Infrared and Ultraviolet Behaviour of Effective Scalar Field Theory

TL;DR

This paper investigates the renormalization properties of a scalar field theory with Z2 symmetry, demonstrating its renormalizability in both massive and massless cases across different momentum regimes, and establishing bounds on Green's functions.

Contribution

It extends previous work by proving the renormalizability of the massless effective scalar field theory away from exceptional momenta and establishes bounds on Green's functions.

Findings

Massless theory is renormalizable away from exceptional momenta.

Established bounds on Green's functions approaching exceptional Euclidean momenta.

Proved Weinberg's Theorem for the massive effective scalar theory.

Abstract

We consider the infrared and ultraviolet behaviour of the effective quantum field theory of a single $Z_2$ symmetric scalar field. In a previous paper we proved to all orders in perturbation theory the renormalizability of massive effective scalar field theory using Wilson's exact renormalization group equation. Here we show that away from exceptional momenta the massless theory is similarly renormalizable, and we prove detailed bounds on Green's functions as arbitrary combinations of exceptional Euclidean momenta are approached. As a corollary we also prove Weinberg's Theorem for the massive effective theory, in the form of bounds on Green's functions at Euclidean momenta much greater than the particle mass but below the naturalness scale of the theory.