Bound state equation in the Wilson loop approach with minimal surfaces

Abstract The large-distance dynamics in quarkonium systems is investigated, in the large- N c limit, through the saturation of Wilson loop averages by minimal surfaces. Using a representation for the quark propagator in the presence of the external gluon field based on the use of path-ordered phase factors, a covariant three-dimensional bound state equation of the Breit–Salpeter type is derived, in which the interaction potentials are provided by the energy–momentum vector of the straight segment joining the quark to the antiquark and carrying a constant linear energy density, equal to the string tension. The interaction potentials are confining and reduce to the linear vector potential in the static case and receive, for moving quarks, contributions from the moments of inertia of the straight segment. The self-energy parts of the quark propagators induce spontaneous breakdown of chiral symmetry with a mechanism identical to that of the exchange of one Coulomb-gluon. In the nonrelativistic limit, long range spin–spin potentials are absent; the moments of inertia of the straight segment provide negative contributions to the spin–orbit potentials going in the opposite direction to those of the pure timelike vector potential. In the ultrarelativistic limit, the mass spectrum displays linear Regge trajectories with slopes in agreement with their classical relationship with the string tension.