Techniques for n-Particle Irreducible Effective Theories

TL;DR

This paper demonstrates that skeleton diagrams in m-Loop nPI effective actions represent an infinite, non-double-counting resummation of perturbative diagrams, and that their equations of motion align with Schwinger-Dyson equations up to symmetry-consistent order.

Contribution

It provides a diagrammatic method to derive the 5-Loop 5PI effective action for scalar theories with cubic and quartic interactions, confirming key theoretical properties.

Findings

Skeleton diagrams correspond to an infinite resummation without double counting.

Variational equations match Schwinger-Dyson equations up to symmetry-consistent order.

Derived the 5-Loop 5PI effective action for scalar theories.

Abstract

In this paper we show that the skeleton diagrams in the m-Loop nPI effective action correspond to an infinite resummation of perturbative diagrams which is void of double counting at the m-Loop level. We also show that the variational equations of motion produced by the n-Loop nPI effective theory are equivalent to the Schwinger-Dyson equations, up to the order at which they are consistent with the underlying symmetries of the original theory. We use a diagrammatic technique to obtain the 5-Loop 5PI effective action for a scalar theory with cubic and quartic interactions, and verify that the result satisfies these two statements.