How Robust is the Froissart Bound?

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.