Open access peer-reviewed chapter - ONLINE FIRST

Quarks Mixing in Chiral Symmetries

By Zbigniew Szadkowski

Submitted: April 8th 2019Reviewed: November 25th 2020Published: April 16th 2021

DOI: 10.5772/intechopen.95233

Downloaded: 51

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 SUnSUnbroken symmetry in the limit of exact SUkSUksymmetry

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 SU3SU3symmetry in the limit of exact SU2SU2symmetry [4] to determine the value of the Cabibbo angle [10]. The transformation of rotation is connected with the seventh generator of the SU3group. After the charmed particles have been discovered the SU3SU3symmetry is no longer adequate to describe the strong interactions. The SU4SU4symmetry 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 SU4SU4symmetry has arisen. It is considered in [5, 6] and the method of calculating the Cabibbo angle in SU4SU4symmetry 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 SU3subspace too.

The problem of chiral SU4SU4symmetry breaking in the limit of exact SU2SU2symmetry is considered in [6]. Symmetry breaking is connected with the transformation of rotation around the tenth axis in SU4space. The rotation angle is determined in [7].

The other variant of the SU4SU4symmetry breaking in the limit of the exact SU3SU3symmetry is described in [8]. It is connected with the rotation around the fourteenth axis in SU4space. In this paper we introduce the general method of rotation angle description in the broken SUnSUnsymmetry. The chiral SUnSUnsymmetry is broken according to the GMOR model. In the first step we introduce the Hamiltonian density breaking SUnSUnsymmetry but invariant under SUkSUksymmetry. In the second step we introduce quark mixing and the resulting exact symmetry is SUk1SUk1The 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 SUnSUnsymmetry breaking Hamiltonian density can be written as a linear combination of diagonal operators ui.

HE=j=1ncj21uj21E1

where the scalar densities ui=q¯λiqand pseudo-scalar densities vi=iq¯λiγ5qsatisfy the equal-time commutation rules

Qiuj=ifijkukQivj=ifijkvkE2
Q¯iuj=idijkvkQ¯ivj=idijkuk

where fijkare the structure constants, dijk- symmetric generators of the SUnSUngroup. If the SUkSUksymmetry is exact then

μVμi=μAμi=0i=12k21E3

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

μVμi=iHEQiμAμi=iHEQ¯iE4

We require that the SUkSUksymmetry be exact, then the following constraints are obeyed

cj21=0j=2kE5
2n c0+j=k+1n2jj+1 cj21=0

The symmetry breaking Hamiltonian density can be written as follows

HE=c0u0n1un21+j=k+1n1cj21uj21nn1jj1un21E6

Using the standard representation of λmatrices one obtains

u0=2nj=1nq¯jqjE7
uj21=2jj1l=1j1q¯lqlj1q¯jqjE8
u0n1un21=2nq¯nqnE9
uj21nn1jj1un21=2jj1n1q¯nqnjq¯jqjl=j+1n1q¯lqlE10

Let us note that the term q¯kqkdoes not exist in Eq. 11.

HE=2nc0+n1j=k+1n1cj212jj1q¯nqnj=k+1n1cj212jj1jq¯jqj+l=j+1n1q¯lqlE11

The chiral SUnSUnsymmetry with the exact SUkSUksub-symmetry is broken by the rotation of the SUkSUkinvariant Hamiltonian density around the axis with the index m=n12+2k1.

HSB=e2QmHEe2QmE12

Only the quarks qkand qnare mixed. The SUkSUksymmetry is no longer exact. Only the term q¯nqnis rotated under transformation (12), because there is no q¯kqkterm in the Hamiltonian density (11).

e2QmHEe2Qm=q¯nqnq¯nqnq¯kqksin2α12q¯kqn+q¯nqksin2αE13

The above consideration is limited to processes not having the change of the quantum number N connected with the SUnsymmetry. So in the broken Hamiltonian density HSBΔN=0the terms q¯nqkand q¯kqndo not appear. The broken Hamiltonian density is a linear combination of the diagonal operators uionly.

HSBΔN=0=HE+Aj=kn1j+12juj+121j12juj21sin2αE14

where

A=2nc0+n1j=k+1n1cj212jj1E15

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

HSBΔN=0=c0u0Ak12ksin2αuk21+j=k+1n1uj21cj21+Asin2α12jj1+un21c0n1+j=k+1n1cj21nn1jj1Aun21n2n1sin2αE16

If the SUkSUksymmetry is exact then the pseudo-scalar mesons corresponding to the indices j=1k21are massless [9]. After the SUkSUkhas been broken, the SUk1SUk1symmetry is still exact, because the operator Qmdoes not mix the quarks q1,,qk1neither with themselves nor with other quarks. The mesons corresponding to the indices j=1k121after symmetry breaking are still massless, while the mesons corresponding to the indices j=k12k21belong to the massive multiplet k1The masses of mesons are determined in the GMOR model. Before the SUnSUnsymmetry is broken the masses are described by the coefficients c0,,cn21from Eq. (1). After the symmetry has been broken the new factors c0,,cn21are obtained as the coefficients standing by the operators uiin the broken Hamiltonian density (16) [7].

c0=c0E17
ck21=Ak12ksin2α
cj21=cj21+Asin2α12jj1k<j<n
cn21=c0n1j=k+1n1cj21nn1jj1+An2(n1)sin2α

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

ma2Ja2δab=2nc0d0ab+j=k+1ncj21dj21ab<u0>0E18

The relation between the indices a, b, j and meson states is described, for example, for the SU4SU4symmetry in [12, 13]. For a=b=k2l, the mass of the (k) meson is given as follows

mk2fk2=2n2nc0+j=k+1ndj21abcj21<u0>0=1k2nAsin2α<u0>0E19

For a=b=m=n12+2k1, the mass of the (n) meson is given by

mn2fn2=2n2nc0+j=k+1ndj21mmcj21<u0>0E20

Because

dj21mm=12jj1j<nE21
dn21mm=2n2nn1E22
mn2fn2=12nA111ksin2α<u0>0E23

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 (<u0>0, c0, ck+121,,cn21,sinα). Nevertheless the angle a is determined by the masses and decay constants of two pseudo-scalar mesons (k) and (n) only.

sin2α=kmk2fk22mn2fn2+k1mk2fk2E24

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 Θassociated with rotation around the seventh axis in SU3subspace [4, 5, 6, 7].

sin2Θ=2mπ2fπ22mK2fK2+mπ2fπ2E25

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

sin2α=2mπ2fπ22mD2fD2+mπ2fπ2E26

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

sin2α=3mK2fK22mD2fD2+mK2fK2E27

In general the determination of the rotation angle (24) in SUnSUnsymmetry 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 ui; only i=j21j=1n. The method of determining the rotation angle, discussing and interpreting the symmetry breaking is described in more detail in [7] on SU4SU4symmetry as an example.

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2. Quarks mixing and the Cabibbo angle in the SU4SU4broken symmetry

It is known that the Cabibbo angle has been introduced into SU3symmetry 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 SU4[6]. The charged weak current is then given as follows

Jμ=q¯γμ1γ5AqE28

where

A=00cosΘsinΘ00sinΘcosΘ00000000E29

The current (28) can be expressed in another form

Jμ=u¯c¯γμ1γ5cosΘsinΘsinΘcosΘdsE30

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=00cosΘsinΘ00sinΘcosΘcosϕsinϕ00sinϕcosϕ00E31

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

Jμ=u¯c¯γμ1γ5cosΘ+ϕsinΘ+ϕsinΘ+ϕcosΘ+ϕdsE32

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 Θand ϕ, can always be substituted the effective angle Θ+ϕ. To solve the problem the Gell-Mann, Oakes, Renner (GMOR) model [9] will be used.

The charged weak current in SU3symmetry can be written as follows

JμΘ=cosΘJμ1+iJμ2+sinΘJμ4+iJμ5ΘCabibboangleE33
Jμ=q¯γμ1γ5λkqq=udscE34

The current (33) can be obtained from the isospin component of the current Jμ1+iJμ2by rotation through an angle 2Θabout the seventh axis in SU3space according to

JμΘ=e2iΘF7Jμ1+iJμ2e2iΘF7E35

where

Fk=d3xq+xλk2qxE36

The charged weak current in SU4symmetry (30) can be expressed in the following form

JμΘ=cosΘJμ1+iJμ2+sinΘJμ4+iJμ5sinΘJμ11iJμ12+cosΘJμ13iJμ14E37

The current (37) can be obtained by rotation of the components ΔS=ΔCthrough an angle 2Θabout the seventh axis in SU4space

JμΘ=e2iΘF7Jμ1+iJμ2+Jμ13iJμ14e2iΘF7E38

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

F7q1=F7q4=0E39

The transformation (38) is connected with the mixing of d and s quarks (as in the case of SU3symmetry). In the SU4symmetry the mixing in electric charge subspace +2/3 can be taken into consideration. This is not possible in the SU3symmetry 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].

Jμϕ=e2F10Jμ1+iJμ2+Jμ13iJμ14e2F10E40
F10q2=F10q3=0E41

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 SU4group 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 SU3SU3symmetry in the limit of the exact SU2SU2symmetry is broken. The SU2SU2sub-symmetry is no longer exact. The symmetry is broken by the rotation of the SU2SU2invariant Hamiltonian density through angle 2Θabout the seventh axis. Then the pion becomes massive. The symmetry breaking is connected with the mixing of d and s quarks. The rotation angle Θ, 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 SU2SU2symmetry, 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 SU4SU4symmetry breaking in the limit of the exact SU2SU2sub-symmetry by the rotation of the SU2SU2invariant Hamiltonian density through an angle 2ϕabout the tenth axis the angle ϕconnected 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 SU2SU2symmetry) 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 SU4SU4symmetry is given in the form

H=c0u0+c8u8+c15u15E42

where c0,c8,c15are constants, ua(a = 0, 1, ..., 15) are the scalar components of the 4¯4+44¯representation of the chiral SU4SU4group. On the grounds of the GMOR model the following relation for masses of the pseudo-scalar mesons can be obtained [12].

i<0Q¯aD¯b0>=δabfa2ma2+dq2q2ρab=E43
=<u0>0c02δab+c82da8b+c152da15b+
+<u8>0c02da8b+c8da8cdb8c+c15da8cdb15c+
+<u15>0c02da15b+c8da15cdb8c+c15da15cdb15c

where

ρab=2π3naδ4pnq<0D¯an><nD¯b0>E44

fa- decay constants, <ui>0- vacuum expectation value of the operator ui. Because the vacuum expectation values of operators u8, u15and the spectral density δabare proportional to the squared parameters of symmetry breaking, they are further neglected [12]. Approximately from Eq. (43) we obtain

ma2fa2δab=12c02+c8da8b+c15da15b<u0>0E45

The masses of the mesons are given as follows

mπ2fπ2=1233c0+2c8+c15<u0>0
mK2fK2=1233c012c8+c15<u0>0E46
mD2fD2=1233c0+12c8c15<u0>0

In the limit of the exact chiral SU2SU2sub-symmetry there is the following constraint

3c0+2c8+c15=0E47

so the pion is massless.

Let us make some remarks. The task of the Cabibbo angle calculation in SU4symmetry using the procedure of symmetry breaking has been done in [6]. In Ebrahim’s earlier paper [5] the parameters of the SU4SU4symmetry breaking have been found.

c8c0=223mK2fK2mπ2fπ2mK2fK2+mD2fD2c15c0=133mD2fD2mK2fK22mπ2fπ2mK2fK2+mD2fD2E48

In [6] the numerical values of parameters (48) have been used to calculate the rotation angle (interpreted as the Cabibbo angle). The SU2SU2invariant Hamiltonian density breaking SU4SU4symmetry has been rotated through an angle 2 Θabout the seventh axis and the coefficients of the operators ua(a = 0, 8, 15) have been identified with the parameters of symmetry breaking

HSBΔS=0=c0u0+32c8sin2Θu3+c8132sin2Θu83c0+2c8u15E49

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

sin2Θ=2mπ2fπ22mK2fK2+mπ2fπ2E50

Then the value of Θis given by

sin2Θ=0.2152E51

instead of

sin2Θ=0.04E52

from Eqs. (10) in [6]. Formula (50) has the same form as in SU3symmetry. In agreement with our expectation the angle Θis described by parameters of the pion and the strange meson.

In Ebrahim’s method the SU4SU4symmetry breaking the Hamiltonian density is parametrized by the factors c0, c8, c15. 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 SU2SU2sub-symmetry the factors c0, c8, c15should satisfy the constraint (47) but it is possible only if mπ=0namely the parameters of symmetry breaking are not expressed by real (measured in experiment) masses of mesons. In [6] Ebrahim breaks the SU4SU4symmetry in the limit of the exact SU2SU2sub-symmetry by the rotation of the SU2SU2invariant Hamiltonian density through an angle 2 Θabout 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 c0, c8, c15dependent 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 SU4SU4symmetry in the limit of the exact SU2SU2sub-symmetry is broken the masses of mesons have been expressed by the factors c0, c8, c15which satisfy the constraint (47). After symmetry breaking a new set of factors c0, cg, c15dependent on the old factors c0, c8, c15and on the rotation angle is introduced. The new factors are identified with the coefficients by the operators uiof the rotated Hamiltonian density (49).

c0=c0cg=c8132sin2Θc15=3c02c8E53

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

mπ2fπ2=1233c0+2cg+c15<u0>0=322c8sin2Θ<u0>0E54
mK2fK2=1233c0c82+c15<u0>0=322c8112sin2Θ<u0>0
mD2fD2=1233c0+c82c15<u0>0=c0+322c8112sin2Θ<u0>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 Θas in Eq. (50). Let us consider the other variant of symmetry breaking described in [6]. Ebrahim, using his method, broke the SU4SU4symmetry in the limit of the exact SU3SU3symmetry by the rotation of SU3SU3invariant Hamiltonian density about the fourteenth axis in the SU4space. The rotation angle Θis identified with the Cabibbo angle. The formula describing the angle Θshould be given as follows

sin2Θ=3mK2fK22mK2fK2+mD2fD2E55

(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 Θwith 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 SU4SU4symmetry is broken in such a way that the SU2SU2sub-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 SU3SU3symmetry 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 SU4SU4symmetry taken as a whole cannot be connected with the Cabibbo angle. This is possible, however, for SU4SU4symmetry breaking in the limit of exact SU2SU2sub-symmetry. Then results are in agreement with our expectation.

Our method described above is used· to calculate an angle ϕwhich is connected with the rotation about the tenth axis in SU4space. Then the SU4SU4symmetry is broken by the rotation of the SU2SU2invariant Hamiltonian density through an angle 2ϕabout the tenth axis.

HSBΔC=0=c0u0+2c0+32c8sin2ϕu3+c8(12243c0+2c8sin2ϕu8++3c0+2c8+43c0+2c8sin2ϕu15E56

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

mπ2fπ2=c0+64sin2ϕ<u0>0E57
mK2fK2=6412c0+64c8sin2ϕ<u0>0
mD2fD2=112sin2ϕc0+64c8<u0>0

so

sin2ϕ=2mπ2fπ22mD2fD2+mπ2fπ2E58

In agreement with our expectation the angle ϕis a function of the mass of the pion (as a measure of the SU2SU2violation) and is connected with the parameters of the charmed meson (mixing in (u, c) doublet). For the mass mD= 1862 MeV and fD/fπ= 0.974 [13] one gets

sin2ϕ=0.076E59

The small value of the angle ϕis 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 ϕcontradicts 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 ϕis relatively small, it is significant: the sum of the angles Θ+ϕis 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 SU2SU2invariant Hamiltonian density is rotated through an angle 2ϕabout the tenth axis and afterwards by an angle 2Θabout the seventh axis. The sequence of the rotations is insignificant, because

F7F10=0E60

The rotated Hamiltonian density is given by

HSBΔS=ΔC=0=c0u0+32c8sin2Θ+2c0+32c8sin2ϕu3+E61
+132c0+32c8sin2ϕ+c8132sin2Θu8+
+3c02c8+2232c0+32c8sin2ϕu15

The meson masses are given as follows

mπ2fπ2=12362c0+32c8sin2ϕ32c8sin2Θ<u0>0E62
mK2fK2=123362c0+32c8sin2ϕ32c8(112sin2Θ<u0>0
mD2fD2=12323c0362c0+32c8sin2ϕ+32c8112sin2Θ<u0>0

Now the angles Θand ϕcannot be described independently. The following relation is obeyed.

2mπ2fπ2+2mK2fK2+mD2fD2sin2Θsin2ϕ=E63
=2mK2fK2+mπ2fπ2sin2Θ+2mD2fD2+mπ2fπ2sin2ϕ

or equivalently

1+1sin2Θ+1sin2ϕ1sin2Θsin2ϕ=sin2Θsin2Θ0+sin2ϕsin2ϕ0E64

where

sin2Θ0=2mπ2fπ22mK2fK2+mπ2fπ2sin2ϕ0=2mπ2fπ22mD2fD2+mπ2fπ2E65

The angles Θand ϕfrom 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 Θand ϕ. The maximal values of the angles Θ0and ϕ0are given by Eq. (65). The value of the function

fΘϕ=sinΘ+ϕE66

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

Θm=0.20452ϕm=0.02575sinΘm+ϕm=0.2282E67

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 SU6SU6chiral symmetry

Jμ=u¯c¯t¯γμ1γ5UdsbE68

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=c1s1c3s1s3s1c2c1c2c3s2s3ec1c2s3+s2c3es1s2c1s2c3c2s3ec1s2s3+c2s3eE69

where si=sinΘi, ci=cosΘi.

The matrix (69) can be expressed as follows

U=1000c2s20s2c210001000ec1s10s1c100011000c3s30s3c3E70
U=U2UδU1U3E71

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=U2sbUδU1dsU3sbE72
B:U=U2ctUδU1dsU3sbE73
C:U=U2ctUδU1ucU3sbE74
D:U=U2ctUδU1ucU3ctE75

where Ukxydenotes the mixing of x and y quarks by the matrix Uk. 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

Jμ=RJμ0R1E76

where

Jμ0=u¯c¯t¯γμ1γ5IdsbE77
R=e2iΘ3Q21e2iΘ1Q7eiδXe2iΘ2Q32E78
X=410Q24115Q35E79

where Qkis the 66matrix representation of the k-th generator of SU6group. To get the values of the angles Θithe 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 SU6SU6symmetry is given as follows

H0=c0u0+c8u8+c15u15+c24u24+c35u35E80

where c0,,c35are the symmetry breaking parameters, ui(i = 0,1, ..., 35) are the scalar components Of the 6¯6+66¯representation of the chiral SU6SU6group. From the GMOR model, neglecting the vacuum expectation values of operators uk(k = 8, 15, 24, 35) and the spectral density ρabas proportional to the squared parameters of the symmetry breaking [12, 16], we get the approximate relation for masses of the pseudo-scalar mesons

ma2fa2δab=13c03+c8da8b+c15da15b+c24da24b+c35da35b<u0>0E81

where faare the decay constants, daib- symmetric constants of the SU6group, <u0>0- the vacuum expectation value of the operator u0. From (81) we obtain

π=mπ2fπ2=13c03+c83+c156+c2410+c3515<u0>0E82
K=mK2fK2=13c03c823+c156+c2410+c3515<u0>0E83
D=mD2fD2=13c03+c823c156+c2410+c3515<u0>0E84
B=mB2fB2=13c03+c823+c15263c24210+c3515<u0>0E85
T=mT2fT2=13c03+c823+c1526+c242102c3515<u0>0E86

By the symmetry breaking, the massless quark xcan 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 SUnSUnsymmetry with the exact SUkSUksub-symmetry is broken to the exact SUk1SUk1symmetry, the rotation angle is a function of masses of a pseudo-scalar meson belonging to n-multiplet of the SUnSUngroup 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 SU4SU4(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 SU3SU3or SU4SU4symmetry, 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 SU4SU4symmetry 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 SU4SU4symmetry is broken to the exact SU2SU2symmetry, in the 2ndstage even the SU2SU2symmetry 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 SU6SU6symmetry will be rotated in the inverse sequence in comparison with the rotation of the weak charged current.

HSB=R1H0R11E87

where

R1=e2iΘ2Q32eiXδe2iΘ1Q7e2iΘ3Q21E88

The exact SU4SU4symmetry implies the following relations

c8=c15=05c0+c35=0E89

So, the SU4SU4invariant Hamiltonian density is given as

HE=c0u05u35+c24u2432u35E90

or equivalently

HE=Pq¯6q6Vq¯5q5E91

where

P=c0+VV=510c24E92

The symmetry-breaking Hamiltonian density

HSB=R1HER11E93

retaining the flavor-conservation part only is given as follows

HSB=q¯6q6Pc22q¯5q5Vc32+q¯4q4Ps22q¯3q3Vc12s32q¯2q2Vs12s32E94

Let us notice that the phase transformation does not produce terms q¯iqisince the operator (79) commutes with the scalar components uk. 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 uk′s are given as

c0=c0E95
c34=V2s12s32E96
c8=V232c12s32s12s32E97
c15=1263Ps22+Vs32E98
c24=12104V5Vs32+Ps22E99
c35=12155P+V6Ps22E100

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

π=ZVs12s32E101
K=ZVs32112s12E102
D=ZPs2212Vs12s32E103
B=ZV1s32112s12E104
T=ZPc2212Vs12s32E105

where

Z=123<u0>0E106

The Cabibbo angle Θ1is expressed in the same form as at four-quark level in the SU4SU4symmetry [7, 16],

s12=π2K+πE107

Because

s12s32=πK+Bs32=K+π2K+BE108

In an agreement with our prediction the angle Θ3connected with the mixing of s and b quarks is expressed by the parameters of the strange and beautiful mesons. The angle Θ1however, connected with mixing d and s quarks and breaking of the SU2SU2symmetry is expressed by the masses of the pion and the kaon. The angle Θ2connected with the mixing in the (c-t) sector is given as

s22=Dπ2D+TπE109

Let us notice that if we do not demand the flavor-conservation on each stage of the symmetry breaking, after the rotation around the 21staxis the terms q¯5q3, q¯3q5in the broken Hamiltonian density arise. In the second stage (the rotation around the 7thaxis) there will be in HSBthe following terms: q¯5q3, q¯3q5, q¯2q3, q¯3q2, q¯5q2, q¯2q5. Because

Xq¯3q5=iδq¯3q5Xq¯5q3=iδq¯5q3E110

after the phase rotation there will arise in the HSBthe following terms: q¯3q5e, q¯5q3e, .... In the variant B the matrix U2has mixed c and t quarks so that in the flavor-conservation part of the broken Hamiltonian density the phase factor ecannot appear. But if the matrix U2mixes s and b quarks again, due to the following relations

e2iΘ2Q21q¯3q5e2iΘ2Q21=q¯3q5c22q¯5q3s22+12q¯5q5q¯3q3sin2Θ2E111
e2iΘ2,Q21q¯5q3e2iΘ2Q21=q¯5q3c22q¯3q5s22+12q¯5q5q¯3q3sin2Θ2E112

the following terms: q¯5q5e, q¯5q5e, q¯3q3e, q¯3q3eappear 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 SU4SU4symmetry. The Hamiltonian density is given by Eq. (91). After symmetry breaking, the flavor conserving part of the broken Hamiltonian density HSBis given as

HΔF=0=q¯6q6Pq¯5q5VαAq¯3q3Vβ+Aq¯2q2VγE113

where

α=c12s22s32+c22c32E114
β=c12c22s32+s22c32E115
γ=s12s32E116
A=12cosΘ1sin2Θ2sin2Θ3cosδE117

Since

α+β+γ=1E118

αcan be eliminated from (113).

The coefficients by the operators ukare as follows

c0=c0E119
c3=V2γE120
c8=V232β+2AγE121
c15=V26β+A+γE122
c24=V21045β+A+γE123
c35=5c032c24E124

Let us notice that the functions βand A occur in Eqs. (121123) as a sum β+Aonly. 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

π=ZVγE125
K=ZVβ+A+γ2E126
B=ZV1βAγ2E127

so we immediately obtain

s12s32=πK+BE128

as in the variant B, but at the moment the angles Θ1and Θ3cannot be calculated separately. Putting the experimental value

cosΘ1=0.9737sinΘ1=0.2278E129

as an input [22], we get

sinΘ3=0.136Θ3=7.8E130

for

mπ=0.139GeVE131
mK=0.495GeVfK=1.2821E132
mB=5.2GeVfB=0.8622E133

The angle Θ3was calculated by Fritzsch [15] also for the following quark masses ratios:

mu:md:ms:mc=1:1.78:35.7:285E134

and the limit for the angle Θ2

Θ2<mcmt=0.33E135

For the assumptions given above Fritzsch obtained the following boundary

sinΘ3<0.09Θ3<5E136

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

mu:md:ms:mc=1:1.1:6.4:23.6E137

Thus for the ratio (137) we get the following limit for the angle Θ3

sinΘ3<0.163E138

The value of the angle Θ3(130) is consistent with the boundary (138). The value (130) is close to the value given by Białas [24] and consistent with results in [25, 26] as well as the experimental boundary:

sinΘ3<0.4224E139
sinΘ3=0.28+0.210.2819E140

Let us consider the relation between the angle Θ2and the phase parameter δ. From (125)(127) we obtain

β+A=Kπ2K+BE141

or equivalently

K+π2K+Bγs12=s221+γ12s12+AE142

denoting

ξ=K+π2πs12K+BE143
η=1+πK+BE144
ρ=12cosΘ1sin2Θ3E145

we get

cosδ=ξs22ηρsin2Θ2E146

It is worth noting that if we take the constraint on the Cabibbo angle Θ1from the four- quark level [7, 16], which is the same as given by Eq. (107), the parameter ξ(143) will be exactly equal to zero, hence we get

cosδ=η2ρtanΘ2E147

Because cosδ1, so from (147)

Θ2<arctan2ρηE148

and we get also a boundary on the angle Θ2

sinΘ2<0.265Θ2<15.4E149

The value (149) is in a good agreement with the results given by Fritzsch [15], Białas [24], Shrock, Treiman, Wang [22], Barger, Long, Pakvasa [25] and experimental limits [27], respectively:

9<Θ2<19E150
sinΘ2=0.23E151
sinΘ2<0.25mt=15GeVE152
sinΘ2<0.5mt=30GeVE153

The Eq. (146) can be written as follows

x2η2+4ρ2cos2δ2xξη+2ρ2cos2δ+ξ2=0E154

where

x=sin2Θ2E155

To get a real value of the angle Θ2the determinant of the square Eq. (154) cannot be negative, so

16ρ2cos2δξη+ρ2cos2δξ20E156

hence

1cos2δξξηρ2E157

For (129, 130) we get

η=0.9647ρ=0.1309E158

If the parameter ξ. 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 δwill be bounded (δshould 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

ξ>0.0175E159

From (147)

signcosδ=signtanΘ2E160

so, for the angle Θ2lying in the first quadrant, it follows π2<δ<πand from (159) there is a lower limit for the phase δ. For an input given by the Eqs. (129, 130, 132) we get

ξ=0.002E161

so there is no boundary on δ, since sign ξ= + 1. Let us notice, that a small change of the fKcan change the sign of the parameter ξ. Following Fuchs [28], in a chiral perturbation theory at the SU3SU3level

fKfπ=1+3mK2mπ264π2fπ2lnΛ4μ2+OεE162

where μ2is the average meson squared mass and Λis a cut-off parameter, which is estimated to be near 4mN2, it implies

fKfπ=1.15E163
ξ=0.00179E164
cos2δ>0.1E165

Taking into account (160) we obtain

109<δ<180E166

Since fKis treated as a variable and can depend on the energy scale via Λparameter and the symmetry breaking parameters ε, the boundary of the phase due to the Eqs. (143, 157) can be expected. For sign (cos δ) = −1 there is a lower limit of the angle Θ2also. A variant cos δ>0is allowed but the angle Θ2corresponding 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 21staxis and next around the 7thone, even the exact SU2SU2symmetry did not remain; however the angle Θ2connected 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 Θ3and to find the relation connecting the angle Θ2and the phase parameter δ. Limits of trigonometric functions values imply boundaries on the angle Θ2and the phase δ. The kaon decay constant is a sensitive parameter, which can introduce CP-non-conservation to the chiral perturbation theory. Boundaries for the angle Θ2and the phase δvs. fKcan 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 θ7common 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 fxfactors (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 θ7, which implies that only quark mixings with mass generation are permitted. Bounds of the phase δhave 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

Jμ=u¯c¯t¯γμ1γ5U+UdsbE167

where

U+=1000c2s20s2c2c5s50s5c5000110001000eiδ21000c6s60s6c6E168
U=1000c7s70s7c710001000eiδ1c1s10s1c100011000c3s30s3c3E169

The matrices U+and Umix 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 U11, U12, U13, U21, U31of the effective matrix to be real and the complex phase to exist in the elements U22, U23, U32, U33only, as in the original K-M matrix. The only solution is

θ6=θ7E170

Hence in the matrix U+Uthere will be effectively only four parameters: θ2, θC=θ1+θ5, θ3and δ=δ1+δ2. The current (167) can be expressed as follows

Jμ=R2R1Jμ0R11R21E171

where

Jμ0=u¯c¯t¯γμ1γ5IdsbE172
R1=e2iθ3Q21e2iθ1Q7eiXδ1e2iθ7Q21E173
R2=e2iθ2Q32eiYδ2e2iθ5Q10e2iθ7Q32E174

where X = (79) and Y=153Q35. Qkis the 66matrix representation of the k-th generator of SU6group. In variants A (72) and B (73), because of quark mixing in (d, s, b) sector only, the electromagnetic mass splitting of uand dquarks was neglected. For the simultaneous mixing in both (d, s, b) and (u, c, t) sectors the calculation of the angles θiexplicitly is not possible (see below formulas (187) and (188). The Hamiltonian density breaking the chiral SU6SU6symmetry is given as follows

H0=j=16cj21uj21E175

where ciare the symmetry breaking parameters, ui- the scalar components of the 6¯6+66¯of the chiral SU6SU6group. From the GMOR model we obtain the following relations for masses of pseudo-scalar mesons for SU6SU6symmetry:

π=mπ2fπ2=Zc03+c83+c156+c2410+c3515E176
K+=mK+2fK+2=Zc03+c32c823+c156+c2410+c3515
K0=mK02fK02=Zc03c32c823+c156+c2410+c3515
D+=mD+2fD+2=Zc03c32+c823c156+c2410+c3515
D0=mD02fD02=Zc03c32+c823c156+c2410+c3515
B+=mB+2fB+2=Zc03+c32+c823+c15263c24210+c3515
T+=mT+2fT+2=Zc03c32+c823+c1526+c242102c3515

In a model with hierarchical symmetry breaking the highest exact symmetry, which can be assumed, is the SU4SU4one. At least one quark in each sector must be massive. Following the procedure described in [29] the Hamiltonian density breaking the chiral SU6SU6symmetry will be rotated in the opposite direction by comparison with the rotation of the weak charged current.

HSB=R21R11HER111R211E177

where

R11=e2iθ7Q21eiXδ1e2iθ1Q7e2iθ3Q21E178
R21=e2iθ7Q32eiYδ2e2iθ5Q10e2iθ2Q32

The exact SU4SU4symmetry implies that

c3=c8=c15=5c0+c35=0E179

The SU4SU4invariant Hamiltonian density is given as

HE=Pq¯6q6Vq¯5q5E180

where

P=12c0+VV=510c24E181

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

HΔF=0=q¯6q6PλMq¯5q5VαA+q¯4q4Pρ+Mq¯3q3Vβ+Aq¯2q2+q¯1q1E182

where

α=c12s32s72+c32c72E183
β=c12s32c72+c32s72
γ=s12s32
A=12sin2θ7sin2θ3c1cosδ1
λ=s22c52s72+c22c72E184
ρ=s22c52c72+c22s72
τ=s22s52
M=12sin2θ7sin2θ2c5cosδ2

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

c0=c0E185
c3=12Pτ+Vγ
c8=123Pτ+V2βγ
c15=126Pτ3ρVβ+γ
c24=1210Pτ+ρV5β+5γ4
c35=12156Pτ+ρ5PV

where

β=β+Aρ=ρ+ME186

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

π=Z2PτVγK+=Z2PτVβK0=Z2Vβ+γD0=Z2Pρ+τD+=Z2PρVγB+=Z2Pτ+Vβ+γ1T+=Z2P1τρVγE187

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

β=K0+K+π2B++3K0K+πγ=K0K++π2B++3K0K+πρ=D0+D+π2T++3D0D+πτ=D0D++π2T++3D0D+πE188

(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 c3= 0 in Eq. (175) as in variants A and B in [29], the parameters γ,βτ,ρcould not be calculated separately. We would obtain only three nonlinear relations connecting these parameters with meson masses). Since

α+β+γ=λ+ρ+τ=1E189

putting (183, 184, 186) to (188) and eliminating θ2and θ3from the obtained set of four equations we get

f1θ1δ1=tanθ7=f5θ5δ2E190

where

f1θ1δ1=B1B12A1C1A1E191
A1=s121βγB1=γs12γc1cosδ1C1=γs12β+γE192
f5θ5δ2=B5B52A5C5A5E193
A5=s521ρτB5=τs52τc5cosδ2C5=τs52ρ+τE194

We considered in [16] the simultaneous mixing in (d, s) and (u, c) sectors in the SU4SU4symmetry. The mixing angles Θand ϕcould not be calculated separately, however the nonlinear formula connecting both angles and pseudo-scalar masses was found

2π+2K+Dsin2Θsin2ϕ=2K+πsin2Θ+2D+πsin2ϕE195

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

fθ1θ5=sinθ1+θ5E196

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

f1θ1δ1+f5θ5δ2=0f1θ1δ1δ1=0E197
f1θ1δ1θ1f5θ5δ2θ5=0f5θ5δ2δ2=0

From (197) we get

C1=0C5=0E198

respectively, which implies that the separation constant

tanθ7=0E199

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 R11and R21on quarks. The operator R11mixes quarks in the negative electric charge subspace in the following sequence: (s-b)(θ3), (d-s)(θ1), a phase rotation (δ1), (s-b)(θ7), however the operator R21mixes quarks as follows: (c-t) (θ2), (u-c)(θ5). a phase rotation (δ2), (c-t)(θ7). By the exact SU4SU4symmetry 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 θ3generated by the operator Q21), the quark d has become massive in the second stage after mixing with the already massive quarks (the rotation on the angle θ1). The next rotation by the angle θ7, 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 θ2and θ5, respectively), however the rotation by the angle θ7and 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

sin2θ1=K0K++π2K0sin2θ5=D0D++π2D0E200
sin2θ3=2K02B++3K0K+πsin2θ2=2D02T++3D0D+π

so θC=θ1+θ5depends on the parameters of mesons belonging only to the SU4multiplet. 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 θ2and θ3. Let us compare the value of the calculated angle θC=θ1+θ5realized for the maximal symmetry breaking with the experimentally measured Cabibbo angle value [22].

cosθ=0.9737±0.0025E201

The well known values of meson masses were taken from [30]. fπ, fK+, ... 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 fx· They widely differ in magnitude, depending on the particular approach to the estimation of the matrix element <0vxx>and so far they have no reliable experimental support. Only in the case of fKthere is a fair consensus that the value is around 1.28 [12, 13, 31, 32]. For a calculation we took as fDfor comparison’s sake values significantly different

fD=0.97410fD=0.6529E202

which gives

cosθ1+θ5=0.9799cosθ1+θ5=0.9709E203

respectively, very close to the experimental value (201), as in the case of the SU4SU4broken symmetry [16]. It seems to us that such a well agreement in both SU4SU4and SU6SU6symmetries 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

cosδ2=ξcosδ1E204

where

ξ=γρ1γβρ+ττβ1τρβ+γE205

The effective phase parameter δ=δ1+δ2is bounded for ξ1. Indeed, the Eq. (204) has solution only for

δ>arccos1ξifξ>1>arccosξifξ<1E206

It is worth noticing that even for ξor ξ0the second and third quadrant for δis still allowed.

Appendix

From the Gell-Mann Oakes Renner model for SU6SU6symmetry we obtain the following relation for masses of pseudo-scalar mesons

ma2fa2δab+dq2q2ρab=i<0[Q¯a,D¯b0>=i=16j=16cj21di21,a,cdj11,b,c<ui21>0E207

where

ρab=2π3naδ4pnq<0D¯an><nD¯b0>E208

dabcsymmetric constants of the SU6group, <ui>0- vacuum expectation value of the operator ui, Qi±Q¯i=d3xV0αx±d3xA0αx- the generators of the SU6SU6group

Da=μVμaxD¯a=μAμax12E209

Because the vacuum expectation values of operators ui: i = 3, 8, 15, 24, 35 and the spectral density ρabare proportion to the squared parameters Of symmetry breaking, they were neglected. Approximately we obtain

ma2fa2=13j=16cj21dj21,a,a<u0>0E210

Because the symmetric constants of SU6group: d113=d223=d333=0the 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

ΔmK=mK0mK+=4.003MeVΔmD=mD0mD+=5.3MeVE211

so

signΔmK=signΔmDE212

Let us notice that from (17613) we get

signK0K+=signD0D+E213

so the direction of the electromagnetic mass splitting by the factor c3responsible for this effect is consistent with the experimental data. On the other hand

signΔmπ=mπ0mπ+=4.603MeVE214

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 c3. This means that such an approximation does not generate error greater than the electromagnetic ma s splitting of pion in meson masses description.

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Zbigniew Szadkowski (April 16th 2021). Quarks Mixing in Chiral Symmetries [Online First], IntechOpen, DOI: 10.5772/intechopen.95233. Available from:

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