# Simple non-perturbative resummation schemes beyond mean-field: case study for scalar $\phi^4$ theory in 1+1 dimensions

## TL;DR

This paper introduces a systematic non-perturbative resummation method for scalar 4 theory in 1+1 dimensions that improves upon mean-field approximation, is computationally simple, and shows promising results near criticality.

## Contribution

A new non-perturbative resummation scheme that extends mean-field theory and is easily applicable to various quantum field theories.

## Key findings

- Approximate agreement with known vacuum energy and mass gap results.
- Numerically simple calculation involving nested one-loop integrals.
- Potential for generalization to other dimensions, fermionic fields, and finite temperature.

## Abstract

I present a sequence of non-perturbative approximate solutions for scalar $\phi^4$ theory for arbitrary interaction strength, which contains, but allows to systematically improve on, the familiar mean-field approximation. This sequence of approximate solutions is apparently well-behaved and numerically simple to calculate since it only requires the evaluation of (nested) one-loop integrals. To test this resummation scheme, the case of $\phi^4$ theory in 1+1 dimensions is considered, finding approximate agreement with known results for the vacuum energy and mass gap up to the critical point. Because it can be generalized to other dimensions, fermionic fields and finite temperature, the resummation scheme could potentially become a useful tool for calculating non-perturbative properties approximately in certain quantum field theories.