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Cascade Textures and SUSY SO(10) GUT

We give texture analyses of cascade hierarchical mass matrices in supersymmetric SO(10) grand unified theory. We embed cascade mass textures of the standard model fermion with right-handed neutrinos into the theory, which gives relations among the mass matrices of the fermions. The related phenomenologies, such as the lepton flavor violating processes and leptogenesis, are also investigated in addition to the PMNS mixing angles.


Introduction
The neutrino oscillation experiments have suggested that there are two large mixing angles among three generations in the lepton sector while all mixing angles in the quark sector are small. It is known that the current experimental data of leptonic mixing angles [1] is well approximated by the tri-bimaximal mixing [2], which is given by Such suggestive form of the generation mixing gives us a strong motivation to study a flavor structure of the lepton sector. Actually, there are a number of proposals based on a flavor symmetry to unravel it and related phenomenologies have been elaborated [3].
It has been pointed out that the neutrino Dirac mass matrix of the cascade form can lead to the tri-bimaximal mixing at the leading order in the framework of type I seesaw mechanism [4]. The mass matrix of the cascade form is parametrized by 2) and v denotes an overall mass scale. We call this kind of hierarchy and the matrix with such hierarchy, "cascade hierarchy" and "cascade matrix", respectively. On the other hand, the down quark mass matrix of a different hierarchical form, which is can give realistic values of CKM matrix elements. The (1, 1) element, ǫ ′ , of this matrix is smaller than all other elements but this hierarchical structure is close to the cascade form except for ǫ ′ . We call this type of hierarchy "hybrid cascade (H.C.) hierarchy", and the matrix with such a hierarchy "hybrid cascade (H.C.) matrix". The neutrino Dirac mass matrix of a cascade form gives nearly tri-bimaximal generation mixing and the down quark mass matrix of a H.C. form realizes the CKM structure. The fact gives us a strong motivation to comprehensively investigate the quark and lepton. Actually, a proposal to embed such cascade textures into a supersymmetric (SUSY) SU(5) grand unified theory and texture analyses have been presented [5]. For comparison with a SU(5) case and its results, we investigate embedding cascade hierarchies into a SUSY SO(10) GUT in this paper, which is also one of fascinating grand unified models.
The paper is organized as follows: In section 2, we give a brief review of cascade hierarchies for the fermion masses and mixing angles. In section 3, we embed the cascade where M u , M d , M νD , M e , and M R are mass matrices of up-and down-type quarks, neutrino Dirac, charged lepton, and right-handed neutrino, respectively.
Next, we comment on the cascade textures in a SU (5) case. The SU (5) GUT predicts a relation between mass matrices for the down-type quark and charged lepton, 6) due to an unification of matter contents. As discussed above, since only mass matrix of a H.C. form are allowed for M d in the study of cascade texture, the mass matrix for charged lepton should also have the H.C. from. On the other hand, some hierarchical structure of the mass matrices for the up-type quark and right-handed neutrino are allowed as long as induced mixing angles from these matrices can be treated as collections for the CKM and PMNS structures, respectively. Therefore, we can parametrize the mass matrices of the cascade or H.C. form for the fermions as (2.10) without O(1) coefficients for all elements.
Here v u and v d are vacuum expectation values (VEVs) of up-and down-type Higgs fields in a supersymmetric scenario, and the overall factor ξ d and ξ ν could be small. We also notice that the Georgi-Jarlskog (GJ) factor [6] is introduced to mass ratio between the down-type quarks and charged leptons for each generation, 3 Cascade hierarchies in SO(10) GUT

A SUSY SO(10) Model
We consider about embedding the (hybrid) cascade hierarchical mass matrices into SO(10) GUT in this paper. A simple SO(10) GUT predicts relations between mass matrices the up-type quark and neutrino Dirac, (3.1) in addition to the relation (2.6). As discussed above, since only the mass matrix of the H.C. form is allowed for M d , the mass matrix for the charged lepton should also have the H.C. form like in SU (5) case. For the up-type quark sector, a simple SO(10) case predicts a GUT relation of the mass matrices M u ≃ M νD , and the up-type quark mass matrix M u should be restricted to a cascade form because the cascade form of neutrino Dirac mass matrix is needed for generating the tri-bimaximal neutrino mixing at the leading order. For the structures of right-handed neutrino mass matrix, some arbitrary matrices are allowed as long as induced mixing angles can be treated as collections for the PMNS matrix.
To demonstrate the idea, we consider a simple SUSY SO(10) model, which the Standard Model (SM) fermions with the right-handed neutrino are included into the spinor 16-dimensional representation, ψ. To give suitable fermion masses, we introduce the Higgs fields, i.e. two Higgs 10-plets, H 1,2 and two Higgs1 26-plets,∆ 1,2 . There are several ways to break SO(10) down to the SM. Here, we consider a minimal framework where the breaking of SO(10) is achieved by the Higgs 210-plet [7,8], Φ, which breaks SO(10) down to Pati-Salam group: SU(4) C × SU(2) L × SU(2) R . We choose that the Pati-Salam group is broken further down to the SM via the VEV of the SM singlet component in∆ 2 and the VEV also gives Majorana masses for the right-handed neutrinos. Since this singlet VEV gives the non-vanishing contribution to D-term in the superpotential resulting in the unwanted source of SUSY breaking at high energy (close to the GUT scale), we introduce a 126-plet, ∆, whose SM singlet component obtains the VEV to cancel the D-term contribution (for instance, see [9,10]). Here we include two 10-plets, H 1,2 because the mass matrices of the up-type quark and down-type quark have to be different in order to predict the correct CKM mixing angles, as well motivated from the previous discussion. Moreover, we also need one1 26-plet,∆ 1 , in order to achieve the GJ relations (2.11), that is, to give the factor of −3 in the (2,2), (2,3), and (3,2) elements of the charged lepton mass matrix with respect to that of the down quark mass matrix. * Another1 26-plet,∆ 2 , is introduced to generate the different texture for the right-handed neutrino masses and also break the Pati-Salam group to the SM. In our setup, there are six pairs of Higgs doublets, φ u = (H 1,u , H 2,u ,∆ 1,u ,∆ 2,u Note the label u, d refer to the SU(2) L doublet component with hypercharge ±1/2 within the GUT multiplet. The mass matrix M H can be diagonalized by U T φu M H U φ d , which U φu , U φ d are unitaty matrices acting on φ u and φ d respectively. In the diagonal basis, the Higgs fields are given by For the sake of the study, we will not specify how SO(10) is broken in detail, but by some doublet-triplet splitting mechanism (for instance see [12,13,14]) we will assume that H u = (φ ′ u ) 1 and H d = (φ ′ d ) 1 have mass at the electroweak scale while the others are so heavy and decoupled from the low energy theory. The Higgs fields, H u,d are the two Higgs doublets of the Minimal Supersymmetric Standard Model (MSSM).
The superpotential of the model is given by which can be written in terms of the SM components as follows [15]: where the doublet component in the GUT multiplet can be written in term of the MSSM For the neutrino sector, we assume that the SU(2) L triplet component,∆ 2,T , and the SM singlet component,∆ 2,S , in∆ 2 , give tiny Majorana masses for the left-handed neutrinos and the heavy Majorana masses for the right-handed neutrinos respectively. This results in the seesaw formula as follow: 26 2 v R and M νD is the Dirac mass term whose structure will be discussed below. Since the triplet VEV ∆ 2,T = v L depends on parameters in Higgs superpotential (for instance, see [9]), we assume that the VEV is tiny such that the second term in Eq (3.4) dominates, resulting in the type I seesaw dominance. Note that the singlet VEV, ∆ 2,S = v R , is of order 10 16 GeV.
After the electroweak symmetry is broken via the doublet VEVs, H u,d = v u,d , the fermion masses are given by where we assume that the main contribution for the up-type quark (Dirac neutrino) masses comes from the coupling to H 1 while for the down-type quark (charged lepton) masses they arise from the H 2 and∆ 1 couplings. These can be achieved through the following assumptions: We impose the hierarchical forms to the Yukawa couplings, The structure ofỸ1 26 2 will be discussed in the following sections in term of the right-handed neutrino mass matrix, M R =Ỹ1 26 2 v R . These lead to the fermion mass matrices at the GUT scale as (3.15) where O(1) coefficients for all elements have been dropped.

Cabibbo fitting of cascade mass matrices
The cascade hierarchical parameters are determined by observed values. It is naturally expected that such hierarchies are originated from a symmetry and/or some dynamics in a high energy regime rather than solely determined by the magnitudes of Yukawa couplings.
Although the origin of the hierarchies is not specified in the analysis, one can estimate and study the relative magnitudes of the hierarchies introducing a small parameter in the mass matrices. In the following, we choose the Cabibbo angle, sin θ c ≃ λ = 0.227, as a fitting parameter, and study significant implications of the cascade SO(10) scenario. Then we 16) for up-quark mass matrix of the cascade form and for down-quark one of the H.C. form at GUT scale, where we utilized values of quark masses listed in [16].
Notice that the ξ d is a parameter, which determines a ratio between (3,3) element of Yukawa matrices for up-and down-type quarks, and thus, it is correlated with the tan β as, .
( 3.18) As the results we can write cascading textures at GUT scale as , (3.20) where k d ≥ 1 is needed to obtain suitable mass eigenvalues after diagonalizing these matrices. It should be remembered that M e ∼ M d but the additional GJ factor −3 is multiplied to the (2,2), (2,3), and (3,2) elements of M e as discussed in the previous section.

Neutrino sector
Next, we consider the structure of neutrino mass matrices. In the cascade model [4,5], cascade parameters are constrained as 21) in order to preserve the tri-bimaximal mixing at the leading order with at the 3σ level [1]. Due to the SO(10) GUT relation M νD ≃ M u , the neutrino mass matrix can be parametrized as where we note that an opposite sign between (2,2) and (2,3) elements is experimentally required to obtain the tri-bimaximal mixing as commented in [4]. †

Diagonal M R case
Let us discuss the case of a diagonal Majorana mass matrix of the right-handed neutrinos, The cascade model requires the normal mass hierarchy of light neutrino mass spectrum in order to realize a nearly tri-bimaximal mixing [4]. The mass eigenvalues can be estimated as with a leading order corrections of O(m 1 ). In order to understand the hierarchical structure of the mass matrix and the constraints on the cascade parameters, we write down the effective neutrino mass matrix as We find that if the terms in the first and second lines are leading contributions, the tribimaximal mixing can be realized at the leading order. In order that the first term in the † Since the Dirac mass matrix and the up-type quark mass matrix are constrained to have the same structure, this opposite sign is also imposed on the up-type quark mass matrix (3.19).
second line does not spoil the structures given in the first line, m 1 ≪ m 2 , m 3 is required.
That is the reason why the neutrino mass spectrum in the cascade model should be the normal hierarchy. In the case, we can well approximated as and m 3 ≃ |∆m 2 31 |. (3.29) Now we can obtain the following four constraints on the cascade parameters: (i) The neutrino masses should satisfy m 1 ≪ m 2 . This means that x 1 ≥ 17 for the parameters by utilizing (3.25) and (3.26). This constraint leads to small mass of the lightest right-handed neutrino as shown later. (ii) In order to be consistent with a experimental data for the neutrino mass squared difference as where m 3 ≃ |∆m 2 31 | is taken. (iv) The hierarchy m 2 ≫ m 3 λ 4 is required in order that the second term in the last line of (3.28) does not spoil the democratic structure in the first line. This gives a constraint x 1 − x 2 ≥ 5. The above four constraints restrict the neutrino Dirac mass matrix of the cascade form and the right-handed one of the diagonal form to textures presented in Tab. 1 and 2. We find that the minimal model for the neutrino mass matrices is described by (x 1 , x 2 ) = (17,10) given in Tab. 1. In this case, mass spectrum of the right-handed neutrinos is estimated as Here we comment on the predicted mixing angles from cascade model. The mixing angles of the cascade model deviate from the exact tri-bimaximal mixing angles even if the right-handed neutrino mass matrix is diagonal. The mixing angles can be estimated as in a perturbative method, ‡ where parameters θ (1) ij indicate deviations from the exact tribimaximal mixing angles. These are elements of the following mixing matrix, (3.39) In our notation, the experimentally observed PMNS mixing matrix is given by V PMNS ≃ V TB V (1) P M , where the P M is a diagonal phase matrix.

Non-diagonal M R case
We discuss the case of non-diagonal M R , which is generically allowed in the context of the cascade textures. First, we define the diagonalized right-handed neutrino mass matrix, ‡ See [5] for a detailed derivation.
where M R is a non-diagonal mass matrix for the right-handed neutrinos but mixing angles among each generation are assumed to be small in order to preserve the tri-bimaximal mixing. If the mixing angles among each generation of the right-handed neutrino are small enough, U νR can be written by (3.41) up to the first order of θ R,ij (i, j = 1 ∼ 3). After the seesaw mechanism, we obtain the Majorana mass matrix of light neutrinos in low-energy as, When we operate the V TB to M ν as V T TB M ν V TB , the neutrino mass matrix is where This mass matrix can be rewritten by where M 0 comes from the diagonal elements of M R , which is given by In (3.45) M off has effects from the off-diagonal elements of M R . In order to obtain experimentally accepted mixing structure without unnatural cancellations, we focus only on a case that the collections from the off-diagonal elements of M R are small enough not to spoil the nearly tri-bimaximal mixing constructed by the cascade neutrino Dirac mass matrix. This means that the resultant structure of neutrino mass matrix given in (3.43) should not be drastically differed from the (3.46), and thus the magnitude of neutrino mass eigenvalues (3.25)- (3.27) and the above four constraints should be satisfied at the leading order even in non-diagonal M R case. These discussions give the following neutrino mass eigenvalues up to the next leading order, § where m R i include effects from the off-diagonal element of M R described by Finally, the PMNS mixing angles including collections from the off-diagonal elements are . (3.61)

Charged lepton and quark sectors
At the end of this section, we study the charged lepton and quark sectors. Under the condition of the SO(10) scenario, we examine quantitative features of the masses and mixing angles.
We take the charged lepton mass matrix as (3.62) in our study. The magnitudes of cascade parameters can be partially evaluated from the observed values of charged lepton masses, and are given by |λ d | ≃ m µ /(3m τ ) and |δ d | ≃ 3 √ m e m µ /m τ . We find that the corrections from the charged lepton sector are generally small; the total leptonic mixing angles can be written as 3.65) at the first order of perturbations.
Next, we comment on the quark sector. One must remember that the mass matrix of the H.C. form is motivated for the mass spectra and mixing angles of quark sector. The mixing matrices for the up-and down-sector are given by the cascading mass matrices (3.19) and (3.20). From the mass matrices, one can estimate the following mixing angles

Phenomenologies
In this section, we perform the numerical study of phenomenologies based on the above analyses of the cascade textures for the quark and lepton sectors: the PMNS mixing angles, lepton flavor violation (LFV), baryon asymmetry of the Universe (BAU) via thermal leptogenesis.

PMNS mixing angles
Firstly, we show numerical analyses of the generation mixing angles of the quark and lepton sectors predicted from the cascade model. Here, we investigate two typical types of minimal texture for the neutrino Dirac and right-handed Majorana neutrino mass matrices, and Model II : 3) (4.4) for the cases of the condition x 1 − x 2 = 7. ¶ Here a ν , b ν c ν , and a R , · · · , f R are complex numbers whose absolute values are taken as 0.4 ∼ 1.4. In both models, the following charged lepton, up and down quark mass matrices are utilized and where a e , · · · , e e are also complex values whose range are the same as a ν , b ν , c ν , and a R , · · · , f R . Moreover, notice that the mass matrices of down and up quarks have the same structures of ones of charged lepton, except for the GJ factor, and neutrino respectively because of the mass relations of SO (10)  m d m s = 0.051 (7), m s m b = 0.018 (2), y b = 0.34 (3), (4.8) (3), λ c = 0.227(1),ρ = 0.22 (6),η = 0.33 (4), (4.9) at the GUT scale, where the numbers in parentheses mean an uncertainty in last digit [16].
We have taken a ν and a e as 0.49 ≤ a ν ≤ 0.53 and 0.31 ≤ a e λ ≤ 0.37 to fit y t and y b in our numerical Here A, λ c ,ρ, andη are the Wolfenstein parameters which correspond to three mixing angles and one phase in the CKM matrix; y t and y b represent largest eigenvalues of Yukawa matrices for the up-type and down-type quarks, respectively. These values are calculated with the 2-loop gauge coupling and 2-loop Yukawa coupling renormalization equation taking tan β, threshold corrections γ t,b,d , and an effective SUSY scale as tan β = 38, 21, γ t = 0, and m SUSY = 500 GeV, respectively. The threshold corrections are approximated by where µ, A t , mq, g 3 , and M 3 indicate supersymmetric Higgs mass µH u H d , soft top quark tri-linear coupling, mass of the squarkq, strong coupling, and gaugino soft breaking mass, respectively [17]. The utilized values of SUSY can lead to the relations at the GUT scale, (4.11) in a good accuracy. Therefore, the above values can automatically reproduce the experimental observed charged lepton masses at the low energy. The numerical calculations of Model I suggest that the predicted region of solar angle covers the experimental upper bound but the model gives a predicted lower bound around 0.29 sin 2 θ 12 . On the other hand, a constrained region is predicted for the reactor angle as 0.002 sin 2 θ 13 0.007. This result is one of important predictions of the present cascade textures. It might be checked by the upcoming DoubleChooz [18], RENO [19], and DayaBay [20] experiments as the reactor experiments in addition to the accelerator experiments such as T2K [21] and NOνA [22]. * * Finally, the atmospheric angle covers the current experimentally allowed region. However, there is a relatively clear correlation between the magnitudes of reactor and atmospheric angles in Fig. 1. On the other hand, the result for Model II can cover the experimental allowed region because of largeness of corrections from the right-handed neutrino mass matrix but there is an upper bound of sin 2 θ 13 which is sin 2 θ 13 0.015. Therefore, we can conclude that the minimal cascade textures in the context of SUSY SO(10) (Model I) can lead to clear predictions, which are 0.29 sin 2 θ 12 and 0.002 sin 2 θ 13 0.007, and relatively sharp correlations between the reactor and atmospheric angles. In the next-to minimal cascade model (Model II)  thus it might be checked by the future experiments. It would be too difficult to distinguish other cascade textures even if the future precision data of neutrino oscillation experiment could be used. It is worth studying a new method to check the models.
At the end of this subsection, we give a brief comparison between our results and ones from a similar SO(10) approach, which utilizes type II seesaw mechanism and a simple ansatz such that the dominant Yukawa matrix has rank one [24]. † † The work gives some predicted regions for leptonic mixing angles based on three typical models in this approach:

Lepton flavor violation
Next, we investigate the branching ratios of LFV process l i → l j γ, in our cascade lepton mass matrices. We suppose that soft SUSY breaking masses of sleptons are universal at the GUT scale, Λ GUT , for simplicity. In the case, the off-diagonal matrix elements are generated by radiative corrections from the Yukawa couplings of neutrinos [25]. The oneloop renormalization group evolution gives the left-handed slepton masses. The leading contribution is estimated by where m 0 and a 0 are the universal SUSY breaking mass and three-point coupling of scalar superpartners given at the GUT scale. The branching fractions of each LFV process are roughly given by  Table 5: Typical magnitudes of branching ratios for lepton flavor violating rare decay process.
where α, M W , and m SUSY are the fine structure constant, W boson mass, and a typical mass scale of superparticles, respectively. These branching ratios are estimated by (4.14) (4.16) for both Model I and II. Here we define B ≡ (M W /m SUSY ) 4 tan 2 β and take m 0 = |a 0 | = m SUSY . Typical magnitudes of the branching ratios are shown in Tab. 5. In these analyses, Λ GUT = 2 × 10 16 GeV is taken. These results are compared with the current experimental upper bounds at 90% confidence level [26,27]: The magnitudes of the branching ratios for the lepton flavor violating process in the model with the applicable heaviest right-handed Majorana neutrino mass are far below the experimental limit. Once one fix the value of tan β, the current experimental limit gives lower bound on m SUSY , which is e.g. m SUSY ≥ 41.6 GeV from Br(τ → µγ) ≤ 4.5 × 10 −8 with tan β = 38. In the present case, the bound can be easily satisfied.

Leptogenesis
At the end of this section, we examine whether the thermal leptogenesis [28] works in our model. The CP asymmetry parameter in the decay process of right-handed neutrino, R i , is given by , (4.18) where L i and H denote the left-handed lepton and Higgs fields. An approximation for ǫ i at low temperature is estimated by [29], and D being a diagonal phase matrix to make the eigenvalues M i real and positive. The function F describes contributions from the oneloop vertex and self-energy corrections, We here define the resultant CP asymmetry, η CP , as the ratio of the lepton asymmetry to the photon number density n γ , η CP = 135ζ(3)κsǫ 1 /(4π 4 g * n γ ). In the equation, κ, s, and g * are the efficiency factor, entropy density, and the effective number of degrees of freedom in thermal equilibrium. They are given by [30], s = 7.04n γ , g * = 228.75, and (4.20) The m eff is the effective light neutrino mass defined as m eff ≡ |(M † νD M νD ) 11 /M 1 |. The BAU, η B , is transferred via spharelon interactions as η B = −8η CP /23. Finally, the baryon asymmetry in our model is predicted as η B ∼ 10 −23 sin θ B , where θ B ≡ θ 3 − θ 1 and θ i = arg(M i ). These results are compared with the current observational data at 68% confidence level from the WMAP 7-year resulting in the standard ΛCDM model [31]. We can see that the baryon asymmetry generated through the leptogenesis is too small to explain the BAU. This is because the hierarchy in the Dirac neutrino mass matrix in the SO(10) model is determined by the up-type quark mass matrix. Therefore, there is no freedom to adjust the Dirac neutrino mass matrix such that the BAU can be generated in the present model. It is expected that enough BAU might be realized by extending our cascade model to the inverse seesaw case, see [32]. In the case, the structure of effective light neutrino mass matrix is slightly changed but the realistic PMNS mixing angels would be obtained.

Discussion
At the end of the paper, we give a comment on phenomenological aspects of proton decay.
In general for SUSY GUTs, there are three sources that mediate the proton decays. The first one comes from the dimension-6 operators, arising from the exchanging of the heavy gauge bosons. Note that this type of operators exists in both non-SUSY and SUSY GUTs. These operators are significantly suppressed by 1/Λ 2 GUT , therefore, we have no problem with the proton decay from these operators if Λ GUT is large enough, i.e. Λ GUT ≥ 10 16 GeV. The second source is from the dimension-5 operators, arising from the exchanging of color triplet Higgsino fields. In this case, the proton decay contribution is suppressed by 1/M H , where M H is the mass of the color triplet Higgsinos. To suppress the proton decay contributions, the mass M H has to be very heavy, which can achieved by some doublet-triplet splitting mechanism [12,13,14]. The third contribution arises from the dimension-4 operators, which are not suppressed by the GUT scale, however, these operators are eliminated by the R−parity. In the class of SO(10) models, which do not contain the spinor 16 or16 as the Higgs fields, the R−parity is conserved. Otherwise, in order to avoid the proton decay contributions from these operators, the R−parity has to be introduced by hands. Since we consider an SO(10) model without using the spinor Higgs representations, we have no problem with these dimension-4 operators [33,34].

Summary
We have done texture analyses of cascade model in supersymmetric SO(10) model. The neutrino Dirac mass matrix of a cascade form can realize the tri-bimaximal mixing at the leading order while the down-quark one of a H.C. form can lead to realistic structure of CKM mixing. This fact gives us a strong motivation to study cascade hierarchical textures in a grand unified theory.
We analytically clarified possible structures of neutrino Dirac, charged lepton, quark, and right-handed neutrino mass matrices by estimating collection from them to the tribimaximal mixing. The numerical analyses based on two typical models have been also presented. The minimal cascade texture in the context of SUSY SO(10) GUT can lead to clear predictions for the PMNS mixing angles, which are 0.29 sin 2 θ 12 and 0.002 sin 2 θ 13 0.007, and relatively sharp correlations between the reactor and atmospheric angles. This result is a hot topic for the upcoming experiments of the reactor neutrino mixing angle. It might be checked by such future experiments. We have also shown that our typical cascade models can pass the constraints from the lepton flavor violation searches. For generating the BAU, we cannot generate enough asymmetry through the thermal leptogenesis mechanism in our model. Therefore, we need other mechanisms to generate the BAU.