A Variational Perturbation Approach to One-Point Functions in QFT

TL;DR

This paper introduces a variational perturbation method for calculating vacuum expectation values in quantum field theories, providing a practical Gaussian smearing formula and validating results against known exact solutions.

Contribution

The paper develops a second-order variational perturbation scheme with Gaussian smearing formulae for local field VEVs in QFT, applicable to sine- and sinh-Gordon models.

Findings

Second-order VP results agree with conjectured exact formulas

Gaussian smearing simplifies path integral calculations

Demonstrates non-perturbability of the VP scheme

Abstract

In this paper, we develop a variational perturbation (VP) scheme for calculating vacuum expectation values (VEVs) of local fields in quantum field theories. For a comparatively general scalar field model, the VEV of a comparatively general local field is expanded and truncated at second order in the VP scheme. The resultant truncated expressions (we call Gaussian smearing formulae) consist mainly of Gaussian transforms of the local-field function, the model-potential function and their derivatives, and so can be used to skip calculations on path integrals in a concrete theory. As an application, the VP expansion series of the VEV of a local exponential field in the sine- and sinh-Gordon field theories is truncated and derived up to second order equivalently by directly performing the VP scheme, by finishing ordinary integrations in the Gaussian smearing formulae, and by borrowing Feynman diagrammatic technique, respectively. Furthermore, the one-order VP results of the VEV in the two-dimensional sine- and sinh-Gordon field theories are numerically calculated and compared with the exact results conjectured by Lukyanov, Zamolodchikov $et al.$, or with the one-order perturbative results obtained by Poghossian. The comparisons provide a strong support to the conjectured exact formulae and illustrate non-perturbability of the VP scheme.