# Asymmetric jet shapes with 2D jet tomography

Yu-Xin Xiao  Key Laboratory of Quark & Lepton Physics (MOE) and Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China  Yayun
He  School of Physics and Optoelectronics, South China University of
Technology, Guangzhou 510640, China  Long-Gang Pang  Key Laboratory of Quark &
Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal
University, Wuhan 430079, China  Hanzhong Zhang<EMAIL_ADDRESS>Key
Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China  Guangdong Provincial Key
Laboratory of Nuclear Science, Institute of Quantum Matter, South China Normal
University, Guangzhou 510006, China  Guangdong-Hong Kong Joint Laboratory of
Quantum Matter, Southern Nuclear Science Computing Center, South China Normal
University, Guangzhou 510006, China  Xin-Nian Wang<EMAIL_ADDRESS>Key
Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China  Nuclear Science
Division, MS 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA

###### Abstract

Two-dimensional (2D) jet tomography is a promising tool to study jet medium
modification in high-energy heavy-ion collisions. It combines gradient
(transverse) and longitudinal jet tomography for selection of events with
localized initial jet production positions. It exploits the transverse
asymmetry and energy loss that depend, respectively, on the transverse
gradient and jet path length inside the quark-gluon plasma (QGP). In this
study, we employ the 2D jet tomography to study medium modification of the jet
shape of $\gamma$-triggered jets within the linear Boltzmann transport (LBT)
model for jet propagation in heavy-ion collisions. Our results show that jets
with small transverse asymmetry ($A_{N}^{\vec{n}}$) or small $\gamma$-jet
asymmetry ($x_{J\gamma}=p_{T}^{\rm jet}/p_{T}^{\gamma}$) exhibit a broader jet
shape than those with larger $A_{N}^{\vec{n}}$ or $x_{J\gamma}$, since the
former are produced at the center and go through longer path lengths while the
later are off-center and close to the surface of the QGP fireball. In events
with finite values of $A_{N}^{\vec{n}}$, jet shapes are asymmetric with
respect to the event plane. Hard partons at the core of the jet are deflected
away from the denser region while soft partons from the medium response at
large angles flow toward the denser part of QGP. Future experimental
measurements of these asymmetric features of the jet shape can be used to
study the transport properties of jets and medium responses.

##  I INTRODUCTION

A few microseconds after the Big Bang, the Universe was filled with a novel
state of matter called quark-gluon plasma (QGP) that consists of deconfined
quarks and gluons. In the past two decades, relativistic heavy-ion collision
experiments have been carried out at the Relativistic Heavy Ion Collider
(RHIC) at BNL and the Large Hadron Collider (LHC) at CERN to produce such hot
and dense QGP matter and study its properties.

Together with the formation of QGP, large transverse momentum partons
($p_{T}$) are also produced in the early stage of relativistic heavy-ion
collisions that will traverse the QGP and encounter multiple scattering with
quarks and gluons inside the medium. These scatterings will also induce gluon
bremsstrahlung and cause a significant amount of energy loss to the
propagating parton, leading to a phenomenon called jet quenching Gyulassy and
Plumer (1990); Wang and Gyulassy (1992); Qin and Wang (2015). Large transverse
momentum ($p_{T}$) hadrons and jets are significantly suppressed due to jet
quenching in heavy-ion collisions, as compared to large $p_{T}$ hadrons and
jets in proton-proton collisions. Jet quenching has been observed in
experiments by PHENIX Adare et al. (2010) and STAR Adler et al. (2002)
Collaboration at RHIC, ALICE Abelev et al. (2013), ATLAS Aad et al. (2012) and
CMS Chatrchyan et al. (2012); Sirunyan et al. (2018) Collaboration at LHC. In
addition, large $p_{T}$ dihadrons, $\gamma$-hadrons and dijets are also
suppressed and exhibit azimuthal decorrelation Wang and Wang (2002); Vitev
(2004); Zhang et al. (2007, 2009); Vitev and Zhang (2010). Full jet structure
is also modified Qin and Muller (2011); He et al. (2012); Young et al. (2011);
Barata et al. (2023a) by the QGP medium as observed at both RHIC and LHC Adams
et al. (2006); Casalderrey-Solana et al. (2011); Aamodt et al. (2010); Aad et
al. (2010); Sirunyan et al. (2019).

Theoretical studies have revealed that the strength of jet quenching is
proportional to jet transport coefficient $\hat{q}$, which is defined as the
transverse momentum squared per unit length along the parton trajectory Baier
et al. (1997a, b, 1998); Guo and Wang (2000); Wang and Guo (2001);
Casalderrey-Solana and Wang (2008); Majumder (2012). This coefficient is
directly related to the medium gluon density. In high-energy heavy-ion
collisions, the spatial distribution of gluon density and the jet transport
coefficient are not uniform inside the QGP fireball. They reach their highest
values in the central region of the QGP where the temperature is also the
highest. The initial spatial distributions are given by the nuclear geometry
of two colliding nuclei and the time evolution can be determined from
hydrodynamic model of heavy-ion collisions. Jet tomographic techniques have
been developed that exploit these non-uniform distributions of the medium and
jet quenching strength to reveal information about the locations of the
initial jet production.

The longitudinal tomography was first proposed in Refs. Zhang et al. (2007,
2009) to localize the initial jet production positions along the direction of
jet propagation in the study of dihadron and $\gamma$-hadron spectra in heavy-
ion collisions. In the case of $\gamma$-jet processes, low $p_{T}$
$\gamma$-associated hadrons are produced primarily through the fragmentation
of $\gamma$-jets created in the central region of the medium (volume emission)
after losing some amount of energy. The high $p_{T}$ $\gamma$-associated
hadrons, however, mainly come from $\gamma$-jets initially produced in the
outer corona region (surface emission) that suffer little or no energy loss.
Furthermore, a significant fraction of the surface emission is tangential to
the medium surface. In a more recent study He et al. (2020), a transverse or
gradient jet tomography was developed. In this case, an asymmetry observable
$A_{N}^{\vec{n}}$ caused by the transverse gradient of the medium for high
$p_{T}$ hadrons inside a jet is introduced to localize the initial position of
jet production perpendicular to the jet propagation direction. Two-dimensional
(2D) jet tomography combines longitudinal and transverse tomography to
localize the initial jet production positions in the transverse plane. Such 2D
jet tomography has been applied to select special classes of events with
localized region of initial jet production so that signals of jet-induced
medium response are enhanced Yang et al. (2021, 2023a).

In this work, we study medium-modified $\gamma$-jet shape in events of jet
production selected with transverse and longitudinal jet tomography in heavy-
ion collisions, using the linear Boltzmann transport (LBT) model Li et al.
(2011); He et al. (2015); Cao et al. (2016); Wang and Zhu (2013); Cao and Wang
(2021); Luo et al. (2023) for jet transport simulations in the QGP medium. We
consider in particular the asymmetric jet shape caused by flow and gradient of
the hot QGP medium along the path of jet propagation in events selected with
the 2D jet tomography.

The remainder of this paper is organized as follows. Sec. II introduces the
LBT model and the $\gamma$-jet shape in p+p and Pb+Pb collisions. Sec. III
describes the technique of transverse and longitudinal tomography for
$\gamma$-jet events. The numerical results for the medium-modified jet shape
via transverse and longitudinal tomography are presented in Sec. IV. In Sec.
V, we investigate the asymmetric jet shape in events with special class of jet
propagation path relative to the geometry of the dense QGP medium as selected
by 2D jet tomography. A summary is given in Sec. VI.

##  II  Jet shape in the LBT model

The linear Boltzmann transport (LBT) model Li et al. (2011); He et al. (2015);
Cao et al. (2016); Wang and Zhu (2013); Cao and Wang (2021) was developed to
study jet propagation in quark-gluon plasma in heavy-ion collisions, and to
describe not only parton energy loss but also jet-induced medium excitation.
It has been used to describe experimental data on suppression of single
hadrons Cao et al. (2016), single jets He et al. (2019), and $\gamma$-hadron
Chen et al. (2018) and $\gamma$-jet correlations Luo et al. (2018) in heavy-
ion collisions at both RHIC and LHC energies.

The transport of jet shower and recoil partons is described by the linear
Boltzmann equations,

$\displaystyle\begin{split}p_{a}&\cdot\partial
f_{a}=\int\prod_{i=b,c,d}\frac{d^{3}p_{i}}{2E_{i}(2\pi)^{3}}\frac{\gamma_{b}}{2}(f_{c}f_{d}-f_{a}f_{b})|\mathcal{M}_{ab\rightarrow
cd}|^{2}\\\ &\times
S_{2}(\hat{s},\hat{t},\hat{u}){2\pi}^{4}\delta^{4}(p_{a}+p_{b}-p_{c}-p_{d})+{\rm
inelastic},\end{split}$ (1)

where $|\mathcal{M}_{ab\rightarrow cd}|$ is the leading-order elastic
scattering amplitude that depends on the Mandelstam variables
$\hat{s},\hat{t}$, and $\hat{u}$.
$f_{i}=(2\pi)^{3}\delta^{3}(\vec{p}-\vec{p}_{i})\delta^{3}(\vec{x}-\vec{x_{i}}-\vec{v_{i}}t)(i=a,c)$
is the phase space density for jet shower partons before and after scattering.
$f_{i}=1/(e^{p_{i}\cdot u/T}\pm 1)(i=b,d)$ is the phase space distribution of
thermal partons in the QGP medium with local temperature $T$ and fluid
4-velocity $u$. The regularization factor,

$S_{2}(\hat{s},\hat{t},\hat{u})=\theta(\hat{s}\geq
2\mu_{D}^{2})\theta(-\hat{s}+\mu_{D}^{2}\leq\hat{t}\leq-\mu_{D}^{2}),$ (2)

is used to regulate the collinear divergence in $|\mathcal{M}_{ab\rightarrow
cd}|$, and $\mu_{D}^{2}=3g^{2}T^{2}/2$ is the Debye screening mass.

The description of inelastic processes of induced gluon radiation in the LBT
model is based on the high-twist approach Guo and Wang (2000); Zhang et al.
(2004) with the radiative gluon spectrum,

$\frac{dN^{a}_{g}}{dzdk^{2}_{\perp}d\tau}=\frac{6\alpha_{s}P_{a}(z)k^{4}_{\perp}}{\pi(k^{2}_{\perp}+z^{2}m^{2})^{4}}\frac{p\cdot
u}{p_{0}}\hat{q}_{a}(x)\sin^{2}\frac{\tau-\tau_{i}}{2\tau_{f}},$ (3)

where $m$ is the mass of the propagating parton $a$, $z$ and $k_{\perp}$ are
the energy fraction and transverse momentum of radiated gluon, respectively.
$\alpha_{s}$ is the strong coupling constant, and $\hat{q}_{a}$ is the jet
transport coefficient, defined as the transverse momentum transfer squared per
unit path for the parton $a$ traveling in the medium. $P_{a}(z)$ is the
splitting function for $a\rightarrow a+g$, and
$\tau_{f}=2Ez(1-z)/(k^{2}_{\perp}+z^{2}M^{2})$ is the formation time of the
radiated gluon Luo et al. (2018). In LBT model, recoiled medium partons are
also transported according to the Boltzmann equation, the same as jet shower
partons, going through further elastic and inelastic scattering once they are
produced. The depletion of the medium due to back-reaction is also taken into
account through “negative” partons whose energy and momentum must be
subtracted from the final observables. Recoil and “negative” partons are
generally referred to as medium-response.

In this study, we employ the multi-phase transport (AMPT) model Lin et al.
(2005) to generate the initial energy density distribution and collision
geometry. These include the distribution of binary collisions $N_{\rm coll}$,
which is used to determine the initial positions of $\gamma$-jet production,
as well as the initial energy density distribution that serves as the initial
condition for CLVisc 3+1D viscous hydrodynamic evolution Pang et al. (2012,
2018) of the bulk QGP medium. PYTHIA 8 Sjostrand et al. (2006, 2008) is
utilized to generate the momenta of the initial jet shower partons in
$\gamma$-jet events. The LBT model is then employed to describe the
propagation and transport of the jet shower partons inside the QGP medium
whose evolution is provided by the CLVisc hydrodynamics with the initial
condition given by the AMPT model. The information on local temperature and
fluid four-velocity from the CLVisc hydrodynamics is used in LBT to determine
the elastic and inelastic scattering rates. The initial time and freeze-out
temperature in CLVisc hydrodynamics are set to $\tau_{0}=0.2$ fm/c and
$T_{f}=137$ MeV, respectively, which also determine the beginning and end of
the parton-medium interaction. We use the FastJet package (Cacciari et al.,
2012; Cacciari and Salam, 2006) with the anti-$k_{t}$ jet algorithm to
reconstruct full jets using the partonic information of the final jets.

The jet shape is defined as the probability density of transverse momentum
distribution inside a jet cone Sirunyan et al. (2019); Luo et al. (2018),

$\rho(r)=\frac{1}{\Delta r}\frac{\sum_{i}^{N_{\rm
jet}}p_{T}^{i}(r-\frac{1}{2}\Delta r,r+\frac{1}{2}\Delta r)}{\sum_{i}^{N_{\rm
jet}}p_{T}^{i}(0,R)},$ (4)

calculated from the parton distribution inside the reconstructed jets, where
the distance $r$ between the track partons and the jet axis is defined as
$r=\sqrt{\left(\eta-\eta_{\rm jet}\right)^{2}+\left(\phi-\phi_{\rm
jet}\right)^{2}}$ in the plane of pseudo-rapidity $\eta$ and azimuthal angle
$\phi$. The total energy from the $i$-th jet in the circular annulus with
inner radius $r_{1}=r-\Delta r/2$ and outer radius $r_{2}=r+\Delta r/2$ is
defined as $p_{T}^{i}(r_{1},r_{2})=\sum_{{\rm assoc}\in\Delta r}p_{T}^{\rm
assoc}$ where $\Delta r=r_{2}-r_{1}$ is the width of the annulus.
Additionally, the total energy inside the jet cone with a radius $R$ for the
$i$-th jet is given by $p_{T}^{i}(0,R)$. The summation is over the total
number of jet $N_{\rm jet}$ analyzed.

Figure 1: (Color online) The ratio of the $\gamma$-jet shape between 0-10%
Pb+Pb and p+p collisions as a function of $r$ at $\sqrt{s_{NN}}=5.02$ TeV,
from LBT calculations (red-solid) as compared with the CMS data (black-
squares) Sirunyan et al. (2019).

In the following calculations, we select the trigger photon with
$p_{T}^{\gamma}>$ 60 GeV/$c$ in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV.
The cone size of the $\gamma$-triggered jets is set as $R=0.3$, and the lower
threshold of transverse momentum for the jets is set at $p_{T}^{\rm jet}>30$
GeV/c, with associated partons selected with $p_{T}^{\rm assoc}>1$ GeV/$c$.
The pseudorapidities for the $\gamma$ and jets are constrained within
$|\eta_{\gamma}|<1.44$ and $|\eta_{\rm jet}|<1.6$, and their azimuthal angle
differences $|\Delta\phi_{j\gamma}|$ are restricted to be larger than
$(7/8)\pi$.

Shown in Fig. 1 is the ratio of $\gamma$-jet shape in 0-10% Pb+Pb over that in
p+p collisions as a function of $r$ from the LBT model simulations as compared
with the CMS data Sirunyan et al. (2019). The LBT result is in good agreement
with experimental data. The numerical ratio indicates an enhancement at large
r region, suggesting that a significant fraction of the transverse momentum is
transported to larger angles with respect to the jet axis in Pb+Pb relative to
p+p collisions. This is partially due to radiated gluons, but most of the
enhancement is caused by jet-induced medium response at large angles Luo et
al. (2018). This medium modification of the jet shape can also be described by
the coupled LBT hydro model Chen et al. (2018); Yang et al. (2021, 2023b)
where the soft mode of jet-induced medium-response is modeled through
hydrodynamics that couples to the Boltzmann transport of hard partons.

##  III Transverse and longitudinal jet tomography of heavy-ion collisions

The path-length dependence of the parton energy loss during the jet
propagation within the QGP medium can be utilized to localize the initial jet
production positions along the longitudinal direction, a technique known as
longitudinal jet tomography Zhang et al. (2009). Additionally, non-zero
spatial gradients of the parton transport coefficient $\hat{q}$ perpendicular
to their propagation directions arise due to temperature gradients in the QGP
fireball. Parton propagation in such a non-uniform medium leads to an
asymmetrical transverse momentum distribution He et al. (2020); Fu et al.
(2023); Barata et al. (2023b, c) of final state hadrons. In a class of events
with a given transverse asymmetry, a jet observable defined to characterize
the asymmetric transverse momentum distribution of hadrons or partons inside a
jet, the distribution of the initial jet transverse positions is found
localized in a region with a corresponding transverse gradient of the jet
transport coefficient. The average transverse position of the initial jet
production has a monotonic dependence on the transverse asymmetry, therefore,
enabling transverse jet tomography. By considering both longitudinal and
transverse jet tomography, one is able to localize the jet production
positions in the transverse plane, resulting in a 2D jet tomography Yang et
al. (2021, 2023a).

Figure 2: (Color online) Illustration of the $\gamma$-jet configuration
relative to the event plane and the azimuthal angle distribution of the jet
shape inside the jet cone.

In our study here, we consider the event plane along the $x$-axis and select
the direction of the trigger photon to be along the negative $x$-axis, while
the correlated jets propagate approximately along the positive $x$-axis as
illustrated in Fig. 2. The asymmetry $A_{N}^{\vec{n}}$ used for transverse jet
tomography is defined as He et al. (2020),

$A_{N}^{\vec{n}}=\frac{\sum_{a}\int
d^{3}rd^{3}kf_{a}\left(\vec{k},\vec{r}\right){\rm
Sign}\left(\vec{k}\cdot\vec{n}\right)}{\sum_{a}\int
d^{3}rd^{3}kf_{a}\left(\vec{k},\vec{r}\right)},$ (5)

where $f_{a}$ represents the phase-space distribution of partons (hadrons),
$\vec{n}$ denotes the normal direction of plane defined by the beam direction
and the direction of the trigger particle, which is the same as the positive
$y$-axis in our set-up in Fig. 2. The summation is over all partons (hadrons)
inside a jet.

Figure 3: (Color online). The $\gamma$-jet production rates $1/Nd^{2}N/dxdy$
as a function of initial jet production positions ($x$, $y$) in the transverse
plane in 0-10% Pb+Pb collisions at 5.02 TeV, selected with 6 different ranges
of transverse asymmetry, $A_{N}^{y}$ = [-0.05, 0.05], [0.05, 0.15], [0.15,
0.25], [0.25, 0.35], [0.35, 0.45] and [0.45, 0.55] in the 6 panels,
respectively. The trigger $\gamma$ is selected along the negative $x$-axis
direction with $p_{T}^{\gamma}>$ 60 GeV/c and $p_{T}^{\rm jet}>$ 30 GeV/c.
Other kinetic cuts are the same as in Fig. 1.

Shown in Fig. 3 are the final $\gamma$-jet production rates $(1/N)d^{2}N/dxdy$
as a function of the initial locations ($x$, $y$) in the transverse plane in
0-10% Pb+Pb collisions at 5.02 TeV with different values of the jet transverse
asymmetry $A_{N}^{\vec{n}}$. The six panels, (a), (b), (c), (d), (e), and (f),
correspond to 6 different bins of the transverse asymmetry, $A_{N}^{y}$ =
[-0.05, 0.05], [0.05, 0.15], [0.15, 0.25], [0.25, 0.35], [0.35, 0.45] and
[0.45, 0.55], respectively. As we can see from the numerical results, the
typical initial jet production transverse position ($y$) shifts from the
center to the outer region of the medium when $A_{N}^{\vec{n}}$ (or
$A_{N}^{y}$) is increased from [-0.05, 0.05] to [0.45, 0.55]. For small values
of $A_{N}^{y}$, $\gamma$-jets originate from the center region and jets have
to traverse the whole volume of the QGP as depicted in panel (a). Conversely,
for large values of $A_{N}^{y}$, $\gamma$-jets emerge from surface areas of
the QGP and jets propagate tangentially to the medium surface, as shown in
panel (f).

It is worth noting that the selected jet production positions also exhibit an
asymmetric distribution along the x-axis. This is attributed to the path
length dependence of $A_{N}^{y}$, the parton energy loss and the azimuthal
angle restriction. The path-length dependence of the parton energy loss biases
the initial production position of the selected jets closer to the surface
(positive $x$ values). The probability to have a smaller value of $A_{N}^{y}$
with a longer path length is also rare as compared to a shorter path length.
Additionally, longer path lengths may result in a larger drift of the jet
direction, outside the azimuthal cut-off $|\Delta\phi_{j\gamma}|<(7/8)\pi$.

The integration of the initial $\gamma$-jet production rates along the
$x$-axis for different selected transverse asymmetry $A_{N}^{y}$ projects the
2-dimensional $\gamma$-jet rate onto the $y$-axis, as shown in Fig. 4. It is
clear from this figure that the peak of the transverse distribution of the
initial jet production position shifts towards large values of $y$ with
increased value of the transverse asymmetry $A_{N}^{y}$. The variance appears
to be similar across different $A_{N}^{y}$ ranges. The shape of the
approximately asymmetric distribution $(1/N)dN/dy$ does not change
significantly when $A_{N}^{y}$ varies between 0 and 0.35, corresponding to the
central regions of QGP. However, the transverse distribution becomes more
asymmetric for large $A_{N}^{y}$ because of the surface bias by the path-
length dependence of $A_{N}^{y}$.

Figure 4: (Color online) The $\gamma$-jet production rates $1/NdN/dy$ as a
function of the initial jet production position $y$, selected with different
asymmetry $A_{N}^{y}$. The kinematic cuts are the same as in Fig. 3.

For the purpose of an illustration, one can also select the initial transverse
jet production positions of $\gamma$-jets and calculate the corresponding
distribution of the transverse asymmetry. Fig. 5 shows the $\gamma$-jet
production rates $(1/N)dN/dA_{N}^{y}$ as a function of $A_{N}^{y}$ with given
bins of $y$ coordinate of the initial jet production. With given initial bins
of $y$ ranging from $y$ = [-0.5, 0.5] fm to [4.5, 5.5] fm, the peak of the
distribution shifts from small to large $A_{N}^{y}$ values. In summary,
according to Figs. 4 and 5, the transverse asymmetry $A_{N}^{y}$ of the final
state particles inside a jet does localize the initial transverse positions of
$\gamma$-jet production.

Figure 5: (Color online) The $\gamma$-jet production rates $1/NdN/dA_{N}^{y}$
as a function of $A_{N}^{y}$, with given initial jet production position $y$.
The corresponding cuts are the same as in Fig. 3. Figure 6: (Color online).
(Upper) The $\gamma$-jet production rates $1/Nd^{2}N/dxdy$ as a function of
initial jet production locations ($x$, $y$) in the transverse plane in 0-10%
Pb+Pb collisions at 5.02 TeV, selected with 3 different jet transverse
momentum, $p_{T}^{\rm jet}$ = [15, 30] GeV/c, [30, 60] GeV/c and $p_{T}^{\rm
jet}>$ 60 GeV/c, respectively. These jets are triggered by photons with
transverse momentum $p_{T}^{\gamma}>$ 60 GeV/c. Other cuts are the same as in
Fig. 3. (Lower) The corresponding projections onto the $x$-axis.

To illustrate the power of longitudinal jet tomography in localizing the
initial jet production positions, we show in the upper panels of Fig. 6 the
contour distributions of the initial $\gamma$-jet production rates
$(1/N)d^{2}N/dxdy$ in the transverse plane ($x$, $y$) for different jet
transverse momentum (a) $p_{T}^{\rm jet}$ = [15, 30] GeV/$c$, (b) $p_{T}^{\rm
jet}$ = [30, 60] GeV/$c$ and (c) $p_{T}^{\rm jet}>$ 60 GeV/$c$, respectively,
for direct photons with the transverse momentum $p_{T}^{\gamma}>60$ GeV/$c$.
The lower panels give the corresponding projections onto the $x$-axis,
$(1/N)dN/dx=(1/N)\int dy(d^{2}N/dxdy)$. For fixed photon transverse momentum
$p_{T}^{\gamma}$, a higher value of $p_{T}^{\rm jet}$ indicates less jet
energy loss due to a shorter path-length of jet propagation. Conversely,
smaller $p_{T}^{\rm jet}$ corresponds to larger jet energy loss and longer
path-length. This is indeed seen in Fig. 6: $\gamma$-jets with smaller
$p_{T}^{\rm jet}$ are primarily produced through volume emissions that undergo
long path lengths and have significant energy loss, while $\gamma$-jets with
larger $p_{T}^{\rm jet}$ are mainly produced through surface emissions with
shorter path lengths and less energy loss. Varying $p_{T}^{\rm jet}$ while
keeping $p_{T}^{\gamma}$ fixed thus provides a method to approximately
localize the initial $\gamma$-jet production locations in the longitudinal
direction along the jet path.

##  IV Jet shape modification with jet tomography

In this section, we investigate the modification of jet shape for
$\gamma$-jets produced in different regions of the QGP fireball as selected by
jet tomography. During the jet propagation, hard partons tend to penetrate the
dilute region of the medium with low gluon density, while soft recoil partons
are more likely to be created in the dense region with high gluon density. By
localizing $\gamma$-jets to different regions, we can study the modification
of jet shape for jets with different path lengths and temperature gradients.
In particular, temperature gradients along the direction perpendicular to the
jet propagation may lead to an asymmetric jet shape. We will employ transverse
tomography, longitudinal tomography, and a combination of these two methods,
namely 2D-tomography, to localize $\gamma$-jets in the transverse plane. The
kinetic cuts used in this section remain the same as before.

###  IV.1 Jet shapes selected with transverse tomography

We first consider jet events selected with the transverse jet tomography and
examine the corresponding modification of the jet shape. Shown in Fig. 7 are
(a) the jet shape as a function of the radius $r$ in 0-10% Pb+Pb collisions
(solid curves), for $\gamma$-jets selected with different ranges of the
transverse asymmetry $A_{N}^{y}$ from [-0.05, 0.05] to [0.65, 0.75], as
compared to the jet shape in p+p collisions (dashed line) at
$\sqrt{s_{NN}}=5.02$ TeV, and (b) the ratios of jet shapes in Pb+Pb and p+p
collisions. In both Pb+Pb and p+p collisions, the jet shape decreases
monotonically with the radius $r$, indicating that most of the jet energy is
concentrated in the core region close to the jet axis. However, the jet shape
in Pb+Pb is in general broader than that in p+p collisions. This suggests that
some fraction of the transverse momentum lost by hard partons at the core of
the jet is transported to large angles by soft partons (radiated gluons and
medium response) with respect to the jet axis in Pb+Pb relative to p+p
collisions. Furthermore, $\gamma$-jets selected with smaller transverse
asymmetry $A_{N}^{y}$ exhibit a broader jet shape, indicating that the path
length is longer and quenching effect is stronger for jets produced in the
central region of QGP as localized by smaller values of $A_{N}^{y}$. According
to the picture we have illustrated in the last section, increasing $A_{N}^{y}$
will shift the $\gamma$-jet production positions toward the outer region of
QGP. This will reduce the jet energy loss and jet quenching effect with
smaller broadening of the jet shape. Therefore, the width of the jet shape in
Pb+Pb collisions decreases monotonically as the values of $A_{N}^{y}$
increase.

Figure 7: (Color online) (a) Jet shape of $\gamma$-jets as a function of $r$
in 0-10% Pb+Pb (solid curves) selected with different ranges of $A_{N}^{y}$
and p+p (dashing curve) collisions at $\sqrt{s_{NN}}=5.02$ TeV. (b) Ratios of
jet shapes in Pb+Pb over p+p collisions as a function of $r$.

For theoretical illustrations, we also show in Fig. 8 the jet shape
modification of $\gamma$-jets selected according to their transverse
coordinates $y$ which shows similar trends for the corresponding $A_{N}^{y}$
dependence of the modification in Fig. 7.

The broadening of jet shape is caused by interaction between the jet and the
medium, in which hard partons at the core of a jet lose energy leading to a
slight suppression of the jet shape at the core $r<0.05$. This lost energy
will be carried by radiated gluons and medium-response to large angles,
therefore transporting the energy towards the outer layer of the jet cone and
leading to the enhancement of the jet shape in large $r$ regions. Selection of
jet events with different values of the transverse asymmetry localizes the
transverse positions of the initial jet production with different average path
lengths and therefore different degree of jet quenching. This naturally leads
to different levels of broadening of the jet shape, decreasing with increasing
values of the transverse asymmetry or the average values of the transverse
coordinate ($y$). This observation is also consistent with previous studies
Luo et al. (2018, 2022).

Show in Fig. 9 is the $\gamma$-jet production rate as a function of
$x_{J\gamma}$ ($=p_{T}^{\rm jet}/p_{T}^{\gamma}$), selected with different
ranges of the transverse asymmetry $A_{N}^{y}$, in 0-10% Pb+Pb at
$\sqrt{s_{NN}}=5.02$ TeV. The dashed curve is the result from p+p collisions
for comparison. One can see that the rate in Pb+Pb shifts toward smaller
$x_{J\gamma}$ relative to p+p collisions due to jet quenching. The shift is
smaller for $\gamma$-jets selected with a larger $A_{N}^{y}$ range, indicating
more bias toward surface emission. For theoretical illustrations, we also show
in Fig. 10 the $\gamma$-jet production rate as a function of $x_{J\gamma}$
with different initial jet production positions in the $y$ axis. The trend is
similar to the corresponding dependence of $A_{N}^{y}$ as shown in Fig. 9.

Although the $x_{J\gamma}$ distributions in Fig. 9 for different $A_{N}^{y}$
selections exhibit the similar trends as in Fig. 10 for the results with
different selection of the initial production position in the $y$ axis, the
$x_{J\gamma}$ distribution in Pb+Pb is observed to move back to the
distribution in p+p collisions only when $A_{N}^{y}$ is sufficiently large. To
understand these trends, we also show in Fig. 11 the average path length of
the leading partons $\left\langle L\right\rangle$ which is defined as,

$\left\langle L(A_{N}^{y})\right\rangle=\frac{\int
dxdyP(x,y,A_{N}^{y})l(x,y,A_{N}^{y})}{\int dxdyP(x,y,A_{N}^{y})},$ (6)

where $l(x,y,A_{N}^{y})$ is the distance between the longitudinal position of
the initial jet production and the boundary of QGP, and $P(x,y,A_{N}^{y})$ is
the initial jet production rates at $(x,y)$ for jet events with $A_{N}^{y}$.
In the results shown in Fig. 11, we assume that the path is parallel to the
$x$-axis and we take the critical temperature ($T_{c}=165$ MeV) at the
boundary of the QGP. The panel (a) in Fig. 11 indicates that with increasing
$A_{N}^{y}$, $\left\langle L(A_{N}^{y})\right\rangle$ remains nearly
unchanged, but rapidly decreases as $A_{N}$ approaches to 1. Meanwhile, panel
(b) shows that $\left\langle L(y)\right\rangle$ decreases smoothly with the
initial jet production position $y$. Since the modification of the
$\gamma$-jet $x_{J\gamma}$ distributions is controlled mainly by jet energy
loss which in turn is controlled by the jet path length, Fig. 11 explains why
modification of $x_{J\gamma}$ distribution is similar across most of the
$A_{N}^{y}$ range where the average path length is insensitive to $A_{N}^{y}$.
It reverts back to the distribution in p+p collisions only when $A_{N}^{y}$ is
sufficiently large.

Figure 8: (Color online) (a) Jet shape of $\gamma$-jets as a function of $r$
in 0-10% Pb+Pb (solid curves) selected with different initial production
position $y$ and p+p (dashing curve) collisions at $\sqrt{s_{NN}}=5.02$ TeV.
(b) Ratios of jet shapes in Pb+Pb over p+p collisions as a function of $r$.
Figure 9: (Color online) $x_{J\gamma}$ distribution, with $\gamma$-jets
selected with different ranges of the transverse asymmetry $A_{N}^{y}$, in in
0-10% Pb+Pb (solid line) and p+p (dashing line) collisions at
$\sqrt{s_{NN}}=5.02$ TeV. Figure 10: (Color online) $x_{J\gamma}$
distribution, with $\gamma$-jets selected with different ranges of the
transverse coordinates y, in 0-10% Pb+Pb (solid line) and p+p (dashing line)
collisions at $\sqrt{s_{NN}}=5.02$ TeV. Figure 11: (Color online) Average
propagating path length of the leading jet partons as a function of
$A_{N}^{y}$ (left) and as a function of initial production transverse position
$y$ (right).

###  IV.2 Jet shapes selected with longitudinal tomography

In the previous subsection, we have seen how selecting different values of the
transverse asymmetry $A_{N}^{\vec{n}}$ localizes the initial transverse
position ($y$) and leads to different jet propagation path length and jet
shape broadening. Alternatively, one can directly vary the jet propagation
path length using longitudinal jet tomography, as illustrated in Fig. 6.

Shown in Fig. 12 are (a) the jet shapes as a function of $r$ for $\gamma$-jets
in 0-10% Pb+Pb and p+p collisions at $\sqrt{s_{NN}}=5.02$ TeV, with different
$p_{T}^{\rm jet}$ and (b) the ratio of jet shapes between Pb+Pb and p+p
collisions. Because of QCD branching in vacuum, the jet shape in p+p becomes
broader with larger jet transverse momentum $p_{T}^{\rm jet}$ as we see in the
figure. This energy-dependence of the vacuum jet shape can lead to jet shape
broadening due to jet energy loss in Pb+Pb collisions. In addition, jet-medium
interaction will also broaden the jet shape because of radiated gluons and
medium-response at large angles. This medium-induced jet shape broadening will
also depend on $p_{T}^{\rm jet}$ because of the path-length bias imposed by
the longitudinal jet tomography.

As we have seen in Fig. 6, for given values of the photon transverse momentum
$p_{T}^{\gamma}$, $\gamma$-jets with small final $p_{T}^{\rm jet}$ normally
comes from jets initially produced at the center of QGP that have gone through
a along path length and lost significant amount of energy. One should also see
large broadening of the jet shapes. On the other hand, $\gamma$-jets with
large $p_{T}^{\rm jet}$ are more likely produced initially close to the
surface of QGP, travel a shorter path length and lose less energy. The
corresponding jet shape broadening is also smaller as we see in Fig. 12.

It is worth noting again that $\gamma$-jets with the same final $p_{T}^{\rm
jet}$ in Pb+Pb collisions usually come from initial jets with higher
transverse momentum than in p+p collisions due to jet energy loss. This energy
loss alone can lead to jet shape broadening due to energy-dependence of the
jet shape in vacuum. Such jet shape broadening due to jet energy-loss is
approximately independent of the final jet energy $p_{T}^{\rm jet}$ as we see
in the LBT results for $r<0.2$. At large radius $r>0.22$, the jet shape
broadening due to radiated gluons and medium-response strongly depends on
$p_{T}^{\rm jet}$ due to path-length bias imposed by the longitudinal jet
tomography.

Figure 12: (Color online) Upper panel: the jet shape as a function of $r$,
selected with different jet transverse momentum, $p_{T}^{\rm jet}<30$ GeV/c,
$30<p_{T}^{\rm jet}<60$ GeV/c and $p_{T}^{\rm jet}>60$ GeV/c in in 0-10% Pb+Pb
and p+p collisions at $\sqrt{s_{NN}}=5.02$ TeV, respectively. The transverse
momentum of the trigger photon is $p_{T}^{\gamma}>60$ GeV/c. Lower panel: the
ratio of jet shapes between Pb+Pb and p+p collisions. Other kinematic cuts are
the same as in Fig. 7.

###  IV.3 Jet shapes selected with 2D tomography

One can combine transverse and longitudinal jet tomography, or 2D jet
tomography to study the jet shape modification in Pb+Pb collisions by
simultaneously selecting both final jet $p_{T}^{\rm jet}$ and transverse
asymmetry $A_{N}^{y}$. We select trigger photons with $p_{T}^{\gamma}>60$
GeV/$c$ along the negative $x$-axis direction. We calculate the jet shape of
the correlated jets with two intervals of transverse momentum: small
$p_{T}^{\rm jet}=[15,30]$ GeV and large $p_{T}^{\rm jet}>60$ GeV. In addition,
we require the $\gamma$-jets have small $A_{N}^{y}$ = [-0.05, 0.05] and large
transverse asymmetry $A_{N}^{y}$ = [0.55, 0.65].

Figure 13: (Color online) Similar to Fig. 12, but simultaneously selected with
the two final observables, $p_{T}^{\rm jet}$ = [15, 30] GeV/c and $A_{N}^{y}$
= [-0.05, 0.05], $p_{T}^{\rm jet}$ = [15, 30] GeV/c and $A_{N}^{y}$ = [0.55,
0.65], $p_{T}^{\rm jet}>60$ GeV/c and $A_{N}^{y}$ = [-0.05, 0.05], $p_{T}^{\rm
jet}>60$ GeV/c and $A_{N}^{y}$ = [0.55, 0.65], respectively.

The jet shapes of these $\gamma$-jets and their corresponding ratios between
0-10% Pb+Pb and p+p collisions at $\sqrt{s_{NN}}=5.02$ TeV are shown Fig. 13
for four different combinations of transverse and longitudinal tomographic
selections. Among these four sets of tomographic selected configurations,
$\gamma$-jets with small $p_{T}^{\rm jet}$ in conjunction with small
$A_{N}^{y}$ have the most broadened jet shapes, since these jets are produced
at the central region of QGP and traverse the longest path length on the
average. Conversely, the configurations that have the least jet shape
broadening are those $\gamma$-jets with large values of both $p_{T}^{\rm jet}$
and $A_{N}^{y}$, that are initially produced in the outer region of QGP and
travel a smaller path-length. If the $\gamma$-jets are selected with small
$p_{T}^{\rm jet}$ and large $A_{N}^{y}$, or large $p_{T}^{\rm jet}$ and small
$A_{N}^{y}$, the medium modification of the jet shape falls between the above
two extreme cases as seen in Figure 13.

##  V Asymmetric jet shapes

Figure 14: (Color online) Comparisons of $\rho_{upper}$ (red dash-dotted) and
$\rho_{lower}$ (blue dashed ) as a function of $r$ in 0-10% Pb+Pb collisions
at $\sqrt{s_{NN}}=5.02$ TeV, with different given $y$ positions of the initial
jet production. Other cuts are the same as in Fig. 3.

The dependence of the jet shape modification on variables defined in different
types of tomography methods in the last section is essentially the
manifestation of the path-length dependence of jet-medium interaction and the
associated medium response. We will study in this section the jet shape
modification specifically caused by the gradient of the parton density or the
jet transport coefficient $\hat{q}$ and the radial flow. We will look at the
azimuthal angle distribution of energy density inside the jet cone relative to
the direction of the transverse gradient of the QGP medium. We consider the
same $\gamma$-jet configurations relative to the event plane of heavy-ion
collisions as illustrated in Fig 2. Because of the gradient of the medium
density and the radial flow, both final jet partons and medium-response
distribution will be asymmetric relative to the direction of the gradient.

For a purely theoretical illustration, we first divide the jet cone into two
equal halves using a plane defined by the beam direction and the direction of
the jet axis as shown in Fig 2. We then calculate the jet shapes as a function
of the radius $r$ for each half cone - one for the upper half and another for
the lower half,

$\displaystyle\rho(r)_{\rm upper/lower}=\frac{1}{\Delta
r}\frac{\sum_{i}^{N_{\rm jet}}p_{T}^{i}(r-\frac{\Delta r}{2},r+\frac{\Delta
r}{2})\Theta(\pm\vec{k}\cdot\vec{n})}{\sum_{i}^{N_{\rm jet}}p_{T}^{i}(0,R)},$
(7)

where $\pm$ in the $\Theta$ function selects associated particles in the upper
half and lower half of the jet cone, respectively, $\vec{k}$ is the momentum
of the associated particle and $\vec{n}$ is the norm of the plane defined by
the beam direction and the momentum of this jet, which coincides with the
event plane in our setup.

Shown in Fig. 14 are the upper (red dash-dotted), lower half (blue dashed),
and the total jet shape (black solid line) for $\gamma$-jets initially
produced in 6 different regions of the transverse coordinate $y$ in 0-10%
Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV. For jets produced in the central
region of the QGP with $y$ = [-0.5, 0.5] fm, the upper and lower jet shapes
are almost identical, while jets produced near the system edge with $y$ =
[4.5, 5.5] fm have a broader lower-half jet shape as compared to the upper-
half due to the asymmetric parton transport of jet partons and medium response
in the non-uniform QGP medium.

We also plot the ratio between $\rho_{\rm lower}$ and $\rho_{\rm upper}$ in
Fig. 15 as a function of the radius $r$ for jets initially produced in
different regions of the QGP. It is clear that the lower-half jet shape is
suppressed at the core of the jet cone but enhanced toward the edge of the jet
cone as compared to the upper-half jet shape. Since the jet core is made up of
the energetic jet partons, their diffusion in the transverse direction is
enhanced toward the dilute region of the QGP medium (opposite direction of the
transverse gradient $d\hat{q}/dy$) leading to the positive value of the
transverse asymmetry $A_{N}(y)$ and the suppression of $\rho_{\rm lower}$
relative to $\rho_{\rm upper}$ at the core of the jet cone. Jet-induced medium
response in the same jet event will compensate such transport of momentum in
the transverse direction and flow in the opposite direction of the transverse
diffusion of hard jet partons (along the direction of the transverse gradient
$d\hat{q}/dy$). This therefore contributes to the enhancement of the lower-
half jet shape $\rho_{\rm lower}$ relative the upper-half $\rho_{\rm upper}$
in the large radius region where medium response dominate. The difference
between $\rho_{\rm lower}$ and $\rho_{\rm upper}$ is more pronounced for jets
produced in the outer region of the QGP where the gradient of $\hat{q}$
increases toward edge due to the initial energy density distribution of the
hot matter.

Figure 15: (Color online) The ratio of $\rho_{\rm lower}$ over $\rho_{\rm
upper}$ as a function of $r$. Kinematic selections and cuts are the same as in
Fig. 14. Figure 16: (Color online) The in-cone transverse momentum
distribution as a function of $\Delta\phi$ and $\Delta\eta$ in 0-10% $Pb+Pb$
collisions at $\sqrt{s_{NN}}=5.02$ TeV with $A_{N}^{y}=[0.55,0.65]$. Kinematic
selections and cuts are the same as in Fig. 3.

Note that in experiments one can only statistically localize the transverse
region of the initial jet production according to the value of the jet
transverse asymmetry $A_{N}^{y}$. However, fluctuations of $A_{N}^{y}$ make it
difficult to determine the transverse location precisely in each event,
especially for small values of $A_{N}^{y}$ as seen in Fig. 5. In order to
observe the asymmetric jet shape as we have illustrated in the about
theoretical study, we introduce a two-dimensional $r$-weighted jet shape
$\tilde{\rho}_{2D}(\Delta\eta,\Delta\phi)$ for the transverse momentum
distributions inside the jet cone in the $\Delta\eta-\Delta\phi$ plane,

$\displaystyle\tilde{\rho}_{2D}(\Delta\eta,\Delta\phi)$
$\displaystyle=\frac{1}{\delta\eta\delta\phi}$ $\displaystyle\frac{\sum_{\rm
jet}\sum_{(\Delta\eta,\Delta\phi)}^{(\Delta\eta+\delta\eta,\Delta\phi+\delta\phi)}r_{i}p_{T}^{\rm
i}/p_{T}^{\rm jet}}{\sum_{\rm jet}\sum_{r_{i}<R}r_{i}p_{T}^{\rm i}/p_{T}^{\rm
jet}},$ (8)

where $r_{i}=\sqrt{\Delta\eta^{2}+\Delta\phi^{2}}$ is the distance between the
$i$-th associated parton and the jet axis, $p_{T}^{\rm i}$ is the transverse
momentum of the $i$-th associated parton, $\Delta\eta=\eta_{i}-\eta_{jet}$ and
$\Delta\phi=\phi_{i}-\phi_{jet}$, the $\delta\eta$ and $\delta\phi$ are the
bin sizes in the $\Delta\eta$ and $\Delta\phi$ directions, respectively. The
2-D jet profile is weighted with the distance $r$ to enhance the contribution
of soft partons in the large $r$ region from medium response.

The 2D jet shape $\tilde{\rho}_{2D}(\Delta\eta,\Delta\phi)$ inside the jet
cone is shown in Fig. 16, for $\gamma$-jets in 0-10% Pb+Pb collisions at
$5.02$ TeV, with $A_{N}^{y}$ in the range $[0.55,0.65]$. The contribution from
the leading hard jet parton to the 2-D jet profile at the core is shown in the
inset, while the remaining contribution (with the leading parton subtracted)
primarily from soft particles to the 2D jet profile inside the jet cone is
displayed in the main figure. This 2D jet shape with given transverse
asymmetry indeed shows the same asymmetrical structure as we have discussed in
the above study with specified region of transverse position of the initial
jet production. The hard jet partons are diffused toward the dilute region of
the QGP medium while soft partons from the medium response tend to move to the
direction of the dense region of the QGP. In other words, the asymmetric jet
shape suggests that hard partons inside the jet are deflected away from dense
regions, while soft partons from medium response are more likely to be created
in dense regions. This asymmetrical behavior is particularly more pronounced
for $\gamma$-jet events with large $A_{N}^{y}$, where the jet is produced
mostly in the outer layer of the medium, and is tangential to the surface.

Figure 17: (Color online) The in-cone transverse momentum distribution as a
function of $\phi_{r}$ in three different circular regions inside
$\gamma$-jets in 0-10% $Pb+Pb$ collisions at $\sqrt{s_{NN}}=5.02$ TeV. Three
kinds of jet events are selected with different $A_{N}^{y}$ values in panels
(a), (b), and (c), respectively.  Figure 18: (Color online) The in-cone
transverse momentum distribution as a function of $\phi_{r}$ in three
different $p_{T}^{\rm assoc}$ ranges in 0-10% $Pb+Pb$ collisions at
$\sqrt{s_{NN}}=5.02$ TeV. Three kinds of jet events are selected with
different $A_{N}^{y}$ values in panels (a), (b), and (c), respectively.

To further illustrate the asymmetrical jet shape and the underlying
contributions from hard and soft particles, we project the 2D jet shape to the
azimuthal angle distribution inside the jet cone,

$\displaystyle\frac{d{\cal P}_{T}}{d\phi_{r}}=\frac{1}{N_{\rm
jet}}\frac{\sum_{\rm jet}\sum_{\phi_{r}}^{\phi_{r}+\Delta\phi_{r}}r_{i}\cdot
p_{T}^{\rm i}}{\Delta\phi_{r}},$ (9)

where $N_{\rm jet}$ is the total number of jet events, and $\phi_{r}$ is the
azimuthal angle in the $\Delta\eta-\Delta\phi$ plane,

$\displaystyle\begin{cases}\begin{aligned}
\phi_{r}&=\arcsin{\left(\frac{\Delta\eta}{r}\right)},&(\Delta\phi\geqslant
0)\\\\[5.0pt]
\phi_{r}&=\pi-\arcsin{\left(\frac{\Delta\eta}{r}\right)},&(\Delta\phi<0,\Delta\eta\geqslant
0)\\\\[5.0pt]
\phi_{r}&=-\pi-\arcsin{\left(\frac{\Delta\eta}{r}\right)},&(\Delta\phi<0,\Delta\eta<0).\end{aligned}\end{cases}$
(10)

with $\phi_{r}=0$ defined as the direction of $y$ or along the positive
direction of $A_{N}^{y}$.

Shown in Fig. 17 are the azimuthal angle $\phi_{r}$ distribution of transverse
energy inside $\gamma$-jets in 0-10% Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$
TeV with three different jet transverse asymmetry (a)
$A_{N}^{y}=[-0.05,0.05]$, (b) $[0.25,0.35]$ (c) and $[0.55,0.65]$ for three
different circular annulus at radius $r$ = [0, 0.1] (solid), [0.1, 0.2]
(dashed) and [0.2, 0.3] (dotted), respectively. We also show in Fig. 18 the
angular $\phi_{r}$ distribution of transverse energy inside $\gamma$-jets with
three different jet transverse asymmetry (a) $A_{N}^{y}=[-0.05,0.05]$, (b)
$[0.25,0.35]$ (c) and $[0.55,0.65]$, for three different regions of parton
transverse momentum, $p_{T}^{\rm assoc}>3$ GeV/c (solid), $2<p_{T}^{\rm
assoc}\leq 3$ GeV/c (dashed) and $p_{T}^{\rm assoc}\leq 2$ GeV/c (dotted),
respectively. Soft partons originating from the medium response have smaller
$p_{T}^{\rm assoc}$, while large $p_{T}^{\rm assoc}$ is associated with hard
partons. We observe different trends in the $\phi_{r}$ distributions for hard
and soft partons, particularly for jet events with a large value of
$A_{N}^{y}$. This conclusion is consistent with the results presented in Fig.
17, except that different circular annuli are used to separate hard from soft
partons: hard jet partons are concentrated around the core of the jet cone
($r=0$) whereas soft partons tend to dominate the region toward the edge of
the jet cone.

With small jet transverse asymmetry $A_{N}^{y}$ ranging from -0.05 to 0.05,
the jet shape is mostly symmetric with an approximately uniform distribution
in $\phi_{r}$ at both small and large radius $r$ or for both small and large
transverse momentum partons as seen in Figs. 17(a) and 18(a). For jets with
sizable or large transverse asymmetry $A_{N}^{y}$ in ranges [0.25, 0.35] or
[0.55, 0.65], they propagate through regions of QGP with large transverse
gradient. These jets have asymmetric jet shapes with non-uniform azimuthal
$\phi_{r}$ distributions as seen in Figs. 17(b-c) and 18(b-c). Hard jet
partons at the core of the jet have a peak in the opposite direction of the
gradient ($\phi_{r}=0$ toward the dilute region) while soft partons at large
radius have a peak in the direction of the gradient ($\phi_{r}=\pi$ toward the
dense region). This is consistent with the findings as illustrated in Fig. 16.

Note that in experimental measurements, one can only know the absolute value
of the transverse asymmetry $|A_{N}^{y}|$ with equal probability of positive
and negative values. If one averages events with both positive and negative
values of $A_{N}^{y}$, the final azimuthal distribution will be uniform. To
circumvent this, one should align the orientation of the hard jet (or soft)
partons in each event before averaging over events. The anti-correlation of
hard and soft partons in asymmetric jets serves to determine the direction of
the gradient of the QGP through which the jets propagate. This is an
especially useful technique to measure the asymmetric jet shape caused by the
transverse gradient of the QGP medium in central heavy-ion collisions where
one cannot determine the event plane from the bulk hadron distributions.

##  VI summary

We have investigated the medium-modified jet shapes of $\gamma$-jets in high-
energy heavy-ion collisions using a two-dimensional jet tomography that
combines gradient (transverse) and longitudinal jet tomography using event-by-
event linear Boltzmann Transport simulations which can reproduce CMS
experimental data. Transverse and longitudinal jet tomography are demonstrated
to be able to localize the initial $\gamma$-jet production locations as
proposed in Zhang et al. (2009) and He et al. (2020). By employing the 2D jet
tomography in which $\gamma$-jets are selected with different jet transverse
asymmetry $A_{N}^{\vec{n}}$ or the longitudinal asymmetry $p_{T}^{\rm
jet}/p_{T}^{\gamma}$, we computed the medium modification for the transverse
momentum distribution inside the jet cone, namely the jet shape. Jet shapes
are in general broadened in Pb+Pb collisions relative to p+p collisions
because of the energy transfer carried by soft partons in medium response at
large angles relative to the jet axis. Our results indicate that jet shape
broadening are enhanced in events with small values of $A_{N}^{\vec{n}}$ and
$p_{T}^{\rm jet}$, corresponding to jets initially produced at the center of
the QGP or with long path lengths. We further illustrated the asymmetrical jet
shapes of these jets that propagate through regions of the QGP with finite
transverse gradient. Hard jet partons at the core of the jet are defected away
from the dense region of QGP while soft partons from the medium response are
produced more in the dense region of QGP. This anti-correlation of hard and
soft partons within the asymmetric jets can be used to determine the direction
of the transverse gradient of the QGP in each jet event. These distinctive
behaviors of hard and soft partons inside the jet cone may pave the way for
new avenues to investigate the properties of the QGP and strong interactions.
Future experimental measurements can validate these distinctive features in
the jet shape, thereby testing the theory of QCD and parton transport.

## Acknowledgments

The authors would like to thank Guang-You Qin, Enke Wang and Zhong Yang for
stimulating discussions. This work is supported by National Natural Science
Foundation of China under Grants Nos. 11935007, 11435004, 12075098, 12305140,
12147134, by Guangdong Basic and Applied Basic Research Foundation No.
2021A1515110817, by Guangdong Major Project of Basic and Applied Basic
Research No. 2020B0301030008, Science and Technology Program of Guangzhou No.
2019050001, and the Director, Office of Energy Research, Office of High Energy
and Nuclear Physics, Division of Nuclear Physics, of the U.S. Department of
Energy under Grant No. DE-AC02-05CH11231 and within the framework of the SURGE
Collaboration, the US National Science Foundation under Grant No. OAC-2004571
within the X-SCAPE Collaboration.

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