# CP violation and quark-lepton complementarity of the neutrino mixing matrix

## Abstract

The comparative analysis of the neutrino mixing in the standard, cobimaximal and exponential parameterizations is performed. With the latest November 2018 data the logarithm of the mixing matrix is computed and the exact entries for the exponential matrix are obtained. The factorization of the real rotation and the CP violation in the exponential form of the mixing matrix is demonstrated. Quark-lepton complementarity hypothesis is reformulated, involving three mixing angles in the framework of the exponential parameterisation of the mixing matrix. It is shown that the cobimaximal parameterization, consistent with recent experimental data on neutrino mixing with the spread \(3\sigma \), can provide exact quark-lepton complementarity based on the data for all three mixing angles. The dependence of the CP violation degree on the parameterization parameters in the standard and the exponential forms is studied with the help of the Jarlskog invariant.

## 1 Introduction

**U**Pontecorvo–Maki–Nakagawa–Sakata (PMNS) [8]:

*i*is proportional to \({\vert }U_{\alpha i}{\vert }^{2}\). Neutrino mixing can be parameterized in several ways by different matrices. While the physical result obviously does not depend on the choice of the parameterization, the latter may be more or less useful and convenient for certain studies. In what follows we will consider three generations of Dirac neutrinos. Among many parameterizations, the most common is the standard parameterization matrix:

**V**. Mixing and CP violation for quarks are smaller than for neutrinos and the quark mixing matrix

**V**is rather close to the unitary matrix. Small deviations from the unitary matrix were described, for example, by Wolfenstein parameters \(\lambda ,A,\rho ,\eta \) [33]. Looking for some more global symmetry in mixing, researchers explored parallels between neutrino and quark mixings. Also for neutrinos there has been a proposal for the deviation parameters from the unitary matrix; these parameters were expressed via the Wolfenstein parameters through empiric relations [34] and the hypothesis of quark-lepton complementarity (QLC) and of self-lepton complementarity (SC) appeared [34, 35]. Quark-lepton complementarity [36, 37, 38] related the mixing angles of quarks to those of neutrinos. Relevant to it is the fact that without CP violation the real part of the mixing matrix remains and it represents the 3D rotation in real space.

In the present work we will consider the cobimaximal matrix (CBM), fit with the existing experimental data, and we will use the Jarlskog invariant to control the degree of the CP violation. We will use the matrix exponential to frame the CBM ansatz in the exponential parameterization and get the value of the rotation angle in space. With the help of the exponential parameterization we will show that neutrino mixing reduces to rotations in real space and further action of the CP violating matrix. We will redefine the quark-lepton complementarity by including all three mixing angles in it, and demonstrate that CBM parameterization provides *exact* quark-lepton complementarity in this sense.

## 2 Exponential parameterization of mixing matrix

**A**ensures that the transforms by the matrix \(\mathbf{U}_{\mathrm{exp}}\) are unitary. The exponential parameterization (7) has several advantages; one of them consists in that the mixing can be presented as the product of matrices in charge of the real rotation, \(\mathbf{P}_{{\mathrm{Rot}}} =e^{\mathbf{A}_{{\mathrm{Rot}}} }\), and the CP violation, \(\mathbf{P}_{CP} =e^{\mathbf{A}_{\mathbf{CP}} }\), which form new unitary parameterization

**A**in the exponent by taking matrix logarithm of the mixing matrix. This procedure can be executed in several different ways (see, for example, [51, 52]); computer programs for analytical calculations, such as

*Mathematica*, allow fast computations. According to the November 2018 data [39], the mixing angles values are as follows:

**A**is given by (10),

*exactly*reproduces the experimental data (5). Moreover, the CBM matrix (12) \(\mathbf{U}_{{\mathrm{best}~ \mathrm{fit}~ \mathrm{CBM}}} =\exp \left[ {\mathbf{A}_{CBM} } \right] \), where \(\mathbf{A}_{CBM}\) is given by (11), is within the allowance (6), determined by the experimental data. Matrix

**A**has zero trace \(\hbox {Tr}{} \mathbf{A}=0\) (same for \(\mathbf{A}_{CBM}\) etc). Apart small diagonal imaginary elements of the matrices

**A**(10) (and \(\mathbf{A}_{CBM}\) (11))

**U**.

*CP*-violating term \(\mathbf{A}_{CP}\) (17). For the recently established most probable value of the CP violating phase \(\delta _{CP} =215^{\circ }\) we therefore get

**H**is the traceless (\(3\times 3\)) Hermitian matrix with the normalization \({\mathrm{tr}}[\mathbf{H}^{2}]=2\), which formally frames neutrinos into SU(3) group. Although at this moment of time there are no obvious physical reasons to frame neutrinos in SU(3) group, the underlying mathematics determines the specific way to distinguish the group parameter \(\theta \) in (26):

*Mathematica*program, we obtain the following explicit matrix form for

**H**:

**H**is traceless and \(\det \left( \mathbf{H} \right) =-0.076\), the latter being the invariant for the SU(3) group. For the CBM parameterization, fitted with the available experimental data [see matrix (12)], we get the following matrix \(\mathbf{H}_{CBM}\):

**A**(23) in (7), note that the angle \(\theta \) is by definition (27) independent on the phases \(\delta _{1,2,3}\) in (23).

## 3 Jarlskog invariant

**U**the following equality holds: \(\mathbf{U}_{e3}^*\mathbf{U}_{\mu 3} =-\mathbf{U}_{e1}^*\mathbf{U}_{\mu 1} -\mathbf{U}_{e2}^*\mathbf{U}_{\mu 2} \), which means that the square of the triangle is \(S=\frac{1}{2}\left| {\hbox {Im}(\mathbf{U}_{e1} \mathbf{U}_{\mu 2} \mathbf{U}_{e2}^*\mathbf{U}_{\mu 1}^*)} \right| =\frac{1}{2}J\). The squares of other triangles can be calculated in similarly; they also equal

*J*/ 2, where

*J*is the Jarlskog invariant [56]. For the matrix

**U**(2) we therefore get the Jarlskog invariant in the following explicit form:

## 4 Complementarity of neutrino and quark mixing

The quark-neutrino complementarity (QLC) hypothesis was expressed in [35, 36, 57]; it is usually understood that the sum of the Cabibbo angle \(\theta _{12}\) for quarks and of the solar mixing angle \(\theta _{12}\) for neutrinos gives \(\pi /4\). Its extended formulation includes also “weak complementarity” relation for \(\theta _{23}\): \(\theta _{23 \nu } +\theta _{23 q} \cong 45^{\circ }\) [58, 59, 60, 61]. It would be natural to assume that the third QLC relation holds also for \(\theta _{13}\). However, it appears that \(\theta _{13 \nu } +\theta _{13 q} <10^{\circ }\), although the angles \(\theta _{13}\) are established with less precision that \(\theta _{12}\), \(\theta _{23}\). Despite the third possible QLC relation in the above formulation is not realized at all appearance, we have obtained another, more general relation for neutrinos and quarks, which involves all three mixing angles, \(\theta _{12}\), \(\theta _{23}\) and \(\theta _{13}\); it yields \(45^{\circ }\) angle for the rotation axes of neutrinos and quarks. This \(45^{\circ }\) space angle holds remarkably well through the years with different from each other data sets. We will demonstrate it in what follows.

*exactly*\(45.0^{\circ }\). So, the angle between the quark and neutrino rotation axes is relatively stable, \(\approx 45^{\circ }\), despite the values of \(\vec {n}\) и \(\Phi \) vary from year to year. For the CBM parameterization this angle \(=45.0^{\circ }\) (see Fig. 6, where all the axes are drawn with the opposite signs \(\vec {n}\rightarrow -\vec {n}\) to show them in first quadrants for clarity); this means

*exact*complementarity of mixing for neutrinos and quarks in CBM parameterization, based on the data for all three mixing angles. This fact in turn makes us look deeper for the underlying symmetries.

## 5 Study of the CP-violation in the exponential parameterization

The plots in Fig. 7 demonstrate that in the likely range of values \(\delta _{CP} \sim [180^{\circ }\)–\(360^{\circ }]\) the angle \(\theta \approx \Phi \) (27) for space rotations for neutrinos is in narrow range \([56^{\circ }\)–\(60^{\circ }]\). The angle \(\phi \) is at its maximum for \(\delta _{CP}=270^{\circ }\).

The relation between \(\delta _{1,2,3}\) values [see matrix \(\mathbf{A}_{1}\) (23)] and the CP violating phase \(\delta _{CP} \) is demonstrated in Fig. 8. The analysis of the behavior of the function \(\delta _1 (\delta _{CP} )\) in the left plot in Fig. 8 shows that its minimal value \(\delta _1^{\min } =147.2^{\circ }\) is reached for \(\delta _{CP} =302^{\circ }\). The phase \(\delta _{CP} =270^{\circ }\) corresponds \(\delta _1 \cong 151.6^{\circ }\) and currently likely value \(\delta _{CP} =215^{\circ }\) corresponds \(\delta _1 =167.9^{\circ }\). Due to large uncertainty for \(\delta _{CP} \), the range for \(\delta _{1}\) is \(\approx [145^{\circ }\)–\(180^{\circ }]\) (see Fig. 8). The behaviors \(\delta _2 (\delta _{CP} )\) and \(\delta _3 (\delta _{CP} )\) are qualitatively similar each other (see Fig. 8); they have their maximums at \(\delta _{CP} \approx 270^{\circ }\), but the range of values for \(\delta _{2}\) is \(\sim 3\) times more than that for \(\delta _{3}\) (see Fig. 8).

*J*. The matrix \(\mathbf{A}_{\mathrm{diag}}\) apparently reminds the Majorano term in the neutrino mixing, beyond the scope of the present paper; \(\mathbf{A}_{\mathrm{diag}}\) comes due to the CP violation. It follows from (10) that the entries of \(\mathbf{A}_{\mathrm{diag}}\) obey with precision relation (14): \(\alpha _1 \cong \alpha _3 \cong -\alpha _2 /2\). Let us then consider the following parameterization for \(\mathbf{A}_{\mathrm{diag}}\):

*J*as the function of the parameter \(\alpha \) in (48), where \(\alpha \) varies in the interval [\(0, \alpha _{2}\)].

The dependence \(J\left( \alpha \right) \) is linear. Moreover, it follows from Fig. 9 that even if we assume \(\mathbf{A}_{{\mathrm{diag}}} =0\), then the absolute value of the Jarlskog invariant changes insignificantly, from 0.01914 to 0.01875. Thus, it is not \(\mathbf{A}_{\mathrm{diag}}\), which determines most of the CP violation, but \(\mathbf{A}_{\mathbf{CP}} \) matrix (17), i.e. the imaginary part of the matrix **A**, except for the diagonal \(\mathbf{A}_{\mathrm{diag}}\). This justifies the approximate equality in (15). The matrices of the real rotation \(\mathbf{A}_{\mathrm{Rot}}\) (16) and of the CP violation \(\mathbf{A}_{\mathbf{CP}} \) (17) together constitute the most important part of the mixing matrix \(\mathbf{A}_{1}=\mathbf{A}_{\mathrm{Rot}}+\mathbf{A}_{CP}\) (15). The CP violation is described primarily by \(\delta _{1}\); small contribution comes from \(\delta _{2}\); the contribution of \(\delta _{3}\) in the exponential matrix (23) is negligible.

## 6 Results and discussion

With the help of the exponential form of the mixing matrix for neutrinos we analyzed the mixing data for November 2018. We compared it with that for January 2018 and for the year 2016. The exponential matrix \(\mathbf{A}=\mathbf{A}_{\mathrm{Rot}} +\mathbf{A}_{CP} +\mathbf{A}_{{\mathrm{diag}}} \) allows separation of the real part \(\mathbf{A}_{\mathrm{Rot}}\) in charge of the rotation, the imaginary non-diagonal part \(\mathbf{A}_{\mathrm{CP}}\) responsible for most of the CP violation, and small imaginary diagonal part \(\mathbf{A}_{\mathrm{diag}}\), whose trace is zero. We factorized in the exponential form \(\mathbf{U}_{\exp } =\exp \mathbf{A}\) the contributions of the rotation \(\mathbf{P}_{\mathrm{Rot}} =e^{\mathbf{A}_{\mathrm{Rot}} }\) and of the CP violation \(\mathbf{P}_{CP} =e^{\mathbf{A}_{CP} }\) in a new unitary matrix \({\tilde{\mathbf{U}}}=\mathbf{P}_{\mathrm{Rot}} \mathbf{P}_{CP} \) compliant with \(3\sigma \) spread of Best Fit. The commutators of \(\mathbf{A}_{\mathrm{Rot}}\), \(\mathbf{A}_{\mathrm{CP}}\), \(\mathbf{A}_{\mathrm{diag}}\) with each other are small, \({\tilde{\mathbf{U}}}\cong \mathbf{U}\) with the accuracy of the order \(10^{-2}\). The exponential argument \(\mathbf{A}_{CP}\) is imaginary and has zero diagonal entries; this matrix describes the CP violation quite well. Additional account for the small imaginary diagonal matrix \(\mathbf{A}_{{\mathrm{diag}}} =i{\mathrm{diag}}\left\{ {\alpha _1 ,\;\alpha _2 ,\;\alpha _3 } \right\} \) provides *exact* match with Best Fit; however, \(\mathbf{A}_{\mathrm{diag}}\) entries, \(\alpha _2 /2\cong -\alpha _1 \cong -\alpha _3 \approx 10^{-2}\), can be omitted without significant sacrifice of precision, when \({\tilde{\mathbf{U}}}\) is compared with Best Fit. Moreover, the latest experimental data yields the smallest ever values for the entries \(\alpha _{1,2,3}\).

We compared the rotation vectors and angles for neutrinos and quarks in the exponential parameterizations fitted with the data for 2016–2018. We have found that the space angle between the rotation axes for quarks and neutrinos remains \(\approx 45^{\circ }\), despite the data varied from year to year. This result has high statistical value since it involves the data for all three mixing angles and thus it can be viewed as an alternative formulation of quark-lepton complementarity hypothesis (QLC), usually based on the relations for \(\theta _{12}\) and \(\theta _{23}\) in its extended formulation. We have demonstrated that the cobimaximal parameterization fits the experimental range \(3\sigma \) and the angle between the rotation axes for quarks and neutrinos in the CBM parameterization equals *exactly* \(45^{\circ }\) (see Fig. 6). Other angles in the CBM parameterization compliant with Best Fit are \(\theta _{12} \cong 33.51^{\circ }\), \(\theta _{13} \cong 8.695^{\circ }\) and it is assumed that \(\theta _{23} =45^{\circ }\), \(\delta _{CP} =-90^{\circ }\). The resulting value of the Jarlskog invariant *J*, which measures the degree of the CP violation, is \(J=-0.034\), which is close to the maximal value of \(\left| {J_{\max } } \right| =0.035\). Thus, the cobimaximal parameterization not only fits the experimental data spread \(3\sigma \), but it provides *exact* complementarity for neutrinos and quarks in the above defined sense. For smaller CP violating phase, \(\delta _{CP}=215^{\circ }\), we get \(J_\nu \cong -0.019\) and the complementarity is satisfied approximately.

Using exponential parameterization of the neutrino mixing, we framed it into SU(3) group [see (31)–(35)] and obtained the value for the rotation angle \(\theta _{\nu 2018}^{Nov} ={58.24^{\circ }}_{-8.62^{\circ }}^{+7.25^{\circ }} ,\) best fitted with the experimental data spread \(3\sigma \). This angle is very close to the rotation angle \(\Phi \) around the axis \(\vec {n}\) in 3D space for the rotation matrix \(\mathbf{P}_{\mathrm{Rot}}\); the small difference is due to the way this rotation matrix was distinguished. We have obtained the relations of \(\theta \) and \(\phi \) with the CP violating phase \(\delta _{CP} \) in the standard parameterization (see Fig. 7). The angle \(\theta \) by definition is independent from \(\delta _{1,2,3}\) in the exponential parameterization. We have shown that in the likely wide range of values \(\delta _{CP} \sim [180^{\circ }\)–\(360^{\circ }]\) the angle \(\theta \) for neutrinos varies in the narrow range: \(\theta \sim [56^{\circ }\)–\(60^{\circ }]\).

The correspondence between the CP violation in the standard and in the exponential parameterizations is established. For the exponential parameterization the hierarchy of the complex entries (1,3), (2,3), (3,2) of the matrix exponential with the phases \(\delta _1 \approx 150^{\circ }\)–\(180^{\circ }\), \(\delta _2 \approx 4^{\circ }\), \(\delta _3 \approx 1^{\circ }\) is established.

With the help of the Jarlskog invariant we measured the CP violation in the framework of the standard, cobimaximal and exponential parameterizations (see Figs. 2, 3, 4, 5, 9, 10). The major variation of the Jarlskog invariant in the exponential parameterization is induced by \(\delta _1 \). For the currently likely value \(\delta _{CP} \approx 215^{\circ }\), the exponential parameter \(\delta _{1}\), which mainly determines the CP violation, is in the middle of its possible range of values. The parameter \(\delta _2 \) and especially \(\delta _3 \) weakly influence the CP violation and depend on \(\delta _{CP}\) in qualitatively similar to each other way (see Fig. 8).

## 7 Conclusions

- 1.
The exponential parameterization \(\mathbf{U}_{\exp } =\exp \mathbf{A}\) of the mixing matrix allows explicit factorization of pure rotation in 3D space, \(\mathbf{P}_{\mathrm{Rot}} =e^{\mathbf{A}_{\mathrm{Rot}} }\), \(\mathbf{A}_{\mathrm{Rot}} \in {\mathrm{Reals}}\), and the CP violation, \(\mathbf{P}_{CP} =e^{\mathbf{A}_{\mathbf{CP}} }\), \(\mathbf{A}_{{\mathrm{CP}}} \in {\mathrm{Imaginaries}}\); the small diagonal elements in the latter matrix can be neglected. The result is consistent with the Best Fit data.

- 2.
The axes of real rotation for neutrinos and quarks in the exponential parameterization constitute with each other the angle \(\approx 45^{\circ }\). This result is based on the data for all three mixing angles \(\theta _{12,23,13}\) and therefore it has higher statistical value than common quark-lepton complementarity (QLC). This can be viewed as an alternative formulation of QLC; it is more general than the common formulation, which involves just the angles \(\theta _{12}\) and in its extended form also \(\theta _{23}\).

- 3.
The quark-neutrino complementarity, understood in the above redefined way, holds quite well through the years 2016–2018, despite the data for the rotation angles and axes varied.

- 4.
In the cobimaximal parameterization (CBM), \(\theta _{23} =\pi /4\), \(\delta _{CP} =-\pi /2\), fitted with most recent experimental data 2018, the angles of rotation for quarks and neutrinos constitute

*exactly*\(45^{\circ }\). Thus QLC is satisfied exactly for it. - 5.
In the exponential parameterization the neutrino mixing can be formally framed in SU(3) group, whose parameter \(\theta \) coincides with the rotation angle \(\Phi \) around the fixed axis in 3D space with the accuracy \(0.5^{\circ }\). The angle \(\theta \) does not depend on the values of \(\delta _{1,2,3}\) in the exponential parameterization; dependently on the Best Fit data with \(\delta _{CP}\) in the range [180\(^{\circ }\)–360\(^{\circ }\)], it varies in very narrow range around \(58^{\circ }\pm 2^{\circ }\).

- 6.
Using Jarlskog invariant we have shown that in the exponential parameterization the value of \(\delta _{1}\) determines most of the CP violation. While the CP violation for \(\nu _{e}\) mixing comes from \(\nu _{3}\) mass state, the degree of CP violation behaves similarly for \(\upnu _{\upmu ,\uptau }\) flavor states and comes from \(\upnu _{1,2}\) mass states. This might indicate possible symmetry of the CP violation in \(\nu _{\mu }\) and in \(\nu _{\tau }\) neutrino mixing.

## Notes

### Acknowledgements

The authors are grateful to Prof. A. V. Borisov for useful advises and discussions.

## References

- 1.S. Weinberg, Phys. Rev. Lett.
**19**, 1264 (1967)ADSCrossRefGoogle Scholar - 2.A. Salam,
*Elementary Particle Theory*, ed. by N. Svartholm (Almquist Forlag AB, 1968)Google Scholar - 3.S.L. Glashow, Nucl. Phys.
**22**, 579 (1961)CrossRefGoogle Scholar - 4.B. Pontecorvo, Zh. Eksp. Teor. Fiz.
**33**, 549 (1957)Google Scholar - 5.B. Pontecorvo, Zh. Eksp. Teor. Fiz.
**53**, 1717 (1967)Google Scholar - 6.M. Gell-Mann, P. Ramond, R. Slansky, in
*Sanibel Talk, CALT-68-709, February 1979, and in Supergravity*(North-Holland, Amsterdam, 1979)Google Scholar - 7.T. Fukuyama, H. Nishiura, Mass matrix of majorana neutrinos. arXiv:hep-ph/9702253
- 8.Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys.
**28**, 870 (1962)ADSCrossRefGoogle Scholar - 9.Y. Zhang, et al., Phys. Rev. D
**86**, 093019 (2012). arXiv:1211.3198 [hep-ph] - 10.P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B
**530**, 167 (2002). arXiv:hep-ph/0202074 ADSCrossRefGoogle Scholar - 11.
- 12.L. Merlo, J. Phys. Conf. Ser.
**335**, 012049 (2011)CrossRefGoogle Scholar - 13.S.F. King, Models of neutrino mass, mixing and CP-violation. J. Phys. G Nucl. Part. Phys.
**42**, 123001 (2015). arXiv:1510.02091 ADSCrossRefGoogle Scholar - 14.E. Ma, G. Rajasekaran, Phys. Rev. D
**64**, 113012 (2001). arXiv:hep-ph/0106291 ADSCrossRefGoogle Scholar - 15.K.S. Babu, E. Ma, J.W.F. Valle, Phys. Lett. B
**552**, 207–213 (2003). arXiv:hep-ph/0206292 ADSCrossRefGoogle Scholar - 16.G. Altarelli, F. Feruglio, Nucl. Phys. B
**720**, 64–88 (2005). arXiv:hep-ph/0504165 ADSCrossRefGoogle Scholar - 17.G. Altarelli, F. Feruglio, Nucl. Phys. B
**741**, 215–235 (2006). arXiv:hep-ph/0512103 ADSCrossRefGoogle Scholar - 18.A. Y. Smirnov, Talk given at DISCRETE’10—Symposium on Prospects in the Physics of Discrete SymmetriesGoogle Scholar
- 19.F. Feruglio, C. Hagedorn, Y. Lin, L. Merlo, Nucl. Phys. B
**775**, 120–142 (2007). arXiv:hep-ph/0702194 ADSCrossRefGoogle Scholar - 20.F. Bazzocchi, L. Merlo, S. Morisi, Nucl. Phys. B
**816**, 204–226 (2009). arXiv:0901.2086 ADSCrossRefGoogle Scholar - 21.F. Bazzocchi, L. Merlo, S. Morisi, Phys. Rev. D
**80**, 053003 (2009). arXiv:0902.2849 ADSCrossRefGoogle Scholar - 22.I. de Medeiros Varzielas, G.G. Ross, Nucl. Phys. B
**733**, 31–47 (2006). arXiv:hep-ph/0507176 - 23.I. de Medeiros Varzielas, S. F. King, G.G. Ross, Phys. Lett. B
**644**, 153–157 (2007). arXiv:hep-ph/0512313 - 24.E. Ma, Phys. Rev. D
**92**, 051301(R) (2015)ADSCrossRefGoogle Scholar - 25.E. Ma, Phys. Lett. B
**752**, 198–200 (2016)ADSMathSciNetCrossRefGoogle Scholar - 26.E. Ma, G. Rajasekaran, EPL 119(3), 31001 (2017)Google Scholar
- 27.E. Ma, Phys. Lett. B
**777**, 332–334 (2018)ADSCrossRefGoogle Scholar - 28.A. Damanik, J. Phys. Conf. Ser.
**909**, 012024 (2017)CrossRefGoogle Scholar - 29.R.N. Mohapatra, C.C. Nishi, Phys. Rev. D
**86**, 073007 (2012)ADSCrossRefGoogle Scholar - 30.P. Chen, C.-Y. Yao, G.-J. Ding, Phys. Rev. D
**92**, 073002 (2015)ADSCrossRefGoogle Scholar - 31.A.S. Joshipura, K.M. Patel, Phys. Lett. B
**749**, 159 (2015)ADSCrossRefGoogle Scholar - 32.H.-J. He, W. Rodejohann, X.-J. Xu,. arXiv:1507.03541 [hep-ph]
- 33.L. Wolfenstein, Phys. Rev. Lett.
**51**(21), 1945 (1983)ADSCrossRefGoogle Scholar - 34.N. Li, B.-Q. Ma, Unified parametrization of quark and lepton mixing matrices. Phys. Rev. D
**71**, 097301 (2005). arXiv:hep-ph/0501226 - 35.H. Minakata, AYu. Smirnov, Phys. Rev. D
**70**, 073009 (2004)ADSCrossRefGoogle Scholar - 36.M. Raidal, Phys. Rev. Lett.
**93**, 161801 (2004)ADSCrossRefGoogle Scholar - 37.X. Zhang, B.-Q. Ma, Phys. Lett. B
**710**, 630 (2012)ADSCrossRefGoogle Scholar - 38.Y. Zhang, X. Zhang, B.-Q. Ma, Phys. Rev. D
**86**, 093019 (2012)ADSCrossRefGoogle Scholar - 39.I. Esteban, M.C. Gonzalez-Garcia, et al., NuFIT 4.0 (2018). http://www.nu-fit.org
- 40.I. Esteban, M.C. Gonzalez-Garcia, et al., NuFIT 3.2 (2018). http://www.nu-fit.org
- 41.M. Gonzalez-Garcia, M. Maltoni, T. Schwetz, http://www.nu.fit.org
- 42.M. Tanabashi, et al. (Particle Data Group), Phys. Rev. D
**98**, 030001 (2018)Google Scholar - 43.A. Strumia, F. Vissani, Neutrino masses and mixings and... (2006). arXiv:hep-ph/0606054
- 44.G. Dattoli, K. Zhukovsky, Eur. Phys. J. C
**50**, 817 (2007)ADSCrossRefGoogle Scholar - 45.G. Datolli, K.V. Zhukovsky, Eur. Phys. J. C
**55**, 547 (2008)ADSCrossRefGoogle Scholar - 46.G. Dattoli, K. Zhukovsky, Eur. Phys. J. C
**52**, 591 (2007)ADSCrossRefGoogle Scholar - 47.K. Zhukovsky, F. Melazzini, Eur. Phys. J. C
**76**, 462 (2016)ADSCrossRefGoogle Scholar - 48.K. Zhukovsky, A. Borisov, Eur. Phys. J. C
**76**, 637 (2016)ADSCrossRefGoogle Scholar - 49.K.V. Zhukovsky, Moscow Univ. Phys. Bull.
**72**(5), 433–440 (2017)ADSCrossRefGoogle Scholar - 50.K.V. Zhukovsky, Phys. Atom. Nucl.
**80**(4), 690–698 (2017)ADSCrossRefGoogle Scholar - 51.A. Wouk, J. Math. Anal. Appl.
**11**, 131 (1965)MathSciNetCrossRefGoogle Scholar - 52.T.A. Loring, Computing a logarithm of a unitary matrix with general spectrum. Numer. Linear Algebra Appl.
**21**, 744 (2014)MathSciNetCrossRefGoogle Scholar - 53.T.L. Curtright, C.K. Zachos, Rep. Math. Anal. Appl.
**76**, 401 (2015)ADSGoogle Scholar - 54.K.V. Zhukovsky, D. Dattoli, Phys. Atom. Nucl.
**71**(10), 1807–1812 (2008)ADSCrossRefGoogle Scholar - 55.F. Ayres Jr.,
*Schaum’s Outline of Theory and Problems of Matrices*(Schaum, New York, 1962), p. 181Google Scholar - 56.C. Jarlskog, Phys. Rev. Lett.
**55**, 1039 (1985)ADSCrossRefGoogle Scholar - 57.H. Minakata, A.Yu. Smirnov, Phys. Rev. D
**70**, (2004). https://doi.org/10.1103/PhysRevD.70.073009 - 58.F. González Canales, A. Mondragón, On quark-lepton complementarity. arXiv:hep-ph/0606175v1
- 59.F. Gonzalez Canales, A. Mondragon, Universal mass texture and quark-lepton complementarity. J. Phys. Conf. Ser.
**171**, 012063 (2009)Google Scholar - 60.G. Sharma, B.C. Chauhan, Quark-lepton complementarity predictions for \(\theta _{23}^{{\rm pmns}}\) and CP violation. JHEP
**07**, 075 (2016)Google Scholar - 61.D. Meloni, Fron. Phys.
**5**, Article 43 (2017). https://doi.org/10.3389/fphy.2017.00043

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}.