Two-loop background field calculations for arbitrary background fields

The ultraviolet divergences of two-loop vacuum graphs in the presence of an arbitrary background field are determined for four-dimensional φ⁴ and gauge theories on flat space. Dimensional regularisation is employed and the heat kernel is used to analyse the short-distance singularities in the products of propagators, in the presence of the background field, that occur for two-loop graphs. The single and double poles in ? = 4 ? d are determined in a concise fashion, giving known results for the β function. A procedure for determining the remaining finite parts in terms of explicit convergent integrals in four dimensions is described.