Composite magnetic monopoles for SU(N) spontaneously broken to U(1)

Abstract

The point monopole solution found by Corrigan, Olive, Fairlie and Nuyts for SU(3) broken down to U(2) by an octet Higgs field is shown to survive when the symmetry is further broken to U(1), and therefore is a magnetic monopole in the traditional sense. The demonstration exploits a singular transformation from the Abelian gauge with a Dirac string, to a non-singular gauge in which the vector field has a manifest rotational symmetry. The procedure is generalized to SU(N) broken to U(1), always yielding the smallest strength monopole consistent with the Dirac quantization condition, and in some cases higher-strength monopoles as well. The manifest symmetry of the vector field corresponds to an angular momentum J0, different in general from the physical charge-pole angular momentum J. The latter generates a symmetry of the whole system of Higgs and vector fields. Even that symmetry fails for the interior of a finite-energy solution unless J and J0 coincide, which happens only for the minimal monopole coupled to a charge in an SU(2) subgroup of SU(N). If SU(N) is not broken all the way to U(1), there can be solutions with exact J0 symmetry whose possible significance is discussed. A theorem of Georgi and Glashow is used to show that SU(3) → U(1) could occur in a natural way if there were two Higgs fields.