Gauge invariance from on-shell massive amplitudes and tree unitarity

TL;DR

This paper derives gauge invariance conditions from on-shell massive amplitude constraints and tree unitarity, revealing how high-energy behavior restricts particle couplings and masses in spontaneously broken gauge theories.

Contribution

It provides a purely on-shell derivation of gauge invariance constraints from tree unitarity for massive particles, without relying on Lagrangian formulations.

Findings

Constraints on three-particle couplings and masses from tree unitarity.

Relations among couplings that match gauge invariance predictions.

On-shell perspective elucidates spontaneously broken gauge theories.

Abstract

We study the three-particle and four-particle scattering amplitudes for an arbitrary, finite number of massive scalars, spinors and vectors by employing the on-shell massive spinor formalism. We consider the most general three-particle amplitudes with energy growing behavior at most of $\mathcal{O} (E)$. This is the special case of the requirement of tree unitarity, which states that the $N$-particle scattering amplitudes at tree-level should grow at most as $\mathcal{O} (E^{4-N})$ in the high energy hard scattering limit, i.e. at fixed non-zero angles. Then the factorizable parts of the four-particle amplitudes are calculated by gluing the on-shell three-particle amplitudes together and utilizing the fact that tree-level amplitudes have only simple poles. The contact parts of the four-particle amplitudes are further determined by tree unitarity, which also puts strong constraints on the possible allowed three-particle coupling constants and the masses. The derived relations among them converge to the predictions of gauge invariance in the UV theory. This provides a purely on-shell understanding of spontaneously broken gauge theories.