The properties of noninteracting σ and π 0 mesons are studied at finite temperature, chemical potential and in the presence of a constant magnetic field. To do this, the energy dispersion relations of these particles, including nontrivial form factors, are derived using a derivative expansion of the effective action of a two-flavor, hot and magnetized Nambu-Jona-Lasinio (NJL) model up to second order. The temperature dependence of the pole and screening masses as well as the directional refraction indices of magnetized neutral mesons are explored for fixed magnetic fields and chemical potentials. It is shown that, because of the explicit breaking of the Lorentz invariance by the magnetic field, the refraction index and the screening mass of neutral mesons exhibit a certain anisotropy in the transverse and longitudinal directions with respect to the direction of the external magnetic field. In contrast to their longitudinal refraction indices, the transverse indices of the neutral mesons are larger than unity.

The axial anomaly of the noncommutative U(1) gauge theory is calculated by a number of methods and compared with the commutative one. It is found to be given by the corresponding Chern class.

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 µ, 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 T Sch =κQeB in the energy dispersion relation of quarks. Here,Q is the quark charge matrix. We consider three different sets forκ, and numerically determine the dependence of the constituent quark mass on T, µ and eB for fixedκ. By exploring the complete phase portrait of this model in T -µ, µ-eB, and T -eB phase spaces for various fixed eB, T , µ andκ, we observe that inverse magnetic catalysis occurs for large enoughκ. Moreover, in the regime of weak magnetic fields, the phenomenon of reentrance of chiral symmetry broken and restored phases occurs for T, µ and eB dependentκ.

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 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 (µ)qqπ 0 and the weak decay constant f (µ) π 0 of neutral pions. The temperature dependence of g (µ)qqπ 0 and f (µ) π 0 are then determined for fixed chemical potential and various constant background magnetic fields. The GT and GOR 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 qqπ 0 = g ⊥ qqπ 0 and f π 0 = f ⊥ π 0 . As it turns out, for fixed T, µ and B, g qqπ 0 > g ⊥ qqπ 0and f π 0 < f ⊥ π 0 .