Missing Partner Mechanism in SO(10) Grand Unification

We present a new possibility for achieving doublet-triplet splitting naturally in supersymmetric SO(10) grand unified theories. It is based on a missing partner mechanism which is realized with the 126 + 126-bar Higgs superfields. These Higgs fields, which are also needed for generating Majorana right-handed neutrino masses, contain a pair of color triplets in excess of weak doublets. This feature enables us to remove the color triplets from the low energy spectrum without fine-tuning. We give all the needed ingredients for a successful implementation of the missing partner mechanism in SO(10) and present explicit models wherein the Higgs doublet mass is protected against possible non-renormalizable corrections to all orders. We also show how realistic fermion masses can be generated in this context.


Introduction
up the colored higgses from the 5 + 5 with those from the 50 + 50. Such a pairing will leave the doublets naturally light. The stability of such a solution against higher order operators requires some additional effort [4].
In SU (6) grand unified theories, the pseudo-Goldstone mechanism [5,6] can solve the DT splitting problem rather elegantly. Here the Higgs doublets are identified as pseudo-Goldstone bosons of a larger global symmetry. The gauge symmetry should be augmented by additional symmetries for this realization. The anomalous U(1) symmetry of string origin is very efficient for this purpose [6].
In this paper we are concerned with grand unified theories based on SUSY SO (10), which are particularly attractive [7]. The spinor representation of SO(10) unifies all matter fermions of a given family in a single multiplet -a feat not achieved in SU (5) or SU (6) GUT. The spinor of SO (10) contains also the right handed neutrino (ν R ), which can generate light neutrino masses via see-saw mechanism [8]. The ν R can also naturally account for the baryon asymmetry of the Universe via leptogenesis [9]. Another nice property of SO(10)-based GUT is that, they can automatically lead to matter parity (or R-parity) [10] which is usually assumed as an ad hoc symmetry in the MSSM. Such a symmetry is needed in order to avoid rapid proton decay, it also provides a natural cold dark matter candidate. In SUSY SO (10), matter parity can be automatic since it contains B − L as a subgroup.
The most widely discussed approach to the DT splitting problem in SO (10) is the Dimopoulous-Wilczek mechanism, or the missing VEV mechanism [11]. Here one employs an adjoint 45-plet of Higgs field with its vacuum expectation value pointing in the B − L conserving direction: 45 = V × diag(1, 1, 1, 0, 0) ⊗ iτ 2 . The MSSM Higgs doublets are contained in 10 of SO(10) (10 = 5 + 5 under SU(5)). If the superpotential contains the terms W ⊃ M 10 10 ′ 10 ′ + λ 10 · 10 ′ · 45, because of the VEV structure of 45, the color triplets from 10, 10 ′ will acquire GUT scale masses, while a doublet pair from 10 will remain massless. Note that this is done without fine-tuning, and is facilitated by the fact that the adjoint of SO (10) is not a traceless matrix, unlike the adjoint of SU(N) groups. A variety of realistic models based on this mechanism have been constructed in the literature [12]. Additional ingredients are usually needed to guarantee the stability of the VEV structure of 45 [13,14]. Realistic models for fermion masses including neutrino oscillations have been constructed based on this mechanism [15].
Although SO(10)-based model building has attracted considerable attention, to date, the missing partner mechanism that works in SUSY SU (5) has not been successfully implemented in SO (10). The purpose of this paper is provide such a realization. We present examples of models with all order stability of the DT hierarchy which also have realistic phenomenology.
In the next sections, first we point out the possibility and some properties of a missing doublet SO(10) scenario and draw the ingredients needed for realistic model building. Then we present explicit models which provide all order stability of the proposed DT splitting mechanism. Then we show how realistic fermion masses can be generated in this context. Finally we comment on the perturbativity of the gauge coupling above the GUT scale.
2 Missing Partner Mechanism in SO (10) First let us recall how the missing partner mechanism, or missing doublet (MD) mechanism, works within SUSY SU(5) GUT. Then, the steps needed for building a missing doublet SO(10) (MDSO10) GUT will be easier to follow. In SUSY SU (5), the pair of the MSSM Higgs doublets h u and h d are embedded in the supermultiplets h(5) andh (5) respectively. The composition of these states where T,T are color triplets. The 50 + 50 representations of SU(5) (which we denote as ψ +ψ) have the curious feature that they contain states with the same quantum numbers as T h andT h , but not the h u , h d states [16]. Thus arranging suitable couplings between h,h and ψ, ψ one can decouple the triplets T h ,T h from 5 + 5 through the mixing with T ψ ,T ψ [3]. For this to be achieved, a scalar φ(75)-plet must be introduced with non-zero After substituting the VEV φ ∼ V φ , the color triplet mass matrix will bē and therefore with M ψ ∼ V φ ∼ M GUT one expects all triplet masses to be near the scale M GUT . At this level the doublets from 5 + 5 are massless since there are no doublets in the 50 + 50. Crucial for this mechanism is the omission of the mass term M hh h. Of course, this must be justified and additional symmetries can be employed for this purpose [4]. Now we turn to models based on SO(10) gauge symmetry and try to see how the missing partner mechanism can be realized. The lowest dimensional Higgs representations which has a missing doublet in SO(10) is the 126 + 126. They contain SU(5) 50 + 50-plets. Indeed, in terms of SU(5) × U(1) ≡ G 51 (one of the maximal subgroups of SO(10)) we have [16] 126 = 1 −10 +5 −2 + 10 −6 + 15 6 where the subscripts stand for U(1) charges. This representation, together with 126, will be used for building MDSO10 model. The MSSM Higgs doublets are embedded (at least partially) in the scalar H(10)-plet of SO (10). This is the lowest representation which admits renormalizable Yukawa couplings -16 · 16 H(10). So, for DT splitting we should arrange the coupling of H with 126-plets.
To do this at the renormalizable level we need to introduce a scalar supermultiplet in the 210 representation of SO (10). Note that it is quite exciting that the multiplets 126 + 126 + 210, which we just mentioned, have been used extensively recently for building renormalizable SO(10) GUT [17,18] with some predictive power in the fermion sector, including neutrino oscillations.
Here we will show the importance of these states in achieving the DT splitting. The couplings of the bi-linears 126 H(10) and 126 H(10) with 210-plet form SO(10) singlets and will be used below. However, we note that some caution is needed for building self-consistent model. From Eq.
(3) we see that the 126, 126-plets contain, besides the 50-plets, 5 and 45-plets. The latter states contain color triplets as well as weak doublets. Thus there is danger that together with color triplets of H(10) the doublets also gain large masses. However, if we introduce a set of Higgs superfields containing in total three pair of weak doublets, it is clear by a simple counting of degrees of freedom that, only two pair of doublets will get masses by mixing with 5 and 45-plets from the 126, 126. Now, one must decide which additional states are most convenient for this purpose together with one H(10) supermultiplet. It turns out that the state Σ(120) can do a very useful job in this regard. The decomposition of 120 in terms of G 51 reads 120 = 5 2 +5 −2 + 10 −6 + 10 6 + 45 2 + 45 −2 .
The multiplets H(10) and Σ(120) together contain three pairs of doublets and three tripletantitriplet pairs. To be more clear, let us consider the multiplets H(10), Σ(120), ∆(126),∆(126), Φ(210) and the superpotential couplings With ∆ = ∆ = 0 and the VEV of Φ in the most general direction that preserves the SM gauge symmetry (see next section for more details) the mass matrices for triplet and doublet states schematically are given byT where the dimensions of the block matrices have been denoted appropriately by subscripts. The dimension of the doublet mass matrix is by one unit less than the dimension of the triplet mass matrix, because there is one missing doublet pair (in states 50∆ +50 ∆ ). Considering now the matrix Eq. (6), we can see that all triplets from H, Σ-plets gain masses through the mixings with the three triplet-antitriplet pairs from ∆(126) +∆(126)-plets. However, according to Eq. (7), two doublet pairs from ∆(126) +∆(126) generate masses for two doublet pairs from H, Σ states. Therefore, the third pair of doublets coming from H, Σ will remain massless. (This is also obvious from the fact that Det(M D ) = 0). The reason is simple: as already was mentioned, there is one missing pair of doublet in ∆(126) +∆ (126). This is a transparent demonstration how the missing partner mechanism can work in SO (10). However, for realistic model building some more elaboration will be required. Namely, one should make sure that the couplings H 2 , Σ 2 , ΦHΣ are absent, i.e. the zero of the 3 × 3 block in Eq. (7) must be guaranteed. Although the supersymmetric non-renormalization theorem guarantees that once set to zero these terms will not be generated perturbatively, we wish to explain the origin of their absence based on some symmetries. In addition, we wish to insure that certain higher order non-renormalizable operators which may be induced by unknown Planck scale effects are absent. Also, in order the for the DT hierarchy to remain intact we need the VEV of either ∆ or∆ to be zero. In the next section we present explicit SO(10) model(s) which address all these issues and show the consistency of the mechanism.
Before closing this section, let us mention that besides the triplet and doublet states Σ(120)-plet contain other vector like states. All of these extra states will acquire masses through the mixings with ∆(126),∆(126) multiplets. Note that the quantum numbers of all the fragments of Σ(120)-plet match with those of the states from ∆,∆ [see Eq. (3) and Eq. (4)]. Therefore, no state (besides the one massless doublet which partially also resides in H) from Σ remains massless.

Explicit Missing Doublet SO(10) Models
From the discussions of the previous section we already got a clear idea of what field content we would need in order to realize the missing doublet mechanism in SO(10). As it will turn out, it is much more convenient if the 126-plets involved in the DT splitting have no VEVs (or at least one, out of ∆(126) and∆(126)-plets, has no VEV). Thus, the symmetry breaking sector should be discussed is some detail. For the rank breaking of SO(10) we can use either a scalar 16 + 16-plets or another 126 + 126-plets. In either case we denote the rank breaking superfields by C,C and distinguish between two possible cases: The states C,C together with Φ(210)-plet break the SO(10) group down to SU(3) c × SU(2) L × U(1) Y . This discussion concludes the selection of the GUT scalar superfields. We wish to build models which preserves DT splitting to all order, i.e. all couplings (including non-renormalizable operators) allowed by symmetries must be taken into account. Thus, we will need to forbid some of the couplings and the easiest way to do so is to introduce an additional gauge U(1) symmetry. As it turns out, this symmetry is anomalous. The anomalous U(1) symmetry of string origin has been applied in GUT model building [6,4,14] and has been shown to be very efficient for stabilizing the DT splitting to all orders. Here we apply this U(1) symmetry in our MDSO10 scenario. The anomalous U(1) factors can appear in effective field theories from string theory upon compactification to four dimensions. The apparent anomaly in this U (1) is canceled through the Green-Schwarz mechanism [19]. Due to the anomaly, a Fayet-Iliopoulos term −ξ d 4 θV A is always generated [20] and the corresponding D A -term has the form [21] g 2 where Q i is the U(1) charge of φ i superfield. For U(1) breaking we introduce an SO(10) singlet scalar superfield X with U(1) charge Q X = 2. With ξ > 0, in Eq. (9) the VEV of the scalar component of X is fixed as X = ξ/2. In Table 1 we list all scalar superfields introduced, the matter 16 i -plets (i = 1, 2, 3) and the corresponding U(1) charges. The fermion sector will be discussed at the end of this section. With this assignment we can write down the superpotential couplings. The part which is important for DT splitting is W DT = Φ∆ (H + Σ) + Φ∆ (H + Σ) + X∆∆ .
In order to carry the detailed analysis, we should first investigate the symmetry breaking and field VEV structure. The superpotential couplings important for the symmetry breaking are Also, there are higher order superpotential couplings (potentially induced by unknown gravity effects) with '∆ − C mixing': Thus, the total symmetry breaking superpotential is In terms of SU (5) group, Φ(210) decomposes as where the dots stand for states which have no SU(3) c × SU(2) L × U(1) Y singlet components. Thus, only the first three fragments of Φ given in Eq. (14) are relevant in studying the VEV structure.
For denoting their VEVs we will introduce the following notations The VEVs of C,C will wind towards the SU(5) singlet direction and will be denoted as Similarly, the SU(5) singlet fragments in ∆ and∆ can have (induced) VEVs and will be denoted as ∆ 1 and∆ 1 respectively. For completeness we will take these induced VEVs also into account. From the F -flatness conditions F X = F ∆ = F∆ = 0 we have the solution .
On the other hand, D-flatness conditions for the anomalous U(1) and the U(1) of SO(10) are: where the U(1) charge Q C is given in Table 1, while q U = −5 and −10 for cases (a) and (b) respectively. In addition, from the F C = FC = 0 conditions we fix while the condition F Φ gives schematically (up to some irrelevant Clebsch factors) One can easily verify that the conditions Eqs. (17)- (20) fix non-zero X , Φ 1,24,75 and C 1 ,C 1 VEVs. For simplicity we can assume that all this VEVs are ∼ M GUT . We also have∆ 1 = 0, ∆ 1 = 0, and all the F and the D-terms vanish, ensuring unbroken supersymmetry. It is important that the operators H 2 , Σ 2 , ΦHΣ are forbidden by U(1) symmetry. U(1) invariance would require that these operators should be multiplied by some field combinations carrying negative U(1) charge. We can readily check that such operator will involve ∆ and since ∆ = 0 they are not relevant. Therefore, quadratic couplings with respect to H, Σ will not give rise to the doublet masses to all orders. There will be additional operators which are linear with respect to H and Σ: The SO(10) × U(1) symmetry determines their structure and for the two cases (a) and (b) we have (cut off scale is ommited). As we will see, these operators do not spoil the DT hierarchy. We will take them into account in order to demonstrate that we are getting successful DT splitting. Since the VEV configuration and all superpotential terms are already fixed, we are ready to discuss the issue of the DT splitting. The relevant coupling matrix in terms of SU(5) fragments is where each subscript indicates where the appropriate superfield fragment is coming from. For the 'vector' states the following notations have been used: and for the two cases we have: The blocks appearing in (23) are given by andΩ has the structure of Ω T , while M F ∝ X diag (1 , 1 , 1). Forms of Γ, M f , q, v andv depend on the case we are dealing with [either (a) or (b)]. For example, for case (a), i.e. when the rank reduction occurs by C(16),C(16)-plets, we have where M C is the mass of 5-plets from C,C arising from symmetry breaking superpotential. These block entries have different dimensions for case (b). However, it is remarkable that the result does not depend on the structure of these entries. This becomes obvious from the whole form of the matrix Eq. (23). The integration of the states f C ,f C and 5 Φ ,5 Φ does not give any contribution to the upper left 3 × 3 zero block matrix of Eq. (23). An important role for this is played by the off-diagonal zero block matrices which are protected by U(1) symmetry. Upon integration of f C ,f C , 5 Φ ,5 Φ states the matrix Eq. (23) reduces to the following 6 × 6 matrix which reproduce the results already discussed briefly in the previous section. From Eq. (29) we see that the triplets gain masses from the integration of F ∆ states with the entries Ω,Ω being crucial. Thus, the 3 × 3 induced mass matrix for the triplets will have form where the subscript T indicates that the appropriate matrices should be derived from matrices appearing in Eq. (29). For example M F,T ≃ M F and for Ω T ,Ω T one should take into account some GUT Clebsch factors. These factors do not play any role in our analysis. It is important that the triplet 3 × 3 mass matrix is generated and all the triplets acquire masses. The situation differs for the doublet fragments. Since the 50-plets do not include the doublets, the appropriate M F,D , Ω D andΩ D matrices will have dimensions 2 × 2, 3 × 2 and 2 × 3 respectively. Up to some irrelevant Clebsch factors its structure is and the induced 3 × 3 doublet mass matrix is Clearly, due to the form of the matrices in Eq. (31), the matrix in Eq. (32) has one zero eigenvalue. The reason is simple: this 3 × 3 mass matrix is effectively induced by integrating out two heavy states (the matrix M F,D is of 2 × 2 dimension). Thus, one doublet pair is light, and should be identified to the MSSM Higgs doublets. Once more we stress that this is a result in both (a) and (b) cases. Let us summarize the role of the anomalous U(1) symmetry used here. It forbids the renormalizable coupling H 2 , Σ 2 , ΦHΣ which would contribute to the MSSM doublet mass. This U(1) symmetry also guarantees that ∆ = 0. This is important because the combination ∆∆ has negative U(1) charge and the allowed operators such as ∆∆ (H 2 + Σ 2 ), which might be induced by Planck scale physics, do not give rise to any contribution to the doublet mass. The condition ∆ = 0 also guarantees that there are no mixings between 5 Φ and 5 H,Σ , 45 Σ states (see Eq. (23)) and thus the integration of heavy 5 Φ , 5 Φ -plets does not destroy the DT hierarchy.
In what follows, we discuss some details of the Yukawa sector of this model.

Yukawa Sector and Fermion Mass Generation
Now we discuss the fermion sector of the model and show that the charge assignments given in Table 1 give a self-consistent picture. For the matter 16 i -plets (i = 1, 2, 3) we take the family where M is some cut-off scale and can be taken close to M Pl . Note that without using Φ insertion, although both H(10) and Σ(120)-plets include light Higgs doublets, only renormalizable couplings 16 · 16H and 16 · 16Σ do not give desirable fermion mass pattern [22]. Thus, at least the first power of Φ in one of the couplings of Eq. (33) is needed. Note that the operators Φ16 · 16H, Φ16 · 16Σ can be generated from renormalizable couplings through integrating some heavy states. For example, introducing the heavy states 16 h and 16 h with U(1) charges −1/2 and 1/2, the relevant couplings are Integration of 16 h , 16 h states induces the effective operators This is shown in Fig. 1. Besides the Yukawa coupling discussed above we need the operator which will generate Majorana masses for the right handed neutrinos. For the case (a), the corresponding coupling is We now discuss the possibility of generating such couplings from renormalizable interactions. Introducing SO(10) singlet states N,N and N 0 with U(1) charges 2, −2 and 0 respectively, the allowed renormalizable couplings are It is easy to check out that after integrating out the states N,N, N 0 , the operator in Eq. (36) is generated with M * ∼ (M 2 N M 0 ) 1/3 . This integrating-out mechanism is shown in Fig. 2. For case (b) the coupling for the Majorana neutrino mass can be from the operator Recall that∆ can have the VEV [see Eq. (17)] due the non-renormalizable coupling in Eq. (12). The operator in Eq. (38) can also be generated from the renormalizable couplings. Introducing three pairs of 16 ′ , 16 ′ -plets with U(1) charges 3/2 and −3/2, the relevant superpotential terms are Integration of 16 ′ , 16 ′ states leads to the operator in Eq. (38), with corresponding diagram in Fig.   3.a. Besides this, we have to make sure that∆ has a non-zero VEV. For this to happen, the presence of the operator in Eq. (12) [case (b)] is important. If we wish to not rely on unknown Planck physics, these coupling can be generated by introducing the scalar superfields Y andȲ with U(1) charges 4 and −4 resp. With couplingsC∆Y + X 2Ȳ + M Pl YȲ , the integration of Y,Ȳ states induce the operator (b) in Eq. (12). This is depicted in Fig. 3.b. Note that the additional fieldȲ with negative U(1) charge has no VEV and therefore is harmless for DT hierarchy.
As wee see, the presented missing doublet SO(10) model(s) is fully consistent with realistic fermion masses and mixings. The remarkable thing is that the whole scenario including the fermion sector can be constructed from renormalizable couplings.  38) and (12) (case (b)) respectively.

Discussion
In this paper we have proposed a new solution of the doublet-triplet splitting problem within SO (10) GUT via a missing partner mechanism. For this mechanism to be realized through renormalizable superpotential couplings we have considered the scalar superfield content 10 + 120 + 126 + 126 + 210 and the SO(10) rank breaking states C,C. For the latter, two possibilities (a): C = 16,C = 16 and (b): C = 126,C = 126 can be considered with equal success. Our scenario is consistent with realistic fermion sector as well as with successful gauge coupling unification. Unification is achieved because below the GUT scale, the light fields are just those of the MSSM. One can also address the issue of gauge coupling perturbativity above the GUT scale. In this respect, let us point out that the chance is not bad. For instance, in case (a), one can consider the SO(10) breaking down to SU(5) at scale M SO(10) ≃ C ≃ C ∼ 10 17 GeV. Below this scale, light scalar states which are needed are fragments from 10, 120 and 75 (from 210). Note that the states 126, 126 can have mass∼ 10 17 GeV and similar masses for the remaining fragments from 210 (apart from the 75-plet). All these can be achieved by a (mild) fine-tuning. Eventually, the VEV of 24-plet (from 210) is somewhat suppressed, but this does not change anything for the considered DT splitting scenario. With this mass spectrum (including light fermion families), the SU(5) gauge coupling interpolated from M SU (5) ≃ 2 · 10 16 GeV up to the M SO(10) ∼ 10 17 GeV is still perturbative α GUT (M SO(10) ) ≃ 1/12.5. Above the scale M SO (10) , all SO(10) states listed above should be included in the RGE study and one finds that the gauge coupling becomes strong near 1.7 · 10 17 GeV. Thus, this scale should be considered as a natural cut-off of the theory. Of course, more detailed study with accurate calculation of the mass spectrum is needed. Besides this question one should also address proton stability and the problem of fermion flavor (mass and mixing hierarchy) within this scenario.
Since the mechanism which we have proposed opens up a wide playground for SO(10) model building, we hope that our proposal will motivate others to address and investigate an array of issues which we have not attempted in this work.