Light-Front QCD(1+1) Coupled to Adjoint Scalar Matter

TL;DR

This paper studies adjoint scalar matter coupled to light-cone QCD in 1+1 dimensions with SU(2) gauge group, revealing topological features, anomalies, and a trivial vacuum structure through Hamiltonian analysis.

Contribution

It introduces a Hamiltonian formalism for light-front QCD(1+1) with adjoint scalars, highlighting the role of zero modes, anomalies, and gauge transformation properties.

Findings

Physical states transform under large gauge transformations.

Vacuum expectation values of key operators vanish.

Spectrum aligns with large N continuum results.

Abstract

We consider adjoint scalar matter coupled to QCD(1+1) in light-cone quantization on a finite `interval' with periodic boundary conditions. We work with the gauge group SU(2) which is modified to ${\rm{SU(2)/Z_2}}$ by the non-trivial topology. The model is interesting for various nonperturbative approaches because it is the sector of zero transverse momentum gluons of pure glue QCD(2+1), where the scalar field is the remnant of the transverse gluon component. We use the Hamiltonian formalism in the gauge $\partial_- A^+ = 0$. What survives is the dynamical zero mode of $A^+$, which in other theories gives topological structure and degenerate vacua. With a point-splitting regularization designed to preserve symmetry under large gauge transformations, an extra $A^+$ dependent term appears in the current $J^+$. This is reminiscent of an (unwanted) anomaly. In particular, the gauge invariant charge and the similarly regulated $P^+$ no longer commute with the Hamiltonian. We show that nonetheless one can construct physical states of definite momentum which are not {\it invariant} under large gauge transformations but do {\it transform} in a well-defined way. As well, in the physical subspace we recover vanishing {\it expectation values} of the commutators between the gauge invariant charge, momentum and Hamiltonian operators. It is argued that in this theory the vacuum is nonetheless trivial and the spectrum is consistent with the results of others who have treated the large N, SU(N), version of this theory in the continuum limit.