Effective potential at finite temperature in a constant magnetic field: Ring diagrams in a scalar theory

We study symmetry restoration at finite temperature in the theory of a charged scalar field interacting with a constant, external magnetic field. We compute the finite-temperature effective potential including the contribution from ring diagrams. We show that in the weak field case, the presence of the field produces a stronger first order phase transition and that the temperature for the onset of the transition is lower, as compared to the case without magnetic field.