Proton decay in a supersymmetric SO(10) model with missing partner mechanism

The extended supersymmetric SO(10) model with missing partner mechanism is studied. An intermediate vacuum expectation value is incorporated which corresponds to the see-saw scale. Gauge coupling unification is not broken explicitly. Proton decay is found to satisfy the present experimental limits at the cost of fine-tuning some parameters.

In the present work, we will study the extended SO10MPM. The SO(10) symmetry is broken directly to the SM gauge group, while an intermediate VEV to account for the see-saw scale is used which is only a small correction to the SO(10) symmetry breaking. The other 126 − 126 need to have large VEVs not only to keep unification, but also to suppress proton decay required by the experimental data.
In section 2 we will review the Minimal SO10MPM briefly. In section 3 the extended SO10MPM is constructed and solved explicitly. Mass matrices of doublets and triplets are shown in section 4. Fermion masses and proton decay are studied in section 5. In section 6 we will summarize.

Review of the minimal SO10MPM
In the SU(5) models, the MPM is realized by introducing a U(1) symmetry to forbid the 5 − 5 to have direct mass term. By coupling 5(5) Higgs to 50(50), which contains a color triplet (anti-triplet) but no weak doublet, through 75 which breaks SU (5), the color triplets in 5 − 5 gain GUT scale masses while the weak doublets are still massless.
In SO(10), the 5 and 5 of SU(5) constitute a 10, and the 50 and 50 of SU(5) are contained in 126 and 126, respectively, as 126 = 1 + 5 + 10 + 15 + 45 + 50, 126 = 1 + 5 + 10 + 15 + 45 + 50. (2.1) In total, a pair of 126 − 126 multiplets contains 2 pairs of doublets and 3 pairs of triplets. In addition, the SU(5)-breaking 75 is contained in 210 of SO(10) which breaks SO (10) symmetry. Then, in principle, the MPM of SU(5) can be embedded into SO (10). This needs to include more Higgs in the model and to impose an extra U(1) symmetry to eliminate unwanted bilinear and trilinear couplings from the superpotential, leaving some elements of the mass matrices to be zero. Denoting the multiplets which cannot(can) get mass through couplings with itself or its conjugate as "light"("heavy") fields, the Higgs fields 10 × 2 + 120 + 126/126 + 210 are required in the minimal version [25], in which the 10s and 120 are light with Q = 1, 126 − 126 are heavy with Q = −1, and 210 is also heavy with Q = 0. The heavy fields 126 − 126 get their masses through couplings with a singlet X with Q = 2, while the light fields get masses only through couplings with 126 − 126 and 210. Fermions are contained in the three 16plets with Q = − 1 2 , coupling only with the light fields. It is easy to count that there are 4 (4) pairs of light doublets (triplets), and 3 (4) pairs of heavy doublets (triplets). The mass matrices are for the doublets, and for the triplets. Here, 0 4×4 stands for a 4 × 4 null matrix, while O(G)s mean matrices whose elements are of the GUT scale. Generally speaking, M D has one zero (light) eigenvalue, and M T has none. Because the determinant of M D contains 7! = 5040 terms, and each term inevitably has one factor that comes from the 4 × 4 null matrix part, this makes the determinant of M D vanished. Since the heavy part of M T is larger than that of M D , all eigenvalues of M T remain superheavy, and the DTS is thus achieved. However, one cannot fit all fermion masses with only 10 and 120 fields, a light 126 must be introduced to couple with the fermions. Meanwhile, the heavy side of fields must be extended to keep the MPM [26]. We will study this extended SO10MPM in the next section which was not solved explicitly in [26].

Extended SO10MPM with pairs of 126 − 126
In the Minimal SO10MPM the see-saw mechanism is not included to explain the neutrino masses. In fact, SO(10) has a basic conflict between incorporating the see-saw mechanism to generate the neutrino masses and realizing the unification of gauge couplings. In the type-I see-saw models [27][28][29][30], the right-handed neutrinos need to have masses at the intermediate scale of the order of O(10 14 )GeV [31][32][33]. Similarly, in the type-II see-saw models, a weak triplet at the intermediate scale is needed. This intermediate scale, however, breaks the gauge coupling unification badly since fields at this scale will change dramatically the running behaviors of the coupling constants.
To incorporate the see-saw mechanism, we use the extended SO10MPM with three pairs of 126 − 126. The GUT scale VEVs are assigned to the MSSM singlets of these 126 − 126 except that of the 126 which couples to fermions has an intermediate VEV.
The SO(10) symmetry is thus directly broken to MSSM symmetry, so that gauge coupling unification can be maintained hopefully as all particles other than the MSSM Higgs doublets are superheavy. All the three pairs of 126 − 126 are needed to keep SUSY unbroken at high energy, as the conditions for the F-and D-flatness are satisfied.
The model includes the following set of states in Table 1, and their U(1) quantum numbers are also assigned to eliminate unwanted masses and couplings. The U(1) contains an anomaly which is canceled through the Green-Schwarz mechanism [34][35][36][37].
The general renormalizable Higgs superpotential is given by where i, j = 1, 2. Repeated indices stand for summation, as usually understood. Following the notation of [38], SO(10) breaks down to the MSSM when the MSSM singlets get VEVs as Here the SU (4) C × SU (2) L × SU (2) R indices are used to specify different singlets of MSSM. Inserting these VEVs into (3.1), we get: where we have defined for further convenience.
To preserve SUSY at high energy, the F-and D-flatness conditions are required. The D-flatness condition requires while the F-flatness conditions are which are explicitly Here we have defined: Now that there are 10 equations for the F-flatness and 1 for the D-flatness in total, but only 9 of these constraints are independent. The situation here is comparable with that of the minimal SUSY SO(10) model in [6,8], where there are 5 equations for the F-flatness and 1 for the D-flatness, but only 5 are independent. We can solve the above equations as follows. The first three equations in the F-flatness conditions, (3.7-3.9), are solved in the same way as in the minimal SUSY SO(10) model which gives For x is not to be taken special values to generate accidental intermediate symmetries, all the VEVs Φ 1,2,3 are at GUT scale [6,8]. The next three equations, (3.10-3.12), are linear homogeneous equations about v R , v 1R and v 2R , so they have non-zero solutions only when Then, among v R , v 1R and v 2R , only one can be considered as free which is chosen to be v 1R without loss of generality. Similarly, equations (3.13-3.15) require to have non-zero solutions for v R , v 1R and v 2R , and v 1R is taken to be free among them. Note that the condition (3.23) is exactly the same as (3.22). However, the transpose relation of the matrices in (3.22) and in (3.23) means that the two sets of equations (3.10-3.12) and When all the superpotential parameters are fixed, this condition (3.22) or (3.23) determines the value of Φ 0 , and thus x can be extracted from (3.4) and (3.18-3.20). The ratio of v 1R to v 1R is determined by the D-flatness condition (3.5). With x being solved already, all these VEVs of 126 − 126 are given.
To realize the see-saw mechanism, an intermediate value for v R is taken. All the other VEVs are taken at the GUT scale. Consequently, there is no Higgs existing at the intermediate scale. The only exceptions are the right-handed neutrinos needed in the type-I see-saw.

Mass matrices for doublets and triplets
In the Extended SO10MPM model, there are 6 (7) pairs of light doublets (triplets) and 5 (7) pairs of heavy doublets (triplets). The mass matrix for the doublets is written as where the bases are for the first 6 columns, for the first 6 rows, for the last 5 columns, and for the last 5 rows. In these bases, the subscripts and the superscripts label the SU (4) C × SU (2) L × SU (2) R and the SM representations, respectively. In the M D , A 11 is a 6 × 6 null matrix, while and where for simplicity we have defined (4.5) It is easy to see that there is one zero-valued eigen-state in the row and another one in the column, which are the MSSM Higgs doublets. It can be also noted that these light doublets contain no components from the doublets in 210 which break B − L quantum numbers. Consequently, the SU (2) L triplets in 126s has no VEV, which excludes the type-II see-saw mechanism [39] in the model.  for the last 7 rows. In the M T , B 11 is a 7 × 7 null matrix, the rest are and where Φ (T ) We can observe that there is no zero eigenvalue of the triplet mass matrix. Also, all the other Higgs are massive except those Goldstone modes. This justifies the realization of the MPM in the extended model.

Fermion masses and proton decay
In the present model, the fermion sector is described by the superpotential In [40,41] the fermion masses can be fitted by using only one 10 H and one 126 H . As in [40] for example, the resultant fermion masses are in accord with their experimental values except the electron mass. In present, although all the Higgs H 1,2 , D and ∆ contribute to the fermion masses, we can also use the H 1 and ∆ as in [40], taking contributions from H 2 and D as small corrections to the electron mass. These corrections are suppressed by a factor of Y ij and are negligible in studying proton decay. At the GUT scale the fermion masses are taken from [42] for tan β(M SU SY ) = 10, µ = 2 × 10 16 GeV. After numerical fitting, we get the Yukawa couplings Y ij 10 1 and Y ij 126 in the u-diagonal basis (5.3) One can easily verify these couplings by calculating the mass eigenvalues of the following matrices where v u = 123.8GeV and v d = 17.87GeV (see Table 5 of [42]). α u = 0.6647, β u = −0.7471, α d = 0.06816 and β d = −0.9977 come from numerical fitting. We also noticed that the normalizations (α u ) 2 +(β u ) 2 = 1 and (α d ) 2 +(β d ) 2 = 1 are not accurate due to the existence of H 2 and D(120), but the deviations are small to be neglected reasonably.
In studying proton decay via dimension-5 operators, we limit our analysis to LLLL operators only, although contributions from RRRR operators are also sizable [43,44]. The resultant operators are contained in the superpotential where the contractions of the indices are understood as In (5.5) we have defined where  .6), Running the dimension-5 operators down to the SUSY scale and dressing them by wino-loops, we get the four-fermion operators. For the dominant decay modes p → K + ν l , the coefficients are
To estimate proton decay rates, we need to analyze the effective triplet mass matrix M ef f in (5.8) in some details. In getting the right-handed neutrino masses, the VEV v R is taken a small value 10 14 GeV compared to the GUT scale 2 × 10 16 GeV. In the limit of neglecting v R , there are three zero eigenvalues in the matrix B 22 , since in this limit  Table 3. Dimensionless parameters. equations (3.8-3.9) require η 11 /η 21 ∼ η 12 /η 22 and thus B 22 has rank-4 instead of rank-7. Consequently, the elements in M ef f are all divergent. However, these elements are correlated, thus elements in the inverse of M ef f are not small even in this limit. Proton decay proceeds fastly whenever the eigenvalues in M ef f are not all very big, being irrelevant to the appearance of all the large entries.
To suppress proton decay, we need to keep all the eigenvalues in B 12 and B 21 at GUT scale while suppressing those in B 22 . The later can be achieved only by fine-tuning slightly the parameters η ij (i, j = 1, 2) and m Φ + λ( Φ 1 3 ). The former, to keep the eigenvalues in B 12 and B 21 all large, requires at least one pair of large VEVs in v iR (see the last row in the matrix of (4.7)) and v iR (see the last column in the matrix of (4.8)). These large v iR and v iR , now required by suppressing proton decay, correspond to the direct breaking of SO(10) into MSSM, and explicitly breaking of the unification is avoided as all the Higgs particles beyond MSSM are at GUT scale.
Numerically, there are too many parameters in the model to analyze. The typical dimensional and dimensionless parameters are taken as in Table 2 and 3, respectively. They lead to the VEVs in Table 4.
Note that the VEVs v R s andv R s are normalized to 1 √ 120 [38], thus the appearance of their large values is artificial. Indeed, all the gauge superfields get masses at the GUT scale through the VEVs listed in Table 4. Equations (3.5-3.16) are satisfied. v R = 0 is taken to study proton decay; in analyzing neutrino masses, however, it should take its practical values like 10 14 GeV which are still negligible compared to the GUT scale.
We have also checked numerically the entire Higgs spectrum, confirming the absence of intermediate state which may otherwise spoil the gauge coupling unification explicitly. There is the problem, however, that the total β−function will be a large number after unification, so the coupling constant of SO(10) will blow up quickly.  Table 5. Proton partial lifetime /years The numerical results of proton partial lifetimes are listed in Table 5. As we can see, they can be well above the current experimental limit, although at the cost of fine-tuning some parameters. According to the recent discussion of [47], by taking decoupling effects of SUSY particles into account, the triangle diagram factors can be even smaller, leading to longer proton lifetime.

Summary and conclusions
The extended SO10MPM is analyzed with the following results. First, type-I see-saw can be realized by introducing an intermediate VEV which couples to fermions. Second, SUSY is maintained at high energy. Third, unification is hopefully realized although a fully adjustment of the parameters are not carried out. Fourth, proton lifetime is consistent with the data if fine-tuning is used slightly. Works we have not done here include a fully numerical calculation of gauge coupling unification including GUT scale threshold effects, and a fully analysis with electron mass corrected by the 120plet Higgs effect. These are big challenges in future researches. We thank Jun-hui Zheng for useful discussions.