Feynman rules for the rational part of the Electroweak 1-loop amplitudes

TL;DR

This paper provides comprehensive Feynman rules for the rational parts of one-loop electroweak amplitudes, enabling fully automated calculations within the Standard Model using the OPP method.

Contribution

It introduces the complete set of R2 Feynman rules for electroweak 1-loop calculations, verified across multiple gauges and regularization schemes, facilitating automation.

Findings

Validated Ward identities up to 4-points for rational amplitudes

Confirmed gauge invariance of Four Dimensional Helicity scheme in the Standard Model

Provided explicit formulae for automated 1-loop computations

Abstract

We present the complete set of Feynman rules producing the rational terms of kind R_2 needed to perform any 1-loop calculation in the Electroweak Standard Model. Our results are given both in the 't Hooft-Veltman and in the Four Dimensional Helicity regularization schemes. We also verified, by using both the 't Hooft-Feynman gauge and the Background Field Method, a huge set of Ward identities -up to 4-points- for the complete rational part of the Electroweak amplitudes. This provides a stringent check of our results and, as a by-product, an explicit test of the gauge invariance of the Four Dimensional Helicity regularization scheme in the complete Standard Model at 1-loop. The formulae presented in this paper provide the last missing piece for completely automatizing, in the framework of the OPP method, the 1-loop calculations in the SU(3) X SU(2) X U(1) Standard Model.