## Abstract

We discuss the neutrino mass matrix based on the Occam’s-razor approach in the framework of the seesaw mechanism. We impose four zeros in the Dirac neutrino mass matrix, which give the minimum number of parameters needed for the observed neutrino masses and lepton mixing angles, while the charged lepton mass matrix and the right-handed Majorana neutrino mass matrix are taken as real diagonal ones. The low-energy neutrino mass matrix has only seven physical parameters. We show successful predictions for the mixing angle $$\theta_{13}$$ and the CP-violating phase $$\delta_{\rm CP}$$ with the normal mass hierarchy of neutrinos by using the experimental data on the neutrino mass-squared differences, the mixing angles $$\theta_{12}$$ and $$\theta_{23}$$. The most favored region of $$\sin\theta_{13}$$ is around $$0.13$$$$0.15$$, which is completely consistent with the observed value. The CP-violating phase $$\delta_{\rm CP}$$ is favored to be close to $$\pm \pi/2$$. We also discuss the Majorana phases as well as the effective neutrino mass for the neutrinoless double-beta decay $$m_{ee}$$, which is around $$7$$$$8$$ meV. It is extremely remarkable that we can perform a “complete experiment" to determine the low-energy neutrino mass matrix, since we have only seven physical parameters in the neutrino mass matrix. In particular, two CP-violating phases in the neutrino mass matrix are directly given by two CP-violating phases at high energy. Thus, assuming leptogenesis, we can determine the sign of the cosmic baryon in the universe from the low-energy experiments for the neutrino mass matrix.

## Introduction

The standard model has been well established by the discovery of the Higgs boson. However, the origin and structure of quark and lepton flavors are still unknown in spite of the remarkable success of the standard model. Therefore, the underlying physics of the masses and mixing of quarks and leptons is one of the fundamental problems in particle physics. A number of models have been proposed based on flavor symmetries, but there is no convincing model at present.

On the other hand, the neutrino oscillation experiments are moving onwards to reveal the CP violation in the lepton sector. The T2K experiment has confirmed the neutrino oscillation in $$\nu_\mu \to\nu_e$$ appearance events [1], which may provide us with new information on the CP violation in the lepton sector. Recent NO$$\nu$$A experimental data [2] also indicate CP violation in the neutrino oscillation. Thus, various pieces of information are now available to discuss Yukawa matrices in the lepton sector.

Recently, the Occam’s-razor approach was proposed to investigate the neutrino mass matrix [3] in the case of two heavy right-handed neutrinos. Because of tight constraints, it was shown that only the inverted mass hierarchy for the neutrinos is consistent with the present experimental data. The quark sector was also successfully discussed in this approach [4] and we found a nice prediction of the Cabibbo angle, for instance.

In this paper, we discuss the seesaw mechanism [5,6] (see also Ref. [7]) with three right-handed heavy Majorana neutrinos, predicting the normal mass hierarchy of the light neutrinos. We impose four zeros in the Dirac neutrino mass matrix, which give the minimum number of parameters needed for the observed neutrino masses and lepton mixing angles in the normal mass hierarchy of neutrinos [8,9]. Here, the charged lepton mass matrix and the right-handed Majorana neutrino mass matrix are taken to be real diagonal ones. The Dirac neutrino mass matrix is given with five complex parameters. Among them, three phases are removed by the phase redefinition of the three left-handed neutrino fields. The remaining two phases are removed by the field–phase rotation of the right-handed neutrinos. Instead, these two phases appear in the right-handed Majorana neutrino mass matrix. After integrating the heavy right-handed neutrinos, we obtain a mass matrix of the light neutrino, which contains five real parameters and two CP-violating phases.

In the present Occam’s-razor approach with the four zeros of the Dirac neutrino mass matrix, we show successful predictions of the mixing angle $$\theta_{13}$$ and the CP-violating phase $$\delta_{\rm CP}$$ with the normal mass hierarchy of neutrinos. We also discuss the Majorana phases and the effective neutrino mass of the neutrinoless double-beta decay.

It is extremely remarkable that we can perform a “complete experiment” to determine the low-energy neutrino mass matrix [10], since we have only seven physical parameters in the neutrino mass matrix. In particular, two CP-violating phases in the neutrino mass matrix are directly related to two CP-violating phases at high energy. Thus, assuming leptogenesis, we can determine the sign of the cosmic baryon in the universe from only the low-energy experiments for the neutrino mass matrix [11].

In Sect. 2, we show a viable Dirac neutrino mass matrix with four zeros, where we take the real diagonal basis of the charged lepton mass matrix and the right-handed Majorana neutrino mass matrix. We also present qualitative discussions of our parameters in order to reproduce the two large mixing angles of neutrino flavors. In Sect. 3, we show the numerical results for our mass matrix. Section 4 is devoted to the summary. In the appendix, we show the parameter relations in our mass matrix.

## Neutrino mass matrix

From the standpoint of the Occam’s-razor approach [3,4], we discuss the neutrino mass matrix in the framework of the seesaw mechanism without assuming any symmetry. We take the real diagonal basis of the charged lepton mass matrix and the right-handed Majorana neutrino mass matrix as

(1)
$ME=(me000mμ000mτ)LR , MR=(M1000M2000M3)RR.$

We reduce the number of free parameters in the Dirac neutrino mass matrix by putting zero at several elements in the matrix. The four zeros of the Dirac neutrino mass matrix give us the minimum number of parameters to reproduce the observed neutrino masses and lepton mixing angles. This is what we call the Occam’s-razor approach.

The successful Dirac neutrino mass matrix with four zeros1 is given as

(2)
$mD=(0A0A′0B0B′C)LR,$
which has five complex parameters.2 The three phases can be removed by the phase rotation of the three left-handed neutrino fields. This phase redefinition does not affect the lepton mixing matrix because the charged lepton mass matrix is diagonal and the phases are absorbed in the three right-handed charged lepton fields. In order to get the real matrix for the Dirac neutrino mass matrix, the remaining two phases are removed by the phase rotation of the two right-handed neutrino fields. Instead, the right-handed Majorana neutrino mass matrix becomes complex diagonal as follows:
(3)
$MR=(M1e−iϕA000M2e−iϕB000M3)RR=M0(1k1e−iϕA0001k2e−iϕB0001)RR,$
where $$M_0\equiv M_3$$, $$k_1=M_3/M_1$$, and $$k_2=M_3/M_2$$. We obtain the left-handed Majorana neutrino mass matrix after integrating out the heavy right-handed neutrinos,
(4)
$mν=mDMR−1mDT=1M0(A2k2eiϕB0AB′k2eiϕB0A′2k1eiϕA+B2BCAB′k2eiϕBBCB′2k2eiϕB+C2),$
in which there are clearly ten parameters. However, this is expressed in terms of seven parameters by the rescaling of parameters. Let us replace the parameters by introducing the new parameters $$a$$, $$b$$, $$c$$, $$k'_1$$, and $$k'_2$$ as follows:
(5)
$A=M0k′2 a, A′=M0k′1 a, B=M0 b, B′=M0k′2 b, C=M0 c.$

Then, the neutrino mass matrix is written as

(6)
$mν=(a2K2eiϕB0abK2eiϕB0a2K1eiϕA+b2bcabK2eiϕBbcb2K2eiϕB+c2),$
where
(7)
$K1=k′1k1=(A′B′AB)2M3M1 , K2=k′2k2=(B′B)2M3M2 .$

Finally, the neutrino mass matrix is expressed by five real parameters, $$a, b,c, K_1,K_2$$ and two phases $$\phi_A, \phi_B$$. Since we can input five pieces of experimental data for the neutrinos—the mass-squared differences $$\Delta m^2_{\rm atm}$$, $$\Delta m^2_{\rm sol}$$ and three lepton mixing angles $$\theta_{23}$$, $$\theta_{12}$$, and $$\theta_{13}$$—there remain two free parameters. These two parameters are determined by the Dirac CP-violating phase $$\delta_{\rm CP}$$ and the effective neutrino mass $$m_{ee}$$ for the neutrinoless double-beta decay [10].

Here we comment on the concern with the texture zero analysis of the left-handed neutrino mass matrix [12]. Actually, some two-zero textures of the left-handed neutrino mass matrix are consistent with the recent data [13]. On the other hand, our neutrino mass matrix of Eq. (6) is a one-zero texture. The two-zero textures are never realized without tuning between the parameters, as seen in Eq. (4), since we start with the seesaw mechanism of the neutrino masses, in which we take the right-handed Majorana neutrino mass matrix to be diagonal [14]. Although there are seven parameters in the neutrino mass matrix in Eq. (6), we can give clear predictions at large $$K_1$$ and $$K_2$$, which correspond to a large mass hierarchy among right-handed Majorana neutrinos.

We can obtain the eigenvectors by solving the eigenvalue equation of Eq. (6). The mass eigenvalues are expressed by $$a, b,c, K_1,K_2$$ and $$\phi_A, \phi_B$$, as seen in the appendix; we then get the lepton mixing matrix, the so-called Maki–Nakagawa–Sakata (MNS) matrix $$U_{\text{MNS}}$$ [15,16]. It is expressed in terms of three mixing angles $$\theta _{ij}$$$$(i,j=1,2,3; i<j)$$, the CP-violating Dirac phase $$\delta _{\rm CP}$$, and two Majorana phases $$\alpha$$ and $$\beta$$ as

(8)
$UMNS≡(c12c13s12c13s13e−iδCP−s12c23−c12s23s13eiδCPc12c23−s12s23s13eiδCPs23c13s12s23−c12c23s13eiδCP−c12s23−s12c23s13eiδCPc23c13)(e−iα000e−iβ0001),$
where $$c_{ij}$$ and $$s_{ij}$$ denote $$\cos \theta _{ij}$$ and $$\sin \theta _{ij}$$, respectively.

There is a CP-violating observable, the Jarlskog invariant $$J_{\rm CP}$$ [17], which is derived from the following relation:

(9)
$i𝒞≡[MνMν†,MEME†] ,det𝒞=−2JCP(m32−m22)(m22−m12)(m12−m32)(mτ2−mμ2)(mμ2−me2)(me2−mτ2) ,$
where $$m_1$$, $$m_2$$, and $$m_3$$ are neutrino masses with real numbers. The predicted one is expressed in terms of the parameters of the mass matrix elements as
(10)
$JCP≃12F1(Δmatm2)2Δmsol2 ,$
where
(11)
$F=2a2b4c2K22{b4K2sinϕB+a4K1K2sin(ϕA−ϕB)+a2c2(K1sinϕA−K2sinϕB)+a2b2K2(K1sin(ϕA+ϕB)−K2sin2ϕB−sinϕB)} .$

We can extract $$\sin \delta _{\rm CP}$$ from $$J_{\rm CP}$$ by using the following relation between the mixing angles, the Dirac phase, and $$J_{\rm CP}$$:

(12)
$sinδCP=JCP/(s23c23s12c12s13c132) .$

The Majorana phases $$\alpha$$ and $$\beta$$ are obtained after diagonalizing the neutrino mass matrix of Eq. (6) as follows:

(13)
$UMNS†mνUMNS*=diag{m1, m2, m3} .$

Then, we can estimate the effective mass that appears in the neutrinoless double-beta decay as

(14)
$mee=c132c122e−2iαm1+c132s122e−2iβm2+s132e−2iδCPm3 .$

The neutrino mass matrix of Eq. (6) becomes a simple one at the $$K_1$$ and $$K_2$$ large limit with $$b^2 K_2$$ being finite. This case corresponds to the large hierarchy of the right-handed neutrino mass ratios $$M_3/M_1$$ and $$M_3/M_2$$. Then, the magnitudes of our parameters are estimated qualitatively to reproduce the two large mixing angles $$\theta_{23}$$ and $$\theta_{12}$$. First, impose the maximal mixing of $$\theta_{23}$$. Then, the $$(2,3)$$ element of Eq. (6) should be comparable to the $$(3,3)$$ one, so that the cancellation must be realized between two terms in the $$(3,3)$$ element, and then we have:

(15)
$K2~c2b2 , ϕB~±π .$

The $$(2,3)$$ element of Eq. (6) is also comparable to the $$(2,2)$$ one, which is dominated by the first term $$a^2 K_1 \exp(i\phi_A)$$ at large $$K_1$$. So, we get

(16)
$K1~bca2 .$

In the next step, we impose a large $$\theta_{12}$$, which requires the $$(1,3)$$ element of Eq. (6) to be comparable to $$(2,2)$$ within a few factors; therefore, we get

(17)
$aK1~bK2r , (r=2−3) .$

By combining Eqs. (15), (16), and (17), we obtain

(18)
$acr~b2 , K1~(cb)3 , K2~(cb)2 , K12~K23 r4 , (r=2−3) .$

Actually, these relations are well satisfied in the numerical result at large $$K_1$$. Then, $$\theta_{13}$$ becomes rather large, roughly of the order of $$\sin\theta_{12}/r$$, since the $$(1,3)$$ element of Eq. (6) is comparable to $$(2,3)$$ within a factor of two or three. Thus, the sizable mixing angle $$\theta_{13}$$ is essentially derived in these textures when the observed mixing angles $$\theta_{23}$$ and $$\theta_{12}$$ are input. This situation is well reproduced in our numerical result.

Furthermore, we expect a large CP-violating phase $$\delta_{\rm CP}$$ in this discussion. As shown in Eq. (15), the real part of the $$(3,3)$$ element of Eq. (6) is significantly suppressed in order to reproduce the almost maximal mixing of $$\theta_{23}$$. Then, the imaginary part of the $$(3,3)$$ element is relatively enhanced even if $$\phi_B$$ is close to $$\pm 180^\circ$$. Actually, $$\phi_B\simeq \pm 175^\circ$$ leads to the $$\delta_{\rm CP}\simeq \pm 90^\circ$$ in the numerical analysis of the next section.

## Numerical analysis

Let us discuss the numerical result with the normal mass hierarchy of neutrinos. In the first step, we constrain the real parameters $$a, b,c, K_1,K_2$$ and two phases $$\phi_A, \phi_B$$ by inputting the experimental data for $$\Delta m^2_{\rm atm}$$ and $$\Delta m^2_{\rm sol}$$ with $$90\%$$ C.L. into the relations of Eq. (A.1) in the appendix. By removing $$c$$, $$\phi_A$$, and $$\phi_B$$ for a fixed $$m_1$$, which is varied in the region of $$m_1=0\sim \sqrt{\Delta m^2_{\rm sol}}$$ , there remain four parameters $$a, b, K_1$$, and $$K_2$$.

In the second step, we scan them in the following regions by generating random numbers in the linear scale as follows:

(19)
$K1=[1−106], K2=[1−104], a=[0−0.03] eV1/2, b=[0−0.2] eV1/2.$

They are constrained by the experimental data for the lepton mixing angles. We then predict $$\delta_{\rm CP}$$, $$m_{ee}$$, and the Majorana phases $$\alpha$$ and $$\beta$$. The input data are given as follows [18]:

(20)
$Δmatm2=(2.457±0.047)×10−3eV2 , Δmsol2=7.50−0.17+0.19×10−5eV2 ,sin2θ12=0.304−0.012+0.013 , sin2θ23=0.452−0.028+0.052 , sin2θ13=0.0218±0.0010 ;$
we adopt these data with an error-bar of $$90\%$$ C.L in our calculations. We assume the normal mass hierarchy of neutrinos. Actually, we have not found the inverted mass hierarchy, in which the three lepton mixing angles are consistent with the observed values in our numerical calculations. Thus, we consider that the normal mass hierarchy is a prediction in the present model as long as there is no extreme fine tuning of the parameters.

Let us show the result for $$K_1=1$$$$5000$$. By inputting the data for the two mixing angles $$\theta_{12}$$ and $$\theta_{23}$$, we present the frequency distribution of the predicted $$\sin\theta_{13}$$ in Fig. 1, where the vertical red lines denote the experimental data for Eq. (20) with a $$3\sigma$$ range. The peak is within the experimental data for the $$3\sigma$$ range. It is remarked that $$\sin\theta_{13}\simeq 0.14$$ is most favored. This prediction is understandable, as discussed below Eq. (18). We also present the frequency distribution of the predicted value of $$\delta_{\rm CP}$$ in Fig. 2, where the vertical red lines denote the NO$$\nu$$A experimental allowed region with a $$1\sigma$$ range, which is obtained by the method of library event matching (LEM) [2]. We see that $$\delta_{\rm CP}$$ is favored to be around $$\pm 2$$ radian, which is consistent with the T2K [1] and NO$$\nu$$A data for the $$1\sigma$$ range.

Fig. 1.

The frequency distribution of the predicted $$\sin\theta_{13}$$ at $$K_1=1$$$$5000$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$. Here the vertical red lines denote the experimental data with $$3\sigma$$.

Fig. 1.

The frequency distribution of the predicted $$\sin\theta_{13}$$ at $$K_1=1$$$$5000$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$. Here the vertical red lines denote the experimental data with $$3\sigma$$.

Fig. 2.

The frequency distribution of the predicted $$\delta_{\rm CP}$$ at $$K_1=1$$$$5000$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$. Here the vertical red lines denote the NO$$\nu$$A allowed region with $$1\sigma$$.

Fig. 2.

The frequency distribution of the predicted $$\delta_{\rm CP}$$ at $$K_1=1$$$$5000$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$. Here the vertical red lines denote the NO$$\nu$$A allowed region with $$1\sigma$$.

Fig. 3.

The allowed region on the $$K_1$$$$K_2$$ plane at $$K_1=1$$$$5000$$ by inputting the data for three mixing angles.

Fig. 3.

The allowed region on the $$K_1$$$$K_2$$ plane at $$K_1=1$$$$5000$$ by inputting the data for three mixing angles.

Fig. 4.

The frequency distribution of the predicted $$\delta_{\rm CP}$$ at $$K_1=1$$$$5000$$ by inputting the data for three mixing angles. Here the vertical red lines denote the NO$$\nu$$A allowed region with $$1\sigma$$.

Fig. 4.

The frequency distribution of the predicted $$\delta_{\rm CP}$$ at $$K_1=1$$$$5000$$ by inputting the data for three mixing angles. Here the vertical red lines denote the NO$$\nu$$A allowed region with $$1\sigma$$.

Let us discuss the $$K_1$$ dependence of $$\delta_{\rm CP}$$, which is shown in Fig. 5. In the region of $$K_1={\cal O}(1$$$$100)$$, the predicted $$\delta_{\rm CP}$$ has a broader distribution. As $$K_1$$ increases, the predicted region gradually becomes narrower. It then becomes consistent with the NO$$\nu$$A experimental allowed region with a $$1\sigma$$ range at high $$K_1$$.

Fig. 5.

The $$K_1$$ dependence of the predicted $$\delta_{\rm CP}$$ at $$K_1=1$$$$5000$$ by inputting the data for three mixing angles. Here the horizontal red lines denote the NO$$\nu$$A experimental allowed region with $$1\sigma$$.

Fig. 5.

The $$K_1$$ dependence of the predicted $$\delta_{\rm CP}$$ at $$K_1=1$$$$5000$$ by inputting the data for three mixing angles. Here the horizontal red lines denote the NO$$\nu$$A experimental allowed region with $$1\sigma$$.

We also predict the effective neutrino mass $$m_{ee}$$, which appears in the amplitude of the neutrinoless double-beta decay. In Fig. 6, we present the frequency distribution of $$m_{ee}$$. The favored $$m_{ee}$$ is around $$7$$ meV.

Fig. 6.

The frequency distribution of the predicted $$m_{ee}$$ at $$K_1=1$$$$5000$$ by inputting the data for three mixing angles.

Fig. 6.

The frequency distribution of the predicted $$m_{ee}$$ at $$K_1=1$$$$5000$$ by inputting the data for three mixing angles.

As shown in Fig. 5, our result depends on $$K_1$$. Actually, the predicted region becomes narrow as $$K_1$$ increases significantly. Let us discuss the result at $$K_1=10^4$$$$10^6$$. We show the $$K_1$$ dependence of the predicted $$\sin\theta_{13}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$ in Fig. 7. The mixing angle $$\sin\theta_{13}$$ is larger than $$0.1$$ in all regions of $$K_1$$, but the large mixing angle $$0.5$$ is allowed below $$K_1=10^5$$. However, it is remarked that $$\sin\theta_{13}$$ decreases gradually and converges on the experimental allowed value.

Fig. 7.

The $$K_1$$ dependence of the predicted $$\sin\theta_{13}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$. Here the horizontal red lines denote the experimental data with $$3\sigma$$.

Fig. 7.

The $$K_1$$ dependence of the predicted $$\sin\theta_{13}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$. Here the horizontal red lines denote the experimental data with $$3\sigma$$.

In Fig. 8, we present the frequency distribution of the predicted $$\sin\theta_{13}$$ by inputting the data for the two mixing angles $$\theta_{12}$$ and $$\theta_{23}$$. The distribution becomes rather sharp compared with the case of $$K_1=1$$$$5000$$. The most favored region of $$\sin\theta_{13}$$ is around $$0.13$$$$0.15$$, which is completely consistent with the experimental data.

Fig. 8.

The frequency distribution of the predicted $$\sin\theta_{13}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$. Here the vertical red lines denote the experimental data with $$3\sigma$$.

Fig. 8.

The frequency distribution of the predicted $$\sin\theta_{13}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for $$\theta_{12}$$ and $$\theta_{23}$$. Here the vertical red lines denote the experimental data with $$3\sigma$$.

In Fig. 9, we show the frequency distribution of the predicted value of $$\delta_{\rm CP}$$ by inputting the data for the three mixing angles. It is remarked that the peak of the frequency distributions of $$\delta_{\rm CP}$$ becomes close to $$\pm \pi/2$$. Moreover, the region of $$\delta_{\rm CP}=-1$$$$1$$ radian is almost excluded. Our result is consistent with the data for the T2K [1] and NO$$\nu$$A [2] experiments.

Fig. 9.

The frequency distribution of the predicted $$\delta_{\rm CP}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles. Here the vertical red lines denote the NO$$\nu$$A allowed region with $$1\sigma$$.

Fig. 9.

The frequency distribution of the predicted $$\delta_{\rm CP}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles. Here the vertical red lines denote the NO$$\nu$$A allowed region with $$1\sigma$$.

The predicted $$m_{ee}$$ of the neutrinoless double-beta decay is not so changed compared with the case of $$K_1=1$$$$5000$$. The favored value of $$m_{ee}$$ is around $$7$$$$8$$ meV. Here, we show the predicted $$\delta_{\rm CP}$$ versus $$m_{ee}$$ by inputting the data for three mixing angles in Fig. 10. They are rather well correlated, as seen in Eq. (A.2) in the appendix. If $$\delta_{\rm CP}$$ is restricted around $$-\pi/2$$ in the neutrino experiment, the allowed region is restricted. The predicted $$m_{ee}$$ is then $$6.5$$$$8$$ meV.

Fig. 10.

The predicted Dirac phase $$\delta_{\rm CP}$$ versus the predicted $$m_{ee}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles. Here the horizontal green dashed line, inserted to guide the eye, denotes $$\delta_{\rm CP}=-\pi/2$$.

Fig. 10.

The predicted Dirac phase $$\delta_{\rm CP}$$ versus the predicted $$m_{ee}$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles. Here the horizontal green dashed line, inserted to guide the eye, denotes $$\delta_{\rm CP}=-\pi/2$$.

Finally, we show the correlation between the Dirac phase $$\delta_{\rm CP}$$ and the Majorana phases $$\alpha$$, $$\beta$$ in Figs. 11,12, and 13. There appears to be a tight correlation between them because we have only two phase parameters in the neutrino mass matrix of Eq. (6).

Fig. 11.

The predicted Dirac phase $$\delta_{\rm CP}$$ versus the predicted Majorana phase $$\alpha$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles.

Fig. 11.

The predicted Dirac phase $$\delta_{\rm CP}$$ versus the predicted Majorana phase $$\alpha$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles.

Fig. 12.

The predicted Dirac phase $$\delta_{\rm CP}$$ versus the predicted Majorana phase $$\beta$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles.

Fig. 12.

The predicted Dirac phase $$\delta_{\rm CP}$$ versus the predicted Majorana phase $$\beta$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles.

Fig. 13.

The predicted Majorana phase $$\alpha$$ versus the predicted Majorana phase $$\beta$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles.

Fig. 13.

The predicted Majorana phase $$\alpha$$ versus the predicted Majorana phase $$\beta$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles.

## Summary

We have presented the neutrino mass matrix based on the Occam’s-razor approach [3,4]. In the framework of the seesaw mechanism, we impose four zeros in the Dirac neutrino mass matrix, which give the minimum number of parameters needed for the observed neutrino masses and lepton mixing angles without assuming any flavor symmetry. Here, the charged lepton mass matrix and the right-handed Majorana neutrino mass matrix are taken to be real diagonal ones. Therefore, the neutrino mass matrix is given with seven parameters after absorbing the three phases into the left-handed neutrino fields.

Then, we obtain the successful predictions of the mixing angle $$\theta_{13}$$ and the CP-violating phase $$\delta_{\rm CP}$$ with the normal mass hierarchy of neutrinos. We also discuss the Majorana phases $$\alpha$$ and $$\beta$$ as well as the effective neutrino mass of the neutrinoless double-beta decay $$m_{ee}$$. In particular, as $$K_1$$ increases to $$10^4$$$$10^6$$, the predictions become sharper. The most favored region of $$\sin\theta_{13}$$ is around $$0.13$$$$0.15$$, which is completely consistent with the experimental data. $$\delta_{\rm CP}$$ is favored to be close to $$\pm \pi/2$$, and the effective mass $$m_{ee}$$ is around $$7$$$$8$$ meV. The reduction of the experimental error-bar of the two mixing angles of $$\theta_{12}$$ and $$\theta_{23}$$ will provide more precise predictions in our neutrino mass matrix.

Finally, it is emphasized that we can perform a “complete experiment” to determine the low-energy neutrino mass matrix, since we have only seven physical parameters in the mass matrix (see Eq. (6)). In particular, two CP-violating phases $$\phi_A$$ and $$\phi_B$$ in the neutrino mass matrix are directly related to two CP-violating phases at high energy. Thus, assuming leptogenesis, we can determine the sign of the cosmic baryon in the universe from the low-energy experiments for the neutrino mass matrix.3 In fact, the sign of the baryon is given by the sign of $$\sin \phi_A$$ for the normal mass hierarchy $$M_1< M_2 < M_3$$, which is suggested from the predicted hierarchy $$K_1>K_2>1$$ shown in Fig. 3.4 Unfortunately, the present experimental data show both signs to be allowed, as shown in Fig. 14.5 We expect precise measurements of the three mixing angles and CP-violating phases in low-energy experiments.

Fig. 14.

The frequency distribution of the predicted $$\sin \phi_A$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles.

Fig. 14.

The frequency distribution of the predicted $$\sin \phi_A$$ at $$K_1=10^4$$$$10^6$$ by inputting the data for three mixing angles.

## Acknowledgements

T.T.Y. thanks Prof. Serguey Petcov for the discussion on CP violation. This work is supported by JSPS Grants-in-Aid for Scientific Research (No. 28.5332; Y.S.), Scientific Research (Nos. 15K05045, 16H00862; M.T.), and Scientific Research (Nos. 26287039, 26104009, 16H02176; T.T.Y.). This work is also supported by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. Y.S. is supported in part by a National Research Foundation of Korea (NRF) Research Grant NRF-2015R1A2A1A05001869.

## Funding

Open Access funding: SCOAP3.

## Appendix

By solving the eigenvalue equation in Eq. (6), the mass eigenvalues are expressed by $$a, b,c, K_1,K_2$$ and $$\phi_A, \phi_B$$. We have three equations among them as follows:

(A.1)
$m12+m22+m32 =c4+b4(1+K22)+a4(K12+K22)+2b2[a2(K22+K1cosϕA)+c2(1+K2cosϕB)] ,m12m22+m22m32+m32m12 =b8K22+a8K12K22+2a6b2K1K22(K1+cosϕA)+a4[c4(K12+K22)+b4K22(1+K12+4K1cosϕA)+2b2c2K2(K2+K12cosϕB)]+2a2b4K2[b2(K2+K1K2cosϕA)+c2(K2+K1cosϕAcosϕB+K1sinϕAsinϕB)] , m12m22m32=a8c4K12K22 .$

Since the neutrino mass matrix in Eq. (6) has one zero, it constrains the observed values. Among the three mixing angles, the three phases, and the neutrino masses, there is one relation:

(A.2)
$0=c12c13(−s12c23−c12s23s13eiδCP)e−2iαm1+ s12c13(c12c23−s12s23s13eiδCP)e−2iβm2+s13s23c13e−iδCPm3 .$

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1Other four-zero textures may be available for lepton mixing. These will be discussed comprehensively in future work.
2$$A'=0$$ corresponds to the case discussed in Ref. [3]. Thus, five-zero textures are not excluded.
3The effect of quantum corrections of the lepton mixing matrix is neglected in the evolution from the GUT scale to the electroweak scale for the normal mass hierarchy [19].
4$$K_{1(2)}$$ is not $$M_3/M_{1(2)}$$ itself, as seen in Eq. (7). However, it is almost $$M_3/M_{1(2)}$$ as long as the Dirac neutrino mass matrix in Eq. (2) is not extremely asymmetric.
5A detailed discussion on this issue will be given in future work.
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