Analytic Bounds on the Spectrum of Crossing Symmetric S-Matrices

TL;DR

This paper establishes rigorous constraints on the spectrum of massive states in weakly coupled gauge theories with massless scalars, revealing relationships between spins and masses and providing bounds relevant to QCD mesons.

Contribution

It introduces novel bounds on the spectrum of massive states in crossing symmetric S-matrices, linking spins and masses and applying these to QCD phenomenology.

Findings

Massive spinning states imply lighter states with lower spins.

Upper bounds on the masses of the lightest states at each spin.

A predicted narrow mass window (~150 MeV) for a hypothetical spin-7 meson.

Abstract

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.