Exponential parameterization of neutrino mixing matrix with account for CP-violation data

The exponential parameterization of Pontecorvo-Maki-Nakagawa-Sakata mixing matrix for neutrino is discussed. The exponential form allows easy factorization and separate analysis of the CP-violating and Majorana terms. Based upon the recent experimental data on the neutrino mixing, the values for the exponential parameterization matrix for neutrinos are determined. The matrix entries for the pure rotational part in charge of the mixing without CP-violation are derived. The complementarity hypothesis for quarks and neutrinos is demonstrated. The comparison of the results, based on most recent and on old data is held. The CP-violating parameter value is estimated, based on the so far imprecise experimental indications, regarding CP-violation for neutrinos. The unitarity of the exponential parameterisation and the CP-violating term transform is confirmed. The transform of the neutrino mass state vector by the exponential matrix with account for CP-violation is shown.


Introduction
One of the paramount achievements of Physics of 20 th century was certainly the formulation of the Standard Model [1]- [3], which unifies the description of electromagnetic and weak interactions in one theory. Important role in the Standard Model is played by neutrinos. In the framework of the Standard Model neutrinos may have three flavours, matching three charged leptons, with which they interact by means of weak interaction. The proper states form full normalized orthogonal basis. Originally the Standard Model assumed massless neutrinos; later it was adopted to incorporate their mass. Existence of mass of neutrinos means the existence of at least three massive neutrino states ν 1 , ν 2 , ν 3 , and, also, it means the existence of the neutrino oscillations [4], i.e., neutrinos flavour change while they are propagating. Evidences for that were found in experiments for mixing of solar neutrinos [5], atmospheric neutrinos [6], and reactor neutrinos [7]. The phenomenon of neutrino mixing was predicted by Pontecorvo [8], [9].
where V PMNS is the unitary PMNS mixing matrix [10]. Note the analogy with the mixing of the low elements of left components of quark spinors  (3) is similar to that the CKM matrix plays in quark mixing [14], [16]- [19]. PMNS matrix is fully determined by four parameters: three mixing angles  12 ,  23 ,  13 and the phase δ, in charge of the CP-violation description [14]. Experimental values of the mixing angles are relatively well determined [14], [15], [20]- [22], [23]: Contrary to quark mixing angles, these are not small and the expansion in series of the only small parameter is not possible. Thus, there is no small parameter, like λ=sin Cabibbo ≈0.22 [24], for neutrino mixing. Experimentally determined absolute values for the elements of PMNS-matrix read as follows [14]: Moreover, there are indications, that the CP-violating phase may have non-zero value; moreover, very approximately it is supposed to be as big as   300  (see, [25], [26]).

Exponential mixing matrix
Exponential parameterization for neutrino mixing matrix was outlined in [27] and then in [28]; it is constructed similarly to that for quarks [29], [30]. Unitarity of the exponential mixing matrix is guaranteed by the anti-Hermitian form of the exponent (see [32] ), which depends on the mixing parameters λ i , and on the CP-violation phase δ СР . Note, that for δ СР =2πn we obtain simply a rotation matrix around the axis in space [30]. The matrix also becomes Real for δ СР =π(2n+1). The most important advantage of the exponential parameterization for the mixing matrix with respect to the commonly known standard parameterization is that the exponential parameterization allows easy separation of the contributions of the rotation part, the CP-violation and possible other terms in stand alone factors. This separation can be made in a variety of modes, which details we omit here; proper discussion was made, for example, in [30], [31]. The following unitary parameterization was proposed in [28]: where the rotation part is given by the Real exponential matrix and it contains imaginary component as follows: and the Majorana part depends on the Majorana phases in the exponential: .
The details of the splitting between CP-conserving and CP-violating terms in the above parameterization can be found in [28] (also compare with [30], [31]). The values of the Majorana phases α 1 and α 2 are at present undetermined; the value of the СР-violating phase δ СР can be figured out from the existing experimental indications and estimations (see, for example, [25], [26]).
The rotation matrix can be conveniently presented in the form of the rotation in the angle  around the axis, given by the vector with the following coordinates: Thus, formulae (17) are different from those in our parameterization (9). Omitting the Majorana part, the exponential parameterization (9) reads as follows: The values of the entries of the rotation matrix ij M can be derived from the following tensor identity (see [33]): where we denote ij  the Kronecker symbol, ijk  is the Levi-Civita symbol, n i are the components of the vector and ) , , (  is the rotation angle. The expressions, relating the entries of the standard parameterization c ij and s ij with n  and  , can be derived from (18), (19), (20) and (3), but they are very cumbersome and we omit them for brevity. From the experimental data [25] by using matrix equations and expansions we obtain for the rotation vector in 3D space the following coordinates: and for the rotation angle around this axis we obtain The precision of the above values is determined by the errors in the experimental data evaluation and is about of 4%. From (15) we note that quark n  and constitute the angle of 44°, as demonstrated in Fig. 1 neutrino n  . This fact is interesting itself and it is the demonstration of the so-called hypothesis of complementarity for neutrinos and quarks [34], [35], according to which, the rotation axes for quarks and neutrinos form the 45° angle; however, this last statement is rather an observation since it does not have solid theoretical fundaments and physical reasons.
Note, that the obtained value of 44 differs from 45 by 2%, which is within the margin of errors of the original experimental data sets, which determines the entries of the exponential mixing matrix and the rotation vectors directions.

Exponential parameterization and CP-violation.
Now let us take advantage of the possibility, given by the exponential parameterization, which allows us to factorize separately the contributions of the rotation, the CP-violation and the Majorana term. We can write the matrix product in the following form, which reminds the rotation in the angle 2Ω with the proper weights for each entry:     .
The above result (32)   We underline that the above transform by V MCP as well as the transform by purely rotational part P Rot , is unitary. It can be verified directly with the help of the Hermiteconjugated matrix: This ensures the unitarity of the whole exponential parameterization (9) of the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix:

Conclusions
The exponential parameterization of the mixing matrix for neutrinos is explored with account for the present experimental data. The proper entries of the exponential mixing matrix are determined; the CP-violating term in the exponential parameterization is estimated. Based upon the accuracy of the experimental data, the range of the values for the parameters of the neutrino mixing matrix is given. Without CP-violation the neutrino mixing represents in fact the geometric rotation in three-dimensional space. In this simple case mixing can be viewed as the rotation in the angle Φ around the axis in three-dimensional space. This interpretation follows straight from the structure of the exponential mixing matrix. Evidently, there is no mixing for Φ=0, when the mixing matrix without CP-violation reduces to the unit matrix I. Based  . The difference in their directions in 3D space is 11°.
Interestingly, the angle between the axes of rotation for quarks and neutrinos remains unchanged and equals ≈45°, despite the change of 11° in the direction of the neutrinos rotation axis, verified in the last 10 years. This demonstrates the hypothesis of complementarity for quarks and neutrinos [34] (23)) and we have estimated the entries of the CP-violating matrix in the exponential parameterization (see (29), (31)) from the current indications on CP-violation: δ CP -60. This value is quite approximate due to uncertain experimental data, regarding CP-violation for neutrinos. We calculated the CP-violating exponential matrix for the extremities of the range of δ CP from 0 to 180. The rotation angle 2 for the CP-violating matrix is rather small: 2 /  (33).
Exponential presentation of the mixing matrix and obtained with its help results and interpretations can be useful for treatment and analysis of new experimental data, regarding the neutrino oscillations in currently running experiments as well as in planned experimental projects.