A Field Theory of Weak Interactions. I

Higher order weak-interaction effects are studied in the framework of vector meson field theory. We find that such effects may be observable even at low energy. The cause of this can be traced precisely to the unrenormalizability of the theory in the sense of conventional perturbation expansions. These expansions are circumvented by new techniques for summing the most singular parts of perturbation graphs. In this first paper we study in detail the infinite subset of uncrossed ladder graphs. Purely leptonic processes are considered to begin with. The corresponding Bethe-Salpeter equation is soluble by a new iteration scheme. In leading order we reproduce the conventional zero-energy results provided $g^2$ is replaced by $\frac{3g^2}{4}$. ($g=\mathrm{bare}\mathrm{meson}\mathrm{lepton}\mathrm{coupling}\mathrm{constant}$). An argument is presented which leads to the conjecture that this result is valid for larger classes of graphs. However, there exist energy-dependent deviations from the conventional second-order results. These are in principle observable in $\mu$ decay. The applicability of the theory to semileptonic and nonleptonic phenomena depends on properties of the baryon and meson currents and on the effects of the strong interactions. Preliminary considerations along these lines are given.