# Unraveling Gluon Jet Quenching through $J/\psi$ Production in Heavy-Ion
Collisions

Shan-Liang Zhang  Key Laboratory of Atomic and Subatomic Structure and Quantum
Control (MOE), Institute of Quantum Matter, South China Normal University,
Guangzhou 510006, China  Guangdong Provincial Key Laboratory of Nuclear
Science, Institute of Quantum Matter, South China Normal University, Guangzhou
510006, China  Guangdong-Hong Kong Joint Laboratory of Quantum Matter,
Southern Nuclear Science Computing Center, South China Normal University,
Guangzhou 510006, China  Jinfeng Liao<EMAIL_ADDRESS>Physics Department
and Center for Exploration of Energy and Matter, Indiana University, 2401 N
Milo B. Sampson Lane, Bloomington, Indiana 47408, USA  Guang-You Qin
<EMAIL_ADDRESS>Key Laboratory of Quark & Lepton Physics (MOE) and
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
China  Enke 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  Guangdong Provincial Key
Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal
University, Guangzhou 510006, China.  Guangdong-Hong Kong Joint Laboratory of
Quantum Matter, Southern Nuclear Science Computing Center, South China Normal
University, Guangzhou 510006, China  Hongxi Xing<EMAIL_ADDRESS>Key
Laboratory of Atomic and Subatomic Structure and Quantum Control (MOE),
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
China  Guangdong Provincial Key Laboratory of Nuclear Science, Institute of
Quantum Matter, South China Normal University, Guangzhou 510006, China
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China  Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern
Physics, Chinese Academy of Sciences, Huizhou 516000, China

( )

###### Abstract

Jet quenching has long been regarded as one of the key signatures for the
formation of quark-gluon plasma in heavy-ion collisions. Despite significant
efforts, the separate identification of quark and gluon jet quenching has
remained as a challenge. Here we show that $J/\psi$ in high transverse
momentum ($p_{\text{T}}$) region provides a uniquely sensitive probe of in-
medium gluon energy loss since its production at high $p_{\text{T}}$ is
particularly dominated by gluon fragmentation. Such gluon-dominance is first
demonstrated for the baseline of proton-proton collisions within the framework
of leading power NRQCD factorization formalism. We then use the linear
Boltzmann transport model combined with hydrodynamics for the simulation of
jet-medium interaction in nucleus-nucleus collisions. The satisfactory
description of experimental data on both nuclear modification factor
$R_{\text{AA}}$ and elliptic flow $v_{2}$ reveals, for the first time, that
the gluon jet quenching is the driving force for high $p_{\text{T}}$ $J/\psi$
suppression. This novel finding is further confirmed by the data-driven
Bayesian analyses of relevant experimental measurements, from which we also
obtain the first quantitative extraction of the gluon energy loss distribution
in the quark-gluon plasma.

Keywords: Relativistic heavy-ion collisions, heavy quarkonium, jet quenching,
elliptic flow.

##  1 Introduction

Since the start of the pioneering high energy nuclear collision experiments at
Relativistic Heavy Ion Collider (RHIC), followed by the cutting edge
measurements at the Large Hadron Collider (LHC), many experimental signatures
have been suggested for the discovery of Quark-Gluon Plasma (QGP), a new form
of matter under extreme high temperature. Among these signatures, jet
quenching has been widely considered as one of the most important probes
Qin:2015srf , which results in the yield suppression of high $p_{\text{T}}$
hadrons and jets Qin:2007rn ; Majumder:2011uk ; ALICE:2010yje , the shift of
two-particle correlations PHENIX:2009cvn ; Zhang:2007ja , the modification of
jet internal structure ATLAS:2019dsv ; Chang:2019sae , as well as azimuthal
anisotropy ($v_{2}$) of hadrons and jets STAR:2003wqp ; CMS:2017xgk  in the
large transverse momentum ($p_{\text{T}}$) region in nucleus-nucleus (AA)
collisions, in comparison with an equivalent number of proton-proton (pp)
collisions.

While comprehensive efforts have been devoted to extract key transport
properties of QGP based on jet quenching frameworks (see e.g. by JET and
JETSCAPE collaborations JET:2013cls ; JETSCAPE:2020shq ), the information on
the jet quenching of specific parton type is still limited. A separate
determination of quark and gluon jet energy loss could play a significant role
in revealing the fundamental color structures of the QGP and testing the color
representation dependence of the jet-medium interaction Frye:2017yrw ;
Gras:2017jty . In particular, such knowledge can help directly access and
verify the unique non-Abelian gauge symmetry of quantum chromodynamics (QCD)
which is a key component of the Standard Model as fundamental laws of nature.
This however proves difficult, as quark and gluon contributions are often
entangled together in final state hadronic observables. A clean method for
identifying quark or gluon energy loss remains a challenge, despite many past
attempts such as the multivariate analysis of jet substructure observables
Chien:2018dfn , the proposal of using the averaged jet charge Chen:2019gqo ;
CMS:2020plq  and electroweak boson tagged jet Zhang:2018urd ; CMS:2017eqd , as
well as data driven analysis of Casimir factors and parametrizations of jet
quenching models Apolinario:2020nyw ; Arleo:2017ntr ; Spousta:2016agr .

In this Letter, we demonstrate that $J/\psi$ production at high $p_{\text{T}}$
can serve as a unique probe of gluon in-medium energy loss. It is well known
that heavy quarkonia production at low $p_{\text{T}}$ have long been studied
as a signature of color deconfinement Matsui:1986dk  and a thermometer of QGP.
Experimental data for low-$p_{\text{T}}$ $J/\psi$ suppression in Pb+Pb
ALICE:2012jsl  and Au+Au collisions PHENIX:2006gsi  are well explained by the
interplay between the dissociation and regeneration of quarkonium states heavy
quark pairs Braun-Munzinger:2000csl ; Thews:2000rj  in hot QGP medium. On the
other hand, $J/\psi$ suppression at large $p_{\text{T}}$ CMS:2016mah ;
ATLAS:2018hqe  was described by the transport models Zhou:2014kka ; He:2021zej
and effective theory approaches Aronson:2017ymv ; Makris:2019ttx . However,
which physics effect “dictates” the suppression of $J/\psi$ in the standard
jet quenching picture at high $p_{\text{T}}$? This question motivates the
present work.

To benchmark the nuclear effect at high $p_{\text{T}}$, we first validate the
pp baseline for $J/\psi$ production by utilizing a theoretical framework
analogous to the gluon fragmentation improved PYTHIA (GFIP) method
Bain:2017wvk , in which the heavy quarkonia are produced by the fragmentation
of shower quarks and gluons. The jet evolution in QGP medium is simulated by
the linear Boltzmann transport (LBT) model He:2015pra ; Cao:2016gvr , which
includes both elastic and inelastic interactions between the fast jet partons
and the medium constituents. We demonstrate, for the first time, that the
suppression of high $p_{\text{T}}$ $J/\psi$ yield in heavy-ion collisions is
dominated by the gluon jet quenching effect. Therefore, measurements of high
$p_{\text{T}}$ $J/\psi$ production can be used as a unique probe to identify
gluon energy loss. Such a novel finding is further verified by the data-driven
Bayesian analysis of the relevant experimental data, from which we perform the
first quantitative extraction of the gluon energy loss distribution in quark-
gluon plasma.

##  2 Theoretical frameworks

In the high transverse momentum region ($p_{\text{T}}\gg m_{\text{c}}$ with
$m_{\text{c}}$ the charm quark mass), one encounters large logarithms of
$p_{\text{T}}^{2}/m_{\text{c}}^{2}$ in fixed order calculations using the
nonrelativistic QCD (NRQCD), a widely adopted approach for studying heavy
quarkonium production. Such large logarithms need to be resumed to preserve
the convergence of perturbation theory. In the leading power of
$p_{\text{T}}^{2}/m_{\text{c}}^{2}$, the cross section for hadroproduction of
$J/\psi$ is schematically given by Collins:1981uw :

$\displaystyle d\sigma[AB\to J/\psi+X]=\sum_{i}d\hat{\sigma}_{AB\to
i+X}\otimes D_{i\to J/\psi},$ (1)

where $d\hat{\sigma}_{AB\to i+X}$ is the cross section for inclusive parton
$i$ in $AB$ collisions, and $D_{i\to J/\psi}$ is the fragmentation function
(FF) for parton $i$ to produce a $J/\psi$ Bodwin:1994jh .

In Eq. (1), a standard method to resum the large logarithms of
$p_{\text{T}}^{2}/m_{\text{c}}^{2}$ is achieved by solving the DGLAP evolution
equation Gribov:1972ri ; Dokshitzer:1977sg ; Altarelli:1977zs of $D_{i\to
J/\psi}$ at scale $\mu_{\text{f}}\sim p_{\text{T}}$, and the corresponding
$d\hat{\sigma}_{AB\to i+X}$ must be evaluated by the convolution of fixed
order hard part coefficient and the parton distribution functions.
Alternatively, the large logs can be effectively resummed using the so-called
GFIP event generator Bain:2017wvk , where the parton $i$ production
$d\hat{\sigma}_{AB\to i+X}$ is simulated in PYTHIA by considering the hard
process followed by parton shower down to scale $2m_{c}$, and the
corresponding $J/\psi$ fragmentation function $D_{i\to J/\psi}$ at $\mu\sim
2m_{\text{c}}$ can be further factorized as follows

$\displaystyle D_{i\to
J/\psi}(z,\mu)=\sum_{n}\hat{d}_{i\to[Q\bar{Q}(n)]}(z,\mu)\langle\mathcal{O}_{[Q\bar{Q}(n)]}^{J/\psi}\rangle,$
(2)

where the quantum numbers $n$ for the intermediate nonrelativistic $Q\bar{Q}$
states are denoted as $n={}^{2S+1}L_{J}^{[1,8]}$ with the superscript $[1]$ or
$[8]$ standing for color singlet or octet state respectively. The short-
distance coefficients $\hat{d}_{i\to[Q\bar{Q}(n)]}$ are perturbatively
calculable within NRQCD and can be found in Refs. Ma:2013yla . Contributions
to $J/\psi$ production from quarks other than $c$ in the hard process are
suppressed, due to two orders of magnitude suppression of light-quark
fragmentation functions Baumgart:2014upa . Therefore, one can neglect their
contribution. $\langle\mathcal{O}_{[Q\bar{Q}(n)]}^{J/\psi}\rangle$ is the
nonperturbative long distance matrix element (LDME) representing the
transition from $[Q\bar{Q}(n)]$ state to $J/\psi$. The GFIP event generator
was shown to give good agreement with analytical calculations at next-to-
leading-log-prime accuracy and remedies the default PYTHIA in describing the
LHCb data for $J/\psi$ production in jets Bain:2017wvk . Here we will extend
the GFIP, as sketched in Eqs. (1,2), to the description of $J/\psi$ production
in high-$p_{\text{T}}$ region. In particular, we use MadGraph Alwall:2014hca
for the hard parton creation and PYTHIA8 for parton shower, with the LDMEs
taken from Bodwin:2014gia . To be consistent with Ref. Bodwin:2014gia  and
ensure the validity of Eq. (1), in principle, the gluon splitting and
fragmentation processes must happen outside the QGP medium. Based on the
formation time argument, the minimum $p_{\text{T}}$ of the gluons (and
similarly $J/\Psi$) is estimated to be $p_{{\rm T,min}}\sim
2m_{\text{c}}^{2}L\sim 50$ GeV for a typical path length $L\sim 3$ fm
(considering the rapid expansion and cooling of the medium). It is interesting
to note that the energy loss of a gluon before splitting into heavy quark
pairs is not very different from the total energy loss experienced by a pair
of splitted heavy quarks. In this paper, we restrict our discussion in the
high $p_{\text{T}}$ region ($p_{\text{T}}>10$ GeV) and focus on direct
$J/\psi$ production. While the feed-down from higher charmonium states
contributes an appreciable fraction in the total prompt $J/\Psi$ production
Faccioli:2008ir , their contribution in jet quenching observable like
$R_{\text{AA}}$ would largely cancel out in the ratio of AA and pp cross
sections.

In relativistic heavy ion collisions, the energetic parton $i$ from hard
interaction encounters multiple scatterings inside the QGP medium before its
fragmentation into high-$p_{\text{T}}$ heavy quarkonium. A Linear Boltzmann
Transport (LBT) model is employed to incorporate both elastic and inelastic
processes for the charm quarks and gluons scattering with medium constituents
He:2015pra ; Cao:2016gvr . For elastic scatterings, the evolution of hard
partons are simulated by the following linear Boltzmann transport equation,

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

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

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

Here $x$ denotes the energy fraction of the radiated gluon relative to a
parent parton with mass $M$, $k_{\perp}$ is the transverse momentum. A lower
energy cut-off $x_{\text{min}}=\mu_{\text{D}}/E$ is applied for the emitted
gluon in the calculation. $P(x)$ is the splitting function in vacuum, and
$\tau_{f}=2Ex(1-x)/(k^{2}_{\perp}+x^{2}M^{2})$ is the formation time of the
radiated gluons in QGP. More details on the implementation of the LBT model
with the HT formalism can be found in Refs. He:2015pra ; Cao:2016gvr . In
principle, one can also use other formalisms to calculate medium-induced gluon
radiation and parton energy loss, such as BDMPS-Z Baier:1996kr ;
Zakharov:1996fv , ASW Wiedemann:2000za , GLV Gyulassy:2000er , AMY
Arnold:2001ms . For a comprehensive comparison of various energy loss
formalisms, the reader is referred to Ref. Armesto:2011ht . The dynamic
evolution of bulk medium is given by 3+1D CLVisc hydrodynamical model
Pang:2012he  with parameters fixed by reproducing hadron spectra from
experimental measurements. In LBT model, the strong coupling constant
$\alpha_{\text{s}}$ is the only free parameter that controls the strength of
parton-medium interaction. We follow the previous study Zhang:2018urd  and set
$\alpha_{s}=0.18$; There is no additional parameter in our calculation. Thus,
the jet transport parameter $\hat{q}$ is the same as previous LBT calculations
for various jet quenching observables, such as light and heavy flavor hadrons
suppression, single inclusive jets suppression, as well as boson-jet
correlation He:2015pra ; Cao:2016gvr ; Zhang:2018urd .

##  3 Numerical results

We first examine the $J/\psi$ production mechanism in p+p collisions. In Fig.
1 (upper panel), we show the transverse momentum spectra of $J/\psi$
production in p+p collisions at 5.02 TeV and 7 TeV from our GFIP simulations,
which compare well with ATLAS ATLAS:2018hqe  and CMS CMS:2017uuv ; CMS:2015lbl
measurements at large $p_{\text{T}}$. The lower panel shows the relative
contributions, $f_{i\rightarrow J/\psi}=\sigma_{i\rightarrow
J/\psi}/\sigma^{\text{total}}_{J/\psi}$, from gluon and charm-quark
fragmentation, respectively. One can see that the contribution from charm
fragmentation to the production of $J/\psi$ is less than $15\%$ over a broad
range of $p_{\text{T}}$, indicating that high-$p_{\text{T}}$ $J/\psi$
production is dominated by gluon fragmentation Baumgart:2014upa ; Liu:2006hc .
This observation is also consistent with the fixed order NRQCD calculations
showing that high-$p_{\text{T}}$ $J/\psi$ is dominated by color-octet
fragmentation Bodwin:2015iua .

Figure 1: (Color online) Transverse momentum spectra of $J/\psi$ production in
p+p collisions from leading power NRQCD calculations and the comparison with
CMS and ATLAS data ATLAS:2018hqe ; CMS:2017uuv ; CMS:2015lbl  . The band is
the uncertainties of LDME. Contributions from different fragmentation channel
are also shown.

In Fig. 2, we present the nuclear modification factor $R_{\text{AA}}$ from our
LBT simulations as a function of $p_{\text{T}}$ for $J/\psi$ production in
0-10$\%$ and 10$\%$-30$\%$ Pb+Pb collisions at 5.02 TeV, compared with CMS
CMS:2017uuv  and ATLAS ATLAS:2018hqe  data. The total $R_{\text{AA}}$ results
(shown in green lines) give a very good description of the experimental data
in high-$p_{\text{T}}$ region for both centrality classes within the current
statistical uncertainties. We further examine the $R_{\text{AA}}$ results
separately from charm quarks and gluons, in blue and red lines, respectively.
A particularly interesting observation is that the strong suppression of high
$p_{\text{T}}$ $J/\psi$ production in AA collisions is mainly driven by the
gluon jet quenching effect. This result can be well explained by the dominance
of gluon fragmentation in $J/\psi$ production (see Fig. 1) combined with the
stronger energy loss for gluon jet than charm jet due to their different color
charges. Such a novel finding is different from the naive expectation that
$J/\psi$ suppression is driven by its constituent charm quarks. Our result
presents a unique opportunity to directly access the gluon jet quenching by
utilizing existing and future heavy ion measurements on $J/\psi$ production at
high $p_{\text{T}}$ region.

Figure 2: (Color online) Nuclear modification factor $R_{\text{AA}}$ evaluated
as a function of $J/\psi$ transverse momentum $p_{\text{T}}$ and the
comparison with CMS CMS:2017uuv  and ATLAS ATLAS:2018hqe  experimental data in
central 0-10$\%$ (top panel) and 10$\%$-30$\%$ (bottom panel) Pb+Pb collisions
at 5.02 TeV.

Another key observable for hard probes is the elliptic anisotropy coefficient
$v_{2}$, which can be evaluated as

$\displaystyle
v_{2}(p_{\text{T}})=\frac{1}{d\sigma^{J/\psi}(p_{\text{T}})}\sum_{i}\int
d\sigma^{i}(\frac{p_{\text{T}}}{z})v_{2}^{i}(\frac{p_{\text{T}}}{z})\otimes
D_{i\rightarrow J/\psi},$ (5)

where $v_{2}^{i}$ is the elliptic flow coefficients for the parent charm
quarks and gluons. As shown in Eq. (5), the elliptic flow coefficient $v_{2}$
of $J/\psi$ should inherit its parent partons that fragment into it. The
numerical results for $v_{2}$ are shown in Fig. 3 and compared to experimental
data CMS:2016mah ; ATLAS:2018xms ; ALICE:2020pvw . Our results show reasonable
agreement with experimental data at high $p_{\text{T}}$ region which though
have large uncertainties. One can see again that the contribution from gluon
quenching effect dominates the $v_{2}$ of high-$p_{\text{T}}$ $J/\psi$.

Figure 3: (Color online) Elliptic flow coefficient $v_{2}$ of $J/\psi$ from
the fragmentation of gluon (Short-dashed red line) and charm (Short-dotted
dark line), and total $J/\psi$ (Solid blue line) as well as the comparison to
experimental measurements CMS:2016mah ; ATLAS:2018xms ; ALICE:2020pvw .

##  4 Bayesian extraction of gluon energy loss distribution

To test the robustness of the above finding based on the framework of leading
power NRQCD and LBT model, we now use an alternative approach, i.e. the data-
driven Bayesian analysis Andrieu , to examine the sensitivity of
high-$p_{\text{T}}$ $J/\psi$ suppression to parton flavors. The Bayesian
analyses have been successfully employed to extract the equation of state of
QCD matter Pratt:2015zsa , heavy quark transport coefficients Xu:2017obm , and
jet energy loss distributions He:2018gks  as well as jet transport coefficient
$\hat{q}$ JETSCAPE:2021ehl  in heavy-ion collisions. Here we also adopt such
an advanced statistical tool to constrain the flavor dependence of energy loss
distribution from relevant experiment data.

To do that, we extend the method in Ref. He:2018gks  to the heavy quarkonium
problem and consider systematically the flavor (charm quark and gluon)
dependence of parton energy loss. The production spectrum can be expressed as
the convolution of its cross section in pp collisions and a flavor-dependent
parton energy loss distribution

$\displaystyle\frac{d\sigma_{\text{AA}}}{dp_{\text{T}}}=\sum_{i}\int\frac{d\Delta
p_{\text{T}}^{i}}{\langle\Delta
p_{\text{T}}^{i}\rangle}\frac{d\sigma_{\text{pp}}^{i}(p_{\text{T}}+\Delta
p_{\text{T}}^{i})}{dp_{\text{T}}}W^{i}(x)\otimes D_{i\rightarrow J/\psi},$ (6)

where $d\sigma_{\text{pp}}^{i}$ is the differential cross section for parton
$i$, $\Delta p_{\text{T}}^{i}$ is the energy loss of parton $i$ with initial
transverse energy $p_{\text{T}}+\Delta p_{\text{T}}^{i}$, $x=\Delta
p_{\text{T}}^{i}/\langle\Delta p_{\text{T}}^{i}\rangle$ is the scaled energy
loss, and the averaged parton energy loss is parametrized as $\langle\Delta
p_{\text{T}}^{i}\rangle(p_{\text{T}}^{i})=\beta_{i}({p_{\text{T}}^{i}}/{p_{\text{T}}^{0}})^{\gamma_{i}}\log({p_{\text{T}}^{i}}/{p_{\text{T}}^{0}})$
with $p_{\text{T}}^{0}=1$ GeV. One can see that parton energy loss directly
controlls the quenching of high $p_{\text{T}}$ hadrons; it is also closely
related to jet transport coefficient $\hat{q}$ (see e.g., Ref. Baier:1996sk ).
In Eq. (6), we assume a general functional form for the scaled energy loss
distribution of parton $i$ He:2018gks :

$W^{i}(x)=\frac{\alpha_{i}^{\alpha_{i}}x^{\alpha_{i}-1}e^{-\alpha_{i}x}}{\Gamma(\alpha_{i})}.$
(7)

where $\Gamma$ is the standard Gamma-function, and the above functional form
can be empirically interpreted as the energy loss distribution resulting from
$\alpha_{i}$ number of jet-medium scatterings He:2018gks . In the end, the
problem is then reduced to the determination of six free parameters
[$\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$], with $i$ standing for gluon and
charm quark. It may be noted that the Bayesian analysis here uses specific
functional form for the parameterzation, thus introducing long-range
correlations in the parameter space which may potentially bias the extracted
parameters. A possible solution to tackle such issue is to use information
field based approach as presented in Ref. Xie:2022ght .

To proceed, a uniform prior distribution $P(\theta)$ in the region
$[\alpha_{i},\beta_{i},\gamma_{i}]\in[(0,10),(0,8),(0,0.8)]$ is used for the
Bayesian analysis. We first run $1\times 10^{6}$ burn-in MCMC steps to allow
the chain to reach equilibrium, and then generate $1\times 10^{6}$ MCMC steps
in parameter space. We show in Fig. 4 the density distribution and
correlations of the parameters [$\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$] for
gluon and charm quark from Bayesian fits to experimental data on inclusive
$J/\psi$ suppression in 0-10$\%$ central Pb+Pb collisions at 5.02 TeV. One can
see that the parameters for gluon energy loss are much better constrained than
that for charm quark. This clearly confirms our earlier finding that $J/\psi$
production is particularly sensitive to the gluon energy loss. The analysis
also shows a strong inverse correlation between $\beta_{i}$ and $\gamma_{i}$
parameters for both gluon and charm quark, in consistency with the pattern for
flavor-averaged jet energy loss seen in Ref. He:2018gks .

Figure 4: (Color online) Density distribution (line) of and correlations
(contour) between the parameters $[\alpha_{i},\beta_{i},\gamma_{i}]$ for gluon
(left) and charm quark (right) from Bayesian fits to experimental data on
inclusive $J/\psi$ suppression in 0-$10\%$ central Pb+Pb collisions at 5.02
TeV CMS:2017uuv ; ATLAS:2018hqe .

The final results from Bayesian analysis of CMS CMS:2017uuv  and ATLAS
ATLAS:2018hqe  data for the inclusive $J/\psi$ $R_{\text{AA}}$ as well as the
extracted $R_{\text{AA}}$ separately from gluon and charm quark are shown in
Fig. 5 (upper panels). The extracted average fractional energy loss
$\langle\Delta p_{\text{T}}\rangle/p_{\text{T}}$ as a function of parton
energy $p_{\text{T}}$ and the energy loss distributions $W(x)$ are shown in
Fig. 5 (lower panels), with the obtained parameters for gluon and charm quark
energy loss distribution summarized in Table 1. Fig. 5 also shows the LBT
results which roughly agree with the Bayesian results. Our flavor-dependent
results clearly demonstrate that the high-$p_{\text{T}}$ $J/\psi$ suppression
is mainly sensitive to gluon energy loss. Owing to this sensitivity, the
extracted average fractional energy loss $\langle\Delta
p_{\text{T}}\rangle/p_{\text{T}}$ for gluons are much better constrained than
the charm quark. As for the energy loss distributions $W(x)$, the
uncertainties for gluons and charm quarks are both quite large. We have
checked that by reducing the uncertainties of experimental data by a factor of
two, the gluon energy loss distribution is much better constrained. In short,
the Bayesian analysis provides a robust way to confirm that high
$p_{\text{T}}$ $J/\psi$ suppression is particularly sensitive to gluon jet
quenching and thus allows a quantitative extraction of the in-medium gluon
energy loss.

Figure 5: (Color online) Nuclear modification factor for inclusive $J/\psi$ suppression, the extracted $R_{\text{AA}}$ of gluon and quarks, the extracted fraction of average energy loss $\langle\Delta p_{\text{T}}\rangle/p_{\text{T}}$ and energy loss distributions $W(x)$ from Bayesian fits to experimental data in 0-10$\%$ central Pb+Pb collisions at 5.02 TeV from CMS CMS:2017uuv  and ATLAS ATLAS:2018hqe . The bands are results of Bayesian analysis with one sigma deviation from the average fits of inclusive $J/\psi$ $R_{\text{AA}}$ data. The solid, dotted and dashed lines are from LBT simulations.  Table 1: Parameters $[\alpha_{i},\beta_{i},\gamma_{i}]$ for gluon and charm quark energy loss from Bayesian analysis of experimental data on inclusive $J/\psi$ suppression CMS:2017uuv ; ATLAS:2018hqe  in 0-10$\%$ Pb+Pb collisions at $\sqrt{s_{\text{NN}}}$ = 5.02 TeV .  |  $\alpha$ |  $\beta$ |  $\gamma$
---|---|---|---
Gluon |  5.25$\pm$1.09 |  0.7$\pm$0.07 |  0.37$\pm$0.03
Charm |  6.33$\pm$2.06 |  0.53$\pm$0.19 |  0.36$\pm$0.09

##  5 Summary and discussions

In this work, based on the leading power NRQCD factorization formalism and the
LBT model, we have presented a novel finding, namely: the high-$p_{\text{T}}$
$J/\psi$ production in AA collisions provides a robust probe to the gluon jet
quenching. This finding is supported by the following two key results: (1) The
agreement between the GFIP event generator and the CMS and ATLAS data for pp
collisions, where the gluon fragmentation dominates ($>$ 85$\%$) the
high-$p_{\text{T}}$ $J/\psi$ production over a wide range of $p_{\text{T}}$;
(2) The agreement on the nuclear modification factor $R_{\text{AA}}$ and
$v_{2}$ between the LBT simulation and CMS and ATLAS data for AA collisions,
where we find that the suppression of high-$p_{\text{T}}$ $J/\psi$ production
is mainly driven by the gluon energy loss effect. To further confirm our
finding and to extract quantitatively the parton energy loss distribution, we
have used the data-driven Bayesian analysis for the experimental data on
high-$p_{\text{T}}$ $J/\psi$ suppression. In consistency with the LBT
simulation, the extracted $R_{\text{AA}}$ is dominated by the gluon energy
loss. We have quantitatively extracted, for the first time, the fraction of
average energy loss $\langle\Delta p_{\text{T}}\rangle/p_{\text{T}}$ as well
as the energy loss distributions $W(x)$ for the gluons in QGP. These results
help explain the latest high-$p_{\text{T}}$ $J/\psi$ data, advance the efforts
to separately extract gluon energy loss, and mark an important step toward
probing the fundamental color structures of QGP medium. One interesting future
direction is to combine high-$p_{\text{T}}$ $J/\psi$ suppression with open
heavy flavor and light hadron suppressions to quantify quark versus gluon
energy loss distributions and test the parton-type dependence of jet-medium
interactions Arleo:2017ntr .

Acknowledgments

We thank W. Chen for providing 3+1D hydro profile of the bulk medium in our
LBT simulation. This research is supported by National Natural Science
Foundation of China (NSFC) under Grants No. 12035007, No. 12022512, No.
12147131, No. 12225503, No. 11890710, No. 11890711, No. 11935007, by Guangdong
Major Project of Basic and Applied Basic Research No. 2020B0301030008, S.Z. is
supported by the MOE Key Laboratory of Quark and Lepton Physics (CCNU) under
Project No. QLPL2021P01. J.L. is supported by NSF Grant No. PHY-2209183.

Author contributions

Hongxi Xing perceived the main idea and designed the overall project. Shan-
Liang Zhang performed the calculations and simulations. All of the authors
participated in the scientific discussion and interpretation of the research
study, and the writing of the manuscript.

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