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Massless spinning fields on the Light-Front: quartic vertices and amplitudes

Mattia Serrani

TL;DR

This work analyzes the quartic light-front closure of the Poincaré algebra for massless spinning fields in four dimensions, focusing on nonholomorphic h–ah sectors and their impact on the spectrum and couplings. By constructing general quartic ansätze and solving coupled PDEs, it reproduces established YM and gravity results while revealing no-go constraints for fully local, unitary higher-spin theories in flat space. It shows that including higher-derivative cubic vertices can yield nontrivial, local quartic vertices and leads to a comprehensive classification of unitary local higher-spin theories and novel quasi-chiral HS sectors that extend self-dual YM/GR. Furthermore, the paper derives all corresponding local four-point amplitudes using spinor-helicity methods and discusses mild non-localities as a possible way to reconcile HS interactions with locality at the amplitude level, suggesting paths toward a consistent nonlocal HS framework in flat space.

Abstract

Within the light-front approach in flat space, we study the closure of the Poincare algebra at the quartic order, specifically the nonholomorphic constraint involving both MHV and anti-MHV vertices. We first recover some well-established results: the existence of Yang-Mills theory and gravity, as well as the inconsistency of interacting multi-graviton theories. We explicitly construct several lower-derivative and lower-spin quartic vertices. We then turn to theories involving massless higher-spin fields. It becomes evident that the quartic constraint does not allow many cubic interactions to survive, in accordance with the well-known no-go results. Nevertheless, once higher-derivative cubic vertices are included, we find nontrivial solutions to the full quartic constraint and determine the corresponding quartic vertices. On this basis, we conjecture the complete set of quartic vertices that solve the light-cone consistency conditions. Exploiting this, we find all allowed unitary local higher-spin theories and identify new families of local quasi-chiral higher-spin theories. We then determine all local higher-spin four-point amplitudes using the spinor-helicity formalism together with locality. We conclude with a short discussion on non-locality and propose a ``local'' (at the amplitude level) higher-spin theory in flat space.

Massless spinning fields on the Light-Front: quartic vertices and amplitudes

TL;DR

This work analyzes the quartic light-front closure of the Poincaré algebra for massless spinning fields in four dimensions, focusing on nonholomorphic h–ah sectors and their impact on the spectrum and couplings. By constructing general quartic ansätze and solving coupled PDEs, it reproduces established YM and gravity results while revealing no-go constraints for fully local, unitary higher-spin theories in flat space. It shows that including higher-derivative cubic vertices can yield nontrivial, local quartic vertices and leads to a comprehensive classification of unitary local higher-spin theories and novel quasi-chiral HS sectors that extend self-dual YM/GR. Furthermore, the paper derives all corresponding local four-point amplitudes using spinor-helicity methods and discusses mild non-localities as a possible way to reconcile HS interactions with locality at the amplitude level, suggesting paths toward a consistent nonlocal HS framework in flat space.

Abstract

Within the light-front approach in flat space, we study the closure of the Poincare algebra at the quartic order, specifically the nonholomorphic constraint involving both MHV and anti-MHV vertices. We first recover some well-established results: the existence of Yang-Mills theory and gravity, as well as the inconsistency of interacting multi-graviton theories. We explicitly construct several lower-derivative and lower-spin quartic vertices. We then turn to theories involving massless higher-spin fields. It becomes evident that the quartic constraint does not allow many cubic interactions to survive, in accordance with the well-known no-go results. Nevertheless, once higher-derivative cubic vertices are included, we find nontrivial solutions to the full quartic constraint and determine the corresponding quartic vertices. On this basis, we conjecture the complete set of quartic vertices that solve the light-cone consistency conditions. Exploiting this, we find all allowed unitary local higher-spin theories and identify new families of local quasi-chiral higher-spin theories. We then determine all local higher-spin four-point amplitudes using the spinor-helicity formalism together with locality. We conclude with a short discussion on non-locality and propose a ``local'' (at the amplitude level) higher-spin theory in flat space.
Paper Structure (54 sections, 216 equations, 22 figures, 11 tables)

This paper contains 54 sections, 216 equations, 22 figures, 11 tables.

Table of Contents

  1. Introduction
  2. No-go's
  3. A-category: global constraints of higher spin symmetry.
  4. B-category: local constraints of higher spin symmetry.
  5. Living on the Light-Front
  6. Cubic vertices.
  7. Higher orders.
  8. Quartic analysis
  9. Chiral theories.
  10. $\mathbf{[H_3,J_3]}$ commutator.
  11. $\mathbf{[H,J^{z-}]}$ and $\mathbf{[H,J^{{\bar{z}}-}]}$ constraints.
  12. Construction of the ansätze
  13. Solving the quartic constraint
  14. Structure of PDEs.
  15. Solving the PDEs.
  16. ...and 39 more sections

Figures (22)

  • Figure 1: Generic $C\bar{C}$ exchange with contact term $C^{\lambda_1,\lambda_2,\lambda_3,\lambda_4}$.
  • Figure 2: Holomorphic constraint.
  • Figure 3: Non-holomorphic constraint.
  • Figure 4: Exchange diagrams for a generic pair $C\bar{C}$.
  • Figure 5: A quartic vertex with a single exchange exists when the conditions \ref{['kinda_trinagular']} are satisfied.
  • ...and 17 more figures