Topology of a Nontopological Magnetic Monopole

Certain nontopological magnetic monopoles, recently found by Lee and Weinberg, are reinterpreted as topological solitons of a non-Abelian gauged Higgs model. Our study makes the nature of the Lee-Weinberg monopoles more transparent, especially with regard to their singularity structure. Submitted to Physics Letters B. e-mail address: [email protected] Address after Sept. 1, 1994: Physics Department, Columbia University, New York, NY 10027, U.S.A. When Dirac [1] founded the theory of magnetic monopoles in 1931, the monopole was not something that people could not live without. Things changed a great deal in the seventies when ’t Hooft and Polyakov [2] showed that magnetic monopoles inevitably occur as solitons of spontaneously broken non-Abelian gauge theories; such as all grand unified theories where an internal semi-simple gauge symmetry is spontaneously broken to U(1). Their existence is understood in terms of the nontrivial topology of the vacuum manifold, and as such the non-Abelian nature of the original gauge group plays a crucial role. In particular, the Dirac quantization rule is naturally enforced by the underlying non-Abelian structure. Recently, however, Lee and Weinberg [3] constructed a new class of finite-energy magnetic monopoles in the context of a purely Abelian gauge theory. Amazingly enough, the corresponding U(1) potential is simply that of a point Dirac monopole (with the monopole strength satisfying the Dirac quantization rule), yet the total energy is rendered finite by introducing a charged vector field of positive gyromagnetic ratio and by fine-tuning a quartic self-interaction thereof. For more general values of the couplings, this theory together with Einstein gravity was found to produce new magnetically charged black hole solutions with hair [3]. One might be tempted to conclude from this that the existence of the magnetic monopoles does not require an underlying non-Abelian structure, let alone a nontrivial topology of the vacuum manifold. We believe this is a bit premature, and is misleading as far as this particular model is concerned. As pointed out by Lee and Weinberg [3], their U(1) theory may be regarded as a gauge-fixed version of the usual SO(3) Higgs model at some special values of the couplings. An important question to ask here is whether there exists such a hidden structure at other values of the couplings as well. In this letter, we will show that there is indeed a hidden non-Abelian gauge symmetry for general values of the couplings, and that subsequently the integer-charged monopoles of Lee and Weinberg may be regarded as topological solitons associated with certain nonrenormalizable deformations of the SO(3) Higgs model. Adopting the radial gauge rather than the unitary gauge, we found that the apparent Dirac string disappears as usual, while the singularity at the origin still needs to be examined. An important byproduct of our study is a topological understanding of the Dirac quantization rule for the integer-charged Lee-Weinberg