Weak decay constant of neutral pions in a hot and magnetized quark matter

The directional weak decay constants of neutral pions are determined at finite temperature T, chemical potential μ 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 partially conserved axial vector current relation of neutral pions and derive the Goldberger-Treiman as well as the Gell--Mann-Oakes-Renner relations consisting of the directional quark-pion coupling constant g_qqπ⁰^(μ) and the weak decay constant f_π⁰^(μ) of neutral pions. The temperature dependence of g_qqπ⁰^(μ) and f_π⁰^(μ) are then determined for fixed chemical potential and various constant background magnetic fields. The Goldberger-Treiman and Gell--Mann-Oakes-Renner relations are also verified at finite T, μ 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_qqπ⁰^∥≠g_qqπ⁰^⊥ and f_π⁰^∥≠f_π⁰^⊥. As it turns out, for fixed T, μ and B, g_qqπ⁰^∥>g_qqπ⁰^⊥ and f_π⁰^∥<f_π⁰^⊥.