Discrete symmetries and isosinglet quarks in low-energy supersymmetry

Many extensions of the minimal supersymmetric standard model contain superfields for quarks which are singlets under weak isospin with electric charge −1/3. We explore the possibility that such isosinglet quarks have low or intermediate scale masses, but do not mediate rapid proton decay because of a discrete symmetry. By imposing the discrete gauge anomaly cancellation conditions, we show that the simplest way to achieve this is to extend the Z3 “baryon parity” of Ibáñez and Ross to the isosinglet quark superfields. This can be done in three distinct ways. This strategy is not consistent with grand unification with a simple gauge group, but may find a natural place in superstring-inspired models, for example. An interesting feature of this scenario is that proton decay is absolutely forbidden. † Address after Sept. 1, 1994: Randall Physics Laboratory, University of Michigan, Ann Arbor MI 48109 Extensions of the standard model with supersymmetry unbroken down to energies comparable with the electroweak-breaking scale can solve the naturalness problem associated with the Higgs scalar boson. It is remarkable that in the minimal supersymmetric standard model [1] (MSSM), the three gauge couplings appear to unify [2] at a scale ∼ 1016 GeV, hinting at a grand unified theory (GUT) or some other organizing principle such as superstring theory. There is a potential phenomenological embarrassment in such theories, however; they contain chiral superfields for quarks which are singlets of weak isospin and carry electric charge −1/3. In GUT models, these isosinglet quarks necessarily appear in the same multiplets as the Higgs doublets of the MSSM, so that generically one might expect them to have masses comparable to the electroweak scale. In superstring-inspired models, the chiral superfields come from remnants of the 27 and 27 representations of E6. The masses of the isosinglet quark superfields are extremely model-dependent, and are typically determined by perturbations from flat directions in the superpotential. Therefore, again in superstring models, the isosinglet quarks can very often have low or intermediate scale masses. The most general superpotential for the MSSM plus the isosinglet quarks is given schematically by: W = W0 +W1 +W2 W0 = QHuū+QHdd̄+ LHdē+ μHuHd + μDDD̄ W1 = QLD̄ + ūēD W2 = QQD + ūd̄D̄ . Here Q, ū, d̄, L, ē are the quark and lepton chiral superfields of the MSSM; Hu, Hd are the Higgs doublet chiral superfields of the MSSM; and D, D̄ are the chiral superfields for the isosinglet quarks. Under the gauge group SU(3)c × SU(2)L × U(1)Y , they transform as Q ∼ (3, 2, 1/6) L ∼ (1, 2,−1/2) Hd ∼ (1, 2,−1/2) ū ∼ (3, 1,−2/3) ē ∼ (1, 1, 1) D ∼ (3, 1,−1/3) d̄ ∼ (3, 1, 1/3) Hu ∼ (1, 2, 1/2) D̄ ∼ (3, 1, 1/3) . For now, we assume the conservation of the usual Z2 matter parity which is given for each chiral superfield in the MSSM by (−1)3(B−L), where B and L are the usual baryon number 2 and total lepton number. This matter parity is trivially related to R-parity by a minus sign for fermions. Thus Q, ū, d̄, L, ē all have matter parity −1, and Hu, Hd, D, D̄ each have matter parity +1. Later we will consider the implications of relaxing this assumption. We assume that there are 3 chiral families of quarks and leptons, and the isosinglet quark superfields D and D̄ may or may not also be replicated, but we suppress all flavor and gauge indices. The most pressing phenomenological problem posed by the existence of isosinglet quarks is the possibility of rapid proton decay. For example, if both of the terms QLD̄ and QQD existed in the superpotential with couplings of order unity, and D, D̄ had a mass μD in the TeV range, then the proton would decay in minutes due to one-loop diagrams with the virtual exchange of an isosinglet quark and a wino. The dominant decay mode would be p → K+ν̄, for which Kamiokande has established the experimental limit [3] τ(p → K+ν̄) > 1032 yrs. More generally, the presence of either term in W1 together with either term from W2 will prevent us from consistently assigning B or L to the chiral superfields in the theory, generically resulting in catastrophic proton decay. If the isosinglet quarks exist at all, then there appear to be two ways out of this disaster; either D and D̄ must be very heavy so that their effects on low energy physics very nearly decouple, or some additional symmetry must be invoked to explain why either W1 or W2 or both are missing. Both of these potential solutions are problematic in a supersymmetric GUT. It is possible to arrange for D and D̄ to obtain a very large mass; however, this requires some cleverness because at least one copy of D and D̄ lives in the same multiplet of the GUT gauge group as Hu and Hd. Various proposals have been put forward to effect a separation in mass scales between D, D̄ and Hu, Hd, including the “sliding singlet” mechanism [4], the “missing partner” mechanism [5], the related “missing VEV” mechanism [6], and Higgses as Nambu-Goldstone bosons [7]. These attempts generally require an intricate system of global symmetries to provide for realistic quark and lepton masses. It is also possible in supersymmetric GUTs to accept light isosinglet quarks but to rely on delicate cancellations among couplings to prevent proton decay [8]. 3 In this paper we consider instead the possibility that isosinglet quarks D, D̄ are light, but a discrete symmetry prohibits the terms from either W1 or W2 or both. This strategy is not consistent with a GUT based on a simple gauge group, but could be useful in superstring-inspired models, for example. The possibilities may then be divided into three cases, as follows: Case A: QLD̄ and ūēD are allowed; QQD and ūd̄D̄ are forbidden. Then we can assign baryon number and lepton number B = 1/3, L = 1 to D and B = −1/3, L = −1 to D̄. Thus D and D̄ are “leptoquarks”. Case B: QQD and ūd̄D̄ are allowed; QLD̄ and ūēD are forbidden. Then we can assign baryon number and lepton number B = −2/3, L = 0 to D and B = 2/3, L = 0 to D̄. Thus D and D̄ are “diquarks”. Case C: QQD, ūd̄D̄, QLD̄, and ūēD are all forbidden. Here it is not yet clear how to assign B and L to the isosinglet quark superfields, since they have no renormalizable superpotential interactions other than a mass term. There has already been much interest [9-15] in the phenomenological implications of each of these three cases, particularly in the context of superstring models based on remnants of E6. In this paper we will examine possibilities for discrete symmetries which can enforce the missing couplings in each of the three cases. Specifically, we consider a ZN symmetry under which each chiral superfield transforms as Φ → exp(2πiαΦ/N)Φ (1) where the αΦ are the additive ZN charges. An operator is allowed if and only if the sum of its ZN charges is 0 [mod N ]. For simplicity, we will assume that the ZN charges are not family-dependent; this seems to be required for the quark superfields anyway in order to allow for the observed Cabibbo-Kobayashi-Maskawa mixing. Since even a small violation of the discrete symmetry could result in catastrophic proton decay, it is strongly suggested that the ZN is a “gauged” discrete symmetry. One way [16] (but perhaps not the only way) to obtain a gauged discrete symmetry is to break a gauged U(1) symmetry with an 4 order parameter whose charge is Nq, where the smallest non-zero U(1) charge assignment in the theory is q. Unlike a global symmetry, a gauged discrete symmetry is automatically protected against violation by Planck-scale and other non-perturbative effects. As shown in [17-20], such gauged discrete symmetries are subject to stringent requirements based on anomaly cancellation. Requiring that the ZN symmetry allow the usual Yukawa couplings and masses in W0, one immediately obtains some relations between the ZN -charges. Thus αHd = −αHu and αD̄ = −αD (2) are required in order to allow for Higgs and isosinglet quark masses respectively and αū = −αQ − αHu, αd̄ = −αQ + αHu , αē = −αL + αHu . (3) in order to allow for Yukawa couplings. Each of these equations is understood to hold modulo N . We get further constraints [mod N ] in each of the three cases: Case A: αD = αQ + αL and 3αQ + αL 6= 0. Case B: αD = −2αQ and again 3αQ + αL 6= 0. Case C: αD 6= αQ + αL and αD 6= −2αQ. Now, cases A and B are in some sense more interesting, because in case C the isosinglet quarks are relatively sterile, having only gauge interactions with the chiral superfields of the MSSM. Therefore, we consider cases A and B first. To further constrain the ZN symmetry, we now consider the discrete anomaly cancellation conditions of Ibáñez and Ross [17]. First we consider the mixed ZN ×SU(n)×SU(n) anomaly cancellation condition, which is given in general by