O(N)-symmetric λφ4theory: The Gaussian-effective-potential approach

Abstract

The O(N)-symmetric generalization of h94 theory is studied in the Gaussian-effective-potential (GEP) approach. It is shown that the GEP encompasses and transcends the leading-order 1 / N expansion results. We find two distinct, nontrivial versions of the (3 + 1)-dimensional theory: (i) "precarious 94 theory," with negative, infinitesimal AB, which coincides with the 1 / N result; and (ii) "autonomous b4 theory," with positive, infinitesimal hB and an infinite wave-function renormalization, which is inaccessible to perturbative or 1 / N methods, and could well have eluded lattice-based approaches. "Autonomous d4 theory" can exhibit spontaneous symmetry breaking, and we speculate on its relevance for the Higgs mechanism.

Effective Hamiltonian for scalar theories in the Gaussian approximation

“…In previous Hamiltonian based work on the O(2)-model a finite Goldstone boson mass m(2) ≃ m(1)/2.06 [9] is obtained, in contradiction to Goldstone's theorem. We will show that a symmetric Gaussian ansatz and a further rediagonalization of the effective Hamiltonian exactly gives a zero mass Goldstone excitation.…”

“…Although spontaneous symmetry breaking for the O(N)-model has been discussed before in the framework of the Gaussian Effective Potential [9], only massive excitations were found. This violation of Goldstone's theorem was ascribed to symmetry breaking "at the operator level", i.e.…”

“…Of course, particles and anti-particles should have the same mass. The problems concerning the ansatz with two mass parameters [9] will be discussed in more detail in section (6).…”

“…Much effort has been put into the renormalization of the Gaussian Effective Potential of scalar λφ 4 [5,7,9]. Generally, it is believed that after the regulator is removed, λφ 4 -theory becomes trivial (see e.g.…”

“…We do not consider here the so called "autonomous" renormalization scheme, which has been developed in ref. [9] and critically examined in ref. [15].…”