Question of the validity of the eikonal approximation

We examine the question of whether, in the scalar-scalar theory, the eikonal approximation reproduces in each order of perturbation the high-energy behavior of the sum of amplitudes for generalized ladder diagrams with scalar-meson exchanges. In the eighth order, which marks the onset of noneikonal asymptotic contributions, one need study for this purpose only seven distinct diagrams, and we derive the asymptotically leading terms of the absorptive parts of the amplitudes for these diagrams. The amplitudes of two of these diagrams show an enhanced behavior $\frac{\left({\ln }^{2}s\right)}{s^3}$ essentially because of the existence of pinch in the coefficient of $s$. The leading terms of these dominant diagrams, however, cancel and we show that the sum of contributions from all the diagrams behaves asymptotically as $\frac{\left(\ln s\right)}{s^3}$ in contradiction with the $\frac{1}{s^3}$ behavior dictated by the eikonal approximation. In the ${\phi }^3$ theory, however, because of the identity of the exchanged meson with the incident ones, the two dominant diagrams get an extra weight factor of 1/2 relative to the remaining diagrams and this leads to the complete cancellation of the $\frac{\left(\ln s\right)}{s^3}$ terms in the sum. Thus the question of validity of the eikonal approximation in the ${\phi }^3$ theory is still open.