}}\frac{{\rm
Tr}\left[\not\\!{k}_{q}\gamma^{\tau}(\not\\!{k}_{q}+\not\\!{k}-\not\\!{p})\gamma^{\alpha}(-\not\\!{p}_{2}+\not\\!{k})\gamma_{\sigma}\not\\!{p}_{2}\gamma^{\beta}(\not\\!{p}-\not\\!{k}_{q})\gamma^{\nu}\right]}{k^{2}(p-k)^{2}(k_{q}+k-p)^{2}(k-p_{2})^{2}(p-k_{2})^{2}}V_{\tau\sigma\mu}(p-k,k,-p),$
(5.18e)

where

$V_{\alpha\beta\gamma}(p_{1},p_{2},p_{3})=(p_{1\gamma}-p_{2\gamma})\eta_{\alpha\beta}+(p_{2\alpha}-p_{3\alpha})\eta_{\beta\gamma}+(p_{3\beta}-p_{1\beta})\eta_{\alpha\gamma}\,.$
(5.19)

Each individual expression can be reduced to scalar integrals using the well-
known Passarino-Veltman reduction algorithm [155]. To carry out this and also
Dirac algebra, we use the FeynCalc [156, 157, 158] plugin for Mathematica
[159]. Explicit results for scalar integrals (including both triangles and
boxes, both of which contribute leading poles) are taken from Ref. [160]. We
may then substitute the Mandelstam invariants of Eqs. (5.8), parameterised
according to Eq. (5.10). Finally, one may expand to in the limit of Eq.
(5.11), keeping leading terms only. We find that the results of all non-zero
diagrams explicitly agree with Eqs. (5.12, 5.17).

To summarise, we have obtained two properties: (i) the generalised soft-
emission lines introduced in Section 3 correctly capture the leading virtual
corrections to the off-diagonal partonic channel in DIS at NLO; (ii) the
additional gluon corrections to this partonic channel exponentiate, and hence
the leading virtual corrections at this order do as well. This confirms the
conjectures presented in Refs. [113, 117], which we thus obtain as a result of
the replica trick.

##  6 Discussion

In this paper, we have addressed whether certain perturbative corrections in
QCD can be shown to exponentiate, and thus be resummed to all orders in
perturbation theory. It is very well-known that this is true for soft gluon
emissions, and a recent series of papers has developed an elegant method for
proving exponentiation properties in (multi-)parton scattering [127, 128, 129,
130, 131, 132, 133, 134, 135], based on a variant of the replica trick that
originated in statistical physics [126]. Here, we have applied this technique
to the exponentiation of soft (anti-)quark emissions, in addition to soft
gluons. Previous replica trick arguments for gluon emissions have relied upon
the fact that the soft function describing soft gluon emissions is expressed
in terms of vacuum expectation values of Wilson lines, where one may define
the soft function at either amplitude or cross-section level. One may then
write a generating functional for these VEVs, whose path integral description
provides the basis for applying the replica trick. The latter involves
considering a modified theory containing $N$ identical non-interacting copies,
or replicas, of the soft gluon field. One may then show that the ${\cal O}(N)$
part of each diagram in the replicated theory exponentiates, such that upon
setting $N\rightarrow 1$ one establishes exponentiation in the original (non-
replicated) theory. Crucially, this argument can be applied at either
amplitude or cross-section level, and we have here provided more rigorous
arguments than have appeared before, regarding how the replica trick works for
real emissions.

As demonstrated in this paper, applying the replica trick to soft (anti-)quark
emission is also possible, provided that one first identifies a generalised
soft-emission line operator that can be used to describe the emission of any
soft parton. The latter may be described using so-called emission factors
[79], valid only in the leading soft limit. Assembling these into something
that resembles a Wilson line is ultimately nothing more than a book-keeping
exercise. It differs from the formal definition of a Wilson line in that it
does not share the usual gauge transformation properties, and that the
resulting object is matrix-valued in not only colour space, but also parton
species/spin. Nevertheless, it allows us to successfully apply the replica
trick, which has some interesting additional features when compared with its
previous incarnation for soft gluons. As in the previous case, one must
introduce a replica-ordering operator as a means to elucidate which diagrams
exponentiate. In the gluon case, this affects the colour factors of each
graph, such that the colour factor in the logarithm of the soft amplitude is
different to the colour factor in the amplitude itself. This remains true in
the case of general soft parton emission, but with the additional complication
that the replica-ordering operator also modifies the kinematic numerator of
each graph. We verified in a simple example that this is indeed what is
required to reproduce the structure of the soft amplitude, upon expanding the
exponential form of the soft function.

To illustrate our arguments, we rederived the exponentiation properties of
certain real emission corrections in DIS, agreeing with the results of Refs.
[142, 123]. Having verified our approach in a known example, we then turned to
the unproven conjectures of Refs. [113, 117], namely that the leading virtual
corrections to the NLO off-diagonal partonic channel in DIS exponentiate. This
necessitated a slightly modified form of the replica trick, in which some but
not all soft parton fields are replicated. This in turn coincides with the
fact that only a single soft quark emission is needed to generate leading
logarithmic threshold effects at NLP in the threshold expansion. We confirm,
however, that the exponentiation of the relevant virtual contributions indeed
follows straightforwardly from this modified replica argument.

We expect that our results are much more general than the specific examples
presented here. That is, the replica trick in principle implies that soft
parton emissions completely exponentiate, such that certain perturbative
contributions of leading logarithmic type would also exponentiate at NNLP
order and beyond in the threshold expansion. It would then be important to
ascertain whether these types overlap with other sources of leading
logarithms. Furthermore, although we have considered DIS in this paper, our
arguments can be applied to any scattering process, including those of direct
phenomenological interest for the ongoing LHC experiments. We thus foresee a
programme of work in applying our methods to resum NLP contributions in a
variety of contexts. We hope our results also inspire further investigations
of what the replica trick may be useful for.

## Acknowledgments

This work has been supported by the UK Science and Technology Facilities
Council (STFC) Consolidated Grant ST/P000754/1 "String Theory, gauge theory
and duality" (CDW), and ST/T000864/1 (MvB). The work of LV has been partly
supported by Fellini - Fellowship for Innovation at INFN, funded by the
European Union’s Horizon 2020 research programme under the Marie Skłodowska-
Curie Cofund Action, grant agreement no. 754496. LV thanks the Erwin-
Schrödinger International Institute for Mathematics and Physics at the
University of Vienna for partial support during the Programme “Quantum Field
Theory at the Frontiers of the Strong Interactions", July 31 - September 1,
2023, and the Galileo Galilei Institute for Theoretical Physics of INFN,
Florenze, for partial support during the Programme “Theory Challenges in the
Precision Era of the Large Hadron Collider", August 28 - October 13, 2023.

##  Appendix A Kinematic factors for soft emissions

Here we summarise our results for the kinematic factors that feature in the
transition matrix elements defined in Eq. (3.12). For an incoming hard parton
we have

$\displaystyle Q_{\rm
in}^{C,BA}(p)=\begin{cases}p^{\mu}\delta^{\hat{b}\hat{a}}&\text{for
$q^{\hat{a}}(p)\to q^{\hat{b}}(p-k)\,g^{\mu}(k)$ \,\,\,(diagram (a)),}\\\
p^{\mu}\delta^{\hat{a}\hat{b}}&\text{for
$\bar{q}^{\hat{a}}(p)\to\bar{q}^{\hat{b}}(p-k)\,g^{\mu}(k)$ \,\,\,(diagram
(b)),}\\\ p^{\mu}\eta^{\alpha\beta}&\text{for $g^{\alpha}(p)\to
g^{\beta}(p-k)\,g^{\mu}(k)$ \,\,\,(diagram (c)),}\\\
-\frac{1}{2}(\gamma^{\beta})_{\hat{c}\hat{a}}&\text{for $q^{\hat{a}}(p)\to
g^{\beta}(p-k)\,q^{\hat{c}}(k)$ \,\,\,(diagram (d)),}\\\
-\frac{1}{2}(\gamma^{\beta})_{\hat{a}\hat{c}}&\text{for
$\bar{q}^{\hat{a}}(p)\to g^{\beta}(p-k)\,\bar{q}^{\hat{c}}(k)$ \,\,\,(diagram
(e)),}\\\ \frac{1}{2}(\gamma^{\alpha}\not\\!{p})_{\hat{c}\hat{b}}&\text{for
$g^{\alpha}(p)\to\bar{q}^{\hat{b}}(p-k)\,q^{\hat{c}}(k)$ \,\,\,(diagram
(f)),}\\\ \frac{1}{2}(\not\\!{p}\gamma^{\alpha})_{\hat{b}\hat{c}}&\text{for