Here we first show the procedure of how we used the rule for the sum of the oscillation probabilities of three neutrinos [5] [6] because we believe that we could also apply such a formal mathematical approach to quarks in order to find a numerical value for their CP violation phase.

For neutrinos, the well-known PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix is used [7] - [11] :

$\begin{array}{c}{U}_{PMNS}=\left(\begin{array}{ccc}{U}_{e1}& U_{e2}& U_{e3}\\ U_{\mu 1}& U_{\mu 2}& U_{\mu 3}\\ U_{\tau 1}& U_{\tau 2}& U_{\tau 3}\end{array}\right)\\ =\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}{\text{e}}^{-i\delta }\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}{\text{e}}^{i\delta }& c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}{\text{e}}^{i\delta }& c_{13}{s}_{23}\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}{\text{e}}^{i\delta }& -c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}{\text{e}}^{i\delta }& c_{13}{c}_{23}\end{array}\right)\end{array}$ (5)

the following relation applies to the probability of neutrino oscillation for all three possible transitions $\left({\nu }_e\to {\nu }_{\mu }\right),\left({\nu }_e\to {\nu }_{\tau }\right),\left({\nu }_e\to {\nu }_e\right)$; $\left({\nu }_{\mu }\to {\nu }_e\right),\left({\nu }_{\mu }\to {\nu }_{\tau }\right),\left({\nu }_{\mu }\to {\nu }_{\mu }\right)$; $\left({\nu }_{\tau }\to {\nu }_e\right),\left({\nu }_{\tau }\to {\nu }_{\mu }\right),\left({\nu }_{\tau }\to {\nu }_{\tau }\right)$:

$\begin{array}{l}P\left({\nu }_{\alpha }\to {\nu }_{\beta }\right)\\ ={\delta }_{\alpha \beta }-4{\displaystyle \underset{i<j}{\sum }{R}_e\left(U_{\alpha i}{U}_{\beta i}^{\ast }{U}_{\alpha j}^{\ast }{U}_{\beta j}\right)}{\sin }^2\left(\frac{\Delta m_{ji}^{2}L{c}^3}{4E\hslash }\right)\\ +2{\displaystyle \underset{i<j}{\sum }\mathrm{Im}\left(U_{\alpha i}{U}_{\beta i}^{\ast }{U}_{\alpha j}^{\ast }{U}_{\beta j}\right)}\sin \left(\frac{\Delta m_{ji}^{2}L{c}^3}{2E\hslash }\right);i,j=1,2,3;\alpha ,\beta =e,\mu ,\tau .\end{array}$ (6)

In general, we can write sum rules for oscillation probabilities for three neutrinos in the form of the following relations:

$P\left({\nu }_e\to {\nu }_{\mu }\right)+P\left({\nu }_e\to {\nu }_{\tau }\right)+P\left({\nu }_e\to {\nu }_e\right)=1$ (7)

$P\left({\nu }_{\mu }\to {\nu }_e\right)+P\left({\nu }_{\mu }\to {\nu }_{\tau }\right)+P\left({\nu }_{\mu }\to {\nu }_{\mu }\right)=1$ (8)

$P\left({\nu }_{\tau }\to {\nu }_e\right)+P\left({\nu }_{\tau }\to {\nu }_{\mu }\right)+P\left({\nu }_{\tau }\to {\nu }_{\tau }\right)=1$ (9)

In further considerations in this paper, we set the CP violation phase for quarks as the main problem that should be solved to the end, and the following chapters are devoted to it.

There are two conventions in quark physics: the first convention refers to down quarks and the second convention refers to up quarks, and both will be included in research.

The first convention, which is generally accepted, is described by the following relations:

$\begin{array}{c}\left(\begin{array}{c}{d}^I\\ s^I\\ b^I\end{array}\right)=V_{CKM}\left(\begin{array}{c}u\\ c\\ t\end{array}\right)=\left(\begin{array}{ccc}{V}_{ud}& V_{us}& V_{ub}\\ V_{cd}& V_{cs}& V_{cb}\\ V_{td}& V_{ts}& V_{tb}\end{array}\right)\left(\begin{array}{c}d\\ s\\ b\end{array}\right)\\ =\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}{\text{e}}^{-i\delta }\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}{\text{e}}^{i\delta }& c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}{\text{e}}^{i\delta }& c_{13}{s}_{23}\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}{\text{e}}^{i\delta }& -c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}{\text{e}}^{i\delta }& c_{13}{c}_{23}\end{array}\right)\left(\begin{array}{c}d\\ s\\ b\end{array}\right)\\ =\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}{\text{e}}^{-i\delta }\\ -A-B{\text{e}}^{i\delta }& C-D{\text{e}}^{i\delta }& c_{13}{s}_{23}\\ E-F{\text{e}}^{i\delta }& -G-H{\text{e}}^{i\delta }& {}_{13}{c}_{23}\end{array}\right)\left(\begin{array}{c}d\\ s\\ b\end{array}\right).\end{array}$ (10)

where the interaction eigenstates $d^I,s^I,b^I$ are connected via the CKM unitary matrix with mass eigenstates $d,s,b$ for down-quarks.

Here are all three possible transitions $d^I\to s^I,d^I\to b^I,d^I\to d^I;s^i\to d^I,$ $s^i\to b^I,s^i\to s^i;b^I\to d^I,b^I\to s^i,b^I\to b^I$ are described with the rule for the sum of quark oscillation probabilities in a similar way as already defined in neutrino physics (7, 8, 9):

$\begin{array}{l}P\left(d^I\to s^I\right)+P\left(d^I\to b^I\right)+P\left(d^I\to d^I\right)=1.\\ P\left(s^i\to d^I\right)+P\left(s^i\to b^I\right)+P\left(s^i\to s^i\right)=1.\\ P{\left(b^I\to d\right)}^I+P\left(b^I\to s^i\right)+P\left(b^I\to b^I\right)=1.\end{array}$ (11)

as well as the general relation which reads:

$\begin{array}{l}P\left(V_{\alpha }\to V_{\beta }\right)\\ ={\delta }_{\alpha \beta }-4{\displaystyle \underset{i<j}{\sum }{R}_e\left(V_{\alpha i}{V}_{\beta i}^{\ast }{V}_{\alpha j}^{\ast }{V}_{\beta j}\right)}{\sin }^2\left(\frac{\Delta m_{ji}^{2}L{c}^3}{4E\hslash }\right)\\ +2{\displaystyle \underset{i<j}{\sum }\mathrm{Im}\left(V_{\alpha i}{V}_{\beta i}^{\ast }{V}_{\alpha j}^{\ast }{V}_{\beta j}\right)}\sin \left(\frac{\Delta m_{ji}^{2}L{c}^3}{2E\hslash }\right);i,j=d,s,b;\alpha ,\beta =u,c,t.\end{array}$ (12)

Here we especially emphasize that we copied the procedure we applied in the research of neutrinos [5] [6] . Namely, in the case of down-quarks, as we can see here the role of mass eigenstates is taken over by the measured values for down-quarks. And we will see their application in the following chapters.

$\begin{array}{l}P\left(d^I\to S^I\right)+P\left(d^I\to b^I\right)+P\left(d^I\to d^I\right)\\ =1-4R\left\{V_{ud}{V}_{cd}^{\ast }{V}_{us}^{\ast }{V}_{cs}{\sin }^2\left(\pi \frac{\Delta m_{sd}^2}{\Delta m_{sd}^2}\right)\right\}+2\mathrm{Im}\left\{V_{ud}{V}_{cd}^{\ast }{V}_{us}^{\ast }{V}_{cs}\sin \left(2\pi \frac{\Delta m_{sd}^2}{\Delta m_{sd}^2}\right)\right\}\\ -4R\left\{V_{ud}{V}_{cd}^{\ast }{V}_{ub}^{\ast }{V}_{cb}{\sin }^2\left(\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\right)\right\}+2\mathrm{Im}\left\{V_{ud}{V}_{cd}^{\ast }{V}_{ub}^{\ast }{V}_{cb}\sin \left(2\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\right)\right\}\\ -4R\left\{V_{us}{V}_{cs}^{\ast }{V}_{ub}^{\ast }{V}_{cb}{\sin }^2\left(\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\right)\right\}+2\mathrm{Im}\left\{V_{us}{V}_{cs}^{\ast }{V}_{ub}^{\ast }{V}_{cb}\sin \left(2\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\right)\right\}\\ -4R\left\{V_{ud}{V}_{td}^{\ast }{V}_{us}^{\ast }{V}_{ts}{\sin }^2\left(\pi \frac{\Delta m_{sd}^2}{\Delta m_{sd}^2}\right)\right\}+2\mathrm{Im}\left\{V_{ud}{V}_{td}^{\ast }{V}_{us}^{\ast }{V}_{ts}\sin \left(2\pi \frac{\Delta m_{sd}^2}{\Delta m_{sd}^2}\right)\right\}\\ -4R\left\{V_{ud}{V}_{td}^{\ast }{V}_{ub}^{\ast }{V}_{tb}{\sin }^2\left(\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\right)\right\}+2\mathrm{Im}\left\{V_{ud}{V}_{td}^{\ast }{V}_{ub}^{\ast }{V}_{tb}\sin \left(2\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\right)\right\}\\ -4R\left\{V_{us}{V}_{ts}^{\ast }{V}_{ub}^{\ast }{V}_{tb}{\sin }^2\left(\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\right)\right\}+2\mathrm{Im}\left\{V_{us}{V}_{ts}^{\ast }{V}_{ub}^{\ast }{V}_{tb}\sin \left(2\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\right)\right\}\\ -4{\left|V_{ud}\right|}^2{\left|V_{us}\right|}^2{\sin }^2\left(\pi \frac{\Delta m_{sd}^2}{\Delta m_{sd}^2}\right)-4{\left|V_{ud}\right|}^2{\left|V_{ub}\right|}^2{\sin }^2\left(\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\right)\\ -4{\left|V_{us}\right|}^2{\left|V_{ub}\right|}^2{\sin }^2\left(\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\right)\\ =1.\end{array}$ (13)

By making appropriate simplifications, we get an equation in which the CP violating phase is an unknown quantity:

$\begin{array}{l}-4W_d\mathrm{Re}\left\{V_{ud}{V}_{cd}^{\ast }{V}_{ub}^{\ast }{V}_{cb}\right\}+2V_d\mathrm{Im}\left\{V_{ud}{V}_{cd}^{\ast }{V}_{ub}^{\ast }{V}_{cb}\right\}-4W_d\mathrm{Re}\left\{V_{us}{V}_{cs}^{\ast }{V}_{ub}^{\ast }{V}_{cb}\right\}\\ +2V_d\mathrm{Im}\left\{V_{us}{V}_{cs}^{\ast }{V}_{ub}^{\ast }{V}_{cb}\right\}-4W_d\mathrm{Re}\left\{V_{ud}{V}_{td}^{\ast }{V}_{ub}^{\ast }{V}_{tb}\right\}+2V_d\mathrm{Im}\left\{V_{ud}{V}_{td}^{\ast }{V}_{ub}^{\ast }{V}_{tb}\right\}\\ -4W_d\mathrm{Re}\left\{V_{us}{V}_{ts}^{\ast }{V}_{ub}^{\ast }{V}_{tb}\right\}+2V_d\mathrm{Im}\left\{V_{us}{V}_{ts}^{\ast }{V}_{ub}^{\ast }{V}_{tb}\right\}-4W_d{\left|V_{ud}\right|}^2{\left|V_{ub}\right|}^2\\ -4W_d{\left|V_{us}\right|}^2{\left|V_{ub}\right|}^2=0.\end{array}$ (14)

Where the corresponding factors are equal to:

$\begin{array}{l}{V}_d=\sin \left(2\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\right)=\sin \left(2\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\right);\\ W_d={\sin }^2\left(\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\right)={\sin }^2\left(\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\right);\\ \Delta m_{bd}^2=m_b^2-m_d^2,\Delta m_{bs}^2=m_b^2-m_s^2,\Delta m_{sd}^2=m_s^2-m_d^2.\end{array}$ (15)

Taking the elements of the CKM matrix (13) into the calculation, we see that in (14) along with $\cos \delta$ and $\sin \delta$ there are the same algebraic expressions that we extract as a common factor, while all the free terms cancel each other out, so we arrive at the equation:

$K\times \left(2W_d\cos \delta -V_d\sin \delta \right)=0$ (16)

where $\delta$ is the CP violation phase as an unknown quantity of this equation.

We are looking for a solution to Equation (16), which we write in the form:

$\left(2W_d\cos \delta -V_d\sin \delta \right)=0$ (17)

In Equation (16), the factor K multiplies the equation and it is obvious that it has no influence on the final solution of the equation, which, as we can see, makes physical sense. However, with a subsequent check, we find that the factor K is equal to zero, so the final form of Equation (16) reads:

$0\times \left(2W_d\cos \delta -V_d\sin \delta \right)=0$ (18)

By inserting explicit values for parameters (15) into Equation (17), it is reduced to the following two identical forms. The first form looks like this:

$\begin{array}{l}\left(2{\sin }^2\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\cos \delta -\sin 2\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\sin \delta \right)=0\\ \to 2\sin \pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\left(\sin \pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\cos \delta -\cos \pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\sin \delta \right)=0\\ \to 2\sin \pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\left[\sin \left(\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\delta \right)\right]=0;\\ \left(2{\sin }^2\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\cos \delta -\sin 2\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\sin \delta \right)=0\\ \to 2\sin \pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\left(\sin \pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\cos \delta -\cos \pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\sin \delta \right)=0\\ \to 2\sin \pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\left[\sin \left(\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}-\delta \right)\right]=0.\end{array}$ (19)

Mathematically speaking, we see that the solution of Equation (18) consists of a general solution and a particular solution. The general solution satisfies all the values from the set $\delta \in \left[0.2\pi \right)$, which are countless and such solutions have no physical meaning.

But the particular Equation (19) have roots and they can be expressed in the following way:

$\begin{array}{l}2\sin \pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\left[\sin \left(\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\delta \right)\right]=0\to \sin \left(\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\delta \right)=0\to \\ \left(\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\delta \right)=0,\pm \pi \to \\ A:\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\delta =0\to \delta =\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2};\\ B:\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\delta =+\pi \to \delta =\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\pi =\pi \left(\frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}+1\right)-\pi =\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2};\\ C:\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\delta =-\pi \to \delta =\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}+\pi =\pi \left(\frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}+1\right)+\pi =\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}+2\pi =\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}.\\ 2\sin \pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\left[\sin \left(\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}-\delta \right)\right]=0\to \sin \left(\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}-\delta \right)=0\to \\ \left(\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}-\delta \right)=0,\pm \pi \to \\ A:\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}-\delta =0\to \delta =\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2};\\ B:\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}-\delta =+\pi \to \delta =\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}-\pi =\pi \left(\frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-1\right)-\pi =\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-2\pi =\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2};\\ C:\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}-\delta =-\pi \to \delta =\pi \left(\frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-1\right)+\pi =\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}-\pi +\pi =\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}.\end{array}$ (20)

We single out two of these solutions and they make physical sense:

${\delta }_{CKM}=\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2},{\delta }_{CKM}=\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}.$ (21)

Note. We especially emphasize that if we do not have the same algebraic expressions for the cosine of the angle and the sine of the angle, then the Equation (16) would look like this:

$K_1\times 2W_d\cos \delta -K_2\times V_d\sin \delta =0,K_1\ne K_2.$ (22)

Assuming that the coefficients K1 and K2 are different from each other, but are each equal to zero, then Equation (22) takes the form:

$0\times 2W_d\cos \delta -0\times V_d\sin \delta =0$ (23)

In this form of the equation, we would have countless solutions from the set $\delta \in \left[0.2\pi \right)$ that satisfy this equation, and such solutions have no physical meaning. Here, too, it is important to emphasize that if we had factors along with $\cos \delta$ and $\sin \delta$, which according to the algebraic structure of mathematical expressions are mutually different, but are each equal to zero, then in that case we would not have solutions that would make physical sense.

As you can see, the decisive role was played by the equality of factors along with cosine and sine, and they could be extracted as a common factor with which we arrived at a particular solution that makes physical sense.

Applying an analogous theoretical procedure for the other two transitions $s^I\to d^I,s^I\to b^I,s^I\to s^I$ and $b^I\to d^I,b^I\to s^I,b^I\to b^I$

We will get equations in which the equations for the CP violation phase look like this:

$2W_d\cos {\delta }_{CKM}+V_d\sin {\delta }_{CKM}=0$ (24)

Solutions of this type of equation for the CP violation phases are roots with opposite signs compared to formulas (21):

${\delta }_{CKM}=-\pi \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2},{\delta }_{CKM}=-\pi \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}.$ (25)

We especially note that formulas (21) and (25) do not essentially represent CP violating phases, but they will be used to determine the coefficients $V_d$ and $W_d$ which will later be used in the equation obtained in Wolfenstein’s parameterization, in which the actual measurements are values for the CP violation phase.

Here, transitions between weak interactions are described in the following form: $u^I\to c^I,u^I\to t^I,u^I\to u^I;c^I\to u^I,c^i\to t^I,c^I\to c^I;t^I\to u^I,t^I\to c^I,t^I\to t^I$, and the sum of oscillation probabilities between them could be described by the rule for the sum of oscillation probabilities in a similar way as already defined in neutrino physics (7, 8, 9):

$\begin{array}{l}P\left(u^I\to c^I\right)+P\left(u^I\to t^I\right)+P\left(u^I\to u^I\right)=1\\ P\left(c^I\to u^I\right)+P\left(c^I\to t^I\right)+P\left(c^I\to c^I\right)=1.\\ P\left(t^I\to u^I\right)+P\left(t^I\to c^I\right)+P\left(t^I\to t^I\right)=1.\end{array}$ (26)

as well as the general relation which reads:

$\begin{array}{l}P\left(V_{\alpha }\to V_{\beta }\right)\\ ={\delta }_{\alpha \beta }-4{\displaystyle \underset{i<j}{\sum }{R}_e\left(V_{\alpha i}{V}_{\beta i}^{\ast }{V}_{\alpha j}^{\ast }{V}_{\beta j}\right)}{\sin }^2\left(\frac{\Delta m_{ji}^{2}L{c}^3}{4E\hslash }\right)\\ +2{\displaystyle \underset{i<j}{\sum }\mathrm{Im}\left(V_{\alpha i}{V}_{\beta i}^{\ast }{V}_{\alpha j}^{\ast }{V}_{\beta j}\right)}\sin \left(\frac{\Delta m_{ji}^{2}L{c}^3}{2E\hslash }\right);i,j=u,c,t;\alpha ,\beta =d,s,b.\end{array}$ (27)

In this case, the role of mass eigenstates is taken over by the measured values for up quarks and then we apply the same methodology as for down quarks using the rule for the sum of quark oscillation probabilities for all three transitions where the corresponding coefficients are determined with the following relations:

$\begin{array}{l}{V}_u=\sin \left(2\pi \frac{\Delta m_{tu}^2}{\Delta m_{cu}^2}\right)=\sin \left(2\pi \frac{\Delta m_{tc}^2}{\Delta m_{cu}^2}\right);W_u={\sin }^2\left(\pi \frac{\Delta m_{tu}^2}{\Delta m_{cu}^2}\right)={\sin }^2\left(\pi \frac{\Delta m_{tc}^2}{\Delta m_{cu}^2}\right);\\ \Delta m_{tu}^2=m_t^2-m_u^2,\Delta m_{tc}^2=m_t^2-m_c^2,\Delta m_{cu}^2=m_c^2-m_u^2.\end{array}$ (28)

In this case too, we get equations that are similar in structure to Equations (21), (25) with finite solutions:

${\delta }_{CKM}=\pi \frac{\Delta m_{tu}^2}{\Delta m_{cu}^2},{\delta }_{CKM}=\pi \frac{\Delta m_{tc}^2}{\Delta m_{cu}^2}.$ (29)

${\delta }_{CKM}=-\pi \frac{\Delta m_{tu}^2}{\Delta m_{cu}^2},{\delta }_{CKM}=-\pi \frac{\Delta m_{tc}^2}{\Delta m_{cu}^2}.$ (30)

Applying the mathematical formalism in applying the rule for the sum of oscillation probabilities for both down and up quarks, we obtained two types of equations:

Down quarks

$\begin{array}{l}2W_d\cos {\delta }_{CKM}-V_d\sin {\delta }_{CKM}=0,\\ 2W_d\cos {\delta }_{CKM}+V_d\sin {\delta }_{CKM}=0.\end{array}$ (31)

Adding these two equations gives the unification equation:

$4W_d\cos {\delta }_{CKM}=0\to {\delta }_{CKM}=\pm \frac{\pi }{2}.$ (32)

Up quarks

$\begin{array}{l}2W_u\cos {\delta }_{CKM}-V_u\sin {\delta }_{CKM}=0,\\ 2W_u\cos {\delta }_{CKM}+V_u\sin {\delta }_{CKM}=0.\end{array}$ (33)

Adding these two equations gives the unification equation:

$4W_u\cos {\delta }_{CKM}=0\to {\delta }_{CKM}=\pm \frac{\pi }{2}.$ (34)

In the following sections, it will be seen what role the coefficients play in Equations (31, 33). The same coefficients figure in the equation that will be derived in the following chapters based on the Wolfenstein parameterization. Therefore, all the research conducted above in the basis of the CKM matrix is intended exclusive for determining these coefficients, while the equation that will be derived in Wolfenstein’s parameterization serves to determine the numerical value for the CP violation phase.

The dilemma about the sign $\pm \pi /2$ will be resolved when, in the next chapter, the equation in Wolfenstein’s parameterization will be derived, in which the mapping in that equation will be seen.

We include the formulas (21) and (25) in the calculations, so we will have:

$\begin{array}{l}{\delta }_{BFbs+}={180}^{\circ }\times \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\\ =\left\{{180}^{\circ }\times \left[\left({4180}^2-{93.4}^2\right)/\left({93.4}^2-{4.67}^2\right)\right]/{360}^{\circ }-1003\right\}\times {360}^{\circ }\approx {164.734}^{\circ }.\\ {\delta }_{BFbs-}=-{180}^{\circ }\times \frac{\Delta m_{bs}^2}{\Delta m_{sd}^2}\\ =\left\{-{180}^{\circ }\times \left[\left({4180}^2-{93.4}^2\right)/\left({93.4}^2-{4.67}^2\right)\right]/{360}^{\circ }+1004\right\}\times {360}^{\circ }\approx {195.266}^{\circ }.\\ {\delta }_{BFbd+}={180}^{\circ }\times \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\\ =\left\{\left[{180}^{\circ }\times \left[\left({4180}^2-{4.67}^2\right)/\left({93.4}^2-{4.67}^2\right)\right]/{360}^{\circ }-1003\right]\right\}\times {360}^{\circ }\cong {344.734}^{\circ }.\\ {\delta }_{BFbd-}=-{180}^{\circ }\times \frac{\Delta m_{bd}^2}{\Delta m_{sd}^2}\\ =\left\{\left[-{180}^{\circ }\times \left[\left({4180}^2-{4.67}^2\right)/\left({93.4}^2-{4.67}^2\right)\right]/{360}^{\circ }+1004\right]\right\}\times {360}^{\circ }\approx {15.266}^{\circ }.\end{array}$ (37)