Automatic Gauged R-Parity

In the minimal supersymmetric standard model, the existence of Rparity is not required for the internal consistency of the theory and might therefore be regarded as ad hoc. I catalog some simple conditions which are sufficient to guarantee that R-parity survives as an unbroken gauged discrete subgroup of the continuous gauge symmetry in certain extensions of the minimal supersymmetric standard model. If these criteria are met, R-parity is automatic. [Based on a talk given at the International Workshop on Supersymmetry and Unification of Fundamental Interactions (SUSY93) at Northeastern University, Boston, March 29-April 1 1993.] Low energy N = 1 supersymmetry has been proposed as a cure for the fine-tuning problem associated with the Higgs scalar boson[1]. However, in the minimal supersymmetric extension of the standard model, proton decay might be expected to proceed at an unacceptable rate due to the virtual exchange of the superpartners of the standard model states[2]. To see this, we can write all of the renormalizable and gauge-invariant terms which might occur in the superpotential: W = W0 +W1 +W2 W0 = μHH̄ + yuQH̄u+ ydQHd+ yeLHe W1 = λ1udd W2 = μ LH̄ + λ2QLd+ λ3LLe . [Here Q and L are chiral superfields for the SU(2)L-doublet quarks and leptons; u, d, e are chiral superfields for the SU(2)L-singlet quarks and leptons, and H, H̄ are the two SU(2)L-doublet Higgs chiral superfields. Family and gauge indices are suppressed. It is possible to eliminate μ′ by a suitable rotation among the superfields H and L; but we choose not to do this because in most extensions of the minimal supersymmetric standard model H and L will not have the same quantum numbers.] The terms in W0 are just the supersymmetric versions of the usual standard model Yukawa couplings and Higgs mass, and they conserve baryon number (B) and lepton number (L). However, W1 violates B by one unit and W2 violates L by one unit. To prevent the proton from decaying in short order, either (λ1) or (μ ′, λ2, λ3) must be very small. (For precise constraints, see refs. [1-3] and others listed in [4].) The simplest and most popular way to save the proton and avoid other phenomenological disasters is to just banish all of the terms occuring in W1 and W2 by means of a discrete Z2 symmetry known as R-parity[2]. All of the standard model states are taken to be even under R-parity and their superpartners are taken to be odd. All interactions are required to have even R-parity. This means that particles with odd R-parity are always produced in pairs, and that the lightest particle with odd R-parity must be stable. At the level of the chiral superfields, this may be implemented by assigning Rp = −1 to Q,L, u, d, e and Rp = +1 to H, H̄. (This Rp is trivially related to R-parity by a factor of −1 for fermions and is usually called matter parity.) Then the terms in W1 and W2 are forbidden because they are Rp-odd, while the terms in W0 are Rp-even and allowed. The Rp symmetry also forbids some B and L-violating operators of dimension five and higher. At the level of the minimal supersymmetric standard model, the assumption of Rparity appears ad hoc, in the sense that nothing goes wrong with the internal consistency of the theory if Rp is not imposed. In contrast, Rp is actually automatic in certain extensions of the minimal supersymmetric standard model which have gauged B − L (e.g.