Vacuum Structure of Two-Dimensional Gauge Theories on the Light Front

TL;DR

This paper investigates the vacuum structure of (1+1)-dimensional gauge theories on the light front, analyzing the Schwinger model and SU(2) Yang-Mills with adjoint fermions, revealing topological sectors and their effects on the vacuum and condensate.

Contribution

It provides a detailed formulation of 1+1D gauge theories on the light front, incorporating zero modes and topological sectors, and computes the vacuum structure and chiral condensate.

Findings

Reproduces the $ heta$-structure of the vacuum in the Schwinger model.

Identifies the $Z_2$ vacuum structure in SU(2) gauge theory with adjoint fermions.

Finds the chiral condensate is nonzero and inversely proportional to the periodicity length.

Abstract

We discuss the problem of vacuum structure in light-front field theory in the context of (1+1)-dimensional gauge theories. We begin by reviewing the known light-front solution of the Schwinger model, highlighting the issues that are relevant for reproducing the $\theta$-structure of the vacuum. The most important of these are the need to introduce degrees of freedom initialized on two different null planes, the proper incorporation of gauge field zero modes when periodicity conditions are used to regulate the infrared, and the importance of carefully regulating singular operator products in a gauge-invariant way. We then consider SU(2) Yang-Mills theory in 1+1 dimensions coupled to massless adjoint fermions. With all fields in the adjoint representation the gauge group is actually SU(2)$/Z_2$, which possesses nontrivial topology. In particular, there are two topological sectors and the physical vacuum state has a structure analogous to a $\theta$ vacuum. We formulate the model using periodicity conditions in $x^\pm$ for infrared regulation, and consider a solution in which the gauge field zero mode is treated as a constrained operator. We obtain the expected $Z_2$ vacuum structure, and verify that the discrete vacuum angle which enters has no effect on the spectrum of the theory. We then calculate the chiral condensate, which is sensitive to the vacuum structure. The result is nonzero, but inversely proportional to the periodicity length, a situation which is familiar from the Schwinger model. The origin of this behavior is discussed.