Higher-spin massless $S$ matrices in four dimensions

On-shell, analytic $S$-matrix elements in massless theories are constructed from a finite set of primitive three-point amplitudes, which are fixed by Poincaré invariance up to an overall numerical constant. We classify 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 ${\varphi }^3$ 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.

Figure 1

Summary of pole-counting results: with few notable exceptions (below) only three-point amplitudes on, or below, the $H=A$ line, for $A\le 2$ can form tree-amplitudes that are consistent with locality and unitarity. Recall $N_p=2H+1-A$, where $A=|\sum _{i=1}^3{h}_i|$ and $H=\max \{|h_i|\}$. 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. (4.6)—define $S$ matrices that cannot couple (in the sense defined in Sec. 4) 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 (3.1), in Sec. 3b, rules out all $A_3$’s with $(H,A)$ above the $N_p=3$ line. More careful pole counting, in Sec. 3c and Appendix pp2, 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 Sec. 4, 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 5c 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 to either GR or YM, save for $(H,A)=(1,3)$ or (2,6). In Sec. 6, 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 pp4.

Figure 2

Factorization necessitates gravitation in theories with massless spin-$3/2$ states. Specifically, (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, (b) shows the two factorization channels present in the amplitude $A_4(3/2,-3/2,+a,-a)$.