Instantaneous Coulomb interaction for a long-range background field in quantum chromodynamics

The instantaneous Coulomb interaction is studied in the SU(2) Yang-Mills theory. The Coulomb Green's function and instantaneous Coulomb potential of a static quark-antiquark pair are evaluated for a background gauge field $A_a^i\left(\overrightarrow{\mathrm{x}}\right)$ that is spherically symmetric and of long range, i.e., that is of order ${\left|\overrightarrow{\mathrm{x}}\right|}^{-1\mathrm{}}$ as $\left|\overrightarrow{\mathrm{x}}\right|\to \infty$. The field $A_a^i\left(\overrightarrow{\mathrm{x}}\right)$ is of the same form as the Wu-Yang magnetic-monopole field. Expansion of the Coulomb Green's function in vector spherical harmonic functions reduces the problem to a radial problem. It is shown that the background field changes the asymptotic form of the instantaneous Coulomb interaction; specifically for the monopole field the correction term is of the same magnitude as the oridinary Coulomb interaction at large distances. In addition, the instanton contribution to the $q\overline{q}$ potential energy is calculated in the temporal-gauge formulation of the theory, and compared to the instantaneous Coulomb interaction. This calculation illustrates the interpretation of instantons as tunneling field configurations. The possibility that long-range field fluctuations with the magnetic-monopole form occur in an ionized meron phase of quantum chromodynamics is discussed.