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Exploring the S-Matrix of Massless Particles

Paolo Benincasa, Eduardo Conde

TL;DR

The paper addresses the question of which tree-level interactions among massless particles in four-dimensional Minkowski space are consistent when built from on-shell data. It extends the BCFW program to a generalized on-shell recursion with weights fixed by soft limits, enabling amplitudes to be reconstructed from three-particle data while enforcing a four-particle consistency test. The analysis classifies interactions by the derivative order of three-particle couplings, reproducing known theories such as YM, GR, QED, Yukawa, and N=1 supergravity, and derives sharp spin-derivative constraints that constrain possible couplings; intriguingly, the framework also signals potential high-spin couplings only if locality is weakened, as evidenced by infinity-contact terms that behave nonlocally. Overall, the work provides a unified S-matrix–based approach to catalog existing theories and systematically search for high-spin interactions, elucidating how locality, soft limits, and factorization jointly shape consistent massless theories.

Abstract

We use the recently proposed generalised on-shell representation for scattering amplitudes and a consistency test to explore the space of tree-level consistent couplings in four-dimensional Minkowski spacetime. The extension of the constructible notion implied by the generalised on-shell representation, i.e. the possibility to reconstruct at tree level all the scattering amplitudes from the three-particle ones, together with the imposition of the consistency conditions at four-particle level, allow to rediscover all the known theories and their algebra structure, if any. Interestingly, this analysis seems to leave room for high-spin couplings, provided that at least the requirement of locality is weakened. We do not claim to have found tree-level consistent high-spin theories, but rather that our methods show signatures of them and very likely, with a suitable modification, they can be a good framework to perform a systematic search.

Exploring the S-Matrix of Massless Particles

TL;DR

The paper addresses the question of which tree-level interactions among massless particles in four-dimensional Minkowski space are consistent when built from on-shell data. It extends the BCFW program to a generalized on-shell recursion with weights fixed by soft limits, enabling amplitudes to be reconstructed from three-particle data while enforcing a four-particle consistency test. The analysis classifies interactions by the derivative order of three-particle couplings, reproducing known theories such as YM, GR, QED, Yukawa, and N=1 supergravity, and derives sharp spin-derivative constraints that constrain possible couplings; intriguingly, the framework also signals potential high-spin couplings only if locality is weakened, as evidenced by infinity-contact terms that behave nonlocally. Overall, the work provides a unified S-matrix–based approach to catalog existing theories and systematically search for high-spin interactions, elucidating how locality, soft limits, and factorization jointly shape consistent massless theories.

Abstract

We use the recently proposed generalised on-shell representation for scattering amplitudes and a consistency test to explore the space of tree-level consistent couplings in four-dimensional Minkowski spacetime. The extension of the constructible notion implied by the generalised on-shell representation, i.e. the possibility to reconstruct at tree level all the scattering amplitudes from the three-particle ones, together with the imposition of the consistency conditions at four-particle level, allow to rediscover all the known theories and their algebra structure, if any. Interestingly, this analysis seems to leave room for high-spin couplings, provided that at least the requirement of locality is weakened. We do not claim to have found tree-level consistent high-spin theories, but rather that our methods show signatures of them and very likely, with a suitable modification, they can be a good framework to perform a systematic search.

Paper Structure

This paper contains 19 sections, 79 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. The S-Matrix and the BCFW construction
  3. S-Matrix
  4. BCFW construction
  5. Generalised On-Shell Recursion Relations
  6. Generalised On-Shell Recursion Relations and Constructibility
  7. Soft limits and Complex-UV Behaviour
  8. Constructibility and Consistency Conditions
  9. Four-Particle Amplitudes
  10. Interactions with $s$-derivatives: Self-interaction of spin-$s$ particles
  11. Interactions with $s$-derivatives: spin-$s$/spin-$s'$
  12. Interactions with $3s$-derivatives
  13. Spin-$s$/spin-$s'$ interactions with $(2s'+s)$-derivatives
  14. Spin-$s$/spin-$s'$ interactions with $|2s'-s|$-derivatives
  15. Case $s\,<\,2s'$
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Generalised on-shell recursion relation. This new recursion relation shows the same structure of the usual BCFW one, with an additional factor which depends on a sub-set of the zeroes of the amplitude. The notation will become clear later.