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How Far Can the SO(10) Two Higgs Model Describe the Observed Neutrino Masses and Mixings ?

Can the SO(10) model with one {\bf 10} and one {\bf 126} Higgs scalars give the observed masses and mixings of quarks and leptons without any other additional Higgs scalars? Recently, at least, for quarks and charged leptons, it has been demonstrated that it is possible. However, for the neutrinos, it is usually said that parameters which are determined from the quark and charged lepton masses cannot give the observed large neutrino mixings. This problem is systematically investigated, and it is concluded that the present data cannot exclude SO(10) model with two Higgs scalars although it cannot give the best fit values of the data.


I. INTRODUCTION
SO(10) GUT model seems to us the most attractive model when we take the unification of the quarks and leptons into consideration. However, in order to reproduce the observed quark and lepton masses and mixings, usually, a lot of Higgs scalars are brought into the model. So it is the very crucial problem to know the minimum number of the Higgs scalars which can give the observed fermion mass spectra and mixings. A model with one Higgs scalar is obviously ruled out for the description of the realistic quark and lepton mass spectra. Two Higgs models were initially discussed by Mohapatra et.al [2].
In the previous paper [1], we discussed 2 Higgs scalars, {10 and 126} case and {10 and 120} case, and showed that they reproduce quark-lepton mass matrices unlike the conventional results [3]. One of new points of our approach is that we adopt general forms of Yukawa couplings allowable in the SO(10) framework. However, we did not argue there about the neutrino mass matrix since it may incorporate additional assumptions like the seesaw mechanism etc.
One of the merits of the SO(10) model is that it includes a right-handed Majorana neutrinos in the fundamental representation and naturally leads to the seesaw mechanism. Also some papers claim that two Higgs model ({10 and 126+126}) does not reproduce the large mixing angle of the atmospheric neutrino deficit [4]. So in this paper we apply our method developed in the previous paper to the neutrino mass matrix, fitting the other parameters of the quark-lepton mass matrices. Our model has the two Higgs scalars {10 and 126} both of which are symmetric with respect to the family index. Therefore mass matrices are symmetric whose entries are complex valued. We do not adopt another choice {10 and 120}. For it does not involve the mass term of the right-handed Majorana neutrinos which are the ingredients of the seesaw mechanism.
We begin with the short review of our previous work [1]. In the case where two Higgs scalars, φ 10 and φ 126 , are incorporated in the SO(10) model, the mass matrices of quarks and charged leptons have the following forms Here M 0 and M 1 are the mass matrices generated by the Higgs scalars φ 10 and φ 126 , respectively. Also c 0 and c 1 are the ratios of VEV's, and φ u and φ d denote Higgs scalar components which couple with up-and down-quarks, respectively. Eliminating M 0 and M 1 from Eq.(1.1), we obtain where Since M u , M d , and M e are complex symmetric matrices, they are diagonalized by unitary matrices U u , U d , and U e , respectively, as where D u , D d , and D e are diagonal matrices given by Since the Cabibbo-Kobayashi-Maskawa (CKM) matrix V q is given by the relation (1.3) is re-written as follows: Therefore, we obtain the independent three equations: where κ = c u /c d . By eliminating the parameter c d , we have two equations for the parameter κ: , (1.13) where (1.9) 3 , for instance, means the right-hand side of Eq. For the relation between the values at µ = m Z and those at µ = Λ X (Λ X is a unification scale), see Ref. [1]. The purpose of the present paper is to investigate whether these solutions can give reasonable values for observed neutrino masses and mixings or not.

II. THE NUMBER OF PARAMETERS IN THE SO(10) MODEL WITH TWO HIGGS SCALARS
As we have discussed in the previous section, among four freedoms of complex {c 0 , c 1 } or {c d , κ}, we have been able to fix the three of them, κ and |c d |. This is not accidental. Let us discuss the situation in detail in the SO(10) two Higgs model.
In the previous paper, by using the relation (1.8), we have investigated whether there is a set of parameters which can give the 13 observable quantities D e , D u , D d , and V q or not. We can rewrite Eq.(1.8) as The quantities D e , D u , D d , and V q are inputs, and the quantities |c d |, κ, and A e are the parameters which should be fixed from those observed quantities. In general, an n × n unitary matrix for n generations has n 2 parameters. Therefore, the number of the parameters is On the other hand, the number of equations is N (eqs) = n(n + 1), (2.5) because Eq.(2.1) is symmetric. Therefore, the number of the unfixed parameters is given by for n = 3, i.e., the 13 observed quantities fix the parameters |c d |, κ, and A e , but 1 parameter σ remains as an unknown parameter.
In the present paper, we will try to predict neutrino masses and mixing matrix by using the observed quantities D e , D u , D d , and V q and the parameter values |c d |, κ, and A e fixed by Eq.(2.1). SO (10) GUT asserts that the Dirac neutrino mass matrix M D is given by the form and Majorana mass matrices of the left-handed and righthanded neutrinos, M L and M R , are proportional to the matrix M 1 : where M 0 and M 1 are related to the quark and charged lepton mass matrices M u , M d , and M e as follows: Then the neutrino mass matrix derived form the seesaw mechanism becomes 13) In the present paper we adopt c L = 0. Also we may ignore the phase of c R which does not affect the observed values. Therefore, we can rewrite Eq.(2.13) as with Differently from the previous work, the quantities D ν and V ℓ are unknown parameters at the present stage. Since and A e is fixed from Eq.(2.1), the number of the unknown parameters in Eq.(2.20) is Of course, the unknown parameters in A ν contain the n unphysical parameters which cannot be determined because of the rephasing in the fields e L . Therefore, the number of the unknown parameters is and from the number of equations N (eqs) = n(n + 1) in Eq.(2.14), we obtain the number of the unfixed parameters as This means that we can predict neutrino masses and mixing completely if we give the two values |c R | and σ. The numerical predictions will be investigated in the next section.

III. NUMERICAL RESULTS
Here we discuss the numerical results of the neutrino mass spectrum and neutrino mass matrix. For example, we use the set in Eq. (1.18). Even if the other sets are used, our results are scarcely changed. The allowed values of neutrino mass square differences and lepton flavor mixing angles depict complicated tracks with moving σ ≡ arg c d (Fig. 2). This figure shows a general tendency that the lepton flavor mixing angles θ 12 and θ 23 get larger as σ approaches to 3π/2. For an illustration we take σ = 149π/100, then these values become There still remain a little bit discrepancies between our results and experiments. However our results are much improved in comparison with those by Babu-Mohapatra [2] in which they obtained sin θ 12 = 0−0.3, sin θ 13 = 0.05, and sin θ 23 = 0.12 − 0.16. The purpose of the present paper is to study the general tendency of the fittings and not to pursuit the precise data fitting, for the data themselves are not affirmative and we have theoretical ambiguities not incorporated in the present data fitting like the renormalization group effect.
In the choice of Eq.(3.1), we have 3) Let us choose the free parameter |c R | so as to result in small neutrino masses, for example when |c R | = 3.2 × 10 14 , we have ∆m 2 23 = 1.5 × 10 −3 [eV 2 ]. Here there arises a question what makes the two flavor mixing angles large. We need to investigate the mixing matrices U e and U ν which diagonalize M e and M ν , respectively. Those are obtained as Here, |U e11 |, |U e12 |, |U e21 |, |U e22 | > ∼ 0.5 for the charged lepton mass matrix and |U ν22 |, |U ν23 |, |U ν32 |, |U ν33 | > ∼ 0.7 for the neutrino mass matrix. Therefore the components of the lepton flavor mixing matrix become |V l11 |, |V l12 |, |V l21 |, |V l22 |, |V l23 |, |V l32 |, |V l33 | > ∼ 0.5: The mixing angle θ 23 becomes larger, while the mixing angle θ 12 smaller, if we take the smaller value of |m t | or |m d |, or larger |m c |, or |m b |, |m s | than their center values. As a simple example, the shift of |m d | and |m s | causes the change of mixing angles and neutrino mass square differences as depicted in Fig.2. Fig.2 shows that the θ 23 and θ 13 can approach the 99%C.L. of SK [7]+CHOOZ [8] but θ 12 and ∆m 2 12 are out of the range of 99% − 99.9%C.L. of SOLAR [9]+CHOOZ.  (Table I (i)). Since the minimum |m d | for (b) gives bad fitting, we have changed σ from 149π/100 to 146π/100 , which is denoted by star (Table I (ii)). Thus our result approaches the 99%C.L. of SK+CHOOZ and 99.9%C.L. of SOLAR+CHOOZ.

IV. DISCUSSION
Since there are only two basic matrix M 0 and M 1 in this model, the number of parameters in Eq.(2.1) and (2.14) is where α and β are (e, µ, τ ). They correspond to neutrinoless double beta decay [11] for α = β = e, µ− e conversion (µ − +(A, Z) → e + +(A, Z −2) for α = µ, β = e, and K decay (K − → π + µ − µ − ) for α = β = µ [12] etc. In Fig.3 we have depicted σ dependence of m ν αβ / ∆m 2 23 . In the case of Eq.(3.1), these values become as follows. For instance, if we input ∆m 2 23 = 1.5 × 10 −3 [eV 2 ], m ee becomes 0.0012 [eV]. This is smaller than the experimental value of the next generation experiments such as GE-NIUS [13], CUORE [14], and MOON [15]. Jarlskog parameter [16] appears in three generations Here we have adopted the notation The σ dependence of J is depicted in Fig.4 However, it needs careful consideration that J drastically changes at σ ≃ 3π/2. m αβ and J in the cases of Table  I (i) and (ii) discussed in Fig.2 are also listed in Table  II (i) and (ii). In this paper we have discussed how far the SO(10) two Higgs scalar model describes the quarklepton masses and mixing parameters. We can conclude that this model cannot be rejected within the existing data. It should be remarked that the whole parameters can be decided from the existing data in principle.

ACKNOWLEDGEMENT
The work of K.M. is supported by the JSPS Research Grant No. 10421.