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

Abstract

We investigate a planar Dirac oscillator coupled to a spatially uniform \(\utwo=\uone\times\su\) Yang--Mills background. The gauge configuration, adapted from the Dossa--Avossevou construction, contains an Abelian magnetic field \(B\), a non-Abelian spatial amplitude \(β\), and a non-Abelian scalar amplitude \(ρ\). Within the Pauli-reduced formulation, the non-Abelian field strength produces a constant operator on \(\mathbb{C}^{2}_{\mathrm{spin}}\otimes\mathbb{C}^{2}_{\mathrm{iso}}\). This operator contains a diagonal internal-Zeeman contribution proportional to \(σ^{3}T^{3}\) and an off-diagonal spin--isospin term proportional to \(σ^{1}T^{1}+σ^{2}T^{2}\). Its diagonalization gives a doubly degenerate aligned branch and two mixed branches with eigenvalues \[ λ_{\mathrm{FM}}=\frac{g^{2}β^{2}}{4m},\qquad λ_{S}=-\frac{g^{2}β(β-2ρ)}{4m},\qquad λ_{T}=-\frac{g^{2}β(β+2ρ)}{4m}. \] Consequently, the aligned internal-Zeeman scale is quadratic in \(β\), whereas the singlet--triplet separation is linear in \(βρ\). 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.