A Numerical Study on Gauge Symmetry of Electroweak Amplitudes

Electroweak (EW) amplitudes in the gauge-Goldstone 5-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 paper, we use the HELAS package to numerically study gauge symmetry in EW amplitudes. First, we directly test gauge symmetry by examining the MWI of amplitudes. Second, we modify the couplings within a vertex and of vertices to check if and how the MWI changes. Third, we test 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 find that if we modify the couplings to deviate from the relations, the MWI is violated. On the other hand, the MWI is restored when the relations between couplings reduce to those in the SMEFT.