A convergent expansion about mean field theory. I. The expansion

We give a convergent expansion for nearly Gaussian quantum field theory in the multiphase region. The expansion combines (1) an expansion in phase boundaries, (2) a cluster expansion, and (3) a perturbation expansion to isolate dominant behavior. We study in detail the ground state of the P(φ)₂ = (λφ⁴ - φ² - μφ)₂ model, with ∥ μ ∥ ≤ λ² ≪ 1. The ground state is close to the classical free field, obtained by replacing P(φ) by the quadratic mean field polynomial P_c(φ), tangent to P at a global minimum. Selecting one minimum gives a pure phase (ergodic ground state) satisfying the Wightman-Osterwalder-Schrader axioms with a positive mass. We also establish analyticity in λ for μ = 0 in the sector ∥ Im λ ∥ < ε{lunate} Re λ ≪ 1, for ε{lunate} ≪ 1.