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To explore how rigid rotation affects the thermodynamic properties of free relativistic bosons, we employ the standard imaginary time formalism of thermal field theory to calculate the free propagator of complex scalar fields under rotation. We introduce the corresponding partition function and explicitly compute it by expanding the modes in cylindrical coordinates. The resulting propagator is in full agreement with similar findings in the existing literature., Comment: 4 pages, talk presented by E. Siri at the 8th Iranian Conference on Mathematical Physics (ICMP) in Qom, Iran, on 7-8 July 2024; Submitted to Transactions in Theoretical and Mathematical Physics (TTMP)

We study the thermodynamic properties of a rigidly rotating relativistic Bose gas. First, we derive the solution of the equation of motion corresponding to a rotating complex Klein-Gordon field and determine the free propagator of this model utilizing the Fock-Schwinger proper-time method. Using this propagator, we then obtain the thermodynamic potential of this model in the zeroth and first perturbative level. In addition, we compute the nonperturbative ring contribution to this potential. Our focus is on the dependence of these expressions on the angular velocity, which effectively acts as a chemical potential. Using this thermodynamic potential, we calculate several quantities, including the pressure, angular momentum and entropy densities, heat capacity, speed of sound, and moment of inertia of this rigidly rotating Bose gas as functions of temperature ($T$), angular velocity ($\Omega$), and the coupling constant ($\alpha$). We show that certain thermodynamic instabilities appear at high temperatures and large couplings. They are manifested as zero and negative values of the above quantities, particularly the moment of inertia and heat capacity. Zero moment of inertia leads to the phenomenon of supervorticity at certain $T$ or $\alpha$. Supervortical temperatures (couplings) decrease with increasing coupling (temperature). We also observe superluminal sound velocities at high $T$ and for large $\alpha$., Comment: V1: 25 pages, 20 figures; V2: Typos corrected, References updated, accepted for publication in PRD

We generalize a recently introduced formulation of relativistic spinful and vortical fluid to relativistic magnetohydrodynamics (MHD). We refer to it as the "Spinful-Vortical MHD" (SVMHD). The aim is to scrutinize the interplay between the vorticity, magnetic field, and spin, which is treated as a quantum object, in contrast to other formulations of spin hydrodynamics. To this purpose, we first perform a standard entropy current analysis up to first-order gradient expansion, $\mathcal{O}\left(\partial\right)$ as well as $\mathcal{O}\left(\hbar\partial\right)$, where $\hbar$ is the Planck constant. In contrast to alternative formulations of spin MHD, in the absence of vorticity, the zeroth-order energy-momentum tensor includes an additional magneto-vorticity mixed term and reduces, as expected, to the energy-momentum tensor of MHD. We show that in the first-order of gradient expansion, $36$ dissipative transport coefficients appear. They satisfy certain constraints that guarantee the positive definiteness of the entropy production rate. We then modify the formulation of SVMHD by replacing the magnetic part of the thermal vorticity tensor with its electric part. Carrying out the same analysis as in the standard formulation, we show that in this case, the first-order constitutive relations consist of $11$ nondissipative Hall-like coefficients, apart from $25$ dissipative coefficients. This difference arises from different behavior of the electric and magnetic part of the thermal vorticity under time-reversal transformation., Comment: 13 pages

The effect of rotation on the formation of the magnetic dual chiral density wave (\MD) in a dense and magnetized cold quark matter is studied. This phase is supposed to exist in the extreme conditions prevailing, e.g., in a neutron star. These conditions are, apart from high densities and strong magnetic fields, a relatively large angular velocity. To answer the question of whether the rotation enhances or suppresses the formation of this phase, we first determine the effect of rotation on the energy dispersion relation of a fermionic system in the presence of a constant magnetic field and then focus on the thermodynamic potential of the model at low temperature $T$ and finite chemical potential $\mu$. The thermodynamic potential consists, in particular, of an anomalous part leading to certain topological effects. We show that in comparison with the nonrotating case, a term proportional to the angular velocity appears in this anomalous potential. We then solve the corresponding gap equations to the chiral and spatial modulation condensates, and study the dependence of these dynamical variables on the chemical potential ($\mu$), magnetic field ($eB$), and angular velocity ($\Omega$). It turns out that the interplay between these parameters suppresses the formation of the \MD~phase in relevant regimes for cold neutron stars. This is interpreted as the manifestation of the inverse magnetorotational catalysis, which is also reflected in the phase portraits $eB$-$\mu$, $eB$-$\Omega R$, and $\mu$-$\Omega R$, explored in this work., Comment: V1: 19 pages, 7 figures, 2 tables; V2: Discussions improved, accepted for publication in PRD

We study the properties of a hot and magnetized quark matter in a rotating cylinder in the presence of a constant magnetic field. To do this, we solve the corresponding Dirac equation using the Ritus eigenfunction method. This leads to the energy dispersion relation, Ritus eigenfunctions, and the quantization relation for magnetized fermions. To avoid causality-violating effects, we impose a certain global boundary condition, and study its effect, in particular, on the energy eigenmodes and the quantization relations of fermions. Using the fermion propagator arising from this method, we then solve the gap equation at zero and nonzero temperatures. At zero temperature, the dynamical mass $\bar{m}$ does not depend on the angular frequency, as expected. We thus study its dependence on the distance $r$ relative to the axis of rotation and the magnetic field $B$, and explore the corresponding finite size effect for various couplings $G$. We then consider the finite temperature case. The dependence of $\bar{m}$ on the temperature $T$, magnetic field $B$, angular frequency $\Omega$, and distance $r$ for various $G$ is studied. We show that $\bar{m}$ decreases, in general, with $B$ and $\Omega$. This is the ''inverse magneto-rotational catalysis (IMRC)'' or the ''rotational magnetic inhibition'', previously discussed in the literature. To explore the evidence of this effect in the phase diagrams of our model, we examine the phase portraits of the critical temperature $T_c$ as well as the critical angular frequency $\Omega_c$ with respect to $G, B,\Omega$, and $r$ as well as $G, B, T$, and $r$, respectively. We show that $T_{c}$ and $\Omega_c$ decrease, in particular, with $B$. This is interpreted as clear evidence for IMRC., Comment: 23 pages, 12 figures, 5 tables

The Chern-Simons magnetohydrodynamics (CSMHD) is introduced using a Maxwell-Chern-Simons (MCS) Lagrangian including an axion-like field $\Theta$. The MCS equation of motion derived from this Lagrangian consists of a modified current, including a chiral magnetic (CM) and an anomalous Hall (AH) current, in addition to the ordinary Ohm current of resistive magnetohydrodynamics (MHD). The former consists of an axial chemical potential, which is given in terms of the temporal comoving derivative of $\Theta$, and the latter arises from the spatial gradient of $\Theta$. As it turns out, the existence of the axial chemical potential is a nonequilibrium effect that plays no role in the linear stability analysis, whereas the AH current arises as in the first-order linear perturbation of the thermal equilibrium. We analyze the linear stability and causality of the CSMHD in a resistive and chiral medium. We show that the Alfven modes propagating sufficiently close to the direction of the magnetic field are unstable but causal. They are also accompanied by a genuine nonhydro mode. A stable mode in a particular direction can correspond to an unstable mode propagating in the exact opposite direction. The AH instability is a manifestation of a breakdown of the parity. A numerical analysis of the phase velocity confirms these results., Comment: 31 pages, 10 figures

By combining the Wigner function formalism of relativistic quantum kinetic theory with fundamental equations of relativistic magnetohydrodynamics (MHD), we present a novel approach to determine the proper time evolution of the temperature and other thermodynamic quantities in a uniformly expanding hot, magnetized, and weakly interacting plasma. The aim is to study the contribution of quantum corrections to this evolution. We first determine the corresponding Wigner function in terms of the solution of the Dirac equation in the presence of a constant magnetic field. Using this function, we then compute the energy-momentum tensor of the above-mentioned plasma, which eventually yields its energy density and pressure. Plugging these quantities in the energy equation of relativistic MHD, we arrive, after choosing an appropriate coordinate system, at a differential equation for the temperature as a function of the proper time. The numerical solution of this equation leads finally to the proper time evolution of the temperature. The latter is then used to determine the evolution of a large number of thermodynamic quantities in this expanding and magnetized plasma. We compare our results with other existing results from relativistic MHD. We also comment on the effect of point to point decaying magnetic fields on the thermodynamic properties of this plasma., Comment: 23 pages, 9 figures

In ultrarelativistic heavy-ion collisions, enormous magnetic fields are generated because of fast-moving charged particles. In the presence of these magnetic fields, the spin of particles is aligned either in the parallel or in the antiparallel direction with respect to the direction of the magnetic field. A finite magnetization is thus produced. It is known that a finite magnetic susceptibility, $\chi_{m}$, changes the evolution of the energy density of the quark-gluon plasma (QGP), which is believed to be created in these collisions. Depending on whether the system under consideration is a paramagnetic ($\chi_{m}>0$) or diamagnetic ($\chi_{m}<0$) fluid, it slows down or speeds up the decay of the energy density, and affects other thermodynamic quantities. In general, one expects that the magnetic susceptibility depends on the magnetic field and temperature. Bearing in mind that these parameters evolve with the evolution of the fluid, a nonuniform magnetic susceptibility in this system is thus expected. In this work, we first determine $\chi_{m}$ by using a certain analogy to the standard anisotropic kinetic theory, where the one-particle distribution function is replaced by the corresponding anisotropic distribution function. We then determine the proper time dependence of the magnetic susceptibility in the framework of the ideal transverse magnetohydrodynamics. We also study the effect of dissipation on the evolution of $\chi_{m}$., Comment: 8 pages, 4 figures, XIII Quark Confinement and the Hadron Spectrum - Confinement2018, 31 July - 6 August 2018, Maynooth University, Ireland

Due to the absence of a transverse expansion with respect to the beam direction, the Bjorken flow is unable to describe certain observables in heavy ion collisions. This caveat has motivated the introduction of analytical relativistic hydrodynamics (RH) solutions with transverse expansion, in particular, the 3+1 self-similar (SSF) and Gubser flows. Inspired by recent generalizations of the Bjorken flow to the relativistic magnetohydrodynamics (RMHD), we present a procedure for a generalization of RH solutions to RMHD. Our method is mainly based on symmetry arguments. Using this method, we find the relation between RH degrees of freedom and the magnetic field evolution in the ideal limit for an infinitely conductive fluid, and determine the proper time dependence of the magnetic field in aforementioned flows. In the case of SSF, a family of solutions are found that are related through a certain differential equation. To find the magnetic field evolution in the Gubser flow, we solve RMHD equations for a stationary fluid in a conformally flat $dS^3\times E^1$ spacetime. The result is then Weyl transformed back into the Minkowski spacetime. In this case, the temporal evolution of the magnetic field exhibits a transmission between $1/t$ to $1/t^3$ near the center of the collision. The longitudinal component of the magnetic field is found to be sensitive to the transverse size of the fluid. We also find the radial evolution of the magnetic field for both flows. The radial domain of validity in the case of SSF is highly restricted, in contrast to the Gubser flow. A comparison of the results suggests that the Gubser RMHD may give a more appropriate qualitative picture of the magnetic field decay in the quark-gluon plasma (QGP)., Comment: 8 pages, 1 figures, XIII Quark Confinement and the Hadron Spectrum - Confinement2018, 31 July - 6 August 2018, Maynooth University, Ireland

The plasma of quarks and gluons created in ultrarelativistic heavy-ion collisions turns out to be paramagnetic. In the presence of a background magnetic field, this paramagnetism thus leads to a pressure anisotropy, similar to anisotropies appearing in a viscous fluid. In the present paper, we use this analogy, and develop a framework similar to anisotropic hydrodynamics, to take the pressure anisotropy caused, in particular, by the nonvanishing magnetization of a plasma of quarks and gluons into account. We consider the first two moments of the classical Boltzmann equation in the presence of an electromagnetic source in the relaxation-time approximation, and derive a set of coupled differential equations for the anisotropy parameter $\xi_0$ and the effective temperature $\lambda_0$ of an ideal fluid with nonvanishing magnetization. We also extend this method to a dissipative fluid with finite magnetization in the presence of a strong and dynamical magnetic field. We present a systematic method leading to the one-particle distribution function of this magnetized dissipative medium in a first-order derivative expansion, and arrive at analytical expressions for the shear and bulk viscosities in terms of the anisotropy parameter $\xi$ and effective temperature $\lambda$. We then solve the corresponding differential equations for $(\xi_0,\lambda_0)$ and $(\xi,\lambda)$ numerically, and determine, in this way, the proper time and temperature dependence of the energy density, directional pressures, speed of sound, and the magnetic susceptibility of a longitudinally expanding magnetized quark-gluon plasma in and out of equilibrium., Comment: V1:23 pages, 16 figures; V2: Typos corrected, version accepted for publication in PRD (2019)

Motivated by the recently found realization of the $1+1$ dimensional Bjorken flow in ideal and nonideal relativistic magnetohydrodynamics (MHD), we use appropriate symmetry arguments, and determine the evolution of magnetic fields arising from the $3+1$ dimensional self-similar and Gubser flows in an infinitely conductive relativistic fluid (ideal MHD). In the case of the $3+1$ dimensional self-similar flow, we arrive at a family of solutions, that are related through a differential equation arising from the corresponding Euler equation. To find the magnetic field evolution from the Gubser flow, we solve the MHD equations of a stationary fluid in a conformally flat $dS^{3}\times E^{1}$ spacetime. The results are then Weyl transformed back into the Minkowski spacetime. In this case, the temporal evolution of the resulting magnetic field is shown to exhibit a transition between an early time $1/t$ decay to a $1/t^{3}$ decay at a late time. Here, $t$ is the time coordinate. Transverse and longitudinal components of the magnetic fields arising from these flows are also found. The latter turns out to be sensitive to the transverse size of the fluid. In contrast to the result arising from the Gubser flow, the radial domain of validity of the magnetic field arising from the self-similar flow is highly restricted. A comparison of the results suggests that the (conformal) Gubser MHD may give a more appropriate qualitative picture of the magnetic field decay in the plasma of quarks and gluons created in heavy ion collisions., Comment: 26 pages, 10 figures

In order to gain deeper insight into the physics of the novel rotating solution of nonideal transverse magnetohydrodynamics (MHD), presented in one of our recent works, we replace the previously considered Maxwell theory with the ${\cal{CP}}$ violating Maxwell-Chern-Simons (MCS) theory. In this way, dissipationless chiral magnetic (CM) and anomalous Hall (AH) currents appear in the MCS equation of motion, that, together with equations of relativistic hydrodynamics, builds the set of constitutive equations of the nonideal transverse Chern-Simons magnetohydrodynamics (CSMHD). We are, in particular, interested in the effect of these currents on the evolution of electromagnetic fields in a uniformly and longitudinally expanding quark-gluon plasma with chirality imbalance. Combining the constitutive equations of CSMHD under these assumptions, we arrive, as expected, at two distinct rotating and nonrotating solutions for electromagnetic fields. The rotation occurs with increasing rapidity and a constant angular velocity $\omega_{0}$. Remarkably, the relative angle between the electric and magnetic fields, $\delta$, turns out to be given by the coefficient of AH current $\kappa_{E}$ and the electric conductivity of the medium $\sigma$, as $\delta=\tan^{-1}(\kappa_{E}/\sigma)$. Whereas the nonrotating solution implies the AH coefficient to be vanishing, and thus nonrotating electric and magnetic fields to be either parallel or antiparallel, the relative orientation of rotating electric and magnetic fields and the evolution of the CM conductivity $\kappa_{B}$ are strongly affected by nonvanishing $\kappa_{E}$. We explore the effect of positive and negative $\omega_{0}$ on the evolution of the CM current, and show, in particular, that a rotation of electromagnetic fields with negative $\omega_{0}$ implies a sign flip of the CM current in a chiral fluid with nonvanishing AH current., Comment: V1: 25 pages, 15 figures, 1 table; V2: Discussions improved. Version accepted for publication in PRD

The evolution of electromagnetic and thermodynamic fields in a non-ideal fluid are studied in the framework of ultrarelativistic transverse magnetohydrodynamics (MHD), which is essentially characterized by electric and magnetic fields being transverse to the fluid velocity. Extending the method of self-similar solutions of relativistic hydrodynamics to the case of non-conserved charges, the differential equations of non-ideal transverse MHD are solved, and two novel sets of self-similar solutions are derived. The first set turns out to be a boost-invariant and exact solution, which is characterized by non-rotating electric and magnetic fields. The second set is a non-boost-invariant solution, which is characterized by rotating electric and magnetic fields. The rotation occurs with increasing rapidity $\eta$, as the angular velocity is defined by $\omega_{0}\equiv\frac{\partial\zeta}{\partial\eta}=\frac{\partial\phi}{\partial\eta}$, with $\zeta$ and $\phi$ being the angles of electric and magnetic vectors with respect to a certain axis in the local rest frame of the fluid. For both sets of solutions, the electric and magnetic fields are either parallel or anti-parallel to each other. Performing a complete numerical analysis, the effects of finite electric conductivity as well as electric and magnetic susceptibilities of the medium on the evolution of rotating and non-rotating MHD solutions are explored, and the interplay between the angular velocity $\omega_{0}$ and these quantities is scrutinized. The lifetime of electromagnetic fields and the evolution of the temperature of the electromagnetized fluid are shown to be affected by $\omega_{0}$., Comment: 33 pages, 18 figures

The effects of finite magnetization and electric polarization on dissipative and non-dissipative (anomalous) transport coefficients of a chiral fluid are studied. First, using the second law of thermodynamics as well as Onsager's time reversal symmetry principle, the complete set of dissipative transport coefficients of this medium is derived. It is shown that the properties of the resulting shear and bulk viscosities are mainly affected by the anisotropy induced by external electric and magnetic fields. Then, using the fact that the anomaly induced currents do not contribute to entropy production, the corresponding algebro-differential equations to non-dissipative anomalous transport coefficients are derived in a certain derivative expansion. The solutions of these equations show that, within this approximation, anomalous transport coefficients are, in particular, given in terms of the electric susceptibility of the medium., Comment: V1: 14 pages (two-column), 1 table, no figure; V2: 20 pages (one-column), 1 table, no figure. Version accepted for publication in New Journal of Physics (2017)

The differential multiplicity of dileptons in a hot and magnetized quark-gluon plasma, $\Delta_{B}\equiv dN_{B}/d^{4}xd^{4}q$, is derived from first principles. The constant magnetic field $B$ is assumed to be aligned in a fixed spatial direction. It is shown that the anisotropy induced by the $B$ field is mainly reflected in the general structure of photon spectral density function. This is related to the imaginary part of the vacuum polarization tensor, $\mbox{Im}[\Pi^{\mu\nu}]$, which is derived in a first order perturbative approximation. As expected, the final analytical expression for $\Delta_{B}$ includes a trace over the product of a photonic part, $\mbox{Im}[\Pi^{\mu\nu}]$, and a leptonic part, ${\cal{L}}_{\mu\nu}$. It is shown that $\Delta_{B}$ consists of two parts, $\Delta_{B}^{\|}$ and $\Delta_{B}^{\perp}$, arising from the components $(\mu,\nu)=(\|,\|)$ and $(\mu,\nu)=(\perp,\perp)$ of $\mbox{Im}[\Pi^{\mu\nu}]$ and ${\cal{L}}_{\mu\nu}$. Here, the transverse and longitudinal directions are defined with respect to the direction of the $B$ field. Combining $\Delta_{B}^{\|}$ and $\Delta_{B}^{\perp}$, a novel anisotropy factor $\nu_{B}$ is introduced. Using the final analytical expression of $\Delta_{B}$, the possible interplay between the temperature $T$ and the magnetic field strength $eB$ on the ratio $\Delta_{B}/\Delta_{0}$ and $\nu_{B}$ is numerically studied. Here, $\Delta_{0}$ is the Born approximated dilepton multiplicity in the absence of external magnetic fields. It is, in particular, shown that for each fixed $T$ and $B$, in the vicinity of certain threshold energies, $\Delta_{B}\gg \Delta_{0}$ and $\Delta_{B}^{\perp}\gg \Delta_{B}^{\|}$. The latter anisotropy may be interpreted as one of the microscopic sources of the macroscopic anisotropies, reflecting themselves, e.g., in the elliptic asymmetry factor $v_{2}$ of dileptons., Comment: V1: 24 pages (two-column), 11 figures, 1 table; V2: 44 pages (one-column), 12 figures, 2 tables. Typos corrected, discussions improved, references updated, appendix B added; V3: 41 pages (one-column), 12 figures, 2 tables, discussions improved. Version accepted for publication in AOP (2017)

Using the standard Green-Kubo formalism, we determine the shear viscosity $\eta$ of a hot and dense Yukawa-Fermi gas. In particular, we study the effect of particle and plasmino excitations on thermal properties of the fermionic part of the shear viscosity, and explore the effects of thermal corrections to particle masses on bosonic and fermionic shear viscosities, $\eta_b$ and $\eta_f$. It turns out that the effects of plasminos on $\eta_f$ become negligible with increasing (decreasing) temperature (chemical potential)., Comment: 8 pages, 7 figures, Poster presented by F. Taghinavaz at "XIth Quark Confinement and the Hadron Spectrum", Saint-Petersburg, Russia, 8-12 September 2014

The complete quasi-particle spectrum of a magnetized electromagnetic plasma is systematically explored at zero and nonzero temperatures. To this purpose, the general structure of the one-loop corrected propagator of magnetized fermions is determined, and the dispersion relations arising from the pole of this propagator are numerically solved. It turns out that in the lowest Landau level, where only one spin direction is allowed, the spectrum consists of one positively (negatively) charged fermionic mode with positive (negative) spin. In contrast, in higher Landau levels, as an indirect consequence of the double spin degeneracy of fermions, the spectrum consists of two massless collective modes with left- and right-chiralities. The mechanism through which these new collective excitations are created in a uniform magnetic field is similar to the production mechanism of dynamical holes (plasminos) at finite temperature and zero magnetic fields. Whereas cold magnetized plasminos appear for moderate magnetic fields and for all positive momenta of propagating fermions, hot magnetized plasminos appear only in the limit of weak magnetic fields and soft momenta., Comment: V1: 24 pages, 10 figures; V2: Typos corrected. Version accepted for publication in PRD

The effect of anomalous magnetic moment of quarks on thermodynamic properties of the chiral condensate is studied, using of a two-flavor Nambu--Jona-Lasinio model at finite temperature $T$, chemical potential $\mu$, and in the presence of a uniform magnetic field $eB$. To this purpose, the Schwinger linear-in-$B$ ansatz for the quark anomalous magnetic moment in term of the nonperturbative Bohr magneton is considered. In a two-dimensional flavor space, it leads to the correction $\hat{T}_{Sch}=\hat{\kappa}\hat{Q} eB$ in the energy dispersion relation of quarks. Here, $\hat{Q}$ is the quark charge matrix. We consider three different sets for $\hat{\kappa}$, and numerically determine the dependence of the constituent quark mass on $T,\mu$ and $eB$ for fixed $\hat{\kappa}$. By exploring the complete phase portrait of this model in $T$-$\mu$, $\mu$-$eB$, and $T$-$eB$ phase spaces for various fixed $eB$, $T$, $\mu$ and $\hat{\kappa}$, we observe that inverse magnetic catalysis occurs for large enough $\hat{\kappa}$. Moreover, in the regime of weak magnetic fields, the phenomenon of reentrance of chiral symmetry broken and restored phases occurs for $T, \mu$ and $eB$ dependent $\hat{\kappa}$., Comment: 22 pages, 14 figures

We determine the shear viscosity of a hot and dense Yukawa-Fermi gas, using the standard Green-Kubo relation, according to which the shear viscosity is given by the retarded correlator of the traceless part of viscous energy-momentum tensor. We approximate this retarded correlator using a one-loop skeleton expansion, and express the bosonic and fermionic shear viscosities, $\eta_{b}$ and $\eta_{f}$, in terms of bosonic and fermionic spectral widths, $\Gamma_{b}$ and $\Gamma_{\pm}$. Here, the subscripts $\pm$ correspond to normal and collective (plasmino) excitations of fermions. We study, in particular, the effect of these excitations on thermal properties of $\eta_{f}[\Gamma_{\pm}]$. To do this, we determine first the dependence of $\Gamma_{b}$ and $\Gamma_{\pm}$ on momentum $p$, temperature $T$, chemical potential $\mu$ and $\xi_{0}\equiv m_{b}^{0}/m_{f}^{0}$, in a one-loop perturbative expansion in the orders of the Yukawa coupling. Here, $m_{b}^{0}$ and $m_{f}^{0}$ are $T$ and $\mu$ independent bosonic and fermionic masses, respectively. We then numerically determine $\eta_{b}[\Gamma_{b}]$ and $\eta_{f}[\Gamma_{\pm}]$, and study their thermal properties. It turns out that whereas $\Gamma_{b}$ and $\Gamma_{+}$ decrease with increasing $T$ or $\mu$, $\Gamma_{-}$ increases with increasing $T$ or $\mu$. This behavior qualitatively changes by adding thermal corrections to $m_{b}^{0}$ and $m_{f}^{0}$, while the difference between $\Gamma_{+}$ and $\Gamma_{-}$ keeps increasing with increasing $T$ or $\mu$. Moreover, $\eta_{b}$ ($\eta_{f}$) increases (decreases) with increasing $T$ or $\mu$. We show that the effect of plasminos on $\eta_{f}$ becomes negligible with increasing (decreasing) $T$ ($\mu$)., Comment: V1: 25 pages, 18 figures; V2: 26 pages, 18 figures. Discussions improved, references added. Version accepted for publication in PRD

The directional weak decay constants of neutral pions are determined at finite temperature T, chemical potential \mu and in the presence of a constant magnetic field B. To do this, we first derive the energy dispersion relation of neutral pions from the corresponding effective action of a two-flavor, hot and magnetized Nambu--Jona-Lasinio model. Using this dispersion relation, including nontrivial directional refraction indices, we then generalize the PCAC relation of neutral pions and derive the Goldberger-Treiman (GT) as well as the Gell--Mann-Oakes-Renner (GOR) relations consisting of directional quark-pion coupling constant g_{qq\pi^{0}}^{(\mu)} and weak decay constant f_{\pi^{0}}^{(\mu)} of neutral pions. The temperature dependence of g_{qq\pi^{0}}^{(\mu)} and f_{\pi^{0}}^{(\mu)}, are then determined for fixed chemical potential and various constant background magnetic fields. The GT and GOR relations are also verified at finite T,\mu and eB. It is shown that, because of the explicit breaking of the Lorentz invariance by the magnetic field, the directional quark-pion coupling and decay constants of neutral pions in the longitudinal and transverse directions with respect to the direction of the external magnetic field are different, i.e. g_{qq\pi^{0}}^{\|}\neq g_{qq\pi^{0}}^{\perp} and f_{\pi^{0}}^{\|}\neq f_{\pi^{0}}^{\perp}. As it turns out, for fixed T,\mu and B, g_{qq\pi^{0}}^{\|}> g_{qq\pi^{0}}^{\perp} and f_{\pi^{0}}^{\|}< f_{\pi^{0}}^{\perp}., Comment: V1: 25 pages, 15 figures; V2: 27 pages, 15 figures, 1 table. Discussions improved, references updated. Version accepted for publication in PRD