Euclidean coordinate-space perturbation theory with a single mass scale

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.