Neutrino masses and mixings in a predictive SO(10) model with

It has recently been shown that a minimal SO(10) model with a single {\bf 10} and single {\bf 126} Higgs field breaking B-L symmetry predicts large solar and atmospheric mixings in agreement with observations if it is assumed that the neutrino mass obeys the type II seesaw formula. No additional symmetries need to be assumed for this purpose. Understanding CP violation in the renormalizable version of the model however requires a significant non-CKM source. In this paper we show that if we extend the model by the inclusion of a heavy {\bf 120} dimensional Higgs field, then it can accomodate CKM CP violation while remaining predictive in the neutrino sector. Among the predictions are:(i) solar mixing angle in the observed range; (ii)\theta_{13} in the range of 0.1 to 0.26; (iii) the Dirac phase close to maximal for the central value of the solar mixing angle.


Introduction
The simplest grand unified model for understanding small neutrino masses appears to be the SO(10) model [1] for the following reasons: (i) it automatically brings in the right handed neutrino, N R , needed to implement the seesaw [2] mechanism since it fits in with other standard model fermions in the 16 dimensional spinor representation (ii) it contains the SU(4) c symmetry [3] which relates the quark and lepton coupling parameters and in turn helps the predictivity of the model in the neutrino sector by reducing the number of parameters; (iii) it also contains the B-L symmetry [3,4] needed to keep the right handed neutrino masses below the Planck scale and provides a group theoretic explanation of why neutrinos are necessarily Majorana particles.
While all these make the SO(10) models appealing for neutrino mass studies, detailed quantitative predictions generally involve too many parameters limiting the predictive power unless extra symmetries (e.g. family symmetries etc) are imposed on the theory. One exception to this is the class of models that uses only one 10 and one 126 Higgs multiplet to generate fermion masses [5]. The original set of papers on this model [5,6] used type I seesaw formula for neutrino mass is given by is the Dirac mass of the neutrinos and M R is the mass matrix of the right handed neutrinos. These predictions are now in contradiction with experiments.
It was subsequently pointed out in ref. [7] that if one uses type II seesaw formula for the neutrino masses [8] instead, the model automatically predicts large atmospheric mixing angle due to the fact that bottom quark and tau lepton masses converge towards each other when extrapolated to the GUT scale. The question remained whether this works for three generations and can lead to a realistic model for neutrinos. It was shown in ref. [9] that the same b − τ mass convergence not only leads to a large solar mixing angle, but also to a small and detectable value for U e3 ≡ sin θ 13 . A detailed numerical analysis was carried out that showed that the model is indeed in agreement with present neutrino data and in particular the prediction of a "large" value for θ 13 which makes this model testable at the current as well as at the proposed long base line neutrino experiments.
In the three generation neutrino discussion in ref. [9], the Yukawa couplings of fermions were assumed to be real and all CP violating effects were assumed to originate from the supersymmetry breaking sector. It is however interesting to check if one can accommodate the CKM phase in the model by introducing phases in the couplings. A detailed investigation of the minimal model where CP violation is introduced through complex Yukawa couplings (as in the standard model) showed [10] that compatibility with neutrino data requires the CKM phase to be outside the first quadrant whereas the standard model CKM phase is in the first quadrant [11]. This would seem to imply that in order to understand observed CP violation in this model, one must invoke a significant non-CKM source for CP violation (as in the model with real Yukawa couplings) e.g. CP violation from the supersymmetry breaking sector. This could very well be true. However, since all observed CP violating phenomena seem to be explainable by the CKM model, it is important to see whether one can explain both CKM CP violation and neutrino mixings by a minimal modification of this SO(10) model. There are also other issues such as SUSY CP problem that one needs to address in the context of supersymmetry and it would be interesting to see how these can be addressed in this model.
In this paper, we propose a very minimal way to incorporate CP violation into the model, which not only leads to a predictive model in the neutrino sector but also seems to have wider implication beyond just explaining CKM CP violation. For instance, the model presents a solution to the SUSY CP problem. In order to attain our goal, we include a heavy 120 field with an extra Z 2 symmetry which we will call "parity" symmetry imposed on the model 1 . At energy scales below the mass of the 120 field, the effect of this field is to appear as a higher dimensional contribution to the Yukawa couplings. This effective theory has the following properties. Despite the fact there are now three extra parameters in the model, the theory still remains predictive in the neutrino sector. Secondly, the mass matrices for quarks and leptons are hermitian, which therefore has the potential to solve other problems of supersymmetric models such as the SUSY CP problem. In this paper we focus only on the neutrino sector.
The main results of this paper are as follows: (i) using type II seesaw formula we are able to accommodate the CKM CP phase while keeping the model predictive in the neutrino sector; for example, we predict the solar mixing angle in the right range and U e3 ≥ 0.1; (ii) the model has the potential to solve the SUSY CP problem and (iii) it predicts the Dirac phase of PMNS matrix to be near maximal for the central value of the solar mixing angle tan 2 θ ⊙ ≃ 0.4.
The paper is organized as follows: in sec. 2, we introduce the model with the inclusion of the 120 Higgs field and write down the fermion mass formulae in the general case; in sec. 3, we impose parity symmetry on the model making it predictive in the neutrino sector; in sec. 4, we discuss the predictions for neutrino mixings and Dirac CP phase for neutrinos; in sec. 5, we present our conclusions and discuss the outlook for the model.

SO(10) Model and CP violation
We start by writing down the Yukawa interactions of our model, which are responsible for the discussion of neutrino masses and mixings. The Yukawa superpotential involves the couplings of the 16-dimensional matter spinor ψ i with 10 (H), 126 (∆), and 120(A) dimensional Higgs fields: The Yukawa couplings, h and f , are symmetric matrices, whereas h ′ is an anti-symmetric matrix due to SO(10) symmetry. They are all complex matrices in general.
Once the SO(10) symmetry breaks down to the standard model symmetry, we have four pairs of Higgs doublets arising from the H, ∆, and A Higgs fields. There may also be other Higgs doublets e.g. in 210 multiplet. Under the G 422 = SU(4) c ×SU(2) L ×SU(2) R decomposition we have the following representations that contain the Higgs doublets of up and down type: one pair arises from H ⊃ (1, 2, 2), one pair comes from ∆ ⊃ (15, 2, 2), and two pairs come form A ⊃ (1, 2, 2) + (15, 2, 2). We assume that one pair of their linear combinations, H u and H d , remains massless (mass is ∼ O(v wk )) and become the MSSM Higgs doublets. As for other pairs, they all have GUT scale masses. Using the light Higgs doublets, the MSSM Yukawa couplings below the GUT scale and the right-handed Majorana neutrino mass terms can be written as 1 Very different extensions of the model that use 120 but no symmetry have been discussed in ref. [13].
where Q, U c , D c , L, E c , N c are the quark and lepton superfields which are all unified to the 16 spinor ψ field. ∆ L is an SU(2) L triplet Higgs field and ∆ 0 R is a neutral component of SU(2) R triplet, both part of the 126 field. Even though both of them have GUT scale masses we have included them with the MSSM superpotential because their vevs lead to light neutrino masses via the seesaw mechanism.
The gauge coupling unification requires that the 126 Higgs field acquires VEV at or close to the GUT scale. We also need to introduce a 126 Higgs field to satisfy the D-flat condition to maintain supersymmetry down to the weak scale. Though the 126 Higgs field does not couple to the fermions, the pair of Higgs doublets in the 126 mix with the doublets arising from the other Higgs multiplets, since 126 couples to the other Higgs multiplets with non-zero coupling. These five pairs of Higgs doublets, H 10 u,d , ∆ u,d , ∆ u,d , A s u,d and A adj u,d are mixed and the light pair of Higgs doublet can be written as where U H and V H are unitary matrices. The superscripts s and adj stand for SU(4) c singlet and adjoint pieces. We have temporarily ignored the doublets that may arise from other multiplets in the theory such as 210. It is important to stress that in order to obtain one pair of MSSM Higgs doublets from five pairs at GUT scale, one needs to do a fine tuning of parameters. We have enough parameters in the Higgs superpotential that this is possible to achieve. We have also checked that we do not have any light color triplet fields. The results given below remain unchanged in their presence. The Dirac mass matrices of quark and leptons are where the three mass matrices in the expression are given by where v u is a vacuum expectation value of MSSM Higgs doublet H u , and c i are Clebsch-Gordon (CG) coefficients. The coefficients r i and A are written as where cot β is a ratio of vacuum expectation values of doublet Higgs fields, cot β = v d /v u . The Majorana mass matrices of left-and right-handed neutrinos prior to seesaw diagonalization are given by where v L and v R are vacuum expectation values of ∆ L and ∆ R , respectively. As already mentioned, since v R is expected to be close to the GUT scale, this implies that v L is ∼v 2 weak /(ηM GU T ) ≪ v weak , where η is a coupling constant in the Higgs potential. The Majorana mass matrix of the heavy right handed neutrino is proportional to M 126 . The light neutrino mass matrix is given by the mixed type II seesaw formula, As discussed in earlier papers [7,9], there are regions of the parameter space in the theory where the first term will dominate; we will call this the pure type II seesaw case. If on the other hand, we consider the parameter space where the second term is dominant we will call this type I seesaw. The bulk of our results will be for the pure type II case.

Parity invariance and a predictive model for neutrinos
In order to see if the model is predictive for neutrinos, let us count the number of parameters in the theory. In the basis, where M 126 is real and diagonal, there are 3 real parameters in M 126 , 6 complex parameters in M 10 and 3 complex parameters in M 120 . We also have 5 complex parameters in the Eqs.(9-13) as well as the vevs of the Altogether, there are 31 real parameters in the fermion sector, and, therefore, we do not have any prediction for the neutrino mixings.
In order to be predictive in the leptonic sector of the model without imposing any flavor symmetry, we require the theory to be invariant under a parity symmetry. As we will see, it makes the Dirac mass matrices Eqs.(5-8) hermitian and leaves a total of 17 real parameters in the fermion sector making the model predictive. If we further require that the 120 Higgs field has a mass much higher than the GUT scale, its only manifestation is as an effective dimension four term in the superpotential. The reduces the number of parameters to 15 increasing the predictive power of the model. We explore both the cases with 17 and 15 parameters in a subsequent section.
We now define the parity transformation in the G 422 basis. We write SU(4) indices by µ, ν, SU(2) L indices by α, β and SU(2) R indices byα,β. The SO(10) spinor ψ and χ are decomposed as Bi-doublet Higgs fields in the 10, 126 and 120 are written as H αα , ∆ µ ν αα , A αα , and A µ ν αα . Then the Yukawa interactions are written in the following (up to overall factors) where c 2 is a CG coefficient. The Lagrangian is written as and We consider the symmetry under the following parity transformation, Of course, χ is also transformed in same manner.
In the Higgs sector, the transformations of the (1, 2, 2) and (15, 2, 2) sub-multiplets under G 422 are: A consequence of the parity symmetry (23), is that the coupling matrices h and f real and symmetric and h ′ antisymmetric and imaginary; the parameters r i (i = 1, 2, 3, 4) and A in the Eqs. (9)(10)(11)(12)(13) are real. This considerably reduces the number of parameters in the theory and further makes the mass matrices for all charged fermions hermitian. Let us clarify our motivation for introducing the 120 Higgs field. When the 120 Higgs field is absent, the fermion mass matrices are complex symmetric matrices in the absence of the parity symmetry and we have the following relation in the pure type II case where m c and m u contributions in (2,2)  . Assuming that the atmospheric mixing is maximal, we obtain the neutrino mass matrix, Eq.(24), as where m 0 and ǫm 0 are eigenvalues of (2-3) block, and ǫ 2 ∼ ∆m 2 sol /∆m 2 A and m 0 ∼ 2m s . Thus, the solar mixing and 13 mixing are proportional to |V ub /V cb − V us | and tan 2θ sol ∼ tan 2θ 13 /ǫ. Therefore, those mixing angles also depend on the KM phase, δ KM . Since V ub = |V ub |e −iδ KM , the KM phase in the first quadrant gives a smaller value for solar mixing angle rather than in the second quadrant. In order to obtain the proper value of solar mixing angle, we have to choose a smaller value of ǫ. However, in the model without 120, the ǫ parameter, which is a function of the strange quark mass, is constrained due to the fitting of three charged-lepton masses (especially electron mass), and we do not have proper fitting of the solar mixing angle data in the case where the KM phase is in the first quadrant. We can verify the situation in a precise analysis in the pure type II case [10]. In the type I case, things are more complicated, but it has been shown that it is not possible to fit the neutrino oscillation data in this model with the above minimal Higgs choice [12]. The mass squared ratio is constrained due to the charged-lepton mass fitting and it cannot be small enough when the KM phase is in the first quadrant. The mass squared ratio is a free parameter in the model with 120 since the additional parameter A in the sum rules Eqs. (5)(6)(7)(8) can fit the electron mass, and therefore we can explain a smaller KM phase. Thus, we employ the 120 Higgs field to explain the large solar mixing angle along with the KM phase in the first quadrant. Interestingly, even though we have introduced a new Higgs field, the number of parameters is less than the minimal Higgs choice with most general CP phases due to the constraint of parity symmetry.
We further note that if the 120 field is heavier than the GUT scale, its effect on the physics at the GUT scale comes from a higher dimensional operator of the form ψΓΓΓψHΦ/M, where Φ is a 210 Higgs field, so that we get the relation r 1 = r 3 = r 4 . In other words, the choice r 1 = r 3 = r 4 is not an adhoc choice but can be guaranteed in a natural manner.
We now note that in the presence of the parity symmetry, since all the mass matrices are hermitian and also the µ-term and the gluino masses are real, the most dangerous graphs contributing to large electric dipole moment of the neutron are absent [18]. Therefore this model has the potential to solve the so-called SUSY CP problem.
We also wish to recognize that the Z 2 CP symmetry we impose is broken at the GUT scale by the VEVs of 126, 210 and 45 Higgs fields which break SO(10) symmetry. The light MSSM doublet Higgs fields are no more CP eigenstates, and thus there is no cosmological domain wall problem at weak scale in the model.

Near Bi-Maximal Solution for Neutrino Oscillation
As already noted in section 3, under the parity symmetry, M 10 and M 126 are real symmetric matrices, M 120 is a pure imaginary anti-symmetric matrix, and the coefficients r i and A are real parameters in the Eqs. (5)(6)(7)(8). We can therefore rewrite the charged-lepton and Dirac neutrino mass matrices in terms of the other mass matrices as follows: where The up-and down-type quark mass matrices are hermitian, and are written as where D u = diag(±m u , ±m c , m t ), D d = diag(±m d , ±m s , m b ) and V is the CKM matrix, and U is a unitary matrix. We note that m t and m b component in the D u and D d can be made to be positive without loss of generality. Because of the parity symmetry, M d − r 3 M u must be a real symmetric matrix. We fix the flavor basis as The unitary matrix U is determined by r 3 , up to phase matrix P , U = PŪ, P = diag(e iφ 1 , e iφ 2 , 1) The parameters are now 6 quark masses (with signatures), 3 mixing and 1 KM phase in the CKM matrix, and the coefficients c d , c u , φ 1 , φ 2 , r 3 , r 4 , A. There are 17 parameters in all. For example, the three charged-lepton masses can be used to determine c d , c u , Ar 4 . The remaining 4 parameters give the neutrino oscillation parameters.
In the case where the 120 Higgs field is heavier than the GUT scale, it manifests itself as an effective dimension four operator of the form ψψHΦ/M. As a result, the Higgs doublets in 120 are decoupled. This leads to a reduction in the number of mixing parameters. This translates into the the relation r 1 = r 3 = r 4 since the vev ratio that defines r 1,3,4 is the same. In this case, there remain only two parameters describing the neutrino sector. They can be determined by two of the parameters from the neutrino oscillation data and the remaining neutrino parameters can then be predicted.
Interestingly, if we have the relation r 1 = r 3 , the matrix M d − r 3 M u is proportional to the light neutrino mass matrix M L and the diagonalizing matrix U become close to MNSP matrix in the pure type II case. Therefore, if r 1 = r 3 ∼ m b /m t and the (3,3) element of M d − r 1 M u is suppressed, we have a large atmospheric mixing. The mass squared ratio is of the order of 10 −2 , which is the right order seen in the experiment, only if ). Furthermore, since the (3,3) element of M 126 is suppressed for that choice of r 1 , the bottom-tau mass unification is satisfied and it is consistent with the renormalization group flow for the case of tan β ∼ 50. Now let us study the prediction of the model in the case where r 1 = r 3 = r 4 . In this case, we have 15 parameters in the model. After fixing the quark masses (with signatures) and the CKM parameters, we are left with 5 parameters, c d , A, φ 1 , φ 2 , and r 1 . Since mass squared ratio is a function of r 1 , we can fix the parameter r 1 by the experimental value of ∆m 2 sol /∆m 2 A . The three charged-lepton masses can be used to fix c d , A, and φ 2 . As a result, the neutrino oscillation parameters, θ A , θ sol , |U e3 |, and one CP phase δ M N SP are predicted by only one phase parameter φ 1 . Interestingly, the atmospheric mixing θ A doesn't depend on the phase φ 1 very much, and the θ A is really predicted when we fix the quark masses and mass squared ratio of light neutrino.
It should also be noted that the other arbitrariness in the model is due to the choice of the signs of different fermion masses, since the sign of a fermion mass is unobservable. We find that only for the two choices of the signs given below, we obtain acceptable solutions: The solutions we present correspond only for these two choices of signatures. In Fig.1, we show the prediction of the atmospheric mixing sin 2 2θ A as a function of the mass squared ratio. The lines (a) and (b) in the figure are for the set of quark and lepton mass signatures given above which give acceptable solutions for neutrinos. We can obtain large atmospheric mixing angles with the proper choice of quark masses in their allowed range, mixings and KM phase [19] with the choice of the set of signatures, especially for case (a). The most important input parameter for obtaining a large atmospheric mixing is the strange quark mass. We show the strange mass dependence of θ A in Fig.2 in the case where the mass squared ratio R ≡ ∆m 2 ⊙ /∆m 2 A is 0.03. The strange mass in the figure is the running mass at 1 GeV. In order to obtain the experimental constraint sin 2 2θ A > 0.9, we need the parameter region where the strange mass has a larger value. The bottom quark mass dependence is not negligible, and a larger value of bottom mass is preferred to obtain maximal atmospheric mixing.
After fixing all quark and lepton data and also the mass squared ratio, R, we can fit the solar mixing angle by choosing the free phase parameter φ 1 . Then, |U e3 | and δ M N SP are predicted. In Fig.3, we show the correlation between solar mixing tan 2 θ sol and |U e3 | by varying the phase parameter φ 1 . We give two lines for the cases (a) and (b), and the two lines correspond to different mass squared ratios, R = 0.02 and R = 0.07. From the figure, we can see that ∆m 2 ⊙ /∆m 2 A = 0.07 is not favored in the 3σ range of the experimental data of solar neutrino and U e3 for the case (a). In Fig.4, we present the prediction of the sin δ M N SP in the case where R = ∆m 2 ⊙ /∆m 2 A = 0.02. The CP phase can be of any value in the range of the experimental data of solar neutrino. If we restrict the mass signatures to the (a) case, sin δ M N SP could be predicted to be ±0. 9.
The most interesting feature of this model is the prediction of |U e3 |. In Fig.5, we show the bottom quark mass dependence of the |U e3 | prediction. The bottom mass is defined as a running mass at m b . Since a large SUSY correction to the bottom quark mass can be induced in the large tan β case, the running bottom mass can be large, and the larger bottom mass gives smaller value of |U e3 |. In a similar way, larger tan β predicts smaller |U e3 | since the larger tan β gives larger bottom quark mass at GUT scale. In Fig.6, we show the plot of |U e3 |. Each point is dotted for different quark mass and mixings which are randomly generated in the experimentally allowed region. We can see that the model predicts a lower limit for |U e3 | of about 0.1.
In Fig.7, we can see the KM phase dependence of the prediction of the model. The lines in the figure are drawn by changing φ 1 for different values of KM phases. The mass squared ration is set to be 0.02. For a smaller KM phase, the lines shift to the smaller solar mixing. In this model, the experimentally allowed solar mixing can be obtained in the first quadrant KM phase, contrary to the minimal model without the 120 Higgs field. We can see that a smaller value of |U e3 | can be allowed for the larger values of KM phase.
Since all the parameters of the model are now determined, it can be used to make other predictions. As an example, we have calculated the magnitude of lepton flavor violating process µ → e + γ in the model (see Fig.8). Note that the predictions are in the range currently being probed by experiments [21]. We use the mSUGRA universal boundary conditions at the GUT scale i.e. m 0 (universal scalar mass), m 1/2 (universal gaugino mass),A 0 (universal trilinear mass ). The other two parameters are sign of µ and tan β. The dots in the plots are produced for various model points generated by fitting the fermion masses and mixing angles. We can see that the BR of µ → eγ can be large for smaller values of m 1/2 . The lightest neutralino is the dark matter candidate in this model and we satisfy the 2σ range of the recent relic density constraint Ω CDM = 0.1126 +0.008 −0.009 [22] in the parameter space. When we satisfy the relic density constraint, the m 0 gets determined. We choose A 0 = 0 and µ > 0. The right handed masses have hierarchies and therefore get decoupled at different scales. The flavor-violating pieces present in Y ν and f induces flavor violations into the charged lepton couplings and into the soft SUSY breaking masses e.g. m 2 terms etc. Also, an additional symmetry between the GUT and the v R scale (type I) helps to induce flavor violation [12]. The electric dipole moment of electron is smaller than the experimental reach in the range of parameter space showed in the figure.
We comment that in the case of type I seesaw, the large mixing solution is a sharp resonance solution and the solution is not stable to predict the mixings contrary to the type II seesaw.

Discussion
In this paper we consider the prediction of neutrino masses and mixings for an SO(10) model where fermion masses receive contribution from the presence of three Higgs multiplets 10, 120 and 126. We impose a parity symmetry on the model, so that it has very few parameters which enables prediction of two mixing angles and all the CP phases in the neutrino mixing matrix.
The advantage of this model over the most minimal SO(10) model is that now the CKM phase is in the first quadrant as required by the standard model analysis of all observed hadronic CP violation. We also wish to emphasize that this one of the few models in the literature that can predict leptonic CP phases. As far as experimental tests of this model are concerned, the parameter U e3 ≡ sin θ 13 is predicted to be large like the most minimal SO(10) model [7,9] but is somewhat smaller i.e. U e3 ≥ 0.1. This can be tested in the next round of planned long baseline experiments. We also predict that the Dirac phase for neutrinos can be maximal. Furthermore the model has the potential to solve SUSY CP problem due to the fact that all fermion masses are hermitian. The model also predicts observable amount of LFV in muon decay.  Figure 3: The relation between solar mixing angle and |U e3 | is plotted. Each line is plotted by varing a free phase parameter φ 1 . The experimentally allowed region in 3σ of recent data fitting is 0.3 < tan 2 θ sol < 0.6 and |U e3 | < 0.26 [20].