Monopoles in Weinberg-Salam Model

We present a new type of spherically symmetric monopole and dyon solutions with the magnetic charge 4π/e in the standard Weinberg-Salam model. The monopole (and dyon) could be interpreted as a non-trivial hybrid between the abelian Dirac monopole and non-abelian ’t Hooft-Polyakov monopole (with an electric charge). We discuss the possible physical implications of the electroweak dyon. Typeset using REVTEX 1 Ever since Dirac [1] has generalized the Maxwell’s theory with his magnetic monopole, the monopoles have been a subject of extensive studies. The Abelian monopole has been generalized to the non-Abelian gauge theory by Wu and Yang [2] who constructed a nonAbelian monopole solution in the pure SU(2) gauge theory, and by ’t Hooft and Polyakov [3] who have shown that the SU(2) gauge theory allows a finite energy monopole solution as a topological soliton in the presence of a triplet scalar source. In the interesting case of the electroweak theory of Weinberg and Salam [4], however, it has generally been believed that there exists no topological monopole of physical interest. The basis for this “non-existence theorem” is, of course, that with the spontaneous symmetry breaking the quotient space SU(2) × U(1)/U(1)em allows no non-trivial second homotopy. This has led many people to conclude that there is no topological structure in the Weinberg-Salam model which can accommodate a magnetic monopole. The purpose of this letter is to show that this conclusion is premature. In the following we establish the existence of a new type of monopole and dyon solutions in the standard Weinberg-Salam model, and clarify the topological origin of the magnetic charge. Clearly the new monopole and dyon will have important implications in the phenomenology of the electroweak theory, which can make them very interesting from the physical point of view. Before we construct the monopole we must understand how one can circumvent the non-existence theorem in the Weinberg-Salam model and obtain the desired solutions. For this it is important to realize that, with the extra hypercharge U(1) degrees of freedom, the standard Weinberg-Salam model could be viewed as a gauged CP 1 model in which the (normalized) Higgs doublet plays the role of the CP 1 field. Viewed as a CP 1 doublet the Higgs field can now admit a topologically non-trivial configuration whose second homotopy is given by π2(CP ) = Z. This clears the way for a genuine topological monopole in the Weinberg-Salam model which can be described by a completely regular SU(2) potential. To construct the desired solutions we start with the Lagrangian which describes (the bosonic sector of) the standard Weinberg-Salam model