Harmless R-parity violation from Z_{12-I} compactification of E_8 X E_8' heterotic string

In a recent $\Z_{12-I}$ orbifold model, an approximate $Z_2$ symmetry which forbids the baryon number violating operators up to sufficiently high orders is found. The dimension-4 $\Delta B\ne 0$ operators of the MSSM fields occur at dimension 10. The effective dimension-5 $\Delta B\ne 0$ operators derived from these are harmless if some VEVs of neutral singlets are O(1/10) times the string scale. The main reason for forbidding these $\Delta B\ne 0$ operators up to such a high order is the large order N=12 of $\Z_N$ since the $H$-momentum rule is $(-1,1,1)$ mod (12, 3, 12). For a lower order $N<12$, the $\Delta B\ne 0$ operators would appear at lower dimensions.


I. INTRODUCTION
The main reason for imposing the R-parity in the minimal supersymmetric standard model (MSSM) is to forbid the dangerous ∆B = 0 superpotential terms. As a bonus, the exact R-parity ensures an absolutely stable lightest supersymmetric particle (LSP) as a candidate for cold dark matter. The R-parity is a simple discrete symmetry in the MSSM to forbid dangerous (renormalizable) ∆B = 0 operators. However, as for the condition on proton longevity, other discrete symmetries in addition to the R-parity can be possible. All possible candidates are classified in Refs. [1,2]. In this paper, we search for a scheme to obtain such a discrete symmetry in compactifications of E 8 ×E ′ 8 heterotic string. The well known R-parity in SO(10) grand unified theory (GUT) is by assigning -1 for the spinor 16 and +1 for the vector 10. This kind of spinor-vector disparity can be adopted in the untwisted sector of heterotic string also, in particular in the phenomenologically attractive E 8 ×E ′ 8 heterotic string [3]. Let us consider only the E 8 part for an illustration. The untwisted sector massless matter spectrum in E 8 can be P 2 = 2 weights distinguished by the spinor or the vector property S : ([+ + + + + + ++]) V : (±1 ± 1 0 0 0 0 0 0) where ± represents ± 1 2 , the notation [ ] means including even number of sign flips inside the bracket, and the underline means permutations of the entries on the underline. Since two spinors in the group space can transform as a tensor in the group space, cubic Yukawa couplings arising from the untwisted sector involving two spinors are of the form SSV, which can be used to assign a kind of matter parity. For this scheme to work, 48 fermions of the three families (including the singlet neutrinos to generate neutrino masses) must belong to the untwisted sector S and Higgs doublets must belong to the untwisted sector V. This requirement is nontrivial as one finds that the standard-like models of [4] do not satisfy this condition. On the other hand, a part of the spectrum of the Z 12 model of [5] satisfies this condition. But this spinor-vector disparity condition alone is not enough to guarantee a harmless discrete symmetry in supersymmetric standard models because there are twisted matter, and Yukawa couplings are more constrained than this simple spinor-vector disparity.
The twisted matter is more complicated because of the form (internal momenta) plus (shift vectors), P + kV (k = 1, 2, · · · , N − 1 in a Z N orbifold). Except in the Z 2 orbifold, there appear fractional numbers such as 1 3 , 1 4 , · · · . So, in general it is very much involved to find a discrete parity if one includes interactions of twisted matter.
GUTs allow harmless proton decay via gauge interactions if the GUT scale M GUT is greater than 10 15 GeV. String models in addition contain proton decay operators through Yukawa couplings which are measured by the string scale M S being considered to be O (10) larger than the GUT scale. If these proton decay operators occur at dimension 4 and 5, they can dominate over the proton decay operators via gauge interactions [6]. The Rparity forbids dimension-4 ∆B = 0 operators, but does not forbid dimension-5 ∆B = 0 operators. So, in some supergravity GUT models dimension-5 ∆B = 0 operators are considered to be the dominant ones of proton decay, predicting p → K + (antilepton). 1 In string compactifications, however, the dimension-5 operators are considered to be dangerous because the coefficients are considered to be O(1) in general [1]. So, if we introduce a kind of matter parity, it must work very ingeniously to forbid the dimension-4 and dimension-5 If a parity is introduced, it is better to be a discrete gauge symmetry [8], otherwise large gravitational corrections such as through wormhole processes may violate it. Even if the discrete symmetry is broken as we consider in this paper, it is better to be a discrete gauge symmetry to free us from gravitational corrections. The reason for anticipating a broken parity in string compactifications is that we will embed the parity in a global U(1) which is not an exact symmetry in string compactifications. But the breaking of the parity will be considered to be harmless if the ∆B = 0 operators derived from breaking that parity is sufficiently suppressed or suppressed by masses greater than 10 17 GeV since anyway ∆B = 0 operators are present in the gauge sector of string GUTs.
Suppose an approximate global symmetry U(1) Γ and its discrete Z 2 subgroup P Γ . It is a kind generalizing the R-parity. In this paper, we use the word 'R-parity' even though we restrict the discussion to the matter parity.
The parity we consider cannot be put in a general form but must be discussed based on specific models. Restricting to specific models is obvious because the approximate global 1 One of simple ways to forbid the dimension-4 and -5 ∆B = 0 operators in supergravity is to introduce a U (1) R-symmetry. However, it should be broken to a discrete symmetry in orbifold string compactifications. It is known that in a supergravity model [2], a Z 6 symmetry also forbids such ∆B = 0 operators. In Ref. [7], for instance, an anomalous U(1) an is employed for the R-parity and also for the proton longevity. representations. + and -represent + 1 2 and − 1 2 , respectively. The Γ values of -2 and -1 for 10 L −1 (T 6) are those for the vectorlike ones and for the t-quark family, respectively. symmetry U(1) Γ must be given in a specific model. This leads us to the discussion of the specific model, Ref. [5].

II. CONTINUOUS U(1) SYMMETRIES
We observe that a discrete subgroup of the anomalous U(1) an of Ref. [5] is not good for the R-parity because neutral singlets carry even and odd U(1) an charges. A good candidate for housing a part of R-parity is U(1) X of flipped SU(5). In Table I, we list X charges of the non-exotic fields as subscripts. We do not list exotic fields and X = 0 singlet fields.
The key representations of the flipped SU(5), i.e. SU(5) × U(1) X , are Higgs : If we restrict to matter and the electroweak scale Higgs fields only, the Z 2 subgroup of U(1) X is a good candidate for the prime source of the R-parity since matter fields, Eq. (1), carry odd X quantum numbers and the electroweak scale Higgs 5 and 5 (the first line of Eq. (2)) carry even X quantum numbers. Singlets with X = 0 (±5) are neutral (Q em =±1) singlets.
To break the flipped SU (5) down to the standard model, the GUT scale Higgs 10 H and 10 H develop GUT scale VEVs.
In fact, the Z 2 subgroup of U(1) X distinguishes the spinor or the vector origin of our spectrum since we assign For example, 10 from spinor of the form (− − + + + · · · ) has an odd X, while 5 from vector of the form (1 0 0 0 0 · · · ) has an even X. This shows that both 10 H and 10 H having odd Xs. Therefore, for the light fields we have a perfect definition for the R-parity, but including heavy fields 10 H and 10 H make us rethink on the R-parity.
The basic difference between Higgs 10 H and matter 10 in T 6 is that the former forms a vectorlike representation with 10 H in T 6 and is removed at the GUT scale, while the latter remains as a chiral one. From Table I, there appear four 10 L −1 s and three 10 L 1 s in the twisted sector T 6. One unmatched 10 L −1 is interpreted as belonging to the t-quark family. The gauge sector of four 10 L −1 s and three 10 L 1 s has the symmetry U(4) × U(3). We factor out one U(1) belonging to the t-quark family. The remaining symmetry of three 10 The U(1) A symmetry is broken by the anomaly and hence we do not consider it. We are interested only in U(1)s because we will assign the parity as a subgroup of a U(1). Thus, we do not consider the nonabelian symmetry SU(3) L × SU(3) R . Then, we are left with the global symmetry U(1) V 10 . This choice of anomaly free U(1) V 10 is consistent with the discrete gauge symmetry [8]. We define V 10 charges of 10 H and 10 H are +1 and -1, respectively. Matter 10 corresponding to the t-quark family carries the vanishing V 10 . Of course, this global symmetry U(1) V 10 , not protected in string models, is broken by Yukawa couplings.
From T 6 sector in Table I, we also find 5 L 3 s and 5 L −3 s carrying odd-X quantum numbers. Again, we call these two vectorlike pairs as 5 H and 5 H , for which we define another vector global symmetry U(1) V 5 . Let V 5 charges of the vectorlike 5 H and 5 H be +1 and -1, respectively. On the other hand, matter 5s which are chiral, carry the vanishing V 5 charge.
Again, this global symmetry U(1) V 5 is broken by Yukawa couplings in our string compactification. For two vectorlike pairs of 5 and 5 of T 4, carrying even-X quantum numbers, we can consider such a U(1) symmetry but the X charges are already even and we do not need another manipulation for these even-X vectorlike pairs.
Before continuing discussion on V 10 and V 5 charges, let us briefly comment on vectorlike representations of exotics. There are two kinds of exotics. One kind is E-exotics which carry X = ± 5 2 charges, and hence they are Q em = ± 1 2 exotics. The other kind is G-exotics which carry SU(5) charges also: 5 + 1 2 and 5 − 1 2 . Being vectorlike, the E-exotics and G-exotics can be assigned respective U(1) V quantum numbers as discussed in the preceding paragraph. But they are more restricted than the global charges we discussed for U(1) V 10 and U(1) V 5 . The These Γ charges are shown in Table I. The U(1) X symmetry is a gauge symmetry and exact, and U(1) V 10 and U(1) V 5 are global symmetries and approximate. When we break these gauge and global symmetries spontaneously by one direction in the U(1) spaces, the gauge symmetry is considered to be broken and a global symmetry remains. The surviving global symmetry is U(1) Γ . This is the so-called 't Hooft mechanism [9]. In our case, the gauge symmetry U(1) X is broken by the VEVs 10 H and 10 H . Because 10 H carries Γ = 2 and 10 H carries Γ = −2, VEVs of 10 H and 10 H do not break the Z 2 subgroup of U(1) Γ . At this level, the continuous symmetry is broken

III. HARMLESS R-PARITY AS A DISCRETE SUBGROUP OF U(1) Γ
Of course, the continuous global symmetry U(1) Γ is not exact, being broken by superpotential terms. But, "Is the discrete subgroup P Γ of U(1) Γ respected by all superpotential terms?" It is not so. However, this is harmless in the proton decay problem. To show this, let us note possible superpotential terms in the MSSM, generating ∆B = 0 operators, where q and l are quark and lepton doublets, respectively. The dimension-4 operator of Eq. (6) alone does not lead to proton decay, but that term together with the ∆L = 0 superpotential qd c l leads to a very fast proton decay and the product of their couplings must satisfy a very stringent constraint, < 10 −26 . The D = 5 operators in (7) are not that much dangerous, but still the couplings must satisfy constraints, < 10 −8 [1,6].
In our model, even with including the possibilities of multiplying a number of neutral singlets, we find that some of the above operators are forbidden up to very high orders.
Since there are numerous neutral singlets which can acquire GUT or string scale VEVs, we must consider all higher order terms also. u c appears from 5 3 in U 3 , U 1 and T 2. obtaining a GUT scale VEV is 10 H . So, effectively, we need a coupling 5·10·10·10 H which breaks the R-parity P Γ (5 and 10 carry P Γ = 1 while 10 H carries P Γ = 0 according to (4) and (5) Since 5 3 has three possible locations U 1 , U 3 and T 2, and 10 −1 has three possible locations U 1 , U 3 and T 6, the possible combinations to be considered are 18 if we require one 10 −1 must be in T 6. These are tabulated in Table II. Now a function of neutral singlets must be found so that the total H-momentum together with those of Table II becomes (-1, 1, 1) mod (12, 3, 12).
Consider the first case (U 1 U 1 U 1 T 6) of Table II. We find a minimum order solution as u 2 = 2, t 2 = 3, which would give a dimension-9 operator. By the computer program, we checked that there is no operators up to dimension-9. So, this solution must be forbidden by the gauge symmetry. To check this, let us note that the gauge U(1) charges of these singlets are [5] u 2 : (0 8 )(1, 1; 0 6 ) ′ , The gauge charge (0 8 )(2, 2; 0 6 ) ′ of two u 2 s, i.e. two s u s of [5], cannot be canceled by three Thus, for Table II the  H ∼ M GUT . The number 1 100 is understood as an average number if we choose the overall coefficient as 1. However, it should be noted that some singlet VEVs can be much smaller than 10 −2 and the overall coefficient can be a relatively small number, 2 in which case 2 If the nonrenormalizable couplings appear as connected cubic diagrams, dimension-10 couplings would other singlet VEVs can be closer to M S . This shows that the dangerous D = 4 operators of Eq. (10) are not harmful in some vacua, S 0 ∼ ( 1 100 )M S , of the Z 12−I compactification of the heterotic string [5]. This proof also shows that the operators O 1 of Eq. (9) is perfectly harmless since one requires the coefficient of O 1 being less than 10 −8 . The operator O 3 is not dangerous since the ∆B = 0 process with O 3 also needs the second operator of Eq. (8).
Thus, for dangerous ∆B = 0 processes we are left with O 2 of Eq. (9). The phenomenological bound on the coefficient of dimension-5 operator O 2 is taken as order 10 −8 . The relevant flipped SU(5) term 5 5 10 1 needs U 1 , U 3 , T 2 (for matter 5), U 1 , U 3 , T 6 (for matter 10), and U 1 , U 3 , T 2 (for matter 1). Possible cases are 54, which are listed in Table III. We analyzed also Γ violating or R-parity violating O 2 terms. In this case, we have the lowest order R-parity violating O 2 terms at dimension 8. They are Therefore, the R-parity violating terms of [5] can be made harmless.
Other possible ∆B = 0 operators may be present. For example, we may consider a P Γ breaking 10 H 10 H 10 10 5 5 5 −2 , 10 H 10 H 10 H 10 5 H 5 5 −2 , etc. The former one leads to dimension-6 operators suppressed by M 4 S , and the second one leads to dimension-5 operators with heavy particle attached. Here, if any R-parity violating operator occurs, it must involve sufficient suppression by M S powers or inclusion of heavy fields. The baryon number violations with such operators involving the heavy field 5 H are safe phenomenologically. It is not any more dangerous than the standard baryon number violation in GUTs.
In sum, the parity P Γ which is a subgroup of an (approximate) continuous global symmetry is an exact symmetry for operators involving the MSSM fields only, but is not an exact symmetry of the full theory. Nevertheless, it is good enough to forbid dangerous proton involve an extra factor of (cubic couplings) 7 power. For cubic couplings of O( 1 5 ), the average VEV ratio can be made small to O( 1 10 ).

IV. CONCLUSION
In the Z 12−I model of Ref. [5], a discrete parity P Γ which is a discrete subgroup of U(1) Γ is found. U(1) Γ contains the U(1) X symmetry of the flipped SU (5). If the SU(5) breaking components 10 H and 10 H are found to carry even X quantum numbers, then we might have achieved an exact discrete parity as a subgroup of U(1) X of flipped SU (5). But in our model these 10 H and 10 H carry odd X quantum numbers and hence we must resort to an approximate U(1) Γ and the resulting approximate P Γ . Nevertheless, the dangerous dimension-4 and dimension-5 ∆B = 0 operators are forbidden up to sufficiently high dimensions, and the discrete R-parity P Γ is harmless. We also noted that this very high order constraint occurs from the high order 12 of Z 12 .