Dirac CP-violating phase from deviation of neutrino mixing from geometric bimaximal approximation 

   Since by evidence the leading bimaximal approximation to neutrino mixing obeys the 3-space Euclidean geometry Metric Equation, its necessary and sufficient condition is Theta_13 = 0.

After the experimental Theta_13 results of Daya Bay and others the world mean values for neutrino mixing angles are

    Theta_12 = (33.7+-1.1)^o, Theta_23 = (40.7+-1.7)^o, Theta_13 = (8.8+-0.4)^o,   (1)

as estimated by e. g. F. Gapozzi et al, arXiv:1601.07777, and it appears a definite though small violation of real neutrino mixing geometric pattern.

   By our suggestion, this violation of neutrino mixing from geometric pattern is a basic phenomenon related to new neutrino physics of CP-violation absent in Euclidean geometry. Several particular model-realizations of that hypothesis are considered with one special common feature – bimaximal neutrino mixing approximation expressed by Theta_13 = 0, is CP-conserving geometric mixing pattern with maximal value of Dirac CP-phase (CPph)-bm = pi/2; realistic neutrino mixing deviations from that approximation are determinate by deviation of CP-violating phase from maximal value.                       

1) Realistic neutrino mixing angles obey not exact, but slightly modified geometric equation

  cos^2 (2Theta_12) + cos^2 (2Theta_23) + cos^2 (2Theta_13) = 1 + cos^2 (CPph).    (2)

The additional not geometric CP term at right in this equation is needed to satisfy condition CPph = pi/2 for geometric CP-conserving bimaximal mixing. At realistic neutrino mixing angles (1), e. g. for the central angle-values, it determines the CP-violating phase

                                                            CPph = ~ 74^o.                                                   (3)

2) The flavor-geometric idea for neutrino mixing prompted a relation between neutrino mixing angles and neutrino masses m1 < m2 < m3. But such relation cannot be exact because of the violation of mixing angle geometric pattern in (2). Slightly modified mass relations are given by

    cos^2 (2Theta_23) = m1/(m1+m2+m3), cos^2 (2Theta_12) = m2/(m1+m2+m3),

                   cos^2 (2Theta_13) - cos^2 (CPph) = m3/(m1+m2+m3).                            (4)

For definiteness we consider here the special simple and possible by known data at 1 --2 sigma case of maximal atmospheric neutrino mixing cos^2 (2Theta_23) = 0. By well known neutrino oscillation mass squared differences it determines the complete specter of absolute neutrino masses

                                            m1 = ~ 0, m2 = ~ 0.009 eV, m3 = ~ 0.05 eV.                       (5)

The second equation in (4) states normal mass hierarchy. At CPph = pi/2 the third equation in (4) is related to bimaximal mixing approximation. At realistic neutrino mixing angles (1) that equation (4) determines realistic CP-violating phase

                                                            CPph = ~ 76^o.                                                   (6)

3) To summarize, the above two model-examples, and some not considered here others, prompted the most economical relation between the neutrino Theta_13 mixing angle and Dirac neutrino CP-violating phase

                                               CPph = + - (pi/2 – Theta_13),                                        (7)

 which at the example of Theta_13 = 8.8^o predicts the magnitude of Dirac CP-phase

                                                             CPph = ~ + - 81^o.                                             (8)

By the experimental T2K indications ICHEP 2016 the mines sign in (8), also in (3) and (6), is more favorable.


New T2K data indications [H. A. Tanaka, talk at Neutrino 2016, 4-9 July, London] -- Theta_13 = ~ 11.9^o -- in contrast to reactor data, are compatible with exact geometric pattern for neutrino mixing angles in both above definition (2) and (4) without the additional CP-violating terms. If that T2K indications being supported by further experimental data, it would be a remarkable confirmation of complete geometric Euclidean 3-space origin of the SM hierarchies.






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