Parameter-Independent Quark Mass Relation in the U(3)$\times$U(3)${}^{\mathrm{\prime }}$ Model

Recently, we have proposed a quark mass matrix model based on U(3)$\times$U(3)${}^{\mathrm{\prime }}$ family symmetry, in which up- and down-quark mass matrices $M_u$ and $M_d$ are described only by complex parameters $a_u$ and $a_d$, respectively. When we use charged lepton masses as additional input values, we can successfully obtain predictions for quark masses and Cabibbo-Kobayashi-Maskawa mixing. Since we have only one complex parameter $a_q$ for each mass matrix $M_q$, we can obtain a parameter-independent mass relation by using three equations for $\mathrm{T}\mathrm{r}[H_q]$, $\mathrm{T}\mathrm{r}[H_q{H}_q]$ and $\mathrm{d}\mathrm{e}\mathrm{t}{H}_q$, where $H_q\equiv M_q{M}_q^{†}$ ($q=u,d$). In this paper, we investigate its parameter-independent feature of the quark mass relation in the model.