Automatic generation of helicity amplitudes in Feynman-Diagram gauge

TL;DR

This paper introduces a method to automatically compute helicity amplitudes in Feynman-Diagram gauge for arbitrary models, streamlining calculations in high-energy physics and ensuring gauge invariance.

Contribution

The authors develop an automated approach to generate helicity amplitudes in FD gauge from FeynRules models, compatible with MadGraph5_aMC@NLO, including complex BSM operators.

Findings

FD gauge and U gauge yield identical cross sections.

High-energy amplitudes are dominated by higher-dimensional operators.

The method accurately reproduces known gauge invariance properties.

Abstract

We develop a method to calculate helicity amplitudes of an arbitrary tree-level process in Feynman-Diagram (FD) gauge for an arbitrary gauge model with MadGraph5_aMC@NLO. We start from the 't Hooft-Feynman gauge Lagrangian in FeynRules and generate scattering amplitudes by identifying the Goldstone boson as the $5^{\rm th}$ component of each weak boson. All the vertices of the 5-component weak bosons are then created automatically by assembling the relevant weak boson and Goldstone boson vertices in the Feynman gauge. The 5-component weak boson vertices are then connected by the $5\times5$ matrix propagator in the FD gauge. As a demonstration of the method we calculate the cross section for the process $\mu^-\mu^+\to\nu_\mu\bar{\nu}_\mu t\bar{t}H$ with complex top Yukawa coupling, which can be obtained by adding a gauge invariant dimension-6 operator to the Standard Model (SM) Lagrangian. The FD gauge and the unitary (U) gauge amplitudes give exactly the same cross section, and subtle gauge theory cancellation among diagrams in the U gauge at high energies is absent in the FD gauge, as has been observed for various SM processes. In addition, we find that the total cross sections at high energies are dominated by a single, or a set of non-vanishing Feynman amplitudes with the higher dimensional vertices in the FD gauge.