Phase transition in the Maxwell Chern-Simons theory with the Higgs term

We consider the Maxwell Chern-Simons theory with the Higgs spontaneous-symmetry-breaking term as a model of high-temperature superconductivity. We find the finite-temperature effective potential in the one-loop approximation. In order to obtain the well-defined real effective potential and critical temperature, we restrict ourselves to the limiting case that the quartic term of the Higgs field vanishes, and choose the Landau gauge. For this case, we renormalize the finite-temperature effective potential by applying the renormalization conditions of Kirzhnits and Linde. We find the condensate of the Higgs field and discuss its stability. In addition, we evaluate the critical temperature Tc of the phase transition into the U(1)-symmetry-preserving state from the U(1)-symmetry-broken state. The critical temperature Tc is determined by the balance between the Higgs effect and the Chern-Simons effect which make the gauge theory massive. It is an increasing function of the coefficient of the Chern-Simons term for the fixed mass of the Higgs field, and vice versa