Muon Number Nonconservation Effects in a Gauge Theory with V A Currents and Heavy Neutral Leptons

Since in gauge theories eigenstates of weak interactions are in general not mass eigenstates, we would not expect flavor conservation. In particular this should also hold for the leptonic flavors: muon numbers and electron numbers, etc. The apparent conservation of muon number in the standard $V \ensuremath{-} A$ theory should be interpreted as reflecting the fact that neutrino masses (if not identically zero) are almost degenerate when viewed on the normal mass scale. In theories containing $V + A$ currents, the right-handed muon and electrons are expected to couple to intermixing heavy leptons in the GeV range. In such theories muon-number-violation effects will be dramatically larger. However, when constructing new models of leptons one should be mindful that flavor-changing neutral-current processes such as $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma}$ and $\ensuremath{\mu}\ensuremath{\rightarrow}\mathrm{ee}\overline{e}$ are suppressed experimentally. This indicates the need for a "leptonic Glashow-Iliopoulos-Maiani cancellation mechanism." We have proposed a way to incorporate these features in an ${\mathrm{SU}}_{2}$ \ifmmode\times\else\texttimes\fi{} ${\mathrm{U}}_{1}$ gauge theory. Basically it involves the addition to the standard Weinberg-Salam theory of right-handed doublets with the electron and muon coupled to orthogonal "heavy neutrinos." This leads to an electronic neutral current which is purely vector and its attendant suppression of parity-violation effects in high-$Z$ atoms. Muon-number-nonconservation effects involving only familiar particles are higher-order weak processes and are naturally of the order $G_{F}^{}{}_{}{}^{2}$. In this paper we give details of our calculations of $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma}$, $\ensuremath{\mu}\ensuremath{\rightarrow}\mathrm{ee}\overline{e}$, ${K}_{L}\ensuremath{\rightarrow}e\overline{\ensuremath{\mu}}$, $K\ensuremath{\rightarrow}\ensuremath{\pi} e\ensuremath{\mu}$, and $\ensuremath{\mu}e$ conversion in a nucleus, etc. In order to have a "natural" theory, we have incorporated recent suggestions made by Bjorken, Lane, and Weinberg about the Higgs structure for such a model. This modification increases the rate for $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma}$ by a factor of 25 but does not materially affect other processes. For a heavy-lepton mass-difference and mixing-angle combination of $sin\ensuremath{\varphi}cos\ensuremath{\varphi}[m{({N}_{1})}^{2}\ensuremath{-}m{({N}_{2})}^{2}]\ensuremath{\simeq}1$ ${\mathrm{GeV}}^{2}$, the branching ratio for $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma}$ is 4 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}10}$, that for $\ensuremath{\mu}\ensuremath{\rightarrow}\mathrm{ee}\overline{e}$ is around ${10}^{\ensuremath{-}11}$; the $\ensuremath{\mu}e$ conversion rate can be as large as ${10}^{\ensuremath{-}9}$ when compared to the ordinary muon capture in the nucleus. If there is a heavy quark $b$ coupled to the $u$ quark through the $V + A$ current, this conversion rate will be decreased by a factor of 30. The rates for the muon-number-nonconserving $K$ decay are more sensitive to the relative lepton masses. For example the branching ratio of ${K}_{L}\ensuremath{\rightarrow}e\overline{\ensuremath{\mu}}$ is about ${10}^{\ensuremath{-}10}$ for $\frac{m({N}_{1})}{m({N}_{2})}\ensuremath{\simeq}4$, and a 1.8-GeV charmed quark.