Towards the minimal seesaw model via CP violation of neutrinos

We study the minimal seesaw model, where two right-handed Majorana neutrinos are introduced, focusing on the CP violating phase. In addition, we take the trimaximal mixing pattern for the neutrino flavor where the charged lepton mass matrix is diagonal. Thanks to this symmetric framework, the $3\times 2$ Dirac neutrino mass matrix is given in terms of a few parameters. Numerical studies reveal that the observation of the CP violating phase can determine the flavor structure of the Dirac neutrino mass matrix in the minimal seesaw model. In particular, new minimal Dirac neutrino mass matrices are proposed in the case of $\rm TM_1$, which is derived by the additional 2-3 family mixing to the tri-bimaximal mixing basis in the normal hierarchy of neutrino masses. Our analyses include the Littlest seesaw model by King {\it et al.}, which is one of the specific one in our results. Furthermore, it is remarked that our $3\times 2$ Dirac neutrino mass matrix is reproduced by introducing gauge singlet flavons with the specific alignments of the VEV's. These alignments suggest the residual symmetry of $S_4$ group.


Introduction
The standard model (SM) has been well established by the discovery of the Higgs boson. However, the origin of flavor in quarks and leptons is still unknown in spite of the remarkable success of the SM. The underlying physics for the flavor in quarks and leptons is one of the fundamental problems in particle physics.
On the other hand, the neutrino oscillation experiments are going on a new step to observe the CP violation in the lepton sector. The T2K experiment has confirmed the neutrino oscillation in the ν µ → ν e appearance events [1], which provides us a new information of the CP violation in the lepton sector. Recent T2K data strongly indicate the CP violation in the neutrino oscillation [2,3]. The NOνA data also suggest the CP violation [4], which is consistent with the T2K result. Since the experimental data of the CP violating phase will be available in the near future, one can develop the flavor structure of Yukawa couplings by taking account of it.
We study the flavor structure in the seesaw model [5]- [7] in order to find a clue of the underlying physics of flavor. It is advantageous to consider the minimum number of parameters needed for reproducing the neutrino mixing angles and CP violating phases completely [8]. Indeed, there are some attempts toward so-called the minimal seesaw model [9]- [21].
In this work, we investigate the minimal seesaw model via the CP violation, where we assume two right-handed Majorana neutrinos. In addition, we take the trimaximal mixing pattern [22,23] of the neutrino flavor to reduce the number of free parameters of the Dirac neutrino mass matrix. The trimaximal mixing pattern is derived from the flavor symmetry.
Before the reactor experiments reported the non-zero value of θ 13 in 2012 [24,25], there has been a paradigm of the tri-bimaximal (TBM) mixing [26,27]. Since the TBM mixing pattern is a highly symmetric, the non-Abelian discrete groups have become the center of attention at the flavor symmetry [28]- [35]. The observation of the non-vanishing θ 13 forces to study the deviation from the TBM mixing. It is remarked that the trimaximal mixing pattern is still available after the observation of θ 13 . The trimaximal mixing is given by the additional rotation of 2-3 (TM 1 ) or 1-3 (TM 2 ) families of neutrinos to the TBM mixing basis [36,37]. The additional rotation of 1-2 families is called as TM 3 , however it still leads to θ 13 = 0. We do not discuss the additional rotation of 1-2 families, TM 3 , since it leads to θ 13 = 0.
One obtains testable relations among the neutrino mixing angles and the CP violating phase, so-called mixing sum rules in TM 1 and TM 2 [38]- [40]. The prediction for the CP violation of neutrinos has a big impact on the study of the flavor structure [41]- [52] because T2K and NOνA experiments are expected to confirm the non-zero CP violating phase in the near future. We discuss the CP violation of neutrinos for both TM 1 and TM 2 in the framework of the minimal seesaw model to restrict the structure of the Dirac neutrino mass matrix. It is found that those Dirac neutrino mass matrices are reproduced by introducing gauge singlet flavons with the vacuum expectation value (VEV) in the non-Abelian discrete symmetry, for example, S 4 symmetry. The specific alignments of the VEV's suggest the residual symmetry of S 4 group in our work.
We investigate the Dirac neutrino mass matrix in the diagonal basis of the charged lepton mass matrix and the 2 × 2 right-handed Majorana neutrino mass matrix focusing the CP violating Dirac phase. Since our results depend on the neutrino mass hierarchies, we discuss both cases of the normal hierarchy (NH) and the inverted hierarchy (IH) for neutrino masses, respectively.
Our analyses include the Littlest seesaw model by King et al. [53]- [55] as the one of the specific cases of our model. The paper is organized as follows. We present our framework of the minimal seesaw model in section 2, where the structure of the Dirac neutrino mass matrix is discussed to reproduce TM 1 and TM 2 for both cases of NH and IH. The Dirac neutrino mass matrix is also discussed in the view of the flavor symmetry. In section 3, the numerical results are presented. The section 4 is devoted to the summary and discussions. Appendix A gives the detail studies of the Dirac neutrino mass matrix, and Appendix B presents the necessary group theory of S 4 .

Realization of minimal seesaw model
Let us start with discussing the structure of the lepton mixing matrix, so-called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix U PMNS [56,57], where neutrinos are supposed to be Majorana particles. It is parametrized in terms of three mixing angles θ ij (i, j = 1, 2, 3; i < j), one CP violating Dirac phase δ CP , and two Majorana phases α, β as follows: where c ij and s ij denote cos θ ij and sin θ ij , respectively. The CP violating measure, Jarlskog invariant [58], is defined by the PMNS matrix elements U αi , and is written in terms of the mixing angles and the CP violating phase as: For the lepton mixing matrix, Harrison-Perkins-Scott proposed a simple form of the mixing matrix, so-called TBM mixing [26,27] as follows: respectively, where a ∼ f are complex parameters and p is the ratio between the two righthanded Majorana neutrino masses. The minus sign in front of M 0 is taken as our sign convention. By using the seesaw mechanism of Type I, the left-handed Majorana neutrino mass matrix M ν is give by parameters of the neutrino mass matrix. Then, the only one phase which leads to the CP violation of neutrinos is the relative phase between the first and the second columns of the Dirac neutrino mass matrix in Eq. (8), that is the phase of c/f . Since we neglect the relative phase between b and c, the effect of it is discussed in subsection 3.2. Parameters in Eq. (9) are defined as where k, j, and B are real. The neutrino mass matrix in Eq. (9) is diagonalized by the rotation of 2-3 families as where V is given in terms of k, j, B, and φ B . Therefore, the PMNS matrix is calculated as follows: which gives three mixing angles, one Dirac phase, and one Majorana phase.
On the other hand, three neutrino masses are related to those parameters as follows: which indicate that the PMNS matrix elements are correlated with neutrino masses. At first, we study the specific Dirac neutrino mass matrices where the relative phase of b/c vanishes. These cases lead to one zero textures for the Dirac neutrino mass matrix (see Appendix A). For case I, we consider the case of j = b/c = −1. The neutrino mass matrix is given as where the Dirac neutrino mass matrix is For case II where c = 0 is put, it is which corresponds to j = ∞ and B = 0 with a finite quantityB ≡ Bj, that is the neutrino mass matrix turns tô Note that the 2-3 family mixing is determined only by k and φ k . If the relative phase between e and f , that is φ k , is zero, the CP symmetry is conserved. It is remarked that the 2-3 family mixing is independent of neutrino masses m 1 and m 2 because the neutrino masses m 1 and m 2 are given in terms of b and f , respectively. This situation makes our numerical analysis simple for the case of IH.
Besides our neutrino mass matrix, the two-zero texture for the Dirac neutrino mass matrix has been discussed in the context of the minimal seesaw model [12,17,18,20]. Especially, for the IH case, it is completely consistent with the experimental data of mixing angles and masses [17]. The prediction of the CP violating phase will be discussed comparing with our result in section 3.

TM 2 : 1-3 family mixing in NH or IH
Let us consider the other case, in which the additional rotation of 1-3 families diagonalizes the neutrino mass matrixM ν . As seen in Appendix A.3, we obtain the Dirac neutrino mass matrix and the left-handed Majorana neutrino mass matrix as follows: respectively. By using the same notation in Eq. (23),M ν is rewritten as: As well as the case of the 2-3 family mixing for IH in subsection 2.2, the 1-3 family mixing is also determined only by k and φ k , namely, it is independent of neutrino masses.

Dirac neutrino mass matrix and vacuum alignment of flavons
The first two VEV's in Eq. (28) preserve the SU (U S) symmetry for 3 and 3, respectively since the generator SU (U S) is expressed as for 3 and 3 , respectively. These VEV's are broken by S, T , and U (see Appendix B). Those flavons have the Z 2 symmetry with elements {1, SU }. This Z 2 symmetry is a residual one after S 4 is broken. The third VEV in Eq. (28) holds the S symmetry for both 3 and 3 , but broken by T . The last VEV in Eq. (28) is not preserved by S, T , nor U unless e = f . Let us reproduce the Dirac neutrino mass matrix in Eq. (8) by introducing those flavons. Suppose that the left-handed lepton L and the flavons φ atm , φ sol are 3 of S 4 , while the righthanded Majorana neutrinos ν R1 and ν R2 are 1 of S 4 . The Higgs field H u is also 1. For the 2-3 family mixing in NH, the Dirac neutrino mass matrix is reproduced by the Yukawa couplings where y atm and y sol are arbitrary coupling constants, Λ is the cut-off scale of S 4 symmetry. The VEV's of φ atm and φ sol are For the case of IH of TM 1 , the assignments of the irreducible representations should be changed from the one of NH. We suppose that L and φ atm are 3, and φ sol is 3 of S 4 , while ν R1 is 1 and ν R2 is 1 of S 4 . It is easily seen that the Dirac neutrino mass matrix is reproduced by the Yukawa couplings in Eq. (30), where the VEV's of φ atm and φ sol are since 3 ⊗ 3 ( ) also implies that L(3)φ(3 ( ) ) = L 1 φ 1 + L 2 φ 3 + L 3 φ 2 for 1 ( ) . Thus, the Dirac neutrino mass matrix is reproduced by the relevant assignments of the flavons. The 2 × 2 diagonal right-handed Majorana neutrino mass matrix is reproduced by help of the auxiliary Z 2 symmetry in NH while an auxiliary Z 2 is not necessary in IH. The diagonal charged lepton mass matrix is realized as well as the Littlest seesaw model [53] by introducing three 3 flavons, which have VEV's: For TM 2 , that is the case of the 1-3 family mixing, the Dirac neutrino mass matrix can be reproduced by the relevant Yukawa couplings in the similar way. However, the situation of symmetry is different from the case of TM 1 because the fourth VEV in Eq. (28) is not preserved in any subgroups of S 4 . In order to obtain desirable Yukawa couplings, we need an auxiliary Z 2 symmetry.

Numerical results
In this section, we discuss the numerical results for the CP violating Dirac phase δ CP as well as neutrino mixing angles. We also discuss the Majorana phases, which contribute the effective mass for the neutrinoless double beta decay (0νββ decay). The neutrino mixing angles are obtained in terms of the PMNS matrix elements U αi of Eq. (1) as follows: The Dirac CP violating phase δ CP can be calculated by using the Jarlskog invariant in Eq. (2): where cos δ CP is fixed by |U µ1 | 2 = 1/6 for the case of 2-3 family mixing, and |U µ2 | 2 = 1/3 for the case of 1-3 family mixing, respectively. The effective mass for the 0νββ decay is given in terms of the Dirac phase δ CP and Majorana phases α and β as follows: where m 1 = 0 or m 3 = 0 for NH or IH, respectively. The Majorana phase β is directly related to the expression of the effective mass |m ee | for NH, while (α − β) is related to |m ee | for IH. Let us explain how to obtain our predictions of the CP violation taking the case I of subsection 2.1 (TM 1 in NH) as an example. The result of global analyses is often used as the inputting data to constrain the unknown parameters [59,60]. In our calculations, we have adopted the result in Ref. [60]. At first, by inputting the data of ∆m 2 13 and ∆m 2 12 within 3σ (1σ) range in Table 1, we remove the two free parameters f 2 /M 0 and B in Eq. (14). The remained free parameter k is scanned in the region of −20 ∼ 20 by generating random numbers in the linear scale. On the other hand, the phase φ B is also scanned in the full region of −π ∼ π in the linear scale. Then, we calculate three neutrino mixing angles. These  Table 1: 1σ and 3σ ranges of the global analysis of the neutrino oscillation experimental data for NH and IH, respectively [60]. calculated values are judged by using the experimental data within 3σ (1σ) range as shown in Table 1. If they are allowed for the experimental data, we keep the point, in which the CP violating phases and |m ee | are calculated. Otherwise, we throw the point. We continue this procedure to obtain enough points for plotting allowed region.
In the next subsections, we show the numerical results in the case of TM 1 and TM 2 for both NH and IH.

Three cases of TM 1 (2-3 family mixing) in NH
Let us show numerical results of case I, II, and III in subsection 2.1. The most interesting case is the case I. The case I corresponds to one of the one-zero texture models, where the (1, 1) element of the Dirac neutrino mass matrix is zero. There are two free parameters, k and φ B after inputting data of neutrino masses in Table 1. As seen in Fig. 1(a), the prediction of δ CP versus sin 2 θ 23 is same as the previous result of TM 1 [45], where the neutrino mass matrix and the neutrino mass hierarchy have not been specified. Unless the magnitude of the dimensionless parameter k is fixed, the predicted sin 2 θ 23 is allowed to take all values within the 3σ range of the experimental data while δ CP is in the region of ±(45 • ∼ 125 • ) depending on sin 2 θ 23 for the inputting data within 3σ. It is helpful to show the predictions for the inputting data within 1σ. The predicted δ CP is in the region of ±(100 • ∼ 115 • ) for the 1σ range of sin 2 θ 23 = 0.412 − 0.450. It is remarked that |δ CP | should be larger than 90 • if the observed θ 23 is in the first octant. This prediction is consistent with the recent data of T2K, which present the allowed 2σ range for the CP violating phase, δ CP , such as (−171 • , −34.4 • ) for NH [3].
As well known, the predicted sin 2 θ 12 is 0.317 ∼ 0.320 [45], which is a characteristic one for TM 1 independent of details of the neutrino mass matrix. This predicted value is inside the region of the experimental data of 1σ range.
We also show the result for the Majorana phase β in Fig. 1(b). For the inputting data within 3σ, β = 0 is allowed, but β = 0 is excluded for the inputting data with 1σ. It is found that β is zero when the Dirac CP violating phase δ CP is ±π/2. The remaining Majorana phase α is arbitrary because of m 1 = 0.
It is remarked that our predictions depend on the parameter k. As seen in Fig. 1(c), it is allowed in two separated region, k −12 ∼ −2 and k −0.1 ∼ −0.5 for the inputting data within 3σ. These regions are related to inverse values each other. The predicted value of sin 2 θ 23 crucially depends on k as seen in Fig. 1(c). Indeed, for the inputting data within We also present k dependence of δ CP in Fig. 1(d). If the observed |δ CP | is larger than 90 • , k is larger than −3. If δ CP is observed accurately as well as sin 2 θ 23 , k is determined with two-fold ambiguity. The k dependence is important because textures of neutrino mass matrix are given by fixing k as discussing later.
We present the sin 2φ B dependence of δ CP in Fig. 1(e), where the prediction is classified by k. It is remarked that the sign of δ CP depends on both sin 2φ B and k. The sign of δ CP is determined by the sign of (sin 2φ B ) for k > −1 while by the sign of (− sin 2φ B ) for k < −1. This result is important for model-buildings.
We show the effective mass for the 0νββ decay |m ee | versus δ CP in Fig. 1(f), where |m ee | is around 1.5 ∼ 3.0 meV. It is minimal if the CP violating phase δ CP is ±π/2. The correlation between |m ee | and δ CP is helpful to test our model for the Dirac neutrino mass matrix.
Finally, we discuss the Littlest seesaw model [53]- [55] in Eq. (20) of subsection 2.1. This corresponds to k = −3 in the case I. As already presented in Ref. [53], this model leads to the predictions around sin 2 θ 23 = 1/2 and the maximal δ CP = ±π/2. We show the results of the Littlest seesaw model by the color of magenta in Figs. 1(a)-(f). As seen in Fig. 1(a), the predicted sin 2 θ 23 and δ CP are 0.451 ∼ 0.544 and ±(80 • ∼ 105 • ), respectively. As seen in Fig. 1(b), the Majorana phase β is in the range from −60 • to 60 • . In both Figs. 1(c) and (d), the vertical magenta lines of k = −3 denote the predictions of the Littlest seesaw model. The effective mass |m ee | is in the minimal regions in Fig. 1(f).
We propose new Littlest seesaw models by observing the result of Fig. 1. For example, k = −5 or −2 (−0.2 or − 0.5) give different predictions for sin 2 θ 23 and δ CP . The Dirac neutrino mass matrices are which will be testable in the future experiments. Thus, our analyses present new Dirac neutrino mass matrices, which suggest new models of the lepton flavors. Next, we discuss the numerical results for the case II, where the (2, 1) element of the Dirac neutrino mass matrix is zero. In Fig. 2(a), the predicted sin 2 θ 23 is restricted near the upper bound of the 3σ range of the experimental data and the CP violating phase is δ CP ±50 • .  The case II may be excluded if the more precise data of sin 2 θ 23 are available in the near future.
Although the prediction of δ CP in Fig. 2(a) seems to be in the partial regions of the one in the case I, the parameter k of the case II is much different from the range of k in case I, that is, k = −1.65 ∼ −1.10 as seen in Fig. 2(b). We find that the sign of δ CP is determined by the sign of (sin 2φ B ) in the region of k = −1.65 ∼ −1.10.
We note the Majorana phase and the effective mass for the 0νββ decay. The Majorana phase β is restricted to the region around ±5 • and ±(140 • ∼ 170 • ). The effective mass |m ee | is predicted to be 2.6 ∼ 3.0 meV.
In the case III, the (3, 1) element of the Dirac neutrino mass matrix is zero. In contrast to the case II, the mixing angle sin 2 θ 23 is strongly restricted to the lower bound of the 3σ range as seen in Fig. 3(a). The case III is almost excluded in the present status of the neutrino oscillation experiments. We find k = −0.86 ∼ −0.71. In this region, the Dirac CP violating phase δ CP is predicted in the region around ±125 • . The Majorana phase β is predicted in the region around ±5 • and ±140 • . It is also remarked that the sign of δ CP is determined by the sign of (− sin 2φ B ) in the region of k = −0.86 ∼ −0.71. The predicted effective mass |m ee | is around 2.7 meV.
Finally in this subsection, we discuss allowed region of the parameters B and φ B in Eq. (10). We show the combined results for cases I, II, III, and the Littlest seesaw model in Fig. 4, where the blue, red, green, and magenta dots denote the case I, II, III, and the Littlest seesaw model, respectively. Thus, the allowed regions of those parameters are clearly different each other.

General case for TM 1 (2-3 family mixing) in NH
In the previous subsection, j ≡ b/c has been fixed. The cases I, II, and III are specific cases for j, which correspond to j = −1, −∞, and 0, respectively. In order to show j dependence of our predictions in detail, we discuss the case in which j is not specified as in Eq. (9).
We show the prediction of δ CP versus j in Fig. 5(a), where j is allowed in negative value. As |j| increases, δ CP reaches ±45 • asymptotically. We show the prediction of sin 2 θ 23 versus j in Fig. 5(b). As far as j is larger than −3, sin 2 θ 23 is allowed to take all values within the 3σ range of the experimental data. However, it is predicted to be larger than 0.5 when j is smaller than −3. It is also easily seen that the cases II (j = −∞) and III (j = 0) are marginal for the prediction of sin 2 θ 23 as seen in Fig. 5(b). We show the prediction of |m ee | versus j in Fig. 5(c). We find that the predicted region of |m ee | never enlarge compared with those predictions of cases I, II, and III even if j is taken to be arbitrary. Finally, we show the allowed region in k and j plane in Fig. 5(d). It is verified numerically that j and k are symmetric for their exchange. The allowed regions within 1σ are rather narrow as seen in Fig. 5. If the error-bar of the inputting data are reduced in the future, both k and j are expected to be constrained considerably.
In these results, we take j = b/c to be real although the relative phase b/c does not vanish in general as discussed below Eq. (9). We have checked numerically that an extra phase of b/c does not contribute the results in Fig.5 while it affects the φ B dependences of δ CP and |m ee | drastically. This situation is easily understood because the experimental data of mixing angles and mass square differences are input.

TM 1 : 2-3 family mixing in IH
Although the case of the NH is rather favored in the experiments of T2K and NOνA [2]- [4], the inverted hierarchy of neutrino masses is still allowed by the experimental data of the neutrino oscillations. Let us show the numerical result of the IH case (m 3 = 0) for the 2-3 family mixing in Eq. (24) since the Dirac neutrino mass matrix is completely different from the one of NH. In this case, the first column of the Dirac neutrino mass matrix is uniquely fixed as (−2, 1, 1) T . There are two free parameters, k and φ k , as seen in Eq. (23). In contrast to the NH case, the ratio e/f is not allowed to be real in order to obtain the CP violation.
We show the predicted δ CP versus sin 2 θ 23 in Fig. 6(a). The prediction is same as the case I of NH in Fig. 1(a), that is ±(45 • ∼ 125 • ). The recent T2K data present δ CP to be in the range of (−88 • , −68 • ) at 2σ range for IH [3]. We also show this range by red lines in the figures as an eye guide. We plot δ CP versus the difference of the two Majorana phases β − α in Fig. 6(b). The difference of two Majorana phases β − α is almost zero. That is α = β.
We also show the k and φ k dependencies of δ CP in Figs. 6(c) and (d). The parameters are allowed in the narrow ranges k = 0.65 ∼ 1.40 and φ k = ±(25 • ∼ 38 • ). Thus, the structure of the Dirac neutrino mass matrix is restricted considerably.
The effective mass |m ee | is predicted to be around 50 meV since β − α is almost zero and |m ee | does not depend on δ CP because of m 3 = 0.
In the context of the minimal seesaw model, the two-zero texture for the Dirac neutrino mass matrix has been examined in some works [12,17,18,20]. This texture is different from our ones. However, this model is completely consistent with the experimental data of mixing angles and masses for IH [17] although this texture is disfavored for NH without some corrections. Since it is interesting to compare the predictions of two models, we add its prediction of δ CP in Figs. 6(a), where (1, 1) and (2, 2) elements of the 3 × 2 Dirac neutrino mass matrix are zero. The predicted δ CP is ±(88 − 93) • in the whole range of allowed sin 2 θ 23 . The case of vanishing (1, 1) and (3, 2) elements is also available, but the numerical prediction is almost unchanged. The prediction of the two-zero texture is a distinctive one from our ones.

TM 2 : 1-3 family mixing in NH or IH
In this subsection, we discuss the case of the 1-3 family mixing. As well known, the predicted sin 2 θ 12 is close to 0.34 [45], which is a characteristic one for TM 2 . The precise data of sin 2 θ 12 provides a crucial test for TM 2 as well as TM 1 .
In Figs. 7 and 8, we show the numerical results for NH and IH, respectively. For both NH and IH cases, δ CP is allowed to take all range in (−π, π) as seen in Figs. 7(a) and 8(a). It is found that the Dirac CP violating phase becomes maximal ±π/2 around the maximal mixing angle of θ 23 = π/4. For NH, the Majorana phase β is allowed to take all values from −π to π as seen in Fig. 7(b). On the other hand, for IH, the difference of two Majorana phases β − α is almost zero as seen in Fig. 8(b).
The predicted value of δ CP is sensitive to the parameter k which behaves differently for NH and IH. In the case of NH, δ CP is 0 or ±π at the lower bound of k, 0.80 or at the upper bound, 1.25, respectively, as seen in Fig. 7(c). In the region of k = 0.80 ∼ 1.25, the predicted δ CP is changed drastically. On the other hand, for the case of IH, δ CP is 0 or ±π at the lower bound k = 0.50 or at the upper bound k = 2.00, respectively, as seen in Fig. 8(c). In the region of k = 0.50 ∼ 2.00, δ CP is predicted for each k. The phase φ k is restricted in the narrow region of ±(165 • ∼ 180 • ) for NH as seen in Fig. 7(d). On the other hand, for IH, the phase φ k is in the region of −37 • ∼ 37 • as seen in Fig. 8(d). In both cases, δ CP and θ 23 become maximal simultaneously if k = 1 is taken.
The effective mass |m ee | is predicted in the region around 1.6 ∼ 4.2 meV and 50 meV for NH and IH, respectively.

Summary and discussions
We have studied the minimal seesaw model where only two right-handed Majorana neutrinos are assumed, focusing on the CP violating phase δ CP . In addition, we have taken the trimaximal mixing pattern for the neutrino flavor (TM 1 or TM 2 ) where the charged lepton mass matrix is diagonal. Owing to this symmetric framework, the flavor structure of the 3 × 2 Dirac neutrino mass matrix is given in terms of a few parameters.
We have examined three cases of the Dirac neutrino mass matrix for TM 1 in NH. It is emphasized that the observation of the CP violating phase determines the flavor structure of the Dirac neutrino mass matrix in the minimal seesaw model. New minimal Dirac neutrino mass matrices have been presented as the result of the numerical study for case I. Our model includes the Littlest seesaw model by King et al. as the one of the specific cases. We have also discussed the case of TM 1 in IH. The structure of the Dirac neutrino mass matrix is restricted considerably. The parameters are determined in the narrow ranges, k = 0.65 ∼ 1.40 and φ k = ±(25 • ∼ 38 • ). We have studied the case of TM 2 in NH and IH. The predicted δ CP is allowed to take all range in (−π, π), but it is sensitive to the parameter k, which is around 1. In both cases, δ CP and the mixing angle θ 23 become maximal simultaneously for k = 1.
Our 3 × 2 Dirac neutrino mass matrix is reproduced by introducing gauge singlet flavons with the VEV's in S 4 flavor symmetry. The specific alignments of the VEV's are derived from the residual symmetry of S 4 group. It is interesting to consider the underlying mechanism which dynamically realizes the required vacuum alignments.
Finally, we add comments. If our Dirac neutrino mass matrices are given at the high energy scale, for example, the GUT scale, one should examine the renormalization group correction for the neutrino mixing matrix. However, it is very small since the lightest neutrino mass vanishes in our framework as seen in Ref. [61].
The sign of the CP violating phase δ CP is not determined in our framework. As well known, CP violating phases in the neutrino mass matrix are related to CP violating phases at the high energy. Since one can discuss the baryon asymmetry of the universe assuming the leptogenesis [62], the sign of δ CP can be predicted in our framework. This work will appear elsewhere.
A.1 TM 1 : Additional 2-3 family rotation in NH At first, we consider the case of NH in TM 1 . Since (1, 1), (1, 2), (2, 1), (1, 3), and (3, 1) entries of the matrix must be zero, conditions for the additional 2-3 rotation to the TBM mixing are where (a, b, c) are supposed to be independent of (d, e, f ). After imposing these conditions on the Eq. (41), the mass matrix is rewritten aŝ where the lightest neutrino mass m 1 is zero. Since the Majorana neutrino mass can be rescaled in the seesaw formula, the right-handed Majorana and Dirac neutrino mass matrices are written by putting p = 1 as respectively. Starting from these textures, we discuss the specific cases which are attractive in the standpoint of the flavor model. We consider one zero textures leading to the additional 2-3 family rotation to the TBM mixing basis. There are three possible patterns of one zero texture as follows: in Eq. (45). Corresponding Dirac neutrino mass matrices can be obtained as One can get another set by exchanging between the first column and second one in the Dirac neutrino mass matrix of Eq. (47). However, the neutrino mass matrixM ν is invariant by this exchange. Therefore, we consider only three cases in Eq. (47). We show the neutrino mass matrixM ν for three cases: where p = 1 is set. The right-handed Majorana and the Dirac neutrino mass matrices are respectively. There is another solution However, this set leads to same structure of the neutrino mass matrix as in Eq. (55). The mass eigenvalue m 1 or m 3 vanishes for NH or IH, respectively.

B Representations of S 4 group
We show the representations of S 4 group in this appendix. All elements of S 4 group are expressed by products of the generators s and t, which satisfy s 4 = t 3 = e, st 2 s = t, sts = ts 2 t, where e is an identity element. These generators are represented on 1 1 , 1 2 , 2, 3 1 , and 3 2 of S 4 group as follows [33,34]: On the other hand, in Ref. [63,64], all elements of S 4 group are also expressed by products of the three generators S, T , and U , which satisfy of S 4 group as follows: