Bimaximal Pattern Links Color Quark and Neutrino Mixing-Angle Approximations    

                          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 n12, n23 and n13,

                                            cos2 n12 + cos2n23 + cos2n13 = 1.                          (1)

The bimaximal mixing pattern solution without free parameters (benchmark) is                                        cos2n12 = cos2n23 = 0, cos2 2θn13 = 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


                                 (cos2 c12; cos2c23; cos2 c13) = (0; 0; 1),

                                  (cos2 c12; cos2c23; cos2 c13) = (0; 1; 0),

                                  (cos2 c12; cos2c23; cos2 c13) = (1; 0; 0).                        (3)   


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

                                (cos2 2θc12; cos2 2θc23; cos2c13) = (1; 1; 1) / 3.                      (4)

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

                                                       θc12 = θc23 = θc13 = ~ 27.4o.                                     (5)

But we have to sum over the three colors in Eq. (4) because the individual color quark mixing angles θcik 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 (cos2 2θc12; cos2c23; cos2c13) = (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 θq ik,                                                                                                                                             

                                                               cos2 2θqik = ∑c cos2 cik.                                (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 θqik (7) in benchmark mixing without free parameters is given by

                  cos2 2θq12 = cos2q23 = cos2q13 = 1,   θq12 = θq23 = θq13 = 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 θcik within the white quarks θqik in (7).

 In contrast to neutrino one solution (2) with singled out θn13 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 θc12, θc23 and  θc13. Note that if one could measure separated color-quark mixing angles, the result would be universal color quark angle ~ 27.4o in (5), not the known observable white angles θqik, 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 θqik (singlets), not color quarks θcik (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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