Symmetric Cabibbo-Kobayashi-Maskawa matrix and quark mass matrices

In this article we aim to find the constraints on the quark mass matrices for the symmetric Cabibbo-Kobayashi-Maskawa (CKM) matrix $V$. We work in the bases, where (i) $M_u$ is diagonal, (ii) $M_d$ is diagonal, and (iii) $M_d=f\left(M_u^{*}\right)$, i.e., $U=D^{*}P$, where $U$ and $D$ are matrices that diagonalize the up- and down-quark mass matrices, respectively, and $P$ is the phase matrix. We find that none of the moduli of the off-diagonal elements of these interesting forms of the quark mass matrices $M_u$ and $M_d$, which lead to the symmetric CKM matrix, are consistent with zero for these Ansätze, which means that such forms for mass matrices are difficult to obtain from any symmetry. We then give the symmetry constraint for $V$ written in terms of the mass eigenvalues in a basis-independent form.