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

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## Abstract

The exponential parameterization of the Pontecorvo–Maki–Nakagawa–Sakata mixing matrix for neutrinos 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. A comparison of the results based on most recent and on old data is presented. 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 parameterization and the CP-violating term transform is confirmed. The transform of the neutrino mass state vector by the exponential matrix with account of CP violation is shown.

## Keywords

Neutrino Oscillation Sterile Neutrino Rotation Vector Dirac Neutrino Majorana Phase## 1 Introduction

**W**-boson can couple to any mass state of charged leptons \(({\varvec{e}},{\varvec{\mu }},{\varvec{\tau }}\)) with any mass state of neutrino. For example, \(\mathbf{V}_{\varvec{\alpha i}}\) would be the amplitude of the boson \(\mathbf{W}^{+}\) decay in a pair of lepton type \({\varvec{\alpha }}\) and neutrino type \({\varvec{i}}\) and, hence, the production of the lepton \({\varvec{\alpha }}\) and neutrino state \({\varvec{\alpha }}\) implies that all neutrino mass states participate in it in a superposition. In the following we do not consider a sterile neutrino, [12, 13, 14], which does not interact with

**W**- and

**Z**-bosons. Thus, we end up with a unitary 3\(\,\times \,\)3 mixing matrix

**V**, factorized by the matrix:

**V**in the parameterization (3) is similar to the one that the CKM matrix plays in quark mixing [15, 17, 18, 19, 20]. The PMNS matrix is fully determined by four parameters: three mixing angles \(\theta _{12}\), \(\theta _{23}\), \(\theta _{13}\), and the phase \(\delta \), in charge of the CP violation description [15]. The experimental values of the mixing angles are relatively well determined [15, 16, 21, 22, 23, 24]:

## 2 Exponential mixing matrix

This fact is interesting itself and it is the demonstration of the so-called hypothesis of complementarity for neutrinos and quarks [35, 36], according to which the rotation axes for quarks and neutrinos form the 45\(^{\circ }\) angle; however, this last statement is rather an observation since it does not have solid theoretical foundation and there are no physical reasons. Note that the obtained value of 44\(^{\circ }\) differs from 45\(^{\circ }\) by \(\approx \)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.

## 3 Exponential parameterization and CP violation

**I**:

## 4 Conclusions

The exponential parameterization of the mixing matrix for neutrinos is explored with account of 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 by the angle \(\Phi \) around the axis in three-dimensional space. This interpretation follows straightforwardly from the structure of the exponential mixing matrix. Evidently, there is no mixing for \(\Phi =0\), when the mixing matrix without CP violation reduces to the unit matrix, **I**. Based upon the recent data, we have obtained the value for the rotation angle \(\Phi \cong 49.8^{\circ }\) and the coordinates of the rotation vector \({\mathbf{\vec {n}}}=(\hbox {0.702, } \hbox {0.394, } \hbox {0.593})\). This value of the rotation angle is somewhat smaller than that based upon the tribimaximal parameterization: \(\Phi _\mathrm{TBM} \cong 56.6^{\circ }\). Moreover, the direction of the rotation vector differs from that of the vector for the TBM matrix: \({\mathbf{\vec {n}}}_\mathrm{TBM} =(\hbox {0.7858, } \hbox {0.2235, } \hbox {0.5777})\). The difference in their directions in 3D space is 11\(^{\circ }\). Interestingly, the angle between the axes of rotation for quarks and neutrinos remains unchanged and equals \(\approx \)45\(^{\circ }\), despite the change of 11\(^{\circ }\) in the direction of the neutrinos rotation axis, verified in the last 10 years. This demonstrates the hypothesis of complementarity for quarks and neutrinos [35, 36].

The exponential parameterization allows factorization of the CP and the Majorana contributions and is evidence that the CP-term can also be viewed as a sort of rotation with different weights for the matrix entries. We have calculated the entries of the rotational mixing matrix \(\lambda \cong 0.516, \mu \cong -0.342\), \(\nu \cong -0.611\) (see (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: \(\delta _\mathrm{CP}\sim \)–60\(^{\circ }\). 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 \(\delta _\mathrm{CP}\) from 0\(^{\circ }\) to 180\(^{\circ }\). The rotation angle 2\(\Omega \) for the CP-violating matrix is rather small: \(\Omega =\mu /2\). By means of the exponential parameterization one can easily transform the neutrino state vector, distinguishing the CP-violating terms for each type of neutrino. In the case of Dirac neutrinos and if \(\delta _\mathrm{CP} =\pm 2\pi n\), we get pure rotation, since \(\mathbf{P}_{\mathbf{CP}} =\mathbf{I}\). If \(\delta _\mathrm{CP}=\pi \pm 2{\pi n}\), then we end with the real \(\mathbf{P}_{\mathbf{CP}}\) matrix for the CP-violating term, which means the absence of CP violation in this case. For Majorana neutrinos the mass state vector \({\varvec{\nu _1}}\), \({\varvec{\nu _2}}\), \({\varvec{\nu _3}}\) is transformed with complex weights, as demonstrated in (33).

We addressed an exponential presentation of the mixing matrix and obtained with its help results and interpretations that can be useful for the treatment and analysis of new experimental data, regarding the neutrino oscillations in currently running experiments as well as in planned experimental projects.

## Notes

### Acknowledgments

We would like to thank professor A.V. Borisov from the Moscow State University, Russia, for his useful suggestions, fruitful discussions, and important advice.

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