Analytic Bounds on the Spectrum of Crossing Symmetric S-Matrices

We derive two rigorous constraints on the spectrum of massive states in weakly coupled theories with massless scalars in the adjoint representation of a large-$N$ gauge group. First, we show that the presence of massive spinning states necessitates the existence of lighter states with lower spins. Explicitly, if there exists a massive state with spin $J > 2$, then there must be a state with spin $J-1$ and a non-zero mass lower than that of the lightest spin-$J$ state, a state with spin $J-2$ and a mass lower than that of the lightest spin-($J-1$) particle and so on until we reach a mass below which only states with spin less than 2 are exchanged. Second, we find strict upper bounds on the masses of the lightest states at any spin. If there are spin-$J$ states in the spectrum, the maximum mass of the lightest spin-($J+1$) state is determined by the masses of the lightest spin-$J$ and $(J-1)$ states. In the approximation that this bound applies to pion scattering in real world QCD, we find it gives a window of only ${\sim}150$ MeV for the expected mass of the yet unmeasured spin-7 meson.