Froissart bound for/from CFT Mellin amplitudes

TL;DR

The paper develops Froissart-like bounds for the absorptive part of CFT Mellin amplitudes by leveraging AdS/CFT and the flat-space limit, producing a forward-limit Froissart-AdS bound that reduces to the standard Froissart bound in 4d when mapped to flat space. It introduces a spectral ell-cutoff and analyzes the number of subtractions required in Mellin dispersion relations across spacetime dimensions, finding $n=2$ for $d ext{(roughly)} le 6$ and $n>2$ growing with $d$. A non-forward analysis and Gegenbauer bound yield dimension- and momentum-dependent bounds with distinct logarithmic structures, highlighting differences from the flat-space Froissart bound for $d>4$. The work also discusses the flat-space correspondence, Tauberian-like aspects, and limitations, proposing future directions such as $1/R$ corrections and deeper connections to known S-matrix constraints. Overall, the results illuminate how strong-analyticity and unitarity constraints in CFT Mellin space encode Froissart-type growth and its dimensional dependence, with implications for AdS/CFT scattering and the flat-space limit.

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 $π/μ^2$, $μ$ 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.

Figures (1)

• Figure 1: Transition from AdS to Flat Space