---
author:
- |
    Lars Kjærgaard and Paul Mansfield\
    Department of Mathematical Sciences\
    University of Durham\
    South Road, Durham DH1 3LE, U.K.\
    \
title: 'Calculating the infra-red limit of quantum field theory using analyticity of correlation functions'
---

Introduction and summary
========================

It is important to get a better understanding of the infra-red limit of quantum field theory. Conventionally this is studied by extrapolating perturbation theory from the ultra-violet using the renormalisation group. The purpose of this paper is to show how to augment this approach with information derived from the analyticity properties of Green functions. These properties are well-known. For example, the Källen-Lehmann spectral representation of the two-point function of a scalar field shows that it has an analytic continuation to the complex momentum plane. We will use this to express the infra-red limit as an integral in the ultra-violet region where perturbation theory is applicable. For illustration assume that the physical quantity $F$ depends on some distance scale $s$, and that $F(s)$ is analytic in the complex plane with the negative axis cut away, then the contour integral $$\label{firsteq}
  \frac{1}{2\pi i}\int_{C}\frac{ds}{s}e^{\rho/s}F(s)=0
\nonumber$$ vanishes, where the contour $C$ is given in figure 1. This means that we can write 