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 λϕ⁴ theory is considered in the spontaneously broken phase. This kind of approach was proposed by three of us in a previous paper [ Yu. B. Ivanov, F. Riek and J. Knoll Phys. Rev. D 71 105016 (2005)] in order to preserve all the desirable features of Φ-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 μ is discussed.