The 2PI effective action at four loop order in $\varphi^4$ theory

TL;DR

This paper extends the 2PI effective action calculations to four loops in $\

Contribution

It provides the first numerical verification of the renormalizability and counterterm structure of the 2PI effective theory at four loops in scalar $\

Findings

Four loop contributions can surpass three loop terms at strong coupling.

The 2PI approach shows signs of breakdown at four loops, indicating the need for higher nPI methods.

The renormalizability of 2PI at four loops is confirmed, with a unique counterterm structure.

Abstract

It is well known that perturbative pressure calculations show poor convergence. Calculations using a two particle irreducible (2PI) effective action show improved convergence at the 3 loop level, but no calculations have been done at 4 loops. We consider the 2PI effective theory for a symmetric scalar theory with quartic coupling in 4-dimensions. We calculate the pressure and two different non-perturbative vertices as functions of coupling and temperature. Our results show that the 4 loop contribution can become larger than the 3 loop term when the coupling is large. This indicates a breakdown of the 2PI approach, and the need for higher order $n$PI approximations. In addition, our results demonstrate the renormalizability of 2PI calculations at the 4 loop level. This is interesting because the counterterm structure of the 2PI theory at 4 loops is different from the structure at $n\le 3$ loops. Two vertex counterterms are required at the 4 loop level, but not at lower loop order. This unique feature of the 2PI theory has not previously been verified numerically.