Anomaly of discrete family symmetries and gauge coupling unification 1 Takeshi Araki

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 ki assume nontrivial 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 at lowenergy, requires multi SU(2)L doublet Higgs fields. As an example we consider a recently proposed supersymmetric model withQ6 family symmetry. In this example, k2 = 1, k3 = 3 satisfies the GS conditions and the gauge coupling unification appears close to the Planck scale. 1 Anomaly of discrete symmetries Anomaly is a violation of symmetry at the quantum level. In the case of a continuous symmetry, anomaly means non-conservation of the corresponding Noether current. For discrete symmetries, however, there are no corresponding Noether currents. But Fujikawa’s method [1], which is based on the calculation of the Jacobian of the path-integral measure, can be used to define anomaly of discrete symmetries. Let us start by considering a Yang-Mills theory with massless fermions ψ in Euclidean space time, which can be described by the following Lagrangian and the path-integral: