11institutetext:  Key Laboratory of Quark and Lepton Physics (MOE) and
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
China 22institutetext: School of Science and Engineering, The Chinese
University of Hong Kong, Shenzhen 518172, China 33institutetext: University of
Science and Technology of China, Hefei, Anhui, 230026, P.R.China
44institutetext: Key Laboratory of Atomic and Subatomic Structure and Quantum
Control (MOE), Guangdong Basic Research Center of Excellence for Structure and
Fundamental Interactions of Matter, Institute of Quantum Matter, South China
Normal University, Guangzhou 510006, China 55institutetext: Guangdong-Hong
Kong Joint Laboratory of Quantum Matter, Guangdong Provincial Key Laboratory
of Nuclear Science, Southern Nuclear Science Computing Center, South China
Normal University, Guangzhou 510006, China

# Momentum and angular correlations in $Z/\gamma$-hadron production in
relativistic heavy-ion collisions

Zhan Gao 11 Lin Chen 2233 Peng-Hui Hu 11 Man Xie 4455 Han-Zhong Zhang 114455
<EMAIL_ADDRESS>
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<EMAIL_ADDRESS>
###### Abstract

We carry out a detailed study of medium modifications on momentum and angular
correlations between a large transverse momentum hadron and a $Z/\gamma$
trigger in relativistic heavy-ion collisions within a perturbative QCD parton
model improved by the Sudakov resummation technique. The total energy loss of
a hard parton propagating inside the medium is employed to modify the
fragmentation function, while the medium-induced transverse momentum
broadening is included in the resummation approach, and both of them are
related to the jet transport parameter and obtained by the high-twist
formalism. We obtain good agreements with the existing data on transverse
momentum and azimuthal angular correlations for the $Z/\gamma$-hadron pairs in
$pp$ and $AA$ collisions, and predict the correlations for the $\gamma$-hadron
in central $PbPb$ collisions at 5.02 TeV. The numerical analyses for the
$Z/\gamma$-hadron in central $PbPb$ collisions show that the normalized
angular distribution is decorrelated due to the medium-induced transverse
momentum broadening, however, the angular correlation is enhanced due to the
parton energy loss, namely anti-broadening. The observed modification of the
angular correlation is a result of the competition between the broadening and
the anti-broadening. This work provides a reliable theoretical tool for a
comprehensive and precise study of jet quenching in relativistic heavy-ion
collisions.

††journal: Eur. Phys. J. C

##  1 Introduction

The strongly-coupled state of deconfined quarks and gluons known as the Quark-
Gluon Plasma (QGP) is one of the most important discoveries in recent high-
energy heavy-ion colliding experiments carried out by the Relativistic Heavy-
Ion Collider (RHIC) Gyulassy:2004zy (1) and the Large Hadron Collider (LHC)
Muller:2012zq (2). High-energy jets when traversing the strongly-coupled
medium, undergo multiple scatterings that induce gluon radiations off the
jets. The jets lose energies and get additional transverse momentum broadening
$\langle p_{\perp}^{2}\rangle$ due to the interactions between the jets and
QGP. This jet quenching Wang:1992qdg (3) phenomenon has become a powerful tool
for the study of transport properties of the QGP. A single parameter called
jet transport coefficient ($\hat{q}$) is defined as the average transverse
momentum broadening squared per unit length, $\hat{q}=\mathrm{d}\langle
p_{\perp}^{2}\rangle/\mathrm{d}L$, which quantifies the strength of momentum
exchanges transverse to the direction of jet parton caused by the in-medium
scatterings Baier:1996kr (4, 5, 6, 7). With the assumption of jet energy loss
being proportional to the jet transport coefficient, the values of $\hat{q}$
have been phenomenologically extracted through model-to-data comparisons on
hadron suppression Qin:2007rn (8, 9, 10, 11, 7, 12, 13, 14, 15). Recently, an
information field approach is applied to the Bayesian (IF-Bayesian) inference
of $\hat{q}$ from combined experimental data on the suppression of
single-/di-/$\gamma$-hadrons in heavy-ion collisions at both RHIC and LHC
energies Xie:2022ght (16, 17). The extracted $\hat{q}/T^{3}$ is free of long-
range correlations, and exhibits a strong temperature ($T$) dependence without
an explicit functional parametrization. Apart from the $\hat{q}$ extraction
using hadron suppression due to jet energy loss, the transverse momentum
asymmetry of di-/$\gamma$-/$Z$-jets and the angular decorrelation of di-
hadrons or hadron-jets are also attempted to constrain the jet transport
coefficient due to medium-induced transverse momentum broadening Chen:2016vem
(18, 19, 20, 21).

The perturbative QCD (pQCD) calculation at next-to-leading order (NLO) works
well for di-/$\gamma$-hadron transverse momentum spectra except for their
angular correlations near the end points in heavy-ion collisions Zhang:2007ja
(22, 23). The NLO pQCD expansion fails to converge for the azimuthal angular
distribution of the back-to-back di-/$\gamma$-hadron because of the appearance
of large Sudakov logarithms. A Sudakov resummation technique is recently
developed to make up for the insufficiency Mueller:2012uf (24, 25, 26). With
the resummation-improved pQCD framework, the angular decorrelation of di-
hadrons, hadron-jets and di-jets are obtained and fitted data well due to the
vacuum soft gluon radiation (the Sudakov effect) as well as the medium-induced
transverse momentum broadening $\langle p_{\perp}^{2}\rangle$ Chen:2016vem
(18, 19). The same framework is then applied to study $\gamma/Z$-jet
transverse momentum correlations Chen:2018fqu (20, 21) in $AA$ collisions, and
gluon saturation phenomenon in $pA$ collisions with small-$x$ formalism
Stasto:2018rci (27, 28). However, the influences of jet energy loss
suppression are absent in the studies of $\langle p_{\perp}^{2}\rangle$
effects for the back-to-back azimuthal angular correlation of di-hadrons,
hadron-jets and di-jets in heavy-ion collisions.

In jet quenching formalism Baier:1996kr (4, 6, 5, 29, 30), jet energy loss
$\Delta E$ and $\langle p_{\perp}^{2}\rangle$ effects are two sides of the
same coin since $\Delta E\propto\hat{q}=\mathrm{d}\langle
p_{\perp}^{2}\rangle/\mathrm{d}L$. The former leads to the yield suppression
Gyulassy:1990ye (31, 3, 32) and the azimuthal anisotropy Gyulassy:2000gk (33)
of large transverse momentum hadrons or jets, while the latter means a medium
kick transverse to the jets and gives slight enhancements of small transverse
momentum hadron spectra Wang:1998ww (34) and angular decorrelations in
di-/$\gamma$-/$Z$-jet production Chen:2016vem (18, 19, 20, 21). In the
resummation-improved pQCD framework, the typical value of di-jet transverse
momentum imbalance ($\vec{q}_{\perp
AA}=\vec{p}_{T\rm{jet}1}+\vec{p}_{T\rm{jet}2}$) in $AA$ collisions is given by
a pocket formula, $q_{\perp AA}^{*2}\sim q_{\perp pp}^{*2}+\langle
p_{\perp}^{2}\rangle$, in which $q_{\perp pp}^{*2}$ gives the Sudakov effect
from the initial di-jet transverse momentum imbalance and dominates the
angular correlations for the large transverse momentum di-jets in high-energy
$AA$ collisions at LHC Chen:2016vem (18). Since $q_{\perp pp}^{*2}$ is
relative to the initial parton energy, parton energy loss effect appearing in
the yield for the $q_{\perp AA}^{*2}$ dependence of angular correlation of
di-/$\gamma$-/$Z$-jets will result in the energy loss dependence of angular
correlation due to the bias of the same final transverse momentum in $pp$ and
$AA$ collisions.

In this paper we will present a tomographic study on heavy-ion collisions via
transverse momentum and azimuthal angular correlations in $Z/\gamma$-hadron
production within a pQCD parton model improved by the Sudakov resummation
technique. The high-twist (HT) energy loss formalism Guo:2000nz (29, 30) is
combined with the pQCD parton model, and the medium-induced transverse
momentum broadening $\langle p_{\perp}^{2}\rangle$ is included into the
Sudakov resummation technique Chen:2016vem (18, 20). We apply the CLVisc 3+1D
hydrodynamics model to simulate the dynamical evolution of the QGP medium
created in $AA$ collisions Chen:2017zte (35, 36, 37, 38).

The $Z/\gamma$-hadron are perfect probes for studying longitudinal energy loss
and transverse momentum broadening since the $Z/\gamma$ bosons do not interact
strongly with the hot and dense medium. Firstly, they do not have geometric
bias compared to dihadrons and dijets, therefore, the associated jets can
probe more inner regions of the QGP fireball and bring out more robust
information about it Zhang:2007ja (22, 23). Secondly, $Z/\gamma$ bosons could
perfectly preserve the initial momentum information of the away-side jets
before getting quenched, thus, we can cleanly investigate the relation between
the suppression or azimuthal angular correlation and the energy of the initial
jet. Finally and importantly, as mentioned above, the longitudinal energy loss
effect appears in the angular correlation of di-particles. To reveal the
medium modification essentially, we should adopt the same kinematics cuts for
final state particles in $pp$ and $AA$ collisions, hence the momentum
information for $Z/\gamma$ is the same in $pp$ and $AA$ collisions. Meanwhile,
the initial momentum of the associated hadron is larger in $AA$ collisions due
to energy loss Baier:2001yt (39, 40, 41, 42), which causes an anti-broadening
effect. In our studies, we will perform a comprehensive and precise study on
the anti-broadening effect resulting from the jet energy loss and the
broadening effect Wang:2003mm (43) caused by medium-induced transverse
momentum broadening by calculating the momentum ($p_{\rm T}^{h}/z_{\rm
T}/\xi_{\rm T}$) and angular correlations ($\Delta\phi$) of $Z/\gamma$-hadron
production in relativistic heavy-ion collisions.

The paper is organized as follows. In section 2, we describe the theoretical
framework used in our calculations. In section 3, we present and discuss the
medium modifications on momentum and angular correlations between a large
transverse momentum hadron and the $Z/\gamma$ trigger in $AA$ collisions at
RHIC and LHC. We end the paper with a summary in section 4.

##  2 Theoretical framework

###  2.1 a Sudakov resummation improved pQCD parton model in $pp$ collisions

In high-energy $pp$ collisions, the invariant cross section for single
inclusive hadron production at large transverse momentum $p_{\rm T}$ within
the parton model can be factorized into a convolution of collinear parton
distribution functions (PDFs), short-distance partonic cross-sections and the
collinear fragmentation functions (FFs). The $Z/\gamma$-hadron production
cross-section within the Sudakov resummation formalism in $pp$ collisions can
be expressed as,

$\displaystyle\frac{\mathrm{d}\sigma_{pp\rightarrow
Vh}}{\mathrm{d}\mathcal{P.S.}}$ $\displaystyle=$
$\displaystyle\sum_{a,b,d}\int\frac{\mathrm{d}z_{d}}{z^{2}_{d}}D_{h/d}\left(z_{d},\mu^{2}\right)\frac{|\bar{\mathcal{M}}_{ab\rightarrow
Vd}|^{2}}{16\pi^{2}\hat{s}^{2}}$ (1) $\displaystyle\times$ $\displaystyle
x_{a}f_{a/p}\left(x_{a},\mu^{2}\right)x_{b}f_{b/p}\left(x_{b},\mu^{2}\right)$
$\displaystyle\times$
$\displaystyle\int\frac{\mathrm{d}^{2}\vec{b}_{\perp}}{(2\pi)^{2}}e^{-i\vec{q}_{\perp}\cdot\vec{b}_{\perp}}e^{-S(Q,b_{\perp})}.$

Here
$\mathrm{d}\mathcal{P.S.}=\mathrm{d}y_{V}~{}\mathrm{d}y_{h}~{}\mathrm{d}^{2}\vec{p}_{\rm
T}^{V}~{}\mathrm{d}^{2}\vec{p}_{\rm T}^{h}$ represents the final state phase
space, $y_{V}$ and $y_{h}$ are the rapidities of the vector boson and hadron,
respectively. $p_{\rm T}^{V}$ and $p_{\rm T}^{h}$ are the transverse momentum
of the vector boson and hadron. Initial parton momentum fractions are defined
as $x_{a,b}=(m_{\rm T}^{V}e^{\pm y_{V}}+m_{\rm T}^{d}e^{\pm y_{h}})/\sqrt{s}$,
with $m_{\rm T}=\sqrt{m^{2}+p_{\rm T}^{2}}$ the transverse mass of the
particle, with $m$ the particle mass and $p_{\rm T}$ its transverse momentum.
While $\sqrt{s}$ denotes the usual collision energy in the proton-proton
Center-of-Mass frame. $f_{a,b}(x,\mu^{2})$ are the PDFs of the incoming parton
species $a$ and $b$, which we take from Cteq (CT18) parametrizations
Hou:2019efy (44). $z_{d}=p_{\rm T}^{h}/p_{\rm T}^{d}$ is the momentum fraction
of the final observed hadron $h$ carried from the parent parton $d$, and we
use the Kretzer (KRE) parametrizations Kretzer:2000yf (45) for the vacuum
fragmentation function $D_{h/d}\left(z_{d},\mu^{2}\right)$.
$\hat{s}=x_{a}x_{b}s=Q^{2}$, $\hat{t}=m_{V}^{2}-x_{a}m_{\rm
T}^{V}\sqrt{s}e^{-y_{V}}$ and $\hat{u}=-x_{a}m_{\rm T}^{d}\sqrt{s}e^{-y_{h}}$
are then the usual partonic Mandelstam variables. The auxiliary
$b_{\perp}$-space integral guarantees transverse momentum conservation of the
radiated gluons, with $\vec{q}_{\perp}\equiv\vec{p}_{\rm T}^{V}+\vec{p}_{\rm
T}^{h}/z_{d}=\vec{p}_{\rm T}^{V}+\vec{p}_{\rm T}^{d}$ defined as the
transverse momentum imbalance of the two out-going final particles.
$\Delta\phi=|\phi_{V}-\phi_{h}|$ is the azimuthal angle difference between the
vector boson and the hadron. The amplitudes squared, averaged over the color
and spin in the initial state and summed over them in the final state, are
given by

$\displaystyle|\bar{\mathcal{M}}_{q\bar{q}\rightarrow Vg}|^{2}$
$\displaystyle=\frac{N_{c}^{2}-1}{N_{c}^{2}}16\pi^{2}\alpha_{\rm s}\alpha_{\rm
em}e_{q}^{2}\frac{\hat{t}^{2}+\hat{u}^{2}+2\hat{s}m_{V}^{2}}{\hat{t}\hat{u}},$
(2) $\displaystyle|\bar{\mathcal{M}}_{qg\rightarrow Vq}|^{2}$
$\displaystyle=-\frac{1}{N_{c}}16\pi^{2}\alpha_{\rm s}\alpha_{\rm
em}e_{q}^{2}\frac{\hat{s}^{2}+\hat{t}^{2}+2\hat{u}m_{V}^{2}}{\hat{s}\hat{t}},$
(3)

where $e_{q}$ is the electric charge of the quarks in the case of photon
production. For $Z$ boson production, $e_{q}$ is replaced by

$e_{q}^{2}\rightarrow\frac{\left(1-2\left|e_{q}\right|\sin^{2}\theta_{W}\right)^{2}+4e_{q}^{2}\sin^{4}\theta_{W}}{8\sin^{2}\theta_{W}\cos^{2}\theta_{W}},$
(4)

with $\theta_{W}$ the weak mixing angle. The values of the $Z$ boson masses
$m_{Z}$ and $\theta_{W}$ are taken from PDG Workman:2022ynf (46). Replacing
$D_{h/d}\left(z_{d},\mu^{2}\right)$ to $\delta(1-z_{d})$ and performing the
integration of $\mathrm{d}y_{h}$ and $\mathrm{d}^{2}\vec{p}_{\rm T}^{h}$, we
can obtain the differential cross-sections of $Z/\gamma$ bosons production in
$pp$ collisions.

The Sudakov form factor is expressed as follows,

$S_{\text{vac}}(Q,b_{\perp})=S^{i}_{\text{pert}}(Q,b_{\perp})+S^{f}_{\text{pert}}(Q,b_{\perp})+S_{\text{NP}}(Q,b_{\perp}).$
(5)

$S^{i}_{\text{pert}}(Q,b_{\perp})$ is the perturbative Sudakov factor of
initial parton, $S^{f}_{\text{pert}}(Q,b_{\perp})$ is perturbative Sudakov
factor of final state hadrons, and $S_{\text{NP}}(Q,b_{\perp})$ is the non-
perturbative Sudakov factor in $pp$ collisions. The perturbative Sudakov
factors are defined as Collins:1984kg (47, 48, 49, 50)

$S^{i}_{\text{pert}}(Q,b_{\perp})=\int^{\mu^{2}_{\rm res}}_{\mu^{2}_{\rm
fac}}\frac{\mathrm{d}\mu^{2}}{\mu^{2}}\left[\left(A_{i}^{(1)}+A_{i}^{(2)}\right)\ln\frac{Q^{2}}{\mu^{2}}+B_{i}\right],$
(6) $S^{f}_{\text{pert}}(Q,b_{\perp})=S^{i}_{\text{pert}}(Q,b_{\perp}),$ (7)

and

| Quark | Gluon
---|---|---
$A_{i}^{(1)}$ | $C_{F}\left[\frac{\alpha_{\rm s}(\mu^{2})}{2\pi}\right]$ | $C_{A}\left[\frac{\alpha_{\rm s}(\mu^{2})}{2\pi}\right]$
$A_{i}^{(2)}$ | $K~{}C_{F}\left[\frac{\alpha_{\rm s}(\mu^{2})}{2\pi}\right]^{2}$ | $K~{}C_{A}\left[\frac{\alpha_{\rm s}(\mu^{2})}{2\pi}\right]^{2}$
$B_{i}$ | $-\frac{3}{2}C_{F}\left[\frac{\alpha_{\rm s}(\mu^{2})}{2\pi}\right]$ | $-2\beta_{0}C_{A}\left[\frac{\alpha_{\rm s}(\mu^{2})}{2\pi}\right]$

with $N_{c}=3$, $C_{A}=N_{c}$, and $C_{F}=\frac{N_{c}^{2}-1}{2N_{c}}$.
$K=\left(\frac{67}{18}-\frac{\pi^{2}}{6}\right)C_{A}-\frac{10}{9}N_{f}T_{R}$
is the coefficient for the $\alpha_{\rm s}^{2}$ term. To ensure the
$q_{\perp}>\Lambda_{\mathrm{QCD}}$ and to separate the Sudakov form factor
into perturbative and non-perturbative parts in $pp$ collisions, the
$b_{\star}=b_{\perp}/\sqrt{1+b_{\perp}^{2}/b_{\max}^{2}}$ prescription is
introduced Davies:1984sp (51, 52, 53, 54, 55), with
$b_{\max}=1.5~{}\mathrm{GeV^{-1}}$. The result is the definition of the
factorization scale $\mu_{\rm fac}=b_{0}/b_{\star}$ with
$b_{0}=2e^{-\gamma_{E}}$ and $\gamma_{E}$ the Euler constant.
$S_{\text{NP}}(Q,b_{\perp})$ is non-perturbative Sudakov factor in $pp$
collisions. It is defined as

$\displaystyle S^{q}_{\text{NP}}(Q,b_{\perp})$
$\displaystyle=\frac{g_{1}}{2}b_{1}^{2}+\frac{g_{2}}{2}\ln\frac{Q}{Q_{0}}\ln\frac{b_{\perp}}{b_{\star}},$
(8) $\displaystyle S^{g}_{\text{NP}}(Q,b_{\perp})$
$\displaystyle=\frac{C_{A}}{C_{F}}S^{q}_{\text{NP}}(Q,b_{\perp}),$ (9)

where $g_{1}=0.212$, $g_{2}=0.84$, and $Q_{0}^{2}=2.4$ GeV2 Sun:2014dqm (55).

There are three distinct scales in this formalism. The factorization scale
$\mu_{\rm fac}$ appearing in the PDF and FF is fixed as $\mu_{\rm
fac}=b_{0}/b_{\star}$. The renormalization scale $\mu_{\rm ren}$ in
$\left|\bar{\mathcal{M}}\right|$ is taken to be $\mu_{\rm ren}=m_{\rm T}^{V}$
Sun:2018icb (56). To minimize the contributions from these possible large
logs, we will set the resummation scale to $\mu_{\rm res}=p_{\rm T}^{d}$
Sun:2016kkh (57).

###  2.2 parton energy loss and transverse momentum broadening in $AA$
collisions

The cross section of $Z/\gamma$-hadron production in $AA$ collisions can be
expressed as,

$\displaystyle\frac{\mathrm{d}\sigma_{AA\rightarrow
Vh}}{\mathrm{d}\mathcal{P.S.}}=$
$\displaystyle\int\mathrm{d}^{2}\vec{b}\mathrm{d}^{2}\vec{r}~{}t_{A}(\vec{r})t_{B}(\vec{r}+\vec{b})\int\frac{\mathrm{d}\phi_{d}}{2\pi}$
(10) $\displaystyle\times$
$\displaystyle\sum_{abd}\int\frac{\mathrm{d}z_{d}}{z^{2}_{d}}\tilde{D}_{h/d}\left(z_{d},\mu^{2},\Delta
E_{d}\right)\frac{|\bar{\mathcal{M}}_{ab\rightarrow
Vd}|^{2}}{16\pi^{2}\hat{s}^{2}}$ $\displaystyle\times$ $\displaystyle
x_{a}\tilde{f}_{a/A}\left(x_{a},\mu^{2},\vec{r}\right)x_{b}\tilde{f}_{b/A}\left(x_{b},\mu^{2},\vec{r}+\vec{b}\right)$
$\displaystyle\times$
$\displaystyle\int\frac{\mathrm{d}^{2}\vec{b}_{\perp}}{(2\pi)^{2}}e^{-i\vec{q}_{\perp}\cdot\vec{b}_{\perp}}e^{-S(Q,b_{\perp})}.$

Where $\phi_{d}$ is the azimuthal angle between the parton’s propagating
direction and the impact-parameter $\vec{b}$. $t_{A}$ and $t_{B}$ are the
nuclear thickness functions of the projectile and target nucleus, shifted by
the impact parameter $\vec{b}$. 111$\vec{b}$ is dependent on the collision
centrality, whereas $\vec{b}_{\perp}$ is the Fourier conjugate to
$\vec{q}_{\rm T}$ in the Sudakov integral. They are calculated by integrating
the Woods-Saxon nuclear density distribution Jacobs:2000wy (58) along the beam
direction and is normalized to the mass number A of each nucleus, e.g.,
$\int\mathrm{d}^{2}\vec{r}~{}t_{A}(\vec{r})={\rm A}$.
$\tilde{f}_{a/A}\left(x_{a},\mu^{2},\vec{r}\right)$ is the nuclear modified
PDF Wang:1996yf (59, 60):

$\displaystyle\tilde{f}_{a/A}\left(x_{a},\mu^{2},\vec{b}\right)$
$\displaystyle=R_{a/A}\left(x_{a},\mu^{2},\vec{b}\right)\left[\frac{\rm Z}{\rm
A}f_{a/p}\left(x_{a},\mu^{2}\right)\right.$ (11)
$\displaystyle\left.+\left(1-\frac{\rm Z}{\rm
A}\right)f_{a/n}\left(x_{a},\mu^{2}\right)\right],$

where Z and A are the charge and mass number of the nucleus, respectively. The
cold nuclear modification factor $R_{a/A}\left(x_{a},\mu^{2},\vec{b}\right)$
of the PDFs are given by the EPPS21 Eskola:2021nhw (61) parametrizations. The
medium-modified fragmentation function
$\tilde{D}_{h/d}\left(z_{d},\mu^{2},\Delta E_{d}\right)$ can be calculated as,
Zhang:2007ja (22, 62, 23)

$\displaystyle\tilde{D}_{h/d}\left(z_{d},\mu^{2},\Delta
E_{d}\right)=\left(1-e^{-N_{g}^{d}}\right)\left[\frac{z_{d}^{\prime}}{z_{d}}D_{h/d}\left(z_{d}^{\prime},\mu^{2}\right)\right.$
(12)
$\displaystyle\left.\quad+N_{g}^{d}\frac{z_{g}^{\prime}}{z_{d}}D_{h/g}\left(z_{g}{}^{\prime},\mu^{2}\right)\right]+e^{-N_{g}^{d}}D_{h/d}\left(z_{d},\mu^{2}\right),$

where ${z_{d}}^{\prime}=p_{\rm T}^{h}/(p_{\rm T}^{d}-\Delta{E_{d}}$) is the
transverse momentum fraction of a hadron from the fragmentation of parton $d$
with initial transverse momentum $p_{\rm T}^{d}$ after losing a total amount
of energy $\Delta{E_{d}}$ in the medium, $z_{d}$ is the momentum fraction of a
hadron from a parton fragmentation in vacuum, and
${z_{g}}^{\prime}=N_{g}^{d}p_{\rm T}^{h}/\Delta{E_{d}}$ is the momentum
fraction of a hadron from the fragmentation of a radiated gluon that carries
an average energy $\Delta{E_{d}}/N_{g}^{d}$. The number of radiated gluons is
assumed to follow the Poisson distribution with the number $N_{g}^{d}$. In
this case, the first weighting factor $1-e^{-N_{g}^{d}}$ in the above equation
is the probability for a parton to radiate at least one gluon induced by
multiple scatterings. The second factor $e^{-N_{g}^{d}}$ is the probability of
no induced gluon radiating.

Within the high-twist (HT) formalism Wang:2001ifa (30, 63, 64, 65), the
radiative energy loss for a parton $d$ with initial energy $E$ can be
calculated as,

$\frac{\Delta E_{d}}{E}=\int\mathrm{d}\tau\int\mathrm{d}l_{\rm
T}^{2}\int\mathrm{d}z\left(z\frac{\mathrm{d}N^{\rm
med}_{d}}{\mathrm{d}z\mathrm{d}l_{\rm T}^{2}\mathrm{d}\tau}\right),$ (13)

and the number of radiated gluons $N_{g}^{d}$ from the propagating parton $d$
is

$N_{g}^{d}=\int\mathrm{d}\tau\int\mathrm{d}l_{\rm
T}^{2}\int\mathrm{d}z\frac{\mathrm{d}N^{\rm
med}_{d}}{\mathrm{d}z\mathrm{d}l_{\rm T}^{2}\mathrm{d}\tau},$ (14)

where the medium-induced gluon spectrum is given by Deng:2009ncl (65)

$\displaystyle\frac{\mathrm{d}N^{\rm med}_{d}}{\mathrm{d}z\mathrm{d}l_{\rm
T}^{2}\mathrm{d}\tau}=\frac{2C_{A}\alpha_{\rm s}}{\pi l_{\rm
T}^{4}}\left[\frac{1+(1-z)^{2}}{z}\right]$ (15)
$\displaystyle\times\hat{q}_{d}\left(\tau,\vec{r}+\left(\tau-\tau_{0}\right)\vec{n}\right)\sin^{2}\left[\frac{l_{\rm
T}^{2}\left(\tau-\tau_{0}\right)}{4z(1-z)E}\right].$

The integration over the quark propagation path starts from the initial
transverse position $\vec{r}$ at an initial time $\tau_{0}=0.6$ fm/$c$ along
the direction $\vec{n}$. $\alpha_{\rm s}$ is the strong coupling constant,
$l_{\rm T}$ is the transverse momentum of the radiated gluon, and $z$ is its
longitudinal (along the jet direction) momentum fraction. The parton energy
loss for a propagating gluon is assumed to be the same as the quark, except
that the jet transport coefficient of a gluon $\hat{q}_{A}$ differs from that
of a quark $\hat{q}_{F}$ by the color factor $C_{A}/C_{F}$. Therefore, the
radiative energy loss of a gluon is $9/4$ times that of a quark Deng:2009ncl
(65).

Using information field-based global Bayesian inference, jet transport
coefficient $\hat{q}$ is extracted from experimental data of the hadron
suppression Xie:2022fak (17, 16). In this work, we only consider parton energy
loss in the QGP phase with a pseudo-critical temperature $T_{c}=0.165$ GeV.
Effectively, we assume the jet transport coefficient $\hat{q}$ in the local
co-moving frame as $\hat{q}=\hat{q}(T)p^{\mu}\cdot u_{\mu}/p^{0}$ that depends
on the local temperature and vanishes below $T_{c}$, where
$p^{\mu}=\left(p_{0},\vec{p}\right)$ is the four-momentum of the parton and
$u^{\mu}$ is the four flow velocity of the fluid. The dynamical evolution of
the QGP medium created in $AA$ collisions is provided by the CLVisc $3+1$-D
hydrodynamics model simulations Pang:2012he (66, 67, 68).

There are two separate contributions to the azimuthal correlation in heavy-ion
collisions: one is the vacuum parton shower, which is referred to as the
Sudakov effect; the other is the $p_{\rm T}$-broadening effect (or medium
effect) due to multiple scatterings and medium-induced radiation when high-
energy partons propagate through the medium Mueller:2012uf (24, 25, 69, 70).
These two effects contribute differently to the transverse momentum broadening
in well-separated regions of their phase space integral. In $AA$ collisions,
the transverse momentum broadening is calculated by combining the vacuum
Sudakov factor $S(Q,b_{\perp})$ with the medium broadening factor Chen:2016vem
(18)

$S_{\mathrm{med}}(Q,b_{\perp})=S_{\mathrm{vac}}(Q,b_{\perp})+\frac{b_{\perp}^{2}}{4}\left\langle
p_{\perp}^{2}\right\rangle_{\rm tot},$ (16)

where $\left\langle p_{\perp}^{2}\right\rangle$ indicates the amount of the
medium-induced broadening experienced by partons, and
$\left\langle...\right\rangle$ means averaging over all different paths. The
medium-induced broadening for quark and gluon jets are related as
$\left\langle p_{\perp}^{2}\right\rangle_{g}=\frac{C_{A}}{C_{F}}\left\langle
p_{\perp}^{2}\right\rangle_{q}$.

An energetic parton propagating through the QGP medium can encounter
$2\rightarrow 2$ elastic scatterings and exchange transverse momentum with the
medium. These “elastic collisions” or “leading order” contribute to transverse
momentum broadening of the hard parton Majumder:2007hx (71, 72, 73). Such
elastic-scattering contribution to the averaged transverse momentum broadening
squared is given by

$\left\langle p_{\perp}^{2}\right\rangle_{\rm
el}=\left\langle\int\mathrm{d}\tau\hat{q}_{d}\left(\tau,\vec{r}+\left(\tau-\tau_{0}\right)\vec{n}\right)\right\rangle.$
(17)

“Radiative corrections” due to the medium inducing soft gluon radiation also
lead to the $p_{\rm T}$-broadening by the recoil effects in the emission of
soft gluons Liou:2013qya (74, 75, 76). We can obtain $\left\langle
p_{\perp}^{2}\right\rangle_{\rm rad}$ from Equation 15 by calculating the
mean-square momentum $l_{\rm T}$ of radiated gluons:

$\left\langle p_{\perp}^{2}\right\rangle_{\rm rad}=\left\langle\int
d\tau\int\mathrm{d}l_{\rm T}^{2}\int\mathrm{d}z\left(l_{\rm
T}^{2}\frac{\mathrm{d}N^{\rm med}_{d}}{\mathrm{d}z\mathrm{d}l_{\rm
T}^{2}\mathrm{d}\tau}\right)\right\rangle.$ (18)

As an approximation, we assume that the total contributions to $\left\langle
p_{\perp}^{2}\right\rangle$ are given by the sum of elastic-collision and
radiation processes,

$\left\langle p_{\perp}^{2}\right\rangle_{\rm tot}=\left\langle
p_{\perp}^{2}\right\rangle_{\rm el}+\left\langle
p_{\perp}^{2}\right\rangle_{\rm rad}.$ (19)

Figure 1: (Color online) Averaged transverse momentum broadening for a quark
with the median $\hat{q}$ (curves) with uncertainties (bands) as a function of
the initial jet transverse momentum $p_{\rm T}$ in 0-10% $Au+Au$ collisions at
$\sqrt{s_{\rm NN}}=200$ GeV and in 0-10% $Pb+Pb$ collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV. The solid, dot-dashing and dashing curves are for the total,
radiation and elastic collision contributions, respectively.

We plot the $\left\langle p_{\perp}^{2}\right\rangle$ as a function of the
partron transverse momentum in Figure 1 for 0-10% $Au+Au$ collisions at
$\sqrt{s_{\rm NN}}=200\mathrm{GeV}$ and 0-10% $Pb+Pb$ collisions at
$\sqrt{s_{\rm NN}}=5.02\mathrm{TeV}$, respectively. Numerical results show
that the radiation contribution is larger than the collision contribution, and
the total contribution increases slightly with the initial jet transverse
momentum at both RHIC and LHC.

##  3 Numerical results and discussions

With the theoretical framework described in section 2, we will present the
transverse momentum ($p_{\rm T}$ or $z_{\rm T}$) and azimuthal angular
($\Delta\phi$) correlation distributions for $Z/\gamma$-hadron production at
RHIC/LHC in details. The production suppression and azimuthal angular
decorrelation of $Z/\gamma$-hadron caused by energy loss and transverse
momentum broadening will be discussed numerically. At last we calculate the
root-mean-square (RMS) width of $\Delta\phi$ distribution to analyze the
medium broadening effects.

The per-trigger yield of $Z/\gamma$-hadron production or the
$Z/\gamma$-triggered fragmentation function in $pp$ collisions is calculated
as

$\begin{split}D_{pp}(p_{\rm
T}^{h},\Delta\phi)&=\frac{1}{N_{V}^{pp}}\frac{\mathrm{d}^{2}N_{Vh}^{pp}}{\mathrm{d}p_{\rm
T}^{h}\mathrm{d}\Delta\phi}\\\ &=\frac{\int\mathrm{d}p_{\rm
T}^{V}(\mathrm{d}^{3}\sigma^{pp}_{Vh}/\mathrm{d}p_{\rm T}^{V}\mathrm{d}p_{\rm
T}^{h}\mathrm{d}\Delta\phi)}{\int\mathrm{d}p_{\rm
T}^{V}(\mathrm{d}\sigma^{pp}_{V}/\mathrm{d}p_{\rm T}^{V})},\end{split}$ (20)

where the numerator is $Z/\gamma$-hadron cross section and the denominator is
the total cross section of $Z/\gamma$ production. Similarly in $AA$
collisions, the per-trigger yield of $Z/\gamma$-hadron production is given by

$\begin{split}D_{AA}(p_{\rm
T}^{h},\Delta\phi)&=\frac{1}{N_{V}^{AA}}\frac{\mathrm{d}^{2}N_{Vh}^{AA}}{\mathrm{d}p_{\rm
T}^{h}\mathrm{d}\Delta\phi}\\\ &=\frac{\int\mathrm{d}p_{\rm
T}^{V}(\mathrm{d}^{3}\sigma^{AA}_{Vh}/\mathrm{d}p_{\rm T}^{V}\mathrm{d}p_{\rm
T}^{h}\mathrm{d}\Delta\phi)}{\int\mathrm{d}p_{\rm
T}^{V}(\mathrm{d}\sigma^{AA}_{V}/\mathrm{d}p_{\rm T}^{V})}.\end{split}$ (21)

The nuclear modification factor $I_{\rm AA}$ is defined as the ratio of
$Z/\gamma$-hadron yields between $AA$ and $pp$ collisions,

$I_{AA}(p_{\rm T}^{h},\Delta\phi)=\frac{D_{pp}(p_{\rm
T}^{h},\Delta\phi)}{D_{AA}(p_{\rm T}^{h},\Delta\phi)}.$ (22)

Defining $z_{\rm T}=p_{\rm T}^{h}/p_{\rm T}^{V}$, we get the
$Z/\gamma$-triggered fragmentation function as Wang:2003mm (43)

$D_{pp}(z_{\rm T},\Delta\phi)=\frac{\int\mathrm{d}p_{\rm T}^{V}(p_{\rm
T}^{V}\mathrm{d}^{3}\sigma^{pp}_{Vh}/\mathrm{d}p_{\rm T}^{V}\mathrm{d}p_{\rm
T}^{h}\mathrm{d}\Delta\phi)}{\int\mathrm{d}p_{\rm
T}^{V}(\mathrm{d}\sigma^{pp}_{V}/\mathrm{d}p_{\rm T}^{V})}.$ (23)

In $AA$ collisions, the $Z/\gamma$-triggered fragmentation function is given
as

$D_{AA}(z_{\rm T},\Delta\phi)=\frac{\int\mathrm{d}p_{\rm T}^{V}(p_{\rm
T}^{V}\mathrm{d}^{3}\sigma^{AA}_{Vh}/\mathrm{d}p_{\rm T}^{V}\mathrm{d}p_{\rm
T}^{h}\mathrm{d}\Delta\phi)}{\int\mathrm{d}p_{\rm
T}^{V}(\mathrm{d}\sigma^{AA}_{V}/\mathrm{d}p_{\rm T}^{V})}.$ (24)

Then the nuclear modification factor is also written as

$I_{AA}(z_{\rm T},\Delta\phi)=\frac{D_{AA}(z_{\rm
T},\Delta\phi)}{D_{pp}(z_{\rm T},\Delta\phi)}.$ (25)

One can deifne $\xi_{\rm T}=\ln(-\left|\vec{p}_{\rm
T}^{V}\right|^{2}/\vec{p}_{\rm T}^{h}\cdot\vec{p}_{\rm T}^{V})$ CMS:2021otx
(77), then get the $Z/\gamma$-triggered fragmentation function in $pp$
collisions as

$D_{pp}(\xi_{\rm T},\Delta\phi)=-\frac{\int\mathrm{d}p_{\rm T}^{V}(p_{\rm
T}^{h}\mathrm{d}^{3}\sigma^{pp}_{Vh}/\mathrm{d}p_{\rm T}^{V}\mathrm{d}p_{\rm
T}^{h}\mathrm{d}\Delta\phi)}{\int\mathrm{d}p_{\rm
T}^{V}(\mathrm{d}\sigma^{pp}_{V}/\mathrm{d}p_{\rm T}^{V})},$ (26)

and in $AA$ collisions as

$D_{AA}(\xi_{\rm T},\Delta\phi)=-\frac{\int\mathrm{d}p_{\rm T}^{V}(p_{\rm
T}^{h}\mathrm{d}^{3}\sigma^{AA}_{Vh}/\mathrm{d}p_{\rm T}^{V}\mathrm{d}p_{\rm
T}^{h}\mathrm{d}\Delta\phi)}{\int\mathrm{d}p_{\rm
T}^{V}(\mathrm{d}\sigma^{AA}_{V}/\mathrm{d}p_{\rm T}^{V})}.$ (27)

The corresponding nuclear modification factor is then written as

$I_{AA}(\xi_{\rm T},\Delta\phi)=\frac{D_{AA}(\xi_{\rm
T},\Delta\phi)}{D_{pp}(\xi_{\rm T},\Delta\phi)}.$ (28)

Figure 2: (Color online) The $p_{\rm T}^{h}$ distributions of per $Z$-trigger
hadrons with three $p_{\rm T}^{Z}$ ranges are calculated in $pp$ collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV within the Sudakov resummation improved pQCD
parton model, and compared with the ATLAS experimental data ATLAS:2020wmg
(78).

###  3.1 transverse momentum correlations of $Z/\gamma$-hadrons

Figure 3: (Color online) The nuclear modification factors $I_{AA}$ of
$Z$-hadron as a function of $p_{\rm T}^{h}$ with three trigger $p_{\rm T}^{Z}$
ranges in 0-10% (upper panels) and 10-30% (lower panels) $PbPb$ collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV, compared with the ATLAS experimental data
ATLAS:2020wmg (78), respectively.

Taking the energy loss and transverse momentum broadening into account
simultaneously, we first calculate the transverse momentum distribution for
$Z/\gamma$-hadron production and show their medium modifications. The
numerical results with the Sudakov resummation improved pQCD parton model are
compared with all available experimental data at the LHC.

Shown in Figure 2 is the transverse momentum spectra of the charged hadrons
triggered by $Z$ bosons in $pp$ collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV.
The numerical results with three trigger transverse momentum ($p_{\rm T}^{Z}$)
ranges, $15<p_{\rm T}^{Z}<30$, $30<p_{\rm T}^{Z}<60$, and $p_{\rm T}^{Z}>60$
GeV, are compared with the ATLAS data ATLAS:2020wmg (78), respectively.
Numerical results describe the data well, especially at large $p_{\rm T}^{h}$.
The theoretical calculations underestimate the data at low $p_{\rm T}^{h}$
because the fragmentation functions are limited with $z_{\min}=0.05$ and the
more higher order corrections should be taken into account.

Figure 4: (Color online) The hadron $p_{\rm T}^{h}$ distributions with the $Z$
trigger $p_{\rm T}^{Z}>30$ GeV in $pp$ (dashing curves) and 0-30%, 30-50%,
50-70% and 70-90% $PbPb$ (solid curves with bands) collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV are shown in the upper panels, while the corresponding
$I_{AA}(p_{\rm T}^{h})$ in the lower panels, compared with the CMS
experimental data CMS:2021otx (77), respectively. Figure 5: (Color online) The
$Z$-hadron $\xi_{\rm T}$ distributions with the $Z$ trigger $p_{\rm T}^{Z}>30$
GeV and the correlated haron $p_{\rm T}^{h}>1$ GeV in $pp$ (dashing curves)
and 0-30%, 30-50%, 50-70% and 70-90% $PbPb$ (solid curves with bands)
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV are shown in the upper panels,
while the corresponding $I_{AA}(\xi_{\rm T})$ in the lower panels, compared
with the CMS experimental data CMS:2021otx (77), respectively. Figure 6:
(Color online) The $\gamma$-hadron $z_{\rm T}$ distributions with the $\gamma$
trigger $12<p_{\rm T}^{\gamma}<20$ GeV and the correlated haron $p_{\rm
T}^{h}>1.2$ GeV in $pp$ (dashing curve) and 0-10% $AuAu$ (solid curve with
band) collisions at $\sqrt{s_{\rm NN}}=200$ GeV are shown in the upper panel,
while the corresponding $I_{AA}(z_{\rm T})$ in the lower panel, compared with
the STAR experimental data STAR:2009ojv (79, 80), respectively. Figure 7:
(Color online) The $\gamma$-hadron $z_{\rm T}$ distributions with the $\gamma$
trigger $12<p_{\rm T}^{\gamma}<40$ GeV and the correlated haron $0.5<p_{\rm
T}^{h}<10.0$ GeV in $pp$ (dashing curve) and 0-10% $PbPb$ (solid curve with
band) collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV are shown in the upper panel,
while the corresponding $I_{AA}(z_{\rm T})$ in the lower panel. The $pp$
spectra are compared with the ALICE experimental data ALICE:2020atx (81).
Figure 8: (Color online) The per-trigger yield of $Z$-hadron pairs as a
function of $\Delta{\phi}$ with the $Z$ trigger $15<p_{\rm T}^{Z}<30$,
$30<p_{\rm T}^{Z}<60$ and $p_{\rm T}^{Z}>60$ GeV (from left to right panels)
and the associated hadron $p_{\rm T}^{h}>4$ GeV in $pp$ (dashing curves) and
0-30% $PbPb$ (solid curves) collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV are
shown in the upper panels, while the corresponding $I_{AA}(\Delta\phi)$ in the
lower panels, compared with the ATLAS experimental data ATLAS:2020wmg (78).

The nuclear modification factors $I_{AA}(p_{\rm T}^{h})$ of $Z$-hadrons in
0-10% (upper panels) and 10-30% (lower panels) $PbPb$ collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV are shown in Figure 3, compared with the ATLAS
data, respectively. From the left to the right panel, the transverse momentum
of trigger $Z$ is set as $15<p_{\rm T}^{Z}<30$ GeV, $30<p_{\rm T}^{Z}<60$ GeV
and $p_{\rm T}^{Z}>60$ GeV. The azimuthal angle between the vector boson and
the hadron is set as $\Delta\phi=|\phi_{V}-\phi_{h}|>3\pi/4$. The solid curves
are the numerical results with the median value of the $\hat{q}/T^{3}$, which
is extracted from hadron suppression at RHIC and the LHC energies using the
IF-Bayesian method Xie:2022ght (16, 17), and the bands are the estimated
uncertainty given by $\hat{q}/T^{3}$. It is worth mentioning that the
suppression uncertainties shown in Figure 3 are contributed by the parton
energy loss, which is because the medium-induced broadening mainly affects the
azimuthal angular of jets and hardly influences their yield Wang:1998ww (34).
Our numerical results are in good agreement with the experimental data at mid
to large $p_{\rm T}^{h}$ for the different trigger $p_{\rm T}^{Z}$.

No matter the experimental data or the numerical results in Figure 3, show
that the $Z$-hadron suppression becomes weak when the transverse momentum of
trigger $Z$ increases. This phenomenon is consistent with our previous studies
on di-/$\gamma$-hadrons Zhang:2007ja (22, 23, 14), by the difference in the
production positions of jets with different momentum. The hadrons triggered by
the large $p_{\rm T}^{Z}$ $Z$ bosons are from the fragmentation of the hard
partons which are more likely to be produced at the surface of the medium and
prefer to emit off the medium perpendicularly with less energy loss, while the
hadrons triggered by the small $p_{\rm T}^{Z}$ $Z$ bosons are contributed by
the small $p_{\rm T}$ jets which are produced at the center of the medium and
emit off the medium with large energy loss fractions. Numerical results
underestimate the suppression with the large trigger $p_{\rm T}^{Z}>60$ GeV,
which is similar to the previous studies on $\gamma$-hadron Xie:2020zdb (82)
and dihadron Xie:2019oxg (14). It might be because the parton energy
dependence was not taken into account in the IF-Bayesian $\hat{q}$ extraction
method Xie:2022ght (16, 17), and the energy loss fraction $\Delta{E}/E$
decrease with the parton energy increasing.

To further study the centrality dependence of $Z$-hadron transverse momentum
correlations, we show in Figure 4 for the $Z$-triggered hadron spectra
$D_{AA}(p_{\rm T}^{h})$ (upper panels) and the corresponding suppression
factors $I_{AA}(p_{\rm T}^{h})$ (lower panels) with the trigger $p_{\rm
T}^{Z}>30$ GeV in 0-30%, 30-50%, 50-70% and 70-90% $PbPb$ collisions at 5.02
TeV, respectively. The azimuthal angle between the $Z$ boson and the hadron is
set as $\Delta\phi>7\pi/8$. The spectra $D_{pp}(p_{\rm T}^{h})$ in $pp$
collisions at 5.02 TeV are also shown in the upper panels. We can see the
numerical results fit CMS data CMS:2021otx (77) well. In most central
collisions, the $Z$-triggered hadrons could be suppressed by about 30-50% due
to jet energy loss. Such suppression becomes weaker and weaker until jet
energy loss almost vanishes from central to peripheral collisions since the
medium size becomes small in peripheral collisions. Meanwhile, the theoretical
uncertainty related to jet energy loss also becomes small in peripheral
collisions.

Figure 9: (Color online) The per-trigger yield of $\gamma$-hadron pairs as a
function of $\Delta{\phi}$ with the $\gamma$ trigger $12<p_{\rm
T}^{\gamma}<40$ GeV and the associated hadron $5.4<p_{\rm T}^{h}<10$ GeV or
$0.45<z_{\rm T}<0.6$ in $pp$ (green curves) and 0-10% $PbPb$ (pink curves)
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV are shown in the upper panel, while
the corresponding $I_{AA}(\Delta\phi)$ in the lower panel. The solid (dashing)
curves are calculated with the resummation-improved (the $\alpha_{s}^{3}$
fixed-order or NLO) pQCD parton model for $pp$ and $AA$ collisions,
respectively. The $D_{pp}(\Delta{\phi})$ data are from the ALICE experiments
ALICE:2020atx (81). Figure 10: (Color online) The per-trigger yield of
$\gamma$-hadron pairs as a function of $\Delta{\phi}$ with the $\gamma$
trigger $12<p_{\rm T}^{\gamma}<20$ GeV and the associated hadron $3<p_{\rm
T}^{h}<5$ GeV in $pp$ (green curves) and 0-12% $AuAu$ (pink curves) collisions
at $\sqrt{s_{\rm NN}}=200$ GeV are shown in the upper panel, while the
corresponding $I_{AA}(\Delta\phi)$ in the lower panel, compared with the STAR
experimental data STAR:2016jdz (80). The solid (dashing) curves are calculated
with the resummation-improved (the $\alpha_{s}^{3}$ fixed-order or NLO) pQCD
parton model for $pp$ and $AA$ collisions, respectively.

Shown in Figure 5 are the $Z$-hadron $\xi_{\rm T}$ distributions
$D_{AA}(\xi_{\rm T})$ (upper panels) and the corresponding suppression factors
$I_{AA}(\xi_{\rm T})$ (lower panels) with $Z$ trigger $p_{\rm T}^{Z}>30$ GeV
and the away-side associated haron $p_{\rm T}^{h}>1$ GeV in 0-30%, 30-50%,
50-70% and 70-90% $PbPb$ collisions at 5.02 TeV, respectively. The azimuthal
angle between the $Z$ boson and the hadron is set as $\Delta\phi>7\pi/8$. The
spectra $D_{pp}(\xi_{\rm T})$ in $pp$ collisions at 5.02 TeV are also shown in
the upper panels. According to the definitions of $\xi_{\rm
T}=\ln(-\left|\vec{p}_{\rm T}^{Z}\right|^{2}/\vec{p}_{\rm
T}^{h}\cdot\vec{p}_{\rm T}^{Z})\approx-\ln z_{\rm T}$ and $z_{\rm T}=p_{\rm
T}^{h}/p_{\rm T}^{Z}$, $\xi_{\rm T}$ can be understood as the magnitude of the
unit vector where the hadron transverse momentum is projected on the trigger
axis, while the negative sign is for the projection in the away-side
hemisphere. We note that a hadron carrying all of the parent parton momentum,
$z_{\rm T}=1$, and moving back-to-back away from the trigger,
$\Delta\phi=\pi$, would have a $\xi_{\rm T}=0$. Because the calculations for
low $p_{\rm T}^{h}$ hadrons are limited by the fragmentation limit
$z_{\min}=0.05$, we focus on the region of $\xi_{\rm T}\leq 2.5$, which
corresponds to $z_{\rm T}\geq 0.08$. An indication of energy loss would mean
that the $Z$-hadron spectrum of $PbPb$ collisions has a shift to a larger
value of $\xi_{\rm T}$ compared to that in $pp$ collisions, which is more
evident when comparing the results from peripheral collisions to central
collisions. Our calculations can provide a good description to the CMS
measurements for $Z$-hadron $\xi_{\rm T}$ distributions and the supperresions
$I_{AA}(\xi_{\rm T})$, especially for most central $AA$ collisions.

Similar calculations were done for $\gamma$-triggered hadrons, as shown in
Figure 6 with the $\gamma$ trigger $12<p_{\rm T}^{\gamma}<20$ GeV and the
correlated haron $p_{\rm T}^{h}>1.2$ GeV in $pp$ and 0-10% $AuAu$ collisions
at $\sqrt{s_{\rm NN}}=200$ GeV, and in Figure 7 with the $\gamma$ trigger
$12<p_{\rm T}^{\gamma}<40$ GeV and the correlated haron $0.5<p_{\rm
T}^{h}<10.0$ GeV in $pp$ and 0-10% $PbPb$ collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV. The jet energy loss parameter $\hat{q}$ and its uncertainty
are all set the same as in Figure 3, Figure 4, and Figure 5. Notice that the
IF-Bayesian extracted $\hat{q}$ is available for different-centrality $AA$
collisions at RHIC and the LHC Xie:2022ght (16, 17). Our numerical results for
the $\gamma$-triggered fragmentation functions $D(z_{\rm T})$ and the nuclear
modification factor $I_{AA}(z_{\rm T})$ both describe the STAR experimental
data STAR:2009ojv (79, 80) very well, which demonstrates the jet quenching
description and the Sudakov resummation improved pQCD model are effective no
matter at the LHC energies or RHIC energies. Besides for the well fitting to
ALICE data ALICE:2020atx (81) for $pp$ collisions in Figure 7, we predict the
$\gamma$-triggered away-side hadron spectra $D_{AA}(z_{\rm T})$ and the
suppression factor $I_{AA}(z_{\rm T})$ in 0-10$\%$ $PbPb$ collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. Numerical results show that
$\gamma$-triggered hadron spectra are suppressed by 40% for low $z_{\rm T}$
and 80% for large $z_{\rm T}$ due to jet energy loss in central $PbPb$
collisions.

We now have a well-established pQCD framework with the Sudakov resummation
technique, which can well describe the transverse momentum correlations of
$Z/\gamma$-hadron pairs via comparisons with the experimental data in $pp$ and
$AA$ collisions at both RHIC and LHC. Next let’s check $\left\langle
p_{\perp}^{2}\right\rangle$ as well as $\Delta E$ effects on the azimuthal
angular ($\Delta\phi$) correlations of $Z/\gamma$-hadron production in central
$AA$ collisions at RHIC and the LHC.

###  3.2 azimuthal angular correlations of $Z/\gamma$-hadrons

Shown in the upper panels of Figure 8 are the per-trigger yield of $Z$-hadron
pairs as a function of the azimuthal angle $\Delta{\phi}$ between the $Z$
trigger (with $15<p_{\rm T}^{Z}<30$, $30<p_{\rm T}^{Z}<60$ and $p_{\rm
T}^{Z}>60$ GeV from left to right panels) and the associated hadron (with
$p_{\rm T}^{h}>4$ GeV) in $pp$ (dashing curves) and 0-30% $PbPb$ (solid
curves) collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, and in the lower panels
are the corresponding nuclear modificatio factors $I_{AA}(\Delta\phi)$. Our
theoretical calculations can provide a very good description to the ATLAS
experimental data ATLAS:2020wmg (78) in both $pp$ and $PbPb$ collisions for
three trigger $p_{\rm T}^{Z}$ ranges. Numerical results show that with the
trigger $p_{\rm T}^{Z}$ increasing, the angular correlation of $Z$-hadron
pairs becomes stronger in both $pp$ and central $AA$ collisions but the yield
suppression changes weaker. The former is mainly because the Sudakov effect
enhances as the $Q^{2}$ increases, as shown in Eq. (6), and the latter is
similar as in Figure 3. The uncertainty from the medium-induced broadening is
almost swallowed up by that from the jet energy loss. The approximate-constant
$I_{AA}(\Delta\phi)$ as a function of $\Delta\phi$, as shown in the lower
panels of Figure 8, means the medium-induced broadening effect is fairly weak.
This is consistent with previous studies on dijet angular correlations due to
the dominant Sudakov effect for the large $p_{T}$ jets, especially at LHC
energies Chen:2016vem (18, 19).

In Figure 9 we present the per-trigger yield of $\gamma$-hadron pairs as a
function of $\Delta{\phi}$ with the $\gamma$ trigger $12<p_{\rm
T}^{\gamma}<40$ GeV and the associated hadron $5.4<p_{\rm T}^{h}<10$ GeV or
$0.45<z_{\rm T}<0.6$ in $pp$ and 0-10% $PbPb$ collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV in the upper panel, while the corresponding
$I_{AA}(\Delta\phi)$ in the lower panel. Here we use the Sudakov resummation-
improved pQCD parton model to focus on the back-to-back region (in fact we
present in the region $2.2<\Delta{\phi}<\pi$) in which the $\alpha_{s}^{3}$
fixed-order or NLO pQCD expansion for $D_{pp(AA)}(\Delta{\phi})$ diverges. For
a comparison, we also give the NLO results Xie:2022ght (16, 17) in the region
$1.65<\Delta{\phi}<2.85$. In fitting to the ALICE data ALICE:2020atx (81) for
$D_{pp}(\Delta{\phi})$, the two models both work well in their respective
$\Delta{\phi}$ regions. We also predict the $\gamma$-triggered away-side
hadron spectra $D_{AA}(\Delta{\phi})$ and the suppression factor
$I_{AA}(\Delta{\phi})$ in 0-10$\%$ $PbPb$ collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. Compared to the $\Delta\phi$ distribution
of $Z$-hadron production in Figure 7 with the same kinematic cuts, the
distribution of $\gamma$-hadron gets almost the same suppression about
$I_{AA}\approx$ 0.25 in Figure 9 when both energy loss and the medium-induced
broadening with median $\hat{q}/T^{3}$ are included in the calculations for
$PbPb$ collisions as opposed to that for $pp$ collisions.

Figure 11: (Color online) Normalized $Z$-hadron angular distributions in $pp$
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with a fixed trigger-$p_{\rm
T}^{Z}$ region and different association $p_{\rm T}^{h}$ regions. The four
fixed trigger-$p_{\rm T}^{Z}$ regions are $15<p_{\rm T}^{Z}<30$, $30<p_{\rm
T}^{Z}<40$, $40<p_{\rm T}^{Z}<50$ and $100<p_{\rm T}^{Z}<120$ GeV from the
left to the right panel, respectively. Figure 12: (Color online) Similar to
Figure 11 but for the $\gamma$-hadron production at $\sqrt{s_{\rm NN}}=200$
GeV. The kinematics cuts are different correspondingly.

Shown in Figure 10 is the $\Delta\phi$ distribution of the $\gamma$-hadron
correlation with the trigger $12<p_{\rm T}^{\gamma}<20$ GeV and the
association $3<p_{\rm T}^{h}<5$ GeV in $pp$ and 0-12% $AuAu$ collisions at
$\sqrt{s_{\rm NN}}=200$ GeV in the upper panel, while the suppression factor
in the lower panel, compared with the STAR measurements STAR:2016jdz (80),
respectively. The numerical results from the resummation-improved pQCD parton
model provide a very well fit to the experimental data for
$D_{pp}(\Delta{\phi})$, $D_{AA}(\Delta{\phi})$ and $I_{AA}(\Delta{\phi})$ in
the region of $2.4<\Delta{\phi}<\pi$, respectively, while the NLO calculations
fit well in $1.6(2.0)<\Delta{\phi}<2.4$.

###  3.3 RMS analysis of the width of $\Delta\phi$ distribution

Figure 13: (Color online) RMS width of angular distributions of $Z$-hadron as
a function of $p_{\rm T}^{h}$ in $pp$ collisions at $\sqrt{s_{\rm NN}}=5.02$
TeV with four different $p_{\rm T}^{Z}$ regions, $30<p_{\rm T}^{Z}<40$,
$40<p_{\rm T}^{Z}<50$, $50<p_{\rm T}^{Z}<60$ and $60<p_{\rm T}^{Z}<70$ GeV
from top to bottom, respectively. Figure 14: (Color online) Similar to Figure
13 but for the $\gamma$-hadron production at $\sqrt{s_{\rm NN}}=200$ GeV. The
kinematics cuts are different correspondingly.

In the above calculations the Sudakov resummation technique is introduced into
the pQCD parton model to get rid of the divergence on the endpoints in the
$\Delta\phi$ distribution of $Z/\gamma$-hadron pairs in $pp/AA$ collisions.
The numerical results show that the Sudakov resummation improved pQCD parton
model works well not only on the transverse momentum correlation but also on
the azimuthal angular correlation when $\Delta\phi$ is constrained near the
back-to-back region of $Z/\gamma$-hadron pairs. Especially, after including
the jet energy loss $\Delta E$ and the medium-induced broadening $\langle
p_{\perp}^{2}\rangle$ in high-energy $AA$ collisions, the model succeeds in
describing the suppression of the large-$p_{\rm T}$ $Z/\gamma$-hadron as well
as the medium-induced broadening effect. In the $Z/\gamma$-hadron
correlations, the strength of the azimuthal angular correlation is of primary
concern. In $pp$ collisions, the $\Delta\phi$ correlation is contributed by
the vacuum Sudakov effect. While in $AA$ collisions, a medium-induced $\langle
p_{\perp}^{2}\rangle$ effect is included (see Eq.(16)) in the vacuum Sudakov
factor and the jet energy loss $\Delta E$ is adopted (see Eq.(12)) to modify
the fragmentation function. In this subsection, we further analyze the angular
correlation by normalizing the $\Delta\phi$ distribution to unity. This
approach allows us to focus on the differences between $pp$ and $AA$
collisions with $\Delta E$ or $\langle p_{\perp}^{2}\rangle$, or both.

Shown in each panel of Figure 11 are the normalized $Z$-hadron angular
distributions in $pp$ collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV with a fixed
trigger-$p_{\rm T}^{Z}$ region and different association $p_{\rm T}^{h}$
regions. The four fixed trigger-$p_{\rm T}^{Z}$ regions are $15<p_{\rm
T}^{Z}<30$, $30<p_{\rm T}^{Z}<40$, $40<p_{\rm T}^{Z}<50$ and $100<p_{\rm
T}^{Z}<120$ GeV from the left to the right panel, respectively. Figure 12 is
similar to Figure 11 but for $\gamma$-hadron production at $\sqrt{s_{\rm
NN}}=200$ GeV with different kinematics cuts correspondingly. The numerical
results at both LHC and RHIC show that the angular correlation is stronger
with the $p_{\rm T}$ increasing of both the trigger $Z/\gamma$ and the
associated hadron. With current selections of $p_{\rm T}$ cuts, the $p_{\rm
T}$ dependence of the angular correlation is weaker at LHC than at RHIC.

The angular correlation strength can be quantified by the root-mean-square
(RMS) width of $\Delta\phi$ distribution Jia:2019qbl (83):

$\Delta\phi_{\mathrm{RMS}}=\sqrt{\frac{\int\mathrm{d}\Delta\phi(\Delta\phi-\pi)^{2}\frac{\mathrm{d}\sigma}{\mathrm{d}\Delta\phi}}{\int\mathrm{d}\Delta\phi\frac{\mathrm{d}\sigma}{\mathrm{d}\Delta\phi}}},$
(29)

where the integral range of $\Delta\phi$ is chosen from 2.4 to $\pi$ to
constrain $Z/\gamma$-hadron pairs near the back-to-back region. For a totally
back-to-back configuration, corresponding to a delta function for the
azimuthal distribution, $\Delta\phi_{\rm RMS}=0$, which gives the strongest
angular correlation. The larger value of $\Delta\phi_{\rm RMS}$ would indicate
a broader azimuthal distribution or a weaker angular correlation.

Shown in Figure 13 (Figure 14) is the RMS width of angular distributions of
$Z$-hadron ($\gamma$-hadron) as a function of $p_{\rm T}^{h}$ in $pp$
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV (200 GeV) with four different
$p_{\rm T}^{Z}$ ($p_{\rm T}^{\gamma}$) regions, respectively. The conclusion
is the same as that in Figure 11 and Figure 12. The RMS width is dependent on
the transverse momentum of both the trigger and the associated hadron, and the
dependence on $p_{\rm T}^{Z}$ or $p_{\rm T}^{\gamma}$ is slightly stronger
than $p_{\rm T}^{h}$. In following calculations we will choose a fixed region
for $p_{\rm T}^{h}$ with varying $p_{\rm T}^{Z}$ or $p_{\rm T}^{\gamma}$.

(a) * (b) * (c) *

Figure 15: (Color online) Normalized $Z$-hadron angular distributions in $pp$
(solid purple) collisions and 0-10% $PbPb$ collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV with (blue dot-dashed) and without (red dashed) the medium-
induced broadening effect. Different $p_{\rm T}$ cuts are set for the triggers
and the correlated hadrons in the panels, respectively. The data are from the
ATLAS experiment ATLAS:2020wmg (78).

(a) * (b) * (c) *

Figure 16: (Color online) Similar to Figure 15 but for the $\gamma$-hadron
production in 0-10% $AuAu$ collisions at $\sqrt{s_{\rm NN}}=200$ GeV. The
kinematics cuts are different correspondingly. The data are from the STAR
experiment STAR:2016jdz (80).

To see the influence of jet quenching effect on the azimuth angular
correlation, we add the energy loss and transverse momentum broadening into
$AA$ collisions progressively and compare the $\Delta\phi$ distribution with
$pp$ results. Shown in Figure 15 are the normalized $Z$-hadron angular
distributions in $pp$ and 0-10% $PbPb$ collisions at $\sqrt{s_{\rm NN}}=5.02$
TeV with only the energy loss and both the energy loss and the medium-induced
broadening effect. The inserted panels are to show the results near $\pi$
clearly. Different $p_{\rm T}$ cuts are set for the triggers and the
correlated hadrons in the different panels, respectively. Figure 16 is similar
to Figure 15 but for the $\gamma$-hadron production in 0-10% $AuAu$ collisions
at $\sqrt{s_{\rm NN}}=200$ GeV. The kinematics cuts are different
correspondingly. Here only the median value of $\hat{q}$ from the IF-Bayesian
extraction is used for the energy loss and and the medium-induced broadening
in central $AA$ collisions at both RHIC and LHC. First of all, the intuitive
overall phenomenon is similar to the $pp$ cases, even in central $AA$
collisions at both RHIC and LHC the angular correlation is stronger with the
$p_{\rm T}$ increasing of both the trigger $Z/\gamma$ and the associated
hadron.

For central $PbPb$ (Figure 15) and $AuAu$ (Figure 16) collisions, we set two
cases, one is only with the energy loss effect but without the medium-induced
broadening effect (red dashed curves), and the other is with both of them
(blue dot-dashed curves). When only considering the jet energy loss effect in
$AA$ collisions, one can observe that the angular correlation is stronger in
$AA$ collisions than that in $pp$ collisions (blue solid curves). After taken
with both the energy loss $\Delta{E}$ and the medium-induced broadening
$\langle p_{\perp}^{2}\rangle$ in $AA$ collisions, the normalized $\Delta\phi$
distributions are then broadened. Compared with $pp$ collisions, in the first
panels of Figure 16 the normalized $\Delta\phi$ distributions in $AA$
collisions are broader than in $pp$ collisions. However, in the other panels
of Figure 16 and Figure 15, despite the medium-induced broadening effect makes
the normalized $\Delta\phi$ distributions broader, the normalized $\Delta\phi$
distributions in $AA$ collisions are narrower than in $pp$ collisions. In a
word, $\langle p_{\perp}^{2}\rangle$ makes a decorrelation to broaden the
$\Delta\phi$ distribution, while $\Delta{E}$ enhances the correlation to
narrow the $\Delta\phi$ distribution, namely anti-broadening. The modification
of the angular correlation is a result of the competition between the
broadening and the anti-broadening. However, the medium modification of the
sum of positive and negative contributions is too small to be observed
noticeably via the comparison with pp and AA data due to large error bars.

Our numerical results fit data very well in Figure 16 and Figure 15. We do not
observe the evident broadening effect in $Z/\gamma$-hadron angular
distributions in $AA$ collisions at both RHIC and LHC, which is different from
the observation of the angular decorrelation in di-hadron/jet production
Chen:2016vem (18, 19). This is because of the weaker medium broadening effect
in $Z/\gamma$-hadron production. The average broadening strength in one
$Z/\gamma$-hadron/jet event is about half of that in one di-hadron/jet event
due to the absence of the strong interaction between the $Z/\gamma$ boson and
the medium. In addition, the fragmentation contribution ratio of hard quarks
over gluons for the $Z/\gamma$-hadron production is larger than for the di-
hadron production, and the $\langle p_{\perp}^{2}\rangle$ value for a quark
jet is $4/9$ times of that for a gluon jet.

To demonstrate the medium modifications, generally, one need to set the same
kinematics cuts for $Z$/$\gamma$-hadron production between $AA$ and $pp$
collisions. Due to jet quenching or “$p_{\rm T}$ shift”, the parent parton for
the observed hadrons in $AA$ collision has a larger $p_{\rm T}$ prior to
quenching than the parent parton $p_{\rm T}$ in $pp$ collision. The more
larger the $p_{\rm T}$ is, the more stronger the angular correlation is from
the Sudakov effects, so the $Z$/$\gamma$-hadron angular correlation is
enhanced by jet energy loss in $AA$ collisions due to the bias of the same
kinematics cuts between $AA$ and $pp$ collisions. This is why the anti-
broadening is brought by $\Delta{E}$. Numerical results in Figure 15 and
Figure 16 show that the positive effect of the $\langle p_{\perp}^{2}\rangle$
broadening can be either stronger or smaller than the negative effect of the
$\Delta{E}$ anti-broadening in the angular correlations. The observed
broadening displayed in the $\Delta\phi$ distribution is a sum of the positive
and negative contributions. Such a competition between the broadening and the
anti-broadening is stronger at RHIC than at LHC.

Figure 17: (Color online) Shown in the upper panel are the RMS widths as a
function of the trigger $p_{\rm T}^{Z}$ with the associated-hadron $p_{\rm
T}>4$ GeV for the $Z$-hadron production in both $pp$ and central $PbPb$
collisions at 5.02 TeV. Either energy loss or medium broadening or both are
set in $PbPb$ collisions. Their differences of the RMS widths against $pp$
collisions are shown in the lower panel. Figure 18: (Color online) Similar to
Figure 17 but as a function of $p_{\rm T}^{\gamma}$ for the $\gamma$-hadron
production in $pp$ and central $AuAu$ collisions at 200 GeV.

To quantify the broadening and the anti-broadening effects, we then plotted in
the upper panel of Figure 17 the RMS width as a function of the trigger
$p_{\rm T}^{Z}$ with the associated-hadron $p_{\rm T}>4$ GeV for the
$Z$-hadron production in both $pp$ and central $PbPb$ collisions at 5.02 GeV.
Three cases are set for the medium effects in $PbPb$ collisions, only with
broadening effect (green square), only with energy loss effect (red triangle),
and with both energy loss and broadening effects (blue pentagon), while the
$pp$ results are given by purple circle. Here we also use the $\hat{q}$-median
for the jet energy loss and the medium-induced broadening. We see that as we
move to the trigger with the larger $p_{\rm T}^{Z}$ ($p_{\rm T}^{\gamma}$),
the $\Delta\phi_{\rm RMS}$ decreases, corresponding to the sharper angular
distribution or the stronger angular correlation. Because both the energy loss
and the medium broadening effects have weak effects at the LHC kinematics, it
is difficult to differentiate them, we thus plot their difference against $pp$
calculations (without both effects) in the lower panel of Figure 17. Figure 18
is similar to Figure 17 but as a function of $p_{\rm T}^{\gamma}$ for the
$\gamma$-hadron production in $pp$ and central $AuAu$ collisions at 200 GeV.
The medium-induced broadening has a positive azimuthal broadening effect on
the RMS width, as shown (green square) in the both lower panels of Figure 17
and Figure 18, which decreases as moving to large $p_{\rm T}^{Z}$ or $p_{\rm
T}^{\gamma}$ due to the overwhelming contribution from vacuum Sudakov effects.
The energy loss effect (red triangle) has a negative azimuthal broadening
effect, so-called anti-broadening, and decreases slightly with $p_{\rm T}^{Z}$
or $p_{\rm T}^{\gamma}$. This is because the parton energy loss in the HT
energy loss model has a logarithmic energy dependence that is weaker than a
linear dependence, so the fractional energy loss $\Delta E/E$ decreases with
jet energy Wang:2016fds (84).

The observed width difference should be the combined broadening effect (blue
pentagon) from the medium-induced broadening and parton energy loss. Our
numerical results show that the combined effect is positive $\gamma$-hadron
pairs with small $p_{\rm T}^{\gamma}$ at RHIC energy in Figure 18. For LHC
energy and the large $p_{\rm T}$ $Z/\gamma$-hadron pairs at RHIC, for example
$p_{\rm T}^{\gamma}>20$ GeV and $p_{\rm T}^{h}>2$ GeV, the combined effect
becomes negative due to the stronger anti-broadening effect from parton energy
loss than the medium-induced broadening effect with given the
$\hat{q}$-median. However, the negative effect is so weak to be difficultly
observed at LHC experiments.

Figure 19: (Color online) The RMS widths of angular correlation distributions
of the $Z$-hadron pairs as a function of the colliding energy $\sqrt{s_{NN}}$
with or without the parton energy loss or the medium-induced broadening
effects. Figure 20: (Color online) The RMS width of angular correlation
distributions of $\gamma$-hadron pairs as a function of the colliding energy
$\sqrt{s_{NN}}$ with or without the parton energy loss or the medium-induced
broadening effects.

Now that we use the RMS width to quantify the dependence of the
$Z/\gamma$-hadron angular correlation on the transverse momentum as well as
the medium “kick” and the energy loss, let’s also check the dependence on the
mass center energy in heavy-ion collisions. Shown in Figure 19 for $Z$-hadron
pairs and Figure 20 for $\gamma$-hadron pairs are the RMS widths as a function
of the mass center energy in central $AuAu$ collisions at
$\sqrt{s_{\mathrm{NN}}}=0.2$ TeV and central $PbPb$ collisions at
$\sqrt{s_{\mathrm{NN}}}=2.76$ TeV and 5.02 TeV, respectively. The
corresponding $pp$ results are also shown for comparisons with the same cuts,
$30<p_{\rm T}^{Z/\gamma}<40$ GeV and $p_{\rm T}^{h}>4$ GeV. We set the same
three cases for $AA$ collisions as in Figure 17 and Figure 18. Numerical
results for the $Z/\gamma$-hadron production show that the width of the
angular distribution becomes larger with the increasing of the mass center
energy in both $pp$ and central $AA$ collisions. The weakening angular
correlation is resulted from the dominance of the Sudakov contribution which
is stronger due to the higher scale $Q^{2}$, as shown in Eqs. (6) and (8).
Comparing to the angular correlation in $pp$ collisions (purple circle), in
$AA$ collisions the parton energy loss effect (red triangle) enhances the
angular correlation while the medium-induced broadening effect (green square)
makes the decorrelation.

Comparing the numerical results above, one can see that the width of the
$\gamma$-hadron angular distribution is smaller than the corresponding width
of the $Z$-hadron. This is because with the same kinematics cuts the
$\gamma$-hadron process has a lower hard scale $Q^{2}$ than the $Z$-hadron
process due to $Z$ mass, and thus encounters a weaker Sudakov effect. When
subtracting $\Delta\phi_{\rm RMS}^{pp}$ from $\Delta\phi_{\rm RMS}^{AA}$, the
Sudakov effect almost cancels out, which makes $(\Delta\phi_{\rm
RMS}^{AA})^{2}-(\Delta\phi_{\rm RMS}^{pp})^{2}$ similar between the
$\gamma$-hadron and $Z$-hadron angular correlations.

##  4 Summary

We have applied the Sudakov resummation improved pQCD parton model, including
the Higher-Twist energy loss $\Delta E$ and medium-induced transverse momentum
broadening $\left\langle p_{\perp}^{2}\right\rangle$, to study medium
modifications on momentum and angular correlations between a large transverse
momentum hadron and the $Z/\gamma$ trigger in relativistic heavy-ion
collisions. The jet transport coefficient extracted by the IF-Bayesian
inference is used to constrain $\Delta E$ as well as $\left\langle
p_{\perp}^{2}\right\rangle$. Numerical results fit data very well for the
suppression of the $Z/\gamma$-hadron yields as a function of $p_{\rm T}^{\rm
h}$, $\xi_{\rm T}$, $z_{\rm T}$ or $\Delta\phi$ in $AA$ collisions at
RHIC/LHC, which reveals a consistent description of jet quenching.

The Sudakov resummation technique used in this study for the $Z/\gamma$-hadron
production gets rid of the divergence of the back-to-back azimuthal angular
distribution in both $pp$ and $AA$ collisions. This is, somehow, similar to
that a $\delta$-function-like distribution is smeared to a Gaussian-like
distribution. Different from $pp$ collisions, in $AA$ collisions the QGP
medium induced broadening effect is approximately cumulated to the smearing
effect Chen:2016vem (18, 19). In another word, such a medium effect results in
the decorrelation in the angular distribution for the $Z/\gamma$-hadron
production in $AA$ collisions. However, jet energy loss which does not change
the jet direction enhances the correlation in the angular distribution for the
$Z/\gamma$-hadron production in $AA$ collisions. Due to the presence of jet
energy loss in $AA$ collisions, with the same $p_{\rm T}$ cuts for the
hadrons, the parent partons in $AA$ collisions have larger initial energy than
that in $pp$ collisions. In addition, the vacuum Sudakov effects increase with
the partons’ initial energy. So jet energy loss not only leads to large
$p_{\rm T}$ $Z/\gamma$-hadron suppression but also presents an anti-broadening
in the angular distribution for the $Z/\gamma$-hadron production in $AA$
collisions.

We use the RMS width of the normalized angular distribution of the
$Z/\gamma$-hadrons to analyze the competition between the positive broadening
from the medium induction and the negative anti-broadening from jet energy
loss. Numerical results show that when the colliding energy or trigger
transverse momentum is large, the vacuum Sudakov effect overpowers the medium-
induced broadening effect, resulting in a weak positive broadening of the
azimuthal distribution. And when the associated hadron has a relatively low
$p_{\rm T}^{\rm h}$, the energy loss mechanism results in a strong negative
broadening. Under this circumstance, the azimuthal angular distribution will
display an anti-broadening phenomena, where the distribution for $AA$ seems to
be sharper than that for $pp$ collisions.

Our numerical results demonstrate that low colliding energy with small trigger
$p_{\rm T}$ is more suitable to study the angular decorrelation of two-
particle correlations, and that both parton energy loss and medium-induced
broadening should be considered simultaneously to provide an accurate
description to the jet quenching effect in heavy-ion collisions.

###### Acknowledgements.

Acknowledgements We thank Guang-You Qin and Xin-Nian Wang for helpful
discussions. This work is supported by Natural Science Foundation of China
(NSFC) under Grant No. 11935007 and by the Guangdong Major Project of Basic
and Applied Basic Research No. 2020B0301030008 and the Science and Technology
Program of Guangzhou No. 2019050001.
Data Availability Statement This manuscript has no associated data or the data
will not be deposited. [Authors’ comment: Data points used to draw the plots
and predictions for the future measurements can be obtained through email upon
request.]

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