CERN-TH-2021-145

The Infrared Structure of Perturbative Gauge Theories111Partly based on
graduate lectures given by L.M. at the Indian Institute of Technology in
Hyderabad, under the GIAN Initiative of the Ministry of Human Resource
Development, Government of India.

Neelima Agarwala, Lorenzo Magneab,c,

Chiara Signorile-Signoriled and Anurag Tripathie

aDepartment of Physics, Chaitanya Bharathi Institute of Technology

Gandipet, Hyderabad, 500075, Telangana, India

bTheoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland

cDipartimento di Fisica and Arnold-Regge Center, Università di Torino

and INFN, Sezione di Torino

Via P. Giuria 1, I-10125 Torino, Italy

dInstitut für Astroteilchenphysik, Karlsruher Institut für Technologie (KIT),

D-76021 Karlsruhe, Germany

eDepartment of Physics, IIT Hyderabad

Kandi, Sangareddy, 502284, Telangana, India

###### Abstract

Infrared divergences in the perturbative expansion of gauge theory amplitudes
and cross sections have been a focus of theoretical investigations for almost
a century. New insights still continue to emerge, as higher perturbative
orders are explored, and high-precision phenomenological applications demand
an ever more refined understanding. This review aims to provide a pedagogical
overview of the subject. We briefly cover some of the early historical
results, we provide some simple examples of low-order applications in the
context of perturbative QCD, and discuss the necessary tools to extend these
results to all perturbative orders. Finally, we describe recent developments
concerning the calculation of soft anomalous dimensions in multi-particle
scattering amplitudes at high orders, and we provide a brief introduction to
the very active field of infrared subtraction for the calculation of
differential distributions at colliders.

###### Contents

  1. 1 Introduction
    1. 1.1 Catastrophe and recovery in QED
    2. 1.2 Observable cross sections are finite: the KLN theorem
    3. 1.3 Saving the $S$-matrix: coherent states
      1. 1.3.1 Collinear divergences in a scalar theory
      2. 1.3.2 Soft divergences in QED
    4. 1.4 What to do in QCD
      1. 1.4.1 Infrared-safe cross sections
      2. 1.4.2 Factorisable cross sections
  2. 2 Finite orders: tools and results
    1. 2.1 Infrared safety: the total cross section
    2. 2.2 A tool: cut diagrams
    3. 2.3 Real radiation at one loop
    4. 2.4 Virtual corrections: the quark form factor
    5. 2.5 A tool: the $d$-dimensional running coupling
    6. 2.6 General infrared-safe observables
    7. 2.7 A tool: eikonal integrals
    8. 2.8 Hadron scattering and collinear divergences
  3. 3 Infrared singularities to all orders
    1. 3.1 Locating potential singularities: the Landau equations
      1. 3.1.1 Singularities of complex functions defined by an integral
      2. 3.1.2 The case of Feynman diagrams: the Landau equations
      3. 3.1.3 The Coleman-Norton physical picture
      4. 3.1.4 Pinch surfaces of massless vertex graphs
    2. 3.2 Identifying actual singularities: infrared power counting
      1. 3.2.1 Intrinsic and normal coordinates, and homogenous integrals
      2. 3.2.2 One-loop examples: the QED vertex graph
      3. 3.2.3 Soft and collinear power counting to all orders
    3. 3.3 Diagrammatic construction of soft and collinear functions
      1. 3.3.1 Soft and collinear approximations at one loop
      2. 3.3.2 Wilson lines and the eikonal approximation
  4. 4 Factorisation, evolution and resummation: form factors
    1. 4.1 Soft-collinear factorisation of a form factor
    2. 4.2 Evolution and resummation of infrared poles
  5. 5 Fixed-angle scattering amplitudes
    1. 5.1 Handling color structures
      1. 5.1.1 Colour tensor bases
      2. 5.1.2 Colour insertion operators
    2. 5.2 The dipole formula and beyond
      1. 5.2.1 Taking the high-energy limit
      2. 5.2.2 A celestial view
    3. 5.3 Computing the infrared anomalous dimension matrix
      1. 5.3.1 Diagrammatic exponentiation of Wilson-line correlators
      2. 5.3.2 From gluon webs to correlator webs
      3. 5.3.3 A replica algorithm to generate web mixing matrices
      4. 5.3.4 The one-loop angle-dependent cusp anomalous dimension
      5. 5.3.5 The two-loop soft anomalous dimension matrix
  6. 6 Real radiation: factorisation and subtraction
    1. 6.1 Factorisation for soft and collinear real radiation
    2. 6.2 Introducing the problem of infrared subtraction at NLO
    3. 6.3 Soft and collinear local counterterms: a general strategy
    4. 6.4 The structure of infrared subtraction at NNLO
  7. 7 Perspectives

##  1 Introduction

Students taking introductory classes in quantum field theory may be forgiven
for believing, after perhaps the first half of their course, that they hold
the keys for a full understanding of particle interactions and countless
phenomenological applications. Granted that coupling constants are not too
big, armed with the Feynman rules they have just derived, they may imagine
that what lies ahead is largely to learn a set of technical tools, to speed up
calculations that they already have a fair idea of how to perform. This
delusion is of course soon to be shattered when they are faced with their
first radiative corrections, and they realise that ‘applying the rules’ leads
to apparently non-sensical results, uncovering serious conceptual problems,
which need to be patiently and creatively addressed. This revelation strikes
students, for example, when they reach Chapter 6 of Ref. [1], or Chapter 9 of
Ref. [2]. Historically, the ultraviolet problem was indeed baffling enough
that Nobel-Prize winning theoreticians went to their grave convinced that the
entire edifice of quantum field theory should be discarded because of it [3].

A more constructive point of view could be described as follows. If we have
reason to believe that a certain quantum field theory is relevant to physics,
either because of its symmetry and aesthetic appeal, or because it is
confirmed by experiment, and yet we find that our calculations yield infinite
answers, the most likely explanation is surely that we made a mistake, and one
of our approximations is failing. It pays to look for the mistake, and try to
patch the approximation, lest we ‘throw out the baby with the bath water’. As
a matter of fact, finding, understanding and fixing such mistakes has
invariably brought great progress, and a wealth of deeper physical
understanding.

To illustrate this fact, consider a quantity $R$, that one assumes can be
computed in perturbation theory, depending on some physical energy scale $Q$
and on a small dimensionless coupling $\alpha$. Our student would expect that
increasingly precise approximations to this quantity would take the form

$\displaystyle
R(Q,\alpha)\,=\,R_{0}\left[1+\frac{\alpha}{\pi}\,c_{1}(Q)+\left(\frac{\alpha}{\pi}\right)^{2}c_{2}(Q)+\ldots\right]\,,$
(1.1)

upon computing successive coefficients $c_{k}(Q)$. If the quantity $R$ really
depends only on a single scale $Q$, one further expects that $c_{k}$ must be
independent of $Q$, on dimensional grounds. As we know, these expectations
fail drastically for almost all interesting theories, where one finds that the
expressions for $c_{k}$ are given by ultraviolet divergent integrals and are
thus meaningless. Artificially introducing an ultraviolet cutoff $\Lambda$
typically reveals that $c_{k}\sim\log^{k}(\Lambda/Q)$. The question then is:
what was our mistake? In this case, it was hybris, an excess of confidence: we
assumed that our theory would remain applicable to arbitrarily large energy
scales, and to arbitrarily small length scales; or, more precisely, we assumed
that the effect of large-energy, short-distance degrees of freedom would be
negligible for our observable. This assumption is defeated by the rules of
quantum mechanics, which require us to sum over all unobserved field
fluctuations, including those that are far from classical – far off-shell.
Such fluctuations are of course suppressed in the path integral, but it turns
out that, in $d=4$ space-time dimensions, there are just too many of them.
This fact looks dangerous indeed: if unknown Planck-scale physics
significantly affects our low-energy laboratory experiments, we could be
facing a complete loss of predictivity. Fortunately, we have a well-
established solution to this problem, provided by renormalisation, and more
generally by the idea of effective field theory (see, for example, [4, 5]). We
exploit the fact that ultraviolet contributions to our integrals correspond to
highly localised fluctuations in space-time, which allows us to parametrise
our ignorance of the UV completion of our theory by absorbing the unknown UV
effects in the lagrangian couplings. Even when this is not sufficient (the
theory is not renormalisable), in most cases we can still understand our
theory in terms of a low-energy expansion, and increase the accuracy of our
predictions by adding higher-dimensional operators to the Lagrangian.
Operationally, we introduce a new scale $\mu$, which we may take to represent
the boundaries of our current knowledge, and we promote Eq. (1.1) to

$\displaystyle
R\left(\frac{Q}{\mu},\alpha(\mu)\right)\,=\,R_{0}\left[1+\frac{\alpha(\mu)}{\pi}\,\,c_{1}\\!\bigg{(}\frac{Q}{\mu}\bigg{)}+\bigg{(}\frac{\alpha(\mu)}{\pi}\bigg{)}^{\\!2}c_{2}\bigg{(}\frac{Q}{\mu}\bigg{)}+\ldots\right]\,.$
(1.2)

The perturbative coefficients now have a new, non-trivial scale dependence,
but this is tamed by renormalisation group equations, enforcing the
independence of physical observables on the artificial boundary set by $\mu$.
Understanding our UV mistake has paid off handsomely: we now see that unknown
high-energy physics does not necessarily spoil predictivity, we have at our
disposal the tools of the renormalisation group to study asymptotic
behaviours, and the powerful arsenal of effective field theories allows us to
tackle systematically many problems that defied earlier techniques.

A similar story can be told about another problem afflicting the naive
expectations expressed by Eq. (1.1), a problem which many students may not
meet in their early courses at all. For almost all interesting quantities, it
is a well-known fact [6, 7] that the asymptotic behavior of the perturbative
coefficients $c_{k}$ in Eq. (1.1) and in Eq. (1.2) for large $k$ is $c_{k}\sim
k!$, implying that the perturbative series is divergent – in fact, it has
vanishing radius of convergence, and it is often not amenable to be handled
with standard summation methods, such as Borel summation. It is not difficult
to find evidence for this factorial growth in large-order Feynman graphs:
first of all, the number of graphs grows factorially with the order, as
illustrated in Fig. 1(a); next, for example, evaluating graphs involving
chains of fermion loops, such as the one shown in Fig. 1(b), also results in
factorial growth [8].

(a)

(b)

Figure 1: Examples of Feynman graphs representative of the factorial growth of
perturbative coefficients at large orders.

Once again, one may legitimately fear a loss of predictivity, and one may ask
– assuming one believes in the consistency of the underlying theory – what was
the mistake that lead to the problem. Once again the answers are rich with
insights and discoveries: in this case, the underlying assumption was that
non-perturbative effects would be negligible, or in some sense decoupled from
the perturbative expansion. This, quite interestingly, turns out not to be the
case. The factorial growth in the number of diagrams can be precisely related
to the existence of non-trivial classical solutions – instantons – which are
not accessible by means of a weak-coupling expansion [9, 10]; renormalon
singularities, which can be detected by means of fermion-loop chains, reveal
contributions to observables originating from vacuum condensates, in many
cases matching the results of the operator product expansion [11]. Remarkably,
perturbation theory seems to know about its own limitations, and can be used
to infer the existence of non-perturbative effects and study their impact.
While these insights are several decades old, new powerful techniques are
currently being developed to refine existing tools and consolidate their
mathematical foundations (see for example [12, 13, 14]).

Finally, we come to the subject of our review. Even after renormalisation, and
even if we confine ourselves to perturbative effects, Eq. (1.2) is still
‘wrong’ for most interesting theories with massless particles (and in
particular for unbroken gauge theories in $d=4$): expressions for the
coefficients $c_{k}(Q/\mu)$ involve integrals that diverge at low energies (in
momentum space) or at large distances (in coordinate space). Furthermore, if
we attempt to compute the probability for the emission of one or more massless
particles in connection with a hard scattering process, this will typically
also diverge. This infrared catastrophe was actually the first problem to be
clearly identified, among those we have discussed. Long before quantum field
theory was fully developed, some of the earliest studies of the interactions
of electrons with electromagnetic radiation showed that the frequency spectra
of emitted photons behave like $d\nu/\nu$, and thus are not integrable at low
frequencies. This emerged from analyses of electron scattering in a Coulomb
field, when allowing for extra photon radiation [15, 16], with results
subsequently refined in [17], where pair production in the Coulomb field was
also considered.

Following the logic we proposed, we may ask one last time what was the mistake
that led to this problem. The answer to this question has several layers of
depth, that we will explore in the rest of this review, but it is interesting,
and truly remarkable, that a precise understanding of the underlying physical
problem was developed as early as 1937, with the seminal paper by Bloch and
Nordsieck [18]. It is worthwhile to reproduce here the Abstract of that paper,
which reads as follows.

Previous methods of treating radiative corrections in non-stationary processes
such as the scattering of an electron in an atomic field or the emission of a
$\beta$-ray, by an expansion in powers of $e^{2}/\hbar c$, are defective in
that they predict infinite low frequency corrections to the transition
probabilities. This difficulty can be avoided by a method developed here which
is based on the alternative assumption that $e^{2}\omega/mc^{3}$,
$\hbar\omega/mc^{2}$ and $\hbar\omega/c\Delta p$ ($\omega=$angular frequency
of radiation, $\Delta p=$change in momentum of electron) are small compared to
unity. In contrast to the expansion in powers of $e^{2}/\hbar c$, this permits
the transition to the classical limit $\hbar=0$. External perturbations on the
electron are treated in the Born approximation. It is shown that for
frequencies such that the above three parameters are negligible the quantum
mechanical calculation yields just the directly reinterpreted results of the
classical formulae, namely that the total probability of a given change in the
motion of the electron is unaffected by the interaction with radiation, and
that the mean number of emitted quanta is infinite in such a way that the mean
radiated energy is equal to the energy radiated classically in the
corresponding trajectory.

In essence, Bloch and Nordsieck argue that, in the soft photon limit,
perturbation theory (in powers of $\alpha=e^{2}/(\hbar c)$) must be abandoned,
and they advocate a different approximation scheme (what we now call eikonal
approximation), valid when the photon energy is much smaller than the other
energy scales in the problem (the electron mass and the momentum trasfer), and
the photon wavelength is much larger than the classical electron radius
$r_{e}=e^{2}/(mc^{2})$. They then show that this approximation is
semiclassical, in that the classical result for the mean radiated energy is
recovered, but this entails the radiation of an infinite number of low-energy
photons.

The Bloch and Nordsieck paper is truly remarkable because it engineers the
cancellation of divergences between virtual corrections and real-radiation
contributions, long before the treatment of virtual corrections could be
formalised, and furthermore it provides the first example of an all-order
summation of perturbation theory. In the following decades, it received
several refinements, which proved the complete generality of the results,
placed it squarely in the context of (renormalised) QED [19, 20], and
significantly streamlined the proof [21]. We will briefly summarise the
argument, in modern language, in Section 1.1. We still need, however, to
properly diagnose the mistake that lies at the origin of the problem: in this
regard, there are two complementary viewpoints.

First, one can argue that the problem is the proper definition of an
observable. In a theory with massless particles, in any scattering process one
can produce particles with infinitesimal energy, as well as particles with
infinitesimal angular separation; on the other hand, every physical detector
has finite energy and angular resolutions. Quantum mechanics prescribes that
we sum over all configurations we do not observe, so, in principle, an
arbitrary number of low-energy or collinear particles must be included in a
proper definition of an observable cross section. When this is done, the
result is expected to be finite, as it corresponds to a truly measurable
quantity. This line of reasoning led to the most general theorem concerning
the cancellation of infrared singularities, the KLN theorem [22, 23]. In
particular, Ref. [23] shows that the cancellation is a completely general
property of any quantum-mechanical system whose Hilbert space contains sets of
energy-degenerate states. We will sketch a proof of the KLN theorem in Section
1.2. Before we continue, however, we need to make the argument a little
sharper: in principle, the probability for emission of soft or collinear
particles could be small, and the effect on the cross section negligible. In
order to understand the physics of infrared enhancements, consider a diagram
for the emission of a photon from an external fermion leg in massless QED,
shown in Fig. 2.

Figure 2: A photon emission diagram in massless QED, possibly responsible for
infrared divergences in soft and collinear configurations.

The QED Feynman rules give an expression of the form

$\displaystyle
e\,\overline{u}(p)\,\gamma^{\mu}\frac{\not{p}+\not{k}}{(p+k)^{2}+{\rm
i}\eta}\;{\cal M}\,,$ (1.3)

where ${\cal M}$ represent the rest of the scattering process, which may
involve many external legs and virtual corrections as well. If the photon is
emitted in the final state, so that $k^{2}=p^{2}=0$, the denominator of the
fermion propagator reads $2p\cdot k+{\rm
i}\eta=E_{p}\omega_{k}(1-\cos\theta_{pk})+{\rm i}\eta$, where $E_{p}=|{\bf
p}|$ and $\omega_{k}=|{\bf k}|$. While the $\eta$ prescription protects from
an outright singularity, one must expect enhancements from three sources: the
soft photon limit $\omega_{k}\to 0$, the soft fermion limit $E_{p}\to 0$, and
the collinear limit $\theta_{pk}\to 0$. Whether these enhancements translate
into actual divergences will depend on the specific observable one is
computing, and more generally on the theory one is considering. To this end,
one will need to develop appropriate power-counting techniques (discussed here
in Section 3), similarly to what is done for ultraviolet enhancements; for
example, we can anticipate that the soft fermion limit never leads to
divergences in renormalisable theories, thanks to an extra power of the energy
arising from the wave functions of massless spinors. Note also that, in case
the fermion were massive, the collinear singularity would be regulated by the
fermion mass, since the angular factor in the denominator would read $(1-|{\bf
v}|\cos\theta_{pk})$, with $|{\bf v}|<1$ the velocity of the fermion.

That being said, the origin of the enhancement is apparent: in the limits
considered, the fermion propagator reaches the mass shell, $(p+k)^{2}=0$;
furthermore, since we are working in covariant perturbation theory, all four
momentum components are conserved at each vertex; thus, the intermediate
fermion with momentum $p+k$ is a physical state in our theory, and can
propagate freely for any length of space and interval of time. When deriving
the momentum-space Feynman rules, we have formally integrated over the
position of the photon emission vertex in spacetime, under the assumption that
emission at long distances should be sufficiently suppressed. Unfortunately,
it is not, a consequence of the fact that QED (like all unbroken gauge
theories) has long-range interactions. The same conclusion is reached, perhaps
in an even more transparent way, if one uses Time-Ordered Perturbation Theory
(TOPT) (see, for example, Ref. [2] for a detailed discussion): within that
framework, all particles in intermediate states are on the mass shell; energy,
on the other hand, is not in general conserved at the vertices, which forces
the interactions to take place in a finite time. For soft or collinear
emissions, the energy deficit at the emission vertex vanishes, so that once
again late-time emissions are unsuppressed. In this sense, both soft and
collinear divergences are properly described as long-distance effects.

Importantly, these conclusions are essentially unaffected if the photon line
in Fig. 2 folds back and attaches to some other fermion line in the amplitude,
forming a loop, instead of being radiated into the final state. In that case,
the denominator of the fermion propagator has an additional $k^{2}$, which
however is negligible with respect to $p\cdot k$ if all components of the
photon momentum become small at the same rate. Clearly, some power counting
tools will have to be developed to weigh the presence of the loop integral (as
opposed to the phase-space integration over the real photon momentum) and of
the photon propagator, but the basic fact remains that, even for virtual
corrections, soft and collinear emissions are dangerously enhanced at large
distances and times. This provides the seeds for the eventual cancellation of
divergences between virtual and real corrections: both are enhanced by the
same mechanism, and both must be included to allow for the finite energy and
angle resolutions of detectors; once a properly defined observable is
constructed, they must, and do, cancel each other.

Let us summarise this first viewpoint on the mistake that led to the rise of
infrared divergences, as it emerges from the Bloch-Nordsieck analysis.
Theories with massless particles have long-range interactions; as a
consequence, late- (and early-) time emissions are not sufficiently
suppressed; in the circumstances, our organisation of perturbation theory in
powers of the coupling, which distinguishes between virtual corrections and
(undetected) real emissions is inadequate, hence individual matrix elements
are ill-defined. The proposed solution is to introduce a temporary fix for the
matrix elements (an infrared regulator such a particle mass), in the knowledge
of the fact that the singular dependence on the regulator will cancel in
properly defined observable cross sections.

The second viewpoint on the origin of the infrared problem is the following:
when constructing our quantum theory, we have mis-identified the asymptotic
states. In most textbook constructions of the $S$-matrix, one finds, somewhere
along the way, a statement on the need to assume that interactions can be
adiabatically switched off at large times. In theories with massless particles
and long-distance interactions, this is simply unrealistic: after a hard
interaction, electrons will continue to emit and absorb photons all the way
into the asymptotic regime. Choosing as asymptotic states the eigenstates of
the free hamiltonian (in QED, Fock states with a fixed number of electrons and
photons) is inadequate, as such states are not a good approximation of the
actual physical asymptotic states. Indeed, splitting the hamiltonian into
‘free’ and ‘interaction’ terms, as usually done, is inadequate, since
interactions persist at late and early times, and such asymptotic interactions
should be moved to the part of the hamiltonian that one will attempt to
diagonalise exactly. With a better choice of asymptotic states, one may hope
not only to improve the algorithm to compute physical observables, but also to
rescue the $S$-matrix program, by defining scattering amplitudes that are
automatically well-defined, even in the presence of massless particles. This
idea was successfully pursued in QED, starting with preliminary studies in
Refs. [24, 25, 26, 27, 28, 29], and culminating with the seminal paper by
Kulish and Faddeev [30], where a separable, Lorentz- and gauge-invariant
Hilbert space of ‘coherent states’ is defined, and the finiteness of the QED
$S$-matrix in the coherent state basis is proved to all orders. The coherent
state approach will be introduced here in Section 1.3.

Even in the relatively simple case of the abelian theory, understanding our IR
mistake has been quite fruitful: the role of long-distance dynamics has been
exposed, a sharper definition of a physical observable has emerged, and a more
accurate characterisation of asymptotic states has rescued the notion of
scattering amplitudes in the presence of massless particles. The
generalisation of these ideas to the much more intricate case of non-abelian
theories will occupy most of our review. It is to be expected that this
generalisation will be far from trivial, given what we know about the long-
distance behaviour of unbroken non-abelian gauge theories: perturbative
asymptotic states have very little to do with their non-perturbative, confined
counterparts, and we can expect and hope that perturbation theory will carry
at least some partial information about this breakdown.

Studies of the infrared problem in non-abelian theories started in the mid-
seventies [31, 32, 33]. It soon became evident that the simple cancellation
mechanism for soft singularities uncovered by Bloch and Nordsieck in QED,
which involves only a sum over degenerate final states, does not work for non-
abelian theories [34, 35, 36, 37, 38, 39], and a full cancellation can happen
only considering degenerate initial states as well222The result was later
extended to general non-abelian theories in [40], and was recently revisited
in modern language in [41]. Early papers discussing infrared cancellations in
the non-abelian theory for special cross sections include [42, 43, 44, 45].. A
more general problem arises because collinear divergences, which are regulated
by fermion masses in QED, are intrinsic and unavoidable for non-abelian
theories, even when matter fields are massive: the three-gluon vertex involves
three strictly massless particles and inevitably leads to collinear problems.
In the presence of collinear divergences, in particular those associated with
radiation from initial-state coloured particles, it is clear from the outset
that a simple pattern of cancellation à la Block-Nordsieck cannot work, since
hard collinear emissions from the initial state drastically alter the
kinematic configuration of the hard scattering, while virtual corrections do
not affect it: the cancellation of singularities must therefore be spoiled. At
the level of scattering amplitudes, this means that the coherent state program
for non-abelian theories is bound to be much more intricate than was the case
for QED. Important results were however obtained through the eighties,
shedding light on many aspects of the fixed-order and all-order structure of
infrared effects, in particular in QCD [46, 47, 48, 49, 50, 51, 52, 53, 54,
55, 56, 57, 58]; these analyses culminated in a formal proof of the finiteness
of the non-abelian $S$-matrix in the coeherent-state basis, including the case
of collinear divergences, in Ref. [59]. At the level of cross-sections, the
problem of initial-state collinear divergences was of course the starting
point for the QCD factorisation program, reviewed for example in [60, 61].

More recent studies have focused on two (overlapping) directions. On the one
hand, the demands of precision phenomenology have required developing
efficient tools for high-order calculations of observable cross sections. In
this regard, the coherent state approach, notwithstanding its physical appeal,
has not been the main way forward: rather, a ‘KLN’ approach has prevailed, in
which virtual corrections and phase-space integrals of unresolved real
radiation are both computed with an infrared regulator, and then combined to
construct finite distributions. Not surprisingly, dimensional continuation,
setting $d=4-2\epsilon$ with $\epsilon<0$, has been the regulator of choice,
since it preserves gauge and Lorentz invariance without adding to the
complexity of the calculation by introducing non-trivial dependence on
unphysical mass scales. Finite-order calculation will be discussed in Section
2, and detailed algorithms for the cancellation of IR poles will be introduced
in Section 6.

On the other hand, perhaps more interestingly from a theoretical viewpoint, a
great deal of activity has been devoted to elucidate the all-order structure
of infrared singularities, on the basis of the ideas of factorisation and
universality. In principle, these ideas are natural, and they apply equally
well to amplitudes and to cross sections: as we argued, IR singularities are
associated with phenomena that take place at large times and distances from
the hard scattering center; this suggests that the singularity structure
should be largely independent on the details of the short-distance process
being considered; one may then hope to identify universal factors responsible
for the singular behaviour of the amplitude (or cross section). In practice,
proving these properties in a non-abelian theory by standard perturbative
methods is very challenging: collinear configurations carry spin correlations
between different particles taking part in the scattering, while soft
configurations are responsible for very interesting but very intricate colour
correlations at large distances. Factorisation emerges only upon summing over
Feynman diagrams, and only after the constraints of gauge invariance have been
properly taken into account by means of Ward identities. At the level of
scattering amplitudes (which will be the focus of most of our discussion) the
factorisation program was started with pioneering work on form factors, first
in QED [62, 63] and subsequently in QCD [64, 65], with the first extensions to
fixed-angle scattering amplitudes coming shortly thereafter [66]. The
generalisation of these results to multi-particle scattering amplitudes in the
modern language of dimensional regularisation will be the central focus of our
review.

In the remainder of this introductory Section, we will present short accounts
of the Bloch-Nordsieck cancellation mechanism in QED, of the KLN theorem, and
of the coherent state method, with an eye to the long history of the subject,
but also to display in some more detail the different underlying viewpoints,
that we have just briefly introduced. In Section 2, we will present some well-
known one-loop results in QCD, in the modern language of dimensional
regularisation: this is textbook material, but it will give us the opportunity
to introduce some tools that will be useful in what follows, including in
particular the $d$-dimensional running coupling and the calculation of eikonal
integrals in dimensional regularisation. In Section 3, we will introduce the
methods required to perform an all-order diagrammatic analysis of the IR
problem, which are a pre-requisite for the proof of any factorisation theorem.
These methods were reviewed in [61], and are presented in much greater detail
in [2, 67]; we decided however to include a pedagogical introduction and work
out some key examples, since this toolbox lies at the foundation of all
subsequent developments. Sections 4 and 5 form the core of our review. First,
in Section 4, we consider the case of form factors, and we show how the tools
developed in Section 3 lead to the formulation of an all-order factorisation
theorem, isolating soft and collinear divergences of form factors in universal
functions, which can be defined in terms of gauge-invariant matrix elements of
fields and Wilson lines. Once factorisation has been achieved, evolution
equations are bound to follow, and solving them leads to an all-order
summation of perturbative contributions, in this case of IR poles [68, 69,
70]. Form factors have the advantage of having a trivial colour structure:
extending the analysis to general non-abelian scattering amplitudes is
therefore less than trivial. This is pursued in Section 5, where the central
concept of IR anomalous dimension matrix is introduced, and the most general
form of the exponentiation of IR singularities is derived; known results up to
three loops for the anomalous dimension matrix are reviewed [71, 72, 73, 74,
75, 76, 77], and the most recent diagrammatic techniques for its calculation
are introduced. It is perhaps worthwhile to emphasise that all the results of
Sections 4 and 5 are valid not only for QCD, but in fact for any massless
gauge theory (with computable corrections in case the gauge fields are coupled
to massive matter fields): for example, they have found important application
in the case of conformal gauge theories, in particular ${\cal N}=4$ Super-
Yang-Mills theory [78, 79]. On the other hand, Sections 4 and 5 focus on
virtual corrections to fixed-angle scattering amplitudes, while the
construction of measurable cross-sections must also include unresolved soft
and collinear real radiation. Factorisation for real radiation is by itself a
vast subject, and we don’t do it justice by providing just a brief summary in
Section 6; there, we also introduce modern subtraction algorithms [80, 81],
which are currently being developed for high-order perturbative calculations
[82], in order to perform the cancellation of IR singularities in a general
and efficient manner, even for the intricate observables currently being
measured at high-energy colliders.

As must be the case when reviewing a broad subject with a long history, many
important developments and lines of research that are closely related to our
topic were left out: we provide here a partial list, with some of the
references that we believe can be useful for further exploration by the
reader. First of all, we say very little in this review about the subject of
resummation of large logarithms that arise in observable QCD cross sections
and that are closely related to infrared singularities (for example threshold
logarithms and transverse momentum logarithms): we only note in passing that
the techniques used for resummations are tightly connected to the ones
reviewed here, which indeed in some cases were developed directly for cross
sections before being applied to amplitudes. QCD tools for resummation are
reviewed, for example, in Refs. [61, 83, 84], while automated methods are
discussed in [85, 86]. Next, we recall that techniques to construct
factorisation theorems in QCD, under certain general assumptions, and to
derive the corresponding resummations, were made systematic in Refs. [87, 88,
89, 90, 91, 92, 93], by the construction of Soft-Collinear Effective Theory
(SCET), a non-local effective field theory333The idea that infrared
divergences could be organised at Lagrangian level in terms of a non-local
effective theory was floated for the first time (to the best of our knowledge)
in Ref. [94]. for soft and collinear modes of quark and gluon fields. While
the bulk of SCET applications concerns cross sections of phenomenological
interest, SCET methods are of course fully applicable to the study of
scattering amplitudes, and indeed some of the general results discussed here
in Section 5 were independently derived within the SCET framework [75, 95].
SCET has generated countless important phenomenological applications, and is
introduced and reviewed in Ref. [96]; comparisons between the SCET approach
and QCD factorisation were carried out, for example, in Refs. [97, 98, 99,
100, 101, 102, 103, 104]. A third direction that we do not explore is the
extension of factorisation theorems, and eventually resummation techniques,
beyond leading power in the singular variables (for example the soft gluon
energy). In QED, a classic result in this regard is the Low-Burnett-Kroll
theorem [105, 106], showing that next-to-leading power (NLP) contributions of
soft photons to QED cross section are still universal, as an effect of gauge
invariance; the theorem was later extended to collinear-enhanced
configurations in Ref. [107]. NLP contributions to amplitudes and cross
sections do not induce divergences in renormalisable theories, but they are
nevertheless very interesting for both theoretical and phenomenological
reasons. In recent years, their factorisation and resummation properties have
been studied intensively, both with direct QCD methods (see, for example,
Refs. [108, 109, 110, 111, 112, 113, 114, 115, 116]) and in the context of
SCET (see, for example, Refs. [91, 117, 118, 119, 120, 121, 122, 123, 124,
125, 126])444For different approaches to the resummation of next-to-leading-
power threshold logarithms, see, for example, [127, 128, 129, 130, 131, 132,
133, 134, 135].. A thorough review of these developments together with a
survey of recent literature can be found in Ref. [136]. A final important
connection that we do not develop here is that between gauge theories and
gravity, first discussed in the pioneering work of Weinberg [137].
Similarities and differences between massless gauge theories and gravity
theories are intriguing, and have been addressed in the context of
factorisation in a number of recent papers [138, 139, 140, 141, 142, 143, 144,
145, 146]. The connection between gauge theories and gravity theories is also
at the root of radical re-interpretation of infrared phenomena, which
originated in the work of Strominger [147, 148]. Within this framework,
infrared divergences are related to infinite-dimensional asymptotic symmetries
of the (gauge or gravity) theory under consideration, acting on the celestial
sphere intersecting the future (or past) light cone at asymptotic distances.
For gravity, these symmetries were uncovered in [149, 150], while for gauge
theories they take the form of ‘large’ gauge transformations with non-trivial
action at future (or past) null infinity [151, 152, 153, 154, 155, 156, 157,
158, 159, 160]. The universal form of soft and next-to-soft factors for tree-
level radiative amplitudes emerges from the Ward identities of these
symmetries. Early developments in this fast-developing field are reviewed in
[161]. While most of the work done so far within this framework concerns tree-
level gravity or gauge-theory amplitudes (see, however, Refs. [162, 163]), the
connection between infrared properties of $d=4$ gauge theories and $d=2$
conformal invariance on the celestial sphere is very intriguing, and
represents a remarkable new point of view on an old problem. In Section 5, we
will see that scale invariance plays an important role for infrared
factorisation at any perturbative order: conformal invariance, however, is
broken by quantum effects. Furthermore, we will see that the celestial
framework allows for a remarkable rephrasing of the all-order expression for
infrared-divergent colour-dipole correlations, and indeed an interpretation
emerges in terms of a specific two-dimensional conformal theory on the
celestial sphere [164]. Further exploration of these ideas in the context of
all-order perturbative calculations is undoubtedly of great interest.

As is perhaps evident from this long introduction, the present review has a
strong pedagogical emphasis, and we hope that it may help students and junior
researchers to gain an orientation in a field with a long history, which
however is still quickly progressing in both theoretical and phenomenological
directions. At the same time, we have included in Section 5 and in Section 6
some very recent developments, which may be interesting to QCD practitioners,
and give a flavour of current research. Possible future directions of research
are briefly discussed in Section 7.

###  1.1 Catastrophe and recovery in QED

In this Section, and in the following Sections 1.2 and 1.3, we take a largely
historical perspective, and we present classic results on the cancellation of
infrared singularities, which also serve to introduce some of the key concepts
to be developed later in the context of non-abelian theories. We begin by
looking at the relatively simple case of QED with massive fermions, and we
derive the cancellation of soft divergences between virtual correction and
final-state real radiation, first at the lowest non-trivial order, and then to
all-orders, where one can rather easily demonstrate the exponentiation of the
lowest-order result. Our emphasis is on the concepts that will later be
developed in greater detail, so the proof that we discuss is heuristic: for a
detailed analysis one should refer, for example, to [21]. Rather than using
directly the arguments of Ref. [18], we will take advantage of the more modern
language of Ref. [137].

Choosing a simple example, which however displays all the general features of
the proof, we consider the process where an electron scatters from another
heavy particle by means of $t$-channel photon exchange. The heavy particle
simply plays the role of a source for an external photon field. We will first
show the cancellation at one loop, and then discuss how the mechanism
generalises to all orders.

At order $\alpha$, the diagrams contributing to virtual corrections are shown
in Fig. 3, while those contributing to real photon radiation (bremsstralung)
are shown in Figure 4.

(a)

(b)

(c)

(d)

Figure 3: Virtual corrections to electron scattering from a heavy particle, at
order $\alpha$.

(a)

(b)

Figure 4: Bremsstrahlung diagrams for electron scattering from a heavy
particle, contributing to the cross section at order $\alpha$.

We begin by considering the differential cross-section for radiating a photon
with momentum $k$, when the electron scatters from a state of momentum $p$ to
a state of momentum $p^{\prime}$, assuming $k^{\mu}\ll
q^{\mu}\equiv(p^{\prime}-p)^{\mu}$. The matrix element for this process is the
sum of two terms of the form depicted in Fig. 2, and greatly simplifies when
one expands in powers of the soft momentum $k$ and considers only the leading
power (this soft approximation will be considered in greater detail in Section
2.6 and in Section 3.3). At this order, referring to Eq. (1.3) but
appropriately introducing the electron mass, one can perform the following
manipulations:

  * •

neglect $k$ in the numerator;

  * •

neglect $k^{2}$ in the denominator;

  * •

commute the hard momentum factor $p\hskip-4.49997pt/$ (or
$p\hskip-4.49997pt/^{\prime}$) in the numerator of the electron propagator
across the emission vertex, and use the Dirac equation to simplify the
expression.

Contracting the result with the photon polarisation vector $\varepsilon(k)$,
and denoting the lowest-order matrix element by ${\cal M}_{0}$, the leading
power result for the radiative matrix element ${\cal M}_{\rm rad}$ takes the
form

$\displaystyle{\cal M}_{\rm
rad}\,=\,e\,\bigg{(}\frac{p^{\prime}\cdot\varepsilon(k)}{p^{\prime}\cdot
k}-\frac{p\cdot\varepsilon(k)}{p\cdot k}\bigg{)}\,{\cal M}_{0}\,,$ (1.4)

where $e$ is the electron charge. The soft approximation in Eq. (1.4) has
several remarkable properties that we will explore later. Here we simply note
that it is gauge invariant (it vanishes for longitudinally polarised photons),
independent of the spin of the emitter555It is instructive to derive the same
result for scalar electrodynamics, and for gluon scattering in the non-abelian
theory. and singular – as expected – for small $k$. It is formally easy to
construct a soft approximation for the total radiative cross section: at
leading power in $k$ the phase space factorises, and the flux factor reduces
to the flux factor for the non-radiative process. Summing over polarisations,
we find that the cross section for the radiation of a single real soft photon
is given by

$\displaystyle\sigma_{r}\big{(}e(p)\rightarrow
e(p^{\prime})+\gamma\big{)}\,=\,\sigma_{0}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,\,{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{E^{2}},\epsilon\bigg{)}\,,$
(1.5)

where we defined the soft factor for real radiation, ${\cal I}_{r}$, by

$\displaystyle{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{E^{2}},\epsilon\bigg{)}\,=\,-\frac{e^{2}(2\pi\mu)^{2\epsilon}}{16\pi^{3}}\\!\int_{0}^{E}d|{\bf
k}||{\bf k}|^{1-2\epsilon}\\!\int
d\Omega_{2-2\epsilon}\bigg{(}\frac{p^{\prime\mu}}{p^{\prime}\cdot
k}-\frac{p^{\mu}}{p\cdot
k}\bigg{)}\bigg{(}\frac{p^{\prime}_{\mu}}{p^{\prime}\cdot
k}-\frac{p_{\mu}}{p\cdot k}\bigg{)}\,.$ (1.6)

In Eq. (1.6) we have introduced dimensional regularisation, setting
$d=4-2\epsilon$, $\epsilon<0$, since the integral is ill-defined in $d=4$,
diverging logarithmically as $|{\bf k}|\rightarrow 0$: this is the original
infrared catastrophe of Ref. [18]. The fact that the divergence is logarithmic
justifies a posteriori our choice to retain only the leading power in the soft
expansion: sub-leading terms will contribute finite corrections. Furthermore,
we integrate over photon energies up to an arbitrary cutoff
$E\ll\sqrt{-q^{2}}$, which represents the minimum energy resolution of our
detector: photons with $|{\bf k}|<E$ are unresolved, and their contribution
must be included. The resulting integral is well known and not difficult to
compute. One finds

$\displaystyle{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{E^{2}},\epsilon\bigg{)}\,=\,-\,\frac{\alpha}{\pi}\,\frac{1}{\epsilon}\bigg{(}\frac{4\pi{\mu}^{2}}{E^{2}}\bigg{)}^{\epsilon}\left[\left(\frac{1-2m^{2}/q^{2}}{\beta}\right)\log\frac{\beta+1}{\beta-1}-1\right]+{\cal
O}\big{(}\epsilon^{0}\big{)}\,,$ (1.7)

where

$\displaystyle\beta\,=\,\sqrt{1-4m^{2}/q^{2}}\,>\,1\,.$ (1.8)

It is interesting to consider explicitly the limit of large momentum transfer,
$-q^{2}\rightarrow\infty$, focusing on terms enhanced by logarithms of
$q^{2}$. We obtain

$\displaystyle{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{E^{2}},\epsilon\bigg{)}\,\simeq\,\frac{\alpha}{\pi}\left[-\frac{1}{\epsilon}\log\left(\frac{-q^{2}}{m^{2}}\right)+\log\left(\frac{E^{2}}{4\pi\mu^{2}}\right)\log\left(\frac{-q^{2}}{m^{2}}\right)\right]\,.$
(1.9)

As we will see, the soft pole in $\epsilon$ will cancel against the virtual
correction; the second term displays the characteristic ‘Sudakov’ double-
logarithmic enhancement, with one logarithm of soft origin (here parametrised
by the resolution scale $E$) and one logarithm of collinear origin (which
becomes a divergence in the massless limit, $m\to 0$).

Let us now turn to the evaluation of virtual corrections. The problem is
significantly more intricate because of the concurrent presence of ultraviolet
divergences, which must be dealt with by means of renormalisation: in
particular, one must make sure that no infrared divergences survive in the UV
counterterms appropriate to the chosen renormalisation scheme. This can be
achieved for example by using a minimal scheme. On the other hand, in the on-
shell scheme typically used in QED, both the electron field renormalisation
constant $Z_{\psi}$ and the vertex renormalisation constant $Z_{1}$ contain IR
divergences, which however cancel in the scattering amplitude thanks to the
QED Ward identity [2]. Armed with this preliminary result, we can concentrate
on the vertex correction diagram of Fig. 3(a), which, as we will see, gives
the only surviving infrared-singular contribution. A straightforward power-
counting argument shows that only terms with no powers of the photon momentum
in the numerator can give infrared singularities, and those will be
logarithmic. In this approximation, diagram (a) in Fig. 3 gives

$\displaystyle(a)_{\rm
soft}\,=\,-\,e^{3}\mu^{3\epsilon}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{\overline{u}(p^{\prime})\gamma^{\alpha}(\not{p}^{\prime}+m)\gamma^{\mu}(\not{p}+m)\gamma_{\alpha}u(p)}{(k^{2}+{\rm
i}\eta)(k^{2}-2p^{\prime}\cdot k+{\rm i}\eta)(k^{2}-2p\cdot k+{\rm
i}\eta)}\,.$ (1.10)

The integral in Eq. (1.10) can be performed with standard methods, yielding a
single infrared pole in $\epsilon$: potentially, a second infrared
catastrophe. Instead of directly computing the integral, in view of the
generalisation to higher orders, it is more instructive to take the soft
approximation further, and perform on the integrand the same steps that we
applied to the real emission diagrams, i.e. neglecting $k^{2}$ in the
denominator, and using the Dirac equation to simplify the numerator.
Recognising that the lowest-order matrix element factorises, we write then

$\displaystyle(a)_{\rm soft}\,=\,\big{(}{\rm
i}e^{2}\mu^{2\epsilon}\big{)}\,{\cal
M}_{0}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{p\cdot p^{\prime}}{(k^{2}+{\rm
i}\eta)(-p^{\prime}\cdot k+{\rm i}\eta)(-p\cdot k+{\rm i}\eta)}\,.$ (1.11)

We note that the step between Eq. (1.10) and Eq. (1.11) has apparently
generated a new problem: the integral in Eq. (1.11) is now also divergent in
the UV, a region where our approximation breaks down. This UV singularity is
not present in the original QED calculation: it is a singularity of a low-
energy effective theory for the infrared sector of QED, and it will be of
great interest for the developments discussed in Section 4. For the moment, we
will simply introduce a UV regulator and focus on the infrared pole.

Examining Eq. (1.11), one begins to see some similarities with the integral in
Eq. (1.6): these can be sharpened by taking two further steps. First of all,
we include the UV counterterms appropriate to the original QED calculation,
picking the on-shell renormalisation scheme, which makes the calculation
particularly transparent. In the on-shell scheme, the self-energy counterterm
for the graphs in Fig. 3(b) and 3(c) is such that the sum of the graph plus
the counterterm vanishes on shell: these graphs therefore do not contribute to
our calculation. The vertex counterterm, on the other hand, is fixed by the
requirement that the renormalised vertex correction vanish for $q^{2}=0$, i.e.
for $p^{\prime}=p$. This can be enforced directly in our soft approximation by
writing [2]

$\displaystyle(a+{\rm counterterm})_{\rm soft}$ $\displaystyle=$
$\displaystyle{\cal M}_{0}\,\,\bigg{(}\\!\\!-\frac{{\rm
i}e^{2}\mu^{2\epsilon}}{2}\bigg{)}\int\frac{d^{d}k}{(2\pi)^{d}}\frac{1}{(k^{2}+{\rm
i}\eta)}\left(\frac{p^{\prime\mu}}{-p^{\prime}\cdot k+{\rm
i}\eta}-\frac{p^{\mu}}{-p\cdot k+{\rm i}\eta}\right)^{2}$ (1.12)
$\displaystyle\equiv$ $\displaystyle{\cal M}_{0}\,\,{\cal
I}_{v}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{q^{2}},\epsilon\bigg{)}\,,$

which manifestly vanishes for $p^{\prime}=p$, and where the cross term in the
square of the parenthesis gives Eq. (1.11), while the squares of the two terms
(which depend only on $m^{2}$) give the counterterm. The final step to bring
Eq. (1.12) to a form directly comparable to Eq. (1.6) is to note (following
Ref. [137]) that the real part of the integral ${\cal I}_{v}$ has support only
on the photon mass shell666In fact, in space-like kinematics $(q^{2}<0)$ the
integral ${\cal I}_{v}$ is real, whereas in time-like kinematics $(q^{2}>0)$
it has an imaginary part, responsible for a divergent phase in the scattering
amplitude [137], and arising from configurations with an off-shell photon, but
on-shell fermion lines., $k^{2}=0$. This can be seen by focusing on the
$k_{0}$ integration, and observing the locations of the poles in the $k_{0}$
complex plane in Eq. (1.11): the poles associated with the fermion lines are
in the upper-half plane, as is one of the two poles associated with the photon
propagator. One can therefore compute the integral by closing the contour in
the lower-half plane, and picking the residue of the photon pole, which
effectively places the photon on the mass shell. The result is now precisely
of the form of Eq. (1.6), except for an overall factor of $-1/2$, and the fact
that the resolution parameter $E$ must be replaced with an ultraviolet cutoff,
which can naturally be chosen as $\sqrt{-q^{2}}$. We have then

$\displaystyle{\cal
I}_{v}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{q^{2}},\epsilon\bigg{)}\,=\,-\frac{1}{2}\,{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{-q^{2}},\epsilon\bigg{)}\,.$
(1.13)

The observed physical cross section is obtained by summing the real-emission
cross section in Eq. (1.5), which gives the leading-order probability for the
emission of an undetected soft photon, with the virtual correction to the
elastic scattering process, which is proportional to the tree-level cross
section, with a factor given by twice the real part of ${\cal I}_{v}$. One
finds

$\displaystyle\sigma_{\rm obs}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,=\,\sigma_{0}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\bigg{[}1+{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{E^{2}},\epsilon\bigg{)}+2\,{\cal
I}_{v}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{q^{2}},\epsilon\bigg{)}\bigg{]}\,.$
(1.14)

Using Eq. (1.13), we see that all singular terms as $\epsilon\to 0$ cancel, as
announced, leaving behind finite logarithms. In the limit of large negative
$q^{2}$, for example, the result is

$\displaystyle\sigma_{\rm obs}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,\simeq\,\sigma_{0}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\left[1-\frac{\alpha}{\pi}\log\left(\frac{-q^{2}}{E^{2}}\right)\log\left(\frac{-q^{2}}{m^{2}}\right)+\mathcal{O}({\alpha}^{2})\right]\,,$
(1.15)

displaying the product of a soft logarithm and a collinear logarithm. It is
appealing that the cancellation we have observed is made possible by the fact
that the divergent part of the loop integral originates from configurations in
which the virtual photon is on-shell: this resonates with the qualitative
arguments given in the Introduction, identifying infrared divergences as long-
distance effects. We note however that the correction we find in Eq. (1.15) is
finite but negative, and it can lead to a negative cross section at very large
momentum transfer. It is clear that the recovery from the IR catastrophe at
NLO is only a partial solution, and we need to explore higher-order
corrections.

A general and detailed proof of the cancellation of infrared divergences in
QED would require developing power-counting tools (discussed here in Section
3), and analysing the impact of renormalisation to all orders [20, 21].
Fortunately, it is possible to understand the general pattern of the
cancellation with simple diagrammatic and combinatorial arguments, discussed
for example in [137], which are sufficient to deal with terms with the largest
logarithmic enhancements at each order of perturbation theory. These tools
will provide us with a first glimpse of two essential features of infrared
enhancements, factorisation and exponentiation, which will be discussed at
length in later Sections.

Figure 5: Emission of $n$ photons attached to outgoing electron line.

Consider first the emission of $n$ photons with momenta $k_{1},\ldots,k_{n}$
and polarisation vectors $\varepsilon_{1},\ldots,\varepsilon_{n}$, attached to
the outgoing electron line carrying momentum $p^{\prime}$, as depicted in Fig.
5. The contribution to the transition amplitude from this graph is given by

$\displaystyle
G\big{(}1,\ldots,n;p^{\prime}\big{)}\,=\,e^{n}\,\overline{u}(p^{\prime})\,\varepsilon\hskip-4.49997pt/_{1}\,\frac{p\hskip-4.49997pt/^{\prime}+k\hskip-4.49997pt/_{1}+m}{2p^{\prime}\cdot
k_{1}}\,\ldots\,\varepsilon\hskip-4.49997pt/_{n}\,\frac{p\hskip-4.49997pt/^{\prime}+k\hskip-4.49997pt/_{1}\ldots
k\hskip-4.49997pt/_{n}+m}{2p^{\prime}\cdot\big{(}k_{1}+\ldots+k_{n}\big{)}}\,{\cal
H}\,,$ (1.16)

where ${\cal H}$ is the hard scattering subgraph represented by the circle in
Fig. 5. At leading power in each gluon momentum, we can iterate the procedure
applied to the single-radiative amplitude, dropping all dependence on $k_{i}$
in the numerator, and repeatedly using the Dirac equation after commuting the
electron momentum $p^{\prime}$ to the left. This leads to

$\displaystyle
G\big{(}1,\ldots,n;p^{\prime}\big{)}\,\simeq\,e^{n}\,\,\frac{p^{\prime}\cdot\varepsilon_{1}}{p^{\prime}\cdot
k_{1}}\,\frac{p^{\prime}\cdot\varepsilon_{2}}{p^{\prime}\cdot\big{(}k_{1}+k_{2}\big{)}}\,\ldots\,\frac{p^{\prime}\cdot\varepsilon_{n}}{p^{\prime}\cdot\big{(}k_{1}+k_{2}+\ldots+k_{n}\big{)}}\,{\cal
M}\,,$ (1.17)

where the outgoing electron spinor has been reabsorbed in ${\cal M}$, the
fixed-order matrix element without photon radiation. Crucially, the
contribution to the physical amplitude where $n$ photons are radiated from the
outgoing electron line is obtained by summing over $n!$ permutations of the
photon lines. Bose symmetry then requires that the result be symmetric under
permutations of the photon labels: this happens as a consequence of the
eikonal identity

$\displaystyle\sum_{\sigma}\Bigg{[}\prod_{i=1}^{n}\frac{1}{p\cdot\sum_{j=1}^{i}k_{\sigma(j)}}\Bigg{]}\,=\,\prod_{i=1}^{n}\,\frac{1}{p\cdot
k_{i}}\,\,,$ (1.18)

where $\sigma$ enumerates photon permutations. The identity is easily verified
for $n=2$, and can subsequently be proved by induction. It leads to

$\displaystyle\sum_{\sigma}G\big{(}\sigma(1),\ldots,\sigma(n);p^{\prime}\big{)}\,\simeq\,e^{n}\,\,\prod_{i=1}^{n}\,\frac{p^{\prime}\cdot\varepsilon_{i}}{p^{\prime}\cdot
k_{i}}\,{\cal M}\,,$ (1.19)

representing the uncorrelated emissions of $n$ soft photons without recoil.
Notice that soft divergences will arise from the phase-space integration of
the emitted photon momenta in Eq. (1.16) only if no hard photons are emitted
further out along the electron line, closer to the outgoing electron spinor:
any such emission would move all propagators closer to the hard scattering off
their mass shell, so that the approximation leading to Eq. (1.16) would not be
applicable777For example, for $n=2$, the last term in
$(p^{\prime}+k_{1}+k_{2})^{2}-m^{2}=2p^{\prime}\cdot(k_{1}+k_{2})+2k_{1}\cdot
k_{2}$ would not be negligible.. We conclude that soft divergences arise from
real photon emission only when photons attach to external lines, defined as
lines connected to the external states from which every emission is soft: that
is why we have factorised the non-radiative matrix element ${\cal M}$ in Eq.
(1.19), rather than its tree-level approximation ${\cal M}_{0}$. In a space-
time picture, this factorisation of ‘external line’ contributions nicely
matches our understanding of soft divergences as long-distance phenomena, and
it provides a first example of more intricate factorisation theorems to be
discussed in later Sections.

Figure 6: Emission of $n$ photons connected in any order to the initial and
final electron lines.

Naturally, an identical argument applies to the incoming electron line, where
however each photon contributes with a relative minus sign, since
$(p-\Sigma_{i}{k_{i}})^{2}-m^{2}=-2p\cdot\Sigma_{i}{k_{i}}$. Considering
together all the diagrams containing $n$ soft photons, connected in any
possible order to the incoming and outgoing electron lines, as shown in Fig.
6, we find that each photon contributes precisely an eikonal factor, as was
the case in Eq. (1.4). The result is then

$\displaystyle{\cal M}_{\rm
rad}^{(n)}\,=\,e^{n}\,\prod_{i=1}^{n}\bigg{(}\frac{p^{\prime}\cdot\varepsilon_{i}}{p^{\prime}\cdot
k_{i}}-\frac{p\cdot\varepsilon_{i}}{p\cdot k_{i}}\bigg{)}\,{\cal M}\,.$ (1.20)

At leading power in each photon momentum $k_{i}$, the phase space for real
radiation factorises into a product of phase spaces for individual
photons888Notice that some care is needed when treating the resolution scale
$E$: here we are allowing each photon to have an energy up to $E$, whereas the
limit should apply to the sum of the energies of individual photons; a more
refined treatment can be found, for example, in Ref. [21].; furthermore, the
phase-space integral must include a factor of $1/n!$ for $n$ identical bosons.
Summing over polarisations one finds

$\displaystyle\sigma_{r}^{(n)}\big{(}e(p)\rightarrow
e(p^{\prime})+n\gamma\big{)}\,=\,\sigma_{0}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,\,\frac{1}{n!}\left[{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{E^{2}},\epsilon\bigg{)}\right]^{n}\,,$
(1.21)

It is now straightforward to sum the series in Eq. (1.21) over the number of
soft photons $n$, which results in exponentiation of the single-photon result,

$\displaystyle\sigma_{r}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,=\,\sigma_{0}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,\exp\Bigg{[}\,{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{E^{2}},\epsilon\bigg{)}\Bigg{]}\,.$
(1.22)

We can now observe that most of the steps leading to Eq. (1.22) also apply to
virtual corrections: only combinatorial factors must be carefully considered.
First of all, only virtual photons attaching to external lines will generate
soft divergences. Next, focusing on vertex corrections, i.e. on photons
connecting the two electron lines, one may apply the eikonal identity in Eq.
(1.18), by simply considering $n!$ copies of all diagrams contributing to the
$n$-photon vertex correction, and relabelling the photon momenta in each copy:
one must then divide the result by $n!$. Notice that one needs to perform this
sum and normalisation on only one of the two electron lines, since repeating
the operation on the second electron line would reproduce the same
diagrams999In case more than two charged lines are present, the combinatorial
factors become somewhat more intricate, but the final exponentiation of the
one-loop result remains true.. This step leads to a product of factors of the
form of Eq. (1.11). Self-energy corrections on each electron line can then be
renormalised to vanish on-shell, and the condition that the vertex correction
should vanish at $q^{2}=0$ turns each eikonal factor in the vertex into the
form of Eq. (1.12), introducing the appropriate factor of $1/2$ for each
photon. We conclude that virtual corrections also exponentiate, and

$\displaystyle\sigma_{v}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,=\,\sigma_{0}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,\exp\Bigg{[}\,2\,{\cal
I}_{v}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{q^{2}},\epsilon\bigg{)}\Bigg{]}\,.$
(1.23)

Combining real and virtual corrections, we see that, thanks to Eq. (1.13), the
cancellation of soft singularities is replicated to all orders in perturbation
theory, and one finds

$\displaystyle\sigma_{\rm obs}\big{(}e(p)\rightarrow e(p^{\prime})\big{)}$
$\displaystyle=$ $\displaystyle\sigma_{0}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,\exp\bigg{[}{\cal
I}_{r}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{E^{2}},\epsilon\bigg{)}+2\,{\cal
I}_{v}\bigg{(}\frac{m^{2}}{q^{2}},\frac{\mu^{2}}{q^{2}},\epsilon\bigg{)}\bigg{]}$
(1.24) $\displaystyle\simeq$ $\displaystyle\sigma_{0}\big{(}e(p)\rightarrow
e(p^{\prime})\big{)}\,\exp\bigg{[}\\!-\frac{\alpha}{\pi}\log\left(\frac{-q^{2}}{E^{2}}\right)\log\left(\frac{-q^{2}}{m^{2}}\right)\bigg{]}\,,$

where in the second line we have reported the leading double-logarithmic
behaviour in the limit of large momentum transfer. Upon combining real and
virtual corrrections, and resumming the perturbative expansion, we achieved a
finite and well-behaved result: the cross section is positive definite, and it
exhibits the classic ‘Sudakov’ behaviour [62], vanishing exponentially at
large momentum transfer, or equivalently for small values of the resolution
parameter $E$ and of the fermion mass $m$.

###  1.2 Observable cross sections are finite: the KLN theorem

We have seen that in quantum electrodynamics soft divergences cancel out order
by order in perturbation theory, when the transition rates are summed over
final states that are physically indistinguishable. From the derivation in
Section 1.1, it is not however clear how general the mechanism is, and indeed
the cancellation may appear fortuitous. As a matter of fact, as briefly
discussed in the Introduction, the Bloch-Nordsieck theorem is specific to QED,
and breaks down for non-abelian gauge theories [34, 35, 36, 37, 38, 40, 41].
It is important, therefore, to develop a more general understanding of the
cancellation, as well as a more intuitive picture of the underlying physical
mechanism. The framework is provided by two general results, known
respectively as the Kinoshita-Poggio-Quinn (KPQ) theorem [31, 32, 33] and the
Kinoshita-Lee-Nauenberg (KLN) theorem [22, 23, 165, 166], which apply directly
to any quantum theory with massless particles. The KPQ theorem establishes
that momentum-space Green functions with external momenta which are off the
mass shell are infrared safe. This is a natural consequence of the physical
picture that we have been developing: the fluctuations of off-shell fields are
confined to limited volumes of space-time, and cannot be affected by long-
distance singularities. The KLN theorem, on the other hand, establishes the
general framework for the cancellation of singularities when on-shell
quantities are evaluated: one finds that, in any theory involving massless
particles, infrared divergences can be traced to the presence of sets of
quantum states that are degenerate in energy, and the divergences cancel when
the transition rates are summed over the sets of degenerate initial and final
states. In what follows, we will sketch a proof of the KLN theorem, following
the line of argument presented in [23]. We find this approach particularly
suited to show the complete generality of the cancellation, which indeed
applies to any quantum system with sets of states which are degenerate in
energy, including non-relativistic theories and effective field theories.

Consider then a quantum-mechanical system, characterised by a Hilbert space of
states $\mathcal{H}$, and by a quantum hamiltonian $H$ which we split into a
solvable, quadratic part, and an interaction term, as

$\displaystyle H(t)\,=\,H_{0}+H_{I}(t)\,,$ (1.25)

Crucially, we work in the interaction picture, where operators evolve in time
by the action of the free hamiltonian $H_{0}$, while the time evolution of
quantum states is dictated by the interaction hamiltonian $H_{I}(t)$. The time
evolution operator is then given by

$\displaystyle U\big{(}t_{2},t_{1}\big{)}\,=\,{\rm T}\exp\left[-{\rm
i}\int_{t_{1}}^{t_{2}}dt\,H_{I}(t)\right]\,,$ (1.26)

where $H_{I}(t)$ is the interaction hamiltonian in the interaction picture,

$\displaystyle H_{I}(t)\,=\,{\rm e}^{{\rm i}H_{0}t}\,H_{I}(0)\,{\rm e}^{-{\rm
i}H_{0}t}\,.$ (1.27)

The $S$-matrix of the theory can then be formally defined by the limit

$\displaystyle
S\,\equiv\,\lim_{t,t^{\prime}\to\infty}U(t^{\prime},0)\,U(0,-t)\,\equiv\,\Omega_{-}^{\dagger}\,\Omega_{+}\,,$
(1.28)

where the $\Omega_{\pm}$ operators, sometimes called Möller operators, are in
turn defined by

$\displaystyle\Omega_{\pm}\,\equiv\,\lim_{t\to\infty}\,T\exp\left[-{\rm
i}\int_{\mp t}^{0}dt\,H_{I}(t)\right]\,.$ (1.29)

Armed with this definition, we can compute transition amplitudes between
incoming and outgoing states in terms of the Möller operators, as

$\displaystyle\big{\langle}b,{\rm out}\big{|}a,{\rm
in}\big{\rangle}\,\equiv\,\big{\langle}b,{\rm in}\big{|}\,S\,\big{|}a,{\rm
in}\big{\rangle}\,=\,\sum_{c}\big{\langle}b\big{|}\,\Omega_{-}^{\dagger}\,\big{|}c\big{\rangle}\big{\langle}c\big{|}\,\Omega_{+}\,\big{|}a\big{\rangle}\,,$
(1.30)

where we used the completeness of the incoming states, and, in the last
expression, we omitted for simplicity the initial-state label, as we will do
in the following. We are tacitly assuming that the asymptotic states are
eigenstates of the free hamiltonian $H_{0}$: if there are sets of energy-
degenerate states, this is in general not a good approximation (as we will see
in detail in the next section), since energy degeneracy signals long-range
interactions. In the present setting, the problem emerges in the form of
singularities in perturbation theory, which we will have to try to cancel. In
order to proceed, we define the transition probabilities (per unit volume and
per unit time)

$\displaystyle P\big{(}a\to
b\big{)}\,=\,\big{|}\left\langle\smash{b}{\vphantom{1}}\right|S\left|\smash{a}{\vphantom{1}}\right\rangle\big{|}^{2}$
$\displaystyle=$
$\displaystyle\sum\limits_{c,d}\left(\left\langle\smash{b}{\vphantom{1}}\right|\Omega_{-}^{\dagger}\left|\smash{c}{\vphantom{1}}\right\rangle\left\langle\smash{c}{\vphantom{1}}\right|\Omega_{+}\left|\smash{a}{\vphantom{1}}\right\rangle\right)\left(\left\langle\smash{b}{\vphantom{1}}\right|\Omega_{-}^{\dagger}\left|\smash{d}{\vphantom{1}}\right\rangle\left\langle\smash{d}{\vphantom{1}}\right|\Omega_{+}\left|\smash{a}{\vphantom{1}}\right\rangle\right)^{*}$
(1.31) $\displaystyle=$
$\displaystyle\sum\limits_{c,d}\left\langle\smash{d}{\vphantom{1}}\right|\Omega_{+}\left|\smash{a}{\vphantom{1}}\right\rangle^{*}\left\langle\smash{c}{\vphantom{1}}\right|\Omega_{+}\left|\smash{a}{\vphantom{1}}\right\rangle\left\langle\smash{b}{\vphantom{1}}\right|\Omega_{-}^{\dagger}\left|\smash{c}{\vphantom{1}}\right\rangle\left\langle\smash{b}{\vphantom{1}}\right|\Omega_{-}^{\dagger}\left|\smash{d}{\vphantom{1}}\right\rangle^{*}$
$\displaystyle\equiv$
$\displaystyle\sum\limits_{c,d}W_{a,cd}^{+}\left(W_{b,cd}^{-}\right)^{*}\,,$

where in the last equality we defined the matrices

$\displaystyle
W_{l,cd}^{\pm}\,\equiv\,\left\langle\smash{d}{\vphantom{1}}\right|\Omega_{\pm}\left|\smash{l}{\vphantom{1}}\right\rangle^{*}\left\langle\smash{c}{\vphantom{1}}\right|\Omega_{\pm}\left|\smash{l}{\vphantom{1}}\right\rangle\,,$
(1.32)

providing a factorisation of the transition rate into two parts, carrying the
dependence on the initial state $\left|\smash{a}{\vphantom{1}}\right\rangle$,
and on the final state $\left|\smash{b}{\vphantom{1}}\right\rangle$,
respectively.

The KLN theorem can be stated in terms of the transition probabilities $P(a\to
b)$ as follows. Consider the eigenstates
$\left|\smash{\alpha}{\vphantom{1}}\right\rangle$ of the hamiltonian $H$, and
let $E_{\alpha}$ be their energies. Focus on the set of eigenstates with
energies falling in a fixed interval
$E_{0}-\epsilon<E_{\alpha}<E_{0}+\epsilon$, and denote that set by
$D_{\epsilon}(E_{0})$. Transition probabilities $P(\alpha\to\beta)$ are in
general singular in perturbation theory if $E_{\alpha}$ or $E_{\beta}$ are
degenerate. In such cases, one must introduce a regulator for the singularity,
for example a mass $m$ for all particles involved in the scattering. In the
presence of the regulator $m$, one can then construct the inclusive transition
probability

$\displaystyle P_{m}\Big{[}D_{\epsilon}(E_{a})\to
D_{\epsilon}(E_{b})\Big{]}\,\equiv\,\sum\limits_{a\in
D_{\epsilon}(E_{a})}\sum\limits_{b\in D_{\epsilon}(E_{b})}P_{m}(a\to b)\,,$
(1.33)

and the theorem states that this quantity remains finite order by order in
perturbation theory when the regulator is removed by taking $m\to 0$. Given
Eq. (1.31), this can be proved by considering the matrices $W_{l,cd}^{\pm}$,
and showing that the combination

$\displaystyle\sum\limits_{a\in
D_{\epsilon}(E_{a})}W^{\pm}_{a,cd}\,\equiv\,W^{\pm}_{cd}(a)$ (1.34)

is finite. In order to see how the cancellation comes about, let us begin by
considering the first non-trivial perturbative order. Expanding Eq. (1.29) to
first order in $H_{I}$ we find, for example

$\displaystyle\left\langle\smash{i}{\vphantom{1}}\right|\Omega_{-}\left|\smash{j}{\vphantom{1}}\right\rangle$
$\displaystyle=$
$\displaystyle\left\langle\smash{i}{\vphantom{1}}\right|\left[1-{\rm
i}\int_{\infty}^{0}dt\,H_{I}(t)\right]\left|\smash{j}{\vphantom{1}}\right\rangle$
(1.35) $\displaystyle=$ $\displaystyle\delta_{ij}+{\rm
i}\\!\int_{0}^{\infty}dt\left\langle\smash{i}{\vphantom{1}}\right|{\rm
e}^{{\rm i}H_{0}t}\,H_{I}{\rm e}^{-{\rm
i}H_{0}t}\left|\smash{j}{\vphantom{1}}\right\rangle$ $\displaystyle=$
$\displaystyle\delta_{ij}+{\rm
i}\left\langle\smash{i}{\vphantom{1}}\right|H_{I}\left|\smash{j}{\vphantom{1}}\right\rangle\int_{0}^{\infty}\\!dt\,{\rm
e}^{{\rm i}\big{[}\left(E_{i}-E_{j}\right)+{\rm i}\epsilon\big{]}t}$
$\displaystyle=$
$\displaystyle\delta_{ij}-\frac{\left\langle\smash{i}{\vphantom{1}}\right|H_{I}\left|\smash{j}{\vphantom{1}}\right\rangle}{E_{i}-E_{j}+{\rm
i}\epsilon}\,,$

where the infinitesimal real constant $\epsilon$ ensures the convergence of
the integral at infinity, and $E_{i}$ and $E_{j}$ are the eigenvalues of
$H_{0}$ for the states $\left|\smash{i}{\vphantom{1}}\right\rangle$ and
$\left|\smash{j}{\vphantom{1}}\right\rangle$, respectively. Eq. (1.35), and
the analogous result for the matrix elements of $\Omega_{+}$, can be
substituted in Eq. (1.32), yielding

$\displaystyle
W^{\pm}_{a,ij}\,=\,\delta_{ia}\delta_{ja}-\frac{\left\langle\smash{i}{\vphantom{1}}\right|H_{I}\left|\smash{a}{\vphantom{1}}\right\rangle}{E_{i}-E_{a}\mp{\rm
i}\epsilon}\,\delta_{ja}-\frac{\left\langle\smash{j}{\vphantom{1}}\right|H_{I}\left|\smash{a}{\vphantom{1}}\right\rangle^{*}}{E_{j}-E_{a}\pm{\rm
i}\epsilon}\,\delta_{ia}\,,$ (1.36)

up to corrections of second order in $H_{I}$. If the energy levels $E_{i}$ and
$E_{j}$ become degenerate when the regulator $m$ is removed, Eq. (1.36)
exhibits singularities whenever $\left|\smash{a}{\vphantom{1}}\right\rangle$
coincides with either $\left|\smash{i}{\vphantom{1}}\right\rangle$ or
$\left|\smash{j}{\vphantom{1}}\right\rangle$. It is easy to verify that the
singularities cancel at this order in the combination $W^{\pm}_{cd}(a)$
defined in Eq. (1.34). Four different cases arise naturally

  * •

Both states $\left|\smash{i}{\vphantom{1}}\right\rangle$ and
$\left|\smash{j}{\vphantom{1}}\right\rangle$ lie in $D_{\epsilon}(E_{a})$.
This is the potentially singular case, but one easily verifies that the
contribution with
$\left|\smash{a}{\vphantom{1}}\right\rangle=\left|\smash{i}{\vphantom{1}}\right\rangle$
cancels the one with
$\left|\smash{a}{\vphantom{1}}\right\rangle=\left|\smash{j}{\vphantom{1}}\right\rangle$.
So long as
$\left|\smash{i}{\vphantom{1}}\right\rangle\neq\left|\smash{j}{\vphantom{1}}\right\rangle$,
one then finds $W_{ij}^{\pm}(a)=0$.

  * •

Only state $\left|\smash{i}{\vphantom{1}}\right\rangle$ belongs to
$D_{\epsilon}(E_{a})$, while $j\notin D_{\epsilon}(E_{a})$. In this case only
one term appears in the sum, but it is not singular

$\displaystyle
W_{ij}^{\pm}(a)\,=\,-\frac{\left\langle\smash{i}{\vphantom{1}}\right|H_{I}\left|\smash{j}{\vphantom{1}}\right\rangle}{E_{j}-E_{i}\pm{\rm
i}\epsilon}\,,\quad E_{j}\neq E_{i}\,,$ (1.37)

  * •

If $i\notin D_{\epsilon}(E_{a})$, $j\in D_{\epsilon}(E_{a})$, by the same
token one finds

$\displaystyle
W_{ij}^{\pm}(a)\,=\,-\frac{\left\langle\smash{i}{\vphantom{1}}\right|H_{I}\left|\smash{j}{\vphantom{1}}\right\rangle}{E_{i}-E_{j}\mp{\rm
i}\epsilon}\,,\quad E_{i}\neq E_{j}\,,$ (1.38)

  * •

Finally, if $i\notin D_{\epsilon}(E_{a}),j\notin D_{\epsilon}(E_{a})$, the sum
receives no contributions, and one simply finds $W_{ij}^{\pm}(a)=0$.

This proves that $W_{ij}^{\pm}(a)$ is infrared-finite to first order in the
perturbation $H_{I}$: the generalisation of the proof to arbitrary
perturbative order can be achieved by induction. To begin with, in order to
keep track of the order of the perturbation, we extract a factor of a (small)
coupling $g$ from the interaction hamiltonian, changing $H_{I}\to gH_{I}$.
Next, we expand both the Möller operators $\Omega_{\pm}$ and the matrices
$W_{a,ij}^{\pm}$ and $W_{ij}^{\pm}(a)$ in powers of $g$. In particular, using
the definitions in Eq. (1.32) and Eq. (1.34), we can write

$\displaystyle W^{\pm}_{ij}(a)$ $\displaystyle\equiv$
$\displaystyle\sum_{n=0}^{\infty}\,g^{n}\,W^{\pm}_{n,\,ij}(a)\,,\qquad\Omega_{\pm}\,\equiv\,\sum_{n=0}^{\infty}\,g^{n}\,\Omega_{\pm,\,n}\,,$
$\displaystyle W_{n,\,ij}^{\pm}(a)$ $\displaystyle=$
$\displaystyle\sum_{r=0}^{n}\,\sum\limits_{a\in
D_{\epsilon}(E_{a})}\left\langle\smash{j}{\vphantom{1}}\right|\Omega_{\pm,\,r}\left|\smash{a}{\vphantom{1}}\right\rangle^{*}\left\langle\smash{i}{\vphantom{1}}\right|\Omega_{\pm,\,n-r}\left|\smash{a}{\vphantom{1}}\right\rangle\,.$
(1.39)

Finally, we need to keep in mind some fundamental properties of the Möller
operators: first of all, unitarity

$\displaystyle\Omega_{\pm}^{\dagger}\,\Omega_{\pm}\,=\,{\bf 1}\,,$ (1.40)

which implies and reflects the unitarity of the $S$ matrix. Next, as we assume
the asymptotic states to be eigenstates of the unperturbed hamiltonian
$H_{0}$, we conclude that the Möller operators must diagonalise the complete
hamiltonian

$\displaystyle\Omega_{\pm}^{\dagger}\,H\,\Omega_{\pm}\,=\,\hat{H}_{0}\,,$
(1.41)

where $\hat{H}_{0}$ is diagonal in the same basis as $H_{0}$, but we allow for
a difference in eigenvalues due to interactions: the effects of
renormalisation, and more generally of virtual corrections, shifting particle
masses, contribute to this difference. Combining Eq. (1.41) with Eq. (1.25) we
can write

$\displaystyle\Big{[}\Omega_{\pm},\hat{H}_{0}\Big{]}\,=\,\left(H-\hat{H_{0}}\right)\Omega_{\pm}\,=\,\big{(}\Delta+gH_{I}\big{)}\Omega_{\pm}\,,$
(1.42)

where $\Delta\equiv H_{0}-\hat{H}_{0}$. The diagonal operator $\Delta$ can
also be expanded in powers of the coupling as
$\Delta=\sum_{n}g^{n}\Delta_{n}$.

With these tools in hand, we can now proceed to build the inductive argument.
We have already shown that $W_{n,\,ij}^{\pm}(a)$ is finite for $n=1$, and we
know that $\Delta$ vanishes at lowest order; furthermore, we can safely assume
that $\Delta_{n}$ is infrared finite, which can be achieved order by order by
a suitable choice of renormalisation scheme. With these results, we can now
prove that, if $W_{n,\,ij}^{\pm}(a)$ is infrared finite for $n\leq N$, then
$W_{N+1,\,ij}^{\pm}(a)$ will also be infrared-finite. As before there are four
cases to be considered, which in fact can be reduced to three.

  * •

Consider first the situation in which
$\left|\smash{i}{\vphantom{1}}\right\rangle\notin D_{\epsilon}(E_{a})$, with
no assumption on $\left|\smash{j}{\vphantom{1}}\right\rangle$. Then, taking
the matrix element of Eq. (1.42) between the states
$\left\langle\smash{i}{\vphantom{1}}\right|$ and
$\left|\smash{a}{\vphantom{1}}\right\rangle$, and using the fact that in this
case $E_{a}\neq E_{i}$, we find

$\displaystyle\Big{[}\,\Omega_{\pm,\,n}\Big{]}_{ia}\,=\,\frac{1}{E_{a}-E_{i}}\left\\{\sum\limits_{k}\big{(}H_{I}\big{)}_{ik}\Big{[}\Omega_{\pm,\,n-1}\Big{]}_{ka}+\sum_{s=1}^{n}\big{(}\Delta_{s}\big{)}_{ii}\Big{[}\Omega_{\pm,\,n-s}\Big{]}_{ia}\right\\}\,,$
(1.43)

where we made use of the fact that $\Delta$ is diagonal and vanishes at lowest
order. Crucially, the $n$-th order contribution to the Möller operator matrix
element is expressed in terms of lower-order terms, multiplied times finite
factors. We can now substitute Eq. (1.43) for one of the two matrix elements
in the definition of $W_{ij}^{\pm}(a)$, Eq. (1.39). Taking the contribution at
order $N+1$, and suitably shifting the summation indices, we find

$\displaystyle
W_{N+1,\,ij}^{\pm}(a)\,=\,\frac{1}{E_{a}-E_{i}}\left[\sum_{k}\big{(}H_{I}\big{)}_{ik}W_{N,\,kj}^{\pm}(a)+\sum_{s=1}^{N}\big{(}\Delta_{s}\big{)}_{ii}W_{N-s,\,ij}^{\pm}(a)\right]\,.$
(1.44)

which is manifestly finite under the induction hypothesis.

  * •

The symmetric case, in which $\left|\smash{j}{\vphantom{1}}\right\rangle\notin
D_{\epsilon}(E_{a})$, with no assumption on
$\left|\smash{i}{\vphantom{1}}\right\rangle$, yields a finite result thanks to
the hermiticity properties of the matrices $W_{ij}^{\pm}(a)$. Indeed,

$\displaystyle
W_{N,\,ij}^{\pm}(a)\,=\,\left[W_{N,\,ji}^{\pm}(a)\right]^{*}\,,$ (1.45)

as easily seen from Eq. (1.39).

  * •

The last case to be considered, when both states
$\left|\smash{i}{\vphantom{1}}\right\rangle$ and
$\left|\smash{j}{\vphantom{1}}\right\rangle$ belong to the set
$D_{\epsilon}(E_{a})$, is potentially the most intricate. It is remarkable and
suggestive that finiteness in this case can be established using the unitarity
of the Möller operators, and thus of the $S$ matrix. Taking the $n$-th
perturbative order in the expansion of the matrix elements of Eq. (1.40), one
finds

$\displaystyle\sum_{r=0}^{n}\,\sum_{a\in
D_{\epsilon}(E_{a})}\Big{[}\,\Omega_{\pm,\,r}\Big{]}_{ja}^{*}\,\Big{[}\,\Omega_{\pm,\,n-r}\Big{]}_{ia}+\sum_{r=0}^{n}\,\sum_{a\notin
D_{\epsilon}(E_{a})}\Big{[}\,\Omega_{\pm,\,r}\Big{]}_{ja}^{*}\,\Big{[}\,\Omega_{\pm,\,n-r}\Big{]}_{ia}\,=\,0\,.$
(1.46)

This allows us to express the matrix elements of $W^{\pm}(a)$ between states
in the degenerate set in terms of those between states that lie outside the
set, which have already been shown to be finite. At order $N+1$,

$\displaystyle
W_{N+1,\,ij}^{\pm}(a)\,=\,-\sum_{r=0}^{N+1}\,\sum\limits_{a\notin
D_{\epsilon}(E_{a})}\Big{[}\,\Omega_{\pm,\,r}\Big{]}_{ja}^{*}\,\Big{[}\,\Omega_{\pm,\,M+1-r}\Big{]}_{ia}\,,$
(1.47)

which completes the proof of the KLN theorem.

The simple quantum-mechanical setup of the proof that we have outlined
highlights the complete generality of the cancellation mechanism. On the other
hand, in a quantum field theory context, a detailed implementation of the
proof requires further work: one must verify for example that the
renormalisation procedure does not interfere with the cancellation, and
identify the relevant contributions in terms of Feynman graphs, where the
splitting of the $S$-matrix in terms of Möller operators is related to
unitarity cuts [165, 166, 167]. Finally, one needs to face the difficulties
associated with summing over initial state degenerate configurations in the
context of scattering experiments. In the next Section, we will explore an
alternative approach, which attempts to construct directly a finite
$S$-matrix, instead of relying upon a cancellation at the level of observed
cross sections.

###  1.3 Saving the $S$-matrix: coherent states

The KLN theorem provides us, in principle, with a practical way out of the
infrared problem: while it remains true that the fundamental theoretical
objects for scattering predictions, the $S$-matrix elements, are ill-defined
in theories with massless particles, a careful construction of observable
cross sections leads to finite predictions, order by order in perturbation
theory. As we will see, however, there is quite some distance between this
solution ‘in principle’ and an effective method to make reliable predictions,
especially for non abelian theories. Furthermore, one feels that it should be
possible to be more ambitious: indeed, the physics of the problem is quite
clear, and suggests directions for refining the quantum field theory framework
in the massless limit, in order to deal with finite quantities at all stages
of the calculations.

To state again the basic fact, infrared singularities arise in massless
theories because emission and absorption probabilities for massless particles
do not decrease fast enough at large distances and long times. This means that
a basic assumption in the construction of the $S$-matrix fails: interactions
never become negligible in the distant past or future, and therefore the
description of the asymptotic states as Fock states built out of isolated non-
interacting particles is (to say the least) not sufficiently precise. This
sounds, hopefully, like a problem that can be fixed: after all, when given a
quantum Hamiltonian $H$, the distinction between what we call the ‘free’, or
‘integrable’ part, $H_{0}$, and the ‘interaction’ part $H_{I}$ has a degree of
arbitrariness. This could be exploited, in principle, by identifying
interactions that are non-negligible at long distances, and re-assigning them
to $H_{0}$, which is supposed to be treated exactly at all distance scales.
This would translate into a more accurate characterisation of the asymptotic
states, which would no longer be defined as eigenstates of the free
Hamiltonian $H_{0}$, but rather as eigenstates of the proper asymptotic
Hamiltonian. Such states have indeed been introduced and extensively studied,
under the label of coherent states: they have been shown to provide a
consistent definition of the $S$-matrix for theories with massless particles,
and their construction has been exploited to uncover deep and interesting
properties of the infrared dynamics of perturbative gauge theories.

After a series of early suggestions and preliminary studies [24, 26, 25, 28,
27, 29], the breakthrough came with the landmark paper by Faddeev and Kulish
[30] in 1970, where the problem of the construction of an infrared-finite
$S$-matrix for QED was formally solved in a definitive manner. In the rest of
this Section, we will provide an introduction to the formalism of coherent
states for massless quantum field theories, following the general logic of
Ref. [30], but using the concrete implementation and examples from Ref. [57],
and we will give a few pointers to later developments along these lines. As
will become clear, coherent states do not (as yet) provide a fully practical
tool for the evaluation of gauge theory observables for collider applications,
but they do give a precise theoretical framework, with a clear and transparent
physical interpretation, which must to some extent underpin and help organise
all subsequent developments.

As was the case for the KLN theorem, the starting point is the Hamiltonian of
our quantum field theory, which we split into a solvable, quadratic part, and
an interaction, according to Eq. (1.25). Time evolution is described in the
interaction picture, so that Eqs. (1.27) and (1.29) hold. The crucial step is
now to use the explicit knowledge of the time dependence of the evolution
operator in the interaction picture to identify and isolate the leading
behaviour of the interaction Hamiltonian at large times (we will see an
explicit example of this step in Section 1.3.1). We need, of course, to
introduce a scale $E$, in order to define what we mean by long and short
times: roughly speaking, we consider times $t\geq\hbar/E$ as asymptotic. We
then write

$\displaystyle H_{I}(t)\,=\,H_{R}^{E}(t)+H_{A}^{E}(t)\,,$ (1.48)

where $H_{A}^{E}(t)$ denotes the asymptotic Hamiltonian, responsible for the
leading large-time behaviour of the evolution operator. In turn, using
$H_{A}^{E}(t)$, we can define asymptotic Möller operators

$\displaystyle\Omega_{A,\,\pm}(E)\,\equiv\,\lim_{t\to\infty}\,T\exp\left[-{\rm
i}\int_{\mp t}^{0}dt\,H_{A}^{E}(t)\right]\,,$ (1.49)

which allows us to isolate the large-time contributions to the $S$-matrix by
writing

$\displaystyle S$ $\displaystyle=$
$\displaystyle\Omega_{A,\,-}^{\dagger}(E)\,\Omega_{R,\,-}^{\dagger}(E)\,\Omega_{R,\,+}(E)\,\Omega_{A,\,+}(E)$
(1.50) $\displaystyle\equiv$
$\displaystyle\Omega_{A,\,-}^{\dagger}(E)\,S_{R}(E)\,\Omega_{A,\,+}(E)\,,$

where we have introduced regular Möller operators $\Omega_{R,\,\pm}(E)$, and
we have (somewhat optimistically) defined a regular $S$-matrix

$\displaystyle
S_{R}(E)\,=\,\Omega_{A,\,-}(E)\,S\,\Omega_{A,\,+}^{\dagger}(E)\,.$ (1.51)

Note that, in general, the regular Hamiltonian $H_{R}^{E}(t)$ does not commute
with the asymptotic one, so the regular Möller operator is not simply computed
by exponentiating $H_{R}^{E}(t)$, but involves commutator terms: the fact that
the regular $S$-matrix is indeed free of infrared singularities must therefore
be checked with explicit definitions in hand for the operators involved.
Specifically, with the definitions above, we expect that the regular
$S$-matrix elements in the usual Fock space will be free of infrared
singularities. Alternatively, and perhaps more appropriately, one may define a
basis of coherent states

$\displaystyle\left|\smash{h,\,\pm}{\vphantom{1}}\right\rangle_{E}\,=\,\Omega_{A,\,\pm}^{\dagger}(E)\,\left|\smash{h}{\vphantom{1}}\right\rangle\,,$
(1.52)

by acting with the asymptotic Möller operators on a generic Fock state
$\left|\smash{h}{\vphantom{1}}\right\rangle$. The expectation is then that the
usual $S$-matrix will be free of infrared singularities in the coherent state
basis.

In QED, with massive fermions, this expectation was turned into a theorem by
Faddeev and Kulish in Ref. [30]. As we will note below, the form of the
coherent state operator is particularly simple in the soft limit for abelian
gauge theories: this allows for a full treatment, and one can prove not only
the perturbative finiteness of the coherent state $S$-matrix, but also that it
is possible to build a Hilbert space of coherent states which is separable,
gauge invariant, and containing a gauge-invariant subspace of positive norm
states. In some sense, for the soft problem in the massive abelian theory, the
book is closed. In the non-abelian case, the situation is considerably more
complicated, because there are unavoidable collinear singularities associated
with the self-interactions of massless gluons: as we will see below, the
coherent state operator is qualitatively much more intricate in the collinear
limit, and this makes it much more difficult to prove all-order statements. A
number of papers between the late seventies and the early eighties explored
the construction of non-abelian coherent states, their gauge properties, and
the mechanism for the cancellation of divergences [46, 47, 48, 49, 51, 50, 52,
53, 55, 54, 56], and finally an elegant formal proof of the finiteness of the
non-abelian coherent-state $S$-matrix, including collinear singularities, was
given by Giavarini and Marchesini in Ref. [59].

In the remainder of this Section, rather than focusing on all-order proofs, we
would like to flesh out the rather formal arguments given above, showing in
concrete examples how the asymptotic Hamiltonian is constructed, and how the
coherent state operator engineers the cancellation of infrared singularities.
To this end, following Ref. [57], we begin by considering the simple case of a
scalar theory, affected by collinear divergences only, and then we briefly
highlight similarities and differences with the physically more interesting
case of four-dimensional gauge theories.

####  1.3.1 Collinear divergences in a scalar theory

Our toy model for the construction of coherent states is a massless scalar
theory with cubic interaction, which we take to live in dimension $d=6$, where
the theory is renormalisable. The Lagrangian is then simply

$\displaystyle{\cal
L}\,=\,\frac{1}{2}\,\partial_{\mu}\phi\,\partial^{\mu}\phi-\frac{\lambda}{6}\,\phi^{3}\,.$
(1.53)

A detailed power-counting argument at the diagrammatic level shows that
scattering amplitudes in this theory can be affected by collinear
singularities, but not by soft ones. Here we will simply assume that this is
the case, and we will study the asymptotic behaviour of scattering amplitudes
focusing on the collinear limit.

As explained in Section 1.3, the construction of coherent states is
particularly simple and transparent in the interaction picture, where quantum
operators evolve with the free Hamiltonian. The first step is then to expand
the scalar field in Fourier modes, each of which evolves independently in
time, with a fixed energy given by the classical mass-shell condition. We
write

$\displaystyle\phi\left({\bf x},t\right)\,=\,\int\widetilde{dk}\,\Big{[}a({\bf
k}){\rm e}^{\,{\rm i}\,{\bf k}\cdot{\bf x}\,-\,{\rm i}\,w({\bf
k})t}\,+\,a^{\dagger}({\bf k}){\rm e}^{-{\rm i}\,{\bf k}\cdot{\bf x}\,+\,{\rm
i}\,w({\bf k})t}\Big{]}\,,$ (1.54)

where we defined

$\displaystyle\widetilde{dk}\,=\,\frac{d^{5}k}{(2\pi)^{5}\,2w({\bf
k})}\,,\qquad w({\bf k})\,=\,\left|{\bf k}\right|\,,$ (1.55)

and the creation and annihilation operators satisfy the normalisation
condition

$\displaystyle\left[a({\bf k}),a^{\dagger}({\bf
k}^{\prime})\right]\,=\,(2\pi)^{5}\,2w({\bf k})\,\delta^{5}\left({\bf k}-{\bf
k}^{\prime}\right)\,.$ (1.56)

It is straightforward to compute the interaction Hamiltonian, which is given
by

$\displaystyle H_{I}(t)$ $\displaystyle=$ $\displaystyle\frac{\lambda}{6}\int
d^{5}x\,\phi^{3}({\bf x},t)$ (1.57) $\displaystyle=$
$\displaystyle\frac{\lambda}{2}\int\frac{\widetilde{dk}_{1}\widetilde{dk}_{2}}{2w_{3}({\bf
k}_{1},{\bf k}_{2})}\Big{[}a^{\dagger}({\bf k}_{1})a({\bf k}_{2})a({\bf
k}_{1}-{\bf k}_{2})\,{\rm e}^{{\rm i}\,\left(w_{1}-w_{2}-w_{3}\right)\,t}$
$\displaystyle\hskip 79.6678pt+\,a^{\dagger}({\bf k}_{1})a^{\dagger}({\bf
k}_{2})a^{\dagger}(-{\bf k}_{1}-{\bf k}_{2})\,{\rm e}^{{\rm
i}\,\left(w_{1}+w_{2}+w_{3}\right)\,t}\,+\,{\it h.c.}\Big{]}\,,$

where we introduced the shorthand notation $w_{i}\equiv w({\bf k}_{i})$, and
where the spatial integration led to enforcing momentum conservation. In the
spirit of time-ordered perturbation theory (TOPT), the Hamiltonian in Eq.
(1.57) mediates interactions between on-shell particles, and, as a
consequence, energy is not conserved at the interaction vertices. The
coefficients of $t$ in the exponential factors are given by the energy
deficits in the corresponding interactions, and each term in Eq. (1.57) has a
transparent physical interpretation: for example, the first term acts on a
two-particle state, replacing it with a one-particle state, whereas the second
term corresponds to a quantum fluctuation in which three particles are created
out of the vacuum.

One now comes to the crucial part of the argument: contributions to $H_{I}(t)$
at large times, $|t|\to\infty$, are suppressed by the rapid oscillations of
the exponential factors, except for configurations where the coefficients of
$t$ become vanishingly small. For the second term in Eq. (1.57), and its
hermitian conjugate, the only candidate configurations with this property are
the rather uninteresting ones in which all particles involved have vanishing
energy. On the other hand, it is clear that the first term and its hermitian
conjugate involve physically relevant configuration with a vanishing energy
deficit: for example, soft configurations, say with $w_{2}=0$, where an
energetic on-shell scalar quantum emits or absorbs a soft scalar without
changing its own energy, and remaining on-shell, or, as we detail below,
collinear configurations.

A power-counting analysis would show that soft configurations in this theory
would not lead to divergences in $S$-matrix elements. Bypassing the detailed
argument, here we concentrate directly on the collinear limit. To study it, we
introduce a simple parametrisation of the momenta entering the interaction
vertex, writing

$\displaystyle{\bf k}_{1}$ $\displaystyle=$ $\displaystyle
w_{1}\,\widehat{{\bf k}}_{1}\,,$ $\displaystyle{\bf k}_{2}$ $\displaystyle=$
$\displaystyle w_{1}\big{(}\alpha\widehat{{\bf k}}_{1}+\beta\,\widehat{{\bf
k}}_{\perp}\big{)}\,,$ $\displaystyle{\bf k}_{3}$ $\displaystyle=$
$\displaystyle w_{1}\big{(}(1-\alpha)\widehat{{\bf
k}}_{1}-\beta\,\widehat{{\bf k}}_{\perp}\big{)}\,,$ (1.58)

where we picked ${\bf k}_{1}$ as collinear direction, and identified the
perpendicular direction in the scattering plane by ${\bf k}_{\perp}$, with
$|\widehat{\bf k}_{1}|=|\widehat{\bf k}_{\perp}|=1$, and $\widehat{\bf
k}_{1}\cdot\widehat{\bf k}_{\perp}=0$. The collinear limit is clearly
identified by $\beta\to 0$, while the limits $\alpha\to 0$ and $\alpha\to 1$
are soft. For small $\beta$, and generic $\alpha$, we easily find

$\displaystyle\left|w_{1}-w_{2}-w_{3}\right|\,=\,\frac{\beta^{2}}{2\alpha(1-\alpha)}\,w_{1}+{\cal
O}\left(\beta^{4}\right)\,.$ (1.59)

At this stage we are assuming that soft configurations are not problematic:
the presence of singularities in the soft limits in Eq. (1.59) is a harbinger
of many future problems, in cases where soft and collinear configurations have
a singular overlap. In the present case, writing the collinear asymptotic
Hamiltonian is immediate: we simply introduce an appropriate angular cutoff,
and take the leading power of the interaction Hamiltonian as $\beta\to 0$. We
find

$\displaystyle
H_{A}^{\Delta}(t)\,=\,\frac{\lambda}{2}\int\frac{\widetilde{dk}_{1}\widetilde{dk}_{2}}{2(1-\alpha)w_{1}}\,\theta_{\Delta}({\bf
k}_{1},{\bf k}_{2})\Big{[}a^{\dagger}({\bf k}_{1})a({\bf k}_{2})a({\bf
k}_{1}-{\bf k}_{2})\,{\rm e}^{{\rm
i}\,\frac{\beta^{2}}{2\alpha(1-\alpha)}w_{1}t}\,+\,{\it h.c.}\Big{]}\,,$
(1.60)

where $\theta_{\Delta}({\bf k}_{1},{\bf k}_{2})$ is any set of $\theta$
functions constraining the three momenta meeting at the vertex to be in the
collinear region, say lying within a cone of angular size $\Delta$. Armed with
the asymptotic Hamiltonian in Eq. (1.60), one can easily compute the coherent
state operators $\Omega_{A,\,\pm}(\Delta)$ order by order in perturbation
theory. At leading order, performing the time integration, one finds

$\displaystyle\Omega_{A,\,\pm}^{(1)}(\Delta)\,=\,\frac{\lambda}{2}\int\widetilde{dk}_{1}\widetilde{dk}_{2}\,\frac{\alpha}{\beta^{2}w_{1}^{2}}\,\theta_{\Delta}({\bf
k}_{1},{\bf k}_{2})\Big{[}a^{\dagger}({\bf k}_{1})a({\bf k}_{2})a({\bf
k}_{1}-{\bf k}_{2})\,+\,{\it h.c.}\Big{]}\,,$ (1.61)

where the collinear enhancement as $\beta\to 0$ is clearly displayed in the
first factor of the integrand. By construction, when applied to a single-
particle Fock state, the operator in Eq. (1.61) simulates the collinearly-
enhanced part of the interaction, and generates a quasi-collinear pair;
similarly, when acting on a Fock state describing two quasi-collinear incoming
particles, the operator merges them into a single one.

In order to visualise more explicitly the mechanism of the cancellation,
following Ref. [57], we can work out the simple example of scattering from a
classical current, mimicking the well-known QCD calculation of collinear
effects in DIS. Focusing on the real radiation correction, we compute the
regular $S$-matrix element

$\displaystyle\left\langle\smash{{\bf p}_{f_{1}},{\bf
p}_{f_{2}}}{\vphantom{1}}\right|S_{R}(\Delta)\left|\smash{{\bf
p}_{i}}{\vphantom{1}}\right\rangle\,=\,\left\langle\smash{0}{\vphantom{1}}\right|a({\bf
p}_{f_{1}})a({\bf
p}_{f_{2}})\,\Omega_{A,\,-}(\Delta)\,S^{(0)}\,\Omega^{\dagger}_{A,\,+}(\Delta)\,a^{\dagger}({\bf
p}_{i})\left|\smash{0}{\vphantom{1}}\right\rangle\,,$ (1.62)

where $S^{(0)}$ represent the scattering by the external current. We expect
collinear enhancements when ${\bf p}_{f_{1}}$ becomes collinear to ${\bf
p}_{f_{2}}$, and when either ${\bf p}_{f_{1}}$ or ${\bf p}_{f_{2}}$ become
collinear to ${\bf p}_{i}$: for the sake of this example, let us pick this
second case, choosing ${\bf p}_{f_{1}}$ along a direction close to ${\bf
p}_{i}$. In this case, ${\bf p}_{f_{1}}$ and ${\bf p}_{f_{2}}$ cannot be
collinear to each other, since the hard scattering by the external current
imparts a large momentum to the system. As a consequence, at ${\cal
O}(\lambda)$ the coherent state operator to the left of the classical
scattering in Eq. (1.62) cannot contribute, and one is left with

$\displaystyle\left\langle\smash{{\bf p}_{f_{1}},{\bf
p}_{f_{2}}}{\vphantom{1}}\right|S_{R}(\Delta)\left|\smash{{\bf
p}_{i}}{\vphantom{1}}\right\rangle^{(1)}$ $\displaystyle=$
$\displaystyle\left\langle\smash{0}{\vphantom{1}}\right|a({\bf
p}_{f_{1}})a({\bf
p}_{f_{2}})\,S^{(0)}\,\Omega^{\dagger\,(1)}_{A,\,+}(\Delta)\,a^{\dagger}({\bf
p}_{i})\left|\smash{0}{\vphantom{1}}\right\rangle$ (1.63) $\displaystyle\hskip
14.22636pt+\,\left\langle\smash{0}{\vphantom{1}}\right|a({\bf
p}_{f_{1}})a({\bf p}_{f_{2}})\,S^{(1)}\,a^{\dagger}({\bf
p}_{i})\left|\smash{0}{\vphantom{1}}\right\rangle$ $\displaystyle\equiv$
$\displaystyle{\cal A}^{(1)}_{\,\Omega}+{\cal A}^{(1)}_{\,S}\,.$

In the first part of the amplitude, ${\cal A}^{(1)}_{\,\Omega}$, the coherent
state operator acts on the initial particle, turning into a collinear pair,
and one member of the pair is then identified with the final-state particle
carrying momentum ${\bf p}_{f_{1}}$: indeed

$\displaystyle\Omega^{\dagger\,(1)}_{A,\,+}(\Delta)a^{\dagger}({\bf
p}_{i})\left|\smash{0}{\vphantom{1}}\right\rangle\,=\,\frac{\lambda}{2}\int\widetilde{dk}\,\frac{\alpha}{\beta^{2}w_{i}^{2}}\,\theta_{\Delta}({\bf
p}_{i},{\bf k})\,a^{\dagger}({\bf k})a^{\dagger}({\bf p}_{i}-{\bf
k})\left|\smash{0}{\vphantom{1}}\right\rangle\,,$ (1.64)

so that one finds

$\displaystyle{\cal A}^{(1)}_{\,\Omega}\,=\,\lambda\,{\cal A}^{(0)}\left({\bf
p}_{f_{2}},{\bf p}_{i}-{\bf
p}_{f_{1}}\right)\frac{\alpha_{f_{1}}}{\beta_{f_{1}}^{2}w_{i}^{2}}\,\theta_{\Delta}({\bf
p}_{i},{\bf p}_{f_{1}})\,,$ (1.65)

where ${\cal A}^{(0)}\left({\bf p},{\bf q}\right)$ is the Born amplitude for
scattering by the external current. The second part of the amplitude in Eq.
(1.63), ${\cal A}^{(1)}_{\,S}$, is just the ordinary $S$-matrix element for
the scattering accompanied by a single radiation: it comprises the two Feynman
diagrams depicted in Fig. 7 and yields

$\displaystyle{\cal A}^{(1)}_{\,S}\,=\,\lambda\,{\cal A}^{(0)}\left({\bf
p}_{f_{2}},{\bf p}_{i}-{\bf
p}_{f_{1}}\right)\,\frac{1}{(p_{i}-p_{f_{1}})^{2}}\,+\,\lambda\,{\cal
A}^{(0)}\left({\bf p}_{f_{1}}+{\bf p}_{f_{2}},{\bf
p}_{i}\right)\,\frac{1}{(p_{f_{1}}+p_{f_{2}})^{2}}\,.$ (1.66)

Figure 7: Single-radiation graphs contributing to scalar scattering from an
external current, at first order in the coupling $\lambda$, denoted by ${\cal
A}_{\,S}$ in the text.

In our chosen configuration, the first term of Eq. (1.66) is collinearly
enhanced, but one easily verifies that the collinear enhancement is precisely
cancelled at leading power in $\beta$ by the contribution of Eq. (1.65): at
this order, we have indeed built a non-singular version of the scattering
matrix in the collinear limit. Ref. [30] showed that this cancellation persist
to all order in perturbation theory for soft enhancements in QED, while Ref.
[59] proved it in general for a non-abelian theory. A few observations are in
order.

  * •

Perhaps most remarkably, in the coherent state picture the cancellation of
singularities does not involve combining real and virtual corrections to
scattering amplitudes. Indeed, as illustrated in the simple example above,
matrix elements for real radiation and virtual corrections to the Born process
are treated separately: for the first, the coherent state contribution
subtracts the leading-power enhancement, that would lead to divergences upon
phase-space integration; for the second, the leading-power enhancement is
subtracted at the level of the loop integrand, so that loop integrals becomes
finite.

  * •

It is clear from the procedure leading to Eq. (1.60) that there is ample
freedom in choosing what to include in the ‘asymptotic’ behaviour of the
Hamiltonian, and therefore in the coherent state operator: depending on the
application one has in mind, one might want to include in the asymptotic
dynamics also terms that do not lead to divergences in the amplitudes, or sub-
leading powers in the resolution parameter [168, 169]. This could provide an
interesting avenue to explore infrared enhancements, and eventually
resummations, beyond leading power, a subject that has recently received
considerable attention, and could lead to relevant phenomenological
applications (see, for example, [136] and references therein).

  * •

It should also be clear from Eq. (1.60) and Eq. (1.61) that the perturbative
expansion of the collinear coherent state operator at higher orders will not
display any immediate simplifications: it will involve increasingly longer
strings of creation and annihilation operators, and, since all particles
meeting at collinear vertices carry non-vanishing momenta, the commutation
properties do not lead to any obvious rearrangement at high orders. This is to
be contrasted with the soft case in QED massive fermions, to be discussed
briefly below, in Section 1.3.2. The point is perhaps best made by noting that
the states generated by acting with $\Omega_{A,\,\pm}^{\dagger}(\Delta)$ on
the Fock states are not really ‘coherent’ in the traditional sense: the
asymptotic operator is not of the form $\exp(\alpha a^{\dagger})$, with
$\alpha$ a $c$-number. Rather, it has a much more intricate structure, akin to
$\exp(a^{\dagger}aa+{\it h.c.})$: in the presence of collinear dynamics, one
should properly talk of generalised coherent states.

####  1.3.2 Soft divergences in QED

When moving on to the much more interesting case of massless gauge theories,
it becomes unavoidable to grapple with issues related to the superposition of
soft and collinear singularities. In QED, the latter are only relevant in the
presence of massless charged particles, while in the non-abelian case they are
unavoidable, as gluons must be massless and are charged. In such cases, the
natural thing to do is to split the asymptotic hamiltonian into soft and
collinear contributions, as

$\displaystyle H_{A}^{E}(t)\,=\,H_{S}^{E}(t)+H_{C}^{E}(t)\,,$ (1.67)

assigning the soft-collinear sector either to $H_{S}^{E}$ or to $H_{C}^{E}$,
and avoiding double counting. To get a feeling of how that might work, notice
that the boundary of the asymptotic region is defined by imposing a ‘small’
violation of energy conservation at the interaction vertex in Eq. (1.57),

$\displaystyle\left|w_{1}-w_{2}-w_{3}\right|\,<\,E\,,$ (1.68)

where, for massless QED, we can for example take $w_{3}$ to be the photon
energy. Upon enforcing momentum conservation in the massless case, this
becomes

$\displaystyle\left|w_{1}-w_{3}-\sqrt{w_{1}^{2}+w_{3}^{2}-2w_{1}w_{3}\cos\theta_{13}}\,\right|\,<\,E$
(1.69)

where $\theta_{13}$ is the angle between the momentum ${\bf k}_{3}$ of the
photon and the momentum ${\bf k}_{1}$ of the charged particle. Considering for
example the case in which the photon is emitted by an incoming ‘electron’, the
physically relevant branch of the inequality in Eq. (1.69) establishes that
the boundary of the asymptotic region is a hyperbola in the
$w_{3}$-$\cos\theta_{13}$ plane, so that Eq. (1.69) becomes

$\displaystyle\cos\theta_{13}\,>\,\left(1+\frac{E}{w_{1}}\right)-\frac{E}{w_{3}}\left(1+\frac{E}{2w_{1}}\right)\,,$
(1.70)

depicted in Fig. 8. Clearly, the precise shape of the boundary between the
asymptotic region and the hard-scattering region is not relevant, and one
could just as well introduce separate cutoffs in energy and angle, as also
suggested in Fig. 8: what matters is that the soft and collinear regions
overlap. As a consequence, $H_{S}^{E}$ and $H_{C}^{E}$ in Eq. (1.67) do not
commute, and the coherent state operator can only be factorised into soft and
collinear contributions up to commutator terms [57].

Figure 8: Asymptotic and hard regions for a massless gauge theory, with a
soft-collinear overlap.

In the case of QED with massive fermions, one need not consider the collinear
region at all; furthermore, one may exploit the fact that the soft part of the
asymptotic hamiltonian is considerably simpler than its collinear counterpart,
since, at leading power in the soft momentum, one may neglect the recoil of
the hard emitting fermion. A straightforward analysis for massive QED [30]
leads to the expression

$\displaystyle
H_{S}^{E}(t)\,=\,\int\widetilde{dk}\,\widetilde{dp}\,\frac{e}{w_{p}}\,\theta\left(E-{\bf
p}\cdot{\bf k}\right){\cal H}_{\rm eik}({\bf p},{\bf k};t)\,,$ (1.71)

where ${\bf p}$ is the momentum of the emitting fermion, ${\bf k}$ is the
photon momentum, and we have introduced the (eikonal) momentum-space
Hamiltonian density

$\displaystyle{\cal H}_{\rm eik}({\bf p},{\bf k};t)\,=\,\rho({\bf
p})\,\sum_{\lambda}{\bf p}\cdot{\bf\epsilon}_{\lambda}({\bf
k})\,\Big{[}a_{\lambda}({\bf k}){\rm e}^{-{\rm i}\,\widehat{{\bf p}}\cdot{\bf
k}\,t}+{\it h.c.}\Big{]}\,,$ (1.72)

where $\lambda$ is the photon polarisation and $\epsilon_{\lambda}$ the
corresponding polarisation vector. Crucially, we have been able to neglect
${\bf k}$ in all factors that are regular as $w_{k}\to 0$: creation and
annihilation operators for the fermions are then evaluated at the same
momentum ${\bf p}$, and they organise themselves to construct the fermion
number operator

$\displaystyle\rho({\bf p})\,=\,\sum_{s}\Big{[}b_{s}^{\dagger}({\bf
p})b_{s}({\bf p})-d_{s}^{\dagger}({\bf p})d_{s}({\bf p})\Big{]}\,.$ (1.73)

This simplification is at the heart of all subsequent developments in QED: it
means, in particular, that for ordinary Fock states involving a fixed finite
number of charged particles, which are eigenstates of $\rho({\bf p})$, the
coherent state operator takes the standard form $\exp(\alpha a^{\dagger})$,
with $\alpha$ a $c$-number. The well-known properties of ordinary coherent
states can then be exploited to perform an all-order analysis. It is important
to notice that the crucial simplification leading to Eq. (1.72) is spin-
independent: minimal coupling of the photon to a conserved matter current
always leads, in the soft limit, to a Hamiltonian expressed in the terms of
the appropriate number operator. The spin-independence of the soft
approximation will play an important role in many of the developments
discussed in the coming sections.

It is clear that the coherent state approach provides in principle a
conceptually satisfactory solution to the infrared problem, directly emerging
from correcting the inadequate approximation of asymptotic dynamics that is at
the root of soft and collinear divergences of ordinary $S$ matrix elements.
Over the years, a number of authors have contributed to elucidating the
interpretation of the method, providing precise definitions of the finite
matrix elements that emerge, and working towards practical applications [170,
171, 172, 173, 174, 175, 176, 177, 178, 179]. As we will see, however,
phenomenological work in QCD has mostly followed a different approach, which
we will sketch in Section 1.4 and develop in the rest of our review. On the
other hand, quite interestingly, the recent symmetry-based interpretation of
infrared limits on the ‘celestial sphere’ [161] can be linked to coherent
states [180]. This has lead to a number of new studies and applications [158,
181, 182, 183, 184, 185], including the case of gravity [186, 187, 188, 189],
and it is to be expected that further developments will be forthcoming.

###  1.4 What to do in QCD

In the previous sections we have sketched three partial solutions to the
infrared problem: an explicit all-order calculation in the abelian case,
displaying exponentiation and cancellation of singular contributions, in
Section 1.1; a general proof of the cancellation of singularities in any
theory with massless particles, in Section 1.2; and the outline of the
construction of an improved Hilbert space, where the $S$-matrix becomes well
defined, in Section 1.3. Unfortunately, none of these arguments is fully
adequate to tackle the practical problems that arise in the most relevant
physical application, QCD at colliders; furthermore, many beautiful and
interesting structural and theoretical aspects of the problem remain hidden.

To be more specific, the QED ‘solution’ does not generalise to the non-abelian
theory because the presence of massless interacting gluons inevitably brings
collinear divergences into the game, even if one couples only massive matter
fields to the Yang-Mills theory. This, in turn, makes it necessary to consider
initial-state degeneracies in the KLN sum. Next, of course, come the problems
related to confinement, and the non-perturbative aspects of the non-abelian
theory. In QED, the true asymptotic states are not Fock states, but they are
not drastically different either, since at least they can be expressed in
terms of the same degrees of freedom, and we expect that a perturbative
construction of asymptotic coherent states, starting from Fock states, will
remain close to the true result. In QCD, the true asymptotic states are
hadrons, and they are not perturbatively related to Fock states built out of
quarks and gluons. This makes it essentially unfeasible to resort to the KLN
theorem to compute physical QCD cross sections: we would need to include in
our calculations multi-parton states, but in practice we do not know how such
multi-parton states contribute to the wave functions of the hadrons we are
colliding101010That being said, the contribution of two-parton states to QCD
collider processes is phenomenologically relevant, and has been under study
for several years, see for example [190, 191, 192, 193].. Similarly, a
perturbative construction of QCD coherent states will mix Fock states with
different parton content, with specific weights and phases dictated by the
perturbative asymptotic hamiltonian: these weights and phases are bound to be
drastically altered by non-perturbative effects, in ways that are not under
control.

With limited help from the general cancellation theorems that we have
outlined, perturbative QCD relies upon the pillar of asymptotic freedom, and a
mix of rigourous all-order perturbative analyses with physical understanding
strongly supported by experiment. There are basically two categories of
hadronic observables that are accessible by means of perturbative tools, and
we briefly outline them below.

####  1.4.1 Infrared-safe cross sections

In a situation without strongly-interacting particles in the initial state (in
other words, at lepton colliders) one can hope to mimic the ‘QED solution’,
looking for observables that are sufficiently inclusive in the soft and
collinear radiation, in order for the associated divergences to cancel. The
first step is then to introduce soft and collinear regulators, in order to
construct a perturbative estimate of the chosen observable. In the
renormalised theory, quark masses $m_{i}(\mu)$, running with the
renormalisation scale $\mu$, would act as a partial collinear regulator, and a
gluon mass $m_{g}$ would regulate all infrared divergences, were it not for
the technical difficulties associated with the breaking of gauge invariance.
In practice, the only workable and universally applied infrared regularisation
scheme for non-abelian theories is dimensional regularisation111111In special
circumstances, other possibilities have been explored, see for example [194,
195, 196, 197, 198, 199]., where, after renormalisation, one takes
$d=4-2\epsilon$, with $\epsilon<0$. One then proceeds to compute a parton-
level prediction for the observable, which we take to depend upon a set of
momenta $\\{p_{i}\\}$. We write it as

$\displaystyle\sigma_{\rm part}\,=\,\sigma_{0}\,{\cal F}_{\rm
part}\left(\frac{p_{i}\cdot
p_{j}}{\mu^{2}},\alpha_{s}(\mu);\frac{m}{\mu},\epsilon\right)\,,$ (1.74)

where $m$ is a parton-level mass scale, used as an infrared regulator. If the
observable is sufficiently inclusive, in other words sufficiently insensitive
to soft and collinear radiation, the perturbative parton-level prediction will
have a finite limit when the infrared regulator(s) are removed. One can then
write [61]

$\displaystyle\sigma_{\rm part}\,=\,\sigma_{0}\,{\cal F}_{\rm
part}\left(\frac{p_{i}\cdot
p_{j}}{\mu^{2}},\alpha_{s}(\mu);0,0\right)\,+\,{\cal
O}\left[\left(\frac{m}{\mu}\right)^{\\!p},\epsilon\right]\,,$ (1.75)

with $p$ a positive integer. Observables with this property are called
infrared safe, and general criteria are available to ascertain the existence
of the finite limit in Eq. (1.75) (some examples will be given in Section
2.6), which can be established to all orders in perturbation theory. At this
point, two further steps are needed in order to treat the leading term of Eq.
(1.75) as a reliable estimate of the corresponding hadronic observable. First,
we need to be able to trust the perturbative expansion, at least as an
asymptotic expansion. This happens in QCD if it is appropriate to choose the
renormalisation scale $\mu$ as a hard scale, $\mu\gg\Lambda_{\rm QCD}$, which
is the case if all Mandelstam invariants $p_{i}\cdot p_{j}$ in Eq. (1.75) are
large in the same sense. The second step is more subtle, and ultimately
involves a degree of assumption: we interpret the weak dependence of
$\sigma_{\rm part}$ on the infrared regulators as indicating a weak dependence
on long-time, long-distance physical processes, happening at length scales
$\Delta x\gg\hbar/\Lambda_{\rm QCD}$. These are the length scales at which
partons hadronise and color is neutralised. We conclude that our parton-level
perturbative estimate of the observable will only be weakly affected by
hadronisation corrections, which will be expected to be of parametric size
$\left(\Lambda_{\rm QCD}/\mu\right)^{q}$, with $q$ a positive integer. In many
well-motivated models of power corrections, where $m$ in Eq. (1.74) is related
to a gluon mass (see for example [200, 201]), one finds that $q=p$, which is
generally supported by arguments based on the OPE (when available), and
consistent with experimental data.

It is worth pointing out that this last step, translating from parton to
hadron language, relies on a picture of confinement which is very strongly
supported by an immense body of experimental data, but cannot be proved within
QCD until better non-pertubative techniques become available. One could, in
principle, imagine a world in which, as partons ‘cool’, flowing away from the
hard scattering, they meet a sharp confining transition which drastically
redistributes their momenta. The data show instead that confinement and
hadronisation after a hard collision happen locally in momentum space, so that
the parton configuration emerging from the hard scattering is closely mirrored
by the hadron distribution detected at large distances, with corrections that
are parametrically negligible at high energies – a picture that was described
as ‘Local Parton-Hadron Duality’ in the early decades of QCD [202, 203].

####  1.4.2 Factorisable cross sections

Whenever strongly interacting particles are present in the initial
state121212Or, for that matter, whenever one considers processes with
identified hadrons, either in the initial or in the final state., the
relatively simple picture of infrared safety discussed in Section 1.4.1 is
bound to break down. As discussed in a simple example in Section 2.8,
processes initiated by massless partons are affected by collinear
enhancements, which can be associated with the radiation of energetic gluons
form the initiating parton. Such radiation changes the kinematics of the
subsequent hard scattering, and therefore the singularity cannot possibly be
cancelled by virtual corrections to the Born process with the given initial
state: the KLN cancellation will require the inclusion of degenerate initial
configurations.

Since, as discussed above, implementing the initial-state KLN sum is, to say
the least, impractical, we would reach an impasse. Fortunately, for high-
energy scattering, a new ingredient comes to the rescue: the separation of
scales between the hard scattering and hadronisation. Intuitively, we
understand that the collinear divergence associated with radiation from the
initial state is due to the fact that emission at very early times is not
sufficiently suppressed, since the particles involved remain close to the mass
shell after the splitting. On the other hand, one would like to argue that
such early emissions should properly be associated with the wave function of
the initial state, rather than with the hard scattering. In general, this
intuitive, semi-classical picture breaks down when quantum corrections are
included, since all sub-processes contributing to the cross section interfere.
If, however, hadrons are formed on space-time scales that are widely separated
from the short-distance hard-scattering process, one may expect the quantum-
mechanical interference to be suppressed. Proving that this is indeed the case
in a relativistic quantum field theory requires an intricate diagrammatic
analysis, which in some cases can be short-circuited with powerful effective
field theory arguments. We will introduce, at amplitude level, some of the
techniques entering factorisation proofs in Section 3. At the level of
observable cross sections, the relevant arguments are summarised in excellent
existing reviews, such as Refs. [60, 61, 96]. In this Introduction, we simply
present very briefly the result of these arguments.

Since the requirements of infrared safety discussed in Section 1.4.1 are
unchanged, we consider again a sufficiently inclusive cross section
$\sigma_{\rm part}$ for a partonic process, this time involving identified
partons in the initial, and possibly the final state. We assume that the
Mandelstam invariants $p_{i}\cdot p_{j}$ of the partonic process are all much
larger that the hadronic scale $\Lambda_{\rm QCD}^{2}$. We then introduce at
parton level a mass parameter $m$, acting as an infrared cutoff, and a
factorisation scale $\mu_{f}$, separating the energies characteristic of the
hard scattering, $\mu^{2}\sim p_{i}\cdot p_{j}$ from the hadronic scale. One
then finds that the partonic cross section can be factorised, in general in
the form of a convolution, which we write, somewhat formally, as

$\displaystyle\sigma_{\rm part}\,=\,{\cal S}_{\rm
long}\left(\frac{m^{2}}{\mu_{f}^{2}}\right)\star{\cal H}_{\rm
short}\left(\frac{p_{i}\cdot
p_{j}}{\mu^{2}},\frac{\mu_{f}^{2}}{\mu^{2}}\right)\,+\,{\cal
O}\left[\left(\frac{m^{2}}{\mu_{f}^{2}}\right)^{\\!p}\,\right]\,,$ (1.76)

where $\star$ denotes the appropriate convolution and, again, $p$ is a
positive integer. Up to power-suppressed corrections, Eq. (1.76) separates the
long-distance dynamics, governed by the factor ${\cal S}_{\rm long}$, which
contains all the singular dependence on the cutoff $m$, from the short-
distance hard scattering factor ${\cal H}_{\rm short}$, which is free of
infrared singularities. Intuitively, ${\cal S}_{\rm long}$ collects
contributions from (soft and) collinear radiation at early and late times,
which build up probability distributions for initial and final states of the
hard scattering: such contributions are expected to be universal, i.e.
independent of the particular hard process being considered; on the other
hand, they are not calculable in perturbative QCD, and must be extracted from
experiments. On the contrary, ${\cal H}_{\rm short}$ collects contributions
from short-distance, off-shell exchanges: these are infrared finite, and thus
we expect them to be reliably calculable in perturbation theory; on the other
hand, they are clearly process-dependent.

The factorisation of un-cancelled infrared singularities described by Eq.
(1.76), and based on separation of scales, has evident analogies with the
handling of ultraviolet singularities by means of renormalisation: also in
this case, singular contributions have been assigned to universal factors,
which must be determined experimentally, and finite contributions from
processes happening at laboratory scales have become calculable; we will
pursue this analogy a little further in Section 4. As was the case for
infrared safe observables, also for factorisable cross sections the
application of Eq. (1.76) to hadronic cross sections relies to some extent on
the assumption, well supported by experiments, that non-perturbative effects
do not drastically disrupt the dynamical pattern emerging at parton level: we
interpret Eq. (1.76) as stating that long-distance dynamics is factorisable
for inclusive cross sections, up to power-suppressed contributions; we then
substitute the divergent contributions to ${\cal S}_{\rm long}$ with the
appropriate finite, experimentally measured, combinations of parton
distributions and fragmentation functions; we expect the result to be accurate
up to corrections suppressed by $(\Lambda_{\rm QCD}/\mu)^{q}$, with strong
arguments suggesting that $q=p$ in many cases. We recall that Eq. (1.76)
follows from the OPE for highly inclusive cross sections such as DIS structure
functions, and it was proved to all orders in perturbation theory for parton
annihilation into colour-singlet final states in Refs. [204, 205]. A crucial
argument to extend this all-order proof to inclusive jet cross sections was
given in Ref. [206], while for general collider observables the debate on the
boundaries of applicability of expression of the form of Eq. (1.76) continues
(see, for example, Ref. [207] and the extensive discussions in [67]).

##  2 Finite orders: tools and results

After discussing the (pre-)history of the infrared problem in our
Introduction, we now turn to the ‘modern’ viewpoint, which is predominantly
driven by the practical requirements of large-scale perturbative calculations
for applications at high-energy colliders. This means that the focus is the
non-abelian theory, and it largely implies that the infrared regulation scheme
is dimensional regularisation. In the present section, we will briefly present
the treatment of soft and collinear singularities in simple one-loop QCD
calculations. This is of course well-known textbook material, and we will only
sketch the relevant calculations: this simple context will, however, allow us
to introduce some of the tools of the trade, that we will employ in a more
abstract context in later Sections of this review. Following the arguments in
Section 1.4, we will first present the case of infrared-safe cross sections at
lepton colliders, introducing concepts relevant to Section 4, and then we will
sketch the derivation of collinear factorisation in one-loop Deep Inelastic
Scattering, which will prepare some grounds for the discussion in Section 6.

###  2.1 Infrared safety: the total cross section

The prototype infrared-safe observable at lepton colliders is the total cross
section for lepton annihilation into hadrons. By definition, it is expected to
be insensitive to the radiation of soft and collinear massless particles, and
the parton-level result bears the simplest possible relation to the hadronic
observable, since all radiated partons must turn into hadrons with unit
probability. These expectations are borne out by the explicit calculation in
dimensional regularisation, which we now outline: the Bloch-Nordsieck-like
cancellation of soft and collinear poles between real and virtual
contributions will be on display.

Working in the perturbative regime, with a center-of-mass squared energy $s$
much larger than hadron masses, we consider the process in which a massless
lepton of momentum $k_{1}$ annihilates with an anti-lepton of momentum
$k_{2}$, with $k_{1}^{\mu}+k_{2}^{\mu}=q^{\mu}$, and $q^{2}=s$. The
annihilation of the two leptons produces a photon which later decays into
massless partons. Since we will remain at leading order in QED, but will later
consider all orders in the strong coupling, it is worthwhile to give general
expressions for the ingredients entering the cross section. We start from the
definition

$\displaystyle\sigma_{\rm
tot}\left(q^{2}\right)\,=\,\frac{1}{2q^{2}}\,\sum_{X}\int
d\Phi_{X}\,\frac{1}{4}\sum_{\rm spin}\Big{|}{\cal
A}\big{(}k_{1}+k_{2}\rightarrow X\big{)}\Big{|}^{2}\,,$ (2.1)

where $X$ is a generic strongly-interacting final state, $\Phi_{X}$ is the
corresponding phase space measure, and $2q^{2}$ is the flux factor for
massless particles. At leading order in QED, one can split the calculation
into leptonic and hadronic tensors, according to

$\displaystyle\sigma_{\rm
tot}\left(q^{2}\right)\,\equiv\,\frac{1}{2q^{2}}\,L_{\mu\nu}(k_{1},k_{2})\,H^{\mu\nu}(q)\,,$
(2.2)

where the leading-order leptonic tensor, including the photon propagator, is

$\displaystyle
L^{\mu\nu}(k_{1},k_{2})\,=\,\frac{e^{2}}{q^{4}}\,\Big{(}k_{1}^{\mu}k_{2}^{\nu}+k_{1}^{\nu}k_{2}^{\mu}-k_{1}\cdot
k_{2}\,g^{\mu\nu}\Big{)}\,.$ (2.3)

The hadronic tensor, on the other hand, can be expressed (in principle non-
perturbatively in the strong interactions) in terms of matrix elements of the
electromagnetic current, as

$\displaystyle H_{\mu\nu}(q)$ $\displaystyle=$ $\displaystyle