Field-Theoretic Formulation of the Optical Model at High Energies

Abstract

Two approximation schemes are studied within the framework of the quantum electrodynamics of massive photons [Q.E.D.($W$)], motivated by the semiclassical approximation in potential theory (i.e., the optical model) rather than Regge pole or cut expectations. In the first scheme we select a series of Feynman graphs (generalized ladder graphs) that form a natural extension of potential scattering at high energies $s$ and finite momentum transfer $t$. (They moreover incorporate the effects of crossing symmetry in the $s$ and $u$ channels). In the second scheme we make a semiclassical approximation, without the use of conventional perturbation theory, involving classical currents in the $T$-product formula for the $S$ matrix. After certain vertex-renormalization and self-energy effects are taken into account, in a general way, we find an $S$ matrix that is identical with that of an optical model whose potential is a superposition of Yukawa potentials. Thus we find a sufficient (and perhaps necessary) condition for covariance of an optical model. Without taking any vertex and self-energy effects into account in either scheme, we have found agreement between the two schemes so far up to fourth order. To study the effect of spin in our models we introduce a spin-zero object "$\Pi$" and a spin-½ object "$N$." Because of the ${\gamma }_5$ invariance of the ${\gamma }_{\mu }$ vertex, the helicity-conserving amplitudes for the reactions $\Pi +N\to \Pi +N$ and $N+N\to N+N$ are the same as for $\Pi +\Pi \to \Pi +\Pi$, namely, $A\left(t\right)s+B\left(t\right)s \ln s.\mathrm{}$ We have verified that $B\left(t\right)\mathrm{}\equiv 0$ up to sixth order. Taken by itself, this form violates unitarity unless $B\left(t\right)\mathrm{}\equiv 0$ to all orders. We also study other ways to save unitarity in the whole of Q.E.D.($W$). Such unitarity considerations are strongly indicative of the importance of an optical term $\mathrm{sf}\left(t\right)\mathrm{}$ with nonzero real and imaginary parts even when many other series of graphs are taken into account (although other contributions may compete with it near $t=0\mathrm{}$). The optical contribution, taken alone, provides a natural qualitative explanation for the experimentally suggested constant behavior in the ratio of the real to imaginary part of the forward $\mathrm{pp}$ scattering amplitude above 8 $\frac{\mathrm{BeV}}{c}$. Although a similar result is predicted for $\mathrm{Kp}$ scattering, it does not apply to $\pi p$.