Rayleigh-Ritz Variational Approximation and Symmetry Nonrestoration

The investigation of symmetry nonrestoration scenarios has led to a controversy, with certain nonperturbative approximation schemes giving indications in sharp disagreement with those found within conventional perturbation theory. A Rayleigh-Ritz variational approach to the problem, which might be useful in bridging the gap between perturbative and nonperturbative viewpoints, is here proposed. As a first application, this approach is used in the investigation of a Z2×Z2-invariant thermal field theory with two scalar fields, placing particular emphasis on the region of parameter space that has been claimed to support symmetry nonrestoration. OUTP-96-44P hep-ph/9610262 July 1996 The subject of temperature-induced phase transitions[1-3] in relativistic quantum field theories has been extensively investigated over the last twenty years. In particular, transitions from a high-temperature symmetric phase to a low-temperature phase in which some symmetries are spontaneously broken are a crucial ingredient in most modern cosmological scenarios, and have been shown to be realized in large classes of thermal field theories. The possibility of symmetry nonrestoration (SNR) at high temperatures[3-7] (or transitions from a high-temperature broken-symmetry phase to a low-temperature symmetric phase[8]) could also have interesting phenomenological implications, most notably allowing to circumvent the monopole problem in certain Grand Unification Theories; however, the investigation of SNR scenarios has led to controversy, with certain nonperturbative approximation schemes giving indications in sharp disagreement with those found within conventional perturbation theory. Specifically, whereas perturbative analyses find that some models, possibly of phenomenological relevance, can support SNR upon appropriate (however ad hoc) choice of the available parameters, the corresponding analyses within certain nonperturbative approximation schemes[9] indicate that symmetry is inevitably restored. Recently, some progress has been made toward bridging the gap between perturbative and nonperturbative results on SNR. This has been attained via the use of improved perturbative techniques[10, 11] in which, while preserving the general structure of the perturbative expansion, some nonperturbative features of the theory are effectively taken into account. The preliminary results of this improved perturbative approaches have indicated[10, 11] that the conventional (unimproved) perturbative techniques overestimate the “SNR parameter space” (the region of parameter space capable of supporting SNR), and it is reasonable to interpret these results as suggesting that further improvements in the accuracy of the approximations would ultimately lead to the conclusion that the SNR parameter space is actually empty, just as predicted within the nonperturbative approximation schemes adopted in Ref.[9]. This expectation is encouraged by the findings of related studies on the lattice[12]. In this Letter, I discuss techniques which can be useful in making further progress in the direction proposed in the Refs.[10, 11], and, as a first application, which also serves as illustrative example, I use them in the investigation of the two-scalar-field theory of Euclidean Lagrange density L = 1 2 (∂μΦ)(∂ Φ) + 1 2 (∂μΨ)(∂ Ψ) + 1 2 mΦ + 1 2 ωΨ + λΦ 24 Φ + λΨ 24 Ψ − λΦΨ 4 ΦΨ , (1) which is Z2×Z2 invariant [(Φ → −Φ)× (Ψ → −Ψ)], and is among the strongest candidates as a model supporting SNR. Within conventional perturbation theory the model (1) is found to support SNR when