Theory of static quark forces

Abstract

Working within the framework of classical algebraic chromodynamics, I develop a theory of the equilibrium color forces acting between stationary quarks. To identify the equations appropriate to describing static forces on a fixed time slice, I assume that the "static" color fields (1) appear as the end point of a most probable tunneling process, and (2) satisfy the principle of virtual work. The resulting equations differ from the vanishing-time-derivative specialization of the Euler-Lagrange equations. The static equations imply certain compatibility conditions on the orientations of discrete quark charges relative to the local color field. Unlike the analogous static equations in Abelian electrodynamics, which take the same form on all time slices, the static equations for chromodynamics take a simple form only at the instant of tunneling through. Static forces and potentials calculated on this favored time slice describe the behavior of the system at all later times because of gluon energy conservation. When the static equations of algebraic chromodynamics for the $q\overline{q}$ color-singlet force problem are rewritten as equations for the overlying SU(2) classical Yang-Mills field, they take the form of the equations of the 't Hooft-Polyakov model, but with the Higgs field reinterpreted as the static potential and, of course, with external source charges present. I develop these equations in a perturbation expansion in the color gluon coupling $g$. I conjecture that the relevant zerothorder solution is the unit-topological-charge solution of 't Hooft and Polyakov, which appears to behave as a quark-confining "bag." This solution contributes a zeroth-order orientation energy to the quark potential, which by the principle of virtual work must be extremal with respect to variations in the position and orientation of the "bag." The orientation energy gives the $q\overline{q}$ potential a repulsive central core, and also leads to a zeroth-order ${\ensuremath{\sigma}}_{(n)}^{m}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}}}_{(n)}^{\mathrm{eff}}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}}^{m}$ interaction between the quark spins and the "bag" color magnetic field. The compatibility conditions guarantee that the orientation energy is invariant under changes of gauge of the zeroth-order solution. Confinement, I believe, comes about in order ${g}^{2}$, where there are strong distortions of the quark color flux lines arising from the presence of the background "bag" field. A test of this hypothesis requires construction of the color gluon propagator in a Prasad-Sommerfield background field. I adapt the methods of Brown et al. to get an expression for the vector propagator in terms of the scalar propagator and to give an explicit contour integral formula for the latter, the detailed evaluation of which is now in progress. Since the compatibility conditions and the minimization involved in constructing the orientation energy make the source current orthogonal to all normalizable zero modes, all zeroth-order degeneracies are resolved and a consistent order-${g}^{2}$ perturbation theory exists. I conclude with a brief discussion of expected limits of validity of the perturbation expansion, and of the extension of the methods developed here to nonstatic problems and to the force problem for three (or more) quarks. In the first appendix, I give technical details of the scalar propagator construction. In the second appendix, I illustrate in the case of the Abelian Higgs model the subtleties involved in finding static equations which satisfy the principle of virtual work. I also sketch an argument which shows that assumption (2) above follows from assumption (1), and give a thermodynamic interpretation for the equation of energy conservation in processes in which confining "bags" are being created.