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

In order to test the general validity of the impact picture which was extracted from quantum electrodynamics at high energies, we make a systematic study of high-energy scattering amplitudes in ${\phi }^3$ theory. The method we use is the one developed in our treatment of quantum electrodynamics, and it enables us to obtain in closed forms the asymptotic amplitudes for multiple exchanges of Regge poles and scalar particles. Our results are nonperturbative and are valid to all orders of the coupling constant. We show in various examples that (i) the longitudinal momentum of the incident particle is always distributed in positive fractions by the particles created; (ii) an exchanged particle, either composite (Regge pole) or elementary, can carry only transverse momentum; (iii) the scattering amplitude satisfies the impact factor representation; (iv) except in the leading order of the coupling constant, the simple-exponentiation form of the eikonal approximation does not appear. The eikonal approximation fails in ${\phi }^3$ theory because of particle creation: More than one particle may share the incident longitudinal momentum and it is impossible to single out a particular particle to be associated with the eikonal path. The above conclusions are reached by a study of the one-ladder amplitude and the amplitude for the double exchange of a ladder and a scalar particle. An integral equation for the Regge parameters of a ladder was derived and many properties of the Regge parameters were given. In particular, in the Wick-Cutkosky model, the Regge pole at zero invariant mass $\alpha \mathrm{}\left(0\right)=1\mathrm{}$ when the coupling constant $g$ is equal to $4\sqrt{6}\pi m$, where $m$ is the mass of the scalar meson. Thus the physical picture of an expanding black disk in quantum electrodynamics is also realized in the Wick-Cutkosky model as long as $g>\mathrm{}4\mathrm{}\sqrt{6}\pi m$. The scattering amplitude for the double exchange of a ladder and a scalar particle is given explicitly in terms of the Regge parameters of the ladder diagrams. Throughout our investigation of ${\phi }^3$ theory at high energies, there is exact correspondence with the results in quantum electrodynamics at every step. This reaffirms our belief of the general validity of the impact picture.