Froissart bound for/from CFT Mellin amplitudes

Abstract

We derive bounds analogous to the Froissart bound for the absorptive part of CFT${}_d$ Mellin amplitudes. Invoking the AdS/CFT correspondence, these amplitudes correspond to scattering in AdS${}_{d+1}$. We can take a flat space limit of the corresponding bound. We find the standard Froissart-Martin bound, including the coefficient in front for $d+1=4$ being $\pi /{\mu }^2$, $\mu$ being the mass of the lightest exchange. For $d>4$, the form is different. We show that while for $CF{T}_{d\le 6}$, the number of subtractions needed to write a dispersion relation for the Mellin amplitude is equal to 2, for $CF{T}_{d>6}$ the number of subtractions needed is greater than 2 and goes to infinity as $d$ goes to infinity.

Comments: v4:31 pages, 1 figure, typos fixed and more clarifications added, prepared for submission to Scipost