Froissart bound for/from CFT Mellin amplitudes

TL;DR

This paper establishes Froissart-like bounds for CFT Mellin amplitudes, connecting them to scattering in AdS space and deriving bounds that depend on the spacetime dimension, with implications for dispersion relations.

Contribution

The work derives Froissart bounds for CFT Mellin amplitudes and analyzes how the number of subtractions in dispersion relations varies with spacetime dimension.

Findings

Standard Froissart-Martin bound recovered in flat space limit for d+1=4.

Number of subtractions for dispersion relations increases with dimension, exceeding 2 for d>6.

Bounds differ for CFTs in dimensions greater than 4, with implications for scattering 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 $CFT_{d\leq 6}$, the number of subtractions needed to write a dispersion relation for the Mellin amplitude is equal to 2, for $CFT_{d>6}$ the number of subtractions needed is greater than 2 and goes to infinity as $d$ goes to infinity.