Numerical study on the gauge symmetry of electroweak amplitudes

TL;DR

This work demonstrates that the GE (Goldstone equivalence) five-component representation of electroweak interactions encodes gauge symmetry directly in amplitudes, as formalized by the massive Ward identity $k^M\mathcal{M}_M=0$. Using HELAS, the authors numerically validate the MWI for tree-level processes and show that deviations from the gauge-determined relations among couplings—whether at the level of three- or four-point vertices or via SMEFT operators—violate the MWI unless those relations are preserved. They systematically test anomalous couplings within vertices and across vertices, revealing that gauge invariance is maintained only when couplings satisfy the precise gauge-derived relations, including those among Goldstone and gauge components. The SMEFT analysis with $\mathcal{O}_6$ and $\mathcal{O}_{t\Phi}$ confirms that gauge symmetry can be restored by enforcing the SMEFT-induced correlations among modified couplings, highlighting the essential role of Goldstone modes in preserving consistency of massive amplitudes. Overall, the paper provides a practical numerical framework to verify gauge invariance in the GE representation and emphasizes the necessity of coherent coupling relations for physical, gauge-invariant predictions.

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.

Figures (17)

• Figure 1: Five-component vector boson propagator in the double-line notation.
• Figure 2: Five-component vertex of $WWh$ in the double-line notation. The minus sign before $\varphi W h$ comes from the $g_{44}=-1$ component of the "metric" $g_{MN}$.
• Figure 3: Testing the MWI by computing $k^M\mathcal{M}_M$ for $W^+W^-\rightarrow t\bar{t}$.
• Figure 4: Testing the MWI by computing $k_M \mathcal{M}^M$ (a), $k_{1M} k_{2N} \mathcal{M}^{MN}$ (b), and $k_{1M} k_{2N} k^*_{3O} k^*_{4P} \mathcal{M}^{MNOP}$ for $W^+W^-\rightarrow W^+W^-$.
• Figure 5: Cross sections of $WW\rightarrow hh$ (upper panels) and $WW\rightarrow t \bar{t}$ (lower panels) at $\sqrt{s} = 10~\mathrm{TeV}$ with varying $\delta^{23}_{WWh}$ and $\delta^4_{WWh}$ in the unitary gauge (blue solid lines) and GE representation (red dotted lines).