800x600
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.
Comment
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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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