# Transport coefficients of hot magnetized QCD matter

###### Abstract

The transport coefficients such as shear viscosity, bulk viscosity, and
thermal conductivity of a magnetized hot QCD matter have been
estimated in the strong field limit. To model the hot QCD matter in the presence of magnetic field,
a quasi-particle description of the hot QCD equation of state has been adopted.
The temperature dependence of viscous coefficients (bulk and shear viscosities) and thermal
conductivity have been obtained by considering,
processes () and 2 2
quark-antiquark scattering processes in the presence of the
strong magnetic field. All this has been done by
setting up a -dimensional effective covariant kinetic theory
for the lowest Landau level quarks in the strong field limit. This enables one to
include the mean-field contributions in terms of non-trivial quasi-particle energy dispersions
to the transport coefficients. Such contributions have significant impact
at temperature regions which are not very far away from the QCD transition point.
To realize the significance of various processes in the medium,
relative behavior of the transport coefficients in the thermal medium has been
investigated through their respective ratios.

Keywords:
Quark-gluon-plasma, Effective kinetic theory, Strong magnetic field, Thermal relaxation time,
Transport coefficients, Effective fugacity.

PACS: 12.38.Mh, 13.40.-f, 05.20.Dd, 25.75.-q

## I Introduction

Relativistic heavy-ion collision (RHIC) experiments have reported the presence of strongly coupled matter- Quark-gluon plasma (QGP) as a near-ideal fluid STAR (); Aamodt:2010pb (). The quantitative estimation of the experimental observables such as the collective flow and transverse momentum spectra of the produced particles from the hydrodynamic simulations involve the dependence upon the transport parameters of the medium. Thus, the transport coefficients are the essential input parameters for the hydrodynamic evolution of the system.

Recent investigations show that intense magnetic field is created in the early stages of the non-central asymmetric collisions Skokov:2009qp (); Zhong:2014cda (); deng (); Das:2016cwd (). This magnetic field affects the thermodynamic and transport properties of the hot dense QCD matter produced in the RHIC. Ref Inghirami:2016iru () describes the extension of ECHO-QGP DelZanna:2013eua (); Becattini:2015ska () to the magnetohydrodynamic regime. The recent major developments regarding the intense magnetic field in heavy-ion collision include the chiral magnetic effect Fukushima:2008xe (); Sadofyev:2010pr (); Huang (), chiral vortical effects Kharzeev:2015znc (); Avkhadiev:2017fxj (); Yamamoto:2017uul () and very recent realization of global -hyperon polarization in non-central RHIC STAR:2017ckg (); Becattini:2016gvu (). This sets the motivation to study the transport coefficients in presence of the strong magnetic field. The transport parameters under investigation are the viscous coefficients (shear and bulk) and the thermal conductivity of the hot magnetized QGP. Importance of the transport processes in RHIC is well studied Luzum:2008cw () and reconfirmed by the recent ALICE results Adam:2016izf (); Abelev:2012pa (); Abelev:2012pp ().

Quantizing quark/antiquark field in the presence of strong magnetic field background gives the Landau levels as energy eigenvalues. The quark/antiquark degrees of freedom is governed by -dimensional Landau level kinematics whereas gluonic degrees of freedom remain intact in the presence of magnetic field Hattori:2017qih (); Kurian:2017yxj (). However, gluons can be indirectly affected by the magnetic field through the quark loops while defining the Debye mass of the system.

Shear and bulk viscosities can be estimated from Green-Kubo formulation both in the presence and absence of magnetic field Hattori:2017qih (); Kharzeev:2007wb (); Moore:2008ws (); Czajka:2017bod (). Viscous pressure tensor quantifies the energy-momentum dissipation with the space-time evolution and is characterized by seven viscous coefficients in the strong magnetic field Tuchin:2011jw (). The seven viscous coefficients consist of two bulk viscosities (both transverse and longitudinal) and five shear viscosities. The present investigations are focused on the longitudinal component (along the direction of ) of shear and bulk viscosity since other components of viscosities are negligible in the strong field limit. Another key transport coefficient under investigation is thermal conductivity of the QGP medium. The temperature dependence of thermal conductivity has been studied in the absence of magnetic field in the Ref.Marty:2013ita (). Relative behavior of the transport coefficients will lead to physical laws and ratios such as Wiedemann-Franz law, Prandtl number , that can provide information about the dynamics and responses of the medium. The ratio of shear viscosity to thermal conductivity can be counted in terms Prandtl number, which is important for understanding the sound attenuation in the fluids Mitra:2017sjo (). The first step towards the estimation of transport coefficients from the effective kinetic theory is to include proper collision integral for the processes in the strong field. This can be done within the relaxation time approximation (RTA). Microscopic processes or interactions are the inputs of the transport coefficients and are incorporated through thermal relaxation times. Note that quark-antiquark pair production ( processes) and quark-antiquark t-channel scattering ( processes) are dominant in the presence of the strong magnetic field Hattori:2016lqx ().

The prime focus of the present article is to estimate the temperature behavior of the transport coefficients such as bulk viscosity, shear viscosity and thermal conductivity, incorporating the hot QCD medium effects in presence of the strong magnetic field. Estimation of the transport parameters can be done in two equivalent approaches the hard thermal loop effective theory (HTL) Arnold:2003zc (); ValleBasagoiti:2002ir (); Moore:2001fga () and the relativistic semi-classical transport theory Fukushima:2017lvb (); Chen:2009sm (); Khvorostukhin:2010cw (); Xu:2007ns (); Thakur:2017hfc (). The present analysis is done with the relativistic transport theory by employing the Chapman-Enskog method. Hot QCD medium effects are encoded in the quark/antiquark and gluonic degrees of freedom by adopting the effective fugacity quasiparticle model (EQPM) Kurian:2017yxj (); Chandra:2011en (); Chandra:2007ca (). The transport coefficients pick up the mean field term (force term) as described in Ref Mitra:2018akk (). The mean field term comes from the local conservations of number current and stress-energy tensor in the covariant effective kinetic theory. In the current analysis, we investigate the mean field corrections in the presence of strong magnetic field and study the temperature behavior and the relative significance of the transport coefficients. Here, the strong magnetic field restricts the calculations to -dimensional (dimensional reduction) covariant effective kinetic theory.

We organize the manuscript as follows. In section II, the mathematical formulation for the estimation of transport coefficients from the effective covariant kinetic theory is discussed along with the quasiparticle description of hot QCD medium in the strong magnetic field. Section III deals with the thermal relaxations for both processes and quark-antiquark scattering in the strong field limit. Predictions of the transport coefficients and their relative behavior are discussed in section IV. Finally, in section V the summary and outlook of the are presented.

## Ii Formalism: Transport coefficients at strong magnetic field

The strong magnetic field constraints the quarks/antiquarks motion parallel to field with the transverse density of states . We are working on the regime . The first inequality means that the regime under consideration is weakly coupled and the second inequality allows us to focus on the lowest Landau state of quarks and antiquarks since the magnetic field background is considerably strong. The formalism for the estimation of transport coefficients includes the quasiparticle modeling of the system away from the equilibrium followed by the setting up of the effective kinetic theory for different processes. Quasiparticle models encode the medium effects, , effective fugacity or with effective mass. The later include self-consistent and single parameter quasiparticle models Bannur:2006js (), NJL and PNJL based quasiparticle models Dumitru (), effective mass with Polyakov loop D'Elia:97 () and recently proposed quasiparticle models based on the Gribov-Zwanziger (GZ) quantization Su:2014rma (); zwig (); Bandyopadhyay:2015wua ().

Here, the analysis is done within the effective fugacity quasiparticle model (EQPM) where the medium interactions are encoded through temperature dependent effective quasigluon and quasiquark/antiquark fugacities, and respectively. The extended EQPM describes the hot QCD medium effects in strong magnetic field Kurian:2017yxj (). We considered the (2+1) flavor lattice QCD equation of state (EoS) (LEoS) Cheng:2007jq (); Borsanyi () and the 3-loop HTLpt EOS Haque (); Andersen () for the effective description of QGP in strong magnetic field Kurian:2017yxj (); Kurian:2018dbn ().

### ii.1 Transport coefficients from effective (1+1)-D kinetic theory

We first need to define the macroscopic quantities that describe the thermodynamic state in the strong magnetic field . The particle four flow can be defined in terms of quasiparticle (dressed) momenta within EQPM as Mitra:2018akk (),

(1) |

in which is the degeneracy factor of the species. The term is the irreducible tensor with as the projection operator. The metric has the form . The quasiquark distribution function in local rest frame with the hydrodynamic four-velocity is given by,

(2) |

with , where is the Landau level energy eigenvalue in the strong magnetic field. Quasiparticle momenta (dressed momenta) and bare particle four-momenta can be related from the dispersion relations as,

(3) |

which modifies the zeroth component of the four-momenta in the local rest frame. Hence, we have

(4) |

The dispersion relation in Eq. (4) encodes the collective excitation of quasiparton along with the single particle energy. In the presence of the strong magnetic field, the leading order contribution to the transport coefficients are coming from LLL () quarks and antiquarks compared to gluons. This is because the thermal density of LLL quarks and antiquarks is whereas for gluons is only . Since we are working in the regime , the thermal density of quarks and antiquarks are much larger than that of gluons. Hence, here we are considering the contribution of LLL quarks/antiquarks to the macroscopic quantities and transport coefficients. In the strong magnetic field, we have

(5) |

where incorporates the longitudinal components and the integration phase factor in the strong field due to dimensional reduction Bruckmann:2017pft (); Tawfik:2015apa (); Gusynin:1995nb () is defined as,

(6) |

Also, for for the quark/antiquark of each flavor.

Next, we can define energy-momentum tensor focusing the energy density and momentum flow in the longitudinal direction of magnetic field. In terms of dressed momenta, in the strong magnetic field has the following form,

(7) |

where . Note that the Eq. (II.1) gives back the expression of energy density and pressure within the EEQPM in the strong magnetic field as shown in Kurian:2017yxj ().

Estimation of the transport coefficients requires the system away from equilibrium. We need to set-up the relativistic transport equation, which quantifies the rate of change of distribution function in terms of collision integral. The thermal relaxation time () linearize the collision term () in the following way,

(8) |

with is the force term from the conservation of particle density and energy momentum Mitra:2018akk (). The local momentum distribution function of quarks can expand as,

(9) |

Here, defines the deviation of the quasiquark distribution function from its equilibrium. The Eq.( 8) gives the effective kinetic theory description of the quasipartons under EEQPM in the strong magnetic field. In order to estimate the transport coefficients, we employ the Chapman-Enskog (CE) method. Applying the definition of equilibrium quasiparton momentum distribution function as in Eq. (2), the first term of Eq. (8) gives the number of terms with thermodynamic forces of the transport processes. The second term of Eq. (8) vanishes for a co-moving frame. Finally, we are left with,

(10) |

in which the conformal factor due to dimensional reduction in the strong field limit is where is the speed of sound. Here, . The bulk viscous force, thermal force and shear viscous force are defined respectively as follows,

(11) | |||

(12) | |||

(13) |

Note that here describes only the longitudinal components in the strong magnetic field. Also, the deviation function that is the linear combination of these forces can be represented as,

(14) |

where the coefficients can be defined from Eq. (10) as,

(15) | |||

(16) | |||

(17) |

with as the enthalpy per particle of the system that can be defined from the basic thermodynamics and the total enthalpy is given as, . Following this formalism, we can estimate the viscous coefficients and thermal conductivity of the QGP medium in the strong magnetic field.

#### ii.1.1 Shear and bulk viscosity

We can define the pressure tensor from the energy-momentum tensor as in the following way,

(18) |

We can decompose the in equilibrium and non-equilibrium components of distribution function as follows,

(19) |

where is the viscous pressure tensor. Following the definition of as in Eq. (II.1), takes the form,

(20) |

In the very strong magnetic field, the pressure tensor has different form as compared to the case without magnetic field. This is due to the 1+1-dimensional energy eigenvalues of the quarks and antiquarks. Hence, and can be 0 or 3 in the strong magnetic field, describing the longitudinal components of the viscous pressure tensor. The form of viscous pressure tensor in the strong magnetic field is described in the recent works by Tuchin Tuchin:2011jw (); Tuchin:2013ie (). Magnetized plasma is characterized by five shear components. Among the five coefficients, four components are negligible when the strength of the magnetic field is sufficiently higher than the square of the temperature Ofengeim:2015qxz (). Here, we are focusing on the non-negligible longitudinal component of shear and bulk viscous coefficients of the hot QGP medium in the strong magnetic field.

Following Mitra:2017sjo (), the longitudinal shear viscous tensor has the following form,

(21) |

Also, the bulk viscous part in the longitudinal direction comes out to be,

(22) |

Substituting from Eq. (14) and comparing with the macroscopic definition , we can obtain the expressions of longitudinal viscosity coefficients in the strong field limit. Note that the longitudinal component of shear viscosity, i.e., in the direction of magnetic field, is defined from Ofengeim:2015qxz (). The longitudinal shear and bulk viscosity are obtained as,

(23) |

and

(24) |

The second term in the Eq. (II.1.1) and Eq. (II.1.1) gives correction to viscous coefficients due to the quasiparton excitations whereas the first term comes from the usual kinetic theory of bare particles.

#### ii.1.2 Thermal conductivity

The heat flow is the difference between the energy flow and enthalpy flow by the particle,

(25) |

In terms of the modified/non-equilibrium distribution function Eq. (25) becomes,

(26) |

in which heat flow retains only non-equilibrium part of the distribution function. After contracting with projection operator and hydrodynamic velocity along with the substitution of from Eq. (8) and comparing with the macroscopic definition of heat flow, we obtain

(27) |

We obtain the thermal conductivity in the strong magnetic field as,

(28) |

The second term with in the heat flow comes from the which encodes the quasiparticle excitation in the thermal conductivity.

### ii.2 Relative momentum and thermal diffusion: The Prandtl number

The Prandtl number for the thermal QGP medium in the strong magnetic field is defined as follows Mitra:2017sjo (),

(29) |

where is the specific heat at constant pressure and is the mass density. In the strong field limit, can obtain from basic thermodynamics within extended EQPM Kurian:2017yxj () as,

(30) |

where . The mass density takes the form,

(31) |

in which is the Debye mass and is the thermal (medium) mass of quarks. The number densities can be obtained as,

(32) |

and

(33) |

Here, the LLL quasiquark momentum distribution is defined as,

(34) |

Also, the quasigluon distribution function has the form,

(35) |

in which for gluons.

#### Medium mass of gluons and quarks

Magnetic field effects are entering into the system through the dispersion relations and the medium (thermal) mass Hattori:2016idp (). The medium mass of gluons and quarks can be obtained in terms of effective coupling constant. Being an essential input for transport processes, the effective coupling controls the behavior of transport parameters critically. There are several investigations on the Debye masses (gluon medium mass) of the QGP in presence of magnetic field Bandyopadhyay:2016fyd (); Bonati:2017uvz (); Singh:2017nfa (). We recently estimated the EoS/medium dependence on the Debye screening mass and hence the effective coupling constant in the Ref. Kurian:2017yxj (). Following the same prescription, the Debye mass with finite can be expressed as,

(36) |

in which is the running coupling constant at finite temperature taken from two-loop QCD gauge coupling constants Laine:2005ai (). Medium mass of dressed quark is defined in terms of the quark/antiquark and gluon momentum distribution function as,

(37) |

From the quantities defined above, we estimated the Prandtl number for the QGP in the strong magnetic field.

## Iii Thermal relaxations in strong magnetic field

Thermal relaxation is the essential dynamical input of the transport processes which counts for the microscopic interaction of the system. In the strong magnetic field, the 2 2 quark-antiquark t-channel scattering and the 1 2 processes (gluon to quark-antiquark pair) are dominant Hattori:2016lqx (). The 2 2 quark-gluon scattering is sub-leading to quark-antiquark scattering. This is because the thermal density of the quark/antiquark dominates over the thermal density of gluons in the regime . The thermal relaxation time , can be defined from the relativistic transport equation in terms of distribution function in the strong magnetic field as,

(38) |

Here, represents the collision integral for the process under consideration. We derived the momentum dependent thermal relaxation time for processes , where primed notation for antiquark) from the extended EQPM in and has the following form Kurian:2018dbn (),

(39) |

where is the Casimir factor of the processes and is the effective coupling constant Kurian:2018dbn () within EQPM and has the form as follows,

(40) |

Here, the estimation of thermal relaxation is done with case. The processes are significant in the regime in which the dominant charge carriers have momenta in the order of Kurian:2018dbn (); Hattori:2016cnt (). Impact of the higher Landau levels and hot QCD medium effects on the relaxation time for processes is explored in Kurian:2018dbn ().

For t-channel quark-antiquark scattering, the collision integral in strong magnetic field is defined in the recent work Hattori:2016lqx (),

(41) |

where is the IR cutoff and the color trace gives the factor of in the square of scattering amplitude for the quark-antiquark scattering processes. The response function (primed notation for antiquark) is defined as

(42) |

Note that we neglected the factor in the strong magnetic field limit. Here,

(43) |

The quark-antiquark scattering is dominant in the regime where and as described in Hattori:2016lqx (). Hence always dominates over in this regime within the strong field limit. Finally, we end up with,

(44) |

We have the diffusion expansion within the approximation,

(45) |

Hence, the collision integral for the quark-antiquark scattering becomes,

(46) |

Following the Eq. (38), thermal relaxation time takes the following form,

(47) |

Hot medium effects are entering through the quasiparton distribution function and the effective coupling defined in Eq. (III). Note that the momentum dependence of the relaxation time is significant in the estimation of transport properties as expressed in Eqs. (II.1.1), (II.1.1) and (II.1.2). The effective thermal relaxation time controls the behavior of transport coefficients critically.

## Iv Results and discussions

We initiate the discussion with the temperature dependence on the bulk viscosity to entropy density ratio in the presence of strong magnetic field with and without the mean field corrections as shown in Fig. 1 (left panel). The bulk viscosity of the hot QCD medium for processes has been obtained from the basic thermodynamics within the relaxation time approximation in Kurian:2018dbn (). The present analysis is done by employing the effective covariant kinetic theory using the Chapman-Enskog or Grad’s 14 method. The mean field force term which emerges from the effective theory indeed appear as the mean field corrections to the transport coefficients. The second term in the Eq. (II.1.1) describes the mean field contribution to the longitudinal bulk viscosity in the strong magnetic field whereas the first term exactly gives back the leading order contribution of for processes as described in Kurian:2018dbn () by the substitution of as in Eq. (39). The mean term consists the term which is the temperature gradient of the effective fugacity . At higher temperature, the region over than , the mean field effects are negligible since the effective fugacity behaves as slowly varying function in that regime. Hence, the mean field corrections due to the quasiparticle excitation are significant at temperature region near .

The longitudinal bulk viscosity to entropy ratio for processes and t-channel quark-antiquark scattering at GeV is depicted in the Fig. 1 (right panel). The behavior of bulk viscosity depends on the term and the relaxation time . The significance of in longitudinal bulk viscosity is discussed in Kurian:2018dbn (). On the other hand, thermal relaxation act as the dynamical input for different processes.