# Transverse momentum balance of dijets in Xe+Xe collisions at the LHC

Yao Li  Key Laboratory of Quark & Lepton Physics (MOE) and Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China  Shuwan
Shen  Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle
Physics, Central China Normal University, Wuhan 430079, China  Sa Wang
<EMAIL_ADDRESS>Key Laboratory of Atomic and Subatomic Structure and
Quantum Control (MOE), Institute of Quantum Matter, South China Normal
University, Guangzhou 510006, China  Ben-Wei Zhang<EMAIL_ADDRESS>Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle
Physics, Central China Normal University, Wuhan 430079, China

( )

###### Abstract

We present a theoretical study of the medium modifications on the $p_{\rm T}$
balance ($x_{\rm J}$) of dijets in Xe+Xe collisions at $\sqrt{s_{\rm
NN}}=5.44$ TeV. The initial production of dijets is carried out by the
POWHEG+PYTHIA8 prescription, which matches the next-to-leading order (NLO) QCD
matrix elements with the parton shower (PS) effect. The in-medium evolution in
nucleus-nucleus collisions is described by the SHELL model with a transport
approach. The theoretical results of the dijet $x_{\rm J}$ in Xe+Xe collisions
exhibit more imbalanced distributions than that in p+p, consistent with the
recently reported ATLAS data. By utilizing the Interleaved Flavor
Neutralisation, an infrared-and-collinear-safe jet flavor algorithm, to
identify the flavor of the reconstructed jets, we classify dijets processes
into three categories: gluon-gluon ($gg$), quark-gluon ($qg$) and quark-quark
($qq$), and investigate the respective medium modification patterns and
fraction changes of the $gg$, $qg$, and $qq$ components of the dijet sample in
Xe+Xe collisions. It is shown that the $qg$ component plays a key role in the
increased imbalance of the dijet $x_{\rm J}$, and especially the $q_{1}g_{2}$
(quark-jet-leading) dijets experience more significant asymmetric energy loss
than the $g_{1}q_{2}$ (gluon-jet-leading) dijets as traversing the QGP. By
comparing the $\Delta\langle x_{\rm J}\rangle$ of inclusive, $c\bar{c}$ and
$b\bar{b}$ dijets in Xe+Xe collisions, we observe $\Delta\langle x_{\rm
J}\rangle_{\rm incl.}>\Delta\langle x_{\rm J}\rangle_{\rm
c\bar{c}}>\Delta\langle x_{\rm J}\rangle_{\rm b\bar{b}}$. Moreover, $\rho_{\rm
Xe,Pb}$, the ratios of nuclear modification factors of dijets in Xe+Xe to that
in Pb+Pb, are calcualted, which indicates that the yield suppression of dijets
in Pb+Pb is more pronounced than that in Xe+Xe due to the larger radius of the
lead nucleus.

##  I introduction

The ultra-relativistic heavy-ion collisions at the Large Hadron Collider (LHC)
and the Relativistic Heavy Ion Collider (RHIC) provide a unique arena to
search for the new form of nuclear matter, the quark-gluon plasma (QGP), in
which the degrees of freedom of the quarks and gluons in the protons and
neutrons are released Wang:1992qdg ; Gyulassy:2003mc ; Mehtar-Tani:2013pia ;
Qin:2015srf ; Wang:2001cs . The strong interactions between the hard-scattered
partons with the medium, referred to as the “jet quenching” phenomenon, open
up new avenues to understand the properties of such a fantastic strongly-
coupled quark matter JET:2013cls ; Xie:2022ght ; Zhang:2021xib ;
JETSCAPE:2020mzn  and test the fundamental theory of quantum chromodynamics
(QCD) at the extremely hot and dense conditions Collins:1974ky ;
Cunqueiro:2021wls ; Cao:2020wlm ; Tang:2020ame ; Shen:2020mgh ; Gao:2020vbh ;
Liu:2022ewr . In the past two decades, a series of tools have been extensively
investigated to reveal this partonic strong interaction, such as the
suppression factor $R_{\rm AA}$ of high-$p_{\rm T}$ hadron/jet ATLAS:2014ipv ;
Wang:1998bha ; Bass:2008rv ; Wang:2002ri ; He:2018xjv ; JETSCAPE:2021ehl , the
momentum asymmetry of dijets ATLAS:2010isq ; Fialkovsky:2011pu ; Zhang:2007ja
; Qin:2010mn ; Kang:2018wrs ; Senzel:2013dta ; Xing:2012ii ; Kang:2011bp ;
Gao:2016ldo ; Renk:2012cb ; Chen:2016cof ; Dai:2018mhw , correlations of the
vector boson associated jets ($\gamma/Z^{0}$+jets) CMS:2012ytf ;
Neufeld:2010fj ; Dai:2012am ; Huang:2015mva ; Yang:2022nei ; Chen:2020tbl ,
the global event geometry Chen:2020pfa ; Kang:2023qxb  and the jet
substructures CMS:2013lhm ; Vitev:2009rd ; Vitev:2008rz ; Zhang:2021sua ;
Chen:2022kic ; Zhang:2018urd ; Wang:2023eer ; JETSCAPE:2023hqn ; Wang:2022yrp
; Kang:2023ycg .

Since the dijets are the dominant QCD processes in the hadron collisions in
the experiment and have less influence from the underlying background, they
show a unique glamour in jet physics. In vacuum, the parton shower effects and
higher-order QCD processes may break the symmetry of the final-state dijets,
which will lead to a deflection from the back-to-back azimuthal angle and an
unequal transverse momentum between the leading and subleading jets. In A+A
collisions, because the two jets usually experience asymmetric energy loss as
traversing the QGP medium, the transverse momentum balance of dijets $x_{\rm
J}\equiv p_{\rm T,2}/p_{\rm T,1}$ ATLAS:2017xfa , defined as the ratio of the
subleading to leading jet $p_{\rm T}$, can be further modified by the in-
medium interactions and show a different sensitivity to the path-length
dependence of jet quenching Qin:2010mn  and jet-by-jet fluctuations of jet-
medium interactions Milhano:2015mng . And more imbalanced $x_{\rm J}$
distributions of dijets have been observed in Pb+Pb collisions relative to p+p
at $\sqrt{s_{\rm NN}}=2.76$ TeV and $\sqrt{s_{\rm NN}}=5.02$ TeV by the ATLAS
ATLAS:2010isq ; ATLAS:2017xfa ; ATLAS:2022zbu  and CMS Collaborations
Fialkovsky:2011pu ; CMS:2012ulu ; CMS:2018dqf , which are extensively
investigated by the theoretical calculations Kang:2018wrs ; Senzel:2013dta ;
Gao:2016ldo ; Renk:2012cb ; Chen:2016cof ; Dai:2018mhw .

Recently, the ATLAS Collaboration has measured the dijet $x_{\rm J}$ in Xe+Xe
collisions at $\sqrt{s_{\rm NN}}=5.44$ TeV for the first time ATLAS:2023xzy ,
however the timely theoretical studies are still in lack. Because the xenon
nucleus has a smaller radius than that of lead, studying the dijet productions
in different collision systems will deepen our understanding of the system
size dependence of jet quenching effect ALICE:2018hza ; ATLAS:2019dct ;
ATLAS:2022zbu ; ATLAS:2017xfa ; Xie:2019oxg ; Li:2021xbd ; Zhang:2022fau .
Furthermore, since the dijet events consist of $gg$, $qg$ and $qq$ components,
while the jet energy loss is closely related to the flavor of hard partons
($\Delta E_{g}/\Delta E_{q}\sim C_{A}/C_{F}$) Gyulassy:2003mc ; Wang:1998bha ,
it is of significance to figure out their respective modification patterns and
assess what roles they played in the overall medium modifications of the dijet
$x_{\rm J}$. Furthermore, the massive heavy quarks are believed to lose less
energy than the light quarks due to the “dead-cone” effect Dokshitzer:2001zm ;
Zhang:2003wk ; Armesto:2003jh ; Djordjevic:2003qk , which leads to a mass
hierarchy of partonic energy loss $\Delta E_{q}>\Delta E_{c}>\Delta E_{b}$
Xing:2019xae ; Xing:2023ciw ; Wang:2023eer ; Zhang:2023oid . It is of
particular interest to explore such mass dependence of the medium modification
on the dijet $x_{\rm J}$ by the comparisons between the light- and heavy-
flavor (such as $c\bar{c}$ and $b\bar{b}$) dijets in high-energy nuclear
collisions.

This paper presents the first theoretical study of medium modifications on the
dijet $p_{\rm T}$-balance $x_{\rm J}$ in Xe+Xe collisions. The initial
production of dijets is carried out by the POWHEG+PYTHIA8 prescription, which
matches the next-to-leading order (NLO) QCD matrix elements with the parton
shower (PS) effect. The transport approach describes dijets’ in-medium
evolution, which considers both the elastic and inelastic partonic
interactions in the quark-gluon plasma (QGP). Firstly, we present the
theoretical results of the dijet $x_{\rm J}$ in Xe+Xe collisions at
$\sqrt{s_{\rm NN}}=5.44$ TeV compared with the recently reported ATLAS
measurements. Specifically, we will discuss the flavor and mass dependence of
the medium modification on the dijet $x_{\rm J}$. We study the respective
medium modification patterns and fraction changes of the $gg$, $qg$, $qq$, as
well as the $q_{1}g_{2}$ and $g_{1}q_{2}$ components in the dijet samples in
both p+p and Xe+Xe collisions. We will also investigate the mass effect of the
$x_{\rm J}$ modifications by comparing the $\Delta x_{\rm J}$ of inclusive,
$c\bar{c}$ and $b\bar{b}$ dijets in Xe+Xe collisions. At last, we present the
calculated results of dijets nuclear modification factor in Xe+Xe at
$\sqrt{s_{\rm NN}}=5.44$ TeV and Pb+Pb at $\sqrt{s_{\rm NN}}=5.02$ TeV
compared to the recent ATLAS data.

##  II Theoretical framework

In this work, we generate the next-to-leading order (NLO) matrix elements for
QCD di-jet processes Alioli:2010xa  in the framework of POWHEG-BOX-V2
Nason:2004rx ; Frixione:2007vw ; Alioli:2010xd  and then simulate the parton
shower (PS) with PYTHIA 8.309 Bierlich:2022pfr  to produce p+p events. The
CT18NLO parton distribution functions (PDF) Hou:2019qau  are chosen in the
computation. Jets are reconstructed using the anti-$k_{\rm T}$ clustering
algorithm and radius parameter $R$ = 0.4 as implemented in the FastJet package
Cacciari:2011ma . Then, the highest two $p_{\rm T}$ jets out of the set of
jets in an event are selected as the dijet candidate. The leading jet
transverse momentum $p_{\rm T,1}$ and subleading jet transverse momentum
$p_{\rm T,2}$ are required to be greater than 100 GeV/c and 32 GeV/c,
respectively. The two jets are required to be nearly back-to-back in azimuth
with $\Delta\phi\equiv|\phi_{1}-\phi_{2}|\geq 7\pi/8$ and to be in the
rapidity region $|y|$ ¡ 2.1. If all these conditions are met, the desired
dijet candidate is accepted.

Figure 1:  (Color online) Normalized $x_{\rm J}$ distributions of dijets in
p+p collisions at $\sqrt{s}=5.02$ TeV for six $p_{\rm{T,1}}$ intervals: [100,
112], [112, 126], [126, 141], [141, 158], [158, 171] and [171, 200] GeV/c are
compared with the ATLAS data ATLAS:2022zbu .

We calculate the normalized $x_{\rm J}$ distributions in p+p collisions and
compare that to the ATLAS data ATLAS:2022zbu  as shown in Fig. 1. We observe
that the $x_{\rm J}$ distributions calculated by POWHEG+PYTHIA8 give decent
descriptions at small value of $x_{\rm J}$ for all six different
$p_{\rm{T,1}}$ intervals compared to the ATLAS data, except for a little
overestimation of $x_{\rm J}$ distributions than the ATLAS data near 1. At
each $p_{\rm T}$ interval, the $x_{\rm J}$ distribution peaks near $x_{\rm
J}\simeq 1$, where the leading and subleading jets are almost balanced.
However, with the higher-order perturbative QCD corrections and the splitting
processes during the parton shower in vacuum, a considerable fraction of
dijets are produced with imbalanced transverse momentum at the smaller $x_{\rm
J}$ region.

The in-medium evolution of both light- and heavy-flavor dijets are simulated
by the SHELL model Dai:2018mhw ; Wang:2019xey ; Wang:2020bqz ; Wang:2020qwe ;
Wang:2020ukj ; Wang:2020ffj ; Wang:2021jgm ; Li:2022tcr ; Wang:2023udp , which
takes into account the elastic and inelastic partonic energy loss within the
hot/dense QGP medium. Since the propagation of massive partons in the QCD
medium can be viewed as “Brownian motion”, the transport of heavy quarks can
be well described by the modified Langevin equations,

$\displaystyle\Delta\vec{x}(t)=\frac{\vec{p}(t)}{E}\Delta t,$ (1)
$\displaystyle\Delta\vec{p}(t)=-\eta_{D}\vec{p}\Delta t+\vec{\xi}(t)\Delta
t-\vec{p}_{\rm g}(t).$ (2)

These two equations describe heavy quarks’ position and momentum updates as
traversing the QGP medium. $\eta_{D}$ is the drag coefficient controlling the
energy dissipation strength of heavy quarks in the medium. The stochastic term
$\xi(t)$ denotes the random kicks as heavy quarks scatter with the thermal
particle, which obeys a Gaussian distribution. The diffusion coefficient
$\kappa$ could be related to the $\eta_{D}$ by the fluctuation-dissipation
theorem $\kappa=2\eta_{D}ET$. Note that the first two terms at the right-hand
side of Eq. 2 represent the collisional energy loss of heavy quarks. In
contrast, the last term $-p_{g}$ is the momentum correction caused by the
medium-induced gluon radiation. In our framework, the Higher-Twist Guo:2000nz
; Zhang:2003wk ; Zhang:2003yn ; Majumder:2009ge  formalism has been employed
to simulate the medium-induced gluon radiation of jet partons, in which the
gluon radiation spectra of an energetic parton in the QGP can be obtained,

$\displaystyle\frac{dN_{g}}{dxdk^{2}_{\perp}dt}=\frac{2\alpha_{s}P(x)\hat{q}}{\pi
k^{4}_{\perp}}\sin^{2}(\frac{t-t_{i}}{2\tau_{f}})(\frac{k^{2}_{\perp}}{k^{2}_{\perp}+x^{2}M^{2}})^{4},$
(3)

where $x$ and $k_{\perp}$ denote the radiated gluon’s energy fraction and
transverse momentum. $P(x)$ is the QCD splitting function for the splitting
processes $g\rightarrow g+g$ and $q(Q)\rightarrow q(Q)+g$ Wang:2009qb ,
$\tau_{f}=2Ex(1-x)/(k^{2}_{\perp}+x^{2}M^{2})$ the formation time of the
daughter gluon. $\hat{q}$ denotes the general jet transport parameter in the
QGP Chen:2010te . The last term in Eq. 3 represents the suppression factor
results from the “dead-cone” effect of heavy quarks Dokshitzer:2001zm ;
ALICE:2019nuy , which suppresses the probability of gluon radiation within a
small cone ($\theta\sim M_{Q}/E$). The collisional energy loss is generally
dominant for low-energy heavy quarks due to the “dead-cone” effect. In
contrast, the radiative energy loss is usually expected to become significant
at $p_{\rm T}^{Q}>5m_{Q}$ Dong:2019byy . For massless partons, the collisional
energy loss is estimated by the pQCD calculations within the Hard-Thermal Loop
approximation Neufeld:2010xi ; Huang:2013vaa , while their radiative
contribution by the same Higher-Twist formalism as for massive partons. The
hydrodynamics time-space evolution of the QGP medium is described by the
CLVisc programs Pang:2016igs ; Pang:2018zzo , which provides the temperature
and velocity of the expanding hot/dense nuclear matter. The SHELL model has
been successfully applied in the study of heavy-flavor jets in high-energy
nuclear collisions, which gives satisfactory descriptions on a series of
experiment measurements, such as $p_{\rm T}$ imbalance Dai:2018mhw , radial
profiles Wang:2019xey ; Wang:2020ukj ; Wang:2020bqz  and fragmentation
functions Li:2022tcr  of heavy-flavor jets, correlations of $Z^{0}+$ HF
hadron/jet Wang:2020qwe ; Wang:2021jgm .

Figure 2:  (Color online) (a) Calculated normalized dijet $x_{\rm J}$
distributions in Xe+Xe collisions at $\sqrt{s_{\rm NN}}=5.44$ TeV are compared
with the ATLAS data for four centrality bins ($0-10\%$, $10\%-20\%$,
$20\%-40\%$ and $40\%-80\%$) and two $p_{\rm{T,1}}$ intervals (green: [100,
126] GeV/c, red: [158, 199] GeV/c). (b) Comparisons of the normalized dijets
$x_{\rm J}$ distributions between Xe+Xe and p+p collisions at $\sqrt{s_{\rm
NN}}=5.44$ TeV for two $p_{\rm{T,1}}$ intervals (left: [100, 126] GeV/c,
right: [158, 199] GeV/c), and the ratios of XeXe/pp are shown in the bottom
panels.

##  III Numerical Results and Discussions

In Fig. 2, we firstly show the dijet $x_{\rm J}$ distributions in Xe+Xe
collisions at $\sqrt{s_{\rm NN}}=5.44$ TeV calculated by the SHELL model as a
comparison to the ATLAS data for four centrality bins ($0-10\%$, $10\%-20\%$,
$20\%-40\%$ and $40\%-80\%$) and two $p_{\rm{T,1}}$ intervals ([100, 126]
GeV/c and [158, 199] GeV/c). We find that the calculations by the SHELL model
give a satisfactory description of the recently reported ATLAS data for almost
all four centrality bins and two $p_{\rm{T,1}}$ intervals, only overestimate
the data at $x_{\rm J}\sim$1 for $158<p_{\rm{T,1}}<199$ GeV/c in the central
$0-10\%$ collisions. Furthermore, our theoretical results show a more balanced
$x_{\rm J}$ distribution for higher $p_{\rm T}$ dijets ($158<p_{\rm{T,1}}<199$
GeV/c) than the lower one, which is consistent with the trend observed in the
ATLAS measurements. To figure out the collision centrality and jet $p_{\rm T}$
dependence of the medium modification on the dijet $x_{\rm J}$ distributions,
we also compare the dijet $x_{\rm J}$ distributions in Xe+Xe collisions with
their p+p baseline for two $p_{\rm T}$ intervals as shown in Fig. 2, and the
ratios of XeXe/pp are shown in the bottom panels. The $x_{\rm J}$
distributions are observed to shift towards smaller $x_{\rm J}$ values in
Xe+Xe collisions compared to their p+p baseline, especially for the most
central collisions. The phenomenon that the dijet transverse momentum gets
more imbalanced in A+A collisions due to the asymmetric energy loss suffered
on the leading and subleading jets as the dijet traversing the QGP Qin:2010mn
; Milhano:2015mng  has been observed by the previous measurements in Pb+Pb
collisions at 2.76 TeV ATLAS:2017xfa  and 5.02 TeV ATLAS:2022zbu . In the
central collisions, such asymmetric energy loss between the leading and
subleading jets can be more significant due to the larger medium size and
higher temperature than the peripheral case. In addition, we observe slightly
weaker $x_{\rm J}$ modification for higher $p_{\rm T}$ dijets than the lower
one, which is consistent with the expected. We can imagine that the $x_{\rm
J}$ of dijets with very high $p_{\rm T}$ is relatively difficult to be
influenced by the medium effects compared to the one with low $p_{\rm T}$.

Figure 3:  (Color online) Calculated normalized $x_{\rm J}$ distributions in
p+p (top) and $0-10\%$ Xe+Xe (middle) collisions as well as their ratios
(bottom) for: inclusive, gluon-gluon, quark-gluon, and quark-quark dijets
(left column); quark-jet-leading and gluon-jet-leading of quark-gluon dijets
(right column). The $x_{\rm J}$ distributions of quark-gluon dijets (dashed
green line) in the right column are the same as in the left column.

The flavor dependence of the medium modification of dijet $p_{\rm T}$ balance
is also an interesting topic in heavy-ion collisions. Due to the different
color factors ($C_{A}=3$ for gluons, $C_{F}=4/3$ for quarks), the gluon-
initiated jets are expected to lose more energy than the quark-initiated one
as traversing the hot and dense nuclear matter, and the “dead-cone” effect
also leads to smaller energy loss of heavy quarks than that of the light one
Gyulassy:2003mc ; Wang:1998bha ; Dokshitzer:2001zm ; Zhang:2003wk ;
Armesto:2003jh ; Djordjevic:2003qk . An optimal step toward this goal is
determining the flavor of the selected jets in nucleus-nucleus collisions with
an infrared and collinear (IRC) safe jet algorithm. In recent years, four IRC-
safe jet-algorithm-based approaches have been developed to identify the flavor
of the underlying hard partons for the final-state reconstructed jets, called
“flavor-$k_{\rm T}$” Banfi:2006hf , “flavor anti-$k_{\rm T}$” Czakon:2022wam ,
“flavor-dressing” Gauld:2022lem  and “Interleaved Flavor Neutralisation (IFN)”
Caola:2023wpj . In this work, we employ the IFN algorithm to identify the
flavor of the reconstructed jets at parton level for the gluon-jets, light-
quark jets and the heavy-flavor jets in both p+p and nucleus-nucleus
collisions. Note that, since the flavor information of the jets is accessible
at each stage of the clustering sequence, the IFN algorithm provides a
consistent flavor identification both to the full jets and their
substructures.

By utilizing the IFN algorithm, the inclusive dijet events can be classified
into three categories: gluon-gluon ($gg$), quark-gluon ($qg$), and quark-quark
($qq$). In particular, by distinguishing the flavor of the leading jet, the
$qg$ dijets can be further divided into $q_{1}g_{2}$ and $g_{1}q_{2}$,
denoting the quark-jet-leading and gluon-jet-leading quark-gluon dijets,
respectively. In the left column of Fig. 3, we show the normalized $x_{\rm J}$
distributions of $gg$, $qg$, $qq$, and inclusive dijets in both p+p and
$0-10\%$ Xe+Xe collisions at $\sqrt{s_{\rm NN}}=5.44$ TeV, and the ratio of
XeXe/pp is plotted in the bottom panel. We find that $gg$ dijets have a more
imbalanced initial distribution than $qq$. Quark and gluon experience
different parton shower processes in vacuum due to their different color
factors and splitting functions Bierlich:2022pfr ; Sjostrand:2006za . In the
bottom panel, it is observed that the $qq$ and $gg$ dijets have almost the
same medium modification, while the $qg$ dijets have the stronger suppression
near $x_{\rm J}\sim 1$ and enhancement near $x_{\rm J}\sim$ 0 than the others.
The different color charge carried by the leading and subleading jets of $qg$
dijets leads to enhanced asymmetric energy loss in Xe+Xe collisions relative
to the $qq$ and $gg$ one. However, we note that the $qg$ dijets contain two
types of subsets, $q_{1}g_{2}$ and $g_{1}q_{2}$, which may have different
modification patterns. In the right column of Fig. 3, we observe stronger
enhancement at small $x_{\rm J}$ and also stronger suppression at $x_{\rm
J}\sim$1 on the $x_{\rm J}$ distribution of $q_{1}g_{2}$ dijets than that of
$g_{1}q_{2}$. Compared to the $g_{1}q_{2}$ dijets, the leading jet of
$q_{1}g_{2}$ dijets loses less energy while the subleading one loses more. In
other words, the flavor configuration of $q_{1}g_{2}$ dijets makes them
experience more significant asymmetric energy loss than $g_{1}q_{2}$ as
traversing the QGP medium.

Figure 4:  (Color online) Calculated fraction of subset dijets in inclusive
dijets as the function of $x_{\rm J}$ in p+p (top) and $0-10\%$ Xe+Xe (middle)
collisions as well as their differences (bottom) for: gluon-gluon, quark-
gluon, and quark-quark dijets (left column); quark-jet-leading and gluon-jet-
leading quark-gluon dijets (right column). The fraction distributions of
quark-gluon dijets in the right column (dashed green line) are the same as in
the left column.

On the other hand, we also estimate the component fractions of dijet sample in
p+p and $0-10\%$ Xe+Xe collisions at $\sqrt{s_{\rm NN}}=5.44$ TeV shown in
Fig. 4, which would be helpful to understand the role of jet flavor played in
the jet-medium interactions. In the left column, we show the fractions of
$gg$, $qg$, and $qq$ in the inclusive dijets in p+p (top) and Xe+Xe (middle)
collisions, as well as their differences (bottom). Firstly, we find that the
$qg$ has the most prominent initial fraction ($\sim 50\%$) in p+p collisions,
and the fraction is increased at small $x_{\rm J}$ but reduced at $x_{\rm
J}\sim 1$ in Xe+Xe collisions. Second, the fraction of $gg$ is overall
reduced. At the same time, that of $qq$ is enhanced because, generally, the
$gg$ dijets lose more energy, which makes it more difficult to survive in jet
selection relative to $qq$. Since the $qg$ has the most significant fraction
in the dijet sample, the enhanced fractions of $qg$ at small $x_{\rm J}$ in
Xe+Xe is the key point that leads to the increased $p_{\rm T}$ imbalance of
inclusive dijets. Furthermore, it is essential to address the fraction changes
of the $q_{1}g_{2}$ and $g_{1}q_{2}$ subsets in A+A collisions. In the right
column of Fig. 4, it is observed that the fractions of $q_{1}g_{2}$ and
$g_{1}q_{2}$ have opposite behaviors in Xe+Xe collisions compared to their
initial values. The fraction of $q_{1}g_{2}$ is significantly enhanced at
small $x_{\rm J}$ after traversing the QGP, while that of $g_{1}q_{2}$ is
decreased at this region.

Figure 5:  (Color online) Averaged $x_{\rm J}$ of the inclusive, $c\bar{c}$,
$b\bar{b}$ dijets (left) and $q_{1}g_{2}$, $g_{1}q_{2}$, $qg$ dijets (right)
in p+p (top) and Xe+Xe (middle) collisions as a function of the collision
centrality, as well as their differences $\Delta\langle x_{\rm J}\rangle$
(bottom).

To quantify the overall shift of the $x_{\rm J}$ distribution in Xe+Xe
collisions relative to p+p, we calculate the average values ($\langle x_{\rm
J}\rangle$) of dijet $x_{\rm J}$ distributions and their differences
($\Delta\langle x_{\rm J}\rangle$) between p+p and A+A collisions, defined as
follows

$\displaystyle\langle x_{\rm J}\rangle$
$\displaystyle\equiv\int\frac{1}{N_{\rm pair}}\frac{dN_{\rm pair}}{dx_{\rm
J}}x_{\rm J}dx_{\rm J},$ (4) $\displaystyle\Delta\langle x_{\rm J}\rangle$
$\displaystyle=\langle x_{\rm J}\rangle_{\rm pp}-\langle x_{\rm J}\rangle_{\rm
AA}.$ (5)

To address the mass dependence of the medium modification of dijet $x_{\rm
J}$, in the left column of Fig. 5 we show the $\langle x_{\rm J}\rangle$ of
inclusive, $c\bar{c}$ and $b\bar{b}$ dijets in p+p and Xe+Xe collisions as a
function of centrality, as well as their differences $\Delta\langle x_{\rm
J}\rangle$. We find that $\Delta\langle x_{\rm J}\rangle$ decreases
monotonously from central to peripheral collisions, as we found in Fig. 2. It
is also observed that the $\Delta\langle x_{\rm J}\rangle$ of these three
kinds of dijets obey the hierarchy $\Delta\langle x_{\rm J}\rangle_{\rm
incl.}>\Delta\langle x_{\rm J}\rangle_{\rm c\bar{c}}>\Delta\langle x_{\rm
J}\rangle_{\rm b\bar{b}}$ in Xe+Xe collisions for the same centrality bin. It
indicates that the massive dijets suffer less asymmetric energy loss in A+A
collisions than the massless light flavor one. The future measurements
focusing on these comparisons will help test the mass effect of jet energy
loss. In the right column of Fig. 5, $\Delta\langle x_{\rm J}\rangle$ of
$q_{1}g_{2}$ and $g_{1}q_{2}$ dijets in Xe+Xe are also plotted, and the former
has significantly large values than the latter for each centrality bin.

Figure 6:  (Color online) Calculated ratios of Xe+Xe and Pb+Pb pair nuclear
modification factors, $\rho_{\rm Xe,Pb}$, evaluated as a function of
$p_{\rm{T,1}}$ (left) and $p_{\rm{T,2}}$ (right) in the same centrality
intervals, and compared to the ATLAS data.

To quantitatively characterize the relative dijet yield suppression between
Xe+Xe and Pb+Pb collisions, the nuclear-modification factors for leading jets,
$\rho_{\rm Xe,Pb}(p_{\rm{T,1}})$ is defined as follows (similarly, $\rho_{\rm
Xe,Pb}(p_{\rm{T,2}})$ can be defined for subleading jets)

$\displaystyle R_{\rm AA}(p_{\rm T,1})$ $\displaystyle=\frac{1}{\langle T_{\rm
AA}\rangle}\frac{d\sigma^{\rm pair}_{\rm AA}/dp_{\rm T,1}}{d\sigma^{\rm
pair}_{pp}/dp_{\rm T,1}},$ (6) $\displaystyle\rho_{\rm Xe,Pb}(p_{\rm T,1})$
$\displaystyle=\frac{R_{\rm XeXe}(p_{\rm T,1})}{R_{\rm PbPb}(p_{\rm T,1})}.$
(7)

In Fig. 6, we show the calculated $\rho_{\rm Xe,Pb}(p_{\rm{T,1}})$ (left) and
$\rho_{\rm Xe,Pb}(p_{\rm{T,2}})$ (right) of dijets for three centrality bins:
$0-10\%$, $10\%-20\%$ and $20\%-40\%$. Note that we use a cut $x_{\rm J}>0.32$
in the calculation to be consistent with ATLAS’s treatment in the
measurements. We observe that the values of $\rho_{\rm Xe,Pb}$ are generally
larger than one for both leading and subleading jets for all centrality bins.
It indicates a weaker yield suppression of dijets in Xe+Xe collisions than
that in Pb+Pb for the same centrality bin, and this phenomenon is still
evident even at peripheral collisions. These findings are consistent with the
previous phenomenological studies Xie:2019oxg ; Li:2021xbd ; Zhang:2022fau .
Since the nucleus of xenon has a smaller radius than that of lead, the system
size and the mean temperature of the QGP medium formed in Xe+Xe collisions is
expected to be smaller than that in Pb+Pb collisions within the same
centrality interval Wang:2023udp . Hence, dijets traverse a longer path-length
medium and experience more effective energy loss in Pb+Pb collisions.

##  IV Conclusion

In this paper, we present the first investigation on the medium modifications
of dijet $p_{\rm T}$ balance ($x_{\rm J}$) in Xe+Xe collisions at
$\sqrt{s_{\rm NN}}=5.44$ TeV. The initial $x_{\rm J}$ distributions of dijets
are calculated by the POWHEG+PYTHIA8 prescription, which matches the NLO QCD
matrix elements with the parton shower effect. The in-medium evolution of
dijets in nucleus-nucleus collisions is described by the SHELL model, which
considers both the elastic and inelastic partonic interactions in the quark-
gluon plasma (QGP). Our theoretical results of the dijet $x_{\rm J}$ in Xe+Xe
collisions exhibit a more imbalanced distribution than that in p+p, consistent
with the recently reported ATLAS data. The dijet becomes increasingly
imbalanced from peripheral to central Xe+Xe collisions, consistent with the
previous measurements in Pb+Pb collisions at the LHC. Furthermore, with the
help of an infrared-and-collinear-safe flavor jet algorithm, we explore the
flavor dependence of medium modification of dijet $p_{\rm T}$ balance in
nucleus-nucleus collisions. We study the respective medium modification
patterns and fraction changes of the $gg$, $qg$, and $qq$ components in the
dijet sample in both p+p and Xe+Xe collisions. We demonstrate that the $qg$
component plays a key role in the increased imbalance of the dijet $x_{\rm
J}$. Especially we find that the $q_{1}g_{2}$ dijets experience more
significant asymmetric energy loss than the $g_{1}q_{2}$ dijets as traversing
the QGP. By comparing the $\Delta\langle x_{\rm J}\rangle$ of inclusive,
$c\bar{c}$ and $b\bar{b}$ dijets in Xe+Xe collisions, we observe
$\Delta\langle x_{\rm J}\rangle_{\rm incl.}>\Delta\langle x_{\rm
J}\rangle_{\rm c\bar{c}}>\Delta\langle x_{\rm J}\rangle_{\rm b\bar{b}}$
consistent with the mass hierarchy of partonic energy loss. In addition, our
calculated nuclear modification factor $\rho_{\rm Xe,Pb}$ of dijets in Xe+Xe
at $\sqrt{s_{\rm NN}}=5.44$ TeV and Pb+Pb at $\sqrt{s_{\rm NN}}=5.02$ TeV are
consistent with the ATLAS data, which indicates that the yield suppression of
dijets in Pb+Pb is more pronounced than that in Xe+Xe due to the larger radius
of the lead nucleus.

Acknowledgments: This research is supported by the Guangdong Major Project of
Basic and Applied Basic Research No. 2020B0301030008, and the National Natural
Science Foundation of China with Project Nos. 11935007, 12035007, 12247127 and
12247132. S. Wang is supported by China Postdoctoral Science Foundation under
project No. 2021M701279.

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