aainstitutetext: Rudolf Peierls Centre for Theoretical Physics, Clarendon
Laboratory, Parks Road, University of Oxford, Oxford OX1 3PU,
UKbbinstitutetext: Institute for Particle Physics Phenomenology, Durham
University, Durham DH1 3LE, United Kingdomccinstitutetext: Institute of High
Energy Physics, Chinese Academy of Sciences, Beijing 100049,
Chinaddinstitutetext: University of Chinese Academy of Sciences, Beijing
100049, Chinaeeinstitutetext: Center of Advanced Quantum Studies, Department
of Physics, Beijing Normal University, Beijing 100875, Chinaffinstitutetext:
Center for High Energy Physics, Peking University, Beijing 100871,
Chinagginstitutetext: Theoretical Division, MS B283, Los Alamos National
Laboratory, Los Alamos, NM 87545, USAhhinstitutetext: Department of Physics,
University of Alberta, Edmonton AB T6G 2J1, Canadaiiinstitutetext: C.N. Yang
Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY
11794, USAjjinstitutetext: Department of Physics and Astronomy, Stony Brook
University, Stony Brook, NY 11794, USAkkinstitutetext: Department of Physics,
Brookhaven National Laboratory, Upton, N.Y., 11973, USAllinstitutetext: INFN,
Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italymminstitutetext:
Nikhef, Science Park 105, NL-1098 XG Amsterdam, The
Netherlandsnninstitutetext: SLAC National Accelerator Laboratory, Stanford
University, Stanford, CA 94039, USAooinstitutetext: School of Physics,
Shandong University, Jinan, Shandong 250100, China

# Snowmass 2021 White Paper: Resummation for future colliders

Melissa van Beekveld b Sebastian Jaskiewicz c,d Tao Liu e,f Xiaohui Liu g Duff
Neill h Alexander Penin i,j Felix Ringer k Robert Szafron l,m Leonardo
Vernazza n Gherardo Vita o Jian Wang<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Resummation techniques are essential for high-precision phenomenology at
current and future high-energy collider experiments. Perturbative computations
of cross sections often suffer from large logarithmic corrections, which must
be resummed to all orders to restore the reliability of predictions from first
principles. The precise understanding of the all-order structure of field
theories allows for fundamental tests of the Standard Model and new physics
searches. In this white paper, we review recent progress in modern resummation
techniques and outline future directions. In particular, we focus on the
resummation beyond leading power, the joint resummation of different classes
of logarithms relevant for jets and their substructure, small-$x$ resummation
in the high-energy regime and the QCD fragmentation process in the
small-$z_{h}$ limit.

††preprint:  IPPP/22/11, SLAC-PUB-17664, YITP-SB-2022-06

Submitted to the Proceedings of the US Community Study

on the Future of Particle Physics (Snowmass 2021)

##  1 Introduction

Tests of the Standard Model and New Physics searches at the Large Hadron
Collider (LHC), Electron-Ion Collider (EIC) and other future colliders require
extremely precise theoretical predictions. Correctly accounting for large
quantum corrections will be indispensable for the High-Luminosity phase of the
LHC, especially since the direct searches for new particles have not resulted
in a discovery so far. Precise measurements of the Higgs boson and top quark
properties, or tests of the Standard Model’s flavor structure, rely on an
excellent theoretical control over the QCD-induced corrections.

The accurate theoretical description of high-energy processes at colliders is
among the most successful applications of perturbative QCD. The relevant QCD
corrections are typically computed as an expansion in the strong coupling
constant $\alpha_{s}$. Currently, state-of-the-art predictions include
corrections up to the third order in $\alpha_{s}$, with most of the processes
being known only up to the next-to-next-to-leading order (NNLO).

A typical process in QCD depends on various energy scales. When the scales are
widely separated, the presence of the scale hierarchy manifests itself through
logarithmically enhanced radiative corrections, which require an all-order
resummation of the series in the strong coupling constant. One of the great
simplifications of perturbative analysis at high energy is that the
contributions suppressed by a power of a ratio of an infrared scale of the
process to its center-of-mass energy are small and can be neglected in the
first approximation, reducing the number of scales and hence the complexity of
the calculations. Thus, the resummation techniques typically target the
leading power (LP) terms in the power expansion. Before we review in more
detail the recent progress in the most challenging fields, such as subleading
power resummation, small-$x$ resummation, fragmentation and small-$z_{h}$
resummation, and the physics of jets and their substructure, let us briefly
mention two of the well-established application of resummation techniques,
which played a crucial role in the development of this research field.

Threshold resummation is one of the flagship applications of factorization and
resummation for collider physics Parisi:1979xd ; Curci:1979am ; Sterman:1986aj
; Catani:1989ne ; Catani:1990rp . Typically, when the phase space available
for radiation is limited, the corrections are dominated by soft gluons.
Therefore, threshold resummation is relevant for the production of heavy
states. In the context of LHC physics, the leading power threshold resummation
is well understood. It has been applied to many processes, such as the
production of the electroweak gauge and Higgs bosons and top quarks as well as
production of new heavy particles, see e.g. Refs. Moch:2005ky ; Laenen:2005uz
; Idilbi:2005ni ; Becher:2007ty ; Kulesza:2008jb ; Beneke:2011mq ;
Bonvini:2012az ; Bonvini:2014joa ; Hinderer:2018nkb .

A second important class of problems involves transverse momentum resummation.
Substantial progress has been achieved in recent years, with the state-of-the-
art results reaching next-to-next-to-next-to leading logarithmic (N3LL)
accuracy combined with NNLO and even N3LO fixed order computations, see e.g.
Li:2016ctv ; Bizon:2017rah ; Coradeschi:2017zzw ; Chen:2018pzu ;
Camarda:2019zyx ; Bacchetta:2019sam ; Luo:2019szz ; Ebert:2020yqt ;
Kallweit:2020gva ; Wiesemann:2020gbm ; Ebert:2020dfc ; Luo:2020epw ;
Becher:2020ugp ; Ebert:2020qef ; Ebert:2020sfi ; Ju:2021lah ; Neumann:2021zkb
; Billis:2021ecs ; Chen:2022cgv . These results are very important components
for precision tests of the Standard Model at the LHC.

The need for a precise understanding of the all-order structure of QCD led to
the development of new theoretical techniques and methods. Today the field of
resummation is exploring new topics crucial for phenomenology and our
understanding of gauge theories. This paper reviews the recent progress in the
theoretical methods of resummation and their relevance for collider
phenomenology. We will discuss the most challenging problems currently under
investigation, which are addressed with soft-collinear effective field theory
(SCET) Bauer:2000yr ; Bauer:2001yt ; Beneke:2002ph ; Beneke:2002ni  and
directly in QCD. In section 2, we discuss recent progress in understanding the
all-order structure of subleading power terms using the traditional
diagrammatic approach and SCET. Section 3 provides an overview of jet physics
and jet substructure observables. Here the focus is in particular on the
simultaneous resummation of several classes of large logarithmic corrections.
This aspect is particularly relevant for jet substructure observables, where
multiple energy scales appear. Next, in section 4, we discuss recent progress
in understanding gluon saturation, which is relevant in the high-energy regime
where the initial-state gluon momentum fraction is small. Section 5 focuses on
the QCD fragmentation process in the limit where the energy fraction of the
identified hadrons becomes small relative to the hard scale of the process.
Finally, we conclude in section 6.

##  2 Resummation beyond leading power

In the leading-power approximation the structure of the logarithmic
corrections is in general well understood and the relevant resummation
techniques are well elaborated. However, in many crucial cases the continually
improving accuracy of the experimental measurements requires the inclusion of
power suppressed terms in the theoretical estimates. As a result, much effort
is currently invested into the study of a diverse class of power corrections,
see Penin:2014msa ; Melnikov:2016emg ; Penin:2016wiw ; Liu:2017axv ;
Liu:2017vkm ; Boughezal:2018mvf ; Bruser:2018jnc ; Moult:2018jjd ;
Ebert:2018lzn ; Liu:2018czl ; Beneke:2018gvs ; Engel:2018fsb ; Ebert:2018gsn ;
Bahjat-Abbas:2019fqa ; Penin:2019xql ; Beneke:2019mua ; Beneke:2019oqx ;
Anastasiou:2020vkr ; Liu:2020wbn ; Broggio:2021fnr ; Liu:2021chn  and
references therein. Incorporating logarithmically enhanced power-suppressed
terms can significantly increase the accuracy and extend the region where the
perturbative calculation is applicable. Besides the phenomenological
importance, the power-suppressed contributions are very interesting from the
general effective field theory point of view since the structure of the
factorization and renormalization group evolution in this case becomes highly
nontrivial already in the leading logarithmic (LL) approximation.

###  2.1 Diagrammatic methods for threshold resummation

#### Large logarithms near threshold.

Power corrections play an important role for precise predictions of physical
observables involving heavy final states. Because of a phenomenon of dynamical
enhancement, these scattering processes receive a large contribution near
their partonic threshold, where the dimensionless partonic cross takes the
following schematic form

$\Delta(\xi)=\sum_{n=0}^{\infty}\alpha_{s}^{n}\Bigg{\\{}\sum_{m=0}^{2n-1}c^{\rm
LP}_{nm}\left(\frac{\ln^{m}\xi}{\xi}\right)_{+}+d_{n}\delta(\xi)+\sum_{m=0}^{2n-1}c^{\rm
NLP}_{nm}\ln^{m}\xi+\dots\Bigg{\\}}\,.$ (1)

In this equation $\xi$ represents a variable, which approaches $\xi\to 0$ near
threshold; for instance, $\xi=1-z\equiv 1-Q^{2}/\hat{s}$ for the Drell-Yan
process, where $Q^{2}$ is the invariant mass of the final state, and $\hat{s}$
the partonic center of mass energy squared. For $\xi\to 0$ the partonic cross
section in Eq. (1) develops large logarithms that need to be resummed to
obtain precise predictions. The terms proportional to $c^{\rm LP}_{nm}$
contribute at leading power, and their resummation has been known for a long
time to a high logarithmic accuracy, since the seminal papers Parisi:1979xd ;
Curci:1979am ; Sterman:1986aj ; Catani:1989ne ; Catani:1990rp . Later, LP
threshold resummation has been reinterpreted and clarified using a wide
variety of methods, including the use of Wilson lines Korchemsky:1992xv ;
Korchemsky:1993uz , the renormalization group Forte:2002ni , the connection to
factorization theorems Contopanagos:1996nh , and soft collinear effective
theory Becher:2006nr ; Schwartz:2007ib ; Bauer:2008dt . The state-of-the-art
for resummation at LP is next-to-next-to-next-to-leading logarithmic (N3LL)
accuracy for color singlet final states, and next-to-next-to-leading
logarithmic (NNLL) accuracy for processes involving colored particles in the
final state.

The resummation of logarithms with $c^{\rm NLP}_{nm}$ coefficients,
contributing at next-to-leading power (NLP), has been considered only more
recently and it is still subject of intense study. Progress has been made by
employing diagrammatic Bahjat-Abbas:2019fqa ; vanBeekveld:2019prq ;
vanBeekveld:2021hhv ; vanBeekveld:2021mxn , effective field theory
Moult:2018jjd ; Beneke:2018gvs ; Beneke:2019mua ; Moult:2019uhz ;
Beneke:2020ibj  and renormalization group methods Ajjath:2020ulr ;
Ajjath:2020sjk . In this section, we provide a brief overview of the
diagrammatic approach, while the application of SCET to the resummation of
large logarithms at NLP will be discussed in the following section.

#### Resummation at NLP for diagonal partonic channels.

Near threshold, additional radiation is constrained to be soft and the
partonic cross section develops a hierarchy of scales, with $Q$ representing
the energy involved in the hard interaction, and $Q(1-z)$ with $z\ll 1$ the
energy of the soft radiation. It can be shown that soft radiation at LP is
described in terms of uncorrelated eikonal emissions. As a consequence, the
partonic cross section factorizes into a soft and a hard function, the latter
representing the central hard scattering, while the former describes soft
radiation in terms of a vacuum expectation value of Wilson lines along the
directions of the hard scattering particles. The exponentiation of eikonal
radiation together with the factorization of the associated phase space (in
Mellin space) is the basis for resummation of the logarithms proportional to
the $c^{\rm LP}_{nm}$ coefficients in Eq. (1).

Beyond LP, the factorization of the partonic cross section becomes more
involved. In general, this happens because soft radiation becomes sensitive to
the nature of the hard scattering particles and their hard interaction. For
instance, soft gluons can be emitted through chromo-magnetic interactions,
which are sensitive to the spin of the emitting particles; furthermore, at NLP
one needs to take into account the emission of soft quarks, too. In this
section we will focus on the emission of soft gluon radiation from collinear
legs, which contributes at LP and NLP to the large logarithms in diagonal
production channels, such as $q\bar{q}$ in Drell-Yan, or $gg$ in Higgs
production. The emission of soft quarks from collinear legs contributes
instead to the so-called off-diagonal partonic channels, such as the $qg$
channel in Drell-Yan. We will briefly discuss these channels in the next
subsection.

Besides being sensitive to the spin of the emitting particles, soft gluon
radiation at NLP begins to reveal the structure of the hard scattering
Low:1958sn ; Burnett:1967km , as well as the structure of virtual collinear
radiation associated to the hard scattering particles of the process
DelDuca:1990gz . These effects can be described in terms of convolutions of
soft and collinear matrix elements multiplying a hard function, which
represents the central non-radiative scattering process. In a diagrammatic
approach, the soft and collinear matrix elements are described in terms of QCD
fields and Wilson lines. For instance, in the case of Drell-Yan, the
$q\bar{q}$ amplitude with an emission of one additional soft gluon is written
(up to NNLO) as Bonocore:2015esa ; Bonocore:2016awd

${\cal A}_{\mu,a}(p_{j},k)=\sum_{i=1}^{2}\left\\{\frac{1}{2}\,\widetilde{\cal
S}_{\mu,a}(p_{j},k)+g\,{\bf
T}_{i,a}\,G^{\nu}_{i,\mu}\,\frac{\partial}{\partial
p_{i}^{\nu}}+\big{[}J_{\mu,a}\left(p_{i},n_{i},k\right)-{\cal
J}_{\mu,a}\left(p_{i},n_{i},k\right)\big{]}\right\\}{\cal A}(p_{j})\,,$ (2)

where $p_{j}$, $j=1,2$ represent the momenta of the initial state partons, and

$G^{\mu\nu}_{i}=g^{\mu\nu}-\frac{(2p_{i}-k)^{\nu}}{2p_{i}\cdot
k-k^{2}}\,k^{\mu}.$ (3)

In Eq. (2), $\widetilde{\cal S}_{\mu,a}(p_{j},k)$ represents a generalized
soft function, which accounts for soft emissions from the external particles
beyond the eikonal approximation Laenen:2008gt ; Laenen:2010uz ; the second
term is given by the orbital angular momentum operator acting on the non-
radiative amplitude ${\cal A}(p_{j})$, and represents the so-called Low-
Burnett-Kroll (LBK) theorem Low:1958sn ; Burnett:1967km . In the last term,
$J_{\mu,a}\left(p_{i},n_{i},k\right)$ represents a radiative jet function,
introduced first in DelDuca:1990gz , while ${\cal
J}_{\mu,a}\left(p_{i},n_{i},k\right)$ subtracts overlapping soft modes between
the radiative and the soft function.

Starting from Eq. (2) it is possible to derive the corresponding factorized
cross section, which has been explicitly verified up to NNLO in perturbation
theory Bonocore:2015esa ; Bonocore:2016awd . The factorization theorem in Eq.
(2) is valid up to this perturbative order, which means that it is not enough
to develop a theory of resummation at NLP, at arbitrary logarithmic accuracy.
While studies of the all-order factorization structure are in development (see
e.g. Gervais:2017yxv ; Laenen:2020nrt ; Bonocore:2021cbv  for recent
progress), it has already been possible to exploit the result of Eq. (2) to
obtain the resummation of NLP large logarithms at leading logarithmic
accuracy. Indeed, it is possible to show that the second and third terms in
Eq. (2) contribute only starting at next-to-leading logarithmic (NLL)
accuracy, i.e., leading logarithms are generated only by momenta
configurations which are maximally soft and collinear (see Bahjat-
Abbas:2019fqa  for a detailed discussion), which are taken into account by the
generalized soft function, i.e. the first term in Eq. (2). As a consequence,
at LL accuracy the factorization theorem simplifies. In Mellin space, the bare
partonic cross section takes the form

$\widehat{\Delta}^{(q\bar{q})}_{\rm
NLP,LL}\left(N,Q^{2},\epsilon\right)\,=\,\left|\mathcal{H}\left(Q^{2}\right)\right|^{2}{\cal
S}\\!\left(N,Q^{2},\epsilon\right)\,.$ (4)

By using tools such as the replica trick Gardi:2010rn , the generalized soft
function ${\cal S}\\!\left(N,Q^{2},\epsilon\right)$ can be shown to
exponentiate, Laenen:2010uz , leading to Bahjat-Abbas:2019fqa

$\int_{0}^{1}dzz^{N-1}\Delta_{\rm\,NLP,LL}^{(q\bar{q})}(z)\,=\,{\Delta}^{(q\bar{q})}_{\rm
LO}(Q^{2})\exp\left[\frac{2\alpha_{s}C_{F}}{\pi}\left(\log^{2}\\!N+\frac{\log
N}{N}\right)\right]\,,$ (5)

where the exponent of the one-loop soft function resums large leading
logarithms, with the second term corresponding to the NLP correction.

It can be proven that the structure of the NLP correction in Eq. (5) is
universal for all electroweak annihilation processes involving a colorless
final state DelDuca:2017twk ; Bahjat-Abbas:2019fqa . The LL resummed cross
section, up to NLP, takes the form of Eq. (5), with the LO partonic cross
section multiplying a universal factor, whose color structure ($C_{F}$ in Eq.
(5)) depends on the initial state particles. In general, in $z$ space this
leads to the resummation formula

$\displaystyle\Delta_{aa}^{\rm dQCD,\,LP+NLP}(N,Q^{2}/\mu^{2})$
$\displaystyle=$ $\displaystyle g_{0}(\alpha_{s})\,{\rm
exp}\Bigg{\\{}\int_{0}^{1}{\rm
d}z\,z^{N-1}\,\Bigg{[}\frac{1}{1-z}D_{aa}\left(\alpha_{s}\left(\frac{(1-z)^{2}Q^{2}}{z}\right)\right)$
(6) $\displaystyle\hskip
28.45274pt+2\int_{\mu^{2}}^{(1-z)^{2}Q^{2}/z}\frac{{\rm
d}k_{T}^{2}}{k_{T}^{2}}P^{\rm
LP+NLP}_{aa}\left(z,\alpha_{s}(k_{T}^{2})\right)\Bigg{]}_{+}\,\Bigg{\\}},$

where $g_{0}(\alpha_{s})$ collects the $N$-independent contributions. Large
logarithms are resummed in the exponent, which contains the diagonal DGLAP
splitting function $P_{aa}(z,\alpha_{s})$ and soft wide-angle contributions,
collected in $D_{aa}(\alpha_{s})$, where $aa=q\bar{q}$ for quark-antiquark
initiated processes, and $aa=gg$ for gluon-gluon initiated processes.
Evaluating these functions to a given order in $\alpha_{s}$ resums the large
logarithms to a corresponding logarithmic accuracy. To date,
$P_{aa}(z,\alpha_{s})$ is known to fourth order in $\alpha_{s}$ and
$D_{aa}(\alpha_{s})$ to third order, which guarantees the resummation (at LP)
up to N3LL accuracy. Both terms are process-independent, to the extent that
they only depend on the color structure of the underlying hard-scattering
process. The NLP correction in Eq. (5) is given by expanding
$P_{aa}(z,\alpha_{s})$ up to NLP in the threshold variable. This ensures that
all LLs are resummed at NLP. In principle, Eq. (6) resums also NLP logarithms
beyond LL accuracy. These originates from keeping higher order contributions
in $P_{aa}(z,\alpha_{s})$ at NLP, as well as from the argument of $\alpha_{s}$
in the function $D_{aa}$ and from the $1/z$-dependence of the upper limit of
the $k_{T}$ integral. However, these contributions give only part of the whole
NLL contribution and beyond, and can be dropped for a consistent resummation
at fixed logarithmic accuracy.

Figure 1: Ratio plot of the total Higgs cross section (left) and the DY
invariant mass distribution (right), normalized to the NNLL′ \+ NNLO result.
The NNLL′ (+NLP LL) result is again shown by the orange solid (red dash-
dotted) lines. The NNLL′ NLP LL result is obtained by adding the N3LO NLP LL
$qg$ result, which is shown by the dotted light-blue line. Plots taken from
vanBeekveld:2021hhv .

An extensive phenomenological analysis investigating the effects of resumming
threshold LLs at NLP in physical observables of interest for the LHC has been
presented in vanBeekveld:2021hhv . There, it has been shown that, in general,
the NLP LL correction becomes competitive with the resummation of LP logarithm
at NNLL accuracy, as can be seen for instance in Fig. 1. Therefore, any
precise prediction in which LP resummation is performed at NNLL accuracy and
beyond should include resummation of NLP logarithms. The analysis of
vanBeekveld:2021hhv  has also shown that the diagrammatic and the effective
field theory (SCET) approach Beneke:2018gvs ; Beneke:2019mua , give almost
identical results, when the same contributions at NLP are taken into account,
thus removing earlier claims of numerical differences between the two
approaches.

#### Resummation at NLP for subleading partonic channels.

Figure 2: Left: ladder diagram contributing to the $g\bar{q}$ channel in DY
production; right: similar but for the $qg$ channel in Higgs production.

The resummation of large threshold logarithms in off-diagonal channels, such
as the $qg$ channel in Drell-Yan, are particularly interesting. These channels
start contributing already at NLP, because near threshold they involve the
emission of a soft quark, which is itself power-suppressed with respect to a
soft gluon emission. Earlier conjectures for Deep Inelastic Scattering (DIS)
and Drell-Yan Vogt:2010cv ; LoPresti:2014ihe  showed that the resummation of
large threshold logarithms in off-diagonal channels has a non-trivial
exponentiation pattern, even at LL accuracy, which involves the color
structure $C_{F}-C_{A}$. The reason for this non-trivial behavior can be
understood, for instance, by investigating the factorization structure of the
Drell-Yan $qg$ channel by means of the method of regions, which shows that
both virtual collinear and soft modes contribute at LL accuracy. Therefore, a
simplification such as in Eq. (4) is not possible for off-diagonal channels.
Despite this complication, it has been possible to derive an all-order
resummation formula for NLP LLs in off diagonal channels of processes such as
DIS, Drell-Yan and Higgs production Beneke:2020ibj ; SCETunpublished ;
vanBeekveld:2021mxn . To this end, a fundamental input is provided by deriving
so-called consistency conditions, obtained by requiring the cancellation of
leading poles in the total cross section. One parameterizes the leading poles
in the total cross section in terms of unknown coefficients multiplying the
relevant scales of the process. Requiring pole cancellation in the bare total
cross section provides a set of equations, whose solution determines the
minimal number of parameters needed to fix all the poles – and thus the
related large logarithms. The derivation is rather technical, and we refer to
Beneke:2020ibj ; vanBeekveld:2021mxn  for further details. The relevant
observation is that a minimal set of parameters is given entirely by those
associated to a single scale in the problem – one parameter for each
perturbative order. It is thus possible to obtain the entire cross section by
determining the full tower of coefficients associated, for instance, to the
hard scale, or to the soft scale in the problem. Determining the all-order
coefficients associated to the hard scale can be done by means of effective
field theory methods, and we refer to Beneke:2020ibj  for further details. The
diagrammatic approach, instead, is particularly suitable to determine at all
orders the coefficients associated to the leading poles in the soft region –
in a suitable gauge, it can be shown that the soft region is determined by
ladder diagrams such as those represented in Fig. 2 vanBeekveld:2021mxn .
These can be calculated to all orders, and subsequently the consistency
condition allow one to fix the entire cross section. For instance, in the case
of the $qg$ channel in Drell-Yan, the resummed NLP LL partonic cross section
takes the form SCETunpublished ; vanBeekveld:2021mxn

$\displaystyle\Delta_{qg}^{\rm dQCD,\,LP+NLP}$ $\displaystyle=$
$\displaystyle\frac{T_{R}}{C_{A}-C_{F}}\frac{1}{2N\ln
N}\bigg{\\{}e^{\frac{2C_{F}\alpha_{s}}{\pi}\ln^{2}N}{\cal
B}_{0}\bigg{[}\frac{\alpha_{s}}{\pi}(C_{A}-C_{F})\ln^{2}N\bigg{]}$ (7)
$\displaystyle\hskip
113.81102pt-\,e^{(2C_{F}+6C_{A})\frac{\alpha_{s}}{4\pi}\ln^{2}N}\bigg{\\}},$

where

${\cal B}_{0}(x)=\sum_{n=0}^{\infty}\frac{B_{n}}{(n!)^{2}}x^{n}\,,$ (8)

and in turn $B_{n}$ are Bernoulli numbers, $B_{0}=1$, $B_{1}=-1/2$, $\ldots$.
Combining together Eq. (6) with Eq. (7) completes the resummation of LLs at
NLP in all the production channels contributing to Drell-Yan. Analogous
equations can be obtained for Higgs boson production in gluon fusion and DIS,
and we refer to Beneke:2020ibj ; vanBeekveld:2021mxn  for a detailed
derivation.

###  2.2 Effective field theory approach for cross-sections

The effective field theory approach can achieve a proper, systematic treatment
of factorization and resummation beyond the leading power. SCET provides a
framework to describe processes involving energetic and soft particles to all
powers. It allows for an operator-based expansion, maintaining homogeneous
power-counting and gauge invariance at every step of the computations. The EFT
formulation provides further insights into the structure of well-established
soft theorems Larkoski:2014bxa ; Beneke:2017mmf ; Beneke:2021umj , which are
the basis for a systematic study of power suppressed effects. The EFT approach
enables the study of subleading power terms in soft and collinear expansion
not only in QCD but also in Gravity Beneke:2021aip  and it would be
interesting to explore further applications.

The SCET expansion parameter is typically denoted by $\lambda$. Its precise
definition depends on the process under consideration, e.g.,
$\lambda\sim\sqrt{\xi}$ as in Eq. (1) above. The theory contains collinear and
soft modes. Each collinear direction has its collinear mode, i.e., a generic
$N$-jet process will be described by $N$ collinear modes. Interactions within
a given collinear sector and interactions with the soft background are
contained in the Lagrangian. The complete Lagrangian can then be written as a
sum over collinear directions

$\displaystyle\mathcal{L}_{\rm
SCET}=\sum_{i=1}^{N}\mathcal{L}_{i}+\mathcal{L}_{s},$ (9)

where $\mathcal{L}_{i}$ is the $i$-collinear Lagrangian and $\mathcal{L}_{s}$
is a pure soft Lagrangian. In principle, one also has to consider the Glauber
Lagrangian Rothstein:2016bsq . However, for many observables, the Glauber
contribution cancels and thus this term is often ignored. The $i$-collinear
Lagrangian can then be written as a series in $\lambda$

$\mathcal{L}_{i}=\mathcal{L}_{i}^{(0)}+\mathcal{L}_{i}^{(1)}+\mathcal{L}_{i}^{(2)}+\mathcal{O}(\lambda^{3})\,.$
(10)

Interactions between different collinear sectors are obtained after
integrating out the hard modes. They are encoded in the currents, which are
constructed from the collinear gauge-invariant building blocks Bauer:2000yr ;
Bauer:2001ct . At leading power, we can only use a single building block

$\displaystyle
J_{i}^{A0}(t_{i})\in\left\\{\chi_{i}(t_{i}n_{i+}),\overline{\chi}_{i}(t_{i}n_{i+}),\mathcal{A}_{i\perp}(t_{i}n_{i+})\right\\},$
(11)

for each collinear direction. Here $\chi_{i}(x)=W_{i}^{\dagger}(x)\xi_{i}(x)$
is a fermionic building block, constructed out of the $i$-collinear field
$\xi_{i}(x)$ and the collinear Wilson line $W_{i}$. Similarly, the gluon
building block is related to the collinear gluon field. Since the building
blocks scale as $\lambda$, adding more $i$-collinear fields in one direction
creates power-suppressed operators. Alternatively, power suppression can be
introduced by the action of a derivative on the collinear fields. Details on
the operatorial basis construction and renormalization of the subleading
operators for various cases can be found in Moult:2015aoa ;
Kolodrubetz:2016uim ; Beneke:2017ztn ; Beneke:2017mmf ; Moult:2017rpl ;
Chang:2017atu ; Feige:2017zci ; Beneke:2018rbh ; Beneke:2019kgv ;
Moult:2019mog .

While the study of power corrections in the context of flavor physics has a
long history Beneke:2003pa ; Hill:2004if ; Lee:2004ja ; Beneke:2004rc ;
Beneke:2004in ; Paz:2009ut ; Benzke:2010js ; Beneke:2015wfa , the first
resummation using SCET for event shape-type observables was achieved
relatively recently for thrust distribution Moult:2018jjd . Subsequently,
threshold resummation has been studied for Drell-Yan Beneke:2018gvs  and Higgs
production Beneke:2019mua . Each of these studies considers only diagonal
channels, i.e., not involving soft quarks and achieved leading logarithmic
accuracy. The LL results have a relatively simple structure for all these
processes. For example, the Drell-Yan threshold resummation leads to the
following expression

$\displaystyle\Delta^{\rm LL}_{\rm NLP}(z,\mu)=\frac{\hat{\sigma}^{\rm
LL}_{\rm NLP}(z,\mu)}{z}=$
$\displaystyle\exp\left[-2\frac{\alpha_{s}C_{F}}{\pi}\ln^{2}\frac{\mu}{\mu_{h}}\right]\,\exp\left[+2\frac{\alpha_{s}C_{F}}{\pi}\ln^{2}\frac{\mu}{\mu_{s}}\right]$
$\displaystyle\times(-4)\frac{\alpha_{s}C_{F}}{\pi}\ln\frac{\mu_{s}}{\mu}\,\theta(1-z)\,.$
(12)

One can recognize the subleading power logarithm appearing at NLO in the
second line, dressed by Sudakov exponents originating from the hard and soft
functions. Interestingly, even these simple results required the development
of new techniques allowing to perform the resummation. These are related to
the observation that the leading logarithms appear due to operator mixing
under renormalization Moult:2018jjd ; Beneke:2018gvs .

The result for the Higgs threshold production Beneke:2019mua  can be obtained
from the Drell-Yan result (2.2) after substituting color factors
$C_{F}\leftrightarrow C_{A}$. Phenomenological study for $13\rm TeV$ proton-
proton collisions shows that the LL NLP corrections can exceed 30% of the NNLL
LP resummed cross section. However, as shown in Fig. 3, the dependence on the
soft scale is substantial and indicates a need for extension for the NLP
resummation beyond LL accuracy, and possibly even further terms in the power
expansion Anastasiou:2015vya .

Figure 3: Soft scale dependence of the NNLL LP and LL NLP resummed Higgs
production cross section. Left panel: $\mu_{h}^{2}=m_{H}^{2}$, right panel:
$\mu_{h}^{2}=-m_{H}^{2}$. Letters $A$ and $B$ refer to different possible
choice of initial conditions, see Beneke:2019mua .

While the general procedure of deriving subleading power factorization
theorems is well understood Beneke:2019oqx ; Moult:2019mog , the resummation
beyond leading logarithmic accuracy for diagonal channels is still an open
problem. The so-called endpoint divergent problem hinders the extension of the
above results beyond LL accuracy or to the off-diagonal contributions
involving soft quarks. The factorization theorems at NLP generically are a sum
over several terms, where each term is a convolution between the hard,
collinear, and soft functions. While these convolution integrals are well
defined for the bare factorization theorem in $d$-dimensions, the limit $d\to
4$ does not exist as was explicitly demonstrated in Beneke:2019oqx ;
Jaskiewicz:2019phy . This problem originates, for example, due to the overlap
between soft and collinear modes. In Beneke:2019kgv , it has been observed
that for certain cases protected by reparameterization invariance, one can
redefine the operator basis to remove the endpoint divergences. In
Moult:2019uhz  the result for the Sudakov factor involving soft quarks has
been conjectured and it has been shown that endpoint divergences are absent
for $\mathcal{N}=1$ QCD, i.e. when the color charge of quark and gluon are
equal. The conjectured soft quark Sudakov factor has been subsequently proven
and generalized in Beneke:2020ibj . This has been possible thanks to the
development of the additional endpoint factorization of the divergent
contributions and the introduction of new auxiliary modes, which parameterize
the overlap between soft and collinear contributions. Using this result, the
short-distance coefficient in the $\overline{{\rm{MS}}}$ scheme in off-
diagonal Higgs induced DIS can be obtained in closed form, yielding the
following result

$\displaystyle\tilde{C}_{\phi,q}^{NLP,LL}\Big{|}_{\epsilon\to 0}$
$\displaystyle=$ $\displaystyle\frac{1}{2N\ln
N}\frac{C_{F}}{C_{F}-C_{A}}\Bigg{(}{\cal
B}_{0}(a)\exp\left[C_{A}\frac{\alpha_{s}}{\pi}\left(\frac{1}{2}\ln^{2}N+\ln
N\ln\frac{\mu^{2}}{Q^{2}}\right)\right]$ (13)
$\displaystyle-\,\exp\left[\frac{\alpha_{s}C_{F}}{\pi}\left(\frac{1}{2}\ln^{2}N+\ln
N\ln\frac{\mu^{2}}{Q^{2}}\right)\right]\Bigg{)}\,,$

where $a=\frac{\alpha_{s}}{\pi}(C_{F}-C_{A})\ln^{2}N\,$ and
$\mathcal{B}_{0}(x)$ is defined in (8). The above result agrees with Eq. (29)
of Vogt:2010cv  for $\mu=Q$, and generalizes it to $\mu\not=Q$.

Dealing with the endpoint divergences is currently at the forefront of EFT
studies. Once a systematic treatment is established, we can expect the
accuracy of resummation to be improved. Not much is known about the properties
of the subleading soft and jet functions. Further progress will require a
better understanding of the renormalization properties of these complicated
non-local objects. Beyond the applications of SCETI that are discussed here,
there are new interesting areas of study in SCETII involving subleading power
rapidity divergences for transverse momentum distributions Ebert:2018gsn ,
Energy-Energy correlators Moult:2019vou , Regge kinematics and forward
scattering Moult:2017xpp ; Bhattacharya:2021zsg , and azimuthal asymmetries in
semi-inclusive deep inelastic scattering Ebert:2021jhy .

Progress on the application of the EFT methods to power expansion for collider
observables will be crucial for achieving the precision required by the High-
Luminosity LHC, EIC and future colliders. Applications of a systematic power
expansion and resummation with the help of the renormalization group equations
will enable us to achieve better precision and has a potential to impact the
development of subtraction methods for fixed order computations Moult:2016fqy
; Boughezal:2016zws ; Moult:2017jsg ; Ebert:2018lzn ; Bhattacharya:2018vph ;
Ebert:2018gsn ; Boughezal:2018mvf ; Boughezal:2019ggi .

###  2.3 Diagrammatic methods for mass- and angle-suppressed amplitudes

Two of the most phenomenologically and conceptually interesting examples of
power corrections are given by the mass-suppressed amplitudes in processes
with massive fermions in the fixed-angle Sudakov limit and the amplitudes
suppressed by the ratio of a characteristic momentum transfer to the total
energy in the small-angle or Regge limit.

In processes with massive fermions, the origin of the logarithmic corrections
and the asymptotic behavior of the amplitudes at next-to-leading power (NLP)
drastically differ from the classical leading-power Sudakov result
Sudakov:1954sw . The double-logarithmic terms in this case are related to the
effect of the eikonal color charge nonconservation in the process with soft
fermion exchange and result in asymptotic exponential enhancement for a wide
class of amplitudes and a breakdown of a formal power counting Penin:2014msa ;
Liu:2017vkm .

To illustrate the general properties of such processes, we consider the
amplitude of light quark mediated Higgs boson production in gluon fusion which
is suppressed by the quark to Higgs boson mass ratio $m_{q}^{2}/m_{H}^{2}$
relative to the leading contribution of the top quark loop. With the targeted
precision of the QCD prediction for the cross section in a percent range
Anastasiou:2016cez  the contributions of bottom quark loop cannot be ignored
and the problem of its theoretical description ultimately goes beyond the
finite order perturbative calculation Lindert:2017pky ; Czakon:2020vql .
Indeed, the radiative corrections in this case are enhanced by the second
power of the large logarithm $\ln(m_{H}/m_{q})$ with the effective expansion
parameter being $\alpha_{s}\ln^{2}(m_{H}/m_{b})\approx 40\alpha_{s}$. In the
leading (double) logarithmic approximation, the light quark mediated $gg\to H$
amplitude reads Liu:2017vkm ; Liu:2018czl

${{\cal M}^{q}}^{LL}_{gg\to
H}=-{\frac{3}{2}}{Z^{2}_{g}}g(z){\frac{m_{q}^{2}}{m_{H}^{2}}}\ln^{2}\left({\frac{m_{H}^{2}}{m_{q}^{2}}}\right)\,{{\cal
M}^{t}}^{(0)}_{gg\to H}\,,$ (14)

where ${{\cal M}^{t}}^{(0)}_{gg\to H}$ is the leading order heavy top quark
mediated amplitude. Eq. (14) reveals a number of characteristic features of
mass-suppressed processes. It includes the standard infrared divergent Sudakov
factor $Z^{2}_{g}$ for the external on-shell gluon states and the function
$g(z)$ of the variable $z=(C_{A}-C_{F}){\alpha_{s}\over
4\pi}\ln^{2}(m_{H}^{2}/m_{q}^{2})$ which accounts for the non-Sudakov mass
logarithms to all orders in $\alpha_{s}$. It can be solved in terms of the
generalized hypergeometric function
$g(z)={}_{2}F_{2}\left(1,1;{3/2},2;{z/2}\right)$ and has the following
asymptotic behavior at $z\to\infty$

$g(z)\sim\left({2\pi e^{z}\over z^{3}}\right)^{1/2}\\!\\!,$ (15)

i.e. in contrast to the Sudakov factor it results in the exponential
enhancement of the amplitude in the high-energy (small quark mass) limit. The
difference $C_{A}-C_{F}$ of the quadratic Casimir operators of the gluon and
quark color group representations in the definition of the variable $z$
accounts for the color charge variation of the highly energetic parton after
emission of a soft quark, which constitutes the physical origin of the non-
Sudakov double logarithms.

Though the leading logarithmic result determines the qualitative behavior of
the amplitude, it cannot be used for a reliable quantitative estimates since
it does not account for the numerically important effects such as the
renormalization scale dependence of the quark mass and coupling constants,
which requires the inclusion of subleading logarithms. In Ref.
Anastasiou:2020vkr , the next-to-leading logarithmic corrections to Eq. (14)
of the form $\alpha_{s}^{n}\ln^{2n-1}(m_{H}/m_{q})$ for all $n$ have been
computed and applied to the analysis of the Higgs boson production in
threshold approximation. For the yet unknown next-to-next-to-leading and next-
to-next-to-next-to-leading bottom quark contribution to the total cross
section this analysis gives $-0.12~{}pb$ and $-0.02~{}pb$, respectively. With
a rather conservative assessment of the accuracy of the next-to-leading
logarithmic and the threshold approximations it provides a rough estimate of
the bottom quark mediated contribution to the total cross section of Higgs
boson production in gluon fusion beyond NLO to be in the range from $-0.32$ to
$0.08~{}pb$, thereby reducing the previous uncertainty estimate
Anastasiou:2016cez  by a factor of two.

The actual accuracy of the logarithmic and threshold approximations, however,
is difficult to estimate, and the above interval has to be further reduced by
evaluating the next-to-next-to-leading logarithmic contribution and getting an
approximation valid beyond the threshold region. The latter requires the
analysis of the logarithmically enhanced corrections to the hard real emission
which is currently not available even in the leading logarithmic approximation
(only the abelian part of the double-logarithmic corrections for the $gg\to
Hg$ amplitude of Higgs plus jet production has been obtained in Ref.
Melnikov:2016emg ). The extension of the method Liu:2017vkm ; Liu:2018czl ;
Anastasiou:2020vkr  beyond the next-to-leading logarithms and to processes
with hard real radiation is one of big challenges for the modern effective
field theory and is of primary phenomenological importance for the high-
precision Higgs physics program at the LHC. At the same time the high-order
next-to-next-to-leading power terms are well under control Liu:2021chn .

The Regge limit of high-energy scattering describes a kinematical
configuration with vanishing ratio of the characteristic momentum transfer to
the total energy of the process. The relevant expansion parameter is given by
the ratio of the Mandelstam variables $t/s$. Despite a crucial simplification
due to decoupling of the light-cone and transversal degrees of freedom, the
gauge interactions in this limit possess highly nontrivial dynamics giving a
rigorous quantum field theory realization of the Regge concept for high-energy
scattering. Major progress has been achieved in the analysis of the leading-
power amplitudes which culminated in the evaluation of the next-to-leading QCD
corrections to the theory of BFKL pomeron Fadin:1998py .

The expansion in $t/s$ is also mandatory for the evaluation of the
nonfactorizable QCD corrections to the Higgs boson production via vector boson
fusion. The leading order of this expansion is equivalent to the eikonal
approximation, which reduces the problem to the effective theory analysis in
the two-dimensional transversal space. This analysis has been performed at
next-to-next-to-leading order in QCD for the cross sections of a single Higgs
boson Liu:2019tuy  and Higgs boson pair production Dreyer:2020urf .

At the same time, very little is known about the structure of the logarithmic
corrections and the asymptotic behavior of the power suppressed amplitudes
which are crucial to control the accuracy and validity of the analysis based
on the eikonal approximation. Only recently the leading logarithmic result for
the power-suppressed part of the fermion scattering amplitude has been
obtained in an abelian gauge theory within an effective field theory framework
Penin:2019xql . In contrast to the single-logarithmic Regge behavior of the
leading-power amplitude, the radiative corrections to the power-suppressed
term exhibit a double-logarithmic enhancement and the result for the amplitude
has the following form

${\cal M}^{LL}=|t/s|z\,g(2z){\cal M}^{\rm Born}\,,$ (16)

where $z={\alpha\over 2\pi}\ln(|t/s|)$ and $g(z)$ is, amazingly, the same
function as in Eq. (14) describing the non-Sudakov corrections to the mass
suppressed amplitude. Due to the exponential growth of this function at large
values of its argument, at $\left|{s/t}\right|\approx e^{2\pi/\alpha}$, the
formally power-suppressed contribution to the scattering cross section becomes
comparable to the leading-power result and the small-angle expansion breaks
down. This, in particular, may be considered as a possible solution of the
long standing unitarity problem of the Regge analysis of the leading-power
scattering amplitudes in QED Frolov:1970ij . The generalization of the above
result to the nonabelian gauge groups and more complex amplitudes is therefore
of primary theoretical interest.

###  2.4 Effective field theory approach for mass suppressed amplitudes

In parallel to the direct QCD approach, the mass suppressed effects on the
amplitude level have been studied in the SCETII framework. It has been shown
that these effects give rise to unexpectedly large QED corrections for the
leptonic decays of $B_{s}$ mesons Beneke:2017vpq . They have been resummed, in
part, in Beneke:2019slt . The resummed corrections were due to a contribution
which is not endpoint divergent thanks to the stronger suppression of the
endpoint region at the hard scale.

The breakthrough has been achieved in the study of the $h\to\gamma\gamma$
amplitude mediated by the light quark. The condition on cancellation of the
rapidity divergences has been used to derive refactorization conditions
Liu:2019oav  and perform the LL resummation. The renormalization properties of
the NLP jet and soft functions have been subsequently studied in Liu:2020eqe ;
Bodwin:2021cpx ; Liu:2020ydl ; Liu:2021mac . These objects have a close
connection to functions appearing earlier in the flavor applications of SCET.
Generally, they are simpler than the corresponding objects defined at the
level of the cross section in SCETI, as their dependence on the convolution
variables can be traced back to a particular external state which defines the
amplitude.

The refactorization conditions were then rigorously derived in Liu:2020wbn  as
operatorial identities. This allowed proving that the reshuffling of the
factorization theorem removes endpoint divergences. Using the previously
computed anomalous dimensions of soft and jet functions led to the first
complete NLL resummation Liu:2020tzd  for mass suppressed effects.

Resummation of power suppressed amplitudes will have a substantial impact on
the precision in studies of the top and Higgs sectors at the LHC. Applications
of the EFT methods will impact also flavor studies and in particular the
evaluation of the QED corrections, which are a necessary ingredient for the
resolution of flavor anomalies.

##  3 Jets and their substructure

Jets are collimated sprays of particles which are observed in the detectors of
high-energy scattering experiments such as the LHC, RHIC, LEP, HERA and the
future EIC. They reflect the underlying quark and gluon degrees of freedom
which emit radiation approximately collinear to their initial direction. In
recent years, significant progress has been made in improving the perturbative
precision of perturbative QCD calculations. For example, fixed order
calculations for inclusive jet production are now available at NNLO
Currie:2016bfm ; Czakon:2019tmo . In addition, logarithmic corrections at
threshold and in the jet radius have been resummed at NLL′ accuracy
Dasgupta:2014yra ; Liu:2017pbb ; Neill:2021std . In the future, it will be
critical to extend these calculations to N3LO and next-to-next-to leading
logarithmic (NNLL) accuracy in order to match the experimental precision. In
particular, the field of jet substructure has received a growing attention
over the last decade Larkoski:2017jix ; Kogler:2018hem ; Marzani:2019hun .
Here the goal is to characterize and utilize the radiation pattern inside
jets. Jet substructure observables are relevant for searches of BSM physics
and various precision measurements of Standard Model processes such as
Transverse Momentum Dependent (TMD) PDFs and fragmentation functions
Procura:2009vm ; Bain:2016rrv ; Kang:2017glf ; Neill:2016vbi ; Neill:2018wtk ;
Gutierrez-Reyes:2019msa , the strong coupling constant $\alpha_{s}$
Proceedings:2018jsb ; Marzani:2019evv , quarkonia Bain:2017wvk ; Kang:2017yde
, and the top quark mass Hoang:2017kmk  and to extract medium properties in
heavy-ion collisions. The need for precision calculations will increase during
the high-luminosity era of the LHC, where the detectors will be optimized for
the first time to perform precision jet substructure measurements. In
addition, the EIC is expected to perform precise measurements of low-energy
jets. The clean environment of the EIC will allow for novel measurements
complimentary to the LHC. In addition, RHIC and LEP data are analyzed to
measure jet energy spectra and jet substructure observables. Jet grooming
techniques Krohn:2009th ; Ellis:2009me ; Dasgupta:2013ihk ; Cacciari:2014gra ;
Larkoski:2014wba  have been developed to systematically remove soft radiation
from identified jets which can be difficult to account for from first
principles in QCD. These techniques can reduce nonperturbative effects and
allow for direct and precise comparisons between theoretical calculations and
data. Jet substructure observables at collider experiments typically involve
the measurement of multi-differential cross sections. This necessitates the
joint resummation of several large logarithmic corrections to all orders in
the QCD strong coupling constant, which is the main theoretical challenge.
Examples include logarithms in the jet mass, jet radius and grooming
parameters. These different large logarithmic corrections are generally not
independent and require a careful identification of relevant energy scales and
corresponding factorization theorems. Depending on the relative scaling of the
involved variables, different factorization theorems need to be derived.
Eventually, the different results need to be merged to allow for a meaningful
comparison to experimental data. Moreover, observables like the groomed
momentum sharing fraction $z_{g}$ are Sudakov safe Larkoski:2015lea  and
require, in addition, the joint resummation of an auxiliary variable like the
groomed jet radius $\theta_{g}$. To illustrate the complexity of factorization
theorems for jet substructure observables, we consider the momentum sharing
fraction $z_{g}$ Larkoski:2015lea . It is of considerable phenomenological
interest as it allows for the most direct measurement of the QCD splitting
function. For inclusive jet production, jet functions ${\cal G}_{i=q,g}$ that
depend on the jet substructure measurement can be factored out, which are then
multiplied by appropriate quark/gluon fractions. In order to resum all
relevant large logarithmic corrections, the jet functions needs to be
refactorized as Cal:2021fla

$\displaystyle{\cal G}_{i}$ $\displaystyle=\Theta(1/2>z_{g}>z_{\rm
cut}\theta_{g}^{\beta})\,\tilde{H}_{i}(p_{T}R,\mu)\,C_{i}^{\in{\rm
gr}}(\theta_{g}p_{T}R,\mu)\,S_{i}^{\notin{\rm gr}}(z_{\rm
cut}p_{T}R,\beta,\mu)\,$ $\displaystyle\quad\tilde{\cal S}_{G}(z_{\rm
cut}\theta_{g}^{1+\beta}p_{T}R,\beta,\mu)\,S_{i}^{\rm NG}(z_{\rm
cut})\bigg{[}\frac{\rm d}{{\rm d}z_{g}}\frac{\rm d}{{\rm
d}\theta_{g}}\tilde{\cal S}_{z_{g}}(z_{g}\theta_{g}p_{T}R,\mu)$
$\displaystyle\quad+\tilde{\cal S}^{\prime\rm
NG}_{i,1}(z_{g}\theta_{g},z_{g})+\tilde{\cal S}^{\prime\rm
NG}_{i,2}\Big{(}z_{g}\theta_{g},\frac{z_{g}}{z_{\rm
cut}\theta_{g}^{\beta}}\Big{)}\bigg{]}\,.$ (17)

Here, $z_{\rm cut},\beta$ denote parameters of the soft drop grooming
procedure and the functions $\tilde{\cal S}^{\prime\rm NG}_{i,j}$ account for
non-global contributions. The factorization in Eq. (3) enables the joint
resummation of four classes of large logarithmic corrections in the jet radius
$R$, the grooming parameter $z_{\rm cut}$ and the variables
$z_{g},\theta_{g}$. Currently, the achieved precision using the factorization
in Eq. (3) is NLL′ accuracy, which can be extended to yet higher accuracy in
the future. These results allow percent level comparisons to the available
data without the need to model nonperturbative effects. As an example, we show
the comparison to ATLAS data ATLAS:2019mgf  in the left panel of Fig. 4.

In recent years, significant progress has been made in developing
factorization formulas and improving the theoretical precision of theoretical
calculations using both direct perturbative QCD methods and SCET. Several
hallmark observables in the field of jet substructure at the LHC have now been
extended systematically to NLL′ and partially NNLL accuracy. Examples include
the jet mass Frye:2016aiz ; Marzani:2017mva ; Kang:2018vgn ; Larkoski:2020wgx
; Benkendorfer:2021unv , jet angularities Ellis:2010rwa ; Larkoski:2014uqa ;
Kang:2018vgn ; Kang:2018qra ; Caletti:2021oor ; Reichelt:2021svh , the groomed
jet radius Kang:2019prh , the momentum sharing fraction $z_{g}$
Larkoski:2015lea ; Cal:2021fla , the jet shape Cal:2019hjc , the angle between
different jet axes Cal:2019gxa , the jet pull Larkoski:2019urm ;
Larkoski:2019urm ; Bao:2019usu , and the primary Lund plane Lifson:2020gua . A
new jet grooming technique called dynamical grooming has been developed in
Ref. Mehtar-Tani:2019rrk ; Caucal:2021bae . Dedicated observables to
distinguish boosted Higgs, $Z$, $W$ or BSM jets have been studied analytically
in Refs. Larkoski:2017cqq ; Napoletano:2018ohv ; Cavallini:2021vot . These
observables make use of the multi-prong structure of jets originating from
boosted resonances compared to single-prong QCD jets. In Refs. Chen:2020vvp ;
Chen:2020adz ; Li:2021zcf , energy-energy correlators in the collinear limit
were discussed with are particularly suitable for extensions to higher
accuracy. Recently, jet tagging has also been proposed as a useful tool to
constrain collinear PDFs at the LHC Caletti:2021ysv . While jet grooming has
been introduced initially to remove soft wide-angle radiation from the jet, it
has recently been utilized to design Infrared-Collinear Safe observables which
are particularly sensitive to soft radiation Chien:2019osu ; Cal:2019gxa .
These observables may lead to a better understanding of the underlying event
contribution and hadronization effects. In the future it will be essential to
extend these calculations to NNLL′ or even N3LL accuracy. This requires
progress in fixed order calculations of relevant collinear and soft functions.
For example, in Ref. Liu:2021xzi , the exclusive collinear jet function for
anti-$k_{T}$ jets Cacciari:2008gp  was computed at NNLO which is an essential
ingredient to extend the precision of existing jet substructure calculations.
See also Refs. Kardos:2020ppl ; Kardos:2020gty . Moreover, future developments
are necessary to resum non-global logarithms. In Ref. Banfi:2021owj , the NLL
resummation of NGLs was achieved for the $e^{+}e^{-}$ hemisphere case which
sets the starting point for extending the precision of relevant jet
substructure observables.

Figure 4: Left: Comparison of numerical results for the momentum sharing
fraction $z_{g}$ at NLL′ accuracy Cal:2021fla  to ATLAS data ATLAS:2019mgf .
Figure taken from Ref. Cal:2021fla . Right: Prediction of the Sivers asymmetry
at the EIC in back-to-back electron-jet production for different bins of the
jet charge $Q_{\kappa}$. Figure taken from Ref. Kang:2020fka

At the EIC, precision calculations of jets and their substructure will be
critical to explore the 3D structure of nucleons and nuclei and study aspect
of the QCD hadronization process. Inclusive jet production has been calculated
up to NNLO in unpolarized and polarized $ep$ scattering which can provide
important constraints for collinear PDFs. See Refs. Hinderer:2015hra ;
Abelof:2016pby ; Boughezal:2018azh ; Borsa:2020yxh ; Borsa:2021afb  for recent
developments. Jets are expected to play an important role to constrain TMD
PDFs and fragmentation functions. Different than Semi-Inclusive Deep-Inelastic
Scattering (SIDIS), jet observables allow for a decorrelation of initial and
final-state TMD dynamics. Back-to-back electron-jet correlations measured in
the laboratory frame Liu:2018trl  can be used to study TMD PDFs which is
independent of TMD fragmentation. In Ref. Kang:2020fka , it was proposed to
use the jet charge Field:1977fa ; Krohn:2012fg  to separate up and down quark
contributions to the Sivers asymmetry in transversely polarized $ep$
scattering. The numerical predictions for the EIC are shown in the right panel
of Fig. 4. In the Breit frame, jets can be clustered in terms of energy
instead of transverse energy or momentum as in the laboratory frame
Arratia:2019vju . Since the photon virtuality $Q$ provides an additional
reference scale, the jet energy spectrum can be accessed. This provides a
unique opportunity to not only measure inclusive jets, but also the energy
loss of leading jets in vacuum and the nuclear environment Neill:2021std .
Breit frame jets also allow for measurements which are sensitive to TMD PDFs
Gutierrez-Reyes:2018qez . Chiral-odd TMDs such as the transversity can be
probed by time-reversal odd jets Liu:2021ewb . The observable and
factorization structure is similar to SIDIS but the final-state TMD
fragmentation function is replaced with a jet function that can be calculated
perturbatively. Moreover, gluon TMD PDFs can be accessed using high transverse
momentum di-jets in the Breit frame delCastillo:2021znl ; Kang:2021ffh . TMD
fragmentation can be studied independent of TMD PDFs using jet substructure
observables. This includes both the unpolarized case and for example the
Collins effect which can be accessed when the initial proton is transversely
polarized Yuan:2007nd ; Arratia:2020nxw ; Kang:2021ffh . Energy-energy
correlators in the back-to-back region can also provide important new insights
into TMD dynamics Gao:2019ojf . Moreover, EIC jets provide a powerful tool to
probe cold nuclear matter effects in electron-nucleus collisions Liu:2018trl ;
Sievert:2018imd ; Barata:2020rdn ; Li:2020rqj ; Zhang:2021tcc . In order to
match the precision of traditional processes such as SIDIS, higher
perturbative accuracy is needed at leading and subleading power.

##  4 Small-$x$ resummation

Gluon saturation has attracted a lot of attention in recent years and is one
of the major scientific goals of the future EIC Accardi:2012qut . Gluon
saturation plays the key role in understanding proton and heavy nuclei
collisions in the high-energy limit, where the gluon momentum fraction $x$ is
very small. In this small-$x$ region, the gluon density grows dramatically and
enters the nonlinear regime where gluon recombination becomes equally
important to the splitting process, and the color glass condensate (CGC)
effective theory McLerran:1993ni ; McLerran:1993ka ; McLerran:1994vd ;
Gelis:2010nm  is the proper framework to describe this regime. The nonlinear
B-JIMWLK (or its infinite color approximation, the BK equation) equation
Mueller:1989st ; Mueller:1993rr ; Balitsky:1995ub ; Kovchegov:1999yj ;
Jalilian-Marian:1997qno ; Jalilian-Marian:1997jhx ; Iancu:2000hn ;
Ferreiro:2001qy  replaces the linear parton evolution equations
Balitsky:1978ic , which inevitably leads to gluon saturation Gribov:1983ivg ;
Mueller:1985wy  with a characteristic scale $Q_{s}$. The saturation scale
$Q_{s}$ features the typical transverse momentum of the gluons inside the
proton or the nucleus and grows as $x$ decreases.

There are experimental results Golec-Biernat:1998zce ; BRAHMS:2004xry ;
PHENIX:2004nzn ; STAR:2006dgg ; PHENIX:2005veb ; Braidot:2010ig ;
ALICE:2018vhm ; Morreale:2021pnn  that are compatible with saturation-model
predictions, but there still exists no definitive evidence to claim a
discovery. In the future, dedicated measurements at the EIC will provide
further information on the regime of gluon saturation.

In order to faithfully and unambiguously establish gluon saturation and its
onset, reliable theoretical predictions for small-$x$ collider phenomena are
crucial. In the small-$x$ regime, the typical semi-hard saturation scale is of
the order of a few GeV and, therefore, $\alpha_{s}(Q_{s})$ is typically not
small enough. As a consequence, calculations beyond the leading order (LO) are
generally required to ensure the convergence of the perturbative results.
Recently, some attempts have been made in realizing the next-to-leading order
(NLO) calculations for small-$x$ physics Dumitru:2005gt ; Altinoluk:2011qy ;
Chirilli:2011km ; Beuf:2011xd ; Beuf:2016wdz ; Beuf:2017bpd ;
Boussarie:2016ogo ; Boussarie:2016bkq ; Balitsky:2012bs ; Roy:2018jxq ;
Liu:2019iml ; Roy:2019cux ; Roy:2019hwr ; Caucal:2021ent . However, systematic
calculations remain challenging and often cut-offs need to be introduced that
violate the small-$x$ factorization theorem and mix non-perturbative and
perturbative quantities that can invalidate the predictions. Meanwhile, there
also lacks a systematic way to resum the large logarithms in the semi-hard NLO
coefficient of the small-$x$ factorization theorem, which leads to the
infamous negative cross section problem Stasto:2013cha  and hinders practical
phenomenological applications of the NLO calculations.

One recent proposal suggests to use power counting arguments and to couple
SCET to the CGC formalism as a possible approach to combine systematic fixed
order results and resummation Kang:2019ysm ; Liu:2020mpy . The idea is applied
to single-inclusive hadron production in proton-nucleus collisions, $pA\to
hX$, at forward rapidity. The CGC factorization for this observable is
confirmed at NLO Chirilli:2011km . However, the observed negative cross
section when the hadron transverse momentum $p_{h,\perp}$ becomes a bit larger
has been quite puzzling to the community Stasto:2013cha . The NLO partonic
cross section contains contributions of the form

$\displaystyle\frac{\mathrm{d}^{2}\hat{\sigma}^{(1)}}{\mathrm{d}z\mathrm{d}^{2}p_{\perp}^{\prime}}\propto-\frac{\alpha_{s}}{2\pi}{\bf
T}^{2}_{i}P_{i\to
i}(z)\ln\frac{r_{\perp}^{2}\mu^{2}}{c_{0}^{2}}\left(1+\frac{1}{z^{2}}e^{i\frac{1-z}{z}p_{\perp}^{\prime}\cdot
r_{\perp}}\right)\,$ $\displaystyle-\frac{\alpha_{s}}{\pi}{\bf T}_{i}^{a}{\bf
T}_{j}^{a^{\prime}}\int\frac{\mathrm{d}x_{\perp}}{\,\pi}\left\\{\frac{1}{z}\frac{2}{(1-z)_{+}}\,e^{i\frac{1-z}{z}p_{\perp}^{\prime}\cdot
r_{\perp}^{\prime}}\frac{r_{\perp}^{\prime}\cdot
r_{\perp}^{\prime\prime}}{{r_{\perp}^{\prime}}^{2}{r_{\perp}^{\prime\prime}}^{2}}\right.$
$\displaystyle\hskip
8.61108pt\left.+\,\delta(1-z)\ln\frac{X_{f}}{X_{A}}\,\left[\frac{r_{\perp}^{2}}{{r_{\perp}^{\prime}}^{2}{r_{\perp}^{\prime\prime}}^{2}}\right]_{+}\right\\}W_{aa^{\prime}}(x_{\perp})+\dots\,,$
(18)

where $p_{\perp}^{\prime}$ is the transverse momentum of the fragmenting
parton. Here, $W_{aa^{\prime}}$ is the CGC Wilson line in the adjoint
representation, $X_{A}$ is the momentum fraction carried by the gluon from the
nucleus and $X_{f}$ is the scale due to the rapidity divergence
Fleming:2014rea ; Rothstein:2016bsq ; Liu:2019iml ; Kang:2019ysm . Moreover,
$P_{i\to i}(z)$ is the QCD splitting function. Other conventions of the
notation can be found in Liu:2020mpy . The terms in the second line are due to
the interference between initial and final state radiation. At forward
rapidity, the interference is not power suppressed. It was explicitly
demonstrated in Liu:2020mpy  that as the hadron transverse momentum
$p_{h,\perp}$ increases, $z$ quickly approaches $1$, and the threshold
logarithms of the form $\frac{1}{(1-z)_{+}}$ overwhelmingly dominate over the
other contributions and drive the cross section to negative values. Therefore,
an appropriate threshold resummation is required to ensure a reliable
prediction.

It is worthwhile to point out that the threshold resummation in the small-$x$
regime is different from the threshold resummation within the collinear
factorization, as is obvious from Eq. (4) where the color structure ${\bf
T}_{i}^{a}{\bf T}_{j}^{a^{\prime}}W_{aa^{\prime}}$ of the single logarithmic
interference term proposes new challenges and can hardly be produced by the
known threshold resummation techniques. Numerically, it is found that it is
this interference term that makes up the dominant bulk of the threshold
contribution Liu:2020mpy . Therefore, a new resummation formalism is required.

In Liu:2020mpy , the couplings between the CGC Wilson line
$W_{aa^{\prime}}(x)$ and the SCET soft and collinear fields are put in by
hand. Invoking SCET power counting and factorization, it is shown that the
cross section for this process can be factorized as

$\displaystyle\frac{\mathrm{d}^{2}\sigma}{\mathrm{d}y_{h}\mathrm{d}^{2}p_{h\perp}}=\sum_{i,j=g,q}\frac{1}{4\pi^{2}}\int\frac{\mathrm{d}\xi}{\xi^{2}}\,\frac{\mathrm{d}x}{x}\,zxf_{i/P}(x,\mu)D_{h/j}(\xi,\mu)$
$\displaystyle\times\int\mathrm{d}^{2}b_{\perp}\mathrm{d}^{2}b_{\perp}^{\prime}\,e^{ip_{\perp}^{\prime}\cdot
r_{\perp}}\,\Big{\langle}\langle{\bf{\cal
M}}_{0}(b_{\perp}^{\prime})|\,\bm{{\cal
J}}(z,\mu,\nu,b_{\perp},b_{\perp}^{\prime})\bm{{\cal
S}}(\mu,\nu,b_{\perp},b_{\perp}^{\prime})|{\bf{\cal
M}}_{0}(b_{\perp})\rangle\Big{\rangle}_{\nu}\,.$ (19)

where $\bm{{\cal J}}$ and $\bm{{\cal S}}$ represent the collinear and soft
contributions, respectively. The LO color space notation $|{\cal
M}_{0}(b_{\perp})\rangle$ Catani:1999ss  is used and encodes the CGC (Glauber)
Wilson line $W_{i_{c}j_{c}}(b_{\perp})$ where $i_{c}$ and $j_{c}$ are the
color indices for the incoming and the outgoing partons using the fundamental
for quarks and the adjoint representation for gluons.

Figure 5: Comparison of the CGC threshold resummation prediction with the LHC
ATLAS:2016xpn  and RHIC BRAHMS:2004xry  data. Figure taken from Ref.
Liu:2020mpy

It is shown in Liu:2020mpy  that the formalism

  * •

reproduces the NLO results in covariant gauge. The soft function reproduces
the so-called kinematic constraint, which was put in by hand in the previous
calculations, but shows up naturally and automatically within the SCET/CGC
hybrid formalism.

  * •

systematically regulates all the divergences by dimensional regularization or
by the rapidity regulator. And the final results are consistent with the
$\overline{\text{MS}}$ scheme; No cutoffs that violate the SCET power counting
are introduced;

  * •

naturally assigns the IR/rapidity poles to either the PDF/FFs or the nucleus
CGC dipole distribution. The scale dependence correctly reproduces the DGLAP
evolution for the proton and the final state hadron, as well as the small-$x$
BK equation for the colliding nucleus.

  * •

allows for the resummation of the threshold logarithms, which formally can be
achieved through the evolution factor of the jet and the soft function

$\displaystyle{\bm{U}}_{{J}}$
$\displaystyle{\bm{U}}_{{S}}=\exp\left[\,-\frac{\alpha_{s}}{\pi}\int\frac{\mathrm{d}x_{\perp}}{\pi}\left(\ln\frac{\nu_{S}}{\nu_{J}}I_{BK,r}+\ln\frac{X_{f}}{X_{A}}I_{BK}\right){\bf
T}^{a}_{i}{\bf T}^{a^{\prime}}_{j}\,W_{aa^{\prime}}(x_{\perp})\right]\,.$ (20)

Here we see clearly that the resummation is not a conventional Sudakov
evolution but it contains a complicated color structures and is highly non-
linear in color. The exponentiation of the CGC Wilson line $W_{aa^{\prime}}$
suggests that multiple structures will arise at higher orders which, again,
cannot be reproduced by conventional threshold resummation. After the
threshold resummation, the problem of the negative cross section is resolved
and the theoretical predictions are found to agree with the known data
Liu:2020mpy . A different approach to the negative cross section problem can
be found in Shi:2021hwx .

The success of handling the threshold logarithms in forward $pA$ collisions
suggests plausible future applications to other small-$x$ phenomena by
combining the SCET and CGC formalism. For instance to apply the formalism to
the forward di-jet/di-hadron production at the EIC. At the moment, the hybrid
formalism is performed in a brute force way by coupling the CGC shock wave
(Glauber Wilson line) to the SCET soft and collinear fields. In the future, a
first principle and more systematic treatment is required along this line.

##  5 Fragmentation and small-$z_{h}$ resummation

Another critical observable requiring resummation is fragmentation.
Fragmentation is simply the inclusive cross-section for a particle to carry a
certain amount of momentum in the event, while ignoring the rest of the
process. These cross-sections have long been foundational to our understanding
of both the perturbative and non-perturbative structure of high energy
scattering. Perturbative physics since, at asymptotically high energies, the
scaling properties with the hard momentum scale of the collision can be
calculated, but non-perturbative physics plays a critical role since the
observable is defined directly on the hadrons exiting the hard interaction.

The canonical example of fragmentation is $e^{+}e^{-}\rightarrow h+X$, where
one asks how many hadrons in the event carry a fraction of the total energy of
the event. This example also enjoys one of the oldest factorization theorems
of QCD, as well as the corresponding processes $ep\rightarrow h+X$ and
$pp\rightarrow h+X$, see Refs. Altarelli:1979kv ; PhysRevD.18.3705 ;
Collins:1981uw ; Furmanski:1981cw . If $Q$ is the center of mass energy of the
electron-positron pair, and $p_{h}$ is the momentum of the hadron, the cross-
section is:

$\displaystyle z_{h}$ $\displaystyle=\frac{2Q\cdot p_{h}}{Q^{2}}\,,\text{ with
}0\leq z_{h}\leq 1,$ (21)
$\displaystyle\frac{1}{\sigma_{0}}\frac{d\sigma}{dz_{h}}$
$\displaystyle=\sum_{i=q,\bar{q},g}\int_{z_{h}}^{1}\frac{dz}{z}C_{i}\Big{(}z,\mu^{2},Q^{2}\Big{)}d_{h/i}\Big{(}\frac{z_{h}}{z},\mu^{2},\Lambda_{QCD}^{2}\Big{)}+\mathcal{O}\Big{(}\frac{\Lambda^{2}_{QCD}}{Q^{2}}\Big{)}\,.$
(22)

Here $C_{i}$ is the coefficient function that describes how the $e^{+}e^{-}$
pair excite the QCD vacuum at high energies, and from this exits a parton of
flavor $i$ carrying energy fraction $z$. From this parton, a hadron is formed
carrying energy fraction $z_{h}$, and this process is captured by the
fragmentation function $d_{h/i}$. The fragmentation function depends upon the
scale of confinement $\Lambda_{QCD}$, and both functions depend on the
factorization scale $\mu$. The cross-section or number density
$d\sigma/\sigma_{0}$, (normalized by the total inclusive cross-section) does
not depend on the factorization scale $\mu$, leading to the renormalization
group equations:

$\displaystyle\bar{C}_{i}(n,\mu^{2},Q^{2})$
$\displaystyle=\int_{0}^{1}\frac{dz}{z}z^{n+1}C_{i}(z,\mu^{2},Q^{2})\,,$ (23)
$\displaystyle\bar{d}_{h/i}(n,\mu^{2},Q^{2})$
$\displaystyle=\int_{0}^{1}\frac{dz}{z}z^{n+1}d_{h/i}(z,\mu^{2},\Lambda_{QCD}^{2})\,,$
(24) $\displaystyle\mu^{2}\frac{d}{d\mu^{2}}\bar{C}_{i}(n,\mu^{2},Q^{2})$
$\displaystyle=-\sum_{j}\gamma^{T}_{ji}(n)\bar{C}_{j}(n,\mu^{2},Q^{2})\,,$
(25) $\displaystyle\mu^{2}\frac{d}{d\mu^{2}}\bar{d}_{h/i}(n,\mu^{2},Q^{2})$
$\displaystyle=\sum_{j}\gamma^{T}_{ij}(n)\bar{d}_{h/j}(n,\mu^{2},Q^{2})\,.$
(26)

All processes have the same fragmentation function $d_{h/i}$, an early and
testable prediction of factorization: the universality of the hadronization
process when the cross-section is sufficiently inclusive. Moreover, the
fragmentation to hadrons can be generalized to fragmentation to jets, where
one replaces the intrinsic cut-off scale of confinement with the jet radius
$R$, as was done in Refs. Dasgupta:2016bnd ; Kang:2016mcy ; Dai:2016hzf . At
fixed order, $C_{i}$ has been calculated to order $\alpha_{s}^{2}$ in Ref.
Rijken:1996vr ; Rijken:1996ns ; Mitov:2006wy , with partial results at the
same order for the DIS fragmentation process Refs. Abele:2021nyo ;
Anderle:2016kwa , and $\gamma^{T}_{ij}$ has been calculated to order
$\alpha_{s}^{3}$ Refs. Mitov:2006ic ; Moch:2007tx ; Almasy:2011eq ;
Chen:2020uvt .

The cross-section is largest in the $z_{h}\rightarrow 0$ limit. That is, the
bulk of hadrons produced have low energy. This has its origins in the fact
that QCD enjoys an enhanced infra-red region of phase-space, and the
$z_{h}\rightarrow 0$ limit probes the soft singularities of QCD. The
importance of this region was recognized early on, Refs. Mueller:1981ex ;
Mueller:1982cq ; Bassetto:1982ma , and a resummation of the logarithms of
$z_{h}$ in the anomalous dimension was accomplished at leading and next-to
leading order, however, with a scheme dependence which did not interface well
with the minimal subtraction schemes of dimensional regularization, making it
difficult to compare to fixed order calculations. At leading log order in the
$z_{h}\rightarrow 0$ limit, which in mellin space corresponds to $n\rightarrow
0$, we have:

$\displaystyle\gamma^{T}_{gg}(n)$
$\displaystyle=\frac{n}{4}\Big{(}-1+\sqrt{1+8\frac{\alpha_{s}C_{A}}{\pi
n^{2}}}\Big{)}+...\,.$ (27)

After performing the inverse mellin transform, one can see then that this is
double logarithmic series, _in the anomalous dimension itself_.

The resummation was further developed to embody the concepts of angular-
ordering and hard cut-offs in transverse momentum, culminating in the so-
called mixed leading log (MLL) approximation, see Refs. Fong:1990nt ;
Dokshitzer:1991ej ; Khoze:1996dn . The MLL has seen surprising phenomological
success (Ref. Akrawy:1990ha ) when coupled to the hypothesis of parton-hadron
duality (_very_ crudely, each parton becomes a hadron), see Ref. Azimov:1984np
, but with no clear connection to traditional fixed order calculations of the
anomalous dimension or the coefficient function.

More recently, two other approaches to small-$z_{h}$ resummation have been
developed, which have been pushed beyond leading logarithmic order, and
include a clear relationship to fixed order calculations.111Though see also
Ref. Albino:2011si , which clarified the role of scheme-dependence in
small-$z_{h}$ resummations, thus enabling the first successful matching of
fixed-order to resummed perturbation theory in the context of fragmentation.
The first approach of Refs. Kom:2012hd ; Vogt:2011jv , which has also been
used to resum transverse-momentum dependent fragmentation functions Ref.
Luo:2020epw , makes use of a set of recursion relations based on an ansatz
that the energy fraction $z_{h}$ must ultimately cut-off soft divergences in
the calculation. This leads to an over-determined set of linear relations,
whose unique solution enables N2LL resummation of the coefficient function,
and N3LL resummation of the anomalous dimension with arbitrary flavor
dependence. Parton-to-pion fragmentation functions have been fit using these
resummed coefficient functions and anomalous dimensions Ref. Anderle:2016czy .

The second approach of Ref. Neill:2020bwv  seeks to put the notion of an
angular-ordered cascade on firmer theoretical grounds, by explicitly writing
an evolution equation for the partonic coefficient function. The evolution
equation seeks to make use of a universal kernel that in principle should be
calculated from the BFKL equation that governs forward scattering and the
resummation of the DIS cross-section as Bjorken $x$ goes to zero. Consistency
of the BFKL equation Kuraev:1976ge ; Lipatov:1976zz ; Balitsky:1978ic ;
Kuraev:1977fs ; Lipatov:1985uk  with the collinear factorization of DIS cross-
section has been known to give the resummation of space-like anomalous
dimensions of twist-two operators, where the consistency in dimensional
regularization was first worked out in Ref. Catani:1993ww ; Catani:1994sq .
However, work on the resummation of the soft physics of NGLs in Refs.
Weigert:2003mm ; Hatta:2008st ; Avsar:2009yb ; Caron-Huot:2015bja ; Caron-
Huot:2016tzz  established an essential connection between forward scattering
physics and the resummation of non-global-logs (NGLs), based on the
correspondence between the BK and BMS equations Balitsky:1995ub ;
Kovchegov:1999yj ; Banfi:2002hw . While the NGL appear in semi-exclusion
processes where radiation is vetoed, in certain angular regions Ref.
Neill:2020bwv  makes the connection between forward scattering and soft parton
cascades explicit for the case of inclusive fragmentation, writing a
deformation of the BFKL equation which resums the _time-like_ anomalous
dimensions in complete analogy to the space-like case. It is worth noting that
Refs. Marchesini:2003nh ; Marchesini:2004ne  also examined a form of the BFKL
equation that could be used for fragmentation, but never explicitly connected
the results with the traditional collinear factorization approach.

In addition to developing a BFKL approach to soft fragmentation, the authors
also showed how one may write DGLAP-style evolution equations that resum the
small-$z_{h}$ logs. Focusing on the case of pure Yang-Mills, one postulates
that in $4-2\epsilon$ dimensions, the cross-section
$d(z_{h},\mu^{2},Q^{2},R^{2})$ for fragmenting a gluon of energy fraction
$z_{h}$ can be defined with an angular cut-off: the descendant parton cannot
approach within angle $R$ of their parent or siblings. This regulates the
collinear divergence. Then one takes the limit $R\rightarrow 0$ using the
evolution equation:

$\displaystyle R^{2}\frac{\partial}{\partial
R^{2}}z_{h}^{1+2\epsilon}d\Big{(}z_{h},\mu^{2},Q^{2},R^{2}\Big{)}$
$\displaystyle=\rho\Big{(}\frac{\mu^{2}}{R^{2}Q^{2}}\Big{)}z_{h}^{1+2\epsilon}d\Big{(}z_{h},\mu^{2},Q^{2},R^{2}\Big{)}$
$\displaystyle\qquad+\int_{z_{h}}^{1}\frac{dz}{z}P\Big{(}\frac{z_{h}}{z};\frac{\mu^{2}}{z^{2}R^{2}Q^{2}}\Big{)}z^{1+2\epsilon}d\Big{(}z,\mu^{2},Q^{2},R^{2}\Big{)}\,,$
(28) $\displaystyle
P\Big{(}\frac{z_{h}}{z};\frac{\mu^{2}}{z^{2}R^{2}Q^{2}}\Big{)}$
$\displaystyle=\sum_{\ell=1}^{\infty}P^{(\ell-1)}\Big{(}\frac{z_{h}}{z};\alpha_{s};\epsilon\Big{)}\,\Big{(}\frac{\mu^{2}}{z^{2}R^{2}Q^{2}}\Big{)}^{\ell\epsilon}\,,$
(29) $\displaystyle\rho\Big{(}\frac{\mu^{2}}{R^{2}Q^{2}}\Big{)}$
$\displaystyle=\sum_{\ell=1}^{\infty}\rho^{(\ell-1)}(\alpha_{s};\epsilon)\Big{(}\frac{\mu^{2}}{R^{2}Q^{2}}\Big{)}^{\ell\epsilon}\,.$
(30)

$P$ is derived from the space-like anomalous dimension in $4-2\epsilon$
dimensions of Refs. Ball:2005mj ; Ciafaloni:2005cg ; Ciafaloni:2006yk , and
$\rho$ is connected to the beta-function. After taking $R\rightarrow 0$, we
recover the bare perturbative cross-section in $4-2\epsilon$ dimensions, so:

$\displaystyle\bar{d}\Big{(}n,\mu^{2},\mu^{2},0\Big{)}$
$\displaystyle=\text{exp}\Big{(}\int_{0}^{\alpha_{s}}\frac{d\alpha}{\beta(\alpha,\epsilon)}\gamma^{T}(\alpha,n)\Big{)}\,\mathcal{R}^{T}(\alpha_{s},n).$
(31)

Both $\gamma^{T}$ and $\mathcal{R}^{T}$ are resummed automatically in the soft
limit, and indeed, $\mathcal{R}^{T}$ is the soft resummed coefficient function
of Eq. (22). The evolution equation (5) can be viewed as an appropriate
generalization of the reciprocity relations of Refs. Dokshitzer:2005bf ;
Basso:2006nk

Figure 6:  The fragmentation spectrum for $e^{+}e^{-}\rightarrow h^{\pm}+X$ at
center of mass energies $Q=91$ and $206$ GeV. The gray region is excluded from
the $\chi^{2}$/d.o.f determination based on hadron mass corrections or
$z_{h}\sim O(1)$, and the quoted $\chi^{2}$/d.o.f. is the goodness-of-fit for
that data set. All curves are fitted at $91$ GeV and evolved to $206$ GeV.
Compared to the NLL curve (red) is the evolved 3 flavor scheme MLLA limiting
spectrum (purple LS) from Ref. Akrawy:1990ha , and a 5 flavor scheme evolving
the moments of a Gaussian according to the MLLA anomalous dimension, as in
Ref. Fong:1990nt  (orange G).

Going into the future, there are multiple directions to study. Firstly, it is
necessary to extend the resummation of soft fragmentation beyond the well-
studied case of $e^{+}e^{-}\rightarrow h+X$ to processes that include initial
state hadrons like $ep\rightarrow h+X$ and $pp\rightarrow h+X$. Secondly,
there was a history of so-called fractal observables that were resummed in the
unsystematic MLL approximation, many of which are outlined in Ref.
Khoze:1996dn . It would be important to revisit these observables with an
improved understanding of the resummation of the soft fragmentation region.
Additionally, fragmentation in a constrained jet (Ref. Procura:2009vm ;
Jain:2011iu ) is critical for many experimental analyses, as is following the
flow of charge within the jet Ref. Waalewijn:2012sv ; Chang:2013rca ;
Li:2021zcf . The extension of the soft resummation to these cases would bring
insights into the flow of charge and momentum in QCD cascades. Lastly, the
approach of Ref. Neill:2020bwv  should be extended to include the full flavor
structure of QCD, and the exact connection to forward scattering physics
should move beyond conjectures and verification order-by-order, but involve
full-scale proofs and examples showing the limits of the correspondence
between time-like and space-like branching processes.

Figure 7:  The fragmentation spectrum extrapolated into the large $z_{h}$
region for $e^{+}e^{-}\rightarrow h^{\pm}+X$ at center of mass energies $Q=91$
and $189$ GeV. The region below ln$\frac{1}{z_{h}}<$0.5 was excluded from the
fit. Data taken from the OPAL, SLD, and ALEPH collaborations Refs. Abe:2003iy
; Akrawy:1990ha ; Heister:2003aj . The NLL result was a global fit to data
from $Q=35$ to $206$ GeV.

On a more phenomological note, Ref. Neill:2020tzl  initiated a study of
parton-hadron duality based off of the improved understanding of the
resummation of the soft physics from Ref. Neill:2020bwv , resumming all
logarithms of $z_{h}$ to NLL order. The results showed a marked improvement on
the older MLL approximation, see Fig. 6, where fragmentation to charged
hadrons is compared to theory predictions making use of parton-hadron-duality
hypothesis. Perhaps even more surprising was the ability to describe the data
in regions outside the soft limit, see Fig. 7. With improved analytical
control of the perturbative structure of the theory, the time seems ripe to
revisit parton-hadron duality, and the ultimate limits of perturbation theory.

##  6 Conclusions

The perturbative approach to calculate QCD scattering processes from first
principles can be spoiled by the presence of large logarithmic corrections
that appear in the power expansion of cross sections in the QCD strong
coupling constant $\alpha_{s}$. These large logarithmic corrections need to be
resummed to all orders in $\alpha_{s}$, which restores the predictive power of
perturbative QCD. In this work, we reviewed recent progress in QCD resummation
techniques. An improved understanding of the all-order structure of QCD is
relevant for the high-luminosity era at the LHC as well as future experiments
such as the Electron-Ion Collider. First, we considered the resummation at
subleading power, which has become possible for several processes. For
example, the threshold resummation for the Drell-Yan process and Higgs
production via gluon fusion has been now been performed at subleading power
and leading logarithmic accuracy. It was found that the numerical impact at
the LHC can be comparable to high-precision resummation at leading power. We
discussed techniques both in soft collinear effective theory and direct QCD.
Both methods still face conceptual challenges which hinder systematic
evaluation of the higher logarithmic orders. Second, we reviewed progress in
the calculation of jet and jet substructure observables. Here the challenging
aspect is the simultaneous resummation of multiple classes of large
logarithmic corrections. Especially, jet substructure observables can be
sensitive to several disparate energy scales that require careful treatment to
identify the relevant energy scales and establish the corresponding
factorization theorems. We discussed jet measurements at the LHC and outlined
how jet physics can provide new insights into the structure of hadrons at the
future EIC. Third, we focused on small-$x$ resummation, which is relevant in
the high-energy limit where the momentum fraction carried by the initial-state
gluon becomes small. We discussed recent work that combines resummation
techniques in SCET and the Color Glass Condensate formalism. The additional
resummation of threshold logarithms allows for the description of forward $pA$
scattering data. Lastly, we focused on the QCD fragmentation process. The
inclusive hadron cross section peaks in the limit where the momentum fraction
of the identified hadrons becomes small $z_{h}\to 0$. Large logarithmic
corrections in the coefficient function and the time-like anomalous dimension
need to be taken into account to achieve a reliable prediction within
perturbative QCD. We discussed recent progress that employs space-time
reciprocity relations, which connect the description of final-state hadrons to
BFKL dynamics. The resummation of small-$z_{h}$ logarithms allows for a
precise comparison to the available data from LEP and SLD colliders. The
topics discussed here provide a starting point for future studies at high
perturbative accuracy both at leading and subleading power. QCD resummation is
a critical ingredient to stress test the Standard Model of particle physics,
better understanding QCD itself, and to improve searches for physics beyond
the Standard Model.

###### Acknowledgements.

MvB acknowledges the UK Science and Technology Facilities Council (STFC) grant
452 number ST/T000864/1 and the Royal Society Research Professorship
(RP$\setminus$R1$\setminus$180112). SJ is supported by the UK Science and
Technology Facilities Council (STFC) under grant ST/T001011/1. XL is supported
by the National Natural Science Foundation of China under Grant No. 12175016.
DN is supported by the U.S. Department of Energy under Contract No.
89233218CNA000001 and by the LDRD program at LANL. FR is supported by the
Simons Foundation under the Simons Bridge program for Postdoctoral Fellowships
at SCGP and YITP, award number 815892 and the NSF, award number 1915093. RS is
supported by the United States Department of Energy under Grant Contract DE-
SC0012704. LV 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. GV is supported by
the United States Department of Energy, Contract DE-AC02-76SF00515. JW is
supported by the National Natural Science Foundation of China under Grant No.
12005117.

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