A Field Theory of Weak Interactions. II

Abstract

The study of the Bethe-Salpeter equation for lepton-lepton interaction mediated by charged vector mesons is continued. Starting from the exact integral equation, an improved approximation procedure is developed. This reproduces the low-energy results for "allowed" processes given in a previous paper. Beyond that, it is now found that to leading order the BS equation gives the value $\frac{3g^2}{4{\pi }^2}$ for the zero-energy ratio between "forbidden" and "allowed" amplitudes, where $g$ is the bare meson-lepton coupling constant. Some information on the momentum dependence of the forbidden amplitude is also obtained. The mathematical methods developed in an earlier paper are then applied to the corresponding Bethe-Salpeter equation of the Fermi field theory. It is shown that the calculated amplitudes for both allowed and forbidden processes are equal to zero. This illustrates the fact that if higher order effects are taken seriously, there is no reason to consider the Fermi field theory as the limiting case of a vector meson theory with a boson mass which tends to infinity.