# Parton splitting scales of reclustered large-radius jets in high-energy
nuclear collisions

Shan-Liang Zhang  Guangdong Provincial Key Laboratory of Nuclear Science,
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
China.  Key Laboratory of Quark & Lepton Physics (MOE) and Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China.  Meng-Quan Yang  Key Laboratory of Quark & Lepton Physics (MOE) and
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
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  Guangdong Provincial Key Laboratory of
Nuclear Science, Institute of Quantum Matter, South China Normal University,
Guangzhou 510006, China.

( )

###### Abstract

We carry out the first theoretical investigation on yields and the hardest
parton splitting of large-radius jets reclustered from small radius ($R=0.2$)
anti-$k_{t}$ jets in Pb+Pb collisions, and confront them with the recent ATLAS
measurements. The Linear Boltzmann Transport (LBT) model is employed for jet
propagation and jet-induced medium excitation in the hot-dense medium. We
demonstrate that, with their complex structures, the medium suppression of the
reclustered large radius jets at $R=1$ is larger than that of inclusive
$R=0.4$ jets defined conventionally. The large radius jet constituents are
reclustered with the $k_{t}$ algorithm to obtain the splitting scale
$\sqrt{d_{12}}$, which characterizes the transverse momentum scale for the
hardest splitting in the jet. The large-radius jet production as a function of
the splitting scale $\sqrt{d_{12}}$ of the hardest parton splitting is overall
suppressed in Pb+Pb relative to p+p collisions due to the reduction of jets
yields. A detailed analyses show that the alterations of jet substructures in
Pb+Pb also make significant contribution to the splitting scale
$\sqrt{d_{12}}$ dependence of the nuclear modification factor $R_{AA}$.
Numerical results for the medium modifications of the jet splitting angle
$\Delta R_{12}$ and the splitting fraction $z$ are also presented.

###### pacs:

13.87.-a; 12.38.Mh; 25.75.-q

##  I introduction

Jet quenching has long been proposed as an excellent probe of the properties
of the Quark-Gluon Plasma(QGP) Wang:1991xy ; Wang:1998ww , formed in high-
energy heavy-ion collisions, such as those experiments at the Large Hadron
Collider (LHC) Aad:2010bu ; Chatrchyan:2012gt ; Chatrchyan:2013kwa ;
Aad:2014bxa  and the Relativistic Heavy Ion Collider(RHIC)Adcox:2001jp ;
Adler:2002xw ; Adler:2002tq . Parton energy loss in the dense QCD medium can
lead to suppression of the single inclusive hadron spectra at large transverse
momentum Adcox:2001jp ; Adler:2002xw ; Adler:2002tq ; Qin:2007rn ; Chen:2011vt
; Buzzatti:2011vt ; Majumder:2011uk ; Aamodt:2010jd ; CMS:2012aa , shift of
$\gamma$-hadron Wang:1996yh ; Renk:2006qg ; Zhang:2009rn ; Qin:2009bk ;
Adare:2009vd ; Abelev:2009gu ; Chen:2017zte  and dihadron transverse momentum
asymmetry Zhang:2007ja ; stardihadron ; Ayala:2009fe ; Ayala:2011ii  in high-
energy heavy-ion collisions. Phenomenological studies of experimental data on
these observables at RHIC and LHC have provided important constraints on the
properties of QGP. Jet transport coefficient $\hat{q}$ indicating the
transverse momentum broadening squared per unit distance due to jet-medium
interaction was extracted from a set of single inclusive hadron spectra by JET
Collaboration Burke:2013yra  and from other observables Dai:2017piq ;
Dai:2017tuy ; Ma:2018swx ; Xie:2019oxg ; Andres:2016iys ; JETSCAPE:2021ehl ;
JETSCAPE:2020shq  .

The fragmentations of large transverse momentum ($p_{T}$) quarks or gluons
result in groups of angularly-correlated particles, referred to as jets, which
pave a new way to study the interactions between jet partons and the medium
constituents. Production and suppression of fully reconstructed single jets
Aad:2014bxa ; Adam:2015ewa ; Khachatryan:2016jfl ; Wang:2016fds ;
Chien:2015hda ; Casalderrey-Solana:2014bpa ; Tachibana:2017syd ; Chen:2019gqo
; Yan:2020zrz ; He:2020iow ; He:2018gks ; CMS:2012ulu ; ALICE:2013dpt , di-
jets Aad:2010bu ; Chatrchyan:2011sx ; Vitev:2009rd ; Qin:2010mn ; Young:2011qx
; He:2011pd ; Renk:2012cx  and $\gamma/Z^{0}$-jets Dai:2012am ; Neufeld:2010fj
; Kang:2017xnc ; Chatrchyan:2012gt ; Luo:2018pto ; Zhang:2018urd ;
Sirunyan:2017jic ; Yang:2021qtl  have also been studied in heavy-ion
collisions and they can provide additional constraints on the jet-medium
interaction and jet transport coefficient. Investigations on the jet
substructures: jet shape Vitev:2008rz ; Chatrchyan:2013kwa ; Sirunyan:2018ncy
; Chang:2019sae ; KunnawalkamElayavalli:2017hxo ; Kang:2017mda ; Ma:2013uqa ,
jet fragmentation functions Chatrchyan:2014ava ; Aaboud:2017bzv ;
Sirunyan:2018qec ; Aaboud:2019oac ; Chen:2020tbl , groomed subjets
Sirunyan:2017bsd ; Sirunyan:2018gct ; Acharya:2019djg ; Andrews:2018jcm ;
Casalderrey-Solana:2019ubu , and jet-novel subjet Apolinario:2017qay  put new
insight into the energy-loss process and, ultimately, the properties of the
QGP itself. The modifications of jet fragmentation functions
Chatrchyan:2014ava ; Aaboud:2017bzv ; Sirunyan:2018qec ; Aaboud:2019oac  as
well as jet shape Chatrchyan:2013kwa ; Sirunyan:2018ncy  show that larger
fraction of jet energy is carried by soft particles at large distances from
the jet axis in Pb+Pb collisions due to the strong interactions between jet
partons and medium constituents. Angular distributions of charged particles
around the jet axis in Pb+Pb and p+p collisions are also studied CMS:2018zze ;
CMS:2016qnj ; Aad:2019igg ; Luo:2021hoo , which indicate that yields of
charged particles with low transverse momenta are observed to be increasingly
enhanced, while charged particles with high transverse momenta have a
suppressed yield in Pb+Pb collisions at larger distances. However, those jet
substructures are related to the particle number density which is rather
sensitive to hadronization processes and underlying background contamination
of soft particles. Recently, jet spectra measurements are extended to large
area jets CMS:2021vui , with an anti-$k_{t}$ distance parameter R up to 1.0.
The new data place further constraints on the underlying jet quenching
mechanisms, as well as a challenge to theoretical models with considerations
of hadronization effects and large background. Besides, the grooming technique
has been applied in Sirunyan:2017bsd ; Sirunyan:2018gct ; Acharya:2019djg ;
Andrews:2018jcm ; Casalderrey-Solana:2019ubu  by removing soft wide-angle
radiation from the jet, and so far focuses on the small radius jet.

In this paper we studied jets in Pb+Pb collisions using a large radius, R =
1.0 with a reclustering procedure adopted by the recent ATLAS measurements
ATLAS:2019rmd . The R = 1.0 jets are clustered from small-radius, R = 0.2,
anti-$k_{t}$ jets with $p_{T}>$ 35 GeV/$c$ reconstructed using the standard
procedure described in Sec. III. This procedure limits the impact of the
underlying event but also does not allow recovering the energy transferred
outside the R = 0.2 subjets via jet quenching. The $k_{t}$ jet finding
algorithm is used to re-cluster R = 1.0 jet constituents to obtain the $k_{t}$
splitting scales. Therefore, this measurement studies the hardest parton
splittings and is not so sensitive to the hadronization processes.

In this paper, a Monte Carlo event generator PYTHIA8 Sjostrand:2014zea  is
employed to simulate initial hard partons with shower partons, and the
framework of Linear Boltzmann Transport (LBT) Monte Carlo model Li:2010ts ;
Wang:2013cia ; He:2015pra ; Cao:2016gvr ; Cao:2017hhk  is used to study the
interaction and propagation of those partons in hot/dense QGP medium. With
this framework, medium modifications on the large-radius jet production as
well as its hardest parton splitting scale in Pb+Pb collisions relative to
that in p+p collisions are investigated.

The remainder of the paper is organized as follows. After providing a brief
introduction to the LBT model in Sec. II, we show in Sec. III results from LBT
on the large-radius jets as well as its hardest splitting. We present the
averaged jet energy loss responsible for the nuclear modification factor for
large-radius and inclusive jets. The splitting scale $\sqrt{d_{12}}$
characterizing the hardest splitting of the large radius jets is investigated
in the following in Sec. III. The $R_{AA}$ sharply goes down with increasing
$\sqrt{d_{12}}$ for small values of the splitting scale followed by flattening
for larger $\sqrt{d_{12}}$. The medium modifications of $\sqrt{d_{12}}$
distributions result from both the reduction of jet yields and the alteration
of jet substructure. We also demonstrate that large radius jets with small
splitting angle and small fragmentation function are less suppressed. A
summary and some discussions are given in Sec. IV.

##  II The Linear Boltzmann Transport model

A Linear Boltzmann Transport (LBT) model is employed to consider both elastic
and inelastic scattering processes of both the initial jet shower partons and
the thermal recoil partons with the quasi-particle in the QGP medium
Wang:2013cia ; He:2015pra ; Cao:2016gvr . The elastic scattering process is
simulated by the linear Boltzmann transport equation,

$\displaystyle p_{1}\cdot\partial
f_{a}(p_{1})=-\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\int\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\int\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}$
$\displaystyle\frac{1}{2}\sum_{b(c,d)}[f_{a}(p_{1})f_{b}(p_{2})-f_{c}(p_{3})f_{d}(p_{4})]|M_{ab\rightarrow
cd}|^{2}$ $\displaystyle\times
S_{2}(s,t,u)(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})$ (1)

where $f_{i=a,b,c,d}$ are the phase-space distributions of jet shower partons,
$|M_{ab\rightarrow cd}|$ are the corresponding elastic matrix elements which
are regulated by a Lorentz-invariant regulation condition
$S_{2}(s,t,u)=\theta(s>2\mu^{2}_{D})\theta(-s+\mu^{2}_{D}\leq
t\leq-\mu^{2}_{D})$. $\mu_{D}^{2}=g^{2}T^{2}(N_{c}+N_{f}/2)/3$ is the Debye
screening mass. The inelastic scattering is described by the higher twist
formalism for induced gluon radiation as Guo:2000nz ; Zhang:2003yn ;
Zhang:2004qm ; Zhang:2003wk ,

$\frac{dN_{g}}{dxdk_{\perp}^{2}dt}=\frac{2\alpha_{s}C_{A}P(x)\hat{q}}{\pi
k_{\perp}^{4}}\left(\frac{k_{\perp}^{2}}{k_{\perp}^{2}+x^{2}M^{2}}\right)^{2}\sin^{2}\left(\frac{t-t_{i}}{2\tau_{f}}\right).$
(2)

Here $x$ denotes the energy fraction of the radiated gluon relative to parent
parton with mass $M$, $k_{\perp}$ is the transverse momentum. A lower energy
cut-off $x_{min}=\mu_{D}/E$ is applied for the emitted gluon in the
calculation. $P(x)$ is the splitting function in vacuum,
$\tau_{f}=2Ex(1-x)/(k^{2}_{\perp}+x^{2}M^{2})$ is the formation time of the
radiated gluons in QGP. The dynamic evolution of bulk medium is given by 3+1D
CLVisc hydrodynamical model Pang:2012he ; Pang:2014ipa  with initial
conditions simulated from A Multi-Phase Transport (AMPT) model Lin:2004en .
Parameters used in the CLVisc are fixed by reproducing hadron spectra with
experimental measurements. In LBT model, $\alpha_{s}$ is the strong coupling
constant which is served as the only one parameter to control the strength of
parton-medium interaction. Based on the previous studies He:2018xjv , we
choose $\alpha_{s}=0.15$ for the following calculations. LBT model has been
well tested that could provide nice description of a series of jet quenching
measurements, from light and heavy flavor hadrons suppression to single
inclusive jets suppression, as well as bosons-jet correlations He:2015pra ;
Cao:2016gvr ; Luo:2018pto ; Zhang:2018urd .

##  III Results and Analysis

In our calculations, initial hard partons with shower partons are generated by
PYTHIA8 Sjostrand:2014zea , with the infrared (IR) cut-off 1 GeV/$c$, and then
those partons will propagate through the hot-dense medium within LBT
Wang:2013cia ; He:2015pra ; Cao:2016gvr . In order to compare with the
experimental data, the jet reconstruction procedures ATLAS:2019rmd  and the
underlying event subtraction are performed following the ATLAS experiment
ATLAS:2019rmd ; Aad:2012vca . The small-radius jets are reconstructed using
the anti-$k_{t}$ algorithm Cacciari:2008gp  implemented in the FastJet
Cacciari:2011ma  software package with $R$ = 0.2. Those small-radius jets are
accepted in the rapidity range of $|y|<3.0$ and in transverse momentum range
of $p_{T}^{jet}>35$ GeV/$c$. The anti-$k_{t}$ algorithm with distance
parameter $R$ = 1 is used to recluster the small-radius, $R=0.2$ jets. These
resulting jets are then referred to as reclustered large-radius (LR) jets in
the following. In the calculation we impose kinematic cut that
$|y^{jet}|<2.0$.

The distributions of reclustered LR jets as a function of jet transverse
momentum $p_{T}^{jet}$ in both p+p and Pb+Pb collisions are calculated at
$\sqrt{s_{NN}}=5.02$ TeV and compared to experimental data ATLAS:2019rmd , as
presented in the top panel of Fig. 1. Distributions of the inclusive $R=0.4$
jet are also plotted with the corresponding experimental data Aaboud:2018twu .
Our numerical calculations show well agreements with the experimental data for
both inclusive jets and reclustered LR jets in both p+p and Pb+Pb collisions.
An obvious spectra suppression of the inclusive jet as well as the reclustered
LR jet in Pb+Pb collisions relative to p+p collisions can be seen. The nuclear
modification factors of those spectra, $R_{AA}$, evaluated as a function of
jet transverse momentum $p_{T}^{jet}$ are compared to experimental data
ATLAS:2019rmd ; Aaboud:2018twu  in the bottom panel of Fig. 1. Our model
calculations give nice descriptions of the experimental data, and show that
$R_{AA}$ goes up slowly with $p_{T}^{jet}$. It is interesting to observe that
the nuclear modification factor for reclustered LR jets with $R=1.0$ is a bit
smaller than the one of inclusive jets with $R=0.4$, which results from the
fact that with its specific jet-finding algorithm reclustered LR jets exhibit
quite different structures from inclusive jets with conventional definition.

Figure 1: (Color online) Distributions of reclustered LR jets and inclusive
jets in both p+p and Pb+Pb collisions with the comparison against the
experimental data ATLAS:2019rmd ; Aaboud:2018twu  (top). Nuclear modification
factor $R_{AA}(p_{T}^{jet})$ of reclustered LR jets and inclusive jets with
comparison against the experimental data ATLAS:2019rmd ; Aaboud:2018twu
(bottom).

Figure 2: (Color online) Number of the subjet of reclustered LR jets before
and after jet quenching and the comparison with ATLAS preliminary data (top
panel). Fraction of N-subjet of reclustered LR jet in Pb+Pb and p+p collisions
(bottom panel).

Figure 3: (Color online) Fraction transverse momentum loss of reclustered LR
jets as well as inclusive jets as a function of jet transverse momentum in
vacuum (top panel). Ratio of transverse momentum loss of reclusteed LR jets
with two subjets both before and after jet quenching as well as its leading
subjet and subleading constituent to initial LR jet $p_{T}$ evaluated as a
function of the vacuum $p_{T}^{jet}$ at $\sqrt{s_{NN}}=5.02$ TeV (bottom
panel).

We plot the fraction of the subjet multiplicity of LR jet in both p+p and
Pb+Pb collisions in Fig. 2. As can be seen, almost 70$\%$ of reclustered LR
jets have only one subjet, and in this case these reclustered LR jets are just
inclusive jets with $R=0.2$. Besides, compared to p+p collisions, the fraction
of the LR jet with only a single subjet is a bit increased, while the fraction
of large-radius jets with complex substructures is decreased. Those
modifications indicate that jets with one subjet may lose on average less
energy than jets with multiple subjets, and the change of jet substructure due
to jet-medium interactions may also play a role, which will be discussed in
more details later.

To illustrate the transverse momentum and jet cone dependence of jet energy
loss, we calculated the average $p_{T}$ reduction of reclustered LR jets with
$R=1$ as well as inclusive jets with $R=0.2$ and $R=0.4$ in the medium as a
function of jet transverse momentum $p_{T}^{jet}$, shown in Fig. 3. In the
calculations, partons within a distance $R=2.0$ of the reconstructed jet in
each event are identified. These partons are then used for the re-construction
of the vacuum leading jet in p+p collisions with a given jet-cone size R. They
propagate through the hot-dense medium through LBT and then are collected as
the medium modified jets with the same jet algorithm.

The comparison between inclusive $R=0.2$ and $R=0.4$ jets in Fig. 3 shows that
the averaged transverse momentum loss is significantly reduced for inclusive
jets with large jet cone as expected in Vitev:2008rz ; Vitev:2009rd ;
He:2011pd ; Chien:2015hda ; Aad:2012vca . While the averaged transverse
momentum attenuation of reclustered LR jet with $R=1.0$ is a little larger
than that of the inclusive jet with $R=0.2$.

The analysis in Fig. 3 shows that the fraction first increases with
$p_{T}^{jet}$ in the region $p_{T}^{jet}<80$ GeV/$c$, as also illustrated in
Refs. Zhang:2018urd ; Zhang:2018kjl ; Zhang:2021oki , and reaches its peak
value (about 20$\%$) near $p_{T}^{jet}\simeq 80$ GeV/$c$, then goes down with
$p_{T}^{jet}$ in the region $p_{T}^{jet}>80$ GeV/$c$. The downtrend in the
large $p_{T}^{jet}$ region leads to slow increasing nuclear modification
factor with $p_{T}^{jet}$.

We notice that the reclustered LR jet with $R=1$ containing only one subjet is
just the same with inclusive $R=0.2$ jet. However, these reclustered LR jets
with two subjets both before and after jet quenching (denoted as
“Reconstructed $2\rightarrow 2$ subjets ” ), will lose much larger fraction of
their energies than inclusive $R=0.2$ jets. The ratios of average energy loss
$\Delta p_{T}$ of the reclustered LR jet with two subjets both before and
after jet quenching, as well as the leading subjet and subleading constituent
to the initial jet energy $p_{T}^{jet}$ before quenching are shown in the
bottom panel of Fig. 3. Though the leading subjets have relative smaller
energy compared to inclusive $R=0.2$ jets, those small radius jets may lose
relative larger fraction of its energy compared to inclusive $R=0.2$ jets with
larger transverse momentum. Therefore we see that the leading subjets lose
almost the same amount of energy as the inclusive $R=0.2$ jets. Besides, the
subleading constituent will also lose some amount of energy. Consequently, the
energy loss of the reclustered LR jets should be the sum of the individual
subjets, and is found to be larger than that of inclusive $R=0.2$ jets at
fixed $p_{T}^{jet}$. As a result, the nuclear modification factor for the
reclustered LR jets with $R=1.0$ is a bit smaller than that for inclusive
$R=0.2$ jets.

Figure 4: (Color online) Distributions of splitting scale in different
reclustered LR jet transverse momentum intervals in p+p collisions as well as
the comparison with the experimental data ATLAS:2019rmd  (top panel). Nuclear
modification factor evaluated as a function of the splitting scale
$\sqrt{d_{12}}$ in different reclustered LR jet transverse momentum intervals
as well as the comparison with the experimental data ATLAS:2019rmd  at
$\sqrt{s_{NN}}=5.02$ TeV (bottom panel).

To obtain a deeper understanding of the medium modification on jet
fragmentation pattern due to jet-medium interactions, we calculated the medium
modification on the hardest jet splitting processes. We recluster the large-R
jet constituents with $k_{t}$ jet algorithm Ellis:1993tq  which combines the
most soft or the closest parton firstly. Therefore, the last step of the
combining procedure should be the hardest splitting process. The splitting
scales of the hardest splitting process are determined by clustering objects
together according to their distance from each other with the following
distance definition:

$\displaystyle d_{12}=min(p_{T1}^{2},p_{T2}^{2})\cdot\Delta R_{12}^{2}/R^{2},\
\ \ \ $ $\displaystyle\Delta R_{12}=\sqrt{\Delta\phi_{12}^{2}+\Delta
y_{12}^{2}}$ (3)

where $p_{T1}$, $p_{T2}$ are the constituent transverse momentum of the
hardest splitting, $\Delta\phi_{12}$, $\Delta y_{12}$, $\Delta R_{12}$ are the
azimuthal angle, rapidity and distance between the two splitting constituents
respectively. For reclustered LR jets with only one single subjet (SSJ),
$\sqrt{d_{12}}$ is equal to zero. In this work, distribution of
$\sqrt{d_{12}}=0$ is shown in the bin $0<\sqrt{d_{12}}<7$, and donated as
“SSJ” in Fig. 4. The distributions of reclustered LR jets as a function of the
splitting scale $\sqrt{d_{12}}$ are calculated in four $p_{T}^{jet}$ intervals
in p+p collisions and compared to experimental data ATLAS:2019rmd  in the top
panel of Fig. 4. Our numerical results show well agreement with the
experimental data in all $p_{T}^{jet}$ intervals.

The nuclear modification factors evaluated as a function of $\sqrt{d_{12}}$ in
four $p_{T}^{jet}$ intervals are also calculated and compared to the
experimental data in the bottom panel of Fig. 4. Our simulation results can
provide well description to the experimental measurements on the suppression
of the splitting scale $\sqrt{d_{12}}$. The distributions are all suppressed
in Pb+Pb collisions compared to p+p collisions in all $p_{T}^{jet}$ intervals.
The $R_{AA}$ sharply decreases with increasing $\sqrt{d_{12}}$ for small
values of the splitting scale followed by flattening for larger
$\sqrt{d_{12}}$.

We emphasize that the medium modifications on jet $\sqrt{d_{12}}$ spectra are
the combined result of both jet yields reduction and the changes of the jet
substructures in Pb+Pb collisions. The magnitude of parton energy loss in the
hot QGP may significantly decrease the LR jet yields (for instance, see Fig.
2), which then leads to an overall suppression of the nuclear modification
factor $R_{AA}(\sqrt{d_{12}})$. It is noted that the selection bias
Brewer:2021hmh ; Connors:2017ptx ; Renk:2012ve  plays an important role for
the reduction of jet yields, for instance, the numerical analysis shows that
for LR jets with $200<p_{T}^{PbPb}<251$ GeV/c in Pb+Pb at $5.02$ TeV, more
than $60\%$ of them originate from unquenched jets with the same momentum
interval $200<p_{T}<251$ GeV/c, whereas less than $40\%$ of them come from
unquenched jets with $p_{T}>251$ GeV/c. Fig. 5 presents the distributions of
splitting scale $\sqrt{d_{12}}$ with $200<p_{T}^{jet}<251~{}$GeV/c in p+p and
Pb+Pb collisions as well as the contribution fractions to $\sqrt{d_{12}}$
distributions from $2\rightarrow 2$ process and $1\rightarrow 2$ process in
Pb+Pb collisions. Here $2\rightarrow 2$ stands for the process which has $2$
subjets both before and after quenching, and gives leading contribution for
those processes without the change of subjet numbers (or jet substructures),
since $\geq 3$ subjets events are much rarer, whereas $1$ subjet events give
$\sqrt{d_{12}}=0$. The process $1\rightarrow 2$ denotes the process which has
$1$ subjet before quenching but $2$ subjets after quenching. We observe in the
bottom panel of Fig. 5 that $2\rightarrow 2$ process gives $>90\%$
contributions in the region with $\sqrt{d_{12}}>30$ GeV/c. However it is noted
that in small $\sqrt{d_{12}}$ region the contribution of $2\rightarrow 2$
process has been reduced to about $50\%$, and the modifications of jet
substructure (for instance, the process $1\rightarrow 2$) will give
significant contributions ( $\sim 50\%$ when $\sqrt{d_{12}}\sim 10$ GeV/c),
which are indispensable in making a decent description of distributions of the
splitting scale $\sqrt{d_{12}}$ in Pb+Pb collisions. Medium modifications on
jet substructure may alter $\sqrt{d_{12}}$ distribution pattern, eg.
$1\rightarrow 2$ process, increasing the yield of jet with small splitting
scale, leads to less suppression in small splitting scale region, while
$2\rightarrow 1$ process, reducing the yield with large splitting scale, leads
to almost constant suppression in large splitting scale region, as we may show
below.

Figure 5: (Color online) Distributions of splitting scale $\sqrt{d_{12}}$ with
$200<p_{T}^{jet}<251~{}$GeV/c in p+p and Pb+Pb collisions as well as the
contributions from different processes in Pb+Pb (top), and the comparison with
experimental data ATLAS:2019rmd . The relative contribution fractions from the
process $2\rightarrow 2$ (without changes of jet substructures) and
$1\rightarrow 2$ (with changes of jet substructures) to jet $\sqrt{d_{12}}$
spectrum in Pb+Pb (bottom).

According to the definition of the hardest splitting scale, $\sqrt{d_{12}}$ is
strongly correlated with the transverse momentum of soft-subjet
$p_{T}^{Subjet}$(subjet with smaller energy) and $\Delta R_{12}$. To
illustrate the $\sqrt{d_{12}}$ dependence suppression of $R_{AA}$, we
calculated the modification of $\Delta R_{12}$ and the fragmentation scale $z$
in Fig. 6. The normalized distributions of $\Delta R_{12}$ and $z$ as well as
their modification factors are presented in Fig. 6. Both $\Delta R_{12}$ and
$z$ will be 0 for the large-radius jets with a single subjet. In our
calculations, we only show the results of reclustered LR jets of a complex
substructure with more than one subjet, giving a non-zero $\sqrt{d_{12}}$. We
note that jets with distance $\Delta R_{12}<0.2$ may overlap with each other
with a small chance, and could be considered as one jet in most of cases. The
kinematic threshold for the subjets is $p_{T}^{jet}>35$ GeV/$c$, thus
$z_{min}=(p_{T}^{Subjet})_{min}/(p_{T}^{jet})_{max}=35/250=0.14$. Besides,
$p_{T}^{subjet}$ can hardly exceed half of the LR jets transverse momentum
$p_{T}^{jet}$, so the value of $z$ is limited to the range 0.14-0.5.

As can be seen, the $\Delta R_{12}$ in the LR jets with complex substructure
are mostly focused on small distance region. An enhancement in small distance
region and a suppression in large $\Delta R_{12}$ region are found in Pb+Pb
collisions compared to p+p collisions. It indicates that reclustered LR jets
with small splitting angle are less suppressed than those with large splitting
angle, which leads to larger value for $R_{AA}$ in small $\sqrt{d_{12}}$
region than that in large $\sqrt{d_{12}}$ region. The modification factor
sharply decreases with $\Delta R_{12}$ for small values of the splitting
angle, followed by flattening for larger $\Delta R_{12}$ due to the
modification of jet fragmentation pattern, as displayed in Fig. 8, which
ultimately will lead to the $\sqrt{d_{12}}$ dependence of $R_{AA}$ in Fig. 4.

Figure 6: (Color online) Normalized LR jet distributions as a function of the
splitting scale $\Delta R_{12}$ in p+p and Pb+Pb collisions as well as its
modification due to jet-medium interactions (top panel). Normalized
distributions calculated as a function of the splitting variable
$z=p_{T}^{min}/p_{T}^{jet}$ in both p+p and Pb+Pb collisions as well as its
modification in Pb+Pb collisions relative to p+p collisions
$\sqrt{s_{NN}}=5.02$ TeV (bottom panel).

Figure 7: (Color online) Average energy of the hardest splitting in different
reclustered jet transverse momentum intervals in p+p and Pb+Pb collisions as a
function of the splitting scale (top panel). The angle of the hardest
splitting in different reclustered LR jet transverse momentum intervals in p+p
and Pb+Pb collisions as a function of the splitting scale at
$\sqrt{s_{NN}}=5.02$ TeV (bottom panel).

In addition, the modification factor for the normalized fragmentation scale
$z$ is greater than one in small $z$ region and is smaller than unity in large
$z$ region as shown in the bottom panel of Fig. 6. The medium modification of
$z$ distribution comes from two sources: the energy shift of jet and subjet
$p_{T}$ in the QGP, and the medium modifications on the internal structure of
the reclustered LR jet. We may come back to this in detail in Fig. 10 by
presenting more discussions.

In order to investigate how the hardest splitting is modified due to jet-
medium interactions, we need to have a detailed understanding of the structure
of the splitting. The average transverse momentum of soft-subjet
$p_{T}^{Subjet}$(subjet with smaller energy) of the hardest splitting as well
as the angle of the hardest splitting $\Delta R_{12}$ in different reclustered
jet transverse momentum intervals in p+p and Pb+Pb collisions as a function of
the splitting scale $\sqrt{d_{12}}$ are presented in Fig. 7. We see, both the
average subjet transverse momentum $\langle p_{T}^{Subjet}\rangle$ and the
distance between the splitting constituents $\langle\Delta R_{12}\rangle$
sharply increase with increasing $\sqrt{d_{12}}$. There is no surprise that
$\langle p_{T}^{Subjet}\rangle$ and $\langle R_{12}\rangle$ are strongly
correlated to $\sqrt{d_{12}}$, however, further detailed analyses show that
$\langle p_{T}^{Sub,jet}\rangle$ is weakly correlated to $\Delta R_{12}$.
Besides, almost no difference of the values $\langle p_{T}^{Subjet}\rangle$
and $\langle\Delta R_{12}\rangle$ between p+p and Pb+Pb collisions with the
same jet transverse momentum $p_{T}^{jet}$ can be observed. While the $\langle
p_{T}^{Subjet}\rangle$ and $\langle\Delta R_{12}\rangle$ are quite different
in different $p_{T}^{jet}$ intervals. $\langle p_{T}^{Subjet}\rangle$ is much
larger for higher energy LR jets, while $\langle\Delta R_{12}\rangle$ is much
smaller for larger energy jets.Therefore, it is the change of the yields of
the value of $p_{T}^{Subjet}$ and $\Delta R_{12}$ that lead to the deviation
of the nuclear modification factor from unity in Pb+Pb collisions relative to
p+p collisions.

Figure 8: (Color online) Contributions from different processes to reclustered
LR jet distributions as a function of the hardest jet splitting angle $\Delta
R_{12}$ before jet quenching(BQ) and after jet quenching(AQ) respectively at
$\sqrt{s_{NN}}=5.02$ TeV(top). Ratio of distributions from different processes
AQ to the distribution BQ(bottom). Detailed descriptions are in the text.

To further explore the medium modification on the hardest jet splitting
pattern, we calculate the contributions from different processes to both
vacuum and medium-modified reclustered LR jet distributions as a function of
the jet splitting angle $\Delta R_{12}$ and show the results in Fig. 8. The
picture shows the distribution of reclustered LR jets with
$200<p_{T}^{jet}<251$ GeV/$c$ after propagating through the hot-dense medium.
The fragmentation pattern of reclustered LR jets with single subjet(denoted as
“1”), giving zero $\Delta R_{12}$, and multi-subjet( donated as “$(\geq 2)$”)
processes,giving non-zero $\Delta R_{12}$, are quite different. The changes of
the fragmentation patterns are separated into four processes: $1\rightarrow 1$
, $1\rightarrow(\geq 2)$ , $(\geq 2)\rightarrow 1$, $(\geq 2)\rightarrow(\geq
2)$ due to propagating through the hot-dense medium via LBT. We don’t plot the
contributions from $1\rightarrow 1$ in the following.

Compared to the total distribution of the LR jets before jet quenching(BQ),
the total yield of medium-modified LR jets after jet quenching(AQ) with
splitting angle $0.2<\Delta R_{12}<0.35$ is significantly increased, while is
decreased in the region $\Delta R_{12}>0.35$, which is similar with the
results in Fig. 6, indicating that jets with small splitting angle is
enhanced, while jets with large splitting angle is suppressed. The
modification patterns are also similar with the results of JEWEL, QPYTHIA and
PYQUEN with wide angle radiation Apolinario:2017qay .

To illustrate the medium effect on jet structures, the comparison between jet
distribution before (BQ) and after (AQ) jet quenching in $(\geq
2)\rightarrow(\geq 2)$ processes is also shown in Fig. 8. The distribution of
$(\geq 2)\rightarrow(\geq 2)$ AQ is a little suppressed in region $0.2<\Delta
R_{12}<0.25$, while is moderately enhanced in the region $0.25<\Delta
R_{12}<0.5$ relative to $(\geq 2)\rightarrow(\geq 2)$ BQ. And no distinctness
is observed in large angle region. The $\Delta R_{12}$ dependence modification
in $(\geq 2)\rightarrow(\geq 2)$ processes is different with the modification
pattern of total distribution(or in Fig. 6), where large-radius jet yields
with large splitting angle jets will be reduced, and jet yields with small
splitting angle will be significantly increased in Pb+Pb compared to p+p
collisions.

For further illustration of the modification mechanism, we also calculate
other contributions to $\Delta R_{12}$ distributions. On the one hand,
$1\rightarrow(\geq 2$) processes, leading to enahncement of reclustered LR jet
yields with $(\geq 2)$ after jet quenching, have considerable contributions to
the reclustered LR jets distributions in small $\Delta R_{12}$ region,
especially, more than 50$\%$ in the region $0.2<\Delta R_{12}<0.25$ . Those
processes mainly come from the hard gluon radiation, which will form a new
subjet with other particles due to coherent enhancement. The closer to the jet
axis, the more energy will be deposited because of jet-medium interactions.
The yields of those processes sharply decrease with increasing $\Delta
R_{12}$.

On the other hand, some subjets will loss energy, leading to that their final
transverse momentum fall below the kinematic cut. As a consequence of which,
some reclustered LR jets with complex substructures in vacuum will have only
one single subjet after propagating through the hot-dense medium, like $(\geq
2)\rightarrow 1$ process. Those processes will reduce reclustered LR jet
yields. The contribution of those processes is widely distributed in the whole
space and leads to the reduction of absolute yields of jet with complex
substructures due to jet-medium interactions, which therefore gives rise to
the suppression of the reclustered LR jets in large splitting angle $\Delta
R_{12}$ region. And the contribution of $1\rightarrow(\geq 2)$ processes is
much larger than that of $(\geq 2)\rightarrow 1$ processes in small angle
region, which results in the enhancement of the reclustered LR jets in small
splitting angle region.

Figure 9: (Color online) Medium modification on the hardest LR jet splitting
angle $\Delta R_{12}$ from $(\geq 2)\rightarrow(\geq 2)$ processes to
reclustered LR jet distributions as a function of the hardest jet splitting
angle $\Delta R_{12}$ of medium-modified reclustered LR jets at
$\sqrt{s_{NN}}=5.02$ TeV.

To go a step further to see the modification of $(\geq 2)\rightarrow(\geq 2)$
processes, we calculate the contributions from $(\geq 2)\rightarrow(\geq 2)$
processes to reclustered LR jet distributions as a function of the hardest jet
splitting angle $\Delta R_{12}$ of medium-modified large-radius jets that
originate from different $\Delta R_{12}^{p+p}$ regions and show the results in
Fig. 9. As mentioned above, large fraction of subjets with very high energy
would radiate hard gluons which would form a new subjet due to coherent
enhancement. The splitting angle of those large-radius jets will be changed
due to the reclustering. As can be seen in Fig. 9, large fraction of jets with
splitting angle $0.2<\Delta R_{12}<0.25$ in vacuum will be much more broadened
after propagating through the medium. While those jets with splitting angle
$\Delta R_{12}>0.5$ will be widely distributed, especially in small $\Delta
R_{12}$ region because the radiated subjets are mainly deposited in
$0.2<\Delta R_{12}<0.35$ region.

Figure 10: (Color online) Contributions from different processes to
reclustered LR jet distributions as a function of the hardest jet splitting
function z of vacuum and medium-modified reclustered LR jets at
$\sqrt{s_{NN}}=5.02$ TeV.

Finally, we present the contributions from different processes to reclustered
LR jet distributions as a function of the jet splitting variable $z$ for both
vacuum and medium-modified reclustered LR jets in Fig. 10. The $z$
distribution after jet quenching is moderately enhanced in small $z$ region
and suppressed in large $z$ region in $(\geq 2)\rightarrow(\geq 2)$ processes
compared to the jets before propagating though the LBT. To clarify the medium
modification of $z$ distribution, we express the splitting variable $z$ in
medium as:

$\displaystyle z^{AQ}$ $\displaystyle=$
$\displaystyle\frac{p_{T,AQ}^{SubJet}}{p_{T,AQ}^{Jet}}=\frac{p_{T,BQ}^{SubJet}\cdot(1-f^{SubJet})}{p_{T,BQ}^{Jet}\cdot(1-f^{Jet})}$
(4) $\displaystyle=$ $\displaystyle z^{BQ}\frac{(1-f^{SubJet})}{(1-f^{Jet})}$

where $z^{BQ}$ and $z^{AQ}$ are the splitting variable $z$ before and after
jet quenching, respectively. $f^{SubJet}$ and $f^{Jet}$ are energy loss
fractions of the reclusted LR jet and its subjet as shown in Fig. 3. In the
very small $z$ region with relatively low subjet $p_{T}$, we have
$f^{SubJet}<f^{Jet}$, thus find $z^{AQ}>z^{BQ}$. Whereas at very large $z$
region with high subjet $p_{T}$, we observe $f^{SubJet}>f^{Jet}$ (see Fig. 3),
which then gives $z^{AQ}<z^{BQ}$.

Furthermore, we study the contribution from $1\rightarrow(\geq 2)$ processes
which increase the jet yields, and $(\geq 2)\rightarrow 1$ processes which
reduce the jet yields. As can be seen, the contribution from
$1\rightarrow(\geq 2)$ processes is larger than that from $(\geq 2)\rightarrow
1$ processes in the whole region, leading to an overall enhancement of jet
yields.

Combining the contributions of medium modifications from three channels:
$(\geq 2)\rightarrow(\geq 2)$ processes, $(\geq 2)\rightarrow 1$ processes,
and $1\rightarrow(\geq 2)$ processes, we find that both the jet energy loss
and the change of jet structure will give rise to the enhancement of jet yield
in the small $z$ region, leading to large deviation in small $z$ region. While
jet energy loss reduces the jet yield in large $z$ region, and the change of
jet structure increases the jet yield. These two mechanisms compete with each
other in large $z$ region, and give a moderate difference in large $z$ region
in Pb+Pb collisions compared to p+p collisions, eventually lead to the
$\sqrt{d_{12}}$ dependence of the nuclear modification factors.

##  IV Summary

We have carried out the first detailed theoretical investigation of the medium
modification on the reclustered LR jets production as well as its the hardest
parton splitting in Pb+Pb collisions relative to that in p+p collisions. The
nuclear modification factor of the reclustered LR jets evaluated as a function
of jet transverse momentum is a little smaller than the value of inclusive
$R=0.4$ jets. A quantitative calculation of the absolute amount of the
transverse momentum missing in the medium shows that reclustered LR jets will
lose larger fraction of its energy than inclusive $R=0.4$ jets with the same
transverse momentum due to its complex structure. The fraction of energy loss
via jet quenching rapidly increases in the region $p_{T}^{jet}<80$ GeV/$c$,
and smoothly decreases with increasing $p_{T}^{jet}$ when $p_{T}^{jet}>80$
GeV/$c$. As a result of which, the nuclear modification factor for reclustered
LR jet is a little smaller than that of inclusive $R=0.4$ jet and increases
smoothly with increasing $p_{T}^{jet}$.

The jet spectrum evaluated as a function of the splitting scale
$\sqrt{d_{12}}$ of the hardest parton splitting obtained from a reclusting
procedure is overall suppressed in Pb+Pb collisions relative to p+p collisions
and the nuclear modification factor $R_{AA}$ sharply decreases with increasing
$\sqrt{d_{12}}$ for small values of the splitting scale followed by flattening
for larger $\sqrt{d_{12}}$. The suppression is a result of the reduction of
jet yields as well as the modification on the jet fragmentation pattern. Jet
energy loss dominates the modification in large splitting scale region, while
the change of jet fragmentation pattern via different modification mechanism
has almost $50\%$ contributions in small splitting scales region.
$\sqrt{d_{12}}$ is strongly correlated to the splitting angle $\Delta R_{12}$
and fragmentation function $z$. A detailed calculation of the splitting angle
$\Delta R_{12}$ and fragmentation function $z$ shows that reclustered LR jet
with small splitting angle or small $z$ is less suppressed, which lead to the
$\sqrt{d_{12}}$ dependence of the nuclear modification pattern.

A further investigation on splitting angle $\Delta R_{12}$ from $(\geq
2)\rightarrow(\geq 2)$ processes shows that jet yield is suppressed in small
$\Delta R_{12}<0.25$ region, while is enhanced in $0.25<\Delta R_{12}<0.5$
region, and keeps unmodified in large $\Delta R_{12}$ region. Further, we find
that $(\geq 2)\rightarrow 1$ processes plays a dominant role in the reduction
of jet yields in large $\Delta R_{12}$ region. And the contribution of
$1\rightarrow(\geq 2)$ processes is much larger than that from $(\geq
2)\rightarrow 1$ processes in small angle region, which gives the enhancement
of jet yield in small $\Delta R_{12}$ region.

Finally, we demonstrate that in $(\geq 2)\rightarrow(\geq 2)$ processes the
$z$ distribution in Pb+Pb is moderately enhanced in small $z$ region and
suppressed in large $z$ region. In addition, contribution from
$1\rightarrow(\geq 2)$ processes is larger than that from $(\geq 2)\rightarrow
1$ processes in the whole region, which may moderately increase the jet
yields. The total contributions of these three processes give rise to a large
deviation in small $z$ region and moderate difference in large $z$ region in
Pb+Pb collisions as compared to p+p collisions, and eventually leading to the
$\sqrt{d_{12}}$ dependence of the nuclear modification factors.

Acknowledgments: The authors would like to thank H Zhang, W Dai, T Luo, P Ru,
and G Ma for helpful discussions. This research is supported by Guangdong
Major Project of Basic and Applied Basic Research No. 2020B0301030008, Natural
Science Foundation of China with Project Nos. 11935007 and 11805167.

## References

  * (1) X. N. Wang and M. Gyulassy, Phys. Rev. Lett. 68, 1480 (1992).
  * (2) X. N. Wang, Phys. Rev. C 61, 064910 (2000)
  * (3) G. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett. 105, 252303 (2010).
  * (4) S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 718, 773 (2013);
  * (5) G. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett. 114, no. 7, 072302 (2015)
  * (6) S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 730, 243 (2014)
  * (7) K. Adcox et al., Phys. Rev. Lett. 88, 022301 (2002).
  * (8) C. Adler et al., Phys. Rev. Lett. 89, 202301 (2002).
  * (9) C. Adler et al. [STAR], Phys. Rev. Lett. 90 (2003), 082302
  * (10) G. Y. Qin, J. Ruppert, C. Gale, S. Jeon, G. D. Moore and M. G. Mustafa, Phys. Rev. Lett. 100, 072301 (2008).
  * (11) X. -F. Chen, T. Hirano, E. Wang, X. -N. Wang and H. Zhang, Phys. Rev. C 84, 034902 (2011).
  * (12) A. Buzzatti and M. Gyulassy, Phys. Rev. Lett. 108, 022301 (2012).
  * (13) A. Majumder and C. Shen, Phys. Rev. Lett. 109, 202301 (2012).
  * (14) K. Aamodt et al. [ALICE Collaboration], Phys. Lett. B 696, 30 (2011).
  * (15) S. Chatrchyan et al. [CMS Collaboration], Eur. Phys. J. C 72, 1945 (2012).
  * (16) X. N. Wang, Z. Huang and I. Sarcevic, Phys. Rev. Lett. 77, 231 (1996)
  * (17) T. Renk, Phys. Rev. C 74, 034906 (2006).
  * (18) H. Zhang, J. F. Owens, E. Wang and X.-N. Wang, Phys. Rev. Lett. 103, 032302 (2009).
  * (19) G. Y. Qin, J. Ruppert, C. Gale, S. Jeon and G. D. Moore, Phys. Rev. C 80, 054909 (2009).
  * (20) A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 80, 024908 (2009).
  * (21) B. I. Abelev et al. [STAR Collaboration], Phys. Rev. C 82, 034909 (2010).
  * (22) W. Chen, S. Cao, T. Luo, L. G. Pang and X. N. Wang, Phys. Lett. B 777, 86 (2018)
  * (23) H. Zhang, J. F. Owens, E. Wang and X. N. Wang, Phys. Rev. Lett. 98, 212301 (2007).
  * (24) C. Adler et al., Phys. Rev. Lett. 90, 082302 (2003).
  * (25) A. Ayala, J. Jalilian-Marian, J. Magnin, A. Ortiz, G. Paic and M. E. Tejeda-Yeomans, Phys. Rev. Lett. 104, 042301 (2010)
  * (26) A. Ayala, J. Jalilian-Marian, A. Ortiz, G. Paic, J. Magnin and M. E. Tejeda-Yeomans, Phys. Rev. C 84, 024915 (2011)
  * (27) K. M. Burke et al. [JET Collaboration], Phys. Rev. C 90, no. 1, 014909 (2014)
  * (28) W. Dai, X. F. Chen, B. W. Zhang, H. Z. Zhang and E. Wang, Eur. Phys. J. C 77 (2017) no.8, 571
  * (29) W. Dai, B. W. Zhang and E. Wang, Phys. Rev. C 98 (2018), 024901
  * (30) G. Y. Ma, W. Dai, B. W. Zhang and E. K. Wang, Eur. Phys. J. C 79 (2019) no.6, 518
  * (31) M. Xie, S. Y. Wei, G. Y. Qin and H. Z. Zhang, Eur. Phys. J. C 79 (2019) no.7, 589
  * (32) C. Andrés, N. Armesto, M. Luzum, C. A. Salgado and P. Zurita, Eur. Phys. J. C 76 (2016) no.9, 475
  * (33) S. Cao et al. [JETSCAPE], Phys. Rev. C 104 (2021) no.2, 024905
  * (34) D. Everett et al. [JETSCAPE], Phys. Rev. Lett. 126 (2021) no.24, 242301
  * (35) J. Adam et al. [ALICE], Phys. Lett. B 746 (2015), 1-14
  * (36) V. Khachatryan et al. [CMS], Phys. Rev. C 96 (2017) no.1, 015202
  * (37) X. N. Wang, S. Y. Wei and H. Z. Zhang, Phys. Rev. C 96, no. 3, 034903 (2017)
  * (38) Y. T. Chien and I. Vitev, JHEP 1605, 023 (2016)
  * (39) J. Casalderrey-Solana, D. C. Gulhan, J. G. Milhano, D. Pablos and K. Rajagopal, JHEP 1410, 019 (2014) Erratum: [JHEP 1509, 175 (2015)]
  * (40) Y. Tachibana, N. B. Chang and G. Y. Qin, Phys. Rev. C 95, no. 4, 044909 (2017)
  * (41) S. Y. Chen, B. W. Zhang and E. K. Wang, Chin. Phys. C 44, no.2, 024103 (2020)
  * (42) J. Yan, S. Y. Chen, W. Dai, B. W. Zhangy and E. Wang, Chin. Phys. C 45, no.2, 024102 (2021)
  * (43) Y. He, L. G. Pang and X. N. Wang, Phys. Rev. Lett. 125, no.12, 122301 (2020)
  * (44) Y. He, L. G. Pang and X. N. Wang, Phys. Rev. Lett. 122, no.25, 252302 (2019)
  * (45) S. Chatrchyan et al. [CMS], Phys. Lett. B 712 (2012), 176-197
  * (46) B. Abelev et al. [ALICE], JHEP 03 (2014), 013
  * (47) S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. C 84, 024906 (2011).
  * (48) I. Vitev and B. W. Zhang, Phys. Rev. Lett. 104, 132001 (2010)
  * (49) G. -Y. Qin and B. Muller, Phys. Rev. Lett. 106, 162302 (2011); 108, 189904 (2012).
  * (50) C. Young, B. Schenke, S. Jeon and C. Gale, Phys. Rev. C 84, 024907 (2011).
  * (51) Y. He, I. Vitev and B. -W. Zhang, Phys. Lett. B 713, 224 (2012).
  * (52) T. Renk, Phys. Rev. C 85, 064908 (2012).
  * (53) W. Dai, I. Vitev and B. -W. Zhang, Phys. Rev. Lett. 110, 142001 (2013).
  * (54) R. B. Neufeld, I. Vitev and B. W. Zhang, Phys. Rev. C 83, 034902 (2011)
  * (55) Z. B. Kang, I. Vitev and H. Xing, Phys. Rev. C 96, no. 1, 014912 (2017)
  * (56) T. Luo, S. Cao, Y. He and X. N. Wang, Phys. Lett. B 782 (2018), 707-716
  * (57) S. L. Zhang, T. Luo, X. N. Wang and B. W. Zhang, Phys. Rev. C 98 (2018), 021901
  * (58) A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 119, no.8, 082301 (2017)
  * (59) Z. Yang, W. Chen, Y. He, W. Ke, L. Pang and X. N. Wang, Phys. Rev. Lett. 127 (2021) no.8, 082301
  * (60) I. Vitev, S. Wicks and B. W. Zhang, JHEP 11 (2008), 093
  * (61) A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 122 (2019) no.15, 152001
  * (62) N. B. Chang, Y. Tachibana and G. Y. Qin, Phys. Lett. B 801 (2020), 135181
  * (63) R. Kunnawalkam Elayavalli and K. C. Zapp, JHEP 07 (2017), 141
  * (64) Z. B. Kang, F. Ringer and W. J. Waalewijn, JHEP 07 (2017), 064
  * (65) G. L. Ma, Phys. Rev. C 89 (2014) no.2, 024902
  * (66) S. Chatrchyan et al. [CMS Collaboration], Phys. Rev. C 90, no. 2, 024908 (2014)
  * (67) M. Aaboud et al. [ATLAS Collaboration], Eur. Phys. J. C 77 (2017) no.6, 379
  * (68) A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 121 (2018) no.24, 242301
  * (69) M. Aaboud et al. [ATLAS], Phys. Rev. Lett. 123 (2019) no.4, 042001
  * (70) W. Chen, S. Cao, T. Luo, L. G. Pang and X. N. Wang, Phys. Lett. B 810, 135783 (2020)
  * (71) A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 120 (2018) no.14, 142302
  * (72) A. M. Sirunyan et al. [CMS], JHEP 10 (2018), 161
  * (73) S. Acharya et al. [ALICE], Phys. Lett. B 802 (2020), 135227
  * (74) H. A. Andrews, D. d’Enterria, L. Apolinario, R. A. Bertens, C. Bierlich, M. Cacciari, Y. Chen, Y. T. Chien, L. C. Mendez and M. Deak, et al. J. Phys. G 47 (2020) no.6, 065102
  * (75) J. Casalderrey-Solana, G. Milhano, D. Pablos and K. Rajagopal, JHEP 01 (2020), 044
  * (76) L. Apolinário, J. G. Milhano, M. Ploskon and X. Zhang, Eur. Phys. J. C 78 (2018) no.6, 529
  * (77) A. M. Sirunyan et al. [CMS], JHEP 05 (2018), 006
  * (78) V. Khachatryan et al. [CMS], JHEP 02 (2016), 156
  * (79) A. Luo, Y. X. Mao, G. Y. Qin, E. K. Wang and H. Z. Zhang, [arXiv:2107.11751 [hep-ph]].
  * (80) G. Aad et al. [ATLAS], Phys. Rev. C 100 (2019) no.6, 064901 [erratum: Phys. Rev. C 101 (2020) no.5, 059903]
  * (81) A. M. Sirunyan et al. [CMS], JHEP 05 (2021), 284
  * (82) [ATLAS], ATLAS-CONF-2019-056.
  * (83) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen and P. Z. Skands, Comput. Phys. Commun. 191 (2015), 159-177
  * (84) H. Li, F. Liu, G. l. Ma, X. N. Wang and Y. Zhu, Phys. Rev. Lett. 106, 012301 (2011)
  * (85) X. N. Wang and Y. Zhu, Phys. Rev. Lett. 111, no. 6, 062301 (2013)
  * (86) Y. He, T. Luo, X. N. Wang and Y. Zhu, Phys. Rev. C 91, 054908 (2015) Erratum: [Phys. Rev. C 97, no. 1, 019902 (2018)]
  * (87) S. Cao, T. Luo, G. Y. Qin and X. N. Wang, Phys. Rev. C 94, no. 1, 014909 (2016)
  * (88) S. Cao, T. Luo, G. Y. Qin and X. N. Wang, Phys. Lett. B 777, 255 (2018)
  * (89) X. F. Guo and X. N. Wang, Phys. Rev. Lett. 85, 3591 (2000).
  * (90) B. W. Zhang and X. N. Wang, Nucl. Phys. A 720, 429-451 (2003)
  * (91) B. W. Zhang, E. Wang and X. N. Wang, Phys. Rev. Lett. 93, 072301 (2004)
  * (92) B. W. Zhang, E. Wang and X. N. Wang, Nucl. Phys. A 757, 493 (2005)
  * (93) L. Pang, Q. Wang and X. N. Wang, Phys. Rev. C 86 (2012), 024911
  * (94) L. G. Pang, Y. Hatta, X. N. Wang and B. W. Xiao, Phys. Rev. D 91 (2015) no.7, 074027
  * (95) Z. W. Lin, C. M. Ko, B. A. Li, B. Zhang and S. Pal, Phys. Rev. C 72, 064901 (2005).
  * (96) Y. He, S. Cao, W. Chen, T. Luo, L. G. Pang and X. N. Wang, Phys. Rev. C 99 (2019) no.5, 054911
  * (97) G. Aad et al. [ATLAS], Phys. Lett. B 719 (2013), 220-241
  * (98) M. Cacciari, G. P. Salam and G. Soyez, JHEP 04 (2008), 063
  * (99) M. Cacciari, G. P. Salam and G. Soyez, Eur. Phys. J. C 72, 1896 (2012).
  * (100) M. Aaboud et al. [ATLAS], Phys. Lett. B 790 (2019), 108-128
  * (101) S. L. Zhang, X. N. Wang and B. W. Zhang, [arXiv:2103.07836 [hep-ph]].
  * (102) S. L. Zhang, T. Luo, X. N. Wang and B. W. Zhang, Nucl. Phys. A 982 (2019), 599-602
  * (103) S. D. Ellis and D. E. Soper, Phys. Rev. D 48 (1993), 3160-3166
  * (104) S. Turbide, C. Gale, S. Jeon and G. D. Moore, Phys. Rev. C 72 (2005) 014906
  * (105) J. Brewer, Q. Brodsky and K. Rajagopal, JHEP 02, 175 (2022) [arXiv:2110.13159 [hep-ph]].
  * (106) M. Connors, C. Nattrass, R. Reed and S. Salur, Rev. Mod. Phys. 90, 025005 (2018) [arXiv:1705.01974 [nucl-ex]].
  * (107) T. Renk, Phys. Rev. C 88, no.5, 054902 (2013) [arXiv:1212.0646 [hep-ph]].