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

Abdelmalek Boumali

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arXiv · 2605.30403

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

Abdelmalek Boumali

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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 $\beta$, and a non-Abelian scalar amplitude $\rho$. Within the Pauli-reduced formulation, the non-Abelian field strength produces a constant operator on $\mathbb{C}^{2}_{\mathrm{spin}}\otimes\mathbb{C}^{2}_{\mathrm{iso}}$. 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 \[ \lambda_{\mathrm{FM}}=\frac{g^{2}\beta^{2}}{4m},\qquad \lambda_{S}=-\frac{g^{2}\beta(\beta-2\rho)}{4m},\qquad \lambda_{T}=-\frac{g^{2}\beta(\beta+2\rho)}{4m}. \] 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.

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Abdelmalek Boumali. 2026-05-28. Non-Abelian Dirac oscillator in a uniform Yang--Mills background: spin--isospin mixing and singlet--triplet splitting. https://arxiv.org/abs/2605.30403

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