**Flavor-Geometric Duality and
Pseudo-Euclidean Symmetry of Neutrino Mixing in Semi-Empirical Phenomenology**

**I.** Suggested early by the author, main equation of neutrino mixing semi-empirical
phenomenology is given by

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

where θik are the three neutrino mixing angles. The left side of this equation obeys the Euclidean 3-space geometric type SO(3)-symmetry if the sine-term on the right is neglected. But with the sine-term virtually present, the non-relativistic geometric symmetry of neutrino mixing angles is violated.

In our semi-empirical methodology based on Eq. (1), symmetry violation effects follow from that initial equation.

It should be underlined, Eq. (1) has a fully symmetric bimaximal solution

(2θ12) = (2θ23) = 90o, (2θ13) = 0. (2)

It is the only one solution that obey both the SO(3)-symmetric equation

cos^2 (2θ12) + cos^2 (2θ23) + cos^2 (2θ13) = 1 (3)

and the violating that symmetry Eq. (1). It means that even by violation of the nonrelativistic geometric-type symmetry the leading bimaximal approximation of neutrino mixing angles (2) obeys flavor-geometric duality – bimaximal equation (2) describes direction angles of a constant vector in outer Euclidean 3-space.

But after evidence of not zero reactor angle in 2011, the bimaximal solution (2) appears not exact, though it remains a good leading approximation.

The realistic neutrino mixing angles do obey SO(3)-symmetry violating Eq. (1) if reactor angle θ13 is not zero, in agreement with data θ13 =~ 8.5o.

**II.** A problem of the considered Eq. (1) is the origin of 3-symmetry
violating sine-term on the right. Here we attempt to answer it. By Einstein,
relativistic generalization of Euclidean 3-space symmetry should be
pseudo-Euclidean 4-space symmetry including time as the fourth dimension. The cause of violation of Euclidean 3-space symmetry of
Eq. (1) may be restoration of pseudo-Euclidean 4-space symmetry. A nearly exact
equation for neutrino mixing angles may exist in 4-space with pseudo-Euclidean
relativistic geometry symmetry.

It seems,**
**such realistic near accurate relation between neutrino mixing angles is Eq.
(1).

The relativistic condition motivates appearance of supported by data second term at right with very needed positive sign (negative sign - if that term is placed at left in Eq. (1), as required by relation to the imaginary fourth coordinate in Minkowski space).

A note. Weakly relativistic pseudo-Euclidean space-like region in the 4-dimensional spacetime is characterized by invariant interval s^2 (e.g. at one space dimension)

c(t2 – t1) << (x2 – x1), s^2 = {(x2 – x1)^2 - [c (t2 – t1)]^2}= ~ (x2 – x1)^2, (4)

where (t2 – t1) and (x2 – x1) are the differences in time and space coordinates between two events in spacetime. In that region, the invariant interval between two events is space-like and nearly equal to the 3-space squared distance between the events.

With pseudo-Euclidean flavor-geometry duality of neutrino mixing angles, the indicated by relations (4) conditions are represented in case of neutrino mixing angles by

s^2 = cos^2 (2 θ12) + cos^2 (2 θ23) + cos^2 (2 θ13) - sin^2 (2 θ13), (5)

sin^2 (2 θ13) << cos^2 (2 θ13), (6)

s^2 = cos^2 (2 θ12) + cos^2 (2 θ23) + cos^2 (2 θ13) = ~ 1, (7)

where unit 3-vector considered.

In spirit of flavor-geometric duality, realistic neutrino mixing angles are represented by a space-like 4-vector which does exist in Minkowski 4 space, namely

[cos(2 θ12), cos(2 θ23), cos(2 θ13), i sin(2 θ13)]. (8)

The space-like condition (7) follows from inequality (6) at small empirical reactor angle θ13 = ~ 8.5^o, that is in good agreement with Eq. (1) prediction.

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