Abstract
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 $V_\text{CKM} = U^{u\dagger} U^d$. 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.
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