Towards solving QCD: The transverse zero modes in light-cone quantization

Abstract

We formulate QCD in d+1 dimensions using Dirac's front form with periodic boundary conditions, that is, within discretized light-cone quantization. The formalism is worked out in detail for SU(2) pure glue theory in 2+1 dimensions which is approximated by restriction to the lowest transverse momentum gluons. The dimensionally reduced theory turns out to be an SU(2) gauge theory coupled to adjoint scalar matter in 1+1 dimensions. The scalar field is the remnant of the transverse gluon. This field has modes of both nonzero and zero longitudinal momentum. We categorize the types of zero modes that occur into three classes, dynamical, topological, and constrained, each well known in separate contexts. The equation for the constrained mode is explicitly worked out. The Gauss law is rather simply resolved to extract physical, namely, color singlet states. The topological gauge mode is treated according to two alternative scenarios related to the elimination of the cutoff. In one, a spectrum is found consistent with pure SU(2) gluons in 1+1 dimensions. In the other, the gauge mode excitations are estimated and their role in the spectrum with genuine Fock excitations is explored. A color singlet state is given which satisfies Gauss' law. Its invariant mass is estimated and discussed in the physical limit.