Phase diagram for charged scalars in a magnetic field at finite temperature

We investigate the nature of the phase transition for charged scalars in the presence of a magnetic background for a theory with spontaneous symmetry breaking. We perform a careful treatment of the negative mass squared as a function of the order parameter and present a suitable method to obtain magnetic and thermal corrections up to ring order for the high temperature limit and the case where the magnetic field strength is larger than the absolute value of the square of the mass parameter. We show that for a given value of the self-coupling, the phase transition is first order for a small magnetic field strength and becomes second order when the magnetic field strength grows. We also show that the critical temperature in the presence of the magnetic field is always below the critical temperature for the case where the field is absent.