Muon-Electron Conversion in a Family Gauge Boson Model

We study the $\mu$-$e$ conversion in muonic atoms via an exchange of family gauge boson (FGB) $A_2^1$ in a $U(3)$ FGB model. Within the class of FGB model, we consider three types of family-number assignments for quarks. We evaluate the $\mu$-$e$ conversion rate for various target nuclei, and find that next generation $\mu$-$e$ conversion search experiments can cover entire energy scale of the model for all of types of the quark family-number assignments. We show that the conversion rate in the model is so sensitive to up- and down-quark mixing matrices, $U^u$ and $U^d$, where the CKM matrix is given by $V_{\text{CKM}}=U^{u†}{U}^d$. Precise measurements of conversion rates for various target nuclei can identify not only the types of quark family-number assignments, but also each quark mixing matrix individually.