Effective-action approach to mean-field non-Abelian statics, and a model for bag formation

Abstract

I propose a simple set of equations for mean-field non-Abelian statics with $c$-number sources, at general inverse temperature $\ensuremath{\beta}$, working from the Euclidean path-integral representation of the Hamiltonian partition function. The problem of finding the background-field configuration, and the mean-field potential, for point sources can be reduced to a classical differential equation problem involving a suitably defined thermal effective action functional. As an application I study the interaction of a pair of static classical sources coupled to a quantized SU(2) gauge field, using the simplified model defined by keeping only the leading-logarithm renormalization-group improvement to the local Euclidean action functional. I prove that the mean-field potential in this model grows at least linearly with the source separation, giving a simple model for bag formation. The use of these methods to construct a leading approximation to the $q\overline{q}$ binding problem in SU(3) quantum chromodynamics is discussed in two appendices. Appendix A describes the use of color-charge-algebra methods to generate an equivalent classical source problem, while Appendix B develops the properties of the transformation to a running coupling constant for which the one-loop renormalization group is exact. As a consistency check, in Appendix C I calculate the total mean-field ground-state energy, with source kinetic terms included, and show that it has the expected form.