where $n_{s}$ and $n_{c,\,i}$ count positive powers of $\lambda$ arising from
numerator factors. With our assignments, one readily verifies that
$\omega_{s}=0$ for the reduced graph in Fig. 22(a), and $\omega_{c}=0$ for the
reduced graph in the second panel of Fig. 22(b). In general, provided the
lines connecting subgraphs have been consistently assigned, superficial
degrees of divergence for subgraphs are additive. For a generic pinch surface
$\Sigma$ for form factor graphs one can then define

$\displaystyle\omega_{\Sigma}\,=\,\sum_{i=1}^{2}\omega_{c,\,i}+\omega_{s}\,.$
(3.44)

The next step in the procedure is to rewrite Eq. (3.43) and Eq. (3.44) in
terms of the numbers of vertices and external lines of the various subgraphs,
using elementary graphical relations, such as the Euler identity connecting
the numbers of loops, lines and vertices. There are a number of subtleties in
the analysis: for example it is necessary to pay attention to the possible
existence of lower-dimensional pinch surfaces in a given homogeneous integral,
and to the fact that suppression factors for collinear numerators are
polarisation-dependent, and thus gauge-dependent. The details of the procedure
were worked out in Refs. [278, 279, 280], and are reviewed, for example, in
Refs. [2, 61, 67, 281, 282]. In what follows, we will just provide a set of
simple and, hopefully, intuitive arguments for the structure of the
constraints emerging from power counting, and we will summarise the final
results.

Consider for example the lines directly connecting the soft and hard subgraphs
in Fig. 24(a): each such line would provide a power suppression, and they
ultimately cannot contribute to divergent momentum configurations. To see
that, let a soft gluon with momentum $k$ attach directly to a hard line
carrying momentum $q\sim Q$, and assume that the graph contributes at leading
power before the soft gluon is inserted. In case the hard line is a gluon, the
corresponding off-shell propagator contributes a factor of $1/Q^{2}$ before
the soft gluon insertion. After the insertion, the hard propagator is doubled,
contributing a factor of $1/Q^{4}$, while the new three-gluon vertex can
contribute at most a factor of $Q$: one has effectively gained a power
suppression, which is sufficient to make the new graph finite if the graph was
logarithmically divergent before the insertion. One can similarly argue that a
graph with a soft fermion line connecting soft and jet subgraphs is suppressed
by one power of the hard scale with respect to the same graph with a soft
gluon, since soft fermion denominators are linear in the soft scaling
parameter.

Turning now to numerator suppressions, we first note that in the soft limit
the only source of suppression is the presence of three-gluon vertices, which
are linear in momenta and thus provide a single power of $\lambda$ in the
numerator: the suppression factor $n_{s}$ thus simply equals the number of
soft three-gluon vertices. The situation concerning numerator suppressions in
the collinear limit is much more subtle and interesting, since suppression
factors are spin-dependent. In order to understand this, consider a fermion
line carrying momentum $p$ and emitting a collinear gluon with momentum $k$.
Both $p$ and $k$ can depend on collinear loop momenta, and thus they can have
transverse components, which will however vanish in the collinear limit, where
both particles reach the mass shell. The numerator factor associated with the
emission vertex and the neighboring propagator reads

$\displaystyle\left(\not{p}+\not{k}\right)\gamma_{\alpha}\,\not{p}\,=\,-\gamma_{\alpha}\left(\not{p}+\not{k}\right)\,\not{p}\,+\,2\left(p+k\right)_{\alpha}\,\not{p}\,.$
(3.45)

In the collinear limit, $p$ and $k$ become light-like and the first term
vanishes; as a consequence, away from the strict limit it will provide a power
suppression proportional to $k_{\perp}$, i.e. a factor of $\sqrt{\lambda}$ for
every collinear vertex. The second term in Eq. (3.45) does not provide any
suppression, but one should keep in mind that the index $\alpha$ will be
contracted with the neighbouring gluon propagator, whose numerator represents
a sum over gluon polarisations; in the collinear limit, assuming the jet to be
in the $+$ direction, the only surviving polarisation coupling to the vertex
will be a $-$ component, which corresponds to an unphysical gluon. Such gluons
would be absent in a physical gauge, and must decouple, after summing over
diagrams, in a covariant gauge [61, 281]. To illustrate what happens in a
physical gauge, note that a gluon connecting the jet subgraph to the hard (or
soft) subgraph will carry a factor of the axial-gauge gluon propagator

$\displaystyle G^{\mu\nu}(k)\,=\,\frac{1}{k^{2}+{\rm
i}\eta}\left(-g^{\mu\nu}+\frac{n^{\mu}k^{\nu}+n^{\nu}k^{\mu}}{n\cdot
k}-n^{2}\,\frac{k^{\mu}k^{\nu}}{(n\cdot k)^{2}}\right)\,,$ (3.46)

where $n^{\mu}$ is the gauge vector, and the gauge condition imposes $n\cdot
A=0$. As illustrated by Eq. (3.45), if the gluon connects the jet to the hard
subgraph, at the vertex where $G^{\mu\nu}(k)$ attaches to the jet, it will be
contracted with a collinear momentum, proportional to the gluon momentum $k$.
This will yield a factor[61]

$\displaystyle k_{\mu}G^{\mu\nu}(k)\,=\,\frac{1}{n\cdot
k}\left(n^{\nu}-\frac{n^{2}k^{\nu}}{n\cdot k}\right)\,.$ (3.47)

This factor has no pole at $k^{2}=0$ (except in the soft limit $k\sim 0$), so
that such a graph is power-suppressed. In covariant gauges, the argument
breaks down, since the contraction $k_{\mu}G^{\mu\nu}(k)$ does not feature the
same suppression. For instance, in Feynman gauge, we have

$\displaystyle k_{\mu}G^{\mu\nu}(k)\,=\,-\frac{k^{\nu}}{k^{2}}\,.$ (3.48)

We see that in Feynman gauge multiple gluons may connect the jets subgraphs to
hard subgraph $\mathcal{H}$, but Eq. (3.48) suggests that such gluons must
carry unphysical polarisations. This, in turn, means that, when computing a
gauge-invariant quantity, these configurations will be suppressed by means of
the Ward Identity, when all the relevant diagrams have been summed. On a
diagram-by-diagram basis, this is not guaranteed, and one needs to further
manipulate the longitudinally polarised gluons to factor $\mathcal{J}_{i}$
from $\mathcal{H}$.

It is hopefully clear from these arguments that only a small subset of the
pinch surfaces depicted in Fig. 24(a) will actually yield divergent
contributions. A detailed analysis [2, 61, 67, 281] yields the following
results.

  1. 1.

The superficial degree of divergence $\omega_{\Sigma}$ is bounded by
$\omega_{\Sigma}\geq 0$, implying that all infrared divergences are
logarithmic. This has important consequences in what follows, since it means
that even a single extra power suppression, such as those discussed above,
makes the diagram superficially finite.

  2. 2.

As a first consequence, pinch surfaces with lines directly connecting the soft
subgraph $\mathcal{S}$ and the hard subgraph $\mathcal{H}$ are finite.

  3. 3.

A second consequence is that only gluons can directly connect the soft
subgraph $\mathcal{S}$ with the jet subgraphs $\mathcal{J}_{i}$.

  4. 4.

Finally, in a physical gauge, in order to generate a divergence, every jet
subgraph $\mathcal{J}_{i}$ must be connected to the hard subgraph
$\mathcal{H}$ only by a single line (which must therefore carry the same
quantum numbers as the external line entering or exiting the graph). In
Feynman gauge, further additional lines linking $\mathcal{J}_{i}$ and
$\mathcal{H}$ are allowed, but they must be scalar-polarised gluons.
Furthermore one finds that, for each jet $\mathcal{J}_{i}$, the number of
external soft gluons plus the number of jet gluons attached to the hard part
cannot be greater than the total number of three-point vertices in that jet.

Graphically, these rules mean that pinch surfaces giving rise to divergences
are considerably simpler than the general configuration depicted in Fig.
24(a), and they can be described by Fig. 24(b) in a covariant gauge (with
double curly lines denoting scalar-polarised gluons), and by Fig. 24(c) in a
physical gauge.

To summarise, we have learnt from power counting that sufficient conditions
for the presence of infrared divergences are considerably more restrictive
that the corresponding necessary conditions arising from the Landau equations
and the CN physical picture. In axial gauges, Fig. 24(c) is already,
effectively, a partially factorised expression for the form factor, separating
the divergent subgraphs $\mathcal{S}$ and $\mathcal{J}_{i}$ from the finite
subgraph $\mathcal{H}$. In Feynman gauge, we intuitively expect that Ward
identities will drive a cancellation among diagrams that will ultimately lead
to to a similar result. We can also hope to apply similar arguments to tame
soft gluons attaching the soft subgraph to jet subgraphs. The third part of
the analysis consists in examining in detail the actual Feynman rules in the
relevant soft and collinear limits: this will enable us to leverage the
universality of long-distance contributions, identifying operator matrix
element that generate soft and collinear divergences for all fixed-angle
scattering amplitudes, regardless of the particular hard process under
consideration.

###  3.3 Diagrammatic construction of soft and collinear functions

Our next task is to turn the “quasi-factorisation” depicted in Fig. 24 (b) or
(c) into an exact statement, writing the form factor (and later more general
scattering amplitudes) as a product of functions, each responsible for the
enhancements associated with specific kinematic configurations. This is still
far from trivial, since what we have achieved so far is true for individual
pinch surfaces, and the task ahead is to organise the contributions of all
these pinch surfaces into universal operator matrix elements, that can be
computed independently of the specific hard process under consideration. In
other words, we want to identify matrix elements which share with the
scattering amplitude specific sets of singularities, soft and/or collinear, so
that the product of these matrix elements will contain all the infrared poles
of the amplitude, leaving behind only a finite ‘matching coefficient’. If we
succeed in identifying such matrix elements, we can then study infrared
singularities to all orders, in a process-independent way, by means of
evolution equations, as we will discuss in Section 4.

One way to identify the matrix elements we need is to consider soft and
collinear approximations at the diagrammatic level, observe the systematic
simplifications that arise in the relevant limits, and build the appropriate
operators accordingly. A prominent role in this game will be played by Wilson-
line operators, as we will see in some detail in Section 3.3.2.

####  3.3.1 Soft and collinear approximations at one loop

We begin by revisiting our one-loop results in Section 3.2.2, in particular
the approximations leading to the soft and collinear homogeneous integrals in
Eq. (3.32) and Eq. (3.2.2). At this level, we can easily generalise the
results to the non-abelian theory, and, this time, we will pay closer
attention to the numerator structures, which carry information about gluon
polarisations, providing a more precise chracterisation of the result in Eq.
(3.45). For this purpose, consider the annihilation process
$q(p_{1})\,\bar{q}(p_{2})\rightarrow\gamma^{*}$. The un-approximated integral
corresponding to the one-loop QCD correction to the QED vertex reads

$\displaystyle\Gamma_{\mu}^{(1)}(p_{1},p_{2})\,=\,-eQ_{q}\int\frac{d^{d}k}{(2\pi)^{d}}\;\frac{\bar{v}(p_{2})\,\big{(}g\,T_{ij}^{a}\,\gamma^{\alpha}\big{)}\,\big{(}-p\hskip-4.49997pt/_{2}-k\hskip-4.49997pt/\big{)}\,\gamma_{\mu}\,\big{(}p\hskip-4.49997pt/_{1}-k\hskip-4.49997pt/\big{)}\,\big{(}g\,T_{jk}^{a}\,\gamma_{\alpha}\big{)}\,u(p_{1})}{\big{(}k^{2}+{\rm
i}\eta\big{)}\,\big{[}(p_{2}+k)^{2}+{\rm
i}\eta\big{]}\,\big{[}(p_{1}-k)^{2}+{\rm i}\eta\big{]}}\,.$ (3.49)

(a)

(b)

Figure 25: IR interactions modelled by eikonal Feynman rules. (a) Soft gluon
exchange: both interaction vertices become eikonal. (b) Collinear exchange,
with $k\parallel p_{1}$: only the vertex involving the antiquark line becomes
eikonal.

In the soft limit, as was discussed already in Section 1.1, the homogeneous
integral arises upon taking the eikonal approximation, amounting to

$\displaystyle\frac{\big{(}\not{p}_{1}-\not{k}\big{)}\big{(}g\,T^{a}\,\gamma_{\alpha}\big{)}u(p_{1})}{(p_{1}-k)^{2}+{\rm
i}\eta}$ $\displaystyle\overset{k^{\mu}\to\,0}{\longrightarrow}$
$\displaystyle\frac{g\,T^{a}\beta_{1,\alpha}}{-\beta_{1}\cdot k+{\rm
i}\eta}\,u(p_{1})\,,$
$\displaystyle\frac{\bar{v}(p_{2})\big{(}g\,T_{a}\,\gamma^{\alpha}\big{)}\big{(}-\not{p}_{2}-\not{k}\big{)}}{(p_{2}+k)^{2}+{\rm
i}\eta}$ $\displaystyle\overset{k^{\mu}\to\,0}{\longrightarrow}$
$\displaystyle\bar{v}(p_{2})\,\frac{-g\,T_{a}\,\beta_{2}^{\alpha}}{\beta_{2}\cdot
k+{\rm i}\eta}\;,$ (3.50)

where the Dirac equation has been used, and we took advantage of the scaling
invariance of the approximation to replace the momenta $p_{1}$ and $p_{2}$
with the corresponding four-velocities using $p_{1}=\mu\beta_{1}$ and
$p_{2}=\mu\beta_{2}$. Analogous replacements can be made for final-state
fermions and anti-fermions, with due attention to overall signs and to the
relative sign of the ${\rm i}\eta$ prescription: the results can be simply
summarised in the eikonal Feynman rules

$\displaystyle{\rm propagator:}\quad\frac{\rm i}{\beta\cdot k+{\rm
i}\eta}\,,\qquad\qquad{\rm vertex:}\quad{\rm i}g\,T_{a}\beta^{\alpha}\,,$
(3.51)

with the convention that $\beta$ is the velocity flowing along the arrow of
the fermion line, and $k$ is the soft momentum flowing in the same
direction242424Thus in the first line of Eq. (3.50) one must take $\beta\to
p_{1}/\mu$, and $k\to-k$, while in the second line $\beta\to-p_{2}/\mu$, and
$k\to-k$.. Applying these Feynman rules one gets directly the expression for
the soft homogeneous integral, Eq. (3.32), with the appropriate non-abelian
prefactor.

As already noted in Section 1.1, these Feynman rules are independent of the
spin and energy of the hard emitter, and only sensitive to its direction and
colour charge. They apply at leading power in all soft momenta, which, by the
power-counting arguments of Section 3.2, is all that is needed to capture soft
divergences. We note now two further important properties of the eikonal
approximation. First, the coupling to the colour charge is universal, in the
sense that it is sufficient to replace the generator $T_{a}$ in the
fundamental representation in Eq. (3.51) with the corresponding matrix
$T^{r}_{a}$ in the representation $r$ of the gauge group, in order to
reproduce the leading-power result if the emitter belongs to $r$ (for example,
$(T_{a})_{bc}\to-{\rm i}f_{abc}$ in the adjoint representation): we will take
advantage of this universality property in Section 5. Second, we note that,
taking the hard momentum in a fixed direction, say
$\beta=\\{\beta^{+},0^{-},{\bf 0}_{\perp}\\}$ in light-cone coordinates, the
eikonal coupling of the soft gluon to the hard line involves only the $-$
component of its polarisation at leading power. Furthermore, if all components
of the soft gluon momentum scale equally in the soft limit, according to Eq.
(3.33), the only component of the gluon momentum which is relevant at leading
power is $k^{-}$. Soft gluons thus couple longitudinally to hard lines, a fact
which will be instrumental in what follows. The simplified coupling of soft
lines to hard lines is shown pictorially in Fig 25 (a), where eikonal vertices
are represented as the merging of gluon propagator with a double line.

In the same spirit, let’s revisit the collinear approximation, leading to the
collinear homogeneous integral in Eq. (3.2.2), focusing on the numerator
structure. In the frame in which $p_{1}^{\mu}$ in in the $+$ direction, while
$p_{2}^{\mu}$ is in the $-$ direction, the integrand reads

$\displaystyle
T^{a}\,T_{a}\,\frac{\bar{v}(p_{2})\gamma^{-}\big{(}p\hskip-4.49997pt/_{2}+\not{k}\big{)}\,\gamma^{\mu}\big{(}p\hskip-4.49997pt/_{1}-k\hskip-4.49997pt/\big{)}\,\gamma^{+}u(p_{1})}{\big{[}k^{2}+{\rm
i}\eta\big{]}\;\big{[}(p_{1}-k)^{2}+{\rm
i}\eta\big{]}\;\big{[}(p_{2}+k)^{2}+{\rm i}\eta\big{]}}\,,$ (3.52)

where we used the massless Dirac equations in this frame, which read
$\bar{v}(p_{2})\gamma^{+}=\gamma^{-}u(p_{1})=0$. By the same token, and using
the fact that, in the region collinear to $p_{1}$, the $k^{-}$ component of
the loop momentum is power-suppressed, we can approximate the numerator of Eq.
(3.52) using

$\displaystyle\bar{v}(p_{2})\gamma^{-}\big{(}p\hskip-4.49997pt/_{2}+\not{k}\big{)}\,\simeq\,(p_{2}+k)^{-}\;\bar{v}(p_{2})\gamma^{-}\gamma^{+}\,=\,2(p_{2}+k)^{-}\,\bar{v}(p_{2})\,\simeq\,2\,p_{2}^{-}\,\bar{v}(p_{2})\,.$
(3.53)

In the collinear approximation, Eq. (3.52) can then be rewritten as

$\displaystyle
T^{a}\,T_{a}\,\frac{\bar{v}(p_{2})\,\gamma^{\mu}\big{(}\not{p}_{1}-\not{k})\gamma_{\alpha}\,u(p_{1})}{k^{2}(p_{1}-k)^{2}}\;\frac{\beta_{2}^{\alpha}}{\beta_{2}\cdot
k}\;.$ (3.54)

Remarkably, the vertex attaching the collinear gluon to the anti-quark has
become eikonal, and can be expressed in terms of the effective Feynman rules
of Eq. (3.51), as represented pictorially in Fig. 25 (b). As before, the gluon
coupling to the anti-quark has become independent of the anti-quark energy;
furthermore, the gluon couples to the antiquark with a polarisation parallel
to its momentum: it is longitudinal, and thus unphysical. On the other hand,
the coupling of the collinear gluon to the quark has not simplified, and
retains its spin and energy dependence.

Figure 26: Factorisation of the hard subgraph after a collinear emission.
Notice that the collinear prefactor is a spin matrix, as discussed in the
text.

In view of generalising this result to higher perturbative orders, it is
useful to note that the sequence of approximations leading to Eq. (3.54) can
be reformulated as the application of a tree-level Ward identity. Let us
assume to have a longitudinally polarised gluon moving in the $+$ direction,
and attaching to a generic hard interaction subgraph $H$. Writing the
(Feynman-gauge) numerator of the gluon propagator as a sum over polarisations,
including unphysical ones, the amplitude for this process is of the form

$\displaystyle\sum_{\lambda}\,\bar{v}(p_{2})\gamma_{\alpha}\frac{\not{p}_{2}+\not{k}}{(p_{2}+k)^{2}+{\rm
i}\eta}\,\varepsilon^{\alpha}_{(\lambda)}(k)\,H\,\varepsilon^{*\beta}_{(\lambda)}(k)\frac{\not{p}_{1}-\not{k}}{(p_{1}-k)^{2}+{\rm
i}\eta}\,\gamma_{\beta}\,u(p_{1})\,,$ (3.55)

where the hard subgraph $H$ is defined here to include the denominator of the
gluon propagator and we can focus on the case of longitudinal polarisation. We
can then substitute
$\not{\varepsilon}_{(\lambda)}(k)=\gamma^{-}\varepsilon^{+}(k)$, and insert a
factor of $1=k^{+}\beta_{2}^{-}/k^{+}\beta_{2}^{-}$, so that Eq. (3.55)
becomes [283]

$\displaystyle\frac{k^{+}\beta_{2}^{-}}{k^{+}\beta_{2}^{-}}\,\bar{v}(p_{2})\,\gamma^{-}\frac{\not{p}_{2}+\not{k}}{(p_{2}+k)^{2}+{\rm
i}\eta}\,\varepsilon^{+}(k)\,H\,\frac{\not{p}_{1}-\not{k}}{(p_{1}-k)^{2}+{\rm
i}\eta}\,\not{\varepsilon}(k)\,u(p_{1})$ (3.56) $\displaystyle=$
$\displaystyle\frac{\varepsilon(k)\cdot\beta_{2}}{k\cdot\beta_{2}}\;\bar{v}(p_{2})\,\not{k}\,\frac{\not{p}_{2}+\not{k}}{(p_{2}+k)^{2}+{\rm
i}\eta}\,H\,\frac{\not{p}_{1}-\not{k}}{(p_{1}-k)^{2}+{\rm
i}\eta}\,\not{\varepsilon}(k)\,u(p_{1})$ $\displaystyle=$
$\displaystyle\bar{v}(p_{2})\,H\,\frac{\varepsilon(k)\cdot\beta_{2}}{k\cdot\beta_{2}}\frac{\not{p}_{1}-\not{k}}{(p_{1}-k)^{2}+{\rm
i}\eta}\,\not{\varepsilon}(k)\,u(p_{1})\,,$

where in the last step we have used
$\not{k}=-\not{p}_{2}+(\not{p}_{2}+\not{k})$: the first term then vanishes by
the Dirac equation, and the second term cancels the denominator of the
antiquark propagator, effectively replacing it with the eikonal denominator.
One then readily recognises Eq. (3.54). Thanks to this ‘Ward identity’, we can
effectively write the amplitude as the product of a spin factor with an
eikonal coupling to the antiquark, multiplying the hard subgraph, as depicted
in Fig. 26.

To summarise the results of this one-loop analysis, we have seen that a soft
gluon couples eikonally to all hard lines, while a collinear gluon couples
eikonally to anti-collinear hard lines. Eikonal couplings are energy- and
spin-independent, and, more interestingly, involve unphysical polarisations at
leading power. This observation is the key to the generalisation of this
result to higher orders in perturbation theory: the only effect of unphysical
gluons is to dress the hard particles with a gauge rotation. The framing of
the one-loop collinear limit in the language of diagrammatic Ward identities
suggests how to proceed at higher orders: in case of multiple radiations, the
number of possible collinear connections to the hard subgraph increases,
however the Ward identity will guarantee that the eikonal approximation on the
anticollinear line still holds, and all emissions can be expressed in terms of
effective Feynman rules. This was already established for abelian soft
radiation in Section 1.1. In the non-abelian theory, further care must be
taken in order to insure that the colour ordering of the attachments is
consistently reproduced: as we will see in the next section, there is a
natural solution to this problem. We must however note an important caveat in
the non-abelian case: at higher orders, soft gluons will couple to a set of
collinear lines (rather than a single line), which in general will have
momenta with small transverse components. Couplings of soft gluons to these
components will be negligible only if the soft limit is taken at the same rate
for all components of the soft momentum $k$, as in Eq. (3.33). In a momentum
region where $k^{-}$ is much smaller that $k_{\perp}$, the approximation that
we have made in order to effectively factorise soft lines from ‘jet’ lines may
fail. We will briefly return to this issue in Section 5. A more detailed
description of the all-order proof of the factorisation of soft lines from
hard collinear ones is given in Ref. [61], while a full analysis can be found
in Ref. [67].

####  3.3.2 Wilson lines and the eikonal approximation

Power counting and diagrammatic analyses have given us a much-simplified
picture of soft and collinear radiation. Specifically, we have seen that only
limited physical information links soft, hard and collinear subgraphs. Our
next step is to identify operator matrix elements that contain this same
information, and therefore share the same singular regions in loop-momentum
space. Before we proceed, however, it is important to pause and focus on the
physical underpinnings of our conclusions so far. A crucial ingredient of the
factorisation is the fact that soft radiation is insensitive to the nature of
hard exchanges, as well as to the internal structure of collinear jets of fast
particles: only the overall direction of the collinear beam and its overall
colour charge can be detected by soft gluons. This is a consequence of the
fact that soft radiation has long wavelength, which cannot discriminate the
short-distance features of the scattering process, or the fine structure of
collimated jets. This simple physical fact is not directly apparent from
individual Feynman diagrams, and only emerges, at high orders, after intricate
cancellations. It has however been well-understood for a long time, under the
heading of colour transparency, first discussed in the non-abelian theory in
Refs. [284, 285], and reviewed for example in Refs. [286, 287]. This in turns
forms the basis for the colour coherence approach, one of the first powerful
tools for all-order analyses in perturbative QCD (see, for example, [288, 289,
290, 291, 292]). For collinear radiation, we have observed a similar
phenomenon: collinear gluons are sensitive to the spin and energy of the
emitter, but cannot resolve the quantum numbers of anti-collinear particles.
The axial-gauge result, showing that only a single particle can connect jets
to the hard scattering, which in turn is reflected by the eikonal couplings of
scalar-polarized collinear gluons in covariant gauges, also implies that hard
collinear radiation depends on the colour charge of the collinear beam, but is
not sensitive to colour exchanges with other hard particles: the only long-
distance colour exchanges between different hard particles are effected by
soft gluons.

We now turn to the construction of matrix elements reproducing these physical
results, expressed by eikonal Feynman rules. In the soft case, a natural guess
leads to the correct ansatz: at leading power in the soft radiation, hard
particles do not recoil, and follow straight-line trajectories from the hard
scattering point out to time-like infinity, in a specified direction. Along
these trajectories, soft interactions can only dress the hard particle with a
gauge phase. This phase in turn is naturally expressed by integrating the
gauge connection along the trajectory. The required operator, for each hard
particle, is therefore a straight Wilson line, defined in general by

$\displaystyle\Phi_{n}(\lambda_{2},\lambda_{1})\,=\,\mathcal{P}\exp\bigg{[}{\rm
i}g\int_{\lambda_{1}}^{\lambda_{2}}d\lambda\;n\cdot A^{a}(\lambda
n)\,T_{a}\bigg{]}\;,$ (3.57)

where $n^{\mu}$ is the direction of the line, and, in the non-abelian case,
one needs to introduce the path ordering operator $\mathcal{P}$. Eq. (3.57) is
a colour matrix in the representation of the selected hard particle. In the
present context, we will only need semi-infinite lines, with $\lambda_{1}=0$
and $\lambda_{2}=\infty$. It is not difficult to convince oneself that Wilson
lines reproduce eikonal Feynman rules, and the path-ordering provides the
correct non-abelian generalisation of the eikonal approximation.

To this end, consider, for example, a graph $G$ featuring an outgoing on-shell
hard quark line carrying momentum $p$, emitting an arbitrary number $n$ of
low-energy gluons with colour indices $a_{i}$ and carrying momenta $k_{i}$. At
leading power in all soft momenta, we can simply adapt Eq. (1.17) to include
color operators, and we obtain the expression

$\displaystyle G^{\,a_{1}\dots
a_{n}}_{\mu_{1}\dots\mu_{n}}\big{(}\,p,\\{k_{i}\\}\big{)}\,\simeq\,\prod_{i=1}^{n}\;\left[\big{(}{\rm
i}g\,T^{a_{i}}\big{)}\frac{i\,p^{\mu_{i}}}{p\cdot\left(\sum_{j=1}^{i}k_{j}\right)}\right]\bar{u}(p)\,{\cal
M}\big{(}p\big{)}\;.$ (3.58)

In the abelian case, it was possible to sum explicitly over the permutations
of the outgoing gluons, so that Eq. (3.58) could be simplified by applying the
eikonal identity, Eq. (1.18). The identity, however, expresses the fact that
successive photon emissions are independent and uncorrelated in the abelian
theory, and is no longer valid in the non-abelian case, since the colour
matrices do not commute. This, in turn, requires the path-ordering
prescription in Eq. (3.58).

(a)

(b)

Figure 27: Different steps of the factorisation procedure. a) Factorisation of
the soft subgraph: multiple soft gluon emissions are modelled via eikonal
Feynman rules. b) Factorisation of the jet subgraph from the hard part:
collinear gluons attach to eikonal vertices

To verify that the momentum-space eikonal Feynman rules are reproduced, it is
sufficient to expand the definition in Eq. (3.57), and Fourier transform the
gauge field, according to

$\displaystyle{A_{\mu}^{a}(\lambda
n)\,=\,\int\frac{d^{d}k}{(2\pi)^{d}}\;e^{{\rm i}k\cdot\lambda
n}\tilde{A}_{\mu}^{a}(k)}\,.$ (3.59)

At leading order the expansion gives

$\displaystyle{\cal P}\exp\bigg{[}\,{\rm i}g\int_{0}^{\infty}d\lambda\,n\cdot
A^{a}(\lambda
n)\,T_{a}\bigg{]}\,=\,1-g\int\frac{d^{d}k}{(2\pi)^{d}}\,\frac{n\cdot\tilde{A}^{a}(k)}{n\cdot
k+{\rm i}\eta}\,\,T_{a}+{\cal O}(g^{2})\,,$ (3.60)

where we used Feynman’s prescription to define the parameter integral at large
distances, resulting in

$\displaystyle\int_{0}^{\infty}d\lambda\,{\rm e}^{{\rm i}(k\cdot
n)\lambda}\,\to\,\int_{0}^{\infty}d\lambda\,{\rm e}^{{\rm i}(k\cdot n+{\rm
i}\eta)\lambda}\,=\,\frac{{\rm i}}{k\cdot n+{\rm i}\eta}\,,$ (3.61)

reflecting the Feynman rules in Eq. (3.51). At the next order in the expansion
the path-ordering prescription becomes relevant, yielding the correct partial
denominators of Eq. (3.58). Indeed one finds

$\displaystyle\big{(}{\rm
i}g\big{)}^{2}\int_{0}^{\infty}\\!d\lambda_{1}\int_{0}^{\lambda_{1}}\\!d\lambda_{2}\,n\cdot
A^{a}(\lambda_{1}n)\,n\cdot A^{b}(\lambda_{2}n)\,T_{a}\,T_{b}$
$\displaystyle=\,\big{(}{\rm
i}g\big{)}^{2}\,\int\frac{d^{d}k_{1}}{(2\pi)^{d}}\frac{d^{d}k_{2}}{(2\pi)^{d}}\int_{0}^{\infty}\\!d\lambda_{1}\int_{0}^{\lambda_{1}}\\!d\lambda_{2}\,{\rm
e}^{{\rm i}(\lambda_{1}k_{1}+\lambda_{2}k_{2})\cdot
n}\,n\cdot\tilde{A}^{a}(k_{1})\,n\cdot\tilde{A}^{b}(k_{2})\,T_{a}\,T_{b}$
$\displaystyle=\,g^{2}\\!\int\frac{d^{d}k_{1}}{(2\pi)^{d}}\frac{d^{d}k_{2}}{(2\pi)^{d}}\,\frac{n\cdot\tilde{A}^{a}(k_{1})}{k_{1}\cdot
n+{\rm i}\eta}\,\frac{n\cdot\tilde{A}^{b}(k_{2})}{(k_{1}+k_{2})\cdot n+{\rm
i}\eta}\,T_{a}\,T_{b}\,,$ (3.62)

which is fully consistent with the diagrammatic expression of a double
emission. The pattern in Eq. (3.62) generalises to all orders, yielding

$\displaystyle\mathcal{P}\exp\left[\,{\rm
i}g\,T_{a}\int_{0}^{\infty}d\lambda\,n\cdot A^{a}(\lambda
n)\right]\,=\,1+\sum_{n=1}^{\infty}\left[\,\prod_{i=1}^{n}\left(\int\frac{d^{d}k_{i}}{(2\pi)^{d}}\frac{g\,T_{a_{i}}\,n\cdot\tilde{A}^{a_{i}}(k_{i})}{\sum_{j=1}^{i}n\cdot
k_{i}+{\rm i}\eta}\right)\,\right]\,,$ (3.63)

which reproduces the leading-power result for soft gluon attachments to a hard
line, exemplified in Eq. (3.58).

These results confirm our intuition, that the interactions of a hard particle
as it propagates in a background of soft gluons without recoil are correctly
reproduced by replacing the particle with an appropriate Wilson-line operator.
Interactions between different hard particles propagating in different
directions and exchanging soft gluons will similarly be reproduced by taking a
vacuum expectation value of a set of Wilson lines, each in the appropriate
representation of the gauge group, and defined along the classical straight-
line trajectory of the hard emitter. The path-integral evaluation of the
resulting correlator will automatically generate all the radiative corrections
building up the generic soft subgraphs discussed in the previous sections. To
illustrate these facts in the simplest case, we can easily reproduce the
expression of the one-loop eikonal integral in Eq. (2.39) by considering the
correlator of two Wilson lines. Writing explicitly the open colour indices, we
find

$\displaystyle\left\langle\smash{0}{\vphantom{1}}\right|{\rm
T}\,\Big{[}\Phi^{f_{1}g_{1}}_{\beta_{1}}(\infty,0)\,\Phi^{f_{2}g_{2}}_{\beta_{2}}(\infty,0)\Big{]}\left|\smash{0}{\vphantom{1}}\right\rangle$
$\displaystyle=\,1\,-\,g^{2}\,\big{(}T_{1}^{a}\big{)}^{f_{1}g_{1}}\,\big{(}T_{2}^{b}\big{)}^{f_{2}g_{2}}\,\beta_{1}^{\mu}\beta_{2}^{\nu}\int_{0}^{\infty}d\lambda_{1}\,d\lambda_{2}\,\left\langle\smash{0}{\vphantom{1}}\right|{\rm
T}\,\Big{[}A_{\mu}^{a}(\lambda_{1}\beta_{1})\,A_{\nu}^{b}(\lambda_{2}\beta_{2})\Big{]}\left|\smash{0}{\vphantom{1}}\right\rangle\,+\,\ldots$
$\displaystyle=\,1\,-g^{2}\,\big{(}T_{1}^{a}\big{)}^{f_{1}g_{1}}\,\big{(}T_{2}^{b}\big{)}^{f_{2}g_{2}}\,\beta_{1}^{\mu}\beta_{2}^{\nu}\int_{0}^{\infty}d\lambda_{1}\,d\lambda_{2}\,\int\frac{d^{d}k}{(2\pi)^{d}}\,\bigg{(}\frac{-{\rm
i}g_{\mu\nu}\,\delta^{ab}}{k^{2}+{\rm i}\eta}\bigg{)}\,{\rm e}^{{\rm
i}k\cdot(\lambda_{1}\beta_{1}-\lambda_{2}\beta_{2})}\,+\,\ldots$
$\displaystyle=\,1\,+\,{\rm i}g^{2}\,T_{1}\cdot
T_{2}\,\beta_{1}\cdot\beta_{2}\int\frac{d^{d}k}{(2\pi)^{d}}\,\frac{1}{(k^{2}+{\rm
i}\eta)(k\cdot\beta_{1}-{\rm i}\eta)(k\cdot\beta_{2}+{\rm
i}\eta)}\,+\,\ldots\,.$ (3.64)

We omitted self-energy corrections, which arise by expanding one of the two
Wilson lines to ${\cal O}(g^{2})$, and vanish in a massless theory, where the
Wilson lines lie on the light cone. In the second line of Eq. (3.3.2) we
recognised the coordinate-space gluon propagator, and in the last line we
performed the parameter integrals, and we defined the scalar product
$T_{1}\cdot T_{2}\equiv T_{1}^{a}T_{2,\,a}$. We note that one could also
perform the calculation by using the explicit expression of the coordinate-
space gluon propagator: this possibility is exploited and discussed in detail
in Section 5. We recognise the result of Eq. (2.39), but this time with open
colour indices, which can later be contracted with open colour indices of the
hard matching function.

The arguments above represent an important step forward on the way to a
complete infrared factorisation for form factors, and later for fixed-angle
scattering amplitudes: having identified a set of operator matrix elements
(such as Eq. (3.3.2)) that reproduce all the leading-power soft singularities
of the form factor, we can promote the diagrammatic factorisation represented
in Fig. 24, where subgraphs are still linked by colour and spin indices, to
the complete factorisation depicted in Fig. 27(a), where we can now interpret
the double lines as Wilson lines, and the soft subgraph is promoted to a soft
function,

$\displaystyle\langle
0|T\Big{[}\Phi_{\beta_{1}}(\infty,0)\,\Phi_{\beta_{2}}(\infty,0)\Big{]}|0\rangle\,,$
(3.65)

responsible for all divergent configuration originating from soft gluons. Hard
and collinear integration regions will of course be misrepresented by the soft
function, and will have to be included in other terms in the factorisation.

As far as collinear regions are concerned, the discussion leading to Eq.
(3.54) provides a clear suggestion for how to proceed: collinear gluons couple
eikonally to anti-collinear lines, so we can build a jet function by replacing
anti-collinear lines by a Wilson line, while retaining spin and colour
information for the emitting collinear line. This results in the further
factorisation depicted in Fig. 27 (b). Tentatively, one could introduce an
object of the form

$\displaystyle\langle
0\,|\,\Phi_{\beta_{2}}(0,\infty)\,\psi(0)\,|p_{1},s_{1}\rangle\,,$ (3.66)

where a quark of momentum $p_{1}$ and spin polarisation $s_{1}$ in the initial
state is annihilated by the quark field at the origin, and a Wilson line
replaces the incoming antiquark. This function, however, has the drawback that
it still contains collinear divergences associated with the antiquark, since
the Wilson line is light-like in the massless case, and gluons attaching to it
will have anti-collinear enhancements, not fully matching those of the
original amplitude. The solution in this case is very simple: we can just
replace the light-like Wilson line along the $\beta_{2}$ direction with a
‘massive’ Wilson line with a direction $n^{\mu}$, with $n^{2}\neq 0$. The
resulting function will have the correct collinear singularities associated
with the quark direction, but only finite contributions from the anti-
collinear momentum region. As before, the resulting function will again
misrepresent the soft and hard momentum integration regions, which are
correctly approximated by other terms in the factorisation. In particular, we
note that gluons that are both soft and either collinear or anti-collinear are
correctly approximated both in the soft and in the collinear factors.
Including both factors will result in double-counting, and will require a
subtraction. In the next section, we will tackle these issues, and we will
provide a complete picture of infrared factorisation for form factors.

##  4 Factorisation, evolution and resummation: form factors

The tools developed in Section 3 have motivated, if not proved in detail, a
factorisation formula for form factors, where all soft and collinear
divergences are accounted for in terms of universal function, which can be
expressed as matrix elements of fields and Wilson lines. This factorisation
formula will be discussed in greater detail in Section 4.1. For the moment, we
wish to emphasise that, once the hard work of proving factorisation has been
completed, there are low-hanging fruits to be reaped: every factorisation
theorem, in fact, implies a set of evolution equations, and the solution of
these equation leads to a partial summation of perturbation theory. Before we
apply this to soft-collinear factorisation of scattering amplitudes, we wish
to recall briefly how this happens in standard applications, following [293].

The simplest and best known example of this process is of course the
renormalisation of UV divergences, which amounts to the factorisation of
singular cutoff dependence into a finite set of universal renormalisation
constants. Assuming that perturbative renormalisability has been proven for
the theory at hand, consider the relation between bare and renormalised Green
functions,

$\displaystyle
G_{0}^{(n)}\big{(}p_{i},\Lambda,g_{0}\big{)}\,=\,\prod_{i=1}^{n}\,Z_{i}^{1/2}\bigg{(}\frac{\Lambda}{\mu},g(\mu)\bigg{)}\,G_{R}^{(n)}\big{(}p_{i},\mu,g(\mu)\big{)}\,,$
(4.1)

where $Z_{i}$ are renormalisation constants for the fields appearing in the
correlator, $g_{0}$ is a bare coupling, $\Lambda$ is an ultraviolet cutoff
(for example the inverse of a lattice spacing), and $\mu$ is the
renormalisation scale. Proving that Eq. (4.1) holds, possibly with the added
burden of showing that other useful properties of the theory, such as
unitarity, or gauge symmetry, are preserved, is in general difficult. When
this has been achieved, however, one can mine Eq. (4.1) for further
information. In order to achieve the factorisation, it is strictly necessary
to introduce a additional energy scale, $\mu$, which, intuitively, we place
somewhere between the laboratory energy scales, given by the Mandelstam
invariants $p_{i}\cdot p_{j}$, and the cutoff scale. Renormalisability means
that all singular cutoff dependence is confined to the field renormalisation
constants $Z_{i}$. Once that is established, a (Callan-Symanzik) evolution
equation immediately follows, by simply noting that the l.h.s of Eq. (4.1)
does not depend on the renormalisation scale $\mu$, which has been introduced
as a necessary artefact of renormalisation. Scale dependence must therefore
cancel between the factors on the r.h.s., so that

$\displaystyle\frac{dG_{0}^{(n)}}{d\mu}\,=\,0\,\qquad\longrightarrow\qquad\frac{d\log
G_{R}^{(n)}}{d\log\mu}\,=\,-\sum_{i=1}^{n}\gamma_{i}\big{(}g(\mu)\big{)}\,,$
(4.2)

where the anomalous dimensions $\gamma_{i}$ are defined by

$\displaystyle\gamma_{i}\big{(}g(\mu)\big{)}\,\equiv\,\frac{1}{2}\,\frac{d\log
Z_{i}}{d\log\mu}\,.$ (4.3)

The simple functional dependence of the anomalous dimensions is dictated by
separation of variables in Eq. (4.2): $\gamma_{i}$ cannot depend on the cutoff
$\Lambda$, because the renormalised Green function $G_{R}^{(n)}$ does not, and
it cannot depend on the momenta $p_{i}$, because $Z_{i}$ does not. Anomalous
dimensions can only depend on arguments that are in common between the l.h.s.
and the r.h.s. of Eq. (4.2), in this case just on the renormalised coupling
$g(\mu)$. Armed with an evolution equation, one can solve it, and, in this
case, ‘resum’ logarithms of the renormalisation scale $\mu$. Since the
renormalised Green function $G_{R}^{(n)}$ depends on $\mu$ through the
coupling and through ratios of the form $p_{i}\cdot p_{j}/\mu^{2}$, solving
the equation gives precious information on the dependence of the correlator on
external energy scales.

A second standard example of the same phenomenon is collinear factorisation in
high-energy scattering, which we briefly introduced in Section 2.8. Eq. (2.62)
is an all-order factorisation theorem, which can be proved in perturbation
theory either by hard diagrammatic work, or by using more sophisticated tools
such as the light-cone expansion. It states that collinearly enhanced
contributions can be factored from the cross section in the form of a
convolution, and can be organised into a set of parton distribution functions
$f_{j/i}$. If desired, the convolution can be diagonalised by a Mellin
transform, using the definition

$\displaystyle\widetilde{W}(N)\,\equiv\,\int_{0}^{1}z^{N-1}\,W(z)\,.$ (4.4)

Applying this definition to Eq. (2.62), for a single particle flavour, results
in the simple factorisation

$\displaystyle\widetilde{W}\bigg{(}N,\frac{Q^{2}}{m^{2}},\alpha_{s}(Q^{2})\bigg{)}\,=\,\widetilde{H}\bigg{(}N,\frac{Q^{2}}{\mu_{f}^{2}},\alpha_{s}(Q^{2})\bigg{)}\,\widetilde{f}\bigg{(}N,\frac{\mu_{f}^{2}}{m^{2}},\alpha_{s}(Q^{2})\bigg{)}\,,$
(4.5)

where $Q$ is the hard scale, we used a particle mass $m$ as a collinear
regulator, replacing dimensional regularisation, and we noted that the
factorisation procedure requires the introduction of a factorisation scale
$\mu_{f}$. In Eq. (4.5), the singular dependence on the collinear cutoff $m$
is collected in the (Mellin-space) parton distribution $\widetilde{f}$, while
the Wilson coefficient $\widetilde{H}$ is free of singularities as $m\to 0$.
Also in this case, DGLAP evolution follows from the observation that the full
partonic structure function does not depend on the factorisation scale. Then

$\displaystyle\frac{d\widetilde{W}}{d\mu_{f}}\,=\,0\,\qquad\longrightarrow\qquad\frac{d\log\widetilde{f}}{d\log\mu_{f}}\,=\,-\,\gamma_{N}\big{(}\alpha_{s}(Q^{2})\big{)}\,,$
(4.6)

where

$\displaystyle\gamma_{N}\big{(}\alpha_{s}(Q^{2})\big{)}\,\equiv\,\frac{d}{d\log\mu_{f}}\,\widetilde{H}\bigg{(}N,\frac{Q^{2}}{\mu_{f}^{2}},\alpha_{s}(Q^{2})\bigg{)}\,.$
(4.7)

Once again, the anomalous dimension $\gamma_{N}$ (the Mellin transform of the
DGLAP splitting function) can depend only on variables that are common to the
two factors on the r.h.s. of Eq. (4.5), in this case $N$ and $\alpha_{s}$.
Solving the DGLAP equation leads to the resummation of collinear logarithms,
which effectively disappear from the calculation of the DIS cross section, as
they are absorbed into the parton distribution evaluated at the proper scale.

Soft-collinear factorisation of scattering amplitudes generalises these
familiar examples. Considering the form factor as a first example, we note
that a double factorisation has been performed, extracting a soft factor, and
a collinear factor for each external leg. We expect therefore two evolution
equations, matching the double-logarithmic nature of the amplitude we are
considering. This is indeed what we will find, after examining the
factorisation in more detail.

###  4.1 Soft-collinear factorisation of a form factor

Let us summarise the path that leads to the soft-collinear factorisation of
the form factor, as outlined in Section 3. First, the Landau equations, as
implemented in the Coleman-Norton physical picture, identify soft and
collinear gluons as the only potential source of divergences in the
(renormalised) massless form factor. Next, power counting reveals that the
connections between hard, soft and collinear subdiagrams are strongly
constrained: at leading power in the normal variables, no lines can directly
connect soft and hard factors; in a physical gauge, only one line can connect
collinear and hard factors (while in a covariant gauge this line can be
supplemented by any number of scalar-polarised gluons, which however decouple
from the hard subgraph after summing over diagrams, by means of the Ward
identity); soft gluons at wide angles couple to collinear lines only with the
anti-collinear components of their momenta, and they are longitudinally
polarised along the anti-collinear direction, so that they effectively couple
to a single Wilson line along the jet direction. The last step in Section 3
was the use of explicit diagrammatic arguments in order to identify operator
matrix elements carrying the same singularities as the form factor in the soft
and collinear sectors. This led to the identification of the soft function as
a matrix element of light-like Wilson lines extending along the classical
trajectories of the incoming particles. Taking as an example an incoming
quark-antiquark pair, we define then

$\displaystyle{\cal
S}\left(\beta_{1}\cdot\beta_{2},\alpha_{s}(\mu^{2}),\epsilon\right)\,\equiv\,\langle
0|T\Big{[}\Phi_{\beta_{1}}(\infty,0)\,\Phi_{\beta_{2}}(\infty,0)\Big{]}|0\rangle\,.$
(4.8)

Similarly, for collinear singularities, after the decoupling from the anti-
collinear line and from wide-angle soft gluons, it becomes clear that
collinear divergences are uniquely associated with each incoming hard
particle, and they can be simulated with a jet function of the form

$\displaystyle{\cal J}\bigg{(}\frac{(p\cdot
n)^{2}}{n^{2}\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}u_{s}(p)\,\,\equiv\,\langle
0\,|\,T\Big{[}\Phi_{n}(\infty,0)\,\psi(0)\Big{]}\,|p,s\rangle\,,$ (4.9)

and a similar one for the antiquark. Contrary to the soft function in Eq.
(4.8), the jet function in Eq. (4.9) does not contain colour correlations
between the quark and the antiquark, but it is spin-dependent. In general, one
will also need a definition for a gluon jet function: in order to provide it,
one cannot simply replace the quark field by a gluon field in Eq. (4.9), since
the result would not be gauge invariant. A possible definition bypassing this
problem was proposed in [294, 295], and is given by

$\displaystyle g\,{\cal J}_{g}^{\mu\nu}\\!\left(\frac{(k\cdot
n)^{2}}{n^{2}\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)\,\varepsilon^{(\lambda)}_{\mu}(k)\,\equiv\,\left\langle\smash{0}{\vphantom{1}}\right|T\Big{[}\Phi_{n}(\infty,0)\,{\rm
i}D^{\nu}\,\Phi_{n}(\infty,x)\Big{]}\left|\smash{k,\lambda}{\vphantom{1}}\right\rangle\Big{|}_{x=0}\,,$
(4.10)

where the Wilson line in the $n$ direction acts directly as the gluon source.

Finally, we turn to the issue of subtracting the potential double counting of
the momentum regions involving gluons that are both soft and collinear to the
incoming partons. Such regions indeed appear both in the soft function in Eq.
(4.8) and in the jets, Eqns. 4.9 and 4.10. It would appear to be non-trivial
to subtract collinear configurations from the soft function, however the
simple form of the jet functions leads to a straightforward proposal for
subtracting their soft limits: one may simply divide each jet by its own soft
approximation, which is naturally given by a very similar matrix element, with
the field of the hard parton replaced by its own Wilson line. Thus we define
the eikonal jet function

$\displaystyle{\cal J}_{\rm E}\bigg{(}\frac{(\beta\cdot
n)^{2}}{n^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}$ $\displaystyle\equiv$
$\displaystyle\langle
0|T\Big{[}\,\Phi_{\beta}(\infty,0)\,\Phi_{n}(\infty,0)\Big{]}\,|0\rangle\,.$
(4.11)

As expected from the general properties of the soft approximation, eikonal
jets are independent of spin, and they depend on colour only through the
representation in which the Wilson lines are defined. The fact that the proper
way to ‘subtract’ the soft-collinear region is to divide by the eikonal jets
will be better justified in what follows, where we will see that jet and
eikonal jet functions exponentiate, and they are normalised to unity at lowest
order: by taking a ratio, we are effectively subtracting their perturbative
exponents.

Not suprisingly, closely related definitions of soft and jet functions arise
in the SCET approach to infrared factorisation (see, for example, Refs. [75,
92, 95, 96, 296]), and they have been extensively investigated both at
amplitude and at cross-section level252525We note in passing a significant
difference between the soft functions we discuss here for fixed-angle
scattering amplitudes and those featuring at cross-section level. For cross
sections, soft functions are not universal, since the phase-space integration
of real soft radiation is weighted with the selected observable. Thus, in
principle, every observable requires a new calculation. Representative
examples beyond NLO can be found in Refs. [297, 298, 299, 300, 301, 302, 303,
304, 305, 306, 307, 308, 309, 310], but the literature is extensive. Cross-
section-level jet functions, on the other hand, retain a degree of
universality, and have been computed to high orders in SCET and in QCD, see
for example Refs. [311, 312, 313, 314, 315]. Finally, we note that, in the
context of resummation, it is useful to define non-universal initial state
functions – essentially modified parton distribution functions – that reflect
the phase-space constraint of the observable under study. This technique was
pioneered in Ref. [226]. In SCET, the appropriate functions are called beam
functions, and they have been extensively studied in recent years [316, 317,
318, 319, 320, 321, 322, 323, 324, 325].. In the context of SCET, soft and jet
functions are defined in terms of soft and collinear fields, which directly
enter the SCET lagrangian. Practical calculations at leading power in the two
formalisms are closely related, but the double-counting problem for the soft-
collinear region is dealt with in a different way, using the so-called zero-
bin subtraction, discussed for example in Refs. [326, 327, 328, 329].

Before making several comments about the physical meaning and implications of
our definitions in Eq. (4.8), Eq. (4.9), and Eq. (4.11), we now pause to write
down a precise form for the infrared factorisation of the form factor in terms
of our soft and jet functions, which is illustrated pictorially in Fig. 28. We
write

$\displaystyle\Gamma\left(\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)$
$\displaystyle=$ $\displaystyle\prod_{i=1}^{2}\,\frac{{\cal
J}_{i}\left(\frac{(p_{i}\cdot
n_{i})^{2}}{n_{i}^{2}\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)}{{\cal
J}_{E,\,i}\,\left(\frac{(\beta_{i}\cdot
n_{i})^{2}}{n_{i}^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)}$
$\displaystyle\hskip 56.9055pt\times\,{\cal
S}\Big{(}\beta_{1}\cdot\beta_{2},\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,{\cal
H}\left(\frac{Q^{2}}{\mu^{2}},\frac{(p_{i}\cdot
n_{i})^{2}}{n_{i}^{2}\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)\,.$

Here the hard function ${\cal H}$, which is finite as $\epsilon\to 0$, plays
the role of a matching coefficient: order by order in perturbation theory,
soft and collinear factors miss or misrepresent finite contributions from hard
exchanges, and these can be reinstated by computing ${\cal H}$, which is
defined by subtraction of the singular contributions from the full form
factor. A detailed illustration of this factorisation at the one-loop order
was given in Ref. [70].

Figure 28: Pictorial representation of soft-collinear factorisation for the
quark form factor.

It is important to spend some time on Eq. (4.1), since it forms the basis for
further generalisations to fixed-angle scattering amplitudes. We begin by
noting that the factorisation of collinear effects into jet functions has
required the introduction of a vector $n_{i}^{\mu}$ ($i=1,2$) for each hard
parton. The introduction of these vectors can be understood in three
complementary ways.

  * •

First of all, such vectors are needed in order to ensure gauge invariance of
the jet definition, Eq. (4.9). In the absence of the Wilson line, the jet
function would transform as the quark field under a gauge transformation. The
Wilson line takes this gauge variation at the origin, and transports it out to
infinite distance, where it vanishes, since we are working perturbatively and
considering only globally trivial gauge fields262626It should be noted that
neglecting ‘large’ gauge transformations, which do not vanish at infinity, can
be considered as part of the problem that originates infrared divergences in
the first place. This point of view is developed in the celestial approach to
infrared singularities, see for example [161]..

  * •

The one-loop collinear calculation leading to Eq. (3.54) illustrates that the
vectors $n_{i}$, de facto, replace the anti-collinear parton. Collinear gluons
absorbed at late times (or emitted at early times) are effectively blind to
the detailed structure of non-collinear parts of the amplitude, so that those
parts can be replaced, at leading-power accuracy, with a Wilson-line absorber.
We recall again that it is important in principle to keep $n_{i}^{2}\neq 0$ (a
typical choice [283] for the form factor being
$n_{1}=\beta_{1}-\beta_{2}=-n_{2}$). This avoids the inclusion of spurious
collinear divergences in the jet function, arising from gluon emission from
the Wilson line, which are not present in the form factor and would need to be
subtracted.

  * •

A third way to understand the $n_{i}$’s is to think of them as factorisation
vectors, generalising the idea of a factorisation scale. Since we are trying
to distinguish collinear emissions from wide-angle ones, it is perhaps not
surprising that one needs to introduce a vector in order to draw a cone around
the direction of the emitter, specified by $\beta_{i}$: in this
interpretation, one may decide to assign a gluon to the collinear region if,
say, $\beta_{i}\cdot k<n_{i}\cdot k$.

The next set of observations on Eqs. (4.8 \- 4.1) concerns their functional
dependences. The fact that jet functions can depend only on the variable
$(p\cdot n)^{2}/(n^{2}\mu^{2})$ (and similarly eikonal jets can only depend on
$(\beta\cdot n)^{2}/n^{2}$) stems from their invariance under rescalings of
the form $n\to\kappa n$, which is a property of semi-infinite straight Wilson
lines stretching along the direction $n$, with $n^{2}\neq 0$. Indeed, a
correlator of two Wilson lines, in directions $n_{i}$, $i=1,2$, with
$n_{i}^{2}\neq 0$, can only depend on the Minkowskian cusp angle between the
two directions, defined in a Lorentz-invariant way by $\gamma_{12}=(n_{1}\cdot
n_{2})^{2}/(n_{1}^{2}n_{2}^{2})$. In the case of light-like Wilson lines, such
as those appearing in the soft function and in the eikonal jet function, the
cusp angles are ill-defined, so that one might expect correlators involving
such lines to be just numbers, free of kinematic dependence. As we have seen
in Section 2.7, however, this is not the case, since the rescaling symmetry of
Wilson lines correlators is broken by the presence of an extra collinear
divergence. As a consequence, the soft function for massless partons acquires
a dependence on $\beta_{1}\cdot\beta_{2}$, as shown in Eq. (4.8), and the
eikonal jets correspondingly acquire a dependence on $(\beta_{i}\cdot
n_{i})^{2}/n_{i}^{2}$. It is clear from the outset that this ‘anomalous’ scale
dependence must be comparatively simple, since it is entirely associated with
the soft-collinear double pole of the form factor. This double singularity is
governed by the light-like cusp anomalous dimension, as we will see in Section
4.2.

A consequence of this analysis, which will be crucial in applications to
multi-parton amplitudes, is the following. If we can construct quantities
where soft-collinear double poles cancel, then, for these quantities, the
anomalous scale dependence must cancel as well. By inspection of Eq. (4.1), we
can readily identify two such quantities.

  * •

The ratio of the partonic jet and the corresponding eikonal jet, ${\cal
J}_{i}/{\cal J}_{E,\,i}$, which is responsible for hard-collinear corrections,
carrying a single collinear pole per loop. In this case, at least at one loop,
it is easy to envisage how the cancellation of the anomalous scale dependence
takes place: both jets depend logarithmically on their arguments, and one can
use

$\displaystyle\log\bigg{(}\frac{(p_{i}\cdot
n_{i})^{2}}{n_{i}^{2}\mu^{2}}\bigg{)}-\log\bigg{(}\frac{(\beta_{i}\cdot
n_{i})^{2}}{n_{i}^{2}}\bigg{)}\,\sim\,\log\bigg{(}\frac{Q^{2}}{\mu^{2}}\bigg{)}\,,$
(4.13)

as the two logarithms must have the same coefficient, which stems from the
soft-collinear double pole. Explicit expressions for the quark jet and eikonal
jet, at one loop, verifying Eq. (4.13), are given in Ref. [70].

  * •

Less trivially, one can define a reduced soft function, by taking the ratio of
the soft function with the product of the two eikonal jets,

$\displaystyle\widehat{\cal
S}\,\big{(}\rho_{12},\alpha_{s}(\mu^{2}),\epsilon\big{)}\,\equiv\,\frac{{\cal
S}\Big{(}\beta_{1}\cdot\beta_{2},\alpha_{s}(\mu^{2}),\epsilon\Big{)}}{{\cal
J}_{E,\,1}\\!\left(\frac{(\beta_{1}\cdot
n_{1})^{2}}{n_{1}^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)\,{\cal
J}_{E,\,2}\\!\left(\frac{(\beta_{2}\cdot
n_{2})^{2}}{n_{2}^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)}\,,$ (4.14)

which describes soft radiation at wide angles with respect to the hard
partons. Also this quantity provides a single (soft) pole per loop, and
therefore must be rescaling invariant under $\beta_{i}\to\kappa_{i}\beta_{i}$.
This is achieved if $\widehat{\cal S}$ is a function of an invariant variable,
which we take here to be272727It is possible, and often useful, to attach to
scalar products in Eq. (4.15) appropriate phases, dictating the rules for
analytic continuation from time-like to space-like kinematics, as done for
example in [74]. Since here we are only interested in scaling properties, for
simplicity we omit the phase information.

$\displaystyle\rho_{12}\,\equiv\,\frac{\big{(}\beta_{1}\cdot\beta_{2}\big{)}^{2}\,n_{1}^{2}\,n_{2}^{2}}{\big{(}\beta_{1}\cdot
n_{1}\big{)}^{2}\big{(}\beta_{2}\cdot n_{2}\big{)}^{2}}\,.$ (4.15)

Once again, this happens naturally if the dependence of each function in the
ratio (4.14) is simply logarithmic, and the logarithms have the same
coefficients, tied to the soft-collinear double pole. Clearly, the
cancellation implied in Eq. (4.14) is more intricate than the jet case:
imposing this constraint will have highly non-trivial consequences in the case
of multi-parton amplitudes.

To conclude our analysis of Eq. (4.1), we note the mechanism of cancellation
for the dependence on the factorisation vectors $n_{i}$: poles in $\epsilon$
carrying $n_{i}$ dependence must cancel between the partonic and the eikonal
jet, through a mechanism akin to Eq. (4.13). On the other hand the hard
function ${\cal H}$, which is finite as $\epsilon\to 0$, must depend on
$p_{i}\cdot n_{i}$ (or equivalently $\beta_{i}\cdot n_{i}$) in order to cancel
the same dependence in the finite parts of the jet functions.

###  4.2 Evolution and resummation of infrared poles

Based on the discussion at the beginning of the present Section, we expect the
soft-collinear factorisation displayed in Eq. (4.1) to lead to an evolution
equation, and thus to a resummation, in this case of infrared poles in
dimensional regularisation. The basic observation is that, in order to achieve
that double factorisation into hard, soft and collinear factors, it was
necessary to introduce a factorisation scale separating soft and hard momenta
(which we can take equal to the renormalisation scale), and also to introduce
the ‘factorisation vectors’ $n_{i}$, separating collinear from wide-angle
radiation. Needless to say, the full form factor does not depend on any of
these quantities, a statement which directly leads to two evolution equations,
which can then be fruitfully combined.

Beginning with the factorisation/renormalisation scale dependence, the fact
that we are considering a matrix element of a conserved current leads directly
to the RG equation

$\displaystyle\mu\,\frac{d}{d\mu}\,\Gamma\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}\,=\,0\,.$
(4.16)

If we now rewrite Eq. (4.1) as

$\displaystyle\Gamma\bigg{(}\frac{Q^{2}}{\mu^{2}}\bigg{)}\,=\,\widehat{\cal
S}\big{(}\rho_{12}\big{)}\,{\cal J}_{1}\bigg{(}\frac{(p_{1}\cdot
n_{1})^{2}}{n_{1}^{2}\mu^{2}}\bigg{)}\,{\cal J}_{2}\bigg{(}\frac{(p_{2}\cdot
n_{2})^{2}}{n_{2}^{2}\mu^{2}}\bigg{)}\,{\cal
H}\bigg{(}\frac{Q^{2}}{\mu^{2}},\frac{(p_{i}\cdot
n_{i})^{2}}{n_{i}^{2}\mu^{2}}\bigg{)}\,,$ (4.17)

where we omitted the dependence on the coupling and on $\epsilon$ for brevity,
we can define three anomalous dimensions for ${\cal J}_{i}$, $\widehat{\cal
S}$, and ${\cal H}$, by taking logarithmic derivatives with respect to the
scale, as

$\displaystyle\mu\frac{d}{d\mu}\ln{\cal J}$ $\displaystyle=$
$\displaystyle-\,\gamma_{\cal J}(\alpha_{s})\,,$
$\displaystyle\mu\frac{d}{d\mu}\ln\widehat{\cal S}$ $\displaystyle=$
$\displaystyle-\,\gamma_{\widehat{\cal
S}}\left(\rho_{12},\alpha_{s}\right)\,,$ (4.18)
$\displaystyle\mu\frac{d}{d\mu}\ln{\cal H}$ $\displaystyle=$
$\displaystyle-\,\gamma_{\cal H}\left(\rho_{12},\alpha_{s}\right)\,.$

Note that all anomalous dimensions are finite, since the functions we are
differentiating have a single pole per loop. Furthermore, note that the jet
anomalous dimension $\gamma_{\cal J}$ depends only on the coupling: the
partonic jet Eq. (4.9) is not scale-less, therefore UV poles have no direct
relation to IR poles, and the anomalous dimension is given by the coefficient
of the pole in the UV counterterm; on the other hand, the anomalous dimension
for the reduced soft function has a residual kinematic dependence, since, for
scale-less quantities, UV singularities are strictly related to IR ones: in
this case, there is dependence on the rescaling-invariant variable
$\rho_{12}$, leftover after the cancellation of soft-collinear double poles.
Finally, the renormalisation group equation (4.16) implies

$\displaystyle\gamma_{\widehat{\cal
S}}\left(\rho_{12},\alpha_{s}\right)+\gamma_{\cal
H}\left(\rho_{12},\alpha_{s}\right)+2\gamma_{\cal J}(\alpha_{s})\,=\,0\,;$
(4.19)

in particular, since $\gamma_{\cal J}$ is independent of $\rho_{12}$, the
residual kinematic dependence must cancel between $\gamma_{\cal H}$ and
$\gamma_{\widehat{\cal S}}$.

Turning now to the dependence on the factorisation vectors $n_{i}$, first of
all we note that this dependence is always through the scalar variables
$x_{i}=\left(\beta_{i}\cdot n_{i}\right)^{2}/n_{i}^{2}$ (recall that
$p_{i}=\mu\beta_{i}$). Since the complete form factor does not depend on
$n_{i}$, we can write

$\displaystyle x_{i}\frac{\partial}{\partial
x_{i}}\ln\Gamma\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}\,=\,0\,,$
(4.20)

where we let the derivative act on the logarithm of the form factor in order
to turn the factorisation in Eq. (4.1) into a sum of terms. Taking into
account the fact that the soft function does not depend upon $n_{i}$, we find
that

$\displaystyle x_{i}\frac{\partial}{\partial x_{i}}\ln{\cal
J}_{i}\,=\,-\frac{1}{2}\,x_{i}\frac{\partial}{\partial x_{i}}\ln{\cal
H}+x_{i}\frac{\partial}{\partial x_{i}}\ln{\cal J}_{E,i}\,.$ (4.21)

Eq. (4.21) is very important, because it achieves the separation of a
complicated problem into two simpler ones. The first term on the r.h.s. of Eq.
(4.21) can be intricate at high orders, since ${\cal H}$ is a matching
function, and thus inherits some of the complexity of the full amplitude: on
the other hand, ${\cal H}$ is finite as $\epsilon\to 0$, so the same must be
true for its derivative in Eq. (4.21). The second term on the r.h.s. of Eq.
(4.21), on the contrary, is divergent as $\epsilon\to 0$ (indeed, it contains
only poles in $\epsilon$, since ${\cal J}$ is a pure counterterm in
dimensional regularisation); however, its functional form must be extremely
simple: indeed, ${\cal J}$ would be just a number, with no kinematic
dependence whatsoever, were it not for the anomalous breaking of rescaling
invariance due to the collinear poles associated with the $\beta$ direction.
Order by order in perturbation theory, these soft-collinear poles are
determined by the cusp anomalous dimension, and the ensuing kinematic
dependence must be single-logarithmic282828For a detailed treatment of the
eikonal jet, see [74]; the first all-order perturbative analysis of this kind
of matrix element was given in [330].. It follows that the logarithmic
derivative of the eikonal jet ${\cal J}_{i}$ in Eq. (4.21) must be independent
of the kinematic variable $x_{i}$. These considerations can be made explicit
by defining two functions, corresponding to the two terms on the r.h.s. of Eq.
(4.21). For the eikonal jet we define

$\displaystyle x_{i}\frac{\partial}{\partial x_{i}}\,\ln{\cal
J}_{E,i}\,=\,\frac{1}{2}\,\mathcal{K}\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,,$
(4.22)

while for the hard matching function ${\cal H}$ we define

$\displaystyle x_{i}\frac{\partial}{\partial x_{i}}\,\ln{\cal H}\,=\,-\,{\cal
G}_{i}\Big{(}x_{i},\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,.$ (4.23)

Eq. (4.21) can then be written as

$\displaystyle x_{i}\frac{\partial}{\partial x_{i}}\,\ln{\cal
J}_{i}\,=\,\frac{1}{2}\,\bigg{[}\,\mathcal{K}\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,+\,{\cal
G}_{i}\Big{(}x_{i},\alpha_{s}(\mu^{2}),\epsilon\Big{)}\bigg{]}\,,$ (4.24)

Eq. (4.24) is an example of a class of equations associated with the soft-
collinear evolution of cross-sections and amplitudes in QCD, which we will
call ‘Collins-Soper equations’ as they were first derived in Ref. [330]; they
were later widely used for resummations of Sudakov logarithms and infrared
singularities (see, for example, [226, 283, 293, 331, 332, 70]). In this form,
Eq. (4.24) appears rather uninteresting, describing evolution with respect to
an unphysical quantity. As is typically the case for RG equations, however,
Eq. (4.24), when combined with Eq. (4.16), leads directly to a much more
powerful and interesting equation for the evolution of the full form factor
with respect to the physical scale $Q^{2}$. Indeed, we can write

$\displaystyle Q^{2}\frac{\partial}{\partial
Q^{2}}\,\ln\Gamma\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}$
$\displaystyle=$ $\displaystyle Q^{2}\frac{\partial}{\partial Q^{2}}\,\ln{\cal
H}\,+\,\sum_{i=1}^{2}Q^{2}\frac{\partial}{\partial Q^{2}}\,\ln{\cal J}_{i}$
(4.25) $\displaystyle=$
$\displaystyle-\,\mu^{2}\frac{\partial}{\partial\mu^{2}}\,\ln{\cal
H}\,+\,\sum_{i=1}^{2}x_{i}\frac{\partial}{\partial x_{i}}\,\ln{\cal J}_{i}\,,$

where in the first line we have used the fact that purely eikonal functions do
not depend on the scale $Q^{2}$, and in the second line we used dimensional
analysis, and the fact that jets depend on $Q^{2}$ only through $(p\cdot
n)^{2}$. Now for the $\mu$ dependence of the hard function we can use Eq.
(4.2) and Eq. (4.19), with the result

$\displaystyle-\,\mu^{2}\frac{\partial}{\partial\mu^{2}}\,\ln{\cal H}$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\mu\frac{d}{d\mu}\,\ln{\cal
H}\,+\,\frac{1}{2}\beta(\epsilon,\alpha_{s})\frac{d}{d\alpha_{s}}\ln{\cal H}$
(4.26) $\displaystyle=$
$\displaystyle\frac{1}{2}\beta(\epsilon,\alpha_{s})\frac{d}{d\alpha_{s}}\ln{\cal
H}-\frac{1}{2}\gamma_{\widehat{\cal S}}-\gamma_{\cal J}\,,$

while Eq. (4.24) gives the $x_{i}$ dependence of the jets. The results can be
summarised in a new, much more powerful Collins-Soper equation for the full
form factor

$\displaystyle Q^{2}\frac{\partial}{\partial
Q^{2}}\,\ln\Gamma\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}\,=\,\frac{1}{2}\,\bigg{[}\,K\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,+\,G\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}\bigg{]}\,,$
(4.27)

where we defined

$\displaystyle K\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}$
$\displaystyle\equiv$ $\displaystyle 2{\cal
K}\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,,$ $\displaystyle
G\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}$
$\displaystyle\equiv$
$\displaystyle\beta(\epsilon,\alpha_{s})\frac{d}{d\alpha_{s}}\ln{\cal
H}-\gamma_{\cal S}-2\gamma_{\cal J}+\frac{1}{2}\sum_{i=1}^{2}{\cal
G}_{i}\Big{(}x_{i},\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,.$ (4.28)

Once again, $K$ is a pure counterterm, and independent of kinematics, while
$G$ retains kinematic information, but is finite as $\epsilon\to 0$. There is
a final nugget of information to be extracted from Eq. (4.16): since the form
factor is not affected by an overall renormalisation, the functions $K$ and
$G$ can renormalise additively, but their renormalisations must cancel. Thus

$\displaystyle\mu\frac{d}{d\mu}\,G\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}$
$\displaystyle=$
$\displaystyle-\,\mu\frac{d}{d\mu}\,K\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}$
(4.29) $\displaystyle=$
$\displaystyle-\,\beta(\epsilon,\alpha_{s})\,\frac{d}{d\alpha_{s}}K\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,\equiv\,\gamma_{K}\big{(}\alpha_{s}(\mu^{2})\big{)}\,,$

where in the second line we used the fact that $K$ has no explicit scale
dependence.

For the purposes of our review, Eq. (4.29) can be taken as an operational
definition of the light-like cusp anomalous dimension
$\gamma_{K}(\alpha_{s})$. This is, however, such an important object for
infrared studies of perturbative gauge theories that it deserves a brief
dedicated discussion, before we continue with our analysis of the
exponentiation of the form factor.

The function $\gamma_{K}(\alpha_{s})$ appears in several cornerstone gauge-
theory calculations. As shown in Ref. [333], it gives the coefficient of the
soft singularity of DGLAP splitting functions (discussed here in Section 2.8
and later in Section 6) as the collinear momentum fraction $z\to 1$, to all
orders in perturbation theory. As a consequence, it resums leading threshold
logarithms in many crucial hadronic cross sections (see, for example, [226,
334, 335, 336]): in that context, it is often denoted by $A(\alpha_{s})$. A
closely related function appears in the resummation of leading transverse
momentum logarithms in the hadronic production of colour-singlet final states
[337, 338], however the two functions begin to differ at the three-loop order
[339]. Finally, $\gamma_{K}(\alpha_{s})$ plays a crucial role for Regge
behaviour in the high-energy limit [340, 341, 342, 343, 344], and of course,
as we discuss below, for the all-order structure of infrared divergences in
gauge-theory amplitudes.

The reason for the ubiquitous and crucial role played by the cusp anomalous
dimension is not difficult to understand, and emerges from our discussion in
Section 3.3.2. In the soft approximation, hard emitters can be replaced by
Wilson lines directed along their classical trajectories; then, by the
mechanism discussed here in Section 2.7, infrared poles of the original
amplitude can be replaced by ultraviolet poles of the corresponding Wilson-
line correlator [243]; such correlators (up to collinear divergences) are
multiplicatively renormalisable [345, 346, 347, 348]; finally, one finds that
the anomalous dimension controlling the ultraviolet divergences arising when
two light-like Wilson lines meet at a cusp is given precisely by
$\gamma_{K}(\alpha_{s})$. More generally, as noted above in Section 4.1, when
two straight Wilson lines directed along non-light-like directions
$n_{1}^{\mu}$ and $n_{2}^{\mu}$ meet, forming a Minkowskian angle given by
$\gamma_{12}=(n_{1}\cdot n_{2})^{2}/(n_{1}^{2}n_{2}^{2})$, the ultraviolet
divergences of the corresponding correlator are controlled by a function
$\Gamma_{\rm cusp}(\gamma_{12},\alpha_{s})$, sometimes called angle-dependent
cusp anomalous dimension [349]. Not surprisingly, this function plays an
essential role for the infrared behaviour of scattering amplitudes and form
factors involving massive particles [350, 351, 352, 353]. To all orders, the
function $\Gamma_{\rm cusp}(\gamma_{12},\alpha_{s})$ displays a collinear
singularity as $n_{i}^{2}\to 0$, or $\gamma_{12}\to\infty$, diverging as the
logarithm of $\gamma_{12}$. The coefficient of this logarithm gives the light-
like anomalous dimension $\gamma_{K}(\alpha_{s})$.

Quite naturally, perturbative calculations of these vital anomalous dimensions
have been the focus of intense activity in the past decades. The light-like
cusp anomalous dimension can be extracted from calculations of collinear
splitting kernels, so the two-loop result can be traced to Ref. [354], and, in
the context of resummation, to Ref. [355]; similarly, the three-loop result
emerges from Ref. [356] (see also [357]). At four loops, the result has been
computed directly from correlation functions with lagrangian insertions in
Ref. [230], and extracted from form factor calculations in Ref. [231]. The
full angle-dependent cusp anomalous dimension was computed at two loops in
Ref. [244] (see also [358]), and at three loops in Refs. [359, 360]. At four
loops, efforts are ongoing, and a wealth of partial information is already
available [361, 362, 363, 364, 365]. Perhaps most remarkably, in the special
case of ${\cal N}=4$ Super-Yang-Mills theory, the integrability of the planar
limit gives access to fully non-perturbative information: gluon amplitudes can
be studied in the strong-coupling regime [366], the light-like cusp anomalous
dimension can be defined [367], and it can be shown that it obeys a non-
perturbative equation [368], which can be analysed both at weak [369, 370,
371] and at strong coupling [372, 373]. We will further discuss the remarkable
role played by the cusp for the infrared limit of gauge theory amplitudes in
Section 5, where we will also perform in detail the one-loop, angle-dependent
calculation.

Returning now to the form factor problem, we note that Eq. (4.29) can be
readily solved for $K$, with the result

$\displaystyle
K\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,=\,-\frac{1}{4}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\,\gamma_{K}\big{(}\overline{\alpha}(\lambda^{2},\epsilon)\big{)}\,.$
(4.30)

Alternatively, it is possible to proceed order by order in perturbation
theory, and recursively determine the perturbative coefficients of $K$ in
terms of those of the $\beta$ function and of the cusp anomalous dimension
$\gamma_{K}$, as was done in Ref. [69]. Up to three loops, the result takes
the form

$\displaystyle K(\alpha_{s},\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}}{\pi}\,\frac{\gamma_{K}^{(1)}}{4\epsilon}\,+\left(\frac{\alpha_{s}}{\pi}\right)^{2}\,\left(\frac{\gamma_{K}^{(2)}}{8\epsilon}-\frac{b_{0}\,\gamma_{K}^{(1)}}{32\epsilon^{2}}\right)$
(4.31)
$\displaystyle\,+\left(\frac{\alpha_{s}}{\pi}\right)^{3}\left(\frac{\gamma_{K}^{(3)}}{12\epsilon}-\frac{b_{0}\,\gamma_{K}^{(2)}+b_{1}\,\gamma_{K}^{(1)}}{48\epsilon^{2}}+\frac{b_{0}^{2}\,\gamma_{K}^{(1)}}{192\epsilon^{3}}\right)\,+{\cal
O}(\alpha_{s}^{4})\,.$

The function $K$ has a long history in the context of perturbative QCD (see,
for example, Ref. [330]), and plays an important role also for multi-particle
scattering amplitudes, and in the high-energy limit, as we will see in Section
5.

Solving Eq. (4.27) is a standard exercise, but in this case the consistent use
of dimensional regularisation yields a very significant simplification:
indeed, working with $\epsilon<0$, one can use the fact that the
$d$-dimensional strong coupling vanishes in the infrared to impose the simple
boundary condition

$\displaystyle\Gamma\Big{(}0,\alpha_{s}(\mu^{2}),\epsilon\Big{)}\,=\,\Gamma\Big{(}1,\overline{\alpha}(0,\epsilon),\epsilon\Big{)}\,=\,1\,,$
(4.32)

where on the l.h.s we have set $Q^{2}=0$ for fixed $\mu$, while in the second
step we have first set $\mu^{2}=Q^{2}$, and then taken $\mu\to 0$. Thanks to
Eq. (4.32), the form factor is a pure exponential with no prefactor, and can
be written as [68]

$\displaystyle\Gamma\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\exp\Bigg{[}\frac{1}{2}\int_{0}^{-Q^{2}}\frac{d\xi^{2}}{\xi^{2}}\Bigg{(}K\Big{(}\alpha_{s}(\mu^{2}),\epsilon\Big{)}+G\Big{(}\\!-\\!1,\overline{\alpha}(\xi^{2},\epsilon),\epsilon\Big{)}$
(4.33) $\displaystyle\hskip
56.9055pt+\,\frac{1}{2}\int_{\xi^{2}}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\,\gamma_{K}\big{(}\overline{\alpha}(\lambda^{2},\epsilon)\big{)}\Bigg{)}\Bigg{]}\,,$

where we have used Eq. (4.29) to evolve $G$ from the scale $\xi^{2}$ to the
scale $\mu^{2}$, and we have chosen to integrate to the scale $(-Q^{2})$ to
emphasise that the form factor is real for negative $Q^{2}$: in this way,
$G(-1)$ is real, and all phases can be explicitly obtained by analytic
continuation. We note that in Eq. (4.33), in its present form, one needs a
non-trivial cancellation of ill-defined contributions between the first term,
which diverges at the lower limit of integration, since $K$ does not depend on
$\xi$, and the contribution of the upper limit of integration of the $\lambda$
integral, which also does not depend on $\xi$. This cancellation can be
performed analytically by using Eq. (4.30), which easily leads to the final
expression for the form factor,

$\displaystyle\Gamma\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\exp\Bigg{[}\frac{1}{2}\int_{0}^{-Q^{2}}\frac{d\xi^{2}}{\xi^{2}}\Bigg{(}\\!G\Big{(}\\!-\\!1,\overline{\alpha}(\xi^{2},\epsilon),\epsilon\Big{)}-\frac{1}{2}\,\gamma_{K}\big{(}\overline{\alpha}(\xi^{2},\epsilon)\big{)}\ln\\!\bigg{(}\frac{-Q^{2}}{\xi^{2}}\bigg{)}\\!\\!\Bigg{)}\Bigg{]}\\!.$
(4.34)

As promised, all the ingredients entering the exponent in Eq. (4.34) are
finite as $\epsilon\to 0$, and all infrared poles, to all orders in
perturbation theory, are generated by the scale integration.

It is worthwhile at this point to pause for a few considerations on Eq.
(4.34), which is the simplest case of exponentiation of infrared poles for
non-abelian scattering amplitudes292929Early references for the exponentiation
of QCD form factors include [374, 64, 65]., anticipating many features that
will be generalised to multi-particle amplitudes in Section 5.

  * •

Perhaps the first point to emphasise is the fact that Eq. (4.34) is
predictive: we are not simply shifting the problem from the calculation of
$\Gamma$ to the calculation of its logarithm. To see this, note that the
direct computation of the form factor order by order yields two infrared poles
per loop, so that one finds $\epsilon^{-2n}$ poles at ${\cal
O}(\alpha_{s}^{n})$. The exponent in Eq. (4.34), on the other hand, has only
one infrared pole per loop beyond one loop: the logarithm of the form factor
has singularities up to $\epsilon^{-n-1}$ at ${\cal O}(\alpha_{s}^{n})$.
Clearly, all poles of the form $\alpha_{s}^{n}\epsilon^{-p}$ with $n+1<p\leq
2n$ are generated at lower orders and can be obtained by exponentiation: for
example, the one-loop calculation of the form factor predicts the leading
poles to all orders in perturbation theory. One can in fact look at Eq. (4.33)
a little closer, and note that a singularity is generated in the exponent by
each integration, so that one immediately detects an exponentiated double
pole. The remaining poles in the exponent, $\epsilon^{-p}$ with $2<p\leq n+1$,
are generated by the running of the the $d$-dimensional coupling: for example,
note that the expansion of Eq. (2.27) in powers of $\alpha_{s}(\mu_{0}^{2})$
is finite order by order as $\epsilon\to 0$, but the integration of each term
over the scale $\mu$ generates poles proportional to powers of $b_{0}$.
Higher-order terms in the $\beta$ function have a similar effect. This
predictive structure has been both verified and exploited in finite-order
calculations up to four loops [375, 376, 230, 231, 377].

  * •

These considerations lead to a second significant observation: the form factor
- perhaps not surprisingly - becomes extremely simple in the case of gauge
theories that are conformal in $d=4$, such as ${\cal N}=4$ Super-Yang-Mills
theory, and several of its possible deformations. For these theories, the
four-dimensional $\beta$ function vanishes to all orders, so that one has

$\displaystyle\beta\big{(}\epsilon,\alpha_{s}\big{)}\,=\,-2\epsilon\alpha_{s}\,,$
(4.35)

and Eq. (2.28) holds exactly. As a consequence, the scale integrals in Eq.
(4.33), or equivalently Eq. (4.34), can be performed explicitly. Expanding the
cusp anomalous dimension and the the function $G$ as

$\displaystyle\gamma_{K}(\alpha_{s})\,=\,\sum_{n=1}^{\infty}\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{n}\gamma_{K}^{(n)}\,,\qquad
G\big{(}-1,\alpha_{s},\epsilon\big{)}\,=\,\sum_{n=1}^{\infty}\bigg{(}\frac{\alpha_{s}}{\pi}\bigg{)}^{n}G^{(n)}(\epsilon)\,,$
(4.36)

one readily finds [79]

$\displaystyle\ln\Bigg{[}\Gamma\bigg{(}\frac{Q^{2}}{\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\bigg{)}\Bigg{]}\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!-\,\frac{1}{2}\sum_{n=1}^{\infty}\bigg{(}\frac{\alpha_{s}(\mu^{2})}{\pi}\bigg{)}^{\\!n}\bigg{(}\frac{\mu^{2}}{-Q^{2}}\bigg{)}^{\\!n\epsilon}\Bigg{[}\frac{\gamma_{K}^{(n)}}{2\,n^{2}\epsilon^{2}}+\frac{G^{(n)}(\epsilon)}{n\epsilon}\Bigg{]}\,.$
(4.37)

Using Eq. (2.28), one easily sees that Eq. (4.37) displays exact RG invariance
(it is independent of $\mu$), as expected. Eq. (4.37) has important
consequences: indeed, it is sufficient to determine the infrared singularity
structure of all scattering amplitudes in ${\cal N}=4$ SYM in the planar
limit. This is easily understood by noting that, for planar gluon amplitudes,
soft gluons can only connect two external states that are consecutive in the
selected cyclic ordering, so that the infrared divergences of the full planar
amplitude are just given by a product of (gluon) form factors. This fact was
exploited in Ref. [79] to constrain the ‘BDS’ ansatz for $n$-point planar
amplitudes in ${\cal N}=4$ SYM, which was later proved to give the exact all-
order result for $n=4$ and $n=5$ [378, 379].

  * •

A non-trivial practical feature of Eq. (4.34) is the fact that it can be
directly compared to finite-order results obtained evaluating Feynman diagrams
in dimensional regularisation. Using different infrared regulators, the
exponential containing singular terms is multiplied by an ‘initial condition’
which depends on a factorisation scale, and in general admits its own
perturbative expansion. Using Eq. (4.34), on the other hand, a finite-order
calculation can directly be compared to the corresponding expansion of the
exponential, thus extracting the perturbative coefficients of $\gamma_{K}$ and
$G$. Non-singular contributions to the form factor are encoded in ${\cal
O}(\epsilon^{p})$ terms of $G$, with $p>0$, and they also ‘exponentiate’,
yielding a weak prediction for higher-order finite contributions [225, 226,
233, 234].

  * •

A final feature of Eq. (4.34) which is worth noting is the fact that it allows
for a straightforward extraction of the analytic continuation of the form
factor from negative values of $Q^{2}$ (where $\Gamma$ is real) to positive
ones [68]. Indeed, the dependence on $Q^{2}$ is just in the upper limit of
integration, so that the ratio of the time-like form factor to the space-like
one is simply given by the exponential of the same integral, stretching from
$+Q^{2}$ to $-Q^{2}$. Since at the origin one only finds integrable
singularities, one can deform the integration contour to a half-circle of
radius $Q^{2}$ in the complex $\xi^{2}$ plane, showing that this ratio is
completely dominated by perturbative contributions in asymptotically free
theories. Furthermore, one can show [68] that the poles of this ratio can be
collected in an overall divergent phase, given by the counterterm function
$K$,

$\displaystyle\frac{\Gamma\big{(}Q^{2},\alpha_{s}\big{)}}{\Gamma\big{(}\\!-\\!Q^{2},\alpha_{s}\big{)}}\Bigg{|}_{\rm
poles}\,=\,\exp\bigg{[}{\rm
i}\,\frac{\pi}{2}\,K\big{(}\alpha_{s}(Q^{2}),\epsilon\big{)}\bigg{]}\,,$
(4.38)

so that the modulus of the ratio is finite. This is relevant for
phenomenological applications, since the modulus of the ratio (or related
quantities) appears in cross-sections for vector boson or Higgs production
[225, 226, 233], and ‘resums’ large constants such as the factor of $\pi^{2}$
appearing in Eq. (2.23). In the case of ${\cal N}=4$ SYM, starting from Eq.
(4.37), one can also get a strikingly simple and elegant expression for finite
terms, which are given by

$\displaystyle\left|\frac{\Gamma\big{(}Q^{2},\alpha_{s}\big{)}}{\Gamma\big{(}\\!-\\!Q^{2},\alpha_{s}\big{)}}\right|^{2}\,=\,\exp\bigg{[}\frac{\pi^{2}}{4}\gamma_{K}(\alpha_{s})\bigg{]}\,.$
(4.39)

Note that the r.h.s. is independent of $Q^{2}$, as expected for a four-
dimensional quantity in a conformal theory. Since all quantities in Eq. (4.39)
are well-defined at the non-perturbative level, and since the r.h.s. provides
a finite, unambiguous resummation of perturbation theory, Eq. (4.39) can be
argued to be an exact non-perturbative result. In some cases, such non-
perturbative results can be tested against strong-coupling calculations based
on the AdS/CFT correspondence [366, 367, 380].

Having completed our survey of the exponentiation of form factors, we now move
on to discuss the general case of fixed-angle scattering amplitudes.

##  5 Fixed-angle scattering amplitudes

We consider now the case of multi-particle, fixed-angle scattering amplitudes
in a general massless gauge theory. We write such amplitudes as

$\displaystyle{\cal A}_{n}^{\,a_{1}\ldots
a_{n}}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu^{2}),\epsilon\right)\,,$ (5.1)

implying that the normalisation has been chosen in order to work with a
dimensionless quantity. The spin structure is understood, but we are
displaying the colour indices, which in general can belong to different
representations of the gauge group. The fixed-angle assumption amounts to the
statement that all Mandelstam invariants $s_{ij}=2p_{i}\cdot p_{j}$ are
parametrically of the same size, say $\left|s_{ij}\right|\sim Q^{2}$,
$\forall\,i,j$. This means that there are no strong scale hierarchies: the
cases in which some momentum $p_{i}$ becomes soft, or two momenta $p_{i}$ and
$p_{j}$ become collinear, must be treated separately.

For amplitudes in this class, it is not too difficult to generalise the
reasoning leading to the soft-collinear factorisation of the form factor, Eq.
(4.1). A diagrammatic analysis, using the Landau equations and the Coleman-
Norton physical picture, confirms that non-integrable singularities arise only
from soft and collinear configurations; soft gluons, and separately collinear
gluons, factorise from the hard part by the same reasoning that was applied to
the form factor; finally, so long as external particles are not collinear to
each other, the factorisation of soft gluons from each jet at leading power is
supported by the same diagrammatic arguments and Ward identities. From a
physical point of view, at leading power, jets moving in different directions
can interact only at the hard scattering, and soft particles cannot resolve
either the details of the hard interaction, or the fine structure of streams
of massless particles moving collinearly. The most delicate issue in this
generalisation is whether different loop momentum regions, other than soft or
collinear, could give raise to singularities. Specifically, recall that the
soft region for loop momentum $k$ is defined by the scaling

$\displaystyle k^{\mu}\,=\,\\{k^{+},k^{-},{\bf
k}_{\perp}\\}\,\quad\to\quad\lambda k^{\mu}\,,\qquad\lambda\to 0\,.$ (5.2)

In the case of collinear scaling, we can separately consider each hard
particle in the amplitude, and in each case define the collinear direction as
one of the two directions spanning the light-cone: for example, for a given
loop momentum $k$, when considering the collinear region $k^{\mu}\parallel
p_{i}^{\mu}$, we can choose a frame where $p_{i}^{\mu}=\\{p_{i}^{+},0^{-},{\bf
0}_{\perp}\\}$; then the collinear scaling is

$\displaystyle k^{\mu}\,=\,\\{k^{+},k^{-},{\bf
k}_{\perp}\\}\,\quad\to\quad\\{k^{+},\lambda k^{-},\sqrt{\lambda}{\bf
k}_{\perp}\\}\,,\qquad\lambda\to 0\,,$ (5.3)

which leads to $k^{2}=2k^{+}k^{-}-{\bf k}_{\perp}^{2}$ vanishing uniformly as
$\lambda$. One must wonder whether different momentum scalings could lead to
singular contributions: a notable possibility303030Finer distinctions are
possible and often useful. One may for example consider soft gluons whose
momentum components scale like the transverse momentum in Eq. (5.3), rather
then the anti-collinear component $k_{-}$. In the context of SCET, such gluons
are referred to as soft, while the scaling in Eq. (5.2) is called ultrasoft.
The version of SCET that distinguishes the two scalings is called SCETI, and
it can mapped to the theory with a single soft scaling (SCETII) by a suitable
matching procedure. One may also consider Coulomb gluons, whose (soft) spatial
momentum components dominate their energy in a selected frame. These scalings
will not be needed in what follows. is given by soft wide-angle gluons,
scaling as

$\displaystyle k^{\mu}\,=\,\\{k^{+},k^{-},{\bf
k}_{\perp}\\}\,,\quad\to\quad\\{\lambda^{p}k^{+},\lambda^{p}k^{-},\lambda{\bf
k}_{\perp}\\}\,,\quad p\geq 2\,,\qquad\lambda\to 0\,,$ (5.4)

with respect to one of the hard parton directions. For fixed-angle amplitudes,
such gluons essentially have vanishing longitudinal components, and are
usually called Glauber gluons. Glauber gluons are known to give important
contributions at cross-section level, and the cancellation of the
corresponding divergences is a crucial step in the proof of collinear
factorisation for collider processes [205, 206]; furthermore, they can
contribute to infrared poles of scattering amplitudes when the external
particles are allowed to become collinear, while some of the Mandelstam
invariants become space-like [207]. For fixed-angle scattering amplitudes,
however, Glauber gluons do not contribute at leading power [276].

With these premises, it is easy to propose a generalisation of Eq. (4.1) for
multi-particle amplitudes. First of all, we expect that collinear dynamics
will be captured by jet functions, which are essentially single-particle
quantities, and thus cannot change the colour content of the outgoing state;
soft gluons, on the other hand, can connect any pair of hard particles, and
they will change the colour of those particles even if they carry a vanishing
energy. We expect then that the soft function in Eq. (4.1) will need to be
promoted to a colour operator acting on the colour indices of all external
particles. The emergent form of soft-collinear factorisation for fixed-angle
multi-particle scattering amplitudes in massless gauge theories is then

$\displaystyle{\cal
A}_{n}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu),\epsilon\right)$
$\displaystyle=$ $\displaystyle\prod_{i=1}^{n}\frac{{\cal
J}_{i}\left(\frac{\left(p_{i}\cdot
n_{i}\right)^{2}}{n_{i}^{2}\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)}{{\cal
J}_{E,i}\left(\frac{\left(\beta_{i}\cdot
n_{i}\right)^{2}}{n_{i}^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)}$ (5.5)
$\displaystyle\times\,{\cal
S}_{n}\big{(}\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\big{)}\,{\cal
H}_{n}\\!\left(\\!\frac{p_{i}\cdot p_{j}}{\mu^{2}},\frac{\left(p_{i}\cdot
n_{i}\right)^{2}}{n_{i}^{2}\mu^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)\,,$

where we introduced a jet function, and its eikonal counterpart, for each
external hard particle, with the definitions given in Eq. (4.9) and Eq.
(4.11), and we defined the $n$-particle soft function as the natural
generalisation of Eq. (4.8),

$\displaystyle{\cal
S}\left(\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\right)\,\equiv\,\langle
0|\,T\bigg{[}\prod_{k=1}^{n}\Phi_{\beta_{k}}(\infty,0)\bigg{]}|0\rangle\,.$
(5.6)

The factorisation in Eq. (5.5) is supported by the exhaustive diagrammatic
analysis carried out in Ref. [381], which leads to a BPHZ-like forest formula
for soft and collinear singularities in fixed-angle scattering amplitudes.
This analysis makes use of the coordinate-space formalism developed in Ref.
[224], and generalises the early results of [330, 66]. An independent
derivation of an analogous factorisation in the context of SCET was given in
[95].

In order to proceed to a more detailed analysis of the factorisation, we first
need to be much more concrete concerning the treatment of colour flow in Eq.
(5.5), clarifying the action of the soft colour operator ${\cal S}$ on the
hard part of the amplitude, ${\cal H}$: this was trivial for form factors,
where particles with opposite colour charges annihilate into a colour-singlet
state, but can be very intricate for general scattering amplitudes. With this
in mind, before proceeding to discuss the dynamical aspects of the soft-
collinear factorisation in Eq. (5.5), we briefly digress to describe the two
main methods commonly used to handle colour in this general case.

###  5.1 Handling color structures

The colour structure of a generic multi-particle non-abelian scattering
amplitude is an interesting group theory problem, solved in principle with
standard tools, but not easy to implement in practice when the particle number
grows. In general, each color index in Eq. (5.1) can belong to a different
representation of the gauge group $r_{i}$, $i=1,\ldots,n$. The amplitude
${\cal A}$ is a tensor in the vector space ${\cal V}_{n}\equiv
r_{1}\otimes\ldots\otimes r_{n}$, and colour conservation implies that it must
be an invariant tensor. The task is then to decompose the reducible
representation ${\cal V}_{n}$ into a sum of irreducible representations, and
impose the colour-conservation constraint on the result. A practical way to do
this is to pick a channel, i.e. a set of particles taken as ‘incoming’, and
build the tensor product of the corresponding representations; one then
repeats the construction for the remaining, ‘outgoing’, particles, and selects
the irreducible representations that appear in both lists: those are the
possible colour states that can be exchanged in that channel. A thorough
analysis of this construction was performed in Ref. [382] (see also [343]):
using Clebsch-Gordan coefficients, one can construct projection operators
identifying each possible colour flow in the selected channel.

In the present context, our concern is to understand how the colour structure
is implemented in the soft-collinear factorisation of the amplitude, Eq.
(5.5). In order to represent the action of the soft operator ${\cal S}_{n}$ on
the hard matching coefficient ${\cal H}_{n}$, two formalisms are commonly
adopted, one based upon a choice of basis in the space of colour tensors
available for the selected process, and one formulated in terms of basis-
independent colour insertion operators. For completeness, we briefly present
below the basics of both formalisms.

####  5.1.1 Colour tensor bases

Using the techniques discussed in Refs. [382, 343], it is possible to
identify, in any selected channel, a basis of colour tensors spanning the
representations that can be exchanged in that channel. Denoting the basis
tensors by $c_{L}^{\,\,a_{1}\ldots a_{n}}$, we can then write the amplitude as

$\displaystyle{\cal A}_{n}^{\,a_{1}\ldots
a_{n}}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu^{2}),\epsilon\right)\,=\,\sum_{L}{\cal
A}_{n}^{L}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu^{2}),\epsilon\right)\,c_{L}^{\,\,a_{1}\ldots
a_{n}}\,.$ (5.7)

The tensors $c_{L}^{\,\,a_{1}\ldots a_{n}}$ are quadratic combinations of the
Clebsch-Gordan coefficients appearing in the decomposition of ${\cal V}_{n}$
in a direct sum of irreducible representations. They can be chosen to be
orthonormal, in the sense that

$\displaystyle\sum_{\\{a_{i}\\}}c_{L}^{\,\,a_{1}\ldots
a_{n}}\,\left(c_{M}^{\,\,a_{1}\ldots a_{n}}\right)^{*}\,=\,\delta_{LM}\,.$
(5.8)

To illustrate this, consider the simple case of $q\bar{q}$ scattering: at tree
level, the contributing diagrams are depicted in Fig. 29. Applying the Feynman
rules, and the Fierz identity

Figure 29: Feynman diagrams for tree-level quark-antiquark scattering, and
their colour flow at leading power in $N_{c}$.

$\displaystyle\big{(}T_{a}\big{)}_{ij}\,\big{(}T^{a}\big{)}_{kl}\,=\,\frac{1}{2}\bigg{(}\delta_{il}\delta_{jk}-\frac{1}{N_{c}}\delta_{ij}\delta_{kl}\bigg{)}\,,$
(5.9)

one immediately identifies a possible set of basis color tensors as

$\displaystyle\hat{c}_{1}^{\,ijkl}\,=\,\delta^{il}\delta^{jk}\,,\qquad\hat{c}_{2}^{\,ijkl}\,=\,\delta^{ij}\delta^{kl}\,,$
(5.10)

corresponding to leading-color flow in the $s$ channel and in the $t$ channel,
respectively. It is clear that this set forms a basis for higher-order
corrections as well, since the third possible combination of Kronecker
$\delta$ functions in the fundamental representation is forbidden, as it would
require a quark to turn into an antiquark. More formally, one would start by
noting that the representations entering the scattering (for example in the
$s$ channel) give $\underline{\bf 3}\otimes\underline{\bf
3}^{*}=\underline{\bf 1}\oplus\underline{\bf 8}$: one then expects a set of
only two basis tensors. The correspondence between the basis tensors and the
representation content is made more transparent by transforming the basis in
Eq. (5.10) to an orthonormal one, according to Eq. (5.8), by using again Eq.
(5.9). One can then pick

$\displaystyle
c_{1}^{\,ijkl}\,=\,\frac{2}{\sqrt{N_{c}^{2}-1}}\big{(}T_{a}\big{)}_{ji}\,\big{(}T^{a}\big{)}_{kl}\,,\qquad
c_{2}^{\,ijkl}\,=\,\frac{1}{N_{c}}\,\delta^{ij}\delta^{kl}\,,$ (5.11)

corresponding to octet and singlet $s$-channel exchanges, respectively. Once a
basis has been selected, the amplitude can be thought of as a vector in the
vector space spanned by the $c_{L}$ tensors. In the chosen basis, the soft
operator is represented by a matrix, acting on a vector of hard matching
coefficients. To be precise, we note that the Wilson lines in Eq. (5.6) have
open colour indices at both ends, and both sets of indices can be projected on
the basis tensors. We define then the matrix elements ${\cal
S}^{\,\,L}_{n\,\,K}$ of the soft operator, in the chosen basis, by

$\displaystyle\sum_{L}c_{L}^{\,\,a_{1}\ldots a_{n}}\,{\cal
S}^{\,\,L}_{n\,\,K}\big{(}\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\big{)}\,=\,\sum_{b_{1},\ldots,b_{n}}\langle
0|\prod_{k=1}^{n}\Big{[}\Phi_{\beta_{k}}(\infty,0)\Big{]}^{a_{k}}_{\,\,\,\,b_{k}}|0\rangle\,c_{K}^{\,\,b_{1}\ldots
b_{n}}\,;$ (5.12)

in Eq. (5.5), the soft matrix thus defined acts on a vector of finite matching
coefficients ${\cal H}^{K}$.

Working with explicit colour bases is useful for practical applications, for
example for the implementation of soft gluon resummations in hadron scattering
[383, 384, 385, 386, 387]. The formalism, however, does not easily lend itself
to the general treatment of $n$-point amplitudes: the size of the matrices
involved grows steeply with the number of particles, and colour conservation
is implemented essentially on a process-by-process basis. For example, the
treatment of $gg\to gg$ scattering in $SU(N_{c})$ requires 9$\times$9 matrices
(although some simplification are possible [388, 389]), while already for
amplitudes involving five gluons ($gg\to ggg$) the matrices involved have
dimension $d=44$ [390]. We now proceed to describe a formalism which is better
suited for generic particle multiplicities.

####  5.1.2 Colour insertion operators

Considering soft-gluon corrections, the factorisation formula in Eq. (5.5)
formalises an intuitive understanding: soft divergences arise when soft gluons
(real or virtual) are emitted from an on-shell particle. Essentially, this
means that soft divergences are associated with emissions from external legs,
so that one can think of the soft operator as acting ‘from the outside’ on the
hard factor, which contains corrections associated with ‘inner’, off-shell
virtual exchanges. It then makes sense, at leading power, to think of the soft
emission as a colour operator, inserting a soft gluon on the non-radiative
amplitude, according to the scheme

$\displaystyle{\cal A}_{n+1}^{\,\,a\,b_{1}\ldots\,b_{n}}\bigg{|}_{\rm
soft}\,\propto\,\sum_{i=1}^{n}\Big{[}{\bf
T}_{i}^{\,a}\Big{]}^{b_{i}}_{\,\,\,c_{i}}{\cal
A}_{n}^{\,\,b_{1}\ldots\,c_{i}\ldots\,b_{n}}\,.$ (5.13)

The colour operators ${\bf T}_{i}$ act on the tensor product space ${\cal
V}_{n}$, and the index $i$ identifies the specific factor in the product on
which the operator acts non-trivially, corresponding to the emission of a
gluon with (adjoint) index $a$ from particle $i$. In order to be more precise,
and to illustrate how this operator formalism emerges [289, 81], consider the
case of soft gluon emission from an outgoing quark, in a generic tree-level
matrix element involving $n$ coloured particles, plus the emitted gluon. Let
the $n$ ‘Born-level’ particles have momenta $p_{i}$, colour indices $c_{i}$
and polarisation $\lambda_{i}$, and let the radiated gluon be characterised by
momentum $k$, colour $c$ and polarisation $\lambda$. The only diagrams
surviving at leading power in $k$ are those where the soft gluon is emitted by
the on-shell external legs, so we can easily isolate the contribution of a
specific outgoing quark, say the one carrying momentum $p_{i}$. The relevant
contribution to the emission amplitude is given by

$\displaystyle
g\mu^{\epsilon}\,\overline{u}_{s_{i}}(p_{i})\,\gamma_{\alpha}\,\frac{p\hskip-4.49997pt/_{i}+k\hskip-4.49997pt/}{2p_{i}\cdot
k}\,\big{(}T^{c}\big{)}_{c_{i}d_{i}}\widehat{\cal
A}^{\,\,\,c_{1}\ldots\,d_{i}\ldots\,c_{n}}_{\,s_{1}\dots
s_{n}}\big{(}\\{p_{j}\\},k\big{)}\,\epsilon_{\lambda}^{*\,\alpha}(k)\,,$
(5.14)

where $\widehat{\cal A}$, while not quite a scattering amplitude itself, since
the $i$-th leg is off-shell for generic $k$, represents all the remaining
factors of the radiative amplitude. Now, taking the soft limit $k^{\mu}\to 0$,
we can apply the eikonal approximation to the factors associated with the
$i$-th particle, and, at leading power in $k$, we can neglect $k$ in the
factor $\widehat{\cal A}$, which thus becomes precisely the ‘Born’ amplitude
for the original $n$ on-shell particles. Following the steps already outlined
in Section 1 in the case of QED, this leads to the expression

$\displaystyle
g\mu^{\epsilon}\,\frac{\beta_{i}\cdot\epsilon_{\lambda}^{*}(k)}{\beta_{i}\cdot
k}\,\big{(}T^{c}\big{)}_{c_{i}d_{i}}\big{(}{\cal
A}_{n}\big{)}^{\,c_{1}\ldots\,d_{i}\ldots\,c_{n}}_{s_{1}\dots
s_{n}}\big{(}\\{p_{j}\\}\big{)}\,\equiv\,g\mu^{\epsilon}\,\frac{\beta_{i}\cdot\epsilon_{\lambda}^{*}(k)}{\beta_{i}\cdot
k}\,\,{\bf T}_{i}\,{\cal A}_{n}\big{(}\\{p_{j}\\}\big{)}\,.$ (5.15)

We have thus identified the precise expression for the colour operator ${\bf
T}_{i}$, when the $i$-th particle is an outgoing quark: in that case, ${\bf
T}_{i}$ is the colour generator $T^{a}_{\,\,cd}$, in the fundamental
representation. Performing the same calculation for an outgoing antiquark, and
an outgoing gluon, one finds the identifications

$\displaystyle{\bf T}_{i}\Big{|}_{\rm
q,\,out}\,\to\,T^{a}_{\,\,cd}\,,\qquad{\bf
T}_{i}\Big{|}_{\rm\bar{q},\,out}\,\to\,-\,T^{a}_{\,\,dc}\,,\qquad{\bf
T}_{i}\Big{|}_{\rm g,\,out}\,\to\,-\,{\rm i}f^{a}_{\,\,cd}\,,$ (5.16)

with the convention that the colour index $d$ is the one to be contracted with
the Born amplitude; the action on incoming particles is defined by crossing
symmetry: it is unchanged for gluons, while the expressions for quarks and
antiquarks are interchanged. Notice that the negative sign for antiquarks in
Eq. (5.16) emerges from the orientation of the fermion line of the Born
amplitude, which is opposite to the (outgoing) momentum flow for an
antiparticle.

In case the soft gluon is emitted into the final state, this discussion
sketches the derivation of the tree-level current for the emission of a soft-
gluon with momentum $k$,

$\displaystyle{\bf
J}^{\mu}(k)\,=\,g\mu^{\epsilon}\sum_{i=1}^{n}\frac{\beta_{i}^{\mu}}{\beta_{i}\cdot
k}\,{\bf T}_{i}\,,$ (5.17)

which will be discussed in greater detail in Section 6. If the soft gluon is
virtual, it will be reabsorbed by another hard particle, leading to the
colour-dipole structure ${\bf T}_{i}\cdot{\bf T}_{j}$, where the product
implies a sum over the soft gluon colour indices. Since the colour operators
${\bf T}_{i}$ act non-trivially only on the colour index of particle $i$,
operators acting on different particles commute, and they satisfy the algebra

$\displaystyle{\bf T}_{i}\cdot{\bf T}_{j}\,=\,{\bf T}_{j}\cdot{\bf
T}_{i}\,\qquad(i\neq j)\,;\qquad{\bf T}_{i}\cdot{\bf T}_{i}\,\equiv\,{\bf
T}_{i}^{2}\,=\,C^{(2)}_{r_{i}}\,,$ (5.18)

where $C^{(2)}_{r_{i}}$ is the quadratic Casimir eigenvalue for the
representation $r_{i}$, so that $C^{(2)}_{A}\equiv C_{A}=N_{c}$ and
$C^{(2)}_{F}\equiv C_{F}=(N_{c}^{2}-1)/2N_{c}$ for the adjoint and the
fundamental representations, respectively. The statement that the scattering
amplitude is an invariant tensor of the colour group is represented in this
formalism by the fact that, when acting on the non-radiative amplitude, the
colour operators satisfy

$\displaystyle\sum_{i=1}^{n}{\bf T}_{i}\,=\,0\,,$ (5.19)

which, as we will see, poses a non-trivial constraint on the structure of the
soft operator at high orders. This brief introduction suggests that the colour
operator notation, while somewhat more formal, and thus perhaps less suited to
concrete applications, is very powerful to construct general statements about
scattering amplitudes with a generic number of particles. In what follows, we
will often leave the colour structure implicit, treating the soft matrix as a
formal colour operator, as in Eq. (5.5): when needed, we will adapt our
notation, either by choosing a basis or by explicitly introducing colour
operators.

###  5.2 The dipole formula and beyond

Having dealt with ways to handle the colour structure of the factorised
amplitude, we now turn back to explore the all-important dynamical
consequences of the factorisation in Eq. (5.5). As we will see, most of the
considerations that were discussed in Section 4.1 in the case of form factors
still apply for the general case of fixed-angle amplitudes, but they turn out
to have much more powerful consequences.

The first point to notice is that the soft function defined in Eq. (5.6), like
its two-line counterpart in Eq. (4.8), obeys a renormalisation group equation,
which however now has a matrix form. Technically, multiplicative
renormalisability of (straight) Wilson-line correlators holds when the lines
are not on the light cone, or in the presence of a collinear regulator:
formally, for the moment, we are going to use dimensional regularisation as a
collinear regulator, while later, for concrete calculations in Section 5.3, it
will turn out to be more practical to tilt Wilson lines off the light cone and
reserve dimensional continuation for the control of UV divergences. With this
understanding, the renormalisation group equation for the soft operator (in a
selected colour basis) reads

$\displaystyle\mu\frac{d}{d\mu}\,{\cal
S}_{LK}\left(\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\right)\,=\,-\,{\cal
S}_{L}^{\,\,\,M}\left(\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\right)\Gamma^{({\cal
S})}_{MK}\left(\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\right)\,,$
(5.20)

which defines the anomalous dimension matrix $\Gamma^{({\cal S})}$. At this
stage, $\Gamma^{({\cal S})}$ is not finite, but carries collinear poles in
$\epsilon$. The solution to Eq. (5.20) can readily be written as

$\displaystyle{\cal
S}\left(\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\right)\,=\,{\cal
P}\exp\left[-\frac{1}{2}\int_{0}^{\mu^{2}}\frac{d\xi^{2}}{\xi^{2}}\,\Gamma^{({\cal
S})}\big{(}\beta_{i}\cdot\beta_{j},\alpha_{s}(\xi^{2},\epsilon),\epsilon\big{)}\right]\,,$
(5.21)

where we used the vanishing of the $d$-dimensional coupling in the infrared,
and we have shuffled the tree-level colour structure into the hard coefficient
function.

It would be nice, of course, to deal with a finite anomalous dimension matrix,
and this is where the full power of the soft-collinear factorisation of the
amplitude can be brought to bear. Indeed, as was the case for form factors, we
can organise Eq. (5.5) either by building the hard-collinear combinations
${\cal J}_{i}/{\cal J}_{E,i}$, which have no soft poles, or, much more
interestingly, by defining the reduced soft matrix

$\displaystyle\widehat{\cal
S}_{LK}\big{(}\rho_{ij},\alpha_{s}(\mu^{2}),\epsilon\big{)}\,\equiv\,\frac{{\cal
S}_{LK}\left(\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\right)}{\prod_{i=1}^{n}{\cal
J}_{E,i}\left(\frac{\left(\beta_{i}\cdot
n_{i}\right)^{2}}{n_{i}^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)}\,.$ (5.22)

which must be free of collinear poles, and is responsible for wide-angle soft
singularities. We see, however, that in this case the cancellation is much
more surprising, since the numerator of Eq. (5.22) is a matrix, while the
denominator is just a number. Clearly, this poses strong constraints on the
soft operator: for example, as the notation suggests, the anomalous dependence
on the light-like vectors $\beta_{i}$ must cancel along with the collinear
poles, which forces the reduced soft function to depend on $n_{i}$ only
through the rescaling-invariant ratios introduced in Eq. (4.15),

$\displaystyle\rho_{ij}\,\equiv\,\frac{\left(\beta_{i}\cdot\beta_{j}\right)^{2}\,n_{i}^{2}\,n_{j}^{2}}{\left(\beta_{i}\cdot
n_{i}\right)^{2}\left(\beta_{j}\cdot n_{j}\right)^{2}}\,.$ (5.23)

The full power of the constraints arising from Eq. (5.22) is best displayed at
the level of the anomalous dimensions. One can write the renormalisation group
equation for the reduced soft function as

$\displaystyle\mu\frac{d}{d\mu}\,\widehat{\cal
S}_{LK}\left(\rho_{ij},\alpha_{s}(\mu^{2}),\epsilon\right)\,=\,-\,\widehat{\cal
S}_{L}^{\,\,\,M}\left(\rho_{ij},\alpha_{s}(\mu^{2}),\epsilon\right)\Gamma^{(\widehat{\cal
S})}_{MK}\left(\rho_{ij},\alpha_{s}(\mu^{2})\right)\,,$ (5.24)

and then note that the matrix $\Gamma^{(\widehat{\cal S})}$ can be written as

$\displaystyle\Gamma^{(\widehat{\cal
S})}_{KL}\left(\rho_{ij},\alpha_{s}(\mu^{2})\right)\,=\,\Gamma^{({\cal
S})}_{KL}\left(\beta_{i}\cdot\beta_{j},\alpha_{s}(\mu^{2}),\epsilon\right)-\delta_{KL}\,\sum_{i=1}^{n}\gamma_{{\cal
J}_{E}}\\!\left(\\!\frac{\left(\beta_{i}\cdot
n_{i}\right)^{2}}{n_{i}^{2}},\alpha_{s}(\mu^{2}),\epsilon\right)\,,$ (5.25)

where $\gamma_{{\cal J}_{E}}$ is the anomalous dimension for the eikonal jet
function, which carries the anomalous dependence on the jet direction
$\beta_{i}$, arising from soft-collinear double poles. It is immediately clear
that three separate powerful constraints on the soft anomalous dimension
matrix $\Gamma^{({\cal S})}$ arise from Eq. (5.25).

  * •

Collinear poles in $\Gamma^{({\cal S})}$ must be confined to the diagonal
entries of the matrix, and they must be determined solely by the light-like
cusp anomalous dimension $\gamma_{K}$, which governs soft-collinear
singularites, and thus fixes $\gamma_{{\cal J}_{E}}$ [74].

  * •

Finite diagonal terms in $\Gamma^{({\cal S})}$ must combine with finite terms
in $\gamma_{{\cal J}_{E}}$ to form the rescaling-invariant ratios $\rho_{ij}$
in Eq. (5.23), as was the case for the form factor.

  * •

Off-diagonal terms in $\Gamma^{({\cal S})}$ must be finite, and furthermore
they must, by themselves, be rescaling-invariant functions of the four-
velocities $\beta_{i}$. Thus, they can only depend on conformal-invariant
cross ratios of the form

$\displaystyle\rho_{ijkl}\,=\,\frac{\beta_{i}\cdot\beta_{j}\,\beta_{k}\cdot\beta_{l}}{\beta_{i}\cdot\beta_{k}\,\beta_{j}\cdot\beta_{l}}\,.$
(5.26)

In order to translate these qualitative statements into a precise mathematical
formulation, we can take a derivative of the matrix $\Gamma^{(\widehat{\cal
S})}$ with respect the arguments $x_{i}=\left(\beta_{i}\cdot
n_{i}\right)^{2}/n_{i}^{2}$, which only occur in the eikonal jets. We find

$\displaystyle x_{i}\frac{\partial}{\partial
x_{i}}\,\Gamma_{KL}^{(\widehat{\cal
S})}\left(\rho_{ij},\alpha_{s}\right)\,=\,-\,\delta_{KL}\,\,x_{i}\frac{\partial}{\partial
x_{i}}\,\gamma_{J_{E}}\,=\,-\,\frac{1}{4}\,\gamma_{K}(\alpha_{s})\,\delta_{KL}\,,$
(5.27)

where the last equality expresses the fact that kinematic dependence in the
eikonal jets can only arise through the superposition of soft and collinear
singularities, and is thus determined by the light-like cusp anomalous
dimension $\gamma_{K}(\alpha_{s})$. The precise form of the equality follows
from a detailed analysis of the RG equation for the eikonal jet [74], as well
as from the general arguments leading to Collins-Soper equations, as first
laid out in Ref. [330]. We know, however, that $\Gamma^{(\widehat{\cal S})}$
can only depend on $x_{i}$ through the combinations $\rho_{ij}$ defined in Eq.
(5.23). Using the chain rule we then conclude that [74, 75]

$\displaystyle\sum_{i=1}^{n}\sum_{j\neq
i}\,\frac{\partial}{\partial\ln\rho_{ij}}\,\Gamma_{KL}^{(\widehat{\cal
S})}\left(\rho_{ij},\alpha_{s}\right)\,=\,\frac{1}{4}\,\gamma_{K}(\alpha_{s})\,\delta_{KL}\,.$
(5.28)

Eq. (5.28) governs the interplay of colour and kinematics in the soft
anomalous dimension matrix to all orders in perturbation theory, and is an
exact result.

Our remaining task is now to solve Eq. (5.28). As we are dealing with a linear
inhomogeneous partial differential equation, the general solution can be
expressed as the sum of a particular solution of the inhomogeneous problem,
plus a generic solution of the corresponding homogeneous equation. In order to
proceed, we now shift from the basis-dependent notation employed thus far to
the colour-operator notation: upon making a single approximation, this will
allow us to write a particular solution of Eq. (5.28) in a transparent and
elegant way. We begin by noting that the cusp anomalous dimension in colour
representation $r$ can be computed from the form factor for particles
trasforming under $r$, and admits a Casimir expansion [391, 392, 393, 394]

$\displaystyle\gamma_{K}^{(r)}(\alpha_{s})\,=\,\sum_{p,i}^{\infty}C^{\,(2p,i)}_{r}\,\,\widehat{\gamma}^{\,(2p,i)}_{K}(\alpha_{s})\,,$
(5.29)

where $C^{\,(n,i)}_{r}$ are eigenvalues of $n$-th order Casimir operators of
the Lie algebra, in representation $r$, and the functions
$\widehat{\gamma}^{\,(2p,i)}_{K}$ are independent of $r$. The index $i$ takes
into account possible dependence on the matter content of the particular gauge
theory under consideration: in ordinary QCD, for example, one finds loop-level
contributions involving both the adjoint and the fundamental representations.
For $p>1$, building a Casimir invariant of order $2p$ requires $2p$ gluon
attachments to the Wilson lines, and a further $2p$ vertices (for example
forming a loop) in order to build a symmetric colour tensor saturating the
open colour indices. Thus, for example, quartic Casimir contributions
(corresponding to $p=2$) will start at four loops, sixth-order Casimir
contributions ($p=3$) at six loops, and so on; quadratic Casimirs are special
because they are built with the Lie algebra metric tensor $\delta_{ab}$, so
that the corresponding contribution starts at one loop. Finite-order
calculations indeed confirm that the cusp anomalous dimension obeys Casimir
scaling up to three loops (i.e. it is proportional to the quadratic Casimir
eigenvalue $C^{(2)}_{r}$), while it develops quartic Casimir components at
four loops, as expected [395, 363, 396, 397, 398, 230, 399, 231].

Given this general structure, it is both phenomenologically and theoretically
interesting to begin by focusing on the quadratic Casimir component of the
cusp, which provides the full answer up to three loops. This component yields
a simple solution to Eq. (5.28), since one can write (using Eq. (5.18))

$\displaystyle\gamma_{K}^{(r)}(\alpha_{s})\,=\,C^{(2)}_{r}\,\widehat{\gamma}_{K}(\alpha_{s})\,=\,\widehat{\gamma}_{K}(\alpha_{s})\,{\bf
T}_{i}\cdot{\bf T}_{i}\,,$ (5.30)

where for simplicity we dropped the Casimir-counting indices in Eq. (5.29). It
is then easy to verify, using Eq. (5.19), that the ‘dipole sum’

$\displaystyle\Gamma_{\rm dip}^{(\widehat{\cal
S})}\left(\rho_{ij},\alpha_{s}\right)\,=\,-\,\frac{1}{8}\,\widehat{\gamma}_{K}(\alpha_{s})\sum_{i=1}^{n}\sum_{j\neq
i}\ln\rho_{ij}\,{\bf T}_{i}\cdot{\bf T}_{j}\,+\,\delta^{\,(\widehat{\cal
S})}(\alpha_{s})\,\sum_{i=1}^{n}{\bf T}_{i}\cdot{\bf T}_{i}\,,$ (5.31)

is a solution of Eq. (5.28), with the function $\delta^{\,(\widehat{\cal S})}$
playing the role of an integration constant. Before highlighting the important
properties of Eq. (5.31), it is useful to embed it in the full factorisation
of the amplitude, in order to get rid of the unphysical dependence on the
auxiliary vectors $n_{i}$, which still feature in Eq. (5.31) through the
variables $\rho_{ij}$. Indeed, as already noted at the end of Section 4.1,
$n_{i}$-dependent singular terms must cancel in the ratio of jets and eikonal
jets, whereas finite $n_{i}$-dependent terms must cancel between the jets and
the hard matching function ${\cal H}$. Performing these cancellations, order
by order in perturbation theory, yields a simplified form for the soft-
collinear factorisation of fixed-angle amplitudes, which can be written as
[75, 76]

$\displaystyle{\cal
A}_{n}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu^{2}),\epsilon\right)\,=\,{\cal
Z}_{n}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu^{2}),\epsilon\right){\cal
F}_{n}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu^{2}),\epsilon\right)\,,$ (5.32)

where ${\cal Z}_{n}$ is a colour operator generating all infrared
singularities, and acting on the finite vector ${\cal F}_{n}$. Since the non-
trivial colour structure of ${\cal Z}_{n}$ arises entirely from the soft
factor, it will obey a matrix renormalisation group equation of the same form
as Eq. (5.24), and the solution will again be an exponential, which we can
write as

$\displaystyle{\cal
Z}_{n}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu^{2}),\epsilon\right)\,=\,{\cal
P}\exp\left[\frac{1}{2}\int_{0}^{\mu^{2}}\frac{d\lambda^{2}}{\lambda^{2}}\,\,\Gamma_{n}\left(\frac{p_{i}}{\lambda},\alpha_{s}\\!\left(\lambda^{2},\epsilon\right)\right)\right]\,,$
(5.33)

with the full infrared anomalous dimension matrix $\Gamma_{n}$ satisfying a
constraint equation inherited from Eq. (5.28). The general solution to that
equation will take the form

$\displaystyle\Gamma_{n}\left(\frac{p_{i}}{\mu},\alpha_{s}(\mu^{2})\right)\,=\,\Gamma_{n}^{\rm
dip}\left(\frac{s_{ij}}{\mu^{2}},\alpha_{s}(\mu^{2})\right)\,+\,\Delta_{n}\left(\rho_{ijkl},\alpha_{s}(\mu^{2})\right)\,,$
(5.34)

with $\Gamma_{n}^{\rm dip}$ a particular solution of the inhomogeneous
equation, akin to Eq. (5.31), and $\Delta_{n}$ the general solution of the
homogeneous problem. Considering first the dipole term, after the cancellation
of of the dependence on the auxiliary vectors, the logarithms of the scaling
variables $\rho_{ij}$ must be replaced with logarithms of the corresponding
Mandelstam invariants. At this point we can also reinstate the analytic
properties, which we have neglected so far, recalling from Section 4.1 that
time-like invariants carry imaginary parts. With these considerations, we can
finally write the dipole formula as [73, 74, 75, 76]

$\displaystyle\Gamma_{n}^{\rm
dip}\bigg{(}\frac{s_{ij}}{\mu^{2}},\alpha_{s}(\mu^{2})\bigg{)}\,=\,\frac{1}{2}\,\widehat{\gamma}_{K}\left(\alpha_{s}(\mu^{2})\right)\sum_{i=1}^{n}\sum_{j=i+1}^{n}\log\left(\frac{s_{ij}\,{\rm
e}^{{\rm i}\pi\lambda_{ij}}}{\mu^{2}}\right){\bf T}_{i}\cdot{\bf
T}_{j}+\sum_{i=1}^{n}\gamma_{i}\left(\alpha_{s}(\mu^{2})\right)\,,$ (5.35)

where $\gamma_{i}$ are colour-diagonal anomalous dimensions arising from jet
functions, and the phases are given by $\lambda_{ij}=1$, if particles $i$ and
$j$ are either both in the final state or both in the initial state, while
$\lambda_{ij}=0$ otherwise.

Turning now to the homogeneous problem, we note that the operator
$\Delta_{n}$, in the language of Eq. (5.28), must satisfy

$\displaystyle\sum_{i=1}^{n}\sum_{j\neq
i}\,\frac{\partial}{\partial\ln\rho_{ij}}\,\Delta_{n}\,=\,0\,.$ (5.36)

As anticipated by the notation in Eq. (5.34), the general solution to this
equation expresses the fact that $\Delta_{n}$ must be scale invariant in each
momentum variable, which in turn implies that it can depend on Mandelstam
invariants only through the conformal cross ratios defined in Eq. (5.26).
Since $\Delta_{n}$ involves correlations of at least four particles, we can
conclude that Eq. (5.35) gives the exact all-order result for the infrared
anomalous dimension for $n=2,3$. Furthermore, for $n\geq 4$, we note that
correlations involving four hard particles can only arise starting at three
loops. We conclude that Eq. (5.35) provides the exact result, up to two loops,
for any number of hard particles: this is non-trivial, since at two loops
three-particle correlations can arise (and indeed they do arise when hard
particles are massive [400], as we briefly discuss in Section 5.3.5). This
confirms, and further explains, the result of the direct calculation performed
in [71, 72]. Further constraints on $\Delta$ at three loops from various
kinematic limits and symmetry arguments were studied in Refs. [401, 402],
highlighting the limited span of the space of functions available for such
corrections.

The lowest order contribution to the conformal correction $\Delta_{n}$, at the
three-loop order, was finally calculated for the first time in Ref. [77], and
verified by an explicit amplitude calculation in ${\cal N}=4$ Super-Yang-Mills
theory in Ref. [403]. It takes the form of quadrupole colour correlations,
with a kinematic dependence expressed in terms of a single combination of
weight-five polylogarithms, of remarkable simplicity and elegance. We write

$\displaystyle\Delta_{n}\big{(}\rho_{ijkl},\alpha_{s}\big{)}\,=\,\left(\frac{\alpha_{s}}{\pi}\right)^{3}\Delta_{n}^{(3)}\big{(}\rho_{ijkl}\big{)}\,+\,{\cal
O}(\alpha_{s}^{4})\,,$ (5.37)

where [77]

$\displaystyle\Delta_{n}^{(3)}(\rho_{ijkl})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\,f_{abe}f^{e}_{{\phantom{e}}cd}\,\Bigg{\\{}-C\,\sum_{i=1}^{n}\,\sum_{\begin{subarray}{c}{1\leq
j<k\leq n}\\\ j,k\neq i\end{subarray}}\left\\{{\rm\bf T}_{i}^{a},{\rm\bf
T}_{i}^{d}\right\\}{\rm\bf T}_{j}^{b}{\rm\bf T}_{k}^{c}$
$\displaystyle+\,{\sum_{1\leq i<j<k<l\leq n}}\bigg{[}{\rm\bf T}_{i}^{a}{\rm\bf
T}_{j}^{b}{\rm\bf T}_{k}^{c}{\rm\bf T}_{l}^{d}\,\,{\cal
F}(\rho_{ikjl},\rho_{iljk})\,+\,{\rm\bf T}_{i}^{a}{\rm\bf T}_{k}^{b}{\rm\bf
T}_{j}^{c}{\rm\bf T}_{l}^{d}\,\,{\cal F}(\rho_{ijkl},\rho_{ilkj})$
$\displaystyle\qquad\qquad+\,\,{\rm\bf T}_{i}^{a}{\rm\bf T}_{l}^{b}{\rm\bf
T}_{j}^{c}{\rm\bf T}_{k}^{d}\,\,{\cal
F}(\rho_{ijlk},\rho_{iklj})\bigg{]}\Bigg{\\}}\,.$

One readily observes that the form of $\Delta$ is heavily constrained by Bose
symmetry. The first line in Eq. (5.2) is a constant (kinematic-independent)
colour tensor, with

$\displaystyle C\,=\,\zeta_{5}+2\zeta_{2}\zeta_{3}\,.$ (5.39)

The specific value of the constant $C$ is crucial to preserve the property of
collinear factorisation, to which we will return in Section 6: in the limit in
which two (or more) hard particles become collinear, the scattering amplitude
is expected to factorise, yielding a lower-multiplicity amplitude, multiplied
times a splitting kernel, and the splitting kernel is expected to depend only
on the quantum numbers of the collinear set. The value of the constant $C$
guarantees the necessary cancellations to enforce this result at three loops
[77]. In order to display the kinematic dependence of $\Delta$, it is useful
to introduce auxiliary variables $\\{z_{ijkl},\bar{z}_{ijkl}\\}$, defined by
the relations

$\displaystyle
z_{ijkl}\,\bar{z}_{ijkl}\,=\,\rho_{ijkl}\,,\qquad\big{(}1-z_{ijkl}\big{)}\big{(}1-\bar{z}_{ijkl}\big{)}\,=\,\rho_{ilkj}\,.$
(5.40)

In terms of these variables, we can express the kinematic function ${\cal F}$
as

$\displaystyle{\cal
F}\big{(}\rho_{ijkl},\rho_{ilkj}\big{)}\,=\,F\big{(}1-z_{ijkl}\big{)}-F\big{(}z_{ijkl}\big{)}\,,$
(5.41)

where, finally, one finds

$\displaystyle F(z)\,={\cal L}_{10101}(z)+2\zeta_{2}\Big{[}{\cal
L}_{001}(z)+{\cal L}_{100}(z)\Big{]}\,.$ (5.42)

The functions ${\cal L}_{w}(z)$ (where $w$ is a word composed of zeroes and
ones) are single-valued harmonic polylogarithms (SVHPL) [404]. These are
special combinations of harmonic polylogarithms [405] with the property of
being single-valued in the kinematic region where $\bar{z}$ is equal to the
complex conjugate of $z$: this region is a subset of the Euclidean region,
where all Mandelstam invariants are spacelike, $s_{ij}<0$. Unitarity of
massless scattering amplitudes dictates that they can only have singularities
due to the vanishing of Mandelstam invariants, which do not occur in the
Euclidean region of fixed angle scattering. Thus, the single-valuedness of the
function $F$ directly reflects the analytic structure of the underlying
amplitude. Interestingly, the landmark result in Eq. (5.2) can be derived
without resorting to the explicit calculation of the relevant Feynman
diagrams, but rather by using bootstrap methods, i.e. by identifying the space
of functions which can contribute to $\Delta^{(3)}$, and subsequently imposing
kinematic constraints arising from known symmetries and limiting behaviours of
the scattering amplitude, such as high-energy and collinear limits. This was
done successfully at the three-loop order in Ref. [406], reproducing Eqs.
(5.2-5.42).

We emphasise that, in this Section, we have chosen to arrive at the general
expression for the infrared anomalous dimension matrix in Eq. (5.34) following
the path of factorisation. Eq. (5.34), with the definition of the dipole
contribution in Eq. (5.35), are thus valid to all orders in perturbation
theory. The historical path was quite different, starting with studies at two
loops. The general structure of infrared divergences in multi-particle
scattering amplitudes for massless non-abelian gauge theories at two-loops was
first displayed in the seminal paper by Catani, Ref. [407], and subsequently
derived from factorisation in Ref. [408]. At that time, the precise form of
soft single-pole contributions at two loops could not be explicitly
determined, and in principle it was natural to expect a colour-tripole
contribution at that level. Explicit calculations of four-point scattering
amplitudes and splitting functions at two loops [409, 410, 411, 412, 413, 414,
415, 416, 417] indeed found single-pole contributions with a colour-tripole
structure. It turns out however that these contributions do not arise at the
level of the soft anomalous dimension (in a minimal scheme), but can be
generated when the infrared factorisation formula is expanded to finite
orders, and the one-loop hard part has been defined in a non-minimal scheme,
such as the one employed in Ref. [407]. The absence of tripole contributions
in the soft anomalous dimension at two loops was later explicitly verified by
direct calculation in Refs. [71, 72]. Finally, Refs. [73, 74, 75, 76] provided
the underlying symmetry argument for the cancellation, which leads to the all-
order structure in displayed in Eq. (5.34). Importantly, it has been
conjectured in [418, 419] that the cancellation extends to all odd colour
multipoles, which would for example forbid the presence of a penta-pole
contribution to $\Delta_{n}$ at four loops.

The frontier of current research is the evaluation of the infrared anomalous
dimension matrix at the four-loop level for massless gauge theories, and at
the three-loop level in the presence of massive coloured particles. At four
loops, the occurrence of quartic Casimir operators in the cusp anomalous
dimensions provides interesting new constraints, in particular arising from
the requirements of collinear factorisation. The infrared anomalous dimension
matrix for multi-particle amplitudes will also contain quartic Casimir
contributions at four loops, with a more intricate colour structure as
compared with the cusp. These two sets of contributions must however be subtly
correlated in order to preserve known all-order properties of the amplitudes
in collinear and high-energy limits. These constraints have been studied in
Refs. [420, 421], where a general ansatz for the infrared matrix was provided.
It is possible that a bootstrap approach will succeed in determining all the
unknown functions in the ansatz in [421], if a sufficient number of
constraints can be imposed. Interestingly, the first direct evidence for the
existence of terms beyond the dipole formula came from the analysis of the
high-energy limit in Ref. [422], which uncovered a single-pole contribution at
four loops, of the quartic Casimir type. More recently, further constraints on