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 Coulombgluon. 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.

QCD coupling below 1 GeV from the quarkonium spectrum

“…5 displays a synthesis of results for s Q 2 defined from bound states in the BS framework with high energy data. Here, low energy results are reported together with a sample of high energy data [2] in a logarithmic scale from 100 MeV to 220 GeV, against the three-loop analytic coupling 3 E Q 2 (5) and its massive modification (8). The perturbative three-loop coupling, which IR behavior is definitively ruled out by the data, is also shown in this figure.…”

Integral equation for gauge invariant quark Green’s function

“…At large distances, Wilson loops are saturated by the minimal surfaces having as supports the contours [8,9,18,19,20,21]. Here also, the dominant contributions come from the lowest-order derivative terms.…”

“…Thus, the dominant part of the integral equation (20) to be solved reduces to a closed form relative to the full Green's function S. The approximate form (21) can be used for a first resolution of the integral equation (20).…”

“…We use the following representation which involves phase factors along straight lines together with the full quark Green's function 5(1) = S [6,21]: (14) where S{x^x']A) is the quark propagator in the presence of the external gluon field A. (Integrations on intermediate points are implicit.)…”

Structure of the Gauge Invariant Quark Green’s Function in QCD2

“…Then one integrates with respect to the gluon field through Wilson loops. To achieve the latter program, we use for the quark propagator in external field a representation which involves phase factors along straight lines together with the full gauge invariant quark Green's function [9,11]. The latter feature allows implicit summation of self-energy effects at each step of the operation.…”