Anomaly of Discrete Symmetries and Gauge Coupling Unification

Anomaly of a discrete symmetry is defined as the Jacobian of the path-integral measure. Assuming that anomaly at low energy is cancelled by the Green-Schwarz (GS) mechanism at a fundamental scale, we investigate possible Kac-Moody levels for anomalous discrete family symmetries. As the first example we consider discrete abelian Baryon number and Lepton number symmetries in the minimal supersymmetric standard model with see-saw mechanism, and find that the ordinary unification of gauge couplings is not consistent with the GS conditions, indicating a possible existence of further Higgs doublets. We consider various recently proposed supersymmetric models with a non-abelian discrete family symmetry. In a supersymmetric example with Q6 family symmetry, the GS conditions are such that the gauge coupling unification appears close to the Planck scale.