Dirac CP phase in the neutrino mixing matrix and the Froggatt-Nielsen mechanism with ${\rm \bf det} \bf [M_\nu]=0$

We discuss the Dirac CP violating phase $\delta_{CP}$ in the Froggatt-Nielsen model for a neutrino mass matrix $M_\nu$ imposing a condition ${\rm det} [M_\nu]=0$. This additional condition restricts the CP violating phase $\delta_{CP}$ drastically. We find that the phase $\delta_{CP}$ is predicted in the region of $\pm (0.4- 2.9)$ radian, which is consistent with the recent T2K and NO$\nu$A data. There is a remarkable correlation between $\delta_{CP}$ and $\sin^2\theta_{23}$. The phase $\delta_{CP}$ converges on $\sim \pm \pi/2$ if $\sin^2\theta_{23}$ is larger than $0.5$. Thus, accurate measurements of $\sin^2\theta_{23}$ are important for a test of our model. The effective mass $m_{ee}$ for the neutrinoless double beta decay is predicted in the rage of $3.3-4.0$ meV.


Introduction
The Froggatt-Nielsen (FN) mechanism [1] is very attractive since it naturally explains the observed masses and mixing angles for quarks and leptons. It is well known that the magnitudes of observed mixing angles for quarks are given by powers of Wolfenstein parameter λ ≃ 0.2 [2]. This is nothing but the feature predicted by the FN mechanism. The lepton flavor mixing matrix, so called MNS matrix [3,4], exhibits two large mixing angles, and one rather small mixing angle of the order of Cabibbo angle. Surprisingly, this lepton mixing matrix is also explained by the FN mechanism [5][6][7][8][9][10][11].
Among various proposals, Ling and Ramond [10] presented a clear phenomenological discussion of neutrino masses and mixing angles in terms of Cabibbo angle λ C ≃ 0.225 [12]. Their texture is still consistent with the recent precise data on the three neutrino mixing angles and two neutrino mass squared differences. However, this texture cannot predict the CP phase as we discuss later in this paper.
The neutrino oscillation experiments are now on a new step to confirm the CP violation in the lepton sector. Actually, the T2K and NOνA experiments indicate a finite CP phase [13][14][15][16].
Therefore, it is very interesting to extend the FN model to predict the Dirac CP violating phase.
In this paper, we discuss the Dirac CP violating phase in the FN model for the neutrino mass matrix M ν imposing an additional condition det[M ν ] = 0 [17]. This flavor-basis independent condition of det[M ν ] = 0 is obtained easily by assuming two families of heavy right-handed neutrinos [18] in the framework of the seesaw mechanism [19,20]. It is also interesting that the Affleck-Dine scenario [21] for leptogenesis [22,23] requires the mass of the lightest neutrino to be m 1 = 10 −10 eV [24,25], which practically leads to our condition det[M ν ] = 0. We show that the phase δ CP is predicted in a narrow region using the presently available data on the mass squared differences and the mixing angles.
In section 2, we discuss a texture of the neutrino mass matrix imposing det[M ν ] = 0 in the FN model, where neutrinos are supposed to be Majorana particles. In section 3, we show numerical results on δ CP , sin 2 θ 23 , sin 2 θ 13 and sin 2 θ 12 . The effective mass m ee that appears in the neutrinoless double beta decay is also discussed. The summary and discussion are given in section 4.

FN texture for leptons
Let us discuss lepton mass matrices in the framework of the FN model. We assign the FN charges of the FN broken U(1) [1] to the three left-handed leptons ℓ Li as ℓ L1 , ℓ L2 , ℓ L3 : n + 1, n, n , where n is a positive integer. Then, the mass matrix of the left-handed Majorana neutrinos is given in terms of the FN parameter λ, which is of the order of the Cabibbo angle λ C , as follows: the charged lepton mass matrix M E is given as which gives the mass ratio in terms of λ as follows: These mass ratios are consistent with observed ones for about λ ≃ 0.2.
We move to the diagonal basis of the charged lepton mass matrix in order to reduce the number of free parameters. Then, the rotation of the left-handed lepton doublets to diagonalize the charged lepton mass matrix does not change the powers of λ in the entries of the neutrino mass matrix of Eq. (2). Therefore, we discuss the following neutrino mass matrix in the diagonal basis of the charged lepton mass matrix: where a − f are dimensionless complex parameters with their magnitudes of the order 1 1 .
By using the freedom of phase redefinition of the left-handed lepton fields, we take the diagonal elements to be real. Then, we have three CP phases in the mass matrix. Now, we parameterize the mass matrix in order to analyze the neutrino mixing numerically as where a − e are redefined as real parameters of the order 1.
Let us determine the magnitude of λ from the observed charged lepton mass ratios in Eq. (5).
We use the m e /m τ ratio to fix λ, since it has the strongest λ dependence among charged lepton mass ratios as seen in Eq. (5). Then, we obtain λ ≃ 0.20 from the m e /m τ ratio. Taking into account the order one coefficients in those mass ratios in Eq.(5), λ = 0.20 can explain all lepton mass ratios consistently. We take λ = 0.18 ∼ 0.22 considering the ambiguity of 10% for λ in our numerical computation 2 . We have now nine parameters m 0 , a − e, φ b , φ c and φ e , where the m 0 has dimension of a mass, but others are dimensionless parameters.
Let us impose a flavor-basis independent condition that the determinant of the neutrino mass matrix vanishes, that is det[M ν ] = 0. This condition gives two constraints on the parameters, and then the neutrino mass matrix has now just seven free parameters which can be fully determined by future feasible experiments [17]. We see below that thanks to this condition, we can predict the CP violating phase δ CP , which is defined in the Particle Data Group [12]. 1 Due to the rotation of the left-handed lepton doublets, the magnitude of the coefficients a − f may be rather enlarged. We address this point in the section 4.
2 The ambiguity of the coefficients a − e due to the rotation of the left-handed lepton doublets is partially absorbed by taking account of the ambiguity of 10% for λ.
Let us present our numerical analysis of the neutrino mass matrix in Eq. (7). The free parameters a−e are of the order one. We scan them in the region of 0.7 ∼ 1.3 by generating random numbers in the liner scale. Our choice of the parameter region of 0.7 ∼ 1.3 is justified later by the predicted mixing of sin 2 θ 23 . The parameter λ is essentially given by the FN model. As discussed in the previous section, the charged lepton mass hierarchy indicates λ ≃ 0.2. In our numerical analysis, we also scan it at random with the liner scale in the region λ = 0.18 ∼ 0.22. Furthermore, the extension of the scanning region, for example, 0.5 ∼ 2 is not favored because the hierarchies between aλ 2 and bλ(cλ), and between bλ(cλ) and d, are no longer distinguishable, and then the FN scheme with λ ≃ 0.2 becomes meaningless.
The CP violating phases φ b , φ c and φ e are also scanned in the full region of −π ∼ π by generating random numbers in the liner scale. Now we explain how to obtain our predictions in our figures. By scanning the parameters of a − e and three phases with λ = 0.18 ∼ 0.22, we generate a neutrino mass matrix. The parameter m 0 is determined to reproduce the observed values of ∆m 2 23 and ∆m 2 12 at 2σ interval in Table 1. In practice, m 0 is also scanned randomly in the linear scale up to the upper bound of the total neutrino mass 0.2 eV, which is given by the cosmology observation [12]. Actually, the obtained m 0 is in the region of (0.025 − 0.035) eV. It is noticed, in the case of det[M ν ] = 0, m 0 is easily determined by the experimental data of ∆m 2 12 and ∆m 2 23 because of m 1 = 0. Then, we obtain the calculated three mixing angles. If these predicted mixing angles are OK for the experimental data in Table 1, we keep the point. Otherwise, we disregard the point. We continue this procedure to obtain 10 4 points, which satisfy the experimental data.   Let us use the constraint from the data sin 2 θ 12 with 2σ error-bar in Table 1 in addition to the data of ∆m 2 23 and ∆m 2 12 . The predicted sin 2 θ 23 is shown in Fig.2. The frequency distribution of sin 2 θ 23 is remarkably changed. It is asymmetric around 0.5 as seen in Fig.2. The region sin 2 θ 23 < 0.5 is favored. It may be interesting to comment that this prediction is not changed even if the data of sin 2 θ 13 is added. Thus, the input of sin 2 θ 12 pushes sin 2 θ 23 toward a region smaller than 0.5. It is interesting that the peak of the frequency distribution is around 0.44, which is the best fit value of the experimental data as seen in Table 1.

Prediction of mixing angles
We add a comment that the distribution plot of Fig.2 covers all region of the experimental interval of ∆m 2 23 and ∆m 2 12 in Table 1. It also covers all region of the experimental interval of sin 2 θ 12 as seen later in Fig. 8. in the T2K experiment [14].
In fact, the predicted region for sin 2 θ 12 contains the region around 0. We present the predicted region on the plane of sin 2 θ 12 and sin 2 θ 13 with the condition of det[M ν ] = 0 in Fig. 4, where the scattered plot is shown in the experimental allowed region with 3σ. It is concluded that the predicted θ 12 and θ 13 are completely consistent with the experimental data. Now, we try to predict the CP violating phase δ CP in the next subsection.

Prediction of δ CP
In order to predict the CP violating phase δ CP precisely, we also use the data of all mixing angles, θ 23 , θ 12 and θ 13 , in addition to ∆m 2 23 and ∆m 2 12 . At first, we show the calculated frequency distribution of δ CP without imposing det[M ν ] = 0 in Fig.5. The vertical dashed lines denote the observed δ CP interval at 90% C.L. in the recent T2K experiment [14]. We see that the predicted δ CP lies in the all region −π ∼ π.
Thus, the condition of det[M ν ] = 0 is essential for the prediction of δ CP .
We also discuss the correlations among mixing angle θ 23 and CP violating phase δ CP . We show the plot δ CP versus sin 2 θ 23 in Fig.7, where det[M ν ] = 0 is imposed. As sin 2 θ 23 increases, the predicted range of δ CP becomes narrow. If sin 2 θ 23 is larger than 0.5, δ CP converges toward ±π/2. Actually, the allowed region of δ CP is ±(0.7 ∼ 2.4) radian. More accurate measurements of sin 2 θ 23 will be important to test our model.
We show the allowed region in the plane of sin 2 θ 12 and sin 2 θ 23 in Fig.8

Prediction of the effective mass m ee
Finally, we discuss the effective neutrino mass responsible for the neutrinoless double beta decay where U ei denotes the MNS mixing matrix element. We show the frequency distribution of the predicted m ee , which lies in the range m ee = 3.35 − 4.00 meV, in Fig.9, where det[M ν ] = 0 is imposed.

Summary and Discussion
We have discussed the mixing angles and the Dirac CP violating phase in the framework of the FN model with the flavor-basis independent condition det[M ν ] = 0. It is remarkable that sin 2 θ 23 is predicted inside of the experimental allowed region of 2σ, where we have used only the data of ∆m 2 23 and ∆m 2 12 . Here, we have taken the order one parameters to be a − e = 0.7 ∼ 1.3 and the FN parameter λ = 0.18 ∼ 0.22. We have found that the predicted sin 2 θ 13 and sin 2 θ 12 are also completely consistent with the experimental data. Our numerical results depend on the scanning region a − e = 0.7 ∼ 1.3. The condition of det[M ν ] = 0 is essential for the nontrivial prediction of δ CP . The allowed region of δ CP is consistent with the recent T2K and NOνA data.
In order to see the effect of the order one parameters a−e on our prediction of δ CP , we present the frequency distributions of δ CP for a − e = 0.5 ∼ 2 in Fig.10. As the region of the parameter a − e expands, the frequency distribution becomes broader. Notice that the hierarchies in the neutrino mass matrix M ν predicted by the FN mechanism becomes obscure with such a large region of the parameters a − e as stressed in section 3. In conclusion, we claim that det[M ν ] = 0 predicts δ CP as seen in Fig.6 if the FN flavor structure is sharp.
It is helpful to comment on why det[M ν ] = 0 rules out δ CP = 0, ±π as seen in Fig. 6.
The five neutrino experimental data, two mass squared differences and three mixing angles It is emphasized that the scenario with the two family heavy right-handed neutrinos is not necessarily required. In practice, we have checked that our prediction of δ CP is not changed in the case of m 1 being smaller than 10 −4 eV. However, we do not address the model with tiny m 1 since it is beyond the scope of our work.
We have also found the remarkable correlation between δ CP and sin 2 θ 23 . If sin 2 θ 23 is larger than 0.5, δ CP converges to around ±π/2. We expect the accurate measurement of sin 2 θ 23 will be done in near future experiments. The effective mass in the neutrinoless double beta decay m ee is also predicted to be m ee = 3.3 − 4.0 meV.
We should note that our results are consistent with the conclusions in [27], where an exchange symmetry between two heavy right-handed neutrinos is further imposed. The CP violating phase δ CP is predicted near by the maximal value ± π 2 due to the exchange symmetry.