Feynman rules for the rational part of the Electroweak 1-loop amplitudes in the R_xi gauge and in the Unitary gauge

TL;DR

The paper derives a complete set of Feynman rules for the rational part ${\rm R_2}$ of electroweak one-loop amplitudes in the general renormalizable $R_\xi$ gauge and in the Unitary gauge, complementing prior 't Hooft–Feynman results. Using the OPP/Generalized Unitarity framework, it expresses ${\rm R_2}$ through tree-level-like effective vertices up to four external legs, and provides explicit classifications and coefficients for 2-, 3-, and 4-point bosonic vertices; fermionic contributions are deferred to prior results. The authors perform extensive checks, confirming gauge independence of $R=R_1+R_2$ in self-energies and in the $H\to\gamma\gamma$ process, and demonstrating consistent limits when taking $\xi\to\infty$ in the Unitary gauge. The full results are made available as FORM form files, enabling seamless integration into 1-loop calculators across gauges and improving numerical stability through cross-gauge consistency tests.

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 formulae are given both in the R_xi gauge and in the Unitary gauge, therefore completing the results in the 't Hooft-Feynman gauge already presented in a previous publication. As a consistency check, we verified, in the case of the process H -> gamma gamma and in a few other physical cases, the independence of the total Rational Part R_1+R_2 on the chosen gauge. In addition, we explicitly checked the equivalence of the limits xi -> infinity after or before the loop momentum integration in the definition of the Unitary gauge at 1-loop.

Figures (3)

• Figure 1: All possible 2-point vertices.
• Figure 2: All possible non vanishing 3-point vertices.
• Figure 3: All possible non vanishing 4-point vertices.