## 2 Problem Definition
\tag{7}\]  
In references [6; 18], it is shown that this matrix can be diagonalized by a unitary matrix as follows:  
\[U_{q}\,M_{q}\,U_{q}^{\dagger} = diag\{\lambda_{1}^{q},\lambda_{2}^{q},\lambda_{3}^{q}\}\] \[\left(O_{q}^{T}\,P_{q}\right)M_{q}\left(O_{q}^{T}\,P_{q}\right)^ {\dagger} = diag\{\lambda_{1}^{q},\lambda_{2}^{q},\lambda_{3}^{q}\}\]  
where \(diag\{\lambda_{1}^{q},\lambda_{2}^{q},\lambda_{3}^{q}\}\) denotes a diagonal matrix and \(\lambda_{i}^{q}\) denotes each of the three eigenvalues of \(M_{q}\). The matrices \(P_{q}\) and \(O_{q}\) are given by:  
\[P_{q}= \begin{pmatrix}1&0&0\\ 0&e^{i\phi_{D_{q}}}&0\\ 0&0&e^{i(\phi_{D_{q}}+\phi_{B_{q}})}\end{pmatrix}  \tag{8}\]  
and  
\[O_{q} = \begin{pmatrix}(O_{q})_{11}&(O_{q})_{12}&(O_{q})_{13}\\ (O_{q})_{21}&(O_{q})_{22}&(O_{q})_{23}\\ (O_{q})_{31}&(O_{q})_{32}&(O_{q})_{33}\end{pmatrix}  \tag{9}\] \[= \begin{pmatrix}\sqrt{\frac{\lambda_{2}^{q}\lambda_{3}^{q}(A_{q}- \lambda_{1}^{q})}{A_{q}(\lambda_{2}^{q}-\lambda_{1}^{q})(\lambda_{3}^{q}- \lambda_{1}^{q})}}&\eta_{q}\sqrt{\frac{\lambda_{1}^{q}\lambda_{2}^{q}(A_{2}^{q }-A_{q})}{A_{q}(\lambda_{2}^{q}-\lambda_{1}^{q})(\lambda_{3}^{q}-\lambda_{2}^{ q})}}&\sqrt{\frac{\lambda_{1}^{q}\lambda_{2}^{q}(A_{q}-\lambda_{2}^{q})}{A_{q}( \lambda_{2}^{q}-\lambda_{1}^{q})(\lambda_{3}^{q}-\lambda_{2}^{q})}}\\ -\eta_{q}\sqrt{\frac{\lambda_{1}^{q}(A_{1}^{q}-A_{q})}{(\lambda_{2}^{q}- \lambda_{1}^{q})(\lambda_{3}^{q}-\lambda_{1}^{q})}}&\sqrt{\frac{\lambda_{2}^{ q}(A_{q}-\lambda_{2}^{q})}{(\lambda_{2}^{q}-\lambda_{1}^{q})(\lambda_{3}^{q}- \lambda_{2}^{q})}}&\sqrt{\frac{\lambda_{2}^{q}(A_{q}^{q}-A_{q})}{(\lambda_{3}^ {q}-\lambda_{1}^{q})(\lambda_{3}^{q}-\lambda_{2}^{q})}}\\ \eta_{q}\sqrt{\frac{\lambda_{1}^{q}(A_{q}-\lambda_{2}^{q})(A_{q}-\lambda_{3}^ {q})}{A_{q}(\lambda_{2}^{q}-\lambda_{1}^{q})(\lambda_{3}^{q}-\lambda_{1}^{q})} }&-\sqrt{\frac{\lambda_{2}^{q}(A_{q}-\lambda_{1}^{q})(\lambda_{3}^{q}-\lambda_ {2}^{q})}{A_{q}(\lambda_{2}^{q}-\lambda_{1}^{q})(\lambda_{3}^{q}-\lambda_{2}^ {q})}}&\sqrt{\frac{\lambda_{3}^{q}(A_{q}-\lambda_{1}^{q})(A_{q}-\lambda_{2}^{q })}{A_{q}(\lambda_{3}^{q}-\lambda_{1}^{q})(\lambda_{3}^{q}-\lambda_{2}^{q})}} \end{pmatrix} \]  
Taking \(\lambda_{3}^{q}>0\) and \(A_{q}>0\), the relations between \(\lambda_{i}^{q}\) and the physical masses of the quarks are:  
\[(\lambda_{1}^{u},\lambda_{2}^{u},\lambda_{3}^{u}) =(-\eta_{u}m_{u},\eta_{u}m_{c},m_{t}) \tag{10}\] \[(\lambda_{1}^{d},\lambda_{2}^{d},\lambda_{3}^{d}) =(-\eta_{d}m_{d},\eta_{d}m_{s},m_{b}) \tag{11}\]  
where \(m_{u}\) is the u quark mass, \(m_{c}\) is the c quark mass, \(m_{t}\) is the t quark mass, \(m_{d}\) is the d quark mass, \(m_{s}\) is the s quark mass, and \(m_{b}\) is the b quark mass, and its experimental value is presented in Appendix A. In this work, the same 4-zero texture structure is considered for the u-quark mass matrix and the d-type quark mass matrix (a parallel mass structure). The \(q\)-index takes two values, \(q=u\) and \(q=d\). From Equation (9), it is noticed that the elements of the matrix \(O_{q}\) depend on the free parameters \(\eta_{q}\) and \(A_{q}\); the first parameter takes values of \(+1\) and \(-1\) and tells us that the eigenvalue of the mass matrix is negative, when \(\eta_{q}=1\) the first eigenvalue \(\lambda_{1}^{q}\) is negative and the second eigenvalue \(\lambda_{2}^{q}\) is positive, and when \(\eta_{q}=-1\) the first eigenvalue \(\lambda_{1}^{q}\) is positive and the second eigenvalue \(\lambda_{2}^{q}\) is negative. The combinations of signs of the \(\eta_{u}\) and \(\eta_{d}\) parameters define the different cases of study to be considered (see Table 1). The second pair of free parameters are \(A_{u}\) and \(A_{d}\), whose values are restricted to the intervals \(m_{c}<A_{u}<m_{t}\) and \(m_{s}<A_{d}<m_{b}\) to ensure that the elements of the \(O_{u}\) and \(O_{d}\) matrices are real.