Asymmetric jet shapes with 2D jet tomography

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.

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,

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,

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,

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),

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.

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.

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).

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),

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.

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 $A_{N}^{y}$.

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.

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.

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.

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,

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.

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,

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.

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,

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,

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.

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.

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.