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Optimized perturbation theory for charged scalar fields at finite temperature and in an external magnetic field

Symmetry restoration in a theory of a self-interacting charged scalar field at finite temperature and in the presence of an external magnetic field is examined. The effective potential is evaluated nonperturbatively in the context of the optimized perturbation theory method. It is explicitly shown that in all ranges of the magnetic field, from weak to large fields, the phase transition is second order and that the critical temperature increases with the magnetic field. In addition, we present an efficient way to deal with the sum over the Landau levels, which is of interest especially in the case of working with weak magnetic fields.