On Zero Modes and the Vacuum Problem -- A Study of Scalar Adjoint Matter in Two-Dimensional Yang-Mills Theory via Light-Cone Quantisation

SU(2) Yang-Mills Theory coupled to massive adjoint scalar matter is studied in (1+1) dimensions using Discretised Light-Cone Quantisation. This theory can be obtained from pure Yang-Mills in 2+1 dimensions via dimensional reduction. On the light-cone, the vacuum structure of this theory is encoded in the dynamical zero mode of a gluon and a constrained mode of the scalar field. The latter satisfies a linear constraint, suggesting no nontrivial vacua in the present paradigm for symmetry breaking on the light-cone. I develop a diagrammatic method to solve the constraint equation. In the adiabatic approximation I compute the quantum mechanical potential governing the dynamical gauge mode. Due to a condensation of the lowest omentum modes of the dynamical gluons, a centrifugal barrier is generated in the adiabatic potential. In the present theory however, the barrier height appears too small to make any impact in this odel. Although the theory is superrenormalisable on naive powercounting grounds, the removal of ultraviolet divergences is nontrivial when the constrained mode is taken into account. The open aspects of this problem are discussed in detail.