Perturbation theory for the $\Phi^4_3$ measure, revisited with Hopf algebras

TL;DR

This paper provides a concise proof that the renormalised $\

Contribution

It introduces a novel approach combining Wiener chaos, Hopf algebras, and BPHZ renormalisation to analyze the $\

Findings

Asymptotic expansion of the partition function is established.

Coefficients of the expansion converge as the UV cut-off is removed.

Exploration of Borel summability of the series.

Abstract

We give a relatively short, almost self-contained proof of the fact that the partition function of the suitably renormalised $\Phi^4_3$ measure admits an asymptotic expansion, the coefficients of which converge as the ultraviolet cut-off is removed. We also examine the question of Borel summability of the asymptotic series. The proofs are based on Wiener chaos expansions, Hopf-algebraic methods, and bounds on the value of Feynman diagrams obtained through BPHZ renormalisation.