Are Charged Leptons in the Simultaneous Eigenstates of Mass and Family?

Abstract

Conventionally, the observed charged leptons are regarded the simultaneous eigenstates of "mass" and "family". Against this view, we discuss a possibility that the observed charged leptons $e_i=(e, \mu, \tau)$ are not identical with the eigenstates of family $e^0_\alpha =(e_1^0, e_2^0, e_3^0)$. Here, we define the eigenstates of family, $e^0_\alpha$, as the states which interact with family gauge bosons in the mass eigenstates of the broken U(3)$_{family}$ gauge symmetry. Although there is at present not any experimental evidence for $e^0_1$-$e^0_2$ mixing, and we have only an upper limit for the mixing from the present experimental data. We will conclude that the $e$-$\mu$ mixing angle $\theta$ must be $\theta \lesssim 10^{-3}$. Thus, we can not exclude a possibility $\theta\neq 0$. If we want more small upper limit of $\theta$, a rare decay search $\mu \rightarrow e + \gamma$ will be useful.