Confinement: Understanding the Relation Between the Wilson Loop and Dual Theories of Long Distance Yang Mills Theory

In this paper we express the velocity dependent, spin dependent heavy quark potential $V_{q\bar q}$ in QCD in terms of a Wilson Loop $W(\Gamma)$ determined by pure Yang Mills theory. We use an effective dual theory of long-distance Yang Mills theory to calculate $W(\Gamma)$ for large loops; i.e. for loops of size $R > R_{FT}$. ($R_{FT}$ is the flux tube radius, fixed by the value of the Higgs (monopole) mass of the dual theory, which is a concrete realization of the Mandelstam 't Hooft dual superconductor mechanism of confinement). We replace $W(\Gamma)$ by $W_{eff}(\Gamma)$, given by a functional integral over the dual variables, which for $R > R_{FT}$ can be evaluated by a semiclassical expansion, since the dual theory is weakly coupled at these distances. The classical approximation gives the leading contribution to $W_{eff}(\Gamma)$ and yields a velocity dependent heavy quark potential which for large $R$ becomes linear in $R$, and which for small $R$ approaches lowest order perturbative QCD. This latter fact means that these results should remain applicable down to distances where radiative corrections giving rise to a running coupling constant become important. The spin dependence of the potential reflects the vector coupling of the quarks at long range as well as at short range. The methods developed here should be applicable to any realization of the dual superconductor mechanism. They give an expression determining $W_{eff}(\Gamma)$ independent of the classical approximation, but semi classical corrections due to fluctuations of the flux tube are not worked out in this paper. Taking these into account should lead to an effective string theory free from the conformal anomaly.