Euclidean coordinate-space perturbation theory with a single mass scale

TL;DR

This paper develops coordinate-space perturbation theory for massive quantum field theories, providing analytic expressions for correlation functions and propagators, useful for finite temperature, volume, and lattice QCD calculations.

Contribution

It introduces a systematic approach using Gegenbauer polynomial expansion to derive analytic expressions for three-point functions involving massive and massless fields in Euclidean space.

Findings

Derived analytic expressions for three-point correlation functions.

Computed the one-loop coordinate-space propagator at finite temperature.

Systematically studied antiderivatives of Bessel function products.

Abstract

We develop elements of coordinate-space perturbation theory for massive quantum field theories in general $d$-dimensional Euclidean space. Using the expansion in Gegenbauer polynomials, we provide analytic expressions for several three-point correlation functions in theories with one massive and one massless field. To this end, a class of antiderivatives of products of two Bessel functions multiplied by a power of their common argument are studied systematically. We expect these results to be useful in perturbative calculations involving vertices of high degree, at finite temperature and/or in finite volume, as well as in auxiliary perturbative computations for a lattice QCD treatment of hadronic effects in precision observables. As an illustration, we compute the one-loop coordinate-space propagator of a massive particle coupled to a massless one, both in the vacuum and at finite temperature.