New limits on "odderon" amplitudes from analyticity constraints

In studies of high energy $pp$ and $\bar{p}p$ scattering, the odd (under crossing) forward scattering amplitude accounts for the difference between the $pp$ and $\bar{p}p$ cross sections. Typically, it is taken as $f_{-}=-\frac{p}{4\pi }D{s}^{\alpha -1}{e}^{i\pi (1-\alpha )\mathrm{/}2}$ ($\alpha \sim 0.5$), which has $\mathrm{\Delta }\sigma ,\mathrm{\Delta }\rho \to 0$ as $s\to \mathrm{\infty }$, where $\rho$ is the ratio of the real to the imaginary portion of the forward scattering amplitude. However, the odd-signatured amplitude can have in principle a strikingly different behavior, ranging from having $\mathrm{\Delta }\sigma \to$non-zero constant to having $\mathrm{\Delta }\sigma \to \ln s\mathrm{/}{s}_0$ as $s\to \mathrm{\infty }$, the maximal behavior allowed by analyticity and the Froissart bound. We reanalyze high energy $pp$ and $\bar{p}p$ scattering data, using new analyticity constraints, in order to put new and precise limits on the magnitude of ``odderon’’ amplitudes.