Gauged Discrete Symmetries and Proton Stability

We discuss the results of a search for anomaly free Abelian Z_N discrete symmetries that lead to automatic R-parity conservation and prevents dangerous higher-dimensional proton decay operators in simple extensions of the minimal supersymmetric extension of the standard model (MSSM) based on the left-right symmetric group, the Pati-Salam group and SO(10). We require that the superpotential for the models have enough structures to be able to give correct symmetry breaking to MSSM and potentially realistic fermion masses. We find viable models in each of the extensions and for all the cases, anomaly freedom of the discrete symmetry restricts the number of generations.


I. INTRODUCTION
Supersymmetry (SUSY) is widely believed to be one of the key ingredients of physics beyond the standard model (SM) for various reasons: (i) stability of the Higgs mass and hence the weak scale; (ii) possibility of a supersymmetric dark matter; (iii) gauge coupling unification, suggesting that there is a grand unified theory (GUT) governing the nature of all forces and matter. The fact that the seesaw mechanism for understanding small neutrino masses also requires a new scale close to the GUT scale adds another powerful reason to believe in this general picture.
There are however many downsides to SUSY. For instance, while the SM guarantees nucleon stability, in its supersymmetric version, there appear two new kinds of problems: (i) There are renormalizable R-parity breaking operators allowed by supersymmetry and standard model gauge invariance, e.g.
Here, Q, L, u c , d c and e c denote the left-chiral quark-doublet, lepton doublet, u-type, d-type and electron superfields. Combination of the last two terms leads to rapid proton decay and present limits on nucleon stability imply (for squark masses of TeV; cf. e.g. [1,2,3]): These terms also eliminate the possibility of any SUSY particle being the dark matter of the Universe. 1 Usually assumptions such as either R-parity or matter parity (cf. [4]) are invoked to forbid these couplings and rescue the proton (as well as dark matter) stability. (ii) A second, more vexing, problem is that even after imposing R-parity, one may have dimension five R-parity conserving operators such as κ ijkℓ Q i Q j Q k L ℓ /M P . Such operators also lead to rapid proton decay. In fact, present nucleon stability limits put an upper limit on κ 1121 , κ 1122 10 −8 [1]. In what follows we will refer to these operators as Q Q Q L.
To understand that these problems can be attributed to SUSY, recall that the R-parity breaking terms are forbidden in the case of SM by Lorentz invariance in the standard model and the Q Q Q L is suppressed by two powers of Planck mass and hence not problematic.
In this study, we focus on the problem of baryon number non-conservation and require the model to satisfy the following constraints: (i) R-parity symmetry is exact so that it prevents catastrophic proton decay; (ii) R-parity conserving dimension five proton decay operators of type Q Q Q L are either forbidden or suppressed to the desired level and (iii) the superpotential of the theory has enough structure for ensuring proper symmetry breaking down to the standard model and give fermion masses. We look for symmetries that allow all desired terms in the superpotential while forbidding the unwanted terms discussed and satisfying the anomaly constraints.
The anomaly constraints can be related to triangle graphs involving the not only discrete symmetry and the gauge symmetries but also gravity. The reason for requiring anomaly freedom is the following: there is a general belief that non-perturbative gravitational effects such as black holes and worm holes break global symmetries of nature [5]. In an effective field theory language, they are parametrized by Planck scale suppressed higher dimensional operators. In fact, the above-mentioned Q Q Q L type operators are a manifestation of the fact that global baryon number symmetry is broken by such effects. On the other hand no hair theorem of general relativity says that any non-perturbative gravitational effect must respect the gauge symmetries. Therefore if there is a gauged discrete symmetry in the theory that prevents the undesirable terms under discussion, they will be absent even after all non-perturbative effects are taken into account.
How does one ensure that a discrete symmetry is a gauge symmetry ? This problem has been extensively studied in the literature in the context of the MSSM [1,2,6,7,8,9,10] and the general procedure is to calculate the various anomaly equations involving the discrete group with gravity and the gauge group i.e. vanishing of D g g, D G 2 anomalies where D stands for the discrete symmetry group in question, g is the graviton and G is the (continuous) gauge symmetry on which the theory is based.
In the context of the MSSM, a discrete 6 symmetry has been identified, dubbed 'proton-hexality' in [2], that contains R-parity as 2 subgroup and forbids Q Q Q L [7]. Remarkably, anomaly freedom of this 6 requires the number of generations to be 3 [1]. Moreover, it has been shown that (given 3 generations) this is the only anomaly free symmetry that allows the MSSM Yukawa couplings and neutrino masses while forbidding the dangerous operators [2]. Such symmetries can be extended such as to also forbid the µ term [11]. (For an approach to ensure proton stability by flavor symmetries see e.g. [12].) On the other hand, the charge assignment is different for different standard model representations. This raises the question whether nucleon stability can be ensured by discrete symmetries in (unified) theories where standard model representations get combined in larger multiplets, and the charge assignment is hence restricted more strongly. We therefore seek discrete symmetries ensuring sufficient proton stability in three gauge extensions of the supersymmetric standard model: (i) the left-right symmetric model [13], (ii) the Pati-Salam model [14] and (iii) SO (10). All these models incorporate the B − L gauge group, which is generally used in the discussion of the seesaw mechanism for neutrino masses [15,16,17,18,19] (for a review see e.g. [20]) and also provides one way to guarantee R-parity conservation [21,22,23]. Due to the higher gauge symmetry, which must be broken down to the SM gauge symmetry, specific terms must be present in the superpotential. This poses constraints on the discrete symmetry.
One main result of the study is a connection between the order of the discrete group and the number of generations in all the cases. We give examples of viable models for all the different gauge groups. In our discussion, we follow a certain 'route of unification'. We start with the left-right symmetric extension of the supersymmetric standard model, proceed via the Pati-Salam model to SO(10) GUTs and finally comment on how our results might be used in string compactifications.
This paper is organized as follows: in section II, we discuss anomaly-free gauge symmetries ensuring proton stability in left-right models; we proceed to the Pati-Salam model in section III, and in section IV we discuss SO(10) models. We give our conclusions in section V.

II. LEFT-RIGHT MODEL AND DISCRETE SYMMETRIES
In this section, we discuss the left-right symmetric extension of MSSM, i.e. the gauge group is SU(3) c × SU(2) L × SU(2) R × U(1) B−L . This amounts to breaking B − L symmetry of the model by either a B − L = 2 triplet or a B − L = 1 doublet. In Table I     the Higgs sector of the triplet model.

A. Left-right symmetric models -doublet Higgs case
The (wanted) superpotential is given by On the other hand, the following couplings must be forbidden (we suppress coefficients): To study the anomaly constraints in this model for an arbitrary N group, we start by giving the charge assignments under N to the various superfields (denoted as q F for the field F in the equation below) and writing down the anomaly constraints. The anomaly constraints for N g g, where N g denotes the number of generations. In the first equation which follows from equation (10) of [7]. The assignments must be consistent with the superpotential (3) and has to forbid the terms in (4). We scanned over possible non-anomalous N ≤12 symmetries, keeping the number of generations N g as a free parameter. Remarkably, the smallest viable N g we found is 3, and the smallest N that works with 3 generations is 6. An example is shown in Table II. qL qQ qLc qQc qΦ qχ qχc qχ qχc qS 1 1 5 5 0 0 2 0 4 0 By giving vacuum expectation values (vevs) to the fields χ c and χ c , the 6 symmetry is broken to a 2 symmetry under which matter is odd while the MSSM Higgs are even. That is, we have obtained an effective R-parity which, although there is a gauged B − L symmetry, originates from an 'external' 6 .

B. Left-right models -triplet case
The transformation properties of the fields under the gauge group are shown in Table I (a) and (c). In this case, the right handed neutrino masses arise from the renormalizable couplings in the theory. We have to forbid Q 3 L and (Q c ) 3 L c . There are many anomaly-free discrete symmetries which do the job. The interesting point is that in this case, the minimum number of generations is N g = 2 with N = 2 for the discrete group. The smallest symmetry that works for 3 generations is 3 .
FIG. 1: Effective proton decay operators. (a) shows the usual triplet exchange diagram [24,25]. (b) illustrates that the amplitude vanishes if the mass partner of the Higgs triplet does not couple to SM particles [27].

III. PATI-SALAM MODEL
We now proceed to the Pati-Salam model, i.e. the gauge group is G PS = SU(4) c × SU(2) L × SU(2) R . The new feature of this model compared to the left right model just discussed is that the quarks and leptons belong to the same representation (see Table III).  The rest of the discussion is completely analogous to the left-right case and in fact the solutions displayed in Table  II for the discrete charges apply to the doublet case, and as in the triplet version of the left-right model there are many solutions. Note that the triplet version of Pati-Salam model does not have proton decay due to SU(3) c ⊂ SU(4) invariance.

IV. SO(10) GUT AND DISCRETE SYMMETRIES
We now turn our attention to the discussion of discrete symmetries in SO(10) GUTs. In SO(10) models, the dimension 5 proton decay operators Q Q Q L have two sources:(i) the higher-dimensional coupling [16 m ] 4 and (ii) effective operators emerging from integrating out Higgs triplets [24,25] (see figure 1). Triplet masses of the order M GUT appear to be too small to be consistent with the observed proton life time [26]. As discussed, the coefficient of the Q Q Q L operators have to be strongly suppressed. This requires an explanation of why both contributions to these operators are simultaneously small. As we shall see, such an explanation might arise from a simple discrete symmetry.
The proton decay via Higgs triplet exchange can be forbidden by eliminating the mass term for the 10 multiplet (H). However one needs to make the color triplets in H heavy so that coupling unification is maintained. This can be done by introducing a second 10-plet (H ′ ) such that it forms a mass term with H but the color triplet field in it does not couple to standard model matter [27]. Therefore, from now on we will consider SO(10) models with 2 10-plets.
We will discuss two classes of models: (i) where B − L is broken by 16-Higgs fields (cf. Table IV (b)) and (ii) where B − L is broken by 126-Higgs fields [28,29] (cf. Table IV (c)). We consider Abelian discrete ( N ) symmetries and require that higher dimensional R-parity conserving leading order ∆B = 0 operators, which in this case are of type 16 4 m (where the subscript m stands for matter), are forbidden as are all R-parity breaking terms while allowing all terms in the superpotential that are needed to break the GUT symmetry down to the MSSM. Our focus is on proton stability, and we leave other issues such as fermion masses and doublet-triplet splitting for future studies.

A. 16-Higgs models
In this class of models, one has an independent motivation for introducing a second 10-plet coming from doublettriplet splitting. Further, apart from a pair of 16 ⊕ 16-Higgses, 45-and 54-plets are required to ensure proper GUT symmetry breaking down to MSSM (cf. [30,31,32,33]). The superpotential terms that must be allowed are:  )). This leads to the following constraints on the N charges: Here, we denote the N charge for a field F by q F . Next, we list the anomaly constraints, 16 (N g q ψm + q ψH + q ψ H ) + 10 (q H + q H ′ ) + 45 q A + 54 q S = 0 mod N ′ (8a) To forbid the dangerous couplings ψ m ψ m H ′ , ψ 4 m , ψ m ψ H and ψ 3 m ψ H , ψ m ψ H H, ψ m ψ H H ′ and ψ m ψ H A, the values of the N charges have to be chosen such that they satisfy the inequalities 2 q ψm + q H ′ = 0 mod N , The smallest symmetry allowing to fulfill all criteria is 6 and requires N g = 3. A possible charge assignments is q ψm = 1, q ψH = −2, q ψ H = +2, q H = −2, q H ′ = +2, q 45 = 0 (cf. tables IV (a) and (c)). This charge assignment allows for seesaw couplings and the possibility of fermion masses from couplings of type ψ m ψ m H. The allowed operator ψ m ψ m ψ 2 H contributes to both the fermion masses as well as to the seesaw. We note that the charge assignment is such that we have 3 × generation + vector-like (10) under SO(10) × 6 . The model also eliminates the dangerous proton decay operator Q Q Q L or operator of type (ψ m ) 4 /M P . We note that, although there are higher-dimensional gauge (and 6 ) invariant operators, they do not give rise to proton decay operators for the following reasons:  (ii) any B − L neutral combination is also 6 invariant because it has to involve as many ψ H as ψ H fields, as only the SM singlets with B − L charge ±1 attain a vev. Therefore the product of the 6 non-invariant combination Q Q Q L with the B − L neutral combination of ψ H and ψ H fields cannot be 6 invariant.
Consider, for instance, the operators 1 In the first operator, the expectation value of [ψ H ] 4 vanishes because of the first argument while the second operator is not 6 invariant. We also observe that operators like ψ 4 H H 2 , which would lead to small diagonal entries in the H − H ′ mass matrix, do not exist for the same reason. That is, in this model, proton decay is only due to dimension 6 operators.
We refrain from spelling out the detailed phenomenological analysis of this model. However, our preliminary studies seem to indicate that one can achieve doublet-triplet splitting and realistic fermion masses while avoiding proton decay problem by extending the Higgs content. A complete analysis of these issues are defered to a future publication.
Let us also comment that, like in the left-right model with doublets, the 6 symmetry gets broken by the ψ H and ψ H vevs down to a 2 which forbids W R , i.e. acts as an R-parity. This means that R-parity in this SO(10) model does not originate from B − L.

B. 126-Higgs models
We now discuss models where the 16 H ⊕ 16 H get replaced by 126 ⊕ 126 -the motivation being that R-parity becomes an automatic symmetry. Such models have been extensively discussed in the literature [34,35,36,37]. In our context it means that the last two of the four inequalities in Eq. (9) do not exist (see Eq. (13) below). Instead we have the following set of constraints on the charges from anomaly freedom The superpotential constraints can be decomposed in analogs of Eq. (7) 2q ψm + q H = 0 mod N , and analogs of Eq. (9) 2 q ψm + q H ′ = 0 mod N , (13a) 4 q ψm = 0 mod N . (13b) Typically in a class of these models, there are only 210 dimensional representations that need to couple to ∆ and ∆ fields among themselves as well as with 10 Higgs [38,39]. These imply that the discrete charge of 210 vanishes and also that of ∆ and ∆ are opposite. Substituting these conditions into Eq. (11), it becomes clear that if there is only a single 10 Higgs in the model (or, if q H ′ = 0), the requirement of anomaly freedom becomes rather constraining. We could only find a 8 symmetry with one generation. However, once we allow for non-trivial q H ′ , it is possible to have simple anomaly free N symmetries which satisfy all constraints, the simplest example being a 3 symmetry with the charge assignment listed in table V. The N ≤12 symmetries with non-trivial q H ′ require N g = 3 or larger.
Another possible symmetry is 6 with the charge assignment listed in table IV (c). This model has the same effective structure as the 126 model of Ref. [34,35,36,37] as far as the discussion of fermion masses go (even though it has 2 10-plets, one of the 10's does not couple to fermions due to 6 charge assignments).
For the same reasons as in the 16-Higgs models, the Q Q Q L type induced proton decay operators are forbidden. Again we refrain from a detailed phenomenological analysis of the way this model leads to MSSM at low energies. We close the discussion with a comment on how easily the MSSM Higgs fields emerge from the superpotential: the relevant part of the superpotential has the form: which has the right linear combination of MSSM doublets to maintain all the simple form for the fermion mass results of Ref. [34,35,36,37]. A detailed analysis of these issues will be given elsewhere.

C. SO(10) GUTs in higher dimensions
Let us now comment on implications of our findings for higher-dimensional models of grand unification, such as 'orbifold GUTs' [40,41,42,43,44,45,46], which provide a simple solution to the doublet-triplet splitting problem. In such models the dimension 5 proton decay can be naturally suppressed [42,43] (while dimension 6 proton decay is slightly enhanced [47]) since here the mass partner of the Higgs triplet has vanishing couplings to matter, as in the discussion above. However, (brane) couplings like ψ 4 m , also leading to proton decay, have not been discussed in this scheme. A reliable discussion of such operators seems hardly possible in the effective higher-dimensional field theory framework.
One possible way to address this question is thus to embed the model into string theory or, in other words, to derive orbifold GUT models from string theory. The first steps for doing so have been performed in Refs. [48,49,50]. This has further lead to the scheme of 'local grand unification' [51,52,53,54], which facilitates the construction of supersymmetric standard models from the heterotic string [52,53,55,56]. Here, the two light MSSM matter generations originate from 16-plets localized at points with SO(10) gauge symmetry. Some of these models can have an R-parity arising as 2 subgroup of a gauged, non-anomalous B − L symmetry [55,56,57], like in ordinary GUTs (however, without the need for 126-plets). On the other hand, Q Q Q L operators remain a challenge [53,56]. The fact that these operators could be eliminated so easily in conventional GUTs by simple symmetries leads to the expectation that similar symmetries will be helpful in the string-derived supersymmetric standard models with (local) GUT structures. One lesson which one might learn from our analysis is that one may derive an effective R-parity and suppress Q Q Q L by a discrete (possibly 6 ) symmetry under which matter 16-plets have a universal charge. One might further hope to get insights about the origin of the discrete symmetries (which remains somewhat obscure in the 4D field-theoretic approach) in string models. These issues will be studied elsewhere.

V. CONCLUSION AND COMMENTS
Motivated by the beauty of the ideas of supersymmetry and unification, we have started a search for discrete symmetries that forbid proton decay operators in gauge extensions of the supersymmetric standard model. We required the symmetries to allow the standard interactions and to be anomaly free. Considering the left-right symmetric, Pati-Salam and SO(10) GUT models with various Higgs contents, we could identify (surprisingly simple) symmetries that satisfy all our criteria. In many cases, there is a connection between the anomaly freedom and the number of generations. Often, simple symmetries exist only for 3 generations (or multiples thereof), as in [1]. In the SO(10) models, our symmetries forbid dimension 5 proton decay operators.
Our findings can be interpreted in the following way. Supersymmetric models with an extended or GUT symmetry are often challenged by proton decay. To rectify this, one might be forced to introduce additional (discrete) symmetries. Our examples then show that R-parity can be a consequence of these additional symmetries rather than being related to B − L. From this one might conclude that the appearance of fields with even B − L charges is not a necessity, and, for instance, 16-Higgs and 126-Higgs SO(10) models can be on the same footing.
It is also interesting that the minimal 126-based SO(10) models become free of all dangerous proton decay operators without losing their ability to be predictive in the fermion sector once we add a simple anomaly free discrete symmetry.