Perturbative renormalization of the lattice regularized $\phi_4^4$ with flow equations

TL;DR

This paper uses flow equations to rigorously prove the renormalizability of 4D lattice $\,\phi_4^4$ theory and demonstrates the restoration of Euclidean symmetry in the continuum limit.

Contribution

It provides a rigorous proof of renormalizability for lattice $\,\phi_4^4$ theory and shows how Euclidean symmetry is restored using flow equations.

Findings

Proof of all-order renormalizability of lattice $\,\phi_4^4$ theory.

Demonstration of Euclidean symmetry restoration.

Control over large momentum and infrared behaviors.

Abstract

The flow equations of the renormalization group allow to analyse the perturbative $n$-point functions of renormalizable quantum filed theories. Rigorous bounds implying renormalizability permit to control large momentum behaviour, infrared singularities and large order behaviour in the number of loops and the number of arguments $n$. In this paper, we analyse the Euclidean $4$-dimensional massive $\phi_4$-theory using lattice regularization. We present a rigorous proof that this quantum field theory is renormalizable, to all orders of the loop expansion based on the flow equations. The lattice is known to break the Euclidean symmetry of the space-time. Our main result is the proof of the restoration of the Euclidean symmetries using the flow equations.