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

Abstract

The renormalization of a gapless Φ-derivable Hartree-Fock approximation to the O(N )-symmetric λφ 4 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.

Renormalisability of the 2PI-Hartree approximation of multicomponent scalar models in the broken symmetry phase

“…We plan to investigate the relation of the presented renormalisation scheme, which requires multiple coupling counterterms with the symmetric and mass-independent renormalisation which in the "symmetryzed" approximation of [19] to the O(N) model was proven to have the counterterm structure expected from the perturbation theory. Another interesting direction will be to see to what extent the proposed single-step procedure can be incorporated in the renormalisation of the momentum-dependent truncations of the 2PI approximation recently studied in [9,1,10,31].…”

“…one finds different cut-off dependences for δλ a 0 and δλ B 0 . In view of the fact that for this modified functional a symmetric and mass-independent renormalisation involving a single quartic counterterm was proven [19], finding the relation of the two renormalisation schemes is an interesting task for future investigation.…”

Symmetry-improved CJT effective action

“…Consequently, within our symmetry-improved CJT formalism, we predict a second-order phase transition already in the HF approximation, in agreement with theoretical expectations to all orders. This should be contrasted with the first-order phase-transition predicted in the HF approximation by the existing approaches [13,14,16].…”

“…In particular, for the case of global O(N ) symmetries that we will be studying here, a naive truncation of the CJT effective action violates the Goldstone theorem [7,8] by higher-order terms, giving rise to a massive Goldstone boson in the Spontaneous Symmetry Breaking (SSB) phase of the theory [9][10][11][12]. Thus far, several studies have been presented in the literature, attempting to provide a satisfactory solution to this problem [11][12][13][14][15][16][17][18][19][20].…”

“…A first attempt in restoring the Goldstone theorem within the CJT formalism was to add a phenomenological term to the 2PI effective action [13,14]. In the HF approximation, such an approach has as major drawback.…”