Weyl-van-der-Waerden formalism for helicity amplitudes of massive particles

TL;DR

This paper develops a practical Weyl–van-der-Waerden spinor framework for helicity amplitudes that handles both massive and massless particles. It systematically rewrites Lorentz structures using two-component spinors, defines spinor products and 4-vector mappings through the $\sigma$ matrices, and provides wave-function constructions for helicity eigenstates, including massive spin-1/2. The approach enables compact, analytically tractable amplitudes and leverages discrete symmetries and pole-avoidance strategies, with explicit results for relevant Standard Model processes and guidance for computer-algebra implementation and crossing relations. Overall, it offers a versatile, efficient toolkit for computing polarized amplitudes in high-energy phenomenology.

Abstract

The Weyl-van-der-Waerden spinor technique for calculating helicity amplitudes of massive and massless particles is presented in a form that is particularly well suited to a direct implementation in computer algebra. Moreover, we explain how to exploit discrete symmetries and how to avoid unphysical poles in amplitudes in practice. The efficiency of the formalism is demonstrated by giving explicit compact results for the helicity amplitudes of the processes gamma gamma -> f fbar, f fbar -> gamma gamma gamma, mu^- mu^+ -> f fbar gamma.