FieldTheoretic 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 Tproduct formula for the S matrix. After certain vertexrenormalization and selfenergy 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 selfenergy 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 spinzero object "Π" and a spin 1/2 object "N." Because of the γ_{5} invariance of the γ_{μ} vertex, the helicityconserving amplitudes for the reactions Π+N>Π+N and N+N>N+N are the same as for Π+Π>Π+Π, namely, A(t)s+B(t)s lns. We have verified that B(t)≡0 up to sixth order. Taken by itself, this form violates unitarity unless B(t)≡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 sf(t) 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). 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 pp scattering amplitude above 8 BeVc. Although a similar result is predicted for Kp scattering, it does not apply to πp.
 Publication:

Physical Review
 Pub Date:
 March 1966
 DOI:
 10.1103/PhysRev.143.1194
 Bibcode:
 1966PhRv..143.1194T