Analytic bounds on the spectrum of crossing symmetric S-matrices

A bstract We derive two rigorous bounds on the spectrum of massive states that contributes to planar amplitudes in weakly coupled theories with massless scalars. 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 that 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 ∼150 MeV for the expected mass of a yet unmeasured spin-7 meson.