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 \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}.