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

TL;DR

This paper explores the possibility that observed charged leptons are not eigenstates of family, analyzing potential mixing with a small angle constrained by experimental data and suggesting rare decay searches for further limits.

Contribution

It proposes a framework where charged leptons are not necessarily eigenstates of family, introducing a mixing angle constrained by current experiments and highlighting rare decay searches for tighter bounds.

Findings

The $e$-$$ mixing angle $ heta$ is constrained to be less than about $10^{-3}$.

No experimental evidence currently supports $e^0_1$-$e^0_2$ mixing.

Rare decay $ ightarrow e + mma$ could provide more stringent limits.

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.