Natural suppression of symmetry violation in gauge theories: Muon- and electron-lepton-number nonconservation

We analyze the circumstances under which the violations of an approximate symmetry in a unified gauge theory of weak interactions are naturally suppressed; in particular, we investigate approximate muon- and electron-type lepton-number conservation as an example of such a symmetry. Extending earlier work, we propose a unified treatment of this symmetry together with strangeness conservation by the weak neutral current and $\mathrm{CP}$ invariance. The rate for the decay $\mu \to e\gamma$ is calculated for a general SU(2) × U(1) gauge model. From this and a similar study of the decay $\mu \to \mathrm{ee}\overline{e}$ we derive a set of conditions which guarantees that the violation of muon- and electron-type lepton-number conservation is naturally strongly suppressed. As part of this, we compute the nondiagonal electromagnetic vertex to one-loop order for an arbitrary SU(2) × U(1) gauge theory. We then focus on the phenomenological predictions of a particular gauge model with three left-handed doublets of leptons and quarks. These include the existence of charged and neutral heavy leptons and of small violations of $\mu$-$e$ universality and the relation $G_F^{\beta }sec{\theta }_C=G_F^{\mu }$. Other muon- and electron-number-violating effects include nonvanishing rates for the decays $K^{\pm }\to {\pi }^{\pm }e\overline{\mu }$ and $K_L\to e\overline{\mu }$ and for the reactions $\mu +N\to e+N$ and ${\nu }_{\mu }+N\to e^{-}+X$.