A Numerical Study on Gauge Symmetry of Electroweak Amplitudes

Abstract: 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). We find that the MWI is violated as long as there are anomalous couplings that deviate from the precise relations fixed by gauge symmetry. On the other hand, as along as those relations hold -- either for the standard model, or for the SMEFT -- modifying the couplings would not violate the MWI.