Some Aspects of Anisotropic Quark-Gluon Plasma

Abstract

We review the various aspects of anisotropic quark-gluon plasma (AQGP) that have recently been discussed by a number of authors. In particular, we focus on the electromagnetic probes of AQGP, inter quark potential, quarkonium states in AQGP, and the nuclear modifications factor of various bottomonium states using this potential. In this context, we will also discuss the radiative energy loss of partons and nuclear modification factor of light hadrons in the context of AQGP. The features of the wake potential and charge density due to the passage of jet in AQGP will also be demonstrated.

1. Introduction

Ever since the possibility of creating the quark-gluon plasma (QGP) in relativistic heavy ion collision was envisaged, numerous indirect signals were proposed to probe the properties of such an exotic state of matter. For example, electromagnetic probes (photon and dilepton) [1], 𝐽/𝜓 suppression [2], jet quenching vis-a-vis energy loss [3–10], and many more. In spite of these, many properties of the QGP are poorly understood. The most pertinent question is whether the matter produced in relativistic heavy ion collisions is in thermal equilibrium or not. Studies on elliptical flow (up to about 𝑝_𝑇∼1.5 GeV) using ideal hydrodynamics indicate that the matter produced in such collisions becomes isotropic with 𝜏_iso∼0.6 fm/c [11–13]. On the other hand, using second-order transport coefficients with conformal field theory, it has been found that the isotropization/thermalization time has sizable uncertainties [14] leading to uncertainties in the initial conditions, such as the initial temperature.

In the absence of a theoretical proof favoring the rapid thermalization and the uncertainties in the hydrodynamical fits of experimental data, it is very hard to assume hydrodynamical behavior of the system from the very beginning. The rapid expansion of the matter along the beam direction causes faster cooling in the longitudinal direction than in the transverse direction [18]. As a result, the system becomes anisotropic with ⟨𝑝_𝐿²⟩≪⟨𝑝_𝑇²⟩ in the local rest frame. At some later time when the effect of parton interaction rate overcomes the plasma expansion rate, the system returns to the isotropic state again and remains isotropic for the rest of the period. If the QGP, just after formation, becomes anisotropic, soft unstable modes are generated characterized by the exponential growth of the transverse chromomagnetic/chromoelectric fields at short times. Thus, it is very important to study the collective modes in an AQGP and use these results to calculate relevant observables. The instability, thus developed, is analogous to QED Weibel instability. The most important collective modes are those which correspond to transverse chromomagnetic field fluctuations, and these have been studied in great detail in [19–26]. This is known as chromo-Weibel instability, and it differs from its QED analogue because of nonlinear gauge self-interactions. Because of this fact, anisotropy driven plasma instabilities in QCD may slow down the process of isotropization whereas, in QED, it can speed up the process [27].

To characterize the presence of initial state of momentum space anisotropy, it has been suggested to look for some observables which are sensitive to the early time after the collision. The effects of preequilibrium momentum anisotropy on various observables have been studied quite extensively over the past few years. The collective oscillations in an AQGP have been studied in [28, 29]. Heavy quark energy loss and momentum broadening in anisotropic QGP have been investigated in [30, 31]. However, the radiative energy loss of partons in AQGP has recently been calculated in [32]. Another aspect of jet propagation in hot and dense medium is the wake that it creates along its path. First, Ruppert and Muller [33] have investigated that when a jet propagates through the medium, a wake of current and charge density is induced which can be studied within the framework of linear-response theory. The result shows the wake in both the induced charge and current density due to the screening effect of the moving parton. In the quantum liquid scenario, the wake exhibits an oscillatory behavior when the charge parton moves very fast. Later, Chakraborty et al. [34] also found the oscillatory behavior of the induced charge wake in the backward direction at a large parton speed using HTL perturbation theory. In a collisional quark-gluon plasma, it is observed that the wake properties change significantly compared to the collisionless case [35]. Recently, Jiang and Li [36, 37] have investigated the color response wake in the viscous QGP with the HTL resummation technique. It is shown that the increase of the shear viscosity enhances the oscillation of the induced charge density as well as the wake potential. The effect of momentum space anisotropy on the wake potential and charge density has recently been considered in [38].

Effects of anisotropy on photon and dilepton yields have been investigated rigorously in [16, 39–43]. Recently, the authors in [44] calculated the nuclear modification factor for light hadrons assuming an anisotropic QGP and showed how the isotropization time can be extracted by comparing with the experimental data.

The organization of the review is as follows. In Section 2 we will briefly discuss various models of space-time evolution in AQGP along with the electromagnetic probes which can be used to extract the isotropization time of the plasma. Section 3 will be devoted to discuss the works on heavy quark potential and related phenomena (such as gluon 𝐽/𝜓 dissociation cross-section in an AQGP) that have been investigated so far. We will also discuss the radiative energy loss of partons in AQGP and nuclear modification factor of light hadrons due to the energy loss of the jet in Section 4. In Section 5, the effect of momentum anisotropy on wake potential and charge density due to the passage of a jet will be presented. Finally we summarize in Section 6.

2. Electromagnetic Probes

Photons and dileptons have long been considered to be the good probes to characterize the initial stages of heavy ion collisions as these interact “weakly” with the constituents of the medium and can come out without much distortion in their energy and momentum. Thus, they carry the information about the space-time point where they are produced. Since anisotropy is an early stage phenomena, photons and dileptons are the efficient probes to characterize this stage. The yield of dileptons (henceforth called medium dileptons/photons) in an AQGP has been calculated using a phenomenological model of space-time evolution in (1 + 1) dimension [42, 43]. This model (henceforth referred to as model I) introduces two parameters: 𝑝_hard and 𝜉 which are functions of time. The former is called the hard momentum scale and is related to the average momentum of the particles in the medium. In isotropic case this can be identified with the temperature of the system. The latter is called the anisotropy parameter and −1<𝜉<∞. Furthermore, the model interpolates between early-time 1 + 1 free streaming behavior (𝜏≪𝜏_iso) and late-time ideal hydrodynamical behavior (𝜏≫𝜏_iso). We first discuss various space-time evolution models of AQGP. In model I the time dependence of 𝜉 is given by [42, 43] 𝜉()=𝜏𝜏,𝛿𝜏_𝑖^𝛿−1, (1) where the exponent 𝛿=2(2/3) corresponds to free streaming (collisionally broadened) preequilibrium state momentum space anisotropy and 𝛿=0 corresponds to thermalization. 𝜏_𝑖 is the formation time of the QGP. For smooth transition from free streaming to hydrodynamical behavior a transition width 𝛾⁻¹ is introduced. The time dependences of various relevant parameters are obtained in terms of a smeared step function [42, 43] as follows: 𝜆(𝜏)=12𝛾tanh𝜏−𝜏_iso𝜏_𝑖+1. (2)

For 𝜏≪𝜏_iso(≫𝜏_iso), we have 𝜆=0(1) which corresponds to free streaming (hydrodynamics). Thus, the time dependences of 𝜉 and 𝑝_hard are as follows [42, 43]: 𝜉()=𝜏𝜏,𝛿𝜏_𝑖^{𝛿(1−𝜆(𝜏))}𝑝−1,_hard(𝜏)=𝑇_𝑖𝒰^1/3(𝜏), (3) where 𝒰(𝜏)≡ℛ𝜏_iso𝜏^𝛿−1^{3𝜆(𝜏)/4}𝜏_iso𝜏^{1−𝛿(1−𝜆(𝜏))/2},𝒰(𝜏)𝒰≡𝒰𝜏_𝑖,ℛ(𝑥)=121()+𝑥+1tan⁻¹√𝑥√𝑥, (4) and 𝑇_𝑖 is the initial temperature of the plasma. In our calculation, we assume a fast-order phase transition beginning at the time 𝜏_𝑓 and ending at 𝜏_𝐻=𝑟_𝑑𝜏_𝑓, where 𝑟_𝑑=𝑔_𝑄/𝑔_𝐻 is the ratio of the degrees of freedom in the two (QGP phase and hadronic phase) phases and 𝜏_𝑓 is obtained by the condition 𝑝_hard(𝜏_𝑓)=𝑇_𝑐, which we take as 192 MeV. We also include the contribution from the mixed phase.

The other model (referred to as model II hereafter) of space-time evolution of highly AQGP is the boost invariant dissipative dynamics in (0 + 1) dimension [45]. This model can reproduce both the hydrodynamics and the free streaming limits. The time evolution of the phase space distribution 𝑓(𝑡,𝑧,𝐩) can be described by Boltzmann equation. Thus, as a starting point, we write the Boltzmann equation in (0+1) dimension in the lab frame as follows: 𝑝^𝑡𝜕_𝑡𝑓(𝑡,𝑧,𝐩)+𝑝^𝑧𝜕_𝑧𝑓(𝑡,𝑧,𝐩)=−𝒞𝑓(𝑡,𝑧,𝐩), (5) where homogeneity in the transverse direction is assumed and 𝒞[𝑓(𝑡,𝑧,𝐩)] is the collision kernel. We assume that the phase-space distribution for the anisotropic plasma is given by the following ansatz [28, 29]: 𝑓𝐩,𝜉(𝜏),𝑝_hard(𝜏)=𝑓_iso𝐩²+𝜉(𝜏)(𝐩⋅𝐧)²𝑝²_hard(𝜏), (6) where 𝐧 is the direction of anisotropy. Note that, in subsequent sections, this distribution function will be used to calculate various observables. Now it is convenient to write (5) in the comoving frame. Introducing space-time rapidity (Θ), particle rapidity (𝑦), and proper time (𝜏) one can write (5) in terms of the comoving coordinates as [45] 𝑝_𝑇𝜕cosh𝑦−Θ+𝑝𝜕𝜏^𝑇sinh𝑦−Θ𝜏𝜕𝜕𝜏×𝑓𝐩,𝜉,𝑝_hard=−Γ𝑝_𝑇cosh𝑦−Θ𝑓𝐩,𝜉,𝑝_hard−𝑓_eq(𝐩,𝑇(𝜏)), (7) where Γ=2𝑇(𝜏)/(5𝜂) and 𝜂=𝜂/𝑠, 𝜂 is the shear viscosity coefficient.

The zeroth-order and first-order moments of the Boltzmann equation give the time dependence for 𝜉 and 𝑝_hard as described in [45]. Without going into further details we simply quote the coupled differential equations that have to be solved to get the time dependence of 𝜉 and 𝑝_hard [45] as follows: 1𝜕1+𝜉^𝜏2𝜉=𝜏(𝜉)ℛ−4Γℛ^3/41+𝜉−12ℛ(𝜉)+3(1+𝜉)ℛ^′(𝜉),111+𝜉𝑝_hard𝜕_𝜏𝑝_hard=2𝜏−4Γℛ^′(𝜉)ℛ^3/41+𝜉−12ℛ(𝜉)+3(1+𝜉)ℛ^′(𝜉). (8) The previous two coupled differential equations have to be solved numerically. The results are shown in Figure 1. It is seen that the anisotropy parameter falls much rapidly compared to the case when model II is used. There is a narrow window in 𝜏 where 𝜉 dominates in case of model I. The cooling is slower in case of model II as can be seen from the right panel of Figure 1. These observations have important consequence on various observables.

(a)

(b)

Figure 1 

(Color online) Time evolutions of (a) the anisotropy parameter 𝜉 and (b) the hard momentum scale 𝑝hard in the two space-time models described in the text. The graphs are taken from [15].

The assumption of boost invariant in the longitudinal direction can be relaxed and such a space-time model (the so called AHYDRO) has been proposed in [46]. As before, the time evolutions of various quantities can be obtained by taking moments of the Boltzmann equation. However, in this case, instead of two one obtains three coupled differential equations. The third variable is the longitudinal flow velocity (see [46] for details). The observations of this work are as follows. It removes the problem of negative longitudinal pressure sometimes obtained in 2nd-order viscous hydrodynamics and this model leads to much slower relaxation towards isotropy. In this review, for the sake of simplicity, the observables of AQGP will be calculated using space-time model I. The same can also be calculated using other space-time models of AQGP, and the results may differ from case to case.

2.1. Photons

We first consider the medium photon production from AQGP. The detail derivation of the differential rate is standard and can be found in [16, 39–41]. Here we will quote only the final formula for total photon yield after convoluting with the space-time evolution. The total medium photon yield, arising from the pure QGP phase and the mixed phase is given by 𝑑𝑁^𝛾𝑑𝑦𝑑²𝑝_𝑇=𝜋𝑅²_⟂^𝜏𝑓_𝜏i𝜏𝑑𝜏𝑑𝜂𝑑𝑁^𝛾𝑑⁴𝑥𝑑𝑦𝑑²𝑝_𝑇+^𝜏𝐻_𝜏𝑓𝑓_QGP(𝜏)𝜏𝑑𝜏𝑑𝜂𝑑𝑁^𝛾𝑑⁴𝑥𝑑𝑦𝑑²𝑝_𝑇, (9) where 𝑓_QGP(𝜏)=(𝑟_𝑑−1)⁻¹(𝑟_𝑑𝜏_𝑓𝜏⁻¹−1) is the fraction of the QGP phase in the mixed phase [47] and 𝑅_⟂=1.2𝐴^1/3 fm is the radius of the colliding nucleus in the transverse plane. The energy of the photon in the fluid rest frame is given by 𝐸_𝛾=𝑝_𝑇cosh(𝑦−Θ), where Θ and 𝑦 are the space-time and photon rapidities, respectively. The anisotropy parameter and the hard momentum scale enter through the differential rate via 𝑑𝑁^𝛾/𝑑⁴𝑥𝑑𝑦𝑑²𝑝_𝑇 (see [16, 39–41] for details).

We plot the total photon yield coming from thermal QGP, thermal hadrons and the initial hard contribution in Figure 2 and compare it with the RHIC data for various values of 𝜏_iso. In the hadronic sector, we include photons from baryon-meson (BM) and meson-meson (MM) reactions. Two scenarios have been considered: (i) pure hydrodynamics from the beginning and (ii) inclusion of momentum state anisotropy. We observe that (i) photons from BM reactions are important, (ii) pure hydro is unable to reproduce the data, that is, some amount of momentum anisotropy is needed, and (iii) exclusion of BM reactions underpredict the data. We note that the value of 𝜏_iso needed to describe the data also lies in the range 1.5 fm/c ≥𝜏_iso≥0.5 fm/c for both values of the transition temperatures.

Figure 2 

(Color online) Photon transverse momentum distributions at RHIC energies. The initial conditions are taken as 𝑇𝑖=440 MeV, 𝜏𝑖=0.1 fm/c, and 𝑇𝑐=192 MeV [16].

2.2. Dileptons

The dilepton production from AQGP has been estimated in [17, 42, 43] using the same space-time model. It is argued in [17] that the transverse momentum distribution of lepton pair in AQGP could provide a good insight about the estimation of 𝜏_iso. We will briefly discuss the high mass dilepton yield along with the 𝑝_𝑇 distribution in AQGP. Here we consider only the QGP phase as in the high mass region the yield from the hadronic reactions and decay should be suppressed. The dilepton production from quark-antiquark annihilation can be calculated from kinetic theory and is given by 𝐸𝑑𝑅𝑑³𝑃=𝑑³𝐩₁(2𝜋)³𝑑³𝐩₂(2𝜋)³𝑓_𝑞𝐩₁𝑓_𝑞𝐩₂𝑣_𝑞𝑞𝜎^𝑙+𝑙−_𝑞𝑞𝛿⁽⁴⁾𝑃−𝑝₁−𝑝₂, (10) where 𝑓_𝑞(𝑞) is the phase space distribution function of the medium quarks (anti-quarks), 𝑣_𝑞𝑞 is the relative velocity between quark and anti-quark, and 𝜎^𝑙+𝑙−_𝑞𝑞 is the total cross-section. Consider 𝜎^𝑙+𝑙−_𝑞𝑞=4𝜋3𝛼²𝑀²1+2𝑚²_𝑙𝑀²1−4𝑚²_𝑙𝑀^21/2. (11)

Using the anisotropic distribution functions for the quark (antiquark) defined earlier the differential dilepton production rate can be written as [43] 𝑑𝑅𝑑⁴𝑃=5𝛼²18𝜋⁵×¹₋₁𝑑cos𝜃_𝑝1×^𝑎−_𝑎+𝑑𝑝₁√𝜒𝑝₁𝑓_𝑞𝐩²₁𝟏+𝜉cos²𝜃_𝐩1,𝑝_hard×𝑓_𝑞𝐄−𝐩₁²+𝜉𝐩₁cos𝜃_𝐩1−𝐏cos𝜃_𝐏²,𝑝_hard. (12)

The invariant mass and 𝑝_𝑇 distributions of lepton pair can be obtained after space-time integration using the evolution model described earlier. The final rates are as follows [17]: 𝑑𝑁𝑑𝑀²𝑑𝑦=𝜋𝑅²_⟂𝑑²𝑃_𝑇^𝜏𝑓_𝜏𝑖^∞_−∞𝑑𝑅𝑑⁴𝑃𝜏𝑑𝜏𝑑𝜂,𝑑𝑁𝑑²𝑃_𝑇𝑑𝑦=𝜋𝑅²_⟂𝑑𝑀^2𝜏𝑓_𝜏𝑖^∞_−∞𝑑𝑅_ann𝑑⁴𝑃𝜏𝑑𝜏𝑑𝜂. (13)

The numerical results are shown in Figure 3 for the initial conditions 𝜏_𝑖=0.88 fm/c and 𝑇_𝑖=845 MeV corresponding to the LHC energies. For 𝜏_iso∼2 fm/c, it is observed that the dilepton yield from AQGP is comparable to Drell-Yan process. The 𝑝_𝑇 distribution shows (Figure 3(b)) that the medium contribution dominates over all the other contributions upto 𝑝_𝑇∼9 GeV. The extraction of the isotropization time can only be determined if these results are confronted with the data after the contributions from the semileptonic decay from heavy quarks and Drell-Yan processes are subtracted from the total yield.

(a)

(b)

Figure 3 

(Color online) Invariant mass (a) and momentum (b) distribution of midrapidity dileptons in central Pb + Pb collisions at LHC. The figures are taken from [17].

3. Heavy Quark Potential and Quarkonium States in AQGP

In this section, we will discuss the heavy quark potential in AQGP that has been calculated in [48]. It is to be noted that this formalism will enable us to calculate the radiative energy loss of both heavy and light quarks, and this will be discussed in Section 4. To calculate the interquark potential one starts with the retarded gluon self-energy expressed as [49] Π^𝜇𝜈(𝐾)=𝑔²𝑑³𝑝(2𝜋)³𝑃^𝜇𝜕𝑓(𝐩)𝜕𝑃_𝛽𝑔^𝛽𝜈−𝑃^𝜈𝐾^𝛽.𝐾⋅𝑃+𝑖𝜀 (14) This tensor is symmetric, Π^𝜇𝜈(𝐾)=Π^𝜈𝜇(𝐾), and transverse, 𝐾_𝜇Π^𝜇𝜈(𝐾)=0. The spatial components of the self-energy tensor can be written as Π^𝑖𝑗(𝐾)=−𝑔²𝑑³𝑝(2𝜋)³𝑣^𝑖𝜕^𝑙𝑓(𝐩)𝛿^𝑗𝑙+𝑣^𝑗𝑘^𝑙,𝐾⋅𝑉+𝑖𝜖 (15) where 𝑓(𝐩) is the arbitrary distribution function. To include the local anisotropy in the plasma, one has to calculate the gluon polarization tensor incorporating anisotropic distribution function of the constituents of the medium. We assume that the phase-space distribution for the anisotropic plasma is given by (6). Using the ansatz for the phase space distribution given in (6), one can simplify (15) to Π^𝑖𝑗(𝐾)=𝑚²_𝐷𝑑Ω()𝑣4𝜋^𝑖𝑣^𝑙+𝜉(𝐯⋅𝐧)𝑛^𝑙1+𝜉(𝐯⋅𝐧)²²𝛿^𝑗𝑙+𝑣^𝑗𝑘^𝑙,𝐾⋅𝑉+𝑖𝜖 (16) where 𝑚_𝐷 is the Debye mass for isotropic medium represented by 𝑚²_𝐷𝑔=−²2𝜋^2∞₀𝑑𝑝𝑝²𝑑𝑓_iso𝑝².𝑑𝑝 (17)

Due to the anisotropy direction, the self-energy, apart from momentum 𝐤, also depends on the anisotropy vector 𝐧, with 𝑛²=1. Using the proper tensor basis [28], one can decompose the self-energy into four structure functions as Π^𝑖𝑗(𝑘)=𝛼𝐴^𝑖𝑗+𝛽𝐵^𝑖𝑗+𝛾𝐶^𝑖𝑗+𝛿𝐷^𝑖𝑗, (18) where 𝐴^𝑖𝑗=𝛿^𝑖𝑗−𝑘^𝑖𝑘^𝑗𝑘²,𝐵^𝑖𝑗=𝑘^𝑖𝑘^𝑗,𝐶^𝑖𝑗=𝑛^𝑖𝑛^𝑗𝑛²,𝐷^𝑖𝑗=𝑘^𝑖𝑛^𝑗+𝑛^𝑖𝑘^𝑗, (19) with 𝑛^𝑖=𝐴^𝑖𝑗𝑛^𝑗 which obeys 𝑛⋅𝑘=0. 𝛼, 𝛽, 𝛾, and 𝛿 are determined by the following contractions: 𝑘^𝑖Π^𝑖𝑗𝑘^𝑗=𝐤²𝛽,𝑛^𝑖Π^𝑖𝑗𝑘^𝑗=𝑛²𝐤²𝑛𝛿,_𝑖Π^𝑖𝑗𝑛^𝑗=𝑛²𝛼+𝛾,TrΠ^𝑖𝑗=2𝛼+𝛽+𝛾. (20) Before going to the calculation of the quark-quark potential let us study the collective modes in AQGP which have been thoroughly investigated in [28, 29], and we briefly discuss this here. The dispersion law for the collective modes of anisotropic plasma in temporal axial gauge can be determined by finding the poles of propagator Δ^𝑖𝑗 as follows: Δ^𝑖𝑗(𝐾)=1𝐤²−𝜔²𝛿^𝑖𝑗−𝑘^𝑖𝑘^𝑗+Π^𝑖𝑗(𝑘.) (21) Substituting (19) in the previous equation and performing the inverse formula [28], one finds Δ(𝐾)=Δ_𝐴[𝐀−𝐂]+Δ_𝐺𝐤²−𝜔²+𝛼+𝛾𝐁+𝛽−𝜔²𝐂−𝛿𝐃. (22) The dispersion relation for the gluonic modes in anisotropic plasma is given by the zeros of Δ_𝐴⁻¹(𝑘)=𝑘²−𝜔²Δ+𝛼=0,_𝐺−1(𝑘)=𝑘²−𝜔²+𝛼+𝛾𝛽−𝜔²−𝑘²𝑛²𝛿²=0. (23) Let us first consider the stable modes for real 𝜔>𝑘 in which case there are at most two stable modes stemming from Δ_𝐺⁻¹=0. The other stable mode comes from zero of Δ_𝐴⁻¹. Thus, for finite 𝜉, there are three stable modes. Note that these modes depend on the angle of propagation with respect to the anisotropy axis. The dispersion relation for the unstable modes can be obtained by letting 𝜔→𝑖Γ in Δ_𝐺⁻¹=0 and Δ_𝐴⁻¹ leading to two unstable modes and these modes again depend the direction of propagation with respect to the anisotropy axis.

The collective modes in a collisional AQGP have been investigated in [50] using Bhatnagar-Gross-Krook collisional kernel. It has been observed that inclusion of the collisions slows down the growth rate of unstable modes and the instabilities disappear at certain critical values of the collision frequency.

In order to calculate the quark-quark potential, we resort to the covariant gauge. Using the previous expression for gluon self-energy in anisotropic medium the propagator, in covariant gauge, can be calculated after some cumbersome algebra [48] as follows: Δ^𝜇𝜈=1𝐾²𝐴−𝛼^𝜇𝜈−𝐶^𝜇𝜈+Δ_𝐺𝐾²𝜔−𝛼−𝛾⁴𝐾⁴𝐵^𝜇𝜈+𝜔²−𝛽𝐶^𝜇𝜈𝜔+𝛿²𝐾²𝐷^𝜇𝜈−𝜆𝐾⁴𝐾^𝜇𝐾^𝜈, (24) where Δ_𝐺⁻¹=𝐾²−𝛼−𝛾𝜔²−𝛽−𝛿²𝐾²−(𝑛⋅𝐾)². (25)

The structure functions (𝛼, 𝛽, 𝛾, and 𝛿) depend on 𝜔, 𝐤, 𝜉, and on the angle (𝜃_𝑛) between the anisotropy vector and the momentum 𝐤. In the limit 𝜉→0, the structure functions 𝛾 and 𝛿 are identically zero, and 𝛼 and 𝛽 are directly related to the isotropic transverse and longitudinal self-energies, respectively [28]. In anisotropic plasma, the two-body interaction, as expected, becomes direction dependent. Now the momentum space potential can be obtained from the static gluon propagator in the following way [32, 48]: 𝑉𝑘_⟂,𝑘_𝑧,𝜉=𝑔²Δ⁰⁰𝜔=0,𝑘_⟂,𝑘_𝑧,𝜉=𝑔²𝐤²+𝑚²_𝛼+𝑚²_𝛾𝐤²+𝑚²_𝛼+𝑚²_𝛾𝐤²+𝑚²_𝛽−𝑚²_𝛿, (26) where 𝑚²_𝛼𝑚=−^2𝐷2𝑘²_⟂𝜉×𝑘²_𝑧tan⁻¹𝜉−𝑘_𝑧𝐤²𝐤²+𝜉𝑘²_⟂×tan⁻¹𝜉𝑘_𝑧𝐤²+𝜉𝑘²_⟂,𝑚²_𝛽=𝑚²_𝐷𝜉+(1+𝜉)tan⁻¹𝜉𝐤²+𝜉𝑘²_⟂+𝜉𝑘_𝑧𝜉𝑘_𝑧+𝐤²(1+𝜉)/𝐤²+𝜉𝑘²_⟂×tan⁻¹𝜉𝑘_𝑧/𝐤²+𝜉𝑘²_⟂×2𝜉(1+𝜉)𝐤²+𝜉𝑘²_⟂⁻¹,𝑚²_𝛾𝑚=−^2𝐷2𝐤²𝐤²+𝜉𝑘²_⟂−1+2𝑘²_𝑧/𝑘²_⟂𝜉tan⁻¹𝜉+𝑘_𝑧𝐤²2𝐤²+3𝜉𝑘²_⟂𝜉𝐤²+𝜉𝑘²_⟂^3/2𝑘²_⟂×tan⁻¹𝜉𝑘_𝑧𝐤²+𝜉𝑘²_⟂,𝑚²_𝛿=−𝜋𝑚²_𝐷𝜉𝑘_𝑧𝑘_⟂|𝐤|4𝐤²+𝜉𝑘²_⟂^3/2, (27) where 𝛼_𝑠=𝑔²/4𝜋 is the strong coupling constant, and we assume constant coupling. The coordinate space potential can be obtained by taking Fourier transform of (26): 𝑉(𝐫,𝜉)=−𝑔²𝐶_𝐹𝑑³𝑘(2𝜋)³𝑒^{−𝑖𝐤⋅𝐫}𝑉𝑘_⟂,𝑘_𝑧,𝜉, (28) which, under small 𝜉 limit, reduces to [48] 𝑉(𝐫,𝜉)≈𝑉_iso(𝑟)−𝑔²𝐶_𝐹𝜉𝑚²_𝐷𝑑³𝑘(2𝜋)³𝑒^{−𝑖𝐤⋅𝐫}2/3−(𝐤⋅𝐧)²/𝐤²𝐤²+𝑚²_𝐷², (29) where 𝑉_iso(𝑟)=−𝑔²𝐶_𝐹𝑒^{−𝑚𝐷𝑟}/(4𝜋𝑟). As indicated earlier, the potential depends on the angle between 𝐫 and 𝐧. When 𝐫‖𝐧 the potential (𝑉_‖) is given by [48] 𝑉_‖(𝐫,𝜉)=𝑉_iso(𝑟2𝑒𝑟̂)1+𝜉−1𝑟²̂−2𝑟̂𝑟̂−1−6, (30) whereas [48] 𝑉_⟂(𝐫,𝜉)=𝑉_iso(𝑟𝑟̂)1+𝜉1−𝑒𝑟²̂+1𝑟̂+12+𝑟̂3, (31) where 𝑟̂=𝑟𝑚_𝐷.

For arbitrary 𝜉, (28) has to be evaluated numerically. It has been observed that because of the lower density of the plasma particles in AQGP, the potential is deeper and closer to the vacuum potential than for 𝜉=0 [48]. This means that the general screening is reduced in an AQGP.

Next, we consider quarkonium states in an AQGP where the potential, to linear order in 𝜉, is given by (29) which can also be written as [51] 𝑉(𝐫,𝜉)=𝑉_iso(𝑟)1−𝜉𝑓₀(𝑟)̂+𝑓₁(𝑟)̂cos2𝜃, (32) where cos𝜃=𝑟̂⋅𝑛 and the functions are given by [51] 𝑓₀(𝑟)̂=61−𝑒𝑟̂+𝑟̂[6−𝑟̂(𝑟̂−3)]12𝑟²̂𝑟̂=−6−𝑟²̂𝑓48+⋯,₁(𝑟)̂=61−𝑒𝑟̂+𝑟̂[6+𝑟̂(𝑟̂+3)]12𝑟²̂𝑟̂=−²16+⋯. (33) With the previous expressions, the real part of the heavy quark potential in AQGP, after finite quark mass correction, becomes [51] 𝑉(𝐫)𝛼=−𝑟1+𝜇𝑟exp−𝜇𝑟+2𝜎𝜇−1−exp−𝜇𝑟−𝜎𝑟exp−𝜇𝑟0.8𝑚²_𝑄𝑟, (34) where 𝜇/𝑚_𝐷=1+𝜉(3+cos2𝜃)/16 and the previous expression is the minimal extension of the KMS potential [52] in AQGP. The ground states and the excited states of the quarkonium states in AQGP have been found by solving the three-dimensional Schrodinger equation using the finite difference time domain method [53]. Without going into further details, we will quote the main findings of the work of [51]. The binding energies of charmonium and bottomonium states are obtained as a function of the hard momentum scale. For a fixed hard momentum scale, it is seen that the binding energy increases with anisotropy. Note that the potential (34) is obtained by replacing 𝑚_𝐷 by 𝜇 in KMS equation [52]. For a given hard momentum scale 𝜇<𝑚_𝐷, the quarkonium states are more strongly bound than the isotropic case. This implies that the dissociation temperature for a particular quarkonium state is more in AQGP. It is found that the dissociation temperature for 𝐽/𝜓 in AQGP is 1.4𝑇_𝑐, whereas for isotropic case it is 1.2𝑇_𝑐 [51].

Quarkonium binding energies have also been calculated using a realistic potential including the complex part in [54] by solving the 3D Schrodinger equation where the potential has the form 𝑉(𝐫,𝜉)=𝑉_𝑅(𝐫,𝜉)+𝑖𝑉_𝐼(𝐫,𝜉), (35) where the real part is given by (34) and the imaginary part is given in [55]. The main results of this calculations are as follows. For 𝐽/𝜓, the dissociation temperature obtained in this case is 2.3𝑇_𝑐 in isotropic case. It has been found that with anisotropy the dissociation temperature increases as the binding of quarkonium states becomes stronger in AQGP.

Thermal bottomonium suppression (𝑅_𝐴𝐴) at RHIC and LHC energies has been calculated in [56, 57] in an AQGP. Two types of potentials have been considered there coming from the free energy (case A) and the internal energy (case B), respectively. By solving the 3D Schrodinger equation with these potentials, it has been found that the dissociation temperature for Υ(1𝑠) becomes 373 MeV and 735 MeV for the cases A and B, respectively, whereas in case of 𝜉=0, these becomes 298 MeV and 593 MeV. Thus, the dissociation temperature increases in case of AQGP irrespective of the choice of the potential. In case of other bottomonium states, the dissociation temperatures increase compared to the isotropic case. The nuclear modification factor has been calculated by coupling AHYDRO [46] with the solutions of Schrodinger equation. Introduction of AHYDRO into the picture makes 𝑝_hard and 𝜉 functions of proper time (𝜏), transverse coordinate (𝐱_⟂), and the space-time rapidity (Θ). As a consequence, both the real and imaginary part of the binding energies become functions of 𝜏,𝐱_⟂, and Θ. Now the nuclear modification factor (𝑅_𝐴𝐴) is related to the decay rate (Γ) of the state in question, where Γ=−2ℐ[𝐸]. Thus, 𝑅_𝐴𝐴 is given by [56, 57] 𝑅_𝐴𝐴𝑝_𝑇,𝐱_⟂,Θ=𝑒^{−𝜁(𝑝𝑇,𝐱⟂,Θ)}, (36) where 𝜁(𝑝_𝑇,𝐱_⟂,Θ) is given by 𝜁𝑝_𝑇,𝐱_⟂,Θ=𝜃𝜏_𝑓−𝜏_form^𝜏𝑓_{𝜏maxform,𝜏𝑖}𝑑𝜏Γ𝑝_𝑇,𝐱_⟂,Θ. (37) Here, 𝜏_form is the formation time of the particular state in the laboratory frame and 𝜏_𝑓 is the time when the hard momentum scale reaches 𝑇_𝑐. To study the nuclear suppression of a particular state, one needs to take into account the decay of the excited states to this particular state (so-called feed down). For RHIC energies, using the value of 𝑑𝑁_ch/𝑑𝑦=620 and various values of 𝜂/𝑆 (note that 𝜂/𝑆 is related to the anisotropy parameter, 𝜉) the corresponding initial temperatures are estimated with 𝜏_𝑖=0.3 fm/c. 𝑝_𝑇 integrated 𝑅_𝐴𝐴 values of individual bottomonium states (direct production) have been calculated using both the potentials A and B [56, 57] as functions of number of participants and rapidity. It is seen that for both potentials, more suppression is observed in case of anisotropic medium than in the isotropic case. However, the suppression is more in case of potential model A. There is also indication of sequential suppression [56, 57]. Similar exercise has also been done in case of LHC energies (√𝑠=2.76 TeV) using a constant value of 𝜂/𝑆. It is again observed that the suppression in case of potential A is more. These findings can be used to constrain 𝜂/𝑆 by comparing with the RHIC and LHC data.

Next, we discuss the effect of the initial state momentum anisotropy on the survival probability of 𝐽/𝜓 due to gluon dissociation. This is important as we have to take into account all possibilities of quarkonium getting destroyed in the QGP to estimate the survival probability and hence 𝑅_𝐴𝐴. In contrast to Debye screening, this is another possible mechanism of 𝐽/𝜓 suppression in QGP. In QGP, the gluons have much harder momentum, sufficient to dissociate the charmonium. Such a study was performed in an isotropic plasma [58]. In this context, we will study the thermally weighted gluon dissociation cross of 𝐽/𝜓 in an anisotropic media.

Bhanot and Peskin first calculated the quarkonium-hadron interaction cross-section using operator product expansion [59]. The perturbative prediction for the gluon 𝐽/𝜓 dissociation cross-section is given by [60] 𝜎𝑞⁰=2𝜋3323²16𝜋3𝑔₂1𝑚²_𝑄𝑞⁰/𝜖₀−1^3/2𝑞⁰/𝜖₀⁵, (38) where 𝑞⁰ is the energy of the gluon in the stationary 𝐽/𝜓 frame; 𝜖₀ is the binding energy of the 𝐽/𝜓 where 𝑞₀>𝜖₀, and 𝑚_𝑄 is charm quark mass. It is to be noted that we have used the constant binding energy of the 𝐽/𝜓 in AQGP at finite temperature. We assume that the 𝐽/𝜓 moves with four momentum 𝑃 given by 𝑃=𝑀_𝑇cosh𝑦,0,𝑃_𝑇,𝑀_𝑇sinh𝑦, (39) where 𝑀_𝑇=𝑀²_𝐽/𝜓+𝑃²_𝑇 is the 𝐽/𝜓 transverse mass and 𝑦 is the rapidity of the 𝐽/𝜓. A gluon with a four momentum 𝐾=(𝑘⁰,𝐤) in the rest frame of the parton gas has energy 𝑞⁰=𝐾⋅𝑢 in the rest frame of the 𝐽/𝜓. The thermal gluon 𝐽/𝜓 dissociation cross-section in anisotropic media is defined as [15, 58] 𝜎(𝐾⋅𝑢)𝑣_rel𝑘=∫𝑑³𝑘𝜎(𝐾⋅𝑢)𝑣_rel𝑓𝑘⁰,𝜉,𝑝_hard∫𝑑³𝑘𝑓𝑘⁰,𝜉,𝑝_hard, (40) where 𝑣_rel=1−(𝐤⋅𝐏)/(𝑘⁰𝑀_𝑇cosh𝑦) is the relative velocity between 𝐽/𝜓 and the gluon. Now changing the variable (𝐾↔𝑄), one can obtain by using Lorentz transformation [15], the following relations: 𝑘⁰=𝑞⁰𝐸+𝑞sin𝜃_𝑝sin𝜃_𝑞sin𝜙_𝑞+cos𝜃_𝑝cos𝜃_𝑞𝑀_𝐽/𝜓,𝐤=𝐪+𝑞𝐸|𝐏|𝑀_𝐽/𝜓×𝑞𝑀_𝑇cosh𝑦−𝑀_𝐽/𝜓×sin𝜃_𝑝sin𝜃_𝑞sin𝜙_𝑞+cos𝜃_𝑝cos𝜃_𝑞+|𝐏|𝐯_𝐉/𝜓, (41) where 𝐯_𝐉/𝜓=𝐏/𝐸, 𝑃=(𝐸,0,|𝐏|sin𝜃_𝑝,|𝐏|cos𝜃_𝑝), and 𝐪=(𝑞sin𝜃_𝑞cos𝜙_𝑞,𝑞sin𝜃_𝑞sin𝜙_𝑞,𝑞cos𝜃_𝑞). In the rest frame of 𝐽/𝜓, numerator of (40) can be written as 𝑑³𝑞𝑀_𝐽/𝜓𝐸𝜎𝑞⁰𝑓𝑘⁰,𝜉,𝑝_hard, (42) while, the denominator of (40) can be written as [30] 𝑑³𝑘𝑓𝑘⁰,𝜉,𝑝_hard=𝑑³𝑘𝑓_iso𝐤²+𝜉(𝐤⋅𝐧)²,𝑝_hard=1(3)𝑝1+𝜉8𝜋𝜁³_hard, (43) where 𝜁(3) is the Riemann zeta function. The maximum value of the gluon 𝐽/𝜓 dissociation cross-section [60] is about 3 mb in the range 0.7≤𝑞⁰≤1.7 GeV. Therefore high-momentum gluons do not see the large object and simply passes through it, and the low-momentum gluons cannot resolve the compact object and cannot raise the constituents to the continuum.

To calculate the survival probability of 𝐽/𝜓 in an anisotropic plasma, we consider only the longitudinal expansion of the matter. The survival probability of the 𝐽/𝜓 in the deconfined quark-gluon plasma is 𝑆𝑃_𝑇=∫𝑑²𝑟𝑅²_𝐴−𝑟²exp−∫^𝜏max_𝜏𝑖𝑑𝜏𝑛_𝑔(𝜏)𝜎(𝐾⋅𝑢)𝑣_rel𝑘∫𝑑²𝑟𝑅²_𝐴−𝑟², (44) where 𝜏_max=min(𝜏_𝜓,𝜏_𝑓) and 𝜏_𝑖 are the QGP formation time. 𝑛_𝑔(𝜏)=16𝜁(3)𝑝³_hard(𝜏)/[𝜋²1+𝜉(𝜏)] is the gluon density at a given time 𝜏. Now the 𝐽/𝜓 will travel a distance in the transverse direction with velocity 𝐯_𝐽/𝜓 given by 𝑑=−𝑟cos𝜙+𝑅²_𝐴−𝑟²1−cos²𝜙, (45) where cos𝜙=𝑣̂_𝐽/𝜓⋅𝑟̂. The time interval 𝜏_𝜓=𝑀_𝑇𝑑/𝑃_𝑇 is the time before 𝐽/𝜓 escapes from a gluon gas of transverse extension 𝑅_⟂. The time evaluation of 𝜉 and 𝑝_hard is determined by (3) and (4).

We now first discuss the numerical result of the thermal averaged gluon dissociation cross-section in the anisotropic system. The results are displayed in Figure 4 for 𝑃_𝑇=0 and 𝑃_𝑇=8 GeV for a set of values of the anisotropy parameter. It is seen that the velocity averaged cross-section decreases with 𝜉 for 𝑝_hard up to ~500 MeV and then increases as compared to the isotropic case (𝜉=0) (see Figure 4(a)). For higher 𝑃_𝑇, a similar feature has been observed in Figure 4(b), where the cross-section starts to increase beyond 𝑝_hard~ 200 MeV. For fixed 𝑝_hard, the dissociation cross-section as a function of 𝑃_𝑇 shows that the cross-section first decreases and then marginally increases [15].

(a)

(b)

Figure 4 

(Color online) The thermal-averaged gluon 𝐽/𝜓 dissociation cross-section as function of the hard momentum scale at central rapidity (𝜃𝑝=𝜋/2) for 𝜉={0,1,3,5}. (a) corresponds to 𝑃𝑇=0 and (b) is for 𝑃𝑇=8 GeV.

Equation (44) has been used to calculate the survival probability. Figure 5 describes the survival probability of 𝐽/𝜓 for various values of the isotropization time 𝜏_iso at RHIC energy. Left (Right) panel corresponds to 𝜃_𝑝=𝜋/2 (𝜃_𝑝=𝜋/3). It is observed that the survival probability remains the same as in the isotropic case upto 𝑃_𝑇=4 GeV in the central region. Beyond that marginal increase is observed with the increase of 𝜏_iso. In the forward rapidity, the results are almost the same as the isotropic case throughout the whole 𝑃_𝑇 region. For this set of initial conditions, the argument of the exponential in (44) becomes similar to that of the isotropic case. Also the dissociation cross-section first decreases with 𝑃_𝑇 and then increases. Because of these reasons, we observe minor change in the survival probability [15]. Therefore, the survival probability is more or less independent of the direction of propagation of the 𝐽/𝜓 with respect to the anisotropy axis, whereas in the case of radiative energy loss of fast partons in anisotropic media the result has been shown to depend strongly on the direction of propagation [32]. The results for the LHC energies for the two sets of initial conditions at two values of the transition temperatures are shown in Figure 6. For set I, we observe marginal increase as in the case for set I at RHIC energies. However, for set II substantial modification is observed for the reason stated earlier.

(a)

(b)

Figure 5 

(Color online) The survival probability of 𝐽/𝜓 in an anisotropic plasma at central and forward rapidity region. The initial conditions are taken as 𝑇𝑖=550 MeV, 𝑇𝑐=200 MeV, and 𝜏𝑖=0.7 fm/c.

(a)

(b)

Figure 6 

(Color online) The survival probability of 𝐽/𝜓 in an anisotropic plasma for the two sets of initial conditions at LHC energies. Left (right) panel corresponds to 𝑇𝑐=200(170) MeV.

4. Radiative Energy Loss and 𝑅_𝐴𝐴 of Light Hadrons

In this section, we calculate the radiative energy loss in an infinitely extended anisotropic plasma. We assume that an on-shell quark produced in the remote past is propagating through an infinite QCD medium that consists of randomly distributed static scattering centers which provide a color-screened Yukawa potential originally developed for the isotropic QCD medium given by [61] 𝑉_𝑛=𝑉𝑞_𝑛𝑒^{𝑖𝐪𝑛⋅𝐱𝑛}=2𝜋𝛿𝑞⁰𝑣𝑞_𝑛𝑒^{−𝑖𝐪𝑛⋅𝐱𝑛}𝑇_𝑎𝑛(𝑅)⊗𝑇_𝑎𝑛(𝑛), (46) with 𝑣(𝐪_𝑛)=4𝜋𝛼_𝑠/(𝑞²_𝑛+𝑚²_𝐷). 𝑥_𝑛 is the location of the 𝑛th scattering center, 𝑇 denotes the color matrices of the parton and the scattering center. It is to be noted that the potential has been derived by using hard thermal loop (HTL) propagator in QGP medium. In anisotropic plasma, the two-body interaction, as expected, becomes direction dependent. The two-body potential required to calculate the radiative energy loss has been calculated in the previous section.

Now the parton scatters with one of the color center with the momentum 𝑄=(0,𝑞_⟂,𝑞_𝑧) and subsequently radiates a gluon with momentum 𝐾=(𝜔,𝑘_⟂,𝑘_𝑧). The quark energy loss is calculated by folding the rate of gluon radiation Γ(𝐸) with the gluon energy by assuming 𝜔+𝑞₀≈𝜔. In the soft-scattering approximation, one can find the quark radiative energy loss per unit length as [62] 𝑑𝐸=𝐸𝑑𝐿𝐷_𝑅𝑥𝑑𝑥𝑑Γ.𝑑𝑥 (47) Here, 𝐷_𝑅 is defined as [𝑡_𝑎,𝑡_𝑐][𝑡_𝑐,𝑡_𝑎]=𝐶₂(𝐺)𝐶_𝑅𝐷_𝑅, where 𝐶₂(𝐺)=3,𝐷_𝑅=3, and 𝐶_𝑅=4/3. 𝑥 is the longitudinal momentum fraction of the quark carried away by the emitted gluon.

Now in anisotropic medium we have [32] 𝑥𝑑Γ=𝐶𝑑𝑥^𝑅𝛼_𝑠𝜋𝐿𝜆𝑚²_𝐷16𝜋²𝛼_𝑠×𝑑²𝑘_⟂𝜋𝑑²𝑞_⟂𝜋𝑉𝑞_⟂,𝑞_𝑧,𝜉²×𝑘_⟂+𝑞_⟂𝑘_⟂+𝑞_⟂²−𝑘+𝜒𝑘²_⟂+^⟂𝜒², (48) where 𝜒=𝑚²_𝑞𝑥²+𝑚²_𝑔 with 𝑚²_𝑔=𝑚²_𝐷/2 and 𝑚²_𝑞=𝑚²_𝐷/6, and 𝑉(𝑞_⟂,𝑞_𝑧,𝜉) is given by (26).

In the present scenario, we assume that the parton is propagating along the 𝑧 direction and makes an angle 𝜃_𝑛 with the anisotropy direction. In such case, we replace 𝑞_⟂ and 𝑞_𝑧 in (26) by 𝑞_⟂→𝑞²_⟂−𝑞²_⟂sin²𝜃_𝑛cos²𝜙 and 𝑞_𝑧→|𝐪|sin𝜃_𝑛cos𝜙. For arbitrary 𝜉 the radiative energy loss can be written as [32] Δ𝐸𝐸=𝐶_𝑅𝛼_𝑠𝜋²𝐿𝑚²_𝐷𝜆×𝑑𝑥𝑑²𝑞_⟂𝑉𝑞_⟂,𝑞_𝑧,𝜉²16𝜋²𝛼²_𝑠×−12−𝑘²_𝑚𝑘²_𝑚+𝑞+𝜒_2⟂−𝑘²_𝑚+𝜒2𝑞⁴_⟂+2𝑞²_⟂𝜒−𝑘²_𝑚+𝑘²_𝑚+𝜒²+𝑞²_⟂+2𝜒𝑞²_⟂1+4𝜒/𝑞²_⟂𝑘×ln^2𝑚+𝜒𝜒×𝑞²_⟂+3𝜒+1+4𝜒/𝑞²_⟂𝑞²_⟂×𝑞+𝜒_2⟂−𝑘²_𝑚+3𝜒+1+4𝜒/𝑞²_⟂×𝑞⁴_⟂+2𝑞²_⟂𝜒−𝑘²_𝑚+𝑘²_𝑚+𝜒²⁻¹. (49) In the previous expression, 𝜆 denotes the average mean-free path of the quark scattering and is given by 1𝜆=1𝜆_𝑔+1𝜆_𝑞, (50) which depends on the strength of the anisotropy. In the last expression 𝜆_𝑞 and 𝜆_𝑔 correspond to the contributions coming from 𝑞-𝑞 and 𝑞-𝑔 scattering, respectively. Consider 1𝜆_𝑖=𝐶_𝑅𝐶₂(𝑖)𝜌(𝑖)𝑁²_𝑐𝑑−1²𝑞_⟂(2𝜋)²𝑉𝑞_⟂,0,𝜉², (51) where 𝐶₂(𝑖) is the Cashimir for the 𝑑_𝑖-dimensional representation and 𝐶₂(𝑖)=(𝑁²_𝑐−1)/(2𝑁_𝑐) for quark and 𝐶₂(𝑖)=𝑁_𝑐 for gluon scatterers. 𝜌_𝑖 is the density of the scatterers. Using 𝜌_𝑖=𝜌^iso_𝑖1+𝜉, we obtain 1𝜆=18𝛼_𝑠𝑝_hard𝜁(3)𝜋²11+𝜉𝑅(𝜉)1+𝑁_𝐹/61+𝑁_𝐹,/4 (52) where 𝑁_𝐹 is the numbers of flavors.

The fractional energy loss in anisotropy medium for the light quark is shown in Figure 7. We consider a plasma at a temperature 𝑇=250 MeV with the effective number of degrees of freedom 𝑁_𝐹=2.5, 𝛼_𝑠=0.3 and the length of the medium is 𝐿=5 fm. The energy loss in the anisotropic media depends on the angle of propagation of the fast parton with respect to the anisotropy axis (𝐧). We see that the fractional energy loss increases in the direction parallel to the anisotropy axis. With the increase of anisotropy parameter 𝜉 the fractional energy loss subsequently increases for 𝜃_𝑛=𝜋/6. However, away from the anisotropy axis (𝜃_𝑛=𝜋/2), the fractional energy loss decreases because the quark-quark potential is stronger in the anisotropy direction [32]. In the perpendicular direction to the anisotropy axis, the fractional energy loss is quite small. The fractional energy loss for the heavy quarks (i.e., for charm and bottom) is shown in Figure 8. The fractional energy loss is enhanced in the direction parallel to anisotropy axis as well as for 𝜃_𝑛=𝜋/6. However, for 𝜃_𝑛=𝜋/2, the fractional energy loss decreases for both heavy and light quarks.

Figure 7 

Color online: fractional energy loss for the light quark for 𝜉=(0.5,1).

(a)

(b)

Figure 8 

(Color online) Same as 7 for charm quark (a) and bottom quark (b) with 𝜉=1 and 𝑇=500 MeV.

Next, we consider the nuclear modification factor of light hadrons incorporating the light quark energy loss in AQGP discussed in the previous paragraphs. When a parton is propagating in the direction of anisotropy it is found that the fractional energy loss increases. In this section, we will apply this formalism to calculate the nuclear modification factor of the light hadrons. Starting with two-body scattering at the parton level, the differential cross-section for the hadron production is [63] 𝐸𝑑𝜎𝑑³𝑝𝐴𝐵⟶jet+𝑋=𝐾_{𝑎𝑏𝑐𝑑}𝑑𝑥_𝑎𝑑𝑥_𝑏𝐺_𝑎/ℎ𝐴𝑥_𝑎,𝑄²𝐺_𝑏/ℎ𝐵×𝑥_𝑏,𝑄²𝑠̂𝜋𝑑𝜎𝑑𝑡̂(𝑎𝑏⟶𝑐𝑑)𝛿𝑠̂+𝑡̂+𝑢. (53) The argument of the 𝛿 function can be expressed in terms of 𝑥_𝑎 and 𝑥_𝑏, and doing the 𝑥_𝑏 integration we arrive at the final expression as follows: 𝐸𝑑𝜎𝑑³𝑝𝐴𝐵⟶jet+𝑋=𝐾_{𝑎𝑏𝑐𝑑}¹_𝑥min𝑑𝑥_𝑎𝐺_𝑎/ℎ𝐴𝑥_𝑎,𝑄²𝐺_𝑏/ℎ𝐵×𝑥_𝑏,𝑄²2𝜋𝑥_𝑎𝑥_𝑏2𝑥_𝑎−𝑥_𝑇𝑒^𝑦𝑑𝜎𝑑𝑡̂(𝑎𝑏⟶𝑐𝑑), (54) where 𝑥_𝑏=(𝑥_𝑎𝑥_𝑇𝑒^−𝑦)/(2𝑥_𝑎−𝑥_𝑇𝑒^𝑦), 𝑥_𝑇=2𝑝_𝑇/√𝑠, and 𝑥_min=𝑥_𝑇𝑒^𝑦/(2−𝑥_𝑇𝑒^−𝑦), and the factor 𝐾 is introduced to take into account the higher-order effects. It should be noted that to obtain single-particle inclusive invariant cross-section, the fragmentation function 𝐷_ℎ/𝑐(𝑧,𝑄²) must be included. To obtain the hadronic 𝑝_𝑇 spectra in A-A collisions, we multiply the result by the nuclear overlap function for a given centrality. The inclusion of jet quenching as a final state effect in nucleus-nucleus collisions can be implemented in two ways: (i) modifying the partonic 𝑝_𝑇 spectra [64] and (ii) modifying the fragmentation function [65] but keeping the partonic 𝑝_𝑇 spectra unchanged. In this calculation we intend to modify the fragmentation function. The effective fragmentation function can be written as 𝐷_ℎ/𝑐𝑧,𝑄²=𝑧^⋆𝑧𝐷_ℎ/𝑐𝑧^⋆,𝑄², (55) where 𝑧^⋆=𝑧/(1−Δ𝐸/𝐸) is the modified momentum fraction. Now we take into account the jet production geometry. We assume that all the jets are not produced at the same point; therefore, the path length transversed by the partons before the fragmentation is not the same. We consider a jet initially produced at (𝑟,𝜙) and leaves the plasma after a proper time (𝑡_𝐿) or equivalently after traversing a distance 𝐿 (for light quarks 𝑡_𝐿=𝐿) where 𝐿𝑟,𝜙=𝑅²_⟂−𝑟²sin𝜙²−𝑅_⟂cos𝜙, (56) where 𝑅_⟂ is the transverse dimension of the system. Since the number of jets produced at 𝐫 is proportional to the number of binary collisions, the probability is proportional to the product of the thickness functions as follows: 𝒫(𝑟)⃗∝𝑇_𝐴(𝑟)⃗𝑇_𝐵(𝑟)⃗. (57) In case of hard sphere 𝒫(𝑟) is given by [66] 𝒫(𝑟)=2𝜋𝑅²_⟂𝑟1−²𝑅²_⟂𝜃𝑅_⟂−𝑟, (58) where ∫𝑑²𝑟𝒫(𝑟)=1. To obtain the hadron 𝑝_𝑇 spectra, we have to convolute the resulting expression over all transverse positions and the expression is 𝑑𝑁^𝜋0(𝜂)𝑑²𝑝_𝑇𝑑𝑦=_𝑓𝑑²𝑟𝒫(𝑟)^𝑡𝐿_𝑡𝑖𝑑𝑡𝑡_𝐿−𝑡_𝑖×𝑑𝑧𝑧²𝐷_{𝜋0(𝜂)/𝑓}𝑧,𝑄²_{𝑧=𝑝𝑇/𝑝𝑓𝑇}𝐸𝑑𝑁𝑑³𝑝^𝑓. (59) The quantity 𝐸(𝑑𝑁/𝑑³𝑝^𝑓) is the initial momentum distribution of jets and can be computed using LO-pQCD. Here, we use average distance traversed by the partons, ⟨𝐿⟩ is given by ∫⟨𝐿⟩=^𝑅𝑇0𝑟𝑑𝑟∫₀^2𝜋𝐿𝜙,𝑟𝑇_𝐴𝐴(𝑟,𝑏=0)𝑑𝜙∫^𝑅𝑇₀𝑟𝑑𝑟∫₀^2𝜋𝑇_𝐴𝐴(𝑟,𝑏=0)𝑑𝜙, (60) where ⟨𝐿⟩∼5.8(6.2) fm for RHIC (LHC). Finally, the nuclear modification factor (𝑅_𝐴𝐴) becomes [44] 𝑅_𝐴𝐴𝑝_𝑇=𝑑𝑁^𝜋0(𝜂)_𝐴𝐴/𝑑²𝑝_𝑇𝑑𝑦𝑑𝑁^𝜋0(𝜂)_𝐴𝐴/𝑑²𝑝_𝑇𝑑𝑦₀, (61) where [𝑑𝑁^𝜋0(𝜂)_𝐴𝐴/𝑑²𝑝_𝑇𝑑𝑦]₀ corresponds to the hadron 𝑝_𝑇 distribution without the energy loss.

For an expanding plasma, the anisotropy parameter 𝑝_hard and 𝜉 are time dependent. The time evaluation is again given by (3) and (4). In the present work, it is assumed that an isotropic QGP is formed at an initial time 𝜏_𝑖 and initial temperature 𝑇_𝑖. Rapid longitudinal expansion of the plasma leads to an anisotropic QGP which lasts till 𝜏_iso. For 0–10% centrality (relevant for our case), we obtain 𝑇_𝑖 = 440 (350) MeV for 𝜏_𝑖=0.147(0.24) fm/c. at RHIC energy [44].

Figure 9 describes the nuclear modification factor for two different initial conditions with various values of isotropization time, 𝜏_iso along with the PHENIX data [67]. It is quite clear from Figure 9(a) that the value of 𝑅_𝐴𝐴 for anisotropic medium is lower than that for the isotropic media as the energy loss in the anisotropy medium is higher [32]. It is also observed that as 𝜏_iso increases the value of 𝑅_𝐴𝐴 decreases compared to its isotropic value [44]. This is because the hard scale decreases slowly as compared to the isotropic case, that is, the cooling is slow. For reasonable choices of 𝜏_iso, the experimental data is well described. It is seen that increasing the value of 𝜏_iso beyond 1.5 fm/c grossly underpredict the data. We find that the extracted value of isotropization time lies in the range 0.5≤𝜏_iso≤1.5 fm/c [44]. This is in agreement with the earlier finding of 𝜏_iso using PHENIX photon data [16]. In order to see the sensitivity on the initial conditions we now consider another set of initial conditions, 𝑇_𝑖=350 MeV and 𝜏_𝑖=0.24 fm/c. The result is shown in Figure 9(b). It is observed that to reproduce the data a larger value of 𝜏_iso is needed as compared to the case of higher initial temperature. We extract an upper limit of 𝜏_iso=2 fm/c [44] in this case.

(a)

(b)

Figure 9 

(Color Online) Nuclear modification factor at RHIC energies. The initial conditions are taken as (a) 𝑇𝑖=440 MeV and 𝜏𝑖=0.147 fm/c and (b) 𝑇𝑖=350 MeV and 𝜏𝑖=0.24 fm/c.

5. Plasma Wakes

It is mentioned earlier that when a jet propagates through hot and dense medium it loses energy mainly by the radiative process. As mentioned earlier, it also creates wake in the charge density as well as in the potential. Now, we calculate the wake in charge density and the wake potential due to the passage of a fast parton in a small 𝜉 limit. Dielectric function contains essentially all the information of the chromoelectromagnetic properties of the plasma. The dielectric function 𝜖(𝐤,𝜔) can be calculated from the dielectric tensor using the following relation: 𝜖()=𝑘𝐤,𝜔^𝑖𝜖^𝑖𝑗(𝐤,𝜔)𝑘_𝑗𝑘², (62) where the dielectric tensor 𝜖^𝑖𝑗 can be written in terms of the gluon polarization tensor as follows (given by (18)): 𝜖^𝑖𝑗=𝛿^𝑖𝑗−Π^𝑖𝑗𝜔². (63) Therefore, the dielectric function is directly related to the structure functions mentioned in Section 3 through (18), (63), and (62). To get the analytic expressions for the structure functions, one must resort to small 𝜉 limit. To linear order in 𝜉, we have [28] 𝛼=Π_𝑇(𝑧)𝑧+𝜉²123+5cos2𝜃_𝑛𝑚²_𝐷−161+cos2𝜃_𝑛𝑚²_𝐷+14Π_𝑇(𝑧)1+3cos2𝜃_𝑛−𝑧²3+5cos2𝜃_𝑛,𝛽=𝑧²Π_𝐿(𝑧)1+𝜉61+3cos2𝜃_𝑛𝑚²_𝐷+Π_𝐿(𝑧)cos2𝜃_𝑛−𝑧²21+3cos2𝜃_𝑛,𝜉𝛾=33Π_𝑇(𝑧)−𝑚²_𝐷𝑧²−1sin²𝜃_𝑛,𝜉𝛿=3𝑘4𝑧²𝑚²_𝐷+3Π_𝑇(𝑧)1−4𝑧²cos𝜃_𝑛, (64) with Π_𝑇(𝐾)=𝑚²_𝐷2𝑧²×11−21𝑧−𝑧ln𝑧+1𝑧−1−𝑖𝜋Θ1−𝑧²,Π_𝐿(𝐾)=𝑚²_𝐷𝑧2ln𝑧+1𝑧−1−𝑖𝜋Θ1−𝑧²−1, (65) where 𝑧=𝜔/𝑘.

In the presence of the test charge particle, the induced charge density and the wake potential depend on the velocity of the external charged parton and also on the distribution of the background particle [68]. When a static test charge is introduced in a plasma, it acquires a shielding cloud. As a result, the induced charge distribution is spherically symmetric. When a charge particle is in motion relative to the plasma, the induced charge distribution no longer remains symmetric. As a result spherical symmetry of the screening cloud reduces to ellipsoidal shape.

The passage of external test charge through the plasma also disturbs the plasma and creates induced color charge density [69]. Therefore, the total color charge density is given as 𝜌^𝑎_tot(𝐤,𝜔)=𝜌^𝑎_ext(𝐤,𝜔)+𝜌^𝑎_ind(𝐤,𝜔), (66) where 𝑎 represents the color index. However, the total color charge density is linearly related to 𝜌^𝑎_ext through the dielectric response function (𝜌^𝑎_tot(𝐤,𝜔)=𝜌^𝑎_ext(𝐤,𝜔)/𝜖(𝐤,𝜔)). Therefore, the induced color charge density is explicitly written as 𝜌^𝑎_ind()=1𝐤,𝜔𝜖()𝜌𝐤,𝜔−1^𝑎_ext(𝐤,𝜔). (67) Now, we consider a charge particle 𝑄^𝑎 moving with a constant velocity 𝐯 and interacting with the anisotropic plasma. The external charge density associated with the test charge particle can be written as [33, 34, 38] 𝜌^𝑎_ext=2𝜋𝑄^𝑎𝛿(𝜔−𝐤⋅𝐯). (68) The delta function indicates that the value of 𝜔 is real and the velocity of the charge particle is restricted between 0<𝑣<1, which is known as the Cerenkov condition for the moving parton in the medium. Therefore, the collective modes are determined in the space-like region of the 𝜔-𝑘 plane [34, 38]. According to Cerenkov condition, there will be two important scenarios which occur due to the interaction of the particle and the plasmon wave: first, the modes which are moving with a speed less than the average speed of the plasmon modes can be excited, but the particle moving slightly slower than the wave will be accelerated. While the charge particle moving faster than the wave will decrease its average velocity [69]. The slowly moving particle absorbs energy from the wave, the faster moving particle transfers its extra energy to the wave. The absorption and emission of energy result in a wake in the induced charge density as well as in the potential. Second, when the charge particle moving with a speed greater than the average phase velocity 𝑣_𝑝, the modes are excited and they may not be damped. Such excited modes can generate Cerenkov-like radiation and a Mach stem [70] which leads to oscillation both in the induced charge density and in the wake potential.

Substituting (68) into (67) and transforming into 𝐫-𝑡 space, the induced charge density becomes 𝜌^𝑎_ind(𝐫,𝑡)=2𝜋𝑄^𝑎𝑑³𝑘(2𝜋)³×𝑑𝜔2𝜋exp^{𝑖(𝐤⋅𝐫−𝜔𝑡)}1𝜖()().𝐤,𝜔−1×𝛿𝜔−𝐤⋅𝐯 (69) First, we consider the case where the fast parton is moving in the beam direction, that is, 𝐯‖𝐧. In spherical coordinate system 𝐤=(𝑘sin𝜃_𝑛cos𝜙,𝑘sin𝜃_𝑛sin𝜙,𝑘cos𝜃_𝑛) and the cylindrical coordinate for 𝐫=(𝜌,0,𝑧); therefore, the induced charge density can be written as 𝜌^𝑎_ind()=𝑄𝐫,𝑡^𝑎𝑚³_𝐷2𝜋^2∞₀𝑑𝑘𝑘²×¹₀𝑑𝜒𝐽₀𝑘𝜌1−𝜒²𝑚_𝐷×()cosΓRe𝜖𝐤,𝜔Δ()−1+sinΓIm𝜖𝐤,𝜔Δ_{𝜔=𝐤⋅𝐯}, (70) where 𝜒 is represented as cos𝜃_𝑛, 𝐽₀ is the zeroth-order Bessel function, Γ=𝑘𝜒(𝑧−𝑣𝑡)𝑚_𝐷, and Δ=(Re𝜖(𝐤,𝜔))²+(Im𝜖(𝐤,𝜔))². To get the previous equation, we use the simple transformation 𝜔→𝜔𝑚_𝐷 and 𝑘→𝑘𝑚_𝐷. It is seen that the charge density 𝜌^𝑎_ind is proportional to 𝑚³_𝐷.

Numerical evaluation of the previous equation leads to the contour plots of the induced charge density shown in Figure 10 with two different speeds of the fast parton. The contour plot of the equicharge lines shows a sign flip along the direction of the moving parton in Figure 10. The left (right) panel shows the contour plot of the induced color charge density in both isotropic and anisotropic plasma with parton velocity 𝑣=0.55(0.99). It is clearly seen that, because of anisotropy, the positive charge lines appear alternately in the backward space which indicates a small oscillatory behavior of the color charge wake (see Figure 10(c)). When the charge particle moves faster than the average plasmon speed, the induced charge density forms a cone-like structure which is significantly different from when the parton velocity is 𝑣=0.55. It is also seen that the induced charge density is oscillatory in nature. The supersonic nature of the parton leads to the formation of Mach cone and the plasmon modes could emit a Cerenkov-like radiation, which spatially limits the disturbances in the induced charge density [71]. In the backward space ((𝑧−𝑣𝑡)<0), induced color charge density is very much sensitive to the anisotropic plasma than that in the forward space ((𝑧−𝑣𝑡)>0). Due to the effect of the anisotropy, the color charge wake is modified significantly and the oscillatory behavior is more pronounced than the isotropic case. It is also seen that the oscillatory nature increases with the increase of the anisotropic parameter 𝜉.

(a)

(b)

(c)

(d)

Figure 10 

(Color online) Left Panel: the plot shows equicharge line with parton velocity 𝑣=0.55 for different 𝜉(0,0.8). Right Panel: same as left panel with parton velocity 𝑣=0.99.

Next, we consider the case when the parton moves perpendicular to the anisotropy direction in which case the induced charge density can be written as 𝜌^𝑎_ind()=𝑄𝐫,𝑡^𝑎𝑚³_𝐷2𝜋^2∞₀𝑑𝑘𝑘²×¹₀𝑑𝜒₀^2𝜋𝑑𝜙()2𝜋cosΩRe𝜖𝐤,𝜔Δ()−1+sinΩIm𝜖𝐤,𝜔Δ_{𝜔=𝐤⋅𝐯}, (71) with Ω=𝑘(𝑧𝜒+(𝜌−𝑣𝑡)1−𝜒²cos𝜙)𝑚_𝐷. Numerical results of the equicharge lines are shown in Figure 11. The left (right) panel shows the contour plots of the induced charge density for the parton velocity 𝑣=0.55(0.99). When 𝑣=0.99, the number of induced charge lines that appear alternately in the backward space is reduced for the anisotropic plasma in comparison to the isotropic plasma. Therefore, the anisotropy reduces the oscillatory behavior of the induced color charge density when the parton moves perpendicular to the anisotropy direction.

(a)

(b)

(c)

(d)

Figure 11 

(Color online) The left (right) panel shows the equicharge lines for 𝑣=0.55(0.99). In this case the parton moves perpendicular to the direction of anisotropy [38].

According to the Poisson equation, the wake potential induced by the fast parton reads as [33] Φ^𝑎()=𝜌𝐤,𝜔^𝑎ext(𝐤,𝜔)𝑘²𝜖().𝐤,𝜔 (72) Substituting (68) into (72) and transforming to the configuration space, the wake potential is given by [38] Φ^𝑎(𝐫,𝑡)=2𝜋𝑄^𝑎𝑑³𝑘(2𝜋)³𝑑𝜔2𝜋exp^{𝑖(𝐤⋅𝐫−𝜔𝑡)}1𝑘²𝜖()𝛿().𝜔,𝐤𝜔−𝐤⋅𝐯 (73) Using similar coordinate system as before, the screening potential turns into Φ^𝑎()=𝑄𝐫,𝑡^𝑎𝑚_𝐷2𝜋^2∞₀×𝑑𝑘₁₀𝑑𝜒𝐽₀𝑘𝜌1−𝜒²𝑚_𝐷×()cosΓRe𝜖𝜔,𝐤Δ()+sinΓIm𝜖𝜔,𝐤Δ_{𝜔=𝐤⋅𝐯}. (74) We solve the wake potential for the two special cases: (i) along the parallel direction of the fast parton, that is, 𝐫‖𝐯 and also 𝜌=0 and (ii) perpendicular to direction of the parton, that is, 𝐫⟂𝐯. The potential for the parallel case is obtained as Φ^𝑎_‖()=𝑄𝐫,𝑡^𝑎𝑚_𝐷2𝜋^2∞₀×𝑑𝑘₁₀()𝑑𝜒cosΓRe𝜖𝜔,𝐤Δ()+sinΓIm𝜖𝜔,𝐤Δ_{𝜔=𝐤⋅𝐯}, (75) whereas that for the perpendicular case, we have Φ^𝑎_⟂()=𝑄𝐫,𝑡^𝑎𝑚_𝐷2𝜋^2∞₀×𝑑𝑘₁₀𝑑𝜒𝐽₀𝑘𝜌1−𝜒²𝑚_𝐷×cosΓ^′Re𝜖(𝜔,𝐤)Δ−sinΓ^′Im𝜖(𝜔,𝐤)Δ_{𝜔=𝐤⋅𝐯}, (76) with Γ^′=𝑘𝜒𝑣𝑡𝑚_𝐷.

Figure 12 describes the scaled wake potential in two specified directions. In these figures, the scaled parameter Φ^𝑎₀ is given by (2𝜋²/𝑚_𝐷)Φ^𝑎. The left panel shows the wake potential along the direction of the moving color charge. In the backward direction, the wake potential for isotropic plasma decreases with the increase of 𝑧−𝑣𝑡 and exhibits a negative minimum when 𝑣=0.55. With the increase of the anisotropic parameter 𝜉, the depth of the negative minimum decreases for 𝑣=0.55. At large 𝜉(0.8), there is no negative minimum and the wake potential behaves like a modified Coulomb potential. For 𝑣=0.99, the wake potential is Lennard-Jones potential type which has a short range repulsive part as well as a long range attractive part [34, 38] in both isotropic and anisotropic plasma. It is also seen that the wake potential is oscillatory in nature in the backward direction. It is clearly visible that the depth of the negative minimum is increased compared to the case when 𝑣=0.55. Because of the anisotropy effect, the oscillation of the wake potential is more pronounced, and it extends to a large distance [38]. In the forward direction, the screening potential is a modified Coulomb potential in both types of plasma. Figure 12(b) describes the wake potential along the perpendicular direction of the moving parton. It can be seen that the wake potential is symmetric in backward and forward direction, no matter what the speed is [38]. In presence of the moving charge particle, the wake potential is Lennard-Jones type. When 𝑣=0.55, the value of negative minimum is increased with increase of 𝜉 but in case of 𝑣=0.99, it decreases with 𝜉. However, with the increase of 𝜉, the depth of negative minimum is moving away from the origin for both the jet velocities considered here.

(a)

(b)

Figure 12 

(Color online) (a): scaled wake potential along the motion of the fast parton, that is, 𝑧-axis for different 𝜉 with two different parton velocity 𝑣=0.55 and 𝑣=0.99. (b) same as (a) but perpendicular to direction of motion of the parton.

Next, we consider the case when the parton moves perpendicular to the beam direction. The wake potential in (74) is also evaluated for the two spacial cases: (i) along the direction of the moving parton, that is, 𝐫‖𝐯 and (ii) perpendicular direction of the parton, that is, 𝐫⟂𝐯. The wake potential for the parallel case can be written as [38] Φ^𝑎_‖()=𝑄𝐫,𝑡^𝑎𝑚_𝐷2𝜋^2∞₀𝑑𝑘¹₀×𝑑𝜒_02𝜋𝑑𝜙×2𝜋cosΩ^′Re𝜖(𝜔,𝐤)Δ+sinΩ^′Im𝜖(𝜔,𝐤)Δ_{𝜔=𝐤⋅𝐯}, (77) where Ω^′=𝑘(𝜌−𝑣𝑡)1−𝜒²cos𝜙𝑚_𝐷. For the perpendicular case it is given by Φ^𝑎_⟂()=𝑄𝐫,𝑡^𝑎𝑚_𝐷2𝜋^2∞₀𝑑𝑘¹₀×𝑑𝜒_02𝜋𝑑𝜙×2𝜋cosΩ^′′Re𝜖(𝜔,𝐤)Δ+sinΩ^′′Im𝜖(𝜔,𝐤)Δ_{𝜔=𝐤⋅𝐯}, (78) with Ω^′′=𝑘(𝑧𝜒−𝑣𝑡1−𝜒²cos𝜙)𝑚_𝐷. The left panel in Figure 13 shows screening potential along the parallel direction of the moving color charge. The behavior of the wake potential is more like a modified Coulomb (Lennard-Jones) potential at parton velocity 𝑣=0.55(0.99). For 𝑣=0.99, the wake potential shows an oscillatory behavior in an isotropic plasma [34] but in anisotropic case, oscillatory structure of the wake potential is smeared out for 𝜉=0.5 and 𝐯⟂𝐧 [38]. But the depth of the negative minimum increases in the case of anisotropic plasma for both the parton velocities considered here. The behavior of the wake potential in the perpendicular direction of the moving parton is shown in the right panel of Figure 13. At 𝑣=0.55, the anisotropy modifies the structure of the wake potential significantly, that is, it becomes modified Coulomb potential instead of Lennard-Jones potential. For 𝑣=0.99, the wake potential is a Lennard-Jones potential type but the depth of the minimum decreases in anisotropic plasma [38].

(a)

(b)

Figure 13 

(Color online) The left (right) panel shows scaled wake potential for 𝜉={0,0.5} with parton velocity 𝑣=0.55(0.99). In this case the parton moves perpendicular to the direction of anisotropy [38].

6. Summary and Discussions

We have reviewed the effect of initial state momentum anisotropy that can arise in an AQGP on various observables. It is shown that electromagnetic probes could be a good signal that can be used to characterize this anisotropic state as this can only be realized in the early stages of heavy ion collisions. It has been demonstrated that the isotropization time of the QGP can be extracted by comparing the photon yield with the experimental data. We further estimate the radiative energy loss of a fast moving parton (both heavy and light flavours) in an AQGP and show that the it is substantially different from that in the isotropic QGP. Moreover, it depends on the direction of propagation of the parton with the anisotropic axis. Related to this is the nuclear modification factor of light hadrons that is produced due to the fragmentation of light partons which lose energy in the medium. Thus,we have also discussed the nuclear modification factor in the context of AQGP and compared it with the RHIC data to extract the isotropization time. The extracted value is compatible with that obtained from photon data.

It might be mentioned here that the presence of unstable modes in AQGP may affect radiative energy loss. However, in [72, 73] the authors have shown that the polarization loss (collisional loss) remains unaffected by the unstable modes, but in a recent paper [74] it is shown that the polarisation loss indeed has strong time and directional dependence and also the nature of the loss is oscillatory. Such effect may be present in radiative energy loss. In this review, we did not consider it.

The heavy quark potential and the quarkonium states in AQGP have also been reviewed with both real and complex valued potential. In all these calculations, it has been found that the dissociation temperature of various quarkonium states increases in comparison with the isotropic case. We have also focused on the nuclear modification factors of various bottomonium states which have been calculated by combining hydrodynamics and solutions of 3D Schrodinger equation using two types of complex valued potentials.

Apart from the energy loss of a jet in a medium, the jet also creates wake in the plasma. We have demonstrated that due to the jet propagation in an AQGP, the wake potential and the charge density are significantly modified in comparison with the isotropic case.

We end by mentioning that the ADS/CFT calculation of the electromagnetic correlator has been performed in strongly coupled 𝒩=4 super Yang-Mills theory using anisotropic momentum distribution [75]. Photon production rate is then estimated and it is concluded that in the weak coupling limit, the rate is consistent with that in [41] with an oblate phase space distribution in momentum space. There are other models that deal with the gravity dual theory for anisotropic plasma with additional bulk fields [76, 77]. Thus, a comparative study of various observables in gravity dual theory should be done in the future.