aainstitutetext: Theoretical Physics Department, CERN, CH-1211 Geneva 23,
Switzerlandbbinstitutetext: Dipartimento di Fisica and Arnold-Regge Center,
Università di Torino,
and INFN, Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy

#  Non-abelian infrared divergences on the celestial sphere

Lorenzo Magnea<EMAIL_ADDRESS>
###### Abstract

We consider the infrared factorisation of non-abelian multi-particle
scattering amplitudes, and we study the form of the universal colour operator
responsible for infrared divergences, when expressed in terms of coordinates
on the ‘celestial sphere’ intersecting the future light-cone at asymptotic
distances. We find that colour-dipole contributions to the infrared operator,
to all orders in perturbation theory, have a remarkably simple expression in
these coordinates, with scale and coupling dependence factorised from
kinematics and colour. Generalising earlier suggestions in the abelian theory,
we then show that the infrared operator can be computed as a correlator of
vertex operators in a conformal field theory of Lie-algebra-valued free bosons
on the celestial sphere. We verify by means of the OPE that the theory
correctly predicts the all-order structure of collinear limits, and the tree-
level factorisation of soft real radiation.

††preprint:
CERN-TH-2021-063

##  1 Introduction

The problem of infrared divergences in gauge theory amplitudes and cross
sections has been studied for close to a century, starting with the seminal
work of Bloch and Nordsieck in QED Bloch:1937pw , and solutions involving an
increasing degree of generality and depth have been put forward across the
decades. One class of proposed solutions is based upon the idea of carefully
constructing well-defined (and thus finite) observable quantities, in the
spirit of the KLN theorem Kinoshita:1962ur ; Lee:1964is . Within this
framework, one accepts that $S$-matrix elements are ill-defined in massless
theories, introduces an infrared regulator, and verifies the cancellation of
singularities for infrared-safe observables Sterman:1978bi ; Sterman:1978bj .
Alternatively, one can try to tackle the infrared problem at the root,
recognising that it originates from an inadequate definition of asymptotic
states. This viewpoint led to the development of the method of coherent
states, by means of which an infrared-finite perturbative $S$-matrix can be
defined, both in QED Kulish:1970ut  and for non-abelian theories
Giavarini:1987ts . Remarkably, the study of infrared singularities for non-
abelian gauge theories continues to this day to be a very active field of
research, both for theoretical reasons, and in view of phenomenological
applications. On the theoretical side, a general understanding of the all-
order infrared factorisation properties of non-abelian multi-particle
scattering amplitudes has been developed over the years Mueller:1979ih ;
Collins:1980ih ; Sen:1981sd ; Sen:1982bt ; Collins:1989bt ; Magnea:1990zb ;
Catani:1998bh ; Sterman:2002qn ; Dixon:2008gr ; Becher:2009cu ; Gardi:2009qi ;
Becher:2009qa ; Gardi:2009zv ; Gardi:2013ita ; Feige:2014wja ; Erdogan:2014gha
; Ma:2019hjq , and the relevant soft and collinear anomalous dimensions have
been computed to very high orders, uncovering interesting new mathematical
structures Gardi:2010rn ; Mitov:2010rp ; Gardi:2011wa ; Gardi:2011yz ;
Vladimirov:2014wga ; Vladimirov:2015fea ; Almelid:2015jia ; Henn:2016jdu ;
Almelid:2017qju ; Henn:2019swt ; vonManteuffel:2020vjv ; Agarwal:2020nyc ;
Agarwal:2021zft ; Agarwal:2021him . On the phenomenological side, a vast
effort has been devoted to the construction of efficient algorithms for the
subtraction of infrared singularities at high perturbative orders, in both
virtual corrections and real radiation contributions to cross sections of
experimental interest at colliders. In this case, the most significant problem
is the treatment of unresolved real radiation for complex observables, in the
presence of intricate experimental cuts. Recent advances in this direction
have been reviewed in Ref. TorresBobadilla:2020ekr .

All the developments briefly summarised above are based on standard, if
advanced, techniques of quantum field theory in four dimensions, including
diagrammatic tools, renormalisation group equations, Ward identities and
effective field theories. In recent years, starting with Refs.
Strominger:2013lka ; Strominger:2013jfa , a remarkable novel formulation of
the infrared problem has been introduced, based on the discovery of enhanced
long-distance symmetries of theories involving massless particles. Within this
framework, the infrared behaviour of scattering amplitudes is determined by
infinite-dimensional asymptotic symmetries of the massless theory under
consideration, acting on the ‘celestial’ sphere intersecting the future (or
past) light cone at asymptotic distances. For gravity, these symmetries have
been known for a long time Bondi:1962px ; Sachs:1962wk , while in the case of
gauge theories they take the form of ‘large’ gauge transformations with non-
trivial action at null infinity Lysov:2014csa ; He:2014cra ; He:2015zea ;
Adamo:2015fwa ; Strominger:2015bla ; Gabai:2016kuf ; Cheung:2016iub ;
Nande:2017dba . In this context, soft factorisation theorems for tree-level
radiative amplitudes, and their next-to-leading-power extensions, emerge as
Ward identities of these asymptotic symmetries. The first few years of work in
this fast-developing field have been reviewed in Strominger:2017zoo , where
one can find ample references to the literature. At the moment, much of the
work done within this framework is focused on tree-level amplitudes for gauge
theories and gravity (see, however, Refs. Albayrak:2020saa ; Gonzalez:2020tpi
, where one-loop effects are analysed), but the connection that has been
uncovered between infrared properties of $d=4$ massless field theories and
$d=2$ conformal invariance on the celestial sphere is intriguing, and
represents a radically innovative point of view on an old problem. An
important new tool introduced in this context has been the use of Mellin
transforms of scattering amplitudes Pasterski:2016qvg ; Pasterski:2017kqt ,
which are constructed to have simple transformation properties under the
conformal group acting on the celestial sphere, while from a four-dimensional
viewpoint they are boost eigenstates. A rapidly growing body of literature
(see, for example, Refs. Pasterski:2017ylz ; Schreiber:2017jsr ;
Stieberger:2018onx ; Himwich:2019dug ; Fan:2019emx ; Pate:2019mfs ;
Nandan:2019jas ; Fotopoulos:2019tpe ; Ebert:2020nqf ; Arkani-Hamed:2020gyp ;
Fan:2021isc ) is exploring the properties of these celestial amplitudes,
including detailed comparisons with known soft and collinear limits.
Interestingly, related ideas have also started to be applied in a more
phenomenological context Neill:2020bwv .

In this paper, we will take a different viewpoint: instead of considering the
full multi-particle scattering amplitude, we will take advantage of its
factorisation properties in the infrared, and focus on the universal infrared
colour operator responsible for generating all soft and collinear
singularities. Colour correlations for this operator are well understood to
all orders in perturbation theory, and they originate from correlators of
straight, semi-infinite Wilson lines, emerging from the hard scattering and
pointing in the light-like directions defined by the momenta of the hard
particles participating in the scattering. The role of scale and conformal
invariance in establishing the form of this infrared operator was already
pointed out in Refs. Gardi:2009qi ; Becher:2009qa , and the conformal
properties of the infrared factor were instrumental in the determination of
the three-loop infrared anomalous dimension by a bootstrap approach in Ref.
Almelid:2017qju . More recently, a very important observation in this
direction was made in Ref. Kalyanapuram:2020epb : examining the abelian case,
Kalyanapuram noticed that the one-loop soft factor for QED-like theories is
very closely connected to a correlator of vertex operators for free bosons on
the celestial sphere. Here, we will pursue this remarkable connection, and
propose an extension to the non-abelian case.

At the outset, such a generalisation appears to be difficult to achieve, for
several reasons: first of all, in QED the exponentiation of the infrared
factor is essentially one-loop exact (as it is in gravity Weinberg:1965nx ),
while, in a non-abelian theory, intricate corrections emerge at all
perturbative orders; next, as a consequence, the scale dependence of the non-
abelian soft operator is non-trivial, and possibly entwined with colour
correlations; finally, and more importantly, the non-abelian infrared factor
is a colour operator and not just a number. These obstacles not withstanding,
we will be able to propose a generalisation of the results of Refs.
Kalyanapuram:2020epb  to non-abelian massless gauge theories, in terms of a
conformal theory of Lie-algebra-valued free bosons on the celestial sphere.

We will begin in Section 2 by providing a bird’s-eye view of our current
understanding of infrared divergences for massless non-abelian scattering
amplitudes, focusing our attention on colour correlations, which arise
exclusively from soft gauge boson exchanges. In Section 3, concentrating on
colour correlations with a dipole structure, which are understood to all
orders, we will translate known results in terms of variables defined on the
celestial sphere. Remarkably, we will find that all dependence on the gauge
coupling, on the scale, and on the dimensional regulator factorises from
kinematic and colour correlations, and can be collected in a single universal
function which is well-known in perturbative QCD, giving, among other things,
the all-order perturbative expression for the gluon Regge trajectory in high-
energy scattering Korchemsky:1993hr ; Korchemskaya:1994qp ;
Korchemskaya:1996je ; DelDuca:2011xm ; DelDuca:2011ae ; dependence on colour
and kinematics is then strikingly simple, and naturally expressed in terms of
distances on the celestial sphere. In Section 4, we introduce a Lie-algebra-
valued free-boson conformal theory on the celestial sphere. Vertex operators
in this theory are colour matrices, in representations corresponding to the
hard particles participating in the scattering. That notwithstanding, they
have a well-defined conformal weight, and they form well-behaved correlation
functions, which could be seen as colour-kinematic duals Bern:2008qj ;
Bern:2019prr  of the integrands of bosonic string amplitudes. These
correlators exactly reproduce the all-order expression for infrared
divergences of gauge amplitudes with dipole colour correlations.

In order to further test the correspondence between the conformal theory and
the gauge theory, we note the presence in the celestial theory of a Lie-
algebra-valued conserved Noether current, whose OPE with vertex operators
reproduces the tree-level soft-gluon current responsible for soft real
radiation in the gauge theory. This provides strong evidence for identifying
this Noether current with the Kac-Moody current constructed in Ref.
Strominger:2013lka ; He:2015zea  by considering asymptotic limits of four-
dimensional gauge fields. Further, we use the conformal OPE of vertex
operators to study collinear limits of the infrared factor, and we recover the
expected expression for the splitting anomalous dimension for dipole
correlations Catani:2003vu ; Becher:2009qa , once again an all-order result.
In Section 5, we discuss some of the many open questions that our proposal
leaves open at this stage. In particular, we discuss the matching between the
celestial coupling and the gauge-theory coupling factor, noting that it would
be of great interest to develop an understanding of the role played by scale
dependence in the correspondence between the celestial and the four-
dimensional theory. We emphasise that, in our view, the theory as presented
here is incomplete, and we expect that a generalisation, or possibly a
deformation, of the conformal theory we propose, should be able to predict,
and indeed compute, the quadrupole correlations which arise in the gauge
theory starting at three loops, and are explicitly known at that order
Almelid:2015jia ; Henn:2016jdu ; Almelid:2017qju . We conclude in Section 6
summarising our results and suggesting several open lines for future research.

##  2 Non-abelian soft anomalous dimensions: a primer

The structure of infrared divergences for on-shell scattering amplitudes in
massless non-abelian gauge theories is understood in remarkable generality. To
summarise what is known, consider an $n$-point amplitude in dimensional
regularisation, which we write as

$\displaystyle{\cal A}^{\,a_{1}\ldots
a_{n}}_{n}\bigg{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu),\epsilon\bigg{)}\,,$ (1)

displaying the colour indices $a_{i}$ for each external particle. Note that we
are allowing for particles in different representations of the gauge group, so
$a_{i}$ is not necessarily an adjoint index; furthermore, renormalisation has
already been performed, and we are working in $d=4-2\epsilon>4$, so that
$\epsilon<0$, to regularise infrared singularities; finally, we are not
displaying polarisation indices, which will be irrelevant in what follows. If
desired, one can select a basis of colour tensors $c_{M}^{\,a_{1}\ldots
a_{n}}$ spanning the space of allowed colour configurations for the process at
hand: the amplitude in Eq. (1) is a vector in that space, whose components in
the chosen basis can be computed by suitable projections.

Infrared divergences arise from long-distance exchanges of virtual particles,
and they can be shown to factorise in the form Sen:1982bt ; Dixon:2008gr ;
Gardi:2009qi ; Becher:2009qa ; Feige:2014wja ; Ma:2019hjq

$\displaystyle{\cal
A}_{n}\bigg{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu),\epsilon\bigg{)}\,=\,{\cal
Z}_{n}\bigg{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu),\epsilon\bigg{)}{\cal
H}_{n}\bigg{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu),\epsilon\bigg{)}\,,$ (2)

where colour indices are understood, ${\cal Z}_{n}$ is a universal colour
operator generating all infrared poles in $\epsilon$, and acting on a colour
vector ${\cal H}_{n}$, which is process-dependent and finite as $\epsilon\to
0$. The divergent factor ${\cal Z}_{n}$ satisfies a renormalisation group
equation, which can be solved in the form

$\displaystyle{\cal
Z}_{n}\bigg{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu),\epsilon\bigg{)}\,=\,P\exp\left[\frac{1}{2}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\,\Gamma_{n}\bigg{(}\frac{p_{i}}{\lambda},\alpha_{s}(\lambda,\epsilon)\bigg{)}\right]\,,$
(3)

where $\alpha_{s}(\lambda,\epsilon)$ is the $d$-dimensional running coupling,
satisfying

$\displaystyle\lambda\frac{\partial\alpha_{s}}{\partial\lambda}\,\equiv\,\beta(\alpha_{s},\epsilon)\,=\,-2\epsilon\alpha_{s}-\frac{\alpha_{s}^{2}}{2\pi}\,\sum_{k=0}^{\infty}\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{\\!k}b_{k}\,.$
(4)

Eq. (3) features the central object of our discussion, the infrared anomalous
dimension matrix, which we denote by $\Gamma_{n}$. Note that the initial
condition in Eq. (3) has been fixed in the infrared, by making use of the fact
that $\alpha_{s}(\lambda=0,\epsilon<0)=0$; note also that the matrix
$\Gamma_{n}$ is finite as $\epsilon\to 0$, and all infrared singularities
arise from the scale integration.

The matrix $\Gamma_{n}$ is clearly a fundamental object for perturbative gauge
theories: for massless theories, it is fully known up to three loops for any
$n$ and for any gauge group and representation content Almelid:2015jia . What
is most remarkable, however, is the fact that its all-order structure is
strongly constrained by an underlying scale invariance of the infrared sector.
In order to describe this structure, it is best not to write $\Gamma_{n}$ is a
specific basis, but rather to adopt the basis-independent colour-operator
notation of Refs. Bassetto:1984ik ; Catani:1996vz . In this language, one
introduces, for each hard particle $i$, $i=1,\ldots,n$, a colour operator
${\bf T}_{i}$, which acts on the finite factor ${\cal H}_{n}$ as a generator
of the gauge algebra in the appropriate representation111Thus, for example,
for an outgoing quark one has ${\bf T}_{i}\to T^{a}_{\,\,ij}$, where $T^{a}$
is a generator in the fundamental representation of $SU(N)$, while for a gluon
one has ${\bf T}_{i}\to-{\rm i}f^{a}_{\,\,cd}$, where $f^{a}_{\,\,cd}$ are the
structure constants.. The colour operators ${\bf T}_{i}$ carry an adjoint
index, and satisfy

$\displaystyle\Big{[}{\bf T}_{i}^{a},{\bf T}_{i}^{b}\Big{]}\,=\,{\rm
i}f^{ab}_{\,\,\,\,\,c}\,{\bf T}_{i}^{c}\,,\qquad{\bf T}_{i}\cdot{\bf
T}_{i}\,\equiv\,{\bf T}_{i}^{a}{\bf
T}_{i}^{b}\,\delta_{ab}\,=\,C_{i}^{(2)}\,,\qquad\sum_{i=1}^{n}{\bf
T}_{i}\,=\,0\,,$ (5)

where $C_{i}^{(2)}$ is the quadratic Casimir eigenvalue of the gauge algebra
in the representation of particle $i$, while the last equation enforces colour
conservation, and must be interpreted as constraint on the action of the
colour operators ${\bf T}_{i}$ on the finite factor ${\cal H}_{n}$, dictated
by gauge invariance. Note that colour operators associated with different hard
particles commute, since ${\cal Z}_{n}$ is an operator in the tensor product
of the colour representations of the $n$ particles, and each ${\bf T}_{i}$
acts on a single factor in that product.

Let us now discuss the all-order structure of the infrared anomalous dimension
matrix. As we will see, a crucial ingredient is the light-like cusp anomalous
dimension $\gamma_{K,r}(\alpha_{s})$, governing the UV divergences of form
factors of Wilson lines in representation $r$ Dotsenko:1979wb ; Brandt:1981kf
; Korchemsky:1985xj ; Korchemsky:1985xu ; Korchemsky:1987wg . For the sake of
simplicity, we will consider the approximation in which the cusp anomalous
dimension satisfies ‘Casimir scaling’, i.e. we will write

$\displaystyle\gamma_{K,r}(\alpha_{s})\,=\,C_{r}^{(2)}\,\widehat{\gamma}_{K}(\alpha_{s})\,,$
(6)

where $\widehat{\gamma}_{K}(\alpha_{s})$ is representation-independent. This
approximation is known to fail at the four-loop level Moch:2018wjh ;
Grozin:2017css ; Henn:2019rmi ; Henn:2019swt ; vonManteuffel:2020vjv ;
Falcioni:2020lvv , where contributions proportional to quartic Casimir
eigenvalues have been shown to appear, with expected consequences on the
structure of $\Gamma_{n}$ at four loops and beyond Ahrens:2012qz ;
Becher:2019avh . With this single simplifying approximation, the soft
anomalous dimension matrix admits the all-order representation

$\displaystyle\Gamma_{n}\bigg{(}\frac{p_{i}}{\mu},\alpha_{s}(\mu)\bigg{)}\,=\,\Gamma_{n}^{\rm\,dipole}\bigg{(}\frac{s_{ij}}{\mu^{2}},\alpha_{s}(\mu)\bigg{)}+\Delta_{n}\Big{(}\rho_{ijkl},\alpha_{s}(\mu)\Big{)}\,.$
(7)

The colour-dipole term $\Gamma_{n}^{\rm\,dipole}$ is the only contribution at
one and two loops, and can be written as

$\displaystyle\Gamma_{n}^{\rm\,dipole}\bigg{(}\frac{s_{ij}}{\mu^{2}},\alpha_{s}(\mu)\bigg{)}\,=\,\frac{1}{2}\,\widehat{\gamma}_{K}\big{(}\alpha_{s}(\mu)\big{)}\sum_{i=1}^{n}\sum_{j=i+1}^{n}\log\frac{-s_{ij}+{\rm
i}\eta}{\mu^{2}}\,\,{\bf T}_{i}\cdot{\bf
T}_{j}\,-\,\sum_{i=1}^{n}\gamma_{i}\big{(}\alpha_{s}(\mu)\big{)}\,,$ (8)

where we assumed for simplicity that all particles are outgoing,
$s_{ij}=2p_{i}\cdot p_{j}$, and $\gamma_{i}(\alpha_{s})$ is the UV anomalous
dimensions for the field corresponding to particle $i$. The conformal
correction $\Delta_{n}$ arises starting at three loops, with at least four
particles, and it is constrained by scale invariance to depend only upon the
conformal cross-ratios

$\displaystyle\rho_{ijkl}\,=\,\frac{p_{i}\cdot p_{j}\,p_{k}\cdot
p_{l}}{p_{i}\cdot p_{l}\,p_{j}\cdot
p_{k}}\,=\,\frac{s_{ij}s_{kl}}{s_{il}s_{jk}}\,.$ (9)

At the three-loop level, $\Delta_{n}$ was computed in Refs. Almelid:2015jia ;
Henn:2016jdu ; Almelid:2017qju , and it is built out of quadrupole
correlations of the form

$\displaystyle
F_{ijkl}\big{(}\\{\rho\\}\big{)}\,f_{abe}f_{cd}^{\,\,\,\,\,e}\,{\bf
T}_{i}^{a}{\bf T}_{j}^{a}{\bf T}_{k}^{c}{\bf T}_{l}^{d}\,,$ (10)

where the kinematic dependence is contained in the functions $F_{ijkl}$; by
Bose symmetry and colour conservation, actually only one function is involved,
and it turns out to be an extremely simple combination of weight-5 single-
valued harmonic polylogarithms Brown:2004ugm ; Schnetz:2013hqa .

In the present context, it is important to note that the dipole term can be
organised in a more useful way by exploiting colour conservation to simplify
the scale dependence, which will need to be integrated. One may write

$\displaystyle\Gamma_{n}^{\rm\,dipole}\left(\frac{s_{ij}}{\lambda^{2}},\alpha_{s}(\lambda,\epsilon)\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\,\widehat{\gamma}_{K}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\sum_{i=1}^{n}\sum_{j=i+1}^{n}\ln\left(\frac{-s_{ij}+{\rm
i}\eta}{\mu^{2}}\right){\bf T}_{i}\cdot{\bf T}_{j}$ (11)
$\displaystyle-\,\sum_{i=1}^{n}\gamma_{i}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\,-\,\frac{1}{4}\,\widehat{\gamma}_{K}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\ln\left(\frac{\mu^{2}}{\lambda^{2}}\right)\sum_{i=1}^{n}C_{i}^{(2)}$
$\displaystyle\equiv$
$\displaystyle\Gamma_{n}^{\rm\,corr.}\bigg{(}\frac{s_{ij}}{\mu^{2}},\alpha_{s}(\lambda,\epsilon)\bigg{)}\,+\,\Gamma_{n}^{\rm\,singl.}\bigg{(}\frac{\mu^{2}}{\lambda^{2}},\alpha_{s}(\lambda,\epsilon)\bigg{)}\,,$

where $\mu$ is a fixed scale, while $\lambda$ is the integration variable in
Eq. (3). In this form, one can show that (aside from poles originating from
the running of the coupling) the first term, which contains dipole colour
correlations, generates only single poles of soft origin, while the remaining
terms, which are colour singlets, generate single poles of hard-collinear
origin, as well as double poles of soft-collinear origin. In order to compute
them explicitly, one may use the explicit expression for the $d$-dimensional
running coupling at the desired order, and perform the scale integration in
Eq. (3). For example, at the one-loop level, one may use the lowest-order
expression for $d$-dimensional running coupling

$\displaystyle\alpha_{s}\left(\lambda,\epsilon\right)\,=\,\alpha_{s}\left(\mu\right)\left(\frac{\lambda^{2}}{\mu^{2}}\right)^{\\!-\epsilon}\,,$
(12)

together with the basic integrals

$\displaystyle\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\,\alpha_{s}\\!\big{(}\lambda,\epsilon\big{)}\,=\,-\,\frac{1}{\epsilon}\,\alpha_{s}(\mu)\,,\quad\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\ln\left(\frac{\lambda^{2}}{\mu^{2}}\right)\alpha_{s}\\!\big{(}\lambda,\epsilon\big{)}\,=\,-\,\frac{1}{\epsilon^{2}}\,\alpha_{s}(\mu)\,,\quad\big{(}\epsilon<0\big{)}\,.$
(13)

Note that in a gauge theory which is conformal in $d=4$, such as ${\cal N}=4$
Super-Yang-Mills theory, the result in Eq. (12) is exact, and indeed the
logarithm of the infrared operator ${\cal Z}_{n}$ has only single and double
poles Bern:2005iz , which are trivially determined in terms of the
perturbative coefficients of the singlet anomalous dimensions
$\widehat{\gamma}_{K}$ and $\gamma_{i}$. In what follows, we will concentrate
on the colour correlated term, $\Gamma_{n}^{\rm\,corr.}$. It is useful to
remember that that it originates from the Wilson-line correlator

$\displaystyle{\cal
S}_{n}\Big{(}\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu),\epsilon\Big{)}\,\equiv\,\langle
0|\prod_{i=1}^{n}\Phi_{\beta_{i}}(\infty,0)\,|0\rangle\,,$ (14)

where $\beta_{i}$ are dimensionless four-velocity vectors parallel to the
momenta $p_{i}$, and $\Phi_{\beta_{i}}$ are semi-infinite Wilson lines along
the classical trajectories of the particles exiting the hard interaction
vertex

$\displaystyle\Phi_{\beta}(\infty,0)\,\equiv\,P\exp\left[{\rm
i}g\int_{0}^{\infty}d\lambda\beta\cdot A(\lambda\beta)\right]\,.$ (15)

Note that the Wilson lines in Eq. (14) have open colour indices at both ends,
so that ${\cal S}_{n}$, as announced, is a colour operator acting on the hard
factor ${\cal H}_{n}$ in Eq. (2). Classically, the correlator in Eq. (14) is
invariant under the rescalings $\beta_{i}\to\kappa_{i}\beta_{i}$, however this
invariance is broken by collinear divergences in the massless case: the dipole
contribution to the anomalous dimension matrix arises as a solution to the
corresponding ‘anomaly equation’, where the ‘anomaly’ is expressed by the cusp
anomalous dimension Gardi:2009qi ; Becher:2009qa .

##  3 Colour correlations on the celestial sphere

Taking inspiration from Refs. Almelid:2017qju ; Kalyanapuram:2020epb , we now
rewrite the colour-correlated part of the infrared anomalous dimension matrix
in terms of coordinates on the celestial sphere. We parametrise the momenta
$p_{i}$ as

$\displaystyle
p_{i}^{\mu}\,=\,\omega_{i}\,\Big{\\{}1+z_{i}\bar{z}_{i},\,z_{i}+\bar{z}_{i},\,-{\rm
i}\big{(}z_{i}-\bar{z}_{i}\big{)},\,1-z_{i}\bar{z}_{i}\Big{\\}}\,,$ (16)

which is real if (as we assume) $\bar{z}_{i}=z_{i}^{*}$, and implies

$\displaystyle s_{ij}\,=\,2p_{i}\cdot
p_{j}\,=\,4\omega_{i}\omega_{j}\big{|}z_{i}-z_{j}\big{|}^{2}\,,$ (17)

so that all momenta are indeed light-like, $p_{i}^{2}=0$. We can define the
four-velocities $\beta_{i}$, for example, by
$p_{i}^{\mu}=\sqrt{2}\omega_{i}\beta_{i}^{\mu}$, so that
$\beta_{i}\cdot\beta_{j}=|z_{i}-z_{j}|^{2}$. Then we notice that (as expected
from Eq. (14)) the energies $w_{i}$ do not contribute to colour correlations,
again as an effect of colour conservation. Indeed one can use

$\displaystyle\log\big{(}\\!-\\!s_{ij}+{\rm
i}\eta\big{)}\,=\,\log\Big{(}\big{|}z_{i}-z_{j}\big{|}^{2}\Big{)}+\log\omega_{i}+\log\omega_{j}+2\log
2+{\rm i}\pi\,,$ (18)

and perform the colour sum on terms that do not depend simultaneously on $i$
and $j$. All these terms can then be shuffled into the colour-singlet
contribution to $\Gamma_{n}$, so that we can write

$\displaystyle\Gamma_{n}^{\rm\,dipole}\left(\frac{s_{ij}}{\lambda^{2}},\alpha_{s}(\lambda,\epsilon)\right)\,\equiv\,\widehat{\Gamma}_{n}^{\rm\,corr.}\Big{(}z_{ij},\alpha_{s}(\lambda,\epsilon)\Big{)}\,+\,\widehat{\Gamma}_{n}^{\rm\,singl.}\bigg{(}\frac{\omega_{i}}{\lambda},\alpha_{s}(\lambda,\epsilon)\bigg{)}\,,$
(19)

where we defined $z_{ij}\equiv z_{i}-z_{j}$. The colour-singlet contribution
is given by

$\displaystyle\widehat{\Gamma}_{n}^{\rm\,singl.}\bigg{(}\frac{\omega_{i}}{\lambda},\alpha_{s}(\lambda,\epsilon)\bigg{)}\,=\,-\,\sum_{i=1}^{n}\gamma_{i}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\,\,-\,\frac{1}{4}\,\widehat{\gamma}_{K}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\sum_{i=1}^{n}\,\ln\left(\frac{-4\hskip
1.0pt\omega_{i}^{2}+{\rm i}\eta}{\lambda^{2}}\right)C_{i}^{(2)}\,,$ (20)

while

$\displaystyle\widehat{\Gamma}_{n}^{\rm\,corr.}\Big{(}z_{ij},\alpha_{s}(\lambda,\epsilon)\Big{)}\,=\,\frac{1}{2}\,\widehat{\gamma}_{K}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\sum_{i=1}^{n}\sum_{j=i+1}^{n}\ln\Big{(}\big{|}z_{ij}|^{2}\Big{)}\,{\bf
T}_{i}\cdot{\bf T}_{j}\,.$ (21)

The remarkable fact about Eq. (21) is that the scale dependence is factorised,
universal, and free from colour correlations. This enables us to write the
colour-correlated part of the infrared operator ${\cal Z}_{n}$ in a strikingly
simple way, as

$\displaystyle{\cal
Z}_{n}^{\rm\,corr.}\Big{(}z_{ij},\alpha_{s}(\mu),\epsilon\Big{)}$
$\displaystyle\equiv$
$\displaystyle\exp\left[\int_{0}^{\mu}\frac{d\lambda}{\lambda}\,\,\widehat{\Gamma}_{n}^{\rm\,corr.}\Big{(}z_{ij},\alpha_{s}(\lambda,\epsilon)\Big{)}\right]$
(22) $\displaystyle=$
$\displaystyle\exp\bigg{[}-K\big{(}\alpha_{s}(\mu),\epsilon\big{)}\,\sum_{i=1}^{n}\sum_{j=i+1}^{n}\ln\Big{(}\big{|}z_{ij}|^{2}\Big{)}\,{\bf
T}_{i}\cdot{\bf T}_{j}\bigg{]}\,,$

where the universal scale-dependent prefactor is given by

$\displaystyle
K\big{(}\alpha_{s}(\mu),\epsilon\big{)}\,=\,-\frac{1}{2}\int_{0}^{\mu}\frac{d\lambda}{\lambda}\,\,\widehat{\gamma}_{K}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\,.$
(23)

The function $K$ is well-known in QCD, where it plays a role in a number of
different settings: importantly, it defines the perturbative Regge trajectory
in the high-energy limit of $2\to 2$ scattering amplitudes Korchemsky:1993hr ;
Korchemskaya:1994qp ; Korchemskaya:1996je ; DelDuca:2011xm ; DelDuca:2011ae ,
and it computes soft-collinear poles for Sudakov form factors Magnea:1990zb ;
Magnea:2000ss . In a gauge theory with conformal symmetry in $d=4$ one finds

$\displaystyle
K\big{(}\alpha_{s},\epsilon\big{)}\,=\,\sum_{n=1}^{\infty}\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{\\!n}\,\frac{\widehat{\gamma}_{K}^{(n)}}{4n\epsilon}\,,$
(24)

where $\widehat{\gamma}_{K}^{(n)}$ are the coefficients of the expansion of
the function $\widehat{\gamma}_{K}(\alpha_{s})$ in powers of $\alpha_{s}/\pi$.
With a non-vanishing $\beta$ function one finds instead DelDuca:2014cya

$\displaystyle K\big{(}\alpha_{s},\epsilon\big{)}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{\pi}\frac{\widehat{\gamma}_{K}^{(1)}}{4\epsilon}+\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{\\!2}\Bigg{(}\frac{\widehat{\gamma}_{K}^{(2)}}{8\epsilon}+\frac{b_{0}\widehat{\gamma}_{K}^{(1)}}{32\epsilon^{2}}\Bigg{)}$
(25) $\displaystyle\hskip
34.14322pt+\,\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{\\!3}\Bigg{(}\frac{\widehat{\gamma}_{K}^{(3)}}{12\epsilon}+\frac{b_{0}\widehat{\gamma}_{K}^{(2)}+b_{1}\widehat{\gamma}_{K}^{(1)}}{48\epsilon^{2}}+\frac{b_{0}^{2}\,\widehat{\gamma}_{K}^{(1)}}{192\epsilon^{3}}\Bigg{)}+{\cal
O}\Big{(}\alpha_{s}^{4}\Big{)}\,,$

and all the higher-order coefficients can be explicitly determined
Magnea:2000ss  in terms of $b_{n}$ and $\widehat{\gamma}_{K}^{(n)}$. A simple
factorised form for the infrared operator ${\cal Z}_{n}$ in terms of the
function $K$ had previously been observed in the high-energy limit
Korchemskaya:1994qp ; Korchemskaya:1996je ; DelDuca:2011xm ; DelDuca:2011ae ,
but the complete generality of Eq. (22) only becomes apparent in the celestial
coordinates of Eq. (16).

As noted in Ref. Kalyanapuram:2020epb  in the abelian case (where the
operators ${\bf T}_{i}$ are replaced by the electric charges $e_{i}$, and the
one-loop approximation to the cusp anomalous dimension suffices), Eq. (22)
bears a striking resemblance to a correlator of vertex operators in a $d=2$
conformal theory. In the non-abelian case, we need to handle the matrix
structure of Eq. (22), but the eventual connection to conformal invariance is
actually strengthened by the knowledge that high-order corrections depend only
on conformal cross-ratios, which in this language read
$\rho_{ijkl}=|z_{ij}|^{2}|z_{kl}|^{2}/\big{(}|z_{il}|^{2}|z_{kj}|^{2}\big{)}$.
In what follows, we will make a proposal for the conformal theory generating
Eq. (22), and speculate about the origin of higher-order corrections.

##  4 A theory of Lie-algebra-valued free bosons

We consider a set of scalar fields on a two-dimensional sphere,
$\phi_{a}(z,\bar{z})$, forming a multiplet in the adjoint representation of a
Lie algebra, which we will take to be $su(N_{c})$, so that
$a=1,\ldots,N_{c}^{2}-1$. For these fields one can naturally choose the action

$\displaystyle S(\phi)\,=\,\frac{1}{2\pi}\int
d^{2}z\,\partial_{z}\phi^{a}(z,\bar{z})\,\partial_{\bar{z}}\phi_{a}(z,\bar{z})\,,$
(26)

so that we have a theory of Lie-algebra-valued free bosons. Since we will have
to pair these free bosons with colour operators, one may also note from the
outset that the scalars could be naturally organised into a matrix field

$\displaystyle\Phi_{r}(z,\bar{z})\,\equiv\,\phi_{a}(z,\bar{z})\,T^{a}_{r,\,z}\,,$
(27)

where $T^{a}_{r,\,z}$ are generators of the group in the irreducible
representation $r$. Considering the gauge-theory structure we are trying to
mimick, the generators must be associated with the point $z$ on the sphere
where the fields are evaluated: in particular, this means that the colour-
space commutator
$\big{[}\Phi(z_{1},\bar{z}_{1}),\Phi(z_{2},\bar{z}_{2})\big{]}$ must vanish,
so long as $z_{1}\neq z_{2}$. Products of matrix fields at different points
are operators on the tensor product of the corresponding irreducible
representations, and indeed one is free to consider products of matrix fields
belonging to different representations. In terms of these matrix fields, one
can write the action in Eq. (26) as

$\displaystyle S(\Phi)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
t_{r}}\int d^{2}z\,{\rm
Tr}\Big{[}\partial_{z}\Phi_{r}(z,\bar{z})\,\partial_{\bar{z}}\,\Phi_{r}(z,\bar{z})\Big{]}\,,$
(28)

where we normalised the generators according to ${\rm
Tr}\left(T_{r}^{a}T_{r}^{b}\right)\,=\,t_{r}\delta^{ab}$. Writing the action
in the form of Eq. (28), with matrices $T^{a}_{r,\,z}$ that are taken to
depend on $z$, suggests that the proper underlying mathematical structure
should be a gauge bundle on the sphere, and one might then expect that the
derivative should be replaced by a covariant derivative. Such an extension is
non-trivial for at least two reasons: first of all, one would need a precise
form for the gauge connection on the celestial sphere; next, we emphasise that
the collinear limits where two punctures $z_{i}$ and $z_{j}$ coincide is
singular in our framework: indeed, Eq. (2), which was our starting point, was
derived for fixed-angle scattering amplitudes, where none of the invariants
$s_{ij}$ vanishes. In any case, we expect that corrections to Eq. (28) arising
from this issue would be of higher order in the gauge coupling, and
furthermore they would contribute only to colour structures beyond dipoles,
involving the structure constants $f_{abc}$. The free action in Eq. (26) will
therefore suffice for our present purposes, and we will take it as the
definition of the theory. We will briefly come back to this issue in Section
5.

It is clear that in this setup the nature of the fields $\phi^{a}(z,\bar{z})$
as fields in a conformal theory on the sphere is quite decoupled from the
matrix structure that we have superimposed. Indeed, Eq. (26) is essentially
the bosonic string action in a conformal gauge, with scalar fields interpreted
as coordinates in a Lie algebra rather than a spacetime – in a sense, it is a
colour-kinematic dual Bern:2008qj  of the tree-level bosonic string. Most of
the well-known formalism for the treatment of two-dimensional free scalar
fields Ginsparg:1988ui  is then simply inherited by our theory with only minor
changes. In particular, the equations of motion are

$\displaystyle\partial_{z}\,\partial_{\bar{z}}\,\phi^{a}(z,\bar{z})\,=\,0\,,$
(29)

implying that the field $\partial_{z}\phi^{a}$ is holomorphic, while
$\partial_{\bar{z}}\phi^{a}$ is anti-holomorphic. In the quantum theory, when
taking matrix elements of products of fields, Eq. (29) leads to

$\displaystyle\partial_{z}\,\partial_{\bar{z}}\,\phi^{a}(z,\bar{z})\,\phi^{b}(w,\overline{w})\,=\,-\,\pi\,\delta^{ab}\delta^{2}(z-w,\bar{z}-\overline{w})\,,$
(30)

which suggests the definition of the normal-ordered product

$\displaystyle:\\!\phi^{a}(z,\bar{z})\,\phi^{b}(w,\overline{w})\\!:\,\,=\,\phi^{a}(z,\bar{z})\,\phi^{b}(w,\overline{w})+\frac{1}{2}\,\delta^{ab}\log\left|z-w\right|^{2}\,,$
(31)

whose derivatives are free of contact terms. Following standard arguments, the
theory defined by Eq. (26) has a traceless conserved energy momentum tensor,
which can be organised into holomorphic and anti-holomorphic components, with

$\displaystyle
T(z)\,=\,-\,:\\!\partial_{z}\phi^{a}(z,\bar{z})\,\partial_{z}\phi_{a}(z,\bar{z})\\!:\,,\qquad\widetilde{T}(\bar{z})\,=\,-\,:\\!\partial_{\bar{z}}\phi^{a}(z,\bar{z})\,\partial_{\bar{z}}\phi_{a}(z,\bar{z})\\!:\,.$
(32)

More interestingly, since the action in Eq. (26) involves only derivatives of
$\phi^{a}$, there is a symmetry under translations in field space, which are
now to be interpreted as translations in the Lie algebra. The Noether current
for this symmetry is Lie-algebra valued, and we will propose that it should be
interpreted as a leading-order contribution to a Kac-Moody current for the
full theory, to be identified with the current constructed in
Strominger:2013lka ; He:2015zea . Its holomorphic and anti-holomorphic
components are simply given by the derivatives of the scalar fields

$\displaystyle
j^{a}(z)\,=\,\partial_{z}\phi^{a}(z,\bar{z})\,,\qquad\tilde{j}^{a}(\bar{z})\,=\,\partial_{\bar{z}}\phi^{a}(z,\bar{z})\,.$
(33)

Following the standard lore for free bosons, we may now introduce vertex
operators for this theory, in the form

$\displaystyle V(z,\bar{z})\,\equiv\,:\,{\rm e}^{{\rm i}\kappa\,{\bf
T}_{\\!z}\cdot\,{\bf\phi}(z,\bar{z})}:\,\,=\,\,:\,{\rm e}^{{\rm
i}\kappa\Phi(z,\bar{z})}:\,,$ (34)

where we dropped the representation label $r$, and we introduced the operator
notation discussed in Section 2. The vertex operators in Eq. (34) are matrices
in the selected representation of the gauge group: one can think of them as
operators defined on the celestial sphere and acting on the bulk colour
degrees of freedom. We have introduced a coupling $\kappa$ in the exponent,
allowing for the fact that the normalisation of the fields at this stage is
arbitrary, and we will need to match our results to the gauge theory. The
vertex operators in Eq. (34) are reminiscent of those used in the vertex-
operator construction of Kac-Moody algebras Goddard:1986bp , but we emphasise
that they are quite different. In the Kac-Moody construction, one works in the
Cartan-Weyl basis for the Lie algebra, and one introduces a (Fubini-Veneziano)
scalar field $Q_{i}(z,\bar{z})$ only for generators $H_{i}$ in the Cartan
subalgebra; one then builds a vertex operator for every root $\alpha$ of the
algebra, with an exponent proportional to $\alpha^{i}Q_{i}$. These operators
build a representation of the Kac-Moody algebra in the Hilbert space of the
conformal theory, but they act as numbers in colour space. In our case, on the
other hand, in order to reproduce the gauge theory results, we need an
expression treating all colour degrees of freedom on the same footing, which
results in colour matrices. Pursuing an alternative analogy, one may note that
a standard construction of Kac-Moody algebras from ordinary Lie algebras
involves promoting the parameters $\theta^{a}$ of the Lie algebra to functions
$\theta^{a}(z)$ on a circle $S^{1}$, and taking Fourier modes: in a sense, Eq.
(34) is similar, with the circle replaced by the sphere $S^{2}$. Finally, we
must note a formal analogy between the fields $\phi^{a}$ in Eq. (34) and the
interpolating fields for Reggeized gluons introduced in Ref. Caron-
Huot:2013fea , where however the gauge-theory Wilson lines are not semi-
infinite but infinite, and the sphere is replaced by the transverse plane. For
the purposes of the present paper, we will simply take Eq. (34) at face value,
and derive its properties.

With the operators in Eq. (34), we can construct correlation functions which
bear a direct analogy with the world-sheet integrands of tree-level bosonic
string amplitudes. There are a number of potential obstacles for the
consistency of this procedure, that we will now consider, but, interestingly,
they can all be bypassed using well-understood properties of the operators
${\bf T}_{\\!z}$.

  * •

First of all, we need to establish that the vertex operator in Eq. (34) is a
good conformal field. This appears far from obvious given its matrix
structure. Once again, following standard procedures, one can write down an
explicit solution of the equations of motions for the fields
$\phi_{a}(z,\bar{z})$ in terms of a Fourier expansion, and proceed to quantise
the theory by interpreting the coefficients of the Fourier modes as (rescaled)
harmonic-oscillator creation and annihilation operators. In this formalism,
the conformal dimension $h\,(=\bar{h})$ of the vertex operator emerges
entirely from the normal ordering, bringing annihilation operators to the left
and creation operators to the right. The colour operators ${\bf T}_{\\!z}$ are
spectators in this calculation, and the standard free-field (‘bosonic string’)
result is reproduced. Pursuing the analogy, we recall that in a string-theory
context one finds

$\displaystyle V_{\rm c.s.}(z,\bar{z})\,\equiv\,:\,{\rm e}^{{\rm
i}k^{\mu}X_{\mu}(z,\bar{z})}:\qquad\longrightarrow\qquad
h\,=\,\frac{1}{4}\,k^{\mu}k^{\nu}\eta_{\mu\nu}\,=\,\frac{k^{2}}{4}\,,$ (35)

where the space-time metric $\eta_{\mu\nu}$ comes from the interpretation of
the fields $X^{\mu}$ as space-time coordinates. In the case of Eq. (34), the
fields $\phi_{a}$ are Lie-algebra coordinates, and the metric is simply the
Cartan-Killing metric $\delta_{ab}$. Therefore

$\displaystyle V(z,\bar{z})\,\equiv\,:\,{\rm e}^{{\rm i}\kappa\,{\bf
T}_{\\!z}\cdot\,{\bf\phi}(z,\bar{z})}:\qquad\longrightarrow\qquad
h\,=\,\frac{\kappa^{2}}{4}\,{\bf T}_{\\!z}\cdot{\bf
T}_{\\!z}\,=\,\frac{\kappa^{2}}{4}\,C_{r}^{(2)}\,,$ (36)

where $C_{r}^{(2)}$ is the quadratic Casimir eigenvalue for representation
$r$, which, crucially, is a number. In a conformal theory, this result must be
consistent with the scaling properties of two-point functions of vertex
operators, which must satisfy

$\displaystyle\big{\langle}V(z_{1},\bar{z}_{1})V(z_{2},\bar{z}_{2})\big{\rangle}\,\sim\,|z_{12}|^{-2\Delta}\,,$
(37)

with $\Delta=h+\bar{h}$. As we will see below, this is indeed the case, upon
enforcing colour conservation, which for the two-point function requires ${\bf
T}_{1}=-{\bf T}_{2}$.

  * •

Of course, the analogy with strings should not be pushed too far: for example,
we note that, in the case of a string theory, the value of $k^{2}$ in Eq. (35)
is fixed, and allowed values for different vertex operators give the mass
spectrum of the string. The origin of the constraint, however, is the fact
that string-theory vertex operators relevant to string amplitudes are
integrated over the world sheet, to preserve diffeomorphism invariance.
Invariance under rescalings of the world-sheet coordinates then imposes $h=1$
for the (‘tachyon’) vertex operator we are considering, and integer values of
$h$ for vertex operators involving prefactors with derivatives of the fields.
In our case, we will never need to integrate over the locations of the
punctures, which represent the (fixed) momenta of the hard particles, so there
are no constraints of this kind on $h$. We are genuinely interested in the
conformal correlation function in a fixed coordinate system.

  * •

We will be evaluating correlation functions of $n$ vertex operators of the
form of Eq. (34). In the case of bosonic strings, arguments from
holomorphicity, or just the evaluation of such correlators from the path
integral on the sphere Polchinski:1998rq , show that the results are
consistent and scale-invariant only if momentum is conserved,
$\sum_{i}k_{i}^{\mu}=0$. For string amplitudes, this has an obvious physical
interpretation. In the present case, colour conservation takes the place of
momentum conservation, as noted in Section 2: the condition $\sum_{i}{\bf
T}_{\\!z_{i}}=0$, which must be satisfied when the correlator is regarded as
an operator acting on the ‘bulk’ gauge theory, guarantees the proper scale-
invariant behaviour. Correlators will be operators with support only on
colour-conserving quantities, consistently with the gauge-invariance of the
four-dimensional theory.

Having established that we have a bona fide conformal field theory on the
sphere, and well-behaved – though matrix-valued – vertex operators, the
obvious next step is to study the correlator

$\displaystyle{\cal
C}_{n}\Big{(}\\{z_{i}\\},\kappa\Big{)}\,\equiv\,\Big{\langle}\prod_{i=1}^{n}V(z_{i},\bar{z}_{i})\Big{\rangle}\,.$
(38)

The evaluation of the correlator ${\cal C}_{n}$ is a textbook exercise (see
for example Chapter 6 in Ref. Polchinski:1998rq ): in a conformal invariant
theory, one is always allowed to use a locally flat metric (as we are doing),
and one simply gets an expression analogous to the integrand of the Virasoro-
Shapiro amplitude for tree-level closed strings. We find

$\displaystyle{\cal
C}_{n}\Big{(}\\{z_{i}\\},\kappa\Big{)}\,=\,C(N_{c})\,\exp\left[\frac{\kappa^{2}}{2}\sum_{i=1}^{n}\sum_{j=i+1}^{n}\ln\Big{(}\big{|}z_{ij}|^{2}\Big{)}\,{\bf
T}_{i}\cdot{\bf T}_{j}\right]\,,$ (39)

where $C(N_{c})$ is a constant dependent on the Lie algebra. This is of course
the result we have been working to achieve: it is precisely of the form of Eq.
(22), which in turn generates all infrared singularities with a colour-dipole
structure for any massless gauge theory and to any perturbative order. It is
therefore tempting at this stage to simply state that we can set the coupling
$\kappa$ to reproduce Eq. (22) exactly. This is certainly possible, but we
note at the outset that this identification raises interesting dynamical
questions, concerning both the scale dependence of the answer, and dependence
on the dimensional regulator $\epsilon$. We discuss this issue and some of the
many important open questions in Section 5. A further observation on Eq. (39)
is warranted: if one computes the correlator in Eq. (38) by means of a path
integral on a curved two-dimensional surface, as done for example in Ref.
Polchinski:1998rq , Eq. (39) is modified by an extra factor arising from the
Weyl anomaly. This factor cancels in the conformal theory, but would be
present, for example, if one were to break scale invariance by considering a
fixed sphere of finite radius $R$, thus introducing a length scale. The Weyl
factor for the correlator in Eq. (38) takes the form

$\displaystyle{\cal
W}_{n}\Big{(}\\{z_{i}\\},\kappa\Big{)}\,=\,\exp\bigg{[}\\!-\frac{1}{2}\,\sum_{i=1}^{n}C_{i}^{(2)}g(z_{i},\bar{z}_{i})\bigg{]}\,,$
(40)

where $g(z_{i},\bar{z}_{i})$ is a scale factor. The interesting point about
Eq. (40) is that the Weyl factor matches the factor arising in the gauge
theory from the energy-sensitive terms that we have shuffled to the colour-
singlet part of the anomalous dimension, $\widehat{\Gamma}_{n}^{\rm\,singl.}$
in Eq. (20). Such ‘diagonal’ contributions to the infrared operator ${\cal
Z}_{n}$, with an exponent proportional to a sum of terms associated with
single points $z_{i}$, cannot be expected to emerge from the celestial CFT: on
the gauge-theory side, they are related to collinear divergences, and do not
contribute to colour correlations, but they are sensitive to particle
energies.

There are further significant tests of the correspondence we have found
between the infrared sector of non-abelian gauge theories and the proposed
conformal theory on the celestial sphere, arising from soft and collinear
limits. On the gauge-theory side, it is well-known that the tree-level
emission of a soft gluon factorises Bassetto:1984ik  from hard $n$-point
scattering amplitudes (and similarly from correlators of $n$ Wilson-lines such
as ${\cal S}_{n}$ in Eq. (14)), and the soft factor is given by the soft
current for the emission of a gluon of momentum $k$

$\displaystyle{\bf J}^{\mu}(k)\,=\,g\,\sum_{i=1}^{n}{\bf
T}_{i}\,\frac{\beta_{i}^{\mu}}{\beta_{i}\cdot k}\,,$ (41)

exposing the singular behaviour of the radiative amplitude as the gluon energy
vanishes, as well as a set of collinear singularities as $k$ becomes parallel
to $\beta_{i}$. The current is gauge-invariant, in the sense that its
longitudinal component vanishes, since

$\displaystyle k\cdot{\bf J}^{\mu}(k)\,=\,g\,\sum_{i=1}^{n}{\bf
T}_{i}\,=\,0\,,$ (42)

by colour conservation. Remarkably, this factorisation theorem for soft non-
abelian radiation can be derived in the conformal theory on the sphere, by
consider a generalised correlator of vertex operators, including an insertion
of the Noether current in Eq. (33). Taking the OPE of the current with the
vertex operators yields the result

$\displaystyle\Big{\langle}\partial_{z}\phi^{a}(z,\bar{z})\,\prod_{i=1}^{n}V(z_{i},\bar{z}_{i})\Big{\rangle}\,\simeq\,-\frac{\rm
i}{2}\sum_{i=1}^{n}\frac{{\bf T}_{i}^{\,a}}{z-z_{i}}\,{\cal
C}_{n}\Big{(}\\{z_{i}\\},\kappa\Big{)}\,.$ (43)

As discussed in detail in Ref. He:2015zea  (see also Refs. Strominger:2013lka
; Cheung:2016iub ; Nande:2017dba ; Fan:2019emx ), the r.h.s. of Eq. (43) is
the representation of the soft-gluon theorem Bassetto:1984ik  on the celestial
sphere, and the poles as $z\to z_{i}$ are collinear poles. As noted above, in
the gauge-theory this pole is superimposed to an angle-independent soft
singularity arising when the energy of the soft gluon becomes vanishingly
small; the soft singularity is absent in Eq. (43), which can be understood in
the present context by recalling that soft-collinear singularities are color-
uncorrelated, as shown in Section 2 for virtual corrections. Colour
conservation emerges in Eq. (43) from the request that the Noether current be
holomorphic as $z\to\infty$: the correlator in Eq. (43) must then vanish as
$z^{-2}$ for large $z$, which indeed requires that $\sum_{i}{\bf T}_{i}=0$.
Importantly, Ref. He:2015zea  finds the same result by employing a current
constructed in terms of asymptotic expressions for the four-dimensional
fields: this provides clear evidence that the two currents are different
representations of the same object, at least to the accuracy of the present
calculation. In the framework of Ref. He:2015zea , the soft pole is absent
because the current is constructed out of field strengths rather than gauge
potentials, providing an extra power of the energy in the numerator. We note
that one finds an equivalent result using the anti-holomorphic current in Eq.
(33): Ref He:2015zea , examining double soft emission, argues that one of the
two currents generates a Kac-Moody symmetry, but there are subtleties in
attempting to extend this to both currents, related to the ordering of soft
limits in the double-emission matrix element. An analysis of this ambiguity
goes beyond the approximation we have taken so far (of including only colour-
dipole correlations), since in QCD the difference between soft orderings is
proportional to colour correlations involving the structure constants
$f_{abc}$, but this is without doubt a very interesting topic for future work.

Next, we turn to collinear limits, when the momenta of two hard particles
become proportional, and consequently two of the punctures on the celestial
sphere become close. As we noted, strict collinear limits, where some of the
Mandelstam invariants $s_{ij}$ vanish, violate the factorisation theorem in
Eq. (2), and therefore lie outside the reach of our discussion222Tools more
suitable to handle collinear limits in this context could possibly be devised
by considering directly the light-ray operators in the $d=4$ theory. These
have a long history in the context of QCD (see, for example, Refs.
Balitsky:1987bk ; Braun:2003rp ), and have been intensively studied recently
in a CFT context (see, for example, Refs. Kravchuk:2018htv ; Kologlu:2019mfz
). Interestingly, phenomenological applications of these tools to QCD have
already stated to appear Chen:2020adz ; Chen:2021gdk .. We can, however, study
the approach to the collinear limit, when two particles $i$ and $j$ are nearly
aligned, so that the invariant $s_{ij}$ is much smaller than all other
invariants. In this limit, the factorisation in Eq. (2) correctly captures all
logarithms of the small invariant that are multiplied times infrared poles.
With this understanding, on the gauge theory side, collinear limits of Eq. (8)
are well understood, and tested up to three loops (where of course the
correction $\Delta_{n}$ must also be taken into account). On the celestial
side of the correspondence, collinear limits must correspond to short distance
limits, $|z_{ij}|\to 0$, and therefore can be probed by considering the OPE of
the vertex operators in Eq. (34). In order to clarify the correspondence, let
us begin by giving a precise definition of the collinear limit (see also Ref.
Ebert:2020nqf ). Following standard practices in QCD, in order to describe the
situation in which two light-like momenta $p_{1}$ and $p_{2}$ become collinear
along a light-like direction $p$, we introduce the Sudakov parametrisation

$\displaystyle p_{1}^{\mu}$ $\displaystyle=$ $\displaystyle
xp^{\mu}+p_{\perp}^{\mu}-\frac{p_{\perp}^{2}}{2xp\cdot n}\,n^{\mu}\,,$
$\displaystyle p_{2}^{\mu}$ $\displaystyle=$
$\displaystyle(1-x)p^{\mu}-p_{\perp}^{\mu}-\frac{p_{\perp}^{2}}{2(1-x)p\cdot
n}\,n^{\mu}\,,$ (44)

where $n^{\mu}$ is a reference light-like vector satisying $n^{2}=n\cdot
p_{\perp}=0$, and $p_{\perp}^{\mu}$ is a space-like vector orthogonal to the
collinear direction, $p\cdot p_{\perp}=0$. The collinear limit is parametrised
by taking $p_{\perp}^{2}\to 0$, and, in the limit, $p_{1}+p_{2}=p$. It is not
difficult to translate the Sudakov parametrisation to the celestial sphere:
using Eq. (16) for the momenta $p_{1}$ and $p_{2}$, and picking a fixed vector
$n^{\mu}$, for example, as

$\displaystyle
n^{\mu}\,=\,\frac{1}{2}\Big{\\{}1,0,0,-1\Big{\\}}\quad\rightarrow\quad n\cdot
p_{i}\,=\,\omega_{i}\,,\quad n\cdot p=\omega_{1}+\omega_{2}\equiv\omega\,,$
(45)

one may determine the celestial expression for $p^{\mu}$, with the result

$\displaystyle p^{\mu}\,=\,\omega\,\Big{\\{}1+z\bar{z},\,z+\bar{z},\,-{\rm
i}(z-\bar{z}),\,1-z\bar{z}\Big{\\}}\,,$ (46)

where $z=xz_{1}+(1-x)z_{2}$, and $\omega_{1}=x\omega$, while
$\omega_{2}=(1-x)\omega$. Furthermore, $p_{\perp}^{\mu}$ can be written as

$\displaystyle p_{\perp}^{\mu}$ $\displaystyle=$ $\displaystyle\omega
x(1-x)\,\Big{\\{}(1-2x)\big{(}z_{1}\bar{z}_{2}+\bar{z}_{1}z_{2})+2xz_{1}\bar{z}_{1}-2(1-x)z_{2}\bar{z}_{2},\,$
(47) $\displaystyle\hskip 8.53581ptz_{12}+\bar{z}_{12},\,-{\rm
i}\big{(}z_{12}-\bar{z}_{12}\big{)},\,-(1-2x)\big{(}z_{1}\bar{z}_{2}+\bar{z}_{1}z_{2})-2xz_{1}\bar{z}_{1}+2(1-x)z_{2}\bar{z}_{2}\Big{\\}}\,.$

The transverse momentum vector $p_{\perp}^{\mu}$ is antisymmetric under the
exchange of the labels 1$\leftrightarrow$2 (which implies $x\leftrightarrow
1-x$), as expected from Eq. (44), and one easily verifies that it satisfies

$\displaystyle p_{\perp}\cdot n\,=\,p_{\perp}\cdot p\,=\,0\,,\qquad\quad
p_{\perp}^{2}\,=\,-x(1-x)s_{12}\,=\,-4x(1-x)\omega_{1}\omega_{2}\big{|}z_{12}\big{|}^{2}\,,$
(48)

providing a precise connection between the short distance limit on the sphere
and the collinear limit in the bulk.

Let us now use the conformal OPE to study the short distance limit of the
correlator ${\cal C}_{n}$ as, say, $z_{1}\to z_{2}$. When the locations of the
two vertex operators are brought together one finds

$\displaystyle:\,{\rm e}^{{\rm i}\kappa\,{\bf
T}_{1}\cdot\,{\bf\phi}(z_{1},\bar{z}_{1})}:\,\,:\,{\rm e}^{{\rm i}\kappa\,{\bf
T}_{2}\cdot\,{\bf\phi}(z_{2},\bar{z}_{2})}:\,\,\,\sim\,\big{|}z_{12}\big{|}^{\kappa^{2}{\bf
T}_{1}\cdot{\bf T}_{2}}\,:\,{\rm e}^{{\rm i}\kappa\,\big{(}{\bf T}_{1}+{\bf
T}_{2}\big{)}\cdot\,{\bf\phi}(z,\bar{z})}:\,\,,$ (49)

where we placed the field on the r.h.s. of Eq. (49) at the point
$z=xz_{1}+(1-x)z_{2}$ associated with the collinear direction $p^{\mu}$, and
corrections are suppressed by powers of $z_{12}$. The $n$-point correlator
${\cal C}_{n}$ in this limit is thus re-expressed in terms of the
$(n-1)$-point correlator ${\cal C}_{n-1}$, where however point $z$ carries the
sum of the colour operators of two merging punctures. This is precisely what
one expects on the gauge theory side Catani:2003vu ; Becher:2009qa , where
this property is a consequence of collinear factorisation. The quantity of
interest is the splitting anomalous dimension, defined by

$\displaystyle\Gamma_{\rm
Sp.}\big{(}p_{1},p_{2}\big{)}\,\equiv\,\Gamma_{n}\big{(}p_{1},p_{2},\ldots,p_{n}\big{)}-\Gamma_{n-1}\big{(}p,p_{3},\ldots,p_{n}\big{)}\Big{|}_{{\bf
T}_{p}\to{\bf T}_{1}+{\bf T}_{2}}\,.$ (50)

The short distance limit on the celestial sphere, by means of Eq. (49),
identifies the form of the correlator that one needs to compute in order to
analyse the collinear limit in the gauge theory. To make the comparison
precise, one needs to extract $\widehat{\Gamma}_{n}^{\rm corr.}$ from ${\cal
C}_{n}$, and the appropriate form of $\widehat{\Gamma}_{n-1}^{\rm corr.}$,
with recombined colour factors, from ${\cal C}_{n-1}$. Next, one needs to
reinstate the dependence on the energies of individual hard particles, since
the collinear limit is sensitive to how the energy of the parent particle is
subdivided between the two particles forming the collinear pair. At that
point, the calculation mirrors exactly the gauge theory procedure: the precise
location of the point $z$ on the arc joining $z_{1}$ and $z_{2}$ is irrelevant
at leading power in $p_{\perp}$, and the result arises entirely from the
scaling of particle energies, $\omega_{1}=x\omega$ and
$\omega_{2}=(1-x)\omega$. One finds

$\displaystyle\Gamma_{\rm Sp.}\big{(}p_{1},p_{2}\big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\widehat{\gamma}_{K}(\alpha_{s})\bigg{[}\ln\bigg{(}\frac{-s_{12}+{\rm
i}\eta}{\mu^{2}}\bigg{)}{\bf T}_{1}\cdot{\bf T}_{2}-\ln x\,\,{\bf
T}_{1}\cdot\big{(}{\bf T}_{1}+\,{\bf T}_{2}\big{)}$ (51) $\displaystyle\hskip
85.35826pt-\,\ln(1-x)\,\,{\bf T}_{2}\cdot\big{(}{\bf T}_{1}+{\bf
T}_{2}\big{)}\bigg{]}\,,$

in agreement with Ref. Becher:2009qa . One may argue that the result in Eq.
(51) was to some extent implicit in our earlier identification of the soft
anomalous dimension matrix with the logarithm of the conformal correlator: we
note however that the emerging colour structure is correctly identified by the
OPE on the celestial sphere. The result is highly non-trivial in the gauge-
theory context: first of all, it embodies the collinear factorisation theorem,
stating that the collinear limit depends only on the particles participating
in the splitting, and colour correlations with the remaining $n-2$ non-
collinear particles vanish at leading power; furthermore, Eq. (51) is an all-
order result, collecting all contributions to $\Gamma_{\rm sp.}$ arising from
dipole colour correlations (interesting corrections do of course arise when
$\Delta_{n}$ is included).

##  5 Some open questions

It is clear that the construction that we have so far proposed represents only
a first step in the exploration of the emerging correspondence, and leaves
many open questions. In this section, we begin to discuss some of these
questions.

First of all, we note, as we did after Eq. (21), that the remarkable all-order
correspondence between the conformal correlator ${\cal C}_{n}$ on the
celestial sphere and the infrared operator ${\cal Z}_{n}^{\,\rm corr.}$ is
made possible by the striking factorisation of the coupling and scale
dependence from angular and colour variables in Eq. (22). This raises the
question of the nature of the correspondence between the coupling $\kappa$ in
the conformal theory and the function $K(\alpha_{s},\epsilon)$ in the gauge
theory. This question has two aspects, one concerning the choice of the gauge
coupling, and the other concerning the integrated scale dependence. With
regards to the coupling, we note that one can bypass the need for a
perturbative expansion of the function $\widehat{\gamma}_{K}(\alpha_{s})$ by
simply choosing a scheme in which $\widehat{\gamma}_{K}$ itself plays the role
of the coupling. The idea that the universal cusp anomalous dimension should
serve as a fundamental gauge coupling, at least in the infrared, is certainly
not new, and it has been raised in several contexts (see, for example, Refs.
Catani:1990rr ; Erdogan:2011yc ; Grozin:2015kna ; Banfi:2018mcq ;
Catani:2019rvy ), providing ample circumstantial evidence that such a choice
is physically sensible.

Having taken this option, one must still note that the celestial coupling
$\kappa$ does not directly correspond to the new gauge coupling,
$\widehat{\gamma}_{K}$, but rather to its ‘average’ over scales, $K$ in Eq.
(23). We believe that this aspect of the correspondence is dynamically
interesting, and deserves closer examination. In a sense, one could argue that
any matching between the coupling $\kappa$ on the celestial sphere and the
gauge theory must share some of the characteristics of $K$. After all,
correlation functions on the sphere are finite and scale-independent, while we
are attempting to reproduce infrared divergences in a scale-dependent theory:
regulator dependence and scale dependence must therefore reside in the
matching coefficient. From the gauge theory side, it is quite surprising that
such a matching is at all possible, since scale dependence is expected in
general to be entangled with colour and kinematics. Even in the case of
conformal gauge theories in $d=4$, the analysis of IR divergences requires
breaking conformal symmetry by the introduction of an appropriate
factorisation scale, much like what happens in the UV case: dimensional
regularisation to $d>4$ simply provides an elegant and computationally
efficient way to perform this symmetry breaking. Indeed, as we noted, all IR
divergences of scattering amplitudes, in any gauge theory, are generated by
the scale integration, which can be taken to extend to the IR region when
$d>4$, while all anomalous dimensions appearing in the scale integrand are
finite. That being said, it would be very interesting and appealing if the
scale integration could be made part of the correspondence, or, in other
words, if one could understand the meaning of the scale integration from the
point of view of the conformal theory on the celestial sphere. We note that,
from the gauge theory point of view, the IR operator ${\cal Z}_{n}$ acts on a
hard, finite coefficient function which is located at the common origin of the
Wilson lines in Eq. (14). Distance along the Wilson lines thus can be
considered as a proxy for the inverse of the factorisation scale, with
infrared scales ($\mu\to 0$) located at large distances ($\lambda\to\infty$ in
Eq. (15)), and ultraviolet scales located at short distances. The scale
integration in Eq. (22), in this picture, could be understood as an
integration over a scale factor for the celestial sphere, as it is brought
from infinite distance to distance $1/\mu$. At this stage, of course, these
considerations are purely speculative, and they could only be made precise
once a more detailed correspondence is constructed.

A second set of considerations must be devoted to the known corrections to Eq.
(22) arising in the gauge theory, i.e. the functions of conformal cross ratios
of kinematic invariants that arise starting at the three-loop level, for the
scattering of at least four hard particles, and build up the operator
$\Delta_{n}$. Clearly, these corrections cannot arise from the free-boson
theory that we have just described: the action in Eq. (26) must be
complemented with interaction terms. The nature of these interaction terms is
strongly constrained by our knowledge on the gauge theory side of the
correspondence. We note, for example, that the three-point correlator in our
free-boson theory is trivial, in the sense that it does not contain any colour
correlations: indeed, all ‘scalar’ products ${\bf T}_{i}\cdot{\bf T}_{j}$
reduce to Casimir eigenvalues, as a consequence of colour conservation, ${\bf
T}_{1}+{\bf T}_{2}+{\bf T}_{3}=0$, which implies that ${\bf T}_{1}\cdot{\bf
T}_{2}=(C^{(2)}_{3}-C^{(2)}_{1}-C^{(2)}_{2})/2$, and similarly for the other
two cyclic permutations333We note in passing that these colour identities
guarantee the proper scaling behaviour of three-point functions of vertex
operators, consistent with Eq. (37).. When more than three hard particles
participate in the scattering, three-point correlations can arise, and one
might expect terms of the form $f_{abc}{\bf T}_{1}^{a}{\bf T}_{2}^{b}{\bf
T}_{3}^{c}$ to appear in $\Gamma_{n}$: indeed, just such correlations do arise
at two loops in the scattering of massive hard particles Becher:2009kw ;
Ferroglia:2009ep ; Ferroglia:2009ii ; Mitov:2009sv ; Kidonakis:2009ev ;
Chien:2011wz . No such corrections are possible, however, in the massless
theory, essentially because conformal cross-ratios necessarily involve at
least four points. The analysis of Refs. Almelid:2015jia ; Almelid:2017qju
furthermore shows that three-point correlations do arise in $\Gamma_{n}$ for
$n>3$, but they are constrained to be independent of kinematics, and their
colour structure has the peculiar form

$\displaystyle f_{abe}f^{e}_{\,\,\,cd}\,\Big{\\{}{\bf T}_{i}^{a},{\bf
T}_{i}^{d}\Big{\\}}\,{\bf T}_{j}^{b}{\bf T}_{k}^{c}\,,$ (52)

with $i\neq j\neq k$; in particular, correlations of this form are necessary
to preserve collinear factorisation. The special form of three-point
correlations is very relevant to possible generalisations of Eq. (26), since
the bootstrap principle for two-dimensional conformal theories holds that
four-point functions of conformal fields are determined once the three-point
functions are given. In the case at hand, of course, the precise form of the
full correction to the four-point function at the three-loop level is also
known, involving the first ‘quadrupole’ correction to Eq. (22). Similar
considerations apply to the soft-gluon current in Eq. (43): in gauge theories,
the current has an all-order definition in terms of matrix element of Wilson
lines, with a gluon state replacing the vacuum state Magnea:2018ebr ; this
matrix element is known both at one Catani:2000pi  and two loops Badger:2004uk
; Duhr:2013msa ; Li:2013lsa , where it exhibits a non-trivial, purely non-
abelian colour structure. It is clear that this abundant and detailed
knowledge on the gauge-theory side of the correspondence will provide powerful
constraints and checks for any attempt to construct the completion of our
matrix-valued free-boson proposal.

##  6 Perspectives

The formulation of the infrared problem for gauge theories and gravity in
terms of asymptotic symmetries of the $S$-matrix, initiated in Refs.
Strominger:2013lka ; Strominger:2013jfa , provides a novel and interesting
viewpoint to investigate long-distance dynamics. One of the most intriguing
aspects of this approach is the emergence of a ‘holographic’ Cheung:2016iub
correspondence between certain dynamical properties of massless gauge theories
in $d=4$ Minkowski space and their asymptotic behaviour on the celestial
sphere at large light-cone times. What emerges is a re-interpretation of the
soft limit of gauge amplitudes in terms of a two-dimensional conformal theory
defined on the sphere: many elements of this conformal theory can be
constructed by taking appropriate limits of gauge-theory quantities, and
infrared factorisation properties of low-order perturbative gauge amplitudes
can be understood, and in fact generalised, in terms of conformal objects
defined on the sphere.

In this paper, inspired by the QED analysis in Refs. Kalyanapuram:2020epb , we
took a different viewpoint, starting from the all-order infrared factorisation
properties of massless gauge theory amplitudes, and attempting to find a
characterisation of the universal infrared operators emerging from the
factorisation in terms of a celestial conformal field theory. For simplicity,
we have focused on colour correlations of dipole form, which are the only ones
appearing up to three loops in the massless theory: we emphasise however that
our results apply to all perturbative orders within this set, and they include
both planar and non-planar correlations. A first strong indication that the
celestial sphere is an appropriate arena for the discussion of infrared
divergences came from the specific form of the exponentiation of kinematic
dependence in Eq. (22): indeed, using the parametrisation in Eq. (16) exposes
a complete factorisation of coupling and scale dependence from kinematic and
colour variables, that had not previously been noticed, except in the high-
energy limit. The simple and transparent expression in Eq. (22) provides a
strong suggestion for the identification of the appropriate celestial
conformal theory, generalising Kalyanapuram:2020epb  to the non-abelian
theory. We find that the all-order gauge-theory result in Eq. (22) is
reproduced by a celestial theory of Lie-algebra-valued free bosons, by
computing the correlator of $n$ vertex operators which are gauge-group
matrices in the representations of the hard particles participating in the
scattering. We have seen that the matrix nature of the vertex operators does
not prevent them from being interpreted as conformal fields of definite
weight, and we have seen that their correlators have good conformal properties
as a consequence of the gauge invariance of the bulk gauge theory. Like ${\cal
Z}_{n}^{\rm\,corr.}$ in Eq. (22), the conformal correlator must be interpreted
as a colour operator, defined on the celestial sphere, and acting on the bulk
colour degrees of freedom, which can be thought of as being located at the
origin.

Both the known gauge-theory results and the structure of our proposed
celestial theory point to the incompleteness of of the picture drawn so far.
On the celestial side, the fact that we have a local matrix theory suggests
that a gauge connection on the sphere might need to be introduced, and
consequently the theory is expected to become interacting. While we leave this
generalisation to future work, we have noted that known multipole colour
correlations arising in the gauge theory starting at the three-loop order will
pose stringent constraints on any extension of Eq. (26). Clearly, the next
very significant goal in this program would be to compute the three-loop value
of the operator $\Delta_{n}$, first determined in Ref. Almelid:2015jia , from
conformal field theory data. This could open the way to the exploration of
even higher-order correlations with novel, and possibly simpler, techniques,
not associated with gauge-theory Feynman diagrams. In particular, it would be
of great interest to understand, in the conformal context, colour correlations
associated with higher-order Casimir operators of the gauge algebra, which
have been neglected here, but are known to be present starting at four loops
Grozin:2017css ; Moch:2018wjh ; Henn:2019rmi ; Henn:2019swt ;
vonManteuffel:2020vjv ; Ahrens:2012qz ; Becher:2019avh , and whose role on the
gauge-theory side is not yet fully understood.

Aside from this first layer of generalisation, several other interesting
questions and possible developments will need to be explored. First of all,
the role of collinear singularities should be clarified: the conformal theory,
as written, determines only colour correlations, which originate exclusively
from soft gluon exchanges at wide angles. Collinear poles are ‘colour-singlet’
quantities, and are associated to single points on the sphere, and not to
arcs. On the other hand, the relatively simple structure of infrared
singularities embodied by Eq. (7) is specific to massless theories and thus it
is inextricably linked to the presence of collinear poles.

A related, and certainly more complex question is to what extent this
framework could be extended to massive hard particles: in a sense, one would
expect drastic complications, since the appropriate asymptotic configuration
space would no longer be the punctured Riemann sphere, but rather a three-
dimensional hyperbolic space; on the other hand, on the gauge theory side,
soft poles are still generated by correlators of semi-infinite Wilson lines,
which are scale-invariant even away from the light-cone, and the modifications
to the soft anomalous dimension for massive particles at two loops are
remarkably simple Becher:2009kw ; Ferroglia:2009ep ; Chien:2011wz . It is not
inconceivable that they might be understood in terms of a deformation of the
conformal theory defined in the massless case.

In a final speculative remark, we note that the basic gauge-theoretical
ingredients of infrared operators – form factors, matrix elements of Wilson
lines and anomalous dimensions – have definitions that apply in principle
beyond perturbation theory, and indeed, when strong-coupling data are
available, they have been shown to smoothly match onto the perturbative
definitions Alday:2007hr ; Alday:2009zf . If the dictionary mapping these
quantities to the celestial sphere can be fully understood, it is possible
that non-perturbative features of the four-dimensional theory could be gleaned
in the simpler context of the two-dimensional celestial framework. To this
end, a more precise matching between the ideas presented here and the well-
developed framework initiated in Strominger:2013lka ; Strominger:2013jfa  will
certainly be useful.

## Acknowledgements

This work was partially supported by the Italian Ministry of University and
Research (MIUR), grant PRIN 20172LNEEZ. I wholeheartedly thank Claude Duhr for
providing the initial motivation for this project, and for important and
enlightening discussions. I thank Michele Caselle for sharing his
understanding of conformal field theories and for several useful insights. I
thank Lance Dixon, Einan Gardi, Marco Meineri, and George Sterman for their
reading of the manuscript prior to publication, and for useful and significant
observations.

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