A Convergent Kinetic-Term Perturbation Expansion for $\phi^4$ Theory

TL;DR

This paper introduces a novel convergent perturbation expansion for scalar  theory that reorganizes the series around the kinetic operator, offering potential advantages for nonperturbative analysis.

Contribution

It develops a reorganized perturbation framework treating the kinetic term as the perturbation, extending it to higher dimensions and lattice models, and compares it with traditional methods.

Findings

The series converges for positive coupling .

Explicit calculations match numerical evaluations.

The expansion relates to weak and strong coupling approaches.

Abstract

We revisit scalar $\phi^4$ theory and construct a reorganized perturbative expansion in which the kinetic operator, rather than the quartic interaction, is treated as the perturbation. Starting from the exactly solvable $0$-dimensional model, we show that the resulting series is convergent for positive coupling and can be written as an expansion in negative powers of the quartic coupling $\lambda$. We extend the construction to higher-dimensional field theory using an auxiliary field, and we formulate a discrete lattice version in which multi-site contributions are systematically organized. We explicitly compute the leading terms in the expansion, study their continuum limit, and compare against brute-force numerical evaluations of the partition function. We discuss the relation of this expansion to standard weak-coupling perturbation theory, strong-coupling expansions, and resummation techniques, and we outline possible applications to nonperturbative studies of scalar field theories.