Exact Solutions of the SU(2) Yang-Mills Equations from a Static Ansatz

Abstract

We present a systematic study of static solutions to the source-free SU(2) Yang-Mills equations, in which the gauge potential explicitly depends on spin operators. By employing the \emph{vector potential extraction approach} -- which requires the total angular momentum operator (orbital plus spin) to satisfy the standard angular momentum algebra -- we derive the most general form of the spin vector potential $\vec{A}$. This leads to the static ansatz $\{\vec{A}=[k_1(\hat{r}\times \vec{\Gamma })+k_2\vec{\Gamma }+k_3(\vec{\Gamma }\cdot \hat{r})\hat{r}]/r,\phi =f_1(r)(\vec{\Gamma }\cdot \hat{r})+f_2(r)\}$, parametrized by three constants $\{k_1,k_2,k_3\}$ and two radial functions $\{f_1(r),f_2(r)\}$. After substituting this static ansatz into the Yang-Mills equations we obtain a set of consistency equations. Solving these equations provides a complete classification of the exact static solutions, including both real and complex families. The known simple SU(2) static solution $\{\vec{A}=\tilde{k}(\hat{r}\times \vec{\Gamma })/r,\phi =\kappa /r\}$ is recovered as a special case. Our classification reveals new static configurations that could be valuable for non-perturbative studies and for models where the internal spin couples to non-Abelian gauge fields.

Comments: Main 20 pages + SM 71 pages, 0 figure. Revised version