A MECHANISM GENERATING AXION HAIR FOR KERR BLACK-HOLES

Recently Campbell et al considered Kalb-Ramond axion fields with a Lorentz Chern-Simons coupling to gravity. They find that in the background metric of a slowly rotating Kerr black hole the axion field strength develops a long range scalar 'hair'. We show that the same phenomenon also occurs with 'ordinary' axions, which are not related to form-valued fields. If these scalars couple to the divergence of some anomalous axial vector current, the Hirzebruch signature density acts as a source for the axion. In a Kerr background this leads to an asymptotic 1/r2 fall off of the axion field, i.e. to a scalar 'hair'. Because of the chiral anomaly of the photon (a local version of the signature index theorem), this can happen even in purely bosonic field theory models.