## 2 Problem Definition
From the above, the mixing matrix \(V_{CKM}\), predicted by the 4-zero texture model, is given by:  
\[V_{CKM}^{th}\equiv O_{u}^{\dagger}P_{u}P_{d}^{\dagger}O_{d} \tag{12}\]  
in an explicit form:  
\[(V_{CKM}^{th})_{ij}=(O_{u})_{1i}(O_{d})_{1j}+(O_{u})_{2i}(O_{d})_{2j}e^{i\phi_{ 1}}+(O_{u})_{3i}(O_{d})_{3j}e^{i(\phi_{1}+\phi_{2})} \tag{13}\]  
where the phases \(\phi_{1}\) and \(\phi_{2}\) are defined as:  
\[\phi_{1}=\phi_{D_{u}}-\phi_{D_{d}},\qquad\phi_{2}=\phi_{\bar{D}_{u}}-\phi_{B_{ d}}. \tag{14}\]  
These are considered to be measured in radians with principal argument \([0,2\pi]\), and the indices \(i\) and \(j\) correspond, respectively, to the indices \((u,c,t)\) and \((d,s,b)\). The magnitude of the elements of the mixing matrix is given by:  
\[|(V_{CKM})_{ij}^{th}| = \left\{[(O_{u})_{1i}(O_{d})_{1j}]^{2}+[(O_{u})_{2i}(O_{d})_{2j}] ^{2}+[(O_{u})_{3i}(O_{d})_{3j}]^{2}\right.\] \[+ 2[(O_{u})_{1i}(O_{d})_{1j}][(O_{u})_{2i}(O_{d})_{2j}]\left.Cos( \phi_{1})\right.\] \[+ 2[(O_{u})_{1i}(O_{d})_{1j}][(O_{u})_{3i}(O_{d})_{3j}]\left.Cos( \phi_{1}+\phi_{2})\right.\] \[+ \left.2[(O_{u})_{2i}(O_{d})_{2j}][(O_{u})_{3i}(O_{d})_{3j}]\left. Cos(\phi_{2})\right\}^{1/2}\right.\]  
At this point, it is essential to emphasize that analytical expressions are obtained for the elements of the mixing matrix predicted by the 4-zero texture formalism; with this information, it is possible to construct a more complete theory of the standard model where the origin of the mixing matrix \(V_{CKM}\) is explained. It only remains to validate the theoretical model of textures with 4 zeros, that is, to find the range of the free parameters \(A_{u}\), \(A_{d}\), \(\phi_{1}\), and \(\phi_{2}\) that agree with the experimental values of the matrix \(V_{CKM}\), and for this, we define a function \(\chi^{2}(X)\) and use a chi-square criterion [45; 46] established by:  
\[0<\frac{\chi^{2}(X)}{4}<1 \tag{16}\]  
where \(X=(A_{u},A_{d},\phi_{1},\phi_{2})\) and  
\[(\lambda_{1}^{d},\lambda_{2}^{d},\lambda_{3}^{d}) =(-\eta_{d}m_{d},\eta_{d}m_{s},m_{b}) \tag{11}\]0 \[(\lambda_{1}^{d},\lambda_{2}^{d},\lambda_{3}^{d}) =(-\eta_{d}m_{d},\eta_{d}m_{s},m_{b}) \tag{11}\]1  
The super-indexes \"\(th\)\" are given by Equation (15) and the quantities without super-index are the experimental data with uncertainty \(\sigma_{V_{th}}^{2}\) (see Equations (A1) and (A2)). \begin{table}
\begin{tabular}{c c c} \hline
**Case** & \(\eta_{u}\) & \(\eta_{d}\) \\ \hline
1 & \(+1\) & \(+1\) \\
2 & \(+1\) & \(-1\) \\
3 & \(-1\) & \(+1\) \\
4 & \(-1\) & \(-1\) \\ \hline \end{tabular}
\end{table} 
Table 1: Cases of study. Although, at first sight, the mathematically constructed function \(\chi^{2}(X)\) turns out to have a simple structure and the amount of free parameters is small, the difficulty of finding the numerical range of the same ones that fulfill the condition given in Equation (16) comes from the cumbersome composition of functions that constitute it. In order to have a notion of the topographic relief of this function, different projections of the function \(\chi^{2}(X)\) in different planes are shown in Figure 1. Figure 1a shows the projection onto the planes \(A_{u}\) and \(A_{d}\) (held fixed values for \(\phi_{1}=c_{1}=const\) and \(\phi_{2}=c_{2}=const\)); that is, the dependence of the function \(\chi^{2}(X)\) on the variables \(A_{u}\) and \(A_{d}\) is shown (right graph) and the corresponding contour lines are illustrated (left graph). Similarly, Figure 1b corresponds to the projection of the function \(\chi^{2}(X)\) onto \(A_{u}\) and \(\phi_{1}\) (setting \(A_{d}=c_{3}=const\) and \(\phi_{2}=c_{4}=const\)). Analogously in the graphical displays, we show the behavior of the function \(\chi^{2}(X)\) in the planes \(A_{u}\) and \(\phi_{2}\) (Figure 1c), in the planes \(A_{d}\) and \(\phi_{1}\) (Figure 1d), in the planes \(A_{d}\) and \(\phi_{2}\) (Figure 1e), and in the \(\phi_{1}\) and \(\phi_{2}\) planes (Figure 1f).