The electroweak effective field theory from on-shell amplitudes

TL;DR

This work develops a complete bottom-up, on-shell framework for the electroweak sector using massive spinor-helicity methods, deriving both renormalizable and non-renormalizable interactions from first principles. By constructing all massive three-point amplitudes and gluing them into four-point functions, it clarifies how electroweak symmetry breaking and the Higgs mechanism are encoded in on-shell amplitudes, and how SMEFT structure emerges in the broken phase. Perturbative unitarity is employed to distinguish renormalizable from non-renormalizable contributions and to derive relations among masses, gauge couplings, and Yukawa couplings, including the Goldstone-boson equivalence theorem. The results provide a robust, gauge-redundancy-free basis for SMEFT computations with on-shell methods, and lay out a path to extend the program to more complex amplitudes while preserving gauge and unitarity constraints. Overall, the paper offers a transparent bridge between bottom-up amplitude techniques and the standard-model EFT paradigm, enabling streamlined, on-shell SMEFT calculations in the broken electroweak phase.

Abstract

We apply on-shell methods to the bottom-up construction of electroweak amplitudes, allowing for both renormalizable and non-renormalizable interactions. We use the little-group covariant massive-spinor formalism, and flesh out some of its details along the way. Thanks to the compact form of the resulting amplitudes, many of their properties, and in particular the constraints of perturbative unitarity, are easily seen in this formalism. Our approach is purely bottom-up, assuming just the standard-model electroweak spectrum as well as the conservation of electric charge and fermion number. The most general massive three-point amplitudes consistent with these symmetries are derived and studied in detail, as the primary building blocks for the construction of scattering amplitudes. We employ a simple argument, based on tree-level unitarity of four-point amplitudes, to identify the three-point amplitudes that are non-renormalizable at tree level. This bottom-up analysis remarkably reproduces many low-energy relations implied by electroweak symmetry through the standard-model Higgs mechanism and beyond it. We then discuss four-point amplitudes. The gluing of three-point amplitudes into four-point amplitudes in the massive spinor helicity formalism is clarified. As an example, we work out the $ψ^c ψZh$ amplitude, including also the non-factorizable part. The latter is an all-order expression in the effective-field-theory expansion. Further constraints on the couplings are obtained by requiring perturbative unitarity. In the $ψ^c ψZh$ example, one for instance obtains the renormalizable-level relations between vector and fermion masses and gauge and Yukawa couplings. We supplement our bottom-up derivations with a matching of three- and four-point amplitude coefficients onto the standard-model effective field theory (SMEFT) in the broken electroweak phase.