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 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(qq pi 0)((mu)) and the 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 Goldberger-Treiman and Gell-Mann-Oakes-Renner 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)(parallel to) not equal g(qq pi 0)(perpendicular to) and f(pi 0)(parallel to) not equal f(pi 0)(perpendicular to). As it turns out, for fixed T, mu and B, g(qq pi 0)(parallel to) > g(qq pi 0)(perpendicular to) and f(pi 0)(parallel to) < f(pi 0)(perpendicular to).