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

The renormalization of a gapless Φ-derivable Hartree–Fock approximation to the O(N)-symmetric λφ theory is considered in the spontaneously broken phase. This kind of approach was proposed by three of us in a previous paper [1] in order to preserve all the desirable features of Φ-derivable DysonSchwinger 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 μ is discussed.