Higher Order Leptonic Weak Interactions

Abstract

Higher order leptonic weak interactions as mediated by a charged intermediate boson are studied on the basis of summing the leading terms in perturbation theory. An important question is whether it is consistent to assume that such a procedure is meaningful and leads to finite results. This is indeed the case for the uncrossed ladder graphs, as shown by Feinberg and Pais. Here an attempt is made to study all higher order graphs. The regularization procedure adopted is the $\ensuremath{\xi}$- limiting formalism of Lee and Yang, and the leading terms taken into account are those which, for a given power of the coupling constant, contain the highest power of ${\ensuremath{\xi}}^{\ensuremath{-}1}$ and also the highest possible power of $\mathrm{ln}\ensuremath{\xi}$ consistent with this maximum power of ${\ensuremath{\xi}}^{\ensuremath{-}1}$. The sum is determined by a comparison with the electrodynamics of vector mesons on the assumption that it is meaningful to sum the leading terms in both theories. It is then found that, for four-fermion allowed processes at energies much below 300 BeV, no inconsistencies are found and the conventional ${g}^{2}$ should be replaced by $\frac{5}{8}{g}^{2}$.