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Can four-zero-texture mass matrix model reproduce the observed quark and lepton mixing angles and CP violating phases?

We reconsider an universal mass matrix model which has a seesaw-invariant structure with four-zero texture common to all quarks and leptons. The CKM quark and MNS lepton mixing matrices of the model are analyzed analytically.We show that the model can be consistent with all the experimental data of neutrino oscillation and quark mixings by tuning free parameters of the model. It is also shown that the model predicts a relatively large value for (1,3) element of the MNS lepton mixing matrix, |(U_{MNS})_{13}|^2 \simeq 2.6 \times 10^{-2}. Using the seesaw mechanism, we also discuss the conditions for the components of the Dirac and the right-handed Majorana neutrino mass matrices which lead to the neutrino mass matrix consistent with the experimental data.


I. INTRODUCTION
The discovery of neutrino oscillation [1] indicates that neutrinos have finite masses and mix one another with near bimaximal lepton mixings in contrast to small quark mixings.
have been investigated in the literature. We think that quarks and leptons should be unified. Therefore, it is an interesting approach to investigate a possibility that all the mass matrices of the quarks and leptons have the same form which can lead to the large lepton mixings and the small quark mixings simultaneously. Since the mass matrix model is intended to be embedded into a grand unified theory (GUT), it is desirable for the model to have the following features: (i) The structure is common to all the mass matrices, M u , M d , M e , and M ν for up quarks (u, c, t), down quarks (d, s, b), charged leptons (e, µ, τ ), and neutrinos (ν e , ν µ , ν τ ), respectively. (ii) Since we assume the seesaw mechanism [32] for neutrino masses, the structure should conserve its form through the relation M ν ≃ −M D M −1 R M T D . We shall call this structure as a seesaw-invariant form. Here M D and M R are, respectively, the Dirac and the right-handed Majorana type neutrino mass matrices, which are also assumed to have the same structure.
In this paper, as typical mass matrices which have the features mentioned above, we reconsider hermitian mass matrices M f for f = u, d, e, and D and symmetric mass matrices M f for f = ν, and M with a universal form given by M f = P † f M f P f , for f = u, d, e, and D, (1.1) Here P f is a diagonal phase matrix given by 3) and the matrix M f is defined by  [33] and the Maki-Nakagawa-Sakata (MNS) lepton mixing matrix [34] of the model in terms of By taking a special value for this free parameter as d f = m f 3 + m f 1 , the model with the same structure has been discussed in Ref. [35]. However, in this special choice, the model predicts a rather smaller value for (1,3) element of the CKM quark mixing matrix than the corresponding observed experimental data. In order to overcome this defect in the quark sector, we treat d f as a free parameter in the present paper and show that the observed small CKM quark mixings as well as large MNS lepton mixings can be well derived by fine tuning of the free parameters.
It has been claimed [36,37,38] that four-zero-texture models for quarks are ruled out at the three σ level from the experimental data for sin2β. However, we shall show from an analysis with use of the free parameter d f that the quark mixing angles and CP violating phase δ q in our model are consistent with the data at one σ level, so that the sin2β is also consistent at the same level.
This article is organized as follows. In Sec. II, we discuss the diagonalization of mass matrix of our model. In Sec. III, approximations we use are presented. The analytical expressions of the quark mixing matrix of the model are given in Sec. IV. In Sec. V, the lepton mixing matrix of the model is given. Sec. VI is devoted to a summary.

II. DIAGONALIZATION OF MASS MATRIX
We now discuss a diagonalization of the mass matrix M f . First we argue a diagonalization of M f given by where R f i (i = 1, 2, and 3) are defined by The expressions of the components a f , b f , and c f in terms of m f 1 , m f 2 , m f 3 , and d f are presented as From the condition that a f , b f , and c f are real, we have the allowed region of d f given by The cases in which 0 < d f < |m f 1 | or |m f 3 | < d f are not allowed. We also have the following sign assignments for the eigenmass m f i : Namely m f 2 should be taken negative while m f 1 and m f 3 are positive for the case in which On the other hand, m f 1 should be taken negative while m f 2 and m f 3

III. APPROXIMATIONS
We present approximated expressions of the orthogonal matrix O f for the normal hierarchy, inverse hierarchy, and quasi degenerate cases for the masses m f i . Here we introduce a x f parameter, instead of using d f , defined by The approximated expressions are obtained as follows: The inverse hierarchy and the quasi degenerate scenarios are unfavorable in our model.

IV. CKM QUARK MIXING MATRIX
Let us discuss the quark sector. The mass matrices M u and M d for the u-and d-quarks are, respectively, given by where P u and P d are diagonal phase matrices and M u and M d are given by Eq. (1.4). The The unitary matrix U Lf is described as Therefore the CKM quark mixing matrix U CKM of the model is given by where P ≡ P u P † d is diagonal phase matrix given by Here we take α d1 = α u1 = 0 without any loss of generality.
By using the expressions of O d and O u in Eq. (2.3), the explicit (i, j) elements of U CKM are obtained as where R ui and R di (i = 1, 2, and 3) are given by reproduce the observed data for (U CKM ) ij as will be shown later. This model can be used for the improvement of the previous model [35] in which a rather small value for |(U CKM ) 13 | is predicted.
In the discussions of the CKM quark mixing matrix, we concentrate our attention on the . In this case, using two free By using the rephasing of the up and down quarks, Eq. (4.5) is changed to the standard representation of the CKM quark mixing matrix, Here ζ q i comes from the rephasing in the quark fields to make the choice of phase convention. By using the expressions of U CKM in Eqs. (4.16)-(4.18), the CP violating phase δ q in the quark mixing matrix is given by  and (ii) the other three parameters α 3 , x u , and x d take values in the allowed regions shown in Fig. 1, Fig. 2, and Fig. 3. The best fit is realized for the following values of the parameters:  Finally let us mention the model in Ref. [35]. It corresponds to our present model with the parameter d f fixed as d f = m f 3 + m f 1 , namely x f = 1 − |m f 1 |/m f 3 ≃ 1 and 1 − x f = |m f 1 |/m f 3 . In this case, the following CKM mixing matrix elements are derived as seen from Eqs. (4.16)-(4.18) and (4.21).
Here P ν and P e are diagonal phase matrices and M ν and M e are given by Eq. (2.1). The charged lepton mass matrix M e is diagonalized as where the unitary matrix U e is described as Since the mass matrix for the Majorana neutrinos is symmetric, M ν is diagonalized as where |m 1 |, m 2 , and m 3 are real positive neutrino masses and the unitary matrix U ν is described as (5.6) Here, in order to make the neutrino masses for the first generation to be real positive, we introduce an additional diagonal phase matrix Q ν defined by In the following discussions, we consider the normal hierarchy 2 for the neutrino masses m i , i.e. |m 1 | < m 2 ≪ d ν < m 3 , and the normal hierarchy 1 for the charged lepton masses, i.e. |m e | ≪ m µ ≪ d e < m τ . In this case, the orthogonal matrix O e and O ν are obtained from Eqs. (3.2) and (3.3) with f = e and ν by replacing |m 1f |, m 2f , and m 3f with |m e |, m µ , and m τ , and with |m 1 |, m 2 , and m 3 , respectively. Therefore we have We now discuss the MNS lepton mixing matrix U M N S of the model, which is given by where P ℓ ≡ P e P † ν is diagonal phase matrix and we take P ℓ = diag(1, e iβ 2 , e iβ 3 ), (5.11) without any loss of generality. Thus we obtain 12) where ξ i are complex quantities defined by 14) From Eqs. (5.18) and (5.21), the neutrino oscillation angles and phases of the model are related to the lepton masses as follows: It should be noted that the present model leads to the same results for θ solar and θ atm as the model in Ref. [27], while a different feature for |(U M N S ) 13 | 2 is derived.
On the other hand, we have [40] a experimental bound for |(U M N S ) 13 | 2 exp from the CHOOZ [41], solar [42], and atmospheric neutrino experiments [1]. From the global analysis of the SNO solar neutrino experiment [40,42], we have ∆m 2 12 and tan 2 θ 12 for the large mixing angle (LMA) Mikheyev-Smirnov-Wolfenstein (MSW) solution [43]. From the atmospheric neutrino experiment [1,40] , we also have ∆m 2 23 and tan 2 θ 23 . These experimental data with 3σ range are given by  Therefore we predict the neutrino masses as follows. The predicted value for |(U M N S ) 13 | in Eq. (5.34) is close to the present experimental constraints Eq. (5.27) in contrast to previously proposed model [25] [27]. Therefore our model will be checked in neutrino factories in near future.
In the preset model, the neutrino mass matrix M ν is given by Now we discuss the requirements for the mass matrix elements of M D and M R to realize the above structure for M ν . In our model we have assumed the seesaw mechanism M ν = −M D M −1 R M T D and following structure for M D and M R . (5.36) where P D = diag (e iα D1 , e iα D2 , e iα D3 ). Here we assume a real symmetric M R for simplicity.
In this case, we have Therefore, the following conditions should be satisfied in order to realize our M ν in Eq. (5.35), Namely, it turns out that the large lepton mixing angle is realized through the seesaw mechanism by using the following M D and M R , with c D = d D and a hierarchy conditon It should be noted that the components b D in M D , b R , and c R in M R which are denoted as asterisks are not important for reproducing the large lepton mixing angle at all.

VI. CONCLUSION
We have reconsidered the mass matrix model with a universal and seesaw-invariant form of four-zero structure given by , and e (6.1) The analytical expressions for the CKM quark mixing maitrix are derived as functions of the four parameters x u , x d , α 2 , and α 3 . We do fine tuning of the parameters so as to reproduce the experimental data. It turns out that the CKM quark mixing matrix can be consistent with the data at the special value of the parameter given by α 2 ≃ π/2 and in the allowed regions among α 3 , x u , and x d as shown in Fig. 1, Fig. 2, and Fig. 3.
We have also analyzed the MNS lepton mixing matrix analytically and shown that it is consistent with the observed large lepton mixings. The model predicts a relatively large which is close to the experimental upper bound at present. Therefore a determination of the finite value for |(U M N S ) 13 | 2 in near future experiment will be expectable in our model.
We have assumed the seesaw mechanism M ν = −M D M −1 R M T D and the same four-zero structure for M D and M R . Within this framework, we have derived the conditions given by Eqs. (5.40) and (5.41) for the components of M D and M R to realize our structure for M ν .