aainstitutetext:  Center for Theoretical Physics, Massachusetts Institute of
Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. bbinstitutetext: The NSF
AI Institute for Artificial Intelligence and Fundamental Interactions

# Pileup and Infrared Radiation Annihilation (PIRANHA): A Paradigm for
Continuous Jet Grooming

Samuel Alipour-fard a Patrick T. Komiske a Eric M. Metodiev a,b and Jesse
Thaler<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Jet grooming is an important strategy for analyzing relativistic particle
collisions in the presence of contaminating radiation. Most jet grooming
techniques introduce hard cutoffs to remove soft radiation, leading to
discontinuous behavior and associated experimental and theoretical challenges.
In this paper, we introduce Pileup and Infrared Radiation Annihilation
(Piranha), a paradigm for continuous jet grooming that overcomes the
discontinuity and infrared sensitivity of hard-cutoff grooming procedures. We
motivate Piranha from the perspective of optimal transport and the Energy
Mover’s Distance and review Apollonius Subtraction and Iterated Voronoi
Subtraction as examples of Piranha-style grooming. We then introduce a new
tree-based implementation of Piranha, Recursive Subtraction, with reduced
computational costs. Finally, we demonstrate the performance of Recursive
Subtraction in mitigating sensitivity to soft distortions from hadronization
and detector effects, and additive contamination from pileup and the
underlying event.

††preprint: MIT–CTP 5540

##  1 Introduction

Jets of hadrons produced in high-energy particle collisions are an important
guide in our understanding of the subatomic universe and quantum
chromodynamics (QCD), our fundamental theory of the strong nuclear force.
Quantifying the internal structure of hadronic jets plays an important role in
translating between experimental results and theoretical predictions. The
toolkit of jet substructure, including techniques from jet grooming
Krohn:2009th ; Ellis:2009me ; Larkoski:2014wba ; Dasgupta:2013ihk ;
Dasgupta:2013via ; Tseng:2013dva ; Dasgupta:2016ktv  and pileup mitigation
Cacciari:2007fd ; Soyez:2012hv ; Komiske:2017ubm ; Bertolini:2014bba ;
Cacciari:2008gn ; Butterworth:2008iy ; Chatrchyan:2012sn ; Alon:2011xb ;
Krohn:2013lba ; ATLAS-CONF-2013-083 ; ATLAS-CONF-2013-085 ; Chatrchyan:2012tt
; CMS-PAS-JME-13-005 ; Sirunyan:2020foa ; ATLAS-CONF-2012-066 ; Monk:2018clo ;
Martinez:2018fwc , helps characterize the radiation patterns produced by jets
in a wide variety of physical processes at colliders. Jet substructure can
distinguish jets produced by light quarks, gluons, and the decays of
electroweak bosons and top quarks Thaler:2008ju ; Thaler:2011gf ; Hook:2011cq
; Gallicchio:2011xq ; Soper:2012pb ; Gallicchio:2012ez ; CMS-PAS-JME-09-001 ;
CMS-PAS-EXO-09-002 ; CMS:2013kfa ; ATL-PHYS-PUB-2009-081 ; ATL-PHYS-
PUB-2010-008 ; Cui:2010km ; ATLAS-CONF-2011-053 ; Chatrchyan:2013rla ;
Larkoski:2013eya ; Dasgupta:2012hg ; Backovic:2013bga ; ATLAS-CONF-2013-084 ;
Komiske:2018vkc ; Komiske:2016rsd ; Metodiev:2018ftz ; Krohn:2012fg ;
MERINO:2013tta ; Bhattacherjee:2016bpy ; Macaluso:2018tck ; Egan:2017ojy ;
Kasieczka:2017nvn ; Pearkes:2017hku ; Butter:2017cot , provide accurate
predictions of the behavior of boosted objects in many contexts Catani:1992ua
; Dokshitzer:1998kz ; Dasgupta:2001sh ; Banfi:2004yd ; Banfi:2005gj ;
Ellis:2009su ; Banfi:2010pa ; Walsh:2011fz ; Chien:2012ur ; Li:2012bw ;
Jouttenus:2013hs ; Hatta:2013iba ; Larkoski:2014tva ; Procura:2018zpn ;
Aaboud:2017aca ; Frye:2016aiz ; Almeida:2008yp ; Larkoski:2017iuy ;
Larkoski:2017cqq ; Thaler:2010tr ; Ellis:2009me ; Abdesselam:2010pt ;
Katz:2010mr ; Gallicchio:2010dq ; Adams:2015hiv ; Sirunyan:2017ezt ;
Moore:2018lsr ; FerreiradeLima:2016gcz ; Rubin:2010fc ; Chatrchyan:2012sn ;
ATLAS:2019kwg ; CMS-PAS-BTV-13-001 ; CMS-PAS-JME-13-006 ; Kribs:2009yh ;
Chen:2010wk ; Hackstein:2010wk ; Kim:2010uj ; Almeida:2011aa ;
Pandolfi:2012ima ; Vernieri:2014wfa ; CMS-PAS-HIG-17-007 ; Procura:2014cba ,
and even reveal non-perturbative features of the infrared structure of QCD
Komiske:2019jim ; Komiske:2019fks ; Hoang:2019ceu ; Mateu:2012nk ;
Wobisch:1998wt .

Jet grooming facilitates our understanding of jet substructure by removing
low-energy pollution from jets. The major goal of jet grooming is to reduce
the sensitivity of jet observables to two conceptually distinct categories of
pollution:

  * •

Soft Distortions are low-energy effects that distort the radiation coming
directly from hard jet production, such as hadronization and detector effects;

  * •

Additive Contamination denotes a backdrop of low-energy radiation that
pollutes a jet but is not directly connected to jet production, including the
pileup (PU) of radiation from overlapping particle collisions and an
underlying event (UE) of secondary parton interactions.

The low-energy phenomena of QCD that produce both soft contamination and
additive contamination are notoriously difficult to model, and jet grooming
has become an important toolkit used to empower both experimental and
theoretical analyses of jet substructure Krohn:2009th ; Ellis:2009me ;
Larkoski:2014wba ; Dasgupta:2013ihk ; Dasgupta:2013via ; Tseng:2013dva ;
Dasgupta:2016ktv ; Thaler:2008ju ; Thaler:2011gf ; Hook:2011cq ;
Gallicchio:2011xq ; Soper:2012pb ; Gallicchio:2012ez ; CMS-PAS-JME-09-001 ;
CMS-PAS-EXO-09-002 ; CMS:2013kfa ; ATL-PHYS-PUB-2009-081 ; ATL-PHYS-
PUB-2010-008 ; Cui:2010km ; ATLAS-CONF-2011-053 ; Chatrchyan:2013rla ;
Larkoski:2013eya ; Dasgupta:2012hg ; Backovic:2013bga ; ATLAS-CONF-2013-084 ;
Komiske:2018vkc ; Komiske:2016rsd ; Metodiev:2018ftz ; Krohn:2012fg ;
MERINO:2013tta ; Bhattacherjee:2016bpy ; Macaluso:2018tck ; Egan:2017ojy ;
Kasieczka:2017nvn ; Pearkes:2017hku ; Butter:2017cot ; Catani:1992ua ;
Dokshitzer:1998kz ; Dasgupta:2001sh ; Banfi:2004yd ; Banfi:2005gj ;
Ellis:2009su ; Banfi:2010pa ; Walsh:2011fz ; Chien:2012ur ; Li:2012bw ;
Jouttenus:2013hs ; Hatta:2013iba ; Larkoski:2014tva ; Procura:2018zpn ;
Aaboud:2017aca ; Frye:2016aiz ; Almeida:2008yp ; Larkoski:2017iuy ;
Larkoski:2017cqq ; Thaler:2010tr ; Abdesselam:2010pt ; Katz:2010mr ;
Gallicchio:2010dq ; Adams:2015hiv ; Sirunyan:2017ezt ; Moore:2018lsr ;
FerreiradeLima:2016gcz ; Rubin:2010fc ; Chatrchyan:2012sn ; ATLAS:2019kwg ;
CMS-PAS-BTV-13-001 ; CMS-PAS-JME-13-006 ; Kribs:2009yh ; Chen:2010wk ;
Hackstein:2010wk ; Kim:2010uj ; Almeida:2011aa ; Pandolfi:2012ima ;
Vernieri:2014wfa ; CMS-PAS-HIG-17-007 ; Procura:2014cba ; ATL-PHYS-
PUB-2019-027 ; Aad:2019vyi ; ATLAS:2020gwe .

A complication of many popular jet grooming procedures is that they are
discontinuous in event space: they may map similar ungroomed jets, which
differ only by small changes in their distributions of energy, into
extensively distinct groomed jets Dasgupta:2013ihk ; Larkoski:2014wba .
Discontinuous behavior in jet grooming algorithms leads in turn to undesirably
sensitive responses to low-energy pollution. Experimentally, discontinuities
in grooming lead to large responses of groomed substructure to the
interactions of jets with experimental detectors, and resulting difficulties
in unfolding ATL-PHYS-PUB-2019-027 ; Aad:2019vyi ; ATLAS:2020gwe .
Theoretically, discontinuities in grooming lead to uncertainties in fixed-
order calculations Larkoski:2014wba  and complications in describing the
effects of hadronization on groomed jet substructure Hoang:2019ceu .

In this paper, we introduce Pileup and Infrared Radiation Annihilation
(Piranha), a paradigm for continuous jet grooming, using recent geometric
perspectives in collider physics. Piranha applies intuition from optimal
transport theory and the Energy Mover’s Distance (EMD) between collider events
Komiske:2019fks ; Komiske:2020qhg  to overcome the discontinuous behavior of
existing jet grooming algorithms. We introduce the new terminology of soft and
angular discontinuities to categorize discontinuous behavior under
infinitesimal changes to the energies and angles of jet constituents,
respectively. We explore the improved continuity of Piranha conceptually
through several toy examples, and more practically by studying the responses
of Piranha grooming to several sources of soft distortions and additive
contamination.

At present, Piranha has three concrete implementations:

  * •

Apollonius Subtraction (P-AS), introduced in Ref. Komiske:2020qhg , is a
continuous, conceptually simple application of the Piranha paradigm with a
computationally expensive implementation;

  * •

Iterated Voronoi Subtraction (P-IVS), also introduced in Ref. Komiske:2020qhg
, is a continuous, computationally efficient modification of P-AS with a
simple geometric implementation;

  * •

Recursive Subtraction (P-RS) is a family of nearly-continuous, computationally
efficient, tree-based algorithms motivated by P-AS and P-IVS. We develop the
concrete implementation of Recursive Subtraction with a Fraction
$\boldsymbol{f}$ (P-RSFf). We discuss how P-RSFf is discontinuous in
suppressed regions of parameter space, and focus on P-RSFf=1/2, which is the
only fully soft-continuous version of P-RSFf.

Table 1 expresses the infrared/collinear safety (using the definition of Ref.
Komiske:2020qhg ) and soft/angular continuity properties of the traditional
and Piranha groomers we discuss in this paper. The soft continuity properties
expressed in Table 1 are simple to understand by comparing the hard-cutoff
framework of popular groomers to the subtractive framework of Piranha. The
angular discontinuities expressed in Table 1 are more subtle and are inherited
from angular-ordered jet clustering. We note that the Recursive Subtraction
algorithms we introduce are discontinuous only in a highly suppressed region
of phase space, and we still find it suitable to call P-RS a family of Piranha
groomers.

| Safety | Continuity
---|---|---
Groomer | Acronym | Infrared | Collinear | Soft | Angular
Apollonius Subtraction | P-AS | ✓ | ✓ | ✓ | ✓
Iterated Voronoi Subtraction | P-IVS | ✓ | ✓ | ✓ | ✓
Recursive Sub. w/Fraction $f=1/2$ | P-RSFf=1/2 | ✓ | ✓ | ✓ |  .13ptyellow!75!black!$\mathbf{\bigtriangleup}$
Recursive Sub. w/Fraction $f\neq 1/2$ | P-RSFf≠1/2 | ✓ | ✓ |  .13ptyellow!75!black!$\mathbf{\bigtriangleup}$ |  .13ptyellow!75!black!$\mathbf{\bigtriangleup}$
Soft Drop (SD), $\beta_{\rm SD}>0$ | SDβ>0 | ✓ | ✓ | ✗ |  .13ptyellow!75!black!$\mathbf{\bigtriangleup}$
SD (grooming mode), $\beta_{\rm SD}\leq 0$ | SDβ≤0 | ✓ | ✗ | ✗ |  .13ptyellow!75!black!$\mathbf{\bigtriangleup}$
Table 1:  Safety and continuity properties of the Piranha and Soft Drop
grooming algorithms studied in this paper. The mark ✓ indicates that a groomer
satisfies the associated condition, while the mark ✗ indicates that it does
not. The mark .13ptyellow!75!black!$\mathbf{\bigtriangleup}$ indicates that a
groomer does not satisfy the condition, but only in a boundary region of phase
space that is exponentially suppressed at leading order in perturbative QCD.
The mark .13ptyellow!75!black!$\mathbf{\bigtriangleup}$ indicates that a
groomer does not satisfy the condition, but only in a boundary region of phase
space that is exponentially suppressed at leading order in perturbative
QCD.111 Soft Drop with $\beta_{\rm SD}\leq 0$ is not collinear safe in
grooming mode, used in this work, where jets consisting of a single particle
are retained as groomed jets. Grooming mode is a natural choice in the context
of pileup mitigation (see Sec. 3.2). Note that Soft Drop with $\beta_{\rm
SD}\leq 0$ is collinear safe in tagging mode, where jets consisting of a
single particle are groomed away completely.

$\mathcal{E}^{+}$$\mathcal{E}^{-}$Ungroomed$z_{\rm cut}$$z_{\rm
cut}$$z^{+}>z_{\rm cut}$$z^{-}<z_{\rm cut}$Soft Drop, $\beta_{\rm SD}=0$P-RSF
(arb. $f_{\rm
soft}$)QCD(Pythia8.244)QCD+Noise$\phi$$\phi$$y$$y$$y$-11-11-11-11-11

Figure 1:  A visualization of how similar ungroomed events (first column, in
black) may be mapped discontinuously to distinct groomed jets by the hard-
cutoff mMDT algorithm (Soft Drop with $\beta_{\rm SD}=0$, or SDβ=0; middle
column, in blue), while the continuous P-RSFf=1/2 algorithm will map them to
similar groomed results (final column, in orange). In the first two rows, we
compare the action of mMDT and P-RSFf=1/2 on simple toy events
$\mathcal{E}^{\pm}$, with particles visualized as arrows whose size
corresponds to their energy. In the final two rows, we compare the groomers
acting on a more realistic QCD event produced with Pythia 8.244, and on the
same QCD event with additional Gaussian noise, respectively.

Popular jet grooming algorithms are soft discontinuous because they introduce
hard cutoffs on the energy of jet constituents, removing particles that are
too low-energy to be classified as important probes of hard physics; we call
such groomers hard-cutoff groomers. The modified Mass Drop Tagger (mMDT)
Dasgupta:2013ihk , for example, requires that certain sub-jets within a jet
carry a sufficient energy fraction, $z>z_{\rm cut}$, to survive the grooming
procedure. If a sub-jet has an energy fraction near $z_{\rm cut}$, low-energy
pollution may push its energy to the other side of the cutoff, leading to soft
discontinuous behavior and associated theoretical and experimental
complications. Fig. 1 demonstrates this soft discontinuous behavior, the
corresponding soft continuity of P-RSFf=1/2, by comparing their action on two
sets of similar events: two toy jet events $\mathcal{E}^{\pm}$ consisting of
two particles each, and a QCD jet produced in Pythia 8.244 along with the same
QCD event with a small amount of gaussian noise applied to the $p_{T}$,
rapidity, and azimuthal angle of each particle. We revisit the events
$\mathcal{E}^{\pm}$ as tools to characterize soft discontinuity more precisely
in Sec. 3.2. The mMDT grooming procedure maps $\mathcal{E}^{+}$ and
$\mathcal{E}^{-}$ to very distinct events, and similarly maps the noiseless
and noisy QCD jets to very distinct groomed jets. We chose an unusually high
value of $z_{\rm cut}=0.25$ when grooming the QCD events to make the
discontinuity of mMDT more apparent. We will also use Soft Drop
Larkoski:2014wba , a generalization of mMDT reviewed in Sec. 3.1 which
reproduces mMDT when a parameter $\beta_{\rm SD}$ is set to zero, as another
example of soft discontinuous hard-cutoff grooming.

Piranha borrows technology from optimal transport to produce soft continuous,
subtractive jet grooming algorithms without hard cutoffs. Piranha groomers may
be roughly imagined as optimally transporting hungry piranhas to eat up
delicious low-energy pollution. Given an ungroomed event with energy density
$\mathcal{E}_{0}$ in the pseudorapidity-azimuth angle plane and any ansatz
$\rho\,\mathcal{C}$ for the energy density of low-energy pollution, one
particularly succinct Piranha algorithm is

$\displaystyle\mathcal{E}_{g}\left[\mathcal{E}_{0}\right]=\underset{\mathcal{E}^{\prime}}{\arg\min}\
{\rm
EMD}_{\beta,R}\left(\mathcal{E}_{0},\mathcal{E}^{\prime}+\rho\,\mathcal{C}\right),$
(1)

where $\mathcal{E}_{g}$ is a groomed energy distribution, $\rho\,\mathcal{C}$
is a free input to the algorithm indicating a model of the energy flow of the
contaminating radiation (see Sec. 2.1), and explicit dependence on the EMD
metric makes the continuity of $\mathcal{E}_{g}\left[\mathcal{E}_{0}\right]$
manifest.

P-AS is a direct, continuous application of Eq. (1) with $\beta=1$, and for
which $\rho\,\mathcal{C}$ is uniform in the pseudorapidity-azimuth plane to
approximate common sources of additive contamination such as PU and UE
Soyez:2018opl ; Monk:2018clo ; Sjostrand:1987su ; Sjostrand:2014zea ;
Dasgupta:2007wa ; Kirchgaesser:2020poq ; Moraes:2007rq ; CDF:2015txs ;
Larkoski:2021hee ; Baron:2020xoi ; Marzani:2017kqd . P-IVS approximates P-AS
and preserves its continuity. Finally, the Recursive Subtraction with a
Fraction 1/2 (P-RSFf=1/2) algorithm we introduce in Sec. 2.5 furnishes, to our
knowledge, the first example of a soft continuous tree-based jet grooming
algorithm; the soft discontinuities of other P-RS algorithms occur only in
highly suppressed regions of phase space.

The rest of the paper proceeds as follows: in Sec. 2, we review the Energy
Mover’s Distance (EMD) of Ref. Komiske:2019fks , propose the Piranha paradigm
for continuous jet grooming, review the Apollonius Subtraction (P-AS) and
Iterated Voronoi Subtraction (P-IVS) algorithms of Ref. Komiske:2020qhg , and
introduce Recursive Subtraction (P-RS) with a Fraction (P-RSF) as a new, tree-
based implementation of Piranha. In Sec. 3, we discuss the soft
discontinuities present in hard-cutoff groomers and P-RSF, demonstrate the
soft continuity of P-RSFf=1/2, and examine angular discontinuity in tree-based
groomers. In the remainder of the paper, we demonstrate the soft insensitivity
enjoyed by Piranha groomers over hard-cutoff groomers through several
phenomenological studies. In Sec. 4, we compare the responses of the hard-
cutoff Soft Drop algorithm and P-RSFf=1/2 to soft distortions from
hadronization and to the removal of neutral particles, as a proxy for detector
effects. In Sec. 5, we compare the ability of Soft Drop and P-RSFf=1/2 to
effectively remove additive contamination from pileup and the underlying
event. We give concluding thoughts and avenues for future exploration in Sec.
6, delegating discussions of Piranha grooming in EMD mode, comparisons of
other hard-cutoff and Piranha groomers, and a first look at the perturbative
structure of Piranha grooming to Apps. A, B, and C, respectively. The Piranha
algorithms used in our analysis are available on GitHub piranhagithub .

##  2 The PIRANHA Paradigm for Continuous Grooming

This section introduces the Piranha paradigm for jet grooming, which uses
optimal transport to continuously remove low-energy pollution from jets. We
first review the Energy Mover’s Distance (EMD) of Ref. Komiske:2019fks , which
provides a framework to introduce the principles of Piranha groomers and to
enforce their continuity. We then discuss three implementations of Piranha:
Apollonius Subtraction (P-AS) and Iterated Voronoi Subtraction (P-IVS) – both
introduced in Ref. Komiske:2020qhg  – and the new tree-based Recursive
Subtraction (P-RS) family of Piranha algorithms. We describe a variant of
Piranha – grooming in EMD mode – in App. A. We end this section with Sec. 2.7,
which may be skipped on a first reading, where we present definitions of
stronger notions of continuity that could potentially pave the way for more
constrained and well-behaved strategies for continuous grooming.

###  2.1 Review of the Energy Mover’s Distance

The Energy Mover’s Distance (EMD) of Ref. Komiske:2019fks  provides a
quantitative measure of the similarity between two jets. We introduce the EMD
here both to facilitate our definition of continuity in jet grooming and as a
useful tool for quantifying the responses of jet groomers to low-energy
pollution. Readers familiar with the EMD should feel free to skip to Sec. 2.2.

The EMD is a metric on the space of collider events. Here and in the remainder
of the paper, we borrow the terminology of Ref. Komiske:2020qhg  and use the
terms “event” and “collider event” to refer to the energy flow of the event,
or the angular distribution of energy of its outgoing radiation.222 For
example, the energy flow of an event $\mathcal{E}$ with $M$ outgoing particles
is the angular distribution
$\mathcal{E}(\hat{n})=\sum_{i=1}^{M}E_{i}\,\delta(\hat{n}-\hat{n}_{i}),$ where
$E_{i}>0$ denotes the energy carried by particle $i$, and $\hat{n}_{i}$
denotes its outgoing angular direction.  The EMD may be thought of as the
amount of “work” required to rearrange one event into another. For events that
consist of a finite number of outgoing particles, the EMD is defined as the
solution to the optimal transport problem

${\rm
EMD}_{\beta,R}\left(\mathcal{E},\mathcal{E}^{\prime}\right)=\min_{f_{ij}>0}\sum_{i=1}^{M}\sum_{j=1}^{M^{\prime}}f_{ij}\left(\frac{\theta_{ij}}{R}\right)^{\beta}+\left|\sum_{i=1}^{M}E_{i}-\sum_{j=1}^{M^{\prime}}E^{\prime}_{j}\right|,$
(2) $\sum_{i=1}^{M}f_{ij}\leq
E_{j}^{\prime},~{}~{}\sum_{j=1}^{M^{\prime}}f_{ij}\leq
E_{i},~{}~{}\sum_{i=1}^{M}\sum_{j=1}^{M^{\prime}}f_{ij}=\min\left(\sum_{i=1}^{M}E_{i},\sum_{j=1}^{M^{\prime}}E^{\prime}_{j}\right),$
(3)

where $i\in\\{1,...,M\\}$ indicates a final-state particle of $\mathcal{E}$
with energy $E_{i}$, $j\in\\{1,...,M^{\prime}\\}$ indicates a final-state
particle of $\mathcal{E}^{\prime}$ with energy $E^{\prime}_{j}$, and
$\theta_{ij}$ denotes an angular distance between particles $i$ and $j$.
$\beta>0$ and $R>0$ are free parameters that control the behavior and relative
importance of the first term on the right-hand side of Eq. (2).333 Strictly
speaking, the EMD defines a metric only if $2R$ is greater than the maximum
possible angular distance $\theta_{ij}$. Furthermore, for $\beta>1$, one must
raise the first term of Eq. (2) to the $1/\beta$ power. We use the EMD without
a subscript, ${\rm EMD}\left(\mathcal{E},\mathcal{E}^{\prime}\right)$ to
denote the EMD with $\beta=1$ and with large enough $R$ that it furnishes a
metric.

In this work, we adopt the hadronic angular measure of Ref. Komiske:2020qhg ,
given by

$\theta_{ij}=\sqrt{-(n_{i}-n_{j})^{2}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n_{i}^{\mu}=\frac{p_{i}^{\mu}}{E_{T\,i}}=(\cosh(y_{i}),\textbf{v}_{T\,i},\sinh(y_{i}))^{\mu}.$
(4)

Here, $n_{i}^{\mu}$ is a vector describing the motion of particle $i$,
parameterized by its rapidity $y_{i}$ and its velocity $\textbf{v}_{T\,i}$ in
the directions transverse to the beampipe. This angular metric reproduces the
rapidity-azimuth distance between particles $i$ and $j$ in the small rapidity-
azimuth distance limit.444 Another popular choice for the form of the
$n_{i}^{\mu}$ is captured by  $\displaystyle
n_{i,~{}{e^{+}e^{-}}}^{\mu}=\frac{p_{i}^{\mu}}{E_{i}}=(1,\textbf{v}_{i})^{\mu},$
(5) which is a natural choice in the study of electron-positron collisions
Komiske:2020qhg . This choice reproduces the real-space opening angle between
particles $i$ and $j$ in the small angle limit.

The EMD between a groomed jet in the presence and absence of low-energy
pollution also provides an observable-independent tool for quantifying the
sensitivity of jet grooming to pollution from soft distortions and additive
contamination. Since the EMD furnishes a metric on the space of events, two
events are separated by zero EMD iff they have identical energy flow, and two
jets separated by zero EMD will yield the same value for every
infrared/collinear (IRC) safe observable (see Lemma 1 of Ref. Komiske:2020qhg
). Further, the EMD between two events also directly bounds the differences in
a large class of IRC-safe observable quantities (Lipschitz continuous
observables) between the events Komiske:2019fks . For example, the EMD between
a parton-level jet and the same jet after hadronization (the parton-hadron
EMD) directly bounds the difference between a large class of parton- and
hadron-level IRC-safe observables. The parton-hadron EMD is therefore a
powerful, observable independent tool for characterizing the response of a jet
to soft distortions from hadronization. We use the EMD in our analysis of
grooming sensitivity to several sources of low-energy pollution in Secs. 4 and
5.

###  2.2 PIRANHA and Continuity

Piranha grooming may be intuitively imagined as the optimal transport of a
group of hungry piranhas towards the low-energy pollution within a jet. In
this analogy, henceforth the piranha analogy, a Piranha groomer populates the
$\eta$-$\phi$ plane with a school of piranhas, distributed according to a
particular model of the polluting energy density. When they are given an event
to groom, the piranhas discuss which piranhas will get to eat which jet
constituent; in the language of optimal transport, we say that the piranhas
decide on a set of transport plans. The piranhas then implement the plan and
feed on the constituents of the jet, or subtract the $p_{T}$ of each jet
constituent.

Piranha grooming uses optimal transport to design continuous grooming
procedures that have reduced sensitivity to low-energy pollution. In
particular, continuity requires that minuscule changes to the jet do not
dramatically change the resulting groomed jet. In the piranha analogy,
continuity emerges when the piranhas’ transport plans do not respond
dramatically to small changes in the constituents of an ungroomed jet. Since
both soft distortions and fluctuations in additive contamination may lead to
small changes in an event, continuity helps ensure that neither dramatically
changes the information carried by groomed jets.

We may use the EMD to define continuity more precisely for maps on energy
flows, in the spirit of Definition 1 of Ref. Komiske:2020qhg :

######  Definition 1.

A map $M$ from energy flows to energy flows is continuous at an event
$\mathcal{E}$ if, for any $\varepsilon>0$, there exists a $\delta>0$ such that
for all $\mathcal{E}^{\prime}$,

${\rm EMD}(\mathcal{E},\mathcal{E}^{\prime})<\delta\quad\implies\quad{\rm
EMD}(M(\mathcal{E}),M(\mathcal{E}^{\prime}))<\varepsilon.$

We say that $M$ is continuous if it is continuous at all events $\mathcal{E}$.

Definition 1 encodes the requirement that a continuous grooming algorithm must
map jets that are infinitesimally nearby in event space to groomed results
that are also infinitesimally close.555 Equivalently, we may say that a
grooming procedure is continuous if it maps nearby jets to nearby groomed
jets. An event $\mathcal{E}$ is near an open ball $\mathcal{B}$ if any
neighborhood of $\mathcal{E}$ contains events in $\mathcal{B}$. On the real
line, for example, the point $E=1$ is near the open ball $B=(-1,1)$. A
grooming procedure $G$ is then continuous if for any event $\mathcal{E}$ and
open ball $\mathcal{B}$,
$\mathcal{E}{\text{~{}}is~{}near~{}}\mathcal{B}\implies
G(\mathcal{E}){\text{~{}}is~{}near~{}}G(\mathcal{B}).$

The EMD itself also furnishes an example of an optimal transport problem,
allowing us to immediately introduce a simple Piranha groomer. To begin, let
us model the radiation that we would like to remove from our jet as a
distribution of energy $\rho\,\mathcal{C}$ in a portion of the rapidity-
azimuth ($\eta$-$\phi$) plane with area $A$. We use $\rho$ to indicate the
mean energy density and $\mathcal{C}$ to indicate a distribution modeling the
shape of the contaminating radiation, normalized such that

$\rho\int\differential y\,\differential\phi\,\mathcal{C}(y,\phi)=\rho A=E$ (6)

gives the total energy of the additive contamination we would like to remove.
Again echoing Ref. Komiske:2020qhg , we may phrase the subtraction of the
energy distribution $\rho\,\mathcal{C}$ from an event $\mathcal{E}$ as the
solution to the optimal transport problem

$\displaystyle\mathcal{E}_{g}[\mathcal{E}_{0}]=\underset{\mathcal{E}^{\prime}}{\arg\min}\
{\rm
EMD}_{\beta,R}\left(\mathcal{E}_{0},\mathcal{E}^{\prime}+\rho\,\mathcal{C}\right).$
(7)

It was shown in Lemma 3 of Ref. Komiske:2020qhg  that Eq. (7) defines a
subtractive algorithm that merely decreases the energy of jet constituents,
and does not add new particles to an event. In the piranha analogy,
$\rho\,\mathcal{C}$ roughly describes the distribution of our piranhas, with a
“piranha energy density” $\rho$, while the minimization problem of Eq. (7)
provides them with the transport plans that guide them to subtract from the
$p_{T}$ of jet constituents. Furthermore, the form of
$\mathcal{E}_{g}[\mathcal{E}_{0}]$ in Eq. (7) is manifestly continuous due to
its dependence on the EMD metric.666 While $\mathcal{E}_{g}[\mathcal{E}_{0}]$
in Eq. (7) is continuous, manifest continuity due to dependence on the EMD
should be treated skeptically. For example, the “grooming procedure”
associated with Eq. (7) may not be continuous if the $\arg\min$ operation is
restricted to a non-convex sub-space of the full space of events. Analogously,
$f(x)=\underset{y}{\arg\min}\ ||x,y+1||=x-1$ is continuous as a map from
$\mathbb{R}$ to $\mathbb{R}$ with the standard topology. However,
$g(x)=\underset{y\in\\{0,1\\}}{\arg\min}\ ||x,y+1||,$ which differs only by
the allowed values of $y$ in the $\arg\min$, is not.

For the more complicated Piranha groomers we introduce in this paper, it will
be helpful to define two additional properties that are required by full
continuity:

  * •

Soft Continuity is the requirement that a groomer is invariant under
infinitesimally soft perturbations of the energies of jet constituents;

  * •

Angular Continuity is the requirement that a groomer is invariant under
infinitesimally small angular changes in the directions of jet constituents.

We will see that hard-cutoff groomers and even many recursive subtraction
algorithms suffer from soft discontinuity in Secs. 3.1 and 3.4 respectively.
In Sec. 3.5, we will explore how tree-based grooming algorithms – hard-cutoff
and Piranha alike – may inherit angularly discontinuous behavior from angular-
ordered jet clustering.777 Collinear safety, or invariance under exact
collinear splittings which replace a final-state particle of momentum
$p^{\mu}$ by two particles with momenta $\lambda p^{\mu}$ and
$(1-\lambda)p^{\mu}$, is a weaker condition. Exact collinear splittings do not
change the energy distribution associated with an event; therefore, any well-
defined map on the space of particle events must be collinear safe
Komiske:2020qhg .

Cartoon of a Piranha groomeracting on the space of energy flows (a)

Visualization of P-AS acting onan event generated in Pythia 8.244 (b)

Figure 2:  (a) A cartoon of a generic implementation of the Piranha paradigm
described by Eq. (7), based on a figure from Ref. Komiske:2020qhg .
$\mathcal{E}\in\Omega$ indicates a particular event $\mathcal{E}$ in the space
$\Omega$ of events to be groomed, $\mathcal{C}$ indicates a model energy
distribution for contaminating radiation, and $\rho\in\mathbb{R}$
parameterizes the strength of the grooming. Eq. (7) uses these ingredients to
produce the groomed event $\mathcal{E}_{g}$. (b) An Apollonius diagram in the
rapidity-azimuth plane, also from Ref. Komiske:2020qhg , used by the
Apollonius Subtraction (P-AS) algorithm. P-AS, an implementation of the
Piranha paradigm described in Sec. 2.3, models contaminating radiation as
uniform in the rapidity-azimuth plane $\mathcal{C}=\mathcal{U}$. The color
intensity of each Apollonius region is proportional to the amount by which the
corresponding particle $i$ is groomed, and thus to the area $A^{\rm
Apoll.}_{i}$ of the region.

###  2.3 Apollonius Subtraction (P-AS)

Apollonius Subtraction (P-AS) is a direct application of Eq. (7), in which we
take contaminating radiation to be uniformly distributed in a region of the
$\eta$-$\phi$ plane bounded by a maximum pseudorapidity, $|\eta|<\eta_{\rm
max}$. We denote the uniform event with energy density $\rho$ by
$\rho\,\mathcal{U}$. This has proven to be an effective model for pileup, the
underlying event, and initial state radiation Soyez:2018opl ; Monk:2018clo ;
Sjostrand:1987su ; Sjostrand:2014zea ; Dasgupta:2007wa ; Kirchgaesser:2020poq
; Moraes:2007rq ; CDF:2015txs ; Larkoski:2021hee ; Baron:2020xoi ;
Marzani:2017kqd , and motivates our approach to jet grooming for the remainder
of the paper.

More precisely, using $\beta=1$, $R=1$, and replacing $\rho\,\mathcal{C}$ with
$\rho\,\mathcal{U}$ in Eq. (7) yields an optimal transport problem known as an
Apollonius problem. The Apollonius problem is well studied in the optimal
transport literature and can be solved with an Apollonius diagram, or
additively weighted Voronoi diagram, which assigns an Apollonius region to
each particle in the event. The precise structure of the Apollonius diagram
corresponding to an event is described in a physics context in Section 5.4 of
Ref. Komiske:2020qhg , and in the context of optimal transport in Refs.
hartmann2017geometrybased ; hartmann2018semidiscrete ; bourne2018semidiscrete
.888 Eq. (7) with $\rho\,\mathcal{C}=\rho\,\mathcal{U}$ and arbitrary positive
$\beta$ also describes a valid optimal transport problem and an associated
Piranha groomer. The solution to the associated optimal transport problem is
described by a generalized Laguerre diagram Komiske:2020qhg ;
bourne2018semidiscrete . We leave the study of Piranha grooming motivated by
generalized Laguerre diagrams for future work.

P-AS may then be phrased as a constituent-level area subtraction procedure
that solves the Apollonius problem for a given event. To solve the Apollonius
problem associated with a particular Apollonius diagram, we subtract from the
$p_{T}$ of each particle an amount proportional to the area of its Apollonius
region, $p_{T\,i}^{\rm AS}=p_{T\,i}-\rho A_{i}^{\rm Apoll.}$. Letting $z_{\rm
cut}=\rho\,A_{\rm tot}/p_{T\,{\rm tot}}$, in analogy to the $z_{\rm cut}$ of
traditional groomers such as Soft Drop, we may equivalently write
$p_{T\,i}^{\rm AS}=p_{T\,i}-z_{\rm cut}\,p_{T\,{\rm tot}}\,A_{i}^{\rm
Apoll.}/A_{\rm tot}$. In the piranha analogy, the transport plans for an event
are determined by its Apollonius diagram, and all of the piranhas within a
given Apollonius region feed on the associated jet constituent. A rigorous
proof of the continuity of P-AS is given in Lemma 3.3 of Ref.
hartmann2017geometrybased , and a visual representation of P-AS, taken from
Ref. Komiske:2020qhg , is depicted in Fig. 2.

P-AS is closely related to constituent-level area subtraction techniques for
pileup mitigation. Indeed, P-AS was initially introduced in the context of
pileup mitigation in Ref. Komiske:2020qhg , where it was shown that in certain
limits, Apollonius Subtraction aligns with the discontinuous Voronoi Area
Subtraction (VAS) procedure for pileup mitigation Cacciari:2007fd ;
Cacciari:2008gn ; Cacciari:2011ma . Thus, P-AS may be thought of as a
continuous analog to existing techniques for constituent-level area
subtraction. P-AS also aligns in certain limits with continuum Constituent
Subtraction (CS), a continuous but computationally expensive algorithm for the
removal of pileup Berta:2014eza ; Komiske:2020qhg .

Unfortunately, the computational cost of finding the Apollonius diagram for a
given particle event is relatively high because we rely on directly solving
the Apollonius problem using numerical ghosts Komiske:2020qhg : we use a
uniform grid of “ghost” particles in the $\eta$-$\phi$ plane to obtain the
Apollonius diagram associated with a particular event computationally. Data
collection and analysis for theoretical and experimental studies at colliders
motivate the computationally efficient analogs of P-AS that we describe in the
remainder of this section.999Another approach to developing analogs of P-AS
that we do not pursue in this work is to use the formalism of linearized
optimal transport, explored mathematically in Refs. cai2022linearized ;
sarrazin2023linearized  and applied to particle collisions in Refs.
Cai:2020vzx ; Cai:2021hnn , which preserves the strengths of the EMD and
significantly reduces the computational costs associated with optimal
transport problems. A comparison of the runtime for our different Piranha
algorithms is given in Fig. 17 of App. B.

###  2.4 Iterated Voronoi Subtraction (P-IVS)

Iterated Voronoi Subtraction (P-IVS) is a Piranha groomer similar to both P-AS
and VAS that overcomes the computational inefficiency of P-AS. P-IVS also uses
the uniform event $\rho\,\mathcal{U}$ as a model for additive contamination,
but crucially uses Voronoi diagrams rather than Apollonius diagrams to
subtract this contamination away.

Like P-AS, the amount of grooming performed by the P-IVS grooming procedure is
encoded in a parameter $\rho$. We may describe P-IVS quite succinctly as the
solution to the iterated series of optimal transport problems Komiske:2020qhg
:

$\displaystyle\mathcal{E}^{(n+1)}_{\rm
IVS}=\underset{\mathcal{E}^{\prime}}{\arg\min}~{}{\rm
EMD}(\mathcal{E}^{(n)},\mathcal{E}^{\prime}+\rho^{(n)}\,\mathcal{U}),$ (8)

with

$\displaystyle\rho^{(n)}=\min\left\\{\rho-\sum_{i=0}^{n-1}\rho^{(i)},~{}\min_{i}\frac{p^{(n)}_{T\,i}}{A^{(n)}_{i}}\right\\}.$
(9)

The final groomed event is given by the limit of the recursive procedure in
Eq. (8), $\mathcal{E}_{\rm IVS}=\lim_{n\to\infty}\mathcal{E}^{(n)}_{\rm IVS}$.
Here, $\rho^{(0)}=0$, $p_{T\,i}^{(n)}$ is the transverse momentum of particle
$i$ after $n$ applications of the recursive algorithm in Eq. (8), and
$A_{i}^{(n)}$ is the Voronoi region for the event after $n$ applications of
Eq. (8). For example, $p_{T\,i}^{(0)}$ and $A_{i}^{(0)}$ describe the
transverse momenta and Voronoi areas of the ungroomed event.

More precisely, we may describe our ungroomed event as a collection of points
in the $\eta$-$\phi$ plane describing the directions of outgoing momenta, each
weighted by its transverse momentum. We may then enumerate the steps of the
IVS procedure as follows:

  1. 1.

P-IVS constructs a Voronoi diagram in the $\eta$-$\phi$ plane for the
particles in the event. We will label the stage of the P-IVS procedure by an
integer $n$, starting at $n=1$ and going up to at most the number of particles
in the event.

  2. 2.

P-IVS modifies the $p_{T}$ of every particle in the event, subtracting $p_{T}$
proportional to the area of the particle’s Voronoi region until a particle
would be removed or the grooming is complete:

$\displaystyle p_{T\,i}^{(n)}=p_{T\,i}^{(n-1)}-\rho^{(n)}A^{(n-1)}_{i},$ (10)

with $\rho^{(n)}$ given by Eq. (9).

  3. 3.

If $\rho-\sum_{i=1}^{n}\rho^{(i)}>0$, P-IVS increments $n$ by 1 and continues
from Step 1 by drawing a new Voronoi diagram for the modified event with a
particle removed. Otherwise, the grooming is complete.

As with P-AS, the simple expression of P-IVS in terms of the EMD already
showcases its continuity and connection to optimal transport. One may worry
that the presence of an infinitesimally soft particle in an event may
dramatically change the Voronoi diagram for an event, and therefore
dramatically change the result of grooming using Voronoi areas. P-IVS
overcomes this challenge by only using Voronoi areas until a particle is
removed; when a particle is removed from the event, P-IVS computes an updated
set of Voronoi areas that do not rely on the removed particle, continuing
recursively until it has removed the correct amount of energy. A rigorous
proof of the continuity of P-IVS is given in Lemma 3.3 of Ref.
hartmann2017geometrybased .

Unweighted Voronoi diagrams for the original event $\mathcal{E}^{(0)}$ can be
found efficiently, unlike the weighted Voronoi/Apollonius diagrams needed for
P-AS. Furthermore, the Voronoi diagrams for the subtracted events
$\mathcal{E}^{(n)}_{\rm IVS}$ used throughout the stages of the P-IVS
algorithm do not need to be computed from scratch, and can be found in
constant (amortized) time Komiske:2020qhg . P-IVS thus retains the continuous
grooming properties of P-AS while remaining amenable to numerical computation
and data collection.

###  2.5 Tree-Based PIRANHA: Recursive Subtraction (P-RS) with a Fraction (F)

Recursive Subtraction (P-RS) is an extension of the Piranha paradigm into the
space of tree-based grooming algorithms and draws on the strengths of both
hard-cutoff and Piranha grooming. We use “Recursive Subtraction” to denote an
algorithm that takes in a binary tree of emissions describing a jet and
recursively subtracts from the momenta of its jet constituents. The dependence
of P-RS on binary trees is reminiscent of many hard-cutoff grooming
strategies, which are computationally efficient and bear a close connection to
the physics of perturbative QCD Larkoski:2014wba ; Collins:2011zzd . The
subtractive nature of P-RS echoes the area subtraction techniques of P-AS and
P-IVS, which have a close connection to optimal transport and enjoy the
advantage of continuity. P-RS combines these strengths and paves the way for
Piranha groomers that are experimentally useful and theoretically tractable.

P-RS acts subtractively on jets associated with a binary tree or clustering
history: a tree structure that emerges from recursively combining jet
constituents, two at a time, until only the full jet remains. In particular,
P-RS is a de-clustering algorithm on the jet tree: it splits the full jet tree
into its two sub-jets, then splits each sub-jet into two sub-jets as dictated
by the clustering history, continuing until only final-state particles remain.
In the discussion that follows, we also refer to sub-jets in the binary tree
as branches or emissions.

Following the procedures of traditional grooming algorithms such as Soft Drop
Larkoski:2014wba , the P-RS grooming algorithms we consider in this paper are
angular-ordered: they act on jets associated with a binary tree whose
splittings are ordered by the angular separation of their sub-jets. In
particular, we consider jets are first clustered using the anti-$k_{t}$
algorithm Cacciari:2008gp  and then re-clustered using the pair-wise angular-
ordered Cambridge-Aachen (C/A) clustering algorithm Dokshitzer:1997in .101010
Starting with the anti-$k_{t}$ algorithm both mitigates fluctuations due to
contamination in jet areas Cacciari:2008gp  and reduces the effects of
clustering logarithms in analytic calculations Larkoski:2014wba .  C/A
recursively clusters the jet constituents that are closest in angle, so that
as we de-cluster each branch of the C/A tree we recover narrower and narrower
emissions. Since the branching structure of an angular-ordered tree of
emissions is comparable to the history of emissions within the parton model, a
phenomenon known as coherent parton branching Collins:2011zzd , angular-
ordered grooming procedures have a closer connection to the physics of
perturbative QCD.

P-RS also takes in a total amount of transverse momenta to subtract from the
clustered jet, analogous to the choice of $\rho A_{\rm jet}$ in the P-AS and
P-IVS algorithms. At each splitting in the tree, P-RS assigns some amount of
the grooming associated with a given branch to its two emissions. When the de-
clustering finally reaches the final-state particles of a jet, P-RS subtracts
from their transverse momenta, much like P-AS and P-IVS, by an amount
proportional to the assigned grooming.

Recursive Subtraction with a Fraction $\boldsymbol{f}_{\rm\bf soft}$ (P-RSFf,
or P-RSF) is a simple implementation of P-RS that does not depend on the
kinematic information of each splitting. P-RSF depends on two real parameters,
each between 0 and 1:

  * •

$z_{\rm cut}$ is the fraction of transverse momentum the grooming will remove
from the entire jet, $p_{T,\,\text{P-RSF}}=(1-z_{\rm cut})p_{T,\,\rm tot}$. We
may write $z_{\rm cut}=z_{\rm cut}^{(0)}=\rho A_{\rm jet}/p_{T,\,\rm jet}$, in
analogy to the “piranha energy density” $\rho$ of P-AS and P-IVS. We denote
the total transverse momentum removed from the jet as $\Delta^{(0)}=z_{\rm
cut}\,p_{T,\,\rm tot}$, and the transverse momentum removed from a branch $i$
of the jet is $\Delta^{(i)}=z_{\rm cut}^{(i)}\,p_{T,\,\rm tot}$;

  * •

$f_{\rm soft}$ is the fraction of the grooming assigned to the softer sub-jet
at each stage of the de-clustering, as detailed below. P-RSF prefers to groom
softer radiation more and more as $f_{\rm soft}$ increases.

In the steps of the algorithm below, we index the current branch of the jet on
which P-RSF is acting by $i$. The softer sub-jet of branch $i$ is denoted
“$i,\,\text{soft}$”, and the harder branch is denoted “$i,\,\text{hard}$”. If
the algorithm is at its starting point and considering the entire jet, we
write $i=0$.

The concrete prescription of P-RSF is as follows:

  1. 1.

P-RSF attempts to de-cluster sub-jet $i$ into two sub-jets. If this is
impossible, the branch corresponds to a final-state particle and P-RSF
proceeds to Step 5.

  2. 2.

P-RSF divides the grooming assigned to branch $i$ between the softer sub-jet
($i,\,\text{soft}$) and the harder sub-jet ($i,\,\text{hard}$):

$\displaystyle\Delta^{(i,\,\text{soft})}$ $\displaystyle=f_{\rm
soft}\,\Delta^{(i)}\overset{\Delta}{=}z_{\rm
cut}^{(i,\,\text{soft})}p_{T,\,\rm tot}$ (11a)
$\displaystyle\Delta^{(i,\,\text{hard})}$ $\displaystyle=(1-f_{\rm
soft})\,\Delta^{(i)}\overset{\Delta}{=}z_{\rm
cut}^{(i,\,\text{hard})}p_{T,\,\rm tot},$ (11b)

so that $\Delta^{(i,\,\text{soft})}+\Delta^{(i,\,\text{hard})}=\Delta^{(i)}$.
In words, P-RSF assigns a fraction $f_{\rm soft}$ of the grooming to the
softer sub-jet, and the remaining fraction $1-f_{\rm soft}$ to the harder sub-
jet. Equivalently,

$\displaystyle z_{\rm cut}^{(i,\,\text{soft})}$ $\displaystyle=f_{\rm
soft}z_{\rm cut}^{(i)}$ (12a) $\displaystyle z_{\rm cut}^{(i,\,\text{hard})}$
$\displaystyle=(1-f_{\rm soft})z_{\rm cut}^{(i)},$ (12b)

so that $z_{\rm cut}^{(i,\,\text{soft})}+z_{\rm
cut}^{(i,\,\text{hard})}=z_{\rm cut}^{(i)}$.

  3. 3.

P-RSF checks if one of the sub-jets of the current branch is removed by the
grooming:

    1. (a)

If the grooming assigned to the softer sub-jet is greater than its actual
transverse momentum, $\Delta^{(i,\,\text{soft})}>p_{T}^{(\rm
i,\,\text{soft})}$, P-RSF removes the softer sub-jet from the event. P-RSF
then assigns the remaining grooming to the harder sub-jet,
$\Delta^{(i,\,\text{hard})}=\Delta^{(i)}-p_{T}^{(\rm i,\,\text{soft})}$.

    2. (b)

Similarly, if the grooming assigned to the harder sub-jet is greater than its
transverse momentum, $\Delta^{(i,\,\text{hard})}>p_{T}^{(\rm
i,\,\text{hard})}$, P-RSF removes the harder sub-jet from the event. It then
assigns the remaining grooming to the softer sub-jet,
$\Delta^{(i,\,\text{soft})}=\Delta^{(i)}-p_{T}^{(i,\,\text{hard})}$. This is
possible only if $f_{\rm soft}<1/2$.

  4. 4.

P-RSF implements Step 1 on the surviving sub-jets of $i$ using the values of
$\Delta^{(i,\,\text{soft})}$ and $\Delta^{(i,\,\text{hard})}$ calculated in
the previous two steps.

  5. 5.

When P-RSF reaches a final-state particle (a branch that cannot be further
divided), it subtracts the grooming from its transverse momentum, effectively
adjusting its energy fraction. Explicitly,

$\displaystyle p^{(i)}_{T,~{}{\rm P-RSF}}=p^{(i)}_{T}-\Delta^{(i)}.$ (13)

Here, $p^{(i)}_{T,~{}{\rm P-RSF}}$ denotes the groomed $p_{T}$ of the final-
state particle associated with the current branch $i$. This may also be
described as a shift in the fraction of the total transverse momentum carried
by the particle $i$,

$\displaystyle z^{(i)}_{\text{P-RSF}}=(z^{(i)}-z_{\rm cut}^{(i)})/(1-z_{\rm
cut}),$ (14)

where the additional factor of $1-z_{\rm cut}$ is due to normalization to the
groomed transverse momentum.

P-RSF builds on the strengths of the P-RS framework, providing a simple
implementation that does not rely on detailed kinematic information. By
combining the advantages of tree-based hard-cutoff and continuous Piranha
grooming paradigms, P-RS and P-RSF offer a promising path for experimentally
useful and theoretically tractable continuous groomers. A visualization of
P-RSF acting on a tree of emissions for $f_{\rm soft}=1/2$ (which we also call
Balanced P-RSF) and $f_{\rm soft}=1$ (an example of P-RSF${}_{f_{\rm soft}\neq
1/2}$ or Unbalanced P-RSF) is shown in Fig. 3.

P-RSFf=1/2 (Balanced P-RSF) (a)

P-RSFf=1 (example of Unbalanced P-RSF) (b)

Figure 3:  Visualization of (a) P-RSFf=1/2 and (b) P-RSFf=1 acting on a toy
jet depicted as an angular-ordered tree of emissions. The jet cone is
indicated by the dashed ellipse, and the groomed final-state particles are
indicated by colored ellipses, in dark orange (P-RSFf=1/2) and green
(P-RSFf=1); the size of each circle is proportional to the groomed $p_{T}$ of
the corresponding particle. The grey highlights behind each emission in the
tree indicate the transverse momentum subtracted from each branch, and
eventually from the final-state particles of that branch, as described in Sec.
2.5. The size of the grey highlights behind each final-state particle
indicates the amount of transverse momentum subtracted from that particle.

###  2.6 P-RSF is (Almost) Continuous

P-RSF is well-defined on the space of energy distributions of particle events
because it is invariant under exactly collinear splittings. To see this, we
first notice that exact collinear splittings are infinitely narrow, and always
correspond to branchings at the final layer of an angular-ordered tree of
emissions. With this in mind, let us implement an exact collinear splitting,
replacing a final-state particle $i$ within a jet by two exactly collinear
final-state particles, “$i,\,\text{soft}$” and “$i,\,\text{hard}$”. Following
the presentation of Sec. 2.5, the transverse momentum subtracted from the new
final state particles adds up to the transverse momentum subtracted from $i$,
$\Delta^{(i,\,\text{soft})}+\Delta^{(i,\,\text{hard})}=\Delta^{(i)}$. Since
$i,\,\text{soft}$ and $i,\,\text{hard}$ are exactly collinear, the energy
distribution of $i$ after grooming is equal to the sum of the energy
distributions for $i,\,\text{soft}$ and $i,\,\text{hard}$ after grooming. The
result of the P-RSF grooming procedure is therefore robust against exactly
collinear splittings for any value of $f_{\rm soft}$.

While the subtractive algorithm of P-RSF cannot be expressed simply in terms
of geometry, P-RSF echoes the features of P-AS and P-IVS that grant their
continuity. For example, Steps 3(a) and 3(b) of the P-RSF algorithm ensure
that P-RSF does not assign an area to any final-state particle that is larger
than its transverse momentum.

However, P-RSF still suffers from discontinuities in suppressed regions of
parameter space. We show in Sec. 3.4 that Unbalanced P-RSF, or P-RSF with
$f_{\rm soft}\neq 1/2$, suffers from soft discontinuities: small changes to
the energy of the jet have the potential to change which emissions of the jet
are softer. The balanced recursive subtraction procedure, P-RSFf=1/2,
overcomes this weakness and is soft-continuous. Furthermore, Sec. 3.5
discusses how recursive tree-based grooming algorithms suffer from
discontinuities inherited from pairwise clustering, leading to angular
discontinuities in both Balanced and Unbalanced P-RSF as well as traditional
grooming algorithms such as Soft Drop.

###  2.7 Stronger Notions of Continuity

As a final note for this section, we point out that there are analogs of
continuity that provide stronger constraints. One example of a stronger form
of continuity is uniform continuity:

######  Definition 2.

A map $M$ from energy flows to energy flows is uniformly continuous if, for
any $\varepsilon>0$, there exists a $\delta>0$ such that for all $\mathcal{E}$
and $\mathcal{E}^{\prime}$,

${\rm EMD}(\mathcal{E},\mathcal{E}^{\prime})<\delta\quad\implies\quad{\rm
EMD}(M(\mathcal{E}),M(\mathcal{E}^{\prime}))<\varepsilon.$

Roughly, we might say that uniform continuity requires that when we pick
$\varepsilon$, $M$ must be “continuous with the same $\delta$ for every
event”.

An even stronger condition is that of Hölder continuity:

######  Definition 3.

A map $M$ from energy flows to energy flows is Hölder continuous with exponent
$\alpha\in\mathbb{R}$ if, for all $\mathcal{E}$ and $\mathcal{E}^{\prime}$,

${\rm EMD}(M(\mathcal{E}),M(\mathcal{E}^{\prime}))<K\cdot\big{(}{\rm
EMD}(\mathcal{E},\mathcal{E}^{\prime})\big{)}^{\alpha},$

with $K\in\mathbb{R}$ a constant. The special case $\alpha=1$ is called
Lipschitz continuity.

Placing stronger constraints on grooming, such as uniform continuity and
Hölder continuity, has the potential to further constrain the effects of soft
contamination and fluctuations in additive radiation on groomed results. For
example, in which regions of parameter space are the hard-cutoff or Piranha
groomers presented in this paper uniformly or Hölder continuous? Are other
methods for continuous grooming, such as EMD-mode Piranha introduced in App. A
or Piranha groomers designed using linearized optimal transport Cai:2020vzx ;
Cai:2021hnn ; cai2022linearized ; sarrazin2023linearized , more amenable to
strongly continuous generalizations? Can this knowledge help us identify
obstacles to achieving these stronger types of continuity and guide us in
designing more powerful, robust observables and groomers?

Though we do not investigate grooming methods that utilize these more
restrictive analogs of continuity in this work, the invention and
understanding of such groomers offer an intriguing and promising avenue for
future research. Instead, the remainder of this work is devoted to introducing
hard-cutoff grooming, examining the types of discontinuities that may emerge
in jet grooming, and studying the sensitivity of the continuous groomers we
introduced in this section to important examples of low-energy pollution that
appear in the study of particle collision data.

##  3 Discontinuities in Grooming

In this section, we begin by reviewing the hard-cutoff Soft Drop grooming
algorithm (Sec. 3.1). We then compare the leading order (LO) soft continuity
properties of Soft Drop and Recursive Subtraction both conceptually (Sec. 3.2)
and through a perturbative calculation (Sec. 3.3).

In Secs. 3.4 and 3.5, we explore potential challenges to continuity that can
arise in more intricate scenarios. Specifically, we discuss suppressed soft
discontinuities that emerge in Unbalanced Recursive Subtraction (Sec. 3.4) and
angular discontinuities induced by angular-ordered jet clustering that emerge
beyond LO in tree-based grooming algorithms (Sec. 3.5). We do not delve into
the mathematical intricacies or perturbation theory of these examples, and
they may be skipped on an initial reading.

###  3.1 Review of Hard-Cutoff Grooming: Soft Drop

A common and elegant approach to jet grooming is to provide a hard cutoff on
the energy of the radiation within a jet; this approach retains high-energy
information, removes radiation that may be sensitive to soft distortions or
due to additive contamination, and even facilitates perturbative calculations
of jet substructure observables. In this section, we review a commonly used
hard-cutoff grooming algorithm: the Soft Drop de-clustering algorithm
Larkoski:2014wba . Readers familiar with Soft Drop may skip to Sec. 3.2, where
we discuss the soft discontinuities of the Soft Drop algorithm.

The Soft Drop de-clustering algorithm is tree-based, computationally
efficient, and serves as a representative example of a hard-cutoff grooming
algorithm. Much like P-RSF, Soft Drop begins with an angular ordered tree of
emissions and grooms at each branch of the tree. For Soft Drop, however, the
parameter $z_{\rm cut}$ provides a hard cutoff, and Soft Drop sometimes maps
similar events into groomed events that are very distinct. The resulting
discontinuity in the grooming procedure leads to theoretical challenges and
uncertainties, unpredictable and non-linear responses to non-perturbative
physics, and difficulties in interpreting experimental data ATL-PHYS-
PUB-2019-027 ; Aad:2019vyi ; ATLAS:2020gwe .

The Soft Drop algorithm depends on two parameters. The first, $z_{\rm cut}$,
is a hard cutoff that parameterizes the strength of the grooming. The second,
$\beta_{\rm SD}$, is a parameter that controls the angular dependence of the
grooming procedure. Soft Drop preferentially removes soft radiation at wider
and wider angles as $\beta_{\rm SD}$ is increased.

Soft Drop acts on a jet with characteristic radius $R_{0}$ that is first
clustered using an algorithm such as the anti-$k_{t}$ algorithm, and then re-
clustered with the Cambridge-Aachen (C/A) algorithm. As in Sec. 2.5, this
produces a binary tree of clustered particles within the jet, such that
particles deeper in the tree tend to be closer in angle. Soft Drop then
implements the following algorithm:

  1. 1.

The most recent stage of the C/A clustering is undone, breaking the jet into
two sub-jets.

  2. 2.

The softer sub-jet is removed from the jet if it is not energetic enough to be
considered relevant to the hard physics under study; in particular, let
$z_{\rm soft}=\min(p_{T,1},p_{T,2})/(p_{T,1}+p_{T,2})$ denote the fraction of
the $p_{T}$ carried by the softer sub-jet, and let $\theta$ denote the angular
distance between the two sub-jets. If the two sub-jets pass the Soft Drop
grooming criterion:

$\displaystyle z_{\rm soft}>z_{\rm
cut}\left(\frac{\theta}{R_{0}}\right)^{\beta_{\rm SD}},$ (15)

the grooming procedure is stopped. The two sub-jets are then re-merged and
labeled as the groomed jet without further modification.

  3. 3.

If the inequality of Eq. (15) is not satisfied, the softer sub-jet is not
energetic enough to be considered relevant to the hard physics of jet
production. In this case, the softer of the two sub-jets is removed from the
event and the procedure is repeated recursively by de-clustering the harder of
the two sub-jets.

In this work, we use Soft Drop in grooming mode: if Soft Drop reaches a final-
state particle that cannot be de-clustered as required by Step 1, we keep it
as the groomed jet. Tagging mode would instead remove it from the jet entirely
and leave a groomed event with no particles Larkoski:2014wba .

Grooming mode is a natural choice in the context of pileup mitigation because
of the extreme behavior of tagging mode, which completely eliminates a single
particle jet even when $z_{\rm cut}=0$. In particular, if we have an event in
which we estimate there is no contamination from pileup (see Sec. 5.1), we
want to avoid grooming the jet at all and correspondingly set $z_{\rm cut}=0$.
For a tree-level jet, consisting of a single quark, grooming mode respects our
wish to leave the jet untouched, while tagging mode would nonetheless remove
it entirely.

Soft Drop de-clustering is a generalization of the modified Mass Drop Tagger
(mMDT) algorithm Dasgupta:2013ihk , and reproduces mMDT when $\beta_{\rm
SD}=0$. Soft Drop has seen success in a wide variety of phenomenological
applications, such as the characterization of boosted objects Thaler:2008ju ;
Thaler:2011gf ; Hook:2011cq ; Gallicchio:2011xq ; Soper:2012pb ;
Gallicchio:2012ez ; CMS-PAS-JME-09-001 ; CMS-PAS-EXO-09-002 ; CMS:2013kfa ;
ATL-PHYS-PUB-2009-081 ; ATL-PHYS-PUB-2010-008 ; ATLAS:2019kwg ; Cui:2010km ;
ATLAS-CONF-2011-053 ; Chatrchyan:2013rla ; Larkoski:2013eya ; Dasgupta:2012hg
; Backovic:2013bga ; ATLAS-CONF-2013-084 ; Komiske:2018vkc ; Komiske:2016rsd ;
Metodiev:2018ftz ; Krohn:2012fg ; MERINO:2013tta ; Bhattacherjee:2016bpy ;
Macaluso:2018tck ; Egan:2017ojy ; Kasieczka:2017nvn ; Pearkes:2017hku ;
Butter:2017cot ; Catani:1992ua ; Dokshitzer:1998kz ; Dasgupta:2001sh ;
Banfi:2004yd ; Banfi:2005gj ; Ellis:2009su ; Banfi:2010pa ; Walsh:2011fz ;
Chien:2012ur ; Li:2012bw ; Jouttenus:2013hs ; Hatta:2013iba ; Larkoski:2014tva
; Procura:2018zpn ; Aaboud:2017aca ; Frye:2016aiz ; Almeida:2008yp ;
Larkoski:2017iuy ; Larkoski:2017cqq ; Thaler:2010tr ; Ellis:2009me ;
Abdesselam:2010pt ; Katz:2010mr ; Gallicchio:2010dq ; Adams:2015hiv ;
Sirunyan:2017ezt ; Moore:2018lsr ; FerreiradeLima:2016gcz ; Rubin:2010fc ;
Chatrchyan:2012sn ; CMS-PAS-BTV-13-001 ; CMS-PAS-JME-13-006 ; Kribs:2009yh ;
Chen:2010wk ; Hackstein:2010wk ; Kim:2010uj ; Almeida:2011aa ;
Pandolfi:2012ima ; Vernieri:2014wfa ; CMS-PAS-HIG-17-007 ; Procura:2014cba
and the extraction of parameters of the Standard Model, such as the top quark
mass Hoang:2017kmk ; ATLAS:2021urs ; Negrini:2022gec  and the strong coupling
constant Marzani:2019evv , from particle collision data.

Soft Drop with $\beta_{\rm SD}>0$ is collinear safe (invariant under exactly
collinear splittings), and is therefore well-defined on the space of energy
flows. However, when $\beta_{\rm SD}\leq 0$, Soft Drop in grooming mode is not
collinear safe. The lack of collinear safety in grooming mode can be quickly
derived when considering a collinear splitting with $z<z_{\rm cut}$ which
fails the Soft Drop criterion. A similar problem emerges if one makes
appropriate definitions for the algorithm in the limit $\theta\to 0$, and Soft
Drop in grooming mode is also not collinear safe when $\beta_{\rm SD}<0$.
Therefore, Soft Drop in grooming mode with $\beta_{\rm SD}\leq 0$ is not well-
defined as a map on the space of energy distributions, and we do not consider
it in the remainder of the paper.

###  3.2 An Invitation to Soft Discontinuities: Soft Drop Versus P-RS

$\mathcal{E}^{+}$$\mathcal{E}^{-}$Ungroomed$z_{\rm cut}$$z_{\rm
cut}$$\color[rgb]{0.0,0.62,0.38}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.62,0.38}z^{+}>z_{\rm
cut}$$\color[rgb]{0.76,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{0.76,0.23,0.13}z^{-}<z_{\rm
cut}$$z_{\rm hard}$$z_{\rm hard}$Soft Drop, $\beta_{\rm
SD}=0$${\color[rgb]{0.0,0.62,0.38}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.62,0.38}\Delta
E^{+}_{\rm
SD}}=0$${\color[rgb]{0.76,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{0.76,0.23,0.13}\Delta
E^{-}_{\rm SD}}\sim
Q{\color[rgb]{0.76,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{0.76,0.23,0.13}z^{-}}$$z_{\rm
hard}^{\prime}<z_{\rm hard}$P-RSF (arb. $f_{\rm
soft}$)${\color[rgb]{0.0,0.62,0.38}\definecolor[named]{pgfstrokecolor}{rgb}{0.0,0.62,0.38}\Delta
E^{+}_{\rm P\mhyphen RSF}}\sim Qz_{\rm
cut}$${\color[rgb]{0.76,0.23,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{0.76,0.23,0.13}\Delta
E^{-}_{\rm P\mhyphen RSF}}\sim Qz_{\rm cut}$$z_{\rm hard}^{\prime}<z_{\rm
hard}$EventSpace$E_{\rm hard}$$E_{\rm soft}$$Qz_{\rm
cut}$$\mathcal{E}^{-}$$\mathcal{E}^{+}$$\mathcal{O}(\delta)$$E_{\rm
hard}$$E_{\rm soft}$$\mathcal{E}^{-}$$\mathcal{E}^{+}$$\mathcal{O}(z_{\rm
cut})\gg\delta$$E_{\rm hard}$$E_{\rm
soft}$$\mathcal{E}^{-}$$\mathcal{E}^{+}$$\mathcal{O}(\delta)$

Figure 4:  A cartoon comparing the discontinuous action of hard-cutoff
grooming to the continuous action of Piranha on the nearly identical two-
particle events with energy $Q$, $\mathcal{E}^{+}$ and $\mathcal{E}^{-}$. We
use Soft Drop (with $\beta_{\rm SD}=0$) and P-RSF (with arbitrary $f_{\rm
soft}$) with a grooming parameter $z_{\rm cut}$ as representative examples of
the hard-cutoff and Piranha paradigms, respectively. The events
$\mathcal{E}^{\pm}$ differ only by the energy fraction of the softer particle,
$z^{\pm}=z_{\rm cut}\pm\delta/2$, and are separated by an infinitesimal EMD,
EMD$(\mathcal{E}^{+},\mathcal{E}^{-})\sim\delta\ll z_{\rm cut}$. Despite their
similarity, Soft Drop maps the events to distinct groomed results separated by
a large EMD: EMD$(\mathcal{E}_{\rm SD}^{+},\mathcal{E}_{\rm SD}^{-})\sim
z_{\rm cut}\gg\delta$. P-RSF instead maps the events continuously to
infinitesimally similar groomed events: EMD$(\mathcal{E}_{\rm P\mhyphen
RSF}^{+},\mathcal{E}_{\rm P\mhyphen RSF}^{-})\sim\delta$. $\Delta E^{\pm}_{G}$
indicates the amount of energy removed from the event $\mathcal{E}^{\pm}$ by
the groomer $G$.

Traditional groomers are discontinuous in the regions of parameter space near
a hard cutoff. For example, Soft Drop is discontinuous in the region of
parameter space where $z_{\rm soft}=z_{\rm cut}\theta^{\beta_{\rm
SD}}/R_{0}^{\beta_{\rm SD}}$ due to the Soft Drop criterion of Eq. (15). In
this subsection, we discuss the discontinuity of hard-cutoff grooming and the
relative continuity of Piranha.

The following discussion centers around the illustrative example shown in Fig.
4, which shows the simplest instance of discontinuous behavior near a hard
cutoff as well as the resolution offered by Piranha. In this example, we
consider two nearly identical events, $\mathcal{E}^{+}$ and $\mathcal{E}^{-}$,
that each contain two particles. The softer constituent of $\mathcal{E}^{+}$
has an energy fraction $z^{+}=z_{\rm cut}+\delta$ slightly above the grooming
parameter $z_{\rm cut}$, while the softer constituent of $\mathcal{E}^{-}$ has
an energy fraction $z^{-}=z_{\rm cut}-\delta$ slightly below $z_{\rm cut}$.
The energy flows of the two events are nearly indistinguishable, in the sense
that ${\rm EMD}(\mathcal{E}^{+},\mathcal{E}^{-})\sim\delta$, and we choose
$\delta$ for consistency with Definition 1. We use Soft Drop with $\beta_{\rm
SD}=0$, or mMDT, as a representative example of hard-cutoff grooming, and
P-RSF with arbitrary $f_{\rm soft}$ as a representative example of the Piranha
paradigm; for each, we use the parameter $z_{\rm cut}$.

Soft Drop treats $\mathcal{E}^{+}$ and $\mathcal{E}^{-}$ very differently: it
does not modify $\mathcal{E}^{+}$, but completely removes the softer particle
of $\mathcal{E}^{-}$. The energy flow of $\mathcal{E}^{+}$ is unchanged, while
that of $\mathcal{E}^{-}$ is changed dramatically, and the EMD between the
groomed jets is relatively large: EMD$(\mathcal{E}^{+}_{\rm
SD},\mathcal{E}^{-}_{\rm SD})\sim z_{\rm cut}\gg\delta$. Direct application of
Definition 1 to our example shows that Soft Drop is discontinuous.111111 We
note that Soft Drop with $\beta\neq 0$ is also soft discontinuous. Indeed, the
above arguments still hold for the case $\beta\neq 0$, verbatim, when the
angle between the particles of $\mathcal{E}^{\pm}$ is fixed to $\theta=R_{0}$.
As expressed in Eq. (15), the Soft Drop criterion at this fixed angle is still
$z_{\rm soft}>z_{\rm cut}$. However, since Soft Drop with $\beta\neq 0$ drops
the softer particle if it does not satisfy $z/\theta^{\beta}>z_{\rm
cut}/R_{0}^{\beta}$, we may also say that Soft Drop with $\beta\neq 0$ is
“$z/\theta^{\beta}$-discontinuous”, where $z$-discontinuity indicates the soft
discontinuity for the case of $\beta=0$ that we discuss above.  Since the
events $\mathcal{E}^{+}$ and $\mathcal{E}^{-}$ differ by small changes to the
energy of jet constituents, we see that Soft Drop is soft discontinuous in the
region of parameter space near the cutoff $z_{\rm cut}$.

The procedure of P-RSF is quite different. When grooming both events,
P-RSFf=1/2 grooms the energy of both emissions by the same amount, leading to
no large discrepancy in the two groomed results. Indeed, after the P-RSFf=1/2
grooming procedure, these two events are still separated by an infinitesimal
EMD: EMD$(\mathcal{E}^{+}_{\rm RSF},\mathcal{E}^{-}_{\rm RSF})\sim\delta$.
This simple example demonstrates how hard-cutoff grooming methods may
discontinuously map two nearly identical events into vastly different groomed
results, and showcases the soft continuous resolution of the P-RSFf=1/2
grooming procedure and other Piranha groomers.

###  3.3 Soft Discontinuities in Perturbation Theory: Soft Drop Versus P-RS

Next, we examine the manifestations of the soft discontinuous behavior of
hard-cutoff grooming more quantitatively by comparing Soft Drop and P-RSF at
leading order in perturbation theory (LO). In particular, we examine the LO
effects of discontinuity on the distributions of the two-prong generalized
energy correlation functions (ECFs) of Ref. Larkoski:2013eya . For jets that
are central ($y=0$) and narrow ($R_{0}\ll 1$), we may write the ECFs in the
form

$C_{1}^{(\varsigma)}\simeq\frac{1}{2}\sum_{i=1}^{M}\sum_{j=1}^{M}z_{i}z_{j}\left(\frac{\theta_{ij}}{R_{0}}\right)^{\varsigma},$
(16)

up to non-singular corrections in powers of the jet radius, which we neglect
Larkoski:2014wba . In Eq. (16), $z_{i}$ represents the fraction of the jet
energy carried by particle $i$, $\theta_{ij}$ indicates the angle between
particles $i$ and $j$, $R_{0}$ indicates the jet radius, and the sum is over
all particles in the groomed jet.

In our discussion below, we focus on the calculation of LO substructure
because it demonstrates the effects of the discontinuity of Soft Drop and the
continuity of Piranha in a simple context. We provide a more detailed
description of our LO calculation of Piranha-groomed substructure in App. C.1,
and extend our substructure analysis to leading logarithmic (LL) accuracy in
App. C.3. The resummed analysis of App. C.3 does not change the qualitative
conclusions of our discussion below. However, it highlights subtleties in
systematically improving our resummed substructure calculations due to the
global, subtractive nature of Piranha grooming. In App. C.2, we also discuss
the calculation of the groomed energy fraction $z_{g}$ – another common
observable used in the study of jet grooming – which exhibits similar
subtleties. Therefore, while our resummation does not change the qualitative
conclusions of the following discussion, it indicates that the computation of
resummed Piranha observables at higher accuracy may require new calculational
tools.

At LO, the emission of a single parton with energy fraction $z$ and angle
$\theta$ is described approximately by the pseudo-probability distribution

$\displaystyle\frac{\alpha_{s}}{\pi}p_{i}(z)\frac{1}{\theta}\approx\frac{2C_{R_{i}}\alpha_{s}}{\pi}~{}\frac{1}{z}~{}\frac{1}{\theta},$
(17)

where $p_{i}(z)$ is a Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP)
splitting function Gribov:1972ri ; Dokshitzer:1977sg ; Altarelli:1977zs
describing the splitting of a mother parton $i$, and $C_{R_{i}}$ is the
quadratic Casimir for the SU(3) color representation $R_{i}$ of the mother
parton. $C_{F}=(N_{C}^{2}-1)/(2N_{C})=4/3$ for quarks and $C_{A}=N_{C}=3$ for
gluons. We are interested in particular in the reduced splitting functions,

$\displaystyle\overline{p}_{i}(z)=p_{i}(z)+p_{i}(1-z),$ (18)

where $z\in(0,1/2)$. Reduced splitting functions are relevant for our groomed
calculations because they describe the energy distribution of the softer of
the two partons after the splitting.

The probability distribution of $C_{1}^{(\varsigma)}$ for Soft Drop was
studied in Ref. Larkoski:2014wba , which found, for $\beta_{\rm SD}>0$, the LO
result

$\displaystyle\rho_{i,\,\,\text{SD}}(C_{1}^{(\varsigma)})\approx\frac{2\alpha_{s}C_{R_{i}}}{\pi\varsigma}\frac{1}{C_{1}^{(\varsigma)}}\times\begin{cases}-\log
C_{1}^{(\varsigma)}+B_{i},&C_{1}^{(\varsigma)}>z_{\rm cut};\\\
-\frac{\beta_{\rm SD}}{\varsigma+\beta_{\rm SD}}\log
C_{1}^{(\varsigma)}-\frac{\varsigma}{\varsigma+\beta_{\rm SD}}\log z_{\rm
cut}+B_{i},&C_{1}^{(\varsigma)}<z_{\rm cut},\end{cases}$ (19)

away from $C_{1}^{(\varsigma)}=0$, up to $\mathcal{O}(\alpha_{s}^{2})$ and
terms that are power suppressed in $C_{1}^{(\varsigma)}$, $z_{\rm cut}$, or
both. $B_{i}$ is a factor due to hard-collinear pieces of the splitting
function that are not singular as $z$ approaches 0: $B_{q}=-3/4$ for quarks
and $B_{g}=-11/12+n_{f}/(6C_{A})$ for gluons, where $n_{f}$ is the number of
active quark flavors. The case $\beta_{\rm SD}<0$ leads to an identical result
up to another constraint on $C_{1}^{(\varsigma)}$, which simply multiplies the
expression above by $\Theta(C_{1}^{(\varsigma)}>z_{\rm
cut}^{\varsigma/|\beta_{\rm SD}|})$.

The piece-wise behavior of Eq. (19) and the associated kink in the Soft Drop
$C_{1}^{(\varsigma)}$ distribution are due to the discontinuous behavior of
Soft Drop. As noted by Ref. Larkoski:2014wba , when $\beta_{\rm SD}\geq 0$ any
two-parton configurations with
$C_{1}^{(\varsigma)}=z\,(\theta/R_{0})^{\varsigma}\,>\,z_{\rm cut}$ must have
$z\,(\theta/R_{0})^{\beta_{\rm SD}}\,>\,z_{\rm cut}$, and are therefore not
affected by Soft Drop. On the other hand, configurations with
$C_{1}^{(\varsigma)}<z_{\rm cut}$ may be Soft Dropped, leading to the piece-
wise change in groomed substructure in this region of phase space.

We discuss the calculation of the $C_{1}^{(\varsigma)}$ distribution for P-RSF
groomed jets in App. C.1, where we find the LO result

$\rho_{i,\,\,\text{P-RSF}}(C_{1}^{(\varsigma)};f)\approx\frac{2\alpha_{s}C_{R_{i}}}{\varsigma~{}\pi}\frac{1}{C_{1}^{(\varsigma)}}\left(-\log\left(C_{1}^{(\varsigma)}+f\,z_{\rm
cut}\right)+B_{i}\right),$ (20)

away from $C_{1}^{(\varsigma)}=0$, using the same factors of $B_{i}$ as for
Soft Drop, and up to terms that are power-suppressed in $C_{1}^{(\varsigma)}$,
$z_{\rm cut}$, or both.121212 We include some additional power-suppressed
terms and additional contributions when $C_{1}^{(\varsigma)}=0$ in the LO
discussion of App. C.1.

Notably, the P-RSF distribution for $C_{1}^{(\varsigma)}$ is smooth and does
not need to be defined in a piece-wise fashion. Even at LO, the subtractive
nature of P-RSF leads to a smooth interpolation between double-logarithmic
behavior at large $C_{1}^{(\varsigma)}$ and single-logarithmic behavior at
small $C_{1}^{(\varsigma)}$.131313The terms “double logarithmic” and “single
logarithmic” here refer to the behavior of the LO cumulative distribution
function. In particular, we note that when $C_{1}^{(\varsigma)}\gg f\,z_{\rm
cut}$, the pseudo-probability distribution $\rho\propto\log\left(C\right)/C$
demonstrates double-logarithmic behavior. On the other hand, when
$0<C_{1}^{(\varsigma)}\ll f\,z_{\rm cut}$, the distribution
$\rho\propto\log\left(f\,z_{\rm cut}\right)/C$ demonstrates single-logarithmic
behavior.  Much as the kink in the Soft Drop distribution can be attributed to
the sudden activation of the grooming procedure in certain regions of phase
space, the smoothness of the P-RSF distribution reflects in part that the
grooming procedure is always active.141414 We say that the smoothness of P-RSF
only reflects the global activity of the grooming in part because we expect
that Piranha groomers that turn on gradually will also have smooth
substructure distributions. While the piece-wise behavior of Soft Drop
observable distributions is smoothed out by all-orders effects
Benkendorfer:2021unv , the smoothness of P-RSF observable distributions at LO
is a manifestation of the subtractive behavior of P-RSF that ensures its
continuity on two-parton jets.

We also note that the substructure distributions of other Piranha grooming
procedures can be calculated at LO with a similar method by upgrading $f_{\rm
soft}$ to a function of $z$ and $\theta$. Further, we expect that such an
$f(z,\theta)$ may be well approximated by $f(0,0)$, with sub-leading
corrections proportional to $\mathcal{O}(z,\theta,z_{\rm cut})$.

In our discussion above, we emphasized that the kinks in Soft Drop groomed
substructure distributions are manifestations of the discontinuity of Soft
Drop itself. Generalizations of these arguments, however, must be verified
carefully. In particular, there is not generically a one-to-one correspondence
between kinks in distributions and discontinuities in the associated
observable. As a toy one-dimensional example, consider a random variable $X$
that is uniformly distributed in $(-1/2,\,1/2)$. The discontinuous, “zig-
zagging” function $f(X)=X\pm 1/2$, choosing the upper sign for $X<0$ and the
lower sign for $X>0$, is uniformly distributed. Therefore, discontinuous
functions on our toy “phase space” – the domain of $X$ – do not necessarily
have kinks in their distributions. Conversely, there is a kink in the
distribution of $f(X)=\mp(1-\sqrt{1\pm 2X})/2$, where we take the upper sign
when $X<0$ and the lower sign for $X\geq 0$, even though $f(x)$ is a
continuous function of $x$ and has a continuous first derivative.

###  3.4 Suppressed Soft Discontinuities in Unbalanced P-RS

Unbalanced P-RSF algorithms are also discontinuous in the suppressed region of
parameter space where $z=1/2$.151515 The probability density of seeing a
softer sub-jet with energy fraction $z$ scales as $1/z$ at leading logarithmic
accuracy in perturbative QCD. Therefore, the region with $z\sim 1/2$ at the
boundary of the jet phase space is far less populated than the regions of
phase space near a hard-cutoff with $z\sim z_{\rm cut}<1/2$.  The
discontinuous behavior emerges because Unbalanced P-RSF algorithms treat
harder and softer sub-jets differently; since an infinitesimally soft
perturbation can lead to either sub-jet having more energy when $z=1/2$, an
infinitesimal change in an ungroomed energy flow can lead to macroscopic
differences in groomed results. In the language of Sec. 2.2, we say that
Unbalanced P-RSF is soft discontinuous in the suppressed, measure zero region
of parameter space where $z=1/2$, and that Balanced P-RSF is the only soft
continuous P-RSF algorithm.

In the following discussion, we explore the soft discontinuity of Unbalanced
P-RS in the context of a jet containing only two partons. We find that the
discontinuity at $z=1/2$ can lead to macroscopic changes to the groomed jet
axis. However, only the orientation of the jet is affected by the soft
discontinuity in our two-parton example, and the soft discontinuity of
Unbalanced P-RS is not reflected at LO by traditional jet substructure
variables. Nonetheless, important features of a jet, such as its thrust axis
Brandt:1964sa ; Farhi:1977sg , are susceptible to discontinuities at LO.
Furthermore, even traditional jet substructure observables are impacted by the
soft discontinuity of Unbalanced P-RS in three-parton jet configurations that
emerge beyond LO.

$\mathcal{E}^{(\rm high)}$$\mathcal{E}^{(\rm low)}$Ungroomed$z_{\rm
high}=\frac{1}{2}+\delta$$z_{\rm low}=\frac{1}{2}+\delta$P-RSF (arb. $f_{\rm
soft}\neq
1/2$)JetAxis$\Delta\theta_{0}/R\sim\mathcal{O}(\delta)$$\Delta\theta_{g}/R\sim
2\,z_{\rm cut}\,\left|f-\frac{1}{2}\right|+\mathcal{O}(\delta)$

Figure 5:  A cartoon demonstrating the soft discontinuity of P-RSF for $f_{\rm
soft}\neq 1/2$ acting on the nearly identical events $\mathcal{E}^{(\rm
high,low)}$, each with two particles with nearly identical energy fractions,
$z\sim 1/2$, and with opening angle $R$. Red denotes the event
$\mathcal{E}^{(\rm high)}$, blue denotes $\mathcal{E}^{(\rm low)}$, and a
thick dotted arrow indicates the jet axis for each event. The hardest particle
of $\mathcal{E}^{(\rm high)}$ is its upper particle, while the hardest
particle of $\mathcal{E}^{(\rm low)}$ is its lower particle. Since P-RSF with
$f_{\rm soft}\neq 1/2$ preferentially subtracts radiation from the softer
particle of each event, the events are mapped discontinuously to distinct
groomed results. The discontinuous behavior of the groomed jet axis is one
manifestation of this soft discontinuous behavior: while the ungroomed energy-
weighted jet axes of $\mathcal{E}^{(\rm high)}$ and $\mathcal{E}^{(\rm low)}$
differ by an infinitesimal amount ($\Delta\theta_{0}\sim R\epsilon$), the
groomed jet axes are widely separated for any $f_{\rm soft}\neq 1/2$.

In the example of Fig. 5, we explore the effect of the soft discontinuity of
Unbalanced P-RSF on the axis of a jet. In particular, we examine nearly
identical two-parton jets in the region of phase space near $z\sim 1/2$ that
differ only by an infinitesimally soft perturbation. We denote these events by
$\mathcal{E}^{(i)}$, where $i\in\\{{\rm high,low}\\}$. The $\mathcal{E}^{(i)}$
each consist of two particles: a “high” and “low” particle separated by an
angular distance $R$. In $\mathcal{E}^{(i)}$, particle $i$ has just slightly
more energy, $z_{i}=1/2+\delta$.

For simplicity, we focus on the discontinuous behavior of the energy-weighted
jet axis,

$\hat{n}_{\rm jet}\propto z_{\rm high}\hat{n}_{\rm high}+z_{\rm
low}\hat{n}_{\rm low},$ (21)

for the groomed events $\mathcal{E}^{(i)}$, and the proportionality holds up
to normalization. Since the ungroomed $\mathcal{E}^{(i)}$ are nearly
identical, the angle between the ungroomed jet axes is infinitesimally small:

$\displaystyle\Delta\theta_{0}$
$\displaystyle=2\,R\,\delta+\mathcal{O}(R^{3},\,\delta^{3}),$ (22a)
$\displaystyle\lim_{\delta\to 0}\Delta\theta_{0}$ $\displaystyle=0,$ (22b)

where $\Delta\theta_{0}$ indicates the angle between the jet axes of
$\mathcal{E}^{(\rm high)}$ and $\mathcal{E}^{(\rm low)}$ before grooming.
After the application of P-RSF, however, the softer branch of each event will
be dramatically diminished, and there is a large angle between the two groomed
jet axes:

$\lim_{\delta\to 0}\Delta\theta_{g}=2\,R\,\frac{z_{\rm cut}}{1-z_{\rm
cut}}\,\left|f_{\rm soft}-1/2\right|+\mathcal{O}(R^{3})\neq 0,$ (23)

where $\Delta\theta_{g}$ indicates the angle between the jet axes of
$\mathcal{E}^{(\rm high)}$ and $\mathcal{E}^{(\rm low)}$ after grooming with
P-RSF. The continuous behavior indicated by $\lim_{\delta\to
0}\Delta\theta_{0}=0$ before grooming is in sharp contrast to the
discontinuous behavior after grooming, $\lim_{\delta\to 0}\Delta\theta_{g}\neq
0$. In Eq. (23), we have assumed that $\max(f,\,1-f)\,z_{\rm cut}<1/2-\delta$,
so that neither particle is fully removed from the event. Of course, when
$f_{\rm soft}=1/2$, the grooming algorithm treats the harder and softer sub-
jets equally, so that we once again have
$\Delta\theta_{g}\to\mathcal{O}(\delta)$ and the soft discontinuous behavior
fades away.

This simple example demonstrates a general principle for designing Recursive
Subtraction algorithms: if they are to be soft-continuous, they must treat the
softer and harder sub-jets identically in the limit $z\to 1/2$. For example,
let us imagine a P-RSF–type grooming algorithm with $f_{\rm soft}$ upgraded to
a function of the energy fraction $z$ and angle $\theta$ of a branch, $f_{\rm
soft}\to f(z,\theta)$. Any P-RSF-type groomer with $f(1/2,\,\theta)=1/2$
overcomes the soft discontinuity discussed of the discussion above. We call
such P-RS groomers that treat emissions in the region $z=1/2$ identically
“Hard-Balanced” Recursive Subtractors, as they are balanced in the region of
phase space where two sub-jets are equally hard. A P-RSF algorithm with
$f(z,\theta)=1-z$, for example, is Hard-Balanced, and still preferentially
grooms infinitesimally soft radiation. We leave the study of Hard-Balanced
Recursive Subtractors to future work.

Finally, we note that while the soft discontinuity we detailed above is an
important formal feature of P-RSF, the regions of phase space where $z\sim
1/2$ contribute far less to observable distributions than those for which
$z\sim z_{\rm cut}$. In particular, the pseudo-probability distribution for
$z$ scales as $1/z$ in perturbative QCD, implying that the soft discontinuity
of Unbalanced P-RSF that we explore above is suppressed relative to the soft
discontinuities of hard-cutoff groomers such as Soft Drop. Indeed, the soft
discontinuities of P-RSF do not manifest in leading logarithmic substructure
distributions, and we may even use P-RSF with $f_{\rm soft}\neq 1/2$ to gain
perturbative insight into the soft continuous analog, P-RSFf=1/2, and into
Piranha grooming in general; we explain this reasoning in more depth and
pursue this goal in App. C.

On the other hand, soft discontinuities associated with hard cutoffs in
traditional grooming procedures lie on the interior of the jet phase space,
straddled by the regions $0<z<z_{\rm cut}$ and $z_{\rm cut}<z<1/2$. As we saw
in Sec. 3.3, the associated soft discontinuities lead to piece-wise
definitions of groomed observable distributions even at LO to isolate the
behavior of radiation above and below the associated hard cutoff.

###  3.5 Clustering Discontinuities Beyond Leading Order in Tree-Based
Grooming

Recursive tree-based grooming algorithms also suffer from clustering
discontinuities, or discontinuities inherited from the procedure of clustering
the jet into a tree of emissions (as discussed in Sec. 2.5). Clustering
discontinuities are inevitable in tree-based grooming because any clustering
algorithm relies on the discontinuous notion of which particles are “closest”
to one another. However, since there is only one way to cluster a two-parton
jet, clustering discontinuities only emerge beyond LO and when a jet contains
three or more partons.

Soft Drop and P-RSF in particular suffer from an angular clustering
discontinuity. Both rely on an angular-ordered jet clustering history obtained
from the angular-ordered C/A clustering algorithm; the notion of the C/A
clustering history is discontinuous because small changes to the angles of jet
constituents may lead to distinct clustering histories. In the language of
Sec. 2.2, we say angular-ordered clustering algorithms and angular-ordered,
tree-based grooming algorithms are angularly discontinuous in a measure zero
region of phase space.

(a)

(b)

Figure 6:  A visualization of angular discontinuity in tree-based grooming in
the style of Fig. 3, with the angular-ordered P-RSFf=1/2 algorithm as a
representative example. The thick dashed line in the center of the jet is
equidistant in angle from the left and right sub-jets of the event, and the
middle emission lies nearly on this line. In (a), the middle emission lies
closer to the left sub-jet, and the right sub-jet is therefore groomed more.
In (b), the middle emission is closer to the right sub-jet, and the left sub-
jet is groomed more. We see that a small change in the angle of the middle
emission may drastically change the clustering history of the jet, and the
result of a tree-based grooming procedure.

In our discussion, we focus on a simple manifestation of angular clustering
discontinuities in the three-parton jet shown in Fig. 6. In particular, we
consider the region of phase space where a particular sub-jet is nearly
equidistant from a sub-jet on the left and another on the right. We call the
left and right sub-jets $L$ and $R$, and we denote the equidistant emission as
$M$, for “middle”.

A small change in the location of emission $M$ may lead to distinct C/A
clustering histories. If $M$ is slightly closer to the left emission $L$, then
$L$ will receive a quarter of the total grooming while $R$ will receive half
(as in Fig. 6a); the result is that $L$ is groomed less than $R$, and much
more of the structure of the left side of the jet is preserved after the
grooming. Alternatively, if $M$ is slightly closer to the right, then $R$ will
be groomed less than $L$; the right side of the jet will be preserved while
much of the left side will be groomed away (as in Fig. 6b). Despite the
similarity of the ungroomed events, the resulting groomed events will have
macroscopic differences and be separated by a relatively large EMD.

To emphasize this point quantitatively, let us consider P-RSF acting on the
simple example where all three emissions lie in a plane, the left and right
emissions each have an energy fraction $z_{\ell}=z_{r}=z_{0}$, and the middle
emission – nearly equidistant from the left and right emissions – has
$z_{m}=1-2z_{0}$.161616We focus on P-RSF in this example to avoid the more
complex analysis required for Soft Drop that may obfuscate the physics at
hand. Soft Drop also exhibits an angular clustering discontinuity that can be
seen in the context of this simple example, but that is complicated by the
possible behaviors of the grooming: $z_{\rm cut}$ and $z_{0}$ may conspire
such that zero, one, or two emissions are groomed from the event. The effects
of the clustering discontinuity on the energy-weighted jet axis must therefore
be considered piece-wise in the $(z_{\rm cut},\,z_{0})$ parameter space.  We
use $R$ to denote the angle between the left and right emissions, and $\delta$
to denote a small perturbation in the angle of the middle emission:
$\theta_{\ell m}=R/2+\delta$ in one event, while $\theta_{\ell m}=R/2-\delta$
in the other. We again examine the discontinuity in the energy-weighted jet
axis, which takes the more general form

$\displaystyle\hat{n}_{\rm jet}\propto\sum_{i}z_{i}\hat{n}_{i},$ (24)

where the proportionality holds up to normalization. A simple computation then
yields

$\displaystyle\Delta\theta_{0}$
$\displaystyle=2\,z_{m}\,\delta+\mathcal{O}\left(R^{2},\delta^{3}\right),$
(25a) $\displaystyle\lim_{\delta\to 0}\Delta\theta_{0}$ $\displaystyle=0,$
(25b)

for the angle $\Delta\theta_{0}$ between the two jet axes before grooming,
while

$\displaystyle\lim_{\delta\to 0}\Delta\theta_{g}$
$\displaystyle=R\,\frac{z_{\rm cut}}{1-z_{\rm
cut}}\,\left|f(f-3)+1\right|+\mathcal{O}(R^{3})\neq 0,$ (26)

for the angle between the two jet axes after the application of P-RSF, where
we have assumed that $z_{\rm cut}$ is sufficiently small that all particles
survive the grooming. We again see a discontinuous behavior in the energy-
weighted axis of the groomed jets, $\lim_{\delta\to 0}\Delta\theta_{g}\neq 0$,
despite the fact that the ungroomed events were nearly identical and
$\lim_{\delta\to 0}\Delta\theta_{0}=0$.

Fortunately, the angular discontinuities associated with tree-based clustering
algorithms are also suppressed in the phase space of perturbative QCD. First,
we note that the angular discontinuities in angular-ordered clustering
algorithms emerge only at higher orders of perturbative accuracy for which a
jet has three or more constituents. These regions of phase space are
generically suppressed in high-energy QCD calculations, where the emission of
extra final-state partons is suppressed by the strong coupling constant
$\alpha_{s}$. Furthermore, perturbative QCD predicts a strong hierarchy in the
angles between different emissions in a jet,
$\theta_{1}\gg\theta_{2}\gg\cdots$ Collins:2011zzd . Therefore, even among
phase space configurations with three or more constituents, it is very
unlikely in perturbative QCD to find a configuration in which a particular
final-state particle lies nearly equidistant in angle between two others.

We also emphasize that the angular discontinuity of the C/A clustering
algorithm is representative of more general discontinuities in tree-based
clustering and in jet algorithms.171717 To mitigate the discontinuities of
tree-based clustering, one can introduce a continuous weighting procedure for
clustering histories. One way to assign continuous weights to different
clustering histories is the Q-jet scheme Ellis:2012sn ; Ellis:2014eya , which
notably leads to reduced statistical fluctuations in observable distributions
compared to the standard scheme of using a single clustering history for jet
substructure calculations. While the Q-jet scheme has not been applied
directly to grooming in the context of energy flows, to our knowledge, it
provides a particularly simple way to overcome the clustering discontinuities
we discuss here: we may simply define an overall groomed energy flow – devoid
of clustering discontinuities – as the weighted average of the groomed energy
flows associated with each possible clustering history.  For example, the
$k_{t}$ and anti-$k_{t}$ algorithms exhibit a similar $k_{t}$ discontinuity by
identical arguments. Discontinuities of different clustering algorithms may be
more or less suppressed in the phase space of perturbative QCD, or more or
less sensitive to particular sources of low-energy pollution. Therefore, a
more detailed discussion of clustering discontinuities and re-clustering
schemes that minimize their effects is a potential direction for future
research.

##  4 Responses of Grooming to Soft Distortions

We now compare the responses of Piranha and traditionally groomed jets to soft
distortions in more detail, focusing first on hadronization and then on the
exclusion of neutral particles as a simplified probe of smearing effects. In
particular,

  * •

We use P-RSFf=1/2 as a representative of Piranha grooming procedures;

  * •

We use Soft Drop with $\beta_{\rm SD}=2$ as a representative of traditional
grooming procedures.

For details regarding other Piranha grooming procedures and Soft Drop with
different $\beta_{\rm SD}$, see App. B.

###  4.1 Samples and Observables

In the studies of this section, we use Pythia 8.244 Sjostrand:2014zea  with
the default 4C tune Corke:2010yf , and work with samples of QCD quark and
gluon jets without multiple parton interactions. We consider jets with
transverse momentum $p_{T}>500$ GeV, maximum pseudorapidity $|\eta|<4$, and
clustered with the anti-$k_{T}$ algorithm Cacciari:2008gp  with $R=1$; we then
re-cluster using the Cambridge-Aachen algorithm Dokshitzer:1997in . Our
analyses focus on Soft Drop with $\beta_{\rm SD}=2$ as a representative for
traditional grooming procedures and on Balanced Recursive Subtraction
(P-RSFf=1/2) as a representative for Piranha grooming; the Balanced Recursive
Subtraction algorithm is available on GitHub piranhagithub .

We study the responses of groomed jets to soft distortion through three
qualitatively different lenses:

  * •

Energy Flow: EMD
To understand the overall response of groomed jet energy flow to soft
radiation, we use the EMD of Sec. 2.1. The EMD between two jets bounds the
difference of a large class of IRC-safe observables between the jets and
provides an observable-independent tool to study the robustness of different
grooming procedures to the presence of soft contamination.181818 In
particular, the EMD of Sec. 2.1 tends to reflect the response of extensive
properties of jets, as we see concretely in our examples; a less concrete way
to see this is by directly looking at Eq. (2), and noticing that the EMD has
units of energy.

  * •

Extensive Properties: $p_{T}$
To understand how the extensive properties of groomed jets are modified by
soft contamination, we examine the responses of the transverse momentum of
groomed jets. We are concerned in particular with the amount by which
transverse momentum in the presence of contamination, $\widetilde{p_{T}}$, is
shifted relative to the associated un-contaminated value, $p_{T}$:

$\displaystyle\Delta p_{T}=\widetilde{p_{T}}-p_{T}.$ (27)

$p_{T}$ indicates transverse momentum in the absence of distortion or
contamination and $\widetilde{p_{T}}$ captures the response of transverse
momentum to different sources of low-energy physics. This is specified more
precisely in each of the subsequent sections.

  * •

Substructure: $C_{1}^{(\varsigma)}$
We use jet substructure to capture the response of the intensive properties of
jets to contaminating radiation. In particular, we study the response of the
two-prong energy correlation functions (ECFs) of Ref. Larkoski:2013eya :

$C_{1}^{(\varsigma)}=\frac{1}{2}\frac{\sum_{i=1}^{M}\sum_{j=1}^{M}p_{T,\,i}\
p_{T,\,j}\ \left(R_{ij}/R_{0}\right)^{\varsigma}}{p_{T,\,{\rm tot.}}^{2}}.$
(28)

$p_{T,\,i}$ represents the fraction of the jet transverse momentum carried by
particle $i$, $R_{ij}$ indicates the rapidity-azimuth distance between
particles $i$ and $j$, and the sum is over all particles in the jet. In the
limit of jets that are central ($y=0$) and narrow (have jet radius $R_{0}\ll
1$), we can write the ECFs in the approximate form of Eq. (16). As a concrete
example of the response of substructure to soft radiation, we focus on $\Delta
C_{1}^{(2)}=\widetilde{C_{1}^{(2)}}-C_{1}^{(2)}\approx\Delta\left(m^{2}/p_{T}^{2}\right)$,
where a tilde again indicates the associated observable in the presence of a
particular source of low-energy physics, while the absence of a tilde
indicates the absence of low-energy effects.

We emphasize that in all of our phenomenological studies (except our studies
of the underlying event in Secs. 5.2 and 5.3), we examine how each source of
contamination leads to changes on a “per-jet” level, rather than at the level
of distributions of observables. In particular, we examine the EMD between
each jet before and after contamination, and the changes in $p_{T}$ and
$C_{1}^{(2)}$ induced by contamination for individual jets. To the best of our
knowledge, the effects of hadronization on a per-jet basis have not been
studied in this way before, as the sensitivity of groomed jets to
hadronization is often studied instead at a statistical level by studying the
distributions of groomed jet observables. Therefore, the results we present
offer a more detailed examination of the effects of the hadronization model of
Pythia 8.244; however, these results may not be precisely reflected solely by
the statistical changes in distributions of groomed jet observables that are
induced by hadronization.

###  4.2 Hadronization

We first provide phenomenological evidence that continuous grooming is less
sensitive than hard-cutoff grooming to the physics of hadronization. A deep
understanding of the physics of traditional jets requires careful
consideration of hadronization corrections, such as in jet masses
Hoang:2019ceu ; Marzani:2017kqd ; Benkendorfer:2021unv  and hadronic event
shapes Dokshitzer:1995zt ; Baron:2020xoi . Hard-cutoff groomed jet observables
in particular can undergo large corrections from the hadronization process due
to the discontinuity of the hard-cutoff paradigm: hadronization can transform
partons with energy below a hard cutoff into hadrons with energy above the
hard cutoff, and vice versa. These subtleties lead to additional complications
in the theoretical calculation of non-perturbative effects on traditionally
groomed jet observables Hoang:2019ceu .

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 7:  Per-jet hadronization responses of (top row) EMD, (middle row)
$\Delta p_{T}$, and (bottom row) $\Delta C_{1}^{(2)}$ using Balanced Recursive
Subtraction (P-RSFf=1/2, orange) and Soft Drop with $\beta_{\rm SD}=2$ (SDβ=2,
blue). We display (left column) the distribution of the response for $z_{\rm
cut}$=0.1, (middle column) the mean response as a function of $z_{\rm cut}$,
and (right column) the standard deviation of the response as a function of
$z_{\rm cut}$. The red arrows indicate the direction corresponding to better
performance.

In Fig. 7, we compare how ungroomed jets, traditionally groomed jets, and
Piranha-groomed jets respond to hadronization in the context of energy flow,
extensive properties, and substructure. Our studies focus on the parton-hadron
EMD (the EMD between parton-level jets and their hadron-level counterparts),
the parton-hadron $\Delta p_{T}$, and the parton-hadron $\Delta C_{1}^{(2)}$.
Piranha-groomed jets typically exhibit smaller and more predictable responses
to hadronization, evinced by the reduced variance in their hadronization
response. One key factor in the improved responses of Piranha-groomed jets is
the scaling-down of distortions due to hadronization in Piranha-groomed jets:
the subtractive approach of Piranha (and in particular P-RSF when $f_{\rm
soft}\neq 0,1$) removes energy from every particle in the event, resulting in
a controlled and uniform hadronization response. On the other hand, Soft Drop
grooming does not implement continuous, event-wide subtraction. Jets groomed
with Soft Drop either retain or remove particles as a result of distortions
due to hadronization, resulting in a larger variance in the properties of jets
groomed with Soft Drop.

We plot distributions of the parton-hadron EMD for groomed QCD jets for the
benchmark value of $z_{\rm cut}{}=0.1$ in Fig. 7a. We see a sharper peak at a
smaller EMD in the parton-hadron EMD distributions for P-RSFf=1/2 and longer
tails for Soft Drop, already indicating that hadronization is more likely to
dramatically change the result of Soft Drop. Figs. 7b and 7c show that the
average parton-hadron EMD and the variance in the parton-hadron EMD are both
smaller for P-RSFf=1/2 groomed jets than for Soft Drop groomed jets for a wide
range of $z_{\rm cut}$ values, again evincing that Piranha-groomed jets have
less sensitive responses to hadronization.

The extensive observable $p_{T}$ and the substructure observable $C_{1}^{(2)}$
also reflect the increased robustness of Piranha-groomed jets to
hadronization. In our discussion of hadronization, we characterize the
response of $p_{T}$ by using Eq. (27) with $p_{T}=p_{T}^{\rm(parton)}$
indicating the parton-level transverse momentum, before the addition of model-
dependent hadronization effects, and $\widetilde{p_{T}}=p_{T}^{\rm(hadron)}$.
In Fig. 7d, we see that for $z_{\rm cut}=0.1$, Piranha-groomed $p_{T}$ again
tends to have a sharper response to hadronization, while Figs. 7e and 7f
indicates that this sharper response holds for a wide range of $z_{\rm cut}$
values. The behavior of the $p_{T}$ closely mimics that of the parton-hadron
EMD, evincing our arguments that the parton-hadron EMD provides a probe of
generic jet observables to hadronization. In Figs. 7g, 7h, and 7i, we make
similar conclusions for $C_{1}^{(2)}\approx m^{2}/p_{T}^{2}$. We first note
that the parton-hadron shift in the $C_{1}^{(2)}$ substructure observable is
larger for P-RSFf=1/2, at least for fixed $z_{\rm cut}$, and the substructure
of P-RSFf=1/2 groomed jets may undergo larger changes due to hadronization
than that of Soft Drop groomed jets. However, the variance in the shift in
P-RSFf=1/2 groomed substructure is smaller than that of Soft Drop, indicating
that the response of Piranha-groomed substructure to hadronization is more
predictable than that of Soft Drop.

Overall, we find that Piranha-groomed jets have more predictable responses to
hadronization than traditionally groomed jets, supported by the observable
independent EMD, as well as by extensive observables and jet substructure. We
have provided evidence that IRC-safe observables of jets groomed with Soft
Drop have generically larger responses to hadronization, while Piranha-groomed
observables are generally less affected by the physics of hadronization. We
hope that the stability of Piranha-groomed jets to the non-perturbative
physics of hadronization may facilitate even further communication between
theoretical predictions and experimental results for groomed jet observables.

###  4.3 All Versus Charged

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 8:  Same as Fig. 7, but for per-jet responses to the exclusion of
neutral particles. We rescale the $p_{T}$ of the charged particles in order to
eliminate the leading $p_{T}$ bias due to the loss of neutral energy.

We next compare the response of hard-cutoff and Piranha grooming to the
exclusion of neutral particles, as a rough analog of the smearing of neutral
particles due to detector effects. Small tweaks in experimental signatures due
to detector effects may produce large changes in hard-cutoff groomed
observables, providing another obstacle in extracting fundamental physics from
traditionally groomed jets. The continuity of Piranha again leads us to expect
that Piranha-groomed observables are more robust to detector effects than
traditionally groomed jet observables.

To counteract the leading bias due to the loss of neutral energy in the jet,
we apply a rescaling to the $p_{T}$ of the charged-only jet constituents:

$\displaystyle\tilde{p}_{T}^{i}=\begin{cases}0,&\text{particle }i\text{ is
neutral}\\\ p_{T}^{i}\frac{\left\langle
p_{T}^{\text{(all)}}\right\rangle}{\left\langle
p_{T}^{\text{(charged)}}\right\rangle},&\text{particle }i\text{ is
charged}\end{cases},$ (29)

where $p_{T}^{i}$ is the transverse momentum of particle $i$ in the original
jet, $\mathcal{J}$, containing both charged and neutral particles, and
$\tilde{p}_{T}^{i}$ is the transverse momentum we assign to particle $i$ in
the charged-only, rescaled jet, $\tilde{\mathcal{J}}$, with neutral particles
excluded. Here, $p_{T}^{\rm(all)}=\sum_{i}p_{T}^{i}$ is the total transverse
momentum including both charged and neutral particles, while
$p_{T}^{\rm(charged)}=\sum_{i\text{ charged}}p_{T}^{i}$ includes only charged
constituents. The rescaling factor
$\big{\langle}p_{T}^{\text{(all)}}\big{\rangle}/\big{\langle}p_{T}^{\text{(charged)}}\big{\rangle}$$\approx$$1.47$
is defined such that the average $p_{T}$ is preserved after removing neutral
particles.191919 We can gain intuition for this rescaling factor by
considering the isospin-preserving limit, where we expect similar numbers of
$\pi^{+}$, $\pi^{-}$, and $\pi^{0}$ mesons to be produced. Events for which we
discard the $\pi^{0}$ particles should have roughly $2/3$ the total transverse
momentum, ignoring subtleties associated with kaons and other heavier states.
This leads to an estimate of
$\big{\langle}p_{T}^{\text{(all)}}\big{\rangle}/\big{\langle}p_{T}^{\text{(charged)}}\big{\rangle}\approx
3/2$ – remarkably close to the numerical value of 1.47 found in Pythia 8.244.

In our discussion of EMD, we compute the response
$\text{EMD}\left(G(\mathcal{J}),G(\tilde{\mathcal{J}})\right)$, where $G$
indicates the grooming algorithm under consideration; we call this the all-
charged EMD. Similarly, in our discussion of Eq. (27), we take
$p_{T}=p_{T}^{\rm(all)}$ to be the transverse momentum including both charged
and neutral particles and
$\widetilde{p_{T}}=\sum_{i}\tilde{p}_{T}^{i}=p_{T}^{\text{(charged,
rescaled)}}$ to be the rescaled contribution from charged particles only. We
refer to the difference as the all-charged $p_{T}$ shift, or the all-charged
$\Delta p_{T}$. Note that the rescaling procedure does not impact
dimensionless charged-only substructure observables like $C_{1}^{(2)}$, since
they are normalized by the jet momentum.

Our comparison of the all-charged response of Piranha grooming to that of
traditional grooming procedures is shown in Fig. 8. As in our study of
hadronization, we begin our study with a discussion of the EMD. The all-
charged EMD bounds changes in IRC-safe observables due to the exclusion of
neutral particles and corresponding rescaling of charged particles. Since
neutral particles are the most susceptible to smearing effects due to detector
responses, the all-charged EMD furnishes an observable-independent probe for
the effects of experimental detectors on groomed jets. We demonstrate
distributions of the all-charged EMD for groomed QCD jets with several choices
of grooming in Fig. 8a. The contrast between Piranha and traditionally groomed
all-charged EMD is not as sharp as for the parton-hadron EMD. However, Piranha
grooming again enjoys smaller and more sharply peaked EMD responses to the
exclusion of neutral particles. Fig. 8b shows that the all-charged EMD is
smaller for P-RSF1/2 groomed jets than for Soft Drop groomed jets for a wide
range of $z_{\rm cut}$ values. Fig. 8c shows that the fluctuations in the all-
charged EMD, and therefore the fluctuations in the response of grooming to our
naive model of smearing, are noticeably smaller for Piranha groomers than for
traditional groomers.

Our results for the all-charged $p_{T}$ shifts are similar to our results for
the all-charged EMD. Fig. 8d demonstrates that the $p_{T}$ response of
P-RSFf=1/2 is smaller and more sharply peaked for $z_{\rm cut}=0.1$, while
Figs. 8e and 8f respectively show that the P-RSFf=1/2 all-charged response
remains small and sharply peaked for a wide range of $z_{\rm cut}$ values.
Indeed, the fluctuations of the all-charged $p_{T}$ response are nearly
identical for each Piranha groomer.

Figs. 8g, 8h, and 8i compare the robustness of P-RSFf=1/2 and Soft Drop for
the substructure observable $C_{1}^{(2)}$. Fig. 8g again shows that the all-
charged response of P-RSFf=1/2 is more sharply peaked than that of Soft Drop.
Similarly, Figs. 8h and 8i demonstrate that the all-charged shifts to Piranha-
groomed jet substructure are generally larger but more predictable than those
of Soft Drop for a wide range of $z_{\rm cut}$ values. The increased
robustness of Piranha grooming procedures compared to Soft Drop is less sharp
than in the case of hadronization, but the Piranha-groomed observables
nonetheless have less spread and a stronger linear correlation in response to
our naive model of smearing.

##  5 Responses of Grooming to Additive Contamination

We examine the ability of Piranha grooming to mitigate the effects of additive
contamination from pileup (PU) and the underlying event (UE), which both
consist of extra soft radiation that sits on top of a hard process.

  * •

We use P-RSFf=1/2 as a representative of Piranha grooming procedures;

  * •

We use SDβ=2 as a representative of traditional grooming procedures;

  * •

In our PU studies, we use Constituent Subtraction (CS), a constituent-level
area subtraction method for pileup mitigation Berta:2014eza , as a
representative of PU mitigation procedures.

The case of additive contamination is qualitatively different than that of
soft distortions; we do not want to examine the robustness of the grooming
procedure to additive contamination, but rather the ability of the grooming to
remove the contamination. In our PU mitigation studies, we find that
P-RSFf=1/2 – which behaves comparably to CS – is a more effective tool than
SDβ=2. In our UE studies, we find that P-RSFf=1/2 and SDβ=2 are comparable
tools for the subtraction of UE effects from jet substructure.

We simulate PU using the dijet and minimum bias samples from Ref.
Soyez:2018opl , produced using Pythia 8.185 with tune 4C for proton-proton
collisions at $\sqrt{s}$ = 14 TeV. We layer minimum bias events on top of the
hard dijet events, taking the number of pileup events to be Poisson
distributed with a mean of $\langle n_{\rm PU}\rangle$ = 50 events.

We simulate UE by turning on multiple parton interactions in Pythia 8.244
Sjostrand:2014zea  with the default 4C tune Corke:2010yf . We begin with a
study of QCD jets and, to roughly echo similar studies in the original Soft
Drop paper Larkoski:2014wba , also explore the groomed mass resolution of jets
produced by boosted $W$ bosons and top quarks in the presence of UE.

###  5.1 Minimum Bias Events and Pileup

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 9:  Properties of per-jet PU-subtracted (top row) EMD, (middle row)
$\Delta p_{T}$, and (bottom row) $\Delta C_{1}^{(2)}$, discussed in Sec. 5.1
for Balanced Recursive Subtraction (P-RSFf=1/2, orange), Soft Drop with
$\beta_{\rm SD}=2$ (SDβ=2, blue), and Constituent Area Subtraction (CS,
green). We display (left column) the distribution of the each PU-subtracted
observable for $\langle n_{\rm PU}\rangle=50$, (middle column) the mean PU-
induced shift in each observable as a function of $\langle n_{\rm PU}\rangle$,
and (right column) the standard deviation of the PU-induced shifts in each
observable as a function of $\langle n_{\rm PU}\rangle$. The red arrows
indicate the direction corresponding to better performance.

We first examine the effectiveness of different groomers in mitigating the
effects of PU on the dijet events of Ref. Soyez:2018opl . To quantify the
ability of different groomers to remove PU, we perform an event-by-event
comparison of hard events to the associated PU-subtracted events:

  * •

A hard event is a dijet event without any PU, representing the physics of a
hard process;

  * •

A PU-subtracted event is produced by simulating the additive contamination due
to PU on top of a hard event and then attempting to groom the PU away.

To produce a PU-subtracted event, we first simulate PU by adding the energy
distributions of a Poisson-distributed number of minimum bias events to the
energy distribution of a given hard event. We then estimate the contaminating
energy density due to PU using the GridMedianBackgroundEstimator (GMBE) method
of FastJet Cacciari:2011ma . We tune an additional correction factor that
scales our estimate $\rho_{\rm est.}$ relative to the GMBE value $\rho_{\rm
GMBE}$ for each grooming algorithm to subtract pileup more effectively, as
discussed in more detail in App. B. Finally, we produce the PU-subtracted
event by removing a fraction $z_{\rm cut}$ of the jet $p_{T}$ consistent with
the estimated pileup density:

$\displaystyle z_{\rm cut}\,p_{T}^{\rm(jet)}=\rho_{\rm est.}\,A_{\rm
jet}\approx p_{T}^{\rm(PU~{}in~{}jet)}.$ (30)

We examine the EMD, $\Delta p_{T}$, and $\Delta C_{1}^{(2)}$ between the
leading jets of the hard and pileup-events in Fig. 9.202020 In some
situations, the pileup contributes enough energy that the leading jet after
the addition of pileup corresponds to the direction of the sub-leading jet
before the addition of pileup. This phenomenon occurs only for a small
fraction of events, and we do not include these events in the following
discussion.  In Figs. 9a, 9b, and 9c, we examine the EMD between the leading
jet in the hard dijet event and the leading jet in the subtracted event with
pileup. Fig. 9a demonstrates that when $\langle n_{\rm PU}\rangle=50$,
P-RSFf=1/2 tends to subtract pileup more accurately and predictably, evinced
by a more sharply peaked EMD distribution and a smaller average PU-subtracted
EMD than found for CS and Soft Drop. This strength of P-RSFf=1/2 persists as
the amount of pileup is increased, as suggested by Figs. 9b and 9c.

Figs. 9d, 9e, and 9f offer similar conclusions for the jet $p_{T}$. In our
study of pileup responses, $p_{T}=p_{T}^{\rm(hard)}$ indicates the transverse
momentum of the hard event, before the layering of minimum bias/pileup events,
and $\widetilde{p_{T}}=p_{T}^{\rm(groomed~{}PU)}$ indicates the transverse
momentum after pileup has been added and then groomed away. RSF1/2 again
reproduces the extensive quantity $p_{T}$ of the jet in the absence of PU for
a wide range of $\langle n_{\rm PU}\rangle$ values. Identical conclusions hold
for the substructure of the jet, as suggested by the behavior of $C_{1}^{(2)}$
shown in Figs. 9g, 9h, and 9i.

Figs. 15 and 16 of App. B evince further that with our tuned procedure for PU
mitigation, Piranha groomers and CS perform more reliably than traditional
groomers in the removal of pileup. Furthermore, P-RSF algorithms are tree-
based and offer orders-of-magnitude faster PU mitigation over CS, P-AS, and
P-IVS (see Fig. 17 of App. B).

###  5.2 Underlying Event

We next examine the ability of grooming to eliminate additive contamination
from UE. Since we model UE by generating events in the presence of multiple
parton interactions in Pythia 8.244, it is less straightforward to study the
effects of UE on an event-by-event basis. We instead work at the level of
distributions: we tune both P-RSFf=1/2 and Soft Drop so that, when acting on
events with UE, they most accurately reproduce the substructure distribution
of events without UE. We focus on substructure alone in our discussion of UE
correction because we cut on jets with $p_{T}>500$ GeV both for events with
and without UE, and it is less meaningful to focus on the associated $p_{T}$
distributions; we leave a more detailed study of the ability of Piranha
groomers to remove UE at an event-by-event level for future work.

We characterize the ability of groomers to correct for the presence of UE by
comparing base distributions to UE-corrected distributions:

  * •

A base distribution is a substructure distribution with a fixed grooming
parameter, $z_{\rm cut}^{(\rm base)}$;

  * •

A UE-corrected distribution is a substructure distribution in the presence of
UE and with slightly more grooming tuned to soak up the additional energy
contribution from UE: $z_{\rm cut}^{(\rm corr.)}=z_{\rm cut}^{(\rm
base)}+\delta z_{\rm cut}$.

We tune $\delta z_{\rm cut}$ by hand, for every $z_{\rm cut}^{(\rm base)}$, to
optimally correct for the presence of UE212121 We note that the optimized
$\delta z_{\rm cut}$ is energy-dependent and process-dependent. For example,
we expect that $\delta z_{\rm cut}$ should be parametrically smaller for
processes at higher energies since the energy removed from a jet scales as
$Qz_{\rm cut}$, where $Q$ is the energy scale of the hard process, while the
energy scales associated with UE are only weakly correlated with the energy of
the hard process (see, for example, Fig. 3 of Ref. Field:2011iq  or Section
7.2 of Ref. Campbell:2017hsr ).  by minimizing the Wasserstein distance
between the two distributions. The Wasserstein distance is a metric on the
space of distribution, much in the same way that the EMD is a metric on the
space of energy flows, and provides a useful and quantitative measure of the
ability of the grooming to correct for UE.

(a)

(b)

(c)

(d)

Figure 10:  The response of groomed substructure distributions of QCD jets to
the underlying event (UE), using P-RSFf=1/2 (top row) and Soft Drop with
$\beta_{\rm SD}=2$ (bottom row). We examine the robustness of groomed
substructure to the presence of UE by comparing “Base” substructure
distributions with and without UE for fixed $z_{\rm cut}=z^{(\rm base)}_{\rm
cut}$. We also examine the possibility of adding additional grooming to remove
the effects of UE, picking a $z^{(\rm corr.)}_{\rm cut}$ that removes UE to
most accurately reproduce substructure distributions using $z_{\rm
cut}=z^{(\rm base)}_{\rm cut}$ in the absence of UE. The left column shows the
substructure distributions for $z^{(\rm base)}_{\rm cut}=0$, while the right
column shows the substructure distributions for $z^{(\rm base)}_{\rm
cut}=0.025$.

Fig. 10 compares base distributions to UE-corrected distributions for
P-RSFf=1/2 and Soft Drop. Figs. 10a and 10c examine the ability of the
groomers to reproduce ungroomed base distributions, with $z_{\rm cut}^{(\rm
base)}=0$. Figs. 10b and 10d examine instead groomed base distributions with
$z_{\rm cut}^{(\rm base)}=0.025$.

(a)

(b)

Figure 11:  The behavior of the Wasserstein distance, in arbitrary units,
between the $C_{1}^{(2)}$ distribution: (a) before UE correction, and (b)
after UE correction, for SDβ=2 and P-RSFf=1/2 as a function of the base amount
of grooming $z_{\rm cut}^{(\rm base)}$. In (b), a lower UE-corrected
Wasserstein distance quantitatively indicates better correction against the
effects of UE: the substructure distribution of UE-corrected jets more closely
resembles the substructure distribution of jets without UE.

Figs. 11a and 11b quantify the effectiveness and robustness of P-RSFf=1/2 and
Soft Drop as a function of $z_{\rm cut}^{(\rm base)}$ by studying the un-
corrected Wasserstein distance and the UE-corrected Wasserstein distance,
respectively. Fig. 11b shows the UE-corrected Wasserstein distance as a
function of $z_{\rm cut}$, and suggests that P-RSFf=1/2 is better at
correcting for UE when $z_{\rm cut}^{(\rm base)}\lesssim.05$, and that SDβ=0
and P-RSFf=1/2 have similar UE correction ability for values of $z_{\rm
cut}^{(\rm base)}$ between 0.05 and .1.

We note that we expect hard-cutoff groomers to be generically more robust to
the presence of UE because even at fixed $z_{\rm cut}$, they can remove an
arbitrary amount of energy due to soft, wide-angle radiation in an event. We
correspondingly expect that hard-cutoff groomers require less additional
grooming to remove the effects of UE. Piranha groomers, on the other hand,
cannot remove arbitrary amounts of energy from an event; we expect that we
must always slightly increase the strength of Piranha grooming to soak up
additive contamination from UE. We demonstrate that hard-cutoff groomers
indeed require less additional $\delta z_{\rm cut}$ for optimal UE correction
in Fig. 18 of App. B.3.

We also point out that Recursive Subtraction techniques that are more geared
towards the removal of the soft and wide-angle radiation which is
characteristic of UE may perform even better in UE correction. In hard-cutoff
grooming, for example, Soft Drop with $\beta_{\rm SD}>0$ preferentially grooms
wide-angle radiation, Soft Drop with positive $\beta_{\rm SD}$ is particularly
well suited to remove UE (see, for example, Fig. 18 of App. B.3). Perhaps P-RS
grooming procedures that upgrade the parameter $f_{\rm soft}$ of P-RSF into a
function of the soft energy fraction $z$ and the splitting angle $\theta$ of a
branch (such as Hard-Balanced P-RSF, as in Sec. 3.4) may be more suited to the
removal of soft wide-angle radiation from UE.

We have argued that P-RSFf=1/2 and SDβ=2 are comparable tools for the
subtraction of UE effects in jet substructure, though P-RSFf=1/2 requires more
tuning to achieve the same level of UE correction. We emphasize that further
investigation of such P-RS algorithms might improve the ability of Piranha in
the tagging of boosted objects and beyond.

###  5.3 Case Study: Mass Resolution for $W$ and Top Jets

As a final case study, we explore the use of Piranha grooming strategies in
the tagging of boosted objects, another useful application of traditional
grooming strategies such as Soft Drop. We must once again overcome the
subtlety that Piranha grooming strategies remove energy proportional to the
grooming parameter (such as $z_{\rm cut}$ in Recursive Safe Subtraction). To
successfully tag boosted objects decaying into jets, we must tune the grooming
parameter for Piranha strategies more than we would for Soft Drop.
Nonetheless, we examine the mass resolution of Piranha-groomed jets from the
decays of boosted $W$ bosons and top quarks and discover that Piranha may
offer some advantages that are complementary to those of traditional grooming
strategies. The tagging procedure we implement in the following discussion is
quite simple, however, and the applications of Piranha in boosted object
tagging should be subjected to a more complete analysis.

(a)

(b)

(c)

(d)

Figure 12:  A comparison of traditional and Piranha grooming procedures in the
tagging of (top row) boosted $W$ bosons and (bottom row) top quarks. The data
in the plots correspond to 100,000 jet events with $p_{T}>500$ GeV in Pythia
8.244. The left column displays the mass distributions of $W$ and top jets
with and without UE, and with UE groomed away using either Soft Drop with
$\beta_{\rm SD}=2$ or P-RSFf=1/2 with additional post-processing discussed in
the text. The right column displays the relationship between background
rejection and signal efficiency associated with $W$ and top tagging,
respectively, produced by considering symmetric windows around the mass of the
relevant boosted object and comparing with a QCD background.

In Fig. 12, we assess the behavior of Piranha in tagging boosted objects. We
focus on jets of $p_{T}\geq 500$ GeV produced from the decays of $W$ bosons
and top quarks in Figs. 12a and 12c, respectively. We then plot the associated
background rejection as a function of signal efficiency in Figs. 12b and 12d.

To produce the groomed mass distributions of Figs. 12a and 12c, we groom jets
in the presence of UE. Afterward, we rescale the mass of the jet to correct
for the shift of the mass peak due to the removal of energy from the jet
during grooming. This rescaling procedure is necessary to reproduce the
correct mass peak for P-RSFf=1/2; however, rescaling Soft Drop groomed mass
distributions has a minimal effect since Soft Drop tends to remove wide-angle
soft radiation without severely affecting radiation from the hard event.

Figs. 12a and 12c show rescaled mass distributions for each grooming algorithm
for the value of $z_{\rm cut}$ that most precisely removes UE, in the sense
that it most closely reproduces the mass distribution of jets without UE. We
find that the best performance is achieved by $z_{\rm
cut}^{(\text{P-RSF}_{1/2})}=0.034$ and $z_{\rm cut}^{(\text{SD}_{2})}=0.025$
when grooming W jets, and $z_{\rm cut}^{(\text{P-RSF}_{1/2})}=0.023$ and
$z_{\rm cut}^{(\text{SD}_{2})}=0.035$ when grooming top jets. In each case,
Soft Drop does not require any additional rescaling, while we find the best
performance if we rescale the energy of P-RSFf=1/2 groomed jet constituents by
about $+4\%$.

In this simple context, P-RSFf=1/2 offers slightly better background rejection
for reasonable signal efficiencies. We evince this in Figs. 12b and 12d, where
we show the signal efficiency versus background rejection curves associated
with comparing signal W or top jets to QCD background, again with $p_{T}\geq
500$ GeV. Our tagging procedure is minimal: we accept events within a
symmetric window of width $\delta m$ around the mass of the $W$ boson or top
quark, respectively, and Figs. 12b and 12d are produced by varying the window
size $\delta m$. For reasonable values of signal efficiency, between around
60% and 90%, the P-RSFf=1/2 algorithm gives greater background rejection than
Soft Drop.

Again, the results in Figs. 12a and 12c and Figs. 12b and 12d suggest that
P-RSFf=1/2 and SDβ=2 behave comparably as tools to remove the effects of UE in
jet mass distributions, and to tag underlying boosted object. These results
are preliminary, and to state these conclusions with more confidence will
require more detailed tagging procedures than the simple algorithm described
above. We leave a more detailed study of the tagging potential of P-RSFf=1/2
and Piranha to future work.

##  6 Discussion and Conclusions

In this work, we proposed the paradigm of Pileup and Infrared Radiation
Annihilation (Piranha) for continuous jet grooming. Motivated by optimal
transport theory and the Energy Mover’s Distance (EMD) of Ref. Komiske:2019fks
, we re-framed the Apollonius Subtraction (P-AS) and Iterated Voronoi
Subtraction (P-IVS) procedures of Ref. Komiske:2020qhg  as implementations of
the Piranha paradigm. We additionally introduced Recursive Subtraction with a
Fraction (P-RSF) as a family of new groomers motivated by the Piranha
paradigm. We showed that a particular Recursive Subtractor, P-RSFf=1/2,
overcomes the soft discontinuities of traditional hard-cutoff grooming
procedures, and that general P-RSF algorithms only have soft discontinuities
in suppressed regions of phase space. We highlighted the unprecedented
robustness of P-RSFf=1/2 to hadronization, detector effects, and pileup. We
showed also that P-RSFf=1/2 may be able to correct for the presence of the
underlying event. Though hard-cutoff groomers may be more robust against
effects from UE without additional tuning, Piranha groomers can be tuned to
remove additive contamination from the underlying event. We used the example
of additive contamination from UE to argue that Piranha may also have
applications in the tagging of boosted objects.

There are several immediately evident avenues for future phenomenological and
theoretical exploration. While we argued that P-RSF had more robust responses
to soft distortions, it will be interesting to quantify this including
theoretical, model-dependent uncertainties. For example, it will be
interesting to study the robustness of P-RSFf=1/2 to hadronization when using
different models of color recombination, with different Pythia tunes, and from
the perspective of effective field theory as in Ref. Hoang:2019ceu .
Similarly, the effects of experimental detectors on P-RSFf=1/2 groomed
quantities must be explored with more realistic models of detector responses
and compared in more detail with traditional grooming techniques. More
detailed studies of these model-dependent uncertainties may facilitate a more
precise, and even process-dependent method to tune the parameters of Piranha
groomers such as P-RSF to the removal of PU and UE in more realistic
scenarios.

Designing and studying variants of P-RSF in which the amount of grooming for a
particular emission depends on the angle of the emission, as in Soft Drop with
$\beta_{\rm SD}\neq 0$, and even the energy fraction, may be another easy way
to improve the robustness and precision of P-RS algorithms in removing
specific models of soft contamination. Indeed, any method that varies $f_{\rm
soft}$, $z_{\rm cut}$, $\rho$, or other parameters of the jet grooming
procedure in a way that depends on more detailed information within the jet
could be useful in more precise removal of contaminating radiation, such as
the removal of PU, UE, or the thermal noise that contaminates jets originating
in heavy ion collisions.

A first-principles calculation of Recursive Subtraction groomed jet
observables will rely on analytic exploration of the elaborately correlated
emissions of P-RS groomed jets, as well as contributions from non-global
configurations. We begin this journey in App. C, but further exploration may
depend on a more precise application of existing tools for jet substructure or
even on newer tools in perturbative QCD, such as the techniques of Ref.
Larkoski:2015zka .

The study of EMD-mode Piranha grooming, introduced in App. A, is another
interesting avenue for the development of continuous grooming techniques.
Unlike the Piranha groomers explored in the main text, EMD-mode Piranha
groomers have the potential to remove an arbitrary amount of $p_{T}$ from an
event. We look forward to future studies of EMD-mode grooming for its
potential to address stochastic fluctuations in the levels of jet
contamination.

Finally, the use of alternative methods for optimal transport, such as the EMD
without the restriction of $\beta=1$ that we chose in this work, or the
flexible and computationally efficient formalism of linearized optimal
transport Cai:2020vzx ; Cai:2021hnn ; cai2022linearized ;
sarrazin2023linearized , may offer new tools and additional insights into