## 2 Problem Definition  
Within the scope of the SM, the masses of the quarks come from hermetic \(3\times 3\) matrices known as mass matrices (one for u-type quarks and another for d-type quarks) [40; 41; 42], as the absolute values of their eigenvalues and the matrix \(V_{CKM}\) as the product of the matrix that diagonalizes the u-type quarks and the matrix that diagonalizes the d-type quarks; however, due to the mathematical formalism used, the mass matrices remain entirely unknown and, consequently, the masses of the quarks and the mixing matrix \(V_{CKM}\) cannot be theoretically predicted. The experiment provides us with the numerical value of these quantities. The mixing matrix \(V_{CKM}\) can be written in a general way as:  
\[V_{CKM}= \begin{pmatrix}(V_{CKM})_{ud}&(V_{CKM})_{us}&(V_{CKM})_{ub}\\ (V_{CKM})_{cd}&(V_{CKM})_{cs}&(V_{CKM})_{cb}\\ (V_{CKM})_{td}&(V_{CKM})_{ts}&(V_{CKM})_{tb}\end{pmatrix}  \tag{1}\]  
and is a unitary matrix containing information about the probability of transition between u-type quarks (left-hand index) and d-type quarks (right-hand index), through the weak interaction [43]. That is, \(|(V_{CKM})_{us}|\) quantifies the transition probability between the quark \(u\) and the quark \(s\) through the interaction with the boson \(W^{\pm}\). Over many years and different collaborations, the magnitudes of the elements of the mixing matrix \(V_{CKM}\) have been experimentally measured with an accuracy up to \(10^{-5}\), and it is well known that four quantities (three angles and one phase) are needed to have a parametrization that adequately describes the matrix \(V_{CKM}\)[3; 44]. In this work, we choose the Chau-Keng parametrization [3], and the three corresponding angles \(\theta_{13}\), \(\theta_{12}\), and \(\theta_{23}\) are obtained by choosing the three magnitudes of the elements \(|(V_{CKM})_{us}|\), \(|(V_{CKM})_{ub}|\), and \(|(V_{CKM})_{cb}|\), and the phase \(\delta_{13}\) is obtained from the Jarslog invariant (\(J\)) through the following relations:  
\[\sin\theta_{13}=|(V_{CKM})_{ub}| \tag{2}\]  
\[\sin\theta_{12}=\frac{|(V_{CKM})_{us}|}{\sqrt{1-|(V_{CKM})_{ub}|^{2}}} \tag{3}\]  
\[\sin\theta_{23}=\frac{|(V_{CKM})_{cb}|}{\sqrt{1-|(V_{CKM})_{ub}|^{2}}} \tag{4}\]  
\[\sin\delta_{13}=\frac{J}{\sqrt{1-(\sin\theta_{12})^{2}}\,\sin\theta_{12}\,[1 -(\sin\theta_{13})^{2}]\,\sin\theta_{13}\,\sqrt{1-(\sin\theta_{23})^{2}}\, \sin\theta_{23}} \tag{5}\]  
where \(J=Im(V_{us}\,V_{cb}\,V_{ub}^{*},\,V_{cs}^{*})\). Hence, we choose \(|(V_{CKM})_{us}|\), \(|(V_{CKM})_{ub}|\), \(|(V_{CKM})_{cb}|\), and \(J\) as independent quantities. Within the SM context, the mixing matrix, \(V_{CKM}\), is defined by [3]:  
\[V_{CKM}=U_{u}^{\dagger}\,U_{d}\]  
where \(U_{u}\) is the matrix that diagonalizes the mass matrix of the u-type quarks and \(U_{d}\) is the matrix that diagonalizes the d-type quarks. It is here where the texture formalism is born, which consists of proposing structures with zeros in some entries of the mass matrix in such a way that we can find the matrices that diagonalize it and be able to calculate analytically the mixing matrix \(V_{CKM}\) and validate the chosen texture structure. Without loss of generality, the mass matrices \(M_{u}\) and \(M_{d}\) are considered hermitian so that the general mass structure has the form:  
\[M_{q}=\begin{pmatrix}E_{q}&D_{q}&F_{q}\\ D_{q}^{*}&C_{q}&B_{q}\\ F_{q}^{*}&B_{q}^{*}&A_{q  
A matrix of the type 4-zero texture [18] is formed from the above matrix by taking the entries \((1,1)\), \((1,3)\), and \((3,1)\) equal to zero. Thus, we arrive at the following matrix structure:  
\[M_{q}= \begin{pmatrix}0&D_{q}&0\\ D_{q}^{*}&C_{q}&B_{q}\\ 0&B_{q}^{*}&A_{q}\end{pmatrix} .