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

Abstract

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 $\ensuremath{\mu}\ensuremath{\rightarrow}e\ensuremath{\gamma}$ is calculated for a general SU(2) \ifmmode\times\else\texttimes\fi{} U(1) gauge model. From this and a similar study of the decay $\ensuremath{\mu}\ensuremath{\rightarrow}\mathrm{ee}\overline{e}$ 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) \ifmmode\times\else\texttimes\fi{} 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 $\ensuremath{\mu}$-$e$ universality and the relation ${G}_{F}^{\ensuremath{\beta}}sec{\ensuremath{\theta}}_{C}={G}_{F}^{\ensuremath{\mu}}$. Other muon- and electron-number-violating effects include nonvanishing rates for the decays ${K}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}e\overline{\ensuremath{\mu}}$ and ${K}_{L}\ensuremath{\rightarrow}e\overline{\ensuremath{\mu}}$ and for the reactions $\ensuremath{\mu}+N\ensuremath{\rightarrow}e+N$ and ${\ensuremath{\nu}}_{\ensuremath{\mu}}+N\ensuremath{\rightarrow}{e}^{\ensuremath{-}}+X$.