  In studies of high energy $pp$ and $\bar pp$ scattering, the odd (under
crossing) forward scattering amplitude accounts for the difference between the
$pp$ and $\bar pp$ cross sections. Typically, it is taken as
$f_-=-\frac{p}{4\pi}Ds^{\alpha-1}e^{i\pi(1-\alpha)/2}$ ($\alpha\sim 0.5$),
which has $\Delta\sigma, \Delta\rho\to0$ as $s\to\infty$, 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 $\Delta\sigma\to$non-zero constant to
having $\Delta\sigma \to \ln s/s_0$ as $s\to\infty$, the maximal behavior
allowed by analyticity and the Froissart bound. We reanalyze high energy $pp$
and $\bar pp$ scattering data, using new analyticity constraints, in order to
put new and precise limits on the magnitude of ``odderon'' amplitudes.
