# Renormalization of the 4PI effective action using the functional renormalization group

## TL;DR

This paper develops a renormalization method for the 4PI effective action at any order using the functional renormalization group, enabling non-perturbative studies of scalar theories beyond the lowest level.

## Contribution

It introduces a novel renormalization approach for n-particle irreducible effective actions at arbitrary order via the functional renormalization group.

## Key findings

- Successfully renormalized the 4-loop 4PI calculation in scalar theory.
- Used only bare parameters without counterterms.
- Demonstrated the method's applicability at higher orders.

## Abstract

Techniques based on $n$-particle irreducible effective actions can be used to study systems where perturbation theory does not apply. The main advantage, relative to other non-perturbative continuum methods, is that the hierarchy of integral equations that must be solved truncates at the level of the action, and no additional approximations are needed. The main problem with the method is renormalization, which until now could only be done at the lowest ($n$=2) level. In this paper we show how to obtain renormalized results from an $n$-particle irreducible effective action at any order. We consider a symmetric scalar theory with quartic coupling in four dimensions and show that the 4 loop 4-particle-irreducible calculation can be renormalized using a renormalization group method. The calculation involves one bare mass and one bare coupling constant which are introduced at the level of the Lagrangian, and cannot be done using any known method by introducing counterterms.