Scattering Amplitudes For All Masses and Spins

TL;DR

This work develops a comprehensive on-shell framework for scattering in four dimensions that unifies massive and massless particles with arbitrary spin by using the SU(2) little group for massive states. It classifies all three-point couplings consistent with Poincaré symmetry, imposes unitarity through factorization to construct four-point amplitudes, and introduces a spinning-polynomial basis to organize them. The analysis recovers known constraints (Yang–Mills for spin-1, gravity for spin-2, SUSY necessities) and explains why high-spin massive particles cannot be elementary, while revealing the Higgs and Super-Higgs mechanisms as infrared unifications of helicity amplitudes. The paper also demonstrates practical on-shell methods at one loop, including g-2, beta functions, and rational terms, and shows how off-shell observables like form factors and correlators fit naturally into the same on-shell language. Overall, it provides a powerful, field-theory-free perspective on fundamental interactions and offers a roadmap for extending on-shell techniques to the full spectrum of massive, spinning particles, with implications for UV completion and string theory.

Abstract

We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin S particles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called "spinning polynomials". Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin can not be "elementary". The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can't be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g-2) as well as the beta function and rational terms in QCD. "Off-shell" observables like correlation functions and form-factors can be thought of as scattering amplitudes with external "probe" particles of general mass and spin, so all these objects--amplitudes, form factors and correlators, can be studied from a common on-shell perspective.

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