Renormalization of a gapless Hartree-Fock approximation to a theory with spontaneously broken O(N)-symmetry

The renormalization of a gapless Phi-derivable Hartree--Fock approximation to the O(N)-symmetric lambda*phi^4 theory is considered in the spontaneously broken phase. This kind of approach was proposed by three of us in a previous paper in order to preserve all the desirable features of Phi-derivable Dyson-Schwinger resummation schemes (i.e., validity of conservation laws and thermodynamic consistency) while simultaneously restoring the Nambu--Goldstone theorem in the broken phase. It is shown that unlike for the conventional Hartree--Fock approximation this approach allows for a scale-independent renormalization in the vacuum. However, the scale dependence still persists at finite temperatures. Various branches of the solution are studied. The occurrence of a limiting temperature inherent in the renormalized Hartree--Fock approximation at fixed renormalization scale mu is discussed.