Numerical study on the gauge symmetry of electroweak amplitudes

TL;DR

This paper numerically investigates gauge symmetry in electroweak amplitudes using the HELAS package, confirming the massive Ward identity and its relation to coupling relations in the standard model and SMEFT.

Contribution

It demonstrates how gauge symmetry manifests in electroweak amplitudes and how modifications to couplings affect the massive Ward identity, validating SMEFT relations.

Findings

MWI is satisfied in standard model couplings

Deviations from gauge relations violate MWI

SMEFT couplings restore MWI

Abstract

Electroweak (EW) amplitudes in the gauge-Goldstone five-component formalism have a distinctive property: gauge symmetry is imprinted in the amplitudes, manifested as the massive Ward identity (MWI) $k^M\mathcal M_M=0$. In this study, we used the HELAS package to numerically study gauge symmetry in EW amplitudes. First, we directly tested gauge symmetry by examining the MWI of amplitudes. Second, we modified the couplings within a vertex and among vertices to check if and how the MWI changes. Third, we tested gauge symmetry by considering the couplings modified by operators from the standard model effective field theory (SMEFT). Similar to the standard model, there are relations between different couplings that are protected by gauge symmetry. We observed that, if we modify the couplings to deviate from the relations, the MWI is violated. In contrast, the MWI is restored when the relations between couplings reduce to those in the SMEFT.