We show that more than two generations of quarks and leptons are required to have an anomaly free discrete $R$ symmetry larger than $R$ parity, provided that the supersymmetric standard model can be minimally embedded into a grand unified theory. This connects an explanation for the number of generations with seemingly unrelated problems such as supersymmetry breaking, proton decay, the $\mu$ problem, and the cosmological constant through a discrete $R$ symmetry. We also show that three generations is uniquely required by a nonanomalous discrete $R$ symmetry in classes of grand unified theories such as the ones based on (semi)simple gauge groups.

• Received 18 November 2011

DOI:https://doi.org/10.1103/PhysRevLett.109.181801