Two-particle irreducible effective actions versus resummation: analytic properties and self-consistency

TL;DR

This paper demonstrates that the main advantage of 2PI effective actions lies in their self-consistency, which enables better capturing of physical amplitude features and instabilities, surpassing traditional resummation techniques in a toy model.

Contribution

It reveals that the strength of 2PI approximations is due to self-consistency rather than diagram resummation, and introduces a hybrid 2PI-Padé method.

Findings

2PI approximations outperform Padé in the toy model.

Self-consistency captures amplitude branch points and instabilities.

Hybrid 2PI-Padé method improves resummation accuracy.

Abstract

Approximations based on two-particle irreducible (2PI) effective actions (also known as $\Phi$-derivable, Cornwall-Jackiw-Tomboulis or Luttinger-Ward functionals depending on context) have been widely used in condensed matter and non-equilibrium quantum/statistical field theory because this formalism gives a robust, self-consistent, non-perturbative and systematically improvable approach which avoids problems with secular time evolution. The strengths of 2PI approximations are often described in terms of a selective resummation of Feynman diagrams to infinite order. However, the Feynman diagram series is asymptotic and summation is at best a dangerous procedure. Here we show that, at least in the context of a toy model where exact results are available, the true strength of 2PI approximations derives from their self-consistency rather than any resummation. This self-consistency allows truncated 2PI approximations to capture the branch points of physical amplitudes where adjustments of coupling constants can trigger an instability of the vacuum. This, in effect, turns Dyson's argument for the failure of perturbation theory on its head. As a result we find that 2PI approximations perform better than Pad\'e approximation and are competitive with Borel-Pad\'e resummation. Finally, we introduce a hybrid 2PI-Pad\'e method.