Spontaneous breaking of the C , P , and rotational symmetries by topological defects in two extra dimensions

We formulate models of complex scalar fields in the space-time that has a twodimensional sphere as extra dimensions. The two-sphere S is assumed to have the Dirac-Wu-Yang monopole as a background gauge field. The nontrivial topology of the monopole induces topological defects, i.e. vortices, in the scalar field. When the radius of S is larger than a critical radius, the scalar field develops a vacuum expectation value and creates vortices in S. Then the vortices break the rotational symmetry of S. We exactly evaluate the critical radius as rq = √ |q|/μ, where q is the monopole number and μ is the imaginary mass of the scalar. We show that the vortices repel each other. We analyze the vacua of the models with one scalar field in each case of q = 1/2, 1, 3/2 and find that: when q = 1/2, a single vortex exists; when q = 1, two vortices sit at diametrical points on S; when q = 3/2, three vortices sit at the vertices of the largest triangle on S. The symmetry of the model G = U(1)×SU(2)×CP is broken to H1/2 = U(1) , H1 = U(1) ′′×CP , H3/2 = D3h, respectively. Here D3h is the symmetry group of a regular triangle. We extend our analysis to the doublet scalar fields and show that the symmetry is broken fromGdoublet = U(1)×SU(2)×SU(2)f×P to Hdoublet = SU(2) ′ × P . Finally we obtain the exact vacuum solution of the model with the multiplet (q1, q2, · · · , q2j+1) = (j, j, · · · , j) and show that the symmetry is broken from Gmultiplet = U(1)×SU(2)×SU(2j+1)f ×CP to Hmultiplet = SU(2)′×CP . Our results caution that a careful analysis of dynamics of the topological defects is required for construction of a reliable model that possesses such a defect structure. PACS: 11.10.Kk; 11.27.+d; 11.30.Ly; 11.30.Qc 1 To be published in Physical Review D. e-mail: [email protected] e-mail: [email protected]