Non-Abelian Dirac oscillator in a uniform Yang--Mills background: spin--isospin mixing and singlet--triplet splitting

We investigate a planar Dirac oscillator coupled to a spatially uniform $\text{utwo}=\text{uone}\times \text{su}$ Yang–Mills background. The gauge configuration, adapted from the Dossa–Avossevou construction, contains an Abelian magnetic field $B$, a non-Abelian spatial amplitude $\beta$, and a non-Abelian scalar amplitude $\rho$. Within the Pauli-reduced formulation, the non-Abelian field strength produces a constant operator on ${\mathbb{C}}_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}^2\otimes {\mathbb{C}}_{\mathrm{i}\mathrm{s}\mathrm{o}}^2$. This operator contains a diagonal internal-Zeeman contribution proportional to ${\sigma }^3{T}^3$ and an off-diagonal spin–isospin term proportional to ${\sigma }^1{T}^1+{\sigma }^2{T}^2$. Its diagonalization gives a doubly degenerate aligned branch and two mixed branches with eigenvalues $${\lambda }_{\mathrm{F}\mathrm{M}}=\frac{g^2{\beta }^2}{4m},{\lambda }_S=-\frac{g^2\beta (\beta -2\rho )}{4m},{\lambda }_T=-\frac{g^2\beta (\beta +2\rho )}{4m}\mathrm{.}$$ Consequently, the aligned internal-Zeeman scale is quadratic in $\beta$, whereas the singlet–triplet separation is linear in $\beta \rho$. The revised formulation makes the sign conventions explicit, verifies the main limiting cases, distinguishes the Pauli-reduced spectrum from a full first-order Dirac diagonalization, and clarifies the physical meaning of the numerical illustrations.