dx_{1}dx_{2}\sum_{ij}f_{i}(x_{1})f_{j}(x_{2})\sigma_{ij}(Q^{2})\delta(Q^{2}-x_{1}x_{2}s),$
(48)

where the sub-process cross section for the $ij$ parton-parton interaction,
where $i$($j$) denotes a parton in the incoming $e^{-}$($e^{+}$) beam, with
$\hat{s}=Q^{2}$ when the $e^{+}e^{-}$ cms energy squared is $s$, is denoted in
a conventional notation for parton densities $\\{f_{j}\\}$. Here,
$\sigma_{ij}(Q^{2})$ is given by the appropriate expression as dictated by the
respective QED electron collinear factorisation calculus as described in Refs.
fad-kur:1985 ; altarelli-mart:1986 ; nicro-trent:1987 ; fad-khz:1987 ;
berends-neerver-burgers:1988 .

Already by comparing Eq.(46) and Eq.(48) we see one important advantage of the
YFS MC approach: the exact phase space for the multiple photon radiation is
realized on an event by event basis to all orders in $\alpha$ whereas in the
collinear factorisation approach the radiation transverse degrees of freedom
that have been integrated to reach the 1-dimensional QED PDF’s have to be
restored and this restoration is inherently approximate, as it was illustrated
in Ref. bhabhacern-lep2:1996 ,for example151515In other words, the
distributions which the QED collinear factorisation approach produces are not
exact for the transverse degrees of freedom which were integrated out to
arrive at the collinear limit represented by the QED PDF’s while our
distributions are exact in these degrees of freedom. We have seen in the LEP
studies bhabhacern-lep2:1996  that the detailed measurements of the exclusive
photon distributions show this deviation from exactness..

Application of Eq. (48) is limited to “academic observables” with a cut-off on
the total photon energy $E_{\max}=(sx_{1}x_{2})^{1/2}$161616We stress that in
any real observable there is always multiple photon radiation to all orders in
$\alpha$. Any fixed-order calculation thus is necessarily academic, and its
usefulness has to determined on a case-by-case basis. In many cases, the
effects of the multi-photons missing from the fixed-order result are small
enough that the fixed-order result can be used to assess the data. Even in the
latter cases, the lack of exactness of the treatment of the transverse degrees
of freedom in Eq. (48) limits its applicability.. All realistic experimental
observables select events using acollinearity and other similar cuts depending
on photon momenta in a complicated way. Only a Monte Carlo with full multiple
photon phase space can provide predictions for the real experiments. On the
other hand, variants of Eq. (48) with added subleading corrections in $f_{l}$
and $\sigma_{ll^{\prime}}$ are quite useful in testing/calibrating Monte Carlo
programs. For instance the BHLUMI program includes the QED collinear
factorisation based program LUMLOG, while ${\cal KK}$MC provides the KKsem and
KKfoam auxiliary programs, which serve for testing/calibrating the main
multiphoton generator, albeit for academic observables.

###  9.3 Summary

We conclude with the following observation: ${\cal KK}$MC-ee is the unique
$e^{+}e^{-}$ annihilation MC which features CEEX exact ${\cal O}(\alpha^{2}L)$
corrections and control over the corresponding EW initial-final interference
(IFI) effects as well the exact ${\cal O}(\alpha)$ pure weak corrections and
complete spin effects both for the initial beams and outgoing fermions. This
uniqueness is important: it proves that we have a unique set of the tools to
extend the CEEX method to the other important processes in the future
$e^{+}e^{-}$ colliders’ precision physics programs in the effort to reach the
new required precision for these processes.

Acknowledgments

The authors thank Prof. G. Giudice for the support and kind hospitality of the
CERN TH Department. S.J. acknowledges funding from the European Union’s
Horizon 2020 research and innovation programme under grant agreement No 951754
and support of the National Science Centre, Poland, Grant No.
2019/34/E/ST2/00457.

##  10 NNLO QED calculations with McMule

_Tim Engel, Marco Rocco, Adrian Signer, Yannick Ulrich_

###  10.1 Introduction

While a huge effort is ongoing to tackle next-to-next-to-leading order (NNLO)
corrections in QCD, fixed-order calculations in QED have received
comparatively little attention in the literature. From a high-energy point of
view this is understandable due to the numerical importance of QCD
corrections, driven by a coupling that is much larger than the QED coupling
$\alpha$. Hence, NNLO QED calculations are typically considered in low-energy
processes involving mainly leptons. However, for electron-positron colliders
initial-state QED corrections are of utmost importance. Often such corrections
are taken into account to high orders in $\alpha$ but with a soft and/or
collinear approximation in the kinematics.

In this contribution we take a slightly different approach in that we start
from complete fixed-order NNLO QED calculations. We will describe a modern
approach for such calculations, based partly on the developments made for QCD,
and explore how these calculations can be adapted to the case of a high-energy
electron-positron collider. In particular, we regularise all singularities
with dimensional regularisation in $d=4-2\epsilon$ dimensions and use a
subtraction method for the phase-space integration. This is a notable
difference to many QED calculations in the literature where a photon-mass
regularisation as well as a slicing method, separating photon radiation into
hard and soft, is used. We stress that the fact that we have thus far
considered only fixed-order calculations and neglected all-order resummation
effects does not imply that the latter cannot be matched with the former – in
fact, we are now working towards this very goal; we shall briefly comments on
this in Section 10.5.

In contrast to most QCD calculations, for QED calculations it is essential to
keep non-vanishing fermion masses $m$. Since QED with massive fermions has
only soft singularities, this leads to a tremendous simplification of the
infrared (IR) structure. In Section 10.2 we will describe how this can be
exploited to devise an efficient subtraction scheme for phase-space
integrations at all orders in $\alpha$.

However, this simplification comes at a cost. The presence of fermion masses
introduces additional scales to the problem and leads to a complication in the
calculation of virtual amplitudes. In fact, often it is not possible to obtain
analytic expressions for two-loop amplitudes with $m\neq 0$. Fortunately, in
many cases fermion masses are much smaller than the other kinematic variables
$Q^{2}$. As described in Section 10.3, we can then use a procedure dubbed
massification. It allows to recover all not polynomially suppressed mass
effects of two-loop virtual amplitudes from the corresponding amplitudes with
a massless fermion.

Another technical difficulty related to small but non-vanishing fermion masses
is the presence of large logarithms $\ln(m^{2}/Q^{2})$. These are the remnants
of collinear singularities regulated by $m\neq 0$. In massless QCD, only
observables that are inclusive in final-state collinear emission are physical,
where these collinear singularities cancel as $1/\epsilon$ poles. Initial-
state collinear emissions are absorbed into the parton distribution functions.
In QED, the $\ln(m^{2}/Q^{2})$ are physical in that they can be measured.
Being less inclusive leads to numerical issues in particular in the phase-
space integration of the real-virtual contributions. Rather than using pure
numerical techniques or brute force to address this issue, we use properties
of radiative QED amplitudes. More precisely, we approximate these amplitudes
in the soft limit at next-to-leading power to enable a stable numerical
integration. This is described in Section 10.4.

The techniques outlined in this contribution are being implemented in the
publicly available Monte Carlo framework McMule mcmule:website . One of the
main motivations for these developments is to obtain a full NNLO prediction of
$\mu$-$e$ scattering. This process has gained a lot of interest
Banerjee:2020tdt  in the context of the MUonE experiment MUonE:LoI . Already
some important contributions have been calculated at NNLO Banerjee:2020rww ;
CarloniCalame:2020yoz ; Fael:2018dmz ; Fael:2019nsf ; Budassi:2021twh  and the
two-loop matrix element is known with $m_{\mu}\gg m_{e}=0$ Bonciani:2021okt .
However, these techniques can also be used for processes at higher energies.
Indeed, they form the basis for our exploration in Section 10.5 where we
discuss first steps towards using McMule at medium- and high-energy electron-
positron colliders.

###  10.2 The FKSℓ subtraction scheme

As with all cross-section calculations beyond leading order in gauge theories,
also in QED with massive fermions IR singularities appearing in the phase-
space integration have to be isolated and properly combined with those of
virtual loop integration. What is particularly simple in our case is that no
collinear $1/\epsilon$ are present and the soft singularities exponentiate. In
other words, the $\ell$-loop matrix element (squared) with $n$ external
particles $\mathcal{M}_{n}^{(\ell)}$ satisfies the YFS formula Yennie:1961ad

$\displaystyle\sum_{\ell=0}^{\infty}\mathcal{M}_{n}^{(\ell)}=e^{-\hat{\mathcal{E}}}\,\sum_{\ell=0}^{\infty}\mathcal{M}_{n}^{(\ell)f}$
(49)

where all $1/\epsilon$ IR poles of $\mathcal{M}_{n}^{(\ell)}$ are contained in
the integrated eikonal factor $\hat{\mathcal{E}}$ and the remainders of the
matrix elements, $\mathcal{M}_{n}^{(\ell)f}$, are finite. This simple
structure can be exploited to convert the FKS subtraction scheme
Frixione:1995ms ; Frederix:2009yq  initially developed for NLO QCD
calculations to FKSℓ Engel:2019nfw  for QED calculations at all orders. At
NNLO for example, the double virtual, real-virtual and double-real corrections
read

$\displaystyle\begin{split}\sigma^{(2)}_{n}(\xi_{c})&=\int\\!\mathrm{d}\Phi_{n}^{d=4}\,\bigg{(}\mathcal{M}_{n}^{(2)}+\hat{\mathcal{E}}(\xi_{c})\,\mathcal{M}_{n}^{(1)}+\frac{1}{2!}\mathcal{M}_{n}^{(0)}\hat{\mathcal{E}}(\xi_{c})^{2}\bigg{)}=\int\\!\mathrm{d}\Phi_{n}^{d=4}\,\mathcal{M}_{n}^{(2)f}(\xi_{c})\,,\end{split}$
(50a) $\displaystyle\sigma^{(2)}_{n+1}(\xi_{c})$
$\displaystyle=\int\\!\mathrm{d}\Phi^{d=4}_{n+1}\left(\frac{1}{\xi_{1}}\right)_{c}\Big{(}\xi_{1}\,\mathcal{M}_{n+1}^{(1)f}(\xi_{c})\Big{)}\,,$
(50b) $\displaystyle\sigma^{(2)}_{n+2}(\xi_{c})$
$\displaystyle=\int\\!\mathrm{d}\Phi_{n+2}^{d=4}\left(\frac{1}{\xi_{1}}\right)_{c}\,\left(\frac{1}{\xi_{2}}\right)_{c}\,\Big{(}\xi_{1}\xi_{2}\,\mathcal{M}_{n+2}^{(0)f}\Big{)}\,.$
(50c)

where $\xi_{i}$ are the (scaled) energies of the radiated photons,
$\mathrm{d}\Phi_{n}^{d=4}$ is the $n$-parton phase space in 4 dimensions, and
the distribution $(1/\xi_{i})_{c}$ acts on a test function $f(\xi_{i})$ as

$\displaystyle\int_{0}^{1}\mathrm{d}\xi_{i}\,\left(\frac{1}{\xi_{i}}\right)_{c}\,f(\xi_{i})$
$\displaystyle\equiv\int_{0}^{1}\mathrm{d}\xi_{i}\,\frac{f(\xi_{i})-f(0)\theta(\xi_{c}-\xi_{i})}{\xi_{i}}\,.$
(51)

All three terms in (50) are individually finite and the dependence on the
unphysical parameter $\xi_{c}$ cancels in the sum. This cancellation is exact
and serves as a test for a correct and numerically stable implementation. The
FKSℓ procedure does not require to split photon radiation into a soft and a
hard part, nor does it introduce a photon mass as regulator. If all matrix
elements are known, numerical integration of (50) combined with a measurement
function to define the IR-safe observable(s) results in a fully-differential
Monte Carlo code. This is the basis of the McMule framework Banerjee:2020rww .

###  10.3 Massification

The statement “if all matrix elements are known” is always easy to make, but
often impossible to realise. Of course, tree-level matrix elements pose no
problem. For one-loop matrix elements we often use OpenLoops Buccioni:2017yxi
; Buccioni:2019sur . However, two-loop matrix elements are a bottleneck.

To alleviate the problem it is possible to start with the two-loop matrix
element computed with massless fermions, $\mathcal{M}_{n}^{(2)}(m\\!=\\!0)$.
This matrix element contains poles up to $1/\epsilon^{4}$ due to double soft-
collinear singularities associated with the massless fermion and photon. The
corresponding matrix element with massive fermions,
$\mathcal{M}_{n}^{(2)}(m)$, instead only contains $1/\epsilon^{2}$ soft poles
and $\ln^{2}(m^{2}/Q^{2})$ as remnants of the collinear singularities. For
small masses $m^{2}\ll Q^{2}$ the idea is tempting to use the massless matrix
element as an approximation of the massive one. This is, however, not possible
due to the different singularity structure. Instead, the massive all-order
result can be related to the massless one via the factorisation formula
Penin:2005eh ; Becher:2007cu ; Engel:2018fsb

$\displaystyle\mathcal{M}_{n}(m)$
$\displaystyle=\Big{(}\prod_{j}{Z(m)}\Big{)}\times
S\times\mathcal{M}_{n}(0)+\mathcal{O}(m)\,.$ (52)

The product is over all external fermion legs with a small mass $m$. The
process-independent factor ${Z(m)}$ is given in (3.12) of Engel:2018fsb  to
NNLO and converts collinear $1/\epsilon$ poles into $\ln(m^{2}/Q^{2})$.
However, it is important to stress that (52) does not only reproduce the
logarithmically enhanced terms but in fact all terms that are not polynomially
suppressed. The soft part $S$ is process dependent, starts at NNLO, and
receives contributions from only fermion loops. Conceptually this is the most
complicated contribution due to the factorisation anomaly Engel:2018fsb . In
practice, it is often advantageous to compute the fermion-loop contributions –
including the hadronic ones – of two-loop matrix elements semi-numerically.
This has the added advantage of eliminating $S$ in (52) and rendering
massification completely universal.

To obtain a perfect cancellation of the $1/\epsilon$ singularities in (50) the
divergent part of $\mathcal{M}_{n}^{(2)}(m)$ has to be taken without
approximation using (49). We also use the full $m$ dependence in the
generation of the momenta and the phase space. Parametrically, the error
induced by (52) at NNLO is of the order $(\alpha/\pi)^{2}\,m^{2}/Q^{2}$,
potentially multiplied by a $\ln(m^{2}/Q^{2})$. For example, in the case of
the electron energy spectrum of the muon decay, massification reproduces the
NNLO correction with full electron-mass dependence at the level of a few
percent Engel:2019nfw . Going to higher energies this approximation is
expected to perform even better.

###  10.4 Next-to-soft stabilisation

Even though all parts in (50) are finite and can in principle be evaluated
numerically in 4 dimensions, in practice numerical instabilities are often
encountered. In particular $\sigma^{(2)}_{n+1}(\xi_{c})$, (50b), is delicate
as it requires a numerically stable evaluation of the real-virtual one-loop
amplitude also in corners of phase space where large hierarchies of scales are
present. OpenLoops shows a remarkable stability and has been used successfully
for the real-virtual corrections of many QCD NNLO calculations. However, the
presence of (small) fermion masses exacerbates the situation. The problem can
be solved by replacing the full one-loop radiative matrix element with a
sufficiently precise approximation in the numerically problematic soft-photon
region. To this end, the next-to-leading-power expression for the limit where
the photon momentum $k$ becomes soft, i.e. $k\to 0$, is needed. This next-to-
soft stabilisation has been used to compute the photonic corrections to Bhabha
scattering Banerjee:2021mty  as well as the full corrections to Møller
scattering Banerjee:2021qvi  at NNLO. The precision of the numerical
evaluation of the radiative matrix elements used in these calculations is
better than $10^{-5}$ for all photon energies.

The leading term in the soft expansion has the celebrated form
$\mathcal{M}_{n+1}^{(\ell)}=\mathcal{E}\,\mathcal{M}_{n}^{(\ell)}$ where the
eikonal factor $\mathcal{E}$ scales as $1/k^{2}$ (and upon integration over
the photon phase space yields $\hat{\mathcal{E}}$). This approximation is not
sufficient for our purpose and has to be improved by including the subleading-
power contributions scaling as $1/k$. At tree-level, the Low-Burnett-Kroll
(LBK) theorem Low:1958sn ; Burnett:1967km ; Adler:1966gc  provides a universal
structure for these next-to-soft contributions. The LBK theorem has been
extended to one loop for massive QED in Engel:2021ccn  and according to its
(3.31) is given by the sum of the hard and soft contributions

$\displaystyle\mathcal{M}_{n+1}^{(1),\text{hard}}$
$\displaystyle\simeq\sum_{l}\sum_{i}Q_{i}Q_{l}\Big{(}-\frac{p_{i}\cdot
p_{l}}{(k\cdot p_{i})(k\cdot p_{l})}+\frac{p_{l}\cdot\tilde{D}_{i}}{k\cdot
p_{l}}\Big{)}\mathcal{M}_{n}^{(1)},$ (53a)
$\displaystyle\mathcal{M}_{n+1}^{(1),\text{soft}}$
$\displaystyle\simeq\sum_{l}\sum_{i\neq
j}Q_{i}^{2}Q_{j}Q_{l}\Big{(}\frac{p_{i}\cdot p_{l}}{(k\cdot p_{i})(k\cdot
p_{l})}-\frac{p_{j}\cdot p_{l}}{(k\cdot p_{j})(k\cdot
p_{l})}\Big{)}\,2S(p_{i},p_{j},k)\,\mathcal{M}_{n}^{(0)},$ (53b)

where the sums run over the external legs with charges $Q_{i}$ and momenta
$p_{i}$. The first term on the r.h.s. of (53a) scaling as $1/k^{2}$ is just
the eikonal approximation mentioned before. The second term of (53a) scales as
$1/k$ and is the expected one-loop generalisation provided by the LBK theorem,
where $\tilde{D}_{i}$ is the (modified) LBK differential operator, see (3.11b)
of Engel:2021ccn . Taken together, these two terms correspond to the tree-
level version of the LBK theorem. Using the method of regions Beneke:1997zp
to perform the expansion in $k$ at the one-loop level, they correspond to the
hard part Bonocore:2014wua . At one loop there is also a soft contribution,
scaling as $1/k$. The important observation of Engel:2021ccn  is that this
soft contribution also has a universal structure, as given in (53b). The
function $S(p_{i},p_{j},k)$ scales as $k$ and can be found in (3.29) of
Engel:2021ccn . Using (53), the next-to-soft limits can be obtained without a
detailed process-specific computation.

There is a subtlety hidden in (53a) in that the non-radiative matrix element
has to be evaluated with radiative kinematics. In particular, the
corresponding momenta only satisfy momentum conservation up to the remnant
$k$. A possible approach on how to deal with this is discussed in
Engel:2021ccn , while an alternative is given in DelDuca:2017twk ;
Bonocore:2021cbv .

###  10.5 Outlook towards higher energies

The techniques described in the previous sections were developed with low-
energy processes in mind. However, they remain applicable and in some cases
become even more important when going to higher energies. On the other hand,
additional features have to be taken into account. As possible applications we
are considering $ee\to\tau\tau$ for $\sqrt{s}\sim 10\,{\rm GeV}$
Kollatzsch:2022xxx  and also planning to extend our work on Bhabha scattering
Banerjee:2021mty  and $ee\to\gamma\gamma$ to energies relevant for luminosity
determinations at electron-positron colliders.

A first extension that is required is the inclusion of electroweak effects. A
complete NLO calculation in the Standard Model can be achieved using OpenLoops
for the matrix elements and FKSℓ for phase-space integrations.

A second issue that will become more prominent at higher energies are
numerical issues related to collinear photon emission. At low energies, it is
sufficient to give special consideration to the soft region, $k\ll m$. At
higher energies, also the hierarchy $m^{2}\ll Q^{2}$ will require some care,
especially in the case of the numerically delicate real-virtual contribution.
To this end, a similar strategy as for the soft region can be applied. This
time the limit where the angle $\sphericalangle(k,p_{i})$ becomes small has to
be considered. Since we keep $p_{i}^{2}=m^{2}$, but now treat $m^{2}\ll
Q^{2}$, we need the limit $k\cdot p_{i}\sim m^{2}\ll Q^{2}$ of one-loop
radiative matrix elements. At leading power in the small quantity and at NLO
in $\alpha$ this leads to a generalisation of massification, (52), that e.g.
for an initial-state collinear configuration reads

$\displaystyle\mathcal{M}_{n+1}(k,p_{j},m)\simeq
J_{\text{ISR}}(x,m)\Big{(}\prod_{j\neq
i}Z(m)\Big{)}\mathcal{M}_{n}(p_{i}-k,p_{j\neq i},0),$ (54)

where $1-x$ is the energy fraction carried off by the photon. Thus, for the
(incoming) leg $i$ to which the photon becomes collinear, the factor $Z(m)$ is
replaced by the splitting function $J_{\text{ISR}}$. The explicit expression
for $J_{\text{ISR}}$ and the crossing related final-state splitting function
$J_{\text{FSR}}$ at one loop can be found in Appendix B of Engel:2021ccn .
This factorisation formula could also turn out to be useful in coping with the
narrow peaks that arise due to the collinear scale hierarchy. In particular,
(54) can be used to subtract these collinear pseudo-singularities pointwise,
in analogy to Dittmaier:1999mb .

Finally, McMule is currently calculating strictly at fixed order. However, for
many applications a combination with resummation, e.g. that achieved by a
parton shower or by analytical techniques, is required. The collinear
configurations mentioned in the previous paragraph give rise to logarithms
$\ln(m^{2}/Q^{2})$. They become more important at higher energies and can be
resummed using fragmentation functions (for final state) or parton
distribution functions (for initial state)171717In the context of the Snowmass
2021 proceedings, the interested reader can find a more extensive discussion
on collinear factorisation and YFS in previous sections and in ref. FERMILAB-
PUB-22-116-SCD-T .. Currently, within McMule an effort is ongoing to include a
YFS parton shower18181817 that is similar to PHOTONS++ Schonherr:2008av  and
matched at NNLO. For the parton shower, instead of $\hat{\mathcal{E}}$ the
exponent of (49) is the YFS form factor, that also includes a virtual
contribution. The subtraction scheme FKSℓ is well suited for this as it is
already based on YFS. This helps dealing with double counting in the matching
of the parton shower to fixed-order calculations.

With these extensions McMule will become a tool for leptonic processes with an
even broader range of applications.

###### Acknowledgements.

TE acknowledges support by the Swiss National Science Foundation (SNF) under
contract 200021_178967.

##  11 Collinear divergences at future $e^{+}e^{-}$ colliders

_Alessandro Vicini_

###  11.1 Precision calculations at an $e^{+}e^{-}$ collider

The high-precision tests of the Standard Model (SM) at an $e^{+}e^{-}$
collider, like the future FCC-ee or CEPC, rely on the possibility to compute
higher-order radiative corrections, covering all the relevant phase-space
regions. Detailed studies have been performed at LEP (cfr.
TwoFermionWorkingGroup:2000nks ; Grunewald:2000ju  and references therein) and
in preparation for the FCC-ee Proceedings:2019vxr  and the CEPC
CEPCStudyGroup:2018ghi , with a systematic discussion of the available codes
and of several aspects related to the treatment of higher-order radiative
corrections. In this note we try to comment about some of the challenges set
by the expected experimental precision at $e^{+}e^{-}$ colliders and about the
possible transfer of the theoretical and computational knowledge developed for
the LHC.

We assume, for the sake of discussion, that the full set of three-loop EW
corrections to the process $e^{+}e^{-}\to f\bar{f}+X$, with $f$ an
electrically charged fermion, is explicitly available. The need for such a
challenging calculation has been discussed in Freitas:2019bre ;
Blondel:2019qlh , in view of high-precision measurements at the $Z$ resonance.
It has been shown that there might be corrections to the lowest-order
couplings, from virtual corrections of weak and mixed QED-weak origin,
numerically relevant e.g. in the prediction of the asymmetries. Their role is
somehow complementary to the one of the QED corrections, in the precise
predictions of the kinematical distributions. It is known, since LEP
Altarelli:1989hv , that initial state QED radiation develops very large
logarithmic corrections, crucial to the high-precision description of the $Z$
lineshape. The resummation to all orders of these classes of corrections is
known to be mandatory, for an accurate prediction of the central values of the
observables and for a reduction of the remaining theoretical uncertainty
Montagna:1993ai ; Jadach:1993yv ; Bardin:1999yd . Few questions191919 We refer
to the three-loop EW case as the ultimate calculation that can be necessary at
the FCC-ee/CEPC, but the same questions could already be addressed, when the
complete set of two-loop EW will become available. arise: 1) Is the three-loop
calculation on its own sufficient to match the precision target, without the
inclusion of additional higher-order contributions? 2) Since the large
logarithmic effects are due to soft or collinear emissions, which of the two
classes of corrections should receive priority in the construction of a
framework matching fixed- and all-order results ? 3) Does the answer to the
previous question depend on the observable under study?

In order to fix some figures, at FCC-ee the total cross section into hadrons,
at the $Z$ peak, can be measured with a relative error at the 0.01% level
Proceedings:2019vxr . We take this number as a illustrative reference for the
needed precision of the theoretical predictions. In the study of the $Z$ boson
lineshape, the center-of-mass energy is $\sqrt{s}\sim m_{\scriptscriptstyle
Z}$. The size of the logarithmic factor, which appears with the initial state
QED radiation upon integration over the photon-emission angle and is dominated
by the phase-space region with the photon collinear to the incoming electron
or positron, is $L\equiv\log(m_{\scriptscriptstyle Z}^{2}/m_{e}^{2})\sim
24.18$, with $m_{e}$ the mass of the electron. The structure of the cross
section, putting in evidence the initial state collinear (also dubbed mass)
logarithm $L$ and the soft logarithm $\ell$, is given, up to first order in
the fine structure constant $\alpha$, by
$\sigma=\sigma_{0}+\alpha\left(a_{11}L\ell+b_{10}L+c_{01}\ell+d_{1}\right)$,
with $\sigma_{0}$ the Born cross section. The soft logarithm $\ell=\log(\Delta
E^{2}/m_{\scriptscriptstyle Z}^{2})$ appears because of the emission of soft,
i.e. low-energy, photons and its precise value depends on the resolution
$\Delta E$ adopted to identify the final state particles. We remark the
presence of a double logarithmic enhancement given by the product of a mass
and a soft logarithms, plus the contributions given by single logarithmic
terms and a subleading term without enhancement. In order to appreciate the
need for a systematic resummation of the large logarithmic corrections, and in
particular those due to collinear emissions, we iterate the initial state
single logarithmic collinear correction factor and find: $\alpha L\sim
0.1764,\,\alpha^{2}L^{2}\sim 0.0311,\,\alpha^{3}L^{3}\sim
0.0055,\,\alpha^{4}L^{4}\sim 0.0010$ . These factors strongly contribute to
enhance the effect of the radiative corrections, at a level relevant for the
precision goal of prediction of the total cross section into hadrons. Even if
we can not draw final conclusions from this numerical inspection, it
nevertheless suggests that the answer to the first question could be negative,
and the matching of resummed and three-loop results could be necessary. One
might ask whether the resummation of terms at the third order in a logarithmic
counting is by itself sufficient to match the precision goals of the future
$e^{+}e^{-}$ colliders, without needing a full three-loop calculation and the
corresponding matching. The estimates presented in Freitas:2019bre ;
Blondel:2019qlh  suggest that both sets of corrections (fixed and to all
orders) will be needed, because of their different impact on the observables.

A detailed review about the structure functions for the resummation of
collinear logarithms, and of the Yennie-Frautschi-Suura (YFS) formalisms
Yennie:1961ad  for the resummation of soft-emissions contributions, has been
presented in WorkingGrouponRadiativeCorrections:2010bjp ; we refer to Section
2.4 of that paper for technical details. In this note we address questions
analogous to those of ref. WorkingGrouponRadiativeCorrections:2010bjp  for the
matching with next-to-leading (NLO) fixed-order results, but, at variance with
that paper, we consider now the problem of a matching which could be needed at
third order.

###  11.2  The collinear singular limit of squared matrix elements

In QED, the amplitude describing the emission of a photon off an external on-
shell fermion $f$ is proportional to

$\frac{1}{(p+k)^{2}-m_{f}^{2}+i\varepsilon}=\frac{1}{2E_{p}E_{k}(1-\beta_{f}\cos\theta)+i\varepsilon}$
(55)

where $p$ and $k$ are the fermion and photon momenta respectively, $k^{2}=0$
and $p^{2}=m_{f}^{2}$, $m_{f}$ is the fermion mass,
$E_{p}=\sqrt{m_{f}^{2}+\vec{p}^{2}}$ is the fermion energy, $E_{k}=|\vec{k}|$
is the photon energy, $\theta$ is the emission angle of the photon with
respect to the fermion $\vec{p}$, and
$\beta_{f}=\sqrt{1-m_{f}^{2}/|\vec{p}|^{2}}$. When we integrate the squared
matrix element over the photon phase space, the interference term of two
Feynman amplitudes with the photon emitted by two charged particles yields a
factor proportional to $\log\left(\frac{1+\beta}{1-\beta}\right)$ which
diverges in the $m\to 0$ limit; this singularity is thus called mass or
collinear divergence, because the relevant phase-space region is the one with
the photon parallel to the emitting particles.

The collinear divergence depends only on the properties (charge,mass,spin) of
the emitting particles, but it is independent of the details of the rest of
the hard scattering process. This universality feature emerges as a
consequence of the factorisation of the squared matrix element. Beyond the
first order, the correct factorisation in the collinear limit has been
explicitly demonstrated to all orders for tree level amplitudes in
Catani:1999ss . Explicit results at second order are known Catani:1998nv , and
explicit checks at third order have been performed Anastasiou:2015vya ;
Duhr:2021vwj  but a formal demonstration of collinear factorization theorems
for a generic process to all orders is lacking.

The appearance, in the first QED calculations of radiative corrections
Kinoshita:1958ru  of mass singularities, and the remark of their cancellation
in inclusive quantities, has led to the formulation, by Kinoshita-Lee-
Nauenberg (KLN) Kinoshita:1962ur ; Lee:1964is , of a theorem which states the
conditions of such cancellation. The sum over all degenerate, i.e.
indistinguishable, final states yields a finite, regular result, although the
individual contributions are separately divergent. The finiteness is achieved
order by order in the perturbative coupling constant expansion, including
contributions due to virtual and real-emission corrections, provided that both
sets are degenerate with respect to the observable definition. This theorem is
fundamental in the physics of hadronic jets and more in general provides the
guiding principle to define infrared-safe and collinear-safe observables.

The emission of collinear radiation from the initial state of a scattering
amplitude modifies the kinematics of its underlying process, because the
emitted particle has in general a finite energy. The non degeneracy of the two
underlying processes, in the virtual and real correction cases, breaks the
hypothesis of the KLN theorem. It is thus possible that a collinear divergence
survives in the combination of the two contributions. Thanks to the
factorisation property of these divergences, the partonic cross section can be
written as the product of a finite cross section times a divergent factor. The
universality of the latter allows us to reabsorb it into the definition of the
incoming physical object starting the scattering: the best known example is
given by the proton and its parameterisation in terms of collinear parton
density functions (PDFs). Thanks to the universality properties, this
redefinition is predictive, with a factorization-scheme ambiguity at the level
of the subdominant finite corrections, formally of a perturbative order higher
with respect to the one of the calculation. The PDFs express the probability
to find a parton that carries a fraction $x$ of the proton momentum, while
probing the latter in a scattering with a momentum transfer $\mu$. In complete
analogy, we can describe a lepton scattering, like e.g. $e^{+}e^{-}$
annihilation, as the interaction of two composite systems, the leptons, whose
internal structure is given in terms of partons (electrons, positrons,
photons, quarks, EW bosons). The definition of the physical lepton is given,
order by order in perturbation theory, only after the factorisation of the
initial state collinear divergences and their reabsorption in the lepton PDFs.
The collinear parton densities $\vec{q}(x,\mu^{2})$ (they can be partons
inside a proton or a lepton), satisfy the integro-differential DGLAP equations
Gribov:1972ri ; Altarelli:1977zs ; Dokshitzer:1977sg

$\frac{\partial}{\partial\log\mu^{2}}\vec{q}(x,\mu^{2})=\frac{\alpha(\mu)}{2\pi}\int_{x}^{1}\frac{dz}{z}\,\mathbf{P}(z)\cdot\vec{q}(\frac{x}{z},\mu^{2})\,,$
(56)

whose solution expresses their dependence on the scale $\mu$ as the
convolution of the densities themselves with the appropriate splitting
functions matrix $\mathbf{P}(z)$. The latter describe the emission responsible
for the scale change, evaluated in the collinear approximation, and can be
computed in perturbation theory. They are know up to 3-loop level Moch:2004pa
; Vogt:2004mw ; Ablinger:2014nga ; Ablinger:2017tan  in QCD. The solution of
the DGLAP equations resums to all orders the contributions due to the
collinear emissions described by the splitting functions. The logarithmic
accuracy of this resummation is determined by the perturbative order of the
splitting function expression.

The solution of the DGLAP equations requires to fix the boundary conditions of
the differential problem. In the QCD case, since the strong interaction does
not allow us to compute these conditions from first principles, the proton
structure has to be modeled in terms of PDFs and the PDF parameters at a given
scale are fit to the available experimental data. This problem has led to a
large number of studies discussing the representation of the uncertainty on
the proton structure induced by the imperfect knowledge of the boundary
conditions Butterworth:2015oua . The consistent determination of the latter
from a fit to several experimental data is possible only if all the relevant
cross sections are known at the perturbative order under discussion. This
latter problem represents a bottleneck in the progress towards a consistent
prediction of the LHC processes at NNLO-QCD and, a fortiori, N3LO-QCD or NNLO
QCD-EW. In summary, the DGLAP evolution of the proton PDFs can be performed if
the splitting function at a given order are known, but the absolute value of
the parton densities can be affected by an inconsistent knowledge of the
boundary conditions. In the lepton case, it is instead possible to compute the
boundary conditions from first principles in QED as shown at NLO-QED in
Frixione:2019lga . An effort towards their determination at NNLO-QED and N3LO-
QED is needed, in order to apply the DGLAP formalism at a lepton collider
Frixione:2021zdp  at least at second, but hopefully also at third order in
perturbation theory.

The solution of the integral-differential DGLAP equations is achieved in QCD
via numerical algorithms, with different evolution codes available in the
literature Whalley:2005nh ; Salam:2008qg , developed to solve the QCD case,
including so far up to NNLO splitting kernels. The peculiar feature, in QED,
of the existence of an integrable singularity at $z=1$, makes an analytic
solution possible in this case. While the results at NLO-QED are available
Bertone:2019hks ; Frixione:2021wzh , the possibility to prepare an evolution
code at NNLO or N3LO in QED requires the two- or three-loop splitting
functions, which could be derived from the QCD expressions, but also the
boundary conditions, discussed above.

###  11.3  Comparison of structure functions and YFS formalisms

We consider the inclusive production of a final state $F$ in the scattering of
two particles $P_{1,2}$, e.g. hadron-hadron or lepton-lepton, in the structure
function approach. We formulate the cross section as the convolution of the
parton densities describing the probability of finding partons $i,j$ inside
the colliding particles, with a partonic cross section.

$\sigma(P_{1}P_{2}\to F+X)=\sum_{i,j}\int
dx_{1}\,dx_{2}\,q_{i}(x_{1},\mu)q_{j}(x_{2},\mu)\,\hat{\sigma}(ij\to F+X)$
(57)

A consistent result is obtained with the partonic cross section computed at a
given fixed perturbative order and subtracted of the initial state collinear
divergent factors, convoluted with the structure functions at the same order.
The latter account, via DGLAP evolution, for the resummed contribution of the
initial state collinear logarithms. For example, we consider in QED the case
of a cross section computed with the full NLO-EW results at parton level
$\hat{\sigma}$ and convoluted with the QED LL solution of the DGLAP equations.
The result has NLO-EW accuracy for the total cross section, it includes to all
orders terms of ${\cal O}(\alpha L)$, but only partially accounts for the
second-order terms proportional to $\alpha^{2}L$: the fixed-order missing
contributions stem from the non-factorisable two-loop corrections. A full
NNLO-EW calculation, convoluted with the corresponding second order structure
functions, will exactly include to all orders terms of ${\cal
O}(\alpha^{2}L)$, but only partially subleading fixed-order terms proportional
to $\alpha^{3}L$. The inclusion of terms proportional to logarithms due to
soft emissions in this formulation is restricted to those present in the
fixed-order expression. In addition, the DGLAP equations obviously include to
all order the contributions of the initial-state soft and collinear phase-
space region.

A different approach to describe multiple radiation is given by the so called
Parton Shower. The latter provides an algorithmic implementation of the
exponentiation of the soft LL contributions. This description can be matched,
avoiding double countings, with the exact matrix elements. The latter improve
the total cross section prediction and the description of hard, large-
transverse-momentum radiation.

$d\sigma=F_{SV}\,\Pi(Q^{2},\varepsilon)\times\sum_{n=0}^{\infty}\frac{1}{n!}\left(\Pi_{i=0}^{n}F_{H,i}\right)|{\cal
M}_{n,LL}|^{2}d\Phi_{n}$ (58)

One possible formulation is given in equation 58, as implemented in the event
generator HORACE CarloniCalame:2007cd . The LL Parton Shower approximation is
corrected at ${\cal O}(\alpha)$ by the infrared and collinear finite factors
$F_{SV}$ and $F_{H,i}$: we have the Sudakov form factor
$\Pi(Q^{2},\varepsilon)$ accounting for the exponentiation of the soft
emissions; we have the real emission matrix elements ${\cal M}_{n,LL}$ written
in terms of the product of $n$ eikonal currents; the factor $F_{SV}$ takes
into account the finite part of soft emissions and virtual corrections, while
the exact matrix element is applied to each hard large-transverse-momentum
emission via the $F_{H,i}$ correction. The factorised formulation guarantees
the correct semiclassical limit, but also introduces universal higher-order
terms, which can partially account, for instance, for a subset of NLL
collinear terms and in particular for part of the subset proportional to
$\alpha^{2}L$, with a significant phenomenological improvement. Extending this
kind of matching beyond NLO is a current topic of investigation, with NNLO
results matched to a LL Parton Shower Monni:2019whf . Also in this case, an
explicit improvement is achieved for the total cross section and for the
observables sensitive to hard large-transverse-momentum radiation, whereas the
Parton Shower resummation includes only LL terms.

The YFS approach Yennie:1961ad  starts from the resummation of the first-order
logarithms in the soft approximation. In the literature, considering the
scattering of two fermions into two fermions, not only the logarithms due to
initial state radiation, but also those stemming from the initial-final
interference and the final-state emissions, have been exponentiated and
matched with fixed order expressions up to second order Jadach:1999vf ;
Jadach:2000ir . The YFS fully exclusive formulation is based on the
factorisation of soft emissions already at the amplitude level. Thanks to the
factorisation of one-loop soft-divergent virtual corrections, it is possible
to obtain a soft-finite one-loop correction factor $Y$ which exponentiates.
The resummed fully differential expression of the cross section reads:

$\displaystyle d\sigma$ $\displaystyle=$ $\displaystyle
d\sigma_{0}\,\exp(Y)\,\exp\left[\frac{dk\,d\cos\theta\,d\phi}{(2\pi)^{2}}k\,\Theta(k-k_{min})\sum_{\varepsilon}\alpha\left(\frac{\varepsilon(k)\cdot
p_{2}}{k\cdot p_{2}}-\frac{\varepsilon(k)\cdot p_{1}}{k\cdot
p_{1}}\right)^{2}\right]$ (59) $\displaystyle Y$ $\displaystyle=$
$\displaystyle\beta\log\frac{k_{min}}{E}+\frac{1}{4}\beta+\frac{\alpha}{\pi}\left(\frac{\pi^{2}}{3}-\frac{1}{2}\right)\quad\quad\beta=\frac{2\alpha}{\pi}\left[\log\left(\frac{s}{m_{e}^{2}}\right)-1\right]\,.$

The second exponential of the first equation is a compact notation for the sum
over all possible additional photon multiplicities $n$ of the product of
eikonal emission factors, valid in the soft limit; the presence of an
integration measure is the symbolic representation of the phase-space factors
for $n$ photons, with arbitrary $n$. The description of photons, with an
energy larger than the cut-off $k_{min}$, is fully differential. We read in
the exponent $Y$ the presence of the first-order double and single logarithmic
terms due to collinear and soft-collinear contributions, evaluated in soft
approximation, integrated in the photon energy up to the cut-off $k_{min}$.
Given the soft limit approximation, for the electromagnetic coupling the fine
structure constant $\alpha$ is a customary choice. The inclusion of higher-
order contributions like the description of additional light fermion pairs
could be simulated with a running coupling, but in this case care should be
paid to avoid double countings with the rest of the hard correction factors.
The matching with the exact ${\cal O}(\alpha)$ results yields, following the
original YFS algorithm, additional corrective factors $\beta_{n}$ called
residuals, free of any soft or collinear divergence, expressing the impact of
the exact matrix element on the $n$-th emission. A modification of the
residuals, without spoiling the formal accuracy of the calculation, allows to
improve the resummed and matched YFS formulation to include also the missing
$\alpha^{2}L$ terms. In summary, the exact exponentiation of the one-loop soft
logarithms is matched to fixed first- and partial second-order results,
covering up to second order all the relevant infrared divergent contributions.

###  11.4 Discussion and conclusions

The estimated experimental precision of future $e^{+}e^{-}$ colliders forces
us to investigate the question of the best formulation of a calculation that
includes fixed- and all-orders results in $e^{+}e^{-}$ collisions, up to
three-loop level. The various approaches can be described with respect to $a)$
the formal inclusion of the various classes of enhanced contributions and $b)$
the possibility of extending the resummation beyond the exponentiation of the
first-order divergent terms. We eventually consider the dependence on the
observable under study.

Any code based on a Parton Shower relies on the algorithmic exponentiation of
the first-order soft emissions, achieving LL accuracy; the angular dependence
of the radiation depends on the specific implementations. After the matching
with NLO matrix elements, it is possible that a subset of the NLL collinear
contributions is correctly included, but the full set of terms proportional to
$\alpha^{2}L$ is under control only with matching at NNLO level. It is
interesting to observe that, in the Parton Shower matching, different recipes,
still preserving the formal accuracy of the results, can lead to significant
numerical differences. This remark shows the role of multiple radiation and
the impact of its “handling”, as ruled by the matching recipe. Important
examples are known from the LHC studies and it would be interesting to check
this point for an $e^{+}e^{-}$ collider, in view of the very high-precision
goal. It is not obvis that an improvement of the accuracy of the resummation
part, going beyond the LL level, can be achieved for a general purpose tool.

The YFS formalism is based on the exclusive exponentiation of first-order
contributions in the soft limit. The inclusion of all the LL terms is achieved
at amplitude level and the full set of $\alpha^{2}L$ collinear contributions
can be included. The full resummation of NLL collinear logarithms is not
available. In the public implementations of the YFS formalism, all the
radiating particles are supposed to be massive, thus separating the issue of
the collinear divergences from the discussion of the soft contributions. While
this is a viable approach in the description of $e^{+}e^{-}$ collisions, it
would introduce a dependence on unphysical quark masses in the description of
quark-initiated processes relevant for the LHC.

In the pure structure functions approach the formulation of the physical cross
section is purely dictated by the factorisation of the initial state collinear
singularities. As a positive outcome, the extension to two- and three-loop
level is conceptually straightforward. A technical effort to translate the
N3LO splitting functions to the QED case and to compute the boundary
conditions to the DGLAP equations is needed, and is feasible. The
contributions which are resummed in this approach are the collinear
logarithms, going up to NNLL, in the case a three-loop result were available.
In the pure structure function approach, the soft logarithms taken into
account are those appearing from the matrix elements, and those of the soft-
collinear phase-space corner.

At a future $e^{+}e^{-}$ collider, two extremely important energy regions in
the precision physics program are the $Z$-boson resonance and the $W^{+}W^{-}$
production threshold. The resummation of the soft logarithms clearly plays a
role in the determination of the precise position of these two points and in
the description of the associated lineshapes: these logarithms in fact control
the exact amount of energy available in the center-of-mass of the hard
scattering process. On the other hand, the measurement takes place in a
fiducial volume, with acceptance cuts also in the plane transverse to the
beam, so that a formalism based on the separation between the longitudinal and
transverse components can be a natural framework for precise predictions.

Exploring the impact of the different resummations, with the currently
available tools, can set the stage to the evaluation of higher-order
corrections and the preparation of an adequate matching formalism.

##  12 Factorization beyond leading power

_Leonardo Vernazza_

###  12.1 Factorization at next-to-leading power: state of the art

Understanding gauge theory scattering amplitudes in the limit where external
radiation becomes soft poses interesting challenges from a conceptual quantum
field theory point of view and has important applications in the phenomenology
of precision particle physics. Let us consider $n$-particle scattering
processes in QED. Given the emission of an additional soft photon with
momentum $k$, the scattering amplitude ${\cal M}_{n+1}$ can be expressed as a
power expansion in the energy $E=k^{0}$ of the photon,

${\cal M}_{n+1}={\cal M}_{n+1}^{\rm LP}+{\cal M}_{n+1}^{\rm
NLP}+\mathcal{O}(E)\,,$ (60)

where the _leading power_ (LP) term has scaling ${\cal M}_{n+1}^{\rm LP}\sim
1/E$, the _next-to-leading power_ (NLP) contribution is of order ${\cal
M}_{n+1}^{\rm NLP}\sim E^{0}$, and so on.

Our interest for the expansion in eq. (60) stems from the fact that the
coefficients in the series _factorize_ , i.e., they can be expressed in terms
of simpler, universal objects, which relate the radiative amplitude to the
non-radiative or _elastic_ amplitude ${\cal M}_{n}$. It is well known that the
LP term in eq. (60) has the universal form Yennie:1961ad ; Grammer:1973db

${\cal M}_{n+1}^{\rm LP}(\\{p_{i}\\},k)=S_{n}\,{\cal
M}_{n}(\\{p_{i}\\})\,,\qquad\qquad
S_{n}=e\sum_{i=1}^{n}q_{i}\,\frac{p_{i}^{\mu}\varepsilon_{\mu}(k)}{p_{i}\cdot
k}\,,$ (61)

where $p_{i}^{\mu}$ and $q_{i}$ represent the momentum and electric charge of
the $i$-th hard particle, and $\varepsilon_{\mu}(k)$ is the polarisation
vector of the soft photon. The soft function $S_{n}$ describes a set of
eikonal interactions between the external particles and the soft photon, which
are sensitive only to the direction and charge of the emitting particle. In
general, $S_{n}$ can be calculated as the vacuum expectation value of a set of
Wilson lines, one for each hard emitting particle.

Relations such as the one in eq. (61) are also known as _soft_ theorems. Their
interest is not just purely theoretical, but also relevant for phenomenology.
The $1/E$ singularity in the soft limit enhances soft radiation in scattering
processes. Measurements that are sensitive to soft radiation involve a small
scale, and the corresponding cross section contains large logarithms of the
ratio of this small scale and the scale of the hard scattering. Such large
logarithms potentially spoil the convergence of the expansion in the coupling
$e$, a problem that can be addressed by resummation. The development of
theorems such as the one in eq. (61) led to the proofs of factorization
Bodwin:1984hc ; Collins:1985ue ; Collins:1988ig , but also constitutes the
first step towards resummation, as it allows one to decompose a multi-scale
amplitude (or cross section) into the product of simpler single-scale
functions.

The physical interpretation of eq. (61) rests upon the long wavelength of soft
radiation not being able to resolve the hard scattering. However, at each
subsequent power in (60) more is revealed, and the factorization structure
becomes more involved. The NLP term in eq. (60) is still under much
investigation. Emitted (next-to-)soft radiation becomes sensitive to the spin
of the hard particles, and starts to reveal details of the internal structure
of the hard interaction. Compared to eq. (61), the factorization of radiative
amplitudes at NLP needs to take into account the following features:

  * •

Emission of soft radiation from the hard partons beyond the eikonal
approximation, for instance involving chromo-magnetic interactions between the
hard particle and the emitted soft photon. This can be taken into account by
introducing generalizations of the soft function $S_{n}$ in eq. (61)
Laenen:2010uz .

  * •

Emission of soft radiation from the non-radiative amplitude. This is one of
the mechanism which probes the structure of the hard interaction, and was
understood a long time ago in papers by Low, Burnett and Kroll Low:1958sn ;
Burnett:1967km : in the case of massive emitting particles, the structure of
the NLP term is dictated by gauge invariance, by means of Ward identities.
This early formulation, now known as the “LBK” theorem, was proven
DelDuca:1990gz  to hold only in the region $k^{0}\ll m^{2}/Q$, with $Q$ the
centre of mass energy.

  * •

For $m^{2}/Q<k^{0}<m$, the LBK theorem must be extended to account for NLP
contributions arising from soft photons emitted from loops in which the
exchanged virtual particles have momenta collinear to the external particles
(i.e. having a small virtuality, while retaining momentum components which are
large compared to the soft radiation). Such type of contributions are
identified as _radiative jets_. The first of such configurations was
investigated for Drell-Yan in DelDuca:1990gz .

Most of the effort in the past few years has been devoted to the derivation of
factorization theorems able to describe consistently such soft and collinear
radiation into well defined field-theoretical matrix elements. This has been
done following mainly two approaches. One, which we will refer to as
_diagrammatic_ , aims at describing such matrix elements in terms of fields,
currents and Wilson lines in QED. The second approach is based on Soft-
Collinear Effective Theory (SCET) Bauer:2000ew ; Bauer:2000yr ; Bauer:2001ct ;
Bauer:2001yt ; Beneke:2002ph .

SCET describes the soft and collinear limits as separate degrees of freedom,
each with their own Lagrangian. The elastic amplitude ${\cal M}_{n}$ in (61)
is encoded in terms of effective $n$-jet operators and their short-distance
coefficients, which capture the contribution from hard loops. At LP, soft
emissions from hard particles can be described by Wilson lines, as in the
diagrammatic picture. Beyond LP, soft emissions originate from time-ordered
non-local operators made out of soft and collinear fields, where the power
suppression follows either from additional insertions of the power-suppressed
soft and collinear Lagrangian, or from subleading operators describing the
hard scattering Larkoski:2014bxa ; Moult:2019mog ; Beneke:2019oqx . SCET
provides a systematic approach, as each operator and Lagrangian term have by
construction a definite power counting. When all operators consistent with
symmetries are included, this completeness ensures that the resulting
factorization is valid to all orders in the coupling constant.

Both approaches present advantages and disadvantages. From a technical point
of view, the calculation of soft and collinear matrix elements within SCET is
designed such as to directly reproduce calculations obtained within the method
of expansion by momentum regions Beneke:1997zp , and can be quite involved
already at one-loop (cf. Beneke:2019oqx ; Broggio:2021fnr ). In this respect,
the calculation of matrix elements within the diagrammatic approach is simpler
(cf. Bonocore:2015esa ; Bonocore:2016awd ; DelDuca:2017twk ), although the
lack of tools such as multipole expansion results into overlaps between soft
and collinear matrix elements, leading to complicated nested subtractions
beyond one loop.

More relevant for applications to the theory of resummation is the fact that
soft and collinear matrix elements involve convolutions, which are well
defined in dimensional regularisation, but often divergent in $d=4$. This
poses a challenge for a standard EFT approach, where one first renormalizes
each function in the factorization theorem to derive the renormalization group
equations needed for resummation, causing these convolution integrals to
become divergent. The problem needs to be investigated case by case, and so
far only one example is known Liu:2019oav ; Liu:2020tzd , where convergence of
the convolution integrals can be achieved after a re-factorization process,
which is non-trivial to be proven in general (see also Moult:2019uhz ;
Beneke:2020ibj ). In this respect, the diagrammatic approach could provide an
easier path to resummation by exploiting exponentiation properties of soft
radiation and the replica trick Laenen:2008gt ; Gardi:2010rn , which can be
carried out within dimensional regularization (cf. Bahjat-Abbas:2019fqa ;
vanBeekveld:2021mxn ).

The classification of soft and collinear matrix elements contributing beyond
LP is rather straightforward in SCET, once the power-counting and field
content of the theory is fixed Larkoski:2014bxa ; Moult:2019mog ;
Beneke:2019oqx . Within the diagrammatic approach, the classification of all
possible soft and collinear matrix elements has been derived for the Yukawa
theory Gervais:2017yxv  and more recently for QED Laenen:2020nrt  (see also
Bonocore:2021cbv ). We will illustrate such derivation in the next section.

###  12.2 Factorization of QED amplitudes: diagrammatic approach

Understanding the factorization structure of ${\cal M}_{n+1}^{\rm NLP}$
requires in turn to determine the factorization of the elastic amplitude
${\cal M}_{n}$. This can be easily seen by recalling the LBK theorem and its
failure to include virtual collinear configurations: one starts by separating
the radiative amplitude in two contributions: one in which the radiation is
emitted from the external legs, plus another in which the radiation is emitted
from a particle within the hard scattering kernel:

${\cal M}_{n+1}={\cal M}_{n+1}^{\rm ext}+{\cal M}_{n+1}^{\rm int}\,.$ (62)

The amplitude ${\cal M}_{n+1}^{\rm int}$ can be obtained by means of the Ward
identity from ${\cal M}_{n+1}^{\rm ext}$. Focusing on the latter, consider the
emission from an outgoing fermion $i$: ${\cal M}_{n+1}^{\rm ext}$ takes the
form

${\cal M}_{n+1}^{\rm
ext}=\bar{u}(p_{i})(ieq_{i}\gamma^{\mu})\,\frac{i(\not{p}_{i}+\not{k}+m)}{(p_{i}+k)^{2}-m^{2}}\,{\cal
M}_{n}(p_{1},\ldots,p_{i}+k,\ldots p_{n})\,,$ (63)

where ${\cal M}_{n}$ represents the elastic amplitude (stripped off the spinor
$\bar{u}(p_{i})$). Within the LBK theorem, one expands the amplitude in the
soft momentum $k$. As discussed, this expansion gives correct results only in
the regime $p_{i}\cdot k/Q\ll m^{2}/Q$. But for parametrically small masses
$m^{2}/Q<p_{i}\cdot k/Q<m$, naively expanding the elastic amplitude in the
soft momentum $k$ misses a contribution in which the soft photon is emitted
from internal particles collinear to the external leg, which is thus included
neither in ${\cal M}_{n+1}^{\rm ext}$, nor in ${\cal M}_{n+1}^{\rm int}$. Eq.
(63) tells us that in order to understand the factorization properties of the
radiative amplitude ${\cal M}_{n+1}$ we also need to obtain the correct
factorization structure of the elastic amplitude ${\cal M}_{n}$, in presence
of a small off-shellness $p_{i}\cdot k\sim m^{2}\ll Q^{2}$. At leading power
the factorization structure is known, see for instance Collins:1989bt ;
Dixon:2008gr , and it takes the following schematic form:

${\cal M}_{n}=H_{n}\times S_{n}\times\prod_{i=1}^{n}\,\frac{J_{i}}{{\cal
J}_{i}}\,.$ (64)

In this equation the jet and soft functions $J_{j}$ and $S$ describe long-
distance collinear and soft virtual radiation in ${\cal M}_{n}$. These
functions are universal, i.e. they depend only on the colour and spin quantum
numbers of the external states, and determine also the structure of collinear
and soft singularities of the elastic amplitude. Note that one must divide
each jet by its eikonal counterpart ${\cal J}_{i}$, to avoid the double
counting of soft and collinear divergences.

The corresponding factorization structure at NLP for QED has been determined
in Laenen:2020nrt . In short, the task consists in obtaining a classification
of the jet-like structures, consisting of virtual radiation collinear to any
of the $n$ external hard particles, contributing at subleading power. This is
obtained by power counting the pinch surfaces that underlie the collinear (and
soft) contributions one wishes to describe in terms of jet (and soft)
functions, for a general QED scattering amplitude. The pinch surfaces are the
solutions of the Landau equations Landau:1959fi  and are represented by
reduced diagrams in the Coleman-Norton picture Coleman:1965xm . In these
diagrams, all off-shell lines are shrunk to a point, while the on-shell lines
are kept and may be organised according to the nature of the singularity they
embody, be it soft or collinear. This gives the general reduced diagram of
fig. 2, in which one distinguishes a soft “blob" containing all lines carrying
solely soft momentum, $n$ jets $J_{i}$ comprised of lines with momenta
collinear to the respective external parton and lastly, a hard blob $H$
collecting all contracted, off-shell lines.

Figure 2: The reduced diagram for a vector boson induced QED process with $n$
hard particles in the final state. Ellipses denote the presence of an
arbitrary number of photons/(anti-)fermions.

The power counting is carried in light-cone coordinates: a generic momentum
$k$ is decomposed as $k^{\mu}=(k^{+},\vec{k}_{\perp},k^{-})$, where the light-
cone components $k^{\pm}$ lie along the direction of two light-like vectors
$n_{i}$ and $\bar{n}_{i}$, normalized such that $n_{i}\cdot\bar{n}_{i}=1$,
with $n_{i}$ along the direction of $p_{i}$. Soft and collinear momenta are
taken to scale as

$\displaystyle\text{Soft:}\hskip 20.0ptk^{\mu}\sim
Q\left(\lambda^{2},\lambda^{2},\lambda^{2}\right)\,,\qquad\quad\text{Collinear:}\hskip
20.0ptk^{\mu}\sim Q\left(1,\lambda,\lambda^{2}\right)\,.$ (65)

One focuses on virtual corrections to a hard scattering configuration, for
which all invariants $s_{ij}=(p_{i}+p_{j})^{2}\sim Q^{2}$ involving external
momenta are large compared to the energy of the radiated soft photon in ${\cal
M}_{n+1}$. Requiring the soft momentum to be of order $\lambda^{2}$ rather
than $\lambda$ guarantees that the photon is soft with respect to all
particles in the elastic amplitude. Within this power counting, a NLP quantity
is suppressed by two powers in $\lambda$ with respect to the leading power
contribution.

Using the momentum scaling in eq. (65), one first derives the superficial
degree of divergence of a particular reduced diagram $\mathcal{G}$ contained
in fig. 2, which is simply the $\lambda$-scaling of this diagram,
$\mathcal{G}\sim\lambda^{\gamma_{\mathcal{G}}}$. In practice,
$\gamma_{\mathcal{G}}$ can be expressed as function of the number of fermion
and photon connections between the hard, soft and collinear subgraphs and, in
presence of massive fermions, on the internal structure of the soft subgraph.
Such a formula expresses, at any perturbative order, which pinch surfaces
contribute up to NLP, which allows one to set up a consistent and complete NLP
factorization framework for QED. The approach is analogous to the one taken
for Yukawa theory in Gervais:2017yxv , and it is a direct application of the
well-known method first developed for deriving factorisation of the elastic
amplitude at LP in Ref. Sterman:1978bi . In general, one has that
$\gamma_{\mathcal{G}}\geq 0$, independent of the number of hard particles in
the final state. The $\gamma_{\mathcal{G}}=0$ diagrams contain at most
logarithmic singularities, while the $\gamma_{\mathcal{G}}>0$ are finite and
give a vanishing contribution in the $\lambda\rightarrow 0$ limit. For small
but non-zero values of $\lambda$, the $\gamma_{\mathcal{G}}=0$ diagrams form
LP contributions, with the $\gamma_{\mathcal{G}}>0$ diagrams acting as power
corrections.

(a)  $\gamma=0$

(b)  $\gamma=1$

Figure 3: Reduced diagrams for the process $\gamma\to f\bar{f}$. (a)
$\gamma=0$ (b) $\gamma=1$, where a similar configuration exists with the
double jet connection on the upper leg instead.

After some work, one finds that the relevant diagrams contributing up to NLP
are represented in figures 3 and 4, where diagram (a) in fig. 3 represents the
LP contribution, diagram (b) contributes starting at $\sqrt{\rm N}$LP, and the
diagrams in fig. 4 contribute at NLP. We see that they involve collinear jets
with up to three collinear partons in the same collinear sector, and
interactions between the collinear and the soft sector mediated by a soft
quark, or a soft quark-antiquark pair in addition to soft gluons.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: Reduced diagrams for $\gamma=2$ and $m=0$. For $m\sim\lambda Q$, (e)
and (f) contribute beyond NLP. (a) Two fermions and a photon connecting a
collinear blob to the hard blob. (b) Triple fermion connection. (c) Two
collinear blobs connected to the hard blob by a fermion and photon each. (d) A
single photon connecting the hard and soft blobs. (e) Two fermions connecting
a collinear blob to the soft blob. (f) Two collinear blobs connected to the
soft blob by a fermion each.

Restricting to the hard-collinear sector, up to NLP the content of figures 3
and 4 translates into the factorization formula

$\displaystyle\mathcal{M}_{\rm coll}^{\mathrm{NLP}}$
$\displaystyle=\bigg{(}\prod_{i=1}^{n}J^{i}_{(f)}\bigg{)}H\,S\,+\sum_{i=1}^{n}\bigg{(}\prod_{j\neq
i}J_{(f)}^{j}\bigg{)}\Big{[}J^{i}_{(f\gamma)}\otimes
H^{i}_{(f\gamma)}+J^{i}_{(f\partial\gamma)}\otimes
H^{i}_{(f\partial\gamma)}\Big{]}\,S$
$\displaystyle+\,\sum_{i=1}^{n}\bigg{(}\prod_{j\neq
i}J_{(f)}^{j}\bigg{)}J^{i}_{(f\gamma\gamma)}\otimes
H^{i}_{(f\gamma\gamma)}\,S+\sum_{i=1}^{n}\bigg{(}\prod_{j\neq
i}J_{(f)}^{j}\bigg{)}J^{i}_{(f\\!f\\!f)}\otimes H^{i}_{(f\\!f\\!f)}\,S$
$\displaystyle+\sum_{1\leq i\leq j\leq n}\bigg{(}\prod_{k\neq
i,j}J_{(f)}^{k}\bigg{)}J^{i}_{(f\gamma)}J^{j}_{(f\gamma)}\otimes
H^{ij}_{(f\gamma)(f\gamma)}\,S\,,$ (66)

where the tensor product $\otimes$ denotes convolution and a contraction of
spinor indices, we suppress the arguments of the functions and introduce the
indices $i,j$, labelling the collinear sectors. For simplicity, we assume that
the potential overlap between the soft and collinear regions has already been
accounted for in a redefinition of the jet functions.

The first term represents the LP contribution from eq. (64). The second term
describes the effect of fig. 3b and starts contributing at order $\lambda$.
This implies that at order $\lambda^{2}$ we may expect a dependence of the
hard function on the perpendicular momentum component of the collinear photon
emerging from it, which can be re-expressed in terms of the
$H_{(f\partial\gamma)}^{i}$ function. The third and fourth terms describe
diagrams $(\rm a)$ and $(\rm b)$ in fig. 4, while diagram $(\rm c)$
corresponds to the last term. These contributions, as well as the
$f\partial\gamma$-term, are strictly $\mathcal{O}(\lambda^{2})$, which implies
that the soft function appearing in those terms is the same appearing at LP.
Eq. (12.2) has been obtained both for the case of massless and massive
fermions, with a small parametric mass $m\sim\lambda Q$, and we refer to
Laenen:2020nrt  for a discussion highlighting differences between the two
cases. The functions in eq. (12.2) can be defined as expectation values of
fields, currents and Wilson lines in QED, in analogy to the LP jet and soft
functions.

Let us conclude by noting that approaches based on SCET Larkoski:2014bxa ;
Beneke:2019oqx  contain similar functions describing amplitudes at the next-
to-leading power. The $J_{f\gamma}$, $J_{f\partial\gamma}$, and
$J_{f\\!f\\!f}$ jets in eq. (12.2) are related to matrix elements of the
operators $J^{B1}$, $J^{B2}$, $J^{C1}$ in the position-space SCET in
Beneke:2019oqx , and to matrix elements of the part of $N$-jet operators
${\cal O}_{N}^{(1,X)}$, ${\cal O}_{N}^{(2,X\delta)}$, ${\cal
O}_{N}^{(2,X^{2})}$ corresponding to a specific collinear direction in the
label formulation of SCET Larkoski:2014bxa .

###  12.3 Outlook

The study of scattering amplitudes in the limit where external radiation
becomes soft is relevant both for obtaining a deeper understanding of quantum
field theory and for phenomenology. In presence of soft radiation, scattering
amplitudes are naturally written as a power expansion in the energy of the
emitted radiation. The coefficients of this expansion factorize. Determining
the factorization structure constitutes the first step toward the resummation
of large logarithms in physical observables which are sensitive to the energy
scale of soft radiation, but it useful for other applications, too. For
instance, determining the factorization properties of soft and collinear
radiation can be useful to derive analytic expression for amplitudes in soft
and collinear limits, which can be used to stabilize and speed up numerical
evaluations of amplitudes in such limits Engel:2021ccn . In this context,
current efforts are devoted to understand the factorization of amplitudes at
next-to-leading power. This can be done following a diagrammatic approach, or
by means of effective field theories.

In this review we have discussed recent developments within the diagrammatic
approach applied to QED, which will be useful for the phenomenological
applications at $e^{+}e^{-}$ colliders. Among various contributions, the
description of soft radiation beyond leading power requires to be able to take
into account the emission of soft radiation from clusters of collinear virtual
particles, which flows in or out from the hard scattering process, and are
known as _radiative jets_. Such contributions are absent at leading power,
where one needs to take into account only the eikonal emission of soft
photons/gluons from the hard particles of the process. The first radiative
jets was identified long ago DelDuca:1990gz , and studied at one loop in
Drell-Yan in Bonocore:2015esa ; Bonocore:2016awd . More recently, a
classification of all possible jets contributing at NLP in QED has been
derived in Laenen:2020nrt , and we have sketched this derivation in section
12.2.

This classification constitutes also the first step toward the development of
a theory for the resummation of NLP (threshold) logarithms. Within a
diagrammatic approach, one envisage to study the combinatorial properties of
soft photons/quarks emitted from the various functions in the factorization
formula by means of tools such as the replica trick, to understand the
exponentiation structure of soft radiation at cross section level.

###### Acknowledgements.

This work is supported by Fellini Fellowship for Innovation at INFN, funded by
the European Union’s Horizon 2020 research programme under the Marie
Skłodowska-Curie Cofund Action, grant agreement no. 754496.

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