How Robust is the Froissart Bound?

TL;DR

This paper reevaluates the Froissart bound's proof, suggesting that its assumptions about high-energy amplitude behavior are overly restrictive and that the actual bounds on total cross sections could be slightly stronger or grow faster depending on asymptotic conditions.

Contribution

It introduces a new perspective on the Froissart theorem by relaxing assumptions and deriving bounds that are slightly stronger, emphasizing the importance of high-energy asymptotics of nonphysical amplitudes.

Findings

Bounds are slightly stronger than Froissart's original estimate.

The scale parameter $s_0$ should grow slowly with energy.

Total cross section growth could surpass $ ext{log}^2 s$ under certain assumptions.

Abstract

Proof of the Froissart theorem is reconsidered in a different way to extract its necessary conditions. Two physical inputs, unitarity and absence of massless intermediate hadrons, are indisputable. Also important are mathematical properties of the Legendre functions. Assumptions on dispersion relations, single or double, appear to be excessive. Instead, one should make assumptions on possible high-energy asymptotics of the amplitude in nonphysical configurations, which have today no firm basis. Asymptotics for the physical amplitude always appear essentially softer than for the nonphysical one. Froissart's paper explicitly assumed the hypothesis of power behavior and obtained asymptotic bound for total cross sections $\sim \log^2{(s/s_0)}$ with some constant $s_0$. Our bounds are slightly stronger than original Froissart ones. They show that the scale $s_0$ should itself slowly grow with $s$. Under different assumptions about asymptotic behavior of nonphysical amplitudes, the total cross section could grow even faster than $\log^2{s}$. The problem of correct asymptotics might be clarified by precise measurements at the LHC and higher energies.