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

A bstract 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 ,open = n + 1 and J n ,closed = 2 n + 2, respectively, at the n th mass level.