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 $\{ \vec{A} = [k_1(\hat{r}\times\vec{\Gamma}) + k_2\vec{\Gamma} + k_3(\vec{\Gamma}\cdot\hat{r})\hat{r}]/r, \varphi = f_1(r)\,(\vec{\Gamma}\cdot\hat{r}) + f_2(r)\}$, 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 $\{\vec{A}=\tilde{k} (\hat{r}\times\vec{\Gamma})/r, \varphi=\kappa/r \}$ 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.