S-Matrix Theory and Higher Order Corrections to the Weak Interactions

Abstract

We have studied the use of dispersive and unitary methods to calculate higher order corrections to the weak interactions, with emphasis on the $W$-meson theory. As a first step, we have examined the ordinary elastically unitary approximation techniques in the $W$ theory of electron-neutrino scattering (namely, $\frac{N}{D}$ with one $W$-meson exchange, Mandelstam iteration procedure, and the strip approximation). We find that these are incapable of generating a nonrenormalizable amplitude; instead, they simply neglect the divergences of the perturbation theory at every order. This is a serious weakness of the methods. Our travail leads to a prescription for the dispersive generation of the nonrenormalizability in the $W$ theory: At least some contributions from many-boson intermediate states must be included as input information. On the basis of this prescription, we have made two distinct nonperturbative attacks on the dynamics. In the first, we have summed the leading absorptive parts of the $W$-meson ladder graphs, using the Cutkosky rules. These absorptive parts correspond to putting all the $W$ mesons on the mass shell, and are finite at each order of perturbation theory (although asymptotically ill-behaved), so that no regulator is needed in the summation. We find an exponentially increasing absorptive part, which contradicts the results of Feinberg and Pais, whose solutions are bounded. In an attempt to discover the source of this discrepancy, we study peratization from a dispersive point of view. This study throws serious doubt on the validity of the Feinberg-Pais program. It is seen explicitly that their non-Hermitian methods (regulator, and an analytic continuation in the coupling) have led them to solutions which violate the positive-definiteness of the mass spectrum. In the second attack, a new family of exact solutions to the $\frac{N}{D}$ equations with singular inputs allows us to propose, and to reduce to quadratures, a systematic, dispersive, unitary and regulator-free program for calculation in nonrenormalizable field theory. We show how to use as input into the $\frac{N}{D}$ equations any set of graphs (regardless of their divergence) whose left-hand absorptive part, although asymptotically illbehaved, is not itself divergent. The program makes calculation possible in a large variety of nonrenormalizable contexts, including $W$-meson theory, Fermi theory, derivative coupling theories, and spin-$\frac{3}{2}$ and higher spin theories in general (for example, linearized gravitational theory). This paper is intended as a sketch of the main results of the work, whereas most of the details will appear in a subsequent series of papers.