**Neutrino Mixing CP-violating
Phenomenology with only Two Free Parameters**

The flavor-geometric semi-empirical phenomenology appears a powerful source of
new basic ideas in the Standard Model Flavor Sector.

Most recent new idea is CP-nonconservation as cause of deviation from exact
Euclidean 3-space geometric symmetry of neutrino bimaximal approximation. It is
represented by cos-squared Dirac CP-phase (CPph) [1. ResearchGate/L, 9/16],

cos^2(2θ12) +cos^2(2θ23) + cos^2(2θ13) = 1 + cos^2(CPph), (1)

so that maximal value (CPph) = pi/2 is related to the exact geometric bimaximal
(θ13 = 0) CP-conserving neutrino mixing approximation.

With the data world average central values of neutrino mixing angles e. g. [2]
F. Gapozzi et al, arXiv:1601.07777,

θ12 = (33.7±1.1)^o, θ23 = (40.7±1.7)^o, θ13 = (8.8 ± 0.4)^o, (2)

the neutrino Dirac CP-violating phase is given by

(CPph) = ~ ± 74^o. (3)

With negative sign it is in agreement with the preliminary experimental data by
the T2K Collaboration ICHEP2016.

It should be noted that there is semi-empirical evidence of a simple
complementary connection between the neutrino Dirac CP-violating phase and the
small reactor Theta13-angle

(CPph) = ± (pi/2 - 2θ13). (4)

By relation (4), the equation (1) describes now small deviation of neutrino
mixing geometric symmetry and CP-conservation of the bimaximal approximation in
a model with only three free parameters – neutrino mixing angles θ12, θ23, and
θ13,

cos^2(2θ12)
+ cos^2(2θ23) + cos^2(2θ13) = 1 + sin^2(2θ13). (5)

It seems, Eq (5) presents a serendipitous discovery in neutrino mixing
phenomenology with specially important inferences.

1) At θ13-angle = 0, geometric symmetry gets restored and bimaximal mixing
approximation appears.

2) At θ13-angle not zero, geometric symmetry of bimaximal mixing in equation
(5) is violated naturally by the small term sin^2(2θ13) << 1.

3) The extra term in (5), sin^2(2θ13), determines small simultaneous deviations
of neutrino mixing from bimaximal geometric and CP symmetries.

4) θ13-angle cannot be large; it must be << pi/4.

5) There is a condition to control whether the symmetry violating term
sin^2(2θ13) on the right side of Eq (5) is the appropriate one. To be sure,
this equation should be solved for the reactor angle θ13, expressed by the data
central values of solar θ12 ~ 34^o and atmospheric θ23 ~ 41^o angles and
compared with θ13-data value. The result

θ13 = ~ 8.3^o (6)

is in accurate agreement with experimental data value θ13 = (8.8 ± 0.4)^o. That
result is an important semi-empirical prediction of the neutrino reactor mixing
angle. It can be tested in the many ongoing experimental searches at reactor
(Daya Bay, Reno etc) and accelerator (T2K, Minos etc) facilities.

6) The CP-violating complex {sinθ13 exp(iCPph)} in the neutrino PMNS mixing
matrix gets expressed by the relation (4) through only one parameter θ13

{sinθ13 exp(iCPph)} = sinθ13 {sin(2θ13) ± i cos(2θ13)}, (7)

and at θ13 = 8.3^o it is given by

{sinθ13 exp(iCPph)} = ~ (0.04 ± i 0.14). (8)

Thus with relations (6)--(8) the new neutrino mixing CP-violating effective
PMNS matrix should contain not four and even not three, but only two free
empirical parameters, solar and atmospheric mixing angle ones.

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