Bimaximal Pattern Links Color Quark
and Neutrino Mixing-Angle** **Approximations
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** **By Emmanuel Lipmanov

In the SM weak interaction phenomenology, the leading approximation benchmark mixing patterns of neutrinos and quarks are different – bimaximal pattern and zero one, respectively.

In contrast, we consider here the bimaximal pattern as the only basic primary mixing link between leptons and quarks. The difference in observable quark and neutrino mixing is mainly related to the symmetry of three quark colors in contrast to neutrino one color. But it’s not all. In addition, the asymmetry of bimaximal neutrino benchmark mixing is related to the singled out θ13 angle. Instead, quark mixing is assumed here fully symmetric of the three mixing angles via general bimaximal pattern (3). Below we illustrate that statement in details.

Consider
the geometric-type SO(3)-symmetric neutrino equation for double mixing angles 2θ^{n}_{12, }2θ^{n}_{23 }and 2θ^{n}_{13},

cos^{2} 2θ^{n}_{12} + cos^{2}
2θ^{n}_{23 }+ cos^{2}
2θ^{n}_{13} = 1. (1)

The bimaximal mixing pattern solution
without free parameters (benchmark) is cos^{2}
2θ^{n}12 = cos^{2} 2θ^{n}23 = 0, cos^{2} 2θ^{n}13 = 1, (2)

it is an exact neutrino mixing solution of Eq. (1) that singles out one mixing angle θ13.

In contrast, quark mixing by definition should be symmetric under all mixing angle bimaximal neutrino-type solutions of geometric equation (1) for each of the color quark states c = 1, 2, 3. A total of nine solutions of Eq. (1) for color quark mixing angles, in contrast to one solution (2) for neutrino mixing angles, is given by

(cos^{2} 2θ^{c}_{12}; cos^{2} 2θ^{c}_{23}; cos^{2}
2θ^{c}_{13}) = (0; 0; 1),

(cos^{2} 2θ^{c}_{12}; cos^{2}
2θ^{c}_{23}; cos^{2} 2θ^{c}_{13})
= (0; 1; 0),

(cos^{2}
2θ^{c}_{12}; cos^{2}
2θ^{c}_{23}; cos^{2} 2θ^{c}_{13})
= (1; 0; 0). (3)

The sum of three solutions for particular color `c` in Eq. (3) is given by

(cos^{2} 2θ^{c}_{12}; cos^{2}
2θ^{c}_{23}; cos^{2}
2θ^{c}_{13}) = (1; 1; 1) / 3. (4)

The universal color quark mixing angle (before uniting the quarks in hadrons) from Eq. (4) is equal

θ^{c}_{12} = θ^{c}_{23} = θ^{c}_{13} = ~
27.4o.
(5)

But we have to sum
over the three colors in Eq. (4) because the individual
color quark mixing angles θ^{c}_{ik} are not observable at least
at low energies. All experimental measurements are made with hadrons, which are
‘white’ symmetric sums related to color quarks. The sum over color in Eq.
(4) is

∑^{c}
(cos^{2} 2θ^{c}_{12}; cos^{2} 2θ^{c}_{23}; cos^{2}
2θ^{c}_{13})
= (1; 1; 1). (6)

The sums over
colors in every term of Eq. (6) should be identified as new quantities in the
SM weak interactions – “white quarks" (one white quark for each flavor instead
of regular three color quarks) for the observable particle mixing angles, denoted
below as θ^{q }_{ik},

cos^{2 }2θ^{q}_{ik}
= ∑^{c} cos^{2
}2θ^{c}_{ik}. (7)

It should be emphasized, the sum over three quark color angles at right in relation (7) is a consistent definition of white quark angles at left, though that definition is not in direct analogy with the composition of white light through color light in optics.

With (6) and
definition (7), the final result for observable leading approximation white quark
angles θ^{q}_{ik}
(7) in benchmark mixing without free parameters is given by

cos^{2} 2θ^{q}_{12} = cos^{2} 2θ^{q}_{23} = cos^{2} 2θ^{q}_{13} =
1, θ^{q}_{12}
= θ^{q}_{23} = θ^{q}_{13} =
0. (8)

This result coincides with empirical leading zero approximation for quark mixing angles. In contrast to neutrinos, there are no large observable white quark mixing angles. In the SM weak interactions participate not color quarks, but the white quarks determined by relation (7).

In the regular weak interaction approach, the mixing is between individual color quarks (three colors for every flavor). In our new approach, the weak interactions mixing is between already made ‘white’ quarks, namely between the sums of three colors in (7).

The different solutions (2) and (8) for neutrinos and quarks respectively mean a connection between quark and neutrino mixing angle benchmark approximations by one phenomenological equation

cos^2 (2θ12) + cos^2 (2θ23) + cos^2 (2θ13) = nc, (9)

where ‘nc’ is the number of primary particle
colors: nc = 1 for neutrinos and nc = 3 for the number of color quarks θ^{c}_{ik} within
the white quarks θ^{q}_{ik}
in (7).

In contrast
to neutrino one solution (2) with singled out θ^{n}13 angle of the bimaximal pattern, the universal color
quark mixing angles in Eq. (3) and (4) require the recognition of a new hidden ‘bimaximal
pattern mixing-angle symmetry’ that accompanies color symmetry in the hadron SM
weak interactions; Eq. (3) system is symmetric of three color quark angles θ^{c}_{12}, θ^{c}_{23} and θ^{c}_{13}.
Note that if one could measure separated color-quark mixing angles, the result
would be universal color quark angle ~ 27.4^{o} in (5), not the known observable
white angles θ^{q}_{ik}, which are given in Eq. (8) at leading
benchmark approximation.

The
quark CKM mixing angles in the low energy Lagrangian of the SM are in fact emergent
angles measured in experiments with hadron particles. The weak interaction SM particle
mixing angles are reinterpreted above – white quarks θ^{q}_{ik} (singlets),
not color quarks θ^{c}_{ik}
(triplets) – participate in the weak interactions.

The leading zero angle approximation of quark mixing in the SM weak interactions is obtained from a ‘new hidden mixing angle symmetry’ of the primary three possible bimaximal mixing patterns at fixed color (3) and (4). This appropriate symmetry is considered by the author for the first time. It leads to leading zero-angle approximation (and beyond, to realistic quark mixing angles via one-parameter epsilon-parametrization considered earlier) in an unbelievably simple semi-empirical methodology.

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