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

TL;DR

This paper investigates the vacuum structure of SU(2) Yang-Mills theory with adjoint scalar matter in 1+1 dimensions using light-cone quantisation, revealing insights into zero modes, constraints, and the impact of gluon condensates.

Contribution

It develops a diagrammatic method to solve the scalar constraint and analyzes the vacuum structure and zero modes in light-cone quantised Yang-Mills theory with scalar matter.

Findings

No nontrivial vacua found in the current paradigm.

A centrifugal barrier arises due to gluon mode condensation, but is too small to affect the model.

Ultraviolet divergence removal is complex when constrained modes are included.

Abstract

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.