# Dynamic Jet Charge

Zhong-Bo Kang  Department of Physics and Astronomy, University of California,
Los Angeles, CA 90095, USA  Mani L. Bhaumik Institute for Theoretical Physics,
University of California, Los Angeles, CA 90095, USA  Center for Frontiers in
Nuclear Science, Stony Brook University, Stony Brook, NY 11794, USA  Xiaohui
Liu  Center of Advanced Quantum Studies, Department of Physics, Beijing Normal
University, Beijing 100875, China  Center for High Energy Physics, Peking
University, Beijing 100871, China  Sonny Mantry  Department of Physics and
Astronomy, University of North Georgia, Dahlonega, GA 30597, USA  M.C. Spraker
Department of Physics and Astronomy, University of North Georgia, Dahlonega,
GA 30597, USA  Tyler Wilson  Department of Physics and Astronomy, University
of North Georgia, Dahlonega, GA 30597, USA

###### Abstract

We propose a modified definition of the jet charge, the “dynamic jet charge”,
where the constant jet momentum fraction weighting parameter, $\kappa$, in the
standard jet charge definition is generalized to be a function of a dynamical
property of the jet or the individual jet constituents. The dynamic jet charge
can complement analyses based on the standard definition and give improved
discrimination between quark and gluon initiated jets and between jets
initiated by different quark flavors. We focus on the specific scenario where
each hadron in the jet contributes to the dynamic jet charge with a
$\kappa$-value dynamically determined by its jet momentum fraction. The
corresponding dynamic jet charge distributions have qualitatively distinct
features and are typically characterized by a multiple peak structure. For
proton-proton collisions, compared to the standard jet charge, the dynamic jet
charge gives significantly improved discrimination between quark and gluon
initiated jets and comparable discrimination between $u$\- and $d$-quark
initiated jets. In Pythia simulations of heavy ion collisions, the dynamic jet
charge is found to have higher jet discrimination power compared to the
standard jet charge, remaining robust against the increased contamination from
underlying event. We also present phenomenological applications of the dynamic
jet charge to probe nuclear flavor structure.

##  I Introduction

Identifying the electric charge of partons that emerge from the hard
scattering process to initiate the formation of jets can be useful in the
search for new physics and testing various aspects of the Standard Model.
Earlier experimental application of jet charge Field:1977fa  was in deep
inelastic scattering studies Berge:1979qg ; Berge:1980dx ; Allen:1982ze ;
Albanese:1984nv ; Barlag:1981wu ; Erickson:1979wa , finding evidence for
quarks in the nucleon. The jet charge observable has also been applied in
measurements of the charge asymmetry Stuart:1989db ; Decamp:1991se , in
tagging the charge of bottom quark jets Braunschweig:1990cv ; Abazov:2006vd ;
Aaltonen:2013sgl ; Aad:2013uza  and hadronically decaying $W$ bosons
Khachatryan:2014vla ; Chen:2019uar , determination of electroweak parameters
Buskulic:1996kx , and in testing aspects of perturbative Quantum
Chromodynamics (QCD) Krohn:2012fg ; Waalewijn:2012sv ; Aad:2015cua ;
Sirunyan:2017tyr ; Fraser:2018ieu . Jet charge has also been used to probe
nuclear medium induced jet quenching on quark and gluon initiated jets in
heavy ion collisions Gyulassy:1993hr ; Wang:1994fx ; Chien:2016led ;
Connors:2017ptx ; Li:2019dre ; Sirunyan:2020qvi ; Du:2020pmp . Most recently,
a new theoretical framework Kang:2020fka ; Kang:2020xez  was introduced to use
jet charge as a probe of the quark flavor structure of the nucleon.

The jet charge is one of a variety of jet substructure tools used for jet
discrimination Gallicchio:2011xq ; Sirunyan:2020qvi ; Brewer:2020och . A
comprehensive review of other jet substructure techniques can be found in Ref.
Larkoski:2017jix . The utility of the jet charge observable for jet
discrimination has also prompted the development of machine learning
techniques Fraser:2018ieu ; Chen:2019uar  for extracting the jet charge.
Recently, a color tagger was introduced Buckley:2020kdp  as another jet
discriminant in order to distinguish between color singlet states decaying
into two jets from dijet backgrounds. In a recent analysis Chien:2018dfn , it
was shown that the quark-gluon jet discrimination power of the various jet
substructure techniques typically worsens in heavy ion collisions, compared to
proton-proton ($pp$) collisions, but suggested a systematically improvable
framework for studying medium modification for quark and gluon initiated jets.

In the context of these various tools developed for jet discrimination, we
introduce a modified definition of the standard jet charge, that we refer to
as the dynamic jet charge. This new definition can complement analyses based
on the standard jet charge and allows for improved discrimination between
quark and gluon initiated jets and between $u$-quark and $d$-quark initiated
jets. We present simulation results to illustrate the characteristic
properties of the dynamic jet charge and its discrimination power at the LHC.
We also present corresponding results for Pythia simulations of heavy ion
PbPb-collisions and find that, unlike for the standard jet charge, the
discrimination power of the dynamic jet charge remains similar to that found
in $pp$-collisions, being largely unaffected by the significantly greater
underlying activity. While Pythia simulations account for the increased
underlying event activity in heavy ion collisions, they do not include nuclear
medium effects and the related jet quenching effects. Further studies of the
dynamic jet charge for heavy ion collisions could be done using the JEWEL
Zapp:2013vla  or JETSCAPE Putschke:2019yrg  simulation which includes hot
nuclear medium (i.e. quark-gluon plasma) effects, and is left for future work.

Figure 1: Normalized standard jet charge distributions for the more forward
(top panel) and more central (bottom panel) of the leading jets in Pythia8
simulations of $pp\to j_{1}j_{2}X$ at $\sqrt{s}=8$ TeV with hadronization and
MPI effects turned on. Selection cuts of 200 GeV $<p_{T_{j_{1},j_{2}}}<$ 300
GeV, $|\eta_{j1,j2}|<2.1$, and $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the
leading ($j_{1}$) and subleading ($j_{2}$) anti-kT jets of jet radius $R=0.4$
are applied. The blue, red, and green curves correspond to $\kappa=0.3,0.5,$
and 0.7, respectively.

Figure 2: Comparison of the ATLAS data Aad:2015cua  for the standard jet
charge with Pythia 8.240 simulations (default tune). The average standard jet
charge is plotted as a function of the jet $p_{T}$ in $pp\to j_{1}j_{2}X$ at
$\sqrt{s}=8$ TeV. The average jet charge of the more forward (top panel) and
more central (bottom panel) of the two leading jets are shown for the values
$\kappa=0.3,0.5,0.7$ corresponding to the blue, red, and green curves
respectively. Selection cuts of 200 GeV $<p_{T_{j_{1},j_{2}}}<$ 300 GeV,
$|\eta_{j1,j2}|<2.1$, and $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading
($j_{1}$) and subleading ($j_{2}$) anti-kT jets of jet radius $R=0.4$ are
applied. The jet charge values correspond to an average over the jet $p_{T}$
bins: [50 GeV,100 GeV], [100 GeV, 200 GeV], [200 GeV, 300 GeV], [300 GeV, 400
GeV], [400 GeV, 500 GeV], [500 GeV, 600 GeV], [600 GeV, 800 GeV], [800 GeV,
1000 GeV], [1000 GeV, 1200 GeV], and [1200 GeV, 1500 GeV].

The standard definition Field:1977fa ; Krohn:2012fg ; Waalewijn:2012sv  of the
jet charge is given by the weighted sum

$\displaystyle Q_{\kappa}^{i}=\sum_{h\in{i\text{-}\rm
jet}}z_{h}^{\kappa}\>Q_{h},$ (1)

for a jet initiated by the parton $i$ and the sum is over all hadrons $h$,
with charge $Q_{h}$, in the jet. $\kappa>0$ is a constant parameter that is
part of the jet charge definition, and $z_{h}$ is the jet transverse momentum
or energy fraction carried by the hadron $h$

$\displaystyle z_{h}=\frac{p_{T_{h}}}{p_{T_{J}}}\qquad\text{or}\qquad
z_{h}=\frac{E_{h}}{E_{J}},$ (2)

for $pp$ and $e^{+}e^{-}$ colliders, respectively. Here $p_{T_{h}}$ and
$E_{h}$ denote the transverse momentum and energy of the hadron $h$ in the
jet, respectively. Similarly, $p_{T_{J}}$ and $E_{J}$ denote the total jet
transverse momentum and energy, respectively. In this work, we focus on the
hadron collider environment and use the $p_{T}$-weighted jet charge
definition.

In this work, we propose the dynamic jet charge, denoted by $Q_{\rm dyn}$,
defined as

$\displaystyle Q_{\rm dyn}^{i}=\sum_{h\in{i\text{-}\rm jet}}z_{h}^{\kappa({\rm
P})}\>Q_{h},$ (3)

for a jet initiated by parton $i$, where the constant parameter $\kappa$ in
the standard definition is promoted to a function $\kappa({\rm P})$ of some
property “$P$” of the jet or each individual jet constituent. Here we focus
exclusively on the scenario where the property $P$ is chosen to be the hadron
momentum fraction $z_{h}$ for each hadron in the jet. Thus, each individual
jet constituent contributes to the dynamic jet charge with the dynamically
determined value $\kappa({z_{h}})$. We demonstrate that this dynamic jet
charge definition can allow for enhanced jet discrimination and complement
analyses based on the standard jet charge definition.

One might also consider scenarios where $\kappa$ is a function of other jet
constituent properties such as the hadron transverse momentum with respect to
the jet axis, $\kappa(k_{h}^{\perp}/p_{T_{J}})$, or more global dynamic jet
properties such as the jet mass, $\kappa(m_{J}/p_{T_{J}})$, or the groomed jet
radius Larkoski:2014wba ; Kang:2019prh , $\kappa(R_{g})$, in defining the
dynamic jet charge. Furthermore, one might generalize other jet observables,
such as jet angularities Berger:2003iw ; Hornig:2009vb ; Kang:2018qra ;
Kang:2018vgn ; Bell:2018gce ; Budhraja:2019mcz ; Bauer:2020npd , by
transforming the constant parameters that appear in their definitions to
dynamic parameters. If and how these dynamic jet charge definitions lead to
increased discrimination power will be explored in future work.

The theoretical prediction for the average standard jet charge is given by
Waalewijn:2012sv

$\displaystyle\langle
Q_{\kappa}^{i}\rangle=\int_{0}^{1}dz\>z^{\kappa}\sum_{h\in{i\text{-}\rm
jet}}Q_{h}\frac{1}{\sigma_{i\text{-}{\rm
jet}}}\frac{d\sigma_{h\in{i\text{-}\rm jet}}}{dz},$ (4)

where $\sigma_{i\text{-}{\rm jet}}$ denotes the cross section for producing a
jet initiated by the parton $i$ and $d\sigma_{h\in{i\text{-}\rm jet}}/dz$ is
the differential cross section for producing the jet in which a hadron $h$
with momentum fraction $z$ is observed. This can be brought into the form
Waalewijn:2012sv

$\displaystyle\langle
Q_{\kappa}^{i}\rangle=\int_{0}^{1}dz\>z^{\kappa}\sum_{h\in{i\text{-}\rm
jet}}Q_{h}\sum_{j}$
$\displaystyle\int_{z}^{1}\frac{dz^{\prime}}{z^{\prime}}{\cal
J}_{ij}(p_{T_{J}},R,z^{\prime},\mu)$ $\displaystyle\times
D_{j}^{h}\left(\frac{z}{z^{\prime}},\mu\right),$ (5)

where the ${\cal J}_{ij}(p_{T_{J}},R,z,\mu)$ are perturbatively calculable
coefficients and the $D_{j}^{h}\left(z,\mu\right)$ are the nonperturbative
fragmentation functions describing the fragmentation of the parton $j$ into
the hadron $h$ which carries away momentum fraction $z$ from the parton $j$.
The argument $R$ in the coefficients ${\cal J}_{ij}$ denotes the jet radius.
The resummation of large logarithms in $\sim p_{T}R/\Lambda_{\rm QCD}$, due to
the disparity in energy scales associated with jet dynamics and hadronization,
is achieved by choosing the renormalization scale $\mu\sim p_{T}R$
Dasgupta:2016bnd ; Kang:2016mcy  in the perturbative coefficients and using
the standard DGLAP evolution to evaluate the fragmentation functions at this
scale. Next-to-Leading Order (NLO) results for the ${\cal J}_{ij}$
coefficients can be found in Ref. Waalewijn:2012sv . At Leading Order (LO),
${\cal J}_{ij}^{(0)}(p_{T_{J}},R,z,\mu)=\delta_{ij}\delta(1-z)$ and the
average jet charge in Eq. (5) becomes

$\displaystyle\langle Q_{\kappa}^{i(0)}\rangle=\sum_{h\in{i\text{-}\rm
jet}}Q_{h}\>\tilde{D}_{i}^{h}\left(\kappa,\mu\right)+{\cal O}(\alpha_{S}),$
(6)

where $\tilde{D}_{i}^{h}\left(\kappa,\mu\right)$ is the Mellin moment of the
fragmentation function $D_{i}^{h}\left(z,\mu\right)$

$\displaystyle\tilde{D}_{i}^{h}\left(\kappa,\mu\right)=\int_{0}^{1}dz\>z^{\kappa}\>D_{i}^{h}\left(z,\mu\right),$
(7)

with a multiplicative renormalization group evolution equation

$\displaystyle\mu\frac{d}{d\mu}\tilde{D}_{i}^{h}\left(\kappa,\mu\right)=\frac{\alpha_{s}}{\pi}\tilde{P}_{ij}(\kappa)\>\tilde{D}_{j}^{h}\left(\kappa,\mu\right),$
(8)

where $\tilde{P}_{ij}(\kappa)$ are the standard splitting functions in Mellin
space.

For simulation results, we use Pythia8 (Pythia 8.240) Sj_strand_2015  for
event generation and FASTJET 3.3.2 Cacciari_2006 ; Cacciari_2012  for
implementing jet algorithms and applying soft drop grooming Larkoski_2014 ;
frye2016precision ; Frye_2016 . Jets are defined using the anti-kT
Cacciari_2008  jet algorithm throughout the manuscript. All Pythia simulation
results presented for $pp$-collisions and PbPb-collisions include underlying
events corresponding to the Multi-Parton Interaction (MPI) switch being turned
on, unless specified otherwise.

For calibration purposes, we perform Pythia simulations to compare with the
ATLAS analysis Aad:2015cua  for $pp$-collisions at $\sqrt{s}=8$ TeV. The two
leading jets, $j_{1}$ and $j_{2}$, denoting the leading and subleading jets,
respectively, are subject to pseudorapidity and transverse momentum selection
cuts of $|\eta_{j1,j2}|<2.1$ and $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$,
respectively. The jets are defined with jet radius of $R=0.4$. These leading
jets are classified as either “more central” or “more forward”, corresponding
to the jet with a smaller or larger magnitude of pseudorapidity, respectively.

In Fig. 1, we show the simulation results for the standard jet charge
distribution with the Pythia parton-level setting ‘HardQCD:all = on’ and
hadronization and underlying event (MPI) turned on. The three curves in each
panel correspond to the values $\kappa=0.3,0.5,$ and 0.7, respectively. The
top (bottom) panel corresponds to jet charge distributions for the more
forward (central) of the two leading jets, each restricted to the jet
$p_{T}$-bin: $[200\>{\rm GeV},300\>{\rm GeV}]$. Following the ATLAS analysis
Aad:2015cua , Fig. 1 uses jet charge bins of size 0.1 in units of the proton
electric charge $e$.

In Fig. 2, we show a comparison of the Pythia simulation results with ATLAS
data Aad:2015cua  for the average jet charge as a function of the jet
$p_{T}$-bin. The top and bottom panels correspond to the more forward and more
central jets, respectively. We see that there is good agreement between the
Pythia simulation results and the ATLAS data.

##  II Dynamic Jet Charge

The properties of the standard jet charge definition in Eq. (1) and its
ability to discriminate between quark and gluon initiated jets and between
quark jets of different flavors have been extensively studied Krohn:2012fg ;
Waalewijn:2012sv ; Fraser:2018ieu . The dynamic jet charge is defined in Eq.
(3) and we focus on the scenario where the constant parameter $\kappa$ in Eq.
(1) is generalized to a function of the momentum fraction $z_{h}$, for each
hadron $h$ in the jet

$\displaystyle Q^{i}_{\rm dyn}=\sum_{h\in{i\text{-}\rm
jet}}z_{h}^{\kappa(z_{h})}\>Q_{h}.$ (9)

Thus, each hadron $h$ contributes to the jet charge with a dynamic
$\kappa$-parameter whose value is determined by its jet momentum fraction
$z_{h}$. Different functional forms of $\kappa(z_{h})$ correspond to different
definitions of the dynamic jet charge.

Figure 3: Normalized dynamic jet charge distributions for the more forward
(top panel) and more central (bottom panel) jets from Pythia8 simulations of
$pp\to j_{1}j_{2}X$ at $\sqrt{s}=13$ TeV with hadronization and MPI effects
turned on. Selection cuts of 200 GeV $<p_{T_{j_{1},j_{2}}}<$ 300 GeV,
$|\eta_{j1,j2}|<2.1$, and $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading
($j_{1}$) and subleading ($j_{2}$) anti-kT jets of jet radius $R=0.4$ are
applied. These distributions are for $\xi_{\rm cut}=0.3$, $k_{<}=1.0$ and the
blue, red, and green curves correspond to $k_{>}=0.3,0.5,$ and 0.7,
respectively.

Figure 4: The average dynamic jet charge from Pythia 8.240 simulations. The
average dynamic jet charge is plotted as a function of the jet $p_{T_{J}}$ in
$pp\to j_{1}j_{2}X$ at $\sqrt{s}=13$ TeV. The average dynamic jet charge of
the more forward (top) and more central (bottom) of the two leading jets are
shown for $\xi_{\rm cut}=0.3,k_{<}=1.0$ and $k_{>}=0.3,0.5,0.7$ corresponding
to the blue, red, and green curves respectively. Selection cuts of 200 GeV
$<p_{T_{j_{1},j_{2}}}<$ 300 GeV, $|\eta_{j1,j2}|<2.1$, and
$p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading ($j_{1}$) and subleading
($j_{2}$) anti-kT jets of jet radius $R=0.4$ are applied. The jet charge
values correspond to an average over the jet $p_{T_{J}}$ bins: [50 GeV,100
GeV], [100 GeV, 200 GeV], [200 GeV, 300 GeV], [300 GeV, 400 GeV], [400 GeV,
500 GeV], [500 GeV, 600 GeV], [600 GeV, 800 GeV], [800 GeV, 1000 GeV], [1000
GeV, 1200 GeV], and [1200 GeV, 1500 GeV]

This dynamic definition is motivated by the observed dependence of the shape
of the standard jet charge distribution on the value of the constant parameter
$\kappa$, as seen in Fig. 1. The standard jet charge distribution is
characterized by a single peak structure that gets narrower for increasing
values of the $\kappa$-parameter. If the parameter $\kappa$ is generalized to
a function of some dynamical property of individual hadrons in the jet, such
as their momentum fraction $z_{h}$, then differences between quark and gluon
jets in the distribution of this particle property will lead to differences in
the average value $\langle\kappa\rangle_{\rm dyn}^{i}$ ,

$\displaystyle\langle\kappa\rangle_{\rm
dyn}^{i}=\int_{0}^{1}dz\>\kappa(z)\sum_{h\in{i\text{-}\rm
jet}}\frac{1}{\sigma_{i\text{-}{\rm jet}}}\frac{d\sigma_{h\in{i-\rm
jet}}}{dz}\,,$ (10)

for a jet initiated by a quark ($i=q$) or a gluon ($i=g$). Thus, such a
dynamic parameter $\kappa(z_{h})$ could give rise to enhanced differences in
the jet charge distributions of quark and gluon jets. For example, gluon jets
are typically characterized by a higher multiplicity of hadrons compared to
quark jets due to the larger color factor of the gluon. As a result, for a jet
of given energy the average value of the momentum fraction $\langle
z_{h}\rangle$ will be larger for quark jets compared to gluon jets.
Correspondingly, the average value of the dynamic parameter
$\langle\kappa\rangle_{\rm dyn}^{i}$ will be different for quark and gluon
jets, leading to enhanced differences in the dynamic jet charge distributions.

The theoretical prediction for the average dynamic jet charge $\langle
Q^{i}_{\rm dyn}\rangle$ is given by replacing the constant parameter $\kappa$
in Eq. (5) by the dynamic function $\kappa(z)$

$\displaystyle\langle Q^{i}_{\rm
dyn}\rangle=\int_{0}^{1}dz\>z^{\kappa(z)}\sum_{h\in{i\text{-}\rm
jet}}Q_{h}\sum_{j}$
$\displaystyle\int_{z}^{1}\frac{dz^{\prime}}{z^{\prime}}{\cal
J}_{ij}(p_{T_{J}},R,z^{\prime},\mu)$ $\displaystyle\times
D_{j}^{h}\left(\frac{z}{z^{\prime}},\mu\right).$ (11)

Once again, using the LO result ${\cal
J}_{ij}^{(0)}(p_{T_{J}},R,z,\mu)=\delta_{ij}\delta(1-z)$, the corresponding LO
expression for the dynamic jet charge is given by

$\displaystyle\langle Q^{i(0)}_{\rm dyn}\rangle$
$\displaystyle=\int_{0}^{1}dz\>z^{\kappa(z)}\sum_{h\in{i\text{-}\rm
jet}}Q_{h}\>D_{i}^{h}\left(z,\mu\right)+{\cal O}(\alpha_{s}).$ (12)

Note that due to the fact that $\kappa(z)$ is not a constant, the LO dynamic
jet charge does not depend on simple Mellin moments of the fragmentation
function, as seen in Eqs. (6) and (7) for the standard jet charge. However,
the standard DGLAP evolution of the fragmentation function in Eq. (12) can
still be done directly in $z$-space.

The properties of the dynamic jet charge can be explored for different
functional forms of $\kappa(z_{h})$, corresponding to different definitions.
For simplicity, in this work we restrict our analysis to the simple functional
form $\kappa(z_{h})=\kappa(z_{h},\xi_{\rm cut},k_{<},k_{>})$

$\kappa(z_{h})=\begin{cases}k_{<},&z_{h}<\xi_{\rm cut}\\\
k_{>},&z_{h}\geq\xi_{\rm cut}\\\ \end{cases}$ (13)

where $\xi_{\rm cut},~{}k_{<}$, and $k_{>}$ are three constant parameters that
are part of the definition of the dynamic jet charge. We choose the default
values to be $\xi_{\rm cut}=0.3,\>k_{<}=1.0,\>k_{>}=0.3$, except when we
explicitly vary these parameters to study their impact on the dynamic jet
charge distributions. Using the functional form in Eq. (13), the dynamic jet
charge definition in Eq. (9) becomes

$\displaystyle Q^{i}_{\rm dyn}$ $\displaystyle=\sum_{h\in{i\text{-}\rm
jet}}\Theta(\xi_{\rm cut}-z_{h})\>z_{h}^{k_{<}}\>Q_{h}$
$\displaystyle+\sum_{h\in{i\text{-}\rm jet}}\Theta(z_{h}-\xi_{\rm
cut})\>z_{h}^{k_{>}}\>Q_{h},$ (14)

in terms of the $\xi_{\rm cut},k_{<},$ and $k_{>}$ parameters. Note that for
$\xi_{\rm cut}=0$ or $\xi_{\rm cut}=1$, the dynamic jet charge reduces to the
standard jet charge with $\kappa=k_{>}$ or $\kappa=k_{<}$, respectively.

Comparing Eqs. (II) and (1), we see that this definition is similar to the
standard jet charge but with the modification that the low momentum hadrons
($z_{h}<\xi_{\rm cut}$) contribute with $\kappa=k_{<}$ and the high momentum
hadrons ($z_{h}>\xi_{\rm cut}$) contribute with $\kappa=k_{>}$. For the
default parameter choices $\xi_{\rm cut}=0.3,k_{<}=1.0,$ and $k_{>}=0.3$, the
contribution of the low momentum hadrons relative to that of the high momentum
hadrons is much more suppressed compared to that in the case of the standard
jet charge. This leads to qualitatively distinct features for the dynamic jet
charge. In Fig. 3, we show Pythia8 simulation results for the more forward
(top panel) and more central (bottom panel) leading jets in $pp$-collisions at
$\sqrt{s}=13$ TeV. We see that the distribution with default parameter choices
(blue) has a central peak and two smaller speaks on either side. The non-
central peaks get smaller for increasing values of $k_{>}$.

The behavior of the peak structure in the dynamic jet charge distribution can
be summarized as follows. The non-central peaks become more prominent when the
high momentum hadrons ($z_{h}>\xi_{\rm cut}$) are given a higher weight
(decreasing $k_{>}$) relative to the low momentum hadrons ($z_{h}<\xi_{\rm
cut}$). The low momentum hadrons tend to smear out the non-central peaks as
seen in the distributions with the larger values of $k_{>}=0.5$ (red) and
$k_{>}=0.7$ (green), corresponding to giving their contribution a larger
weight.

The theoretical prediction for the LO average dynamic jet charge in Eq. (12),
for the functional form in Eq. (13), is given by

$\displaystyle\langle Q^{i(0)}_{\rm dyn}\rangle$
$\displaystyle=\int_{0}^{\xi_{\rm cut}}dz\>z^{k_{>}}\sum_{h\in{i\text{-}\rm
jet}}Q_{h}\>D_{i}^{h}\left(z,\mu\right)$ $\displaystyle+\int^{1}_{\xi_{\rm
cut}}dz\>z^{k_{<}}\sum_{h\in{i\text{-}\rm
jet}}Q_{h}\>D_{i}^{h}\left(z,\mu\right)+{\cal O}(\alpha_{S}).$ (15)

Analogous to Fig. 2, we show Pythia simulation results for the average dynamic
jet charge as a function of the jet $p_{T}$-bin in $pp$-collisions at
$\sqrt{s}=13$ TeV in Fig. 4.

##  III Quark-Gluon Discrimination

In this section, we explore the use of the dynamic jet charge observable to
discriminate between quark and gluon initiated jets. We consider dijet events
in $pp$-collisions at $\sqrt{s}=13$ TeV and heavy ion collisions (Pb-Pb) at
$\sqrt{s}=2.76$ TeV within Pythia. Note that Pythia uses the so-called
Angantyr model Bierlich:2016smv ; Bierlich:2018xfw  for heavy ion collisions,
whose main idea is to stack parton level events, corresponding to individual
nucleon-nucleon sub-collisions, on top of each other and hadronize them
together. The model is able to give a good description of general final-state
properties such as multiplicity and transverse momentum distributions in heavy
ion collisions. However, it cannot describe any observables sensitive to
collective behaviour, because the model generates events without assuming
production of the thermalized medium after the collision, and thus the result
is a quark-gluon-plasma-free simulation of heavy ion collisions. Nevertheless,
such Angantyr/Pythia simulations account for the increased underlying event
activity in heavy ion collisions, and are still very useful to test the
robustness of the dynamic jet charge as a first step.

The jets are found using the anti-kT Cacciari_2008  jet algorithm with a jet
radius, $R=0.4$. The two leading jets are subject to pseudorapidity cuts
$|\eta_{j_{1},j_{2}}|<2.1$ and $|\eta_{j_{1},j_{2}}|<0.9$ for $pp$-collisions
and PbPb-collisions, respectively. They are also restricted to the
$p_{T}$-bins $200\>{\rm GeV}<p_{T_{j_{1},j_{2}}}<300\>{\rm GeV}$ and
$80\>\>{\rm GeV}<p_{T_{j_{1},j_{2}}}<150\>{\rm GeV}$ for $pp$-collisions and
PbPb-collisions, respectively. Finally, the two leading jets are subject to
the additional constraint $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$, following the
ATLAS analysis Aad:2015cua . All results are shown for the more central jet.
Similar results are found for the more forward jet charge distributions and do
not affect the discussion here.

Figure 5: The standard (top) and dynamic (middle) jet charge distributions in
$pp$-collisions at $\sqrt{s}=$13 TeV, for quark (red) and gluon (blue)
initiated jets with selection cuts $p_{T_{J}}=[200,300]$ GeV,
$|\eta_{J}|<2.1$. A comparison of the corresponding quark-gluon discrimination
ROC curves for the standard (purple) and dynamic (green) jet charge
distributions is given in the bottom panel.

Figure 6: The standard (top) and dynamic (middle) jet charge distributions in
Heavy Ion (Pb-Pb) collisions at $\sqrt{s}=$2.760 TeV, for quark (red) and
gluon (blue) initiated jets with selection cuts $p_{T_{J}}=[80,150]$ GeV,
$|\eta_{J}|<0.9$. A comparison of the corresponding quark-gluon discrimination
ROC curves for the standard (purple) and dynamic (green) jet charge
distributions is given in the bottom panel.

We compare jet charge distributions between quark-initiated and gluon-
initiated jets, obtained through simulations of dijet events generated by the
Pythia8 partonic channel settings ‘HardQCD:qq2qq=on’ and ‘HardQCD:gg2gg=on’,
respectively. In Fig. 5, we show simulation results of the normalized jet
charge distributions for the standard (top panel) and dynamic (middle panel)
jet charge definitions for quark (red) and gluon (blue) initiated jets in
$pp$-collisions. We see that while the standard jet charge distributions are
very similar for quark and gluon jets, there are distinct qualitative
differences between their dynamic jet charge distributions. Compared to the
gluon jets, the quark jets have a relatively smaller peak height in the
central region near zero jet charge and have relatively larger peak heights in
the region of non-zero jet charge. This is found to be a consistent feature
when evaluated over a wide range of kinematics and phase space selection cuts.

These qualitative differences between quark and gluon jets can be quantified
by the Receiver Operating Characteristic (ROC) curve as shown in the bottom
panel of Fig. 5. The ROC curves show the background (gluon jet) rejection as a
function of the signal (quark jet) efficiency, based on a cut on the absolute
value of the jet charge $|Q_{J}|$. The background rejection is given by the
fraction of gluon jets rejected and the signal efficiency corresponds to the
fraction of quark jets kept, as a function of the cut on $|Q_{J}|$. The
reference diagonal line (dashed black) corresponds to an ROC curve that shows
no discrimination between the signal and background. ROC curves that lie above
the diagonal correspond to non-zero discrimination power of signal over
background. ROC curves that lie below the diagonal dashed line also indicate
non-zero discrimination power and can be made manifest by interchanging what
we define as the signal and background. We see that the ROC curve for the
dynamic jet charge (green) shows significantly improved discrimination between
quark and gluon jets compared to the ROC curve (purple) for the standard jet
charge.

In Fig. 6, we show the same plots as in Fig. 5 but for the heavy ion PbPb-
collisions. We see that the standard jet charge distribution (top panel) is
much broader compared to that in $pp$-collisions. On the other hand, the
dynamic jet charge distribution (middle panel) is quite similar in PbPb-
collisions and $pp$-collisions. The same distinguishing qualitative features
between quark and gluon jets appear even for heavy ion collisions.
Correspondingly, the ROC curve (bottom panel) for the dynamic jet charge in
heavy ion collisions looks similar to that in $pp$-collisions and gives better
discrimination compared to the standard jet charge. Thus, the quark-gluon
discrimination power of the dynamic jet charge is largely unaffected by the
significantly greater underlying event activity in heavy ion collisions.

Figure 7: The local ROC curves for the standard (purple) and dynamic (green)
jet charge distributions described in Fig. 5 for for $pp$-collisions at
$\sqrt{s}=$13 TeV, with selection cuts $p_{T_{J}}=[200,300]$ GeV,
$|\eta_{J}|<2.1$. The top and bottom panels correspond to the local ROC curves
for the jet charge bins $|Q_{J}|<0.5$ and $|Q_{J}|>0.5$, respectively.

We note that in PbPb-collisions the ROC curve for the standard jet charge
(purple) in Fig. 6 is below the dashed diagonal line, indicating slightly
better discrimination (if one interchanges what is defined as signal and
background) than in the case of $pp$-collisions. This can be understood as the
result of the standard jet charge distribution for gluon jets being slightly
broader than that for quark jets in PbPb-collisions, as seen in the top panel
of Fig. 6.

While the ROC curve provides one quantitative measure of the discrimination
power of the dynamic jet charge, it may not fully capture it due to the
multiple peak structure of the dynamic jet charge distributions. The ROC curve
best describes the discrimination power when comparing two distributions with
a single peak structure and quantifies how well-separated the peaks of the two
distributions are. For distributions with multiple peaks, other ways of
quantifying the discrimination power can be useful.

Due to the multiple peak structure of the dynamic jet charge distribution,
additional quark-gluon discrimination is possible through an analysis of the
jet data binned according to the dynamic jet charge value. In particular, the
multiple peak structure naturally suggests binning the jet charge data such
that the bins are centered around the peaks with bin-widths corresponding to
the width of the peaks. A discrimination analysis can then be separately
performed in each jet charge bin. For example, in Fig. 5, the dynamic jet
charge distribution (middle panel) suggests the jet charge data be divided
into two bins: $|Q_{J}|\leq 0.5$ and $|Q_{J}|>0.5$. In Fig. 7, we show the
local ROC curves for the $|Q_{J}|\leq 0.5$ bin (top panel) and the
$|Q_{J}|>0.5$ bin (bottom panel) for the standard (purple) and dynamic (green)
jet charge distributions. We define the local ROC curve in each jet charge bin
using only the events in that bin. This can be distinguished from the global
ROC curve which is defined using the data in all bins, as was done Figs. 5 and
6. The local ROC curve affords the standard interpretation for the dynamic jet
charge since each jet charge bin is selected to have at most a single peak. We
see from Fig. 7 that some additional discrimination is possible using the
dynamic jet charge in a binned analysis. While the ROC curve for the dynamic
jet charge shows improved discrimination between quark and gluon jets, the
improvement is only marginal. This can be understood as a result of the fact
that in each bin, while there are significant differences between the peak
heights for the quark and gluon dynamic jet charge distributions, the location
of the peaks are about the same and ROC curves tend to probe differences in
the peak positions. However, we will show later on that such a binned analysis
with local ROC curves can provide significant discrimination between $u$-quark
and $d$-quark jets.

Noticing that the dynamic jet charge distribution characterizes the gluon
(quark) jets with a higher (lower) central peak and lower (higher) non-central
peaks, one can quantify this difference through fractional counts in each bin.
We introduce the definitions

$\displaystyle\epsilon_{<}=\frac{N(\>|Q_{J}|\leq 0.5\>)}{N_{\rm
total}},\qquad\epsilon_{>}=\frac{N(\>|Q_{J}|>0.5\>)}{N_{\rm total}},$ (16)

where $\epsilon_{<}$ and $\epsilon_{>}$ give the fraction of events in the
$|Q_{J}|\leq 0.5$ and $|Q_{J}|>0.5$ jet charge bins, respectively. In Table 1,
we show the values of $\epsilon_{<}$ and $\epsilon_{>}$ for quark and gluon
jets for the standard and dynamic jet charge distributions shown in Figs. 5
and 6 for $pp$-collisions and heavy ion collisions, respectively.

Jet Charge |  |
---|---|---
($pp$-collisions) | $\>\>\>\>\>\epsilon_{<}\>\>\>\>\>$ | $\>\>\>\>\>\epsilon_{>}\>\>\>\>\>$
Standard: quark | 0.62 | 0.38
gluon | 0.63 | 0.37
Dynamic: quark | 0.75 | 0.25
gluon | 0.94 | 0.06

Jet Charge |  |
---|---|---
(PbPb-collisions) | $\>\>\>\>\>\epsilon_{<}\>\>\>\>\>$ | $\>\>\>\>\>\epsilon_{>}\>\>\>\>\>$
Standard: quark | 0.37 | 0.63
gluon | 0.32 | 0.68
Dynamic: quark | 0.85 | 0.15
gluon | 0.98 | 0.02

Table 1: Fraction of events for quark and gluon jets in the $|Q_{J}|\leq 0.5$
and $|Q_{J}|>0.5$ jet charge bins. The definitions of $\epsilon_{<}$ and
$\epsilon_{>}$ are given in Eq. (16). The numbers correspond to the
distributions in Fig. 5 and 6 for $pp$-collisions and PbPb-collisions,
respectively.

We see that for the dynamic jet charge in $pp$-collisions only $\sim 6\%$ of
gluon jets are found in the $|Q_{J}|>0.5$ jet charge bin compared to $\sim
25\%$ for quark jets. On the other hand, $\sim 94\%$ of gluon jets and $\sim
75\%$ quark jets are found in the $|Q_{J}|\leq 0.5$ bin. This can be
contrasted with the standard jet charge where almost the same fraction of
quark and gluon jets are present in each jet charge bin. Similarly, as seen in
Table 1, the same overall behavior is observed for PbPb-collisions where the
fraction of quark and gluon jets is about the same in each bin for the
standard jet charge but different for the dynamic jet charge. Thus, using the
differences in the expected fraction of quark and gluon jets in each jet
charge bin, the dynamic jet charge can allow for improved quark and gluon
discrimination simply by sorting the jet data into jet charge bins.

##  IV Quark Flavor Discrimination

In this section, we explore the use of the dynamic jet charge to discriminate
between $u$\- and $d$-quark jets in Pythia8 simulations of $pp$-collisions at
$\sqrt{s}=13$ TeV and heavy ion collisions (Pb-Pb) at $\sqrt{s}=2.76$ TeV. The
$u$-quark and $d$-quark jet samples were generated using the partonic channels
$dg\to W^{-}u$ and $ug\to W^{+}d$, respectively. Selection cuts of
$|\eta_{J}|<2.1$ and $|\eta_{J}|<0.9$, and $200\>{\rm GeV}<p_{T_{J}}<300\>{\rm
GeV}$ and $80\>{\rm GeV}<p_{T_{J}}<150\>{\rm GeV}$ were used for
$pp$-collisions and heavy ion collisions. The leading anti-kT jet radius was
set to $R=0.4$.

In Fig. 8, we show the standard (top panel) and dynamic (middle panel) jet
charge distributions for $u$-quark (red) and $d$-quark (blue) jets. Once again
we see that the dynamic jet charge distributions have qualitatively distinct
features compared to the corresponding standard jet charge distributions.

Figure 8: The standard (top) and dynamic (middle) jet charge distributions in
$pp$-collisions at 13 TeV, for $u$-quark (red) and $d$-quark (blue) initiated
jets with selection cuts $p_{T_{J}}=[200,300]$ GeV, $|\eta_{J}|<2.1$. The
channels $dg\to W^{-}u$ and $ug\to W^{+}d$ partonic channels where used to
select the $u$-quark and $d$-quark initiated jets respectively. A comparison
of the corresponding “$u$” vs. “$d$” discrimination ROC curves for the
standard (purple) and dynamic (green) jet charge distributions is given in the
bottom panel.

Figure 9: The standard (top) and dynamic (middle) jet charge distributions in
PbPb-collisions at 2.760 TeV, for $u$-quark (red) and $d$-quark (blue)
initiated jets with selection cuts $p_{T_{J}}=[80,150]$ GeV, $|\eta_{J}|<0.9$.
The channels $dg\to W^{-}u$ and $ug\to W^{+}d$ partonic channels where used to
select the $u$-quark and $d$-quark initiated jets respectively. A comparison
of the corresponding “$u$” vs. “$d$” discrimination ROC curves for the
standard (purple) and dynamic (green) jet charge distributions is given in the
bottom panel.

The standard jet charge distribution for the $u$-quark ($d$-quark) jet has a
single peak shifted to the right (left) of center, corresponding to the
positive (negative) charge of the jet-initiating quark. The dynamic jet charge
distributions are characterized by a multiple peak structure. A shift of the
central peak to the right (left) is also seen for the $u$-quark ($d$-quark)
jet but is less pronounced. However, the secondary peak on the right (left)
for the $u$-quark ($d$-quark) jet is much higher than that for the $d$-quark
($u$-quark) jet. In Fig. 8, we also show the global ROC curves (bottom panel)
for the standard and dynamic jet charge distributions. We see that in this
case the global ROC curve for the standard jet charge shows better, although
comparable, discrimination power compared to the dynamic jet charge.

Figure 10: The local ROC curves for the standard (purple) and dynamic (green)
jet charge distributions described in Fig. 8 for for $pp$-collisions at 13
TeV, with selection cuts $p_{T_{J}}=[200,300]$ GeV, $|\eta_{J}|<2.1$. The top,
middle, and bottom panels correspond to the local ROC curves for the jet
charge bins $Q_{J}<-0.5$, $|Q_{J}|<0.5$, and $Q_{J}>0.5$, respectively.

Figure 11: The local ROC curves for the standard (purple) and dynamic (green)
jet charge distributions described in Fig. 9 for PbPb-collisions at 2.760 TeV,
with selection cuts $p_{T_{J}}=[80,150]$ GeV, $|\eta_{J}|<2.1$. The top,
middle, and bottom panels correspond to the local ROC curves for the jet
charge bins $Q_{J}<-0.5$, $|Q_{J}|<0.5$, and $Q_{J}>0.5$, respectively.

In Fig. 9, we show the same plots as in Fig. 8 but for heavy ion PbPb-
collisions. We see that the standard jet charge distribution (top panel) is
much broader compared to that in $pp$-collisions. Note that the jet charge
distribution spans over $-4<Q_{J}<4$ in the plot for PbPb-collisions, while
only over $-2<Q_{J}<2$ for $pp$-collisions. On the other hand, the dynamic jet
charge distribution (middle panel) is quite similar in heavy ion and
$pp$-collisions. Correspondingly, the global ROC curve (bottom panel) for the
dynamic jet charge (green) in heavy ion collisions looks similar to that in
$pp$-collisions. Thus, the $u$\- vs. $d$-quark jet discrimination power of the
dynamic jet charge is largely unaffected by the significantly greater
underlying event activity in heavy ion collisions. By contrast, there is
significant degradation in the discrimination power for the standard jet
charge in PbPb-collisions compared to $pp$-collisions, as seen by comparing
the global ROC curves in Figs. 8 and 9.

In fact, for PbPb collisions, the dynamic jet charge shows better, but
comparable, discrimination power compared to the standard jet charge, as seen
in the bottom panel of Fig. 9.

Furthermore, once again, due to the multiple peak structure of the dynamic jet
charge distribution, a binned analysis with local ROC curves can be performed.
We revisit the dynamic jet charge distribution (middle panel) in Fig. 8 for
$pp$-collisions. The peak structure suggests that the jet data be divided into
three bins: $(i)\>Q_{J}<-0.5,(ii)\>-0.5<Q_{J}<0.5,$ and $(iii)\>Q_{J}>0.5$. We
show the local ROC curves for the standard and dynamic jet charge
distributions in each of these three jet charge bins in Figs. 10 and 11 for
$pp$\- and PbPb-collisions, respectively. We see that the local ROC curves for
the dynamic jet charge shows better discrimination compared to standard jet
charge in all bins for PbPb-collisions. For $pp$-collisions, the dynamic jet
charge gives better discrimination in the leftmost ($Q_{J}<-0.5$) and
rightmost ($Q_{J}>0.5$) bins and about the same discrimination in the central
bin ($-0.5<Q_{J}<0.5$), compared to the standard jet charge.

Jet Charge |  |  |
---|---|---|---
($pp$-collisions) | $\>\>\>\>\>\epsilon_{L}\>\>\>\>\>$ | $\>\>\>\>\>\epsilon_{C}\>\>\>\>\>$ | $\>\>\>\>\>\epsilon_{R}\>\>\>\>\>$
Standard: $u$-quark | 0.05 | 0.59 | 0.36
$d$-quark | 0.28 | 0.65 | 0.08
Dynamic: $u$-quark | 0.05 | 0.73 | 0.21
$d$-quark | 0.16 | 0.77 | 0.08

Jet Charge |  |  |
---|---|---|---
(PbPb-collisions) | $\>\>\>\>\>\epsilon_{L}\>\>\>\>\>$ | $\>\>\>\>\>\epsilon_{C}\>\>\>\>\>$ | $\>\>\>\>\>\epsilon_{R}\>\>\>\>\>$
Standard: $u$-quark | 0.19 | 0.40 | 0.41
$d$-quark | 0.38 | 0.43 | 0.21
Dynamic: $u$-quark | 0.04 | 0.79 | 0.17
$d$-quark | 0.12 | 0.82 | 0.06

Table 2: Fraction of events for $u$-quark and $d$-quark jets in the
$Q_{J}<-0.5,-0.5<Q_{J}<0.5,$ and $Q_{J}>0.5$ jet charge bins. The
$\epsilon_{L},\epsilon_{C},$ and $\epsilon_{R}$ fractions are defined in Eq.
(IV). The numbers correspond to the jet charge distributions in Fig. 8 and 9
for $pp$-collisions and PbPb-collisions, respectively.

As done for the quark-gluon discrimination analysis, one can also quantify the
$u$-quark vs. $d$-quark jet discrimination using the fraction of events in
each jet charge bin. Based on the dynamic jet charge distributions in Figs. 8
and 9, we define the fraction of the total number of events in three jet
charge bins as

$\displaystyle\epsilon_{L}$ $\displaystyle=\frac{N(\>Q_{J}<-0.5\>)}{N_{\rm
total}},$ $\displaystyle\epsilon_{C}$ $\displaystyle=\frac{N(-0.5\leq
Q_{J}\leq 0.5\>)}{N_{\rm total}},$ $\displaystyle\epsilon_{R}$
$\displaystyle=\frac{N(\>Q_{J}>0.5\>)}{N_{\rm total}},$ (17)

where $\epsilon_{L},\epsilon_{C},$ and $\epsilon_{R}$ give the fraction of the
total number of events in the jet charge bins $Q_{J}<-0.5$, $-0.5<Q_{J}<0.5$,
and $Q_{J}>0.5$, respectively.

In Table 2, we show the values of $\epsilon_{L},\epsilon_{C}$, and
$\epsilon_{R}$ for $u$-quark and $d$-quark jets for the standard and dynamic
jet charge distributions shown in Figs. 8 and 9 for $pp$-collisions and heavy
ion collisions, respectively. We see that for both $pp$-collisions and PbPb-
collisions, there is a substantial difference in the event fractions of
$u$-quark and $d$-quark jets in the left-most ($Q_{J}<-0.5$) and right-most
($Q_{J}>0.5$) jet charge bins. On the other hand, the event fractions are
about the same for both $u$-quark and $d$-quark jets in the central jet charge
bin ($-0.5<Q_{J}<0.5$). Thus, just as in the case of quark-gluon
discrimination, sorting the data into jet charge bins can be used to
discriminate between $u$-quark and $d$-quark jets in the left-most and right-
most jet charge bins. We note that in this method, the standard jet charge
seems to give better discrimination than the dynamic jet charge for
$pp$-collisions in the leftmost and rightmost bins. On the other hand, for
PbPb-collisions the dynamic jet charge seems to give better discrimination in
the leftmost and rightmost bins.

##  V Characteristics of the Dynamic Jet Charge

Figure 12: A comparison of the dynamic jet charge distributions for quark
(red) and gluon (blue) jets in $pp\to j_{1}j_{2}X$ at 13 TeV. Selection cuts
of $|\eta_{j1,j2}|<2.1$ and $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading
($j_{1}$) and subleading ($j_{2}$) anti-kT jets of jet radius $R=0.4$ are
applied. The top and bottom panels correspond to jets in the $p_{T_{J}}$-bins
[100 GeV, 200 GeV] and [1200 GeV, 1500 GeV], respectively.

Figure 13: A comparison of the dynamic jet charge distributions for different
choices of the dynamic jet charge parameters, $\xi_{\rm cut},k_{<}$, and
$k_{>}$, for quark (red) and gluon (blue) jets in $pp\to j_{1}j_{2}X$ at 13
TeV in the jet $p_{T}$-bin: [200,300] GeV. Selection cuts of
$|\eta_{j1,j2}|<2.1$ and $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading
($j_{1}$) and subleading ($j_{2}$) anti-kT jets of jet radius $R=0.4$ are
applied.

In this section, we explore the typical behavior of the dynamic jet charge
distribution for different choices of kinematic settings and jet charge
parameters. We also present simulation studies of the impact of underlying
event and jet grooming, on both the standard and dynamic jet charge
distributions.

In Fig. 12, we show a comparison of the dynamic jet charge distribution for
quark and gluon jets in the two different $p_{T_{J}}$-bins. The top and bottom
panels correspond to $p_{T_{J}}=$[100 GeV, 200 GeV] and $p_{T_{J}}=$[1200 GeV,
1500 GeV], respectively. We see that the distinctive multiple peak structure
remains across a wide range of $p_{T_{J}}$. As seen in the bottom panel, at
large $p_{T_{J}}$ an asymmetry between the peaks at negative and positive jet
charge develops for the quark jets. This asymmetry can be understood as a
consequence of the dominance of valence $u$-quark PDFs at large Bjorken-$x$,
corresponding to large $p_{T_{J}}$, so that the quark jet sample has a larger
fraction of $u$-quark jets compared to $d$-quark jets. As seen in Fig. 8, the
dynamic jet charge distribution for $u$-quark and $d$-quark jets is tilted
more towards positive and negative jet charge, respectively. The combined
effect of the dominance of the $u$-quark PDFs at large $p_{T_{J}}$, and the
bias of $u$-quark jets towards positive jet charge is reflected in the
asymmetry of the overall quark jet distribution in the bottom panel of Fig.
12.

Figure 14: The standard (top) and dynamic (bottom) jet charge distributions
with MPI switch turned on (solid) and off (dashed) for quark (red) and gluon
(blue) jets in $pp\to j_{1}j_{2}X$ at $\sqrt{s}=13$ TeV and in the jet
$p_{T_{J}}$-bin: [50,100] GeV. Selection cuts of $|\eta_{j1,j2}|<2.1$ and
$p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading ($j_{1}$) and subleading
($j_{2}$) anti-kT jets of jet radius $R=0.4$ are applied.

In Fig. 13, we show the behavior of the dynamic jet charge distribution for
different parameter choices. In particular, we set $k_{<}=1.0,k_{>}=0.3$ and
vary $\xi_{\rm cut}$ over the three values $\xi_{\rm cut}=0.2$ (top panel),
$\xi_{\rm cut}=0.4$ (middle panel), and $\xi_{\rm cut}=0.7$ (bottom panel).
The plot for the default choice of $\xi_{\rm cut}=0.3$ is already shown in
Fig. 5 (middle panel). We see that for increasing values of $\xi_{\rm cut}$,
the secondary non-central peaks in the distribution become less pronounced.
For the largest value shown, $\xi_{\rm cut}=0.7$, the non-central peaks
completely disappear. This can be understood as a consequence of the fact that
there are typically very few hadrons in the jet with $z_{h}>\xi_{\rm
cut}=0.7$. As a result, as seen from Eq. (II), the jet charge distribution
effectively reduces to the standard jet charge distribution with
$\kappa=k_{<}=1.0$. Similarly, in the limit of $\xi_{\rm cut}\to 0$, the
dynamic jet charge distribution will approach the standard jet charge
distribution with $\kappa=k_{>}$.

Figure 15: The standard (top) and dynamic (bottom) jet charge distributions
for ungroomed (solid) and groomed (dashed) quark and gluon jets in $pp\to
j_{1}j_{2}X$ at $\sqrt{s}=13$ TeV and in the jet $p_{T_{J}}$-bin: [50,100]
GeV. Selection cuts of $|\eta_{j1,j2}|<2.1$ and
$p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading ($j_{1}$) and subleading
($j_{2}$) anti-kT jets of jet radius $R=0.4$ are applied.

Figure 16: The standard (top) and dynamic (bottom) jet charge distributions
for ungroomed (solid) and groomed (dashed) quark and gluon jets in PbPb$\to
j_{1}j_{2}X$ at $\sqrt{s}=2.760$ TeV and in the jet $p_{T_{J}}$-bin: [80,150]
GeV. Selection cuts of $|\eta_{j1,j2}|<0.9$ and
$p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading ($j_{1}$) and subleading
($j_{2}$) anti-kT jets of jet radius $R=0.4$ are applied.

Figure 17: The standard (top) and dynamic (bottom) jet charge distributions
for $u$-quark (red) and $d$-quark (blue) jets, with (dashed) and without
(solid) grooming, in $pp$-collisions at $\sqrt{s}=13$ TeV with $R=0.4$,
$p_{T_{J}}=[50,100]$ GeV, and $|\eta_{J}|<2.1$. The partonic channels $dg\to
W^{-}u$ and $ug\to W^{+}d$ generated the $u$-quark and $d$-quark jets,
respectively.

Figure 18: The standard (top) and dynamic (bottom) jet charge distributions
for $u$-quark (red) and $d$-quark (blue) jets, with (dashed) and without
(solid) grooming, in PbPb-collisions at $\sqrt{s}=2.760$ TeV with $R=0.4$,
$p_{T_{J}}=[80,150]$ GeV, and $|\eta_{J}|<0.9$. The partonic channels $dg\to
W^{-}u$ and $ug\to W^{+}d$ generated the $u$-quark and $d$-quark jets.

As seen in Fig. 14, both the standard and dynamic jet charge distributions for
quark and gluon initiated jets in $pp$-collisions are relatively unaffected by
underlying event. This can be understood as the result of the fact that, as
seen in Eqs. (1) and (9) the contributions of the individual jet constituents
to the total jet charge are weighted by their jet transverse momentum or
energy fractions. Thus, soft contamination effects on the jet charge are
suppressed. As seen in Fig. 14, the soft contamination effects appear to be
slightly more suppressed in the dynamic jet charge compared to the standard
jet charge. This is simply understood as a result of the fact that for the
choice of parameters $k_{<}=1.0$ and $k_{>}=0.3$, compared to the standard jet
charge, the dynamic jet charge gives a much lower weighting to the low energy
($z_{h}<\xi_{\rm cut}$) jet constituents compared to those with higher energy
($z_{h}>\xi_{\rm cut}$). Thus, the dynamic jet charge allows the flexibility
to choose parameters that make the jet charge even more robust against soft
contamination.

As a further demonstration of the relative insensitivity of the jet charge to
soft contamination, in Figs. 15 and 16, we show the effect of jet grooming on
the standard and dynamic jet charge distributions in $pp$-collisions and PbPb-
collisions, respectively. We choose typical soft drop grooming parameters of
$z_{\rm cut}=0.1$ and $\beta=2$ Larkoski:2014wba . Once again, we see that the
standard and dynamic jet charge distributions for quark and gluon initiated
jets are very similar for both groomed and ungroomed jets. As seen in Figs. 17
and 18, the same is true of $u$-quark and $d$-quark initiated jets. Thus,
these simulation results indicate that the soft drop grooming of jets does not
have much impact on the jet discrimination power of the standard and dynamic
jet charge observables.

Figure 19: Top panel: The contribution to the dynamic jet charge from the soft
(dashed) and hard (solid) particles in quark (red) and (gluon) jets for the
choice of parameters: $\xi_{\rm cut}=0.3,k_{<}=1.0,k_{>}=0.3$. Middle Panel:
The contribution to the dynamic jet charge from the soft (dashed) and hard
(solid) particles in quark (red) and (gluon) jets for the choice of
parameters: $\xi_{\rm cut}=0.3,k_{<}=0.3,k_{>}=1.0$. Bottom Panel: The total
dynamic jet charge distribution for $\xi_{\rm cut}=0.3,k_{<}=0.3,k_{>}=1.0$.
All the distributions above were generated for the more central jet in dijet
production of $pp$-collisions at $\sqrt{s}=$13 TeV, with selection cuts
$p_{T_{J}}=[200,300]$ GeV, $|\eta_{J}|<2.1$, and
$p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading ($j_{1}$) and subleading
($j_{2}$) anti-kT jets of jet radius $R=0.4$.

Finally, we discuss in more detail the properties of the dynamic jet charge
distribution that underly its discrimination power and characteristic shape
with multiple peaks. The main idea is based on the observation (see Fig. 1)
that the standard jet charge distribution gets broader (narrower) for smaller
(larger) values of $\kappa$. This can be understood as the result of a large
$\kappa$-value washing out the charge information of the individual particles,
in the standard jet charge definition of Eq. (1), giving values closer to
zero. This feature is used in the dynamic jet charge to cleanly separate out
the contributions of the hard ($z_{h}>\xi_{\rm cut}$) particles in the jet
from the soft ($z_{h}<\xi_{\rm cut}$) particles. This is done by using
$\kappa=k_{>}$ and $\kappa=k_{<}$ for the hard and soft particles,
respectively, so that the dynamic jet charge can be defined as the sum of the
contributions from the soft and hard particles in the jet:

$\displaystyle Q_{\rm dyn}^{i}$ $\displaystyle=$ $\displaystyle Q_{\rm
dyn}^{i,\text{soft}}+Q_{\rm dyn}^{i,\text{hard}},$ (18)

where we have defined these contributions as:

$\displaystyle Q_{\rm dyn}^{i,\text{soft}}$ $\displaystyle=$
$\displaystyle\sum_{h\in i\text{-jet}}\Theta(\xi_{\rm
cut}-z_{h})\>z_{h}^{k_{<}}\>Q_{h},$ $\displaystyle Q_{\rm
dyn}^{i,\text{hard}}$ $\displaystyle=$ $\displaystyle\sum_{h\in
i\text{-jet}}\Theta(z_{h}-\xi_{\rm cut})\>z_{h}^{k_{>}}\>Q_{h}.$ (19)

In Fig. 19, we show separately the distributions of the $Q_{\rm
dyn}^{i,\text{soft}}$ and $Q_{\rm dyn}^{i,\text{hard}}$ contributions to the
total dynamic jet charge for the default choice, $\xi_{\rm
cut}=0.3,~{}k_{<}=1.0,~{}k_{>}=0.3$ (top panel) and for a different choice,
$\xi_{\rm cut}=0.3,~{}k_{<}=0.3,~{}k_{>}=1.0$ (middle panel). The $Q_{\rm
dyn}^{i,\text{soft}}$ distributions are normalized to unity. On the other
hand, the $Q_{\rm dyn}^{i,\text{hard}}$ distributions are weighted by the
relative fraction of events in which the jet contains at least one hard
particle ($z_{h}>\xi_{\rm cut}$). Thus, the normalization of the $Q_{\rm
dyn}^{i,\text{hard}}$ distribution relative to the unit normalization of
$Q_{\rm dyn}^{i,\text{soft}}$, corresponds to the fraction of events in which
the jet contains at least one hard particle ($z_{h}>\xi_{\rm cut}$).

The default choice, $\xi_{\rm cut}=0.3,k_{<}=1.0,k_{>}=0.3$ (top panel), which
gives a higher weight to the hard particles relative to the soft particles in
the jet, causes the soft particles to be distributed in a narrow central peak.
This corresponds to the narrow shape of the standard jet charge distribution
with $\kappa=k_{<}=1.0$. On the other hand, the hard particles are accumulated
near larger non-zero values of the jet charge, corresponding to the tails of
the broader standard jet charge distribution with $\kappa=k_{>}=0.3$. Note the
clean separation of the effects of the hard and soft particles in the jet
charge distribution. The choice of $k_{<}=1.0,k_{>}=0.3$, localizes the effect
of the soft particles to the central peak and the hard particles to the non-
central peaks, giving rise to the characteristic multiple peak structure of
the corresponding total dynamic jet charge, $Q_{\rm dyn}^{i}=Q_{\rm
dyn}^{i,\text{soft}}+Q_{\rm dyn}^{i,\text{hard}}$, as seen in Fig. 5 (middle
panel).

Since gluon jets have a higher multiplicity of particles than quark jets, it
contains a higher fraction of soft particles compared to quark jets, for a jet
of given energy. For this reason, the gluon (quark) jets are characterized by
a higher (lower) central peak and lower (higher) non-central peaks, as seen in
Fig. 5 (middle panel). These differences give rise to the enhanced
discrimination power of the dynamic jet charge between quark and gluon jets.
On the other hand, for $u$-quark vs. $d$-quark jet discrimination, the higher
weighting given to hard particles in the jet makes the dynamic jet charge more
resilient at discrimination in heavy ion collisions where significant soft
background activity can contaminate the jet.

The middle panel of Fig. 19 shows the separate distributions of the $Q_{\rm
dyn}^{i,\text{soft}}$ and $Q_{\rm dyn}^{i,\text{hard}}$ contributions to the
dynamic jet charge for the different choice of parameters, $\xi_{\rm
cut}=0.3,k_{<}=0.3,k_{>}=1.0$. For this choice, the contribution of the soft
particles to the jet charge is given a higher weighting compared to the hard
particles. Once again, $Q_{\rm dyn}^{i,\text{soft}}$ is characterized by a
single peak structure, but this time corresponds to the much broader
$\kappa=k_{<}=0.3$ standard jet charge distribution. The distribution of
$Q_{\rm dyn}^{i,\text{hard}}$ is still characterized by accumulations near
non-zero jet charge values but which are now closer to the $Q_{J}\sim 0$
region, corresponding to the tails of the now much narrower standard jet
charge distribution with $\kappa=k_{>}=1.0$. Thus, in this case, we no longer
have a clean separation between the contributions of the soft and hard
particles. The $Q_{\rm dyn}^{i,\text{soft}}$ is much broader and the $Q_{\rm
dyn}^{i,\text{hard}}$ distribution gives less pronounced non-central peaks
that also appear closer to the $Q_{J}\sim 0$ region. Thus, we no longer have a
clean separation in the dynamic jet charge of the effects of the hard and soft
particles, and correspondingly reduced discrimination power. This is seen in
Fig. 19 (bottom panel), where the corresponding total dynamic jet charge
distributions for the quark and gluon jets are nearly identical.

Thus, the default choices, $\xi_{\rm cut}=k_{<}=1.0$ and $k_{>}=0.3$, which
enhance and cleanly separate the effect of the hard particles over the soft
particles is better suited for jet discrimination. The most optimal choice
could be found through a more detailed scan of dynamic jet charge parameters.

##  VI Phenomenology

In this section, we consider some phenomenological applications of the dynamic
and standard jet charges.

###  VI.1 Flavor Separation of Nucleon TMDs using Dynamic Jet Charge

Recently, a theoretical framework was introduced Kang:2020fka  to use the
standard jet charge observable as a unique tool to probe the flavor structure
in the nucleon spin program at the future Electron-Ion Collider (EIC). Similar
directions are currently being explored Aschenauer:2015eha ; bnltalk
experimentally at the Relativistic Heavy Ion Collider (RHIC).

The unpolarized electron-proton scattering process

$\displaystyle e+p\to e+{\rm jet}+X,$ (20)

in the back-to-back limit where the electron-jet transverse momentum imbalance
$q_{T}$ is small, is sensitive to the unpolarized nucleon transverse momentum
dependent parton distribution functions (TMD PDFs) Liu:2018trl ;
Arratia:2020nxw . The TMD PDFs provide 3D imaging of the nucleon in momentum
space Accardi:2012qut . The polarized scattering counterpart,
$e+p(S_{\perp})\to e+{\rm jet}+X$, in the small $q_{T}$ limit, where
$S_{\perp}$ denotes the proton transverse spin vector, receives an additional
contribution from a term that is sensitive to the Sivers function
Sivers:1989cc , the polarized TMD PDF, that encodes additional quantum
correlations between the motion of partons and the spin of the proton. The
Sivers function can be directly accessed via the Sivers asymmetry
$A_{UT}=[d\sigma(S_{\perp}^{\uparrow})-d\sigma(S_{\perp}^{\downarrow})]/[d\sigma(S_{\perp}^{\uparrow})+d\sigma(S_{\perp}^{\downarrow})]$.

Figure 20: The relative size of contributions from the unpolarized $u$-quark
(red), $d$-quark (blue), and sea quark (green) TMDs to the
$q_{T}$-distribution integrated over all jet charge bins, corresponding to no
jet charge measurement being made.

While the electron-proton scattering cross section, in the small $q_{T}$
limit, probes polarized and unpolarized TMDs, it receives contributions from
various partonic channels. Typically, the cross section is dominated by the
$u$-quark channel so that it is sensitive primarily to the $u$-quark TMDs, as
seen in Fig. 20 for the normalized $q_{T}$-distribution in unpolarized
electron-proton scattering. Throughout this analysis we work in the center-of-
mass (CM) frame with CM energy $\sqrt{s}=105\>{\rm GeV}$, and with event
selection cuts of $0.1\leq y\leq 0.85$, and $15\>{\rm GeV}\leq p_{T}^{e}\leq
20\>{\rm GeV}$, $q_{T}<2.5\>{\rm GeV}$, and $Q^{2}>10$ GeV2, where $y$ denotes
the inelasticity. Jets are constructed using the anti-$k_{T}$ jet algorithm
Cacciari_2008  with radius parameter $R=1$.

One can enhance sensitivity to the $d$-quark TMD PDFs by making an additional
measurement of the standard jet charge and restricting to a particular jet
charge bin. For example, restricting to the standard jet charge bin
$Q_{J}<-0.25$ increases the relative contribution of the $d$-quark TMD PDFs to
the $q_{T}$-distribution, as seen in the bottom panel of Fig. 2 of Ref.
Kang:2020fka .

Figure 21: The relative size of contributions from the unpolarized $u$-quark
(red) and $d$-quark (blue) TMDs to the $q_{T}$-distribution in different jet
charge bins, for the dynamic (solid) and standard (dashed) jet charges. The
top and bottom panels correspond to the jet charge bins $Q_{J}<0.0$ and
$Q_{J}<-0.25$, respectively.

Here we show that the dynamic jet charge can also be used to probe the flavor
structure of TMDs and can complement the standard jet charge analysis. In Fig.
21, we show the normalized $q_{T}$ distribution for the unpolarized electron-
proton scattering process in Eq. (20), restricted to specific jet charge bins.
Here, the standard and dynamic jet charges are defined by restricting the sum
over hadrons in the jet, in Eqs. (1) and (9), to only include the charged
pions $\pi^{\pm}$, so as to improve the sensitivity to the $u$\- and $d$\-
quark TMDs. The top and bottom panels show the $q_{T}$-distribution restricted
to the jet charge bins $Q_{J}<0.0$ and $Q_{J}<-0.25$, respectively. The solid
and dashed curves in Fig. 21 correspond to jet charge bin restrictions based
on the dynamic and standard jet charge definitions, respectively. We see that
the $q_{T}$-distribution receives a significantly higher relative contribution
from the $d$-quark channel in these restricted jet charge bins, compared to
Fig. 20 where no jet charge measurement is made. This enhanced sensitivity to
$d$-quark TMD PDFs using the dynamic jet charge is similar to that found using
the standard jet charge.

Thus, by sorting the data into jet charge bins, the dynamic jet charge can be
used for flavor separation of nucleon TMDs and complement the standard jet
charge analysis. The same strategy can be employed to use the dynamic jet
charge for flavor separation of the Sivers function and we leave this for
future work.

###  VI.2 Quark Jet Flavor Separation in Photon-Tagged Jet Production

The RHIC spin program Aschenauer:2015eha ; Aschenauer:2016our ; bnltalk  aims
to further constrain the proton spin structure using longitudinally and
transversely polarized proton beams to measure single and double spin
asymmetries. The standard and dynamic jet charges can be used to disentangle
the flavor structure of proton structure functions that arise in jet-
production-based spin asymmetries. In addition, in heavy ion collisions, a jet
charge analysis can probe the flavor dependence (quark vs. gluon or $u$-quark
vs. $d$-quark) of jet modification Li:2019dre ; Chen:2019gqo ;
Sirunyan:2020qvi  due to parton propagation and showering in the nuclear
medium.

Figure 22: Global ROC curves for discrimination between $u$\- or
$\bar{u}$-jets and $d$\- or $\bar{d}$-jets in $pp\to\gamma+{\rm jet}+X$ (top
panel) and PbPb$\to\gamma+{\rm jet}+X$ (bottom panel) based on Pythia
simulations. The collision center of mass energy and selection cuts used are:
$\sqrt{s}=200$ GeV, $p_{T}>10.0$ GeV, $p_{T}^{\gamma}>30.0$ GeV, and
$\Delta\phi>7/8\pi$.

As an example, we consider the process of photon-tagged jet production in
$pp$\- and PbPb-collisions:

$\displaystyle p$ $\displaystyle+$ $\displaystyle p\to\gamma+{\rm jet}+X,$
$\displaystyle{\rm Pb}$ $\displaystyle+$ $\displaystyle{\rm Pb}\to\gamma+{\rm
jet}+X,$ (21)

A comparison of these processes provides valuable insights Chatrchyan:2012vq ;
Dai:2012am ; Kang:2017xnc ; Aaboud:2018anc ; Sirunyan:2018ncy  into jet
modification since the tagged photon leaves the strongly interacting medium
relatively unaffected.

Figure 23: Local ROC curves for discrimination between $u$\- or $\bar{u}$-jets
and $d$\- or $\bar{d}$-jets in $pp\to\gamma+{\rm jet}+X$ based on Pythia
simulations. The collision center of mass energy and selection cuts used are:
$\sqrt{s}=200$ GeV, $p_{T}>10.0$ GeV, $p_{T}^{\gamma}>30.0$ GeV, and
$\Delta\phi>7/8\pi$. The top, middle, and bottom panels correspond to the jet
charge bins $Q_{J}<-0.5,|Q_{J}|<0.5$, and $Q_{J}>0.5$, respectively.

Figure 24: Local ROC curves for discrimination between $u$\- or $\bar{u}$-jets
and $d$\- or $\bar{d}$-jets in PbPb$\to\gamma+{\rm jet}+X$ based on Pythia
simulations. The collision center of mass energy and selection cuts used are:
$\sqrt{s}=200$ GeV, $p_{T}>10.0$ GeV, $p_{T}^{\gamma}>30.0$ GeV, and
$\Delta\phi>7/8\pi$. The top, middle, and bottom panels correspond to the jet
charge bins $Q_{J}<-0.5,|Q_{J}|<0.5$, and $Q_{J}>0.5$, respectively.

At the parton level, photon-tagged jet production is mediated by the channels:

$\displaystyle q$ $\displaystyle+$ $\displaystyle g\to\gamma+q,$
$\displaystyle\bar{q}$ $\displaystyle+$ $\displaystyle g\to\gamma+\bar{q},$
$\displaystyle q$ $\displaystyle+$ $\displaystyle\bar{q}\to\gamma+g.$ (22)

The $u,d$-quark jet production dominates due to the dominance of the quark and
gluon PDFs in the initial state. In the limit of small $q_{T}$, photon-tagged
jet production can be a probe of the $u$-quark, $d$-quark, and gluon TMD PDFs.
Similarly, asymmetries constructed from longitudinally and transversely
polarized proton beams, can probe helicity PDFs deFlorian:2014yva ;
Nocera:2014gqa  and the Sivers functions Buffing:2018ggv ; Kang:2020xez ,
respectively. The use of jet charge to discriminate between the production of
$u$-quark and $d$-quark jets can thus provide complementary tools for flavor
separation of proton structure functions.

We demonstrate the $u$\- vs. $d$-quark jet discrimination power of the dynamic
jet charge in the photon-tagged jet production processes in Eq. (VI.2) through
Pythia simulation results. We simulate $pp$\- and PbPb-collisions at
$\sqrt{s}=200$ GeV with selection cuts of $p_{T}^{\gamma}>30.0$ GeV,
$p_{T}^{\rm jet}>10.0$ GeV, and $\Delta\phi_{\gamma-\rm jet}>\frac{7}{8}\pi$,
corresponding to the tagged photon transverse momentum, the leading jet
momentum, and the azimuthal angular separation between the photon and the
leading jet.

In Fig. 22, we show the global ROC curves for discrimination between $u$\- or
$\bar{u}$-jets and $d$\- or $\bar{d}$-jets in $pp\to\gamma+{\rm jet}+X$ (top
panel) and PbPb$\to\gamma+{\rm jet}+X$ (bottom panel), using the standard jet
charge (purple) and the dynamic jet charge (green). The global ROC curves
indicate that while the standard jet charge provides better discrimination in
$pp$-collisions, the dynamic jet charge gives better discrimination in PbPb-
collisions. Once again, this can be understood as a result of the robustness
of the dynamic jet charge discrimination power in the presence of increased
underlying activity in heavy-ion collisions.

However, even in $pp$-collisions, the dynamic jet charge can provide improved
discrimination when the data is sorted into jet charge bins. As seen in Fig.
23, the dynamic jet charge gives better discrimination in the $Q_{J}<-0.5$ and
$Q_{J}>0.5$ bins and comparable discrimination in the $|Q_{J}|<0.5$ bin. Thus,
the dynamic jet charge can be a complementary probe of the flavor structure of
the proton spin. Fig. 24 shows the same jet charge binned ROC curves for PbPb-
collisions. In this case, the dynamic jet charges provides improved
discrimination in all jet charge bins.

##  VII Likelihood Ratio Analysis of Jet Charge DIscrimination

In our analysis so far, we have made use of ROC curves to quantify the jet
charge discrimination power. This is standard practice in papers exploring new
jet discrimination observables where the signal and background distributions
are typically characterized by a single peak that may be translated relative
to each other. In the presence of multiple peaks, as in the case of the
dynamic jet charge, a sliding cut in the signal and background distributions
may not be monotonically related to a cut in the likelihood ratio, as is often
used in experimental analyses. Due to this multiple peak structure of the
dynamic jet charge distribution, we also present here an analysis of signal
versus background likelihood ratio distributions to quantify the absolute
discrimination power. We find that the underlying conclusions derived from
such an analysis of likelihood distributions are consistent with those based
on the ROC analysis.

We follow the procedure outlined in Refs. Cousins:JHEP2005 ; Cousins:2018tiz .
For quark vs. gluon jet discrimination, we treat the quark jets as signal
events and gluon jets as background events. For $u$-quark vs. $d$-quark jet
discrimination, we treat the $u$-quark jets as signal events and $d$-quark
jets as background events. We apply the procedure below:

  * •

From the Pythia simulated standard and dynamic jet charge distribution data,
we compute the binned jet charge probability values for quark, gluon,
$u$-quark, and $d$-quark jets. This simply corresponds to the bin heights of
the normalized jet charge distributions and provides us with the underlying
signal and background standard and dynamic jet charge probability
distributions.

  * •

From these underlying signal and background probability distributions, we
generate 20,000 data signal and background samples, each of size $N=50$.

  * •

For each of these samples, we calculate the likelihood ratio statistic:

$\displaystyle-2\>\ln\lambda=\sum_{i=1}^{N}-2\>\ln\left[\frac{p(Q_{J}|\text{signal})}{p(Q_{J}|\text{background})}\right],$
(23)

where $p(Q_{J}|\text{signal})$ and $p(Q_{J}|\text{background})$ give the
probability for a given jet of charge $Q_{J}$ to be a signal jet or background
jet, respectively.

  * •

We generated the likelihood ratio distribution of $-2\>\ln\lambda$ for 20,000
samples taken from the signal and background data, and compared them. The
likelihood ratio statistic approaches a Gaussian distribution as one increases
the sample size, as expected from the Central Limit Theorem.

  * •

One criteria Cousins:JHEP2005 ; Cousins:2018tiz  for determining whether a
given data set should be classified as data corresponding to signal events or
background events, is based on the value:

$\displaystyle-2\ln\lambda_{\rm cut}=\frac{M_{\rm sig.}S_{\rm bkg.}+M_{\rm
bkg.}S_{\rm sig.}}{S_{\rm sig.}+S_{\rm bkg.}},$ (24)

where $M_{\rm sig.}$ and $M_{\rm bkg.}$ are the mean values of the signal and
background distributions, respectively. $S_{\rm sig.}$ and $S_{\rm bkg.}$ are
the standard deviations of the signal and background distributions,
respectively. The value of $-2\ln\lambda_{\rm cut}$ corresponds to the value
such that the number of standard deviations, $S_{\rm sig.}$, it is away from
$M_{\rm sig.}$ is the same as the number of standard deviations, $S_{\rm
bkg.}$, it is away from $M_{\rm bkg.}$. If the calculated value of
$-2\ln\lambda$ for a given data sample, lies to the left (right) of
$-2\ln\lambda_{\rm cut}$, the data sample is classified as corresponding to
signal (background) events.

  * •

Finally, we note that the values of $M_{\rm sig.}$ and $M_{\rm bkg.}$ scale
with the sample size $N$ and the values of $S_{\rm sig.}$ and $S_{\rm bkg.}$
scale with $\sqrt{N}$ Cousins:JHEP2005 ; Cousins:2018tiz . Thus, as one
increases the sample size, the signal and background likelihood Gaussian
distributions will move further apart from each other, on either side of
$-2\ln\lambda_{\rm cut}$, corresponding to a cleaner separation of these
distributions. Thus, the smaller the required sample size to achieve a clean
separation of signal and background likelihood distributions, the greater the
discrimination power of the observable.

In Fig. 25, we show the standard (top panel) and dynamic (bottom panel) jet
charge likelihood distributions for quark and gluon jet samples in
$pp$-collisions. We see clearly that with a sample size of $N=50$, the dynamic
jet charge already gives clean separation of the quark and gluon jet
likelihood distributions. On the other hand, there is still significant
overlap of the quark and gluon jet likelihood distributions for the standard
jet charge. This indicates that the dynamic jet charge has higher
discrimination power between quark and gluon jets, consistent with what was
found using the ROC curve analysis, as seen in the bottom panel of Fig. 5.

Figure 25: The standard (top panel) and dynamic (bottom panel) jet charge
likelihood distributions for quark (red) and gluon (blue) jet samples. The
$-2\ln\lambda_{\rm cut}$ values for the Standard and Dynamic likelihood
distributions are 0.103 and 1.90, respectively. The distributions above were
generated for the more central jet in dijet production of $pp$-collisions at
$\sqrt{s}=$13 TeV, with selection cuts $p_{T_{J}}=[200,300]$ GeV,
$|\eta_{J}|<2.1$, and $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading
($j_{1}$) and subleading ($j_{2}$) anti-kT jets of jet radius $R=0.4$.

Similar results are observed for heavy-ion PbPb-collisions. In Fig. 26, we
show the standard (top panel) and dynamic (bottom panel) jet charge likelihood
distributions for quark and gluon jet samples in PbPb-collisions.

Figure 26: The standard (top panel) and dynamic (bottom panel) jet charge
likelihood distributions for quark (red) and gluon (blue) jet samples. The
$-2\ln\lambda_{\rm cut}$ values for the Standard and Dynamic likelihood
distributions are -0.0308 and 2.17, respectively. The distributions above were
generated for the more central jet in dijet production of PbPb-collisions at
$\sqrt{s}=$2.760 TeV, with selection cuts $p_{T_{J}}=[80,150]$ GeV,
$|\eta_{J}|<0.9$, and $p_{T_{j_{1}}}/p_{T_{j_{2}}}<1.5$ on the leading
($j_{1}$) and subleading ($j_{2}$) anti-kT jets of jet radius $R=0.4$.

Once again, we see that with $N=50$, the dynamic jet charge already gives
clean separation of the quark and gluon jet likelihood distributions while
there is still significant overlap for the standard jet charge. This
conclusion is consistent with the ROC curves in Fig. 6.

Figure 27: The standard (top panel) and dynamic (bottom panel) jet charge
likelihood distributions for $u$-quark (red) and $d$-(blue) jet samples. The
$-2\ln\lambda_{\rm cut}$ values for the Standard and Dynamic likelihood
distributions are 1.34 and 0.340, respectively. The distributions above were
generated for the more central jet in dijet production of $pp$-collisions at
$\sqrt{s}=13$ TeV with $R=0.4$, $p_{T_{J}}=[200,300]$ GeV, and
$|\eta_{J}|<2.1$. The partonic channels $dg\to W^{-}u$ and $ug\to W^{+}d$
generated the $u$-quark and $d$-quark jets, respectively.

Figure 28: The standard (top panel) and dynamic (bottom panel) jet charge
likelihood distributions for $u$-quark (red) and $d$-(blue) jet samples. The
$-2\ln\lambda_{\rm cut}$ values for the Standard and Dynamic likelihood
distributions are -0.0129 and 0.887, respectively. The distributions above
were generated for the more central jet in dijet production of PbPb-collisions
at $\sqrt{s}=2.760$ TeV with $R=0.4$, $p_{T_{J}}=[80,150]$ GeV, and
$|\eta_{J}|<0.9$. The partonic channels $dg\to W^{-}u$ and $ug\to W^{+}d$
generated the $u$-quark and $d$-quark jets, respectively.

In Fig. 27, we show the standard (top panel) and dynamic (bottom panel) jet
charge likelihood distributions for $u$-quark and $d$-quark jet samples in
$pp$-collisions. In this case, we see that the standard jet charge has better
(but comparable) discrimination between $u$-quark and $d$-quark jets, compared
to the dynamic jet charge. This can be seen from the greater separation
between the $u$-quark and $d$-quark likelihood distributions for the standard
jet charge compared to the dynamic jet charge. Once again, this conclusion is
consistent with the ROC curves in the bottom panel of Fig. 8.

However, as seen in Fig. 28, in heavy ion collisions the dynamic jet charge
seems to give better (but comparable) discrimination between $u$-quark and
$d$-quark jets, compared to the standard jet charge. We see that for $N=50$,
the dynamic jet charge gives a cleaner (but comparable) separation between the
likelihood distributions of the $u$-quark and $d$-quark jets in heavy ion
collisions. This is consistent with the ROC curves in the bottom panel of Fig.
9 and is the result of the relative insensitivity of the dynamic jet charge to
the significantly increased soft contamination in heavy ion collisions.

##  VIII Conclusions

We proposed a modified definition of the jet charge observable, the dynamic
jet charge, which makes the parameter $\kappa$ that appears in the standard
definition to be a function of some dynamic property of either the jet or the
individual jet constituents. We focused on the specific scenario where each
hadron in the jet contributes to the dynamic jet charge with a dynamic
parameter, $\kappa(z_{h})$, determined by its jet transverse momentum or
energy fraction. We have shown that the dynamic jet charge can complement
analyses based on the standard jet charge and allow for improved jet
discrimination.

The behavior of the dynamic jet charge was studied using Pythia8 simulations
of $pp$-collisions and heavy ion PbPb-collisions. It was found that the
discrimination power of the dynamic jet charge in PbPb-collisions is about the
same as in $pp$-collisions, being largely unaffected by the significantly
greater underlying event activity. The multiple peak structure observed in the
dynamic jet charge distribution, also allows for a binned analysis where a
discrimination analysis can be performed on jet data binned according to the
jet charge. The jet charge bins are centered around the peaks and with size
corresponding to the width of the peaks. Such a binned analysis could provide
additional discrimination compared to an unbinned analysis alone. We defined
local Receiver Operating Characteristic (ROC) curves, in the presence of
multiple peak structures, to quantify the discrimination power within local
jet charge bins that only involve a single peak structure, as opposed to
global ROC curves that use data from all jet charge bins. Due to the multiple
peak structure of the dynamic jet charge, we also presented a signal vs.
background likelihood distribution analysis for quantifying the absolute
discrimination power, even in the presence of multiple peak structures.

Pythia simulations indicated that the dynamic jet charge provides
significantly improved discrimination between quark and gluon initiated jets,
in both proton-proton and heavy ion collisions, compared to the standard jet
charge. For discrimination between $u$-quark and $d$-quark initiated jets in
proton-proton collisions, the global ROC curves and the likelihood
distributions indicated that the standard jet charge has better, but
comparable, discrimination power. However, the local ROC curves indicated that
the dynamic jet charge can provide improved discrimination when the data is
sorted into jet charge bins. Thus, the dynamic jet charge could complement jet
flavor discrimination based on the standard jet charge analysis. For heavy ion
collisions, it was found that the dynamic jet charge gives better, but
comparable, discrimination as quantified by both the global and local ROC
curves, and the likelihood analysis. This is a consequence of the improved
resilience of the dynamic jet charge against contamination effects of the
underlying event activity in heavy ion collisions. Future work includes
studying the dynamic jet charge in heavy ion collisions using the JEWEL or
JETSCAPE Monte Carlo simulation which, unlike the Pythia8 simulation, includes
hot nuclear medium effects.

We also presented phenomenological studies to show that the dynamic and
standard jet charges can be a unique probe of the nuclear flavor structure
through simulation studies of deep inelastic scattering in the back-to-back
region between the final electron and the leading jet, and in photon-tagged
jet production in proton-proton and heavy ion collisions. We envision many
other phenomenological applications of both the standard and dynamic jet
charges to probe various aspects of nuclear structure. We leave such
explorations for future work. Finally, the underlying idea of using a dynamic
parameter for the jet charge can, in principle, be applied to parameters in
the definitions of other observables, such as jet angularities.

## Acknowledgments

We thank Robert D. Cousins for helpful discussions on the likelihood ratio
analysis. Z.K. is supported by the National Science Foundation under CAREER
award PHY-1945471. X.L. is supported by National Natural Science Foundation of
China under Grant No.11775023. S.M., M.S., and T.W. thank UNG for support
through the Faculty Undergraduate Summer Engagement (FUSE) grant.

## References

  * (1) R. Field and R. Feynman,  Nucl. Phys. B 136, 1 (1978).
  * (2) Fermilab-Serpukhov-Moscow-Michigan, J. Berge et al.,  Phys. Lett. B 91, 311 (1980).
  * (3) J. Berge et al.,  Nucl. Phys. B 184, 13 (1981).
  * (4) Aachen-Bonn-CERN-Munich-Oxford, P. Allen et al.,  p. 307 (1982).
  * (5) European Muon, J. Albanese et al.,  Phys. Lett. B 144, 302 (1984).
  * (6) Amsterdam-Bologna-Padua-Pisa-Saclay-Turin, S. Barlag et al.,  Z. Phys. C 11, 283 (1982),  [Erratum: Z.Phys.C 14, 281 (1982)].
  * (7) R. Erickson et al.,  Phys. Rev. Lett. 42, 822 (1979),  [Erratum: Phys.Rev.Lett. 42, 1246 (1979)].
  * (8) AMY, D. Stuart et al.,  Phys. Rev. Lett. 64, 983 (1990).
  * (9) ALEPH, D. Decamp et al.,  Phys. Lett. B 259, 377 (1991).
  * (10) TASSO, W. Braunschweig et al.,  Z. Phys. C 48, 433 (1990).
  * (11) D0, V. Abazov et al.,  Phys. Rev. Lett. 98, 041801 (2007), hep-ex/0608044.
  * (12) CDF, T. Aaltonen et al.,  Phys. Rev. D 88, 032003 (2013), 1304.4141.
  * (13) ATLAS, G. Aad et al.,  JHEP 11, 031 (2013), 1307.4568.
  * (14) CMS, V. Khachatryan et al.,  JHEP 12, 017 (2014), 1410.4227.
  * (15) Y.-C. J. Chen, C.-W. Chiang, G. Cottin, and D. Shih,  Phys. Rev. D 101, 053001 (2020), 1908.08256.
  * (16) ALEPH, D. Buskulic et al.,  Z. Phys. C 71, 357 (1996).
  * (17) D. Krohn, M. D. Schwartz, T. Lin, and W. J. Waalewijn,  Phys. Rev. Lett. 110, 212001 (2013), 1209.2421.
  * (18) W. J. Waalewijn,  Phys. Rev. D 86, 094030 (2012), 1209.3019.
  * (19) ATLAS, G. Aad et al.,  Phys. Rev. D 93, 052003 (2016), 1509.05190.
  * (20) CMS, A. M. Sirunyan et al.,  JHEP 10, 131 (2017), 1706.05868.
  * (21) K. Fraser and M. D. Schwartz,  JHEP 10, 093 (2018), 1803.08066.
  * (22) M. Gyulassy and X.-n. Wang,  Nucl. Phys. B 420, 583 (1994), nucl-th/9306003.
  * (23) X.-N. Wang, M. Gyulassy, and M. Plumer,  Phys. Rev. D 51, 3436 (1995), hep-ph/9408344.
  * (24) Y.-T. Chien and I. Vitev,  Phys. Rev. Lett. 119, 112301 (2017), 1608.07283.
  * (25) M. Connors, C. Nattrass, R. Reed, and S. Salur,  Rev. Mod. Phys. 90, 025005 (2018), 1705.01974.
  * (26) H. T. Li and I. Vitev,  Phys. Rev. D 101, 076020 (2020), 1908.06979.
  * (27) CMS, A. M. Sirunyan et al.,  JHEP 07, 115 (2020), 2004.00602.
  * (28) Y.-L. Du, D. Pablos, and K. Tywoniuk,  JHEP 03, 206 (2021), 2012.07797.
  * (29) Z.-B. Kang, X. Liu, S. Mantry, and D. Y. Shao,  Phys. Rev. Lett. 125, 242003 (2020), 2008.00655.
  * (30) Z.-B. Kang, K. Lee, D. Y. Shao, and J. Terry,  JHEP 02, 066 (2021), 2008.05470.
  * (31) J. Gallicchio and M. D. Schwartz,  Phys. Rev. Lett. 107, 172001 (2011), 1106.3076.
  * (32) J. Brewer, J. Thaler, and A. P. Turner,  Phys. Rev. C 103, L021901 (2021), 2008.08596.
  * (33) A. J. Larkoski, I. Moult, and B. Nachman,  Phys. Rept. 841, 1 (2020), 1709.04464.
  * (34) A. Buckley, G. Callea, A. J. Larkoski, and S. Marzani,  SciPost Phys. 9, 026 (2020), 2006.10480.
  * (35) Y.-T. Chien and R. Kunnawalkam Elayavalli,  (2018), 1803.03589.
  * (36) K. C. Zapp,  Eur. Phys. J. C 74, 2762 (2014), 1311.0048.
  * (37) J. Putschke et al.,  (2019), 1903.07706.
  * (38) A. J. Larkoski, S. Marzani, G. Soyez, and J. Thaler,  JHEP 05, 146 (2014), 1402.2657.
  * (39) Z.-B. Kang, K. Lee, X. Liu, D. Neill, and F. Ringer,  JHEP 02, 054 (2020), 1908.01783.
  * (40) C. F. Berger, T. Kucs, and G. F. Sterman,  Phys. Rev. D 68, 014012 (2003), hep-ph/0303051.
  * (41) A. Hornig, C. Lee, and G. Ovanesyan,  JHEP 05, 122 (2009), 0901.3780.
  * (42) Z.-B. Kang, K. Lee, and F. Ringer,  JHEP 04, 110 (2018), 1801.00790.
  * (43) Z.-B. Kang, K. Lee, X. Liu, and F. Ringer,  Phys. Lett. B 793, 41 (2019), 1811.06983.
  * (44) G. Bell, A. Hornig, C. Lee, and J. Talbert,  JHEP 01, 147 (2019), 1808.07867.
  * (45) A. Budhraja, A. Jain, and M. Procura,  JHEP 08, 144 (2019), 1903.11087.
  * (46) C. W. Bauer, A. V. Manohar, and P. F. Monni,  (2020), 2012.09213.
  * (47) M. Dasgupta, F. A. Dreyer, G. P. Salam, and G. Soyez,  JHEP 06, 057 (2016), 1602.01110.
  * (48) Z.-B. Kang, F. Ringer, and I. Vitev,  JHEP 10, 125 (2016), 1606.06732.
  * (49) T. Sjöstrand et al.,  Computer Physics Communications 191, 159 (2015).
  * (50) M. Cacciari and G. P. Salam,  Physics Letters B 641, 57 (2006).
  * (51) M. Cacciari, G. P. Salam, and G. Soyez,  The European Physical Journal C 72 (2012).
  * (52) A. J. Larkoski, S. Marzani, G. Soyez, and J. Thaler,  Journal of High Energy Physics 2014 (2014).
  * (53) C. Frye, A. J. Larkoski, M. D. Schwartz, and K. Yan,  Precision physics with pile-up insensitive observables, 2016, 1603.06375.
  * (54) C. Frye, A. J. Larkoski, M. D. Schwartz, and K. Yan,  Journal of High Energy Physics 2016 (2016).
  * (55) M. Cacciari, G. P. Salam, and G. Soyez,  Journal of High Energy Physics 2008, 063 (2008).
  * (56) C. Bierlich, G. Gustafson, and L. Lönnblad,  JHEP 10, 139 (2016), 1607.04434.
  * (57) C. Bierlich, G. Gustafson, L. Lönnblad, and H. Shah,  JHEP 10, 134 (2018), 1806.10820.
  * (58) E.-C. Aschenauer et al.,  (2015), 1501.01220.
  * (59) H. Liu,  Talk given at RIKEN BNL Workshop Jet Observables at the Electron-Ion Collider, 2020.
  * (60) X. Liu, F. Ringer, W. Vogelsang, and F. Yuan,  Phys. Rev. Lett. 122, 192003 (2019), 1812.08077.
  * (61) M. Arratia, Z.-B. Kang, A. Prokudin, and F. Ringer,  Phys. Rev. D 102, 074015 (2020), 2007.07281.
  * (62) A. Accardi et al.,  Eur. Phys. J. A 52, 268 (2016), 1212.1701.
  * (63) D. W. Sivers,  Phys. Rev. D 41, 83 (1990).
  * (64) E.-C. Aschenauer et al.,  (2016), 1602.03922.
  * (65) S.-Y. Chen, B.-W. Zhang, and E.-K. Wang,  Chin. Phys. C 44, 024103 (2020), 1908.01518.
  * (66) CMS, S. Chatrchyan et al.,  Phys. Lett. B 710, 256 (2012), 1201.3093.
  * (67) W. Dai, I. Vitev, and B.-W. Zhang,  Phys. Rev. Lett. 110, 142001 (2013), 1207.5177.
  * (68) Z.-B. Kang, I. Vitev, and H. Xing,  Phys. Rev. C 96, 014912 (2017), 1702.07276.
  * (69) ATLAS, M. Aaboud et al.,  Phys. Lett. B 789, 167 (2019), 1809.07280.
  * (70) CMS, A. M. Sirunyan et al.,  Phys. Rev. Lett. 122, 152001 (2019), 1809.08602.
  * (71) D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang,  Phys. Rev. Lett. 113, 012001 (2014), 1404.4293.
  * (72) NNPDF, E. R. Nocera, R. D. Ball, S. Forte, G. Ridolfi, and J. Rojo,  Nucl. Phys. B 887, 276 (2014), 1406.5539.
  * (73) M. G. Buffing, Z.-B. Kang, K. Lee, and X. Liu,  (2018), 1812.07549.
  * (74) R. Cousins, J. Mumford, J. Tucker, and V. Valuev,  Journal of High Energy Physics 2005 (2005).
  * (75) R. D. Cousins,  arXiv:1807.05996 (2018).