# Effect of magnetic field on jet transport coefficient $\hat{q}$

Debjani Banerjee<EMAIL_ADDRESS>Department of Physics, Bose
Institute, EN 80, Sector-V, Salt Lake, Kolkata 700091, West Bengal, India
Souvik Paul<EMAIL_ADDRESS>Department of Physical Sciences,
Indian Institute of Science Education and Research, Kolkata, Mohanpur 741246,
West Bengal, India  Prottoy Das<EMAIL_ADDRESS>Department of
Physics, Bose Institute, EN 80, Sector-V, Salt Lake, Kolkata 700091, West
Bengal, India  Abhi Modak<EMAIL_ADDRESS>Department of Physics, Bose
Institute, EN 80, Sector-V, Salt Lake, Kolkata 700091, West Bengal, India
Ankita Budhraja<EMAIL_ADDRESS>Tata Institute of
Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, Maharashtra,
India  Sabyasachi Ghosh<EMAIL_ADDRESS>Indian Institute of Technology
Bhilai, GEC Campus, Sejbahar, Raipur 492015, Chhattisgarh, India  Sidharth K.
Prasad<EMAIL_ADDRESS>Department of Physics, Bose Institute, EN 80,
Sector-V, Salt Lake, Kolkata 700091, West Bengal, India

###### Abstract

We report the effect of magnetic field on estimation of jet transport
coefficient, $\hat{q}$ using a simplified quasi-particle model. Our adopted
quasi-particle model introduces temperature and magnetic field dependent
degeneracy factors of partons, which are tuned by fitting the magneto-
thermodynamical data of lattice quantum chromodynamics. In absence of magnetic
field, $\hat{q}$ is estimated by using the temperature dependent degeneracy
factor. At finite magnetic field, ${\hat{q}}$ splits into parallel and
perpendicular components, whose magnetic field dependent part has two sources.
One is field dependent degeneracy factor and another is phase space part,
guided from shear viscosity to entropy density ratio. Their collective role
provides an enhanced jet transport coefficients, which should be considered in
detailed jet quenching phenomenology in presence of magnetic field.

###### pacs:

25.75.-q,12.38.Mh

##  I Introduction

In high energy hadron-hadron, hadron-nucleus and nucleus-nucleus collisions,
partons produced in large-momentum transfer (large $\rm Q^{\rm 2}$) processes,
undergo multiple fragmentations (forming shower of partons) and form a
collimated spray of energetic colorless hadrons (due to color charge
confinement property of quantum chromodynamics (QCD)) known as jets Connors et
al. (2018). The production and properties of jets in vacuum (e.g. in proton-
proton (pp) and in proton-nucleus (pA)) can be very well understood within the
theoretical framework of perturbative quantum chromodynamics (pQCD) et al.
(2015, 2011a, 2013); Abe et al. (1993); et al. (2011b); Adam et al. (2016).
Jets are produced very early in the collision due to their small formation
time Aref’eva (2015); Chatterjee and Srivastava (2009). In nucleus-nucleus
(AA) collisions, where a thermalized (expanding) medium - quark gluon plasma
(QGP) is produced, these jets lose their energy via elastic and inelastic
interactions with the medium partons while passing through it resulting in the
suppression of high energy jets in AA collisions relative to that in pp and pA
collisions. The phenomenon is known as jet quenching Gyulassy and Plumer
(1990); Wang and Gyulassy (1992); Majumder and Van Leeuwen (2011); Burke et
al. (2014) and can be quantified experimentally by a well known observable,
nuclear modification factor $R_{AA}$, defined as the ratio of high momentum
inclusive hadron or jet yields in AA collisions to that in $N_{coll}$ (number
of binary collisions) scaled pp collisions. The suppression of high momentum
hadrons and jets are experimentally measured previously by experiments at SPS,
RHIC and recently by experiments at LHC Laszlo (2011); et al. (2010);
Chatrchyan et al. (2012a); Khachatryan et al. (2016); Abelev et al. (2014).
Within the framework of theoretical models, jet quenching can be studied to
extract and put constrain on the jet transport coefficient ${\hat{q}}$ of the
QGP which is defined as the mean square of the momentum transfer between the
propagating hard jet and the soft medium per unit length.

There are different theoretical models Burke et al. (2014), popularly known as
GLV-CUJET Gyulassy et al. (2001); Vitev and Gyulassy (2002); Buzzatti and
Gyulassy (2012), MARTINI Schenke et al. (2009), MCGILL-AMY Qin et al. (2008),
HT-M Majumder and Van Leeuwen (2011); Majumder (2012), HT-BW Guo and Wang
(2000); Wang and Guo (2001); Chen et al. (2010), which have well explored this
jet quenching phenomena by calculating the theoretical quantity ${\hat{q}}$
and then proceed for experimental quantity $R_{AA}$. In evolution of bulk
medium, different models follow different strategy but for jet-medium
interaction, they have considered pQCD calculations. In GLV-CUJET model
Gyulassy et al. (2001); Vitev and Gyulassy (2002); Buzzatti and Gyulassy
(2012), an effective quark-gluon cross section in terms of strong coupling
constant, electric and magnetic screening mass is used. MARTINI Schenke et al.
(2009) and MCGILL-AMY Qin et al. (2008) models have considered hard thermal
loop (HTL) based resummed elastic collision rate. HT-M Majumder and Van
Leeuwen (2011); Majumder (2012), HT-BW Guo and Wang (2000); Wang and Guo
(2001); Chen et al. (2010) models grossly assume ${\hat{q}}$ as proportional
to (local) entropy or number density by normalizing with its cold nuclear
matter value ${\hat{q}}_{N}\approx 0.02$ GeV${}^{2}/$fm Airapetian et al.
(2007); Deng and Wang (2010) and its initial values ${\hat{q}}_{0}$. To
explain PHENIX data Adare et al. (2008), Ref. Chen et al. (2010) has found
${\hat{q}}_{0}\approx 0.9$ GeV${}^{2}/$fm. Other possible values Burke et al.
(2014) are found as ${\hat{q}}_{0}\approx 1.2$ GeV${}^{2}/$fm to explain
PHENIX data Adare et al. (2008, 2013)[77,78] and ${\hat{q}}_{0}\approx 2.2$
GeV${}^{2}/$fm to explain combined ALICE Abelev et al. (2013) and CMS
Chatrchyan et al. (2012b). Collecting those different possible values of
${\hat{q}}$, we will get a numerical range Burke et al. (2014), which may be
well accepted.

The present article is intended to explore the role of magnetic field on this
jet quenching parameter as RHIC or LHC matter may face $eB=1-10m_{\pi}^{2}$
magnetic field in non-central collisions Tuchin (2013). Hence, without going
to the microscopic calculation of ${\hat{q}}$ in absence of magnetic field,
here we have calibrated it with existing known values and then extended it for
finite magnetic field picture. Following Ref. Chen et al. (2010), firstly we
have attempted to map the transition from ${\hat{q}}_{0}$ to ${\hat{q}}_{N}$
through a quasi-particle type description Satapathy et al. (2020), based on
lattice QCD (LQCD) thermodynamics Borsányi et al. (2014); Bazavov et al.
(2014). Latter it is extended for finite magnetic field picture based on the
LQCD magneto-thermodynamics Bali et al. (2012, 2014). Similar kind of magnetic
field extension of quasi-particle models is addressed by Refs. Li et al.
(2019); Tawfik et al. (2018, 2016); Farias et al. (2017), based on effective
QCD models. In that context, our approach is not at all so rich to deal with
effective dynamics but an easy-dealing parametric based model to map LQCD
magneto-thermodynamics, which has revealed inverse magnetic catalysis (IMC)
property of quark-hadron phase transition in presence of magnetic field. To
add anisotropic phase-space part at finite temperature $T$ and magnetic field
$B$ into the ${\hat{q}}$, we take the help of shear viscosity $\eta$
expression in presence of magnetic field since ${\hat{q}}$ and $\eta$ can be
connected Majumder et al. (2007); Casalderrey-Solana and Wang (2008); Xu et
al. (2015, 2016); Shuryak (2017); Ayala et al. (2016). Our investigation
focuses mainly on observing the changes in the value of jet quenching
parameter due to magnetic field, which is found to be quite noticeable.

The present article is organized as follows. Sec. II describes the mapping of
LQCD thermodynamic quantities in presence of magnetic field by introducing
temperature and magnetic field dependent degeneracy factor. The formalism to
estimate jet quenching parameter using quasi-particle description is discussed
in Sec. III. In Sec. IV, we have attempted to calculate magnetic field
dependent jet quenching parameter using the knowledge of temperature and
magnetic field dependent phase space part of shear viscosity. The obtained
results in this work are discussed in Sec. V and the comparison with non-
perturbative toolbox of AdS/CFT results is outlined in Sec. VI. Finally, we
conclude with a summary in Sec. VII.

##  II Mapping LQCD magneto-thermodynamics

Figure 1:  Top 3 panels: LQCD data points (with error bars) Bali et al. (2014)
and fitted curves using Eq. 7 (solid line) of normalized entropy density,
normalized by its SB limit ($s_{\rm SB}$) vs $T$ at $eB=0$ (blue line),
$eB=0.1$ GeV2 (green line) and $eB=0.2$ GeV2 (cyan line). Bottom panel:
Corresponding $T$ dependent fraction g(T) (Eq. 7) at different $eB$ (to be
multiplied with total degeneracy factor).

This section is aimed to map LQCD thermodynamics, which will be used to
calculate ${\hat{q}}$ in Sec. III. For mapping LQCD thermodynamics, we will
follow the steps of Ref. Satapathy et al. (2020), which is attempted to map
LQCD thermodynamics of Refs. Borsányi et al. (2014); Bazavov et al. (2014).
Here, we will attempt to map LQCD thermodynamics in presence of magnetic
field, given by Ref. Bali et al. (2014), but following the same steps, adopted
in Ref. Satapathy et al. (2020).

In terms of Fermi-Dirac (FD) distribution function of quarks and Bose-Einstein
(BE) distribution function of gluons, the energy density and pressure of QGP
system can be expressed as

$\epsilon_{QGP}=g_{g}\int_{0}^{\infty}\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{\omega_{g}}{e^{\beta\omega_{g}}-1}\\\
+g_{Q}\int_{0}^{\infty}\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{\omega_{Q}}{e^{\beta\omega_{Q}}+1}$
(1)

and

$P_{QGP}=g_{g}\int_{0}^{\infty}\frac{d^{3}\vec{k}}{(2\pi)^{3}}\Big{(}\frac{\vec{k}^{2}}{3\omega_{g}}\Big{)}\frac{1}{e^{\beta\omega_{g}}-1}\\\
+g_{Q}\int_{0}^{\infty}\frac{d^{3}\vec{k}}{(2\pi)^{3}}\Big{(}\frac{\vec{k}^{2}}{3\omega_{Q}}\Big{)}\frac{1}{e^{\beta\omega_{Q}}+1}$
(2)

where $\omega_{Q}$, $\omega_{g}$ are energies and

$\displaystyle g_{Q}=(\rm spin)\times(\rm particle/antiparticle)\times(\rm
color)\times(\rm flavor)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=2\times 2\times
3\times 3=36,$ $\displaystyle g_{g}=(\rm spin)\times(\rm flavor)=2\times
8=16,$ (3)

are degeneracy factors of quarks and gluons respectively. At zero net quark
density or quark chemical potential, entropy density can be expressed in terms
of $P$ and $\epsilon$ as

$s_{QGP}=\frac{P_{QGP}+\epsilon_{QGP}}{T}~{}.$ (4)

For massless QGP system,

$\displaystyle P_{QGP}$ $\displaystyle=$
$\displaystyle\Big{[}g_{g}+g_{Q}\Big{(}\frac{7}{8}\Big{)}\Big{]}\frac{\pi^{2}}{90}T^{4}\approx
5.2~{}T^{4}~{},$ $\displaystyle\epsilon_{QGP}$ $\displaystyle=$
$\displaystyle\Big{[}g_{g}+g_{Q}\Big{(}\frac{7}{8}\Big{)}\Big{]}\frac{3\pi^{2}}{90}T^{4}\approx
15.6~{}T^{4}~{},$ $\displaystyle s_{QGP}$ $\displaystyle=$
$\displaystyle\Big{[}g_{g}+g_{Q}\Big{(}\frac{7}{8}\Big{)}\Big{]}\frac{4\pi^{2}}{90}T^{3}\approx
20.8~{}T^{3}~{}.$ (5)

Now, if we see the LQCD data of $P(T)$, $\epsilon(T)$, $s(T)$ at $eB=0$ from
Ref. Bali et al. (2014), then one can notice that the data points always
remain lower than their massless limits i.e. $\frac{P}{T^{4}}<5.2$,
$\frac{\epsilon}{T^{4}}<15.6$, $\frac{s}{T^{4}}<20.8$. A rich QCD interaction
in non-perturbative domain is responsible for this suppression and it is LQCD
calculation that provides useful insights in this domain, where pQCD does not
work well. One can map this temperature ($T$) dependent suppression of $P(T)$,
$\epsilon(T)$, $s(T)$ Satapathy et al. (2020) by imposing a temperature
dependent fraction $g(T)$, multiplied with the total degeneracy factor of QGP.
So, this picture assumes that as we go from high temperature to low
temperature, the degeneracy factor of massless QGP system is gradually
reduced.

Now, let us proceed with a finite magnetic field picture. In presence of
magnetic field along z-axis, we can get an anisotropy in pressure but if we
consider the pressure along z-direction ($P_{z}$), then the thermodynamic
relation

$s=\frac{\epsilon+P_{z}}{T}$ (6)

remains same Bali et al. (2012). So, the earlier expressions can still be used
for finite $B$ picture by introducing a $T$, $B$ dependent parameter $g(T,B)$.
By matching LQCD data of Bali et. al. Bali et al. (2012, 2014), who have
provided $P(T,B)$, $\epsilon(T,B)$, $s(T,B)$ one can get the parametric form
of $g(T,B)$ as

$g(T,B)=a_{0}-\frac{a_{1}}{e^{a_{2}(T-0.17)}+a_{3}}~{},$ (7)

where values of $a_{0,1,2,3}$ for different $eB$’s are given in Table. 1.

Table 1: Different values of $a_{0,1,2,3}$ given in Eq. (7) for different magnetic field strengths $\mathbf{eB(GeV^{2})}$ | $\mathbf{a_{0}}$ | $\mathbf{a_{1}}$ | $\mathbf{a_{2}}$ | $\mathbf{a_{3}}$
---|---|---|---|---
0.0 | -0.26 | -1.49 | -21.58 | 1.49
0.1 | -0.16 | -1.20 | -22.61 | 1.33
0.2 | -0.47 | -2.82 | -16.51 | 2.26

Now, an exact (quantitative) matching of all the LQCD thermodynamical
quantities through a single parameter tuning is probably an impossible task.
In this context, our mapping might not be considered so robust but can surely
be considered as a simplified and easy-dealing tool. Since it is difficult for
exact fitting of $P$, $\epsilon$ and $s$ simultaneously via a single set of
$g(T,B)$, therefore, we have to focus on any one of them for exact fitting,
doing which a little bad fitting of the remaining thermodynamical quantities
can be expected. Among all the thermodynamical quantities, we have chosen
entropy density as one of our aims is to estimate $\eta/s$ later.

##  III Jet quenching parameter

Following Ref. Chen et al. (2010), the jet quenching parameter of hadronic
matter can be proportionally related to its density as:

$\hat{q}(T)=\frac{\hat{q_{N}}}{\rho_{N}}\rho_{h}(T),$ (8)

where ${\hat{q}}_{N}\approx 0.02$ GeV2/fm Airapetian et al. (2007); Deng and
Wang (2010) is the jet transport parameter at the center of the cold nuclear
matter in a large nucleus, $\rho_{N}=0.17$ fm-3 is nuclear saturation density
and $\rho_{h}$ is hadronic matter density.

Figure 2: Temperature dependence of ${\hat{q}}/T^{3}$ by using Ref. Chen et
al. (2010) (red solid line), Eq. (LABEL:qh_rho_qN) (green solid line) and Eq.
(LABEL:qh_rho_q0) (green dash line) for $eB=0$. ${\hat{q}}/T^{3}$ by using
$g(T,eB=0.2$ GeV${}^{2})$ in Eq. (LABEL:qh_rho_qN) (blue solid line) and Eq.
(LABEL:qh_rho_q0) (blue dash line) for $eB=0.2$ GeV2.

In Ref. Chen et al. (2010), $\rho_{h}(T)$ is obtained from hadron resonance
gas (HRG) model within the hadronic temperature range. As an alternative way,
here we have replaced $\rho_{h}(T)$ by $\rho_{G}(T)$, which is density of
gluon and it can be extended beyond quark-hadron transition temperature. The
$\rho_{G}(T)$ can be calculated by using $g(T)$, as we did for other
thermodynamical quantities for QGP. Similar to Eqs (1), (2) etc., the gluon
number density $\rho_{G}$ can be expressed as

$\rho_{G}=g_{g}\int_{0}^{\infty}\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{1}{e^{\beta\omega_{g}}-1},$
(9)

whose massless limit will be

$\displaystyle\rho_{G}$ $\displaystyle=$
$\displaystyle\Big{[}g_{g}\frac{\zeta(3)}{\pi^{2}}\Big{]}T^{3}$ (10)
$\displaystyle=$ $\displaystyle 1.94~{}T^{3}~{}.$

When this massless gluon density is multiplied with $g(T)$, then one may get a
rough estimation of gluon density in non-pQCD domain. So, Eq. (8) can be
converted as

$\displaystyle{\hat{q}}(T)$ $\displaystyle=$
$\displaystyle\frac{{\hat{q}}_{N}}{\rho_{N}}\times g(T)\times
g_{g}\frac{\zeta(3)}{\pi^{2}}~{}T^{3}$ $\displaystyle=$ $\displaystyle
3.03\times\Big{[}a_{0}-\frac{a_{1}}{e^{a_{2}(T-0.17)}+a_{3}}\Big{]}\times
1.94~{}T^{3}$

The above equation is calibrated by cold nuclear matter data and journey is
from hadronic (low) to QGP (high) temperature domain. The information of LQCD
thermodynamics carried via $g(T)$ may result in a cross-over type transition
in ${\hat{q}}(T)$. The green solid line of Fig. (2) exposes it when we compare
to first order phase transition type or discrete type curve taken from Ref.
Chen et al. (2010). Since Ref. Chen et al. (2010) has considered ${\hat{q}}$
in hadronic temperature range with density from hadron resonance gas (HRG)
model and ${\hat{q}}_{0}\approx 0.9$ GeV2/fm (jet transport parameter at
initial time of QGP) in quark temperature range, therefore, a discrete pattern
in ${\hat{q}}/T^{3}$ is seen. One can find that HRG estimation of Ref. Chen et
al. (2010) and our estimation in hadronic temperature range are quite
comparable. However, our extended estimation in quark temperature range
saturates around 4.5 ($\sim T=0.370$ GeV), which is close to the corresponding
initial value – ${\hat{q}}_{0}/T^{3}=0.9\times 0.197/(0.370^{3})\approx 3.46$
Chen et al. (2010).

Next, a reverse journey from high to low temperature domain is also possible,
where ${\hat{q}}_{0}\approx 0.9$ GeV2/fm at initial time will be our reference
point. Assuming an equivalence between ${\hat{q}}_{0}$ and massless limit of
${\hat{q}}$:

$\displaystyle{\hat{q}}_{m=0}$ $\displaystyle=$
$\displaystyle\lim_{m\rightarrow
0}{\hat{q}}(T)=\frac{{\hat{q}}_{N}}{\rho_{N}}16\frac{\zeta(3)}{\pi^{2}}~{}T^{3}$
(12) $\displaystyle\approx$ $\displaystyle 5.87\times T^{3},$

we can redefine Eq. (8) as

$\displaystyle{\hat{q}}(T)$ $\displaystyle=$
$\displaystyle\frac{{\hat{q}}_{N}}{\rho_{N}}\rho_{h}(T)\Big{(}\frac{{\hat{q}}_{0}}{{\hat{q}}_{m=0}}\Big{)}$
$\displaystyle=$ $\displaystyle g(T)\times{\hat{q}}_{0}$ $\displaystyle=$
$\displaystyle\Big{[}a_{0}-\frac{a_{1}}{e^{a_{2}(T-0.17)}+a_{3}}\Big{]}\times
0.9~{}{\rm GeV}^{2}/{\rm fm}~{}.$

The green dash line curve in Fig. (2), based on Eq. (LABEL:qh_rho_q0),
saturates below the pQCD-based value ${\hat{q}}_{0}/T^{3}$. One can find a
similar kind of picture where LQCD values of different thermodynamical
quantities always saturate around smaller values than their corresponding SB
limits or leading order pQCD values. This equivalence between LQCD
thermodynamics and jet quenching parameter is coming through the concept of
temperature dependent degeneracy factor Satapathy et al. (2020).

So, either by using Eq. (LABEL:qh_rho_qN) or Eq. (LABEL:qh_rho_q0), one can go
for a rough estimation of ${\hat{q}}(T)$. Interestingly, ${\hat{q}}/T^{3}$
from Eq. (LABEL:qh_rho_q0) shows a mild peak structure near transition
temperature, as shown by green dash line in Fig. (2). Analyzing $T$ dependency
of Eq. (LABEL:qh_rho_qN) and Eq. (LABEL:qh_rho_q0), one can recognize
${\hat{q}}\propto g(T)\times T^{3}$ and ${\hat{q}}\propto g(T)$ respectively,
so their normalizing function will be $\frac{\hat{q}}{T^{3}}\propto g(T)$ and
$\frac{\hat{q}}{T^{3}}\propto\frac{g(T)}{T^{3}}$ respectively. It is due to
the collective roles of increasing $g(T)$ and decreasing $\frac{1}{T^{3}}$,
the peak structure is forming for the latter case.

One can also be able to estimate ${\hat{q}}(T,B)$ by using finite $B$
extension of our quasi-particle model, which has fitted LQCD IMC data via a
simple parametric form of $T$ and $B$ dependent degeneracy factor $g(T,B)$.
Replacing $g(T)$ by $g(T,B)$ in Eq. (LABEL:qh_rho_qN) and Eq.
(LABEL:qh_rho_q0) for $eB=0.2$ GeV2, we have obtained blue solid and dash
curves respectively. However, this straight forward extension in finite $B$
case might not be a complete picture. Similar to other transport coefficients,
jet transport parameter can have a $T$, $B$ dependent phase space part, which
is not yet considered. In the next section, we have attempted to include this
structure.

##  IV Phase space component

According to Refs. Majumder et al. (2007); Casalderrey-Solana and Wang (2008);
Xu et al. (2015, 2016); Shuryak (2017), jet quenching parameter is connected
with the shear viscosity coefficient. Here, we attempt to calculate
${\hat{q}}(T,B)$ from the knowledge of $\eta(T,B)$ profile. The advantage of
this approach lies in the fact that the $T$, $B$ dependent phase space part of
$\eta(T,B)$ are well studied in recent Refs. Lifshitz and L.P. (1981); Huang
et al. (2010, 2011); Tuchin (2012); Li and Yee (2018); Mohanty et al. (2019);
Ghosh et al. (2019); Nam and Kao (2013); Dey et al. (2019a); Dash et al.
(2020); Das et al. (2019); Denicol et al. (2018); Chen et al. (2020); Dey et
al. (2019b) . So, based on that knowledge, we have invoked the $T$, $B$
dependent phase space part into ${\hat{q}}(T,B)$.

Let us take a quick revisit of shear viscosity expressions at finite
temperature and magnetic field. We first consider the case of zero magnetic
field, then we move to the non-zero magnetic field picture. According to the
macroscopic fluid definition, shear viscosity $\eta$ is the proportionality
constant between viscous stress tensor $\pi_{ij}$ and velocity gradient tensor
$C_{ij}$ Dey et al. (2019b), i.e.

$\pi^{ij}=\eta C_{ij}~{},$ (14)

which is relativistic and tensor form of the so-called Newton’s law of
viscosity. In microscopic kinetic theory approach, the viscous stress tensor
can be connected with the deviation (from equilibrium distribution function
$f_{0}$) $\delta f=Ck^{k}k^{l}C_{kl}\beta f_{0}(1\mp f_{0})$ as

$\displaystyle\pi^{ij}$ $\displaystyle=$ $\displaystyle
g\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{i}k^{j}}{\omega}\delta f$ (15)
$\displaystyle=$ $\displaystyle
g\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{i}k^{j}}{\omega}Ck^{k}k^{l}C_{kl}\beta
f_{0}(1-af_{0})~{},$

where $f_{0}=1/[e^{\beta\omega}+a]$ denotes the Fermi-Dirac (FD) and Bose-
Einstein (BE) distribution functions for $a=\pm 1$ respectively and
$\omega=\sqrt{\vec{k}^{2}+m^{2}}$ is the energy of medium constituent with
mass $m$ and degeneracy factor $g$.

Connecting the macroscopic Eq. (14) and microscopic Eq. (15), we can get the
shear viscosity tensor:

$\eta^{ijkl}=g\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{k^{i}k^{j}k^{k}k^{l}}{\omega^{2}}\tau_{c}\beta
f_{0}(1-af_{0})~{},$ (16)

whose isotropic expression will be

$\eta=\frac{g}{15}\int\frac{d^{3}\vec{k}}{(2\pi)^{3}}\frac{\vec{k}^{4}}{\omega^{2}}\tau_{c}\beta
f_{0}(1-af_{0})~{},$ (17)

The unknown constant $C$ in Eq. (15) has been obtained in terms of relaxation
time $\tau_{c}$ with the help of relaxation time approximation (RTA) based
relativistic Boltzmann equation (RBE) Dey et al. (2019b).

Now, in presence of magnetic field, instead of a single traceless velocity
gradient tensor $C_{ij}$, we get 5 independent traceless tensors - one set is
proposed in Ref. Lifshitz and L.P. (1981) and another set is proposed by Refs.
Huang et al. (2010, 2011). Therefore, we get two sets of 5 shear viscosity
components ${\tilde{\eta}}_{0,1,2,3,4}$ Dey et al. (2019b); Mohanty et al.
(2019); Dash et al. (2020) and $\eta_{0,1,2,3,4}$ Denicol et al. (2018); Dey
et al. (2019b). They are interconnected and the two main components with
respect to the direction of applied magnetic field are as follows:

$\displaystyle\frac{\eta_{\parallel}}{\eta}$ $\displaystyle=$
$\displaystyle\frac{{\tilde{\eta}}_{2}}{\eta}=\frac{(\eta_{0}+\eta_{2})}{\eta}=\frac{1}{1+(\tau_{c}/\tau_{B})^{2}}$
$\displaystyle\frac{\eta_{\perp}}{\eta}$ $\displaystyle=$
$\displaystyle\frac{{\tilde{\eta}}_{1}}{\eta}=\frac{\eta_{0}}{\eta}=\frac{1}{1+4(\tau_{c}/\tau_{B})^{2}}~{},$
(18)

where another time scale $\tau_{B}=\frac{\omega_{av}}{qB}$ ($\omega_{av}$ is
the average energy) will come into the picture along with the relaxation time
$\tau_{c}$. In massless limit,

$\displaystyle\omega_{av}$ $\displaystyle=$
$\displaystyle\Big{\\{}\frac{\zeta(4)}{\zeta(3)}\Big{\\}}3T~{}~{}{\rm
for~{}BE}$ (19) $\displaystyle=$
$\displaystyle\Big{\\{}\frac{7\zeta(4)}{2\zeta(3)}\Big{\\}}3T~{}~{}{\rm
for~{}FD}~{},$

and

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{4g\>\tau_{c}}{5\pi^{2}}\zeta(4)\>T^{4}~{}~{}{\rm
for~{}BE}$ (20) $\displaystyle=$
$\displaystyle\Big{(}\frac{7}{8}\Big{)}\frac{4g\>\tau_{c}}{5\pi^{2}}\zeta(4)\>T^{4}~{}~{}{\rm
for~{}FD}~{}.$

Now, for massless 3 flavor QGP at $B=0$, we can get

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\Big{[}16+\frac{7}{8}36\Big{]}\frac{4\>\tau_{c}}{5\pi^{2}}\zeta(4)\>T^{4}~{},$
$\displaystyle s$ $\displaystyle=$
$\displaystyle\Big{[}16+\frac{7}{8}36\Big{]}\frac{4\>\tau_{c}}{\pi^{2}}\zeta(4)\>T^{3}~{},$
$\displaystyle\Rightarrow\frac{\eta}{s}$ $\displaystyle=$
$\displaystyle\frac{\tau_{c}T}{5}~{}.$ (21)

For $B\neq 0$, we get

$\displaystyle\frac{\eta_{\parallel}}{s}$ $\displaystyle=$
$\displaystyle\frac{1}{\Big{[}16+\frac{7}{8}36\Big{]}}\Big{[}16+\frac{7}{8}12\sum_{u,d,s}\frac{1}{1+(\tau_{c}/\tau_{B})^{2}}\Big{]}\frac{\tau_{c}T}{5}$
$\displaystyle\frac{\eta_{\perp}}{s}$ $\displaystyle=$
$\displaystyle\frac{1}{\Big{[}16+\frac{7}{8}36\Big{]}}\Big{[}16+\frac{7}{8}12\sum_{u,d,s}\frac{1}{1+4(\tau_{c}/\tau_{B})^{2}}\Big{]}\frac{\tau_{c}T}{5}~{}.$

Now if we roughly connect ${\hat{q}}\propto s/\eta$ at $B=0$ and
${\hat{q}}_{\parallel,\perp}\propto s/\eta_{\parallel,\perp}$ at $B\neq 0$,
then we may write:

$\displaystyle\frac{{\hat{q}}_{\parallel}(B)}{{\hat{q}}(B=0)}$
$\displaystyle=$
$\displaystyle\frac{s/\eta_{\parallel}}{s/\eta}=\frac{47.5}{\Big{[}16+\frac{7}{8}12\sum_{u,d,s}\frac{1}{1+(\tau_{c}/\tau_{B})^{2}}\Big{]}}$
$\displaystyle\frac{{\hat{q}}_{\perp}(B)}{{\hat{q}}(B=0)}$ $\displaystyle=$
$\displaystyle\frac{s/\eta_{\perp}}{s/\eta}=\frac{47.5}{\Big{[}16+\frac{7}{8}12\sum_{u,d,s}\frac{1}{1+4(\tau_{c}/\tau_{B})^{2}}\Big{]}}~{}.$

Now realizing ${\hat{q}}(B=0)$ in terms of a simple quasi-particle form,
either ${\hat{q}}(B=0)=g(T)\times 5.87T^{3}$ from Eq. (LABEL:qh_rho_qN) or
${\hat{q}}(B=0)=g(T)\times{\hat{q}}_{0}$ from Eq. (LABEL:qh_rho_q0), we can
express ${\hat{q}}_{\parallel,\perp}(T,B)$ as:

$\displaystyle{\hat{q}}_{\parallel}(T,B)=\frac{47.5}{\Big{[}16+\frac{7}{8}12\sum_{u,d,s}\frac{1}{1+(\tau_{c}/\tau_{B})^{2}}\Big{]}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Big{[}g(T)\times
5.87T^{3}~{}{\rm or,}~{}g(T){\hat{q}}_{0}\Big{]}$
$\displaystyle{\hat{q}}_{\perp}(T,B)=\frac{47.5}{\Big{[}16+\frac{7}{8}12\sum_{u,d,s}\frac{1}{1+4(\tau_{c}/\tau_{B})^{2}}\Big{]}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Big{[}g(T)\times
5.87T^{3}~{}{\rm or,}~{}g(T){\hat{q}}_{0}\Big{]}~{}.$ (24)

For more rich $B$-dependent structure, we can replace $g(T)$ by $g(T,B)$.

##  V Results and discussions

Figure 3: Dependence of $\hat{q}/T^{3}$ on temperature (T in GeV) in absence
of magnetic field — estimations based on $\hat{q}_{N}$ (green solid line) and
$\hat{q}_{0}$ (green dashed line), pQCD + HTL approach for quark jet of
initial energy $E_{i}$ = 2 GeV (red dash-dotted line) and $E_{i}$ = 10 GeV
(red solid line) and semi-QGP + magnetic monopoles approach for $E_{i}$ = 2
GeV (blue dash-dotted line) and $E_{i}$ = 10 GeV (blue solid line) Xu et al.
(2015). Black dash-dotted and solid lines represent $\hat{q}$ of an isotropic
$\cal{N}$ = 4 SYM plasma at zero magnetic field for $\lambda$ = 4$\pi$ and
8$\pi$ respectively [Eq. (25)].

Figure 4: Dependence of $\hat{q}/T^{3}$ on temperature (T in GeV) —
$\hat{q}_{\parallel,\perp}/T^{3}$ (black dash line and blue dash-dotted line)
at eB = 0.2 GeV2, $\tau_{c}$ = 1 fm and $\hat{q}/T^{3}$ at eB = 0 (green solid
line) using Eq. (24) & (LABEL:qh_rho_q0) (Top panel) and using Eq. (24) &
(LABEL:qh_rho_qN) (Bottom panel). Red solid line represents $\hat{q}$ of an
isotropic $\cal{N}$ = 4 SYM plasma at zero magnetic field for $\lambda$ =
6$\pi$ [Eq. (25)].

In this section, we discuss the results for the estimation of $\hat{q}$
numerically using quasi-particle model. Fig. (3) shows the temperature
dependence of $\hat{q}/T^{3}$ in absence of magnetic field and the comparison
with some of the earlier results Xu et al. (2015, 2014); Hidaka and Pisarski
(2008, 2010); Dumitru et al. (2011); Lin et al. (2014); Liao and Shuryak
(2007, 2008, 2012). We notice that the result obtained using Eq.
(LABEL:qh_rho_qN) ($\hat{q_{N}}$-based) is quite close with the calculation
based on pQCD + HTL approximation Xu et al. (2014). So, we may relate our
$\hat{q_{N}}$-based estimation with a weakly coupled QGP system as
traditionally described by pQCD calculation. On the other hand, to incorporate
the non-pQCD outcome, semi-QGP + magnetic monopoles results Xu et al. (2015)
are also shown which are quite enhanced than HTL results. This non-pQCD
estimation corresponds to strongly coupled system and our estimation from Eq.
(LABEL:qh_rho_q0) ($\hat{q_{0}}$-based) is found to follow similar trend. Both
have a peak structure near transition temperatures of their respective models
and also their peak strengths are close to the boundary, predicted for
$\cal{N}$ = 4 SYM plasma. In the absence of magnetic field, the jet quenching
parameter for an isotropic $\cal{N}$ = 4 SYM plasma has the form Liu et al.
(2006):

$\hat{q}_{0}=\frac{\pi^{\frac{3}{2}}\Gamma(\frac{3}{4})}{\Gamma(\frac{5}{4})}\sqrt{\lambda}\,T^{3}\,$
(25)

where $\sqrt{\lambda}=\sqrt{g_{YM}^{2}\,N_{c}}$ with $g_{YM}$ denoting the
coupling strength of the strongly coupled plasma with color degeneracy factor
$N_{c}$. The black dash-dotted line and solid line in Fig. (3) denote
$\lambda$ = 4$\pi$ and 8$\pi$ respectively. Therefore, estimations of Eq.
(LABEL:qh_rho_qN) and (LABEL:qh_rho_q0) may be considered as order of strength
of $\hat{q}$ for weakly and strongly coupled systems respectively. It is a
quite interesting finding that by playing with existing experimental knowledge
of jet quenching parameter for cold nuclear matter (${\hat{q}}_{N}\approx
0.02$ GeV2/fm Airapetian et al. (2007); Deng and Wang (2010)) and hot QGP
(${\hat{q}}_{0}\approx 0.9$ GeV2/fm Chen et al. (2010)), we are getting two
directional aspects of QCD matter. When we are approaching from hadronic to
QGP phase with ${\hat{q}}_{N}$, we are getting results close to weakly coupled
QGP system. On the other hand, when we are approaching from QGP to hadronic
phase with ${\hat{q}}_{0}$, we are getting results close to strongly coupled
QGP system. Interestingly, a cross-over type LQCD based temperature dependent
gluon density is used for both cases.

Next, we have considered the effect of finite magnetic field on
$\hat{q}/T^{3}$. As mentioned in Refs. Majumder et al. (2007); Casalderrey-
Solana and Wang (2008); Xu et al. (2015, 2016); Shuryak (2017), there exists a
correspondence between s/$\eta$ and $\hat{q}/T^{3}$. On this basis, we can
have the equivalence of phase space in both quantities. In presence of
magnetic field, $\eta/s$ splits into parallel and perpendicular components
which can be connected to the corresponding components of jet quenching
parameter as shown in Eq. (LABEL:qh_q0) and (24). Due to magnetic field, a
cyclotron motion of charged particles introduces a time scale $\tau_{B}$
(inverse of cyclotron frequency) along with the relaxation time $\tau_{c}$
which collectively constitute the phase space part in $\eta/s$ as given in Eq.
(18). The derived results from Eq. (24) are shown in Fig. 4 which incorporate
the same phase space. The temperature dependence of $\hat{q}/T^{3}$ and its
components at eB = 0.2 GeV2, $\tau_{c}$ = 1 fm are shown. The red solid lines
in both figures denote $\hat{q}$ of an isotropic $\cal{N}$= 4 SYM plasma at
zero magnetic field by putting $\lambda$ = 6$\pi$ in Eq. (25). In presence of
magnetic field, the components get enhanced and the difference between the two
components decreases with increasing T due to the phase space part. If one
analyzes detailed anatomy of ${\hat{q}}/T^{3}$ expression in Eq. (24), then we
can identify two sources of $B$-dependent components. One is $B$-dependent
degeneracy factor, for which jet quenching parameter will be enhanced and
another is phase-space part made of $\tau_{B}$, for which jet quenching
parameter will be enhanced further.

This impact of magnetic field on ${\hat{q}}$ might be very challenging to
observe experimentally, however one can think of measuring nuclear suppression
factor, $R_{AA}$ separately for quark and gluon initiated jets. The quark
being charged will be affected by presence of magnetic field while gluon will
not. Therefore, if one can design the experiment for these two separate
measurements, then the difference between two measurements can be linked with
this $B$-dependent jet quenching parameter.

##  VI Aspects from AdS/CFT

The jet quenching parameter $\hat{q}$ for the strongly coupled QCD plasma in
the presence of a magnetic field has also been computed directly using the
non-perturbative toolbox of AdS/CFT correspondence Maldacena (1998); Gubser et
al. (1998); Aharony et al. (2000); Li et al. (2016); Zhang and Ma (2018); Mamo
(2015); Rougemont (2020).

Figure 5: Comparison of the results for the ratio of $\hat{q}(B)$ to
$\hat{q}(B=0)$ obtained from the quasi-particle like approach against AdS/CFT
techniques. The results for the quasi-particle approach are shown as solid
lines and those from AdS/CFT are shown by the dashed lines. The above results
are shown for $T=0.3$ GeV and $\tau_{c}=1$ fm.

It has been shown that the magnitude of $\hat{q}(B=0)$ calculated via AdS/CFT
correspondence is closer to that extracted from RHIC data Eskola et al.
(2005); Dainese et al. (2005) in comparison to the pQCD result. If we consider
a magnetic field along the $z$-direction, the parton that forms the jet while
traversing the medium can suffer a momentum broadening either along or
transverse to the magnetic field direction, giving rise to three different
components of jet quenching parameter: $\hat{q}_{\parallel(\perp)}$,
$\hat{q}_{\perp(\parallel)}$ and $\hat{q}_{\perp(\perp)}$, where
$\hat{q}_{\parallel}\equiv\hat{q}_{\parallel(\perp)}$ and
$\hat{q}_{\perp}\equiv\hat{q}_{\perp(\parallel)}+\hat{q}_{\perp(\perp)}$. In
this notation, the first symbol denotes the direction of the magnetic field
and the second one denotes the direction in which the momentum broadening of
the jet occurs. The result for these distinct $\hat{q}$ values as obtained in
Ref. Zhang and Ma (2018) are

$\hat{q}_{||(\perp)}=\frac{4\sqrt{\lambda}{\cal B}T}{3\log({\cal
B}/T^{2})}\,,$ (26)

$\hat{q}_{\perp(||)}=\frac{2\pi}{3}\sqrt{\lambda}\sqrt{\cal B}T^{2},$ (27)

and

$\hat{q}_{\perp(\perp)}=\frac{\sqrt{\lambda}{\cal B}^{3/2}}{3\pi\,\log({\cal
B}/T^{2})}.$ (28)

From these results, one finds that in general the jet quenching parameter is
enhanced in the presence of a magnetic field in comparison to the isotropic
SYM case. Moreover, one gets
$\hat{q}_{\perp(\perp)}>\hat{q}_{||(\perp)}>\hat{q}_{\perp(||)}>\hat{q}_{0}$
which shows that the values of the jet quenching parameter depends strongly on
the direction of the moving quark as well as the direction in which the
momentum broadening occurs.

We find an interesting comparison for the ratio
$\frac{\hat{q}(B)}{\hat{q}(B=0)}$ obtained in Eq. (LABEL:qh_q0) against the
AdS/CFT results outlined above, for values of $\frac{eB}{T^{2}}$ up to 50,
shown in Fig. (5). Both find $\hat{q}_{\perp}>\hat{q}_{\parallel}$ and support
enhance trend of quenching parameter in presence of magnetic field, although
exact quantitative comparison might not be so good. A better upgradation of
our present estimation can be possible by using the quantum aspect of magnetic
field, where phase space will be proportional to $eB$ due to Landau
quantization. In that picture, we might get a proportional $eB$-dependence as
we grossly notice for AdS/CFT results.

##  VII Summary

In summary, we have estimated the jet quenching parameter, $\hat{q}$ in
absence and presence of magnetic field using quasi-particle model where
temperature and magnetic field dependent degeneracy factors of partons are
considered for fitting the magneto-thermodynamical data of lattice quantum
chromodynamics. Using the parametric degeneracy factor we have estimated the
temperature ($T$) dependent gluon density to obtain $\hat{q}(T)$ from hadronic
to QGP phase by restricting low and high temperature values of ${\hat{q}}$ by
their standard experimental values. During the restriction by low temperature
cold nuclear matter data, temperature dependent $\hat{q}(T)$ from hadron to
QGP phase becomes quite close to hard thermal loop or perturbative QCD
results, indicating transportation of jet in a weakly interacting gas. While
$\hat{q}(T)$ estimation from QGP to hadron phase is close to earlier
estimations, which corresponds to strongly interacting liquid picture.

Next, we go for the calculation of jet transport coefficient at finite
magnetic field. Here, ${\hat{q}}$ splits into parallel and perpendicular
components, whose magnetic field dependent part has two sources. One is field
dependent degeneracy factor and another is phase space part, guided from shear
viscosity to entropy density ratio. The values of jet transport coefficients
are found to be enhanced due to the collective role of these two sources.
These results provide additional information towards our current understanding
and phenomenology of jet quenching in presence of magnetic field. Experimental
measurement and comparison of quark and gluon initiated jet suppression could
possibly shed more light on it as the former being charged is expected to
experience the effect of magnetic field while the latter does not.

Acknowledgment: The authors acknowledge to few members of TPRC-IITBH \-
Sarthak Satapathy, Jayanta Dey, Anki Anand, Ranjesh Kumar, Ankita Mishra,
Prasant Murmu, who worked on quasi-particle picture Satapathy et al. (2020);
Dey et al. (2019a), which is used in present work. D. Banerjee acknowledges
Inspire Fellowship research grant [DST/INSPIRE Fellowship/2018/IF180285]. A.
Modak and P. Das acknowledge the Institutional Fellowship research grant of
Bose Institute. A. Budhraja thanks Dr. Rishi Sharma for useful discussions.

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