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Higher-Spin Interactions: four-point functions and beyond

M. Taronna

TL;DR

Massless higher-spin interactions in flat space are constructed through an infinite class of gauge-consistent four-point functions, derived from linearized gauge invariance using a Free Differential Algebra (FDA) framework. The work recasts the Noether procedure into binary and ternary products and introduces open-string-like and closed-string-like kernel structures, including non-local quartics and a minimal scheme that controls locality while preserving correct factorization. Detailed analyses of Yang-Mills and HS four-point amplitudes reveal how current exchanges are canceled by quartic counterterms, with Weinberg's no-go revisited to show the potential need for non-localities or an infinite HS tower. The results illuminate deep links to string theory and Vasiliev's system and chart a path to n-point functions and mixed-symmetry fields, with future exploration in curved backgrounds and non-local quantum field theory.

Abstract

In this work we construct an infinite class of four-point functions for massless higher-spin fields in flat space that are consistent with the gauge symmetry. In the Lagrangian picture, these reflect themselves in a peculiar non-local nature of the corresponding non-abelian higher-spin couplings implied by the Noether procedure that starts from the fourth order. We also comment on the nature of the colored spin-2 excitation present both in the open string spectrum and in the Vasiliev system, highlighting how some aspects of String Theory appear to reflect key properties of Field Theory that go beyond its low energy limit. A generalization of these results to n-point functions, fermions and mixed-symmetry fields is also addressed.

Higher-Spin Interactions: four-point functions and beyond

TL;DR

Massless higher-spin interactions in flat space are constructed through an infinite class of gauge-consistent four-point functions, derived from linearized gauge invariance using a Free Differential Algebra (FDA) framework. The work recasts the Noether procedure into binary and ternary products and introduces open-string-like and closed-string-like kernel structures, including non-local quartics and a minimal scheme that controls locality while preserving correct factorization. Detailed analyses of Yang-Mills and HS four-point amplitudes reveal how current exchanges are canceled by quartic counterterms, with Weinberg's no-go revisited to show the potential need for non-localities or an infinite HS tower. The results illuminate deep links to string theory and Vasiliev's system and chart a path to n-point functions and mixed-symmetry fields, with future exploration in curved backgrounds and non-local quantum field theory.

Abstract

In this work we construct an infinite class of four-point functions for massless higher-spin fields in flat space that are consistent with the gauge symmetry. In the Lagrangian picture, these reflect themselves in a peculiar non-local nature of the corresponding non-abelian higher-spin couplings implied by the Noether procedure that starts from the fourth order. We also comment on the nature of the colored spin-2 excitation present both in the open string spectrum and in the Vasiliev system, highlighting how some aspects of String Theory appear to reflect key properties of Field Theory that go beyond its low energy limit. A generalization of these results to n-point functions, fermions and mixed-symmetry fields is also addressed.

Paper Structure

This paper contains 21 sections, 265 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Symbol calculus
  3. Structure of the interactions
  4. HS covariant equations and gauge transformations
  5. Lagrangian Form
  6. Binary product
  7. Ternary product
  8. Four-point scattering amplitudes
  9. The Yang-Mills example
  10. The HS case
  11. Off-shell ternary product
  12. Weinberg's theorem revisited
  13. Non-local field theories?
  14. Conclusions
  15. Comments on N-point functions, Fermions and Mixed Symmetry fields
  16. ...and 6 more sections

Figures (10)

  • Figure 1: Building blocks for the ${\cal G}^{\,(i)}_{12\ldots n}$'s.
  • Figure 2: Cubic couplings for scalars and gauge boson corresponding to the pieces in which ${\cal G}_{123}$ splits.
  • Figure 3: Decoupling condition for unphysical polarizations at the level of the amplitude.
  • Figure 4: The decoupling condition can be read as an associator equation in which the cubic coupling is considered as a product and the quartic coupling as a trilinear product.
  • Figure 5: Associative bi-product.
  • ...and 5 more figures