Towards Solving QCD - The Transverse Zero Modes in Light-Cone Quantization

TL;DR

This paper develops a formulation of QCD in lower dimensions using light-cone quantization, focusing on zero modes and their impact on the spectrum, providing insights into non-perturbative aspects of gauge theories.

Contribution

It introduces a detailed treatment of zero modes in light-cone QCD, including their classification and effects on the physical spectrum in a reduced dimensional setting.

Findings

Spectrum consistent with pure SU(2) gluons in (1+1) dimensions.

Explicit resolution of Gauss law for physical states.

Estimated invariant mass of a color singlet state.

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 {\it transverse} momentum gluons. The dimensionally-reduced theory turns out to be 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 non-zero and zero {\it 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 In the 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.