High-Energy Scattering inφ3Theory and the Breakdown of Eikonal Approximation. II

Abstract

In this paper we study, in ${\ensuremath{\varphi}}^{3}$ theory, the high-energy amplitudes for the exchange of two or more ladders. We obtain these amplitudes in closed forms to all orders of the coupling constant. The conclusions are as follows: (i) All of the scattering amplitudes satisfy the impact-factor representation. (ii) Except for the leading term (in the coupling constant) the scattering amplitude of two-ladder exchange is not in the form dictated by the eikonal approximation. (iii) The scattering amplitudes for the exchange of $N$ ladders, $N\ensuremath{\ge}3$, are never in the form dictated by the eikonal approximation - even for the leading term. (When the coupling constant approaches zero, all of them are of the order of ${s}^{\ensuremath{-}3}$ instead of ${s}^{\ensuremath{-}2N+1}$, as dictated by the eikonal approximation.) Thus the eikonal approximation is not valid in ${\ensuremath{\varphi}}^{3}$ theory. The amplitude for the exchange of $n$ scalar mesons, $n>4$, is also given. Contrary to the popular notion, it is not of the order of ${s}^{\ensuremath{-}n+1}$ when $n>4$. Summing over such amplitudes does not lead to the exponentiation form commonly conceded in the past.