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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 2θn12, 2θn23 and 2θn13,
cos2 2θn12 + cos2
2θn23 + cos2
2θn13 = 1. (1)
The bimaximal mixing pattern solution
without free parameters (benchmark) is cos2
2θn12 = cos2 2θn23 = 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 2θc12; cos2 2θc23; cos2
2θc13) = (0; 0; 1),
(cos2 2θc12; cos2
2θc23; cos2 2θc13)
= (0; 1; 0),
(cos2
2θc12; cos2
2θc23; cos2 2θc13)
= (1; 0; 0). (3)
The sum of
three solutions for particular color `c` in Eq. (3) is given by
(cos2 2θc12; cos2
2θc23; cos2
2θc13) = (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; cos2 2θc23; cos2
2θc13)
= (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
2θcik. (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 = cos2 2θq23 = cos2 2θq13 =
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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