Symmetries, observables and solutions of induced QCD

The only viable theory of the strong interactions is an SU(3) gauge theory, namely quantum chromodynamics (QCD). This theory describes many of the quantitative features of the interactions of hadrons at high energies. However, at energies lower than the hadronic scale it is strongly coupled and gives only qualitative information. Part of the reason is that perturbative QCD has only one dimensionless constant, the gauge coupling which is changed by renormalization from a dimensionless constant into a mass scale. Another parameter of QCD which could be varied is the number of colors of quarks. QCD is known to simplify somewhat in the limit where the number of colors, N, is large. Recently, Kazakov and Migdal have proposed a novel approach to the large N limit which has the hope of providing an exact solution of infinite N QCD. They consider induced QCD which is obtained by integrating over the scalar fields in the lattice gauge theory with a particular partition function. In order to familiarize the reader with the Kazakov-Migdal model, it is shown how it can be solved in the simple case of a lattice with a single site and subsequently the case where the lattice is one-dimensional and periodic. These simple models are interesting in that one can obtained critical behavior for the scalar fields when the scalar field action is quadratic. Later is shown that, for that case, a wide array of correlation functions of both scalar and gluon fields can be calculated analytically. 32 refs