Effective potential and mass behavior of a self-interacting scalar field theory due to thermal and external electric and magnetic fields effects

In this article we address the subject of finding the behavior of a charged scalar field $\phi$ under the influence of external constant magnetic and electric fields, perpendicular to each other, including also thermal effects. For this purpose we derive an expression for the corresponding bosonic propagator. As an application, we explore, in the weak field sector, the mass correction for the self interacting $\lambda \phi ^4$ theory. Our results show that the mass diminishes when the magnetic field appears, for small values of temperature, staring to increase then when the strength of the field rises. In the case when we have only an electric field, the mass always grow with the field intensity. We also analyze the phase diagram associated to spontaneous symmetry breaking of the theory finding inverse magnetic catalysis (IMC) or inverse electric catalysis (IEC) for the cases where only a magnetic field or only an electric field are present, respectively. In both cases, taken separately, we have a scenario where the critical temperature associate to symmetry restoration diminishes as function of the corresponding field strengths. A similar situation happens when both type of fields are simultaneously present. We have dubbed this case as inverse magnetic -electric catalysis (IMEC). In this situation, both fields cooperate for the occurrence of IMEC.