Lorentz Constraints on Massive Three-Point Amplitudes

TL;DR

This work extends the on-shell, Lorentz-based analysis of three-point amplitudes from exclusively massless states to cases with massive external legs by using the helicity-spinor formalism and a covariant massive little-group. The authors derive the general Lorentz-consistent forms of massive three-point amplitudes, uncover the remaining undetermined constants, and show that physical requirements—notably a UV massless limit and invariance under a fake little-group transformation—reduce these to a finite set of constants; in a special Elvang frame the amplitudes acquire a universal, frame-fixed structure. They validate the framework by matching to a variety of tree-level Lagrangian vertices (Higgs-gluon, vector-fermion, electroweak decays, QED) and illustrate how the Landau–Yang constraint and angular-momentum conservation emerge naturally. The results provide a non-perturbative seed for on-shell constructions of higher-point amplitudes via recursion and offer a path to understanding massive high-spin interactions and potential higher-dimensional origins. The framework thus bridges foundational symmetry constraints with concrete, testable amplitude structures across realistic theories.

Abstract

Using the helicity-spinor language we explore the non-perturbative constraints that Lorentz symmetry imposes on three-point amplitudes where the asymptotic states can be massive. As it is well known, in the case of only massless states the three-point amplitude is fixed up to a coupling constant by these constraints plus some physical requirements. We find that a similar statement can be made when some of the particles have mass. We derive the generic functional form of the three-point amplitude by virtue of Lorentz symmetry, which displays several functional structures accompanied by arbitrary constants. These constants can be related to the coupling constants of the theory, but in an unambiguous fashion only in the case of one massive particle. Constraints on these constants are obtained by imposing that in the UV limit the massive amplitude matches the massless one. In particular, there is a certain Lorentz frame, which corresponds to projecting all the massive momenta along the same null momentum, where the three-point massive amplitude is fully fixed, and has a universal form.