Proton decay suppression in a supersymmetric SO(10) model

We propose a mechanism for sufficient suppression of dimension-5 operators for proton decay in a supersymmetric SO(10) model. This mechanism is analogue to the double seesaw mechanism in studying neutrino masses. Only an intermediate VEV instead of an intermediate scale is required so that gauge coupling unification is maintained. The VEV is generated by introducing an anomalous U(1) symmetry whose breaking is at higher scale. The proton decay amplitudes are suppressed by this VEV over the GUT scale. We use $\textbf{45+54}$ in breaking GUT symmetry. $\textbf{120}$ is included so that fermion sector is fully realistic. Assuming a minimal fine-tuning in the Higgs doublet sector, $\textrm{tan}\beta$ of order one is predicted.

It has been recognized recently [17] that instead of an intermediate seesaw scale, in SUSY SO(10) models with several pairs of 126 +126, only an intermediate vacuum expectation value (VEV) of the SM singlet in one 126 is needed which couples with the matter superfields. Consequently, the spectra of this kind of models do not contain particles at intermediate scale so that gauge coupling unification is maintained, meanwhile the seesaw mechanism still works. This mechanism is further incorporated in models aiming at sufficiently suppressing proton decay [18], where the seesaw VEV is related to the VEV of an SO (10) singlet which breaks an extra global U(1) symmetry. Proton decay amplitudes are found to be suppressed in [18] by a factor M I M G , where M I ∼ 10 14 GeV is the seesaw VEV which is much smaller than the GUT scale M G ∼ 10 16 GeV. This suppression of proton decay is archived by the enhancements of the effective triplet masses through an inversely analogue to the mass texture in the seesaw mechanism, or a lever mechanism.
In the present work we will extend the observation made by the previous study that the seesaw VEV might be related to the suppression of proton decay in other models. Instead of using 210 to break SO(10), we will use 45+54. The global U(1) will be replaced by an anomalous U(1) whose breaking is generated by an SO(10) singlet through the Green-Shwarz mechanism [19,20]. The enhancements of the effective triplet masses responsible for proton decay are through an inversely analogue to the mass texture in the double-seesaw mechanism [21,22]. We will not improve on either the running behavior of the SO(10) gauge coupling above the GUT scale or on the minimal fine-tuning for the weak doublets which is implicitly assumed.
In the next section, we will give a simple overview on proton decay suppression. Then, we will propose in Section 3 a renormalizable model and show its consistency with high energy supersymmetry. Proton decay suppression mechanism in this model is shown in Section 4. The discussion on the weak doublets of the MSSM and the prediction of small tanβ are followed in Section 5. We will summarize in Section 6.
2 General consideration on proton decay suppression Consider a simplified model with two pairs of color triplets-anti-triplets, with only one pair of them couple with fermions. The mass term for the triplets can be written as where i, j run from 1 to 2. We need to rotate to the mass eigenstates in order to calculate proton decay amplitudes. Two 2 × 2 unitary matrices U and V are introduced as Then the dimension-5 operators mediated by the color triplet higgsinos are proportional to [23] 3) The inverse of (M −1 ) 11 is called the effective triplet mass which mimics the role of the color triplet higgsino in the simplest models with only one pair of color triplet-anti-triplet. Eq.(2.3) is easy to be generalized to models with more pairs of color triplets-antitriplets. The proton decay amplitudes are proportional to sums of specific elements in the inverse of the triplet mass matrix. Algebraically, these matrix elements in the inverse mass matrix can be written as where M * ij represents M with the ith row and the jth column eliminated, whose determinant is called as the algebraic complement, and i and j are the labels of those color triplets-antitriplets which can couple with the fermions.
There are two possible ways to get small (M −1 ) ij 's following (2.4). We can construct a mass matrix either with all small algebraic complements for the elements which couple with fermions, or with a large determinant of the entire mass matrix. In the previous work [18] the first approach is used where the color triplet mass matrix can be symbolically expressed as Here M G stands for a GUT scale mass while M I is the intermediate seesaw VEV. Only the up-left block couples with matter fields, so it is clear that M * 11 = M I is smaller than M G . In this work, we are trying to realize the second possibility. The mass matrix for the color triplets is written as Again, the matter fields couple with the up-left block only. Here M P represents a mass at a scale higher than the GUT scale, or at the Plank scale. Then Det(M T ) ∼ M P M 2 G is enhanced to give large effective triplet masses.
As the texture in (2.5) is analogue to the neutrino mass matrix in the seesaw mechanism, the present texture in (2.6) is analogue to the neutrino mass matrix in the double-seesaw mechanism. The mass texture either in (2.5) or in (2.6) is sufficient to suppress proton decay.

The model and SUSY preserving
The particle content of the present model is as follows. First, it contains three generations of fermion fields which are embedded into three 16-plet (ψ 1,2,3 ) superfields as usual. Second, two pairs of 45+54 (A, A ′ and E, E ′ ) are introduced for different purposes, one pair is to break SO(10), the other pair is used as auxiliary to generate small VEV for the seesaw mechanism. In order to give satisfied fermion masses and mixing, Higgs in 120 (D) is introduced together with those in 10 (H) and in 126-126 (∆/∆). They are further copied for suppression of proton decay. An extra U(1) symmetry, whose breaking is realized by the SO(10) singlets S 1 and S 2 , is introduced to distinguish these Higgs.
Only H 1 , D 1 , ∆ 1 /∆ 1 couple with matter fields due to the U(1) charges. The Yukawa sector is given as which is general enough to fit all fermion masses and mixing [24][25][26][27]. The general renormalizable Higgs superpotential is given by Labeled by the representations under the SU (4) C × SU (2) L × SU (2) R subgroup of SO(10), the following components get VEVs responsible for the SO(10) symmetry breaking Inserting these VEVs into (3.2), we get where we have defined for later convenience.
To preserve SUSY at high energy, the F-and D-flatness conditions are required. The D-flatness condition requires where Similarly, equations (3.10)-(3.12) can be rewritten as . Taking S 1 as free, the remaining variables are now A 1 , E, S 2 , v 1 with equations (3.16)), (3.18), (3.19) and (3.20) left. Given the parameters in (3.2), all the VEVs are now determined. Numerically, A 1 , A 2 and E are taken as GUT scale VEVs in order to break SO(10) down to MSSM.
According to the analysis in [28], the extra U(1) symmetry is naturally related to string theory, and it is appropriate to take the VEV of breaking this U(1) at After inserting (3.13) into (3.18-3.19), the last terms will change into They are naturally at the same scale as other terms, i.e. M 2 G , which indicates that Thus we get from (3.22) and (3.23) Now that all the constrains on SUSY preserving have been satisfied, all the VEVs can be determined and all their scales are known. The seesaw VEV v 1 ∼ M I is naturally generated at 10 −2 M G , which differs from [18] where it is introduced a VEV of a SO(10) singlet which breaks a global U(1) symmetry.

Proton decay suppression
To demonstrate the effectiveness of the present model on solving the proton decay problem, we need to write down the color triplet mass matrix. The color triplets are ordered as The mass term of the Higgs color triplets is given by (ϕ T ) a (M T ) ab (ϕ T ) b , with the 16 × 16 matrix M T written as and Here The mass matrix can be also expressed symbolically as (4.9) Note that the texture in (4.9), constrained by the F-and D-flatness conditions, differs slightly from that in (2.6). However, as will be seen in the rest of this Section, the mechanism of suppressing proton decay following (2.6) will not change.
In SUSY GUTs, the dominant channels inducing proton decay are through the dimension-5 operators [29,30] which are called the LLLL and RRRR operators, respectively, obtained by integrating out the color triplet and anti-triplet Higgs superfields in the interactions in (3.1). Both C ijkl L and C ijkl R are inversely proportional to the effective mass of the colored Higgsino. Since only B 11 part couples with fermions, we can get the effective mass by integrating out the uncoupled parts. From (4.9), such a mass matrix is similar to the mass matrix in the double seesaw models for neutrino masses [21,22] which is used to generate the small neutrino masses. In the present model, the effective masses are large instead of small because B 23 ∼ B 32 ≪ B 12 ∼ B 21 ≪ B 33 . Similarly, this proton decay suppression mechanism requires two steps of integrations. Since S 1 is ten times of the GUT scale, the B 33 part can be integrate out first. Then the mass matrix becomes where is a matrix with all elements of the order 10 −2 M G . Then after the second step, Note that it is the lightest eigenvalues that dominates the proton decay rates, while the two infinitely heavy masses do not contribute. The suppression can be better understood if we write down the dimension-5 operators explicitly. The coefficients C L s at the GUT scale M G are [31] (4.15) Here the Yukawa couplings are strongly constrained by fitting the fermion masses and mixing [24][25][26][27]. The elements of M −1 T are of the order 1 M G in usual SUSY GUT models, but in our model, We can see clearly that the elements contributing to dimension-5 operators, i.e. elements in the up-left most block, are of the order 10 −2 M G . This is the same conclusion drawn in (4.13). This conclusion applies for both the LLLL and RRRR operators.

The weak doublets
Like in [18], the doublet-triplet splitting (DTS) problem requires a minimal fine-tuning, and similar results can be reached. The up-type doublets are ordered as while down-type doublets are The mass terms of the weak doublets are given by (ϕ d ) a (M D ) ab (ϕ u ) b , with the 13 × 13 matrix M D written as If we chose Det(A 21 ) = 0, we will further get the massless doublet can be expressed as The large ratio of α i u α i d (i ≤ 5) is consistent with the ratio of mt m b ∼ 100 at high energy [24][25][26][27]. It also gives the constrain on tanβ Equation (5.8) suggests that a small tanβ is favored in the present model, which is also the same conclusion drawn in [18].

Summery and conclusions
In the present work we have presented a renormalizable SUSY SO(10) model with sufficient suppression of proton decay. Similar to [18], gauge coupling unification is maintained due to the absence of intermediate scales, and the seesaw VEV, proton decay and tanβ are found to be all related, Thus the main conclusions are quite general in a class of models which follow the mechanisms of suppressing proton decay through constructing seesaw-like textures in the color triplet mass matrices. Different from the previous study, we use 45+54 instead of 210 to break SO (10). Instead of a global U(1) used in [18], we use an anomalous U(1) to generate the seesaw VEV through Green-Schwarz mechanism. We have also included 120-plet Higgs to couple with fermions so that the model is highly realistic. We have, however, two main problems untouched. The first is the DTS problem which we simply use an assumed fine-tuning in the weak doublets. The second is the perturbative difficulty for the gauge coupling above the GUT scale which is also common to all realistic SUSY GUT models.