Topologically nontrivial solutions to Yang-Mills equations with axisymmetric external sources.

Abstract

We present a new set of solutions to Yang-Mills equations with axially symmetric external charge sources. Our solutions for the gauge fields are not explicitly axisymmetric, but the noninvariance of the fields under a rotation about the symmetry axis can be compensated by a gauge transformation about a symmetry axis in gauge space. All gauge-invariant quantities are therefore axisymmetric. Our solutions are characterized by a gauge-invariant integer winding number $n$, and all winding numbers are allowed. We prove that the total gauge-invariant charge of the system (source plus gauge fields) vanishes identically in our solutions for $n\ensuremath{\ne}0$, even if the source has net charge. We explicitly solve the equations of motion for a spherical shell of charge. The solution depends on the gauge coupling $g$, the total charge of the shell ${Q}_{S}$, and the topological number $n$. We use perturbative methods to obtain the solution in closed form for $\overline{\ensuremath{\alpha}}\ensuremath{\equiv}\frac{{g}^{2}{Q}_{S}}{(4\ensuremath{\pi})}\ensuremath{\ll}1$. We show analytically that in this limit the energy ${\mathcal{E}}_{n}$ of the system satisfies the bound ${\mathcal{E}}_{n}\ensuremath{\le}[\frac{{g}^{2}{Q}_{S}^{2}}{(8\ensuremath{\pi}a)}]\ifmmode\times\else\texttimes\fi{}\frac{1}{(2n+1)}$, where $a$ is the radius of the shell. Using relaxation methods to find the exact solution to the equations of motion numerically for arbitrary $\overline{\ensuremath{\alpha}}$, we establish that this bound is satisfied for all $g$, ${Q}_{S}$, and $n$.