Rational Terms of UV Origin at Two Loops

• Published in: arXiv.org
• Main Authors: Pozzorini, Stefano, Zhang, Hantian, Zoller, Max F
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 04.06.2020
• Subjects: Algorithms; Automation; Integrals; Mathematical analysis; Physics - High Energy Physics - Phenomenology; Physics - High Energy Physics - Theory; Poles; Subtraction
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2001.11388

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 \(R_{\xi}\)-gauge.