Collective modes of a massive fermion in a magnetized medium with finite anomalous magnetic moment

We calculate, in a systematic way, the general structure of the self-energy of light massive fermions and the effective propagator in a thermomagnetic medium with the inclusion of anomalous magnetic moment (AMM) of the fermion in the weak field approximation. It is found that the self-energy of a massive fermion in this case consists of five non-trivial structure factors in contrast to the massless case where the self-energy contains only four. We employ the real time formalism (RTF) of thermal field theory within the ambit of hard thermal loop (HTL) approximation in the evaluation of the structure factors. The collective modes are obtained from the poles of the effective propagator of the fermion. The investigation of the dispersion relations for non-degenerate ground state shows that the effect of the magnetic field is more for up quark than the down quark because of the larger charge of the former. The important observation is that in the first excited state the degeneracy, which exists for non-zero magnetic field is lifted due to the inclusion of the AMM. It is also observed that the first excited state becomes less dispersive compared to the case when AMM is not considered, whereas the second excited state becomes more dispersive when both the magnetic field and the AMM are non-zero in comparison to the case with vanishing AMM. These effects are observed in both particle and hole-like excitations. Qualitatively similar behaviour is also seen in the case of down quarks.