Models for hadron-hadron scattering at high energies and rising total cross sections

The recent results of rising total cross sections for ${\pi }^{\pm }p$, $K^{\pm }p$, and $pp$ scattering are analyzed by two simple analytic models for high-energy forward scattering which are derived from analyticity, crossing symmetry, and the unitarity constraints of the rigorous results. The numerical fit to the data for $p_L\ge 10$ GeV/c does not rule out the possibility that the crossing-odd amplitude may not be negligible at high energies.