Next-to-soft radiation from a different angle
Melissa van<EMAIL_ADDRESS>Abhinava
<EMAIL_ADDRESS>Eric<EMAIL_ADDRESS>
Sourav<EMAIL_ADDRESS>Anurag<EMAIL_ADDRESS>and Chris D<EMAIL_ADDRESS>
aRudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory,
Parks Road, University of Oxford, Oxford OX1 3PU, United Kingdom
bDepartment of Physics, Indian Institute of Technology Hyderabad, Kandi,
Sangareddy, Telangana State 502285, India
cInstitute of Physics, University of Amsterdam, Science Park 904,
1098 XH Amsterdam, The Netherlands
dNikhef, Theory Group, Science Park 105, 1098 XG Amsterdam,
The Netherlands
eInstitute for Theoretical Physics, Utrecht University, Leuvenlaan 4,
3584 CE Utrecht, The Netherlands
fTheoretical Physics Division, Physical Research Laboratory,
Navrangpura, Ahmedabad 380009, India
gCentre for Theoretical Physics, Department of Physics and Astronomy,
Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

###### Abstract

Soft and collinear radiation in collider processes can be described in a
universal way, that is independent of the underlying process. Recent years
have seen a number of approaches for probing whether radiation beyond the
leading soft approximation can also be systematically classified. In this
paper, we study a formula that captures the leading next-to-soft QCD radiation
affecting processes with both final- and initial-state partons, by shifting
the momenta in the non-radiative squared amplitude. We first examine W+jet
production, and show that a previously derived formula of this type indeed
holds in the case in which massive colour singlet particles are present in the
final state. Next, we develop a physical understanding of the momentum shifts,
showing precisely how they disrupt the well-known angular ordering property of
leading soft radiation.

##  1 Introduction

The calculation of scattering amplitudes in perturbative quantum field theory
continues to be an area of intense activity, due to its many applications to
current and future collider experiments. Whilst it is often possible to obtain
complete amplitudes at a given order in the coupling constant, we sometimes
wish to consider approximate results, particularly where these can be resummed
to all orders in perturbation theory. A particularly well-studied case is the
emission of soft and / or collinear radiation dressing an underlying
scattering amplitude. This generates infrared singularities, which will cancel
for suitably inclusive observables, such as total hadronic cross-sections.
However, large contributions remain in perturbation theory, typically
involving large logarithms of dimensionless energy ratios. A variety of
methods have been developed for resumming such contributions (see e.g. refs.
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]), all of which rely on the tight
relationship between kinematically enhanced terms and infrared singularities,
plus the fact that soft and collinear factorisation can be described in terms
of universal functions acting on arbitrary amplitudes. The latter property has
a simple quantum mechanical interpretation: soft radiation has zero momentum,
and thus an infinite Compton wavelength by the uncertainty principle. Thus, it
cannot resolve the details of the underlying scattering amplitude that
produced the hard outgoing particles. A similar story applies to collinear
radiation, which instead has a zero transverse momentum relative to a given
outgoing particle.

Heuristic arguments such as these are also useful for understanding the wider
implications of soft radiation. Crucial for this paper will be a particular
property of soft radiation that is emitted from pairs of (colour) charges, or
dipoles, in QED or QCD. Including all possible quantum interference
contributions in the squared amplitude, one finds in QED that radiation is
confined to a cone around each charged particle, whose half-angle coincides
with the angle between the two charged particle momenta. This is known as the
Chudakov effect, and textbook treatments may be found in refs. [14, 15]. A
corresponding effect holds in QCD, where for two colour charges there is no
radiation (at leading soft level) outside the cones surrounding each particle.
For more complicated configurations of partons, clusters of particles radiate
according to their combined colour charge at sufficiently large angles. All of
these phenomena have a common quantum mechanical origin similar to that
already mentioned above: at large angles, the wavelength of the emitted
radiation is such that it can only notice the combined colour charge of a
given subset of partons. If this combined charge happens to be zero (or colour
singlet in the QCD case), then there is no radiation at large angles.

So much for soft radiation, whose properties are already well-known. Until
recently, much less has been known about how to systematically classify the
properties of radiation at subleading order in a systematic expansion in the
total radiated momentum. The frontier of such attempts is at next-to-leading
power (NLP), and the last few years have seen an increasing number of
techniques aimed at clarifying whether any universal statements can be made
about such radiation, including its possible resummation. The range of methods
[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69] – some of them
inspired by the much earlier work of refs. [70, 71, 72] – mirrors that used
for soft radiation, and this body of work is ultimately motivated by the fact
that the numerical impact of such contributions may be needed to increase the
theoretical precision of collider physics observables [73, 74, 75, 76, 77, 78,
79, 44]. As well as studies aiming to develop new resummation formulae, there
is also scope for case studies that look at well-defined consequences of next-
to-soft radiation, in order to build up our collective intuition of how it
behaves. The aim of this paper is to carry out such a case study.

Our starting point is to consider a formula – first derived in ref. [31] and
extended in ref [32] – that states that leading next-to-soft radiative
contributions can be expressed in a particularly compact and elegant form.
That is, the squared amplitude including such radiation can be written in
terms of non-radiative amplitudes, but where distinct pairs of partonic
momenta are shifted in a prescribed way. The shifted squared amplitudes are
then dressed by overall factors which are identical to those that occur in the
leading soft limit. Potential uses of such formulae include increasing the
precision of numerical NLO calculations, and similar comments, independently
and using different methods, have been made in refs. [57, 36, 37]. However,
the obvious similarity of these momentum-shift formula to their leading-soft
counterparts means they are an excellent starting point for examining the
physics of next-to-soft radiation in a particularly transparent way.

In this paper, we will first review the momentum-shift formulae of refs. [31,
32], and introduce them by considering a process that has not previously been
considered before in this approach. That is, we will consider radiative
corrections to $W$ production in association with an additional hard jet777For
previous work on this process in the context of next-to-soft physics, see
refs. [37, 80].. This is more general than either of the processes considered
previously in this approach. Reference [31] looked only at colour-singlet
final states, whereas ref. [32] considered only final states with massless
particles. As we will yet again see, leading next-to-soft radiative
corrections take the form of a series of dipole-like terms 888Very recently,
ref. [81] has provided alternative formulae for capturing next-to-soft
radiation, which also work at loop level..

Next, we consider the effect of the momentum-shift formulae on the emission of
soft radiation from a single final-state dipole. We will briefly review the
well-known calculation of how soft gluon emission is confined to cones
surrounding each hard particle, before correcting this to include the effects
of the momentum-shifts, and hence leading next-to-soft effects. We will show
explicitly that the next-to-soft corrections break the angular ordering
property, in that they lead to emission outside of the usual angular region.
That this property is not preserved beyond leading soft level will perhaps not
surprise anyone. However, the mechanism by which this happens, including the
details of the calculation, are interesting. Furthermore, given that the
origin of the momentum shifts is well-understood as arising from orbital
angular momentum effects [31, 32], we will be able to precisely interpret the
physics of how angular ordering breaks down. We believe that this story offers
novel insights into the physics of next-to-soft radiation that, as well as
being compelling in themselves, may be of broader use.

The structure of our paper is as follows. In section 2, we examine $W$ \+ jet
production at NLO, showing that the inclusion of radiative corrections up to
next-to-soft level reproduces the same momentum-shift formula as was found for
prompt production in ref. [32]. After drawing attention to the dipole-like
nature of this formula, in section 3 we show that next-to-soft corrections
lead to radiation outside the cone regions associated with leading soft
radiation, and intepret the physics of this effect in detail. In section 4, we
discuss our results and conclude.

##  2 A momentum-shift formula for $W$ plus jet production

###  2.1 $W$ plus jet production up to NLO

We start by considering the LO process

$q(p_{1})+\bar{q}(p_{2})\rightarrow W(p_{3})+g(p_{4}),$ (1)

whose Feynman diagrams are shown in figure 1. This is itself a correction to
the Drell-Yan production of a $W$ boson, but we will consider that the final-
state gluon is constrained to be hard (e.g. through a non-zero transverse
momentum requirement), such that no infrared singularities are present. Our
aim is to show how next-to-soft corrections to this process can be written
according to a certain formula, and we will be able to illustrate our point
without having to consider the alternative partonic channel $qg\rightarrow
W+g$, which in any case can be obtained from crossing. Denoting the $W$ boson
mass by $m$, the various momenta satisfy

Figure 1: Leading order diagrams for $W$ plus jet production in the $q\bar{q}$
channel.

$p_{1}^{2}=p_{2}^{2}=p_{4}^{2}=0,\quad p_{3}^{2}=m^{2},$ (2)

and we also define the Mandelstam invariants

$s=(p_{1}+p_{2})^{2},\quad t=(p_{1}-p_{3})^{2},\quad u=(p_{2}-p_{3})^{2}.$ (3)

It is also conventional to define the alternative invariants

$t_{1}=t-m^{2},\quad u_{1}=u-m^{2},$ (4)

which obey

$s+t_{1}+u_{1}+m^{2}=0$ (5)

as a consequence of momentum conservation. With this notation, the squared LO
amplitude, summed (averaged) over final (initial) colours and spins, is given
by

$\overline{|{\cal
M}^{(0)}|}^{2}=g_{s}^{2}g_{w}^{2}\frac{C_{F}}{N_{c}}\frac{(2m^{4}-2m^{2}(t+u)+t^{2}+u^{2})}{2tu},$
(6)

where $g_{s}$ and $g_{w}$ are the strong and electroweak coupling constants
respectively, $C_{F}$ the quadratic Casimir in the fundamental representation,
and $N_{c}$ the number of colours.

Let us now consider the radiation of an additional gluon, for which there are
two types of diagram. First, there is radiation of a quark or antiquark, as
shown in figure 2. These diagrams would also be present in the case of
$W\gamma$ production, which was first calculated at NLO in ref. [82]. Next,
there are diagrams in which the gluon is radiated off the final state hard
gluon, as in figure 3.

Figure 2: NLO diagrams to $W$ plus jet production, in which a gluon is
radiated off an (anti-)quark.

Figure 3: NLO diagrams to $W$ plus jet production, in which a gluon is
radiated off the hard gluon.

Although the full set of NLO diagrams for $W$+jet production (including all
partonic channels) has been calculated before [83], full analytic expressions
are rarely reported due to their cumbersome nature. Thus, we have recalculated
these diagrams independently in FORM [84] and FeynCalc [85], finding
agreement. Here, we will report analytic results for the squared and summed /
averaged matrix element expanded to first subleading order in the emitted
gluon momentum. To do so, we can introduce the Mandelstam invariants

$\displaystyle s$ $\displaystyle=(p_{1}+p_{2})^{2},\quad
q_{1}=(p_{1}-p_{3})^{2},\quad q_{2}=(p_{2}-p_{4})^{2},$ $\displaystyle s_{2}$
$\displaystyle=(p_{3}+p_{4})^{2},\quad\hat{q}_{1}=(p_{1}-p_{4})^{2},\quad\hat{q}_{2}=(p_{2}-p_{3})^{2},$
(7)

and

$t_{k}=(p_{1}-k)^{2},\quad u_{k}=(p_{2}-k)^{2},\quad w_{1}=(p_{3}+k)^{2},\quad
w_{2}=(p_{4}+k)^{2},$ (8)

where we have adopted notation for ease of comparison with ref. [31] (see also
ref. [86]). The various Mandelstam invariants in eqs. (7, 8) can be expressed
in terms of five independent invariants, using the relations

$\displaystyle\hat{q}_{1}$ $\displaystyle=m^{2}-s-t_{k}-q_{1};$ $\displaystyle
s_{2}$ $\displaystyle=s+t_{k}+u_{k};$ $\displaystyle\hat{q}_{2}$
$\displaystyle=m^{2}-s-u_{k}-q_{2};$ $\displaystyle w_{1}$
$\displaystyle=m^{2}-q_{1}-q_{2}-t_{k};$ $\displaystyle w_{2}$
$\displaystyle=q_{1}-q_{2}-u_{k}.$ (9)

Next, we can perform the next-to-soft expansion by introducing a book-keeping
parameter $\lambda$ via

$t_{k}\rightarrow\lambda t_{k},\quad u_{k}\rightarrow\lambda u_{k},$ (10)

before performing a Laurent expansion in $\lambda$ to first subleading order.
Finally, one sets $\lambda=1$. Compact results are then obtained upon using a
particular Lorentz frame for the final-state momenta. Following the case of
$W\gamma$ production in ref. [82], we can choose the centre of mass frame of
the $W$ boson and hard gluon, for which an explicit parametrisation is

$\displaystyle p_{1}$ $\displaystyle=(E_{1},0,\ldots,0,E_{1});$ $\displaystyle
p_{2}$ $\displaystyle=(E_{2},0,\ldots,0,\omega\sin\psi,\omega\cos\psi-E_{1});$
$\displaystyle k$
$\displaystyle=(\omega,0,\ldots,0,\omega\sin\psi,\omega\cos\psi);$
$\displaystyle p_{3}$
$\displaystyle=(\omega_{W},\ldots,-\omega^{\prime}\sin\theta_{1}\sin\theta_{2},-\omega^{\prime}\sin\theta_{1}\cos\theta_{2},-\omega^{\prime}\cos\theta_{1});$
$\displaystyle p_{4}$
$\displaystyle=(\omega^{\prime},\ldots,\omega^{\prime}\sin\theta_{1}\sin\theta_{2},\omega^{\prime}\sin\theta_{1}\cos\theta_{2},\omega^{\prime}\cos\theta_{1});$

with

$\displaystyle E_{1}$ $\displaystyle=\frac{s+t_{k}}{2\sqrt{s_{2}}},\quad
E_{2}=\frac{s+u_{k}}{2\sqrt{s_{2}}},\quad\cos\psi=\frac{(u_{k}s-s_{2}t_{k})}{(t_{k}+u_{k})(s+t_{k})},\quad$
$\displaystyle\omega$
$\displaystyle=-\frac{(t_{k}+u_{k})}{2\sqrt{s_{2}}},\quad\omega^{\prime}=\frac{(s_{2}-m^{2})}{2\sqrt{s_{2}}},\quad\omega_{W}=\frac{(s_{2}+m^{2})}{2\sqrt{s}_{2}}.$
(12)

Then the (next-to)leading power contributions to the squared matrix element
(summed / averaged over colours and spins) are

$\displaystyle\overline{|{\cal M}^{(1)}|}^{2}\Big{|}_{\rm
LP}=\frac{1}{N_{c}}\frac{g_{s}^{4}g_{w}^{2}}{t_{k}u_{k}}\Big{\\{}-\frac{4C_{F}^{2}\,s\,((\rho^{2}-1)^{2}\cos^{2}\theta_{1}+(\rho^{2}+1)^{2})}{(\rho^{2}-1)(\cos^{2}\theta_{1}-1)}$
$\displaystyle-\frac{4C_{A}C_{F}\,s\,\sqrt{t_{k}u_{k}}}{(\rho^{2}-1)^{2}(\cos^{2}\theta_{1}-1)}\frac{((\rho^{2}-1)^{2}\cos^{2}\theta_{1}+(\rho^{2}+1)^{2})\sin\theta_{1}\cos\theta_{2}}{(-2\sin\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}+\cos\theta_{1}(t_{k}-u_{k})+t_{k}+u_{k}}\Big{\\}};$
$\displaystyle\overline{|{\cal M}^{(1)}|}^{2}\Big{|}_{\rm
NLP}=\frac{1}{N_{c}}\frac{g_{s}^{4}g_{w}^{2}}{t_{k}u_{k}}\Big{\\{}-\frac{16C_{F}^{2}}{(\rho^{2}-1)^{3}(\cos^{2}\theta_{1}-1)^{2}}$
$\displaystyle\times\left[-(\rho-1)(\rho+1)(\rho^{4}+1)\sin\theta_{1}\cos\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}+(\rho^{2}+1)\rho^{2}\cos^{2}\theta_{1}(t_{k}+u_{k})-(\rho^{2}+1)\rho^{2}(t_{k}+u_{k})\right]$
$\displaystyle+\frac{2C_{F}C_{A}}{(\rho^{2}-1)^{3}(\cos^{2}\theta_{1}-1)^{2}(-2\sin\theta_{1}\cos\theta_{1}\sqrt{t_{k}u_{k}}+\cos\theta_{1}(t_{k}-u_{k})+t_{k}+u_{k})^{2})}$
$\displaystyle\times\Big{[}-(\rho^{2}+1)(\sin\theta_{1}\cos\theta_{2}(\sqrt{t_{k}u_{k}}(-4\rho^{2}(t_{k}^{2}+6t_{k}u_{k}+u_{k}^{2})+\rho^{4}(-(t_{k}-u_{k})^{2})$
$\displaystyle+(t_{k}-u_{k})^{2})+4t_{k}u_{k}\sin\theta_{1}\cos\theta_{2}(\rho^{4}t_{k}+2\rho^{2}(t_{k}+u_{k})-t_{k}))-2(\rho^{4}-4\rho^{2}-1)t_{k}u_{k}(t_{k}+u_{k}))$
$\displaystyle+2\cos^{2}(\theta_{1})(4\sin\theta_{1}\cos\theta_{2}(t_{k}u_{k}\sin\theta_{1}\cos\theta_{2}(2\rho^{6}t_{k}+2\rho^{4}(t_{k}+u_{k})+\rho^{2}(t_{k}+u_{k})-t_{k}+u_{k})$
$\displaystyle-\rho^{2}\sqrt{t_{k}u_{k}}(\rho^{4}(t_{k}^{2}+u_{k}^{2})+t_{k}^{2}+\rho^{2}(t_{k}+u_{k})^{2}+4t_{k}u_{k}+u_{k}^{2}))+t_{k}u_{k}((\rho^{6}+\rho^{4}+7\rho^{2}-1)u_{k}$
$\displaystyle-(\rho^{6}+\rho^{4}-9\rho^{2}-1)t_{k}))+(\rho^{2}-1)\cos^{4}(\theta_{1})(-(\rho^{2}-1)^{2}\sin\theta_{1}\cos\theta_{2}((t_{k}-u_{k})^{2}\sqrt{t_{k}u_{k}}$
$\displaystyle-4t_{k}^{2}u_{k}\sin\theta_{1}\cos\theta_{2})-2t_{k}u_{k}(\rho^{4}(t_{k}+u_{k})+2\rho^{2}(u_{k}-3t_{k})+t_{k}+u_{k}))$
$\displaystyle-(\rho^{2}-1)^{3}\cos^{5}(\theta_{1})(\sin\theta_{1}\cos\theta_{2}(t_{k}-u_{k})\sqrt{t_{k}u_{k}}(t_{k}+u_{k})-4t_{k}u_{k}^{2})$
$\displaystyle+4\rho^{2}\cos^{3}(\theta_{1})(2t_{k}u_{k}(\rho^{2}(t_{k}+u_{k})+t_{k}-3u_{k})-(\rho^{2}+1)^{2}\sin\theta_{1}\cos\theta_{2}(t_{k}-u_{k})\sqrt{t_{k}u_{k}}(t_{k}+u_{k}))$
$\displaystyle+\cos\theta_{1}(\sin\theta_{1}\cos\theta_{2}(8(\rho^{4}+1)t_{k}u_{k}\sin\theta_{1}\cos\theta_{2}((2\rho^{2}-1)t_{k}-2(\rho^{2}-1)\sin\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}-u_{k})$
$\displaystyle-(\rho^{2}-1)^{2}(3\rho^{2}+1)(t_{k}-u_{k})\sqrt{t_{k}u_{k}}(t_{k}+u_{k}))-4(\rho^{2}+1)t_{k}u_{k}(2\rho^{2}t_{k}+(\rho^{4}-2\rho^{2}-1)u_{k}))$
$\displaystyle+2(\rho^{2}-1)^{3}t_{k}u_{k}\cos^{6}(\theta_{1})(t_{k}-u_{k})))\Big{]}\Big{\\}},$
(13)

where we have introduced the dimensionless parameter

$\rho^{2}=\frac{m^{2}}{s}.$ (14)

A cross-check of this result can be obtained by considering only the
$C_{F}^{2}$ terms, which arise from the diagrams of figure 2. These diagrams
would also arise in abelian gauge theory, where $C_{F}^{2}$ would be replaced
by the appropriate squared electromagnetic charge of the incoming
(anti-)quarks. Then, one may verify that taking $\rho\rightarrow 0$ (i.e. the
limit of zero $W$ mass) reproduces the case of $\gamma\gamma$ production
examined in ref. [31] 999It is not immediately obvious that the limit of a
massless $W$ boson should reproduce the photon process. However, this is
ultimately related to the fact that parity-violating terms in the squared
amplitude for $W$+jet production turn out to be absent. Furthermore, the
longitudinal polarisation state does not contribute due to QED Ward
identities..

###  2.2 A momentum shift formula for $Wg$ production

Having obtained the gluonic (next-to)-soft contributions to the NLO $Wg$
matrix element, let us now see how these can be obtained from a momentum-shift
formula analogous to those presented in refs. [31, 32]. Schematically, we can
write the contribution of a next-to-soft gluon emission from a given Born
amplitude as

${\cal M}^{a(1)}_{\rm NLP}=\sum_{i}g_{s}{\bf
T}^{a}_{i}\epsilon^{\dagger}_{\mu}(k)\left[\frac{p_{i}^{\mu}}{\eta_{i}p_{i}\cdot
k+i\varepsilon}-\frac{ik^{\nu}J^{(i)}_{\nu\mu}}{\eta_{i}p_{i}\cdot
k+i\varepsilon}\right]\otimes{\cal M}^{(0)}(\\{p_{i}\\}).$ (15)

Here the sum is over all external parton legs in the Born amplitude, ${\bf
T}_{i}^{a}$ is a colour generator on line $i$, where we have adopted the
Catani-Seymour notation of refs. [87, 88], and $\eta=\mp 1$ for an incoming or
outgoing particle respectively. There is a polarisation vector
$\epsilon^{\dagger}(k)$ for the outgoing (next-to-soft) gluon, and we also
introduced the total angular momentum generator for each parton leg, which can
be further decomposed into its respective spin and orbital contributions as

$J^{(i)}_{\nu\mu}=\Sigma^{(i)}_{\nu\mu}+L^{(i)}_{\nu\mu}.$ (16)

We have used the symbol $\otimes$ in eq. (15) to mean that the various terms
must be sandwiched (where necessary) between the external wavefunction of line
$i$, and the non-radiative amplitude ${\cal M}^{(0)}(\\{p_{i}\\})$. Examples
can be found throughout refs. [31, 32], and we will see how this works in
detail below. Equation (15) follows from the classic works of refs. [70, 71,
72]101010For off-shell gluons, there is an additional term in eq. (15) that is
$\propto k^{\mu}$. It vanishes for on-shell gluons, however, after contraction
with the polarisation vector.. In more modern literature on scattering
amplitudes, it is known as the next-to-soft theorem [89, 90] (see e.g. refs.
[91, 92] for details of how things are related), and has led to the discovery
of interesting mathematical ideas relating bulk spacetime physics to a
conformal field theory living on the celestial sphere at null infinity [93,
94]. Here, we will be much more applied, and show how eq. (15) leads to a
simple formula for next-to-soft gluon emission, whose physical interpretation
can be elucidated further.

Let us now consider the explicit case of $Wg$ production. In applying eq.
(15), we must start with the non-radiative amplitude whose Feynman diagrams
are given in figure 1. It will be convenient to write the (gauge-dependent)
sub-amplitude ${\cal M}_{X}$ corresponding to a given Feynman diagram $X$,
where the latter spans the labels in figures 1–3. In book-keeping all possible
next-to-soft contributions, we will then follow refs. [31, 32] in separating
the three different kinds of effect appearing in eqs. (15, 16).

####  2.2.1 Scalar terms

The first term in the square brackets in eq. (15) acts multiplicatively on the
whole non-radiative amplitude, with no additional spin structure. The physics
of this is that this term corresponds to leading soft level, and hence cannot
be sensitive to the spin (or orbital angular momentum) of a given hard
particle. Acting on the non-radiative amplitude, one finds that the scalar
terms sum to

${\cal M}_{\rm
scal.}=g_{s}\epsilon_{\mu}^{\dagger}(k)\left(\frac{p_{1}^{\mu}}{-p_{1}\cdot
k}{\bf T}_{1}^{a}+\frac{p_{2}^{\mu}}{-p_{2}\cdot k}{\bf
T}_{2}^{a}+\frac{p_{4}^{\mu}}{p_{4}\cdot k}{\bf T}_{4}^{a}\right)({\cal
M}_{a}+{\cal M}_{b}).$ (17)

Upon squaring the amplitude, we may evaluate all colour factors using the
relations

$\displaystyle{\cal C}\left[{\bf T}_{1}^{a}{\cal M}_{a}\right]$
$\displaystyle={\cal C}\left[{\bf T}_{1}^{a}{\cal
M}_{b}\right]=C_{(A)}=C_{(D)};$ $\displaystyle{\cal C}\left[{\bf
T}_{2}^{a}{\cal M}_{a}\right]$ $\displaystyle={\cal C}\left[{\bf
T}_{2}^{a}{\cal M}_{b}\right]=C_{(C)}=C_{(F)};$ $\displaystyle{\cal
C}\left[{\bf T}_{4}^{a}{\cal M}_{a}\right]$ $\displaystyle={\cal C}\left[{\bf
T}_{4}^{a}{\cal M}_{b}\right]=C_{(G)}=C_{(H)},$ (18)

where ${\cal C}[\ldots]$ denotes taking the colour factor of a given diagram,
and we have recognised the colour factors $\\{C_{(X)}\\}$ of specific diagrams
appearing in figures 2 and 3. Then, evaluating all colour traces in the
squared amplitude before summing / averaging over final / initial colours
yields

$\overline{|{\cal M}_{\rm
scal}|^{2}}=\left[C_{F}\left(C_{F}-\frac{C_{A}}{2}\right)\frac{2p_{1}\cdot
p_{2}}{p_{1}\cdot k\,p_{2}\cdot
k}+\frac{C_{A}^{2}C_{F}}{2}\left(\frac{2p_{1}\cdot p_{4}}{p_{1}\cdot
k\,p_{4}\cdot k}+\frac{2p_{2}\cdot p_{4}}{p_{1}\cdot k\,p_{4}\cdot
k}\right)\right]\overline{|{\cal M}^{(0)}|^{2}}.$ (19)

Here, we recognise the usual form of leading soft corrections to a squared
amplitude, where the individual terms that appear correspond to separate pairs
of colour charges that are linked by soft gluon emission. For each pair or
dipole, there is an appropriate colour factor, plus a kinematic prefactor that
results upon combining the eikonal Feynman rules for the gluon. As is well-
known [14, 15], this kinematic factor leads to a pronounced radiation pattern,
including the angular ordering property described in the introduction, and
that we will see in more detail in section 3. For now, we simply note that the
remaining (next-to)soft gluon corrections will lead to corrections to this
simple radiation pattern, and our next task is to write them in a manageable
way.

####  2.2.2 Spin terms

Given that a coupling of the emitted gluon to the spin angular momentum of a
given hard particle is already next-to-soft level, for the spin contributions
to the total squared matrix element, we need only worry about the interference
contribution

$2{\rm Re}\left[{\cal M}_{\rm scal.}{\cal M}^{\dagger}_{\rm spin}\right]$

up to next-to-leading power (NLP), where ${\cal M}_{\rm spin}$ collects all
the spin effects at amplitude level. To find the latter, we need the explicit
forms of the Lorentz generators associated with different parton legs. These
are

$\Sigma^{\alpha\beta}_{ab}=\frac{i}{4}\left[\gamma^{\alpha},\gamma^{\beta}\right]_{ab}$
(20)

and

$\Sigma^{\alpha\beta}_{\mu\nu}=i\left(\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}-\delta^{\alpha}_{\nu}\delta^{\beta}_{\mu}\right)$
(21)

for a spin-1/2 and spin-1 particle respectively, where lower indices in these
equations are spin-indices that must be contracted along the line (n.b. lower-
case Latin letters denote spinor indices). From eq. (15), one then finds that
the spin contribution to the NLO amplitude up to NLP level is given by

$\displaystyle{\cal M}_{\rm spin}$
$\displaystyle=g_{s}\bar{v}(p_{2})\left\\{\frac{{\bf T}^{a}_{1}}{4p_{1}\cdot
k}M^{(0)\rho\gamma}[\gamma^{\mu},\not\\!{k}]-\frac{{\bf
T}_{2}^{a}}{4p_{1}\cdot k}[\not\\!{k},\gamma^{\mu}]M^{(0)\rho\gamma}\right.$
$\displaystyle\left.\quad+\frac{{\bf T}_{4}^{a}}{p_{4}\cdot
k}M^{(0)\tau\gamma}\left[\delta^{\mu}_{\tau}\delta^{\nu}_{\rho}-\delta^{\mu}_{\rho}\delta^{\nu}_{\tau}\right]k_{\nu}\right\\}u(p_{1})\epsilon_{\mu}^{\dagger}(k)\epsilon^{\dagger}_{\rho}(p_{4})\epsilon^{\dagger}_{\gamma}(p_{3}),$
(22)

where $M^{(0)}_{\sigma}$ represents the non-radiative amplitude, stripped of
external wavefunctions. This must then be combined with the scalar amplitude
of eq. (19) and summed over polarisations and colours to find the interference
contribution. Colour factors may again be evaluated using eq. (18), and one
may also simplify the result by repeated use of anticommutation relations for
Dirac matrices. One finally obtains an interference contribution

$2{\rm Re}\left[{\cal M}_{\rm spin}{\cal M}^{\dagger}_{\rm
scal.}\right]=-2N_{c}C_{F}^{2}\frac{(p_{1}+p_{2})\cdot k}{p_{1}\cdot
k\,p_{2}\cdot k}|{\cal M}^{(0)}|^{2}.$ (23)

####  2.2.3 Orbital angular momentum terms

For the orbital angular momentum contributions, we need the explicit form of
the angular momentum operator associated with leg $i$ in momentum space:

$L^{\alpha\beta}=x_{i}^{\alpha}p_{i}^{\beta}-x_{i}^{\beta}p_{i}^{\alpha}=i\left(p_{i}^{\alpha}\frac{\partial}{\partial
p_{1\beta}}-p_{i}^{\beta}\frac{\partial}{\partial p_{1\alpha}}\right).$ (24)

Then, similarly to the spin case, the orbital angular momentum contribution
can be written as

$\displaystyle{\cal M}_{\rm orb.}$
$\displaystyle=ig_{s}\bar{v}(p_{2})\left[\frac{{\bf T}^{a}_{1}}{p_{1}\cdot
k}\left(k\cdot p_{1}\frac{\partial}{\partial
p_{1,\mu}}-p_{1}^{\mu}k\cdot\frac{\partial}{\partial
p_{1}}\right)M^{(0)\rho\gamma}\right.$ $\displaystyle\left.\quad+\frac{{\bf
T}^{a}_{2}}{p_{2}\cdot k}\left(p_{2}\cdot k\frac{\partial}{\partial
p_{2,\mu}}-p_{2}^{\mu}k\cdot\frac{\partial}{\partial
p_{2}}\right)M^{(0)\rho\gamma}\right.$ $\displaystyle\left.\quad+\frac{{\bf
T}^{a}_{4}}{p_{4}\cdot k}\left(p_{4}\cdot k\frac{\partial}{\partial
p_{4,\mu}}-p_{4}^{\mu}k\cdot\frac{\partial}{\partial
p_{4}}\right)M^{(0)\rho\gamma}\right]u(p_{1})\epsilon_{\mu}^{\dagger}(k)\epsilon_{\rho}^{\dagger}(p_{4})\epsilon^{\dagger}_{\gamma}(p_{3}).$
(25)

Once again, this must be combined with the scalar amplitude of eq. (19) in
order to find the relevant interference term. Applying similar steps to those
outlined in ref. [32], we find

$\displaystyle 2{\rm Re}\left[{\cal M}_{\rm orb.}{\cal M}^{\dagger}_{\rm
scal}\right]=\left\\{N_{c}C_{F}^{2}\frac{2p_{1}\cdot p_{2}}{p_{1}\cdot
k\,p_{2}\cdot k}\left[\left(\delta p_{1,2}\cdot\frac{\partial}{\partial
p_{1}}+\delta p_{2,1}\cdot\frac{\partial}{\partial p_{2}}\right){\rm
Tr}\left(\not\\!{p}_{2}M^{(0)\rho\gamma_{1}}\not\\!{p}_{1}M^{(0){\dagger}}_{\rho\gamma_{2}}\right)\right.\right.$
$\displaystyle\left.-{\rm
Tr}\left(\delta\not{p}_{2,1}M^{(0)\rho\gamma_{1}}\not\\!{p}_{1}M^{(0)}_{\rho\gamma_{2}}\right)-{\rm
Tr}\left(\not{p}_{2}M^{(0)\rho\gamma_{1}}\delta\not{p}_{1,2}M^{(0)}_{\rho\gamma_{2}}\right)\right]$
$\displaystyle+\quad\frac{C_{A}C_{F}}{2}\left[-\frac{2p_{1}\cdot
p_{2}}{p_{1}\cdot k\,p_{2}\cdot
k}\left(\not{p}_{1,2}\cdot\frac{\partial}{\partial
p_{1}}+\not{p}_{2,1}\cdot\frac{\partial}{\partial
p_{2}}\right)+\frac{2p_{1}\cdot p_{4}}{p_{1}\cdot k\,p_{4}\cdot
k}\left(\not{p}_{1,4}\cdot\frac{\partial}{\partial
p_{1}}-\not{p}_{4,1}\cdot\frac{\partial}{\partial p_{4}}\right)\right.$
$\displaystyle\left.\left.\quad+\frac{2p_{2}\cdot p_{4}}{p_{2}\cdot
k\,p_{4}\cdot k}\left(\not{p}_{2,4}\cdot\frac{\partial}{\partial
p_{2}}-\not{p}_{4,2}\cdot\frac{\partial}{\partial p_{4}}\right)\right]{\rm
Tr}\left[\not\\!{p}_{2}M^{(0)\rho\gamma_{1}}\not\\!{p}_{1}M^{(0){\dagger}}_{\rho\gamma_{2}}\right]\right\\}{P^{\gamma_{2}}}_{\gamma_{1}},$
(26)

where ${P^{\gamma_{2}}}_{\gamma_{1}}$ denotes the W boson polarisation sum. We
have used the chain rule where necessary, and also introduced the momentum
shifts

$\delta p^{\alpha}_{i,j}=-\frac{1}{2}\left(k^{\alpha}+\frac{p_{j}\cdot
k}{p_{i}\cdot p_{j}}p_{i}^{\alpha}-\frac{p_{i}\cdot k}{p_{i}\cdot
p_{j}}p_{j}^{\alpha}\right).$ (27)

Equation (26) looks cumbersome, but we have yet to combine it with the spin
contribution of eq. (23). To do so, we may first introduce a Sudakov
decomposition for the emitted gluon momentum:

$k^{\mu}=\frac{p_{1}\cdot k}{p_{1}\cdot p_{2}}p_{2}^{\mu}+\frac{p_{2}\cdot
k}{p_{1}\cdot p_{2}}p_{1}^{\mu}+k_{T}^{\mu},$ (28)

where

$k_{T}\cdot p_{1}=k_{T}\cdot p_{2}=0.$ (29)

Equation (27) then implies

$\delta\not{p}_{1,2}=-\frac{1}{2}\left(\not\\!{k}_{T}+2\frac{p_{2}\cdot
k}{p_{1}\cdot
p_{2}}\not\\!{p}_{1}\right),\quad\delta\not{p}_{2,1}=-\frac{1}{2}\left(\not\\!{k}_{T}+2\frac{p_{1}\cdot
k}{p_{1}\cdot p_{2}}\not\\!{p}_{2}\right),$ (30)

where upon substituting this into the second line of eq. (26), we may ignore
the terms $\sim{\cal O}(k^{\mu}_{T})$: upon carrying out all Dirac traces,
$k^{\mu}_{T}$ will only ever be contracted with hard momenta in the process,
to first order in soft momentum, and all such contractions vanish. We then
find

$\displaystyle N_{c}C_{F}^{2}\frac{2p_{1}\cdot p_{2}}{p_{1}\cdot k\,p_{2}\cdot
k}\left[-{\rm
Tr}\left(\delta\not{p}_{2,1}M^{(0)\rho\gamma_{1}}\not\\!{p}_{1}M^{(0)}_{\rho\gamma_{2}}\right)-{\rm
Tr}\left(\not{p}_{2}M^{(0)\rho\gamma_{1}}\delta\not{p}_{1,2}M^{(0)}_{\rho\gamma_{2}}\right)\right]{P^{\gamma_{2}}}_{\gamma_{1}}$
$\displaystyle=2\frac{(p_{1}+p_{2})\cdot k}{p_{1}\cdot k\,p_{2}\cdot k}|{\cal
M}^{(0)}|^{2}.$ (31)

Thus, upon combining eq. (26) with eq. (23), the second line of eq. (26) is
cancelled. Up to next-to-soft level, one may then absorb the various momentum
shift terms into a redefinition of the squared nonradiative amplitude, to give
the final result

$\displaystyle\overline{|{\cal M}_{\rm
NLP}|^{2}}=C_{F}\left(C_{F}-\frac{C_{A}}{2}\right)\frac{2p_{1}\cdot
p_{2}}{p_{1}\cdot k\,p_{2}\cdot k}\overline{|{\cal M}^{(0)}|^{2}}(p_{1}+\delta
p_{1,2},p_{2}+\delta p_{2,1},p_{3},p_{4})$
$\displaystyle\quad+\frac{C_{A}C_{F}}{2}\frac{2p_{1}\cdot p_{4}}{p_{1}\cdot
k\,p_{4}\cdot k}\overline{|{\cal M}^{(0)}|^{2}}(p_{1}+\delta
p_{1,4},p_{2},p_{3},p_{4}-\delta p_{4,1})$
$\displaystyle\quad+\frac{C_{A}C_{F}}{2}\frac{2p_{2}\cdot p_{4}}{p_{2}\cdot
k\,p_{4}\cdot k}\overline{|{\cal M}^{(0)}|^{2}}(p_{1},p_{2}+\delta
p_{2,4},p_{3},p_{4}-\delta p_{4,2}).$ (32)

This is our final result for the NLP matrix element, and it agrees with a
similar formula derived for prompt photon production in ref. [32], thus
showing that this is more general than previously thought. Given eq. (6), we
may implement the momentum shifts as in eqs. (27, 32), and then expand to
next-to-leading power using a similar method to that outlined in the previous
section. Explicit results for the three shift terms appearing in eq. (32) are
respectively as follows:

$\displaystyle\overline{|{\cal M}_{\rm
NLP}|^{2}}\Big{|}_{1}=\frac{4g_{s}^{4}g_{w}^{2}}{t_{k}u_{k}}\frac{C_{F}}{N_{c}}\left(C_{F}-\frac{C_{A}}{2}\right)\left[-\frac{s(\rho^{2}-1)^{2}\cos^{2}\theta_{1}+(\rho^{2}+1)^{2}}{(\rho^{2}-1)^{2}(\cos^{2}\theta_{1}-1)}\right.$
$\displaystyle\left.\quad+\frac{4(\rho-1)(\rho+1)(\rho^{4}+1)\sin\theta_{1}\cos\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}-(\rho^{2}+1)\rho^{2}(1-\cos^{2}\theta_{1})(t_{k}+u_{k})}{(\rho^{2}-1)^{3}(\cos^{2}\theta_{1}-1)^{2}}\right];$
$\displaystyle\overline{|{\cal M}_{\rm
NLP}|^{2}}\Big{|}_{2}=\frac{2g_{s}^{4}g_{w}^{2}C_{F}C_{A}}{N_{c}}\Big{[}\frac{s((\rho^{2}-1)^{2}\cos^{2}\theta_{1}+(\rho^{2}+1)^{2})}{(\rho^{2}-1)^{2}t_{k}(\cos\theta_{1}+1)(-2\sin\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}+\cos\theta_{1}(t_{k}-u_{k})+t_{k}+u_{k})}$
$\displaystyle\quad+\Big{(}4\cos^{2}\theta_{1}(\rho^{2}(2\rho^{4}t_{k}^{2}+t_{k}^{2}+\rho^{2}(t_{k}+u_{k})^{2}+u_{k}^{2})-\sin\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}(-2\rho^{6}(u_{k}-2t_{k})+\rho^{4}(t_{k}+3u_{k})$
$\displaystyle\quad-
t_{k}+2\rho^{2}u_{k}+u_{k}))+\cos\theta_{1}((3\rho^{6}-\rho^{4}+5\rho^{2}+1)t_{k}^{2}+4\sin\theta_{1}\cos\theta_{2}(2(\rho-1)(\rho+1)(2\rho^{4}+1)t_{k}u_{k}$
$\displaystyle\quad\times\sin\theta_{1}\cos\theta_{2}+\sqrt{t_{k}u_{k}}(-2\rho^{6}(2t_{k}+u_{k})+\rho^{4}(3t_{k}+u_{k})-2\rho^{2}(t_{k}+u_{k})+t_{k}+u_{k}))+(\rho-1)(\rho+1)$
$\displaystyle\quad\times(7\rho^{4}+6\rho^{2}+3)t_{k}u_{k}+4\rho^{2}(\rho^{2}+1)u_{k}^{2})+4\rho^{2}\cos^{3}\theta_{1}((\rho^{4}+\rho^{2})t_{k}^{2}-\
(\rho^{4}+2\rho^{2}-3)t_{k}u_{k}-(\rho^{2}+1)u_{k}^{2})$
$\displaystyle\quad+(\rho^{2}-1)^{3}\cos^{4}\theta_{1}(\sin\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}(u_{k}-3t_{k})+t_{k}(t_{k}+u_{k}))-(\rho^{2}+1)(\sin\theta_{1}\cos\theta_{2}$
$\displaystyle\quad\times(8\rho^{2}t_{k}u_{k}\sin\theta_{1}\cos\theta_{2}-\sqrt{t_{k}u_{k}}(\rho^{4}(3t_{k}-u_{k})+4\rho^{2}(t_{k}+3u_{k})-3t_{k}+u_{k}))+(t_{k}+u_{k})$
$\displaystyle\quad\times((\rho^{4}-1)t_{k}+4\rho^{2}u_{k}))+(\rho^{2}-1)^{3}t_{k}\cos^{5}\theta_{1}(t_{k}-3u_{k})\Big{)}$
$\displaystyle\quad\times\frac{1}{(\rho^{2}-1)^{3}t_{k}(\cos\theta_{1}-1)(\cos\theta_{1}+1)^{2}(-2\sin\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}+\cos\theta_{1}(t_{k}-u_{k})+t_{k}+u_{k})^{2}}\Big{]};$
$\displaystyle\overline{|{\cal M}_{\rm
NLP}|^{2}}\Big{|}_{3}=\frac{2g_{s}^{4}g_{w}^{2}C_{F}C_{A}}{N_{c}}\Big{[}-\frac{s((\rho^{2}-1)^{2}\cos^{2}\theta_{1}+(\rho^{2}+1)^{2}}{(\rho^{2}-1)^{2}u_{k}(\cos\theta_{1}-1)(-2\sin\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}+\cos\theta_{1}(t_{k}-u_{k})+t_{k}+u_{k})}$
$\displaystyle\quad+\Big{(}-4\cos^{3}\theta_{1}(\rho^{4}(t_{k}^{2}-2t_{k}u_{k}-u_{k}^{2})+(\rho^{2}-1)^{3}u_{k}\sin\theta_{1}\cos\theta_{2}(\sqrt{t_{k}u_{k}}-t_{k}\sin\theta_{1}\cos\theta_{2})$
$\displaystyle\quad+\rho^{6}u_{k}(t_{k}-u_{k})+\rho^{2}t_{k}(t_{k}+u_{k}))-4\cos^{2}\theta_{1}(\rho^{2}(t_{k}^{2}+\rho^{2}(t_{k}+u_{k})^{2}+2\rho^{4}u_{k}^{2}+u_{k}^{2})$
$\displaystyle\quad+\sin\theta_{1}\cos\theta_{2}((\rho^{2}-1)^{3}t_{k}u_{k}\sin\theta_{1}\cos\theta_{2}-\sqrt{t_{k}u_{k}}(\rho^{4}t_{k}+\rho^{2}(4t_{k}+u_{k})-t_{k}+3\rho^{6}u_{k})))$
$\displaystyle\quad+\cos\theta_{1}(4(\rho^{4}+\rho^{2})t_{k}^{2}+4\sin\theta_{1}\cos\theta_{2}((\rho^{6}+\rho^{4}-3\rho^{2}+1)t_{k}u_{k}\sin\theta_{1}\cos\theta_{2}$
$\displaystyle\quad-\sqrt{t_{k}u_{k}}(\rho^{4}t_{k}+t_{k}+3\rho^{6}u_{k}-\rho^{2}u_{k}))+(\rho^{2}-1)^{2}(3\rho^{2}+1)t_{k}u_{k}+(3\rho^{6}-\rho^{4}+5\rho^{2}+1)u_{k}^{2})$
$\displaystyle\quad-(\rho^{2}-1)^{3}\cos^{4}\theta_{1}(\sin\theta_{1}\cos\theta_{2}(t_{k}-3u_{k})\sqrt{t_{k}u_{k}}+u_{k}(t_{k}+u_{k}))+(\rho^{2}+1)(\sin\theta_{1}\cos\theta_{2}$
$\displaystyle\quad\times(\sqrt{t_{k}u_{k}}(\rho^{4}(t_{k}+u_{k})-4\rho^{2}(3t_{k}+u_{k})-t_{k}-u_{k})-4(\rho^{4}-2\rho^{2}-1)t_{k}u_{k}\sin\theta_{1}\cos\theta_{2})+(t_{k}+u_{k})$
$\displaystyle\quad\times(4\rho^{2}t_{k}+(\rho^{4}-1)u_{k}))+(\rho^{2}-1)^{3}u_{k}\cos^{5}\theta_{1}(t_{k}+u_{k})\Big{)}$
$\displaystyle\quad\times\frac{1}{(\rho^{2}-1)^{3}u_{k}(\cos\theta_{1}-1)^{2}(\cos\theta_{1}+1)(-2\sin\theta_{1}\cos\theta_{2}\sqrt{t_{k}u_{k}}+\cos\theta_{1}(t_{k}-u_{k})+t_{k}+u_{k})^{2}}\Big{]}.$
(33)

Upon adding these results and simplifying, we find precise agreement with the
truncated NLO squared amplitude of eq. (13), thus confirming the validity of
eq. (32).

Comparing eq. (32) with eq. (19), we see that the effect of the next-to-soft
corrections is to modify the leading power soft gluon squared amplitude by
shifting the momenta of the nonradiative amplitude. Crucially, however, the
corrections do not modify the dipole-like nature of the result: in each term,
the momenta that are shifted in the nonradiative amplitude are the same hard
momenta that appear in the accompanying dipole radiation pattern. This
suggests a particularly nice physical interpretation of the next-to-soft
corrections, which we explore in the following section.

##  3 The physics of angular-ordering breakdown

As discussed above, a well-known property of soft radiation from pairs of
(colour) charges is that it is confined to certain cones, centered around the
hard particles that emit the radiation. Put another way, successive soft gluon
emissions from the same pair of charged particles are strongly ordered in
angle, and this effect is built into angular-ordered parton-shower algorithms
to incorporate soft-gluon interference effects in a straightforward way [14,
15]. Given that the leading next-to-soft gluon radiation that is captured by
eq. (32) preserves a dipole-like form, it is natural to ask whether the
momentum-shift contributions lead to a breaking or otherwise of the angular-
ordering property. We will see that, unsurprisingly, angular ordering indeed
does not persist at next-to-soft level. However, the origin of the breaking
can be traced very directly to the momentum-shift formula of eq. (32), which
allows us to understand in physical terms how it happens.

###  3.1 Angular ordering of soft radiation

Let us first recap the arguments leading to angular ordering of soft
radiation, where we will follow closely the presentation in refs. [14, 15].
These arguments are reproduced here to make our presentation self-contained,
as well as being necessary for the next-to-soft generalisation to be discussed
below. We will consider a final state dipole in QED, consisting of e.g. an
electron-positron pair, as shown in figure 4(a).

(a)

(b)

Figure 4: (a) A dipole consisting of two oppositely charged particles,
emitting a photon; (b) similar situation, taking into account a non-zero
displacement of each fermion from the origin.

In the limit in which the emitted photon momentum is soft ($k^{\mu}\rightarrow
0$), the NLO squared amplitude for this process assumes the form

$|{\cal M}_{\rm LP}|^{2}\sim W_{ij}|{\cal M}^{(0)}|^{2},$ (34)

where ${\cal M}^{(0)}$ is the Born amplitude, and we have introduced the
radiation function

$W_{ij}=\frac{E_{k}^{2}p_{i}\cdot p_{j}}{p_{i}\cdot k\,p_{j}\cdot k}.$ (35)

This consists of the eikonal dressing factor that we see in e.g. eq. (19),
multiplied by the square of the photon energy $E_{k}$ to make the radiation
function dimensionless. The energy dependence will be compensated elsewhere in
the total squared amplitude, but it is the radiation function that controls
all angular dependence of the emitted radiation. To probe the latter, it is
standard to write

$W_{ij}=W_{ij}^{[i]}+W_{ij}^{[j]},$ (36)

where the modified radiation functions appearing on the right-hand side are
given by

$\displaystyle W_{ij}^{[i]}$
$\displaystyle=\frac{1}{2}\left(W_{ij}+\frac{1}{1-\cos\theta_{ik}}-\frac{1}{1-\cos\theta_{jk}}\right);$
$\displaystyle W_{ij}^{[j]}$
$\displaystyle=\frac{1}{2}\left(W_{ij}+\frac{1}{1-\cos\theta_{jk}}-\frac{1}{1-\cos\theta_{ik}}\right).$
(37)

The reason for this – which is not necessarily obvious a priori – is that the
modified radiation functions have precisely the angular ordering property
noted above. That is, the soft radiation captured by $W^{[l]}_{ij}$ is
confined to a cone around particle $l$, with a half-angle given by the angle
between particles $i$ and $j$. To see this, we can choose a Lorentz frame such
that the 3-momentum $\vec{p}_{i}$ is oriented along the $z$ direction, and the
3-momentum $\vec{p}_{j}$ lies in the $(x,z)$ plane:

$\displaystyle p_{i}^{\mu}$ $\displaystyle=E_{i}(1,0,0,1);$ $\displaystyle
p_{j}^{\mu}$ $\displaystyle=E_{j}(1,\sin\theta_{ij},0,\cos\theta_{ij});$
$\displaystyle k^{\mu}$
$\displaystyle=E_{k}(1,\sin\theta_{ik}\cos\phi_{ik},\sin\theta_{ik}\sin\phi_{ik},\cos\theta_{ik}).$
(38)

Here $\\{\theta_{ab}\\}$ and $\\{\phi_{ab}\\}$ denote the polar and azimuthal
angles between particles $a$ and $b$ in a conventional spherical polar
coordinate system. Then

$p_{j}\cdot k=E_{j}E_{k}(a-b\cos\phi_{ik}),\quad
a=1-\cos\theta_{ij}\cos\theta_{ik},\quad b=\sin\theta_{ij}\sin\theta_{ik}.$
(39)

However, by choosing an alternative frame in which the 3-momentum of particle
$j$ defines the polar axis, one may also surmise

$p_{j}\cdot k=E_{j}E_{k}(1-\cos\theta_{jk}),$ (40)

such that comparing eqs. (39, 40) implies

$1-\cos\theta_{jk}=a-b\cos\phi_{ik}.$ (41)

The integration over the final-state phase space will include an integral over
the azimuthal angle $\phi_{ik}$ of the emitted photon, and a useful
intermediate step in integrating $W_{ij}^{[i]}$ is to consider the integral

$I_{ij}^{[i]}=\int_{0}^{2\pi}\frac{d\phi_{ik}}{2\pi}\frac{1}{1-\cos\theta_{jk}},$
(42)

which occurs in the first and third terms of eq. (37), whose explicit form in
angular coordinates is

$W_{ij}^{[i]}=\frac{1}{2}\left(\frac{1-\cos\theta_{ij}}{(1-\cos\theta_{ik})(1-\cos\theta_{jk})}+\frac{1}{1-\cos\theta_{ik}}-\frac{1}{1-\cos\theta_{jk}}\right).$
(43)

Following refs. [14, 15], we may transform to $z=e^{i\phi_{iq}}$, such that
eq. (42) becomes a contour integral around the unit circle in the complex
$z$-plane:

$I_{ij}^{[i]}=\frac{1}{i\pi b}\oint\frac{dz}{(z-z_{+})(z-z_{-})},\quad
z_{\pm}=\frac{a}{b}\pm\sqrt{\frac{a^{2}}{b^{2}}-1}.$ (44)

Only the pole at $z=z_{-}$ lies inside the unit circle, such that one may
carry out the integral using Cauchy’s theorem, yielding

$I_{ij}^{[i]}=\frac{1}{\sqrt{a^{2}-b^{2}}}=\frac{1}{|\cos\theta_{ik}-\cos\theta_{ij}|}.$
(45)

Substituting this result into eq. (43), one may rearrange to give

$W_{ij}^{[i]}=\frac{1}{2(1-\cos\theta_{ik})}\left[1+\frac{(\cos\theta_{ik}-\cos\theta_{ij})}{|\cos\theta_{ik}-\cos\theta_{ij}|}\right].$
(46)

A plot of this function is shown in figure 5 in blue, for
$\cos\theta_{ij}=0.2$. We see that there is non-zero radiation only for polar
angles around particle $i$ which are less than or equal to the opening angle
$\theta_{ij}$ between the dipole, as expected. Furthermore, the divergence at
$\cos\theta_{ik}\rightarrow 1$ coincides with the emitted photon becoming
collinear with particle $i$.

Figure 5: Blue: distribution of soft photon radiation from a dipole with
opening angle $\cos\theta_{ij}=0.2$; Orange: corresponding result coming from
the leading next-to-soft correction, as captured by the momentum shifts in eq.
(32). The dashed line shows the non-angular ordered contribution to the next-
to-soft result.

Here we have explicitly considered QED, where this pronounced radiation
pattern is known as the Chudakov effect. The same arguments apply to QCD in
the case of single gluon radiation from a dipole, where this is an overall
colour singlet [14, 15]. In both cases, the simple quantum mechanical argument
for the suppressed radiation outside the cone is that the wavelength of the
emitted photon becomes such that it cannot resolve the individual (colour)
charges in the dipole.

###  3.2 Next-to-soft radiation from a dipole

Having recalled how angular ordering arises from soft radiation, let us now
examine how the additional momentum-shift corrections in eq. (32) modify the
picture. To this end, we may again restrict ourselves to the simplest possible
case of a final-state dipole in gauge theory, namely the electron-positron
pair of figure 4(a). Then the effect of an additional photon emission up to
next-to-soft level is to modify eq. (34) to

$|{\cal M}_{\rm NLP}|^{2}\sim W_{ij}|{\cal M}^{(0)}(p_{i}-\delta
p_{i,j},p_{j}-\delta p_{j,i})|^{2},$ (47)

using the momentum shift definitions of eq. (27).111111An explicit derivation
of this formula can be found in ref. [32], and uses similar steps to those
presented here in section 2.2. Up to NLP level, we can expand to first order
in the momentum shifts, and also use the fact that the squared Born
interaction depends only on the Mandelstam invariant $s=2p_{i}\cdot p_{j}$, to
get

$|{\cal M}_{\rm NLP}|^{2}\sim W_{ij}f(s)-2\Big{[}\delta p_{i,j}\cdot
p_{j}+\delta p_{j,i}\cdot p_{i}\Big{]}W_{ij}f^{\prime}(s),$ (48)

where

$f(s)=|{\cal M}^{(0)}|^{2}$ (49)

is the squared Born amplitude with unshifted kinematics, and the prime denotes
its first-order derivative. To examine the angular properties of the next-to-
soft term, we define the dimensionless radiation functions

$\displaystyle\widetilde{W}^{[l]}_{ij}$ $\displaystyle=-\left(\frac{\delta
p_{i,j}\cdot p_{j}}{E_{k}E_{j}}+\frac{\delta p_{j,i}\cdot
p_{i}}{E_{k}E_{i}}\right)W^{[l]}_{ij},$ (50)

such that

$\widetilde{W}_{ij}=\widetilde{W}^{[i]}_{ij}+\widetilde{W}^{[j]}_{ij}=-\left(\frac{\delta
p_{i,j}\cdot p_{j}}{E_{k}E_{j}}+\frac{\delta p_{j,i}\cdot
p_{i}}{E_{k}E_{i}}\right)W^{[l]}_{ij}$ (51)

controls the total next-to-soft correction to the radiation pattern. Using the
parametrisation of eq. (38), we have

$\displaystyle\widetilde{W}^{[i]}_{ij}=(1-\cos\theta_{ik})W_{ij}^{[i]}+(1-\cos\theta_{jk})W_{ij}^{[i]}.$
(52)

In the first term, the prefactor is independent of the azimuthal angle
$\phi_{iq}$, and thus does not affect the integration over the latter. We can
thus reuse the previous result for the azimuthal integration of $W_{ij}^{[i]}$
when calculating the radiation pattern. In the second term of eq. (52), the
prefactor cancels the singularity in $\cos\theta_{jk}$, such that one finds

$\displaystyle\int_{0}^{2\pi}\frac{d\phi_{ik}}{2\pi}\widetilde{W}^{[i]}_{ij}$
$\displaystyle=\frac{1}{2}\left[1+\frac{\cos\theta_{ik}-\cos\theta_{ij}}{|\cos\theta_{ik}-\cos\theta_{ij}|}\right]$
$\displaystyle\quad+\frac{1}{2}\int_{0}^{2\pi}\frac{d\phi_{ik}}{2\pi}\left[\frac{1-\cos\theta_{ij}}{1-\cos\theta_{ik}}+\frac{a-b\cos\phi_{ik}}{1-\cos\theta_{ik}}-1\right]$
$\displaystyle=\frac{1}{2}\left[1+\frac{\cos\theta_{ik}-\cos\theta_{ij}}{|\cos\theta_{ik}-\cos\theta_{ij}|}\right]+\sin^{2}\left(\frac{\theta_{ij}}{2}\right)\cot^{2}\left(\frac{\theta_{ik}}{2}\right).$
(53)

This is our final result for the azimuthally-averaged next-to-soft radiation
pattern from one particle in a dipole. Interestingly, it has the form of a sum
of an angular-ordered term, analogous to the pure soft case, plus a breaking
term, which has a remarkably compact analytic form. We show this function in
figure 5, and can see clearly the effect of the angular-ordering term, in that
there is a discontinuity at $\cos\theta_{ik}=\cos\theta_{ij}$. However, for
$\cos\theta_{ik}<\cos\theta_{ij}$, corresponding to gluon emission outside the
cone region, there is indeed a non-zero radiation distribution. For
completeness, we show the effect of the non-angular-ordered term (i.e. the
second term in eq. (53)) inside the cone. This is shown as the dashed line in
figure 5, and we see that it smoothly joins the radiation outside the cone, as
it should. There remains a (hard) collinear singularity around particle $i$,
which acts as a next-to-soft correction to a collinear emission which is also
strictly soft. In interpreting the figure, we must remember that the LP and
NLP radiation functions ($W^{[i]}_{ij}$ and $\tilde{W}^{[i]}_{ij}$
respectively) are defined with different energy ratios to make them
dimensionless. Thus, the overall normalisation between the two curves in
figure 5 is not particularly meaningful. Rather, figure 5 illustrates the
significant qualitative difference between the angular distributions of soft
and next-to-soft radiation: the latter breaks angular ordering.

Note that a similar effect appears for massive emitters at leading soft level,
as is well-known [14, 15]. Denoting the energy-normalised velocity of the two
dipole legs with $v_{i}$ and $v_{j}$ respectively, one obtains for the
azimuthally-averaged emission pattern

$\displaystyle\int^{2\pi}_{0}\frac{d\phi_{ik}}{2\pi}W_{ij}^{[i]}=\frac{v_{i}}{2(1-v_{i}\cos\theta_{ik})}\left[\frac{v_{i}-\cos\theta_{ik}}{1-v_{i}\cos\theta_{ik}}+\frac{\cos\theta_{ik}-v_{j}\cos\theta_{ij}}{\sqrt{(\cos\theta_{ik}-v_{j}\cos\theta_{ij})^{2}+\sin\theta_{ik}^{2}(1-v_{j}^{2})}}\right].$
(54)

This reduces to the massless form when $v_{i}=v_{j}=1$. However, when
$v_{j}\neq 1$ (but $v_{i}=1$) the sharp transition is replaced by a smooth
damping of the total radiation from the dipole, which extends to angles larger
than the opening angle of the dipole. This is shown in figure 6, which
contrasts the massless and massive cases. Here, the presence of an intrinsic
momentum scale results in the breaking of angular-ordering, and it would be
interesting to further examine the interplay between next-to-soft and massive
effects.

Figure 6: Blue: distribution of soft photon radiation from a dipole of
massless particles with opening angle $\cos\theta_{ij}=0.2$; Orange:
corresponding result if particle $j$ is massive, such that $v_{j}=0.5$ in eq.
(54).

Let us now understand the next-to-soft effect in more physical terms. We can
do this given that radiation outside the cone is associated with the specific
momentum shifts in eq. (27), whose origin is the orbital angular momentum
contributions to the squared matrix element. One can only generate such an
orbital angular momentum if the two worldlines of the fermions in the dipole
are mutually displaced. In particular, let us choose the origin of spacetime
such that both lines are displaced to 4-positions $x^{\mu}_{i}$ and
$x^{\mu}_{j}$, as shown in figure 4(b). The relationship between such
displacements and next-to-soft corrections was considered in ref. [21], which
used Schwinger proper time methods to write the scattering amplitude for hard
particles emitting radiation in terms of quantum mechanical (first-quantised)
path integrals over their spacetime trajectories. More specifically, one can
show that such an amplitude is given by

${\cal A}(p_{1},\ldots,p_{n})=\int{\cal D}A_{\mu}\left[\prod_{i=1}^{n}\int
d^{d}x_{i}e^{-ip_{i}\cdot x_{i}}f(x_{i},p_{i};A_{\mu})\right]H(x_{1}\ldots
x_{n};A_{\mu})e^{iS[A_{\mu}]},$ (55)

where $H(x_{1},\ldots x_{n};A_{\mu})$ is a hard function that produces the
outgoing particles at initial positions $\\{x_{i}\\}$ with final momenta
$\\{p_{i}\\}$. The formal definition of this quantity can be found in ref.
[21], and will not be needed in what follows. The path integral in eq. (55) is
over the (next-to) soft gauge field, and includes the usual dependence on the
action $S[A_{\mu}]$, where we suppress the dependence on the matter fields for
brevity. Associated with each hard particle is an integral over its initial
position $x_{i}$, a certain exponential factor, and further factor
$f(x_{i},p_{i};A_{\mu})$, which for scalar particles is

$f(x_{i},p_{i};A_{\mu})=\int{\cal
D}z_{i}\exp\left[i\int_{0}^{\infty}d\tau\left(\frac{\dot{z}_{i}^{2}}{2}+q_{i}(\beta+\dot{x})\cdot
A(x_{i}+\beta_{i}\tau+z_{i})+\frac{iq_{i}}{2}\partial\cdot
A(x_{i}+\beta_{i}\tau+z_{i})\right)\right].$ (56)

In this equation, we have parametrised the spacetime trajectory of the $i^{\rm
th}$ particle via

$x^{(i)}(t)=x_{i}+\beta_{i}\tau+z_{i}(\tau),$ (57)

where $\beta_{i}$ is the 4-velocity associated with the final momentum
$p_{i}$, and $\tau$ the proper time. The quantity $z_{i}(\tau)$ then
constitutes a fluctuation about the classical straight-line trajectory, and
the path integral over $z_{i}(\tau)$ corresponds to summing over all possible
fluctuations. This path integral is carried out subject to the boundary
conditions of fixed initial position $x_{i}$ and final momentum $p_{i}$ for
each particle. Finally, $q_{i}$ is the electric charge of hard particle $i$.

As shown in ref. [21], the path integral over worldline trajectories in eq.
(57) can be carried out perturbatively. The leading term – corresponding to
keeping the classical trajectory only – amounts to the hard particle not
recoiling, and thus emitting pure soft radiation only. Expanding order-by-
order in $z_{i}(\tau)$ then amounts to including all possible wobbles in the
spacetime trajectory which, by the uncertainty principle, amounts to including
the emission of radiation at progressively subleading orders in the momentum
of the emitted radiation. By keeping the first-order term only, ref. [21]
found a set of Feynman rules corresponding to the emission of next-to-soft
radiation from hard particles. Repeating the analysis for fermionic emitting
particles leads to an extra term in eq. (56), that corresponds to the spin-
dependent part of the next-to-soft theorem of eqs. (15, 16).

Now let us focus on the contribution to eq. (55) that stems from the initial
separation of the dipole members, namely the non-zero initial positions
$\\{x_{i}\\}$. At next-to-soft level, keeping track of these non-zero
positions means that the path integral in eq. (56) can be replaced by its
leading term (i.e. the classical trajectory only). The hard particle factor of
eq. (56) then reduces to the well-known Wilson line describing the emission of
soft radiation [21]:

$\displaystyle f(x_{i},p_{i};A_{\mu})$
$\displaystyle\rightarrow\exp\left[iq_{i}\int_{0}^{\infty}d\tau\beta_{i}^{\mu}A_{\mu}(x_{i}+\beta_{i}\tau)\right]$
$\displaystyle=1-q_{i}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{p_{i}^{\mu}\tilde{A}_{\mu}(k)}{p_{i}\cdot
k}e^{ip_{i}\cdot k}+{\cal O}(q_{i}^{2}),$ (58)

where we have transformed to momentum space in the second line, and expanded
in the coupling so as to isolate the effect of a single photon emission in the
second term. Next, we can expand the exponential appearing in the $k$
integral, where the first subleading correction corresponds to the next-to-
soft contribution. Collecting these factors on all lines, the effect of the
non-zero initial positions in the path integral of eq. (55) is

$\displaystyle\left[\prod_{i=1}^{n}\int d^{d}x_{i}e^{-ix_{i}\cdot
p_{i}}\right]\left[\sum_{j=1}^{n}q_{j}\frac{d^{d}k}{(2\pi)^{d}}(-ix_{j}\cdot
k)\frac{p_{j}\cdot A(k)}{p_{j}\cdot
k}\right]H(p_{1},\ldots,p_{n})=\Gamma^{\mu}A_{\mu},$ (59)

where carrying out the integrals over the positions $\\{x_{i}\\}$ yields

$\Gamma^{\mu}=\int\frac{d^{d}k}{(2\pi)^{d}}\sum_{j=1}^{n}q_{j}\left(\frac{p_{j}^{\mu}}{p_{j}\cdot
k}k_{\nu}\frac{\partial}{\partial p_{j\nu}}\right)H(p_{1},\ldots,p_{n}).$ (60)

In the path integral over the gauge field, this looks like an additional
Feynman rule for the emission of a single photon from each line, which
involves a derivative acting on the hard function. In fact, the result of eq.
(59) is incomplete. In eq. (55), the hard function depends upon the gauge
field, as it must. Expanding this order-by-order in the coupling amounts to
including the effects of soft gluon emissions from inside the hard interaction
(see ref. [21] for a detailed explanation). As shown in the very early work of
ref. [70], such contributions can be fixed by gauge invariance. The most
straightforward way to implement this here is to note that the factor
$\Gamma^{\mu}$ will form part of a complete scattering amplitude ${\cal
A}^{\mu}$ for the emission of a photon of momentum $k$, which must satisfy the
Ward identity $k_{\mu}{\cal A}^{\mu}=0$. This in turn implies that we must
modify

$\Gamma^{\mu}\rightarrow\tilde{\Gamma}^{\mu},\quad
k_{\mu}\tilde{\Gamma}^{\mu}=0.$ (61)

Requiring a local combination of momenta yields the unique result (see also
ref. [95])

$\tilde{\Gamma}^{\mu}=\int\frac{d^{d}k}{(2\pi)^{d}}\sum_{j=1}^{n}q_{j}k_{\nu}\left(\frac{p_{j}^{\mu}}{p_{j}\cdot
k}\frac{\partial}{\partial p_{j\nu}}-\frac{p_{j}^{\nu}}{p_{j}\cdot
k}\frac{\partial}{\partial p_{j\mu}}\right)H(p_{1},\ldots,p_{n}).$ (62)

Comparison with eq. (24) allows to explicitly recognise the form of the
orbital angular momentum of each hard particle, a fact which was not clarified
in ref. [21]. However, it makes precise the above expectation, that non-zero
initial positions of the dipole members will indeed give rise to the orbital
angular momentum part of the next-to-soft theorem.

The physics of angular-ordering breaking is then as follows. Soft radiation
has an infinite Compton wavelength, and thus is unable to see the separation
between the two fermion worldlines, as they emanate from a given hard
interaction. Next-to-soft radiation, on the other hand, is able to resolve the
length scale corresponding to the difference in initial particle positions,
which manifests itself in the orbital angular momentum contributions being
non-zero, as captured by the momentum shifts in eq. (32). The fact that wide-
angle radiation now sees the initial “size” of the dipole means that it will
no longer see a zero net charge. Hence, radiation can be present outside the
cone.

##  4 Conclusion

In this paper, we have performed a case study looking at the physical
interpretation of next-to-soft radiation. The characterisation of such
radiation is of great interest in furthering the precision frontier at current
collider experiments, as well as addressing interesting conceptual questions
in field theory. In addition to building up new methods and techniques, it is
important to build intuition about next-to-soft physics, that may in turn
inform further developments. With this motivation in mind, we have here
focused on a particular formula for incorporating gluon radiation using a
dipole-like formula that incorporates next-to-soft effects through shifts of
the momenta appearing in the non-radiative amplitude. This formula first
appeared for colour-singlet final states in ref. [31], and was extended to
particular processes with partons in the final state in ref. [32]. We have
here checked its validity in another process ($Wg$ production). Next, we
looked at the physical consequences of this formula, which stem from the fact
that it has the form of a sum of dipole-like contributions.

One of the most well-known properties of soft emission from dipoles is that
interference effects lead to suppression of the radiation outside cones
surrounding each hard particle, whose half-angle corresponds to the opening
angle between the constituents of the dipole. Faced with eq. (32), then, we
can ask if the inclusion of next-to-soft corrections breaks the angular-
ordering property. Indeed it does, and the physical mechanism of this is that
the momentum shifts capture precisely that part of the next-to-soft physics –
orbital angular momentum contributions – that is associated with an initial
separation between the dipole constituents. This provides a new length scale,
which the radiation is then able to resolve. Although this creates some
radiation outside the cones described above, there still remains a significant
discontinuity in the radiation distribution at the edge of the cone.

We hope that our results provide useful physical intuition to researchers
working in this area, as well as inspiring further similar studies.

## Acknowledgments

EL and AT would like to thank the MHRD Government of India, for the SPARC
grant SPARC/2018-2019/P578/SL, entitled Perturbative QCD for Precision Physics
at the LHC. CDW is supported by the UK Science and Technology Facilities
Council (STFC) grants ST/P000258/1 and ST/T000759/1. MvB was supported by a
Royal Society Research Professorship (RP$\backslash$R1$\backslash$180112), and
by the STFC grant ST/T000864/1. SP would like to thank Satyajit Seth for
useful discussions. SP would like to thank the Physical Research Laboratory,
Department of Space, Govt. of India, for a Post Doctoral Fellowship. AD would
like to thank CSIR, Govt. of India, for an SRF fellowship (09/1001(00
49)/2019- EMR-I).

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