# Two-Loop Corrections to Large Order Behavior of $\varphi^4$ Theory

## TL;DR

This paper analyzes the large order behavior of the scalar  theory's perturbative series, computing two-loop corrections in two and three dimensions, with detailed numerical evaluation of eigenvalues.

## Contribution

It provides the first two-loop order calculations of the large order behavior in  theory, including renormalization and numerical eigenvalue analysis.

## Key findings

- Explicit two-loop corrections to large order behavior
- Numerical eigenvalues and eigenfunctions for the analysis
- Renormalization treatment for higher-dimensional cases

## Abstract

We consider the large order behavior of the perturbative expansion of the scalar $\varphi^4$ field theory in terms of a perturbative expansion around an instanton solution. We have computed the series of the free energy up to two-loop order in two and three dimension. Topologically there is only an additional Feynman diagram with respect to the previously known one dimensional case, but a careful treatment of renormalization is needed. The propagator and Feynman diagrams were expressed in a form suitable for numerical evaluation. We then obtained explicit expressions summing over $O(10^3)$ distinct eigenvalues determined numerically with the corresponding eigenfunctions.