Properties of Hadronic Amplitudes in an Absorptive Model

Hadronic two-body amplitudes involve two components. The imaginary part of the non-diffractive component $R$ is dominated by the most peripheral impact parameters ($b\sim r$). The imaginary part of the diffractive component $P$ has substantial contributions from all impact parameters $b\lesssim r$. We study the energy dependence of the $b$ representations of both components as well as various possible forms for the corresponding real parts. We show that the following three assumptions are mutually inconsistent: (i) $\mathrm{Im}R(s,t)\mathrm{}$ is always dominated by $b\sim r$ terms; (ii) $\mathrm{Im}R(s,t)\mathrm{}$ shrinks indefinitely as $s\to \infty$; (iii) $r$ approaches a constant as $s\to \infty$. We define three classes of models obtained by abandoning, one at a time, these three assumptions. We discuss the complex-$J$-plane structure as well as the asymptotic phase of the $R$ amplitude for each of these classes and propose various experimental ways of distinguishing between the models. A detailed analysis of $\mathrm{Re}R$ indicates that, while in certain cases it reaches its asymptotic phase at relatively low energies, in other cases the asymptotic phase is approached very slowly and it has no resemblance to the observed phase at present energies.