Effective quark-antiquark potential for the constituent quark model.

Abstract

We use Dual QCD to derive an effective potential to order (quark mass${)}^{\mathrm{\ensuremath{-}}2}$ for a constituent quark and antiquark. This is done by expanding the dual QCD Lagrangian to second order in the qq\ifmmode\bar\else\textasciimacron\fi{} spins and velocities around the static central potential, in which the quarks are both spinless and stationary. The field equations are then used to eliminate the dual gluon fields and the Higgs fields of dual QCD in favor of quark variables for an arbitrary but slowly moving qq\ifmmode\bar\else\textasciimacron\fi{} pair with a Dirac string of arbitrary shape connecting them. The result is a Lagrangian, and therefore, a potential, which depends only on the qq\ifmmode\bar\else\textasciimacron\fi{} positions, velocities, and spins. Dual QCD contains only three parameters, which can be determined from the vacuum energy density, the string tension, and the strength of the Coulomb singularity of the central potential. The only free parameters in the spin- and velocity-dependent part of the effective potential are, therefore, the masses of the c and b quarks. When inserted into a Schr\"odinger equation these potentials provide a complete effective constituent quark theory which can be used to calculate qq\ifmmode\bar\else\textasciimacron\fi{} energy levels in terms of the masses and the masses can thereby be fixed (agreement with experiment is excellent). The various potential, spin-spin, spin-orbit, and spinless velocity dependent, can also in principle be compared to lattice calculations of the same quantities. For the spin-orbit case, for example, the agreement is good, although lattice results are not yet precise enough for a real comparison to be made. For the potentials proportional to the velocity squared lattice results do not yet exist. We also attempt to extend the use of these potentials to heavy-light quark-antiquark systems through use of the Salpeter equation and the Dirac equation. The results of this effort are described in two Appendices.