Convergence properties of the equal-time connected Green function approach for temporal gauge SU(2)_{2+1} Yang-Mills theory

TL;DR

This paper investigates the convergence and limitations of the equal-time connected Green function approach in SU(2) Yang-Mills theory in 2+1 dimensions, highlighting issues with UV divergences and gauge invariance.

Contribution

It introduces a truncation scheme for Green functions in temporal gauge SU(2) Yang-Mills theory and analyzes its convergence and gauge invariance properties.

Findings

Approach is UV divergent at the 4-point level

Violates gauge invariance and energy conservation in truncations

Restoring Gauss law constraints does not fully resolve issues

Abstract

The hierarchy of equations of motion for equal-time Green functions in temporal gauge SU(N) Yang-Mills theory is truncated using an expansion in terms of connected Green functions. A second hierarchy of constraint equations arises from Gauss law and can be truncated in a similar way. Within this approximation scheme we investigate SU(2) Yang-Mills theory on a torus in 2+1 spacetime dimensions in a finite basis of plane wave states and focus on infrared and ultraviolet properties of the approach. We study the consequences of restoring the hierarchy of Gauss law constraints and of different momentum cutoffs for the 2- and the 3-point functions. In all truncation schemes considered up to the 4-point level the connected Green function approach is found to be UV divergent and either violating gauge invariance and/or energy conservation. The problems associated with adiabatically generating a perturbed ground state are discussed as well.