1. Introduction
1.1 Quarks mixing in chiral SUn∗SUn broken symmetry in the limit of exact SUk∗SUk 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 SU3∗SU3 symmetry in the limit of exact SU2∗SU2 symmetry [4] to determine the value of the Cabibbo angle [10]. The transformation of rotation is connected with the seventh generator of the SU3 group. After the charmed particles have been discovered the SU3∗SU3 symmetry is no longer adequate to describe the strong interactions. The SU4∗SU4 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 SU4∗SU4 symmetry has arisen. It is considered in [5, 6] and the method of calculating the Cabibbo angle in SU4∗SU4 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 SU3 subspace too.
The problem of chiral SU4∗SU4 symmetry breaking in the limit of exact SU2∗SU2 symmetry is considered in [6]. Symmetry breaking is connected with the transformation of rotation around the tenth axis in SU4 space. The rotation angle is determined in [7].
The other variant of the SU4∗SU4 symmetry breaking in the limit of the exact SU3∗SU3 symmetry is described in [8]. It is connected with the rotation around the fourteenth axis in SU4 space. In this paper we introduce the general method of rotation angle description in the broken SUn∗SUn symmetry. The chiral SUn∗SUn symmetry is broken according to the GMOR model. In the first step we introduce the Hamiltonian density breaking SUn∗SUn symmetry but invariant under SUk∗SUk symmetry. In the second step we introduce quark mixing and the resulting exact symmetry is SUk−1∗SUk−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 SUn∗SUn symmetry breaking Hamiltonian density can be written as a linear combination of diagonal operators ui.
HE=∑j=1ncj2−1uj2−1E1
where the scalar densities ui=q¯λiq and pseudo-scalar densities vi=iq¯λiγ5q satisfy the equal-time commutation rules
Qiuj=ifijkukQivj=ifijkvkE2
Q¯iuj=idijkvkQ¯ivj=−idijkuk
where fijk are the structure constants, dijk - symmetric generators of the SUn∗SUn group. If the SUk∗SUk symmetry is exact then
∂μVμi=∂μAμi=0i=12…k2−1E3
In the GMOR model the divergences of currents can be calculated as follows
∂μVμi=iHEQi∂μAμi=iHEQ¯iE4
We require that the SUk∗SUk symmetry be exact, then the following constraints are obeyed
cj2−1=0j=2…kE5
2n c0+∑j=k+1n2jj+1 cj2−1=0
The symmetry breaking Hamiltonian density can be written as follows
HE=c0u0−n−1un2−1+∑j=k+1n−1cj2−1uj2−1−nn−1jj−1un2−1E6
Using the standard representation of λ matrices one obtains
u0=2n∑j=1nq¯jqjE7
uj2−1=2jj−1∑l=1j−1q¯lql−j−1q¯jqjE8
u0−n−1un2−1=2nq¯nqnE9
uj2−1−nn−1jj−1un2−1=2jj−1n−1q¯nqn−jq¯jqj−∑l=j+1n−1q¯lqlE10
Let us note that the term q¯kqk does not exist in Eq. 11.
HE=2nc0+n−1∑j=k+1n−1cj2−12jj−1q¯nqn−∑j=k+1n−1cj2−12jj−1jq¯jqj+∑l=j+1n−1q¯lqlE11
The chiral SUn∗SUn symmetry with the exact SUk∗SUk sub-symmetry is broken by the rotation of the SUk∗SUk invariant Hamiltonian density around the axis with the index m=n−12+2k−1.
HSB=e−2iαQmHEe2iαQmE12
Only the quarks qk and qn are mixed. The SUk∗SUk symmetry is no longer exact. Only the term q¯nqn is rotated under transformation (12), because there is no q¯kqk term in the Hamiltonian density (11).
e−2iαQmHEe2iαQm=q¯nqn−q¯nqn−q¯kqksin2α−12q¯kqn+q¯nqksin2αE13
The above consideration is limited to processes not having the change of the quantum number N connected with the SUn symmetry. So in the broken Hamiltonian density HSBΔN=0 the terms q¯nqk and q¯kqn do not appear. The broken Hamiltonian density is a linear combination of the diagonal operators ui only.
HSBΔN=0=HE+A∑j=kn−1j+12juj+12−1−j−12juj2−1sin2αE14
where
A=2nc0+n−1∑j=k+1n−1cj2−12jj−1E15
In more detail Eq. (14) is given as follows:
HSBΔN=0=c0u0−Ak−12ksin2αuk2−1+∑j=k+1n−1uj2−1cj2−1+Asin2α12jj−1+−un2−1c0n−1+∑j=k+1n−1cj2−1nn−1jj−1−Aun2−1n2n−1sin2αE16
If the SUk∗SUk symmetry is exact then the pseudo-scalar mesons corresponding to the indices j=1…k2−1 are massless [9]. After the SUk∗SUk has been broken, the SUk−1∗SUk−1 symmetry is still exact, because the operator Qm does not mix the quarks q1,…,qk−1 neither with themselves nor with other quarks. The mesons corresponding to the indices j=1…k−12−1 after symmetry breaking are still massless, while the mesons corresponding to the indices j=k−12…k2−1 belong to the massive multiplet k1 The masses of mesons are determined in the GMOR model. Before the SUn∗SUn symmetry is broken the masses are described by the coefficients c0,…,cn2−1 from Eq. (1). After the symmetry has been broken the new factors c0′,…,cn2−1′ are obtained as the coefficients standing by the operators ui in the broken Hamiltonian density (16) [7].
c0′=c0E17
ck2−1′=Ak−12ksin2α
cj2−1′=cj2−1+Asin2α12jj−1k<j<n
cn2−1′=−c0n−1−∑j=k+1n−1cj2−1nn−1jj−1+An2(n−1)sin2α
The masses of the mesons are determined as follows [12].
ma2Ja2δab=2nc0d0ab+∑j=k+1ncj2−1′dj2−1ab<u0>0E18
The relation between the indices a, b, j and meson states is described, for example, for the SU4∗SU4 symmetry in [12, 13]. For a=b=k2−l, the mass of the (k) meson is given as follows
mk2fk2=2n2nc0+∑j=k+1ndj2−1abcj2−1′<u0>0=1k2nAsin2α<u0>0E19
For a=b=m=n−12+2k−1, the mass of the (n) meson is given by
mn2fn2=2n2nc0+∑j=k+1ndj2−1mmcj2−1′<u0>0E20
Because
dj2−1mm=12jj−1j<nE21
dn2−1mm=2−n2nn−1E22
mn2fn2=12nA1−1−1ksin2α<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+12−1,…,cn2−1,sinα). Nevertheless the angle a is determined by the masses and decay constants of two pseudo-scalar mesons (k) and (n) only.
sin2α=kmk2fk22mn2fn2+k−1mk2fk2E24
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 Θ associated with rotation around the seventh axis in SU3 subspace [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 SUn∗SUn 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 ui; only i=j2−1j=1…n. The method of determining the rotation angle, discussing and interpreting the symmetry breaking is described in more detail in [7] on SU4∗SU4 symmetry as an example.
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2. Quarks mixing and the Cabibbo angle in the SU4∗SU4 broken symmetry
It is known that the Cabibbo angle has been introduced into SU3 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 SU4 [6]. The charged weak current is then given as follows
Jμ=q¯γμ1−γ5AqE28
where
A=00cosΘsinΘ00−sinΘcosΘ00000000E29
The current (28) can be expressed in another form
Jμ=u¯c¯γμ1−γ5cosΘsinΘ−sinΘcosΘdsE30
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=00cosΘsinΘ00−sinΘcosΘcosϕsinϕ00−sinϕcosϕ00E31
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¯γμ1−γ5cosΘ+ϕsinΘ+ϕ−sinΘ+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 Θ and 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 SU3 symmetry can be written as follows
JμΘ=cosΘJμ1+iJμ2+sinΘJμ4+iJμ5Θ−CabibboangleE33
Jμ=q¯γμ1−γ5λkqq=udscE34
The current (33) can be obtained from the isospin component of the current Jμ1+iJμ2 by rotation through an angle 2Θ about the seventh axis in SU3 space according to
JμΘ=e−2iΘF7Jμ1+iJμ2e2iΘF7E35
where
Fk=∫d3xq+xλk2qxE36
The charged weak current in SU4 symmetry (30) can be expressed in the following form
JμΘ=cosΘJμ1+iJμ2+sinΘJμ4+iJμ5−sinΘJμ11−iJμ12+cosΘJμ13−iJμ14E37
The current (37) can be obtained by rotation of the components ΔS=ΔC through an angle 2Θ about the seventh axis in SU4 space
JμΘ=e−2iΘF7Jμ1+iJμ2+Jμ13−iJμ14e2iΘF7E38
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 SU3 symmetry). In the SU4 symmetry the mixing in electric charge subspace +2/3 can be taken into consideration. This is not possible in the SU3 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].
Jμϕ=e−2iϕF10Jμ1+iJμ2+Jμ13−iJμ14e2iϕF10E40
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 SU4 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 SU3∗SU3 symmetry in the limit of the exact SU2∗SU2 symmetry is broken. The SU2∗SU2 sub-symmetry is no longer exact. The symmetry is broken by the rotation of the SU2∗SU2 invariant Hamiltonian density through angle 2Θ 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 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 SU2∗SU2 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 SU4∗SU4 symmetry breaking in the limit of the exact SU2∗SU2 sub-symmetry by the rotation of the SU2∗SU2 invariant Hamiltonian density through an angle 2ϕ about the tenth axis the angle ϕ 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 SU2∗SU2 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 SU4∗SU4 symmetry is given in the form
H=c0u0+c8u8+c15u15E42
where c0,c8,c15 are constants, ua (a = 0, 1, ..., 15) are the scalar components of the 4¯4+44¯ representation of the chiral SU4∗SU4 group. On the grounds of the GMOR model the following relation for masses of the pseudo-scalar mesons can be obtained [12].
i<0∣Q¯aD¯b∣0>=δabfa2ma2+∫dq2q2ρab=E43
=<u0>0c02δab+c82da8b+c152da15b+
+<u8>0c02da8b+c8da8cdb8c+c15da8cdb15c+
+<u15>0c02da15b+c8da15cdb8c+c15da15cdb15c
where
ρab=2π3∑n≠aδ4pn−q<0∣D¯a∣n><n∣D¯b∣0>E44
fa - decay constants, <ui>0 - vacuum expectation value of the operator ui. Because the vacuum expectation values of operators u8, u15 and the spectral density δab are 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=1233c0−12c8+c15<u0>0E46
mD2fD2=1233c0+12c8−c15<u0>0
In the limit of the exact chiral SU2∗SU2 sub-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 SU4 symmetry using the procedure of symmetry breaking has been done in [6]. In Ebrahim’s earlier paper [5] the parameters of the SU4∗SU4 symmetry breaking have been found.
c8c0=−223mK2fK2−mπ2fπ2mK2fK2+mD2fD2c15c0=−133mD2fD2−mK2fK2−2mπ2fπ2mK2fK2+mD2fD2E48
In [6] the numerical values of parameters (48) have been used to calculate the rotation angle (interpreted as the Cabibbo angle). The SU2∗SU2 invariant Hamiltonian density breaking SU4∗SU4 symmetry has been rotated through an angle 2 Θ about 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+c81−32sin2Θu8−3c0+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 Θ is given by
sin2Θ=0.2152E51
instead of
sin2Θ=−0.04E52
from Eqs. (10) in [6]. Formula (50) has the same form as in SU3 symmetry. In agreement with our expectation the angle Θ is described by parameters of the pion and the strange meson.
In Ebrahim’s method the SU4∗SU4 symmetry 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 SU2∗SU2 sub-symmetry the factors c0, c8, c15 should satisfy the constraint (47) but it is possible only if mπ=0 namely the parameters of symmetry breaking are not expressed by real (measured in experiment) masses of mesons. In [6] Ebrahim breaks the SU4∗SU4 symmetry in the limit of the exact SU2∗SU2 sub-symmetry by the rotation of the SU2∗SU2 invariant Hamiltonian density through an angle 2 Θ 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 c0, c8, c15 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 SU4∗SU4 symmetry in the limit of the exact SU2∗SU2 sub-symmetry is broken the masses of mesons have been expressed by the factors c0, c8, c15 which satisfy the constraint (47). After symmetry breaking a new set of factors c0′, cg′, c15′ dependent on the old factors c0, c8, c15 and on the rotation angle is introduced. The new factors are identified with the coefficients by the operators ui of the rotated Hamiltonian density (49).
c0′=c0cg′=c81−32sin2Θc15′=−3c0−2c8E53
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=1233c0′−c8′2+c15′<u0>0=−322c81−12sin2Θ<u0>0
mD2fD2=1233c0′+c8′2−c15′<u0>0=c0+322c81−12sin2Θ<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 Θ as in Eq. (50). Let us consider the other variant of symmetry breaking described in [6]. Ebrahim, using his method, broke the SU4∗SU4 symmetry in the limit of the exact SU3∗SU3 symmetry by the rotation of SU3∗SU3 invariant Hamiltonian density about the fourteenth axis in the SU4 space. The rotation angle Θ′ is identified with the Cabibbo angle. The formula describing the angle Θ′ should 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 Θ′ 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 SU4∗SU4 symmetry is broken in such a way that the SU2∗SU2 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 SU3∗SU3 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 SU4∗SU4 symmetry taken as a whole cannot be connected with the Cabibbo angle. This is possible, however, for SU4∗SU4 symmetry breaking in the limit of exact SU2∗SU2 sub-symmetry. Then results are in agreement with our expectation.
Our method described above is used· to calculate an angle ϕ which is connected with the rotation about the tenth axis in SU4 space. Then the SU4∗SU4 symmetry is broken by the rotation of the SU2∗SU2 invariant Hamiltonian density through an angle 2ϕ about 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=−64−12c0+64c8sin2ϕ<u0>0
mD2fD2=1−12sin2ϕc0′+64c8<u0>0
so
sin2ϕ=2mπ2fπ22mD2fD2+mπ2fπ2E58
In agreement with our expectation the angle ϕ is a function of the mass of the pion (as a measure of the SU2∗SU2 violation) 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 ϕ 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 ϕ 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 ϕ is relatively small, it is significant: the sum of the angles Θ+ϕ 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 SU2∗SU2 invariant Hamiltonian density is rotated through an angle 2ϕ about the tenth axis and afterwards by an angle 2Θ about 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+c81−32sin2Θu8+
+−3c0−2c8+2232c0+32c8sin2ϕu15
The meson masses are given as follows
mπ2fπ2=12362c0+32c8sin2ϕ−32c8sin2Θ<u0>0E62
mK2fK2=123362c0+32c8sin2ϕ−32c8(1−12sin2Θ<u0>0
mD2fD2=12323c0−362c0+32c8sin2ϕ+32c81−12sin2Θ<u0>0
Now the angles Θ and ϕ cannot be described independently. The following relation is obeyed.
2mπ2fπ2+2mK2fK2+mD2fD2sin2Θsin2ϕ=E63
=2mK2fK2+mπ2fπ2sin2Θ+2mD2fD2+mπ2fπ2sin2ϕ
or equivalently
1+1sin2Θ+1sin2ϕ−1sin2Θsin2ϕ=sin2Θsin2Θ0+sin2ϕsin2ϕ0E64
where
sin2Θ0=2mπ2fπ22mK2fK2+mπ2fπ2sin2ϕ0=2mπ2fπ22mD2fD2+mπ2fπ2E65
The angles Θ and ϕ 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 Θ and ϕ. The maximal values of the angles Θ0 and ϕ0 are 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).
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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 SU6∗SU6 chiral symmetry
Jμ=u¯c¯t¯γμ1−γ5UdsbE68
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=c1s1c3s1s3−s1c2c1c2c3−s2s3eiδc1c2s3+s2c3eiδs1s2−c1s2c3−c2s3eiδ−c1s2s3+c2s3eiδE69
where si=sinΘi, ci=cosΘi.
The matrix (69) can be expressed as follows
U=1000c2s20−s2c210001000eiδc1s10−s1c100011000c3s30−s3c3E70
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=U2s−bUδU1d−sU3s−bE72
B:U=U2c−tUδU1d−sU3s−bE73
C:U=U2c−tUδU1u−cU3s−bE74
D:U=U2c−tUδU1u−cU3c−tE75
where Ukx−y denotes 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=RJμ0R−1E76
where
Jμ0=u¯c¯t¯γμ1−γ5IdsbE77
R=e−2iΘ3Q21e−2iΘ1Q7e−iδXe2iΘ2Q32E78
X=410Q24−115Q35E79
where Qk is the 6∗6 matrix representation of the k-th generator of SU6 group. To get the values of the angles Θi 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 SU6∗SU6 symmetry is given as follows
H0=c0u0+c8u8+c15u15+c24u24+c35u35E80
where c0,…,c35 are the symmetry breaking parameters, ui (i = 0,1, ..., 35) are the scalar components Of the 6¯6+66¯ representation of the chiral SU6∗SU6 group. From the GMOR model, neglecting the vacuum expectation values of operators uk (k = 8, 15, 24, 35) and the spectral density ρab as 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 fa are the decay constants, daib - symmetric constants of the SU6 group, <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=13c03−c823+c156+c2410+c3515<u0>0E83
D=mD2fD2=13c03+c823−c156+c2410+c3515<u0>0E84
B=mB2fB2=13c03+c823+c1526−3c24210+c3515<u0>0E85
T=mT2fT2=13c03+c823+c1526+c24210−2c3515<u0>0E86
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 SUn∗SUn symmetry with the exact SUk∗SUk sub-symmetry is broken to the exact SUk−1∗SUk−1 symmetry, the rotation angle is a function of masses of a pseudo-scalar meson belonging to n-multiplet of the SUn∗SUn 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 SU4∗SU4 (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 SU3∗SU3 or SU4∗SU4 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 SU4∗SU4 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 SU4∗SU4 symmetry is broken to the exact SU2∗SU2 symmetry, in the 2nd stage even the SU2∗SU2 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 SU6∗SU6 symmetry will be rotated in the inverse sequence in comparison with the rotation of the weak charged current.
HSB=R1H0R1−1E87
where
R1=e2iΘ2Q32e−iXδe−2iΘ1Q7e−2iΘ3Q21E88
The exact SU4∗SU4 symmetry implies the following relations
c8=c15=05c0+c35=0E89
So, the SU4∗SU4 invariant Hamiltonian density is given as
HE=c0u0−5u35+c24u24−32u35E90
or equivalently
HE=Pq¯6q6−Vq¯5q5E91
where
P=c0+VV=510c24E92
The symmetry-breaking Hamiltonian density
HSB=R1HER1−1E93
retaining the flavor-conservation part only is given as follows
HSB=q¯6q6Pc22−q¯5q5Vc32+q¯4q4Ps22−q¯3q3Vc12s32−q¯2q2Vs12s32E94
Let us notice that the phase transformation does not produce terms q¯iqi since 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′=V232c12s32−s12s32E97
c15′=−1263Ps22+Vs32E98
c24′=12104V−5Vs32+Ps22E99
c35′=−12155P+V−6Ps22E100
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=ZVs321−12s12E102
D=−ZPs22−12Vs12s32E103
B=ZV1−s321−12s12E104
T=−ZPc22−12Vs12s32E105
where
Z=−123<u0>0E106
The Cabibbo angle Θ1 is expressed in the same form as at four-quark level in the SU4∗SU4 symmetry [7, 16],
s12=π2K+πE107
Because
s12s32=πK+B⇒s32=K+π2K+BE108
In an agreement with our prediction the angle Θ3 connected with the mixing of s and b quarks is expressed by the parameters of the strange and beautiful mesons. The angle Θ1 however, connected with mixing d and s quarks and breaking of the SU2∗SU2 symmetry is expressed by the masses of the pion and the kaon. The angle Θ2 connected with the mixing in the (c-t) sector is given as
s22=D−π2D+T−πE109
Let us notice that if we do not demand the flavor-conservation on each stage of the symmetry breaking, after the rotation around the 21st axis the terms q¯5q3, q¯3q5 in the broken Hamiltonian density arise. In the second stage (the rotation around the 7th axis) there will be in HSB the 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 HSB the following terms: q¯3q5eiδ, q¯5q3e−iδ, .... In the variant B the matrix U2 has mixed c and t quarks so that in the flavor-conservation part of the broken Hamiltonian density the phase factor eiδ cannot appear. But if the matrix U2 mixes s and b quarks again, due to the following relations
e−2iΘ2Q21q¯3q5e2iΘ2Q21=q¯3q5c22−q¯5q3s22+12q¯5q5−q¯3q3sin2Θ2E111
e−2iΘ2,Q21q¯5q3e2iΘ2Q21=q¯5q3c22−q¯3q5s22+12q¯5q5−q¯3q3sin2Θ2E112
the following terms: q¯5q5eiδ, q¯5q5e−iδ, q¯3q3eiδ, q¯3q3e−iδ 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 SU4∗SU4 symmetry. The Hamiltonian density is given by Eq. (91). After symmetry breaking, the flavor conserving part of the broken Hamiltonian density HSB is given as
HΔF=0=q¯6q6P−q¯5q5Vα−A−q¯3q3Vβ+A−q¯2q2VγE113
where
α=c12s22s32+c22c32E114
β=c12c22s32+s22c32E115
γ=s12s32E116
A=12cosΘ1sin2Θ2sin2Θ3cosδE117
Since
α+β+γ=1E118
α can be eliminated from (113).
The coefficients by the operators uk are as follows
c0′=c0E119
c3′=V2γE120
c8′=V232β+2A−γE121
c15′=−V26β+A+γE122
c24′=V2104−5β+A+γE123
c35′=−5c0−32c24E124
Let us notice that the functions β and A occur in Eqs. (121–123) as a sum β+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
π=ZVγE125
K=ZVβ+A+γ2E126
B=ZV1−β−A−γ2E127
so we immediately obtain
s12s32=πK+BE128
as in the variant B, but at the moment the angles Θ1 and Θ3 cannot be calculated separately. Putting the experimental value
cosΘ1=0.9737sinΘ1=0.2278E129
as an input [22], we get
sinΘ3=0.136Θ3=7.8∘E130
for
mπ=0.139GeVE131
mK=0.495GeVfK=1.2821E132
mB=5.2GeVfB=0.8622E133
The angle Θ3 was 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<5∘E136
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+0.21−0.2819E140
Let us consider the relation between the angle Θ2 and the phase parameter δ. From (125)–(127) we obtain
β+A=K−π2K+BE141
or equivalently
K+π2K+B−γs12=s221+γ1−2s12+AE142
denoting
ξ=K+π2−πs12K+BE143
η=1+πK+BE144
ρ=12cosΘ1sin2Θ3E145
we get
cosδ=ξ−s22ηρsin2Θ2E146
It is worth noting that if we take the constraint on the Cabibbo angle Θ1 from 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 ∣cosδ∣≤1, so from (147)
∣Θ2∣<arctan2ρηE148
and we get also a boundary on the angle Θ2
sinΘ2<0.265Θ2<15.4∘E149
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<19∘E150
sinΘ2=0.23E151
∣sinΘ2∣<0.25mt=15GeVE152
sinΘ2<0.5mt=30GeVE153
The Eq. (146) can be written as follows
x2η2+4ρ2cos2δ−2xξη+2ρ2cos2δ+ξ2=0E154
where
x=sin2Θ2E155
To get a real value of the angle Θ2 the determinant of the square Eq. (154) cannot be negative, so
16ρ2cos2δξη+ρ2cos2δ−ξ2≥0E156
hence
1≥cos2δ≥ξξ−ηρ2E157
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 δ will be bounded (∣δ∣ 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
ξ>−0.0175E159
From (147)
signcosδ=−signtanΘ2E160
so, for the angle Θ2 lying in the first quadrant, it follows π2<δ<π and 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 fK can change the sign of the parameter ξ. Following Fuchs [28], in a chiral perturbation theory at the SU3∗SU3 level
fKfπ=1+3mK2−mπ264π2fπ2lnΛ4μ2+OεE162
where μ2 is the average meson squared mass and Λ is 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<180∘E166
Since fK is treated as a variable and can depend on the energy scale via Λ parameter 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 Θ2 also. A variant cos δ>0 is allowed but the angle Θ2 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 21st axis and next around the 7th one, even the exact SU2∗SU2 symmetry did not remain; however the angle Θ2 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 Θ3 and to find the relation connecting the angle Θ2 and the phase parameter δ. Limits of trigonometric functions values imply boundaries on the angle Θ2 and 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 Θ2 and the phase δ vs. fK can be also found.
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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 θ7 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 fx 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 θ7, which implies that only quark mixings with mass generation are permitted. Bounds of the phase δ 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
Jμ=u¯c¯t¯γμ1−γ5U+U−dsbE167
where
U+=1000c2s20−s2c2c5s50−s5c5000110001000eiδ21000c6s60−s6c6E168
U−=1000c7s70−s7c710001000eiδ1c1s10−s1c100011000c3s30−s3c3E169
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 U11, U12, U13, U21, U31 of the effective matrix to be real and the complex phase to exist in the elements U22, U23, U32, U33 only, as in the original K-M matrix. The only solution is
θ6=−θ7E170
Hence in the matrix U+U− there will be effectively only four parameters: θ2, θC=θ1+θ5, θ3 and δ=δ1+δ2. The current (167) can be expressed as follows
Jμ=R2R1Jμ0R1−1R2−1E171
where
Jμ0=u¯c¯t¯γμ1−γ5IdsbE172
R1=e−2iθ3Q21e−2iθ1Q7e−iXδ1e−2iθ7Q21E173
R2=e−2iθ2Q32e−iYδ2e−2iθ5Q10e−2iθ7Q32E174
where X = (79) and Y=153Q35. Qk is the 6∗6 matrix representation of the k-th generator of SU6 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 θi explicitly is not possible (see below formulas (187) and (188). The Hamiltonian density breaking the chiral SU6∗SU6 symmetry is given as follows
H0=∑j=16cj2−1uj2−1E175
where ci are the symmetry breaking parameters, ui- the scalar components of the 6¯6+66¯ of the chiral SU6∗SU6 group. From the GMOR model we obtain the following relations for masses of pseudo-scalar mesons for SU6∗SU6 symmetry:
π=mπ2fπ2=Zc03+c83+c156+c2410+c3515E176
K+=mK+2fK+2=Zc03+c32−c823+c156+c2410+c3515
K0=mK02fK02=Zc03−c32−c823+c156+c2410+c3515
D+=mD+2fD+2=Zc03−c32+c823−c156+c2410+c3515
D0=mD02fD02=Zc03−c32+c823−c156+c2410+c3515
B+=mB+2fB+2=Zc03+c32+c823+c1526−3c24210+c3515
T+=mT+2fT+2=Zc03−c32+c823+c1526+c24210−2c3515
In a model with hierarchical symmetry breaking the highest exact symmetry, which can be assumed, is the SU4∗SU4 one. At least one quark in each sector must be massive. Following the procedure described in [29] the Hamiltonian density breaking the chiral SU6∗SU6 symmetry will be rotated in the opposite direction by comparison with the rotation of the weak charged current.
HSB=R21R11HER11−1R21−1E177
where
R11=e−2iθ7Q21e−iXδ1e−2iθ1Q7e−2iθ3Q21E178
R21=e−2iθ7Q32e−iYδ2e2iθ5Q10e2iθ2Q32
The exact SU4∗SU4 symmetry implies that
c3=c8=c15=5c0+c35=0E179
The SU4∗SU4 invariant Hamiltonian density is given as
HE=Pq¯6q6−Vq¯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¯6q6Pλ−M−q¯5q5Vα−A+q¯4q4Pρ+M−q¯3q3Vβ+A−q¯2q2Vγ+q¯1q1PτE182
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 uk are as follows
c0′=c0E185
c3′=12Pτ+Vγ
c8′=123Pτ+V2β′−γ
c15′=126Pτ−3ρ′−Vβ′+γ
c24′=1210Pτ+ρ′−V5β′+5γ−4
c35′=12156Pτ+ρ′−5P−V
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=Z2Pτ−VγK+=Z2Pτ−Vβ′K0=−Z2Vβ′+γD0=Z2Pρ′+τD+=Z2Pρ′−VγB+=Z2Pτ+Vβ′+γ−1T+=Z2P1−τ−ρ′−VγE187
From (183), (184), (186) and (187) we get
β′=K0+K+−π2B++3K0−K+−πγ=K0−K++π2B++3K0−K+−πρ′=D0+D+−π2T++3D0−D+−πτ=D0−D++π2T++3D0−D+−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,β′τ,ρ′ could 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 θ2 and θ3 from the obtained set of four equations we get
f1θ1δ1=tanθ7=−f5θ5δ2E190
where
f1θ1δ1=B1∓B12−A1C1A1E191
A1=s121−β′−γB1=γs12−γc1cosδ1C1=γ−s12β′+γE192
f5θ5δ2=B5∓B52−A5C5A5E193
A5=s521−ρ′−τB5=τs52−τc5cosδ2C5=τ−s52ρ′+τE194
We considered in [16] the simultaneous mixing in (d, s) and (u, c) sectors in the SU4∗SU4 symmetry. The mixing angles Θ and ϕ could 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+ϕm 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
fθ1θ5=sinθ1+θ5E196
with condition (190). The following set of equations must be obeyed
f1θ1δ1+f5θ5δ2=0∂f1θ1δ1∂δ1=0E197
∂f1θ1δ1∂θ1−∂f5θ5δ2∂θ5=0∂f5θ5δ2∂δ2=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 R11 and R21 on quarks. The operator R11 mixes 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 R21 mixes quarks as follows: (c-t) (θ2), (u-c)(θ5). a phase rotation (δ2), (c-t)(θ7). By the exact SU4∗SU4 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 θ3 generated 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 θ2 and θ5, respectively), however the rotation by the angle θ7 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
sin2θ1=K0−K++π2K0sin2θ5=D0−D++π2D0E200
sin2θ3=2K02B++3K0−K+−πsin2θ2=2D02T++3D0−D+−π
so θC=θ1+θ5 depends on the parameters of mesons belonging only to the SU4 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 θ2 and θ3. Let us compare the value of the calculated angle θC=θ1+θ5 realized 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 <0∣vx∣x> and so far they have no reliable experimental support. Only in the case of fK there is a fair consensus that the value is around 1.28 [12, 13, 31, 32]. For a calculation we took as fD for 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 SU4∗SU4 broken symmetry [16]. It seems to us that such a well agreement in both SU4∗SU4 and SU6∗SU6 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
cosδ2=ξcosδ1E204
where
ξ=γρ′1−γ−βρ′+ττβ′1−τ−ρ′β′+γE205
The effective phase parameter δ=δ1+δ2 is bounded for ξ≠1. Indeed, the Eq. (204) has solution only for
∣δ∣>arccos1ξifξ>1>arccosξifξ<1E206
It is worth noticing that even for ξ→∞ or ξ→0 the second and third quadrant for δ is still allowed.
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From the Gell-Mann Oakes Renner model for SU6∗SU6 symmetry we obtain the following relation for masses of pseudo-scalar mesons
ma2fa2δab+∫dq2q2ρab=i<0∣[Q¯a,D¯b∣0>=∑i=16∑j=16cj2−1di2−1,a,cdj1−1,b,c<ui2−1>0E207
where
ρab=2π3∑n≠aδ4pn−q<0∣D¯a∣n><n∣D¯b∣0>E208
dabc symmetric constants of the SU6 group, <ui>0 - vacuum expectation value of the operator ui, Qi±Q¯i=∫d3xV0αx±∫d3xA0αx - the generators of the SU6∗SU6 group
Da=∂μVμaxD¯a=∂μAμax12E209
Because the vacuum expectation values of operators ui: i = 3, 8, 15, 24, 35 and the spectral density ρab are proportion to the squared parameters Of symmetry breaking, they were neglected. Approximately we obtain
ma2fa2=13∑j=16cj2−1dj2−1,a,a<u0>0E210
Because the symmetric constants of SU6 group: d113=d223=d333=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
ΔmK=mK0−mK+=4.003MeVΔmD=mD0−mD+=−5.3MeVE211
so
signΔmK=−signΔmDE212
Let us notice that from (17613) we get
signK0−K+=−signD0−D+E213
so the direction of the electromagnetic mass splitting by the factor c3 responsible for this effect is consistent with the experimental data. On the other hand
signΔmπ=mπ0−mπ+=−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.