Higher-spin massless S-matrices in four-dimensions

TL;DR

This work uses on-shell, four-dimensional spinor-helicity methods to classify all massless three-point amplitudes and then tests four-point amplitudes via locality and unitarity (pole-counting). The authors show that, except for a small set of theories (phi^3, Yang–Mills, and General Relativity/Supergravity), higher-spin massless interactions cannot be consistently constructed, and no massless state with helicity >2 can couple to gravitons. Supersymmetry emerges naturally as a consistency requirement when spin-3/2 states are included, tying gravitino interactions to gravity and limiting the amount of SUSY to at most ${\cal N}=8$. The results reinforce the uniqueness of GR and YM in the massless sector and illustrate how on-shell constraints encode deep structure like the equivalence principle and SUSY, without relying on off-shell Lagrangians.

Abstract

On-shell, analytic S-matrix elements in massless theories are constructed from a finite set of primitive three-point amplitudes, which are fixed by Poincare invariance up to an overall numerical constant. We classify \emph{all} such three-point amplitudes in four-dimensions. Imposing the simplest incarnation of Locality and Unitarity on four-particle amplitudes constructed from these three-particle amplitudes rules out all but an extremely small subset of interactions among higher-spin massless states. Notably, the equivalence principle, and the Weinberg-Witten theorem, are simple corollaries of this principle. Further, no massless states with helicity larger than two may consistently interact with massless gravitons. Chromodynamics, electrodynamics, Yukawa and $φ^3$-theories are the only marginal and relevant interactions between massless states. Finally, we show that supersymmetry naturally emerges as a consistency condition on four-particle amplitudes involving spin-3/2 states, which must always interact gravitationally.

Figures (2)

• Figure 1: Summary of pole-counting results. Recall $N_p = 2H + 1 - A$, where $A = |\sum_{i = 1}^3 h_i |$ and $H = {\rm max}\{ |h_i|\}$. (Color online.) In short: black-dots represent sets of three-point amplitudes that define self-consistent S-matrices that can couple to gravity; green-dots represent sets of three-point amplitudes which---save for two exceptions explicitly delineated in Eq. \ref{['remnant']}---define S-matrices that cannot couple (in the sense defined in section \ref{['UniqueLaws']}) to any S-matrix defined by the black-dots; red-dots represent sets of three-point amplitudes that cannot ever form consistent S-matrices. Straightforward application of constraint \ref{['Constraint1']}, in subsection \ref{['Big3']}, rules out all $A_3$s with $(H,A)$-above the $N_p = 3$-line. More careful pole-counting, in subsection \ref{['Np32']} and appendix \ref{['Np2explicit']}, rules out all interactions above the $N_p = 1$ line, save for those with $(H,A) = (1/2,0)$, $(1,1)$, $(3/2,2)$, and $(2,2)$. Further, in section \ref{['UniqueLaws']}, a modified pole-counting rules out interaction between the $(H,A) = (2,2)$ gravity theory and any other theory with a spin-2 particle, save the unique $(H,A) = (2,6)$-theory. Similar results hold for gluon self-interactions: vectors present in any higher-spin amplitude with $A > 3$, save the unique $(H,A) = (1,3)$-theory, cannot couple to the vectors interacting via leading $(H,A) = (1,1)$ interactions. Section \ref{['weirdtheory']} rules out the $(H,A) = (1/2,0)$-interaction. Amplitudes in the grey-shaded regions can never be consistent with locality and unitarity. Higher-spin, $A > 3$, amplitudes between the $H = A/2$ and $A = A/3$ lines may be consistent. However, they cannot be coupled either to GR or YM, save for $(H,A) = (1,3)$ or $(2,6)$. In section \ref{['SUSY']}, we show inclusion of leading-order interactions between massless spin-3/2 states, at $A = 2$, promotes gravity to supergravity. Supergravity cannot couple to even these two $A > 2$ interactions, as seen in appendix \ref{['SUGRAexclude']}.
• Figure 2: Factorization necessitates gravitation in theories with massless spin-3/2 states. Specifically, figure (a) represents the two factorization channels in the minimal four-point amplitude, $A_{4}(1^{+\frac{3}{2}},2^{+\frac{3}{2}},3^{-\frac{3}{2}},4^{-\frac{3}{2}})$ in an S-matrix involving massless spin-3/2 states. Further, figure (b) shows the two factorization channels present in the amplitude $A_4(3/2,-3/2, +a,-a)$.