Connection between proton decay suppression and seesaw mechanism in supersymmetric SO(10) models

We propose a mechanism to suppress proton decay induced by dimension-5 operators in a supersymmetric SO(10) model. Proton lifetime is directly connected with the intermediate vacuum expectation value which is responsible for the seesaw mechanism. The model shows many consistencies with the present theoretical results such as the components of the two Higgs doublets in the minimal supersymmetric standard model.


Introduction
The Grand Unified Theory (GUT) [1,2] has never failed to fascinate the particle physicists since it was proposed. Besides unifying the gauge interactions in a simple group, it also unifies quarks and leptons in same multiplets which, however, are very different in SU(3) C × SU(2) L × U(1) Y of the standard model (SM). Consequently, lepton and baryon numbers are not conserved separately. While lepton number violation is strongly supported by the observation of neutrino oscillations which are usually explained through the seesaw mechanism [3][4][5][6][7][8][9][10][11], the baryon number violation is strongly constrained by the proton decay experiments which need a natural explanation.
Supersymmetric (SUSY) GUT model based on SO(10) [12,13], among the various GUT models, is very attractive due to its several advantages. Firstly, as protected by supersymmetry, it has no problem in naturalness. Secondly, having all the fermions of a generation contained in one 16 dimensional representation which contains the right-handed neutrino, the model can naturally explain the neutrino oscillations through the seesaw mechanism. Thirdly, in the minimal [14][15][16] and the next-to-minimal [17,18] versions of SUSY SO (10), the theories are renormalizable and R-parity is conserved which prohibits the most dangerous dimension-4 operators for proton decay.
Nevertheless, the SUSY SO(10) models have also difficulties to overcome. To realize the seesaw mechanism, an intermediate scale is usually introduced. This will generally bring in new particles at this scale and break down gauge coupling unification badly [16]. However, as noticed recently in [19], there is actually no need to introduce an intermediate seesaw scale above which new gauge interactions begin. Instead, the seesaw mechanism requires only an intermediate vacuum expectation value (VEV) which contributes only a small portion of the new gauge bosons whose masses are still around the GUT scale. As a consequence, the gauge coupling unification will not be broken down as in those models, e.g., the minimal SUSY SO(10) model (MSSO10).
Furthermore, in the SUSY models, the dimension-5 operators dominate the proton decay rates and therefore strongly need to be suppressed by a mechanism. In the literature, since these operators are related to the Yukawa couplings, careful adjustments of the Yukawa couplings [20] are common which however are not sufficient as the lower limit on the proton lifetime from experiments is increasing.
In this work, instead of strictly solving the doublet-triplet splitting problem labored by many groups [21,22], we simply assume that the Higgs doublet pair in the minimal supersymmetric standard model (MSSM) are achieved by fine-tuning which will not be performed explicitly. Our efforts are mainly focused on proposing a mechanism to sufficiently suppress the proton decay rates. We will extend the MSSO10 to achieve this goal. The effective triplet mass (ETM) [23,24], to which the dimension-5 operators are inversely proportional, is enhanced due to the special structure of the color-triplet Higgs mass matrix. This suppression of proton decay is found to be directly related to the intermediate VEV required by the seesaw mechanism. We also find that the massless MSSM doublets obtained by the assumed fine-tuning are also related to the intermediate VEV, and that these doublets conform to the results from simply fitting the fermion sector in SO(10) models without considering other stringent constraints.
In the next section we will present the model, followed by the realization of the seesaw mechanism in Section 3. The solution of the model required by SUSY is presented in Section 4. The mechanism of suppression proton decay follows in Section 5. Predictions on the MSSM Higgs doublets are given in Section 6. We will summarize finally.

The present model
The present model contains the following particles in the spectrum. First, each generation of the matter superfields are contained in a 16-plet superfields ψ i (i = 1, 2, 3) as in most of the SO(10) models. Second, we use 210-plet Higgs to break GUT symmetry. To further break U(1) R × U(1) B−L symmetry down to U(1) Y , two pairs of 126+126-plet Higgs (denoted by ∆ i + ∆ i , i = 1, 2) are introduced. Two Higgs doublets in 10 (H 1,2 ), together with those in the 126+126s, are used to break down the electroweak symmetry. Third, we will introduce a U(1) symmetry to differentiate these Higgs into those couple with the matter fields and those do not. These U(1) quantum numbers Q are listed in Table 1.
Here we will simply treat the U(1) symmetry as a global one broken by the VEV of a SO(10) singlet S which is taken as In Section 3 this VEV S 0 will naturally generate the seesaw VEV and thus the model has no mass larger than the GUT scale explicitly. The value S 0 in (2.1) is also of the order of M 2 G M P lank , which may suggest alternatively that it is possible to be realized through an analogue of a seesaw mechanism, if we treat the U(1) as an anomalous symmetry broken by a Planck scale VEV generated by the Green-Schwarz mechanism [25][26][27][28]. For simplicity, this later possibility will not be discussed further.
The matter fields are negative in U(1) charges, so the Yukawa superpotential is which is just the same as in the MSSO10. The most general renormalizable superpotential in the Higgs sector is given by

On the seesaw mechanism
The small but non-vanishing neutrino masses can be naturally explained using the seesaw mechanism. In a model where the type-I seesaw dominates, the mass matrix of neutrinos is given as  [19]. The D-flatness required by SUSY at high energy scales is where the vs and vs are the VEVs of the SU(2) R triplets in 126s and 126s, respectively. Eq. (3.1) can be fulfilled even if v 1R is small compared to the GUT scale. Then the seesaw mechanism does not conflict with gauge coupling unification if the other VEVs are taken at the GUT scale.

SUSY preserving at high energy
When the SO(10) breaks down to the MSSM, only the MSSM singlets can get VEVs, Substituting these VEVs into (2.3), we get where we have defined In the presence of all the VEVs in (4.1), to preserve SUSY at high energy, besides the D-flatness condition in (3.1), the F-flatness conditions are also required, i.e., Then we get and that for v 1R and v 2R is Here for simplicity we defined which means we can naturally get the seesaw VEV v 1R by considering the F-flatness conditions because of the intermediate VEV S 0 . It is not that both v 1R and v 2R get independent VEV, but only a combination of them gets VEV whose main component comes from v 2R . For a vanishing v 2R , we define Φ 3 = 6m Φ x/λ following [16] and get The x is then determined by M 21 = 0 and thus determines v 2R ∼ v 1R which are generally at the GUT scale. If a vanishing M 12 after (4.8) is taken instead, we can not get the wanted seesaw VEV and the further results of fermion masses are inconsistent with experiments. For these reasons, the M 12 = 0 case will not be discussed further below.
In summary, SUSY at high energy and the seesaw mechanism choose to satisfy for the SO(10) symmetry breaking and thus following (4.10). All Higgs superfields are given masses at the GUT scale except the two doublets in MSSM whose masses require a minimal fine-tuning of the parameters as done in the MSSO10 [16]. Then gauge coupling unification can be realized by adjusting other parameters of the model.

The triplet mass matrix and suppression of proton decay
All the Higgs multiplets in Table 1 contain color triplet-antitriplet pairs. The color triplets are ordered as while the color antitriplets are The mass term of the Higgs color triplets is given by (ϕ T ) a (M T ) ab (ϕ T ) b , with the 9 × 9 matrix M T written as The B 11 is a 4 × 4 null matrix, and the rests are [17,18] and where for simplicity we have defined The determinant of M T is nonzero and consequently M T is reversible with all eigenvalues at GUT scale. In SUSY GUTs, the dominant mechanism inducing proton decays is through the dimension-5 operators [23,24] The Yukawa couplings have been rather constrained by fitting the fermion masses and mixing, thus suppressing proton decay rates needs some detailed investigations on the matrix elements in M T . From For general values of parameters of SO(10) GUTs, it is definitely sufficient to suppress the proton decay rates to satisfy the current experimental limits. This mechanism of suppression of proton decay can be equivalently achieved by another method. The 210-plet does not couple to the matter fields thus its color triplet-antitriplet components can be integrated out first. In result, the reduced mass matrix for the color triplet-antitriplet Higgs is now 8×8 whose four blocks are all 4×4: (i) B 11 keeps unchanged as a null matrix; (ii) B 12 has its leftmost column eliminated, while the other elements remain at M G ; (iii) B 21 has its uppermost row eliminated, while the other elements remain at M G ; (iv) B 22 has its leftmost column and lowest row eliminated, while the other elements are the order O( v 1R M Φ v 1R ) ∼ M I . In the limit M I → 0, this structure is an analogue to the mass matrix for the Higgs color triplets in the flipped SU(5) model [30][31][32][33] which, as is well known, has negligible contributions of dimension-5 operators to proton decay. With the M I elements kept, the inducing suppressed proton decay amplitudes would be of the order O( M I M 2 G ), same as (5.13).

The doublets
To get the almost massless MSSM doublets H u and H d , we need a minimal fine-tuning in the mass matrix of the doublets. In the present model, we have Higgs doublets as follows: got by simply fitting the fermion parameters in SO(10) models from many groups [15,[34][35][36] without considering other constraints. In the present model, the ratios on the R.H.S. of (6.8), however, are predicted to be related to the ratio M G M I . Also, (6.5) holds exactly, showing that there is no Φ u component in H u which, following [37], suggests that it is the type-I instead of type-II seesaw mechanism that works in the present model.

Comments and conclusion
In this work we have proposed a SUSY SO(10) model for sufficient suppression of proton decay. The suppression is found to be linked with the intermediate VEV required by the seesaw mechanism. The seesaw mechanism turns out to be type-I. Assuming that the two doublets in MSSM are achieved by fine-tuning, we find the components of these doublets agree in magnitudes with those got by just fitting the fermion masses and mixing. Again, the ratios of components are linked to the ratio of the GUT scale versus the intermediate VEV. Since all the Higgs particles beyond the MSSM doublets are at GUT scale, the unification of coupling constants will be maintained by adjusting the parameters. Above the GUT scale, the gauge coupling of SO(10) will increase fast into the non-perturbative region, as many of the SUSY SO(10) models do. This, besides the required fine-tuning in the doublet sector, is another unsatisfactory aspect of the model.
Extensions of the present model are straightforward. More realistic SO(10) models usually require 120-plet Higgs to fit the fermion sector [15,[34][35][36]. By adding a pair of Higgs of 120-plets with U(1) charges as +1 and −1, respectively, none of the above conclusions fails. The new prediction is αu α d tanβ ∼ 10 2 for the new components from 120plet which, again, agrees with the result by simply fitting the data [15,[34][35][36].
Alternatively, if we use Higgs multiplets in 45+54 instead of 210 to break SO(10), proton decay can also be suppressed at the same level. However, since in this case, the 10 Higgs cannot couple with 126 or 126, to produce the correct contents of the doublets H u and H d , a pair of Higgs in 120-plets are needed to be included at the beginning, because the 120-plet can both couple with the 10-plet and the 126/126 through 45+54.