Lattice Gauge Fixing as Quenching and the Violation of Spectral Positivity

TL;DR

This paper investigates the violation of spectral positivity in lattice gauge fixing, attributing it to the quenched nature of auxiliary fields, and uses models and simulations to analyze the phase structure and gluon propagator behavior.

Contribution

It introduces a formalism linking spectral positivity violation to quenching effects and provides a detailed analysis of the phase diagram and gluon propagator in lattice gauge theories.

Findings

Spectral positivity violation arises from quenched auxiliary fields.

The phase diagram resembles that of a Higgs model, with connected phases.

Lattice simulations support a quenched mass-mixing model for the gluon propagator.

Abstract

Lattice Landau gauge and other related lattice gauge fixing schemes are known to violate spectral positivity. The most direct sign of the violation is the rise of the effective mass as a function of distance. The origin of this phenomenon lies in the quenched character of the auxiliary field $g$ used to implement lattice gauge fixing, and is similar to quenched QCD in this respect. This is best studied using the PJLZ formalism, leading to a class of covariant gauges similar to the one-parameter class of covariant gauges commonly used in continuum gauge theories. Soluble models are used to illustrate the origin of the violation of spectral positivity. The phase diagram of the lattice theory, as a function of the gauge coupling $\beta$ and the gauge-fixing parameter $\alpha$, is similar to that of the unquenched theory, a Higgs model of a type first studied by Fradkin and Shenker. The gluon propagator is interpreted as yielding bound states in the confined phase, and a mixture of fundamental particles in the Higgs phase, but lattice simulation shows the two phases are connected. Gauge field propagators from the simulation of an SU(2) lattice gauge theory on a $20^4$ lattice are well described by a quenched mass-mixing model. The mass of the lightest state, which we interpret as the gluon mass, appears to be independent of $\alpha$ for sufficiently large $\alpha$.