Theory of Weak Interactions without Divergences

Abstract

A theory of weak interactions is suggested which is free of the divergences associated with the fourfermion theory and the usual intermediate-vector-boson theory, without the use of cutoffs. The interaction Lagrangian is a universal, local coupling of the usual weak currents to a vector field operator which is a nonentire function of a minimal set of vector-meson fields of large mass (\ensuremath{\gtrsim}5 GeV). A summation procedure related to that of Efimov and Fradkin is developed for the analysis of such Lagrangians, and $S$-matrix elements are computed thereby. It is shown that the (two) parameters of the Lagrangian may be adjusted to produce finite fermion self-energies and finite second- and fourth-order graphs. A power-counting argument is given that this finiteness then persists to all orders without the use of cutoffs. It is further shown that the parameters may be chosen to produce agreement with the predictions of the Fermi theory at low-energies and momentum transfers. The high-energy deviations from Fermi theory predicted by this theory are analyzed. The problem of ambiguities associated with nonpolynomial Lagrangians is discussed. A prescription for the resolution of the ambiguity is given which is expected to ensure the unitarity of the theory in all orders.