# Froissart bound for/from CFT Mellin amplitudes

### Submission summary

 Authors (as Contributors): Parthiv Haldar · Aninda Sinha
Submission information
Date accepted: 2020-06-29
Date submitted: 2020-06-02 02:00
Submitted by: Haldar, Parthiv
Submitted to: SciPost Physics
Ontological classification
Specialties:
• High-Energy Physics - Theory
• High-Energy Physics - Phenomenology
Approach: Theoretical

### 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.

Published as SciPost Phys. 8, 095 (2020)

### List of changes

Various clarifications and comments are added in accordance with the referee reports. These are listed below.
1. Following equation (3.11), it has been mentioned that the flat space limit we are also considering is where $R/\ell_{strings}\gg 1$. This is because in the flat space limit we are considering a massive quantum field theory.
2. In the sentence under (3.2), we have added that the general Mellin amplitude for $n-$point conformal correlator depends upon $n(n-3)/2$ independent variables.
3. In section 4.1, various footnotes have been added to add clarifications regarding various assumptions and computational steps. These are delineated in footnotes 5-9.

Typos Fixed
1. In equation (3.26) upper bound of the integral at infinity is fixed.
2. Consistently "flat space" has been used instead of "Flat-space".
3. Notational inconsistency between $\Gamma^2(x)$ and $\Gamma(x)^2$ has been rectified to stick with $\Gamma^2(x)$.
4. On page 8 penultimate line appendix number has been corrected to G from E.
5. In (B.4) the missing division sign is restored, in (C.9) the $45m^2$ is corrected to $4m^2$ and the unmatched parenthesis in (D.7) is taken care of.
6. Spellings corrected at various places and the grammar is improved at places as par the referee recommendation.

### Submission & Refereeing History

Resubmission scipost_202004_00044v5 on 2 June 2020
Submission scipost_202004_00044v1 on 3 April 2020

## Reports on this Submission

### Report

The authors have completed the suggested edits, so I would recommend that the article be published.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Report

The paper looks ready to be published now. The recent changes are much appreciated.

• validity: top
• significance: high
• originality: high
• clarity: high
• formatting: excellent
• grammar: good