Path integral formulation of mean-field perturbation theory

Abstract

We develop a convenient functional integration method for performing mean-field approximations in quantum field theories. This method is illustrated by applying it to a self-interacting φ 4 scalar field theory and a J μ J μ four-Fermion field theory. To solve the φ 4 theory we introduce an auxiliary field χ and rewrite the Lagrangian so that the interaction term has the form χφ 2. The vacuum generating functional is then expressed as a path integral over the fields χ and φ. Since the χ field is introduced to make the action no more than quadratic in φ, we do the φ integral exactly. Then we use Laplace's method to expand the remaining χ integral in an asymptotic series about the mean field χ 0. We show that there is a simple diagrammatic interpretation of this expansion in terms of the mean-field propagator for the elementary field φ and the mean-field bound-state propagator for the composite field χ. The φ and χ propagators appear in these diagrams with the same topological structure that would have been obtained by expanding in the same manner a χφ 2 field theory in which χ and φ are both elementary fields. We therefore argue that by renormalizing these theories so that the mean-field propagators are equivalent, the two theories are described by the same renormalized Green's functions containing the same three parameters, μ 2, m 2, and g. The quartic theory is completely specified by the renormalized masses μ 2 and m 2 of the χ and φ fields. These two masses determine the coupling constant g = g( μ 2, m 2). The cubic theory depends on μ 2 and m 2 and a third parameter g 0, g = g( μ 2, m 2, g 0), where g 0 is the bare coupling constant. We indicate that g( μ 2, m 2, g 0) ⩽ g( μ 2, m 2) with equality obtained only in the limit g 0 → ∞. When g 0 → ∞ the wave function renormalization constant for the χ field in the cubic theory vanishes, and the cubic theory becomes identical to the quartic theory. Our approach guarantees that all quartic theories have the same graphical topology in the mean-field approximation. To illustrate this we show that the mean-field expansion of the four-Fermion current-current interaction theory is renormalizable and reproduces the results of the usual vector meson theory. A coupling-constant eigenvalue condition is derived which could serve to distinguish current-current interactions from normal electrodynamics.