Two-loop vacuum amplitudes in a non-Abelian gauge theory on S4

Abstract

In a series of papers, the authors have developed a matrix-element approach to quantum field theory on S4, and used a complex l-plane method of performing the resulting angular momentum sums. The methods have been used to reproduce previous results for both one- and two-loop vacuum amplitudes for an Abelian gauge theory, including Dirac fermions and scalar bosons. In this paper we extend the formalism to non-Abelian gauge theory and show how the new complex l-plane methods allow the calculation of two-loop vacuum amplitudes involving ghost and gauge fields. We show how it is possible to overcome previous calculational limitations, resulting from the form of the ghost and gauge propagators, by performing the entire calculation in angular momentum space without reference to any particular configuration-space form of the propagators. We derive, for the first time, the two-loop interacting vacuum amplitude in a non-Abelian gauge theory on a curved background and give the required counterterm to renormalize the background gravitational action on S4.