Analytic bootstrap bounds on masses and spins in gravitational and non-gravitational scalar theories

TL;DR

This paper derives analytic bounds on the spectrum of weakly-coupled scalar theories, applicable to gravitational and non-gravitational cases, based on crossing symmetry and polynomial bounds in the S-matrix bootstrap framework.

Contribution

It introduces Sequential Spin and Mass Constraints that relate the ordering of spins and masses in the spectrum of such theories, generalizing previous results for planar amplitudes.

Findings

Superstring spectra saturate the Sequential Spin Constraints.

Derived bounds apply to theories with crossing symmetric four-point amplitudes.

Results include constraints on the ordering of masses and spins in the spectrum.

Abstract

We derive analytic constraints on the weakly-coupled spectrum of theories with a massless scalar under the standard assumptions of the S-matrix bootstrap program. These bootstrap bounds apply to any theory (with or without gravity) with fully crossing symmetric (i.e. $stu$-symmetric) four-point amplitudes and generalize results for color- or flavor-ordered (i.e. $su$-symmetric) planar amplitudes recently proved by one of the authors. We assume that the theory is weakly-coupled below some cut-off, that the four-point massless scalar amplitude is polynomially-bounded in the Regge limit, and that this amplitude exchanges states with a discrete set of masses and a finite set of spins at each mass level. The spins and masses must then satisfy ``Sequential Spin Constraints" (SSC) and ``Sequential Mass Constraints" (SMC). The SSC requires the lightest spin-$j$ state to be lighter than the lightest spin-$(j+1)$ state (in the $su$-symmetric case) or the lightest spin-$(j+2)$ state (in the $stu$-symmetric case). The SMC requires the mass of the lightest spin-$j$ state to be smaller than some non-linear function of the masses of lower-spin states. Our results also apply to super-gluon and super-graviton amplitudes stripped of their polarization dependence. In particular, the open and closed superstring spectra saturate the SSC with maximum spins ${J_{n,\text{open}} = n+1}$ and ${J_{n,\text{closed}} = 2n+2}$, respectively, at the $n^\text{th}$ mass level.