Neutrino mass and low-scale leptogenesis in a testable SUSY SO(10) model

It is shown that a supersymmetric SO(10) model extended with fermion singlets can accommodate the observed neutrino masses and mixings as well as generate the desired lepton asymmetry in concordance with the gravitino constraint. A necessary prediction of the model is near-TeV scale doubly-charged Higgs scalars which should be detectable at the LHC.

Within the SO(10) GUT framework, the intermediate symmetry breaking scales are fixed through the Renormalization Group (RG) equations which reflect the gauge couplings' evolution with energy. In the simplest SO(10) GUT it is well-known that M R turns out to be ∼ 10 16 GeV. (c) In a SUSY context there is an additional constraint, namely, to ensure that there is no overabundance of gravitinos in the universe. To maintain consistency with this, it has been demonstrated [6,7,8] that the lepton asymmetry must be generated through the decay of a heavy neutrino whose mass does not exceed ∼ 10 7−9 GeV in order to prevent a washout, whereas leptogenesis through the canonical Type-I seesaw mechanism sets the lower bound 4.5 × 10 9 GeV. These conflicting requirements have acted as obstacles to a successful implementation of this attractive possibility.
In this letter we propose a remedy for these maladies confining ourselves to the SUSY SO(10) GUT. If sterile -i.e., SO(10) singlet -leptons are introduced, one for each generation [9,10,11,12], then a novel way can be found to meet the demands outlined in the previous paragraph.
The uncharged fermions in this model, per generation, are the following: a left-handed neutrino ν, a right-handed neutrino, N , and a sterile neutrino, S. For the three generation neutral fermion system, the mass matrix on which we focus is: where m D , M N , M X , and µ are all 3×3 matrix blocks.
It is not unreasonable to expect that the mass matrix in eq. (1) will alleviate the tension, summarised earlier, between light neutrino masses and adequate low-scale thermal leptogenesis. As discussed below, the double see-saw structure for the light neutrino masses, arising from eq. (1), also decouples it to some extent from low-scale leptogenesis; M N , M X , and µ appear in different fashions in the expressions. Utilizing an extension of the Minimal Supersymmetric Standard Model (MSSM) by the addition of RH neutrinos and extra fermion singlets, which results in a neutrino mass matrix of the structure of eq. (1), Kang and Kim [12] have found solutions to both the above issues. There, m D has been identified, as is done in the MSSM, with the charged lepton mass matrix. On the other hand, in the SO(10) model which is espoused here, quark-lepton symmetry [2] identifies the neutrino Dirac mass matrix m D with the up-quark mass matrix whose 33 element is nearly 100 times heavier. This, along with other GUT constraints, pose additional hurdles in addressing the problems in SUSY SO (10).
We work in a basis in which the down-quark and charged lepton mass matrices are diagonal. This ensures that the entire mixings in the quark and lepton sectors can be ascribed to the mass matrices of the up-type quarks and the neutrinos, respectively. Using quark-lepton unification, the quark masses, and the Cabibbo-Kobayashi-Maskawa mixing angles, one therefore obtains m D , upto O(1) effects due to RG evolution.
Although we do not assign any direct vev to the RH-triplets, through a ∆ R exchange involving a trilinear coupling in the superpotential, where f is a typical Yukawa coupling of Majorana type. If m ∆ R is around 1 TeV, which can be arranged by a tuning of the D-parity breaking term in the Lagrangian, the entries of M N are O(10 11 ) GeV. Without any loss of generality, M N can be chosen to be diagonal.
GeV while x R ∼ 10 7 GeV. The states ∆ + R and Re(∆ 0 R ) are eaten up as Goldstone bosons by the W ± R and W 0 R fields and ∆ ++ R and Im(∆ 0 R ) survive as physical states with mass ∼ 1 TeV. The Type-II seesaw contribution to the light neutrino mass matrix is damped out in this case because of the large masses of the left-handed Higgs triplet leading to m II ≃ 10 −5 eV -10 −6 eV [14,15]. Further, the vev of χ L is zero or negligible.
Block diagonalization of the mass matrix in eq. (1) in the limit in which we are working (i.e., where m ν , M S , and M are 3×3 matrices. The light neutrino masses are in a double see-saw pattern and µ is determined once M X is fixed. It may be noted that the mass matrix structure in eq. (1) ensures that the type I see-saw contribution is absent and M N remains unconstrained by the light neutrino masses. This freedom in M N -a hallmark of the model -is vital to ensure adequate leptogenesis. The eigenstates of M S , which we denote by T i , (i = 1, 2, 3), are superpositions of the sterile neutrinos S (predominant) and the right-handed ones N . These states are found to lie well-below 10 9 GeV, consistent with the gravitino constraint. In fact we show that this model allows successful leptogenesis at a temperature T ≃ M T ≃ 5 × 10 5 GeV, which is nearly 4 orders below the maximum allowed value. Further, the singlet fermions decay through their mixing with the N i which is controlled by the ratio M X /M N . The latter, which have masses O(10 11 ) GeV and are off-shell, decay to a final lφ state, where l is a lepton doublet and φ the up-type MSSM Higgs doublet. This two-step process -for which tree and loop diagrams are depicted in Fig. 1 (SUSY contributions are small) -results in a lepton asymmetry of the correct order. Because of the large value of M N ≫ M X , a small S − N mixing results naturally in the T i which in turn guarantees the out-of-equilibrium condition to be realized near temperatures T ≃ M T .
A quantitative analysis of this programme has been carried out using the Boltzmann equations determining the number densities in a co-moving volume Y T = n T /n S and Y L = n L /n S , where n T , n L and n S are respectively the number densities of the decaying neutrinos, leptons and the entropy: where Γ T D , Γ T s and Γ ℓ W represent the decay, scattering, and wash-out rates, respectively, that take part in establishing a net lepton asymmetry. We refrain from presenting their detailed expressions here [16]. The Hubble expansion rate H(z), where z = M T /T, and the CP-violation parameter are given by Our target is to use eqs. (2) and (3) to obtain an acceptable solution within the framework of SUSY SO (10). Through an exhaustive analysis we find an appropriate choice of the block matrices appearing in eq. (1) which guarantees adequate leptogenesis while maintaining full consistency with the observed neutrino masses and mixing as well as the gravitino constraint. The mass scales are fixed as dictated by the RG evolution of gauge couplings in SUSY SO(10) when effects of two dim.5 operators scaled by the Planck mass are included [16,17,18]. The strategy we follow is to choose the matrix M X first.
To minimize the number of independent parameters, we take the matrix F to be real and diagonal, which is reflected in M X . Then using m D , as fixed by quark-lepton unification, µ is determined from the double see-saw formula given in eq. (2). Using these inputs, one has to examine, by trial and error, different choices of M N for adequate lepton asymmetry generation. The results for the development of the leptonic asymmetry as the universe evolves are shown in Fig. 2. They are obtained with the choice M X = diag (0. 2, 0.3, 0.4)x R with x R = 6 × 10 6 GeV. The neutrino Dirac mass matrix, m D , is constructed utilizing quark-lepton symmetry; the up-type quark mass eigenvalues and the CKM mixings are taken at the PDG [19] values with the CKM-phase as 1 radian. The neutrino masses are fixed so as to satisfy ∆m 2 21 = 8.0 × 10 −5 eV 2 and ∆m 2 32 = 2.5 × 10 −3 eV 2 with the lightest neutrino taken massless. The neutrino mixing angles used are θ 23 = 45 • , θ 12 = 32 • , and θ 13 = 7 • . No other CP-phases are introduced in the lepton sector except the one through the CKM matrix for quarks. With these inputs the matrix µ is calculated following eq. (2). For the RH-neutrino we use the mass matrix M N = diag(0.1, 0.5, 0.9) × 10 11 GeV. This is consistent with m ∆ R ∼ 1 TeV. We assume that in the very initial stages the number densities, Y T i , i = 1, 2, 3, and the leptonic asymmetry, Y L , are zero. The chosen input values of the mass parameters result in a T i mass spectrum such that only one state -T 1 -is above the kinematic threshold for lφ production (m T 1 = 3.9 × 10 5 GeV) and the lepton asymmetry results through its decay. This ensures that the leptogenesis is consistent with the gravitino bound. It is seen from Fig. 2 that T 1 decays fall out of equilibrium as the universe expands (inset) and Y L achieves the right order (∼ 10 −10 ) starting off from a vanishing initial value while Y T steadily tends towards Y eq T . We stress again that an important outcome of the symmetry breaking is that out of the triplet ∆ R the components ∆ ± R and Re∆ • R are absorbed as longitudinal modes of the broken generators of SU (2) R × U (1) B−L . The physical states are ∆ ++ R and Im∆ • R and their superpartners. They will be within striking range of the LHC and the ILC with m ∆ ≃ 300 GeV -1 TeV.
Finally, we briefly discuss the mechanism of SUSY SO (10) breaking [16,17,18]: The SO (10) Higgs multiplets 210 and 54 are utilized to break the symmetry at M U . Within the 210 there are two components which develop vevs; one breaks SO (10) to G 3221 while the other is responsible for D-parity breaking. The vev of the singlet under the Pati-Salam group contained in 54 ensures that there are no light pseudo-goldstone bosons arising from the 210 to upset perturbative gauge coupling evolution. As already discussed, SU(2) R × U(1) B−L is broken by the vevs of RHtriplets in 126 ⊕ 126. This induced vev, v R ∼ 10 11 GeV, is also responsible for the masses of the N i . The last step of breaking in eq. (5) relies on the electroweak vev of the weak bi-doublet in 10. We have carried out an analysis of the RG evolution of the gauge couplings to determine the intermediate mass scales. We find that M R ∼ 10 9−11 GeV can be obtained through the introduction of effective dim.5 operators scaled by the Planck mass, M P l [18]. It is noteworthy that both 210 and 54 are necessary for a viable SUSY SO(10) breaking pattern and that the resulting two dim.5 operators are instrumental in alleviating the problem of leptogenesis under the gravitino constraint: The details of this analysis will be presented elsewhere [16]. Suffice it to state that |η 1,2 | ∼ O(1) and the interactions in eq. (6) lead to finite corrections to the gauge couplings at the GUT-scale. The couplings of the left-right gauge group thus emerge from one effective GUT-gauge coupling. The upshot of this is that with these additional contributions it is possible to lower M R to as low as 10 9 − 10 11 GeV as required in this model. The grand unification scale is high: M U ∼ 10 17−18 GeV and the model predicts a stable proton for all practical purposes.
We expect that this model will have a natural extension to an E(6)-GUT wherein the matter multiplets and the singlet fields will constitute the fundamental 27 representation of the gauge group.
In conclusion, we have presented a SUSY SO(10)-based model relying on a double see-saw mechanism which is (a) consistent with the known neutrino masses and mixing, and (b) can lead to a correct lepton asymmetry via the decays of sterile, i.e., SO(10) singlet, neutrinos while remaining in concordance with the gravitino constraint. The intermediate scales are obtained through an RG analysis of the gauge coupling running and are consistent with a long-lived proton. The model is falsifiable through its prediction of doubly-charged Higgs bosons within the reach of the LHC.