## 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.

### Keywords

- quarks mixing
- chiral symmetries
- Cabibbo angle
- Kobayashi-Maskawa mixing matrix
- symmetry breaking

## 1. Quarks mixing in chiral SU n ∗ SU n broken symmetry in the limit of exact SU k ∗ 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

The problem of chiral

The other variant of the

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

where the scalar densities

where

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

We require that the

The symmetry breaking Hamiltonian density can be written as follows

Using the standard representation of

Let us note that the term

The chiral

Only the quarks

The above consideration is limited to processes not having the change of the quantum number N connected with the

where

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

If the

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

The relation between the indices a, b, j and meson states is described, for example, for the

For

Because

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 (

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

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

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

In general the determination of the rotation angle (24) in

## 2. Quarks mixing and the Cabibbo angle in the SU 4 ∗ SU 4 broken symmetry

It is known that the Cabibbo angle has been introduced into

where

The current (28) can be expressed in another form

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

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

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

The charged weak current in

The current (33) can be obtained from the isospin component of the current

where

The charged weak current in

The current (37) can be obtained by rotation of the components

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

The transformation (38) is connected with the mixing of d and s quarks (as in the case of

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

If the electromagnetic mass splitting of u-d quarks is neglected the Hamiltonian density breaking the chiral

where

where

The masses of the mesons are given as follows

In the limit of the exact chiral

so the pion is massless.

Let us make some remarks. The task of the Cabibbo angle calculation in

In [6] the numerical values of parameters (48) have been used to calculate the rotation angle (interpreted as the Cabibbo angle). The

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

Then the value of

instead of

from Eqs. (10) in [6]. Formula (50) has the same form as in

In Ebrahim’s method the

Before the

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

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

(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

Our method described above is used· to calculate an angle

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

so

In agreement with our expectation the angle

The small value of the angle

The rotated Hamiltonian density is given by

The meson masses are given as follows

Now the angles

or equivalently

where

The angles

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

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

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.

where

The matrix (69) can be expressed as follows

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:

where

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

where

where

where

where

By the symmetry breaking, the massless quark

where

The exact

So, the

or equivalently

where

The symmetry-breaking Hamiltonian density

retaining the flavor-conservation part only is given as follows

Let us notice that the phase transformation does not produce terms

Now, after the symmetry breaking, the pseudo-scalar masses (82–86) will be described as functions of the coefficients

where

The Cabibbo angle

Because

In an agreement with our prediction the angle

Let us notice that if we do not demand the flavor-conservation on each stage of the symmetry breaking, after the rotation around the

after the phase rotation there will arise in the

the following terms:

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

where

Since

The coefficients by the operators

Let us notice that the functions

so we immediately obtain

as in the variant B, but at the moment the angles

as an input [22], we get

for

The angle

and the limit for the angle

For the assumptions given above Fritzsch obtained the following boundary

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

Thus for the ratio (137) we get the following limit for the angle

The value of the angle

Let us consider the relation between the angle

or equivalently

denoting

we get

It is worth noting that if we take the constraint on the Cabibbo angle

Because

and we get also a boundary on the angle

The value (149) is in a good agreement with the results given by Fritzsch [15], Bia

The Eq. (146) can be written as follows

where

To get a real value of the angle

hence

For (129, 130) we get

If the parameter

From (147)

so, for the angle

so there is no boundary on

where

Taking into account (160) we obtain

Since

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

## 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

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

where

The matrices

Hence in the matrix

where

where X = (79) and

where

In a model with hierarchical symmetry breaking the highest exact symmetry, which can be assumed, is the

where

The exact

The

where

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 onlyÂ· the flavor-conserving part is given as follows

where

The broken Hamiltonian density (182) can be expressed as a function of operators

where

After symmetry breaking the pseudo-scalar masses (176) will be described as functions of the coefficients

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

(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

putting (183, 184, 186) to (188) and eliminating

where

We considered in [16] the simultaneous mixing in (d, s) and (u, c) sectors in the

A numerical calculation showed that there is an extremum (a maximum) of the function (66) with condition (195) for the angles

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

From (197) we get

respectively, which implies that the separation constant

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

so

The well known values of meson masses were taken from [30].

which gives

respectively, very close to the experimental value (201), as in the case of the

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

where

The effective phase parameter

It is worth noticing that even for

## Appendix

From the Gell-Mann Oakes Renner model for

where

Because the vacuum expectation values of operators

Because the symmetric constants of

The experimental data [30] gives

so

Let us notice that from (17613) we get

so the direction of the electromagnetic mass splitting by the factor

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