Rational terms of UV origin at two loops

• Published in: The journal of high energy physics Vol. 2020; no. 5; pp. 1 - 37
• Main Authors: Pozzorini, Stefano, Zhang, Hantian, Zoller, Max F.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.05.2020; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Automation; Classical and Quantum Gravitation; Elementary Particles; High energy physics; Integrals; Mathematical analysis; Perturbative QCD; Physics; Physics and Astronomy; Poles; Precision QED; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Subtraction; Precision QED; Perturbative QCD
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP05(2020)077

A bstract The advent of efficient numerical algorithms for the construction of one-loop amplitudes has played a crucial role in the automation of NLO calculations, and the development of similar algorithms at two loops is a natural strategy for NNLO automation. Within a numerical framework the numerator of loop integrals is usually constructed in four dimensions, and the missing rational terms, which arise from the interplay of the ( D − 4)-dimensional parts of the loop numerator with 1 / ( D − 4) poles in D dimensions, are reconstructed separately. At one loop, such rational terms arise only from UV divergences and can be restored through process-independent local counterterms. In this paper we investigate the behaviour of rational terms of UV origin at two loops. The main result is a general formula that combines the subtraction of UV poles with the reconstruction of the associated rational parts at two loops. This formula has the same structure as the R-operation, and all poles and rational parts are described through a finite set of process-independent local counterterms. We also present a general formula for the calculation of all relevant two-loop rational counterterms in any renormalisable theory based on one-scale tadpole integrals. As a first application, we derive the full set of two-loop rational counterterms for QED in the R ξ -gauge.