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In this work, dispersion relations of π 0 and π ± have been studied in vacuum in the limit of weak external magnetic field using a phenomenological pion-nucleon (πN ) Lagrangian. For our purpose, we have calculated the results up to one loop order in self energy diagrams with the pseudoscalar (P S) and pseudovector (P V ) pion-nucleon interactions. By assuming weak external magnetic field it is seen that the effective mass of pion gets explicit magnetic field dependence and it is modified significantly for the case of PS coupling. However, for the PV coupling, only a modest increase in the effective mass is observed. These modified dispersion relations due to the presence of the external field can have substantial influence in the phenomenological aspect of the mesons both in the context of neutron stars as well as relativistic heavy ion collisions. * souvikpriyam.adhya@saha.ac.in †

A phenomenological pion-nucleon interaction is used to obtain pionic mass modification in presence of constant homogeneous magnetic field background at finite temperature and chemical potential in the real time formalism of thermal field theory. The magnetically modified propagator in its complete form is used to obtain the one loop self-energy for pions. For charged pions we find that the effective mass increases with the magnetic field at given temperature and chemical potential. Since the transverse momentum of charged pion is quantized and its contribution to Dyson-Schwinger Equation is large compared to the loop correction, the charged pion mass remains constant with both temperature and chemical potential for a given landau level. In order to unveil the role of the real part of the self-energy, we also calculate the effective mass neglecting the trivial shift. The effective mass for charged pions shows an oscillatory behaviour which is attributed to the thermal contribution of the self-energy. It is argued that the magnetic field dependent vacuum contribution to the self-energy influences the behaviour of the effective mass both qualitatively and quantitatively. We also find that very large field is necessary for neutral pions to condense.