# Machine learning-based jet and event classification at the Electron-Ion
Collider
with applications to hadron structure and spin physics

Kyle Lee<EMAIL_ADDRESS>Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA  Center for Theoretical Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  James
Mulligan<EMAIL_ADDRESS>Nuclear Science Division, Lawrence
Berkeley National Laboratory, Berkeley, California 94720, USA  Physics
Department, University of California, Berkeley, CA 94720, USA  Mateusz Płoskoń
<EMAIL_ADDRESS>Nuclear Science Division, Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA  Felix Ringer<EMAIL_ADDRESS>C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony
Brook, NY 11794, USA  Department of Physics and Astronomy, Stony Brook
University, Stony Brook, NY 11794, USA  Department of Physics, Old Dominion
University, Norfolk, VA 23529, USA  Thomas Jefferson National Accelerator
Facility, Newport News, VA 23606, USA  Feng Yuan<EMAIL_ADDRESS>Nuclear
Science Division, Lawrence Berkeley National Laboratory, Berkeley, California
94720, USA

###### Abstract

We explore machine learning-based jet and event identification at the future
Electron-Ion Collider (EIC). We study the effectiveness of machine learning-
based classifiers at relatively low EIC energies, focusing on (i) identifying
the flavor of the jet and (ii) identifying the underlying hard process of the
event. We propose applications of our machine learning-based jet
identification in the key research areas at the future EIC and current
Relativistic Heavy Ion Collider program, including enhancing constraints on
(transverse momentum dependent) parton distribution functions, improving
experimental access to transverse spin asymmetries, studying photon structure,
and quantifying the modification of hadrons and jets in the cold nuclear
matter environment in electron-nucleus collisions. We establish first
benchmarks and contrast the estimated performance of flavor tagging at the EIC
with that at the Large Hadron Collider. We perform studies relevant to aspects
of detector design including particle identification, charge information, and
minimum transverse momentum capabilities. Additionally, we study the impact of
using full event information instead of using only information associated with
the identified jet. These methods can be deployed either on suitably accurate
Monte Carlo event generators, or, for several applications, directly on
experimental data. We provide an outlook for ultimately connecting these
machine learning-based methods with first principles calculations in quantum
chromodynamics.

††preprint: JLAB-THY-22-3739, MIT-CTP-5473, YITP-SB-2022-34

###### Contents

  1. I Introduction
  2. II Machine learning applications to hadron structure and spin physics
    1. II.1 Maximizing jet flavor tagging performance
    2. II.2 Maximizing the size of spin asymmetries
  3. III Simulation and training setup
    1. III.1 Event generation
    2. III.2 Machine learning algorithms
  4. IV Jet flavor tagging
    1. IV.1 $u$ vs. $d$ quark jets
    2. IV.2 Out-of-jet information
    3. IV.3 Strange and charm
  5. V Hard process tagging
    1. V.1 Quark vs. gluon jet tagging
    2. V.2 Direct vs. resolved processes and improved constraints on photon structure
  6. VI Conclusions and outlook

##  I Introduction

The future Electron-Ion Collider (EIC) will map out the structure of nucleons
and nuclei with unprecedented precision and allow for novel studies of quantum
chromodynamics (QCD) including explorations of the mechanism of hadronization
and the properties of cold nuclear matter [1]. The EIC center-of-mass energy
of up to $\sqrt{s}=141$ GeV will allow for jets to play a major role in the
EIC science program. Jets are collimated sprays of particles observed in
detectors that arise from the multiple soft and collinear emissions of highly
energetic scattered quarks and gluons. The measured energy and the direction
of a jet represent good proxies of the corresponding quantities at the level
of quarks and gluons that initiate the observed jet. This close correspondence
between partons and jets is expected to hold at the EIC [2, 3]. In this work,
we explore the use of machine learning to classify different flavors of jets
as well as the underlying hard processes in events using machine learning.
Because of the close connection between jets and partons, accurate
classification of jets as well as the underlying hard processes has important
implications for the major scientific goals of the EIC. We will outline
specific applications of these methods that we envision at the future EIC.
Moreover, we expect that the tools and applications discussed in this work are
also relevant to the ongoing experimental program at the Relativistic Heavy
Ion Collider (RHIC).

The rapid progress in artificial intelligence and machine learning over the
last decade has led to various applications in nuclear and high energy
physics, including in the areas of classification, generative modeling,
regression, and inference [4, 5, 6, 7, 8]. In the context of the EIC, machine
learning techniques have been proposed to determine kinematic variables in
Deep Inelastic Scattering (DIS) [9, 10] and to extract quantum correlation
functions such as parton distribution functions (PDFs), transverse momentum
dependent distribution functions (TMDs), generalized parton distributions
(GPDs), and fragmentation functions [11, 12, 13, 14, 15].

In high-energy collider physics, the classification of jets (quark vs. gluon,
QCD vs. $W/Z$, proton-proton vs. heavy-ion) has been studied with increasing
sophistication over the past years [16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32]. An important goal of these studies is to increase
the sensitivity to potential signals of physics beyond the Standard Model. At
the Large Hadron Collider (LHC), machine learning based jet taggers have
succeeded in significantly outperforming traditional jet taggers [33],
including architectures based on Convolutional Neural Networks [19, 34], deep
sets [35], and transformer models [36]. Compared to tagging techniques based
on traditional observables, algorithms based on supervised machine learning
algorithms have both benefits and drawbacks. The principal benefit is the
ability of the machine learning algorithm to take advantage of the full
information at hand and thereby significantly outperform algorithms based on
simple closed-form observables. The drawbacks in doing so are that the results
can be difficult to interpret and connect to first principles since it is
unclear what the machine is learning and that the (simulated or experimental)
data used in the training process may contain biases. Numerous efforts have
been taken to address the limitations of machine learning based algorithms,
both to improve the interpretability of machine learned results and bring
under control the biases of the training data set [33]. For example, in Refs.
[37, 38, 39, 40] complete bases of jet substructure observables were
introduced which span the phase space of emissions inside jets. This large set
of perturbatively calculable observables can rival the performance of
successful machine learning algorithms, which has allowed for insights into
how machines learn and an improved understanding of their good performance. In
this way, machine learning based algorithms have driven progress both in the
performance of the classification tasks but also in pushing analytical
approaches forward. A key question at the EIC is understanding how large an
improvement in performance machine learning based tagging algorithms can
provide, and to identify applications of machine learning based algorithms
that can drive forward the physics goals of the EIC, as it has at the LHC.

In this work, we explore the application of machine learning-based classifiers
at the comparatively low EIC energies. Typically, jet classification studies
are carried out at LHC energies where the jet transverse momentum
$p_{T}^{\rm{jet}}$ is ${\cal O}(100-1000~{}\text{GeV})$. Instead, at the EIC,
jets will be produced predominantly with $p_{T}^{\rm{jet}}$ of 10-30 GeV.
Moreover, an important aspect of machine learning for jets is the sparsity of
the data compared to typical tasks in computer vision. Due to the relatively
low number of hadrons that make up EIC jets, we expect an increased level of
sparsity compared to LHC jets. We will explore if the reduced energy and
increased sparsity of jets at the EIC affect the performance of machine
learning based classifiers. Additionally, at the low EIC energies Monte Carlo
event generators must rely on a larger amount of non-perturbative modeling,
which can cause biases in the simulated data used to train machine learning
classifiers. For the study of spin physics and cold nuclear matter effects, we
will propose an approach that can mitigate this issue by training directly on
experimental data.

We will explore both quark flavor jet tagging and quark vs. gluon jet tagging,
as well as tagging the underlying hard process in the event. For jet flavor
tagging, as examples we consider the binary classification tasks: $u$ vs. $d$
quark [41], $ud$ vs. $s$ and $uds$ vs. $c$ jet tagging and we compare to the
jet charge [42, 43] as a reference. We expect that the flavor tagging of jets
will be an important component to constrain collinear and transverse momentum
dependent PDFs, which we discuss more below. For quark vs. gluon tagging and
tagging the underlying hard process in events, we consider the classification
of $qq,q\bar{q}$ vs. $gg$ di-jet topologies as well as the classification of
direct vs. resolved photon contributions to the photoproduction di-jet cross
section. The latter provides a new opportunity to enhance constraints on the
parton-in-photon PDF.

In addition to using the in-jet information to train the classifiers, we will
investigate how the performance can be improved by not only making use of the
particles inside the jet but also out-of-jet particles in the event. By making
use of this additional information, we extend the jet classification task to
the classification of the underlying hard process in the event. For typical
applications that we foresee at the future EIC, the classification of the
underlying hard process can improve measurements that are not necessarily
limited to in-jet dynamics. Event-wide classification algorithms can help to
improve the measurement of spin asymmetries and studies of cold nuclear
matter. The machine-learned event-wide information can also be mapped to
traditional observables like $N$-jettiness ratios [44] or the jet pull [45,
46].

In addition to baseline studies of the jet flavor tagging performance, we will
gauge the importance of several aspects of detector design within the future
detector setup at the EIC [1]. We will consider the role of particle
identification (PID) information, charge information, and minimum transverse
momentum capabilities on the jet flavor tagging performance. In these studies,
which serve as a first step, we retain the ideal efficiency and PID
capabilities for the detected particle species, leaving the implementation of
a simulated detector response and the impact on the jet energy scale and
resolution for future work. Nonetheless, by varying the minimum particle
$p_{T}$ of the particles input to the classifier as well as comparing the
performance when PID or charge information are included, we will elucidate
baseline considerations on the importance of reconstructing low $p_{T}$
particles and reconstructing PID vs. charge information. For strange quark
initiated jets, we additionally study the impact of identifying the weakly
decaying strange hadrons from their decay products by comparing the jet flavor
tagging performance when the classifier is supplied with the undecayed strange
hadrons vs. their decay products.

We foresee several specific applications of machine learning based jet and
event classification to the major physics goals of the EIC, several aspects of
which we will discuss in further detail in Section II:

(i) Strengthening constraints on (transverse momentum dependent) PDFs. The
flavor tagging of jets will be an important component to constrain collinear
and transverse momentum dependent PDFs. For example, charm-tagged jets can
increase the sensitivity to the (collinear) strange quark PDF in charged
current events [47]. In addition, di-jet events with charm and anti-charm
tagging can also help to constrain the gluon TMD, including the gluon Sivers
function at the EIC [48]. Jet substructure observables have also been proposed
to constrain the gluon PDF at the LHC [49]. New opportunities relevant to RHIC
and the EIC include the gluon helicity distribution, the parton-in-photon PDF,
spin-dependent TMD PDFs and fragmentation functions some of which we explore
quantitatively in this work. In our studies, we find that machine learning-
based classifiers outperform traditional observables like the jet charge and
therefore we expect that machine learning can significantly enhance the
constraints on PDFs. Machine learned classifiers provide an upper bound on the
information content contained in the jet or event [38] and can be used to
design closed-form observables using symbolic regression techniques that are
calculable in perturbative QCD [50, 36, 51]. Additionally, machine-learned
event-by-event classifiers may eventually be directly included in global
analyses of quantum correlation functions like PDFs.

(ii) Enhancing the sensitivity to transverse single spin asymmetries.
Transverse Single Spin Asymmetries (TSSAs) constitute some of the hallmark
measurements at RHIC and the future EIC and they provide constraints on the
spin structure of the proton. TSSAs are defined as the difference of cross
sections where the incoming protons have different transverse spin
$(\uparrow\downarrow)$ orientations

$A_{UT}=\frac{{\rm d}\sigma^{\uparrow}-{\rm d}\sigma^{\downarrow}}{{\rm
d}\sigma^{\uparrow}+{\rm d}\sigma^{\downarrow}}\,.$ (1)

However, it has generally been challenging to measure non-zero TSSAs, in
particular, for those associated with jets [52]. Recently, the STAR
Collaboration used the jet charge as an additional measurement to increase the
size of the asymmetry [53], which was also proposed in Ref. [54]. In this
paper, we propose that an enhancement of the TSSA signal

$\max_{\theta}\,|A_{UT}(\theta)|\,,$ (2)

can be achieved by including an additional machine-learned measurement, which
is here given by the parameters $\theta$. This can be achieved by formulating
the regression task in Eq. (2) as a classification problem of jets (or events)
that are obtained in scattering processes with differently polarized protons
$(\uparrow\downarrow)$ in the initial state. By applying a classifier trained
to distinguish jets in events with different initial spin orientations as an
additional measurement, similar to the jet charge, larger spin asymmetries may
be obtained, which can provide better constraints on the corresponding quantum
correlation functions in global analyses.

(iii) Elucidating cold nuclear matter effects. One of the goals of the EIC is
to achieve an understanding of the transport properties of nuclear matter such
as the jet transport coefficient $\hat{q}$, which denotes the mean square of
the momentum transfer between a propagating hard jet and the nuclear medium
[55, 56, 57, 58, 59]. This can be achieved by comparing jet observables in
$eA$ collisions to those in $ep$ collisions, similar to the jet quenching
program comparing $AA$ and $pp$ collisions at RHIC and the LHC. The entire
basis for extracting such properties of nuclear matter is the difference
between $eA$ and $ep$ observables. By training machine learning methods to
distinguish these two classes of events, one can use interpretable machine
learning methods to gain insight into the type of information responsible for
these differences, and thereby make connections to calculable observables in
perturbative QCD [50]. Additionally, by tagging quark and gluon jets
separately, one can achieve a more detailed understanding of the jet quenching
interaction. This has remained challenging in the $AA$ jet quenching program
[30, 60], and the $eA$ program will offer a cleaner environment where such
techniques may be more likely to succeed.

The remainder of this work is organized as follows. In Section II, we propose
several applications of machine learning based jet identification we carry out
in this paper to the scientific program of the EIC. In Section III, we discuss
the event generation setup of our studies and present the different machine
learning algorithms used in this work. In Section IV we present results for
jet flavor classification at the EIC, and in Section V we extend the
classification to identify underlying hard processes in full events. In
Section VI, we draw conclusions and provide an outlook.

##  II Machine learning applications to hadron structure and spin physics

In this Section, we propose several applications of machine learning based jet
classification algorithms to the scientific program of the EIC and the ongoing
RHIC program. In Section II.1 we provide a high-level description of
applications of jet tagging that can gain stronger constraints with machine
learning methods. In Section II.2, we outline a proposal to enhance spin
asymmetries with machine learning that is largely independent of model biases.

###  II.1 Maximizing jet flavor tagging performance

The measurement of longitudinal and transverse spin asymmetries provides
constraints on the spin decomposition of the proton. Using QCD factorization,
initial and final state spin effects can be disentangled within global
analyses of the available data. As a representative example, we consider TSSAs
where one of the incoming protons is transversely polarized, see Eq. (1).
These asymmetries are generally small due to the cancellation of different
parton contributions with opposite sign. In particular, measurements of Sivers
[61] and Collins [62] asymmetries are often close to zero due to large
cancellations between different PDFs and fragmentation functions.

As discussed also in Section I, since these asymmetries are small,
experimental measurements at RHIC and the future EIC are challenging. Due to
the relatively large experimental uncertainties, the measured asymmetries are
often small or consistent with zero. Measurements have been performed by STAR
using di-jet correlations $p^{\uparrow}+p\to{\rm dijets}+X$ [52, 53] as well
as with single-inclusive measurements by PHENIX and STAR using pions
$p^{\uparrow}+p\to\pi+X$, jets, open heavy-flavor mesons and photons [63, 64,
65, 66, 67]. Theory calculations corresponding to the di-jet measurements have
also recently been performed [68, 69, 70].

The reason for the small size of the asymmetries is due to approximate
cancellations which can be understood from momentum sum rules. In the
following, we will consider the Schäfer-Teryaev sum rule [71, 72] and the
Burkardt [73, 74, 75] sum rule that are satisfied by the Collins and Sivers
functions, respectively. Both of these sum rules state that average transverse
momentum should sum to zero when summed over either the outgoing hadron
flavors (Collins) or incoming quark flavors (Sivers). We note that the
derivation of these sum rules involves bare quark and gluon operators and it
is therefore unknown how much the sum rules are violated due to
renormalization. Nevertheless, they provide an intuitive understanding of the
large cancellations between different quark flavors to first order.

First, we consider the Sivers function $f_{1T}^{\perp a}(x,\vec{k}_{T}^{2})$,
which describes the longitudinal $x$ and transverse momentum $k_{T}$
anisotropy of partons inside a transversely polarized proton. Here the
superscript $a=q,\bar{q},g$ denotes the parton inside the proton. Including
appropriate prefactors and formally integrating over the transverse momentum
dependence, we find

$f_{1T}^{\perp(1)a}(x)=\int{\rm
d}^{2}\vec{k}_{T}\frac{\vec{k}_{T}^{2}}{2M^{2}}f_{1T}^{\perp
a}(x,\vec{k}_{T}^{2})\,,$ (3)

where $M$ is the proton mass. The Burkardt sum rule for the Sivers function
states that the following integral vanishes [74]

$\sum_{a=q,\bar{q},g}\int_{0}^{1}{\rm d}xf_{1T}^{\perp(1)a}(x)=0\,.$ (4)

Under the assumption that the valence quark distributions dominate, the
Burkardt sum rule leads to $u$ and $d$-quark Sivers functions that have
opposite signs and similar magnitudes. This expected behavior of the $u$ and
$d$ quark Sivers functions has been confirmed by recent global analyses [76,
77, 78]. At the EIC, the Sivers function can be measured for example in
lepton-jet correlations [79, 80]. In order to obtain larger spin asymmetries,
and hence a greater sensitivity to the underlying Sivers function, we propose
that machine learning-based classifiers can be included to isolate the
contribution of different quark flavors.

Figure 1: Illustration of the jet production processes considered in this
work. Left: High-$Q^{2}$ electron-proton scattering. At leading order, the
final state consists of the scattered electron and a single jet originating
from different quark flavors. Right: Low-$Q^{2}$ photoproduction, where we
include both the direct and the resolved contribution. At leading order, the
final state consists of the scattered electron in the forward direction close
to the beam axis and a di-jet pair, which can be initiated by both quarks and
gluons. In both cases, the transverse momentum of the jets is measured
relative to the beam axis in the laboratory frame.

Second, we consider the Collins fragmentation function
$H_{1,h/q}^{\perp}(z,\vec{P}_{\perp}^{2})$ as an example of spin-dependent
dynamics in the final state where large flavor cancellations are expected. It
describes the longitudinal $z$ and transverse momentum $P_{\perp}$
distribution of a final-state hadron that originates from a transversely
polarized parton. After integrating out the transverse momentum dependence, we
find

$H_{1,h/q}^{\perp(3)}(z)=\int{\rm
d}^{2}\vec{P}_{\perp}\frac{\vec{P}_{\perp}^{2}}{M_{h}}H_{1,h/q}^{\perp}(z,\vec{P}_{\perp}^{2})\,,$
(5)

where $M_{h}$ is the mass of the observed hadron. The Schäfer-Teryaev sum rule
for the Collins function states that the integral over the longitudinal
momentum fraction vanishes after we sum over all hadron species [71, 72]

$\sum_{h}\int_{0}^{1}{\rm d}z\,H_{1,h/q}^{\perp(3)}(z)=0\,.$ (6)

The cancellation that results here from summing over all hadrons is typically
avoided by measuring identified hadrons in the final states e.g. by measuring
$\pi^{+}$ and $\pi^{-}$ cross sections separately [66]. Nevertheless, there
can be further cancellations, which can be seen as follows. For simplicity, we
now assume that isospin symmetry holds and we assume that the light parton-to-
pion fragmentation process dominates. In this case, only two fragmentation
channels remain. The favored fragmentation functions are pion fragmentation
functions for a valence parton $u$ or $d$, i.e. $H_{1,\pi^{+}/u}^{\perp}$ or
$H_{1,\pi^{-}/d}^{\perp}$, respectively, and the unfavored fragmentation
functions are pion fragmentation functions of $u$ or $d$ that are not a
valence parton, i.e. $H_{1,\pi^{-}/u}^{\perp}$ or $H_{1,\pi^{+}/d}^{\perp}$.
Assuming that the fragmentation of light partons to pions dominates, this then
implies that the favored and unfavored contributions cancel according to the
Schäfer-Teryaev sum rule in Eq. (6) for a given parton. If we now choose to
measure an identified hadron, say $\pi^{+}$, we expect

$\int_{0}^{1}{\rm
d}z\,\Big{(}H_{1,\pi^{+}/u}^{\perp(3)}(z)+H_{1,\pi^{+}/d}^{\perp(3)}(z)\Big{)}\approx
0\,.$ (7)

As the flavor of the jet corresponds to the flavor of the fragmenting parton,
up to higher order corrections in QCD, this cancellation is relevant when we
consider for example the distribution of identified hadrons inside jets [81,
82, 83, 84, 85]. In particular, one can study azimuthal asymmetries that
involve the correlation of the transversity PDF and the Collins fragmentation
function of the hadron inside the jet [86, 87]. Therefore, in order to avoid
the cancellation in Eq. (7), we propose that a machine learned classifier can
be used to tag the flavor of the observed jet. Moreover, recently Refs. [88,
89] proposed to measure spin asymmetries similar to the Collins asymmetry
directly using jets instead of identified hadrons. Analogous sum rules as in
Eq. (6) apply that are expected to lead to small spin asymmetries. In order to
address this problem, Refs. [54, 88, 89] proposed the use of the jet charge as
an additional measurement, which avoids large cancellations between different
quark flavors. Here we also propose that the use of machine learning-based
classification of the jet flavor can enhance the size of the asymmetry
compared to more traditional observables.

We expect that various other measurements and science goals of the EIC and
RHIC will greatly benefit from machine learned classifiers that can identify
the jet flavor or the hard-scattering event. While some of them will be
discussed in this work, we leave more detailed quantitative studies of the
following topics for future work:

  * •

Quark flavor and quark vs. gluon jet identification can help to improve the
sensitivity to the longitudinally polarized gluon distribution $\Delta g$. In
particular, it may be possible to distinguish the positive and negative
solutions for $\Delta g$ that were found in recent global analyses [90]. See
also Refs. [91, 92, 93, 94, 95] for recent discussions and experimental
results.

  * •

Quark vs. gluon jet classification may help to improve measurements of the
gluon Sivers function at RHIC and the future EIC [79, 96].

  * •

The techniques discussed here may also improve searches of physics beyond the
Standard Model at the EIC [97, 98, 99]. For example, in Ref. [100] jet charge-
weighted TSSAs were proposed in this context.

  * •

Exclusive / diffractive processes involving jets can provide constraints on
GPDs and Wigner functions [101, 102, 103]. We expect that machine learning
based classifiers may help to better pin down these higher-dimensional parton
distribution functions along with knowledge about the exact kinematics of the
di-jet events [104, 105].

###  II.2 Maximizing the size of spin asymmetries

In the previous Section, we discussed several areas where machine learning-
based jet and event flavor tagging can play an important role to support the
EIC and RHIC science programs. We implicitly adopted a “UV definition” of the
jet flavor. In this case, the flavor of a jet is defined as the hard parton
that initiates the jet and it can directly be used in Monte Carlo event
generators. The machine learning algorithm is then trained to recover the
assigned flavor label from the IR physics, i.e. the hadrons that make up the
jet [106]. There are theoretical ambiguities associated with this approach
[107] and since the UV label of the jet flavor is not accessible
experimentally, machine learning algorithms have to be trained on simulated
data. This definition has been widely used for machine learning studies of jet
classification at the LHC and various approaches have been developed to
minimize the biases of this approach. For example, data-driven methods [108]
and weakly supervised learning [109] have been introduced, which are tailored
toward the physics goals at the LHC.

In the spin physics context, we propose an alternative approach to directly
train the machine learning algorithms at the observable level. We expect that
several of the science goals discussed in the previous Section can be achieved
without explicitly relying on simulated data. Spin asymmetries such as TSSAs
in Eq. (1) are measured as the difference between cross sections with
different (longitudinal or transverse) spin orientations of the particles in
the initial state. Therefore, we can train a machine learning algorithm that
directly maximizes the size (positive or negative sign) of the spin asymmetry
as repeated in Eq. (2):

$\max_{\theta}\,|A_{UT}(\theta)|\,.$ (8)

The machine learning algorithm is given here in terms of the set of parameters
$\theta$. The optimization of a machine learning algorithm using Eq. (8) only
requires experimentally accessible / hadron-level information. The training
does not explicitly require a UV definition of the jet flavor and it does not
rely on simulated data. As discussed above, spin asymmetries are often small
due to cancellations between different flavor combinations. By including a
machine learning algorithm that is using the size of the spin asymmetry as an
optimization metric or loss function, we can effectively achieve a flavor
separation. Within QCD factorization, the sum rules in Eqs. (4) and (6)
provide the direct connection of the machine learning algorithm that is
optimized using the objective function in Eq. (8) to parton level quantities.

Instead of solving the regression problem in Eq. (8) directly, we can
formulate the task as a classification problem where the machine learning
techniques discussed in the following Sections can be applied. By training a
classifier that distinguishes jets produced in events where the incoming
proton has opposite transverse or longitudinal spin orientation, we can find a
classifier that maximizes the corresponding spin asymmetry. We note that this
approach is similar to other discrimination tasks where the training labels
are known such as between jets in proton-proton and heavy-ion collisions where
the trained classifier can be used to maximize the deviation of the nuclear
modification factor from unity:

$\max_{\theta}\,|R_{AA}(\theta)-1|=\max_{\theta}\,\left|\frac{{\rm
d}\sigma_{AA}}{{\rm d}\sigma_{pp}}(\theta)-1\right|\,.$ (9)

See Ref. [50] for more details. The identification of a machine learned-
classifier can be performed directly on data before unfolding [50] or using
corrected full events [110]. Subsequently, an observable can be identified
that is calculable in perturbative QCD and that approximates the performance
of the machine learned-classifier. For example, this can be achieved using
complete sets of observables such as the $N$-subjettiness basis or Energy Flow
Polynomials (EFPs) that will be discussed below. Note that in the case where
the classifier is trained on uncorrected detector-level inputs, one must
carefully account for systematic differences in detector conditions between
the two data samples. In this case, the designed observables will not be
optimal but rather approximate maximally discriminating observables – where
the deviation of the approximate observable from the optimal observable is
driven by the extent to which detector conditions are understood. These
identified observables can then be measured with traditional techniques that
correct for detector and background effects. In this way, one can verify –
without any reliance on ML – whether the designed observables maximize spin
asymmetries. Ultimately, the observables can then be included in a global
analyses of quantum correlation functions.

##  III Simulation and training setup

To perform the studies in the remainder of this article, we generate simulated
events using the Monte Carlo event generator PYTHIA6 [111], which serves as
the training data for the (supervised) machine learning based classification
algorithms. In the following, we describe the simulated event sample and the
machine learning architecture.

###  III.1 Event generation

We generate two data sets for the following studies, both using PYTHIA6 [111]
with the eRHIC tune [112]. We use CTEQ6.1 [113] and SAS 1D-LO [114] proton and
photon PDFs, respectively. See Figure 1 for an illustration of the two
processes in the respective data sets.

First, we generate jet samples using leading-order (LO) DIS as the hard-
scattering process for the jet flavor tagging studies discussed in Section IV.
At LO, the final state consists of the scattered electron and a single jet
originating from different quark flavors. The LO DIS process is given the
process number $99$ according to PYTHIA6. We then identify the jet flavor with
the flavor of the underlying quark in the LO DIS process $(\gamma^{*}q\to q)$.
We require the photon virtuality and inelasticity to be in the range
$25<Q^{2}<1000$ GeV2 and $0.1<y<0.85$, respectively.

Since gluons do not contribute at LO in DIS, we generate a second data set for
our studies in Section V of quark vs. gluon jet tagging using di-jet events in
low-$Q^{2}$ photoproduction events, including both the direct and resolved
photon contributions. At LO, the final state consists of the scattered
electron in the forward direction close to the beam axis and a di-jet pair,
which can be initiated by both quarks and gluons. We require low
$10^{-5}<Q^{2}<1$ GeV2, while maintaining the same cut on the inelasticity
$0.1<y<0.85$. We identify quark and gluon jets in the photoproduction events
using the PYTHIA6 resolved processes $11\,(qq\to qq),12\,(q\bar{q}\to
q\bar{q}),53\,(gg\to q\bar{q})$ and $13\,(q\bar{q}\to gg),68\,(gg\to gg)$ and
the direct photon-gluon fusion processes $135\,(\gamma^{*}_{T}g\to
q\bar{q}),136\,(\gamma^{*}_{L}g\to q\bar{q})$, where the subscripts $T$ and
$L$ denote the transverse and longitudinal polarization contributions,
respectively. For the quark vs. gluon jet studies in Section V.1, we neglect
the resolved process $28\,(qg\to qg)$ and the direct QCD Compton processes
$131\ (\gamma^{*}_{T}q\to qg)$ and $132\ (\gamma^{*}_{L}q\to qg)$ in order to
avoid ambiguity in labeling processes with $qg$ final states, whereas in
Section V.2 we include them. We note that jet cross sections and jet
substructure observables in EIC photoproduction events were considered in Ref.
[115, 116, 117]. In addition, a comparison of PYTHIA6 results to jet
substructure data from HERA was performed in Ref. [117].

In accordance with experimental particle detection capabilities, we include
all particles in the event and in the jet reconstruction with a lifetime of
$c\tau>1$ cm. This includes

$\displaystyle\gamma,\,e^{-},\,\mu^{-},\,\pi^{-},p,\,n,\,K^{0}_{L},\,K^{0}_{S},\,K^{-},$
$\displaystyle\Lambda^{0},\,\Xi^{0},\,\Xi^{-},\,\Sigma^{\pm},\,\Omega^{-}\,,$
(10)

and the corresponding anti-particles. Particles with $c\tau<1$ cm, such as
neutral pions $\pi^{0}$, are decayed until daughters with $c\tau>1$ cm are
produced. The scattered electron is identified as the leading electron in the
event and removed before we run the jet clustering algorithm. From the
scattered electron we determine the virtuality $Q^{2}$ of the exchanged
photon. We leave the implementation of a simulated detector response for
future work, and as a first step we will instead examine the impact of PID,
charge information, and minimum transverse momentum requirements on the
performance of these taggers in Section IV.

Jets are reconstructed in the laboratory frame with the anti-kT algorithm
[118]. We choose a jet radius parameter of $R=1.0$ and we consider the
rapidity range $|\eta_{\rm lab}|<4$. For the single jets identified in the DIS
events, we require the transverse momentum of the identified jets to be
$p_{T}^{\rm jet}>10$ GeV. We consider a sample of $14$M events satisfying
these criteria. For the di-jets identified in the low-$Q^{2}$ photoproduction
events, we require the transverse momentum of the leading jet to be
$p_{T1}^{\rm jet}>8$ GeV and the subleading jet to be $p_{T2}^{\rm jet}>5$
GeV, as well as the third leading jet to have $p_{T3}^{\rm jet}<4$ GeV. We
consider a sample of $1.4$M events satisfying these criteria, corresponding to
an integrated luminosity of approximately 1 fb-1 [117], corresponding to
approximately one month of EIC runtime (estimated at approximately 10 fb-1 per
year [1]). The size of the sample is determined by studying when the
classification performance approximately saturates as the statistics are
increased.

We use supervised training throughout this work. In order to deploy supervised
models on real data, one can either perform training on simulations or in some
cases on experimental data itself. Training models on Monte Carlo simulations
may be viable at the EIC if event simulations based on parton showers are
sufficiently reliable in the sense that the machine learning algorithms can be
trained on a suite of simulations and deployed on data. In addition, some of
the studies proposed here can be performed on experimental data before
unfolding of detector effects. For example, classifying different spin
orientations or classifying $eA$ vs. $ep$ observables are cases where the
training labels are known experimentally, unlike the case in quark flavor or
quark vs. gluon classification. This will allow for the identification of
suitable observable for which unfolding can be performed afterwards [50].
Furthermore, recent advances in unfolding methods may allow one to unfold
entire events [110]. We leave a more detailed exploration of these aspects for
future work.

Figure 2: The jet charge distribution for EIC jets with $p_{T}^{\rm jet}>10$
GeV produced in high-$Q^{2}$ events as shown on the left side of Figure 1. The
three panels show the results for different flavor discrimination: $u$ vs. $d$
(left), $ud$ vs. $s$ (middle), and $uds$ vs. $c$ (right) for a jet charge
parameter of $\kappa=0.5$, see Eq. (15). The jet charge is able to distinguish
$u$ from $d,s$ reasonably well, whereas it is a relatively poor discriminator
for $u$ vs. $c$ or $q$ vs. $g$. Note that a peak at $Q_{\kappa}=0$ arises from
jets that contain only neutral particles, which happens more frequently
compared to its counterpart at the LHC due to lower particle multiplicity at
the EIC.

###  III.2 Machine learning algorithms

In the next Sections, we will study the binary classification of jets with
different quark flavor and quark vs. gluon jets using machine learning
algorithms. For this task we choose deep sets [119, 120, 121] as our default
classifier, which were introduced as Particle Flow Networks (PFNs) [35] for
data obtained in high-energy collisions in particle and nuclear physics. The
information about the particles in a jet or collider event can be considered
as a set of four vectors with variable length event-by-event. A PFN is a
neural network, which is invariant with respect to permutations of the input
variables and it can naturally handle input with different length. This choice
of machine learning architecture appears to be natural for data in collider
physics since the number of particles varies event-by-event and there is no
inherent ordering of the particles inside a jet or of the particles in the
entire event. We note that other machine learning based classifiers were found
to perform similarly or worse for analogous tasks at LHC energies [35, 36].
The PFNs take as input the information of all the particles inside a
reconstructed jet. We represent the per-particle input variables as

$p_{i}=(z_{i},\eta_{i},\phi_{i},{\rm PID}_{i})\,,$ (11)

where $z_{i}=p_{Ti}/p_{T}^{\rm jet}$ is the normalized transverse momentum of
particle $i$ with respect to the beam axis, and $\eta_{i},\phi_{i}$ are its
rapidity and azimuthal angle. Following Ref. [35], we take $\eta_{i},\phi_{i}$
relative to the ($E$-scheme [122]) jet axis. Lastly, PIDi in Eq. (11) denotes
the particle identification number. See Eq. (III.1) for the different
particles in our data set. We map the PIDs to numerical values in an interval
centered around zero. The numerical values are separated by $0.1$ and
particles (anti particles) are assigned positive (negative) values.

A PFN denoted $f(p_{1},\ldots,p_{N})$ takes as input the kinematics of the $N$
particles in the event. It is constructed such that it satisfies
$f(p_{\pi(1)},\ldots,p_{\pi(N)})$, where $\pi$ denotes the permutation
operator. The required permutation invariance can be achieved by expressing
$f$ as [119]

$f(p_{1},\ldots,p_{N})=F\bigg{(}\sum_{i=1}^{N}\Phi(p_{i})\bigg{)}\,.$ (12)

Here $\Phi,F$ denote fully connected feed-forward neural networks with a
certain number of hidden layers. The connections between hidden layers are
parametrized in terms of weights and each node has a bias term. The per-
particle neural network $\Phi:\mathbb{R}^{4}\to\mathbb{R}^{d}$ maps the input
to a $d$-dimensional latent space. The summation operation in latent space
leads to the permutation invariance of $f$. The second neural network is a map
between the latent space and the final output of the binary classifier
$F:\mathbb{R}^{d}\to\mathbb{R}$.

For quark vs. gluon jet classification, we additionally consider Energy Flow
Networks (EFNs), which are Infrared-Collinear (IRC) safe architectures closely
related to PFNs [35]. IRC safety is built into the permutation invariant
neural network in Eq. (12) by constructing an EFN denoted

$\tilde{f}\left(p_{1},\ldots,p_{M}\right)=F\left(\sum_{i=1}^{M}z_{i}\Phi\left(\hat{p}_{i}\right)\right)\,,$
(13)

where every particle inside a jet is written in terms of its transverse
momentum momentum fractions $z_{i}$ and a 2-component vector which contains
the angular variables $\hat{p}_{i}=(y_{i},\phi_{i})$. Due to the weighting of
$\Phi$ with the momentum fraction $z_{i}$, the resulting expression is IRC
safe [35].

We parametrize the functions $\Phi$ and $F$ in Eqs. (12) and (13) in terms of
DNNs, using the EnergyFlow package [35] with Keras [123]/TensorFlow [124]. For
$\Phi$ we use two hidden layers with 100 nodes each and a latent space
dimension of $d=256$. For $F$ we include three layers with 100 nodes each. For
each dense layer we use the ReLU activation function [125] and we use the
softmax activation function for the final output layer of the classifier. We
train the neural networks using the Adam optimizer [126] and the binary cross
entropy loss function [127], and train for 10 epochs with a batch size of 500.
We do not perform additional hyperparameter optimization for the PFNs or EFNs.
We reserve 20% of the training sample as a validation set, and an additional
20% as a test set on which all metrics are reported. We train the models using
an NVIDIA A100 GPU on the Perlmutter supercomputer.

For quark vs. gluon tagging, we will also consider dense neural networks
(DNNs) that take as input a list of observables that are IRC safe and
generally calculable within perturbative QCD. The resulting classifier is
generally Sudakov safe [38]. The observables that are taken as input to the
DNN form a complete basis of observables. As an example we consider Energy
Flow Polynomials (EFPs) [37]. Alternately, one could consider the
$N$-subjettiness basis [38, 39, 40]. The EFPs constitute a linear basis of jet
substructure observables and they are defined as

$\text{EFP}_{G}=\sum_{i_{1}}\cdots\sum_{i_{V}}z_{i_{1}}\cdots
z_{i_{V}}\prod_{(k,l)\in E}\theta_{i_{k}i_{l}}\,$ (14)

where we sum over all particles inside the jet and $z_{i},\theta_{ij}$ denote
the longitudinal momentum fraction of particle $i$ and the relative angle
between particles $i$ and $j$, respectively. The subscript $G=(V,E)$ indicates
that EFPs are defined in terms of a graph that specifies which terms are
included on the right hand side of Eq. (14). See Ref. [37] for more details.
We note that this basis is insensitive to quark flavor differences but
provides a powerful discriminant for quark vs. gluon jet tagging. In addition,
they provide an increased degree of interpretability compared to PFNs. For the
EFP DNNs, we use 3 hidden layers containing between 32-512 nodes, each with a
ReLU activation function [125], followed by a sigmoid activation for the final
output layer. We train the neural network with the Adam optimizer [126] and a
learning rate ranging from 0.01 to 0.001 and batch size 1000, with the binary
cross entropy loss function of Ref. [127]. We use Keras [123]/TensorFlow [124]
for the implementation, and determine the number of nodes in each hidden layer
and the learning rate using a hyperparameter optimization with the Hyperband
algorithm [128] implemented in Keras Tuner [129]. As in the case of the PFNs,
we reserve 20% of the training sample as a validation set (on which the
hyperparameter tuning is performed), and an additional 20% as a test set on
which all metrics are reported.

The performance of a classifier can be assessed by various metrics quantifying
the rates of correct and incorrect identification of the two classes. There
are four possible outcomes of a classifier’s prediction, “True/False
Positive/Negative”, where “True/False” denotes whether the classifier
prediction was correct, and “Positive/Negative” refers to the predicted class
label. In this work, we will use the following conventions for the positive
class:

  * •

$u$ vs. $d$ classification (Section IV.1): $d$

  * •

$ud$ vs. $s$ classification (Section IV.3): $s$

  * •

$uds$ vs. $c$ classification (Section IV.3): $c$

  * •

$qq/q\bar{q}$ vs. $gg$ classification (Section V.1): $qq/q\bar{q}$

  * •

direct vs. resolved classification (Section V.2): direct

We will consider two metrics in this work, the Receiver Operating
Characteristic (ROC) curve and the Precision-Recall (PR) curve. These metrics
are defined in terms of the following quantities each of which are cumulative
distribution functions of corresponding probability distributions:

  * •

True Positive Rate (TPR, also known as Recall):

$\frac{\mathrm{True\;Positives}}{\mathrm{Total\;Positives}}$

  * •

False Positive Rate (FPR):

$\frac{\mathrm{False\;Positives}}{\mathrm{Total\;Negatives}}$

  * •

Precision:

$\frac{\mathrm{True\;Positives}}{\mathrm{True\;Positives}+\mathrm{False\;Positives}}$

Figure 3: Illustration of particles inside the jet (black) and out-of-jet
radiation (red), which we also take into account to classify both the flavor
of the jet as discussed in Section IV.2 and the underlying hard process in the
event as discussed in Section V. As an example, we show jet production in a
high-$Q^{2}$ DIS scattering process.

The ROC curve shows the TPR vs. the FPR as the decision threshold is varied. A
random classifier follows a diagonal line with an area under the curve (AUC)
of $0.5$ and the better a classifier is, the closer the curve is to the upper
left edge of the plot, with a perfect classifier having AUC $=1$. The ROC
curve does not depend on the relative proportions of the two classes, and we
will use it for classification tasks where there is not a large imbalance in
the proportions of the two classes, such as $u$ vs. $d$ and $q$ vs. $g$
classification.

The PR curve shows the precision vs. recall as the decision threshold is
varied. The PR curve explicitly depends on the relative proportions of the two
classes, since the precision is a measure of the purity of the predicted
positive class. A random classifier based solely on the relative proportions
of the two classes follows a line of constant precision. The larger the
precision and recall, the better the classifier is. A classifier with high
precision but low recall returns only a small fraction of positive cases (low
efficiency) but most of them being identified correctly (high purity), whereas
a classifier with low precision but high recall returns a large fraction of
positive cases (high efficiency) but with many of them being identified
incorrectly (low purity). We will use the PR for classification tasks where
there is a large imbalance in the proportions of the two classes, such as
strange and charm jet classification.

##  IV Jet flavor tagging

Figure 4: ROC curve for $u$ vs. $d$ jet flavor tagging using the jet charge
and PFNs for jets with $p_{T}^{\rm jet}>10$ GeV and $p_{T,\rm{particle}}>0.1$
GeV. We consider three variations of the input to the PFN, providing either
PID information for all particles, charge information for all particles, or
neither.

Using the LO DIS events described in Section III.1, we now study various
binary classifications of quark-jet flavors. We consider several different
classification groupings: $u$ vs. $d$, $ud$ vs. $s$, and $uds$ vs. $c$ quark
jets. We will study the role of PID information, charge information, and
minimum particle transverse momentum thresholds on the performance of the
classifiers, as well as the role of both in-jet and out-of-jet particles.

We will benchmark our machine learning-based algorithms against the energy-
weighted jet charge [130]

$Q_{\kappa}=\sum_{i\in{\rm jet}}z_{i}^{\kappa}Q_{i}\,,$ (15)

where $z_{i}=p_{Ti}/p_{T}^{\rm jet}$ denotes the longitudinal momentum
fraction of the hadrons $i$ inside the jet and $Q_{i}$ is their electric
charge. The weighting factor $z_{i}^{\kappa}$ reduces the sensitivity to
experimental uncertainties and $\kappa$ is a free parameter that we will vary
in our numerical studies below. The jet charge is soft safe but collinear
unsafe, which means that theoretical calculations require a nonperturbative
input that needs to be determined from experiment. Theoretical calculations of
the jet charge were performed in Ref. [42, 43]. Extensions of the jet charge
definition in Eq. (15) were proposed in Refs. [41, 131]. Theoretical work on
defining the flavor of jets can be found in Refs. [132, 133, 134, 135, 136].
Experimental measurements at the LHC can be found in Refs. [137, 138, 139]. In
Figure 2, we show the jet charge distributions for the LO DIS jets considered
in this Section. The jet charge is able to distinguish jets initiated by
quarks of different electric charge reasonably well, such as $u$ from $d,s$,
whereas it is a relatively poor discriminator for $u$ vs. $c$ since they have
the same electric charge, and similarly for $q$ vs. $g$ (not shown here). The
jet charge thereby serves as a reference to which the performance of our
machine learning-based algorithms can be compared.

Figure 5: ROC curves for $u$ vs. $d$ jet flavor tagging using using PFNs with
PID information for jets with $p_{T}^{\rm jet}>10$ GeV and different cuts on
the minimum $p_{T,\rm{particle}}$ required of jet constituents.

In order to study the role of PID information and charge information, we
consider three variations of the information input to the PFN training:

  * •

“PFN w/ PID”: $p_{i}=(z_{i},\eta_{i},\phi_{i},{\rm PID}_{i})\,,$

  * •

“PFN w/ charge”: $p_{i}=(z_{i},\eta_{i},\phi_{i},Q_{i})\,,$

  * •

“PFN w/o PID, charge”: $p_{i}=(z_{i},\eta_{i},\phi_{i})\,.$

We note that the “PFN w/ charge” classifier uses the same experimental
information as the jet charge, whereas the “PFN w/ PID” uses full PID
information, which is not used by the jet charge. Similarly, we consider
varying the minimum transverse momentum of jet constituents input to the PFN
training, varying between $p_{T,\rm{particle}}=0.1-0.4$ GeV. While we do not
consider the exact PID capabilities or single-particle efficiencies of the
proposed EIC detectors, these variations provide a first-order estimate of the
importance of PID and minimum particle transverse momentum detection
capabilities and serve as an initial quantification of the value that may be
gained in jet tagging performance by investing in improved PID or minimum
particle transverse momentum capabilities.

###  IV.1 $u$ vs. $d$ quark jets

Figure 6: ROC curves for $u$ vs. $d$ jet flavor tagging using PFNs with PID
information for jets with $p_{T}^{\rm jet}>10$ GeV, using either in-jet
information as input or using both in-jet and out-of-jet information as input.
We consider two different cuts on the minimum $p_{T,\rm{particle}}$ required
of both the in-jet and out-of-jet particles, which illustrate that soft out-
of-jet particles play a significant role in boosting the classification
performance.

Figure 7: ROC curve (left) and PR curve (right) for $ud$ vs $s$ jet flavor
tagging using the jet charge and PFNs for jets with $p_{T}^{\rm jet}>10$ GeV
and $p_{T,\rm{particle}}>0.1$ GeV. We consider several variations of the input
to the PFN, providing either PID information for all particles, charge
information for all particles, or neither. All curves are constructed from
particles with a decay length $c\tau>1$ cm (in which the weakly-decaying
strange hadrons
$K^{0}_{S},\;\Lambda^{0},\;\Xi^{0},\;\Xi^{-},\,\Sigma^{\pm},\,\Omega^{-}$ and
their associated antiparticles are undecayed), except the curve labeled
$c\tau>10$ cm, which is constructed from particles with a decay length
$c\tau>10$ cm (in which the above weakly-decaying strange hadrons are
decayed). The dashed black lines correspond to a random classifier.

To begin, we consider the classification of $u$ vs. $d$ initiated jets. Our
results are shown in Figure 4. We find that while the jet charge is a fairly
good discriminator of $u$ vs. $d$ jets, the PFN (which uses the full four-
vector information of the final-state particles) improves the performance when
either charge information is included or even more so when PID information is
included. When neither PID nor charge information is included, the classifier
cannot significantly distinguish $u$ jets from $d$ jets in PYTHIA6. The
increase in performance when adding PID information rather than charge
information is fairly small, especially noting that experimental PID
capabilities are not perfectly efficient as assumed in our studies. We will
see in Section IV.3, however, that for strange and charm quark jet
identification, PID information provides a substantial improvement in
performance.

Next, we consider the role of the minimum transverse momentum of jet
constituents input to the PFN training. Figure 5 shows the results when
varying the minimum threshold between $p_{T,\rm{particle}}>0.1-0.4$ GeV. We
find only a minor difference in the classifier performance when varying the
minimum $p_{T,\rm{particle}}$ between 0.1 GeV and 0.4 GeV, suggesting that the
minimum $p_{T,\rm{particle}}$ detector requirements are not essential for
classifying jet flavor using the in-jet information. We will see, however, in
the next Section that this has a stronger impact when considering the out-of-
jet particles.

###  IV.2 Out-of-jet information

The motivation of machine learned-jet classification at the EIC and RHIC is
quite different compared to the LHC. For example, at the LHC di-jet reference
processes can be used as calibration and the resulting classifier can be
applied to identify jets in multi-jet events to search for physics beyond the
Standard Model. Instead, at RHIC and the EIC the focus will be on improving
for example measurements of spin asymmetries as discussed above or to improve
constraints on cold nuclear matter effects. Therefore, at RHIC and the EIC,
the classifier does not need to be limited to the particles inside the
identified jet. We note that event-wide information was also used in
classification studies at the LHC, see for example Refs. [140, 141]. In this
Section, we investigate how the performance can be improved by not only making
use of the particles inside the jet but also out-of-jet particles to classify
the jet flavor, as shown in Figure 3. While we have used a relatively large
jet radius $R=1.0$ in these studies, this choice is somewhat arbitrary and
neglects the role of large-angle radiation and correlations across the entire
event. We therefore compare the performance of a PFN supplied with only in-jet
particles to that of a PFN supplied with both in-jet and out-of-jet particles.

Figure 6 shows the results of this comparison. We show the comparison for two
different minimum $p_{T,\rm{particle}}$ thresholds, $0.1$ GeV and $0.4$ GeV.
We find that the difference between the in-jet classifier and the in-jet +
out-of-jet classifier is significant for the case $p_{T,\rm{particle}}>0.1$
GeV, whereas the difference is almost negligible for $p_{T,\rm{particle}}>0.4$
GeV. This suggests that the soft out-of-jet particles play a significant role
in boosting the classification performance – despite that the soft in-jet
particles had little impact (see Figure 5). This motivates further study of
the origin and role of out-of-jet radiation, since our results suggest it can
provide a significant boost in jet (or event) flavor tagging performance. In
Section V we will revisit the role of out-of-jet particles in order to
classify the underlying hard process of the event.

###  IV.3 Strange and charm

Figure 8: ROC curve (left) and PR curve (right) for $uds$ vs $c$ jet flavor
tagging using the jet charge and PFNs for jets with $p_{T}^{\rm jet}>10$ GeV
and $p_{T,\rm{particle}}>0.1$ GeV. We consider several variations of the input
to the PFN, providing either PID information for all particles, charge
information for all particles, or neither. The dashed black lines correspond
to a random classifier.

We now turn to the identification of strange- and charm-quark initiated jets.
Since strange- and charm-initiated jets are considerably more rare than up- or
down-initiated jets (for our kinematics, the relative
$u\\!:\\!d\\!:\\!s\\!:\\!c$ ratios are approximately
$33\\!:\\!5\\!:\\!1\\!:\\!2$), we quantify the classification performance
using both the ROC curve and the precision-recall curve. In fact, strange jets
are even more rare than charm jets, since despite that the proton PDF contains
a larger quantity of strange than charm, the overall cross section for charm
is larger due to its larger electric charge.

Strange and charm jets also differ from up and down jets in that strange and
charm hadrons have limited decay lifetimes. In the case of strange quarks,
there are a variety of weakly decaying strange hadrons with lifetimes
$1\;\rm{cm}<c\tau<10$ cm (namely
$K^{0}_{S},\;\Lambda^{0},\;\Xi^{0},\;\Xi^{-},\,\Sigma^{\pm},\,\Omega^{-}$ and
their associated antiparticles) which therefore decay on a length scale
comparable to the size of the innermost tracking layers of collider
experiments [142]. We therefore will contrast the classification performance
depending on whether the PFN is provided the undecayed strange hadrons or only
the decay products of these hadrons. In the case of charm quarks, on the other
hand, all charm hadrons decay with lifetimes much shorter than $c\tau=1$ cm,
and cannot be directly detected by experiments but rather must be
reconstructed using the invariant mass of decay products of exclusive charm
hadrons or by tagging displaced vertices. A large literature exists on charm-
jet tagging algorithms, but we will not pursue performance comparisons here
[143, 144, 145, 146].

Figure 7 shows the results for $ud$ vs. $s$ jet classification with final-
state particle decay lifetimes of $c\tau>1$ cm and $c\tau>10$ cm,
respectively. We find several notable differences compared to the $u$ vs. $d$
classification. First, the PFN with PID dramatically outperforms the jet
charge. We also provide as a reference a simple “Leading strange tagger” which
classifies the jet flavor purely based on whether the highest $p_{T}$ particle
in the jet is a strange hadron. The PFN also dramatically outperforms this.
This provides a clear illustration of the value of machine learning-based jet
flavor identification. Second, the overall performance of $ud$ vs. $s$ tagging
is significantly improved when PID information is provided relative to charge
information, especially when the weakly decaying strange hadrons with
$c\tau>1$ cm are included as input to the PFN. If only charge information is
supplied, the performance decreases substantially. This provides a clear
illustration that PID information is highly valuable to obtaining the best
possible strange-jet tagging performance. We leave further study, such as
whether providing PID information of the leading particle rather than all
particles, which could substantially lessen the experimental efforts, to
future work. Third, if neither PID nor charge information is provided, the
performance is yet again substantially worse – however it is still notably
better than in the $u$ vs. $d$ case. This illustrates the relative importance
of particle identification vs. fragmentation in determining the jet flavor –
since when neither PID nor charge information is provided the machine learning
algorithm can only learn from the differences in fragmentation between $ud$
and $s$ jets.

Figure 8 shows the results for $uds$ vs. $c$ jet classification. In this case,
the jet charge is not expected to be a good discriminator, since $u$ (which
dominates the $uds$ sample) and $c$ jets have the same electric charge. We
find similarly strong performance of the PFN classifier when PID information
is included, with an even larger benefit of providing PID information relative
to charge information. Additionally, we note that the PFN that is supplied
with neither PID nor charge information performs better than the previous
cases, illustrating that the amount of information in the fragmentation
pattern unrelated to particle PID or charge plays an increasing role for
heavier quarks, as expected [147, 148].

##  V Hard process tagging

The classification of the underlying hard process is often of primary interest
instead of the classification of a single jet. To do this, we propose to not
only utilize the particles inside the reconstructed jet but to also take as
input particles outside the jet, similar to the studies done in Section IV.2
and shown in Figure 3. Note that we still require a jet with a given
transverse momentum to identify the entire event to ensure the presence of a
hard-scale, which allows for the interpretation or applicability of
perturbative techniques in QCD. The additional information contained in the
dynamics of particles outside the reconstructed jet can generally increase the
performance of the machine learning algorithm. We note that event type
classification using triggers and machine learning was discussed in Ref. [4]
and references therein. Different than Ref. [4], we aim here at identifying
the underlying hard process in the event at parton level. As discussed in
Section III.2 above, the in-jet information that is used to train machine
learned classifiers can be captured by complete sets of observables like
$N$-subjettiness and EFPs. Similar observable bases can be constructed for
out-of-jet information and correlations between jets (such as in
photoproduction events, see Figure 1) can be captured by observables like the
jet pull [45, 46].

Figure 9: ROC curves for quark vs. gluon jet tagging at the EIC using the
leading jet information from quark and gluon di-jets in low-$Q^{2}$
protoproduction events containing $qq,q\bar{q},gg$ topologies. The leading jet
is required to be $p_{T1}^{\rm jet}>8$ GeV, the subleading jet to be
$p_{T2}^{\rm jet}>5$ GeV, and the third leading jet to be $p_{T3}^{\rm jet}<4$
GeV, see Figure 1. We consider several models: (i) PFN including PID
information, (ii) EFN, (iii) DNN with EFPs for two different dimensions $d$,
and (iv) jet mass.

We consider two examples of event classification in this Section. In both
cases, we use low-$Q^{2}$ photoproduction events that contain a di-jet signal
with the transverse momentum of the leading jet required to be $p_{T1}^{\rm
jet}>8$ GeV and the subleading jet to be $p_{T2}^{\rm jet}>5$ GeV, as
described in Section III.1. First, we consider the classification quark vs.
gluon jet topologies by discriminating $qq$ or $q\bar{q}$ di-jet topologies
from $gg$ topologies. Second, we consider the classification of direct vs.
resolved photoproduction processes. Similar to the in-jet particles, we
normalize the transverse momentum of out-of-jet particles relative to the
leading jet transverse momentum $z_{i}=p_{Ti}/p_{T1}^{\rm jet}$. Since we
divide by the transverse momentum of the leading jet in the event, we have
$z_{i}<1$ for both in-jet and out-of-jet particles. Moreover, we count the
values $(\eta_{i},\phi_{i})$ of out-of-jet particles relative to the leading
jet axis.

###  V.1 Quark vs. gluon jet tagging

We consider events with quark and gluon di-jet topologies by considering both
direct and resolved processes that result in $qq$-, $q\bar{q}$-, or
$gg$-initiated di-jets, as described in Section III.1. We then train PFNs
using either (i) the particles in the leading jet, (ii) the particles in both
the leading and subleading jet, or (iii) all particles in the event with
$p_{T,\rm{particle}}>0.1$ GeV.

Figure 10: ROC curves for quark vs. gluon event tagging at the EIC with PFNs
including PID information and considering as input to the PFN either the
leading jet particles, the leading and subleading jet particles, or all
particles in the event with $p_{T,\rm{particle}}>0.1$ GeV. Here we consider
quark and gluon di-jets in low-$Q^{2}$ protoproduction events containing
$qq,q\bar{q},gg$ topologies. The leading jet is required to be $p_{T1}^{\rm
jet}>8$ GeV, the subleading jet to be $p_{T2}^{\rm jet}>5$ GeV, and the third
leading jet to be $p_{T3}^{\rm jet}<4$ GeV, see Figure 1.

Figure 9 shows the classification performance of quark vs. gluon jet event
topologies when trained with only the leading jet particles. Models trained
with the leading jet particles correspond most closely to previous studies of
the classification of quark vs. gluon single jets [34, 27]. We consider a PFN
trained with PID information, as well as an IRC-safe EFN, which performs
slightly worse than the PFN, which is typical in quark vs. gluon jet
classification at the LHC. While the performance at the low EIC jet energies
considered here is lower than quark vs. gluon classification with
high-$p_{T}^{\rm jet}$ jets at the LHC, the PFN and EFN still are able to
achieve substantial classification performance and large improvements compared
to single observables such as the jet mass. Additionally, we compare the a DNN
that uses IRC-safe EFPs with dimension $d=7$ as input, which gives a
performance that approaches that of the deep set models.

Figure 10 shows the classification performance of quark vs. gluon jet event
topologies as the subleading jet particles and out-of-di-jet particles are
added to the PFN training input. As the subleading jet particles and out-of-
di-jet particles are added to the PFN training input, the performance
significantly increases.

###  V.2 Direct vs. resolved processes and improved constraints on photon
structure

Next, we consider discriminating events that arise from direct vs. resolved
photoproduction processes. The direct processes correspond to those where the
low-$Q^{2}$ quasi-real photon directly participates in the hard-scattering
process. Instead, the resolved process corresponds to the case where the
parton content of the photon is resolved and only quarks and gluons are
involved in the hard-scattering process that produces the di-jet pair in the
final state. The resolved process provides access to the nonperturbative
parton-in-photon PDF, which is of particular interest at the future EIC. While
the unpolarized di-jet photoproduction cross section has been measured at HERA
by the H1 Collaboration [149], the polarized cross section will be measured
for the first time at the EIC and the helicity parton-in-photon PDFs are
currently unconstrained from experimental data. See Refs. [116, 150, 151, 117,
152] for theoretical work on the partonic structure of photons in the context
of the future EIC.

An approximate separation of the direct and resolved contribution can
significantly enhance the sensitivity to the nonperturbative parton-in-photon
PDF. Traditionally, this separation has been achieved by measuring a multi-
differential cross section including the kinematic variable $x_{\gamma}$,
which is defined in terms of the electron energy and the di-jet transverse
momenta and rapidities [149, 150]. At LO in QCD, $x_{\gamma}$ corresponds to
the momentum fraction of the parton inside the photon such that for
$x_{\gamma}\to 1(0)$, the direct (resolved) process dominates. Instead, here
we propose that the performance can be augmented using a machine learning-
based binary classifier that can make use of the full event information.

In order to explore this approach, we train a PFN to distinguish the direct
vs. resolved photoproduction processes using the leading and subleading jet
information. We consider events with the same quark and gluon di-jet
topologies described in the previous Section except additionally including
$qg$-initiated di-jets in addition to $qq$-, $q\bar{q}$-, and $gg$-initiated
di-jets. Figure 11 shows the classification performance of direct vs. resolved
photoproduction processes. We find that the performance is worse than the
quark vs. gluon di-jet topology classification shown in the previous Section,
which is unsurprising given that the direct and resolved contributions contain
both quark and gluon jets. We furthermore find that the impact of supplying
PID information to the PFN is almost negligible in this case. It would be
instructive to combine the information of the “QCD-inspired” variable
$x_{\gamma}$ with the machine-learned classifier described here. Note that in
Fig. 11 we have not included information from the electron. In addition, it
would be interesting to combine the tagging of the direct vs. resolved process
with jet flavor identification as discussed in previous Sections. This would
allow for a flavor separation of the parton-in-photon PDFs. We note that a
flavor separation based on identified hadrons inside the di-jets was employed
in Ref. [150]. We leave the exploration of these topics as well as
quantitative impact studies at the EIC for future work.

Figure 11: ROC curves for direct vs. resolved process tagging at the EIC. Here
we consider quark and gluon jets produced in low-$Q^{2}$ photoproduction
events, see Figure 1.

##  VI Conclusions and outlook

We have presented first studies of machine learning based jet and event
classification using simulated events at the EIC. While the performance of jet
flavor classification is more challenging than with high transverse momentum
jets at the LHC, machine learning-based classification algorithms offer
important advantages in performance and many prospects for interpretability.
We found that machine learning algorithms outperform traditional observables
used to identify jet flavor, such as the jet charge. In order to provide input
to the detector specifications at the EIC, we investigated the impact of PID
information, charge information, and minimum particle transverse momentum
requirements. We found that providing charge information is sufficient for $u$
vs. $d$ jet classification, but that PID information gives a large improvement
to strange and charm jet tagging capabilities. We found that soft particles
with $0.1<p_{T}<0.4$ GeV have only minor impact when jet flavor classification
is performed using in-jet particles, but that out-of-jet soft particles give
substantial improvement to the classification performance. The studies
performed here can be extended to a full detector simulation and additional
kinematics. Future detailed studies on the impact of PID for strange and charm
quark-initiated jets may be useful, such as investigating whether it is
important for the all particles to be identified or whether a small number of
leading particles is responsible for most of the flavor tagging performance.
Another future direction is the exploration of different machine learning
architectures. In this work, we limited ourselves to algorithms that have been
known to perform well in the LHC environment. The jet quark flavor
classification could also be extended from binary classification to multi-
label classification for all quark flavors simultaneously.

We have proposed several applications in which these methods will play an
important role in the EIC science program. Jet flavor tagging can lead to
improved constraints of collinear and transverse momentum dependent parton
distribution functions, such as the strange quark PDF and the gluon TMD.
Eventually, machine-learned event-by-event classifiers may be directly
included in global analyses of quantum correlation functions like PDFs. Flavor
tagging also provides opportunities to increase experimental access to
transverse single spin asymmetries, since one can use the magnitude of the
asymmetry as an objective function for the machine algorithm to maximize.
Tagging the underlying hard process in events can enhance constraints on the
parton-in-photon PDF when combined with previous work [150]. Additionally, the
methods outlined here can be applied to classify $ep$ vs. $eA$ collisions and
provide new insight to cold nuclear matter effects, and can be extended to
produce increasingly interpretable results [50]. The supervised machine
learning methods presented here to classify jet flavor require suitably
accurate Monte Carlo event generators to produce training data. This will
require repeating these studies with additional and increasingly sophisticated
Monte Carlo event generators. On the other hand, the applications we proposed
for (i) maximizing spin asymmetries and (ii) classifying $ep$ vs. $eA$ can be
performed directly on experimental data since the training labels are known.
While the latter can currently only be explored using simulations, the former
can be deployed immediately at RHIC.

## Data availability

The data sets used in this work can be found at:

10.5281/zenodo.7538810

We provide the particle four vectors for both the LO DIS sample and
photoproduction sample described above.

###### Acknowledgements.

We would like to thank Roli Esha, Wenqing Fan, Zhongbo Kang, Kolja Kauder,
Brian Page, Nobuo Sato, and Fanyi Zhao for helpful discussions. KL was
supported by the LDRD program of LBNL, the U.S. DOE under contract number DE-
AC02-05CH11231 and DE-SC0011090. JM, MP, FY are supported by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, under the
contract DE-AC02-05CH11231. FR was supported by the Simons Foundation under
the Simons Bridge program for Postdoctoral Fellowships at SCGP and YITP award
number 815892; the NSF, award number 1915093; the DOE Contract No. DE-
AC05-06OR23177, under which Jefferson Science Associates, LLC operates
Jefferson Lab and Old Dominion University. This research used resources of the
National Energy Research Scientific Computing Center, which is supported by
the Office of Science of the U.S. Department of Energy under Contract No. DE-
AC02-05CH11231.

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