Regge bounds and eikonal scaling in a four-dimensional scalar EFT

Abstract

Positivity bounds rest on twice-subtracted dispersion relations, which assume that the amplitude grows more slowly than $s^2$. For the massless shift-symmetric scalar with $g_4(\partialϕ)^4/Λ^4$ and $g_6(\partialϕ)^6/Λ^8$, we derive the dispersion relation, whose cuts start at zero energy, and study the large-$s$ growth at fixed $t<0$ of the tree, the one-loop bubble and the three-loop banana. The partial-wave sum is split at a transition spin: low spins are bounded by unitarity, channel by channel, and high spins are evaluated with the eikonal phase of the crossed-channel logarithms by stationary phase. The same construction is applied in the forward limit required by the dispersion relation, and we use the one- and three-loop absorptive parts below the cutoff to constrain $g_4$ and $g_6$.