Quarks Mixing in Chiral Symmetries

Abstract

We discuss a subject of the quarks mixing in SU4∗SU4 and SU6∗SU6 symmetries trying to calculate the quarks mixing angles and the complex phase responsible for the CP non-conservation on the basis of the Gell-Mann Oakes Renner model. Assuming symmetry breaking in a limit of exact sub-symmetries for simultaneous quarks rotations in both electric charge sub-spaces we can estimate all mention above parameters. A perfect agreement of the experimental value of the Cabibbo angles with a sum of simultaneous quarks mixing angles in doublets (u,c) and (d,s) in the SU4∗SU4 symmetry suggests that a quarks mixing is realized in a maximal allowed range. The same assumption used for the SU6∗SU6 and a simultaneous maximal allowed quarks mixing in both electric charge triplets (u,c,t) and (d,s,b) gives a perfect agreement with the experimental value of the Cabibbo angle and estimation on the angles Θ2 and Θ3 as well as a bond for the complex phase δ.

1. Introduction

1.1 Quarks mixing in chiral ${{SU}}_n\ast {{SU}}_n$ broken symmetry in the limit of exact ${{SU}}_k\ast {{SU}}_k$ symmetry

The hierarchy of chiral symmetry breaking [1, 2, 3] has been investigated since seventies of the previous century [4, 5, 6, 7, 8]. The symmetry breaking and mixing of quarks are connected with the rotation of quark currents and Hamiltonian densities. The determination of the rotation angle becomes an important problem. For the first time the procedure of chiral symmetry breaking, based on the Gell-Mann, Oakes, Renner (GMOR) model [9] has been used in ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ symmetry in the limit of exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry [4] to determine the value of the Cabibbo angle [10]. The transformation of rotation is connected with the seventh generator of the ${\mathit{SU}}_3$ group. After the charmed particles have been discovered the ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ symmetry is no longer adequate to describe the strong interactions. The ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry introduced earlier [11] to explain the behavior of charged and neutral currents becomes quite satisfactory model describing the hadron world. The problem of determining the Cabibbo angle in ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry has arisen. It is considered in [5, 6] and the method of calculating the Cabibbo angle in ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry is described in [7]. It is known that the formula describing the rotation angle is not changed if the symmetry is extended. This is not unexpected because the Cabibbo angle is connected with the mixing of the d and s quarks and the rotation is performed around the seventh axis in ${\mathit{SU}}_3$ subspace too.

The problem of chiral ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry breaking in the limit of exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry is considered in [6]. Symmetry breaking is connected with the transformation of rotation around the tenth axis in ${\mathit{SU}}_4$ space. The rotation angle is determined in [7].

The other variant of the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry breaking in the limit of the exact ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ symmetry is described in [8]. It is connected with the rotation around the fourteenth axis in ${\mathit{SU}}_4$ space. In this paper we introduce the general method of rotation angle description in the broken ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ symmetry. The chiral ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ symmetry is broken according to the GMOR model. In the first step we introduce the Hamiltonian density breaking ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ symmetry but invariant under ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ symmetry. In the second step we introduce quark mixing and the resulting exact symmetry is ${\mathit{SU}}_{k-1}\ast {\mathit{SU}}_{k-1}$ The particular investigation of cases like the above is not necessary.

The generalized GMOR model is used. It is assumed that by enlargement to a higher symmetry the new quantum numbers are the charges (as for example: electric charge, strangeness, charm but not isospin). Then the ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ symmetry breaking Hamiltonian density can be written as a linear combination of diagonal operators $u^i$.

$$H_E=\sum _{j=1}^n{c}_{j^2-1}{u}^{j^2-1}$$E1

where the scalar densities $u^i=\overline{q}{\lambda }^{i}q$ and pseudo-scalar densities $v^i=i\overline{q}{\lambda }^i{\gamma }_{5}q$ satisfy the equal-time commutation rules

$$\left[Q^i,u^j\right]={\mathit{\text{if}}}_{\mathit{ijk}}{u}^k\left[Q^i,v^j\right]={\mathit{\text{if}}}_{\mathit{ijk}}{v}^k$$E2

$$\left[{\overline{Q}}^i,u^j\right]={\mathit{id}}_{\mathit{ijk}}{v}^k\left[{\overline{Q}}^i,v^j\right]=-{\mathit{id}}_{\mathit{ijk}}{u}^k$$

where $f_{\mathit{ijk}}$ are the structure constants, $d_{\mathit{ijk}}$ - symmetric generators of the ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ group. If the ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ symmetry is exact then

$${\partial }^{\mu }{V}_{\mu }^i={\partial }^{\mu }{A}_{\mu }^i=0\left(i=1,2,\dots ,k^2-1\right)$$E3

In the GMOR model the divergences of currents can be calculated as follows

$${\partial }^{\mu }{V}_{\mu }^i=i\left[H_E,Q^i\right]{\partial }^{\mu }{A}_{\mu }^i=i\left[H_E,{\overline{Q}}^i\right]$$E4

We require that the ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ symmetry be exact, then the following constraints are obeyed

$$c_{j^2-1}=0\left(j=2,\dots ,k\right)$$E5

$$\sqrt{\frac{2}{n}} c_0+\sum _{j=k+1}^n\sqrt{\frac{2}{j\left(j+1\right)}} c_{j^2-1}=0$$

The symmetry breaking Hamiltonian density can be written as follows

$$H_E=c_0\left(u^0-\sqrt{n-1}{u}^{n^2-1}\right)+\sum _{j=k+1}^{n-1}{c}_{j^2-1}\left(u^{j^2-1}-\sqrt{\frac{n\left(n-1\right)}{j\left(j-1\right)}}{u}^{n^2-1}\right)$$E6

Using the standard representation of $\lambda$ matrices one obtains

$$u^0=\sqrt{\frac{2}{n}}\sum _{j=1}^n{\overline{q}}_j{q}_j$$E7

$$u^{j^2-1}=\sqrt{\frac{2}{j\left(j-1\right)}}\left(\sum _{l=1}^{j-1}{\overline{q}}_l{q}_l-\left(j-1\right){\overline{q}}_j{q}_j\right)$$E8

$$u^0-\sqrt{n-1}{u}^{n^2-1}=\sqrt{2n}{\overline{q}}_n{q}_n$$E9

$$u^{j^2-1}-\sqrt{\frac{n\left(n-1\right)}{j\left(j-1\right)}}{u}^{n^2-1}=\sqrt{\frac{2}{j\left(j-1\right)}}\left(\left(n-1\right){\overline{q}}_n{q}_n-j{\overline{q}}_j{q}_j-\sum _{l=j+1}^{n-1}{\overline{q}}_l{q}_l\right)$$E10

Let us note that the term ${\overline{q}}_k{q}_k$ does not exist in Eq. 11.

$$H_E=\left(\sqrt{2n}{c}_0+\left(n-1\right)\sum _{j=k+1}^{n-1}{c}_{j^2-1}\sqrt{\frac{2}{j\left(j-1\right)}}{\overline{q}}_n{q}_n\right)-\sum _{j=k+1}^{n-1}{c}_{j^2-1}\sqrt{\frac{2}{j\left(j-1\right)}}\left(j{\overline{q}}_j{q}_j+\sum _{l=j+1}^{n-1}{\overline{q}}_l{q}_l\right)$$E11

The chiral ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ symmetry with the exact ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ sub-symmetry is broken by the rotation of the ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ invariant Hamiltonian density around the axis with the index $m={\left(n-1\right)}^2+2k-1$.

$$H_{\mathit{SB}}=e^{-2\mathit{i\alpha }{Q}^m}{H}_E{e}^{2\mathit{i\alpha }{Q}^m}$$E12

Only the quarks $q_k$ and $q_n$ are mixed. The ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ symmetry is no longer exact. Only the term ${\overline{q}}_n{q}_n$ is rotated under transformation (12), because there is no ${\overline{q}}_k{q}_k$ term in the Hamiltonian density (11).

$$e^{-2\mathit{i\alpha }{Q}^m}{H}_E{e}^{2\mathit{i\alpha }{Q}^m}={\overline{q}}_n{q}_n-\left({\overline{q}}_n{q}_n-{\overline{q}}_k{q}_k\right){sin}^2\alpha -\frac{1}{2}\left({\overline{q}}_k{q}_n+{\overline{q}}_n{q}_k\right)sin2\alpha$$E13

The above consideration is limited to processes not having the change of the quantum number N connected with the ${\mathit{SU}}_n$ symmetry. So in the broken Hamiltonian density $H_{\mathit{SB}\left(\mathrm{\Delta }N=0\right)}$ the terms ${\overline{q}}_n{q}_k$ and ${\overline{q}}_k{q}_n$ do not appear. The broken Hamiltonian density is a linear combination of the diagonal operators $u^i$ only.

$$H_{\mathit{SB}\left(\mathrm{\Delta }N=0\right)}=H_E+A\sum _{j=k}^{n-1}\left(\sqrt{\frac{j+1}{2j}}{u}^{{\left(j+1\right)}^2-1}-\sqrt{\frac{j-1}{2j}}{u}^{j^2-1}\right){sin}^2\alpha$$E14

where

$$A=\sqrt{2n}{c}_0+\left(n-1\right)\sum _{j=k+1}^{n-1}{c}_{j^2-1}\sqrt{\frac{2}{j\left(j-1\right)}}$$E15

In more detail Eq. (14) is given as follows:

$$\begin{array}{l}{H}_{\mathit{SB}\left(\mathrm{\Delta }N=0\right)}=c_0{u}^0-A\sqrt{\frac{k-1}{2k}}{sin}^2\alpha u^{k^2-1}+\sum _{j=k+1}^{n-1}{u}^{j^2-1}\left(c_{j^2-1}+A{sin}^2\alpha \sqrt{\frac{1}{2j\left(j-1\right)}}\right)+\\ -u^{n^2-1}\left(c_0\sqrt{n-1}+\sum _{j=k+1}^{n-1}{c}_{j^2-1}\sqrt{\frac{n\left(n-1\right)}{j\left(j-1\right)}}\right)-A{u}^{n^2-1}\sqrt{\frac{n}{2\left(n-1\right)}}{sin}^2\alpha \end{array}$$E16

If the ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ symmetry is exact then the pseudo-scalar mesons corresponding to the indices $\left(j=1,\dots ,k^2-1\right)$ are massless [9]. After the ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ has been broken, the ${\mathit{SU}}_{k-1}\ast {\mathit{SU}}_{k-1}$ symmetry is still exact, because the operator $Q^m$ does not mix the quarks $q_1,\dots ,q_{k-1}$ neither with themselves nor with other quarks. The mesons corresponding to the indices $\left(j=1,\dots ,{\left(k-1\right)}^2-1\right)$ after symmetry breaking are still massless, while the mesons corresponding to the indices $\left(j={\left(k-1\right)}^2,\dots k^2-1\right)$ belong to the massive multiplet ${\left(k\right)}^1$ The masses of mesons are determined in the GMOR model. Before the ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ symmetry is broken the masses are described by the coefficients $c_0,\dots ,c_{n^2-1}$ from Eq. (1). After the symmetry has been broken the new factors $c_0^{\prime },\dots ,c_{n^2-1}^{\prime }$ are obtained as the coefficients standing by the operators $u^i$ in the broken Hamiltonian density (16) [7].

$$c_0^{\prime }=c_0$$E17

$$c_{k^2-1}^{\prime }=A\sqrt{\frac{k-1}{2k}}{sin}^2\alpha$$

$$c_{j^2-1}^{\prime }=c_{j^2-1}+A{sin}^2\alpha \sqrt{\frac{1}{2j\left(j-1\right)}}\left(k<j<n\right)$$

$$c_{n^2-1}^{\prime }=-c_0\sqrt{n-1}-\sum _{j=k+1}^{n-1}{c}_{j^2-1}\sqrt{\frac{n\left(n-1\right)}{j\left(j-1\right)}}+A\sqrt{\frac{n}{2(n-1)}}{sin}^2\alpha$$

The masses of the mesons are determined as follows [12].

$$m_a^2{J}_a^2{\delta }_{\mathit{ab}}=\sqrt{\frac{2}{n}}\left(c_0{d}_{0\mathit{ab}}+\sum _{j=k+1}^n{c}_{j^2-1}^{\prime }{d}_{\left(j^2-1\right)\mathit{ab}}\right)<u^0{>}_0$$E18

The relation between the indices a, b, j and meson states is described, for example, for the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry in [12, 13]. For $a=b=k^2-l$, the mass of the (k) meson is given as follows

$$m_{\left(k\right)}^2{f}_{\left(k\right)}^2=\sqrt{\frac{2}{n}}\left(\sqrt{\frac{2}{n}}{c}_0+\sum _{j=k+1}^n{d}_{\left(j^2-1\right)\mathit{ab}}{c}_{j^2-1}^{\prime }\right)<u^0{>}_0=\frac{1}{k}\sqrt{\frac{2}{n}}A{sin}^2\alpha <u^0{>}_0$$E19

For $a=b=m={\left(n-1\right)}^2+2k-1$, the mass of the (n) meson is given by

$$m_{\left(n\right)}^2{f}_{\left(n\right)}^2=\sqrt{\frac{2}{n}}\left(\sqrt{\frac{2}{n}}{c}_0+\sum _{j=k+1}^n{d}_{\left(j^2-1\right)\mathit{mm}}{c}_{j^2-1}^{\prime }\right)<u^0{>}_0$$E20

Because

$$d_{\left(j^2-1\right)\mathit{mm}}=\sqrt{\frac{1}{2j\left(j-1\right)}}\left(j<n\right)$$E21

$$d_{\left(n^2-1\right)\mathit{mm}}=\frac{2-n}{\sqrt{2n\left(n-1\right)}}$$E22

$$m_{\left(n\right)}^2{f}_{\left(n\right)}^2=\sqrt{\frac{1}{2n}}A\left(1-\left(1-\frac{1}{k}\right){sin}^2\alpha \right)<u^0{>}_0$$E23

In formulas (19) and (23) to determine the masses of (k) and (n) mesons one has (n-k + 3) unknown quantities with which to deal ($<u^0{>}_0$, $c_0$, $c_{{\left(k+1\right)}^2-1},\dots ,c_{n^2-1},sin\alpha )$. Nevertheless the angle a is determined by the masses and decay constants of two pseudo-scalar mesons (k) and (n) only.

$${sin}^2\alpha =\frac{k{m}_{\left(k\right)}^2{f}_{\left(k\right)}^2}{2m_{\left(n\right)}^2{f}_{\left(n\right)}^2+\left(k-1\right)m_{\left(k\right)}^2{f}_{\left(k\right)}^2}$$E24

All the cases of symmetry breaking considered in [4, 5, 6, 7, 8] can be described by formula (24). Let us give simple examples: a) for k = 2, n = 3 a is the original Cabibbo angle $\mathrm{\Theta }$ associated with rotation around the seventh axis in ${\mathit{SU}}_3$ subspace [4, 5, 6, 7].

$${sin}^2\mathrm{\Theta }=\frac{2m_{\pi }^2{f}_{\pi }^2}{2m_K^2{f}_K^2+m_{\pi }^2{f}_{\pi }^2}$$E25

b) for k = 2, n = 4 and rotation around the tenth axis [6, 7] one obtains

$${sin}^2\alpha =\frac{2m_{\pi }^2{f}_{\pi }^2}{2m_D^2{f}_D^2+m_{\pi }^2{f}_{\pi }^2}$$E26

c) for k = 3, n = 4 and rotation around the fourteenth axis [7, 8] one obtains

$${sin}^2\alpha =\frac{3m_K^2{f}_K^2}{2\left(m_D^2{f}_D^2+m_K^2{f}_K^2\right)}$$E27

In general the determination of the rotation angle (24) in ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ symmetry is possible only if the new quantum numbers introduced by a transition to the higher symmetry are scalars of the charge type (additiv). So the Hamiltonian density (1) can be constructed as a linear combination of the diagonal operators $u^i$; only $\left(i=j^2-1,j=1,\dots ,n\right)$. The method of determining the rotation angle, discussing and interpreting the symmetry breaking is described in more detail in [7] on ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry as an example.

2. Quarks mixing and the Cabibbo angle in the ${{SU}}_4\ast {{SU}}_4$ broken symmetry

It is known that the Cabibbo angle has been introduced into ${\mathit{SU}}_3$ symmetry to explain the suppression of processes in which strangeness is not conserved [4]. The Cabibbo angle is connected with the mixing of d and s quarks for weak interactions of hadrons. Its value, calculated by Oakes, does not contradict the experimental data. Before the charmed particles were discovered Glashow, Iliopoulos and Maiani [11] have suggested the generalization of a strong interaction symmetry to ${\mathit{SU}}_4$ [6]. The charged weak current is then given as follows

$$J_{\mu }=\overline{q}{\gamma }_{\mu }\left(1-{\gamma }_5\right)Aq$$E28

where

$$A=\left(\begin{array}{cccc}0& 0& cos\mathrm{\Theta }& sin\mathrm{\Theta }\\ 0& 0& -sin\mathrm{\Theta }& cos\mathrm{\Theta }\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$$E29

The current (28) can be expressed in another form

$$J_{\mu }=\left(\overline{u},\overline{c}\right){\gamma }_{\mu }\left(1-{\gamma }_5\right)\left(\begin{array}{cc}cos\mathrm{\Theta }& sin\mathrm{\Theta }\\ -sin\mathrm{\Theta }& cos\mathrm{\Theta }\end{array}\right)\left(\begin{array}{c}d\\ s\end{array}\right)$$E30

so, quark mixing is described by an orthogonal matrix. On the grounds of Eq. (30) we cannot come to a conclusion about quarks in which the doublets are mixed. If the matrix A is generalized to the following form

$$A=\left(\begin{array}{cccc}0& 0& cos\mathrm{\Theta }& sin\mathrm{\Theta }\\ 0& 0& -sin\mathrm{\Theta }& cos\mathrm{\Theta }\\ cos\varphi & sin\varphi & 0& 0\\ -sin\varphi & cos\varphi & 0& 0\end{array}\right)$$E31

the quarks in the doublets (u, c) and (d, s) are mixed independently. The zeros in Eq. (31) are associated with the fact that the neutral currents which change the strangeness and/or charm are not observed. So, the current (28) can be given in the following form

$$J_{\mu }=\left(\overline{u},\overline{c}\right){\gamma }_{\mu }\left(1-{\gamma }_5\right)\left(\begin{array}{cc}cos\left(\mathrm{\Theta }+\varphi \right)& sin\left(\mathrm{\Theta }+\varphi \right)\\ -sin\left(\mathrm{\Theta }+\varphi \right)& cos\left(\mathrm{\Theta }+\varphi \right)\end{array}\right)\left(\begin{array}{c}d\\ s\end{array}\right)$$E32

If the currents only are taken into consideration we cannot solve the problem if the quarks are mixed in one or both doublets. This is not unexpected because the currents are built as a bi-linear combination of quark states and the angles $\mathrm{\Theta }$ and $\varphi$, can always be substituted the effective angle $\left(\mathrm{\Theta }+\varphi \right)$. To solve the problem the Gell-Mann, Oakes, Renner (GMOR) model [9] will be used.

The charged weak current in ${\mathit{SU}}_3$ symmetry can be written as follows

$$J_{\mu }\left(\mathrm{\Theta }\right)=cos\mathrm{\Theta }\left(J_{\mu }^1+i{J}_{\mu }^2\right)+sin\mathrm{\Theta }\left(J_{\mu }^4+i{J}_{\mu }^5\right)\mathrm{\Theta }-\text{Cabibbo}\text{angle}$$E33

$$J_{\mu }=\overline{q}{\gamma }_{\mu }\left(1-{\gamma }_5\right){\lambda }^{k}qq=\left(\begin{array}{c}u\\ d\\ s\\ c\end{array}\right)$$E34

The current (33) can be obtained from the isospin component of the current $\left(J_{\mu }^1+i{J}_{\mu }^2\right)$ by rotation through an angle 2$\mathrm{\Theta }$ about the seventh axis in ${\mathit{SU}}_3$ space according to

$$J_{\mu }\left(\mathrm{\Theta }\right)=e^{-2i\mathrm{\Theta }{F}^7}\left(J_{\mu }^1+i{J}_{\mu }^2\right)e^{2i\mathrm{\Theta }{F}^7}$$E35

where

$$F^k=\int d^{3}x{q}^{+}\left(x\right)\frac{{\lambda }^k}{2}q\left(x\right)$$E36

The charged weak current in ${\mathit{SU}}_4$ symmetry (30) can be expressed in the following form

$$J_{\mu }\left(\mathrm{\Theta }\right)=cos\mathrm{\Theta }\left(J_{\mu }^1+i{J}_{\mu }^2\right)+sin\mathrm{\Theta }\left(J_{\mu }^4+i{J}_{\mu }^5\right)-sin\mathrm{\Theta }\left(J_{\mu }^{11}-i{J}_{\mu }^{12}\right)+cos\mathrm{\Theta }\left(J_{\mu }^{13}-i{J}_{\mu }^{14}\right)$$E37

The current (37) can be obtained by rotation of the components $\mathrm{\Delta }S=\mathrm{\Delta }C$ through an angle 2$\mathrm{\Theta }$ about the seventh axis in ${\mathit{SU}}_4$ space

$$J_{\mu }\left(\mathrm{\Theta }\right)=e^{-2i\mathrm{\Theta }{F}^7}\left(J_{\mu }^1+i{J}_{\mu }^2+J_{\mu }^{13}-i{J}_{\mu }^{14}\right)e^{2i\mathrm{\Theta }{F}^7}$$E38

The transformation (38) changes the strangeness but not the charm because

$$\left[F^7,q_1\right]=\left[F^7,q_4\right]=0$$E39

The transformation (38) is connected with the mixing of d and s quarks (as in the case of ${\mathit{SU}}_3$ symmetry). In the ${\mathit{SU}}_4$ symmetry the mixing in electric charge subspace +2/3 can be taken into consideration. This is not possible in the ${\mathit{SU}}_3$ symmetry where only one state with the +2/3 charge exists. The possibility of expressing the current (37) by the transformation which changes charm but not strangeness should exist. The transformation has been described by Ebrahim in [6].

$$J_{\mu }\left(\varphi \right)=e^{-2\mathit{i\varphi }{F}^{10}}\left(J_{\mu }^1+i{J}_{\mu }^2+J_{\mu }^{13}-i{J}_{\mu }^{14}\right)e^{2\mathit{i\varphi }{F}^{10}}$$E40

$$\left[F^{10},q_2\right]=\left[F^{10},q_3\right]=0$$E41

The transformation (40) is connected with the mixing of u and c quarks. The fact that there exist two transformations giving the current (37) but connected with different generators of the ${\mathit{SU}}_4$ group changing strangeness or charm respectively suggests that independent mixing in both doublets is possible. It is known that the Cabibbo angle is connected with strangeness non-conservation in weak interactions. The formula describing the value of the Cabibbo angle has been obtained by Oakes [4] in the procedure of symmetry breaking. Namely the ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ symmetry in the limit of the exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry is broken. The ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ sub-symmetry is no longer exact. The symmetry is broken by the rotation of the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ invariant Hamiltonian density through angle 2$\mathrm{\Theta }$ about the seventh axis. Then the pion becomes massive. The symmetry breaking is connected with the mixing of d and s quarks. The rotation angle $\mathrm{\Theta }$, as a measure of symmetry violation, is a function of the mass and the decay constant of the pion and of the mass and the decay constant of the kaon as well (it is connected with the mixing of the strange quark and the strangeness non-conservation). If the breaking of the chiral ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry, the mass of the pion, the Cabibbo angle as well as a strangeness and charm non-conservation have a common origin then it seems that as a result of ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry breaking in the limit of the exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ sub-symmetry by the rotation of the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ invariant Hamiltonian density through an angle 2$\varphi$ about the tenth axis the angle $\varphi$ connected with the mixing of u and c quarks as a measure of a symmetry violation should be a function of the mass and decay constant of the pion (breaking of the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry) and a function of the mass and decay constant of a charmed meson (charm non-conservation). The cases of the separate and then simultaneously mixing of quarks in the sub-spaces of electric charge will be considered below.

If the electromagnetic mass splitting of u-d quarks is neglected the Hamiltonian density breaking the chiral ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry is given in the form

$$H=c_0{u}^0+c_8{u}^8+c_{15}{u}^{15}$$E42

where $c_0,c_8,c_{15}$ are constants, $u^a$ (a = 0, 1, ..., 15) are the scalar components of the $\left(\overline{4},4\right)+\left(4,\overline{4}\right)$ representation of the chiral ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ group. On the grounds of the GMOR model the following relation for masses of the pseudo-scalar mesons can be obtained [12].

$$i<0\mid \left[{\overline{Q}}^a,{\overline{D}}^b\right]\mid 0>={\delta }^{\mathit{ab}}{f}_a^2{m}_a^2+\int \frac{{\mathit{dq}}^2}{q^2}{\rho }^{\mathit{ab}}=$$E43

$$=<u^0{>}_0\left(\frac{c_0}{2}{\delta }^{\mathit{ab}}+\frac{c_8}{\sqrt{2}}{d}_{a8b}+\frac{c_{15}}{\sqrt{2}}{d}_{a15b}\right)+$$

$$+<u^8{>}_0\left(\frac{c_0}{\sqrt{2}}{d}_{a8b}+c_8{d}_{a8c}{d}_{b8c}+c_{15}{d}_{a8c}{d}_{b15c}\right)+$$

$$+<u_{15}{>}_0\left(\frac{c_0}{\sqrt{2}}{d}_{a15b}+c_8{d}_{a15c}{d}_{b8c}+c_{15}{d}_{a15c}{d}_{b15c}\right)$$

where

$${\rho }^{\mathit{ab}}={\left(2\pi \right)}^3\sum _{n\ne a}{\delta }^4\left(p_n-q\right)<0\mid {\overline{D}}^a\mid n><n\mid {\overline{D}}^b\mid 0>$$E44

$f_a$ - decay constants, $<u^i{>}_0$ - vacuum expectation value of the operator $u^i$. Because the vacuum expectation values of operators $u^8$, $u^{15}$ and the spectral density ${\delta }^{\mathit{ab}}$ are proportional to the squared parameters of symmetry breaking, they are further neglected [12]. Approximately from Eq. (43) we obtain

$$m_a^2{f}_a^2{\delta }^{\mathit{ab}}=\frac{1}{\sqrt{2}}\left(\frac{c_0}{\sqrt{2}}+c_8{d}_{a8b}+c_{15}{d}_{a15b}\right)<u^0{>}_0$$E45

The masses of the mesons are given as follows

$$m_{\pi }^2{f}_{\pi }^2=\frac{1}{2\sqrt{3}}\left(\sqrt{3}{c}_0+\sqrt{2}{c}_8+c_{15}\right)<u^0{>}_0$$

$$m_K^2{f}_K^2=\frac{1}{2\sqrt{3}}\left(\sqrt{3}{c}_0-\frac{1}{\sqrt{2}}{c}_8+c_{15}\right)<u^0{>}_0$$E46

$$m_D^2{f}_D^2=\frac{1}{2\sqrt{3}}\left(\sqrt{3}{c}_0+\frac{1}{\sqrt{2}}{c}_8-c_{15}\right)<u^0{>}_0$$

In the limit of the exact chiral ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ sub-symmetry there is the following constraint

$$\sqrt{3}{c}_0+\sqrt{2}{c}_8+c_{15}=0$$E47

so the pion is massless.

Let us make some remarks. The task of the Cabibbo angle calculation in ${\mathit{SU}}_4$ symmetry using the procedure of symmetry breaking has been done in [6]. In Ebrahim’s earlier paper [5] the parameters of the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry breaking have been found.

$$\frac{c_8}{c_0}=-\frac{2\sqrt{2}}{\sqrt{3}}\frac{m_K^2{f}_K^2-m_{\pi }^2{f}_{\pi }^2}{m_K^2{f}_K^2+m_D^2{f}_D^2}\frac{c_{15}}{c_0}=-\frac{1}{\sqrt{3}}\frac{3m_D^2{f}_D^2-m_K^2{f}_K^2-2m_{\pi }^2{f}_{\pi }^2}{m_K^2{f}_K^2+m_D^2{f}_D^2}$$E48

In [6] the numerical values of parameters (48) have been used to calculate the rotation angle (interpreted as the Cabibbo angle). The ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ invariant Hamiltonian density breaking ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry has been rotated through an angle 2 $\mathrm{\Theta }$ about the seventh axis and the coefficients of the operators $u^a$ (a = 0, 8, 15) have been identified with the parameters of symmetry breaking

$$H_{\mathit{SB}}\left(\mathrm{\Delta }S=0\right)=c_0{u}^0+\frac{\sqrt{3}}{2}{c}_8{sin}^2\mathrm{\Theta }{u}^3+c_8\left(1-\frac{3}{2}{sin}^2\mathrm{\Theta }\right)u^8-\left(\sqrt{3}{c}_0+\sqrt{2}{c}_8\right)u^{15}$$E49

It seems to us that there are some errors in the numerical calculations of the author. The use of the numerical values of the parameters (48) has not been necessary. On the grounds of theoretical formulas only, indeed from the Eq. (7) in Ref. [5] and the Eq. (10) in A3-Ebrahim, it follows that

$${sin}^2\mathrm{\Theta }=\frac{2m_{\pi }^2{f}_{\pi }^2}{2m_K^2{f}_K^2+m_{\pi }^2{f}_{\pi }^2}$$E50

Then the value of $\mathrm{\Theta }$ is given by

$${sin}^2\mathrm{\Theta }={\left(0.215\right)}^2$$E51

instead of

$${sin}^2\mathrm{\Theta }=-0.04$$E52

from Eqs. (10) in [6]. Formula (50) has the same form as in ${\mathit{SU}}_3$ symmetry. In agreement with our expectation the angle $\mathrm{\Theta }$ is described by parameters of the pion and the strange meson.

In Ebrahim’s method the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry breaking the Hamiltonian density is parametrized by the factors $c_0$, $c_8$, $c_{15}$. The parameters of symmetry breaking are expressed by the masses and decay constants of the mesons and they are fixed (Eq. (7) in [5]). In the limit of the exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ sub-symmetry the factors $c_0$, $c_8$, $c_{15}$ should satisfy the constraint (47) but it is possible only if $m_{\pi }=0$ namely the parameters of symmetry breaking are not expressed by real (measured in experiment) masses of mesons. In [6] Ebrahim breaks the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry in the limit of the exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ sub-symmetry by the rotation of the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ invariant Hamiltonian density through an angle 2 $\mathrm{\Theta }$ about the seventh axis. The factors of the rotated Hamiltonian density are identified with the parameters of symmetry breaking (Eq. (7) in [5]). Solving a set of equations the author gets the factors $c_0$, $c_8$, $c_{15}$ dependent on the rotation angle and on the real mesons masses already. The masses of mesons standing in the formula which describes the parameters of symmetry breaking are determined by the method of symmetry breaking and they have a real value for the real realization of the symmetry breaking only. In this case the rotation angle does not matter a parameter of the symmetry violation. It seems to us that such an interpretation is not satisfactory. The expression of meson masses as a function of the rotation angle (as a measure of symmetry violation) seems to be more natural. In the present paper the other interpretation of the symmetry breaking and the method of calculating the rotation angle is proposed. We describe our method as follows.

Before the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry in the limit of the exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ sub-symmetry is broken the masses of mesons have been expressed by the factors $c_0$, $c_8$, $c_{15}$ which satisfy the constraint (47). After symmetry breaking a new set of factors $c_0^{\prime }$, $c_g^{\prime }$, $c_{15}^{\prime }$ dependent on the old factors $c_0$, $c_8$, $c_{15}$ and on the rotation angle is introduced. The new factors are identified with the coefficients by the operators $u^i$ of the rotated Hamiltonian density (49).

$$c_0^{\prime }=c_0{c}_g^{\prime }=c_8\left(1-\frac{3}{2}{\mathit{sin}}^2\mathrm{\Theta }\right)c_{15}^{\prime }=-\sqrt{3}{c}_0-\sqrt{2}{c}_8$$E53

Meson masses are expressed by new factors and they are the function of the rotation angle as a measure of symmetry violation.

$$m_{\pi }^2{f}_{\pi }^2=\frac{1}{2\sqrt{3}}\left(\sqrt{3}{c}_0^{\prime }+\sqrt{2}{c}_g^{\prime }+c_{15}^{\prime }\right)<u^0{>}_0=-\frac{\sqrt{3}}{2\sqrt{2}}{c}_8{sin}^2\mathrm{\Theta }<u^0{>}_0$$E54

$$m_K^2{f}_K^2=\frac{1}{2\sqrt{3}}\left(\sqrt{3}{c}_0^{\prime }-\frac{c_8^{\prime }}{\sqrt{2}}+c_{15}^{\prime }\right)<u^0{>}_0=-\frac{\sqrt{3}}{2\sqrt{2}}{c}_8\left(1-\frac{1}{2}{sin}^2\mathrm{\Theta }\right)<u^0{>}_0$$

$$m_D^2{f}_D^2=\frac{1}{2\sqrt{3}}\left(\sqrt{3}{c}_0^{\prime }+\frac{c_8^{\prime }}{\sqrt{2}}-c_{15}^{\prime }\right)<u^0{>}_0=\left(c_0+\frac{\sqrt{3}}{2\sqrt{2}}{c}_8\left(1-\frac{1}{2}{sin}^2\mathrm{\Theta }\right)\right)<u^0{>}_0$$

It seems to be more natural that the meson masses are functions of the parameters of symmetry breaking (54) than inversely the parameters of symmetry breaking are functions of meson masses which are not consistent with the experimental data and are dependent on the method of symmetry breaking. This interpretation is consistent with the fact that the mass generation of the mesons is a consequence of symmetry breaking. From Eq. (54) we obtain the formula for the angle $\mathrm{\Theta }$ as in Eq. (50). Let us consider the other variant of symmetry breaking described in [6]. Ebrahim, using his method, broke the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry in the limit of the exact ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ symmetry by the rotation of ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ invariant Hamiltonian density about the fourteenth axis in the ${\mathit{SU}}_4$ space. The rotation angle ${\mathrm{\Theta }}^{\prime }$ is identified with the Cabibbo angle. The formula describing the angle ${\mathrm{\Theta }}^{\prime }$ should be given as follows

$${sin}^2{\mathrm{\Theta }}^{\prime }=\frac{3m_K^2{f}_K^2}{2\left(m_K^2{f}_K^2+m_D^2{f}_D^2\right)}$$E55

(in Eqs. (4a) in [8] there is the factor 3/2). The rotation of the Hamiltonian density about the fourteenth axis is considered in [14] too. The D meson is interpreted as a Goldstone boson. Putting aside the agreement of the numerical value of the angle ${\mathrm{\Theta }}^{\prime }$ with the experimental data it seems to us that the angle connected with the rotation about the fourteenth axis cannot be interpreted as the Cabibbo angle, because the rotation is performed inside the doublet (s, c). Then the states with the different electric charges are mixed. The interpretation that the D meson is a Goldstone boson is also unsatisfactory. If the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry is broken in such a way that the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ sub-symmetry is still exact, so the K meson becomes massive but the pion is still massless. Such a symmetry breaking cannot be accepted, results contradict the experimental data. The next breaking of the exact ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ symmetry is connected with the mixing of s and c quarks. The rotation angle cannot be interpreted as the Cabibbo angle for the reasons given above. It seems that the hierarchy of symmetry breaking is extended and the breaking of the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry taken as a whole cannot be connected with the Cabibbo angle. This is possible, however, for ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry breaking in the limit of exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ sub-symmetry. Then results are in agreement with our expectation.

Our method described above is used· to calculate an angle $\varphi$ which is connected with the rotation about the tenth axis in ${\mathit{SU}}_4$ space. Then the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry is broken by the rotation of the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ invariant Hamiltonian density through an angle $2\varphi$ about the tenth axis.

$$\begin{array}{c}{H}_{\mathit{SB}}\left(\mathrm{\Delta }C=0\right)=c_0{u}^0+\left(\sqrt{2}{c}_0+\frac{\sqrt{3}}{2}{c}_8\right){sin}^2\varphi u^3+\left(c_8(\frac{1}{2\sqrt{2}}\left(\frac{4}{\sqrt{3}}{c}_0+\sqrt{2}{c}_8\right){sin}^2\varphi \right)u^8+\\ +\left(-\sqrt{3}{c}_0+\sqrt{2}{c}_8+\left(\frac{4}{\sqrt{3}}{c}_0+\sqrt{2}{c}_8\right){sin}^2\varphi \right)u^{15}\end{array}$$E56

Using the factors from the Hamiltonian density (56) the masses of mesons are given as follows

$$m_{\pi }^2{f}_{\pi }^2=\left(c_0+\frac{\sqrt{6}}{4}\right){sin}^2\varphi <u^0{>}_0$$E57

$$m_K^2{f}_K^2=-\left(\frac{\sqrt{6}}{4}-\frac{1}{2}\left(c_0+\frac{\sqrt{6}}{4}{c}_8\right){sin}^2\varphi \right)<u^0{>}_0$$

$$m_D^2{f}_D^2=\left(1-\frac{1}{2}{sin}^2\varphi \right)\left(c_0^{\prime }+\frac{\sqrt{6}}{4}{c}_8\right)<u^0{>}_0$$

so

$${sin}^2\varphi =\frac{2m_{\pi }^2{f}_{\pi }^2}{2m_D^2{f}_D^2+m_{\pi }^2{f}_{\pi }^2}$$E58

In agreement with our expectation the angle $\varphi$ is a function of the mass of the pion (as a measure of the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ violation) and is connected with the parameters of the charmed meson (mixing in (u, c) doublet). For the mass $m_D$ = 1862 MeV and $f_D/f_{\pi }$ = 0.974 [13] one gets

$${sin}^2\varphi =0.076$$E59

The small value of the angle $\varphi$ is the effect of the large mass of the charmed quark. From (59) results that only mixing in the (u, c) system is excluded, the value of the angle $\varphi$ contradicts the experimental data. The simultaneous mixing in both doublets are, however, still possible. Fritzsch [15] considers also the mixing in (u, c) system. The mixing angle is calculated on the grounds of quark masses and does not contradict the results obtained above. Although the value of the angle $\varphi$ is relatively small, it is significant: the sum of the angles $\mathrm{\Theta }+\varphi$ is larger than the value of the angle measured experimentally, called Cabibbo angle. This fact cannot be explained by the limits of experimental errors. Let us note that the angles (50) and (58) are calculated for the case where quarks are mixed separately. The angles from formula (32) cannot be identified with those from Eqs. (50) and (58). In the case of simultaneous mixing in both doublets the relation between the angles is more complicated. To find the relation, the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ invariant Hamiltonian density is rotated through an angle $2\varphi$ about the tenth axis and afterwards by an angle $2\mathrm{\Theta }$ about the seventh axis. The sequence of the rotations is insignificant, because

$$\left[F^7,F^{10}\right]=0$$E60

The rotated Hamiltonian density is given by

$$H_{\mathit{SB}}\left(\mathrm{\Delta }S=\mathrm{\Delta }C=0\right)=c_0{u}^0+\left(\frac{\sqrt{3}}{2}{c}_8{sin}^2\mathrm{\Theta }+\left(\sqrt{2}{c}_0+\frac{\sqrt{3}}{2}{c}_8\right){sin}^2\varphi \right)u^3+$$E61

$$+\left(\frac{1}{\sqrt{3}}\left(\sqrt{2}{c}_0+\frac{\sqrt{3}}{2}{c}_8\right){sin}^2\varphi +c_8\left(1-\frac{3}{2}{sin}^2\mathrm{\Theta }\right)\right)u^8+$$

$$+\left(-\sqrt{3}{c}_0-\sqrt{2}{c}_8+\frac{2\sqrt{2}}{\sqrt{3}}\left(\sqrt{2}{c}_0+\frac{\sqrt{3}}{2}{c}_8\right){sin}^2\varphi \right)u^{15}$$

The meson masses are given as follows

$$m_{\pi }^2{f}_{\pi }^2=\frac{1}{2\sqrt{3}}\left(\sqrt{6}\left(\sqrt{2}{c}_0+\frac{\sqrt{3}}{2}{c}_8\right){sin}^2\varphi -\frac{3}{\sqrt{2}}{c}_8{sin}^2\mathrm{\Theta }\right)<u^0{>}_0$$E62

$$m_K^2{f}_K^2=\frac{1}{2\sqrt{3}}\left(\frac{3}{\sqrt{6}}\left(\sqrt{2}{c}_0+\frac{\sqrt{3}}{2}{c}_8\right){sin}^2\varphi -\frac{3}{\sqrt{2}}{c}_8(1-\frac{1}{2}{sin}^2\mathrm{\Theta }\right)<u^0{>}_0$$

$$m_D^2{f}_D^2=\frac{1}{2\sqrt{3}}\left(2\sqrt{3}{c}_0-\frac{3}{\sqrt{6}}\left(\sqrt{2}{c}_0+\frac{\sqrt{3}}{2}{c}_8\right){sin}^2\varphi +\frac{3}{\sqrt{2}}{c}_8\left(1-\frac{1}{2}{sin}^2\mathrm{\Theta }\right)\right)<u^0{>}_0$$

Now the angles $\mathrm{\Theta }$ and $\varphi$ cannot be described independently. The following relation is obeyed.

$$2m_{\pi }^2{f}_{\pi }^2+2\left(m_K^2{f}_K^2+m_D^2{f}_D^2\right){sin}^2\mathrm{\Theta }{sin}^2\varphi =$$E63

$$=\left(2m_K^2{f}_K^2+m_{\pi }^2{f}_{\pi }^2\right){sin}^2\mathrm{\Theta }+\left(2m_D^2{f}_D^2+m_{\pi }^2{f}_{\pi }^2\right){sin}^2\varphi$$

or equivalently

$$1+\left(\frac{1}{{sin}^2\mathrm{\Theta }}+\frac{1}{{sin}^2\varphi }-1\right){sin}^2\mathrm{\Theta }{sin}^2\varphi =\frac{{sin}^2\mathrm{\Theta }}{{sin}^2{\mathrm{\Theta }}_0}+\frac{{sin}^2\varphi }{{sin}^2{\varphi }_0}$$E64

where

$${sin}^2{\mathrm{\Theta }}_0=\frac{2m_{\pi }^2{f}_{\pi }^2}{2m_K^2{f}_K^2+m_{\pi }^2{f}_{\pi }^2}{sin}^2{\varphi }_0=\frac{2m_{\pi }^2{f}_{\pi }^2}{2m_D^2{f}_D^2+m_{\pi }^2{f}_{\pi }^2}$$E65

The angles $\mathrm{\Theta }$ and $\varphi$ from Eq. (64) concern a simultaneous mixing in doublets (d, s) and (u, c) respectively and they can be identified with those from Eq. (32). The condition (64) limits the values of the angles $\mathrm{\Theta }$ and $\varphi$. The maximal values of the angles ${\mathrm{\Theta }}_0$ and ${\varphi }_0$ are given by Eq. (65). The value of the function

$$f\left(\mathrm{\Theta },\varphi \right)=sin\left(\mathrm{\Theta }+\varphi \right)$$E66

is also limited. A numerical calculation shows that there is an extremum (a maximum) of function (66) on the condition (64) for

$${\mathrm{\Theta }}_m=0.20452{\varphi }_m=0.02575sin\left({\mathrm{\Theta }}_m+{\varphi }_m\right)=0.2282$$E67

It is worth noticing that the extremum of function (66) on condition (64) can be identified with the measured Cabibbo angle. It is not excluded that symmetry breaking is realized in the maximal allowed case, so the effective angle of mixing would correspond to the maximum of the function (66).

3. Bonds for the Kobayashi-Maskawa mixing parameters in a model with hierarchical symmetry breaking

A simultaneous mixing in (d, s) and (u, c) sectors has also been taken into account [15, 16], but due to the large mass of the c quark, the influence of the mixing in the (u, c) sector can be treated as a perturbation. At the six-quark level the quark mixing is described by three Cabibbo-like flavor mixing angles and the phase parameter responsible for CP-non-conservation [17]. The charged weak current in the ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ chiral symmetry

$$J_{\mu }=\left(\overline{u},\overline{c},\overline{t}\right){\gamma }_{\mu }\left(1-{\gamma }_5\right)U\left(\begin{array}{c}d\\ s\\ b\end{array}\right)$$E68

is described by a unitary matrix U, which can be put in 21 different forms [18], however only the standard Kobayashi-Maskawa matrix [19] will be used further.

$$U=\left(\begin{array}{ccc}{c}_1& s_1{c}_3& s_1{s}_3\\ -s_1{c}_2& c_1{c}_2{c}_3-s_2{s}_3{e}^{\mathit{i\delta }}& c_1{c}_2{s}_3+s_2{c}_3{e}^{\mathit{i\delta }}\\ s_1{s}_2& -c_1{s}_2{c}_3-c_2{s}_3{e}^{\mathit{i\delta }}& -c_1{s}_2{s}_3+c_2{s}_3{e}^{\mathit{i\delta }}\end{array}\right)$$E69

where $s_i=sin{\mathrm{\Theta }}_i$, $c_i=cos{\mathrm{\Theta }}_i$.

The matrix (69) can be expressed as follows

$$U=\left(\begin{array}{ccc}1& 0& 0\\ 0& c_2& s_2\\ 0& -s_2& c_2\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& e^{\mathit{i\delta }}\end{array}\right)\left(\begin{array}{ccc}{c}_1& s_1& 0\\ -s_1& c_1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& c_3& s_3\\ 0& -s_3& c_3\end{array}\right)$$E70

$$U=U_2{U}_{\delta }{U}_1{U}_3$$E71

and it can mix quarks either in the negative or in the positive electric charge subspace.

A simultaneous mixing in both spaces was also considered [10]. From the form of the matrix (70) the following variants of the quark mixing are allowed:

$$A:U=U_2\left(s-b\right)U_{\delta }{U}_1\left(d-s\right)U_3\left(s-b\right)$$E72

$$B:U=U_2\left(c-t\right)U_{\delta }{U}_1\left(d-s\right)U_3\left(s-b\right)$$E73

$$C:U=U_2\left(c-t\right)U_{\delta }{U}_1\left(u-c\right)U_3\left(s-b\right)$$E74

$$D:U=U_2\left(c-t\right)U_{\delta }{U}_1\left(u-c\right)U_3\left(c-t\right)$$E75

where $U_k\left(x-y\right)$ denotes the mixing of x and y quarks by the matrix $U_k$. It is known that the Cabibbo angle cannot be explained by the mixing in the (u-c) sector only [15, 16], so the variants C and D must be rejected. Let us examine the variant B.

The charged weak current (68) with the matrix (69) for the variant B can be expressed as follows

$$J_{\mu }=R{J}_{\mu }\left(0\right)R^{-1}$$E76

where

$$J_{\mu }\left(0\right)=\left(\overline{u},\overline{c},\overline{t}\right){\gamma }_{\mu }\left(1-{\gamma }_5\right)I\left(\begin{array}{c}d\\ s\\ b\end{array}\right)$$E77

$$R=e^{-2i{\mathrm{\Theta }}_3{Q}^{21}}{e}^{-2i{\mathrm{\Theta }}_1{Q}^7}{e}^{-i\mathit{\delta X}}{e}^{2i{\mathrm{\Theta }}_2{Q}^{32}}$$E78

$$X=\frac{4}{\sqrt{10}}{Q}^{24}-\frac{1}{\sqrt{15}}{Q}^{35}$$E79

where $Q^k$ is the $6\ast 6$ matrix representation of the k-th generator of ${\mathit{SU}}_6$ group. To get the values of the angles ${\mathrm{\Theta }}_i$ the Gell-Mann-Oakes-Renner model will be used [9]. If the electromagnetic mass splitting of u-d quarks is neglected the Hamiltonian density breaking the chiral ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ symmetry is given as follows

$$H_0=c_0{u}^0+c_8{u}^8+c_{15}{u}^{15}+c_{24}{u}^{24}+c_{35}{u}^{35}$$E80

where $c_0,\dots ,c_{35}$ are the symmetry breaking parameters, $u^i$ (i = 0,1, ..., 35) are the scalar components Of the $\left(\overline{6},6\right)+\left(6,\overline{6}\right)$ representation of the chiral ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ group. From the GMOR model, neglecting the vacuum expectation values of operators $u^k$ (k = 8, 15, 24, 35) and the spectral density ${\rho }^{\mathit{ab}}$ as proportional to the squared parameters of the symmetry breaking [12, 16], we get the approximate relation for masses of the pseudo-scalar mesons

$$m_a^2{f}_a^2{\delta }^{\mathit{ab}}=\frac{1}{\sqrt{3}}\left(\frac{c_0}{\sqrt{3}}+c_8{d}_{a8b}+c_{15}{d}_{a15b}+c_{24}{d}_{a24b}+c_{35}{d}_{a35b}\right)<u^0{>}_0$$E81

where $f_a$ are the decay constants, $d_{\mathit{aib}}$ - symmetric constants of the ${\mathit{SU}}_6$ group, $<u^0{>}_0$ - the vacuum expectation value of the operator $u^0$. From (81) we obtain

$$\pi =m_{\pi }^2{f}_{\pi }^2=\frac{1}{\sqrt{3}}\left(\frac{c_0}{\sqrt{3}}+\frac{c_8}{\sqrt{3}}+\frac{c_{15}}{\sqrt{6}}+\frac{c_{24}}{\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)<u^0{>}_0$$E82

$$K=m_K^2{f}_K^2=\frac{1}{\sqrt{3}}\left(\frac{c_0}{\sqrt{3}}-\frac{c_8}{2\sqrt{3}}+\frac{c_{15}}{\sqrt{6}}+\frac{c_{24}}{\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)<u^0{>}_0$$E83

$$D=m_D^2{f}_D^2=\frac{1}{\sqrt{3}}\left(\frac{c_0}{\sqrt{3}}+\frac{c_8}{2\sqrt{3}}-\frac{c_{15}}{\sqrt{6}}+\frac{c_{24}}{\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)<u^0{>}_0$$E84

$$B=m_B^2{f}_B^2=\frac{1}{\sqrt{3}}\left(\frac{c_0}{\sqrt{3}}+\frac{c_8}{2\sqrt{3}}+\frac{c_{15}}{2\sqrt{6}}-\frac{3c_{24}}{2\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)<u^0{>}_0$$E85

$$T=m_T^2{f}_T^2=\frac{1}{\sqrt{3}}\left(\frac{c_0}{\sqrt{3}}+\frac{c_8}{2\sqrt{3}}+\frac{c_{15}}{2\sqrt{6}}+\frac{c_{24}}{2\sqrt{10}}-\frac{2c_{35}}{\sqrt{15}}\right)<u^0{>}_0$$E86

By the symmetry breaking, the massless quark $x$ can become massive if it is mixed with the other massive $y$. The rotation angle is then described by the masses of pseudo-scalar mesons. If the ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ symmetry with the exact ${\mathit{SU}}_k\ast {\mathit{SU}}_k$ sub-symmetry is broken to the exact ${\mathit{SU}}_{k-1}\ast {\mathit{SU}}_{k-1}$ symmetry, the rotation angle is a function of masses of a pseudo-scalar meson belonging to n-multiplet of the ${\mathit{SU}}_n\ast {\mathit{SU}}_n$ group and the meson which has become massive [19]. We demand the quarks to become massive due to the hierarchical symmetry breaking, so the highest exact symmetry of the Hamiltonian density, which Can be assumed, is ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ (at least one quark in the each sector must be massive). Oakes and the others [4, 20, 21] in order to get the Cabibbo angle value in the ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ or ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry, have rotated the Hamiltonian density breaking the chiral symmetry in the same way as the weak charged current. In a model with hierarchical symmetry breaking such a procedure cannot be used. Let us notice that from the form (5) of the rotation operator R it follows that the quarks are mixed in the following sequence: (c-t), a phase rotation, (d-s), (s-b), so for the exact ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry the massless quarks d and s would be mixed as the first (in the negative electric charge subspace) and then the generation of their masses would not be possible. The quark s would become massive in the next stage of the symmetry breaking after the mixing with the massive quark b. So, in order to get the massive both d and s quarks, they should be mixed in the inverse sequence. In the first stage of the symmetry breaking the exact ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry is broken to the exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry, in the $2^{\mathit{nd}}$ stage even the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry is no longer exact. The next mixing stages are connected either with the mass generation of the c quark (variant B) or with the repeated mixing of massive s and b quarks (variant A). In our procedure the Hamiltonian density breaking the chiral ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ symmetry will be rotated in the inverse sequence in comparison with the rotation of the weak charged current.

$$H_{\mathit{SB}}=R_1{H}_0{R}_1^{-1}$$E87

where

$$R_1=e^{2i{\mathrm{\Theta }}_2{Q}^{32}}{e}^{-\mathit{iX}\delta }{e}^{-2i{\mathrm{\Theta }}_1{Q}^7}{e}^{-2i{\mathrm{\Theta }}_3{Q}^{21}}$$E88

The exact ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry implies the following relations

$$c_8=c_{15}=0\sqrt{5}{c}_0+c_{35}=0$$E89

So, the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ invariant Hamiltonian density is given as

$$H_E=c_0\left(u^0-\sqrt{5}{u}^{35}\right)+c_{24}\left(u^{24}-\sqrt{\frac{3}{2}}{u}^{35}\right)$$E90

or equivalently

$$H_E=P{\overline{q}}_6{q}_6-V{\overline{q}}_5{q}_5$$E91

where

$$P=\sqrt{c_0}+VV=\frac{5}{\sqrt{10}}{c}_{24}$$E92

The symmetry-breaking Hamiltonian density

$$H_{\mathit{SB}}=R_1{H}_E{R}_1^{-1}$$E93

retaining the flavor-conservation part only is given as follows

$$H_{\mathit{SB}}={\overline{q}}_6{q}_6{\mathit{Pc}}_2^2-{\overline{q}}_5{q}_5{\mathit{Vc}}_3^2+{\overline{q}}_4{q}_4{\mathit{Ps}}_2^2-{\overline{q}}_3{q}_3{\mathit{Vc}}_1^2{s}_3^2-{\overline{q}}_2{q}_2{\mathit{Vs}}_1^2{s}_3^2$$E94

Let us notice that the phase transformation does not produce terms ${\overline{q}}_i{q}_i$ since the operator (79) commutes with the scalar components $u^k$. The flavor-conservation on each stage of the symmetry breaking has been assumed. The Hamiltonian density (94) can be written as a function of the operators if, so the coefficients of $u^k$′s are given as

$$c_0^{\prime }=c_0$$E95

$$c_{34}^{\prime }=\frac{V}{2}{s}_1^2{s}_3^2$$E96

$$c_8^{\prime }=\frac{V}{2\sqrt{3}}\left(2c_1^2{s}_3^2-s_1^2{s}_3^2\right)$$E97

$$c_{15}^{\prime }=-\frac{1}{2\sqrt{6}}\left(3P{s}_2^2+V{s}_3^2\right)$$E98

$$c_{24}^{\prime }=\frac{1}{2\sqrt{10}}\left(4V-5V{s}_3^2+P{s}_2^2\right)$$E99

$$c_{35}^{\prime }=-\frac{1}{2\sqrt{15}}\left(5P+V-6P{s}_2^2\right)$$E100

Now, after the symmetry breaking, the pseudo-scalar masses (82–86) will be described as functions of the coefficients $c_i$′ (i = 0, 3, 8, 15, 24, 35) [7, 16].

$$\pi =ZV{s}_1^2{s}_3^2$$E101

$$K=ZV{s}_3^2\left(1-\frac{1}{2}{s}_1^2\right)$$E102

$$D=-Z\left(P{s}_2^2-\frac{1}{2}V{s}_1^2{s}_3^2\right)$$E103

$$B=ZV\left(1-s_3^2\left(1-\frac{1}{2}{s}_1^2\right)\right)$$E104

$$T=-Z\left(P{c}_2^2-\frac{1}{2}V{s}_1^2{s}_3^2\right)$$E105

where

$$Z=-\frac{1}{2\sqrt{3}}<u^0{>}_0$$E106

The Cabibbo angle ${\mathrm{\Theta }}_1$ is expressed in the same form as at four-quark level in the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry [7, 16],

$$s_1^2=\frac{\pi }{2K+\pi }$$E107

Because

$$s_1^2{s}_3^2=\frac{\pi }{K+B}\Rightarrow s_3^2=\frac{K+\frac{\pi }{2}}{K+B}$$E108

In an agreement with our prediction the angle ${\mathrm{\Theta }}_3$ connected with the mixing of s and b quarks is expressed by the parameters of the strange and beautiful mesons. The angle ${\mathrm{\Theta }}_1$ however, connected with mixing d and s quarks and breaking of the ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry is expressed by the masses of the pion and the kaon. The angle ${\mathrm{\Theta }}_2$ connected with the mixing in the (c-t) sector is given as

$$s_2^2=\frac{D-\frac{\pi }{2}}{D+T-\pi }$$E109

Let us notice that if we do not demand the flavor-conservation on each stage of the symmetry breaking, after the rotation around the ${21}^{\mathit{st}}$ axis the terms ${\overline{q}}_5{q}_3$, ${\overline{q}}_3{q}_5$ in the broken Hamiltonian density arise. In the second stage (the rotation around the $7^{\mathit{th}}$ axis) there will be in $H_{\mathit{SB}}$ the following terms: ${\overline{q}}_5{q}_3$, ${\overline{q}}_3{q}_5$, ${\overline{q}}_2{q}_3$, ${\overline{q}}_3{q}_2$, ${\overline{q}}_5{q}_2$, ${\overline{q}}_2{q}_5$. Because

$$\left[X,{\overline{q}}_3{q}_5\right]=i\delta {\overline{q}}_3{q}_5\left[X,{\overline{q}}_5{q}_3\right]=-i\delta {\overline{q}}_5{q}_3$$E110

after the phase rotation there will arise in the $H_{\mathit{SB}}$ the following terms: ${\overline{q}}_3{q}_5{e}^{\mathit{i\delta }}$, ${\overline{q}}_5{q}_3{e}^{-\mathit{i\delta }}$, .... In the variant B the matrix $U_2$ has mixed c and t quarks so that in the flavor-conservation part of the broken Hamiltonian density the phase factor $e^{\mathit{i\delta }}$ cannot appear. But if the matrix $U_2$ mixes s and b quarks again, due to the following relations

$$e^{-2i{\mathrm{\Theta }}_2{Q}^{21}}{\overline{q}}_3{q}_5{e}^{2i{\mathrm{\Theta }}_2{Q}^{21}}={\overline{q}}_3{q}_5{c}_2^2-{\overline{q}}_5{q}_3{s}_2^2+\frac{1}{2}\left({\overline{q}}_5{q}_5-{\overline{q}}_3{q}_3\right)sin2{\mathrm{\Theta }}_2$$E111

$$e^{-2i{\mathrm{\Theta }}_2,Q^{21}}{\overline{q}}_5{q}_3{e}^{2i{\mathrm{\Theta }}_2{Q}^{21}}={\overline{q}}_5{q}_3{c}_2^2-{\overline{q}}_3{q}_5{s}_2^2+\frac{1}{2}\left({\overline{q}}_5{q}_5-{\overline{q}}_3{q}_3\right)sin2{\mathrm{\Theta }}_2$$E112

the following terms: ${\overline{q}}_5{q}_5{e}^{\mathit{i\delta }}$, ${\overline{q}}_5{q}_5{e}^{-\mathit{i\delta }}$, ${\overline{q}}_3{q}_3{e}^{\mathit{i\delta }}$, ${\overline{q}}_3{q}_3{e}^{-\mathit{i\delta }}$ appear in the broken Hamiltonian density.

We assume that the symmetry is broken by the quarks mixing in the following sequence: (s-b), (d-s), a phase rotation, (s-b) and the flavor will not be conserved in the intermediate stages of the symmetry breaking, but it will be conserved in the broken symmetry taken as a whole. The assumptions given above are consistent with the variant A. Let us take it into account.

We assume the exact ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry. The Hamiltonian density is given by Eq. (91). After symmetry breaking, the flavor conserving part of the broken Hamiltonian density $H_{\mathit{SB}}$ is given as

$$H_{\left(\mathrm{\Delta }F=0\right)}={\overline{q}}_6{q}_{6}P-{\overline{q}}_5{q}_{5}V\left(\alpha -A\right)-{\overline{q}}_3{q}_{3}V\left(\beta +A\right)-{\overline{q}}_2{q}_{2}V\gamma$$E113

where

$$\alpha =c_1^2{s}_2^2{s}_3^2+c_2^2{c}_3^2$$E114

$$\beta =c_1^2{c}_2^2{s}_3^2+s_2^2{c}_3^2$$E115

$$\gamma =s_1^2{s}_3^2$$E116

$$A=\frac{1}{2}cos{\mathrm{\Theta }}_{1}sin2{\mathrm{\Theta }}_{2}sin2{\mathrm{\Theta }}_{3}cos\delta$$E117

Since

$$\alpha +\beta +\gamma =1$$E118

$\alpha$ can be eliminated from (113).

The coefficients by the operators $u^k$ are as follows

$$c_0^{\prime }=c_0$$E119

$$c_3^{\prime }=\frac{V}{2}\gamma$$E120

$$c_8^{\prime }=\frac{V}{2\sqrt{3}}\left(2\beta +2A-\gamma \right)$$E121

$$c_{15}^{\prime }=-\frac{V}{2\sqrt{6}}\left(\beta +A+\gamma \right)$$E122

$$c_{24}^{\prime }=\frac{V}{2\sqrt{10}}\left(4-5\left(\beta +A+\gamma \right)\right)$$E123

$$c_{35}^{\prime }=-\sqrt{5}{c}_0-\sqrt{\frac{3}{2}}{c}_{24}$$E124

Let us notice that the functions $\beta$ and A occur in Eqs. (121–123) as a sum $\beta +A$ only. So, only two functions can be expressed independently. Because there is no mixing in the positive electric charge subspace we shall not use relations describing mesons D and T. The following relations are obeyed

$$\pi =ZV\gamma$$E125

$$K=ZV\left(\beta +A+\frac{\gamma }{2}\right)$$E126

$$B=ZV\left(1-\beta -A-\frac{\gamma }{2}\right)$$E127

so we immediately obtain

$$s_1^2{s}_3^2=\frac{\pi }{K+B}$$E128

as in the variant B, but at the moment the angles ${\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_3$ cannot be calculated separately. Putting the experimental value

$$cos{\mathrm{\Theta }}_1=0.9737\left(sin{\mathrm{\Theta }}_1=0.2278\right)$$E129

as an input [22], we get

$$sin{\mathrm{\Theta }}_3=0.136\left({\mathrm{\Theta }}_3={7.8}^{\circ }\right)$$E130

for

$$m_{\pi }=0.139\mathit{GeV}$$E131

$$m_K=0.495\mathit{GeV}{f}_K=1.28\left[21\right]$$E132

$$m_B=5.2\mathit{GeV}{f}_B=0.86\left[22\right]$$E133

The angle ${\mathrm{\Theta }}_3$ was calculated by Fritzsch [15] also for the following quark masses ratios:

$$m_u:m_d:m_s:m_c=1:1.78:35.7:285$$E134

and the limit for the angle ${\mathrm{\Theta }}_2$

$${\mathrm{\Theta }}_2<\sqrt{\frac{m_c}{m_t}}=0.33$$E135

For the assumptions given above Fritzsch obtained the following boundary

$$sin{\mathrm{\Theta }}_3<0.09\left({\mathrm{\Theta }}_3<5^{\circ }\right)$$E136

However there is no agreement between descriptions of the quark masses ratios. The other authors [23] give smaller difference between quark masses

$$m_u:m_d:m_s:m_c=1:1.1:6.4:23.6$$E137

Thus for the ratio (137) we get the following limit for the angle ${\mathrm{\Theta }}_3$

$$sin{\mathrm{\Theta }}_3<0.163$$E138

The value of the angle ${\mathrm{\Theta }}_3$ (130) is consistent with the boundary (138). The value (130) is close to the value given by Bia$\mathrm{ł}$as [24] and consistent with results in [25, 26] as well as the experimental boundary:

$$sin{\mathrm{\Theta }}_3<0.42\left[24\right]$$E139

$$\mid sin{\mathrm{\Theta }}_3\mid =0.28\left\{\begin{array}{c}+0.21\\ -0.28\end{array}\right.\left[19\right]$$E140

Let us consider the relation between the angle ${\mathrm{\Theta }}_2$ and the phase parameter $\delta$. From (125)–(127) we obtain

$$\beta +A=\frac{K-\frac{\pi }{2}}{K+B}$$E141

or equivalently

$$\frac{K+\frac{\pi }{2}}{K+B}-\frac{\gamma }{s_1^2}=s_2^2\left(1+\gamma \left(1-\frac{2}{s_1^2}\right)\right)+A$$E142

denoting

$$\xi =\frac{\left(K+\frac{\pi }{2}\right)-\frac{\pi }{s_1^2}}{K+B}$$E143

$$\eta =1+\frac{\pi }{K+B}$$E144

$$\rho =\frac{1}{2}cos{\mathrm{\Theta }}_{1}sin2{\mathrm{\Theta }}_3$$E145

we get

$$cos\delta =\frac{\xi -s_2^2\eta }{\rho sin2{\mathrm{\Theta }}_2}$$E146

It is worth noting that if we take the constraint on the Cabibbo angle ${\mathrm{\Theta }}_1$ from the four- quark level [7, 16], which is the same as given by Eq. (107), the parameter $\xi$ (143) will be exactly equal to zero, hence we get

$$cos\delta =-\frac{\eta }{2\rho }tan{\mathrm{\Theta }}_2$$E147

Because $\mid \mathit{cos}\delta \mid \le 1$, so from (147)

$$\mid {\mathrm{\Theta }}_2\mid <\left|arctan\frac{2\rho }{\eta }\right|$$E148

and we get also a boundary on the angle ${\mathrm{\Theta }}_2$

$$sin{\mathrm{\Theta }}_2<0.265\left({\mathrm{\Theta }}_2<{15.4}^{\circ }\right)$$E149

The value (149) is in a good agreement with the results given by Fritzsch [15], Bia$\mathrm{ł}$as [24], Shrock, Treiman, Wang [22], Barger, Long, Pakvasa [25] and experimental limits [27], respectively:

$$9^{\circ }<{\mathrm{\Theta }}_2<{19}^{\circ }$$E150

$$sin{\mathrm{\Theta }}_2=0.23$$E151

$$\mid sin{\mathrm{\Theta }}_2\mid <0.25\left(m_t=15\mathrm{GeV}\right)$$E152

$$sin{\mathrm{\Theta }}_2<0.5\left(m_t=30\mathrm{GeV}\right)$$E153

The Eq. (146) can be written as follows

$$x^2\left({\eta }^2+4{\rho }^2{cos}^2\delta \right)-2x\left(\mathit{\xi \eta }+2{\rho }^2{cos}^2\delta \right)+{\xi }^2=0$$E154

where

$$x={sin}^2{\mathrm{\Theta }}_2$$E155

To get a real value of the angle ${\mathrm{\Theta }}_2$ the determinant of the square Eq. (154) cannot be negative, so

$$16{\rho }^2{cos}^2\delta \left(\mathit{\xi \eta }+{\rho }^2{cos}^2\delta -{\xi }^2\right)\ge 0$$E156

hence

$$1\ge {cos}^2\delta \ge \frac{\xi \left(\xi -\eta \right)}{{\rho }^2}$$E157

For (129, 130) we get

$$\eta =0.9647\rho =0.1309$$E158

If the parameter $\xi$. which can be identified with a change of the Cabibbo angle description by a transition to the higher symmetries, is slightly less than zero, the phase parameter $\delta$ will be bounded ($\mid \delta \mid$ should be nearly zero, as the Cabibbo angle description should not change strongly by a transition to higher symmetries, on the other hand the Eq. (157) gives a boundary on the parameter

$$\xi >-0.0175$$E159

From (147)

$$sign\left(cos\delta \right)=-sign\left(tan{\mathrm{\Theta }}_2\right)$$E160

so, for the angle ${\mathrm{\Theta }}_2$ lying in the first quadrant, it follows $\frac{\pi }{2}<\delta <\pi$ and from (159) there is a lower limit for the phase $\delta$. For an input given by the Eqs. (129, 130, 132) we get

$$\xi =0.002$$E161

so there is no boundary on $\delta$, since sign $\xi$ = + 1. Let us notice, that a small change of the $f_K$ can change the sign of the parameter $\xi$. Following Fuchs [28], in a chiral perturbation theory at the ${\mathit{SU}}_3\ast {\mathit{SU}}_3$ level

$$\frac{f_K}{f_{\pi }}=1+\frac{3\left(m_K^2-m_{\pi }^2\right)}{64{\pi }^2{f}_{\pi }^2}ln\frac{\mathrm{\Lambda }}{4{\mu }^2}+O\left(\epsilon \right)$$E162

where ${\mu }^2$ is the average meson squared mass and $\mathrm{\Lambda }$ is a cut-off parameter, which is estimated to be near $4m_N^2$, it implies

$$\frac{f_K}{f_{\pi }}=1.15$$E163

$$\xi =-0.00179$$E164

$${cos}^2\delta >0.1$$E165

Taking into account (160) we obtain

$${109}^{\circ }<\delta <{180}^{\circ }$$E166

Since $f_K$ is treated as a variable and can depend on the energy scale via $\mathrm{\Lambda }$ parameter and the symmetry breaking parameters $\epsilon$, the boundary of the phase due to the Eqs. (143, 157) can be expected. For sign (cos $\delta$) = −1 there is a lower limit of the angle ${\mathrm{\Theta }}_2$ also. A variant cos $\delta >0$ is allowed but the angle ${\mathrm{\Theta }}_2$ corresponding to this variant is too severely limited and it is not consistent with the experimental data [27].

We have shown that the weak mixing angles at the six-quark level can be estimated in terms of the masses of pseudo-scalar mesons. The calculation of mixing angles is possible by using the hierarchical symmetry breaking leading to a quark masses generation. A number of independent mixing angles that can be calculated on the ground of the given above model is equal to a number of degrees of freedom connected with the symmetry breaking and the quarks mixing in the fixed electric charge subspace (let us notice that in the variant B after the rotation around the ${21}^{\mathit{st}}$ axis and next around the $7^{\mathit{th}}$ one, even the exact ${\mathit{SU}}_2\ast {\mathit{SU}}_2$ symmetry did not remain; however the angle ${\mathrm{\Theta }}_2$ connected with the mixing in the positive electric charge subspace could be calculated). An assumption that in the hierarchical symmetry breaking the flavor does not have to be conserved on each stage of the symmetry breaking, while it is conserved in the broken symmetry taken as a whole, has allowed the author to introduce to the broken Hamiltonian density a phase angle responsible for CP-non-conservation. The experimental value of the Cabibbo angle treated as an input has allowed the author to calculate the angle ${\mathrm{\Theta }}_3$ and to find the relation connecting the angle ${\mathrm{\Theta }}_2$ and the phase parameter $\delta$. Limits of trigonometric functions values imply boundaries on the angle ${\mathrm{\Theta }}_2$ and the phase $\delta$. The kaon decay constant is a sensitive parameter, which can introduce CP-non-conservation to the chiral perturbation theory. Boundaries for the angle ${\mathrm{\Theta }}_2$ and the phase $\delta$ vs. $f_K$ can be also found.

4. The standard six-quark model with a hierarchical symmetry breaking

The simultaneous mixing of quarks in both negative and positive electric charge sub-spaces is considered. Quark mixing in each space is described by the Kobayashi-Maskawa matrix. In order to get a right number of independent mixing parameters only one angle ${\theta }_7$ common for both sub-spaces has been adjusted. Since the electromagnetic mass splitting of u and d quarks has been taken into account the real K-M mixing angles can be calculated explicitly. As an input only meson masses and $f_x$ factors (treated as factors in matrix elements between one meson state and vacuum according to PCAC) are needed. Physical quark mixing is realized for maximal allowed symmetry breaking and it corresponds to vanishing of ${\theta }_7$, which implies that only quark mixings with mass generation are permitted. Bounds of the phase $\delta$ have been also found.

The Kobayashi-Maskawa mixing matrix (69) is usually considered to mix quarks in the negative electric charge subspace. It can be written also as (70) and it can mix quarks either in the negative or in the positive electric charge subspace. A simultaneous mixing in both spaces [13] was also taken onto account.

Let us assume that quarks are mixed in both sub-spaces simultaneously, then the charged weak current is expressed as follows

$$J_{\mu }=\left(\overline{u},\overline{c},\overline{t}\right){\gamma }_{\mu }\left(1-{\gamma }_5\right)U_{+}{U}_{-}\left(\begin{array}{c}d\\ s\\ b\end{array}\right)$$E167

where

$$U_{+}=\left(\begin{array}{ccc}1& 0& 0\\ 0& c_2& s_2\\ 0& -s_2& c_2\end{array}\right)\left(\begin{array}{ccc}{c}_5& s_5& 0\\ -s_5& c_5& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& e^{i{\delta }_2}\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& c_6& s_6\\ 0& -s_6& c_6\end{array}\right)$$E168

$$U_{-}=\left(\begin{array}{ccc}1& 0& 0\\ 0& c_7& s_7\\ 0& -s_7& c_7\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& e^{i{\delta }_1}\end{array}\right)\left(\begin{array}{ccc}{c}_1& s_1& 0\\ -s_1& c_1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 0& 0\\ 0& c_3& s_3\\ 0& -s_3& c_3\end{array}\right)$$E169

The matrices $U_{+}$ and $U_{-}$ mix quarks in spaces with charges +2/3 and − 1/3 respectively. The Kobayashi-Maskawa mixing matrix is parametrized by four independent parameters only, while in the product of the matrices (168) and (169) in the current (167) there are eight mixing angles. In order to get the effective mixing matrix in the K-M form with the right number of independent mixing parameters, we must adjust the angles in such a way as to get the effective matrix with only four independent angles. We shall demand the following elements $U_{11}$, $U_{12}$, $U_{13}$, $U_{21}$, $U_{31}$ of the effective matrix to be real and the complex phase to exist in the elements $U_{22}$, $U_{23}$, $U_{32}$, $U_{33}$ only, as in the original K-M matrix. The only solution is

$${\theta }_6=-{\theta }_7$$E170

Hence in the matrix $U_{+}{U}_{-}$ there will be effectively only four parameters: ${\theta }_2$, ${\theta }_C={\theta }_1+{\theta }_5$, ${\theta }_3$ and $\delta ={\delta }_1+{\delta }_2$. The current (167) can be expressed as follows

$$J_{\mu }=R_2{R}_1{J}_{\mu }\left(0\right)R_1^{-1}{R}_2^{-1}$$E171

where

$$J_{\mu }\left(0\right)=\left(\overline{u},\overline{c},\overline{t}\right){\gamma }_{\mu }\left(1-{\gamma }_5\right)I\left(\begin{array}{c}d\\ s\\ b\end{array}\right)$$E172

$$R_1=e^{-2i{\theta }_3{Q}^{21}}{e}^{-2i{\theta }_1{Q}^7}{e}^{-\mathit{iX}{\delta }_1}{e}^{-2i{\theta }_7{Q}^{21}}$$E173

$$R_2=e^{-2i{\theta }_2{Q}^{32}}{e}^{-\mathit{iY}{\delta }_2}{e}^{-2i{\theta }_5{Q}^{10}}{e}^{-2i{\theta }_7{Q}^{32}}$$E174

where X = (79) and $Y=\frac{\sqrt{15}}{3}{Q}^{35}$. $Q^k$ is the $6\ast 6$ matrix representation of the k-th generator of ${\mathit{SU}}_6$ group. In variants A (72) and B (73), because of quark mixing in (d, s, b) sector only, the electromagnetic mass splitting of $u$ and $d$ quarks was neglected. For the simultaneous mixing in both (d, s, b) and (u, c, t) sectors the calculation of the angles ${\theta }_i$ explicitly is not possible (see below formulas (187) and (188). The Hamiltonian density breaking the chiral ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ symmetry is given as follows

$$H_0=\sum _{j=1}^6{c}_{j^2-1}{u}^{j^2-1}$$E175

where $c_i$ are the symmetry breaking parameters, $u^i$- the scalar components of the $\left(\overline{6},6\right)+\left(6,\overline{6}\right)$ of the chiral ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ group. From the GMOR model we obtain the following relations for masses of pseudo-scalar mesons for ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ symmetry:

$$\pi =m_{\pi }^2{f}_{\pi }^2=Z\left(\frac{c_0}{\sqrt{3}}+\frac{c_8}{\sqrt{3}}+\frac{c_{15}}{\sqrt{6}}+\frac{c_{24}}{\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)$$E176

$$K^{+}=m_{K^{+}}^2{f}_{K^{+}}^2=Z\left(\frac{c_0}{\sqrt{3}}+\frac{c_3}{2}-\frac{c_8}{2\sqrt{3}}+\frac{c_{15}}{\sqrt{6}}+\frac{c_{24}}{\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)$$

$$K^0=m_{K^0}^2{f}_{K^0}^2=Z\left(\frac{c_0}{\sqrt{3}}-\frac{c_3}{2}-\frac{c_8}{2\sqrt{3}}+\frac{c_{15}}{\sqrt{6}}+\frac{c_{24}}{\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)$$

$$D^{+}=m_{D^{+}}^2{f}_{D^{+}}^2=Z\left(\frac{c_0}{\sqrt{3}}-\frac{c_3}{2}+\frac{c_8}{2\sqrt{3}}-\frac{c_{15}}{\sqrt{6}}+\frac{c_{24}}{\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)$$

$$D^0=m_{D^0}^2{f}_{D^0}^2=Z\left(\frac{c_0}{\sqrt{3}}-\frac{c_3}{2}+\frac{c_8}{2\sqrt{3}}-\frac{c_{15}}{\sqrt{6}}+\frac{c_{24}}{\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)$$

$$B^{+}=m_{B^{+}}^2{f}_{B^{+}}^2=Z\left(\frac{c_0}{\sqrt{3}}+\frac{c_3}{2}+\frac{c_8}{2\sqrt{3}}+\frac{c_{15}}{2\sqrt{6}}-\frac{3c_{24}}{2\sqrt{10}}+\frac{c_{35}}{\sqrt{15}}\right)$$

$$T^{+}=m_{T^{+}}^2{f}_{T^{+}}^2=Z\left(\frac{c_0}{\sqrt{3}}-\frac{c_3}{2}+\frac{c_8}{2\sqrt{3}}+\frac{c_{15}}{2\sqrt{6}}+\frac{c_{24}}{2\sqrt{10}}-\frac{2c_{35}}{\sqrt{15}}\right)$$

In a model with hierarchical symmetry breaking the highest exact symmetry, which can be assumed, is the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ one. At least one quark in each sector must be massive. Following the procedure described in [29] the Hamiltonian density breaking the chiral ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ symmetry will be rotated in the opposite direction by comparison with the rotation of the weak charged current.

$$H_{\mathit{SB}}=R_{21}{R}_{11}{H}_E{R}_{11}^{-1}{R}_{21}^{-1}$$E177

where

$$R_{11}=e^{-2i{\theta }_7{Q}^{21}}{e}^{-\mathit{iX}{\delta }_1}{e}^{-2i{\theta }_1{Q}^7}{e}^{-2i{\theta }_3{Q}^{21}}$$E178

$$R_{21}=e^{-2i{\theta }_7{Q}^{32}}{e}^{-\mathit{iY}{\delta }_2}{e}^{2i{\theta }_5{Q}^{10}}{e}^{2i{\theta }_2{Q}^{32}}$$

The exact ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry implies that

$$c_3=c_8=c_{15}=\sqrt{5}{c}_0+c_{35}=0$$E179

The ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ invariant Hamiltonian density is given as

$$H_E=P{\overline{q}}_6{q}_6-V{\overline{q}}_5{q}_5$$E180

where

$$P=\sqrt{12}{c}_0+VV=\frac{5}{\sqrt{10}}{c}_{24}$$E181

We shall assume that in the model with hierarchical symmetry breaking the flavor will not be conserved in the intermediate stages of the symmetry breaking, but it will be conserved in the broken symmetry taken as a whole. The symmetry breaking Hamiltonian density retaining only· the flavor-conserving part is given as follows

$$H_{\left(\mathrm{\Delta }F=0\right)}={\overline{q}}_6{q}_{6}P\left(\lambda -M\right)-{\overline{q}}_5{q}_{5}V\left(\alpha -A\right)+{\overline{q}}_4{q}_{4}P\left(\rho +M\right)-{\overline{q}}_3{q}_{3}V\left(\beta +A\right)-{\overline{q}}_2{q}_2\mathit{V\gamma }+{\overline{q}}_1{q}_1\mathit{P\tau }$$E182

where

$$\alpha =c_1^2{s}_3^2{s}_7^2+c_3^2{c}_7^2$$E183

$$\beta =c_1^2{s}_3^2{c}_7^2+c_3^2{s}_7^2$$

$$\gamma =s_1^2{s}_3^2$$

$$A=\frac{1}{2}sin2{\theta }_{7}sin2{\theta }_3{c}_{1}cos{\delta }_1$$

$$\lambda =s_2^2{c}_5^2{s}_7^2+c_2^2{c}_7^2$$E184

$$\rho =s_2^2{c}_5^2{c}_7^2+c_2^2{s}_7^2$$

$$\tau =s_2^2{s}_5^2$$

$$M=-\frac{1}{2}sin2{\theta }_{7}sin2{\theta }_2{c}_{5}cos{\delta }_2$$

The broken Hamiltonian density (182) can be expressed as a function of operators $u^k$ (k = 0, 3, 8, 15, 24, 35). The coefficients of the operators $u^k$ are as follows

$$c_0^{\prime }=c_0$$E185

$$c_3^{\prime }=\frac{1}{2}\left(P\tau +V\gamma \right)$$

$$c_8^{\prime }=\frac{1}{2\sqrt{3}}\left(P\tau +V\left(2{\beta }^{\prime }-\gamma \right)\right)$$

$$c_{15}^{\prime }=\frac{1}{2\sqrt{6}}\left(P\left(\tau -3{\rho }^{\prime }\right)-V\left({\beta }^{\prime }+\gamma \right)\right)$$

$$c_{24}^{\prime }=\frac{1}{2\sqrt{10}}\left(P\left(\tau +{\rho }^{\prime }\right)-V\left(5{\beta }^{\prime }+5\gamma -4\right)\right)$$

$$c_{35}^{\prime }=\frac{1}{2\sqrt{15}}\left(6P\left(\tau +{\rho }^{\prime }\right)-5P-V\right)$$

where

$${\beta }^{\prime }=\beta +A{\rho }^{\prime }=\rho +M$$E186

After symmetry breaking the pseudo-scalar masses (176) will be described as functions of the coefficients $c_i$’ [16].

$$\begin{array}{l}\pi =\frac{Z}{2}\left(P\tau -V\gamma \right)\\ K^{+}=\frac{Z}{2}\left(P\tau -V{\beta }^{\prime }\right)K^0=-\frac{Z}{2}V\left({\beta }^{\prime }+\gamma \right)\\ D^0=\frac{Z}{2}P\left({\rho }^{\prime }+\tau \right)D^{+}=\frac{Z}{2}\left(P{\rho }^{\prime }-V\gamma \right)\\ B^{+}=\frac{Z}{2}\left(P\tau +V\left({\beta }^{\prime }+\gamma -1\right)\right)T^{+}=\frac{Z}{2}\left(P\left(1-\tau -{\rho }^{\prime }\right)-V\gamma \right)\end{array}$$E187

From (183), (184), (186) and (187) we get

$$\begin{array}{l}{\beta }^{\prime }=\frac{K^0+K^{+}-\pi }{2B^{+}+3K^0-K^{+}-\pi }\gamma =\frac{K^0-K^{+}+\pi }{2B^{+}+3K^0-K^{+}-\pi }\\ {\rho }^{\prime }=\frac{D^0+D^{+}-\pi }{2T^{+}+3D^0-D^{+}-\pi }\tau =\frac{D^0-D^{+}+\pi }{2T^{+}+3D^0-D^{+}-\pi }\end{array}$$E188

(contrary to the case of mixing in (d, s, b) sector only (variant A in [29] the electromagnetic mass splitting of u and d quarks cannot be neglected; if we put arbitrarily $c_3$ = 0 in Eq. (175) as in variants A and B in [29], the parameters $\gamma ,{\beta }^{\prime }\tau ,{\rho }^{\prime }$ could not be calculated separately. We would obtain only three nonlinear relations connecting these parameters with meson masses). Since

$$\alpha +\beta +\gamma =\lambda +\rho +\tau =1$$E189

putting (183, 184, 186) to (188) and eliminating ${\theta }_2$ and ${\theta }_3$ from the obtained set of four equations we get

$$f_1\left({\theta }_1,{\delta }_1\right)=tan{\theta }_7=-f_5\left({\theta }_5,{\delta }_2\right)$$E190

where

$$f_1\left({\theta }_1,{\delta }_1\right)=\frac{B_1\mp \sqrt{B_1^2-A_1{C}_1}}{A_1}$$E191

$$A_1=s_1^2\left(1-{\beta }^{\prime }\right)-\gamma B_1=\sqrt{\gamma }\sqrt{s_1^2-\gamma }{c}_1\mathit{cos}{\delta }_1{C}_1=\gamma -s_1^2\left({\beta }^{\prime }+\gamma \right)$$E192

$$f_5\left({\theta }_5,{\delta }_2\right)=\frac{B_5\mp \sqrt{B_5^2-A_5{C}_5}}{A_5}$$E193

$$A_5=s_5^2\left(1-{\rho }^{\prime }\right)-\tau B_5=\sqrt{\tau }\sqrt{s_5^2-\tau }{c}_5\mathit{cos}{\delta }_2{C}_5=\tau -s_5^2\left({\rho }^{\prime }+\tau \right)$$E194

We considered in [16] the simultaneous mixing in (d, s) and (u, c) sectors in the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry. The mixing angles $\mathrm{\Theta }$ and $\varphi$ could not be calculated separately, however the nonlinear formula connecting both angles and pseudo-scalar masses was found

$$2\pi +2\left(K+D\right){sin}^2\mathrm{\Theta }{sin}^2\varphi =\left(2K+\pi \right){sin}^2\mathrm{\Theta }+\left(2D+\pi \right){sin}^2\varphi$$E195

A numerical calculation showed that there is an extremum (a maximum) of the function (66) with condition (195) for the angles ${\mathrm{\Theta }}_m+{\varphi }_m$ very close to the experimentally measured Cabibbo angle. This fact suggests that the symmetry breaking is realized in the maximal allowed case, so the effective angle of mixing would correspond to the maximum of function (66). As in [16] we shall look for the extremum of the function

$$f\left({\theta }_1,{\theta }_5\right)=sin\left({\theta }_1+{\theta }_5\right)$$E196

with condition (190). The following set of equations must be obeyed

$$f_1\left({\theta }_1,{\delta }_1\right)+f_5\left({\theta }_5,{\delta }_2\right)=0\frac{\partial f_1\left({\theta }_1,{\delta }_1\right)}{\partial {\delta }_1}=0$$E197

$$\frac{\partial f_1\left({\theta }_1,{\delta }_1\right)}{\partial {\theta }_1}-\frac{\partial f_5\left({\theta }_5,{\delta }_2\right)}{\partial {\theta }_5}=0\frac{\partial f_5\left({\theta }_5,{\delta }_2\right)}{\partial {\delta }_2}=0$$

From (197) we get

$$C_1=0C_5=0$$E198

respectively, which implies that the separation constant

$$tan{\theta }_7=0$$E199

This means that the maximal allowed symmetry breaking occurs only for independent mixing of quarks in both sectors.

Let us consider the action of the operators $R_{11}$ and $R_{21}$ on quarks. The operator $R_{11}$ mixes quarks in the negative electric charge subspace in the following sequence: (s-b)(${\theta }_3$), (d-s)(${\theta }_1$), a phase rotation (${\delta }_1$), (s-b)(${\theta }_7$), however the operator $R_{21}$ mixes quarks as follows: (c-t) (${\theta }_2$), (u-c)(${\theta }_5$). a phase rotation (${\delta }_2$), (c-t)(${\theta }_7$). By the exact ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ symmetry only b and t quarks are massive. After the symmetry breaking a massless quark can become massive if it mixes with the other massive one. By the mixing in the sector with the charge - 1/3 the quark s has become massive in the first stage of the hierarchical symmetry breaking, after mixing with the quark b (the rotation on the angle ${\theta }_3$ generated by the operator $Q^{21}$), the quark d has become massive in the second stage after mixing with the already massive quarks (the rotation on the angle ${\theta }_1$). The next rotation by the angle ${\theta }_7$, and mixing of s and b quarks are not connected with the symmetry breaking, because the mixing quarks have been already massive. There is analogical situation in the sector with the charge +2/3. The c and u quarks have become massive due to the hierarchical symmetry breaking (rotations on angles ${\theta }_2$ and ${\theta }_5$, respectively), however the rotation by the angle ${\theta }_7$ and mixing of c and t quarks are also not connected with the symmetry breaking. Thus, from (199) it results that the physical quark mixing is realized only in the symmetry breaking with the quark masses generation. Putting (199) to (183) and (184) and comparing with (188) we get

$${sin}^2{\theta }_1=\frac{K^0-K^{+}+\pi }{2K^0}{sin}^2{\theta }_5=\frac{D^0-D^{+}+\pi }{2D^0}$$E200

$${sin}^2{\theta }_3=\frac{2K^0}{2B^{+}+3K^0-K^{+}-\pi }{sin}^2{\theta }_2=\frac{2D^0}{2T^{+}+3D^0-D^{+}-\pi }$$

so ${\theta }_C={\theta }_1+{\theta }_5$ depends on the parameters of mesons belonging only to the ${\mathit{SU}}_4$ multiplet. Let us notice that in comparison to the variant A in [29], taking into account the quark mixing in the (u, c, t) sector allowed the author to calculate the Cabibbo angle from the model and the angles ${\theta }_2$ and ${\theta }_3$. Let us compare the value of the calculated angle ${\theta }_C={\theta }_1+{\theta }_5$ realized for the maximal symmetry breaking with the experimentally measured Cabibbo angle value [22].

$$cos\theta =0.9737\pm 0.0025$$E201

The well known values of meson masses were taken from [30]. $f_{\pi }$, $f_{K^{+}}$, ... were assumed as the factors in the matrix elements between one meson state and the vacuum according to PCAC, so for meson multiplets with the isospin 1/2 the factors for charged and neutral mesons are the same [31]. There exist many conjectures concerning the values of $f_x$· They widely differ in magnitude, depending on the particular approach to the estimation of the matrix element $<0\mid v_x\mid x>$ and so far they have no reliable experimental support. Only in the case of $f_K$ there is a fair consensus that the value is around 1.28 [12, 13, 31, 32]. For a calculation we took as $f_D$ for comparison’s sake values significantly different

$$f_D=0.974\left[10\right]f_D=0.65\left[29\right]$$E202

which gives

$$cos\left({\theta }_1+{\theta }_5\right)=0.9799cos\left({\theta }_1+{\theta }_5\right)=0.9709$$E203

respectively, very close to the experimental value (201), as in the case of the ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ broken symmetry [16]. It seems to us that such a well agreement in both ${\mathit{SU}}_4\ast {\mathit{SU}}_4$ and ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ symmetries is not accidental and the symmetry breaking is indeed realized for the maximal allowed case.

Putting (198) to (197) we find the relation connecting both phase parameters

$$cos{\delta }_2=\xi cos{\delta }_1$$E204

where

$$\xi =\sqrt{\frac{\gamma {\rho }^{\prime }\left(1-\gamma -\beta \right)\left({\rho }^{\prime }+\tau \right)}{\tau {\beta }^{\prime }\left(1-\tau -{\rho }^{\prime }\right)\left({\beta }^{\prime }+\gamma \right)}}$$E205

The effective phase parameter $\delta ={\delta }_1+{\delta }_2$ is bounded for $\xi \ne 1$. Indeed, the Eq. (204) has solution only for

$$\mid \delta \mid \left\{\begin{array}{cc}>arccos\frac{1}{\xi }& \text{if}\left(\xi >1\right)\\ >arccos\left(\xi \right)& \text{if}\left(\xi <1\right)\end{array}\right.$$E206

It is worth noticing that even for $\xi \to \infty$ or $\xi \to 0$ the second and third quadrant for $\delta$ is still allowed.

From the Gell-Mann Oakes Renner model for ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ symmetry we obtain the following relation for masses of pseudo-scalar mesons

$$m_a^2{f}_a^2{\delta }^{\mathit{ab}}+\int \frac{{\mathit{dq}}^2}{q^2}{\rho }^{\mathit{ab}}=i<0\mid [{\overline{Q}}^a,{\overline{D}}^b\mid 0>=\sum _{i=1}^6\left(\sum _{j=1}^6{c}_{j^2-1}{d}_{i^2-1,a,c}{d}_{j^1-1,b,c}\right)<u^{i^2-1}{>}_0$$E207

where

$${\rho }^{\mathit{ab}}={\left(2\pi \right)}^3\sum _{n\ne a}{\delta }^4\left(p_n-q\right)<0\mid {\overline{D}}^a\mid n><n\mid {\overline{D}}^b\mid 0>$$E208

$d_{\mathit{abc}}$ symmetric constants of the ${\mathit{SU}}_6$ group, $<u^i{>}_0$ - vacuum expectation value of the operator $u^i$, $Q^i\pm {\overline{Q}}^i=\int d{3}^x{V}_0^{\alpha }\left(x\right)\pm \int d^3{\mathit{xA}}_0^{\alpha }\left(x\right)$ - the generators of the ${\mathit{SU}}_6\ast {\mathit{SU}}_6$ group

$$D^a={\partial }^{\mu }{V}_{\mu }^a\left(x\right){\overline{D}}^a={\partial }^{\mu }{A}_{\mu }^a\left(x\right)\left[12\right]$$E209

Because the vacuum expectation values of operators $u^i$: i = 3, 8, 15, 24, 35 and the spectral density ${\rho }^{\mathit{ab}}$ are proportion to the squared parameters Of symmetry breaking, they were neglected. Approximately we obtain

$$m_a^2{f}_a^2=\frac{1}{\sqrt{3}}\left(\sum _{j=1}^6{c}_{j^2-1}{d}_{j^2-1,a,a}\right)<u_0{>}_0$$E210

Because the symmetric constants of ${\mathit{SU}}_6$ group: $d_{113}=d_{223}=d_{333}=0$ the masses of neutral and charged pions are not differentiated, however there is the electromagnetic mass splitting of the other meson multiplets (see Eq. (176)).

The experimental data [30] gives

$$\mathrm{\Delta }{m}_K=m_{K^0}-m_{K^{+}}=4.003\mathit{MeV}\mathrm{\Delta }{m}_D=m_{D^0}-m_{D^{+}}=-5.3\mathit{MeV}$$E211

so

$$\mathit{sign}\mathrm{\Delta }{m}_K=-\mathit{sign}\mathrm{\Delta }{m}_D$$E212

Let us notice that from (17613) we get

$$\mathit{sign}\left(K^0-K^{+}\right)=-\mathit{sign}\left(D^0-D^{+}\right)$$E213

so the direction of the electromagnetic mass splitting by the factor $c_3$ responsible for this effect is consistent with the experimental data. On the other hand

$$\mathit{sign}\mathrm{\Delta }{m}_{\pi }=m_{{\pi }^0}-m_{{\pi }^{+}}=-4.603\mathit{MeV}$$E214

so the electromagnetic mass splitting of pions is of the same order as kaons or D mesons. It suggests that the neglected terms in approximate formula (207) are of the order of the factor $c_3$. This means that such an approximation does not generate error greater than the electromagnetic ma s splitting of pion in meson masses description.