Quark energy in a three-dimensional QCD model with a constant background field

In an $SU(2)$ Yang-Mills field model with a topological mass $\theta$ in the space with dimensionality $d=2+1$ and with account for an external background field, the one-loop quark energy correction is calculated. As a background field, the exact constant solutions of the Yang-Mills field equations are chosen. It is shown that, at the quark mass $m=\frac{1}{4}\theta$, the effective quark mass remains equal to zero even when the calculated one-loop vacuum correction is taken into account. This fact may be caused by a certain internal symmetry of the set of the background gauge field solutions (constant non-Abelian fields) and solutions of the Dirac equation in these fields.