Dependence of the critical temperature on the Higgs field parametrization

We show that, despite the reparametrization symmetry of the Lagrangian describing the interaction between a scalar field and gauge vector bosons, the dynamics of the Higgs mechanism can be really affected by the representation gauge chosen for the Higgs field at temperatures T not equal 0. This can be the case when the physical system described by the Higgs field chooses to undergo a condensation with a given degree of freedom rather than with other possible ones, as apparently observed in exotic superconductor systems. Actually, we study in detail such peculiar conditions and find that, varying the parametrization (without changing any physical parameter) for the two degrees of freedom of the complex scalar field, we obtain different expressions for the Higgs mass at T not equal 0: in its turn this entails different expressions for the critical temperatures, ranging from zero to a maximum value, as well as different expressions for other basic thermodynamical quantities.