Note on the Intermediate Field Representation of Phi^2k Theory in Zero Dimension

TL;DR

This paper refines the intermediate field representation of Phi^2k theory in zero dimension, correcting previous errors, proving Borel-Leroy summability, and offering an improved Gaussian integral-based representation for k=3 in the complex case.

Contribution

It provides a corrected and detailed proof of Borel-Leroy summability and introduces a new convergent Gaussian integral representation for Phi^6 theory in zero dimension.

Findings

Corrected previous misprints in the representation

Proved Borel-Leroy summability for the theory

Developed an improved Gaussian integral representation for k=3

Abstract

This expanded version corrects some misprints of the first version, details completely the poof of Borel-Leroy summability and for $k=3$ in the complex case provides a new improved representation which relies on ordinary convergent Gaussian integrals rather than oscillatory integrals.