the infrared matrix, arising from the high-energy limit, were uncovered in
Refs. [423, 424, 425, 426, 427, 428]: the high-energy limit of the infrared
matrix is now known to all-orders at next-to-leading logarithmic (NLL)
accuracy, while NNLL contributions constrain the quartic Casimir component at
four loops.

Before moving on to illustrating techniques which are useful for explicit
computation of the infrared matrix in Section 5.3, we conclude this Section by
highlighting some interesting consequences of the dipole approximation, Eq.
(5.35), first in the high-energy limit, and then in the language of the
celestial approach to the infrared limit.

####  5.2.1 Taking the high-energy limit

Our discussion of scattering amplitudes so far has been limited by the fixed-
angle assumption, which in principle excludes very interesting and important
configurations, such as the cases of soft and collinear radiation (to be
discussed in Section 6), and the high-energy limit, where the total energy $s$
of the process is much larger than all $t$\- and $u$-channel Mandelstam
invariants involving both initial and final state particles. It is important
to note, however, that once we have achieved a prediction for infrared poles
for a generic configuration of momenta, we are free to approach (though not to
reach) singular limits, where certain Mandelstam invariants are much larger
than others. Near these limits, the corresponding logarithms will become
dominant, but the predictions concerning infrared poles will continue to
apply: indeed, selecting dominant terms in an expansion in external invariants
cannot generate new infrared singularities. The failure of the infrared
factorisation that we have described so far will, instead, manifest itself in
the fact that we will be unable to predict the infrared-finite parts of the
dominant logarithmic terms in the relevant limit. This simple consideration
opens the way for cross-fertilisation between factorisations, and
resummations, obtained in different limiting configurations. In what follows,
we will briefly summarise an example of such a cross-fertilisation, examining
how infrared factorisation constrains and complements the high-energy limit.

The high-energy, or Regge limit of gauge amplitudes has been the subject of a
vast body of studies over a span of several decades (see, for example, [263,
429], and, more recently, [430, 431]). For the sake of illustration, here we
will focus on the case of the four-gluon amplitude in a massless gauge theory,
although the results generalise both to quark amplitudes and to multi-particle
scattering. As with any renormalised massless four-point amplitude, the four-
gluon amplitude is a function of the Mandelstam invariants $s$, $t$ and $u$,
satisfying $s+t+u=0$, and of the renormalisation scale $\mu$. The high-energy
limit is defined, in the physical region, by taking $s\gg-t>0$. In this limit,
the amplitude is dominated by the exchange of a spin-1 gluon in the $t$
channel, and loop corrections generate large logarithms of the ratio $s/(-t)$.
Leading and next-to-leading logarithms can be resummed to all orders thanks to
the process of Reggeisation, which roughly amounts to the replacement of the
$t$-channel gluon propagator in the tree-level amplitude according to the rule
[432]

$\displaystyle\frac{1}{t}\,\,\to\,\,\frac{1}{t}\,\left(\frac{s}{-t}\right)^{\alpha(t)}\,,$
(5.43)

where $\alpha(t)$ is the Regge trajectory, which can be computed in
perturbation theory, and is characterised by the presence of infrared poles.
At one-loop, for example, one can write

$\displaystyle\alpha(t)\,=\,\left(\frac{\mu^{2}}{-t}\right)^{\\!\epsilon}\,\frac{\alpha_{s}(\mu^{2})}{\pi}\,\,\alpha^{(1)}+{\cal
O}\left(\alpha_{s}^{2}\right)\,,$ (5.44)

with $\alpha^{(1)}=C_{A}/(2\epsilon)$. At the $n$-loop order, $\alpha^{(n)}$
has poles up to $1/\epsilon^{n}$, as well as finite parts.

To be more precise, consider the scattering process $g(k_{1})+g(k_{2})\to
g(k_{3})+g(k_{4})$. At leading-logarithmic (LL) accuracy [433], and next-to-
leading logarithmic (NLL) accuracy for the real part of the amplitude [434],
one can write the factorised expression

$\displaystyle{\cal
A}_{4g}\left(\frac{s}{\mu^{2}},\frac{t}{\mu^{2}},\alpha_{s},\epsilon\right)$
$\displaystyle=$ $\displaystyle
4\pi\alpha_{s}\,\,\frac{s}{t}\,\,\bigg{[}\Big{(}{\bf
T}^{a}\Big{)}_{a_{1}a_{3}}C_{\lambda_{1}\lambda_{3}}(k_{1},k_{3})\bigg{]}$
(5.45)
$\displaystyle\times\left[\left(\frac{s}{-t}\right)^{\alpha(t)}+\left(\frac{-s}{-t}\right)^{\alpha(t)}\right]\bigg{[}\Big{(}{\bf
T}_{a}\Big{)}_{a_{2}a_{4}}C_{\lambda_{2}\lambda_{4}}(k_{2},k_{4})\bigg{]}\,,$

where $s=(k_{1}+k_{2})^{2}$, $t=(k_{1}-k_{3})^{2}$, $a_{i}$ and $\lambda_{i}$
are respectively the colour and polarisation labels of the gluons, and the
colour operators ${\bf T}^{a}_{bc}=-{\rm i}f^{a}_{bc}$ in the adjoint
representation. The functions $C_{\lambda_{i}\lambda_{j}}(k_{i},k_{j})$ depend
on gluon helicities, but not on the center-of-mass energy $\sqrt{s}$, and are
called impact factors: they describe the separate dynamical evolution of the
two colliding particles as they are scattered by a small angle through the
exchange of a reggeized gluon. The central factor in Eq. (5.45), which is
responsible for the $s$ dependence, takes into account the symmetry of the
amplitude under $s\leftrightarrow u$, and the fact that $u=-s$ at leading
power in the high-energy limit.

The fact that the Regge trajectory $\alpha(t)$ is infrared singular implies
that the coefficients of the high-energy logarithms $\log(s/t)$ will contain
infrared poles. These poles can be predicted by infrared factorisation: we
expect therefore that the high-energy limit of Eq. (5.33) will reflect the
structure of the factorisation in Eq. (5.45). Focusing on the dipole
contribution to the infrared anomalous dimension, Eq. (5.35), it is easy to
verify that this is indeed the case. At leading power in $t/s$, one finds that
the infrared operator in Eq. (5.33), in the case of the four-gluon amplitude,
factorises in the form [344, 343]

$\displaystyle{\cal
Z}_{4}\left(\frac{s}{\mu^{2}},\frac{t}{\mu^{2}},\alpha_{s}\right)\,=\,\exp\big{[}\\!-2\pi{\rm
i}\,K(\alpha_{s})\big{]}\,{\cal
Z}_{4,\,C}\left(\frac{t}{\mu^{2}},\alpha_{s}\right)\,\widetilde{\cal
Z}\left(\frac{s}{t},\alpha_{s}\right)+{\cal O}\left(\frac{t}{s}\right)\,,$
(5.46)

where $K(\alpha_{s})$ was defined in Eq. (4.30). The energy-independent factor
${\cal Z}_{4,\,C}$ in Eq. (5.46) is a product of four ‘jet’ factors,
associated with each one of the external particles, and is naturally
interpreted as a divergent contribution to the impact factors in Eq. (5.45).
It can be written as

$\displaystyle{\cal
Z}_{4,\,C}\left(\frac{t}{\mu^{2}},\alpha_{s}\right)\,=\,\exp\Bigg{\\{}2\Bigg{[}K(\alpha_{s})\log\left(\frac{-t}{\mu^{2}}\right)+D\left(\alpha_{s}\right)\Bigg{]}+4B_{g}\left(\alpha_{s}\right)\Bigg{\\}}\,,$
(5.47)

where we defined the functions

$\displaystyle D\left(\alpha_{s}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\,\gamma_{K}\big{(}\alpha_{s}(\lambda^{2})\big{)}\log\left(\frac{\mu^{2}}{\lambda^{2}}\right),$
$\displaystyle B_{g}\left(\alpha_{s}\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\,\gamma_{g}\big{(}\alpha_{s}(\lambda^{2})\big{)}\,,$
(5.48)

and $\gamma_{g}$ is the gluon jet anomalous dimension. Recalling Eq. (2.32),
one sees that the function $D(\alpha_{s})$ encodes double poles of soft-
collinear origin, which do not display colour correlations, as expected from
the general structure of infrared factorisation. Energy dependence and colour
correlations in the high-energy limit are confined to the factor
$\widetilde{\cal Z}$ in Eq. (5.46). In order to write it explicitly, it is
useful to define colour charges associated with exchanges in the $s$, $t$ and
$u$ channels, according to [388]

$\displaystyle{\bf T}_{s}$ $\displaystyle\equiv$ $\displaystyle{\bf
T}_{1}+{\bf T}_{2}\,=\,-\big{(}{\bf T}_{3}+{\bf T}_{4}\big{)}\,,$
$\displaystyle{\bf T}_{t}$ $\displaystyle\equiv$ $\displaystyle{\bf
T}_{1}+{\bf T}_{3}\,=\,-\big{(}{\bf T}_{3}+{\bf T}_{4}\big{)}\,,$
$\displaystyle{\bf T}_{u}$ $\displaystyle\equiv$ $\displaystyle{\bf
T}_{1}+{\bf T}_{4}\,=\,-\big{(}{\bf T}_{2}+{\bf T}_{3}\big{)}\,.$ (5.49)

Due to colour conservation, these charges obey a sum rule formally similar to
the sum of Mandelstam invariants, with masses replaced by quadratic Casimir
eigenvalues. One verifies that

$\displaystyle{\bf T}_{s}^{2}+{\bf T}_{t}^{2}+{\bf
T}_{u}^{2}\,=\,\sum_{i=1}^{4}C^{(2)}_{i}\,.$ (5.50)

In terms of these colour operators, one finds

$\displaystyle\widetilde{{\cal
Z}}\left(\frac{s}{t},\alpha_{s}\right)\,=\,\exp\left\\{\widehat{K}(\alpha_{s})\left[\log\left(\frac{s}{-t}\right){\bf
T}_{t}^{2}+i\pi{\bf T}_{s}^{2}\right]\right\\}\,,$ (5.51)

where $\widehat{K}(\alpha_{s})$ is the function in Eq. (4.30) with the
quadratic Casimir of the appropriate representation scaled out (in the present
case, $\widehat{K}(\alpha_{s})=K(\alpha_{s})/C_{A}$). Leading high-energy
logarithms accompanied by infrared poles are generated exclusively from the
first term in the exponent, and a direct matching with Eq. (5.45) yields an
expression for the divergent part of the Regge trajectory $\alpha(t)$. Indeed,
when the colour operator $\widetilde{\cal Z}$ acts on a tree-level exchange
dominated at leading power by $t$-channel octet exchange (as is the case for
gluon scattering), one can replace the colour operator ${\bf T}_{t}^{2}$ by
its eigenvalue $C_{A}$. One then finds that

$\displaystyle\alpha(t)\,=\,K\big{(}\alpha_{s}(-t),\epsilon\big{)}\,,$ (5.52)

where we naturally chose the renormalisation scale as $\mu^{2}=|t|$. Eq.
(5.52) readily reproduces Eq. (5.44) at one loop, and provides an all-order
expression for infrared poles at leading-logarithmic accuracy. Note that an
expression analogous to Eq. (5.52) was derived much earlier, in Refs. [340,
341, 342], with a different approach to the high-energy limit. Recognising
that high-energy scattering for $s\gg-t$ corresponds to small scattering
angles, the colliding particles are approximated with infinite (as opposed to
semi-infinite) Wilson lines, corresponding to the classical trajectories for
forward scattering, and separated by a fixed impact parameter in the
transverse plane. Arguments closer to the ones discussed here were also given
in Ref. [435].

In recent years, constraints linking the high-energy limit of scattering
amplitudes with infrared factorisation have been successfully exploited to
gain a number of important insights. For example, Refs. [436, 437] showed how
infrared information can be used to go beyond the (next-to-)leading
logarithmic approximation of Eq. (5.45), uncovering contributions that are not
described by simple gluon reggeisation, and in particular explaining the
breakdown of Eq. (5.45) for the real part of the amplitude at NLL accuracy,
first observed in Ref. [438]. The case of ${\cal N}=4$ Super-Yang-Mills theory
was considered in Refs. [439, 440], focusing in particular on an interesting
set of sub-leading colour contributions [441]. On the other hand, powerful
tools are available to study the high-energy limit of scattering amplitudes
with a much greater accuracy and generality than the approximation described
by Eq. (5.45). In particular, over the years, an effective theory for the
high-energy limit in terms of multiple parallel Wilson lines interacting at
finite impact parameters has been developed, leading to the Balitsky-JIMWLK
equation [442, 443, 444, 445, 446, 447, 448]. This framework was recast in
Ref. [422] in a formulation allowing for a much more direct comparison with
infrared factorisation: this made possible the first direct calculation of a
contribution to the soft anomalous dimension matrix going beyond the dipole
formula, at the four-loop level. A systematic development of the framework
proposed in [422] has led to the determination of the soft anomalous dimension
matrix for $2\to 2$ scattering amplitudes at NLL accuracy to all orders in
perturbation theory [423, 424, 425], and at NNLL accuracy up to four loops
[426, 427, 428]. This result, in turn, establishes for the first time the
presence of quartic Casimir contributions to the soft anomalous dimension
matrix, beyond those induced by the cusp anomalous dimension.

####  5.2.2 A celestial view

We conclude this Section by providing an example of how the novel approach to
the infrared limit introduced in Refs. [148, 147], and reviewed in Refs. [161,
449], can provide fresh and surprising insights on the physics and mathematics
of soft interactions. We will use the dipole formula, Eq. (5.35), as an
example. Following Ref. [164], we will begin by rewriting the dipole formula
in coordinates appropriate to the celestial sphere. We will see that this
formulation leads to a remarkable simplification of the infrared operator
${\cal Z}_{n}$, defined in Eq. (5.33), in the dipole approximation. We will
then observe that colour correlations, which originate exclusively from wide-
angle soft radiation, can be computed in terms of a simple two-dimensional
conformal field theory on the celestial sphere: a remarkable result, which
might well lead to further insights beyond the dipole approximation, and new
computational techniques.

As a first step, it is useful to disentangle the explicit dependence on the
running renormalisation scale $\lambda$ in the integrand in Eq. (5.33) from
colour correlations. To this end, note that the soft anomalous dimension
matrix, given by Eq. (5.35) in the dipole approximation, must be evaluated at
the running scale $\lambda$ when substituted in Eq. (5.33), where $\lambda$ is
the integration variable. It is then useful to rewrite the arguments of the
logarithms in Eq. (5.33) in terms of the fixed scale $\mu$, and then use
colour conservation, expressed by Eq. (5.19). This leads to

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

Eq. (5.2.2) reflects the fact that colour correlations arise only from the
exchange of soft, wide-angle gluons, and therefore they involve only single
soft poles, generated by the first line in Eq. (5.2.2). Collinear effects must
be free of colour correlations, as dictated by the form of the factorisation
in Eq. (5.5): this is reflected in the second line in Eq. (5.2.2), which is a
sum of contributions from each external particle, including double-pole
contributions arising from the last term.

The second step, which is natural in the approach of Refs. [148, 147], is to
parametrise particle momenta in celestial coordinates, using

$\displaystyle
p_{i}^{\mu}\,=\,\omega_{i}\,\Big{\\{}1+z_{i}\bar{z}_{i},\,z_{i}+\bar{z}_{i},\,-{\rm
i}\big{(}z_{i}-\bar{z}_{i}\big{)},\,1-z_{i}\bar{z}_{i}\Big{\\}}\,.$ (5.54)

Eq. (5.54) describes a massless momentum $p_{i}^{\mu}$ in terms of a light-
cone energy $\omega_{i}$ and a dimensionless complex coordinate $z_{i}$
identifying the direction of $p_{i}^{\mu}$. More precisely, note that the set
of light rays emanating from the origin in four-dimensional Minkowsky space is
in one-to-one correspondence with the points of a sphere ${\cal S}_{2}$. At
asymptotically large light-cone times, the sphere can be mapped to the complex
plane, and $z_{i}$ identifies the intersection of the light ray corresponding
to $p_{i}^{\mu}$ with this asymptotic celestial sphere. Importantly, the
Lorentz group $SO(1,3)$ acts on the celestial coordinates $z_{i}$ as
$SL(2,{\bf C})$, providing a first suggestion that the theory describing the
asymptotic dynamics could be a two-dimensional conformal field theory. For our
present purposes, an important immediate consequence of the parametrisation in
Eq. (5.54) is the expression for the Mandelstam invariants, which are given by

$\displaystyle s_{ij}\,=\,2p_{i}\cdot
p_{j}\,=\,4\omega_{i}\omega_{j}\big{|}z_{i}-z_{j}\big{|}^{2}\,.$ (5.55)

This implies that the logarithms appearing in the colour-correlated part of
Eq. (5.2.2) will decompose as

$\displaystyle\log\left(\frac{-s_{ij}+{\rm
i}\eta}{\mu^{2}}\right)\,=\,\log\Big{(}\big{|}z_{i}-z_{j}\big{|}^{2}\Big{)}+\log\left(\frac{\omega_{i}}{\mu}\right)+\log\left(\frac{\omega_{j}}{\mu}\right)+2\log
2+{\rm i}\pi\,.$ (5.56)

We can now neatly separate colour-correlated from colour-singlet contributions
to the soft anomalous dimension matrix, writing

$\displaystyle\Gamma_{n}^{\rm\,dip}\left(\frac{s_{ij}}{\lambda^{2}},\alpha_{s}(\lambda,\epsilon)\right)\,\equiv\,\Gamma_{n}^{\rm\,corr}\Big{(}z_{ij},\alpha_{s}(\lambda,\epsilon)\Big{)}\,+\,\Gamma_{n}^{\rm\,singl}\bigg{(}\frac{\omega_{i}}{\lambda},\alpha_{s}(\lambda,\epsilon)\bigg{)}\,,$
(5.57)

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

$\displaystyle\Gamma_{n}^{\rm\,singl}\bigg{(}\frac{\omega_{i}}{\lambda},\alpha_{s}(\lambda,\epsilon)\bigg{)}\,=\,-\,\sum_{i=1}^{n}\gamma_{i}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\,\,-\,\frac{1}{4}\,\widehat{\gamma}_{K}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\sum_{i=1}^{n}\,\ln\left(\frac{-4\hskip
1.0pt\omega_{i}^{2}+{\rm i}\eta}{\lambda^{2}}\right)C_{i}^{(2)}\,,$ (5.58)

while colour correlations are generated by

$\displaystyle\Gamma_{n}^{\rm\,corr}\Big{(}z_{ij},\alpha_{s}(\lambda,\epsilon)\Big{)}\,=\,\frac{1}{2}\,\widehat{\gamma}_{K}\big{(}\alpha_{s}(\lambda,\epsilon)\big{)}\sum_{i=1}^{n}\sum_{j=i+1}^{n}\ln\Big{(}\big{|}z_{ij}|^{2}\Big{)}\,{\bf
T}_{i}\cdot{\bf T}_{j}\,.$ (5.59)

Eq. (5.59) is remarkable, showing that the dependence on the running scale
$\lambda$, the coupling, and the infrared regulator is universal, and
completely factorised from colour correlations. As a consequence, one can
write the colour-correlated part of the infrared operator ${\cal Z}_{n}$ in a
strikingly simple way, as

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

The celestial parametrisation given in Eq. (5.54) thus shows that the
factorisation of infrared singularities in the universal function
$\widehat{K}$ is completely general, and not limited to the high-energy limit
of four-point amplitudes given in Eq. (5.51). This universal role of the
function $\widehat{K}$, which can be seen as the scale average of the light-
like cusp anomalous dimension in the infrared range, provides support for a
long-standing suggestion: that the light-like cusp should indeed be taken as a
proper definition of the gauge coupling in the infrared regime [450, 451, 359,
360, 452, 453].

Taking now a more directly celestial viewpoint, we note that the expression in
Eq. (5.60), regarded as a function on the punctured Riemann sphere, bears a
striking similarity to a correlator of primary fields in a two-dimensional
conformal field theory. This similarity was noticed, in the case of QED, in
[454], and, in greater detail, in [455]. Note however that in QED the infrared
operator ${\cal Z}_{n}$ is one-loop exact, whereas Eq. (5.60) organises highly
non-trivial all-order corrections in the non-abelian theory. In order to
explore this remarkable connection, following Ref. [164], consider a theory of
free bosons, spanning the adjoint representation of the gauge algebra, with
the action

$\displaystyle S[\phi]\,=\,\frac{1}{2\pi}\int
d^{2}z\,\partial_{z}\phi^{a}(z,\bar{z})\,\partial_{\bar{z}}\phi_{a}(z,\bar{z})\,,$
(5.61)

with $a=1,\ldots,N_{c}^{2}-1$ for $SU(N)$. As a conformal theory on the
sphere, the action in Eq. (5.61) is of course well-known [456], not least
because it maps to the action for the free bosonic string, if the adjoint
index $a$ is replaced by a space-time index $\mu$ (see, for example, [457]).
The theory is completely solvable: the energy-momentum tensor is traceless and
conserved, and furthermore the manifest global translation invariance of Eq.
(5.61) implies the existence of two conserved Noether currents, respectively
holomorphic and anti-holomorphic, given by

$\displaystyle
j^{a}(z)\,=\,\partial_{z}\phi^{a}(z,\bar{z})\,,\qquad\tilde{j}^{a}(\bar{z})\,=\,\partial_{\bar{z}}\phi^{a}(z,\bar{z})\,.$
(5.62)

By analogy with the bosonic string, and generalising the QED results of Refs.
[454, 455] to the non-abelian theory, one can introduce matrix-valued vertex
operators of the form

$\displaystyle V(z,\bar{z})\,\equiv\,:{\rm e}^{{\rm i}\kappa\,{\bf
T}_{\\!z}\cdot\,{\bf\phi}(z,\bar{z})}:\,\,,$ (5.63)

where the colon denotes normal ordering, $\kappa$ is a normalisation constant,
and ${\bf T}_{z}$ is a colour operator as defined in Section 5.1.2. Note that
colour operators are associated with points on the sphere: operators acting at
different points commute, as dictated by the structure of the gauge-theory
factorisation that we are trying to reproduce.

It is far from obvious that Eq. (5.63) defines a proper conformal primary
field, since $V(z,\bar{z})$ is a matrix in colour space, however it can be
readily verified that the vertex operator has a well-defined conformal weight,
given by $h=\kappa^{2}C_{r}^{(2)}/4>0$, where $C_{r}^{(2)}$ is the quadratic
Casimir eigenvalue of the representation chosen for the operator ${\bf
T}_{z}$. Similarly, one can verify that the two-point function of two vertex
operators has the appropriate power-law behaviour

$\displaystyle\big{\langle}V(z_{1},\bar{z}_{1})V(z_{2},\bar{z}_{2})\big{\rangle}\,\sim\,|z_{12}|^{-4h}\,.$
(5.64)

Computing correlators of vertex operators of the form of Eq. (5.63) in the
theory defined by Eq. (5.61) is a straighforward exercise, and one finds

$\displaystyle{\cal
C}_{n}\Big{(}\\{z_{i}\\},\kappa\Big{)}\,\equiv\,\Big{\langle}\prod_{i=1}^{n}V(z_{i},\bar{z}_{i})\Big{\rangle}\,=\,C(N_{c})\,\exp\left[\frac{\kappa^{2}}{2}\sum_{i=1}^{n}\sum_{j=i+1}^{n}\ln\Big{(}\big{|}z_{ij}|^{2}\Big{)}\,{\bf
T}_{i}\cdot{\bf T}_{j}\right]\,,$ (5.65)

where $C(N_{c})$ is an overall normalisation depending only on the dimension
of the gauge algebra. Manifestly, the correlator ${\cal C}_{n}$ reproduces Eq.
(5.60), with the identification
$\kappa^{2}=-2\widehat{K}(\alpha_{s},\epsilon)$.

Further checks of the power of the celestial approach are possible. One may
consider limits where one of the $n$ hard particles becomes soft, or where two
particles become collinear. Strictly speaking, these limits lie outside the
range of validity of the fixed-angle approximation, as was the case for the
high-energy limit discussed in Section 5.2.1. One may however study how these
limits are approached, and the predictions for infrared poles will remain
valid near the limits. We will discuss the factorisation properties of real
radiation in soft and collinear limits in greater detail in Section 6,
however, for completeness, we will describe here how some crucial aspects of
that factorisation emerge from the celestial viewpoint.

The first interesting case is the limit in which a single hard particle
becomes soft. At tree level, this limit is described by a simple
factorisation, discussed already in [137] for the abelian theory, and
presented here in Eq. (6.2) of Section 6. As expected from the general
reasoning of Ref. [147], this tree-level soft factorisation can be derived in
the celestial context as a Ward identity for the translation symmetry of the
action in Eq. (5.61). Indeed, computing a correlator of vertex operators with
the insertion of a holomorphic current, by means of the conformal OPE, yields
the result

$\displaystyle\Big{\langle}\partial_{z}\phi^{a}(z,\bar{z})\,\prod_{i=1}^{n}V(z_{i},\bar{z}_{i})\Big{\rangle}\,\simeq\,-\frac{\rm
i}{2}\sum_{i=1}^{n}\frac{{\bf T}_{i}^{\,a}}{z-z_{i}}\,{\cal
C}_{n}\Big{(}\\{z_{i}\\},\kappa\Big{)}\,,$ (5.66)

which reproduces Eq. (6.2), upon projecting on one of the two physical
polarisations for the soft gluon, with the tree-level soft current given by
Eq. (5.17). The poles as $z\to z_{i}$ are collinear poles, corresponding to
the singularities of the soft current as the soft momentum $k$ becomes
parallel to one of the hard momenta $p_{i}$. A Ward identity equivalent to Eq.
(5.66) was derived in Ref. [154] (see also Refs. [458, 454, 459]) using a
current built with asymptotic expressions for the gauge fields near null
infinity.

Collinear limits for the soft anomalous dimension matrix in the dipole
approximation were studied in Ref. [75], and are discussed here in Section 6.
In such limits, it is useful to construct a splitting anomalous dimension,
defined by

$\displaystyle\Gamma_{\rm
Sp}\big{(}p_{1},p_{2}\big{)}\,\equiv\,\Gamma_{n}\big{(}p_{1},p_{2},\ldots,p_{n}\big{)}-\Gamma_{n-1}\big{(}p,p_{3},\ldots,p_{n}\big{)}\Big{|}_{{\bf
T}_{p}\to{\bf T}_{1}+{\bf T}_{2}}\,,$ (5.67)

where $p_{1}$ and $p_{2}$ are the momenta becoming collinear, and $p$ is their
common collinear limit. The statement of collinear factorisation at amplitude
level313131The statement is strictly valid when all coloured particles are
outgoing [207]. is the fact that the splitting anomalous dimension
$\Gamma_{\rm Sp}$ depends only on the quantum numbers of the two collinear
particles. Using a precise definition of the collinear limit on the celestial
sphere [164], one may compute collinear limits of the celestial correlator
${\cal C}_{n}$ by means of the conformal OPE. Upon reinstating the energy
dependence that was factored in the singlet contribution, Eq. (5.58), one may
verify that collinear factorisation is built in the conformal correlation
function, and one recovers the known all-order expression for the splitting
anomalous dimension in the dipole approximation, first derived in Ref. [75],

$\displaystyle\Gamma_{\rm Sp.}\big{(}p_{1},p_{2}\big{)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\widehat{\gamma}_{K}(\alpha_{s})\bigg{[}\ln\bigg{(}\frac{-s_{12}+{\rm
i}\eta}{\mu^{2}}\bigg{)}{\bf T}_{1}\cdot{\bf T}_{2}-\ln x\,\,{\bf
T}_{1}\cdot\big{(}{\bf T}_{1}+\,{\bf T}_{2}\big{)}$ (5.68)
$\displaystyle\hskip 85.35826pt-\,\ln(1-x)\,\,{\bf T}_{2}\cdot\big{(}{\bf
T}_{1}+{\bf T}_{2}\big{)}\bigg{]}\,,$

where $x$ and $1-x$ are the energy fractions carried by the two collinear
particles $p_{1}$ and $p_{2}$, respectively.

The emerging connection between the infrared properties of massless non-
abelian gauge theories and two-dimensional conformal field theories is
certainly one of the most striking aspects of the approach pioneered in Refs.
[148, 147]. It can be more precisely formalised by taking a Mellin transform
of momentum-space scattering amplitudes with respect to light-cone energies,
which yields the so-called celestial amplitudes (see, for example, [460, 461,
462, 463]), which have simple transformation properties under the two-
dimensional conformal group. Within this approach, results similar, and in
part complementary, to the ones discussed in this Section, were obtained in
Ref. [464], and further extended in Ref. [465], studying special colour
configurations and the high-energy limit of four-point
amplitudes323232Furthermore, an intense research activity continues on the
general properties of celestial gauge amplitudes and their connections to the
underlying conformal theory: see, for example, Refs. [466, 467, 468, 469, 470,
471, 472, 473, 474, 475, 476, 477].. In principle, existing gauge-theory data
can be used to explore the non-linear generalisation of Eq. (5.61), which
would be needed in order to extend this analysis beyond the dipole
approximation. If this generalisation can be identified, conformal field
theory techniques may well provide powerful new tools for the study of
infrared factorisation, at high orders in perturbation theory, and possibly
beyond.

###  5.3 Computing the infrared anomalous dimension matrix

It should be clear from the arguments of Section 5.2 that the infrared
anomalous dimension matrix defined in Eq. (5.34) is the cornerstone for the
discussion of soft and collinear divergences in gauge theory scattering
amplitudes, as well as a crucial ingredient for the resummation of large
logarithms in many cross sections of phenomenological interest; furthermore,
it gives a perturbative window to explore the connection between colour
exchanges and kinematic configurations, providing insights that have
implications to all orders in perturbation theory. It is clear therefore that
it is very worthwhile to develop systematic tools to compute $\Gamma_{n}$ to
high orders in perturbation theory.

To this end, one may first note that the hard collinear components of the
infrared matrix, essentially contained in the jet anomalous dimensions
$\gamma_{i}$ in Eq. (5.35), are colour-diagonal, and can therefore be
extracted in a straightforward manner from form factor data. Non-trivial
colour structures, and the subtle correlations between colour and kinematics,
arise from the soft operator defined in Eq. (5.6), and this Section will
therefore focus on the properties of correlators of semi-infinite Wilson
lines, and on techniques to evaluate them.

As discussed already in Section 2.7 at the one-loop level, soft operators of
the form of Eq. (5.6) are highly singular, being affected by ultraviolet,
soft, and, in case $\beta_{i}^{2}=0$, collinear divergences. As a consequence,
special care is required to evaluate them [242, 478, 479, 480]. In Section
2.7, it was possible to exploit the simple form of the one-loop correction in
order disentangle the different singularities, and thus extract the relevant
UV counterterm, which still contains a collinear pole, as displayed in Eq.
(2.43). At higher orders, this simple-minded treatment becomes very difficult,
as different singular regions in the relevant integrals overlap in an
intricate way. It is then necessary to introduce different regulators in order
to safely extract the anomalous dimension from the correlator.

As a first step, it is natural to regulate collinear divergences (for massless
amplitudes) by setting $\beta_{i}^{2}\neq 0$, i.e. tilting the Wilson lines
off the light cone. This clearly has an intrinsic interest, since infrared
divergences arising from soft gluon exchanges affect also amplitudes involving
massive particles, and they are described precisely by this kind of soft
function. Results relevant for the massless case can then be extracted by
carefully taking the $\beta_{i}^{2}\to 0$ limit [77]. An immediate consequence
of the introduction of the collinear regulator is that all issues related to
collinear divergences, and in particular the anomalous dependence of the soft
function on the invariants $\beta_{i}\cdot\beta_{j}$, are eliminated. In the
massive case, the soft function can only depend on rescaling-invariant
variables, which can be expressed in terms of the Minkowskian angles

$\displaystyle\gamma_{ij}\,\equiv\,\frac{\beta_{i}\cdot\beta_{j}}{\sqrt{\beta_{i}^{2}\beta_{j}^{2}}}\,=\,\frac{p_{i}\cdot
p_{j}}{\sqrt{p_{i}^{2}p_{j}^{2}}}\,.$ (5.69)

Even for massive Wilson lines, it remains true that all loop corrections to
the soft function in Eq. (5.6), before renormalisation, vanish in dimensional
regularisation, since they are given by scale-less integrals. We must
therefore concentrate on the calculation of UV counterterms, as was done in
Section 2.7. To this end, it is useful to introduce an infrared (soft)
regulator, while retaining dimensional regularisation only for ultraviolet
singularities. The bare, infrared-regulated correlator will have the same UV
divergences as its unregulated counterpart: it can then be evaluated and
renormalised, yielding the desired answer.

An elegant way to proceed was proposed in Ref. [199, 242], and exploited for
explicit calculations up to 3 loops in Refs. [77, 480]. In this approach, one
introduces the infrared regulator at the level of individual Wilson lines,
forcing an exponential suppression of gluon emission at large distances by
means of the definition

$\displaystyle\Phi_{\beta_{i}}^{(m)}\,=\,{\cal P}\exp\left[\,{\rm
i}g\mu^{\epsilon}\int_{0}^{\infty}d\lambda\,\beta_{i}\cdot
A\left(\lambda\beta_{i}\right)\,{\rm
e}^{-m\lambda\sqrt{\beta_{i}^{2}}}\,\right]\,,$ (5.70)

and then defining the regulated soft function by

$\displaystyle{\cal
S}_{n}^{\,(m)}\left(\gamma_{ij},\alpha_{s}(\mu),\epsilon,\frac{m}{\mu}\right)\,\equiv\,\left\langle
0\left|\,T\left[\Phi_{\beta_{1}}^{(m)}\otimes\Phi_{\beta_{2}}^{(m)}\otimes\ldots\otimes\Phi_{\beta_{L}}^{(m)}\right]\right|0\right\rangle\,,$
(5.71)

so that one recovers the unregulated Wilson line as $m\to 0$. The crucial
advantage of the regulator defined in Eq. (5.70) is that it preserves the
rescaling invariance of the correlator under
$\beta_{i}\to\kappa_{i}\beta_{i}$, so that all the ensuing constraints on its
functional form also apply to the regulated definition. If one were to take
seriously the $m$ dependence, this kind of long-distance suppression would
raise issues of gauge invariance, however, in a minimal renormalisation
scheme, all dependence on $m$ cancels in the renormalised correlator, so the
proposed regularisation remains viable.

In order to proceed, we make use of the fact that Wilson line correlators are
multiplicatively renormalisable [244, 345, 346, 481, 347], which, in the
present case, means that we can define the renormalised correlator through the
matrix equation

$\displaystyle{\cal S}_{n}^{\,(\rm
ren.)}\left(\gamma_{ij},\alpha_{s}(\mu^{2}),\epsilon,\frac{m}{\mu}\right)\,=\,{\cal
S}_{n}^{\,(m)}\left(\gamma_{ij},\alpha_{s}(\mu^{2}),\epsilon,\frac{m}{\mu}\right)\,{\cal
Z}_{n}^{\,\cal S}\left(\gamma_{ij},\alpha_{s}(\mu^{2}),\epsilon\right)\,.$
(5.72)

We can now focus on the calculation of ${\cal Z}_{n}^{\cal S}$, which, as we
will verify, is independent of $m$. Indeed, as stated above, the UV
divergences of ${\cal S}_{n}^{\,(m)}$ coincide with those of the original soft
function; furthermore, as we discussed, in the limit $m\to 0$ the bare
correlator reduces to the unit matrix (since all loop corrections vanish in
dimensional regularisation), so that, in that limit, ${\cal Z}_{n}^{\cal S}$
coincides with the renormalised soft function that we wish to evaluate.

The renormalization matrix ${\cal Z}_{n}^{\cal S}$ satisfies a renormalisation
group equation of the same form as Eq. (5.20), which we write as

$\displaystyle\mu\frac{d}{d\mu}\,{\cal Z}_{n}^{\,\cal
S}\left(\gamma_{ij},\alpha_{s}(\mu^{2}),\epsilon\right)\,=\,-\,{\cal
Z}_{n}^{\,\cal
S}\left(\gamma_{ij},\alpha_{s}(\mu^{2}),\epsilon\right)\Gamma_{n}^{\,\cal
S}\left(\gamma_{ij},\alpha_{s}(\mu^{2})\right)\,,$ (5.73)

which defines the soft anomalous dimension matrix $\Gamma_{n}^{\,\cal S}$: it
is a finite matrix, independent of $\epsilon$, with a direct physical meaning
for the scattering of massive coloured particles, while in the massless case
it provides a regularised version of the singular matrix introduced in Eq.
(5.20); it compactly encodes the ultraviolet singularities of ${\cal
Z}_{n}^{\cal S}$, and thus of ${\cal S}_{n}$. We note that the ordering on the
r.h.s of Eq. (5.73) is important, since both ${\cal Z}_{N}^{\cal S}$ and
$\Gamma_{n}^{\,\cal S}$ are matrix valued, and therefore do not commute in
general. This has important consequences on the explicit form of the solution
of Eq. (5.73), which are worth discussing before getting into the details of
the evaluation.

The non-commutativity of the factors in Eq. (5.72) and in Eq. (5.73)
influences the calculation of the anomalous dimension matrix in two different
ways. To illustrate them, let us for the moment assume, as suggested by Eq.
(5.21) and Eq. (5.33), and as will be discussed in detail in Section 5.3.1,
that all factors in Eq. (5.72) can be written in exponential form, as

$\displaystyle{\cal S}_{n}^{\,(\rm ren.)}\,=\,\exp\left[W_{n}^{\,(\rm
ren.)}\right]\,,\qquad{\cal
S}_{n}^{\,(m)}\,=\,\exp\left[W_{n}^{\,(m)}\right]\,,\qquad{\cal Z}_{n}^{\,\cal
S}\,=\,\exp\big{[}\zeta_{n}\big{]}\,.$ (5.74)

Now the first thing to notice is that the matrix nature of ${\cal
Z}_{n}^{\,\cal S}$ complicates the relationship between the perturbative
coefficients of $\Gamma_{n}^{\,\cal S}$ and those of $\zeta^{\,\cal S}_{n}$.
To see this one may write [199]

$\displaystyle\Gamma_{n}^{\,\cal S}$ $\displaystyle=$
$\displaystyle-\,\left({\cal Z}_{n}^{\,\cal S}\right)^{-1}\,\frac{d{\cal
Z}_{n}^{\,\cal S}}{d\ln\mu}\,=\,\int_{0}^{1}d\tau\,{\rm
e}^{-\tau\zeta_{n}}\,\frac{d\zeta_{n}}{d\ln\mu}\,{\rm e}^{\tau\zeta_{n}}$
(5.75) $\displaystyle=$
$\displaystyle-\,\frac{d\zeta_{n}}{d\ln\mu}+\frac{1}{2}\left[\zeta_{n},\frac{d\zeta_{n}}{d\ln\mu}\right]-\frac{1}{6}\left[\zeta_{n},\left[\zeta_{n},\frac{d\zeta_{n}}{d\ln\mu}\right]\right]+\ldots\,.$

Next one may use the fact that, in a minimal subtraction scheme, $\zeta_{n}$
can depend on the renormalisation scale only through the coupling; defining
the perturbative coefficients by

$\displaystyle\Gamma_{n}^{\,\cal
S}\,=\,\sum_{k=1}^{\infty}\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{k}\Gamma_{\\!{\cal
S},n}^{\,(k)}\,,\qquad\zeta_{n}\,=\,\sum_{k=1}^{\infty}\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{k}\zeta_{n}^{\,(k)}\,,$
(5.76)

and using Eq. (2.26) and Eq. (5.75), one easily finds at low orders

$\displaystyle\Gamma_{\\!{\cal S},n}^{\,(1)}\\!$ $\displaystyle=$
$\displaystyle\\!2\epsilon\zeta_{n}^{\,(1)}\,,$ $\displaystyle\Gamma_{\\!{\cal
S},n}^{\,(2)}\\!$ $\displaystyle=$
$\displaystyle\\!4\epsilon\zeta_{n}^{\,(2)}+\frac{b_{0}}{2}\zeta_{n}^{(1)}\,,$
$\displaystyle\Gamma_{\\!{\cal S},n}^{\,(3)}\\!$ $\displaystyle=$
$\displaystyle\\!6\epsilon\zeta_{n}^{\,(3)}-\epsilon\left[\zeta_{n}^{(1)},\zeta_{n}^{(2)}\right]+b_{0}\zeta_{n}^{(2)}+\frac{b_{1}}{2}\zeta_{n}^{(1)}\,,$
(5.77)

with further nested commutators appearing at higher orders. Inverting Eq.
(5.77), one finds, as customary in minimal schemes, that the $k$-loop
anomalous dimension matrix is proportional to the single pole of
$\zeta_{n}^{\,(k)}$; furthermore, $\zeta_{n}^{\,(k)}$ is found to have UV
poles up to $\epsilon^{-k}$, with coefficients which are determined
recursively from lower-order results [199].

The second non-trivial consequence of the matrix nature of soft anomalous
dimensions is the relation between the logarithm of the renormalised
correlator and that of its regularised counterpart. In fact, substituting Eq.
(5.74) into Eq. (5.72), one finds that [482, 199]

$\displaystyle W_{n}^{\,(\rm ren.)}\,=\,W_{n}^{\,(m)}+\zeta_{n}+\ {\cal
C}\Big{(}W_{n}^{\,(m)},\zeta_{n}\Big{)}\,,$ (5.78)

where ${\cal C}$ is a series of nested commutators, determined as usual by the
Baker-Campbell-Haussdorf formula. As is often the case when employing
dimensional regularisation and minimal schemes, Eq. (5.78) provides non-
trivial constraints: indeed, the renormalised matrix $W_{n}^{\,(\rm ren.)}$
must be finite as $\epsilon\to 0$, while the matrix $\zeta_{n}$ contains only
poles in $\epsilon$. In the presence of the commutator terms building up the
function ${\cal C}$, this means, remarkably, that terms proportional to
positive powers of $\epsilon$ in the regularised matrix $W_{n}^{\,(m)}$ can
interfere with the poles in $\zeta_{n}$ to give finite contributions to the
renormalised correlator, and thus to the anomalous dimension matrix.
Concretely, one may expand the regularised matrix $W_{n}^{\,(m)}$ in powers of
$\alpha_{s}$ and $\epsilon$, as

$\displaystyle
W_{n}^{\,(m)}\,=\,\sum_{k=1}^{\infty}\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{\\!k}\sum_{p=-k}^{\infty}\epsilon^{p}\,W_{n}^{\,(k,p)}\,.$
(5.79)

Using the finiteness of Eq. (5.78), as well as Eq. (5.77), one can finally
express the perturbative coefficients of the anomalous dimension matrix
directly in terms of the regularized matrix $W_{n}^{\,(m)}$, which is the
object one actually computes. At low orders, one finds [482, 199]

$\displaystyle\Gamma_{\\!{\cal S},n}^{\,(1)}$ $\displaystyle=$
$\displaystyle-2W_{n}^{(1,-1)}$ $\displaystyle\Gamma_{\\!{\cal S},n}^{\,(2)}$
$\displaystyle=$
$\displaystyle-4W_{n}^{(2,-1)}-2\left[W_{n}^{(1,-1)},W_{n}^{(1,0)}\right]$
$\displaystyle\Gamma_{\\!{\cal S},n}^{\,(3)}$ $\displaystyle=$
$\displaystyle-6W_{n}^{(3,-1)}+\frac{3}{8}\,b_{0}\left[W_{n}^{(1,-1)},W_{n}^{(1,1)}\right]+3\left[W_{n}^{(1,0)},W_{n}^{(2,-1)}\right]+3\left[W_{n}^{(2,0)},W_{n}^{(1,-1)}\right]$
(5.80)
$\displaystyle+\left[W_{n}^{(1,0)},\left[W_{n}^{(1,-1)},W_{n}^{(1,0)}\right]\right]-\left[W_{n}^{(1,-1)},\left[W_{n}^{(1,-1)},W_{n}^{(1,1)}\right]\right]\,,$

which can be straightforwardly (if tediously) extended to higher orders.
Having established this general framework, one can now proceed to evaluate the
soft anomalous dimension matrix $\Gamma_{n}^{\,\cal S}$ from the regularised
correlator, with diagrammatic methods. The problem has two aspects: the
determination and analysis of the possible colour structures appearing in
$\Gamma_{n}^{\,\cal S}$, and the calculation of the corresponding kinematic
factors. We will first study colour structures in Section 5.3.1 and in Section
5.3.2, and then we will give two simple examples of the calculation of
kinematic factors in Section 5.3.4 and in Section 5.3.5.

####  5.3.1 Diagrammatic exponentiation of Wilson-line correlators

So far in our analysis we have shown that Wilson-line correlators exponentiate
on the basis of factorisation and renormalisation group arguments. The
resulting exponentiation has proven to be non-trivial: the logarithm of the
correlator is significantly less singular than the correlator itself, and it
can be expressed as a scale integral of a finite anomalous dimension colour
matrix. We now turn to the discussion of a second, independent method to prove
exponentiation, which is based on purely diagrammatic arguments, and leads to
a systematic way to compute directly the perturbative exponent.

We begin by stating the most general form of the result, which holds not only
for correlators of straight, semi-infinite Wilson lines, such as the ones we
have been discussing so far, but actually extends to generic Wilson lines,
following curved paths $\gamma$ which can be open or closed. For the purposes
of the present discussion we consider then correlators of the form

$\displaystyle{\cal
S}_{n}\big{(}\\!\left\\{\gamma_{i}\right\\}\\!\big{)}\,\equiv\,\left\langle\smash{0}{\vphantom{1}}\right|\,T\bigg{[}\prod_{k=1}^{n}\Phi_{r_{k}}\left(\gamma_{k}\right)\bigg{]}\left|\smash{0}{\vphantom{1}}\right\rangle\,,$
(5.81)

where

$\displaystyle\Phi_{r}\left(\gamma\right)\,\equiv\,P\exp\left[{\rm
i}g\\!\int_{\gamma}dx\cdot{\bf A}_{r}(x)\right]$ (5.82)

is the Wilson line operator defined on the curve $\gamma$ and in the
representation $r$ of the gauge group. All such correlators can be written in
exponential form as

$\displaystyle{\cal
S}_{n}\big{(}\\!\left\\{\gamma_{i}\right\\}\\!\big{)}\,=\,\exp\Big{[}{W}_{n}\big{(}\\!\left\\{\gamma_{i}\right\\}\\!\big{)}\Big{]}\,.$
(5.83)

We already know, from renormalisation group arguments, that the exponentiation
is non trivial. The new statement now is that the exponent $W_{n}$ can be
directly computed in terms of a proper subset of the Feynman diagrams that
form the perturbative expansion of the original correlator ${\cal S}_{n}$;
these diagrams will however appear in the exponent with modified colour
factors, which can be determined from the original Feynman rules, as discussed
below.

This general result has a long history. In the abelian theory, diagrammatic
exponentiation is implicit in the original arguments on the cancellation of
divergences [18, 20, 21], but can actually be derived in general on the basis
of simple combinatoric arguments [110]. The result is that the diagrams
contributing to the exponent are those building up connected photon
correlators, and they appear in the exponent with unchanged weights (since
‘colour’ is trivial). In the non-abelian theory, the case of two Wilson lines
in a colour-singlet configuration was the first to be tackled [483, 484, 485].
In that case, diagrams contributing to the exponent were called webs, and they
are characterised topologically as being two-eikonal-irreducible: they are
diagrams that are not partitioned into disjoint subdiagrams when each Wilson
line is cut exactly once. Web diagrams contribute to the exponent with
modified colour factors, which will emerge from our general treatment below.
Exponentiation in the two-line case is reviewed in [110, 281]. The general
result that we are discussing, for an arbitrary number of generic Wilson
lines, was derived with essentially combinatoric arguments in [486, 482], and
further refined in [199, 487, 488] (see also [489]), culminating in the non-
abelian exponentiation theorem proved in [490], which we will summarise below:
as we will see, this result involves a non-trivial generalisation of the
concept of web333333For introductory reviews of the web idea, see for example
[491, 492].. In what follows, for ease of notation, we will drop the subscript
$n$ that indicates the number of Wilson lines, and we will also leave the
dependence on the contours $\gamma_{i}$ implicit.

Let $D$ be a Feynman diagram contributing to the correlator ${\cal S}$. Each
such diagram is a product of a kinematic factor ${\cal K}(D)$ and a colour
factor $C(D)$. For straight Wilson lines, the kinematic factor is a function
of the four velocities $\beta_{i}$; more generally, it is a functional of the
contours $\gamma_{i}$. Diagrammatic exponentiation means that the logarithm of
the correlator, $W$, can be written as a linear combination of the same
Feynman diagrams, however with modified colour factors. We write

$\displaystyle W\,=\,\sum_{D}{\cal K}(D)\,\widetilde{C}(D)\,,$ (5.84)

where $\widetilde{C}(D)$ is referred to as Exponentiated Colour Factor (ECF)
for diagram $D$. The crucial point in Eq. (5.84) is of course that a large
number of diagrams have vanishing ECFs, and therefore do not contribute to
$W$: for example, for $n=2$, Refs. [483, 484, 485] show that all two-eikonal-
reducible diagrams have $\widetilde{C}(D)=0$. In the general case involving
$n$ Wilson lines, ECFs are linear combinations of the ordinary color factors
of sets of diagrams that differ only by the order of their gluon attachments
to the Wilson lines. This naturally groups the Feynman diagrams that appear at
any given order into subsets, and each such subset343434Note that in the two-
line case the term web was used for individual diagrams, not for sets of
diagrams. For an abelian theory, webs are (sets of) connected photon diagrams.
is called a web, which we denote by $w$. We then use the definition [486]

  * Web: A set of Feynman diagrams contributing to a Wilson-line correlator that can be obtained from any representative diagram by permuting the gluon attachments to the Wilson lines.

Consequently, the sum in Eq. (5.84) can be organized as a sum over webs, and,
for each diagram $D$ of a given web $w$, the ECF can be expressed as

$\displaystyle\widetilde{C}(D)\,=\,\sum_{D^{\prime}\in
w}R_{w}(D,D^{\prime})\,C(D^{\prime})\,,$ (5.85)

where the sum is over all diagrams belonging to web $w$, and
$R_{w}(D,D^{\prime})$ is called web mixing matrix. Thus, each web $w$ can be
written as

$\displaystyle w\,=\,\sum_{D\in w}{\cal
K}(D)\,\widetilde{C}(D)\,=\,\sum_{D,D^{\prime}\in w}{\cal
K}(D)\,R_{w}(D,D^{\prime})\,C(D^{\prime})\,.$ (5.86)

In this language, $W=\sum w$, where the sum extends to all webs, order by
order in perturbation theory, and now we can express the correlator in Eq.
(5.83) as

$\displaystyle{\cal S}\,=\,\exp\left[\sum_{w}\sum_{D,D^{\prime}\in w}{\cal
K}(D)\,R_{w}(D,D^{\prime})\,C(D^{\prime})\right]\,.$ (5.87)

A simple example of a two-loop, three-line web involving two diagrams is
presented in Fig. 30. There are only 2 diagrams in this web, as two
permutations are possible only on the second Wilson line.

(a)

(b)

Figure 30: A simple two-loop, three-line web involving two Feynman diagrams.
Note that Wilson lines are oriented, and labelled by integers in green;
multiple gluon attachments to a given line (whose permutations generate the
web) are labelled by capital letters in blue.

Web mixing matrices are clearly crucial quantities for the purpose of
computing Wilson-line correlators, and therefore, in particular, the soft
anomalous dimension matrix. Their properties were extensively studied in Refs.
[199, 486, 482, 487, 488, 489, 490, 493], while an interesting alternative
approach to exponentiation was developed in Refs. [494, 495]. We conclude this
subsection by listing the properties of the mixing matrices that appear in Eq.
(5.85), while in Section 5.3.3 we will describe an algorithm to compute them
explicitly, after introducing a more suitable diagrammatic organisation of the
perturbative exponent in Section 5.3.2. Properties 1 to 3 below were proved in
Refs. [199, 487, 490], while Property 4 was conjectured in [199], and verified
to hold up to four loops in Refs. [496, 497].

  1. 1.

Idempotency. Web mixing matrices are idempotent, i.e.

$\displaystyle R_{w}^{2}\,=\,R_{w}\,.$ (5.88)

This property implies that the eigenvalues can only be either 1 or 0: in other
words, web mixing matrices are projection operators, selecting a subset of the
available colour factors. If we denote by $Y_{w}$ the matrix diagonalising
$R_{w}$, we have then

$\displaystyle Y_{w}R_{w}Y_{w}^{-1}\,=\,{\rm
diag}\left(\lambda_{1},...,\lambda_{p_{w}}\right)\,=\,{\bf
1}_{r_{w}}\oplus{\bf 0}_{p_{w}-r_{w}}\,,$ (5.89)

where $p_{w}$ is the number of diagrams for web $w$, and thus the dimension of
the matrix $R_{w}$, while $r_{w}<p_{w}$ is the rank of $R_{w}$. We can now
write Eq. (5.86) as

$\displaystyle w$ $\displaystyle=$ $\displaystyle\big{(}{\cal
K}_{w}^{T}\,Y_{w}^{-1}\big{)}\,Y_{w}R_{w}Y_{w}^{-1}\,\big{(}Y_{w}\,C_{w}\big{)}$
(5.90) $\displaystyle=$ $\displaystyle\sum_{h=1}^{r_{w}}\big{(}{\cal
K}_{w}^{T}Y_{w}^{-1}\big{)}_{h}\big{(}Y_{w}C_{w}\big{)}_{h}\,,$

where ${\cal K}_{w}$ is the vector of kinematic factors and $C_{w}$ the vector
of colour factors for web $w$. It is clear that only $r_{w}$ ECFs will
contribute to web $w$.

  2. 2.

Non-abelian exponentiation. The non-abelian exponentiation theorem [490]
identifies the colour factors which survive the projection discussed above: to
all orders in perturbation theory, they are the colour factors which, by the
Feynman rules, would be associated to connected gluon subdiagrams.

  3. 3.

Row sum rule. The elements of web mixing matrices obey the row sum rule

$\displaystyle\sum_{D^{\prime}}R_{w}(D,D^{\prime})\,=\,0\,,$ (5.91)

for any web $w$ with $p_{w}>1$, implying that at least one of the eigenvalues
of any non-trivial web mixing matrix must vanish.

  4. 4.

Column sum rule. One may also envisage the web mixing matrix as acting on the
vector of kinematic factors for the diagrams of web $w$. Taking this
viewpoint, the projection effected by the matrix $R_{w}$ has another non-
trivial effect: it selects combinations of kinematic factors that do not
contain ultraviolet sub-divergences353535We refer here to UV divergences
arising from sub-diagrams involving the Wilson lines: interactions away from
the Wilson lines will still involve the usual gauge-theory UV divergences,
which are dealt with by means of ordinary renormalisation techniques.. This is
crucial for the consistency of the method we are outlining: thanks to this
property, at each successive order in perturbation theory webs contribute a
single extra UV pole, so that the corresponding counterterm can be extracted,
in a minimal scheme, with no dependence on the infrared regulator. The absence
of sub-divergences was established for the two-line case in Refs. [483, 484,
485]; for $n>2$ Wilson lines a complete all-order proof is still lacking, but
the property holds for all webs up to four loops [199, 496, 497]. A necessary
ingredient for the proof is that the elements of web mixing matrices obey a
column sum rule, which can be characterised as follows.

Given a diagram $D$, consider the set of its connected sub-diagrams (after
removing the Wilson lines), $\\{D_{\rm c}^{i}\subset D\\}$; we say that a
connected sub-diagram $D_{\rm c}^{i}$ can be shrunk to the common origin of
the Wilson lines if all the vertices connecting the sub-diagram to the Wilson
lines can be moved to the origin without encountering vertices associated with
other sub-diagrams.

For any given diagram $D$, we then define the column weight of diagram $D$,
$s(D)$, as the number of different ways in which the connected sub-diagrams
$D_{\rm c}^{i}$ of $D$ can be sequentially shrunk, so that all gluon
attachments to Wilson lines in $D$ are moved to their common origin. Thus, if
all gluon attachments are entangled, so that no subdiagram can be shrunk
without shrinking the whole diagram, then $s(D)=0$. On the other hand, if, for
example, a single subdiagram $D_{\rm c}^{i}$ can be shrunk without affecting
the others, this provides a non-trivial sequence for the the shrinking of the
whole diagram, so that $s(D)=1$: this is the situation for the two diagrams
portrayed in Fig. 30.

Armed with these definitions, we can state the (conjectured) column sum rule
for web mixing matrices as

$\displaystyle\sum_{D\in w}\,s(D)\,R_{w}(D,D^{\prime})\,=\,0\,.$ (5.92)

The connection of this rule to UV sub-divergences is clear in a coordinate-
space picture [224]: in this picture, one understands UV divergences as
arising from short distances between interaction vertices, and ‘shrinkable’
sub-diagrams are naturally associated to UV sub-divergences. Eq. (5.92)
guarantees that these sub-diagrams are projected out of the webs.

####  5.3.2 From gluon webs to correlator webs

Even from the preliminary description given in Section 5.3.1, it should be
apparent that the diagrammatic structure of the perturbative exponent has two
independent components: on the one hand, connected gluon subdiagrams with
external (off-shell) gluons, and on the other hand the arrangement of possible
attachments of these gluons to the Wilson lines. Motivated by this
observation, Ref. [496] proposed to organise the perturbative exponent not in
terms of sets of diagrams, but in terms of sets of connected gluon
correlators, which were called Cwebs. Conceptually, Cwebs are the proper
building blocks for the logarithm of the Wilson-line correlator: indeed, their
introduction simplifies considerably the counting and organisation of
diagrammatic contributions, especially at high orders, where radiative
corrections to gluon sub-diagrams become important and proliferate;
furthermore, using Cwebs does not affect the definition and structure of web
mixing matrices, which are derived exclusively from the ordering of gluon
attachments to the Wilson lines, and are not affected by gluon interactions
away from the Wilson lines. In this Section, we will briefly review the
properties of Cwebs, while in Section 5.3.3 we will describe an algorithm to
compute Cweb mixing matrices, which is a simple adaptation of the algorithm
given in Ref. [486], implicitly showing that Cwebs are a natural and proper
generalisation of webs.

Figure 31: A four-line Cweb containing one four-gluon correlator and one two-
gluon correlator.

We begin with the definition

  * Correlator Web (Cweb): A set of skeleton diagrams, built out of connected gluon correlators attached to Wilson lines, and closed under permutations of the gluon attachments to each Wilson line.

As an example, a four-line Cweb built with two connected gluon correlators is
shown in Fig. 31. Clearly, Cwebs are not fixed-order quantities, but admit
their own perturbative expansion in powers of the gauge coupling $g$. Below,
we will use the notation $W_{n}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})$
for a Cweb constructed out of $c_{m}$ $m$-point connected gluon correlators
($m=2,\ldots,p$), where $k_{i}$ is the number of gluon attachments to the
$i$-th Wilson line363636At sufficiently high-orders, this notation does not
uniquely identify Cwebs, since different Cwebs can be constructed out of the
same set of correlators, and with the same number of attachments to each
Wilson line. The proposed notation will however suffice for our purposes
here.. Note that there is an obvious degeneracy in the counting of Cwebs,
since Cwebs that differ only by a permutation of their Wilson lines are
structurally identical, and it is trivial to include them in the calculation
of the full correlator, simply by summing over Wilson-line labels. For the
sake of classification, we can then focus on a specific Wilson-line ordering,
choosing for example $k_{1}\leq k_{2}\leq\ldots\leq k_{n}$. As an example, in
this notation the Cweb in Fig. 31 is denoted by $W_{4}^{(1,0,1)}(1,1,2,2)$.
Taking into account the fact that the perturbative expansion for an $m$-point
connected gluon correlator starts at ${\cal O}(g^{m-2})$, while each
attachment to a Wilson line carries a further power of $g$, the perturbative
expansion for a Cweb can be written as

$\displaystyle
W_{n}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})\,=\,g^{\,\sum_{i=1}^{n}k_{i}\,+\,\sum_{r=2}^{p}c_{r}(r-2)}\,\sum_{j=0}^{\infty}\,W_{n,\,j}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})\,g^{2j}\,,$
(5.93)

which defines the perturbative coefficients
$W_{n,\,j}^{(c_{2},\ldots,c_{p})}(k_{1},\ldots,k_{n})$. With this in mind, it
is natural to classify Cwebs based on the perturbative order where they
receive their lowest-order contribution, which is given by the power of $g$ in
the prefactor of Eq. (5.93); one may then easily design a recursive procedure
for construction of Cwebs.

Starting at lowest order, we notice that only two Cwebs appear at order
$g^{2}$: the self-energy insertion with a two-point connected gluon correlator
attached to a single Wilson line, which we denote by $W_{1}^{(1)}(2)$, and the
configuration where the two-gluon correlator joins two Wilson lines, denoted
by $W_{2}^{(1)}(1,1)$. These two Cwebs are depicted in Fig. 32.

(a)

(b)

Figure 32: The only two Cwebs whose perturbative expansion starts at ${\cal
O}(g^{2})$.

Notice that, if the Wilson lines are light-like, the self-energy Cweb vanishes
identically, since, by the eikonal Feynman rules, it is proportional to the
square of the Wilson-line four-velocity vector, $\beta^{2}$: in that case, one
is left with just one non-vanishing ${\cal O}(g^{2})$ Cweb373737Webs where at
least one gluon begins and ends on the same (non-light-like) Wilson line have
been recently studied in detail in Ref. [498].. Starting from this initial
condition, one may recursively construct all Cwebs starting at higher
perturbative orders: indeed, having constructed all Cwebs at ${\cal
O}(g^{2n})$, there are three ways of adding a power of $g^{2}$.

  * •

Add a two-gluon connected correlator connecting any two Wilson lines
(including ‘new’ Wilson lines that had no attachments in lower-order Cwebs).

  * •

Connect an existing $m$-point correlator to any Wilson line (again, including
Wilson lines with no attachments at lower orders), turning it into an
$(m+1)$-point correlator.

  * •

Connect an existing $m$-point correlator to an existing $n$-point correlator,
resulting in an $(n+m)$-point correlator.

Executing all possible moves is clearly redundant, since the same Cweb is
generated more than once through different sequences of moves: upon performing
all moves, one must therefore remove Cwebs that have been counted more than
once. The procedure can be considerably streamlined by taking into account
known properties of webs, which naturally generalise to Cwebs. For example,
webs (and thus Cwebs) that are given by the product of two or more
disconnected lower-order webs (so that there are subsets of Wilson lines not
joined by any gluons) do not contribute to the logarithm of the correlator,
${\cal W}_{n}$; furthermore, as mentioned above, in a massless theory all
self-energy Cwebs, where all gluon lines attach to the same Wilson line,
vanish as a consequence of the eikonal Feynman rules. Thus, any Cweb
containing a connected gluon correlator attaching to a single Wilson line will
vanish. Using these recursive rules, it is easy to enumerate inequivalent
Cwebs at low orders. In the massless theory, for example, one finds four
inequivalent Cwebs at ${\cal O}(g^{4})$, fourteen new Cwebs at ${\cal
O}(g^{6})$, and a total of sixty new Cwebs at ${\cal O}(g^{8})$: they are
discussed in detail in Ref. [496].

At each order in $g$, Cwebs contain webs, all sharing the same mixing matrix,
since the colour algebra associated with permutations of attachments to the
Wilson lines is unaffected by radiative corrections to the connected
correlators comprising the Cweb. We can thus construct Cweb mixing matrices by
the same technique devised in [486] for webs, which we now discuss.

####  5.3.3 A replica algorithm to generate web mixing matrices

As is the case for other combinatorial problems involving exponentiation, such
as the construction of effective actions, Wilson-line correlators can be
studied by means of algorithms constructed with the replica method [499]. In
this section, we will summarise the application of the replica method to the
construction of Cweb mixing matrices, following Refs. [109, 486].

In order to introduce the method, consider the path integral expression for
the Wilson-line correlator in Eq. (5.81)

$\displaystyle{\cal
S}_{n}\left(\gamma_{i}\right)\,=\,\exp\Big{[}W_{n}(\gamma_{i})\Big{]}\,=\,\int{\cal
D}A_{\mu}^{a}\,\,{\rm e}^{{\rm
i}S\left(A_{\mu}^{a}\right)}\,\prod_{k=1}^{n}\Phi\left(\gamma_{k}\right)\,,$
(5.94)

where $S$ is the classical action383838We display only gluon fields as
arguments of the action, since matter fields are irrelevant to the following
arguments: they can be integrated out, and their effects can be included in
$S$, as only gluons couple to Wilson lines, and matter fields appear only in
loops.. In order to compute $W_{n}(\gamma_{i})$, one starts by building a
replicated theory, replacing the single gluon field $A_{\mu}^{a}$ with $N_{r}$
identical copies $A_{\mu}^{a,\,i}$ ($i=1,\ldots,N_{r}$), which do not interact
with each other. Under this replacement, one has
$S\left(A_{\mu}^{a}\right)\rightarrow\sum_{i=1}^{N_{r}}S\left(A_{\mu}^{a,\,i}\right)$;
if, furthermore, we replace each Wilson line in Eq. (5.94) by the product of
$N_{r}$ Wilson lines, each belonging to one replica of the theory, one readily
sees that the replicated correlator is given by

$\displaystyle{\cal S}_{n}^{\,{\rm
repl.}}\left(\gamma_{i}\right)\,=\,\Big{[}{\cal
S}_{n}\left(\gamma_{i}\right)\Big{]}^{N_{r}}\,=\,\exp\Big{[}N_{r}\,W_{n}(\gamma_{i})\Big{]}\,=\,{\bf
1}+N_{r}\,W_{n}(\gamma_{i})+{\cal O}(N_{r}^{2})\,.$ (5.95)

Eq. (5.95) provides a straightforward method to compute $W_{n}(\gamma_{i})$:
one may compute the replicated correlator (essentially a combinatorial
problem, as we will see), and then extract from the result the term of order
$N_{r}$.

An important point is that, while gluon fields in different replicas do not
interact, they all belong to the same gauge group: therefore, the colour
matrices associated with their attachments to a Wilson line do not commute,
and their ordering is relevant. On the other hand, in a Cweb, each connected
gluon correlator can be assigned a unique replica number, since there are no
interaction vertices connecting different replicas. One can then relate the
contributions of different skeleton diagrams to the replicated correlator in
Eq. (5.95) to those of the same diagrams in the un-replicated theory, by means
of combinatorial factors, counting the multiplicities associated with the
presence of different replicas. The computation of $W_{n}(\gamma_{i})$ in
terms of the skeleton diagrams contributing to ${\cal S}_{n}(\gamma_{i})$ is
thus reduced to the computation of these combinatorial factors. The necessary
steps, listed below, were identified for webs in Refs. [109, 486], and
naturally adapt to the language of Cwebs.

  * •

For any given Cweb, assign a replica number $i$ ($1\leq i\leq N_{r}$) to each
connected gluon correlator present in the Cweb.

  * •

Define a replica ordering operator $R$, acting on color generators on each
Wilson line as

$\displaystyle R\Big{(}{\bf T}_{k}^{(i)}{\bf T}_{k}^{(j)}\Big{)}\\!$
$\displaystyle=$ $\displaystyle\\!{\bf T}_{k}^{(j)}{\bf
T}_{k}^{(i)}\quad\textrm{if}\,\,i>j\,,$ (5.96) $\displaystyle R\Big{(}{\bf
T}_{k}^{(i)}{\bf T}_{k}^{(j)}\Big{)}\\!$ $\displaystyle=$
$\displaystyle\\!{\bf T}_{k}^{(i)}{\bf T}_{k}^{(j)}\quad\textrm{if}\,\,i\leq
j\,,$

where ${\bf T}_{k}^{(i)}$ is the group generator associated with the emission
of a gluon in replica $i$ from the $k$-th Wilson line. The operator $R$
effectively replaces the selected skeleton diagram with another one drawn from
the same Cweb: as a consequence, one may verify that the colour factor in the
replicated theory is a linear combination of the colour factors of all
skeleton diagrams in the Cweb, with multiplicities given by the number of
possible replica orderings of the gluon attachments on every Wilson line.

  * •

In order to proceed, one needs to list all possible hierarchies of the replica
numbers associated with each correlator in the Cweb: these hierarchies in fact
define the action of the $R$ operator on each skeleton diagram of the Cweb.
For a Cweb built out of $m$ connected correlators, the number $h(m)$ of
possible hierarchies between the $m$ replica numbers of the correlators grows
rapidly with $m$, and is not entirely trivial to determine, since the case of
equal replica numbers must be treated separately. The result is however well
known [500]: $h(m)$ are the so-called Fubini numbers393939The Fubini numbers
admit a generating function, and they can be defined by
$\displaystyle\frac{1}{2-\exp(x)}-1\,\equiv\,\sum_{m=1}^{\infty}h(m)\,\frac{x^{m}}{m!}\,\,.$
(5.97) . In the first instances, for $m=\\{1,2,3,4,5,6\\}$ one finds
$h(m)=\\{1,3,13,75,541,4683\\}$. For every fixed hierarchy $h$, the number
relevant for the determination of the colour factor in the replicated theory
is the multiplicity with which that hierarchy can occur in the presence of
$N_{r}$ replicas, which we denote by $M_{N_{r}}(h)$. It is not difficult to
see that it is given by

$\displaystyle
M_{N_{r}}(h)\,=\,\frac{N_{r}!}{\big{(}N_{r}-n_{r}(h)\big{)}!\,\,n_{r}(h)!}\,,$
(5.98)

where $n_{r}(h)$ is the number of distinct replicas present for hierarchy $h$.
To give a concrete example, for $m=5$, labelling the replica numbers of the
five correlators with $r_{k}$, $(k=1,\ldots,5)$, and picking the hierarchy
$r_{1}=r_{2}<r_{3}=r_{4}<r_{5}$, we have $n_{r}(h)=3$, and thus
$M_{N_{r}}(h)=N_{r}(N_{r}-1)(N_{r}-2)/6$.

  * •

Finally, the exponentiated color factors for a skeleton diagram $D$ is given
by

$\displaystyle C_{N_{r}}^{\,{\rm
repl.}}(D)\,=\,\sum_{h}M_{N_{r}}(h)\,R\big{[}C(D)\big{|}h\big{]}\,,$ (5.99)

where $R\big{[}C(D)\big{|}h\big{]}$ is the color factor of the skeleton
diagram obtained from $D$ through the action of the replica-ordering operator
$R$, in the case of hierarchy $h$. The Cweb mixing matrix is built by picking
terms that are linear in $N_{r}$ out of the coefficients $M_{N_{r}}(h)$, which
are polynomials in $N_{r}$. Note that in the presence of $m$-point
correlators, with $m\geq 4$, each correlator can contribute different
‘internal’ colour factors, for example different permutations of products of
structure constants. Since, however, the information on the mixing matrix is
encoded in the coefficients $M_{N_{r}}(h)$, the different colour factors
arising from the internal structure of the correlators can be treated one by
one, without affecting the mixing of the diagrams.

To conclude our discussion of exponentiated colour factors, we briefly comment
on the dimensionality of Cweb mixing matrices. One would naïvely imagine that
it should be given by the product of the numbers of possible gluon
permutations on each Wilson line. This however neglects two important
subtleties. First, if a set of, say, $k$ gluons emerging from a given
correlator attaches on a single Wilson line, their permutations are
irrelevant, since the correlator is Bose symmetric. One should not then count
permutations on each line, but rather shuffles of the subsets of gluons
emerging from each correlator and attaching to a given line. Furthermore,
there is another important degeneracy, which already appears in the simple
two-line Cweb $W^{(2)}_{2}(2,2)$ at two loops: counting shuffles separately on
each line would yield, in that case, the result $d=4$ for the dimensionality
of the mixing matrix, which is wrong, because the two shuffles on the second
Wilson line can be obtained from the shuffles on the first line by exchanging
the two gluons, which is manifestly a symmetry of the Cweb. In order to take
into account this degeneracy, one must divide by the number of available
permutations of subsets of $m$-point correlators that have the property of
being attached to the same sets of Wilson lines.

The discussion in the last three sections has been general and rather formal,
but also incomplete, since of course much of the dynamical content of the soft
anomalous dimension matrix resides in the kinematic factors, which emerge from
loop integrals of rapidly increasing complexity at high orders. To fill these
gaps, we now give two simple examples of the evaluation of these kinematic
factors at one and two loops. At the two-loop level, this will also serve to
illustrate in a concrete case the interplay of colour and kinematics which has
been discussed formally in the previous sections.

####  5.3.4 The one-loop angle-dependent cusp anomalous dimension

We begin by describing the calculation of the only non-vanishing one-loop
diagram contributing to the soft anomalous dimension matrix in the massless
case. The relevant diagram corresponds to the tree-level contribution to the
Cweb in Fig. 32(b), and of course it is a web on its own. Clearly, this
diagram is very simple, and it can be computed in a number of ways - indeed,
we performed an equivalent calculation in Section 3.3.2 with light-like Wilson
lines. We display the complete calculation here, with collinear divergences
regulated by using ‘massive’ Wilson lines, and soft divergences regulated by
the exponential suppression defined in Eq. (5.70), in order to illustrate
techniques that can be profitably used at higher orders as well. Besides, the
calculation yields the one-loop result for the angle-dependent cusp anomalous
dimension, which is a fundamental quantity for the scattering of massive
particles and in many other contexts [370]. Unlike Section 3.3.2, we will
proceed by recognising in the second line of Eq. (3.3.2) the coordinate-space
gluon propagator, $D_{\mu\nu}(x)$, evaluated for
$x=\lambda_{i}\beta_{i}-\lambda_{j}\beta_{j}$. The expression for this
propagator in $d=4-2\epsilon$ is well known, and is given by

$\displaystyle
D_{\mu\nu}^{ab}(x)\,=\,-\,\frac{\Gamma(1-\epsilon)}{4\,\pi^{2-\epsilon}}\,g_{\mu\nu}\,\delta^{ab}\,\frac{1}{(-x^{2}+i\eta)^{1-\epsilon}}\,.$
(5.100)

Using this expression leaves only two parameter integrals to be performed.
Arguably, this ‘coordinate space’ approach to eikonal integrals simplifies
their calculations also at higher orders, and it was indeed adopted for the
state-of-the-art calculation of Ref. [77].

In Cweb notation, the diagram in Fig. 32 (b), including radiative corrections
to the gluon propagator, can be written as

$\displaystyle W_{2,\,ij}^{(1)}(1,1)\,=\,{\bf T}_{i}\cdot{\bf T}_{j}\;{\cal
K}^{(1)}_{2}\bigg{(}\gamma_{ij},\frac{\mu^{2}}{m^{2}},\epsilon\bigg{)}\,,$
(5.101)

where ${\cal K}^{(1)}_{2}(\gamma_{ij},\mu^{2}/m^{2},\epsilon)$ gives the
regularised kinematic factor. The color structure is straightforward and there
is no mixing matrix involved, since this is a single-diagram web. With the
infrared regulator in place, and having introduced the gluon propagator in Eq.
(5.100), the one-loop kinematic factor reads

$\displaystyle{\cal
K}^{(1)}_{2,1}\bigg{(}\gamma_{ij},\frac{\mu^{2}}{m^{2}},\epsilon\bigg{)}=g_{s}^{2}\mu^{2\epsilon}\,\frac{\Gamma(1-\epsilon)}{4\,\pi^{2-\epsilon}}\,\beta_{i}\cdot\beta_{j}\int_{0}^{\infty}\\!d\lambda_{i}\int_{0}^{\infty}\\!d\lambda_{j}\,\frac{{\rm
e}^{-m\lambda_{i}\sqrt{\beta_{i}^{2}}-m\lambda_{j}\sqrt{\beta_{j}^{2}}}}{\Big{(}\\!-\left(\lambda_{i}\beta_{i}-\lambda_{j}\beta_{j}\right)^{2}+i\eta\Big{)}^{1-\epsilon}}\,.$
(5.102)

To perform the integral, it is clearly useful to rescale the variables and
define

$\displaystyle\sigma\,=\,\lambda_{i}\sqrt{\beta_{i}^{2}}\,,\qquad\tau\,=\,\lambda_{j}\sqrt{\beta_{j}^{2}}\,.$
(5.103)

We then introduce coordinates $\lambda$ and $x$, which are respectively normal
and intrinsic to the pinch surface (in this case a point),

$\displaystyle\lambda\,=\,\sigma+\tau\,,\qquad\quad
x\,=\,\frac{\sigma}{\sigma+\tau}\,;$ (5.104)

the normal coordinate $\lambda$ gives the “overall distance” of the gluon from
the cusp, and ranges from 0 to $\infty$. The intrinsic angular coordinate
ranges between 0 and 1, and measures the degree of collinearity of the gluon
to either Wilson line. The integral over $\lambda$ can be readily performed
and gives a factor of $\Gamma(2\epsilon)$ containing the ultraviolet pole. The
integration over the intrinsic variable $x$ remains, and we can write

$\displaystyle{\cal
K}^{(1)}_{2,1}\bigg{(}\gamma_{ij},\frac{\mu^{2}}{m^{2}},\epsilon\bigg{)}\,=\,-\bigg{(}\frac{\mu^{2}}{m^{2}}\bigg{)}^{\epsilon}g_{s}^{2}\,\,\frac{\Gamma(1-\epsilon)}{8\pi^{2-\epsilon}}\,\Gamma(2\epsilon)\,\gamma_{ij}\int_{0}^{1}dx\,P\big{(}x,\gamma_{ij}\big{)}\,,$
(5.105)

where

$\displaystyle
P\big{(}x,\gamma_{ij}\big{)}\,=\,\Big{[}x^{2}+(1-x)^{2}-x(1-x)\gamma_{ij}-{\rm
i}\eta\Big{]}^{-1+\epsilon}\,.$ (5.106)

The quadratic polynomial in $x$ in Eq. (5.106) can be factored with the change
of variables

$\displaystyle\gamma_{ij}\,\equiv\,-\,\alpha_{ij}-\frac{1}{\alpha_{ij}}\,,$
(5.107)

where we can take $\alpha_{ij}$ to lie inside the unit circle in the complex
plane, thanks to the symmetry under $\alpha_{ij}\to 1/\alpha_{ij}$. The $x$
integral can then be performed exactly, and gives a hypergeometric function,

$\displaystyle{\cal
K}^{(1)}_{2,1}\bigg{(}\gamma_{ij},\frac{\mu^{2}}{m^{2}},\epsilon\bigg{)}\,=\,-\left(\frac{\mu^{2}}{m^{2}}\right)^{\epsilon}\,g_{s}^{2}\,\frac{\Gamma(1-\epsilon)}{8\pi^{2-\epsilon}}\,\,\Gamma(2\epsilon)\,\gamma_{ij}\,{}_{2}F_{1}\bigg{(}1,1-\epsilon,\frac{3}{2},\frac{1}{2}+\frac{\gamma_{ij}}{4}\bigg{)}\,.$
(5.108)

Expanding Eq. (5.108) around $\epsilon=0$ we can easily obtain the coefficient
of the simple pole in Eq. (5.105). Reinstating then the colour factor, making
use of Eq. (5.80) at one loop, and summing over available colour dipoles, we
obtain the soft anomalous dimension matrix at one loop, for massive Wilson
lines. In terms of the $\alpha_{ij}$ variables, it is given by

$\displaystyle\Gamma^{(1)}_{\cal S}\,=\,\sum_{i<j}\,{\bf T}_{i}\cdot{\bf
T}_{j}\,\frac{1+\alpha_{ij}^{2}}{1-\alpha_{ij}^{2}}\,\ln\alpha_{ij}\,.$
(5.109)

In this representation, the limit of light-like Wilson lines is reached as
$\alpha_{ij}\to 0$, corresponding to $\gamma_{ij}\to\infty$. The logarithmic
singularity in that limit corresponds to the collinear divergence in Eq.
(3.3.2), and the coefficient of the logarithm in the $\alpha_{ij}\to 0$ limit
gives the light-like cusp anomalous dimensions discussed in earlier Sections.
An alternative representation is often given, in terms of the Minkowski-space
cusp angles between Wilson lines $i$ and $j$, which we denote by $\xi_{ij}$,
defined by

$\displaystyle\xi_{ij}\,=\,\cosh^{-1}\\!\bigg{(}\\!-\frac{\gamma_{ij}}{2}\bigg{)}\,.$
(5.110)

Using this variable one finds

$\displaystyle\Gamma^{(1)}_{\cal S}\,=\,-\,\sum_{i<j}{\bf T}_{i}\cdot{\bf
T}_{j}\,\,\xi_{ij}\coth\big{(}\xi_{ij}\big{)}\,.$ (5.111)

Figure 33: A two-loop, three-line web involving a 3-gluon vertex.

As we will see below, the one-loop angle-dependent cusp anomalous dimension,
given here in Eq. (5.109) and in Eq. (5.111) plays an important role also for
higher-order corrections to the soft anomalous dimension matrix for massive
particles.

####  5.3.5 The two-loop soft anomalous dimension matrix

$i$, $j$ | Replica-ordered colour factor | Multiplicity
---|---|---
$i=j$ | $C(a)$ | $N_{r}$
$i<j$ | $C(b)$ | $N_{r}(N_{r}-1)/2$
$i>j$ | $C(a)$ | $N_{r}(N_{r}-1)/2$
Table 1: Replica analysis of diagram $(a)$ of Fig. 30, as described in the
text.

We now proceed to summarise the calculation of the two-loop soft anomalous
dimension matrix, which will give us an opportunity to illustrate the
application of the replica method discussed in Section 5.3.3 with the simplest
non-trivial example.

At two-loops, a maximum of three Wilson lines can be connected, and we get a
total of two distinct webs: one was shown in Fig. 30, while the other one,
involving a single diagram with a three-gluon vertex, is shown in Fig. 33.
Here we will discuss in some detail the web in Fig. 30, which is the lowest-
order contribution to the Cweb $W_{3}^{(2)}(1,2,1)$, and is also an example of
the class of webs called “multiple gluon exchange webs” (MGEW), discussed in
Ref. [480]. The contribution of this two-loop web can be written schematically
as

$\displaystyle w_{121}^{(2)}\,=\,{\cal K}(a)\,\widetilde{C}(a)+{\cal
K}(b)\,\widetilde{C}(b)\,,$ (5.112)

where ${\cal K}(a)$ and ${\cal K}(b)$ are the kinematic contributions of the
two diagrams of the web and $\widetilde{C}(a)$ and $\widetilde{C}(b)$ are the
corresponding exponentiated color factors. Considering as an example diagram
$(a)$ in Fig. 30, and introducing $N_{r}$ replicas, we note that the replica-
ordering operator will replace the colour factor of $(a)$ with that of $(b)$
in Fig. 30, in the $N_{r}(N_{r}-1)/2$ cases in which the replicas assigned to
the two gluons are in the ‘wrong’ order, as reported in Table 1. The colour
factor of diagram $(a)$ in the replicated theory then reads

$\displaystyle C^{\,\rm
repl.}_{N_{r}}(a)\,=\,N_{r}C(a)+\frac{N_{r}(N_{r}-1)}{2}\big{(}C(a)+C(b)\big{)}\,=\,N_{r}\left(\frac{C(a)-C(b)}{2}\right)+{\cal
O}(N_{r}^{2})\,,$ (5.113)

and similarly for diagram $(b)$. The coefficients of ${\cal O}(N_{r})$ terms
give us then the exponentiated color factors

$\displaystyle\widetilde{C}(a)\,=\,\left(\frac{C(a)-C(b)}{2}\right)\,=\,-\,\widetilde{C}(b)\,.$
(5.114)

Thus

$\displaystyle
w_{121}^{(2)}\,=\,\frac{1}{2}\,\big{(}C(a)-C(b)\big{)}\big{(}{\cal K}(a)-{\cal
K}(b)\big{)}\,.$ (5.115)

corresponding to the web mixing matrix

$\displaystyle\begin{split}&R_{121}\,=\,\frac{1}{2}\,\displaystyle{\left(\\!\\!\begin{array}[]{cc}1&-1\\\
-1&1\\\ \end{array}\\!\\!\right)}\,.\end{split}$ (5.116)

We note in passing that, as expected, $R_{121}$ satisfies the row sum rule in
Eq. (5.91). In order to check the column sum rule in Eq. (5.92), we must
construct the vector $s(D)$ for the two diagrams in Fig. 30. Taking into
account the orientation of the Wilson lines, one sees that for both diagrams
there is only one way in which the two gluons can be sequentially shrunk to
the origin, so that we get

$\displaystyle s(a)\,=\,s(b)\,=\,1\,,$ (5.117)

and, also as expected, the column sum rule is verified. Concerning the
exponentiated colour factors, one easily sees that the non-abelian
exponentiation theorem is verified for this simple web. Indeed, the color
factors of the diagrams (a) and (b) are

$\displaystyle C(a)\,=\,T_{1}^{a}T_{2}^{a}T_{2}^{b}T_{3}^{b}\,,\qquad
C(b)\,=\,T_{1}^{a}T_{2}^{b}T_{2}^{a}T_{3}^{b}\,.$ (5.118)

Recalling that color generators pertaining to the same Wilson line do not
commute, and using the colour algebra, we can write the exponentiated color
factor as

$\displaystyle\widetilde{C}(a)\,=\,-\frac{1}{2}\,{\rm
i}f_{abc}T_{1}^{a}T_{2}^{b}T_{3}^{c}\,=\,-\,\widetilde{C}(b)\,,$ (5.119)

a result which is proportional to the colour factor of the fully connected
diagram in Fig. 33.

We now turn to the computation of the kinematic factor of this web, following
Ref. [242]. This will allow us to illustrate the dynamical effects of the
column sum rule on leading infrared poles. We follow the same strategy as in
Section 5.3.4, associating directly to each gluon its coordinate-space
propagator, and thus leaving only parameter integrals to be performed. We
parametrise the positions of gluon attachments on Wilson lines $1,2$ and $3$
by $s\beta_{1}$, $t_{1}\beta_{2}$, $t_{2}\beta_{2}$ and $u\beta_{3}$,
respectively, and introduce exponential suppressions for each Wilson line. In
diagram (a), the gluon that attaches at $t_{1}$ is further away from the
origin than the one at $t_{2}$, which is accounted for by including step
function $\theta(t_{1}-t_{2})$. We get then, for diagram (a),

$\displaystyle{\cal K}(a)\\!$ $\displaystyle=$
$\displaystyle\\!g_{s}^{4}\mu^{4\epsilon}\,\frac{\Gamma^{2}(1-\epsilon)}{16\,\pi^{4-2\epsilon}}\,\beta_{1}\cdot\beta_{2}\,\beta_{2}\cdot\beta_{3}\,\int_{0}^{\infty}ds\,dt_{1}\,dt_{2}\,du\,\,\theta(t_{1}-t_{2})$
(5.120)
$\displaystyle\times\,\frac{\exp\Big{[}{-m\Big{(}s\sqrt{\beta_{1}^{2}}+u\sqrt{\beta_{3}^{2}}+\big{(}t_{1}+t_{2}\big{)}\sqrt{\beta_{2}^{2}}\Big{)}}\Big{]}}{\Big{(}-(s\beta_{1}-t_{1}\beta_{2})^{2}+{\rm
i}\eta\Big{)}^{1-\epsilon}\Big{(}-(u\beta_{3}-t_{2}\beta_{2})^{2}+i\eta\Big{)}^{1-\epsilon}}\,,$

while the contribution of diagram (b) is the same as the above, except that
one needs to replace the step function by its complement,
$\theta(t_{2}-t_{1})$. As before, we can exploit the rescaling invariance of
the integrand to scale the variables according to

$\displaystyle\sigma\,=\,s\sqrt{\beta_{1}^{2}}\,,\qquad\tau_{1,2}\,=\,t_{1,2}\sqrt{\beta_{2}^{2}}\,,\qquad\upsilon\,=\,u\sqrt{\beta_{3}^{2}}\,,$
(5.121)

and we can define variables $\lambda_{1}$ and $\lambda_{2}$, corresponding to
the average distance of the two gluons from the origin, as well as ‘angular’
variables $x_{1}$ and $x_{2}$, using

$\displaystyle\lambda_{1}\,=\,\sigma+\tau_{1}\,,\qquad\lambda_{2}\,=\,\upsilon+\tau_{2}\,,$
(5.122) $\displaystyle x_{1}\,=\,\frac{\tau_{1}}{\sigma+\tau_{1}}\,,\qquad
x_{2}\,=\,\frac{\tau_{2}}{\upsilon+\tau_{2}}\,.$ (5.123)

The overall UV divergence can then be extracted considering the overall
distance of the two-gluon system from the origin, using

$\displaystyle\lambda\,=\,\lambda_{1}+\lambda_{2}\,,\qquad\omega\,=\,\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}\,,$
(5.124)

Integration over $\lambda$ is now immediate, and gives the overall divergence
in the form of a factor $\Gamma(4\epsilon)$. In this case, however, the
integral over $\omega$ gives a sub-divergence for each graph: for graph $(a)$,
one finds a factor of

$\displaystyle\int_{0}^{1}d\omega\frac{1}{\big{[}\omega(1-\omega)\big{]}^{1-2\epsilon}}\,\,\theta\bigg{(}\frac{\omega}{1-\omega}>\frac{x_{2}}{x_{1}}\bigg{)}\,=\,\frac{1}{2\epsilon}+\phi(a)\,$
(5.125)

where

$\displaystyle\phi(a)\,=\,-\ln\left(\frac{x_{2}}{x_{1}}\right)+\left[4\textrm{Li}_{2}\left(-\frac{x_{1}}{x_{2}}\right)+\ln^{2}\left(\frac{x_{2}}{x_{1}}\right)\right]\epsilon+\mathcal{O}(\epsilon^{2})\,,$
(5.126)

whereas for diagram (b) one finds

$\displaystyle\int_{0}^{1}d\omega\frac{1}{\big{[}\omega(1-\omega)\big{]}^{1-2\epsilon}}\,\,\theta\bigg{(}\frac{\omega}{1-\omega}<\frac{x_{2}}{x_{1}}\bigg{)}\,=\,\frac{1}{2\epsilon}+\phi(b)\,$
(5.127)

where

$\displaystyle\phi(b)\,=\,-\phi(a)-4\zeta_{2}\epsilon+\mathcal{O}(\epsilon^{2})\,.$
(5.128)

The presence of subdivergences is related to the fact that, for each diagram,
one can shrink the inner gluon line to the origin, while leaving the other
gluon untouched. As a consequence, the kinematic contributions of each diagram
have a double pole, which, however, cancels as expected when we combine the
two diagrams to compute the kinematic factor of the whole web. We get

$\displaystyle{\cal K}(a)-{\cal
K}(b)\,=\,2\kappa^{2}\,\Gamma(4\epsilon)\,\gamma_{12}\gamma_{23}\int_{0}^{1}dx_{1}dx_{2}\,P(x_{1},\gamma_{12})\,P(x_{2},\gamma_{23})\,\phi(x_{1},x_{2};\epsilon)\,,$
(5.129)

where

$\displaystyle\kappa\\!$ $\displaystyle\equiv$
$\displaystyle\\!-\left(\frac{\mu^{2}}{m^{2}}\right)^{\epsilon}g_{s}^{2}\,\frac{\Gamma(1-\epsilon)}{8\pi^{2-\epsilon}}\,,$
$\displaystyle\phi(x_{1},x_{2};\epsilon)\\!$ $\displaystyle=$
$\displaystyle\\!\ln\left(\frac{x_{1}}{x_{2}}\right)+\Big{[}4\textrm{Li}_{2}\left(-\frac{x_{1}}{x_{2}}\right)+\ln^{2}\left(\frac{x_{1}}{x_{2}}\right)+2\zeta_{2}\Big{]}\epsilon+\mathcal{O}(\epsilon^{2})\,.$
(5.130)

This can be substituted into Eq. (5.115), together with Eq. (5.119), to obtain
the complete expression for the web $w_{121}^{(2)}$. In order to recover the
corresponding contribution to the soft anomalous dimension, one step still
needs to be performed: we need to use Eq. (5.80), combining the result for
$w_{121}^{(2)}$ with the appropriate commutator involving the one-loop result,
evaluated to ${\cal O}(\epsilon^{0})$. We need the combination

$\displaystyle\Gamma^{(2)}_{{\cal
S},121}\,=\,-4w_{121}^{(2,-1)}-2\Big{[}w^{(1,-1)},w^{(1,0)}\Big{]}\,,$ (5.131)

where $w^{(1,k)}$ denotes the contribution of order $\epsilon^{k}$ to the one-
loop web discussed in Section 5.3.4. The remaining parameter integrals can be
performed with standard techniques [242, 480], and the result has a relatively
simple form, which can be written as

$\displaystyle\Gamma^{(2)}_{{\cal S},121}\,=\,\frac{{\rm
i}}{4}\,f^{abc}T_{1}^{a}T_{2}^{b}T_{3}^{c}\,\,r(\alpha_{12})\,r(\alpha_{23})\Big{[}\ln(\alpha_{12})U(\alpha_{23})-\ln(\alpha_{23})U(\alpha_{12})\Big{]}\,,$
(5.132)

where $\alpha_{ij}$ is defined in Eq. (5.107), and the functions $r(\alpha)$
and $U(\alpha)$ are given by

$\displaystyle r(\alpha)$ $\displaystyle=$
$\displaystyle\frac{1+\alpha^{2}}{1-\alpha^{2}}\,,$ $\displaystyle U(\alpha)$
$\displaystyle=$ $\displaystyle
2\textrm{Li}_{2}(\alpha^{2})+4\ln\alpha\ln(1-\alpha^{2})-2\ln^{2}\alpha-2\zeta_{2}\,.$
(5.133)

In addition to the contribution in Eq. (5.131), which we get from diagrams
with two gluon attachments on leg number two, we also need to add the
contributions arising from diagrams where legs number one and three have two
gluon attachments. The result of this sum can be written as

$\displaystyle\Gamma^{(2)}_{{\cal S},\,{\rm MGEW}}\,=\,\frac{{\rm
i}}{4}\,f^{abc}T_{1}^{a}T_{2}^{b}T_{3}^{c}\,\sum_{i,j,k}\epsilon_{ijk}\,\,r(\alpha_{ij})\,r(\alpha_{jk})\ln(\alpha_{ij})\,U(\alpha_{jk})\,,$
(5.134)

where the subscript MGEW denotes the contribution of multiple-gluon-exchange
webs, in the sense of Ref. [480]. If $n$ Wilson lines are present, a further
sum over all possible triples of Wilson lines yields

$\displaystyle\Gamma^{(2)}_{{\cal S},\,{\rm MGEW}}\,=\,\frac{{\rm
i}}{4}\,\sum_{i>j>k=1}^{n}f^{abc}T_{i}^{a}T_{j}^{b}T_{k}^{c}\,\sum_{I,J,K\in\\{i,j,k\\}}\\!\\!\\!\epsilon_{IJK}\,\,r(\alpha_{IJ})\,r(\alpha_{JK})\ln(\alpha_{IJ})\,U(\alpha_{JK})\,.$
(5.135)

The full two-loop soft anomalous dimension for three particles receives a
further contribution, in addition to MGEWs, from the single-diagram web with a
three-gluon vertex depicted in Fig. 33. As a consequence of the dipole
formula, this web vanishes in light-like limit, as first discovered in Refs.
[71, 72]. The case of Wilson lines off the light cone, of course, has great
interest of its own, since it governs infrared divergences for amplitudes
involving massive particles. At the one-loop level, the general form of these
divergences was given in Ref. [501]; at two-loops, the methods discussed here
become relevant, and the calculation is far from trivial, requiring state-of-
the-art integration techniques. The result, however, is remarkably simple
[502, 503, 358, 504, 505]. For completeness, we give here the complete
outcome, which can be written as

$\displaystyle\Gamma^{(2)}_{\cal S}\,=\,\Gamma^{(2)}_{{\cal
S},\,3g}+\Gamma^{(2)}_{{\cal S},\,{\rm MGEW}}\,,$ (5.136)

with $\Gamma^{(2)}_{{\cal S},{\rm MGEW}}$ given in Eq. (5.132), while

$\displaystyle\Gamma^{(2)}_{{\cal S},\,3g}\,=\,\frac{{\rm
i}}{2}\,\sum_{i>j>k=1}^{n}f_{abc}T_{i}^{a}T_{j}^{b}T_{k}^{c}\,\sum_{I,J,K\in\\{i,j,k\\}}\\!\\!\\!\epsilon_{IJK}\,\ln^{2}(\alpha_{JK})\ln(\alpha_{IJ})\,r(\alpha_{IJ})\,,$
(5.137)

a strikingly compact and elegant expression. Comparing with Eq. (5.109), we
see that the result is a weighted sum of the three available angle-dependent
cusp anomalous dimensions, strongly constrained by the overall Bose symmetry:
an intriguing result, which is not yet fully understood.

##  6 Real radiation: factorisation and subtraction

The main focus of our report is the infrared singularity structure of virtual
corrections to scattering amplitudes, which were discussed in considerable
detail in the previous Sections. We learnt that infrared singularities arise
from specific momentum configurations, corresponding to the exchange of soft
or collinear virtual massless particles. Ultimately, we saw that the long-
distance sensitivity of a massless gauge amplitude is ruled by a small set of
universal quantities, that are in turn determined by their soft and collinear
anomalous dimensions. As a consequence, the IR content of virtual corrections
is universal, i.e. it is independent of the details of the underlying hard
process.

In this Section, for completeness, we will discuss the factorisation
properties of soft and collinear real radiation, with a two-fold motivation.
On the one hand, factorisation for real radiation is a necessary complement of
the factorisation of virtual corrections, in light of the KLN theorem: in
fact, the theorem states the cancellation of IR singularities among real and
virtual corrections, regardless of the details of the resolved subprocess.
Given the factorisation and universality properties of virtual corrections, it
is natural to expect to be able to also identify and extract singular
contributions to real radiation matrix elements in terms of universal kernels,
multiplying lower-multiplicity matrix elements, which should be the only
process-dependent component. This requires a delicate interplay between
virtual and real corrections, with virtual singularities providing ‘sum-rule’
constraints on real radiation kernels, whose phase-space integrals must effect
the predicted cancellation.

A second motivation is the fact that the factorisation of soft and collinear
real radiation is a fundamental tool for phenomenological applications to
collider observables. As we will discuss is some detail in Section 6.2,
constructing a precise theoretical prediction for an infrared-safe QCD
observable requires being able to implement the cancellation of infrared poles
in an automatic and efficient way. This is the goal pursued by subtraction
methods, which have been the focus of a great deal of theoretical activity in
the past decades.

In what follows, we will begin in Section 6.1 by sketching the factorisation
properties of gauge amplitudes in soft and collinear limits, giving explicit
examples at tree level and referring to the literature for known higher-order
corrections. The simple and well-known results at lowest order will suggest
the possibility of computing the factorisation kernels by means of matrix
elements of fields and Wilson lines, closely related to the matrix elements
discussed earlier in the context of virtual corrections. We will then proceed,
in Section 6.2, to outline the basic ideas underlying the subtraction methods,
which propose to combine real and virtual corrections in a universal way,
delivering directly finite predictions for IR-safe observables, that can be
evaluated numerically. In this context, in Section 6.3, we will discuss a
strategy to define and compute the real radiation kernels (and thus the local
universal infrared counterterms required in the context of subtraction) in a
systematic way, in principle at arbitrary order, so that the cancellation of
real and virtual singularities can be made explicit from the outset. We will
illustrate the results of this approach in Section 6.4, up to NNLO in
perturbation theory, paying particular attention to strongly ordered multiple
soft and collinear limits, that play an important and intricate role for
subtraction beyond NLO.

###  6.1 Factorisation for soft and collinear real radiation

In the past decades, many important studies have focus on extracting the
singular contributions to gauge amplitudes and cross sections in the limits
where a number of massless particles become soft and/or collinear. This vast
research endeavour would deserve a review on its own, since the results were
achieved with a variety of innovative techniques, and in many cases they are
of great phenomenological relevance.

As expected from the KLN theorem, and from general properties of $S$-matrix
elements, under infrared singular limits scattering amplitudes quasi-
completely factorise into universal kernels multiplying lower-point
amplitudes. To be precise, when a subset of particles becomes soft or
collinear, certain Mandelstam invariants will vanish: factorisation is a
feature of leading-power contributions in those invariants; we describe it as
quasi-complete in the sense that soft and collinear kernels are still linked
to lower-point amplitudes by either colour correlations (in the soft case) or
spin correlations (in the collinear case). At cross-section level, the basic
lowest-order factorisation properties have of course been well-known since the
early days of perturbative QCD [248, 249, 250, 506, 287, 289, 507, 238], and
they are directly relevant for the cancellation of infrared divergences at
NLO. Collinear splitting kernels for hadronic cross sections are also
necessary ingredients for all applications of collinear factorisation in QCD,
and they have by now been determined to three loops [354, 508, 356, 509, 510],
with remarkable partial results at four and five loops [511, 395, 397, 512,
513]. Detailed studies at amplitude level are somewhat more recent: the
general form of the factorisation of tree-level gauge amplitudes in soft and
collinear limits was established in the late eighties, using recursion
relations [514, 515] or general expression for the amplitudes derived from
string theory [516, 517]; a combined factorisation including both soft and
collinear limits was later proposed in [518, 519]. Amplitude-level
factorisation was extended to single-unresolved radiation at one-loop in [520,
521, 522, 523, 524, 525, 526], and to double-unresolved radiation at tree
level in [527, 528, 529]: these results are necessary ingredients to implement
the cancellation of infrared divergences at NNLO. At yet higher orders, a
growing body of results is available, including an all-order proof of
collinear factorisation in the planar limit [530], and explicit results for
multiple collinear limits at tree level [531, 532, 533] and one loop [534,
535, 536, 537], as well as for multiple soft limits at tree level [538, 539]
and one loop [540, 541], single-soft limits at two loops [542, 543], and
single-collinear limits at two loops [544, 417]. The organisation of colour
degrees of freedom in multiple soft emissions has been further explored in
Refs. [545, 546].

Of special interest for both the theory and the applications of factorisation
are the results of Ref. [207]: they show that for scattering amplitudes
involving space-like as well as time-like collinear splittings, the naïve form
of collinear factorisation is violated; specifically, one finds that the
splitting kernel in such cases does not depend exclusively on the quantum
numbers of the two collinear particles, but there are leftover colour
correlations with the other particles participating in the scattering process.
This effect can be traced back to the existence of non-vanishing phases in the
divergent parts of the amplitude in the space-like case (see for example Eq.
(5.35)) and to the non-commutativity of the collinear limit with the
$\epsilon$ expansion in dimensional regularisation (see for example the
discussion in Ref. [530]). This breakdown of simple collinear factorisation
does not affect the discussion of fixed-angle scattering amplitudes developed
in Section 5, but it has important phenomenological consequences, since it is
related to the appearance of unconventional logarithmic enhancements in QCD
cross sections with phase-space constraints, such as non-global logarithms
[547], or super-leading logarithms [548, 549, 550]. The finiteness of
sufficiently inclusive hadronic cross sections relies upon the cancellation of
these effects, and is proved to all orders for processes with electroweak
final states [205, 60], with techniques that extend to inclusive jet cross
sections [206]. The precise boundaries of applicability of these techniques to
more intricate or less inclusive observables have not however been determined
in detail.

In what follows, we will not try to review in further detail this great body
of results. Rather, we will focus on some simple examples at tree-level, and
later at one loop, and we will infer a general proposal for the definition of
soft and collinear real-radiation kernels, bearing a close correspondence to
the virtual soft and jet functions defined in the previous section. As we will
see, this will lead to a natural procedure for the construction of local
subtraction counterterms [295].

Following [81], we start by introducing a ket notation for scattering
amplitudes, which is particularly efficient for the discussion of real
infrared emission. Consider a generic scattering process involving massless
final-state QCD partons with momenta $p_{1},p_{2},\dots$. Non-QCD particles
carrying a total momentum $Q$ are always understood. To express colour, spin
and flavour degrees of freedom we introduce different sets of indices:
$\\{c_{1},c_{2},\dots\\}$ for colour, $\\{s_{1},s_{2},\dots\\}$ for spin, and
$\\{f_{1},f_{2},\dots\\}$ for flavour. Given a set of basis vectors in colour
and spin spaces
$\\{\left|\smash{c_{1},c_{2},\dots}{\vphantom{1}}\right\rangle\otimes\left|\smash{s_{1},s_{2},\dots}{\vphantom{1}}\right\rangle\\}$,
the scattering amplitude and the corresponding transition probability, with
spin and colour indices summed, can be written as

$\displaystyle\mathcal{A}_{\,f_{1}f_{2}\dots}^{\,c_{1}c_{2}\dots;\,s_{1}s_{2}\dots}\,(p_{1},p_{2},\dots)$
$\displaystyle\equiv$
$\displaystyle\Big{(}\left\langle\smash{c_{1}c_{2}\dots}{\vphantom{1}}\right|\otimes\left\langle\smash{s_{1}s_{2}\dots}{\vphantom{1}}\right|\Big{)}\left|\smash{\mathcal{A}_{\,f_{1}f_{2}\dots}(p_{1},p_{2},\dots)}{\vphantom{1}}\right\rangle\,,$
$\displaystyle\big{|}\mathcal{A}_{\,f_{1}f_{2}\dots}(p_{1},p_{2},\dots)\big{|}^{2}$
$\displaystyle=$
$\displaystyle\langle\mathcal{A}_{\,f_{1}f_{2}\dots}(p_{1},p_{2},\dots)|\mathcal{A}_{\,f_{1}f_{2}\dots}(p_{1},p_{2},\dots)\rangle\,.$
(6.1)

The effects of radiation on the colour content of the amplitude will be
treated by introducing colour-insertion operators, as discussed in Section
5.1. If, for example, we consider the tree-level matrix element
$\mathcal{A}_{\,g,\,a_{1}\dots a_{n}}(k,p_{1},\dots,p_{n})$, where the
outgoing gluon $g$ carries momentum $k$, colour $c$ and spin polarisation
$\lambda$, we know from our earlier discussions that the leading singular
contribution to this matrix element in the limit when the gluon becomes soft
is given by

$\displaystyle\langle
c|\otimes\langle\lambda\\!\left|\smash{\,\mathcal{A}_{\,g,f_{1}\dots
f_{n}}(k,p_{1},\dots,p_{n})}{\vphantom{1}}\right\rangle_{\rm
soft}\,=\,\epsilon_{\lambda}(k)\cdot
J^{c}(k)\left|\smash{\mathcal{A}_{\,f_{1}\dots
f_{n}}(p_{1},\dots,p_{n})}{\vphantom{1}}\right\rangle\,,$ (6.2)

where $J^{c}$ is the appropriate colour component of the eikonal current
defined in Eq. (5.17),

$\displaystyle{\bf J}^{\mu}(k)\,=\,g\mu^{\epsilon}\,\sum_{i=1}^{n}{\bf
T}_{i}\;\frac{p_{i}^{\mu}}{p_{i}\cdot k}\,,$ (6.3)

which is transverse by colour conservation

$\displaystyle k\cdot{\bf J}(k)\,=\,g\mu^{\epsilon}\sum_{i=1}^{n}{\bf
T}_{i}\,=\,0\,.$ (6.4)

Squaring the soft matrix element in Eq. (6.2), and exploiting colour
conservation, one recovers the soft contribution to the cross section, as in
Eq. (2.36), which we rewrite here as

$\displaystyle\Big{|}\mathcal{A}_{\,g,\,f_{1}\dots
f_{n}}(k,p_{1},\dots,p_{n})\Big{|}_{\rm
soft}^{2}\,=\,-8\pi\alpha_{s}\,\mu^{2\epsilon}\sum_{i\neq
j}\,\mathcal{I}^{(k)}_{ij}\;\Big{|}\mathcal{A}^{\,ij}_{\,f_{1}\dots
f_{n}}(p_{1},\dots,p_{n})\Big{|}^{2}\,,\quad$ (6.5)

where the eikonal prefactor is given by

$\displaystyle\mathcal{I}^{(k)}_{ij}\,\equiv\,\frac{s_{ij}}{s_{ki}\,s_{kj}}\,,$
(6.6)

with $s_{ki}=2k\cdot p_{i}$, and where the squared amplitude on the r.h.s. of
Eq. (6.5) is the colour-connected Born amplitude (without radiation), which in
the present notation reads

$\displaystyle\big{|}\mathcal{A}^{\,ij}_{\,f_{1}\dots f_{n}}\big{|}^{2}$
$\displaystyle\equiv$
$\displaystyle\left\langle\smash{\mathcal{A}_{\,f_{1}\dots
f_{n}}}{\vphantom{1}}\right|{\bf T}^{i}\cdot{\bf
T}^{j}\left|\smash{\mathcal{A}_{\,f_{1}\dots
f_{n}}}{\vphantom{1}}\right\rangle\;.$ (6.7)

Based on our experience with virtual corrections in Section 3.3 and in Section
5, we would expect this result to emerge from a matrix element of Wilson
lines, replacing the hard particles appearing in the Born amplitude: the
expectation is borne out by the observation that the current ${\bf J}^{\mu}$
is a sum of eikonal vertices, and thus can be easily modelled in terms of
Wilson lines. A simple explicit computation indeed shows that the matrix
element

$\displaystyle{\cal
S}_{\,n,\,1}\big{(}k;\\{\beta_{i}\\}\big{)}\,\equiv\,\left\langle\smash{k,\lambda}{\vphantom{1}}\right|\,T\bigg{[}\prod_{i=1}^{n}\Phi_{\beta_{i}}(0,\infty)\bigg{]}\left|\smash{0}{\vphantom{1}}\right\rangle\,,$
(6.8)

computed at tree level, precisely reproduces the soft factor in Eq. (6.2), and
thus, upon squaring, Eq. (6.5).

Figure 34: Double soft gluon emission from a generic hard amplitude.

The analysis performed for virtual corrections, leading to the conclusion that
soft emissions can be described by means Wilson lines, makes us confident that
the simple argument leading to Eq. (6.2), and to the identification of the
single soft emission operator in Eq. (6.8), will generalise to the emission of
multiple soft gluons at leading power. Indeed, the limit of QCD tree-
amplitudes when two gluons become simultaneously soft was studied by Berends
and Giele [515] and by Catani [551], showing precisely a factorisation of this
kind. Following Ref. [529], we introduce the double-soft emission current by
writing

$\displaystyle\langle
a_{1}a_{2}|\otimes\langle\lambda_{1}\lambda_{2}\left|\smash{\mathcal{A}_{\,gg,f_{1}\ldots
f_{n}}(k_{1},k_{2},p_{1},\dots,p_{n})}{\vphantom{1}}\right\rangle_{\rm
soft}\,=\,\epsilon^{\mu_{1}}_{\lambda_{1}}(k_{1})\epsilon^{\mu_{2}}_{\lambda_{2}}(k_{2})$
$\displaystyle\hskip
142.26378pt\times\,J^{a_{1}a_{2}}_{\mu_{1}\mu_{2}}(k_{1},k_{2})\left|\smash{\mathcal{A}_{\,f_{1},\dots,f_{n}}(p_{1},\dots,p_{n})}{\vphantom{1}}\right\rangle\,.$
(6.9)

The double-soft current receives contribution from three different diagram
topologies, depicted in Fig. 34. The explicit expression for the double-soft
current $J_{a_{1}a_{2}}^{\mu_{1}\mu_{2}}$ is given by [529]

$\displaystyle J^{a_{1}a_{2}}_{\mu_{1}\mu_{2}}(k_{1},k_{2})$ $\displaystyle=$
$\displaystyle 4\pi\alpha_{s}\mu^{2\epsilon}\Bigg{\\{}\sum_{\ell\neq
m}T_{m}^{a_{1}}\,T_{\ell}^{a_{2}}\,\frac{p_{m}^{\mu_{1}}}{p_{m}\cdot
k_{1}}\,\frac{p_{\ell}^{\mu_{2}}}{p_{\ell}\cdot k_{2}}$ $\displaystyle\hskip
28.45274pt+\,\sum_{m}\,\bigg{[}\,\frac{p_{m}^{\mu_{1}}\,p_{m}^{\mu_{2}}}{p_{m}\cdot\left(k_{1}+k_{2}\right)}\left(\frac{T_{m}^{a_{2}}T_{m}^{a_{1}}}{p_{m}\cdot
k_{2}}\,+\,\left(1\leftrightarrow 2\right)\right)$ $\displaystyle\hskip
28.45274pt+\,\,{\rm
i}f^{\,\,\,a_{1}a_{2}}_{a}\,T_{m}^{a}\,\,\frac{p_{m}\cdot\left(k_{2}-k_{1}\right)g^{\mu_{1}\mu_{2}}+2\,p_{m}^{\mu_{1}}\,k_{1}^{\mu_{2}}-2\,p_{m}^{\mu_{2}}\,k_{2}^{\mu_{1}}}{2k_{1}\cdot
k_{2}\,p_{m}\cdot\left(k_{1}+k_{2}\right)}\bigg{]}\Bigg{\\}}\,.$

The correspondence between the expression in Eq. (6.1) and the diagrams in
Fig. 34 is straightforward: the first term stems from diagrams of type (a),
the second contribution reflects diagrams of type (b), and finally the last
term, proportional to $f_{a}^{\;\;a_{1},a_{2}}$, clearly emerges from diagrams
of type (c). At squared-amplitude level, the factorisation formula for the
configuration at hand can be written as

$\displaystyle\Big{|}\mathcal{A}_{\,gg,f_{1}\ldots
f_{n}}(k_{1},k_{2},p_{1},l\dots,p_{n})\Big{|}^{2}_{\rm
soft}\,=\,2\left(4\pi\alpha_{s}\mu^{2\epsilon}\right)^{2}$ (6.11)
$\displaystyle\qquad\times\,\Bigg{[}\sum_{i\neq j}\sum_{k\neq
l}\,\mathcal{I}_{ij}^{(1)}\,\mathcal{I}_{kl}^{(2)}\,\Big{|}\mathcal{A}_{\,f_{1}\ldots
f_{n}}^{ijkl}(p_{1},\dots,p_{n})\Big{|}^{2}+\sum_{i\neq
j}\,\mathcal{I}_{ij}^{(12)}\Big{|}\mathcal{A}_{\,f_{1}\ldots
f_{n}}^{ij}(p_{1},\dots,p_{n})\Big{|}^{2}\Bigg{]}\,,$

where the first term in square brackets is the factorised contribution,
proportional to the double-colour-connected matrix element

$\displaystyle\big{|}\mathcal{A}_{\,f_{1}\ldots
f_{n}}^{ijkl}\Big{|}^{2}\,\equiv\,\left\langle\smash{\mathcal{A}_{\,f_{1}\ldots
f_{n}}}{\vphantom{1}}\right|\Big{\\{}{\bf T}^{i}\cdot{\bf T}^{j},{\bf
T}^{k}\cdot{\bf T}^{l}\Big{\\}}\left|\smash{\mathcal{A}_{\,f_{1}\ldots
f_{n}}}{\vphantom{1}}\right\rangle\,,$ (6.12)

while the double emission from a single colour dipole si expressed by the
kernel $\mathcal{I}_{ij}^{(12)}$, which can be read off from Eq.(109) in
[529], after appropriate relabelling of the momenta.

As before, the double-soft current can be extracted from a matrix element of
Wilson lines: quite naturally, we propose

$\displaystyle{\cal
S}_{\,n,\,2}\big{(}k_{1},k_{2};\\{\beta_{i}\\}\big{)}\,\equiv\,\left\langle\smash{k_{1},\lambda_{1};k_{2},\lambda_{2}}{\vphantom{1}}\right|\,T\bigg{[}\prod_{i=1}^{n}\Phi_{\beta_{i}}(0,\infty)\bigg{]}\left|\smash{0}{\vphantom{1}}\right\rangle\,.$
(6.13)

It is straightforward to show that Eq. (6.13), evaluated at tree level, indeed
reproduces the current $J^{a_{1}a_{2}}_{\mu_{1}\mu_{2}}$, contracted with
gluon polarisation vectors as in Eq. (6.9). Indeed, expanding the product of
Wilson lines to order $g^{2}$, there are three different configurations that
return non-vanishing contributions: one may expand to first order in $g$ two
of the $n$ Wilson lines, obtaining the first term in Eq. (6.1), or one may
expand only one Wilson line to order $g$, and then include a Lagrangian
interaction, to get the non-abelian component; finally, one can expand a
single line up to order $g^{2}$: in this case, the path ordering that defines
$\Phi(0,\infty)$ precisely returns the remaining contributions in the first
line of Eq. (6.1).

It is important to stress that the singular soft configurations we have just
described have been extracted by taking the two soft momenta to vanish at the
same rate, formally replacing $k_{1}\to\lambda k_{1}$ and $k_{2}\to\lambda
k_{2}$, taking $\lambda\to 0$ and retaining the leading power in $\lambda$.
This we describe as a democratic IR limit. In the context of subtraction, as
we will see, it is also important to identify hierarchical, or strongly-
ordered limits, corresponding to configurations where one gluon is much softer
than the other, i.e. one of the two momenta vanishes at a faster rate with
respect to the other. The expression of the corresponding current can be
easily deduced by taking the leading term in the $k_{1}$ ($k_{2}$) expansion
of $J_{a_{1}a_{2}}^{\mu_{1}\mu_{2}}$ at amplitude level, and of
$\mathcal{I}_{cd}^{(12)}$ at cross-section level. One gets

$\displaystyle\big{(}J^{\,\rm
s.o.}\big{)}^{a_{1}a_{2}}_{\mu_{1}\mu_{2}}(k_{1},k_{2})\\!$ $\displaystyle=$
$\displaystyle\\!g_{s}\mu^{\epsilon}\Bigg{(}\sum_{m=1}^{n}T^{a_{2}}_{m}\,\frac{p_{m,\,\mu_{2}}}{p_{m}\cdot
k_{2}}\,\delta^{a_{1}a}+{\rm
i}\,f^{a_{1}a_{2}a}\,\frac{k_{1,\,\mu_{2}}}{k_{1}\cdot
k_{2}}\Bigg{)}\sum_{\ell=1}^{n}\,T_{\ell,\,a}\,\frac{p_{\ell,\,\mu_{1}}}{p_{\ell}\cdot
k_{1}}\,,$ $\displaystyle\mathcal{I}_{ij}^{(12)\,{\rm s.o.}}\\!$
$\displaystyle=$
$\displaystyle\\!-2C_{A}\,\mathcal{I}_{ij}^{(2)}\;\Big{[}\mathcal{I}_{i2}^{(1)}+\mathcal{I}_{j2}^{(1)}-\mathcal{I}_{ij}^{(1)}\Big{]}\,.$
(6.14)

For the purposes of constructing subtraction counterterms in a systematic way,
it would be interesting to be able to derive strongly ordered limits in terms
of matrix elements of Wilson lines, as done above for the democratic case.
This approach will be briefly discussed in Section 6.4. To conclude our
discussion of soft factorisations for real radiation at tree level, we recall
that the eikonal approximation is spin-independent, so the formulas that we
have described can be applied to soft gluon radiation from hard gluons, simply
by placing the colour generators in the appropriate (adjoint) representation;
furthermore, the case of massless quark radiation (through an intermediate
gluon which splits into a quark-antiquark pair) can be treated similarly and
does not present new difficulties. We remark also that the soft factorisation
is quasi complete, as noted at the beginning of this Section, in the sense
that eikonal kernels and Born-like matrix elements are not entirely
independent of each other, but are still connected by colour correlations.

To complete the picture of tree-level infrared factorisation for real
radiation, we must consider collinear splittings as well. In this case, we
will also find a quasi complete factorisation, but with colour correlations
replaced by spin correlations. The trivial colour structure of tree-level
collinear factorisation is best displayed by picking the most suitable colour
basis: in this case, a natural choice is the colour trace basis of Ref. [517],
later extensively employed at loop level [552, 521]. For multi-gluon
amplitudes, this organisation is intuitively motivated by string theory, where
tree-level gluon amplitudes are computed as expectation values of conformal
vertex operators inserted on the boundaries of the tree-level string world
sheet, which can be conformally mapped to a circle. Colour degrees of freedom
are carried by Chan-Paton factors, which are generators for the fundamental
representation of the gauge group associated with each external state. The
full tree-level amplitude is then constructed as a sum of sub-amplitudes, each
corresponding to a cyclic ordering of the vertex operators on the boundary of
the circle, and each accompanied by the corresponding trace of Chan-Paton
factors. For $n$ gluons, one may write

$\displaystyle{\cal A}_{n}^{a_{1}\ldots
a_{n}}(p_{1},\ldots,p_{n})\,=\,\sum_{\sigma\in S_{n}/Z_{n}}{\rm
Tr}\Big{[}T^{a_{\sigma(1)}}\ldots
T^{a_{\sigma(n)}}\Big{]}\,A_{n}\left(p_{\sigma(1)},\ldots,p_{\sigma(n)}\right)\,,$
(6.15)

where the sum is over $(n-1)!$ non-cyclic permutations of the $n$ external
gluons, and we omitted polarisation indices for simplicity. This string-
inspired organisation of gauge amplitudes can be generalised to loop
amplitudes, where at $\ell$ loops one must allow for products of up to
$\ell+1$ trace factors, and to amplitudes with massless fermions.

The sub-amplitudes $A_{n}$ are gauge invariant, and they enjoy a number of
properties that make them very useful for practical calculations [517]. In the
present context, the most notable aspect is the fact that, for each sub-
amplitude, collinear poles arise only from configurations where the two
particles that become collinear are adjacent in the selected cyclic ordering.
Thus, for example, $A_{5}(p_{1},p_{2},p_{3},p_{4},p_{5})$ will have a
collinear pole when $p_{2}$ becomes collinear to $p_{3}$, but not when $p_{2}$
becomes collinear to $p_{5}$: indeed, within the set of diagrams contributing
to this ordering, there is no diagram where $p_{2}$ and $p_{5}$ emerge from
the same internal vertex, so that no propagator becomes singular as $p_{2}$
becomes parallel to $p_{5}$. Picking then a sub-amplitude where two momenta
$p_{i}$ and $p_{j}$ are adjacent, and the corresponding gluons carry
helicities $s_{i}$ and $s_{j}$, one finds that, in the collinear limit, the
sub-amplitude factorises as

$\displaystyle A_{n}^{\,\ldots
s_{i}s_{j}\ldots}(\ldots,p_{i},p_{j},\ldots)\Big{|}_{p_{i}||\,p_{j}}\,=\,\sum_{s=\pm}{\rm
Split}_{-s}^{(0)}(p_{i},s_{i};p_{j},s_{j})\,A_{n-1}^{\,\ldots
s\ldots}(\ldots,p_{i}+p_{j},\ldots)$ (6.16)

where the parent gluon carrying momentum $p_{i}+p_{j}$ is taken to carry
outgoing helicity $(+s)$, so that it enters the tree-level splitting amplitude
Split${}_{-s}^{(0)}$ with incoming helicity $(-s)$. The factorisation in Eq.
(6.16) encodes the singular behavior of $A_{n}$ as $s_{ij}=2p_{i}\cdot
p_{j}\to 0$, which turns out to be proportional to $s_{ij}^{-1/2}$ (giving the
expected $1/s_{ij}$ singularity at cross-section level). Tree-level splitting
amplitudes in this basis were derived in [514] and in [517], and are listed in
those references for all flavour and spin pairings consistent with QCD
couplings. As an example, one can write the splitting amplitude for a gluon
branching into two gluons as [530, 534]

$\displaystyle{\rm Split}_{-s}^{(0)}(p_{i},s_{i};p_{j},s_{j})\\!$
$\displaystyle=$
$\displaystyle\\!g\mu^{\epsilon}\,\frac{2}{s_{ij}}\,\Big{[}\epsilon_{s_{i}}(p_{i})\cdot\epsilon_{s_{j}}(p_{j})\,\,p_{i}\cdot\epsilon_{s}(p_{i}+p_{j})$
$\displaystyle+\,\epsilon_{s_{i}}(p_{i})\cdot\epsilon_{s}(p_{i}+p_{j})\,\,p_{i}\cdot\epsilon_{s_{j}}(p_{j})\,-\,\epsilon_{s_{j}}(p_{j})\cdot\epsilon_{s}(p_{i}+p_{j})\,\,p_{j}\cdot\epsilon_{s_{i}}(p_{i})\Big{]}\,,$

where the parent gluon is taken on shell (i.e. in the exact collinear limit),
and the $1/s_{ij}$ singularity is softened to a square root by the products of
polarisation vectors in square brackets.

If one does not wish to be constrained by the choice of a specific colour
basis, it is possible to rephrase the collinear factorisation of massless
multi-particle amplitudes in terms of colour operators (which can then be
evaluated in any basis), as was done for soft emissions. In this case, one can
write the tree-level factorisation directly in colour space as

$\displaystyle\left|\smash{\,\mathcal{A}^{\ldots
s_{i}s_{j}\ldots}_{n}(\ldots,p_{i},p_{j},\ldots)}{\vphantom{1}}\right\rangle_{p_{i}||p_{j}}\,=\,{\bf
Sp}^{(0)}_{-s}(p_{i},s_{i};p_{j},s_{j})\left|\smash{\mathcal{A}^{\ldots
s\ldots}_{n-1}(\ldots,p_{i}+p_{j},\ldots)}{\vphantom{1}}\right\rangle\,,$
(6.18)

where the splitting matrix ${\bf Sp}^{(0)}_{-s}$ is now a colour operator
acting on the right on the colour index of the parent particle, and on the
left on the colour indices of the collinear pair. At tree level, the relation
between the operator ${\bf Sp}^{(0)}_{-s}$ and the colourless splitting
amplitude Split${}_{-s}^{(0)}$ is trivial, since each possible flavour
splitting can be proportional to only a single colour tensor. Thus, for
example, for a gluon with colour index $a$ splitting into gluons with colour
indices $b$ and $c$, the splitting operator is obtained from Eq. (6.1) by
simply inserting a factor of $(-{\rm i}f_{abc})$. More complicated colour
structures will, of course, arise for multiple collinear splittings at tree
and loop level.

To discuss collinear factorisation at cross-section level, taking into account
the non-trivial spin structure, it is useful to define a squared amplitude
that is summed over all the spin indices, except for those of a select
particle of flavour $f_{1}$, which will be interpreted as the parent particle
in the subsequent collinear splitting. Following Ref. [529], we define

$\displaystyle\mathcal{T}_{\,f_{1}\dots
f_{n}}^{\,s_{1}s^{\prime}_{1}}(p_{1},\ldots,p_{n})\,=\,\sum_{s_{2},\ldots,s_{n}}\sum_{c_{i}}\mathcal{A}_{\,f_{1}\ldots
f_{n}}^{\,c_{1}\ldots c_{n};\,s_{1}\ldots
s_{n}}(p_{1},\dots,p_{n})\Big{[}\mathcal{A}_{\,f_{1}\ldots
f_{n}}^{\,c_{1}\dots c_{n};\,s^{\prime}_{1}\ldots
s_{n}}(p_{1},\dots,p_{n})\Big{]}^{\dagger}\,.$ (6.19)

Next, we need to be more precise in parametrising the collinear limit. For two
particles of flavour $f_{i}$ and $f_{j}$, carrying momenta $p_{i}$ and
$p_{j}$, we identify a light-like momentum $p^{\mu}$ as the collinear
direction, and we pick an auxiliary light-like vector $n^{\mu}$. How the
collinear direction is approached is specified by the transverse momentum
vector $k^{\mu}_{\bot}$, by definition orthogonal to both $p^{\mu}$ and
$n^{\mu}$. Each collinear parton carries a collinear momentum fraction which
can be defined by $z_{a}=s_{an}/(s_{in}+s_{jn})$, with $a=i,j$ and
$s_{an}=2p_{a}\cdot n$, so that $z_{i}+z_{j}=1$; hence, one may use $z\equiv
z_{i}$ and $z_{j}=1-z$. This results in the Sudakov parametrisation

$\displaystyle
p_{i}\,=\,z\,p^{\mu}+k_{\bot}^{\mu}-\frac{k_{\bot}^{2}}{z}\,\frac{n^{\mu}}{2p\cdot
n}\,,\quad\\!p_{j}\,=\,(1-z)\,p^{\mu}-k_{\bot}-\frac{k_{\bot}^{2}}{1-z}\,\frac{n^{\mu}}{2p\cdot
n}\,,\quad\\!s_{ij}\,=\,-\frac{k_{\bot}^{2}}{z(1-z)}\,.$ (6.20)

The collinear limit is approached as $k_{\bot}\rightarrow 0$, and the leading-
power behaviour of the squared matrix element is encoded by [529]

$\displaystyle\Big{|}\mathcal{A}_{\,f_{1}\ldots
f_{n}}(p_{1},\dots,p_{n})\Big{|}^{2}_{p_{i}||p_{j}}\,=\,\frac{8\pi\alpha_{s}\mu^{2\epsilon}}{s_{ij}}\,\,\mathcal{T}^{ss^{\prime}}_{f,f_{1}\ldots
f_{n}}(p,p_{1},\ldots,p_{n})\,\widehat{P}_{f_{i}f_{j}}^{\,ss^{\prime}}\big{(}z,k_{\bot};\epsilon\big{)}\,,$
(6.21)

where on the r.h.s. it is understood that particles $f_{i}$ and $f_{j}$,
carrying momenta $p_{i}$ and $p_{j}$, are omitted from the list, and they have
been replaced by a single particle $f$ carrying momentum $p$. The possible
flavour combinations are determined by the Feynman rules of the theory under
consideration. For QCD, the kernels $\widehat{P}_{a_{i}a_{j}}^{ss^{\prime}}$
are the $d$-dimensional DGLAP splitting functions, represented as spin
operators acting on the spin indices $s,s^{\prime}$ of the spin tensor
$\mathcal{T}$. The splitting functions are given by squares of the splitting
amplitudes ${\bf Sp}^{(0)}_{-s}$, summed over colours, after extracting the
coupling and the factor of $1/\sqrt{s_{ij}}$. Their explicit expressions are
of course well known, and in the present normalisation they are given by

$\displaystyle\widehat{P}_{qg}^{ss^{\prime}}(z,k_{\bot};\epsilon)\\!$
$\displaystyle=$
$\displaystyle\\!\delta_{ss^{\prime}}\,C_{F}\bigg{[}\frac{1+z^{2}}{1-z}-\epsilon(1-z)\bigg{]}\,,$
$\displaystyle\widehat{P}_{gq}^{ss^{\prime}}(z,k_{\bot};\epsilon)\\!$
$\displaystyle=$
$\displaystyle\\!\delta_{ss^{\prime}}\,C_{F}\bigg{[}\frac{1+(1-z)^{2}}{z}-\epsilon
z\bigg{]}\,,$ (6.22)
$\displaystyle\widehat{P}_{q\bar{q}}^{\mu\nu}(z,k_{\bot};\epsilon)\\!$
$\displaystyle=$
$\displaystyle\\!T_{R}\bigg{[}-g^{\mu\nu}+4z(1-z)\frac{k_{\bot}^{\mu}k_{\bot}^{\nu}}{k_{\bot}^{2}}\bigg{]}\,,$
$\displaystyle\widehat{P}_{gg}^{\mu\nu}(z,k_{\bot};\epsilon)\\!$
$\displaystyle=$
$\displaystyle\\!2C_{A}\bigg{[}-g^{\mu\nu}\Big{(}\frac{z}{1-z}+\frac{1-z}{z}\Big{)}-2(1-\epsilon)z(1-z)\frac{k_{\bot}^{\mu}k_{\bot}^{\nu}}{k_{\bot}^{2}}\bigg{]}\,.$

Notice that all the kernels are symmetric under the exchange of a quark with
an anti-quark, i.e. $\widehat{P}_{Xq}=\widehat{P}_{X\bar{q}}$. From Eq. (6.1)
it is evident that the spin sensitivity of the kernels, and, consequently, the
one of the spin tensor, is trivial in the case of a parent fermion, while
gluon splittings retain a non-trivial azimuthal dependence. Also in the
collinear sector, we are interested in expressing the singular kernels in
terms of universal, gauge invariant operators. Let us begin by by considering
gluon radiation from an outgoing quark: in this case, we may take inspiration
from the virtual jet function defined in Eq. (4.9), and implement collinear
radiation, as was done for the soft current, by including an extra gluon in
the final state. At amplitude level, this amounts to introducing the matrix
element

$\displaystyle\overline{u}_{s}(p){\cal
J}_{q,\,1}^{\lambda}\big{(}k;p,n\big{)}\,\equiv\,\left\langle\smash{p,s;k,\lambda}{\vphantom{1}}\right|\,T\Big{[}\Phi_{n}(\infty,0)\,\psi(0)\Big{]}\left|\smash{0}{\vphantom{1}}\right\rangle\,.$
(6.23)

The function ${\cal J}_{q,\,1}^{\lambda}$ is a spin matrix, and at tree level
it must reproduce the collinear singularity of the splitting amplitudes,
stripped of colour information. The dependence on the vector $n^{\mu}$
defining the Wilson line mimicks the gauge dependence within the families of
axial gauges, and choosing $n^{2}=0$ one can reproduce the Sudakov
parametrisation of the collinear region. To test our ansatz, we can square Eq.
(6.23), take a Fourier transform, and introduce the (unpolarised) cross-
section-level radiative jet function

$\displaystyle J_{q,\,1}\big{(}k;l,p,n\big{)}\equiv\int d^{d}x\,{\rm e}^{{\rm
i}l\cdot
x}\,\sum_{\lambda,s}\left\langle\smash{0}{\vphantom{1}}\right|\,T\,\Big{[}\Phi_{n}(\infty,x)\,\psi(x)\Big{]}\left|\smash{p,s;k,\lambda}{\vphantom{1}}\right\rangle$
$\displaystyle\hskip
85.35826pt\times\,\left\langle\smash{p,s;k,\lambda}{\vphantom{1}}\right|\,\overline{T}\,\Big{[}\overline{\psi}(0)\,\Phi_{n}(0,\infty)\Big{]}\left|\smash{0}{\vphantom{1}}\right\rangle\,,$
(6.24)

where the complex-conjugate matrix element is anti-time-ordered by the action
of the operator $\overline{T}$, and the vector $l^{\mu}$ plays the role of the
total momentum flowing into the final state. We can now perform a simple test
of the correctness of our assumptions by computing, in Feynman gauge, the
lowest perturbative order for Eq. (6.1). It receives contributions from three
different Feynman diagrams, displayed in Fig. 35.

Figure 35: Feynman graphs contributing to the radiative quark jet function,
defined in Eq. (6.1), at lowest order.

Taking, for simplicity, $n^{2}=0$, we find

$\displaystyle
J_{q,\,1}^{(0)}\big{(}k;l,p,n\big{)}\,=\,\frac{4\pi\alpha_{s}}{(l^{2})^{2}}\,C_{F}\,(2\pi)^{d}\delta^{d}\left(l-p-k\right)\left[-l\hskip-4.49997pt/\gamma_{\mu}p\hskip-4.49997pt/\gamma^{\mu}l\hskip-4.49997pt/+\frac{l^{2}}{k\cdot
n}\big{(}\,l\hskip-4.49997pt/n\hskip-4.49997pt/p\hskip-4.49997pt/+p\hskip-4.49997pt/n\hskip-4.49997pt/l\hskip-4.49997pt/\,\big{)}\right]\,.$
(6.25)

It is easy to verify the correspondence between the contributions of the three
diagrams in Fig. 35 and the axial-gauge calculation of tree-level splitting
kernel in Ref. [529]. Notice however that in Eq. (6.25) the collinear limit
for $k$ and $p$, corresponding to $l^{2}\to 0$, still has to be performed.
This can be done exploiting the Sudakov parametrisation in Eq. (6.20), and
truncating the expressions for the two collinear momenta at leading order in
the transverse momentum, as

$\displaystyle p^{\mu}\,=\,zl^{\mu}+{\cal O}\left(l_{\perp}\right)\,,\qquad
k^{\mu}\,=\,(1-z)l^{\mu}+{\cal O}\left(l_{\perp}\right)\,,\qquad
n^{2}\,=\,0\,.$ (6.26)

At leading power in $l_{\bot}$ we easily find

$\displaystyle
J_{q,\,1}^{(0)}\big{(}k;l,p,n\big{)}\,=\,\frac{8\pi\alpha_{s}}{l^{2}}\,C_{F}\,(2\pi)^{d}\,\delta^{d}\left(l-p-k\right)\,l\hskip-4.49997pt/\,\left[\frac{1+z^{2}}{1-z}-\epsilon\left(1-z\right)\right]\,,$
(6.27)

up to corrections of relative order $l_{\bot}^{2}$. In the square bracket, as
expected, we recognise the leading order unpolarised DGLAP splitting function
$P_{qg}$, as reported in Eq. (6.1).

Clearly, the number and complexity of splitting functions at amplitude and
cross-section level rapidly increases at higher orders, both because loop
corrections must be included, and because multiple real radiation comes into
play; furthermore, when multiple collinear particles are radiated, the
identification of strongly-ordered limits becomes relevant. The operator
matrix element expressions that we have identified in these simple examples,
however, point the way to a useful systematic approach, linking together
virtual corrections and real radiation. We will now proceed by first
introducing the problem of infrared subtraction is some detail, and then
discussing a set of possible definitions for real-radiation soft and collinear
counterterms to all orders in perturbation theory.

###  6.2 Introducing the problem of infrared subtraction at NLO

The discussion in Section 6.1 lays the groundwork for constructing a practical
and general implementation of the cancellation of IR divergences in IR-safe
observables, which follows from the KLN theorem discussed in Section 1.2.
After our discussion of the general cancellation theorems, and our tour of
many technical aspects of IR factorisation, the reader may well wonder why
this cancellation should still be described as a ‘problem’. Indeed, so long as
one is computing total cross sections, or highly inclusive observables, the
cancellation between divergences coming from virtual corrections and those
stemming from the phase space integration of unresolved radiation can be
performed analytically, even to rather high orders in perturbation theory.
What changes the game, and raises the stakes, is the nature of typical
collider observables. Even the definition of IR-safe observables, and
verifying their IR-safety to all orders, become non-trivial questions.
Experimental observables are very often defined in terms of particle jets, and
jets in turn are typically defined in terms of clustering algorithms;
furthermore, experimental constraints imply complex phase space cuts on
kinematic variables, often arising at the level of triggers for data
collection: these must be implemented in efficient and versatile numerical
codes. In practice, this means that phase-space integrals of real-radiation
matrix elements must be performed numerically, and this in turn requires
subtracting their singular contributions before performing the integration in
$d=4$. Another way to state the problem is that, in a collider environment,
versatility means being able to provide fully differential distribution for
interesting observables, in order to be able to cope with different sets of
experimental cuts. As a consequence, it becomes necessary to perform the
extraction of singular contributions in a universal way, for arbitrary
configurations of the resolved particles. The factorisation properties of real
radiation matrix elements, sketched in Section 6.1, suggest that completing
this program is possible in principle, however a general and efficient
practical implementation at high orders is far from easy to achieve.

In this section, we will present the basic ideas that have been applied to the
cancellation problem, in the relatively simple context of NLO calculations,
where, arguably, the problem is satisfactorily solved. In the past decades,
two main strategies have been developed to solve the problem at NLO: the
slicing [553, 554, 555] approach and the subtraction [80, 556, 81, 557, 558,
559] approach. In what follows, we will mainly develop the subtraction
viewpoint, but, in order to present the basic idea behind these methods, we
illustrate them in the context of a toy model. Start by considering the
integral

$\displaystyle I(F)\,=\,\lim_{\epsilon\rightarrow
0}\Bigg{[}\int_{0}^{1}\frac{dx}{x}\,x^{\epsilon}\,F(x)-\frac{1}{\epsilon}\,F(0)\Bigg{]}\,,$
(6.28)

where $F(x)$ is a generic function admitting a Taylor expansion around $x=0$,
and the first term mimicks the phase-space integration of the real radiation,
while the second term plays the role of the virtual correction. The goal of
both slicing and subtraction schemes is computing $I(F)$ without relying on
the analytic evaluation of the integral over $x$, which involves the
arbitrarily complicated function $F$. The slicing approach tackles the problem