Asymptotic Behavior of Scattering Amplitudes in the Relativistic Eikonal Approximation

Using techniques developed in an earlier paper, a new one-parameter eikonal representation is derived and its high-energy behavior is studied. It is shown that by also taking $t$ large a Regge-like behavior is obtained in the crossed channel. The procedure of first making $s$ large and then $t$ large is investigated more closely by comparing the exact fourth-order amplitudes with those obtained in the relativistic eikonal approximation. It is found that the latter predicts the correct behavior in the region of large $s$, for all values of $t$. When a similar procedure is applied to the full $M_{\mathrm{eik}}(s, t)$, one obtains in an appropriate domain an asymptotic amplitude having a dual character: The bound states appear as poles in the $s$ channel and lie on the Regge trajectory in the $t$ channel. In particular, for electron-positron scattering the corresponding energy levels coincide with those obtained recently by Brezin et al. The asymptotic behavior of the eikonal function itself is also studied. DOI: http://dx.doi.org/10.1103/PhysRevD.2.1716 © 1970 The American Physical Society