Nuclear Physics B337 (1990) 695—708 North-Holland DISCRETE QUANTUM HAIR ON BLACK HOLES AND THE NON-ABELIAN AHARONOV-BOHM EFFECT M.G. ALFORD and John MARCH-RUSSELL Lyman Laboratory of Physics, Harvard University, Cambridge, M4 02138, USA Frank WILCZEK Institute for Advanced Study, Princeton, NJ 08540, USA Received 5 December 1989 In an abelian Higgs model where U(1) is broken to 7Z~,by a condensate of charge pe, the U(l) charge Q~ in a finite volume V is an observable, but charge is screened, so (QV) falls exponentially to zero as V-. ~. It is demonstrated that the ZL,., charge, Q~ modulo pe, can be cast as a surface integral by evaluating exp(2~riQ~/pe) in states containing a shell of unbroken vacuum around the volume, and its value is unaffected by the presence of the condensate inside the shell. Thus in these states Q~ modulo pe is not screened. This shows that black holes can indeed have hair. The extension to a non-abelian discrete gauge charge is discussed, and the detection of this charge by its non-ahelian Aharonov—Bohm interaction with cosmic strings is described. 1. Introduction It is well known that in a Higgs phase of a gauge theory any charge placed at the origin is screened by a rearrangement of the charged condensate, the associated Coulomb field being exponentially damped. One might therefore be led to believe that all long-range effects of a charge at the origin disappear in a Higgs phase. As we shall demonstrate explicitly for a simple model where U(1) is spontaneously broken to a ~ symmetry this is not necessarily the case. It has been argued recently [1, 2] that the ZL~charge is not shielded by the condensate. Thus if there is a particle of charge qe at the origin then the fractional part of qe relative to pe (where pe is the charge of the condensing field) is a long-range observable, the appropriate operator being ~vp(2~rjQ~/pe), (1.1) where Q~ is the charge in volume V. 0550-3213/90/$03.50 © Elsevier Science Publishers By. (North-Holland)