* Durrett and Iglehart [1977] R.T. Durrett and D.L. Iglehart.  Functionals of Brownian meander and Brownian excursion.  _The Annals of Probability_ , 5:130–135, 1977.
  * Evans [2005] S.N. Evans.  _Probability and real trees, École d’Été de Probabilités de Saint-Flour XXXV-2005_ , volume 1920 of _Lecture Notes in Mathematics_.  Springer, 2005.
  * Falconer [1986] K. J. Falconer.  _The Geometry of Fractal Sets_ , volume 85 of _Cambridge Tracts in Mathematics_.  Cambridge University Press, Cambridge, 1986.
  * Falconer [1990] Kenneth Falconer.  _Fractal Geometry: Mathematical Foundations and Applications_.  John Wiley & Sons Ltd., Chichester, 1990.
  * Frilet [2021] N. Frilet.  _Metric coalescence of homogeneous and inhomogeneous random graphs_.  Phd thesis, Université Grenoble-Alpes, 2021.  https://hal.inria.fr/tel-03667362v1.
  * Graf et al. [1988] Siegfried Graf, R Daniel Mauldin, and Stanley C Williams.  _The exact Hausdorff dimension in random recursive constructions_ , volume 381.  American Mathematical Soc., 1988.
  * Greven et al. [2009] A. Greven, P. Pfaffelhuber, and A. Winter.  Convergence in distribution of random metric measure spaces ($\lambda$-coalescent measure trees).  _Probability Theory and Related Fields_ , 145(1):285–322, 2009.
  * Groeneboom [1983] P. Groeneboom.  The concave majorant of Brownian motion.  _The Annals of Probability_ , 11:1016–1027, 1983.
  * Hoeffding [1963] W. Hoeffding.  Probability inequalities for sums of bounded random variables.  _Journal of the American Statistical Association_ , 58:13–30, 1963.
  * Janson [2007] S. Janson.  Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas.  _Probability Surveys_ , 4:80–145, 2007.
  * Karatzas and Shreve [1988] I. Karatzas and S.E. Shreve.  _Brownian Motion and Stochastic Calculus_.  Graduate Texts in Mathematics. Springer, New York, 1988.
  * Kennedy [1976] D.P. Kennedy.  The distribution of the maximum Brownian excursion.  _Journal of Applied Probability_ , 13:371–376, 1976.
  * Knuth [1973] D. E. Knuth.  _The Art of Computer Programming: Sorting and Searching_ , volume 3.  Addison-Wesley, Reading, MA, 1973.
  * Konheim and Weiss [1966] Alan G Konheim and Benjamin Weiss.  An occupancy discipline and applications.  _SIAM Journal on Applied Mathematics_ , 14(6):1266–1274, 1966.
  * Kortchemski and Thévenin [2023] Igor Kortchemski and Paul Thévenin.  Coupling Bertoin’s and Aldous-Pitman’s representations of the additive coalescent.  _arXiv preprint arXiv:2301.01153_ , 2023.
  * Kruskal [1956] J.B. Kruskal.  On the shortest spanning subtree of a graph and the traveling salesman problem.  _Proceedings of the American Mathematical Society_ , 2:48–50, 1956.
  * Le Gall [1991] J.-F. Le Gall.  Brownian excursions, trees and measure-valued branching processes.  _The Annals of Probability_ , 19:1399–1439, 1991.
  * Le Gall [1993] J.-F. Le Gall.  The uniform random tree in a Brownian excursion.  _Probability Theory and Related Fields_ , 96:369–383, 1993\.
  * Le Gall and Le Jan [1998] J.F. Le Gall and Y. Le Jan.  Branching processes in Levy processes: Laplace functionals of snakes and superprocesses.  _The Annals of Probability_ , 26:1407–1432, 1998.
  * Marckert and Wang [2018] Jean-François Marckert and Minmin Wang.  A new combinatorial representation of the additive coalescent.  _Random Structures & Algorithms_, 54(2):340–370, Apr 2018.  doi: 10.1002/rsa.20775.
  * Martin and Ráth [2017] James B. Martin and Balázs Ráth.  Rigid representations of the multiplicative coalescent with linear deletion.  _Electronic Journal of Probability_ , 22(0), 2017.  doi: 10.1214/17-ejp100.
  * Mattila [1999] P. Mattila.  _Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability_ , volume 44 of _Cambridge Studies in Advanced Mathematics_.  Cambridge Univ Press, 1999.
  * Mauldin and Williams [1986] R.D. Mauldin and S.C. Williams.  Random recursive constructions: Asymptotic Geometric and Topological Properties.  _Transactions of the American Mathematical Society_ , 295:325–346, 1986.
  * Miermont [2003] G. Miermont.  Self-similar fragmentations derived from the stable tree I: Splitting at heights.  _Probability Theory and Related Fields_ , 127:423–454, 2003\.
  * Miermont and Sen [2022] Grégory Miermont and Sanchayan Sen.  On breadth-first constructions of scaling limits of random graphs and random unicellular maps.  _Random Structures & Algorithms_, 2022.
  * Nachmias and Peres [2008] A. Nachmias and Y. Peres.  Critical random graphs: diameter and mixing time.  _The Annals of Probability_ , 36:1267–1286, 2008.
  * Perkins [1981] E. Perkins.  The exact Hausdorff measure of the level sets of Brownian motion.  _Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete_ , 58:373–388, 1981.
  * Pitman and Ross [2011] Jim Pitman and Nathan Ross.  The greatest convex minorant of brownian motion, meander, and bridge.  _Probability Theory and Related Fields_ , 153(3-4):771–807, Aug 2011.  doi: 10.1007/s00440-011-0385-0.
  * Pitman [1983] J.W. Pitman.  Remarks on the convex minorant of Brownian motion.  In E. Çinlar, K.L. Chung, and Getoor R.K., editors, _Seminar on Stochastic Processes_ , volume 5 of _Progress in Probability and Statistics_ , pages 219–227, Boston, 1983. Birkhäuser.
  * Prim [1957] R. C. Prim.  Shortest connection networks and some generalizations.  _Bell Syst. Tech. J._ , 36:1389–1401, 1957.
  * Rogers and Williams [2000] L.C.G. Rogers and D. Williams.  _Diffusions, Markov Processes and Martingales: Itô Calculus_ , volume 2.  Cambridge University Press, Cambridge, UK, 2 edition, 2000.
  * Rossignol [2021] R. Rossignol.  Scaling limit of dynamical percolation on critical Erdős–Rényi random graphs.  _The Annals of Probability_ , 49:322–399, 2021.
  * Taylor and Wendel [1966] S.J. Taylor and J.G. Wendel.  The exact Hausdorff measure of the zero set of a stable process.  _Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete_ , 6:170–180, 1966.

##  Appendix A Auxiliary technical results

######  Lemma A.1.

Let $\omega\in\mathcal{C}(\mathbb{R}_{+},\mathbb{R})$. Then

  1. i)

the process $(Z^{\lambda}(\omega))_{\lambda\in\mathbb{R}}$ non-increasing and
right-continuous with left-limits;

  2. ii)

for every $\lambda\in\mathbb{R}$, $Z^{\lambda}(\omega)$ and $Z^{\lambda-}$ are
both closed;

  3. iii)

the set $\\{\lambda\in\mathbb{R}:Z^{\lambda-}(\omega)\setminus
Z^{\lambda}(\omega)\neq\varnothing\\}$ is countable.

###### Proof.

_i)_ The monotony is a consequence of Lemma 4.1, this implies the existence of
the left and right limits $\cap_{h>0}Z^{\lambda-h}$ and
$\cup_{h>0}Z^{\lambda+h}$, respectively. The right-continuity follows by
continuity of the maps $\lambda\mapsto\omega^{\lambda}$ and
$\underline{\omega}^{\lambda}$: if $s\in Z^{\lambda-h}$ for all $h>0$, then
$\omega(s)+(\lambda-h)s=\inf\\{\omega(r)+(\lambda-h)r:0\leq r\leq s\\}$ for
all $h>0$, and thus this also holds for $h=0$. _ii)_ The fact that
$Z^{\lambda}$ is closed is an easy consequence of the continuity of $\omega$.
The monotony shows that $Z^{\lambda-}$ is a decreasing limit of closed sets,
and is thus closed. _iii)_ Since $Z^{\lambda}$ and $Z^{\lambda-}$ are both
closed for every $\lambda\in\mathbb{R}$, if $\lambda$ is such that
$Z^{\lambda-}\setminus Z^{\lambda}\neq\varnothing$, then there exists
$\epsilon>0$ and $x=x_{\lambda}\in Z^{\lambda-}$ with
$d(x,Z^{\lambda})>\epsilon$. It follows that

$\\{\lambda\in\mathbb{R}:Z^{\lambda-}\setminus
Z^{\lambda}\neq\varnothing\\}=\bigcup_{n\geq
1}\\{\lambda\in\mathbb{R}:\operatorname{d}_{\text{{h}}}(Z^{\lambda-},Z^{\lambda})>1/n\\}\,.$

For each $n\geq 1$, there must exist for each $\lambda$ a ball of radius
$1/n$, and the collection of these balls must be disjoint. For each $n\geq 1$,
any collection of open balls of radius $1/n$ must be countable, and therefore
any set in the right-hand side above is countable. The claim follows. ∎

######  Lemma A.2 (Continuity properties of the metric $d$).

Let $d(\cdot,\cdot)$ be the pseudo-metric on $[0,1]$ defined from the pair
$({\textup{e}},{\mathbf{U}})$ used in the construction of
$\operatorname{CMT}({\textup{e}},{\mathbf{U}})$. Almost surely,

  1. i)

the map $d(0,\cdot)$ is continuous almost everywhere, but

  2. ii)

for every $x\in\mathscr{L}({\textup{e}})\cap(0,1)$, the map $d(0,\cdot)$ is
not left-continuous at $x$, and

  3. iii)

for every $x\in\mathscr{L}({\textup{e}})\cap(0,1)$, the map $d(0,\cdot)$ is
neither left- nor right-continuous at ${{\text{\bsifamily r}}}(x)$.

###### Proof.

_i)_ Let $x$ be uniformly random in $[0,1]$, then a.s. the vertices of the
convex minorant of e on $[0,x]$, $(t_{i}(x))_{i\geq 0}$ and the corresponding
intercepts $(z_{i}(x))_{i\geq 0}$ are such that $t_{i}(x)<x<z_{i}(x)$.
Furthermore, there exists a sequence of local minima $z_{n}>x$ with
$z_{n}\downarrow x$ such that the vertices of the convex minorant of e on
$[0,z_{n}]$ are precisely
${\mathcal{V}}_{z_{n}}={\mathcal{V}}_{x}\cup\\{z_{n}\\}$. Then, for any $i\geq
1$, $\sup\\{d(x,t):t\in(t_{i},z_{i})\\}\leq D_{i}\cdot|t_{i}-z_{i}|^{1/2}$,
where $(D_{i})_{i\geq 1}$ are random variables distributed like the diameter
of a continuum random tree of unit mass (which are not independent). It
follows that

$\displaystyle{\mathbf{P}}(\sup\\{d(x,t):t\in(t_{i},z_{i})\\}>|z_{i}-t_{i}|^{1/4})$
$\displaystyle\leq{\mathbf{P}}(D_{i}>|z_{i}-t_{i}|^{-1/4})$
$\displaystyle\leq\exp(-|z_{i}-t_{i}|^{-1/4}/2v)\,,$

for some constant $v>0$. It follows by the Borel–Cantelli Lemma that a.s.
$\sup\\{d(x,t):t\in(t_{i},z_{i})\\}\leq|z_{i}-t_{i}|^{1/4}$ for all but
finitely many values of $i$, so that $|d(0,t)-d(0,x)|\leq d(x,t)\to 0$ as
$t\to x$.

_ii)_ Let $x\in\mathscr{L}\cap(0,1)$; then a.s. there are only finitely many
vertices in ${\mathcal{V}}_{x}$, and $x=t_{i}(x)$ for some $i\geq 1$. Let
$z_{n}$ be a sequence of local minima with $z_{n}\in(t_{i-1},t_{i})$ and
$z_{n}\uparrow t_{i}$ as $n\to\infty$. Then, for all $n_{0}$ large enough, the
vertices of the convex minorant of e on $[0,z_{n}]$ are exactly
$\\{t_{j},j<i\\}\cup\\{z_{n}\\}$. For each $n\geq n_{0}$, the point
${{\text{\bsifamily j}}}(z_{n})$ is uniform in $(t_{i-1},z_{n})$ and
${{\text{\bsifamily j}}}(x)$ is uniform in $(t_{i-1},t_{i})$. With probability
one, there exists a subsequence $(n_{j})_{j\geq 1}$ such that
$0<{{\text{\bsifamily j}}}(z_{n_{j}})-t_{i-1}<\tfrac{1}{2}({{\text{\bsifamily
j}}}(x)-t_{i-1})$. In particular, since $\llbracket 0,x\rrbracket$ a.s. has an
accumulation point at ${{\text{\bsifamily j}}}(x)$, we have $\sup
d(0,z_{n_{j}})=\sup d(0,{{\text{\bsifamily
j}}}(z_{n_{j}}))<d(0,{{\text{\bsifamily j}}}(x))=d(0,x)$. It follows that, for
any $\epsilon>0$, $\inf\\{d(0,s):s\in(x-\epsilon,x)\\}<d(0,x)$.

_iii)_ For $x\in\mathscr{L}\cap(0,1)$, the point ${{\text{\bsifamily r}}}(x)$
is some intercept, and the proof that $d(0,\cdot)$ is not left-continuous at
${{\text{\bsifamily r}}}(x)$ is the same as in _ii)_. For the lack of right-
continuity at ${{\text{\bsifamily r}}}(x)$, this is also similar, but relies
on the fact that one may find a sequence of local minima $z_{n}$ in
$({{\text{\bsifamily r}}}(x),1)$ with $z_{n}\downarrow{{\text{\bsifamily
r}}}(x)$ such that ${\mathcal{V}}_{z_{n}}={\mathcal{V}}_{x}\cup\\{z_{n}\\}$.
The same argument as above can then be used by considering the random points
${{\text{\bsifamily j}}}(z_{n})$, which are independent, and uniform in
$[x,z_{n}]$. ∎

######  Lemma A.3 (Surplus and area under the curve).

Let e be a Brownian excursion. Consider the subset $D$ of
$[0,1]^{2}\times\mathbb{R}$ of points $(x,y,\lambda)$ such that $[x,y]\cap
Z^{\lambda}({\textup{e}})=\varnothing$. Then, the 3-dimensional volume of $D$
is equal to $\int_{0}^{1}{\textup{e}}(x)dx$.

###### Proof.

Recall the recursive decomposition of Section 4.4. Then, the set $D$ can be
decomposed into countably many portions (with disjoint interior) $D_{u}$,
$u\in{\mathcal{U}}$, as follows:

$D_{u}:=\\{(x,y,\lambda):a_{u}\leq
x<y<R_{-\lambda}(a_{u}),\lambda\leq-\tau_{u}\\}\,.$

There is a corresponding decomposition of the set $\\{(s,t):s\in[0,1],0\leq
t\leq{\textup{e}}(s)\\}$ also into portions with disjoint interior,
$E_{u}=\\{(s,{\textup{e}}(a_{u})-\lambda(s-a_{u})):a_{u}\leq s\leq
R_{-\lambda}(a_{u}),-\lambda\geq\tau_{u}\\}$, for $u\in{\mathcal{U}}$. We show
that, for each $u\in{\mathcal{U}}$,

$\int_{E_{u}}dsdt=\int_{D_{u}}dxdyd\lambda\,.$

We treat the case $u=\varnothing$, the others are just the same, up to the
more complicated notation. First observe that the left-hand side above with
$u=\varnothing$ is precisely the area under the function $f$ given by, for
$i\geq 1$,

$f(s)={\textup{e}}(a_{i})+\tau_{i}(s-a_{i})\qquad a_{i}=R_{\tau_{i}}\leq
s<R_{\tau_{i}-}\,.$

Now, since each point $z=(s,f(s))$ can be represented in polar coordinates as
$z=\rho(\theta)e^{i\theta}$, or alternatively by the pair
$(-\lambda,R_{-\lambda}(0))$, where $-\lambda=f(s)/s$ is the slope of the line
from $0$ to $z$, we have

$\int_{E_{\varnothing}}dsdt=\int_{0}^{1}f(s)ds=\frac{1}{2}\int_{0}^{\pi/2}\rho(\theta)^{2}d\theta=\int_{D_{\varnothing}}dxdyd\lambda\,.$

The claim follows by summing the contributions for $u\in{\mathcal{U}}$. ∎


# PAIReD jet: A multi-pronged resonance tagging strategy across all Lorentz
boosts

Spandan Mondal Brown University, Providence, USA Gaetano Barone11footnotemark:
1 Alexander Schmidt RWTH Aachen University, Aachen, Germany

###### Abstract

We propose a new approach of jet-based event reconstruction that aims to
optimally exploit correlations between the products of a hadronic multi-
pronged decay across all Lorentz boost regimes. The new approach utilizes
clustered small-radius jets as seeds to define unconventional jets, referred
to as PAIReD jets. The constituents of these jets are subsequently used as
inputs to machine learning-based algorithms to identify the flavor content of
the jet. We demonstrate that this approach achieves higher efficiencies in the
reconstruction of signal events containing heavy-flavor jets compared to other
event reconstruction strategies at all Lorentz boost regimes. Classifiers
trained on PAIReD jets also have significantly better background rejections
compared to those based on traditional event reconstruction approaches using
small-radius jets at low Lorentz boost regimes. The combined effect of a
higher signal reconstruction efficiency and better classification performance
results in a two to four times stronger rejection of light-flavor jets
compared to conventional strategies at low Lorentz-boosts, and rejection rates
similar to classifiers based on large-radius multi-pronged jets at high
Lorentz-boost regimes.

##  1 Introduction

Jets are ubiquitous in collider physics experiments. A jet may originate from
a single quark/gluon generated in the hard scattering, or from the hadronic
decay of a heavy Lorentz-boosted object, such as a W, Z, or Higgs (H) boson.
In the former case, the jet is expected to have a single-pronged
structure333The top quark is an exception. It decays into a bottom quark and a
W boson before hadronizing, resulting in top quark jets having a multi-pronged
structure., while in the latter, it is expected to have a multi-pronged
structure. While looking at a jet as a single object does not let us
distinguish between these categories in general, investigation into jet
substructures [1, 2, 3] provides additional handles to predict the nature of
the particle that resulted in the jet. More recent developments use machine
learning architectures to achieve unprecedented accuracies in jet
identification tasks [4, 3, 5, 6].

The choice of one or more jet reconstruction algorithms in analysis of
collision data depends on several factors, such as the composition of the
expected final state and the Lorentz-boosts of physics objects. For example, a
search for the Higgs boson decaying to heavy-flavor (bottom or charm) quarks
may be divided into two distinct reconstruction-strategy regimes [7, 8]: (i) a
resolved-jet regime, when the Higgs boson has a low Lorentz boost (transverse
momentum, $p_{\text{T}}\apprle$ 200 GeV) and the two quarks arising from the
Higgs boson have a large angular separation (opening angle), giving rise to
two separate small-radius, single-pronged jets, and (ii) a merged-jet or
boosted regime, when the Higgs boson has a high Lorentz boost
($p_{\text{T}}\apprge$ 200 GeV), and the decay products of the Higgs boson can
be reconstructed with either two small-radius jets or one large-radius, multi-
pronged jet, owing to a smaller angular separation. Heavy-flavor decays of the
Higgs boson can then be selected using jet flavor identification algorithms.
In case of single-pronged small-radius jets, jet identification usually
involves identifying the flavor of the quark or gluon that initiated the jet
(flavor tagging) [9]. Hence, flavor-tagging classifiers that discriminate
between bottom (b), charm (c) and light-flavor (up, down, strange, gluon) jets
are employed. On the other hand, for multi-pronged jets, either flavor
identification or heavy object identification algorithms are employed [10]. In
the discussed example, one might use a multiclassifier that discriminates
between H(H$\rightarrow$b$\bar{\text{b}}$),
H(H$\rightarrow$$\text{c}\bar{\text{c}}$), and light-flavor jets.

The choice of using one large-radius jet instead of two small-radius jets in
the boosted regime is motivated by several factors. When a heavy object
identification or jet flavor identification algorithm is used to identify a
large-radius jet, it can exploit not only the information of the two (or more)
subjets contained in the jet, but also the correlations that exist between the
kinematics of the hadronization products of the two (or more) quarks. In
addition, the large-radius jet potentially encompasses soft radiations that
may lie spatially between the subjets. The presence and kinematics of these
particles may provide additional handles to discriminate resonant subjet pairs
against non-resonant ones, as the former are expected to exhibit electrical
charge and Quantum Chromodynamics (QCD) color recombination. Small-radius jets
can also be used to reconstruct events in the boosted regime, but each jet
must be flavor-tagged individually. This means that the flavor identification
algorithm has access to the substructures of the small-radius jets
individually and cannot take advantage of correlations between the
constituents of the two jets, which inevitably results in loss of information.
On the other hand, the large-radius jet approach is not suitable at low
Lorentz-boosts of the Higgs boson as the decay products of the two quarks are
expected to have a large angular separation. The event is therefore
reconstructed with two small-radius jets. It can be noted that the resolved
regime usually caters to a large fraction of potentially interesting physics
processes (e.g. nearly 95% of the signal in Ref. [8]) and hence the small-
radius jet approach has the advantage of probing a large part of the signal
phase space.

We refer to the choice of using either two small-radius (e.g. AK4) jets [11],
or one large-radius (e.g. AK15) jet [8] per event as a choice of the event
reconstruction strategy. This is different from choosing a jet tagging
algorithm (e.g. Particle Transformer [12], ParticleTransformerAK4 [13] used in
the CMS experiment, GN1 [14] used in the ATLAS experiment, etc.) to identify
the jet content. In this paper, we examine several reconstruction strategies
and evaluate their performance relative to a fixed tagging algorithm. We
propose a new event reconstruction strategy called Particle Angular-separation
Independent Resonant Di-jet (PAIReD jet) tagging. The goal of this strategy is
to optimally exploit the available information in pairs of small-radius jets
to identify resonant decays of heavy particles irrespective of the angular
separation of the two decay products. This enables the flavor-tagging
algorithms to exploit as much information as large-radius jet tagging
algorithms typically have access to, without being restricted to a narrow
range of the parent particle momentum.

Section 2 outlines the PAIReD jet tagging strategy. Section 3 describes the
simulation of several physics processes which are used to compare the
performances of different event reconstruction strategies. Sections 4 and 5
describe the architecture and implementation of machine learning algorithms
that are used to train classifiers to identify heavy-flavor Higgs boson
decays. A mock ZH(H$\rightarrow\text{c}\bar{\text{c}}$) analysis is outlined
in Section 6 as an example and the reconstruction efficiencies and classifier
performances in the context of reconstructing
H$\rightarrow\text{c}\bar{\text{c}}$ decays are compared across various event
reconstruction strategies. Section 7 provides an outlook and Sec. 8 provides a
summary and conclusion of the topics discussed in the paper.

##  2 The PAIReD jet strategy

A PAIReD jet is defined by the union of all the reconstructed particles that
lie within a radius of $\Delta R=R_{0}$ around the centers of two clustered
small-radius jets present in the event, where $R_{0}$ is the distance
parameter used to cluster the small-radius jets. Thus, for an event with $n$
($n\geq 2$) clustered AK4 jets444For the sake of simplicity, we use “AK4 jets”
and “small-radius jets” interchangeably henceforth. The PAIReD jet strategy
can nevertheless be extended to jets clustered with any clustering algorithm,
with distance parameters suitable for single-pronged decays. , one can define
${n\choose 2}$ PAIReD jets in the event, where ${n\choose 2}$ denotes the
number of ways one can choose 2 objects from a set of $n$ objects.

This can be implemented by calculating distances ($\Delta R_{j_{1}}$, $\Delta
R_{j_{2}}$) of each reconstructed particle in the event from the center of the
two small-radius jets; the particles satisfying $\min(\Delta R_{j_{1}},\Delta
R_{j_{2}})<R_{0}$ constitute the list of constituents of the PAIReD jet. As
some tagging algorithms may require ordered sets of particles, a reordering of
the list of constituents, e.g. in descending order of particle $p_{\text{T}}$,
may be necessary.

Since a PAIReD jet is composed of the hadronization products of both quarks, a
tagging algorithm employed to identify the flavor of the quarks has access to
correlations between the two subjet constituents. However, unlike large-radius
jets, the PAIReD jet lacks information of the reconstructed particles that lie
spatially between the two subjets. This leads to the introduction of the
second version of the PAIReD jet strategy, which we refer to as the
PAIReDEllipse strategy.

The PAIReDEllipse strategy defines a non-conventional jet using two clustered
AK4 jets as “seeds”. The area of the PAIReDEllipse jet is defined by an
elliptical region in the $\eta$–$\phi$ plane, where $\eta$ refers to the
pseudorapidity coordinate of the detector and $\phi$ refers to the azimuthal
angle in radians. The ellipse is defined such that the centers of the two seed
AK4 jets lie on the major axis of the ellipse. The length $b$ of the minor
axis of the ellipse is fixed to 3 units in the $\eta$–$\phi$ plane, motivated
by the largest jet radius ($\Delta R=1.5$) used in recent physics analyses
[8]. With the distance between the centers of the two AK4 jets denoted by
$\Delta R_{jj}$, the length of the major axis is defined as

$a=\max(3,\Delta R_{jj}+2)$

units in the $\eta$–$\phi$ plane. We have tested that increasing these numbers
does not significantly alter the final performance metrics of the
PAIReDEllipse strategy.

All the particles reconstructed within the perimeter of the ellipse thus
defined are considered constituents of the PAIReDEllipse jet. Thus, for a pair
of AK4 jets, one can uniquely define one PAIReDEllipse jet. It can be noted
that the definition of the PAIReDEllipse jet does not involve any sequential
clustering algorithm, except for the ones necessary to define the centers of
the seed AK4 jets. As a result, the advantages that commonly-used sequential
clustering algorithms provide, such as infrared and collinear (IRC) safety
[15], are not realized in this approach. On the other hand, the computational
expense of running an additional sequential clustering algorithm is saved.

In presence of additional jets arising from initial- and final-state
radiations, standard resolved-jet approaches typically involve sorting all AK4
jets in the event by certain predetermined parameters ($p_{\text{T}}$, b/c
tagger score) to select the pair of jets that are most likely to arise from
the decay of a heavy particle. Analogously in the PAIReD(Ellipse) jet
strategy, the PAIReD(Ellipse) jet with the highest “bb/cc-jet” score can be
treated as the decay of interest.

Figure 1 demonstrates the effective areas in the $\eta$–$\phi$ plane covered
by the different jet construction strategies, using simulated
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) events. The details of the simulation
are presented in Section 3. It can be seen from the figure that the AK15
reconstruction strategy covers nearly the entire Higgs boson decay when the
angular separation $\Delta R_{\text{cc}}$ between the two generated c quarks
from the Higgs boson decay is ~2 units or smaller. At higher separations, a
potential AK15 jet would be reconstructed around only one of the quarks (or
two separate AK15 jets around two quarks).

On the other hand, the AK4 reconstruction strategy can be implemented at all
angular separation regimes. However, it can be seen from Fig. 1 that a
significant fraction of the generated particles arising from the Higgs boson
decays lie outside the maximum combined area covered by the two AK4 jets. The
PAIReD approach, which simply concatenates the list of particles lying in a
0.4 radius around the centers of two AK4 jets, would analogously fail to cover
a significant fraction of the generated particles. Finally, the PAIReDEllipse
approach covers a larger area and contains most of the generated particles
arising from the Higgs boson decay at all angular separations.

Figure 1: Simulated ZH(H$\rightarrow\text{c}\bar{\text{c}}$) events with the
$\eta$–$\phi$ coordinate system translated and rotated in a way such that the
c quark with the higher $p_{\text{T}}$ is located at (0,0), while the second c
quark lies on the $\eta$ axis ($\phi=0$). Each subfigure shows generated
particles in 800 events, allowing particles across events to overlap. The
shade of green denoting each particle is chosen according to the (logarithm of
the) particle $p_{\text{T}}$ (darker represents higher $p_{\text{T}}$). The
upper-left, upper-right, lower-left, and lower-right subfigures show events
with the angular separation $\Delta R_{\text{cc}}$ between the two c quarks
lying in a window of 0.05 unit around 1, 2, 3, and 4, respectively. Each of
the magenta circles represents the area of a potential AK4 jet that might be
clustered centering one of the quarks. The two magenta circles together
represent the net area covered by a jet defined using the PAIReD jet strategy.
The blue ellipse represents the area of a jet defined using the PAIReDEllipse
strategy. The dashed orange line represents the typical area a hypothetical
AK15 jet would cover, had it been constructed at the center of the 2-quark
system (see discussion in text).

Even though the higher area covered by the PAIReDEllipse jet strategy allows
capturing the entire Higgs boson decay regardless of the angular separation of
the decay products, it comes at the cost of capturing additional unwanted
particles, such as those arising from pileup interactions, underlying event,
and other unrelated hadronic activities. However, with the advent of high-
level, attention-based [16] networks used for jet identification, it is
expected that such spurious particles would be identified and assigned less
importance while performing classification tasks downstream. A more elaborate
discussion is presented in Appendix D.

Another potential complication to consider as we define high-level physics
objects spanning a large part of the detector is the ability to calibrate the
momentum and other kinematic features of the object. In the case of
PAIReD(Ellipse) jets, this can be overcome by suitably adjusting the standard
methods [17, 18] used to calibrate jet energies, for instance, using event
samples enriched in leptonically-decaying Z bosons recoiling against two
small-radius jets. The flavor tagging efficiency of PAIReD(Ellipse) jets can
similarly be compared in simulation and data using standard calibration
methods used for large-radius jets, for example, using the kinematics of soft-
muons or secondary vertices contained inside the PAIReD(Ellipse) jets to
select a subset of QCD multijet events in data that mimic b$\bar{\text{b}}$ or
$\text{c}\bar{\text{c}}$ jets in terms of tagger score distributions [9, 19].

##  3 Simulation

We simulate proton–proton collision events to train, validate, and compare the
performance metrics across various event reconstruction strategies. We focus
on the 2-lepton channel of
H$\rightarrow\text{b$\bar{\text{b}}$}/\text{c}\bar{\text{c}}$ searches, where
the Higgs boson is produced in association with a vector boson (Z/W) [20, 21,
11, 8]and the Z boson decays to a pair of electrons or muons. We refer to the
ZH(H$\rightarrow\text{b$\bar{\text{b}}$}/\text{c}\bar{\text{c}}$) process as
the signal process. The major background of these searches are Z+jj events,
where the Z boson decays to a pair of electrons or muons and is generated in
association with two additional jets. Only the major background process is
simulated in this study.

All processes are generated with MadGraph5_aMC@NLO (MG5) [22] at leading order
accuracy in QCD, with upto 3 additional partons in the event. The center of
mass energy used is $\sqrt{s}=13$ TeV. The Higgs boson decay into heavy-flavor
quarks, along with parton showering and hadronization, is simulated in Pythia8
[23]. The MLM prescription [24] is used to match jets from the matrix element
calculations and the parton shower description. Finally, the detector response
is simulated in Delphes [25] using the default CMS configuration available in
Delphes.

The effect of pileup is simulated within Delphes with a mean pileup value set
to 50. Particles are reconstructed using the E-Flow algorithm in Delphes. Jets
are clustered using the AK algorithm implemented in the FastJet package [26].
Distance parameters of 0.4 (AK4), 0.8 (AK8), and 1.5 (AK15) are used, and only
jets with $p_{\text{T}}$ greater than 20, 100, and 50 GeV, respectively, are
stored. Charged hadrons coming from pileup vertices are discarded and the
residual event pileup density is used along with the jet area to perform
pileup subtraction on jets [25]. The true flavor of jets is defined
differently for different strategies:

  * •

AK4: If at least one b quark is found within the jet radius, the jet is
labeled as a true b jet. If no b quark is found, but at least one c quark is
found, the jet is labeled c. If neither a b nor a c quark is found, the jet is
labeled udsg or light-flavor.

  * •

AK8 and AK15: If both generated b or c quarks arising from the Higgs boson
decay lie within a radius of 0.8 (1.5) units around the jet center, the AK8
(AK15) jet is labeled as a true bb or true cc jet. Otherwise, it is labeled as
an ll jet.

  * •

PAIReD jet: If both generated b or c quarks arising from the Higgs boson decay
lie within a radius of 0.4 units around the center of either of the two
constituent AK4 jets, the PAIReD jet is labeled as a true bb or cc jet.
Otherwise, it is labeled as an ll jet.

  * •

PAIReDEllipse jet: If both generated b or c quarks arising from the Higgs
boson decay lie within the perimeter of the ellipse defined by the jet, the
PAIReDEllipse jet is labeled as a true bb or cc jet. Otherwise, it is labeled
as an ll jet.

In addition to the standard Higgs boson samples discussed above, alternative
samples with a variable Higgs boson mass are also generated. In these “flat-
mass” samples, the mass of the Higgs boson ($m_{\text{H}}$) is set to integer
values between 10 and 500 GeV with a uniform distribution, while the Higgs
boson decay width is set to 4 MeV as expected from the SM [27]. These samples
are used in the classifier trainings in order to achieve mass decorrelation
[28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. The technique of using flat-mass
sample to achieve decorrelation has previously been implemented in hadron
collider experiments [32, 38]. Using mass decorrelated taggers ensures a fair
comparison among different reconstruction strategies, since algorithms used in
some strategies (e.g. AK4-based reconstruction) do not have access to the
Higgs boson mass information.

An additional complexity arises in the flat-mass Higgs boson samples. The
$p_{\text{T}}$ spectrum of the Higgs boson is dependent on the generated mass
of the Higgs boson; the fraction of events containing at least one bb/cc AK8
or AK15 jet diminishes at high masses of the Higgs boson. To mitigate this
effect, an empirical Higgs boson mass-dependent selection criterion is
applied555The parameter pt_min_pdg in MG5 is used. The resulting Higgs boson
$p_{\text{T}}$ spectrum, however, starts at a value slightly below the applied
selection criterion. on the minimum $p_{\text{T}}$ of the generated Higgs
boson ($p_{\text{T}}^{\text{H}}$) in the physics processes used to sample AK8
and AK15 jets. This criterion ranges between $p_{\text{T}}^{\text{H}}>0$ GeV
at Higgs boson mass $m_{\text{H}}=10$ GeV to $p_{\text{T}}^{\text{H}}>390$ GeV
at $m_{\text{H}}=500$ GeV for AK15 jets, and between
$p_{\text{T}}^{\text{H}}>0$ GeV at $m_{\text{H}}=10$ GeV to
$p_{\text{T}}^{\text{H}}>900$ GeV at $m_{\text{H}}=500$ GeV for AK8 jets.
These selections ensure that a sufficient number of AK15 and AK8 jets are
sampled throughout the Higgs boson mass range, without generating events that
are unlikely to have any AK15 or AK8 jets. The distributions of AK15 and AK8
jets as functions of the generated Higgs boson mass and $p_{\text{T}}$ are
shown in Fig. 2 after the empirical selection criteria on
$p_{\text{T}}^{\text{H}}$ are applied.

Figure 2: The fraction of AK15 (left) and AK8 (right) cc jets sampled at
different values of the generated Higgs boson mass (x-axis) and $p_{\text{T}}$
(y-axis) from the flat-mass ZH(H$\rightarrow\text{c}\bar{\text{c}}$) sample.

##  4 Flavor tagging algorithm

The algorithm used in this paper to identify the flavor of jets defined using
different reconstruction strategies is the Particle Transformer (ParT)
algorithm [12]. The choice of the tagging algorithm is largely independent of
the choice of event reconstruction strategy and one can evaluate the
performance of various reconstruction strategies relative to any one, fixed
tagging algorithm.

The ParT algorithm makes use of a number of features of jet constituents as
well as interactions corresponding to every pair of constituents. The network
comprises of several particle and class attentions blocks. Detailed
description of the network architecture can be found in Ref. [12]. As inputs,
we use kinematics and trajectory displacement information of particles, but
not the particle identification information. This choice is consistent with
the standard methods for jet tagging used in particle physics experiments
which intend to minimize potential biases in subsequent independent validation
steps that exploit dedicated soft lepton information [39].

In addition to the classification tasks that have been discussed so far, the
ParT algorithm is also implemented for mass regression (Appendix A) with minor
changes in the architecture. For regression, the default settings were
modified to use a PReLU activation function [40] and the logarithm of the
hyperbolic cosine (log cosh) [41] as the loss function. In addition, a
gradient clipping feature [42] was implemented to prevent gradients from
becoming too large owing to the output node containing mass values of
$\mathcal{O}(100)$ in units of GeV.

##  5 Neural network training

Several networks corresponding to the various reconstruction strategies are
trained using the weaver framework [43] based on PyTorch [44], implementing
the ParT architecture [12]. Jets are classified into three classes, namely bb,
cc, and ll (b, c, and udsg for AK4 jets). The bb (cc) jets used in the
trainings are sampled from the flat-mass
ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$)
(ZH(H$\rightarrow\text{c}\bar{\text{c}}$)) processes. The ll jets are sampled
from ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$),
ZH(H$\rightarrow\text{c}\bar{\text{c}}$), and Z+jj processes. Wrong
combinations of jets, such as multi-pronged jets containing only one b/c
parton in the ZH processes, are also labeled ll (cf. Sec. 3). The number of ll
jets sampled from each of ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$),
ZH(H$\rightarrow\text{c}\bar{\text{c}}$), and Z+jj processes ranges between
25–30 million. In case of AK4 jet trainings, all jets labeled b, c, and udsg
are used regardless of the physics process they were sampled from.

Jets are assigned weights during the training such that each class contains an
equal number of jets. In case of AK8, AK15, PAIReD, and PAIReDEllipse
trainings, the jets are weighted such that a mass metric has the same
distribution for all labels, using the mass distribution of the ll jets as the
reference. For the purpose of defining a uniform mass metric, mass regression
techniques, discussed in Appendix A, are used. Only jets with regressed masses
($m_{\text{reg}}$) of 50–400 GeV are used to train the classifier. In case of
AK4 jets, the jets are weighted such that the $p_{\text{T}}$ and $\eta$ of the
jets in the b and c classes have the same distributions as those in the
reference udsg class.

About a tenth of the jets are used as an independent validation dataset to
check for overtraining. Each training is run for 80 epochs. The model from the
epoch with the highest classification accuracy on the validation dataset is
selected as the final classifier model. The trainings are then evaluated on an
independent test dataset comprising jets from
ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$) and
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) processes with $m_{\text{H}}=125$
GeV, and an independent Z+jj sample. The performance comparison of the various
classifiers corresponding to different reconstruction strategies is presented
in Sec. 6.3.

##  6 Performance evaluation on H$\rightarrow$$\text{c}\bar{\text{c}}$ events

The various event reconstruction strategies outlined in this paper can be
compared and contrasted using H$\rightarrow\text{c}\bar{\text{c}}$ events. In
this paper, we demonstrate a mock ZH(H$\rightarrow\text{c}\bar{\text{c}}$)
analysis as an example. The analysis strategies corresponding to the various
reconstruction strategies are outlined in Sec. 6.2. In all cases, we ignore
the reconstruction of the leptonic Z decay and focus on reconstructing the
hadronic Higgs boson decay. Another example using
Z$\rightarrow\text{c}\bar{\text{c}}$ events is presented in Appendix C.

###  6.1 Performance metrics

The performances of jet flavor taggers are usually quantified using metrics
such as accuracy, area under the ROC curve (AUC), background rejection at a
given signal efficiency, etc. While these metrics convey useful information
when evaluating the performance of different algorithms on the same type of
reconstructed jets (e.g. public benchmark jet datasets [45, 46, 47, 48]) they
are not very interpretable when comparing across various event reconstruction
strategies and across various Lorentz-boost regimes.

We define end-to-end selection efficiency as the fraction of simulated events
that are selected and correctly reconstructed after all selection criteria and
classifier score requirements are applied. This includes the requirement that
a certain number of jets be reconstructed per event, as well as any
requirements that are applied to the jet tagger scores to achieve a target
signal efficiency. This quantity is, therefore, dependent on the
reconstruction strategy being used, as well as the tagging performance
achievable using that strategy. Events that cannot be reconstructed using a
given reconstruction strategy, and events that are incorrectly reconstructed
(e.g. wrong choice/selection of jets) contribute to a decline in the end-to-
end efficiency metric. This is in contrast to the standard metrics that are
used to evaluate jet tagging algorithm performances—standard metrics focus on
the background rejection for a specific type of jets, disregarding the
fraction of events that contain such jets and the fraction of events where the
correct jet(s) is (are) chosen. This makes the end-to-end efficiency a
suitable metric to compare between event reconstruction strategies and across
Lorentz-boost regimes.

The end-to-end selection efficiency can be defined for both signal and
background events. They can be interpreted as the end-to-end true positive and
false positive rates, respectively. A more natural metric for background
events is the background rejection, which can be defined as the inverse of the
background selection efficiency. Thus, one can define an end-to-end background
rejection rate at a given end-to-end signal selection efficiency.
Additionally, a rough estimation of the differential signal sensitivity as a
function of the Higgs boson $p_{\text{T}}$ is shown in Appendix B.

###  6.2 Event selection strategies

####  6.2.1 Analysis using AK4 jets

We follow the strategy adopted by the CMS H$\rightarrow\text{c}\bar{\text{c}}$
search [8] in the resolved-jet regime. Only events with at least 2 well-
reconstructed AK4 jets with $p_{\text{T}}>20$ GeV lying within $|\eta|<2.5$
are selected. In each event, jets are sorted by the so-called CvsL score,
which is defined as

$\text{CvsL}=\frac{P(\text{c)}}{P(\text{c})+P(\text{udsg)}},$

where $P(\text{c})$ and $P(\text{udsg})$ are the tagger scores obtained at the
c and udsg output nodes of the classifier prediction, respectively. We
verified that using $P(\text{c})$ instead of CvsL to sort the jets did not
change the results significantly. The two jets with the highest CvsL values in
each event are used to reconstruct the Higgs boson candidate. The invariant
mass $m_{jj}$ of the Higgs boson candidate is required to lie within the range
50–200 GeV. The same procedure is repeated for the background sample.

Only signal events where the correct pair of AK4 jets is chosen are counted
towards the “pass” events in the signal selection efficiency calculation. To
obtain the end-to-end background rejection rate at a fixed end-to-end signal
selection efficiency (in line with the strategy outlined in Sec. 6.1), the
$P(\text{c})$ score of the subleading jet in signal events for which exactly
X% of the signal events pass all the selection criteria is first determined.
Next, the same requirement on the $P(\text{c})$ score is applied on the
subleading jet of the background events. The fraction of background events
thus selected gives the end-to-end background mistag rate at an end-to-end
signal efficiency of X%. The inverse of this quantity gives the end-to-end
background rejection rate.

####  6.2.2 Analyses using AK8 and AK15 jets

Events with at least one AK15 (AK8) jet with $p_{\text{T}}>50$ (100) GeV and
$m_{\text{reg}}\in(50,400)$ GeV, lying within the tracker acceptance
($|\eta|<2.5$) are selected. The preselection on $m_{\text{reg}}$ is necessary
since the classifiers are trained only in a limited $m_{\text{reg}}$ range. In
each event, the jet with the highest $P(\text{cc})$ score obtained from the
classifier prediction is considered as the Higgs boson candidate. As before,
only signal events where the selected jet contains both c quarks from the
Higgs boson decay are counted towards the “pass” events in the signal
selection efficiency calculation. Finally, an additional requirement of
$m_{\text{reg}}\in(50,200)$ GeV is applied for the selected jet.

As before, a requirement on the $P(\text{cc})$ score for which a certain
fraction of the signal events pass the selection criteria is applied to the
selected jet of the background events to calculate the end-to-end background
rejection rate.

####  6.2.3 Analyses using PAIReD and PAIReDEllipse jets

The analyses using PAIReD and PAIReDEllipse jets follow the same strategy as
the ones with AK8 and AK15 jet. No minimum $p_{\text{T}}$ requirement is
applied for the jets, but the AK4 jets from which the PAIReD(Ellipse) jets are
defined are required to have $p_{\text{T}}>20$ GeV each. Events with at least
one PAIReD(Ellipse) jet with $m_{\text{reg}}\in(50,400)$ GeV lying within the
tracker acceptance ($|\eta|<2.5$) are preselected. In each event, the jet with
the highest $P(\text{cc})$ score obtained from the classifier prediction is
considered as the Higgs boson candidate. Finally, a more stringent requirement
of $m_{\text{reg}}\in(50,200)$ GeV is applied for the selected jet.

Similarly as before, the same requirement on the $P(\text{cc})$ score is
applied to the selected jet of the background events and the end-to-end
background rejection rate is calculated.

####  6.2.4 Summary

The hadronic part of the ZH(H$\rightarrow\text{c}\bar{\text{c}}$) final state
is reconstructed with 5 different approaches. The fraction of signal events
obtained after progressively applying each selection criterion is shown in a
so-called cutflow plot in Fig. 3. This cutflow excludes the final requirement
on the classifier scores applied to obtain a fixed signal efficiency.

Figure 3: Plot showing fraction of signal (left) and background (right) events
obtained after each selection criterion is progressively applied. The last two
selections in the left plot involve using generator-level flavor information
in signal samples to investigate how many correctly reconstructed events pass
the selection criteria. Selections are, in general, different for small-radius
(AK4) and large-radius (AK8, AK15, PAIReD, PAIReDEllipse) jets, and are marked
using $\dagger$ and $\ddagger$, respectively. The masses are in units of GeV.

###  6.3 Performance comparisons

The end-to-end selection efficiency metric can be factorized into two
components: (a) the reconstruction efficiency, and (b) the classifier
performance. The combined effect of the signal reconstruction efficiency and
the classifier performance determines the final analysis sensitivity. For
instance, a poorer classifier performance can offset any gains achieved from
having a high signal reconstruction efficiency. The two efficiencies are
investigated separately to study how each component contributes to the final
gains.

####  6.3.1 Signal reconstruction efficiency

The signal reconstruction efficiency can be defined as the ratio of the
maximum number of signal events that can be reconstructed using a given
strategy, to the total number of events generated. This involves calculating
the signal efficiency in each of the analyses without applying the requirement
on the classifier output score (cf. Sec. 6). This metric can also be
considered as the maximum signal efficiency that can be achieved using a
particular reconstruction strategy.

The signal reconstruction efficiency is evaluated using the
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) process at various ranges of the
generated Higgs boson $p_{\text{T}}$. A comparison of the yield of
reconstructed signal events and the signal reconstruction efficiency across
various reconstruction strategies is presented in Fig. 4. The PAIReDEllipse
strategy improves the signal reconstruction efficiency by about 40–50%
compared to AK4-based event reconstruction at low Lorentz-boosts ($\apprle$200
GeV) of the Higgs boson. The efficiency of the PAIReDEllipse strategy is
slightly higher (~2–14%) compared to that of the AK15-based strategy at higher
boosts. The PAIReD strategy has efficiencies intermediate to that of the AK4
and PAIReDEllipse approaches, as expected. The AK4, PAIReD, and PAIReDEllipse
approaches reconstruct a significantly larger number of signal events compared
to the AK8 and AK15-based approaches. The gain comes mainly from the low
Lorentz-boost regime.

Figure 4: The number of reconstructed ZH(H$\rightarrow\text{c}\bar{\text{c}}$)
signal events out of the 10M signal events generated (left) and the maximum
signal efficiency or the signal reconstruction efficiency (right) as functions
of the generated Higgs boson $p_{\text{T}}$, for various reconstruction
strategies.

####  6.3.2 Classifier performance

The ability of classifiers to distinguish between signal and background jets
is also expected to vary across different reconstruction strategies.
Classifiers trained using AK4 jets, for instance, evidently have access to
less information than classifiers trained with AK15 jets and hence are
expected to have lower background rejection rates.

The classifier performances are evaluated by calculating the AUC using all the
signal and background events that pass the selection criteria (except the
requirement on the classifier output score). Furthermore, a proxy Higgs boson
$p_{\text{T}}$ needs to be defined for the background events in order to
evaluate the AUC performances at different Lorentz-boosts of the generated
Higgs boson. For this purpose, a diparton $p_{\text{T}}$ is defined by adding
the four-vectors of the two MG5-level partons produced in association with a Z
boson in the background Z+jj events. The diparton $p_{\text{T}}$ is used as
the proxy for Higgs boson $p_{\text{T}}$ in the background events. The
motivation behind this choice is that the fake Higgs boson candidate defined
in the background events passing the selection criteria is most likely
reconstructed from the hadronization products of the two associated MG5-level
partons in the Z+jj events. Therefore, the generator-level diparton system in
background events is a suitable proxy for the generator-level Higgs boson in
signal events.

The AUCs of the classifiers corresponding to the various reconstruction
strategies are compared in Fig. 5. The classifiers corresponding to the PAIReD
and PAIReDEllipse strategies have comparable performances and outperform those
corresponding to other strategies in the low boost regimes. In the high boost
regimes, the AK15- and AK8-based classifiers outperform other classifiers. The
improvement compared to PAIReD and PAIReDEllipse approaches at high boosts can
be attributed to the fact that AK8 and AK15 jets are constructed using IRC-
safe sequential clustering algorithms and are less sensitive to presence of
pileup particles. This is discussed in more detail in Appendix D.

Figure 5: The area under the ROC curves (AUC) obtained when classifiers
corresponding to various reconstruction strategies are evaluated on the signal
and background events passing the respective selection criteria, plotted as a
function of the Higgs boson $p_{\text{T}}$ (for signal events) or the diparton
$p_{\text{T}}$ (for background events).

####  6.3.3 End-to-end efficiencies

The end-to-end background rejection rates at a given target end-to-end signal
efficiency for different reconstruction strategies are discussed in this
subsection. Figure 4 (right) shows the maximum signal efficiency achievable at
different boosts of the generated Higgs boson. As AK4 jets achieve a maximum
signal efficiency of around ~0.37 at low Lorentz-boosts, requiring a target
signal efficiency above ~0.37 is expected to yield no sensitivity with the
AK4-based reconstruction approach at low Lorentz-boosts. Nevertheless, end-to-
end signal efficiencies of 0.4, 0.3, 0.2, and 0.1 are explored and the
corresponding background rejection rates as a function of the Higgs/diparton
$p_{\text{T}}$ are shown in Fig. 6. The end-to-end signal efficiencies and
background rejections are calculated per bin, i.e. exclusively in the
Higgs/diparton $p_{\text{T}}$ range covered by each bin.

Figure 6: The end-to-end background rejection at end-to-end signal
efficiencies of 0.4 (upper-left), 0.3 (upper-right), 0.2 (lower-left), and 0.1
(lower-right) as a function of the generated Higgs boson $p_{\text{T}}$ (for
signal) or diparton $p_{\text{T}}$ (for background), plotted for various
reconstruction strategies. End-to-end signal efficiencies and background
rejections are calculated per bin. The background rejection is set to 0
wherever the target signal efficiency cannot be achieved. Bins with very low
yield of background events are either merged with adjacent bins or are
omitted.

At low Lorentz-boosts of the Higgs boson, the PAIReDEllipse strategy performs
the best with around 2.5-4 times higher background rejection compared to the
AK4 strategy. The PAIReD strategy has a performance intermediate to that of
AK4 and PAIReDEllipse strategies, with around 2-3 times improvement compared
to AK4-based reconstruction. At higher boosts, the PAIReDEllipse and AK15
approaches have similar performances. The AK8 approach has comparable
performances at low signal efficiencies, but performs slightly worse than
PAIReDEllipse and AK15 at higher signal efficiency requirements.

##  7 Outlook

The PAIReD(Ellipse) approaches potentially leverage information across a
larger volume of the detector, especially at low boosts, while the related
observables are still calibratable using modified versions of the standard jet
calibration techniques used in hadron collider experiments. The approach can
be generalized to multi-pronged decays as well, as long as suitable event
samples can be identified in collision data for calibration. A three-pronged
tagger, for example, may be calibrated using hadronically-decaying top quark
events.

Furthermore, more complex final states, like 4b, 6b, 8b, and so on, can be
factorized into two or more two-pronged PAIReD(Ellipse) jets for
reconstruction. The classifier output scores are expected to be higher for the
correct pairs of b quark resonances, which can help identify the correct
combinations of small-radius jets, and subsequently reject background events.
Tagger calibration for such complex final states can be achieved by evaluating
correction factors for simulated events as a function of the efficiency
correction factors of all PAIReD(Ellipse) jets reconstructed in the event. The
strategy may also be used to discriminate events containing resonant heavy
flavor decays (H$\rightarrow$b$\bar{\text{b}}$, for example) from those
containing non-resonant decays (b and $\bar{\text{b}}$ from
$\text{t}\bar{\text{t}}$ decay) at low boosts, without using the invariant
mass information.

The method can also be suitably modified to identify, for example, charged
Higgs boson decays into c$\bar{\text{s}}$, by introducing additional cs flavor
nodes in the outputs. Such a classifier can be calibrated using hadronically-
decaying W bosons from $\text{t}\bar{\text{t}}$ events, for instance. Thus,
the PAIReD and PAIReDEllipse approaches can be extended to several kinds of
physics analyses.

##  8 Conclusion

We propose a new method to reconstruct events containing hadronically-decaying
heavy particles. The two versions of the method, namely PAIReD and
PAIReDEllipse jet based reconstruction, define unconventional jets using
clustered small-radius jets as seeds. The former is limited to the area
defined by the small-radius jets, whereas the latter defines an elliptical
area in the $\eta$–$\phi$ plane based on the centers of the small-radius jets
and considers all reconstructed particles lying in this area as jet
constituents.

We have demonstrated this method in the context of di-pronged resonances,
where the PAIReD(Ellipse) jets are constructed from exactly two AK4 jets. We
have examined two rare physics processes, namely Higgs bosons decaying to a
bottom or charm quark-antiquark pair, where the Higgs boson is produced in
association with a Z boson
(ZH(H$\rightarrow$b$\bar{\text{b}}$/$\text{c}\bar{\text{c}}$)). We have used
the ParticleTransformer jet tagging algorithm for regression and
classification tasks. We have trained mass-decorrelated multi-classifiers to
discriminate b$\bar{\text{b}}$, $\text{c}\bar{\text{c}}$, and light-flavor
jets (b, c and udsg, for AK4 jets).

The PAIReDEllipse strategy has the highest signal reconstruction efficiency at
all Lorentz-boosts of the Higgs boson. The AK4, PAIReD, and PAIReDEllipse
approaches reconstruct significantly more signal events than AK15 and AK8
approaches, while the AK15 and AK8 surpass the AK4-based approach in terms of
signal efficiency at high boosts. In terms of classifier performance, the
PAIReDEllipse and PAIReD perform better than AK4 at all boosts, while the AK15
and AK8 classifiers surpass all others at high boosts. The combined effect
results in the PAIReDEllipse approach having a background rejection about
2.5–4 times higher than the AK4-based approach at low boosts, while the PAIReD
approach performs about 2–3 times better than the AK4-based approach. At
higher boosts, the PAIReDEllipse approach has a performance at par with the
AK15-based approach.

The previous state-of-the-art analysis techniques involve using AK4-based jet
reconstruction at low Higgs boson boosts ($p_{\text{T}}^{\text{H}}\apprle$ 300
GeV) and using AK15-based reconstruction at high boosts. Our proposed
PAIReDEllipse tagger improves the background rejection at low boosts by upto 4
times compared to AK4-based taggers, while maintaining the same performance as
AK15-based taggers at high boosts. The main gain in performance at low boosts
comes from the ability of PAIReD(Ellipse)-based classifiers to exploit
correlations between the hadronization products of the two quarks—a feat that
was so far achieved only at high boosts using the substructure information of
large-radius jets. The new method is expected to significantly improve the
sensitivities of physics searches in the low-boost regime and enable searches
that were previously considered inaccessible at low boosts.

## Acknowledgements

We are grateful to the authors of Ref. [12], H. Qu, C. Li, and S. Qian for
many helpful discussions. We are also grateful to them for publishing and
sharing codes which we adapted for this work. We would also like to thank U.
Heintz and J. Luo for many fruitful discussions. This research was supported
by the Deutsche Forschungsgemeinschaft (DFG) under grant 400140256 - GRK 2497:
The physics of the heaviest particles at the LHC, and the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences Energy Frontier
Research Centers program under Award Number DE-SC0010010. Part of this
research was conducted using computational resources and services at the
Center for Computation and Visualization, Brown University.

## References

  * [1] J. Thaler and K. Van Tilburg, “Identifying Boosted Objects with N-subjettiness”,  JHEP 03 (2011) 015, doi:10.1007/JHEP03(2011)015, arXiv:1011.2268.
  * [2] D. Adams et al., “Towards an Understanding of the Correlations in Jet Substructure”,  Eur. Phys. J. C 75 (2015) 409, doi:10.1140/epjc/s10052-015-3587-2, arXiv:1504.00679.
  * [3] R. Kogler et al., “Jet Substructure at the Large Hadron Collider: Experimental Review”,  Rev. Mod. Phys. 91 (2019) 045003, doi:10.1103/RevModPhys.91.045003, arXiv:1803.06991.
  * [4] D. Guest et al., “Jet Flavor Classification in High-Energy Physics with Deep Neural Networks”,  Phys. Rev. D 94 (2016) 112002, doi:10.1103/PhysRevD.94.112002, arXiv:1607.08633.
  * [5] D. Guest, K. Cranmer, and D. Whiteson, “Deep Learning and its Application to LHC Physics”,  Ann. Rev. Nucl. Part. Sci. 68 (2018) 161–181, doi:10.1146/annurev-nucl-101917-021019, arXiv:1806.11484.
  * [6] A. J. Larkoski, I. Moult, and B. Nachman, “Jet Substructure at the Large Hadron Collider: A Review of Recent Advances in Theory and Machine Learning”,  Phys. Rept. 841 (2020) 1–63, doi:10.1016/j.physrep.2019.11.001, arXiv:1709.04464.
  * [7] CMS Collaboration, “A search for the standard model Higgs boson decaying to charm quarks”,  JHEP 03 (2020) 131, doi:10.1007/JHEP03(2020)131, arXiv:1912.01662.
  * [8] CMS Collaboration, “Search for Higgs boson decay to a charm quark-antiquark pair in proton-proton collisions at $\sqrt{s}$ = 13 TeV”,  Phys. Rev. Lett. 131 (2023) 061801, doi:10.1103/PhysRevLett.131.061801, arXiv:2205.05550.
  * [9] CMS Collaboration, “Identification of heavy-flavour jets with the CMS detector in pp collisions at 13 TeV”,  JINST 13 (2018) P05011, doi:10.1088/1748-0221/13/05/P05011, arXiv:1712.07158.
  * [10] CMS Collaboration, “Identification of heavy, energetic, hadronically decaying particles using machine-learning techniques”,  JINST 15 (2020) P06005, doi:10.1088/1748-0221/15/06/P06005, arXiv:2004.08262.
  * [11] ATLAS Collaboration, “Direct constraint on the Higgs-charm coupling from a search for Higgs boson decays into charm quarks with the ATLAS detector”,  Eur. Phys. J. C 82 (2022) 717, doi:10.1140/epjc/s10052-022-10588-3, arXiv:2201.11428.
  * [12] H. Qu, C. Li, and S. Qian, “Particle Transformer for jet tagging”, in  Proceedings of the 39th International Conference on Machine Learning, pp. 18281–18292.  2022\.  arXiv:2202.03772.
  * [13] CMS Collaboration, “Transformer models for heavy flavor jet identification”, CMS Detector Performance Note CMS-DP-2022-050, 2022.
  * [14] ATLAS Collaboration, “Graph Neural Network Jet Flavour Tagging with the ATLAS Detector”, ATLAS Note ATL-PHYS-PUB-2022-027, 2022.
  * [15] A. Banfi, G. P. Salam, and G. Zanderighi, “Principles of general final-state resummation and automated implementation”,  JHEP 03 (2005) 073, doi:10.1088/1126-6708/2005/03/073, arXiv:hep-ph/0407286.
  * [16] A. Vaswani et al., “Attention is All you Need”, in  Advances in Neural Information Processing Systems, I. Guyon et al., eds., volume 30.  Curran Associates, Inc., 2017.
  * [17] CMS Collaboration, “Jet energy scale and resolution in the CMS experiment in pp collisions at 8 TeV”,  JINST 12 (2017) P02014, doi:10.1088/1748-0221/12/02/P02014, arXiv:1607.03663.
  * [18] ATLAS Collaboration, “Jet energy scale and resolution measured in proton–proton collisions at $\sqrt{s}=13$ TeV with the ATLAS detector”,  Eur. Phys. J. C 81 (2021) 689, doi:10.1140/epjc/s10052-021-09402-3, arXiv:2007.02645.
  * [19] CMS Collaboration, “Calibration of the mass-decorrelated ParticleNet tagger for boosted $\mathrm{b}\bar{\mathrm{b}}$ and $\mathrm{c}\bar{\mathrm{c}}$ jets using LHC Run 2 data”, CMS Detector Performance Note CMS-DP-2022-005, 2022.
  * [20] ATLAS Collaboration, “Observation of $H\rightarrow b\bar{b}$ decays and $VH$ production with the ATLAS detector”,  Phys. Lett. B 786 (2018) 59–86, doi:10.1016/j.physletb.2018.09.013, arXiv:1808.08238.
  * [21] CMS Collaboration, “Observation of Higgs boson decay to bottom quarks”,  Phys. Rev. Lett. 121 (2018) 121801, doi:10.1103/PhysRevLett.121.121801, arXiv:1808.08242.
  * [22] J. Alwall et al., “The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations”,  JHEP 07 (2014) 079, doi:10.1007/JHEP07(2014)079, arXiv:1405.0301.
  * [23] T. Sjöstrand et al., “An introduction to PYTHIA 8.2”,  Comput. Phys. Commun. 191 (2015) 159–177, doi:10.1016/j.cpc.2015.01.024, arXiv:1410.3012.
  * [24] J. Alwall et al., “Comparative study of various algorithms for the merging of parton showers and matrix elements in hadronic collisions”,  Eur. Phys. J. C 53 (2008) 473–500, doi:10.1140/epjc/s10052-007-0490-5, arXiv:0706.2569.
  * [25] DELPHES 3 Collaboration, “DELPHES 3, A modular framework for fast simulation of a generic collider experiment”,  JHEP 02 (2014) 057, doi:10.1007/JHEP02(2014)057, arXiv:1307.6346.
  * [26] M. Cacciari, G. P. Salam, and G. Soyez, “FastJet User Manual”,  Eur. Phys. J. C 72 (2012) 1896, doi:10.1140/epjc/s10052-012-1896-2, arXiv:1111.6097.
  * [27] Particle Data Group Collaboration, “Review of Particle Physics”,  PTEP 2022 (2022) 083C01, doi:10.1093/ptep/ptac097.
  * [28] J. Dolen et al., “Thinking outside the ROCs: Designing Decorrelated Taggers (DDT) for jet substructure”,  JHEP 05 (2016) 156, doi:10.1007/JHEP05(2016)156, arXiv:1603.00027.
  * [29] G. Louppe, M. Kagan, and K. Cranmer, “Learning to Pivot with Adversarial Networks”, arXiv:1611.01046.
  * [30] C. Shimmin et al., “Decorrelated Jet Substructure Tagging using Adversarial Neural Networks”,  Phys. Rev. D 96 (2017), no. 7, 074034, doi:10.1103/PhysRevD.96.074034, arXiv:1703.03507.
  * [31] ATLAS Collaboration, “Performance of mass-decorrelated jet substructure observables for hadronic two-body decay tagging in ATLAS”, ATLAS Note ATL-PHYS-PUB-2018-014, 2018\.
  * [32] CMS Collaboration, “Identification of highly Lorentz-boosted heavy particles using graph neural networks and new mass decorrelation techniques”, CMS Detector Performance Note CMS-DP-2020-002, 2020.
  * [33] L. Bradshaw, R. K. Mishra, A. Mitridate, and B. Ostdiek, “Mass Agnostic Jet Taggers”,  SciPost Phys. 8 (2020) 011, doi:10.21468/SciPostPhys.8.1.011, arXiv:1908.08959.
  * [34] G. Kasieczka and D. Shih, “Robust Jet Classifiers through Distance Correlation”,  Phys. Rev. Lett. 125 (Sep, 2020) 122001, doi:10.1103/PhysRevLett.125.122001.
  * [35] K. Benkendorfer, L. L. Pottier, and B. Nachman, “Simulation-assisted decorrelation for resonant anomaly detection”,  Phys. Rev. D 104 (2021) 035003, doi:10.1103/PhysRevD.104.035003, arXiv:2009.02205.
  * [36] O. Kitouni, B. Nachman, C. Weisser, and M. Williams, “Enhancing searches for resonances with machine learning and moment decomposition”,  Journal of High Energy Physics 2021 (2021) 70, doi:10.1007/JHEP04(2021)070.
  * [37] S. Klein and T. Golling, “Decorrelation with conditional normalizing flows”, arXiv:2211.02486.
  * [38] CMS Collaboration, “Mass regression of highly-boosted jets using graph neural networks”, CMS Detector Performance Note CMS-DP-2021-017, 2021.  CMS-DP-2021-017.
  * [39] CMS Collaboration, “A new calibration method for charm jet identification validated with proton-proton collision events at $\sqrt{s}$ =13 TeV”,  JINST 17 (2022) P03014, doi:10.1088/1748-0221/17/03/P03014, arXiv:2111.03027.
  * [40] K. He, X. Zhang, S. Ren, and J. Sun, “Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification”,  2015 IEEE International Conference on Computer Vision (ICCV) (2015) 1026–1034.
  * [41] Q. Wang, Y. Ma, K. Zhao, and Y. Tian, “A Comprehensive Survey of Loss Functions in Machine Learning”,  Annals of Data Science 9 (2022) 187–212, doi:10.1007/s40745-020-00253-5.
  * [42] T. Mikolov, “Statistical language models based on neural networks”.  Ph.D. thesis, Brno University of Technology, Faculty of Information Technology, 2012.
  * [43] H. Qu and C. Li, “weaver-core”. https://pypi.org/project/weaver-core/.
  * [44] A. Paszke et al., “PyTorch: An Imperative Style, High-Performance Deep Learning Library”, in  Advances in Neural Information Processing Systems, H. Wallach et al., eds., volume 32.  Curran Associates, Inc., 2019.
  * [45] P. Komiske, E. Metodiev, and J. Thaler, “Pythia8 Quark and Gluon Jets for Energy Flow”, May, 2019.  doi:10.5281/zenodo.3164691, https://doi.org/10.5281/zenodo.3164691.
  * [46] G. Kasieczka, T. Plehn, J. Thompson, and M. Russel, “Top quark tagging reference dataset”, March, 2019.  doi:10.5281/zenodo.2603256, https://doi.org/10.5281/zenodo.2603256.
  * [47] H. Qu, C. Li, and S. Qian, “JetClass: A large-scale dataset for deep learning in jet physics”, 2022.  doi:10.5281/zenodo.6619768, https://doi.org/10.5281/zenodo.6619768.
  * [48] Y. Chen et al., “A FAIR and AI-ready Higgs boson decay dataset”,  Scientific Data 9 (2022) 31, doi:10.1038/s41597-021-01109-0.
  * [49] A. J. Larkoski, S. Marzani, G. Soyez, and J. Thaler, “Soft Drop”,  JHEP 05 (2014) 146, doi:10.1007/JHEP05(2014)146, arXiv:1402.2657.
  * [50] CMS Collaboration, “A Deep Neural Network for Simultaneous Estimation of b Jet Energy and Resolution”,  Comput. Softw. Big Sci. 4 (2020) 10, doi:10.1007/s41781-020-00041-z, arXiv:1912.06046.
  * [51] CERN, “Cern yellow reports: Monographs, vol 2 (2017): Handbook of lhc higgs cross sections: 4. deciphering the nature of the higgs sector”, 2017.  doi:10.23731/CYRM-2017-002, https://e-publishing.cern.ch/index.php/CYRM/issue/view/32.
  * [52] Y. L. Dokshitzer, G. D. Leder, S. Moretti, and B. R. Webber, “Better jet clustering algorithms”,  JHEP 08 (1997) 001, doi:10.1088/1126-6708/1997/08/001, arXiv:hep-ph/9707323.
  * [53] M. Wobisch and T. Wengler, “Hadronization corrections to jet cross-sections in deep inelastic scattering”, in  Workshop on Monte Carlo Generators for HERA Physics (Plenary Starting Meeting), pp. 270–279.  4, 1998.  arXiv:hep-ph/9907280.
  * [54] Y. S. Lai, J. Mulligan, M. Płoskoń, and F. Ringer, “The information content of jet quenching and machine learning assisted observable design”,  JHEP 10 (2022) 011, doi:10.1007/JHEP10(2022)011, arXiv:2111.14589.
  * [55] Y. Lu et al., “Resolving extreme jet substructure”,  JHEP 08 (2022) 046, doi:10.1007/JHEP08(2022)046, arXiv:2202.00723.
  * [56] D. Athanasakos et al., “Is infrared-collinear safe information all you need for jet classification?”, arXiv:2305.08979.
  * [57] D. Bertolini, P. Harris, M. Low, and N. Tran, “Pileup Per Particle Identification”,  JHEP 10 (2014) 059, doi:10.1007/JHEP10(2014)059, arXiv:1407.6013.
  * [58] P. T. Komiske, E. M. Metodiev, B. Nachman, and M. D. Schwartz, “Pileup Mitigation with Machine Learning (PUMML)”,  JHEP 12 (2017) 051, doi:10.1007/JHEP12(2017)051, arXiv:1707.08600.
  * [59] P. Hansen, J. W. Monk, and C. Wiglesworth, “A Wavelet Based Pile-Up Mitigation Method for the LHC Upgrade”, arXiv:1812.07412.
  * [60] J. Arjona Martínez et al., “Pileup mitigation at the Large Hadron Collider with graph neural networks”,  Eur. Phys. J. Plus 134 (2019) 333, doi:10.1140/epjp/i2019-12710-3, arXiv:1810.07988.
  * [61] S. Alipour-fard, P. T. Komiske, E. M. Metodiev, and J. Thaler, “Pileup and Infrared Radiation Annihilation (PIRANHA): A Paradigm for Continuous Jet Grooming”, arXiv:2305.00989.
  * [62] I. W. Stewart et al., “XCone: N-jettiness as an Exclusive Cone Jet Algorithm”,  JHEP 11 (2015) 072, doi:10.1007/JHEP11(2015)072, arXiv:1508.01516.
  * [63] J. Thaler and T. F. Wilkason, “Resolving Boosted Jets with XCone”,  JHEP 12 (2015) 051, doi:10.1007/JHEP12(2015)051, arXiv:1508.01518.
  * [64] T. Lapsien, R. Kogler, and J. Haller, “A new tagger for hadronically decaying heavy particles at the LHC”,  Eur. Phys. J. C 76 (2016) 600, doi:10.1140/epjc/s10052-016-4443-8, arXiv:1606.04961.
  * [65] B. Mukhopadhyaya, T. Samui, and R. K. Singh, “Dynamic radius jet clustering algorithm”,  JHEP 04 (2023) 019, doi:10.1007/JHEP04(2023)019, arXiv:2301.13074.
  * [66] A. J. Larkoski, D. Rathjens, J. Veatch, and J. W. Walker, “Jet SIFT-ing: a new scale-invariant jet clustering algorithm for the substructure era”, arXiv:2302.08609.
  * [67] X. Ju and B. Nachman, “Supervised Jet Clustering with Graph Neural Networks for Lorentz Boosted Bosons”,  Phys. Rev. D 102 (2020) 075014, doi:10.1103/PhysRevD.102.075014, arXiv:2008.06064.

##  Appendix A Mass regression

Before flavor classifiers are trained (cf. Sec. 5), regression networks are
trained to define a regressed jet mass for AK8, AK15, PAIReD, and
PAIReDEllipse jets. Defining a jet mass is necessary so that the
classification networks can be trained to be uncorrelated with respect to this
mass quantity [32]. While regular mass terms like SoftDropped mass [49] can be
used in the case of AK8 and AK15 jets, they are not well-defined for jets that
are not sequentially clustered, such as the PAIReD and PAIReDEllipse jets.
Therefore, for a fair comparison in subsequent steps, the regressed mass
($m_{\text{reg}}$) is devised as a mass term that can be uniformly defined for
all four kinds of jets. Moreover, using the same technique to regress mass
values for all reconstruction strategies allows us to compare the accuracy and
resolution of the jet mass across all strategies.

A mass regression network is trained each for AK8, AK15, PAIReD, and
PAIReDEllipse jets. Jets with labels bb and cc (cf. Sec. 3) sampled from the
flat-mass ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$) and
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) processes are used in the training.
The mass of the generated Higgs boson is set as the target of the regression.
The jets are further assigned weights to ensure that the distribution of the
generated Higgs boson mass is flat in the entire range of 10–500 GeV.

About 13–15 million jets with label bb, and a similar number of jets with
label cc, are used in the training process. The exact number varies across
reconstruction strategies. About 90% of these jets are used in the training
while the remaining are used as an independent validation dataset to avoid
overtraining. Each training is run for 50 epochs, which is deemed enough for
the training to converge. The model from the epoch with the lowest validation
loss is chosen as the final regression model.

The trainings are evaluated on an independent test dataset comprising jets
from ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$) and
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) processes with $m_{\text{H}}=125$
GeV, and Z+jj processes. The evaluation results for the PAIReDEllipse jet
training are shown in Fig. 7 as an example. As expected, the distribution of
$m_{\text{reg}}$ peaks at around the Higgs boson mass for the bb and cc
samples. A Double-sided Crystal Ball (DCB) fit is performed to extract the
mean and spread of these distributions. A mass resolution of about 15 GeV is
obtained. The distribution of $m_{\text{reg}}$ assumes a smoothly falling
shape for ll jets taken from Z+jj processes, as expected.

Figure 7: The distribution of the regressed mass obtained from the
PAIReDEllipse jet regression training evaluated on bb jets from
ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$) (left), cc jets from
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) (center), and light-flavor jets from
Z+jj (right) processes. The blue histograms represent the network outputs,
while the orange lines in the first two plots show the results of a Double-
sided Crystal Ball (DCB) fit. The mean ($\bar{x}$) and spread ($\sigma$)
parameters of the fit are also displayed on the first two plots. The
distributions are truncated between 40–240 GeV for bb and cc jets, and 40-500
GeV for ll jets.

It was observed that the mean and spread of the $m_{\text{reg}}$ distribution
had a moderate dependence on the generated Higgs boson $p_{\text{T}}$. The
mass distributions were hence plotted for different ranges of the generated
Higgs boson $p_{\text{T}}$ and the DCB fit was performed separately for every
range. The mean and spread parameters obtained from the fits are plotted as
functions of the generated Higgs boson $p_{\text{T}}$ in Fig. 8. In general, a
better mass resolution is obtained at higher values of the Higgs boson
$p_{\text{T}}$. In all cases, the mean is compatible with the true mass of 125
GeV within the respective resolutions.

Figure 8: The mean (left) and spread (right) fit parameters of the regressed
mass as a function of the generated Higgs boson $p_{\text{T}}$ for different
reconstruction strategies are shown. Uncertainty bands indicate fit
uncertainties. The binning for each strategy is chosen in a way such that each
bin contains approximately an equal number of jets. The upper (lower) row
shows results for bb (cc) jets taken from
ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$)
(ZH(H$\rightarrow\text{c}\bar{\text{c}}$)) processes. The fit results of the
dijet invariant mass (the light-green dashed lines) are also shown for the
AK4-based reconstruction strategy. The corresponding y-axis is shown on the
right edge of the mean plot (left) in light-green text.

Figure 8 also shows the fit results for the dijet invariant mass in signal
events using the AK4-based reconstruction strategy, for reference. The
AK4-based strategy exhibits a poorer mass resolution compared to other
strategies employing dedicated mass regression techniques. However, a better
mass resolution in AK4-based reconstruction is usually achieved in physics
analyses [21, 8] using techniques like b and c jet mass regression [50] that
evaluate per-jet corrections. Such techniques have not been used in this
paper.

The PAIReD approach achieves the best mass resolution at lower Lorentz-boosts
of the generated Higgs boson. The PAIReDEllipse approach has a slightly poorer
resolution, since PAIReDEllipse jets are expected to contain a larger number
of reconstructed particles from pileup and unrelated hadronic activities
compared to the PAIReD approach at high opening angles. The AK8 and AK15
approaches either do not yield sufficient jets or have a poor mass resolution
at low boosts. This is because they fail to encompass the full decay at low
$p_{\text{T}}$’s. On the other hand, the PAIReDEllipse, AK8, and AK15
approaches exhibit similar mass resolutions at high Lorentz-boosts.

The mass regression technique was also applied on Cone15 jets (cf. Appendix
D). The results of the DCB fit to the regressed masses for AK15 and Cone15
jets are shown in Fig. 9. The mass resolutions obtained from the two methods
are similar. Regression using the Cone15 approach, however, yields means
closer to the generated Higgs boson mass both at low and high boosts of the
Higgs boson.

Figure 9: The mean (left) and spread (right) fit parameters of the regressed
mass as a function of the generated Higgs boson $p_{\text{T}}$ for AK15 and
Cone15 jets are shown. Uncertainty bands indicate fit uncertainties. The upper
(lower) row shows results for bb (cc) jets taken from
ZH(H$\rightarrow\text{b$\bar{\text{b}}$}$)
(ZH(H$\rightarrow\text{c}\bar{\text{c}}$)) processes.

##  Appendix B Differential signal sensitivity ($\nicefrac{{S}}{{\sqrt{B}}}$)
in a ZH(H$\rightarrow$$\text{c}\bar{\text{c}}$) analysis

We define a signal sensitivity metric as the ratio of the number of selected
signal events ($S$) to the square-root of the number of passing background
events ($B$) after all selection criteria and classifier score requirements
are applied (cf. Sec. 6). This can be evaluated at a target end-to-end signal
efficiency value, as well as across various reconstruction strategies. As
discussed in Sec. 3, only the major background (Z+jj) has been considered in
this study. To estimate the approximate signal and background yields, the
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) and Z+jj processes are assigned cross
sections of 0.0027 and 270 pb, respectively. The former value is taken from
the LHC Higgs Cross Section Working Group report [51], while the latter is
obtained from Pythia8 after MLM matching is performed during the sample
generation. The simulations are scaled to an integrated luminosity of 200
$\text{fb}^{-1}$, corresponding to the expected data to be delivered by the
LHC in Run-3.

The $\nicefrac{{S}}{{\sqrt{B}}}$ values for four different target signal
efficiencies are shown in Fig. 10. The signal and background yields in each
bin are normalized to reflect a fixed Higgs/diparton $p_{\text{T}}$ range,
specifically a range of 10 GeV, so that the $\nicefrac{{S}}{{\sqrt{B}}}$ value
is independent of the choice of bin width.

Figure 10: The $\nicefrac{{S}}{{\sqrt{B}}}$ values at end-to-end signal
efficiencies of 0.4 (upper-left), 0.3 (upper-right), 0.2 (lower-left), and 0.1
(lower-right) as a function of the generated Higgs boson $p_{\text{T}}$ (for
signal) or diparton $p_{\text{T}}$ (for background), plotted for various
reconstruction strategies. Signal and background yields in each bin are
normalized to reflect a fixed Higgs/diparton $p_{\text{T}}$ range of 10 GeV.
The target end-to-end signal efficiencies are calculated per bin. The
$\nicefrac{{S}}{{\sqrt{B}}}$ value is set to 0 wherever the target signal
efficiency cannot be achieved.

The trends observed in Fig. 10 closely follow those observed in Fig. 6. At low
Lorentz-boosts, the PAIReDEllipse strategy achieves upto twice the signal
sensitivity that the AK4-based strategy achieves, while the PAIReD strategy
has an intermediate performance. At higher boosts, the PAIReDEllipse and AK15
strategies have similar performances. The maximum $\nicefrac{{S}}{{\sqrt{B}}}$
is achieved in Higgs boson $p_{\text{T}}$ ranges of 50–80 GeV at per-bin
signal efficiencies of 0.2–0.3, using the PAIReDEllipse strategy. This
reinforces the hypothesis that novel reconstruction and tagging strategies
employed at the resolved-jet regime can significantly improve the overall
sensitivity of such analyses. However, these studies rely only on the major
background simulation and do not consider any systematic uncertainties, and
can therefore be regarded as rough estimates only.

##  Appendix C Validation with ZZ(Z$\rightarrow$$\text{c}\bar{\text{c}}$)
analysis

Additional ZZ(Z$\rightarrow$$\text{c}\bar{\text{c}}$) processes are simulated
as per the description in Sec. 3, with the difference that the
Z$\rightarrow$$\text{c}\bar{\text{c}}$ decay is performed in MG5 instead of
Pythia8. A mock ZZ(Z$\rightarrow$$\text{c}\bar{\text{c}}$) analysis is
designed in a way similar to the ZH(H$\rightarrow$$\text{c}\bar{\text{c}}$)
analysis described in Sec. 6.

The mass regression trainings obtained from using the Higgs boson samples are
used to predict the regressed mass values of the hadronically-decaying Z boson
in the ZZ(Z$\rightarrow$$\text{c}\bar{\text{c}}$) samples. The regressed mass
and its spread for different reconstruction strategies are shown in Fig. 11.
The decay width of the Z boson ($\Gamma_{\text{Z}}\approx 2.5$ GeV [27]) is
neglected as it is small compared to the best mass resolution
($\sigma_{\text{Z}}\approx 8$ GeV) achieved using this method
($\Gamma_{\text{Z}}^{2}\ll\text{$\sigma_{\text{Z}}^{2}$}$). The relative mass
resolution ($\nicefrac{{\sigma}}{{\bar{x}}}$) values are consistent with the
ones obtained for the Higgs boson sample (Fig. 8).

Similarly, the existing classifier trainings are used to perform the
ZZ(Z$\rightarrow$$\text{c}\bar{\text{c}}$) analysis. The signal reconstruction
efficiencies and the classifier AUCs as a function of the hadronic Z boson (or
diparton) $p_{\text{T}}$ are shown in Fig. 12. The background rejection and
differential signal sensitivity estimates ($\nicefrac{{S}}{{\sqrt{B}}}$) at
different values of the end-to-end signal efficiencies are shown in Figs. 13
and 14, respectively. A cross section of 0.155 pb is used for the
ZZ(Z$\rightarrow$$\text{c}\bar{\text{c}}$) process. This cross section is
predicted by Pythia8 after MLM jet matching is performed.

Figure 11: The mean (left) and spread (right) fit parameters of the regressed
mass as a function of the generated Z boson $p_{\text{T}}$ for different
reconstruction strategies are shown. Uncertainty bands indicate fit
uncertainties. The binning for each strategy is chosen in a way such that each
bin contains approximately an equal number of jets. The cc jets from the
ZZ(Z$\rightarrow\text{c}\bar{\text{c}}$) processes are used. The y-axis
indicates the dijet invariant mass for the AK4-based reconstruction strategy.

Figure 12: The number of reconstructed
ZZ(Z$\rightarrow\text{c}\bar{\text{c}}$) signal events out of the ~11M signal
events generated (upper left) and the maximum signal efficiency or the signal
reconstruction efficiency (upper right) as functions of the generated Z boson
$p_{\text{T}}$, for various reconstruction strategies. The lower plot shows
the area under the ROC curves (AUC) obtained when classifiers corresponding to
various reconstruction strategies are evaluated on the signal and background
events passing the respective selections, plotted as a function of the Z boson
$p_{\text{T}}$ (for signal events) or the diparton $p_{\text{T}}$ (for
background events).

Figure 13: The end-to-end background rejection at end-to-end signal
efficiencies of 0.4 (upper-left), 0.3 (upper-right), 0.2 (lower-left), and 0.1
(lower-right) as a function of the generated Z $p_{\text{T}}$ (for signal) or
diparton $p_{\text{T}}$ (for background), plotted for various reconstruction
strategies. End-to-end signal efficiencies and background rejections are
calculated per bin. The background rejection is set to 0 wherever the target
signal efficiency cannot be achieved. Bins with very low yield of background
events are either merged with adjacent bins or are omitted.

Figure 14: The $\nicefrac{{S}}{{\sqrt{B}}}$ values at end-to-end signal
efficiencies of 0.4 (upper-left), 0.3 (upper-right), 0.2 (lower-left), and 0.1
(lower-right) as a function of the generated Z boson $p_{\text{T}}$ (for
signal) or diparton $p_{\text{T}}$ (for background), plotted for various
reconstruction strategies. Signal and background yields in each bin are
normalized to reflect a fixed Z/diparton $p_{\text{T}}$ range of 10 GeV. The
target end-to-end signal efficiencies are calculated per bin. The
$\nicefrac{{S}}{{\sqrt{B}}}$ value is set to 0 wherever the target signal
efficiency cannot be achieved.

The trends observed in the ZZ(Z$\rightarrow$$\text{c}\bar{\text{c}}$) analysis
closely follow those observed in the
ZH(H$\rightarrow$$\text{c}\bar{\text{c}}$) analysis. The overall signal
efficiency is lower in case of the ZZ(Z$\rightarrow$$\text{c}\bar{\text{c}}$)
analysis owing to the regressed mass (or dijet invariant mass) requirement of
$m_{\text{reg}}\in[50,200]$ GeV, tuned for the Higgs boson analysis. However,
the observations highlight the generalizability of the reconstruction
algorithms and classifier trainings to a wider set of final states.

##  Appendix D Effects of clustering algorithms

The PAIReD and PAIReDEllipse jets used throughout this paper are constructed
without using any sequential clustering algorithms. Their areas are defined
using the centers of clustered jets (in this case, AK4 jets) as seeds, and all
reconstructed particles lying in the defined area are considered constituents
of the jet. This is akin to non-sequential cone algorithms, and in contrast to
the commonly-used sequential jet clustering algorithms like the AK and
Cambridge-Aachen [52, 53] algorithms in which particles are added to the jet
one at a time starting from a seed.

Jet clustering algorithms like AK offer some advantages such as IRC safety
[15], and less sensitivity to pileup and unrelated hadronic activities.
Infrared safety means that the presence of soft radiation does not affect the
results of the algorithm. Collinear safety means that the results of the
algorithm remain unaffected after introducing a collinear splitting of one of
the inputs. These advantages are lacking when we define PAIReD and
PAIReDEllipse jets. On the other hand, previous studies [54, 55, 56] have
shown that classifiers leveraging IRC-unsafe information outperform
classifiers with only IRC-safe inputs in a variety of jet classification
tasks. It has been hypothesized that the gain comes from the facts that IRC-
unsafe classifiers leverage very-soft radiations from jets, which IRC-safe
approaches typically do not have access to, and IRC-unsafe classifiers take
exact position of the constituents as inputs, as opposed to IRC-safe
approaches taking only relative positions as inputs [56]. The presence of
pileup and underlying event, however, mitigates the gains [54].

The working hypothesis in our paper is that the use of high-level deep
learning networks employing attention mechanisms [16] such as the ParT
algorithm, can implicitly identify relevant jet constituents and assign less
importance to particles arising from unrelated hadronic activities and pileup.
Thus, the advantages that a physics-motivated sequential clustering algorithm
provides can be partially replicated in classification and regression tasks
performed downstream using high-level networks, in addition to leveraging
relevant very-soft radiations. Under this hypothesis, we attribute the
superior performances of AK8 and AK15-based classifiers at high Higgs boson
boosts, for example in Fig. 5, to the fact that the AK jets retain some
advantages that the network cannot replicate in the absence of advanced
clustering algorithms, and that unclustered soft radiations are less important
for classification at higher boosts.

However, the geometrical differences between the PAIReD(Ellipse) jets and AK15
jets make it difficult to make a direct conclusion on the effects of
clustering based on the observed differences. Therefore, to validate the above
observations, we additionally define a cone-based jet (Cone15) with radius 1.5
units, for every AK15 jet reconstructed in the generated events. The axis of
every Cone15 jet is defined to be at the center of the corresponding AK15 jet
and all particles in a radius of $\Delta R=1.5$ units around the jet axis are
considered its constituents. This makes the Cone15 jets geometrically similar
to the AK15 jets with the difference that the former does not use the AK
algorithm, except for defining the jet center. The regression and
classification trainings (Sec. 5) and the mock
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) analysis (Sec. 6) are repeated
replacing AK15 jets with Cone15 jets.

The maximum signal efficiencies, classifier AUCs, end-to-end background
rejections, and signal sensitivity estimates in the context of a
ZH(H$\rightarrow\text{c}\bar{\text{c}}$) analysis are shown in Fig. 15. The
Cone15 and AK15 have similar signal reconstruction efficiencies, except for at
high Higgs boson boosts where the Cone15 approach reconstructs about 2% more
signal events. The Cone15 approach selects the correct jet as the Higgs boson
candidate in a slightly higher number of events, resulting in this observed
gain in signal reconstruction efficiency. However, this gain is offset by the
classifier performance in the next step, where the AK15-based classifier
performs better only at high boosts ($p_{\text{T}}\apprge$ 350 GeV) of the
Higgs boson. The Cone15-based classifier has a significantly better
classification performance at lower boosts. The combined effect can be seen in
the end-to-end background rejection and signal sensitivity plots, where the
two approaches show similar performances, with the Cone15 performing
marginally better at low boosts ($\apprle$ 300 GeV), while the AK15-based
approach performs better at higher Higgs boson $p_{\text{T}}$’s ($\apprge$ 350
GeV).

Figure 15: The maximum signal efficiency (upper-left), the classifier AUCs
(upper-right), and the end-to-end background rejection rate (lower-left) and
signal sensitivity (lower-right) at an end-to-end signal efficiency of 0.4 are
plotted as a function of Higgs boson $p_{\text{T}}$ (or diparton
$p_{\text{T}}$) comparing the AK15- and Cone15-based approaches.

The improvement in overall background rejection at low Higgs boson boosts
using the Cone15 approach suggests that the presence of particles (mostly soft
radiations) in addition to the ones clustered by the AK algorithm within the
jet radius helps improve the classifier’s accuracy at higher angular
separation of the partons. However, the difference in the final end-to-end
background rejection is marginal, as few AK15 jets are reconstructed at low
boosts. On the other hand, a better performance of the AK15-based classifier
at higher boosts suggests that the Cone15-based classifier fails to reject
spurious particles as efficiently as the AK algorithm, at small angular
separation of the partons. The advantages of using sequential jet clustering
algorithms are, therefore, more prominent and inimitable when the
hadronization products of the two partons are highly boosted, closeby, and
overlapping.

The study in this section is limited to only one example and the conclusions
drawn here are empirical. A detailed description and a theoretical analysis of
the synergistic effects of various jet clustering methods and attention-based
deep learning networks are beyond the scope of this paper. Nevertheless, the
performance of the PAIReDEllipse strategy can potentially be improved at high
boosts in the future by utilizing advanced jet grooming and pileup mitigation
techniques [57, 58, 59, 60, 61], and potentially at all boosts by replacing
the AK4 seeds with jets constructed with newer algorithms, such as XCone [62,
63], dynamic-radius algorithms [64, 65], SIFT [66], or supervised jet
clustering [67]. These comparisons could be investigated in future studies.