Instanton Structure from 325-Loop 3D $ϕ^3$ Perturbation Theory

Abstract

The high order behavior of perturbation theory provides insight into non-perturbative phenomenon in quantum field theory. While extensively studied in simplified models, high order behavior is challenging to access in more realistic field theories due to the complexity of the standard Feynman diagrammatic expansion. Building on recent progress in global sampling on the moduli space of tropical curves, we introduce an annealed importance sampling framework for tropical field theory. Our approach overcomes the exponential weight-variance barrier, allowing precise numerical calculations to high orders. We apply our technique to compute the perturbative series for the two-point function in the massive three-dimensional $ϕ^3$ theory through 325 loops. These coefficients determine the instanton action in $ϕ^3$ theory to 0.03% precision, confirm the collective-coordinate exponent, and extract the one-loop determinant and the instanton fluctuation corrections through four loops, in remarkable agreement with direct semiclassical calculations. Our approach is applicable to a variety of related field theories, facilitating the quantitative exploration of their high loop behavior.