Broken phase scalar effective potential and Φ-derivable approximations

We study the effective potential of a real scalar φ4 theory as a function of the temperature T within the simplest Φ-derivable approximation, namely, the Hartree approximation. We apply renormalization at a "high" temperature T_* where the theory is required to be in its symmetric phase and study how the effective potential evolves as the temperature is lowered down to T=0. In particular, we prove analytically that no second order phase transition can occur in this particular approximation of the theory, in agreement with earlier studies based on the numerical evaluation or the high temperature expansion of the effective potential. This work is also an opportunity to illustrate certain issues on the renormalization of Φ-derivable approximations at finite temperature and nonvanishing field expectation value and to introduce new computational techniques which might also prove useful when dealing with higher order approximations.