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Spinor-Helicity Three-Point Amplitudes from Local Cubic Interactions

Eduardo Conde, Euihun Joung, Karapet Mkrtchyan

TL;DR

This work establishes a concrete bridge between local covariant cubic interactions and on-shell four-dimensional three-point amplitudes. By employing spinor-helicity variables and covariant TT vertices, it shows how massless and massive cubic couplings reproduce the full set of three-point structures, including parity-even and parity-odd sectors, up to subtleties tied to momentum conservation. The authors extend the analysis to cases with one massive or two equal massive legs, deriving explicit vertex-to-amplitude dictionaries and clarifying where locality constraints limit the realized amplitudes. The results illuminate the relationship between local field theory and on-shell amplitude methods, offering a perspective on flat-space limits and hints toward AdS/CFT-inspired structures.

Abstract

We make an explicit link between the cubic interactions of off-shell fields and the on-shell three-point amplitudes in four dimensions. Both the cubic interactions and the on-shell three-point amplitudes had been independently classified in the literature, but their relation has not been made explicit. The aim of this note is to provide such a relation and discuss similarities and differences of their constructions. For the completeness of our analysis, we also derive the covariant form of all parity-odd massless vertices.

Spinor-Helicity Three-Point Amplitudes from Local Cubic Interactions

TL;DR

This work establishes a concrete bridge between local covariant cubic interactions and on-shell four-dimensional three-point amplitudes. By employing spinor-helicity variables and covariant TT vertices, it shows how massless and massive cubic couplings reproduce the full set of three-point structures, including parity-even and parity-odd sectors, up to subtleties tied to momentum conservation. The authors extend the analysis to cases with one massive or two equal massive legs, deriving explicit vertex-to-amplitude dictionaries and clarifying where locality constraints limit the realized amplitudes. The results illuminate the relationship between local field theory and on-shell amplitude methods, offering a perspective on flat-space limits and hints toward AdS/CFT-inspired structures.

Abstract

We make an explicit link between the cubic interactions of off-shell fields and the on-shell three-point amplitudes in four dimensions. Both the cubic interactions and the on-shell three-point amplitudes had been independently classified in the literature, but their relation has not been made explicit. The aim of this note is to provide such a relation and discuss similarities and differences of their constructions. For the completeness of our analysis, we also derive the covariant form of all parity-odd massless vertices.

Paper Structure

This paper contains 16 sections, 108 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Spinor Realization of Massless and Massive UIRs of Poincaré Algebra
  3. Massless Representations
  4. Massive Representations
  5. Massless Interactions
  6. Three-point Amplitudes
  7. Cubic vertices
  8. Parity-even vertices
  9. Parity-odd vertices
  10. Match
  11. Massive Interactions
  12. Generalities
  13. Two Massless and One Massive
  14. One Massless and Two Equal Massive
  15. Conclusion
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Schematic relation among flat/AdS Local Field Theories and amplitudes/correlators. The shaded region in the left-down corner corresponds to the amplitudes which satisfy the invariance condition but are not realized by Local Field Theory.