Critical Temperature Associated to Symmetry Breaking of Klein--Gordon fields versus Condensation Temperature in a Weakly interacting Bose--Einstein Gas

We deduce the relation between the critical temperature associated to the symmetry breaking of scalar fields with one--loop correction potential immersed in a thermal bath and the condensation temperature of the aforementioned system, assuming a harmonic oscillator type potential. We show that these two temperatures are related through the \emph{scale} associated to the system. In this aim, we infer the order of magnitude for the \emph{scale} as a function of the corresponding healing length, in order to give a criterium to compare both temperatures. Additionally, we prove that the condensation temperature is independent of the thermal bath within the semiclassical approximation, for a positive coupling constant, assuming that the thermal bath contribution is the lowest energy associated to the system.