The quark anti-quark potential in the color-dielectric formulation of the transverse lattice

The Color-Dielectric formulation of the Transverse Lattice seems to be a promising tool to describe Quantum Chromodynamics. The Transverse Lattice blends the advantages of Light Front field theories, with their intuitive interpretation of physical observables, and that of Euclidean Lattice Monte Carlo theories with their powerful numerical capabilities. The Color-Dielectric formalism regards the Transverse Lattice fields as being smeared variables much like the electric field is regarded as being an average field in condensed matter physics. The advantage of this approach is that long distance physics (eg. masses) can be represented using fewer, simpler degrees of freedom. The main drawback is that, because of the difficulty in determining the exact V_eff, one is forced into making an ansatz for V_eff which has free parameters. One can determine these by either fitting these parameters to physical observables or determining them from first principles. One physical observable that is used to fit the parameters is the static QQ¯ potential. The QQ¯ potential is an ideal starting point as it probes a symmetry that is not manifest on the Light Front, namely rotational invariance. In addition, the heavy quark potential is well known hence there is a dearth of data, both from theoretical calculations and from experimental measurements. I begin my study of the Color-Dielectric formulation of the Transverse Lattice by investigating the QQ¯ potential in 2 + 1 dimensions. I then expand my work to 3 + 1 dimensional Transverse Lattice QCD. Finally, I work on the Euclidean Lattice to directly calculate the parameters of the Color-Dielectric theory from a set of Schwinger-Dyson equations.