Gauge invariance from on-shell massive amplitudes and tree-level unitarity

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 $\mathcal{O}({E}^{4\ensuremath{-}N})$ in the high-energy hard-scattering limit, i.e., at fixed nonzero 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.