The Low Energy Effective Theory and Nucleon Stability

We show that the Standard Model Lagrangian, including small neutrino masses, has an anomaly-free discrete Z6 symmetry. Anomaly cancellation requires the number of family to be 3 mod 6. This symmetry can ensure the stability of the nucleon even when the threshold of new physics Λ is low as 10 GeV. All ∆B = 1 and ∆B = 2 (B is the baryon number) effective operators are forbidden by the Z6 symmetry. ∆B = 3 operators are allowed, but they arise only at dimension 15. We suggest a simple mechanism for realizing reasonable neutrino masses and mixings even with such a low scale for Λ.