Improved analyses for ${\mu }^{-}{e}^{-}\to e^{-}{e}^{-}$ in muonic atoms by contact interactions

The charged lepton flavor violating (CLFV) processes of ${\mu }^{-}{e}^{-}\to e^{-}{e}^{-}$ decay by four Fermi contact interactions in a muonic atom for various atoms are investigated. The wave functions of bound and scattering state leptons are properly treated by solving Dirac equations with Coulomb interaction of the finite nuclear charge distributions. This new effect contributes significantly in particular for heavier atoms, where the obtained decay rate is about one order of magnitude larger than the previous estimation for ${}^{208}\mathrm{Pb}$. We find that, as the atomic number $Z$ increases, the ${\mu }^{-}{e}^{-}\to e^{-}{e}^{-}$ decay rates increase more rapidly than the result of the previous work of $Z^3$, suggesting this decay as one of the promising processes to search for CLFV interaction.

Figure 1

The transition density $\rho (r)r^2$ for the ${\mu }^{-}(1S)+e^{-}(1S)\to e^{-}(\kappa =-1)+e^{-}(\kappa =-1)$. The dashed, dash-dotted, dash-two-dotted, and solid curves show the transition density in models I–IV, respectively.

Figure 2

The atomic number ($Z$) dependence of the ratio of the decay rate $\mathrm{\Gamma }/{\mathrm{\Gamma }}_0$. The dashed, dash-dotted, dash-two-dotted, and solid curves show the transition density in models I–IV, respectively. We note that the factor $1/3$ has not been multiplied to the dash-two-dotted curve.

Figure 3

The energy and angular distribution of emitted electrons for the ${}^{208}\mathrm{Pb}$ by using $g_1$ type interaction.

Figure 4

The normalized energy distribution of the electron for the ${}^{208}\mathrm{Pb}$. The solid (red) curve and the dashed (blue) curve are obtained by using models IV and I, respectively. The $g_1$ term is included and all bound $S$ state electrons are taken into account.

Figure 5

The normalized angular distribution of emitted electrons for the ${}^{208}\mathrm{Pb}$. The other features are the same as those in Fig. 4.

Figure 6

The angular distribution of emitted electrons for the ${}^{208}\mathrm{Pb}$. The angular distribution due to the ($g_1\text{−}{g}_4$) terms is shown in the solid (red) curve, and that of the ($g_5\text{−}{g}_6$) terms is shown in the dashed (black) curve.

Figure 7

Upper limits of $Br({\mu }^{-}{e}^{-}\to e^{-}{e}^{-})$. The dashed (blue) curve shows the result of previous work [4]. Our results including only $1S$ electrons and all $1S$ electrons are shown by the solid (red) curve and the dotted (orange) curve, respectively.