# Transport coefficients of hot magnetized QCD matter beyond the lowest Landau level approximation

## Abstract

In this article, shear viscosity, bulk viscosity, and thermal conductivity of a QCD medium have been studied in the presence of a strong magnetic field. To model the quark–gluon plasma, an extended quasi-particle description of the hot QCD equation of state in the presence of the magnetic field has been adopted. The effects of higher Landau levels on the temperature dependence of viscous coefficients (bulk and shear viscosities) and thermal conductivity have been obtained by considering the \(1\rightarrow 2\) processes in the presence of the strong magnetic field. An effective covariant kinetic theory has been set up in (1+1)-dimensional that includes mean field contributions in terms of quasi-particle dispersions and magnetic field to describe the Landau level dynamics of quarks. The sensitivity of these parameters to the magnitude of the magnetic field has also been explored. Both the magnetic field and mean field contributions have seen to play a significant role in obtaining the temperature behaviour of the transport coefficients of the medium.

## 1 Introduction

Relativistic heavy-ion collision (RHIC) experiments have reported the presence of strongly coupled matter-Quark–gluon plasma (QGP) as a near-ideal fluid [1, 2, 3, 4, 5]. 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 [6, 7, 8, 9]. This magnetic field affects the thermodynamic and transport properties of the hot dense QCD matter produced in the RHIC. Reference [10] describes the extension of ECHO-QGP [11, 12] to the magnetohydrodynamic regime. The recent major developments regarding the intense magnetic field in heavy-ion collision include the chiral magnetic effect [13, 14, 15, 16, 17, 18, 19], chiral vortical effects [20, 21, 22] and very recent realization of global \(\Lambda \)-hyperon polarization in non-central RHIC [23, 24]. 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.

The dissipative effects are not only significant in the hydrodynamical evolution of QGP, but also in particle production, final-state hadron spectra and other observables derived from them. In the recent works [25, 26], the authors described the properties of matter produced in the energetic heavy-ion collisions with the identified hadrons. Importance of the transport processes in RHIC is well studied [27] and reconfirmed by the recent ALICE results [28, 29, 30, 31]. There have been several attempts to evaluate the viscous coefficients in the confined phase using effective models of the hadron gas [32, 33, 34]. Furthermore, the significance of viscous effects in the evolution of the Hubble parameter in the QCD era of the early Universe is described in [35, 36, 37].

Quantizing quark/antiquark field in the presence of magnetic field gives the Landau levels as energy eigenvalues. The quark/antiquark dynamics is governed by \((1+1)\)-dimensional Landau level kinematics whereas gluonic degrees of freedom remain intact in the presence of magnetic field [38, 39]. However, gluonic dynamics can be indirectly affected by the magnetic field through the Debye mass of the system.

Viscous coefficients can be estimated from Green-Kubo formulation both in the presence and absence of magnetic field [38, 40, 41, 42]. Lattice results for viscosities to entropy ratio are also well investigated [43, 44, 45]. Viscous pressure tensor quantifies the energy-momentum dissipation with the space-time evolution and is characterized by seven viscous coefficients in the presence of magnetic field [46]. 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 \(\vec {B}\)) of shear and bulk viscosities since other components of viscosities are negligible in the strong magnetic field. Another key transport coefficient under investigation is the thermal conductivity of the QGP medium. The temperature dependence of thermal conductivity has been studied in the absence of magnetic field in the Ref. [47]. The equations of state (EoS) dependance on the viscous coefficients, electric and thermal conductivities have been studied in Ref. [48]. 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 the \(1\rightarrow 2\) processes such as quark–antiquark pair production/annihilation are dominant in the presence of strong magnetic field [49, 50].

The prime focus of the present article is to estimate the temperature behaviour of the transport coefficients such as bulk viscosity, shear viscosity and thermal conductivity, incorporating the EoS effects in the presence of the strong magnetic field. Estimation of the transport parameters can be done in two equivalent approaches *viz*., the hard thermal loop effective theory (HTL) [51, 52, 53] and the relativistic semi-classical transport theory [49, 54, 55, 56, 57]. The present analysis is done with the relativistic transport theory by employing the Chapman–Enskog method. Thermal medium effects are encoded in the quark/antiquark and gluonic degrees of freedom by adopting the effective fugacity quasiparticle model (EQPM) [39, 58, 59, 60]. The transport coefficients pick up the mean field term (force term) as described in Ref. [61]. 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 behaviour of the transport coefficients. Here, the strong magnetic field restricts the calculations to \((1+1)\)-dimensional (dimensional reduction) covariant effective kinetic theory for quarks and antiquarks.

The manuscript is organized as follows. In Sect. 2, 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 3 deals with the thermal relaxation for the \(1\rightarrow 2\) processes in the strong magnetic field. Predictions of the transport coefficients in the magnetic field are discussed in Sect. 4. Finally, in Sect. 5 the summary and outlook of the are presented.

## 2 Formalism: Transport coefficients at strong magnetic field

The strong magnetic field \(\vec {B}=B\hat{z}\) constraints the quarks/antiquarks motion parallel to field with a transverse density of states. The viscous coefficients [38, 62] and heavy quark diffusion coefficient [63] have been perturbatively calculated under the regime \(\alpha _{s}\mid q_feB\mid \ll T^{2}\ll \mid q_feB\mid \) with the lowest Landau level (LLL) approximation. But the validity of LLL approximation is questionable since higher Landau level contributions are significant at \(\mid eB\mid =10 m_{\pi }^2\) in the temperature range above 200 MeV. Here, we are focusing on the more realistic regime \(gT \ll \sqrt{\mid q_feB\mid }\) in which higher Landau level (HLL) contributions are significant. In the very recent work [49], Fukushima and Hidaka have been estimated the longitudinal conductivity of magnetized QGP with full Landau level resummation in the regime \(gT \ll \sqrt{\mid q_feB\mid }\).

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 EoS effects, *viz*., effective fugacity or with effective mass. The later include self-consistent and single parameter quasiparticle models [64, 65, 66], NJL and PNJL based quasiparticle models [67, 68, 69, 70, 71], effective mass with Polyakov loop [72, 73, 74, 75] and recently proposed quasiparticle models based on the Gribov–Zwanziger (GZ) quantization [76, 77, 78, 79]. 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, \(z_{g}\) and \(z_{q}\) respectively. The extended EQPM describes the QGP medium effects in strong magnetic field [39]. We considered the (2+1) flavor lattice QCD EoS (LEoS) [80, 81] and the 3-loop HTLpt EOS [82, 83] for the effective description of QGP in the strong magnetic field [39, 62].

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

*h*is the total enthalpy defined as \(h=\sum _{k=0}^{N}h_k\) and

*n*is the total number density of the system. Note that here \(\mu =0,3\) describes only the longitudinal components in the strong magnetic field. Also, the deviation function \(\phi _k\) that is the linear combination of these forces can be represented as,

#### 2.1.1 1. Shear and bulk viscosity

#### 2.1.2 2. Thermal conductivity

## 3 Thermal relaxation in the strong magnetic field

*M*is the matrix element for the process under consideration.

## 4 Results and discussions

The present analysis is done by employing the effective covariant kinetic theory using the Chapman–Enskog method including the effects of HLLs. The mean field force term which emerges from the effective theory indeed appears as the mean field corrections to the transport coefficients. The second term in the Eqs. (32) and (33) describes the mean field contribution to the longitudinal shear viscosity and bulk viscosity in the presence of magnetic field, respectively. The mean field term consists of the term \(\delta \omega \) which is the temperature gradient of the effective fugacity \(z_{g/q}\). The temperature behaviours of the viscous coefficients (bulk and shear viscosities) in the magnetic field with and without the mean field corrections are shown in Fig. 4 (left panel). At higher temperature, the effects are negligible since the effective fugacity behaves as a slowly varying function of temperature there. Hence, the mean field corrections due to the quasiparticle excitations are significant at temperature region closer to \(T_c\). The magnetic field dependence of the bulk viscosity and shear viscosity have been plotted in the Fig. 4 (right panel). In the strong magnetic field limit, the viscous coefficients could be computed within LLL approximation. The inclusion of HLLs reflects the non-trivial (non-monotonic) magnetic field dependence of the transport coefficients. Similar non-monotonic structure in the magnetic field dependence of longitudinal conductivity with HLLs is described in [49].

Mean field corrections to the thermal conductivity is explicitly shown in Eq. (37) in which thermal relaxation incorporates the microscopic interactions. We depicted the temperature behaviour of \(\lambda /T^2\) in Fig. 5. The HLL effects of the transport coefficients are entering through the thermal relaxation time and the quasiparticle distribution function. These effects are significant in the estimation of transport coefficients in the presence of a magnetic field. The temperature behaviour of the dimensionless quantity \(\lambda /T^2\) in the absence of the magnetic field is well investigated [47, 48] and is in the order of \(100-25\) within the temperature range \((1-4)\frac{T}{T_c}\), which is quantitatively consistent with our result.

The viscous coefficients of the strongly interacting matter could be employed to obtain the viscous corrections to the experimental observables (hadron spectra, dilepton spectra etc.) in the RHIC. The dissipative effects and EoS dependence of the confined phase have been estimated within lattice QCD [25, 26]. These aspects along with the estimation of electric conductivity [95] within our model while including the HLLs is beyond the scope of the present analysis and is a matter of future investigations.

## 5 Conclusion and outlook

In conclusion, we have computed the temperature behaviour of the transport parameters such as longitudinal viscous coefficients (shear and bulk viscosities) and thermal conductivity for the \(1\rightarrow 2\) processes in the strong magnetic field background while including the effects of HLLs. Thermal relaxation time is computed in magnetized QGP incorporating the HLL contributions. Setting up an effective covariant kinetic theory within EQPM in the magnetic field induces mean field contributions to the transport coefficients. We employed the Chapman–Enskog method in the effective kinetic theory for the computation of transport coefficients. The transport coefficients that have been estimated are influenced by the thermal medium and magnetic field. Hot QCD effects are incorporated through the quasiparton degrees of freedom along with effective coupling and the medium effects are found to be negligible at very high temperature. We focused on the weakly coupled regime of the perturbative QCD within the limit \(gT \ll \sqrt{\mid q_feB\mid }\) in which HLL contributions are significant. Notably, the inclusion of HLL contributions are essential to explain the transport processes at the high temperature regimes. Furthermore, effects of the mean field term are seen to be quite significant as fas as the temperature behavior of the above mentioned transport coefficients is concerned (for the temperatures which are not very far away from \(T_c\)).

An immediate future extension of the work is to investigate the aspects of non-linear electromagnetic responses of the hot QGP with the mean field contribution along with the effective description of magnetohydrodynamic waves in the QGP medium. In addition, the estimation of all transport coefficients from covariant kinetic theory within the effective fugacity quasiparticle model using more realistic collision integral, for example, BGK (Bhatnagar, Gross and Krook) collision term, in the strong magnetic field would be another direction to work.

## Notes

### Acknowledgements

VC would like to acknowledge Science and Engineering Research Board (SERB), Govt. of India for the Early Career Research Award (ECRA/2016/000683) and Department of Science and Technology (DST), Govt. of India for INSPIRE-Faculty Fellowship (IFA-13/PH-55). SG would like to acknowledge the Indian Institute of Technology Gandhinagar for the postdoctoral fellowship. SM would like to acknowledge SERB-INDO US forum to conduct the Postdoctoral research in USA. We record our sincere gratitude to the people of India for their generous support for the research in basic sciences.

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