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

Abstract

We formulate models of complex scalar fields in a space-time that has a two-dimensional sphere as extra dimensions. The two-sphere ${S}^{2}$ 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 ${S}^{2}$ is larger than a critical radius, the scalar field develops a vacuum expectation value and creates vortices in ${S}^{2}.$ Then the vortices break the rotational symmetry of ${S}^{2}.$ We exactly evaluate the critical radius as ${r}_{q}=\sqrt{|q|}/\ensuremath{\mu},$ where q is the monopole number and $\ensuremath{\mu}$ 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 ${S}^{2},$ and when $q=3/2,$ three vortices sit at the vertices of the largest triangle on ${S}^{2}.$ The symmetry of the model $G=\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}CP,$ is broken to ${H}_{1/2}=\mathrm{U}{(1)}^{\ensuremath{'}}, {H}_{1}=\mathrm{U}{(1)}^{\ensuremath{''}}\ifmmode\times\else\texttimes\fi{}CP, {H}_{3/2}{=D}_{3h},$ respectively. Here ${D}_{3h}$ is the symmetry group of a regular triangle. We extend our analysis to the doublet scalar fields and show that the symmetry is broken from ${G}_{\mathrm{doublet}}=\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{f}\ifmmode\times\else\texttimes\fi{}P$ to ${H}_{\mathrm{doublet}}=\mathrm{SU}{(2)}^{\ensuremath{'}}\ifmmode\times\else\texttimes\fi{}P.$ Finally, we obtain the exact vacuum solution of the model with the multiplet ${(q}_{1}{,q}_{2},\dots{}{,q}_{2j+1})=(j,j,\dots{},j)$ and show that the symmetry is broken from ${G}_{\mathrm{multiplet}}=\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2j+1)}_{f}\ifmmode\times\else\texttimes\fi{}\mathrm{CP}$ to ${H}_{\mathrm{multiplet}}=\mathrm{SU}{(2)}^{\ensuremath{'}}\ifmmode\times\else\texttimes\fi{}{\mathrm{CP}}^{\ensuremath{'}}.$ Our results caution that a careful analysis of the dynamics of the topological defects is required for construction of a reliable model that possesses such a defect structure.