Rational terms in two-loop calculations

TL;DR

This paper extends the renormalisation procedure to two loops, enabling the construction of four-dimensional numerators and full reconstruction of rational terms from universal counterterms, with applications to QED.

Contribution

It introduces a method to compute two-loop rational counterterms from massive tadpole integrals, extending the concept of rational terms to two loops.

Findings

Full set of two-loop rational counterterms for QED provided.

Method to derive counterterms from one-scale tadpole integrals established.

Rational terms of type R2 extended to two-loop calculations.

Abstract

We present an extension of the renormalisation procedure based on the R-operation in $D$ dimensions at two-loop level, in which the numerators of all Feynman diagrams can be constructed in four dimensions, and the rational terms stemming from the interplay of $(D-4)$-dimensional numerator parts and UV poles are fully reconstructed from a finite set of universal local counterterms. This represents an extension of the concept of rational terms of type $R_2$ to two loops. We provide a general method to compute one and two-loop rational counterterms from massive one-scale tadpole integrals. Finally, we present the full set of rational counterterms of UV origin for QED up to two-loop order.