Anomaly of discrete family symmetries and gauge coupling unification

Abstract

Anomaly of discrete symmetries can be defined as the Jacobian of the path-integral measure. We assume that an anomalous discrete symmetry at low energy is remnant of an anomaly free discrete symmetry, and that its anomaly is cancelled by the Green-Schwarz(GS) mechanism at a more fundamental scale. If the Kac-Moody levels k_i assume non-trivial values, the GS cancellation conditions of anomaly modify the ordinary unification of gauge couplings. This is most welcome, because for a renormalizable model to be realistic any non-abelian family symmetry, which should not be hardly broken low-energy, requires multi SU(2)_L doublet Higgs fields. As an example we consider a recently proposed supersymmetric model with Q_6 family symmetry. In this example, k_2=1, k_3=3 satisfies the GS conditions and the gauge coupling unification appears close to the Planck scale.