# Hadron productions and jet substructures associated with $Z^{0}/\gamma$
in Pb+Pb collisions at the LHC

Shan-Liang Zhang<EMAIL_ADDRESS>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>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.
Ben-Wei Zhang<EMAIL_ADDRESS>Institute of Particle Physics and Key
Laboratory of Quarks and Lepton Physics (MOE), 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 a detailed study of medium modifications on
$Z^{0}$/$\gamma$+hadron correlations as well as jet substructures in
association with $Z^{0}/\gamma$ in Pb+Pb collisions with $\sqrt{s_{NN}}=5.02$
TeV at the LHC. We utilize the Linear Boltzmann transport (LBT) model to
simulate the jet-medium interactions and medium response, and an extended
cluster hadronization model to investigate the non-perturbative transition of
quarks and gluons into final hadrons in heavy-ion collisions. Including
hadronization effect, we can well describe $Z^{0}/\gamma$+hadrons correlations
and $Z^{0}/\gamma$-tagged jet substructures in both p+p and Pb+Pb collisions
simultaneously. Medium modification on jet profile and jet fragmentation
functions indicate that particles carrying a large fraction of the jet
momentum are generally closely aligned with the jet axis, whereas low-momentum
particles are observed to have a much broader angular distribution relative to
jet axis in Pb+Pb collisions due to jet-medium interactions. In particular, we
find that $Z^{0}/\gamma$-tagged hadron correlations are sensitive to the soft
particles from the dense medium and medium response, while jet-substructures
show weak dependence on those soft hadrons with only a fraction of them
falling inside the jet area.

##  I Introduction

A novel state of the QCD matter with deconfined quarks and gluons, the quark-
gluon plasma (QGP), is believed to be produced on a fleeting timescale,
$\tau\sim$1 fm/c, in high energy nucleus-nucleus collisions at the
Relativistic Heavy-ion Collider (RHIC) and the Large Hadron Collider (LHC) .
These quarks and gluons, known as partons, created prior to the formation of
the QGP, traverse the hot-dense medium and experience elastic and inelastic
scatterings with the constituents of the medium. The interactions between the
hard jets and the soft medium usually reduce the energy of the incoming jet
partons, which leads to the suppression of final observed hadron and full jet
yield at high transverse momentum relative to elementary proton-proton (p+p)
collisions at the same colliding energy. This process, called jet quenching,
has been long regarded as one of the best hard probes to study the properties
of QGP Wang:1991xy ; Gyulassy:2003mc ; Qin:2015srf ; Vitev:2008rz ;
Vitev:2009rd ; Qin:2010mn ; CasalderreySolana:2010eh ; Young:2011qx ;
He:2011pd ; ColemanSmith:2012vr ; Zapp:2012ak ; Ma:2013pha ; Senzel:2013dta ;
Casalderrey-Solana:2014bpa ; Milhano:2015mng ; Chang:2016gjp ;
Majumder:2014gda ; Chen:2016cof ; Ru:2014yma ; Dai:2018mhw ; Chen:2019gqo ;
Zhang:2021sua ; Chen:2020pfa .

Early studies of jet quenching focused on high momentum hadrons and there have
been tremendous theoretical/phenomenological studies and experimental
measurements on the suppression of high momentum hadrons Adams:2003kv ;
Adler:2003qi ; Back:2004bq ; Aamodt:2010jd ; CMS:2012aa . In recent years at
the RHIC and especially with the increasing collision energies at the LHC,
productions of full jet, reconstructed from collimated clusters in the final-
state particles within a given radius defined as $\Delta
R=\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}$, such as dijet $p_{T}$
imbalanceATLAS:2010isq ; CMS:2011iwn ; STAR:2016dfv , modifications of the jet
yield in the medium Vitev:2008rz ; Adam:2015ewa ; Khachatryan:2016jfl ;
Aad:2014bxa , electroweak boson-jet correlations Dai:2012am ; Wang:2013cia ;
Chen:2018fqu ; Sirunyan:2017qhf ; Sirunyan:2017jic ; Neufeld:2010fj ;
Neufeld:2012df ; Casalderrey-Solana:2015vaa ; KunnawalkamElayavalli:2016ttl ;
Kang:2017xnc ; Zhang:2018urd ; Zhang:2021oki , missing $p_{T}$ in dijet
systemsCMS:2015hkr , have been intensively explored to extract the properties
of QGP. Though medium modifications on high energy inclusive hadrons are
mainly attributed to the interaction of the leading partons with the medium
constituents, full jets are modified by the interactions of all of the
daughter partons of a given parton with the hot-dense medium, and also by
induced medium response, the result of the excitation of the thermal partons
from jet-medium interaction Cao:2020wlm ; Chen:2017zte . Since full jets are
composite objects with complex internal structures, jet quenching may leave
prints on jet substructures as well as jet yields.

Inclusive jet shape and jet fragmentation functions had been measured by ATLAS
and CMS at the LHC Chatrchyan:2013kwa ; Chatrchyan:2014ava ; Aad:2014wha ;
Chatrchyan:2012gw . Later, $\gamma$ triggered jet shape CMS:2018jco  and
fragmentation function Sirunyan:2018qec ; Chen:2020tbl  have also been
measured and studied, which show different modification patterns compared to
that of inclusive jet Chang:2019sae ; Aaboud:2019oac . The productions of
charged particles in Pb+Pb collisions tagged with $Z^{0}$ boson at
$\sqrt{s}=5.02$ TeV ATLAS:2019gif ; CMS:2020rxf ; Yang:2021qtl  provide
additional information on the jet-medium interaction. It has been noticed that
these jet observables, related to the particle number density and hadron
components, are rather sensitive to hadronization processes and underlying
background contamination of soft particles. In addition, nuclear modification
factors of charged jet recently measured by ALICE alice:jetcone ,
$\gamma$-tagged charged jet at STARstar:jetcone , and inclusive jet at CMS
CMS:2021vui  and at ATLAS ATLAS:2012tjt , show different jet radius $R$
dependence. This very interesting contrast places a challenge to theoretical
models of jet quenching, and also attaches great importance to hadronization
process in heavy-ion collisions, with the quite distinct features between
charged jets and full jets in the measurements.

The above observations motivate us to develop an improved model of non-
perturbative hadronization in high-energy nucleus-nucleus (A+A) collisions, to
interface with parton-jet propagation in hot QCD medium. The non-perturbative
hadronization process in p+p reactions are usually studied via effective
mechanism, such as Lund string fragmentation mechanism Andersson:1983ia  by
PYTHIA and cluster model Gottschalk:1982yt ; Gottschalk:1983fm ;
Gottschalk:1986bv ; Webber:1983if  by SHERPA and HERWIG event generators. In
A+A collisions, it has been shown that hadronizations with recombination play
an important role in describing particle ratios and correlations at the low
$p_{T}$ in relativistic heavy-ion collisions (HIC) Fries:2003vb ; Greco:2003xt
; Hwa:2004ng ; Shao:2004cn . Recently Colored Hadronization Putschke:2019yrg ;
JETSCAPE:2019udz , Colorless Hadronization Putschke:2019yrg ; JETSCAPE:2019udz
have been developed to study particle productions at large transverse
momentum. A hybrid hadronization Kordella:2020nwi ; Zhao:2020wcd  is
introduced as a combination of Lund string fragmentation and recombination. To
investigate heavy-flavor hadron productions in heavy-ion collisions a model
with color reconnections Beraudo:2022dpz  has been proposed with cluster
mechanism, in which partons are no longer connected by strings according to
their color charge in the final stage, in order to minimize the invariant mass
of the strings.

In this paper, we generalize the cluster hadronization mechanism
Gottschalk:1982yt ; Gottschalk:1983fm ; Gottschalk:1986bv ; Webber:1983if  to
non-perturbative particle productions in heavy-ion collisions and develop an
extended cluster model of hadronization, which is then utilized to study
medium modification on $Z^{0}$-hadron correlations and jet substructures. We
simulate the $Z^{0}/\gamma$-tagged parton-jet events in p+p collisions with a
MC event generator SHERPA2.24 Gleisberg:2008ta , and these events will be
embedded into the Linear Boltzmann Transport (LBT) model Li:2010ts ;
He:2015pra ; Cao:2016gvr  to take into account jet quenching effect in the
QGP. In the numerical calculations, we keep track of the whole evolution
history for elastic and inelastic scattering during the jet propagation in hot
dense medium step by step based on LBT model. After leaving the fireball the
escaping partons will hadronize into baryons and mesons via the extend cluster
hadronization model, and we then study in detail $Z^{0}/\gamma$+hadron
correlations as well as $Z^{0}/\gamma$ tagged jet substructures, such as jet
shape, hadron number density, and jet fragmentation function.

The outline of this paper is as follows. In Sec. II , we introduce our
framework, where SHERPA is used to generate the reference jet in p+p
collisions, LBT is applied to study jet evolution and medium response in hot-
dense medium, and the extend cluster hadronization model is adopted to
simulate the transition of partons to hadrons. In Sec. III, we first present
our numerical results for various $Z^{0}/\gamma$+hadron correlations at 5.02
TeV in both p+p and Pb+Pb collisions and compare them with experimental data,
including the azimuthal angle correlations $\Delta\phi_{Z,ch}$, charged hadron
transverse momentum spectra, as well as the fragmentation pattern respect to
the $Z^{0}$ boson. Finally, we study the medium modification on jet
substructures, such as jet shape, charged hadron number density as well as jet
fragmentation functions. A short conclusion is made in Sec. IV.

##  II framework

###  II.1 Jet production in p+p collisions

In our computation, $Z^{0}/\gamma$-tagged jet events in p+p collisions are
simulated with a MC event generator SHERPA2.24 Gleisberg:2008ta , which can
perform next-to-leading order (NLO) matrix element (ME) matched to the
resummation of parton shower (PS) calculations with several merging schemes.
AMEGIC++ Krauss:2001iv  and COMIX Gleisberg:2008fv  are SHERPA’s original
matrix-element generators which provide tree-level matrix-element and create
the phase-space integration. The MC program OpenLoops Cascioli:2011va  is
interfaced with SHERPA to provide the virtual matrix elements. The MEPS@NLO
merging method Hoeche:2009rj ; Hoche:2010kg ; Hoeche:2012yf  is then applied
to yield an improved multi+jets production with NLO matrix elements matched to
the resummation of parton showers Gleisberg:2007md ; Schumann:2007mg . The
parton distribution set“CT14 NLO” Dulat:2015mca  is utilized to obtain the
hard cross section in p+p collisions.

Figure 1:  Jet shape (top panel) and jet fragmentation function (bottom panel)
are calculated at hadron level and parton level by SHERPA and compared to
experimental data in p+p collisions CMS:2018jco ; Sirunyan:2018qec  .

SHERPA has its own fragmentation/hadronization model Gleisberg:2008ta , which
can give nice descriptions on many experimental observables in p+p collisions.
Here we consider two examples of jet substructure observables. Firstly we
consider jet shape, defined as:

$\rho(r)=\frac{1}{\delta r}\frac{\sum_{jets}\sum_{r_{a}\leq
r<r_{b}}(p_{T}^{ch,h}/p^{jet}_{T})}{\sum_{jets}\sum_{0\leq
r<r_{f}}(p_{T}^{ch,h}/p^{jet}_{T})},$ (1)

where $\delta_{r}=r_{b}-r_{a}$ is the width of the annulus of inner and outer
radii $r_{a}$ and $r_{b}$ with respect to the jet axis, $p_{T}^{ch,h}$,
$p^{jet}_{T}$ is the transverse momentum of charged hadrons and jets
respectively, $r=\sqrt{(\phi^{ch,h}-\phi^{j})^{2}+(\eta^{ch,h}-\eta^{j})^{2}}$
is the distance between the charged hadron and jet in the $\eta-\phi$ plane,
$r_{f}$ is used to normalize the jet shape. We fix $r_{f}=\Delta R=0.3$ to
match the setup of CMS measurement CMS:2018jco . in the top panel of Fig. 1 we
plot jet shapes tagged with isolated photons at parton level and hadron level
by SHERPA, which are also confronted with experimental data. It is observed
that though the theoretical results with hadronization effects describe nicely
CMS data in p+p CMS:2018jco , the results at parton level overshoot the
experimental data in the the region $r<$0.05 and undershoot CMS data in the
region $r>0.05$.

Secondly, we investigate jet fragmentation fraction, which presents the
fragmentation pattern of jet constituents with respect to the transverse
momentum of the reconstructed jet and is defined as:

$\xi^{jet}=\ln\frac{|\overrightarrow{p}^{jet}|^{2}}{\overrightarrow{p}^{ch,h}\cdot\overrightarrow{p}^{jet}}$
(2)

The distributions of jet fragmentation at partan level and hadron level by
SHERPA with CMS data in p+p Sirunyan:2018qec  are shown in the bottom panel of
Fig. 1. It shows that numerical results by SHERPA at hadron level give a
decent description of CMS data on the jet fragmentation distribution, whereas
the simulation at parton level is a far cry from the measurement. These two
examples demonstrated clearly that hadronization effects are indispensable in
precise calculations of some jet substructures, especially when related to
particle number density and hadron constituents.

###  II.2 Parton propagation in heavy-ion collisions

When an energetic parton passing through the QGP, it may interact with the
thermal medium and lose energy. In this work, parton propagation and the
corresponding medium response in hot/dense QCD medium due to jet-medium
interactions are simulated by a Linear Boltzmann transport (LBT) model
Li:2010ts ; He:2015pra ; Cao:2016gvr  , based on a Boltzmann equation:

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

$f_{i}$ are phase-space distributions of partons. $S_{2}(s,t,u)$ is Lorentz-
invariant regulation condition to regulate all soft and collinear divergency.
Elastic scattering is introduced by the corresponding $|M_{ab\rightarrow cd}|$
which includes the complete set of leading order 2-2 elastic scattering
processes He:2015pra .

The induced gluon radiation from inelastic scattering is numerically
incorporated into LBT based on High-Twist formalism Guo:2000nz ; Zhang:2003yn
; 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)

in which, $x$ and $k_{\perp}$ give the energy fraction and transverse momentum
of the radiated gluon respectively, $P(x)$ denotes splitting functions.
$\hat{q}$ stands for the jet transport coefficient and is extracted from the
elastic scattering. $\tau_{f}=2Ex(1-x)/(k_{\perp}^{2}+x^{2}M^{2})$
characterizes the formation time of the radiated gluon. The medium information
is provided by 3+1D CLVisic hydrodynamics Pang:2012he ; Pang:2014ipa  and the
initial condition is provided by AMPT Lin:2004en . LBT has been successful in
describing the suppression of experimental data on hadrons He:2015pra ;
Cao:2016gvr , inclusive jets He:2018xjv , $\gamma+$hadron/jets Chen:2017zte ;
Luo:2018pto  correlations and $Z^{0}$+jet production Zhang:2018urd .

The original version of LBT have an excellent description of the momentum
exchange between jet shower partons and hot dense medium. In fact, jet shower
partons not only exchange momentum with the medium partons, but also exchange
color-charge with the medium constituents during its propagation in the
medium. To have the whole evolution history information of a event in Pb+Pb as
that in p+p collisions, we future tracked the color flow of the event during
its propagation in the medium based on LBT model. The color index of shower
partons can be read from SHERPA. While we sample the color index of the
thermal partons randomly. Then we track the color flow as illustrated in
Boos:2001cv  and in the bottom panel of Fig. 2 in elastic scattering by
identifying all types of gluon lines in each elastic process according to the
flavor of the incoming and outgoing partons. Furthermore, we also keep track
of the color flow in gluon radiation process.

###  II.3 Hadronization in p+p and A+A

Once the partons escape the medium or the temperature fall below the
transition temperature, they will quickly hadronize into color neutral bound
states. In this section, we will present an extended cluster model of
hadronization, which generalizes the Cluster model for p+p reactions in SHERPA
to consider hadronization processes in heavy-ion collisions.

For p+p collisions, the non-perturbative transition of these partons into
primary hadronic matter is accomplished similar to the original cluster model
Gottschalk:1982yt ; Gottschalk:1983fm ; Gottschalk:1986bv ; Webber:1983if  by
the following steps:

Figure 2:  Schematic illustration of the procedure used to isolate color
singlet clusters in p+p (top) and A+A(bottom) collisions. $G_{i}$ is the color
index carried by the parton.

  * •

All the partons are intermediate states, so all partons should be brought to
their mass-shell: $M_{(\mu,d)}=0.3$ GeV/$c^{2}$, $M_{s}=0.45$ GeV/$c^{2}$,
$M_{c}=1.5$ GeV/$c^{2}$, $M_{b}=5$ GeV/$c^{2}$, while keeping the parton
momentum unchanged.

  * •

Clusters are formed by quark pairs, so gluons split into $q\bar{q}$ pair
following the probability(splitting) function $f(z)=z^{2}+(1-z)^{2}$, where
$z$ denotes the energy fraction carried by splited quarks. The color flow of
gluon is shared by $q\bar{q}$ correspondingly.

  * •

All final partons are color connected, the color connected quark-antiquarks
are combined to form a color neutral cluster which is uncorrelated to other
clusters and can be seen as an independent string. Schematic illustration of
the procedure used to isolate color singlet is shown in Fig. 2.

  * •

For a cluster C with invariant mass $M_{C}>M_{f}$, where $M_{f}$ is a
parameter need to be fixed via comparison with experimental data, formed from
a (di)quark of momentum $p_{1}$ and an anti(di)quark of momentum $p_{2}$, the
fission C $\rightarrow$ X + Y is assumed to yield

$\begin{split}&p_{x}=(1-x)p_{1}+xp_{2}\\\ &p_{y}=(1-x)p_{2}+xp_{1}\end{split}$
(5)

$x$ is sampled according to the distribution $f(x)=x^{2}+(1-x)^{2}$. The
flavour of the produced pair is taken to be $q\bar{q}$ and $D\bar{D}$, where
$q$ and $D$ represent quark and di-quark. If necessary the process is repeated
for the fission fragments X and Y, until all cluster invariant mass are below
the fission threshold $M_{f}$ Gottschalk:1983fm .

  * •

The colour neutral clusters can decay independently into hadrons. The cluster
which consists of $q_{i}\bar{q_{j}}$ or $q_{i}\bar{D_{j}}$ will be combined
with $q_{i^{\prime}}\bar{q_{j^{\prime}}}$,
$D_{i^{\prime}}\bar{D_{j^{\prime}}}$ coming from vacuum to form hadrons:
$q_{i}\bar{q_{j}^{\prime}}\ \ q_{i^{\prime}}\bar{q_{j}}$,
$q_{i}\bar{D_{j^{\prime}}}\ \ D_{i^{\prime}}\bar{q_{j}}$,
$q_{i}\bar{q_{j^{\prime}}}\ \ q_{i^{\prime}}\bar{D_{j}}$,
$q_{i}\bar{D_{j^{\prime}}}\ \ D_{i^{\prime}}\bar{D_{j}}$. The PDG of the
hadron is determined by the flavor and spin of the daughter quarks(sampled by
hand), as well as the mass of the cluster and the candidate hadron. At
present, only spin 1/2 and 3/2 baryon and spin 0 and 1 meson are considered in
our simulation. The energy and momentum of the final hadrons are the solution
of the energy-momentum conservation equation.

  * •

And then, all the unstable hadrons will decay into final stable particles
according to its properties: branching ratio, number of decay products, and
PDG of decay produces etc… If necessary the process is repeated until all
final particles become stable.

Figure 3: Distribution of cluster invariant mass in p+p and 0-10$\%$ Pb+Pb
collisions of inclusive jet, $\gamma$+jet and $Z^{0}$+jet process at
$\sqrt{s}=5.02$ TeV.

In Pb+Pb collisions, jet shower partons will exchange color-charge with the
medium constituents. As a consequence, some jet shower partons and jet induced
thermal partons may carry the color index which is carried originally by the
initial thermal parton, therefore, those partons are color connected with
medium partons(opposite color index of initial thermal partons) rather than
other shower partons as in p+p collisions due to elastic scatterings as
illustrated in the bottom picture of Fig. 2. While initial thermal partons are
also color connected to other medium constituents with opposite color index.
In the following, jet shower partons and recoil partons are called positive
partons, while thermal partons referred to negative partons as defined in LBT
He:2015pra . Schematic illustration of the procedure used to isolate color
singlet in Pb+Pb collisions is shown in the bottom panel of Fig. 2.

We simulate the hadronization in heavy-ion collisions with the extended
cluster model as:

  * •

We combine the positive (denoted as “posi” in the figures) and negative
(denoted as “nega” in the figures) quarks into clusters separately, similar as
JETSCAPE JETSCAPE:2022jer , and neglect the contributions from mixture of
positive partons with negative partons.

  * •

Those positive partons, which have positive color partners will be treated in
the same approach as in p+p collisions with the original Cluster model. While
those positive partons that do not have positive color partners (color
connected to negative partons) may be combined into clusters with each other
according to the distance $r=\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}$ between
them.

  * •

A negative parton is requested to form a cluster with its closest negative
parton. As a consequence, those constructed clusters, whose daughter partons
are not color connected to its anti-color partners (the later two cases), may
be not color neutral, different from the color neutral clusters in p+p
collisions. Later, these clusters will radiate some soft gluons to become
color neutral.

  * •

The four-momentum of the cluster is the sum of the daughter partons.

  * •

The negative parton distributions will be subtracted form positive parton
distributions within the kinematic region they appear (denoted as “posi+nega”
in the figures) .

In Fig. 3, we present normalized distributions of the cluster invariant mass
$M_{C}=\sqrt{E_{C}^{2}-p_{C}^{2}}$ for three different processes, i.e.,
inclusive jet and $\gamma$+jet as well as Z+jet in p+p and Pb+Pb collisions.
It is illustrated that the $M_{C}$ distributions for three processes in p+p
(or Pb+Pb) overlap with each other, and show negligible dependence on the
processes both in p+p and Pb+Pb, though large deviations of the $M_{C}$
distributions in Pb+Pb from that in p+p are observed. The $M_{C}$
distributions of positive partons in A+A are shifted to smaller value as
compared to p+p, which results from two underlying reasons. First, the
distance of the color connected positive partons will become closer in Pb+Pb
collisions due to medium-induced radiation. Some jet shower partons tend to be
connected to its radiated gluon as illustrated in the bottom panel of Fig. 2.
Second, the clusters formed according to the distance will also have smaller
$M_{C}$. The $M_{C}$ distributions of negative partons in A+A spread mainly in
small $M_{C}$ region because of their much lower energies.

It is noted that the cluster with invariant mass $M_{C}>M_{f}$ may split and
it is found that results with $M_{f}=3.0$ GeV can excellently describe the
experimental data of the correlations between $Z^{0}$ boson and its tagged
charged hadrons, and $\gamma$-tagged jet substructures in p+p collisions. In
A+A collision we also fix $M_{f}=3.0$ GeV for positive partons, though in
principle the parameter $M_{f}$ for positive partons in Pb+Pb collisions could
be different from the one in $p+p$ collisions due to their different $M_{C}$
distributions. The only varying parameter in our extended cluster model of
hadronization is $M_{f}$ for negative parton, which is chosen to be 0.25 GeV
to give optimal descriptions of experimental data in heavy-ion collisions, as
to be discussed in Sec. III.

##  III Numerical Results

###  III.1 Parton-medium interaction effect on $Z^{0}/\gamma$-hadron
correlations

With the framework described in Sec. II, we present a detailed study of
$Z^{0}$-hadron correlation as well as hadron spectra tagged with $Z^{0}$ boson
in both p+p and Pb+Pb collisions. In order to compare with experimental
measurements, we select the $Z^{0}$ boson in association with jets according
to the kinematic cuts adopted by CMS experiment CMS:2020rxf . To compare with
CMS results, the electrons are constrained in the phase space $p_{T}>20$
GeV/$c$ and $|\eta|<2.1$ and are rejected in the transition region
$1.44<|\eta|<1.57$; muons are required to have $p_{T}>20$ GeV/$c$ and
$|\eta|<2.4$; the $Z^{0}$ candidates are defined as electron or muon pairs,
with a reconstructed invariant mass ($M^{ll}$) in the phase space $60-110$
GeV/$c^{2}$ and $p_{T}^{Z}>$30 GeV/$c$. The charged hadrons are required in
the rapidity region $|\eta^{ch,h}|<2.4$ and in the transverse momentum region
$p_{T}^{ch,h}>1$ GeV/$c$.

Figure 4: (Color online) Top: the azimuthal angle correlation
$\Delta\phi_{Z,ch}$ between charged hadron and the recoiling $Z^{0}$ boson in
p+p and Pb+Pb collisions with experimental data CMS:2020rxf . Bottom: the
difference of the $\Delta\phi_{Z,ch}$ distribution between Pb+Pb and p+p
collisions.  Figure 5: (Color online) The distributions of the azimuthal angle
difference $\Delta\phi_{Z,ch}$ between charged particles from positive
(circle), negative (half filled circle) as well as positive minus negative
(filled circle) and the recoiling $Z^{0}$ bosons, normalized by the number of
$Z^{0}$ bosons in different $p_{T}^{ch,h}$ regions in p+p and Pb+Pb
collisions.

Firstly, the distributions of the azimuthal angle difference
$\Delta\phi_{Z,ch}=|\phi_{Z}-\phi_{ch,h}|$ between charged hadrons and the
recoiling $Z^{0}$ bosons, normalized by the number of $Z^{0}$ bosons both in
p+p and Pb+Pb collisions as well as the comparison with CMS measurements data
CMS:2020rxf  are shown in the top panel of Fig. 4. Excellent agreements are
obtained for $\Delta\phi_{Z,ch}$ distributions in both p+p and Pb+Pb
collisions. Our calculations with only hadrons from positive partons overshoot
the measured distribution of $\Delta\phi_{Z,ch}$ in Pb+Pb collisions. However,
when hadronization effects from negative partons are taken into account, our
result presents a very nice description of the experimental data in Pb+Pb.
Compared to p+p collisions, a significant enhancement of $\Delta\phi_{Z,ch}$
distribution in Pb+Pb collisions is observed, indicating that the azimuthal
angle deviations between $Z^{0}$ boson and its associated charged hadrons are
significantly increased due to jet-medium interactions in Pb+Pb collisions.

To quantify the relative shift of $\Delta\phi_{Z,ch}$ in Pb+Pb collisions due
to jet-medium interactions, we show the difference of $\Delta\phi_{Z,ch}$
between p+p and Pb+Pb collisions in bottom panel of Fig.4. The distribution of
$\Delta\phi_{Z,ch}$ is enhanced with the largest magnitude at
$\Delta\phi_{Z,ch}\sim 2.5$ region, while with the smallest magnitude at
$\Delta\phi_{Z,ch}\sim\pi$. Besides, the beam remnants Sjostrand:1987su ;
Beraudo:2022dpz , the remnant of the projectile moving in the same direction
as beam with forward rapidity, which have significant contributions to the
soft hadron yields, especially for soft hadrons in the same-side
($\Delta\phi_{Z,ch}<\pi/2$) region, should also be included. Because the beam
remnants would be color connected to jet partons, the cluster mass would be
very large and they will experience huge amount of string splitting processes.
As a consequence, they would contribute significantly to soft hadrons yields
in the whole region.

Figure 6: (Color online) Top: Comparison between our model calculations of
$Z^{0}$ boson tagged charged hadron $p_{T}^{ch,h}$ distributions in p+p and
Pb+Pb collisions with CMS experimental data CMS:2020rxf . Bottom: the ratio of
the $p_{T}^{ch,h}$ distributions in Pb+Pb to that in p+p collisions.
Figure 7: (Color online) Top : Theoretical results of $\xi_{T}^{Z,ch}$
distributions in p+p and Pb+Pb collisions, confronted against CMS experimental
data CMS:2020rxf . Bottom: the ratio of $\xi_{T}^{Z,ch}$ distributions in
Pb+Pb to that in p+p collisions.

To have a detailed illustration of the modification of the azimuthal angle
correlation, we calculated the $\Delta\phi_{Z,ch}$ distributions in four
$p_{T}^{ch,h}$ intervals: 0.5-1 GeV/$c$, 1-2 GeV/$c$, 2-8 GeV/$c$, $>$8
GeV/$c$ both in p+p and Pb+Pb collisions and show the results in Fig. 5. The
contributions from negative partons and positive partons to
$\Delta\phi_{Z,ch}$ are also calculated. We find that contributions from
negative partons hadronization are uniformly distributed in the whole phase
space and are independent on the $\Delta\phi_{Z,ch}$, while those from
positive partons hadronization are Gaussian distributed with mean value
$\Delta\phi_{Z,ch}=\pi$. It is shown that contributions from negative partons
are essential for soft hadron yields, but become negligible when high
$p_{T}^{ch,h}$ hadrons are concerned. In $p_{T}^{ch,h}>2$ GeV/$c$ region, the
distributions of $\Delta\phi_{Z,ch}$ are overall suppressed in Pb+Pb
collisions (with or without contribution of negative partons hadronization)
relative to p+p, while $p_{T}^{ch,h}<2$ GeV/$c$ they are enhanced in Pb+Pb
collisions. The enhancement of $\Delta\phi_{Z,ch}$ mainly comes from the
increasing of soft hadron yields which is rather sensitive to medium recoils.

In the top panel of Fig. 6, we present the charged hadron transverse momentum
$p_{T}^{ch,h}$ spectra in p+p and Pb+Pb collision and compare them to CMS
experimental data CMS:2020rxf  . In the simulation, the charged hadrons are
required to satisfy $\Delta\phi_{Z,ch}>7\pi/8$. Our calculations are in good
agreement with the experimental data in both p+p and Pb+Pb collisions, while
numerical results only from positive parton in Pb+Pb collisions may
overestimate the soft hadron ($p_{T}^{ch,h}<2$ GeV/$c$), The ratio of charged
hadron $p_{T}$ distribution in Pb+Pb collisions to that in p+p collisions is
compared to experimental data in the bottom panel of Fig. 6. We see that the
particle production in Pb+Pb is suppressed at high $p_{T}^{ch,h}$ while
enhanced at at low $p_{T}^{ch,h}$ (1 $<\ p_{T}^{ch,h}\ <$ 2 GeV/$c$), as
compared to the p+p reference.

Recently the fragmentation pattern of the parton recoiling from a $Z^{0}$
boson,

$\xi^{Z,ch}_{T}=\ln[-|\overrightarrow{p}^{Z}_{T}|^{2}/(\overrightarrow{p}^{ch,h}_{T}\cdot\overrightarrow{p}^{Z}_{T})]$
(6)

has been measured in Pb+Pb collisions by CMS Collaboration CMS:2020rxf  . The
results of $\xi^{Z,ch}_{T}$ distributions in our model and their comparison
with experimental data CMS:2020rxf  are illustrated in the top panel of Fig.
7. The ratio of $\xi^{Z,ch}_{T}$ in Pb+Pb to that in p+p collisions are shown
in the bottom panel of Fig. 7. To match the corresponding experiment
kinematics, charged hadrons are required to satisfy $\Delta\phi_{Z,ch}>7\pi/8$
and $p_{T}^{ch,h}>1$ GeV/$c$. We see our simulations for both p+p and Pb+Pb
collisions are consistent with the experimental data. The definition of
$\xi^{Z,ch}_{T}$ in Eq. 6 tells that small $\xi^{Z,ch}_{T}$ region corresponds
to high energy hadrons and large $\xi^{Z,ch}_{T}$ region comes from soft
hadrons. A significant suppression in small $\xi^{Z,ch}_{T}$ region and a
large enhancement in large $\xi^{Z,ch}_{T}$ region are obtained in Pb+Pb
collisions relative to p+p collisions, a trend in nice agreement with CMS
data. We notice that the contribution from negative parton to $\xi^{Z,ch}_{T}$
is significant in large $\xi^{Z,ch}_{T}$ region (with small $p^{ch,h}_{T}$).
Including contributions from medium response, our calculations can better
describe the experiment for soft hadron productions. We emphasize that
compared to traditional jet fragmentation $\xi^{jet,ch}_{T}$ with dependence
on jet energy, $\xi^{Z,ch}_{T}$ has nothing to do with jet energy, which may
also be altered due to jet-medium interactions in the QGP. So it is
understandable that the shift of the $\xi^{Z,ch}_{T}$ in Pb+Pb will be more
pronounced with a better sensitivity to parton energy lose and medium
response.

###  III.2 Medium-Induced Modifications of Jet substructure

We now turn to medium effects on full jet observables, and investigate the
redistribution of jet substructure associated with a $Z^{0}/\gamma$ in Pb+Pb
relative to the reference p+p collisions by including hadronization effects
with the extended cluster model.

In the numerical compuation, $\gamma$ is required in the region
$|\eta^{\gamma}|<$1.44, and $p_{T}^{\gamma}>$60 GeV/$c$. Background
contributions from fragmentation and decay photons are rejected by imposing
the isolation requirements: the $p_{T}$ sum of final hadrons in a cone of
radius 0.4 with respect to the centroid of the photon is required to be less
than 1 GeV/$c$. The anti-$k_{T}$ jet finding algorithm Cacciari:2011ma ;
Cacciari:2008gp  is used to cluster the resulting particles using a distance
parameter R= 0.3. Jets with $|\eta^{jet}|<$1.6 and $p_{T}^{jet}>$30 GeV/$c$
are selected. An azimuthal separation of
$\Delta\phi_{j\gamma}=|\Delta\phi_{j}-\Delta\phi_{\gamma}|>7\pi/8$ is applied
to select the most balanced photon-jet pairs. The charged hadrons are
collected with requirement that $p_{T}^{ch,h}>$1 GeV/$c$, $|\eta^{ch,h}|<$
2.4, and must fall within a cone of radius $\Delta
r=\sqrt{(\phi^{ch,h}-\phi^{j})^{2}+(\eta^{ch,h}-\eta^{j})^{2}}<$0.3 around the
jet direction.

Figure 8: (Color online) Top: jet shapes $\rho(r)$ for $\gamma$ tagged jets in
p+p and Pb+Pb collisions. Bottom: the ratio of this jet shape in Pb+Pb to that
in p+p collisions. Experimental data are taken from CMS measurement
CMS:2018jco .
Figure 9: (Color online) Top: distributions of particle number density of jets
associated with a photon as a function of the distance to the jet axis in p+p
and Pb+Pb collisions. Bottom: the ratio of charged number density in Pb+Pb to
that in p+p collisions.

To have a detailed understanding of how the energy is redistributed inside and
outside the jet in Pb+Pb, we first calculate the differential jet shape for
jets associated with an isolated photon as defined in Eq. 1. The top panel in
Fig. 8 shows the differential jet shape of isolated photon tagged jet
calculated by our model for both p+p and Pb+Pb collisions and the comparison
with experimental data CMS:2018jco . The results are normalized to unity over
$r<0.3$. Our model calculations can well describe the jet shapes both in p+p
and Pb+Pb collisions. The bottom panel in Fig. 8 presents the ratio of the jet
shape in Pb+Pb to that in p+p collisions. A depletion is observed in the
region $r<0.05$ and an enhancement in the region $r>0.1$ in the central Pb+Pb
collisions relative to p+p. Jet-medium interactions lead to a larger fraction
of jet energy carried by soft particle at large distance from the jet axis,
and result in medium modified jet energy distribution in heavy-ion collisions.
We also see that negative partons make rather insignificant contributions to
jet shape in Pb+Pb. This is because, jet shape is the radial average
transverse momentum distributions inside the jet. In the calculation of jet
shape, only charged hadrons with transverse momentum $p_{T}>1$ GeV/$c$ is
considered and the hadrons from negative partons that fall into the jet cone
are rather few. Furthermore, the momentum of those hadrons from negative
partons is relatively small compared to the jet momentum.

In addition to the transverse momentum distribution within and out of jet
cone, we predict the particle number density
$n_{ch}(r)=\frac{1}{N_{jet}}\frac{dN_{ch}}{2\pi rdr}$, where $r$ is the
distance between changed particles and jet axis in the $\phi-y$ plane.
Distributions of $n_{ch}(r)$ in p+p and Pb+Pb collisions are shown in the top
panel of Fig. 9. The number density distributions is quite similar with jet
shape. Most of the particles are distributed in the jet core. Compared to p+p
collisions, charged hadrons density is suppressed near jet axis, and enhanced
away from the jet axis. The ratio of the number density in Pb+Pb collisions to
that in p+p collisions goes up smoothly as the increasing of $r$.

In top panel of Fig. 10 we plot the fragmentation function of jet (see the
definition in Eq. 2) in association with $\gamma$ for p+p and Pb+Pb
collisions. The ratio of the fragmentation function in Pb+Pb to that in p+p
collisions is shown in Fig. 10. Our model describes the experimental data well
both in p+p and Pb+Pb collisions. One can see a moderate suppression in the
region $\xi^{jet}<$2 and a significant enhancement in region $\xi^{jet}>$2\.
The enhancement in region $\xi^{jet}>$2 indicates that soft hadrons yields are
significantly increased.

Figure 10: (Color online) Top: Distributions of jet fragmentation function
$\xi^{jet}$ in p+p and Pb+Pb collisions and the ratio of $\xi^{jet}$ in Pb+Pb
to that in p+p collisions. Bottom: distributions of jet fragmentation function
$\xi^{\gamma}$ in p+p and Pb+Pb collisions (up panel) and the ratio of
$\xi^{\gamma}$ in Pb+Pb to that in p+p collisions. Experimental data are taken
from Ref. Sirunyan:2018qec

Figure 11:  $Z^{0}$-jet fragmentation functions respect to jet momentum(up)
and $Z^{0}$ boson transverse momentum(bottom) in p+p and Pb+Pb collisions and
the ratios in Pb+Pb to that in p+p collisions.

In photon tagged jet production, since photon does not participate in strong
interaction directly and has a mean free path much longer than the size of the
QGP, its energy will be intact during the propagation in the medium. The
energy loss of the jet partons will be better charectorized with the medium
modification of the jet fragmentation function as Wang:2013cia ; Chen:2020tbl
:

$\xi^{\gamma}_{T}=\ln\frac{-|\overrightarrow{p}^{\gamma}_{T}|^{2}}{\overrightarrow{p}^{ch,h}_{T}\cdot\overrightarrow{p}^{\gamma}_{T}}$
(7)

The variable $\xi^{\gamma}_{T}$ gives the fragmentation pattern with respect
to the transverse momentum of the initial parton. The bottom picture in Fig.
10 shows the $\gamma$-jet fragmentation functions $\xi^{\gamma}_{T}$ for p+p
and Pb+Pb collisions as well as their ratio, which are in a good agreement
with the experimental data. Compared to $\xi^{jet}$ in p+p collisions, in the
Pb+Pb collisions an enhancement is observed in the $\xi^{\gamma}_{T}>3$
region, while a suppression in the region $\xi^{\gamma}_{T}<3$. The medium
modification is much more significant on $\xi^{\gamma}_{T}$ distributions than
$\xi^{jet}$ distributions. This suppression and enhancement of the jet
fragmentation pattern are directly evidence of energy loss of high energy
partons propagating through the hot-dense medium created in heavy-ion
collisions. Hadron yields from negative partons hadronization give moderate
contributions to large $\xi^{\gamma}_{T}$ and $\xi^{jet}$ due to their low
energies.

Finally, we provide theoretical predictions on $Z^{0}$ associated jet
fragmentations, and plot in Fig 11 fragmentation distributions with respect to
the momentum of the reconstructed jet $\xi^{jet}$ and with respect to the
transverse momentum of recoiling $Z^{0}$ boson $\xi^{Z}_{T}$ in p+p and Pb+Pb
collisions. Relative to p+p collisions, charged hadron yields with large
transverse momentum are suppressed and soft charged hadron yields are enhanced
in Pb+Pb collisions.

##  IV Conclusions

We have made a detailed study of medium modification on $Z^{0}/\gamma$+hadron
correlations and $Z^{0}/\gamma$ tagged jet substructures in Pb+Pb collisions
relative to p+p collisions at 5.02 TeV. The initial reference jet in p+p
collisions is generated by SHERPA with NLO+PS, jet-medium interactions and
medium response in the hot-dense medium are simulated by LBT, and the non-
perturbative transition of quarks and gluons into final hadrons are taken into
account within an extend cluster hadronization model. Based on this framework,
we can well describe the experimental data of $Z^{0}/\gamma$+hadrons
correlations and jet substructures in both p+p and Pb+Pb collisions.

Azimuthal angle correlations $\Delta\phi_{Z,ch}$ between charged hadrons and
the recoiling $Z^{0}$ boson are enhanced in Pb+Pb collisions relative to p+p
collisions. Further investigations on the $\Delta\phi_{Z,ch}$ in different
$p_{T}^{ch,h}$ intervals show that the soft hadron yields are significantly
increased in Pb+Pb collisions and sensitive to medium response. Charged
hadrons from negative partons are uniformly distributed in the whole phase
space. While charged hadrons from positive partons are Gaussian distributed
with mean value $\Delta\phi_{Z,ch}=\pi$. Distributions for high energy charged
hadron is narrowed and suppressed in Pb+Pb collisions relative to p+p
collisions, whereas distributions for soft hadrons are broadened and enhanced
due to jet-medium interactions. The increasing of soft hadron yields lead to
the enhancement of $\Delta\phi_{Z,ch}$ in Pb+Pb collisions. And transverse
momentum distributions $p_{T}^{ch}$ of charged hadrons are enhanced in low
$p_{T}$ region and suppressed for high $p_{T}$ hadrons,
$\xi^{Z,ch}_{T}=\ln[-|\overrightarrow{p}^{Z}_{T}|^{2}/(\overrightarrow{p}^{ch,h}_{T}\cdot\overrightarrow{p}^{Z}_{T})]$
distributions are shifted to larger value.

Furthermore, the medium-modification on jet energy density $\rho(r)$, jet
hadron number density $n_{ch}(r)$, and jet fragmentation functions
$\xi^{\gamma}_{T}$ and $\xi^{jet}$ are studied in Pb+Pb collisions. Our
numerical results demonstrate an excellent agreement with experimental data.
Medium modification on jet profile and jet fragmentation functions indicate
that particles carrying a large fraction of the jet momentum are generally
closely aligned with the jet axis, whereas low-momentum particles are observed
to have a much broader angular distribution relative to jet axis in Pb+Pb
collisions. For completeness, jet fragmentation functions respect to jet
momentum and $Z^{0}$ boson transverse momentum for $Z^{0}$ boson tagged jet
are predicted.

Our work provides a valuable attempt to generalize the cluster hadronization
mechanism to study jet productions in heavy-ion collisions and pave a new way
to studying the non-perturbative hadronization processes with the existence of
the QGP. With some simple assumptions of hadronization of positive partons and
negative partons in medium, our results can well describe many observables of
the charged hadrons and jet substructures in both p+p and Pb+Pb collisions. It
will be of great interest to investigate different types of hadron yields,
such as $\pi$, $p$, $K$, $\eta$ Ma:2018swx ; Zhang:2022fau  and heavy flavor
hadrons Wang:2021jgm  in heavy-ion collisions with the extended cluster model,
which can be utilized to further constrain and improve the hadronization model
presented here.

##  V Acknowledgements

This research is supported by Guangdong Major Project of Basic and Applied
Basic Research No. 2020B0301030008, by National Natural Science Foundation of
China (NSFC) under Grants No. 12147131, No. 11935007, No. 12035007 and No.
12022512. S.Z. is also supported by the MOE Key Laboratory of Quark and Lepton
Physics (CCNU) under Project No. QLPL2021P01.

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