International Journal of Advanced Research in Physical Science (IJARPS) Volume 3, Issue 5, 2016, PP 4-14 ISSN 2349-7874 (Print) & ISSN 2349-7882 (Online) www.arcjournals.org SU(3) Polyakov linear-sigma model: bulk and shear viscosity of QCD matter in finite magnetic field Abdel Nasser Tawfik1,2,∗, Abdel Magied Diab1,2,†, M. T. Hussein3,‡ 1Egyptian Center for Theoretical Physics (ECTP), Modern University for Technology and Information (MTI), 11571 Cairo, Egypt 2World Laboratory for Cosmology And Particle Physics (WLCAPP), 11571 Cairo, Egypt 3Physics Department, Faculty of Science, Cairo University, 12613 Giza, Egypt ∗a.tawfi[email protected], †[email protected], ‡[email protected] Abstract: Due to off-center relativistic motion of the charged spectators and the local momentum-imbalance of the participants, a short-lived huge magnetic field is likely generated, especially in relativistic heavy-ion collisions. In determining the temperature dependence of bulk and shear viscosities of the QCD matter in vanishing and finite magnetic field, we utilize mean field approximation to the SU(3) Polyakov linear-sigma model (PLSM). We compare between the results from two different approaches; Green-Kubo correlation and Boltzmann master equation with Chapman-Enskog expansion. We find that both approaches have almost identical results, especially in the hadron phase. In the temperature dependence of bulk and shear viscosities relative to thermal entropy at the critical tem- perature, there is a rapid decrease in the chiral phase-transition and in the critical temperature with increasing magnetic field. As the magnetic field strength increases, a peak appears at the critical temperature (Tc). This can be understood from the small drop on the thermal entropy at Tc, which can be interpreted due to instability in the hydrodynamic flow of the quark-gluon plasma and soft statistical hadronization. It is obvious that, increasing magnetic field accelerates the transition from hadron to QGP phases (inverse catalysis), i.e., taking place at lower temperatures. Keywords: Chiral transition, Magnetic fields, Magnetic catalysis, Critical temperature, Viscous properties of QGP 1. INTRODUCTION Recently, the study of the influence of strong magnetic field on Quantum Chromodynamics (QCD) apparently gains increasing popularity among particle physicists. Such a strong magnetic field can be reproduced in various high-energy regimes such as early universe and non-central heavy-ion collisions (HIC) [1, 2]. In the heavy-ion experiments, a huge magnetic field can be created due to the relativistic motionof charged spectators and the local momentum-imbalanceof the 2 2 2 participants. At SPS, RHIC and LHC energies, the expected magnetic field ranges between 0.1 mπ, mπ and 10 15 mπ, 2 8 − respectively [1, 3], where mπ 10 Gauss. The influence on QCD doesn’t∼ only cause catalysis of the chiral symmetry breaking [4, 5] but also modifies the chiral phase structure of the hadron production. Also, it changes the nature of the chiral phase-transition [6–8] and the energy loss due to quark synchrotron radiation [3, 9]. Furthermore, the magnetic field does not only come up with essential effects during the early stages of HIC, but also during the later ones, where the response of the magnetic effect is assumed to have a large in-medium-dependence. The latter depends on the variation of the magnetic diffusion time [3, 9] and the electrical conductivity which are medium depending [10, 11]. The description of the chiral and deconfinement phase-structure of the hadrons, the characterization of the QGP properties and the definition of the critical endpoint (CEP) are examples on significant researches conducted during last decades. The transport properties are particularly helpful in characterizing strongly interacting QCD matter, such as the phase transition, the critical endpoint,etc. [12]. The viscous transportpropertieshave been reviewed in Ref. [13]. The responseof the QCD matter to an external magnetic field can be described by the transport coefficients, such as bulk and shear viscosities. In the present study, we extend our previous work [14], where the temperature dependence of bulk and shear viscosities was deduced from SU(3) PLSM to a finite magnetic field [15]. The bulk [ζ(T,eB)] and shear [η(T,eB)] viscosity normalized to the entropy density s(T,eB) shall be calculated at finite temperatures and magnetic field strengths. We also address the chiral and deconfinement phase-transitions in finite magnetic field. c ARC Page 4 Abdel Nasser Tawfik, et al. First, we recall that so-far various LSM-calculations have been performed in order to determine the viscous properties of the QCD matter [16–18]. Based on Boltzmann-Uehling-Uhlenbeck (BBU) equation and Green-Kubo (GK) correlation, η/s has been estimated in the large-N limit [16]. Also, ζ/s in the large-N limit has been calculated from Boltzmann- Uehling-Uhlenbeck [17]. From relaxation time approximation (RTA) and BUU equation, the shear and bulk viscosity have been calculated in SU(2) LSM [18]. Second, from BUU equation with relaxation time approximation, some of such dissipative properties haven been studied from the hadron resonance gas (HRG) model with excluded-volume corrections as function of temperature and baryon chemical potential [19]. In the present work, it is assumed that the temperature dependence of QCD viscous properties such as bulk and shear viscosity are strongly affected by the huge short-lived magnetic field, which can be generated in relativistic heavy-ion collisions. We study their dependence on various magnetic field strengths. We present a direct estimation for both types of viscosity coefficients from PLSM by using BUU and GK approaches. For the first time, a systematic study in SU(3) PLSM in vanishing and nonzero magnetic field is presented. Such a way we can compare between the results from these two different approaches. A rapid decrease in the chiral phase-transition and in the critical temperature with increasing magnetic field is observed. Increasing magnetic field is accompanied by phase transitions that take place at lower critical temperatures relative to the ones at vanishing magnetic fields. In other words, increasing magnetic field leads to a decrease in the corresponding critical temperature (inverse catalysis). This paper is organizedas follows, we briefly describe PLSM in mean field approximationin section in which information about hadron matter in the presence of magnetic field is included. BUU and GK approaches are introduced in section and elaborated in Appendices and , respectively. The temperature dependence of the relaxation time and the bulk and shear viscosities normalized to the thermal entropy at finite magnetic field strength and vanishing chemical potential shall be elaborated in section . This is followed by the conclusions in section . 2. REMINDER TO SU(3) LINEAR-SIGMA MODEL WITH MEAN FIELD APPROXIMATION The exchange of energy between particle and antiparticle at temperature (T ) and baryon chemical potential (µf ) can be included in the grand canonical partition function ( ), Z 0 = σa πa ψ ψ¯exp ( + µf ψ¯f γ ψf ) , (1) Z Z D D Z D D Z L Ya x Xf 1 where i /T dt d3x and V is the volume of the system of interest. The subscript f refers to quark flavors x ≡ 0 V and thereforeR µRf is theR chemical potential for quark flavors f = (l, s, ¯l, s¯). One can define a uniform blind chemical potential µf µu,d = µs [20–22] as a result of the assumption of symmetric quark matter and degenerate light quarks. ≡ ∗ is a Lagrangian coupled the chiral LSM Lagrangian with the Polyakov loops potential, = chiral (φ, φ ,T ). MoreL details about the PLSM model can be found in Refs. [23–25]. Moreover, the free energyL L can be−U given as = T log[ ]/V or F − · Z ∗ = U(σ , σ )+ (φ, φ ,T )+Ω¯ (T,µ ,B)+ δ0 Ω¯ (T,µ ). (2) F l s U qq f ,eB qq f The purely mesonic potential is given as • 2 m 2 2 c 2 U(σl, σs) = hlσl hsσs + (σ + σ ) σ σs − − 2 l s − 2√2 l λ (2λ + λ ) (λ + λ ) + 1 σ2σ2 + 1 2 σ4 + 1 2 σ4. (3) 2 l s 8 l 4 s In the present work, we implement the polynomial form of the Polyakov loop potential [26–29], • ∗ 2 (φ, φ ,T ) b2(T ) 2 2 b3 b4 2 2 U = φ + φ∗ φ3 + φ∗3 + φ + φ∗ , (4) T 4 − 2 | | | | − 6 16 | | | | 2 3 where b2(T ) = a0 + a1 (T0/T )+ a2 (T0/T ) + a3 (T0/T ) . With the parameters a0 = 6.75, a1 = 1.95, − a2 =2.625, a3 = 7.44, b3 =0.75 and b4 =7.5 [26], the pure gauge QCD thermodynamics is well reproduced. − For a better agreement with lattice QCD simulations, the critical temperature T0 is fixed at 187 MeV for Nf = 2+1 [28]. The quarks and antiquark contribution to the medium potential can be divided into two regimes. • International Journal of Advanced Research in Physical Science (IJARPS) Page 5 SU(3) Polyakov Linear-Sigma Model: Bulk and Shear Viscosity of QCD Matter in Finite Magnetic Field – In vanishing magnetic field (eB =0) but at finite T and µf [30], ∞ d3~p Ωqq¯ (T,µf ) = 2 T ff (T,µ). (5) − Z (2π)3 Xf 0 When introducing Polyakov-loop corrections to the quark’s degrees of freedom, then the quark Fermi-Dirac distribution function becomes E −µ E −µ E −µ ∗ − f f − f f −3 f f f (T,µ) = ln 1+3 φ + φ e T e T + e T , (6) f × 2 2 1/2 ∗ where Ef = (mf +p ) is the dispersion relation of f-th quark flavor. For antiquarks, φ and φ are replaced with each other and the chemical potential µ should be replaced by µ.