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Lattice Gauge Fixing and the Violation of Spectral Positivity

Spectral positivity is known to be violated by some forms of lattice gauge fixing. The most notable example is lattice Landau gauge, where the effective gluon mass is observed to rise rather than fall with increasing distance. We trace this violation to the use of quenched auxiliary fields in the lattice gauge fixing process, and show that violation of spectral positivity is a general feature of quenching. We illustrate this with a simple quenched mass-mixing model in continuum field theory, and with a quenched form of the Ising model. For lattice gauge fixing associated with Abelian projection and lattice Landau gauge, we show that spectral positivity is violated by processes similar to those found in quenched QCD. For covariant gauges parametrized by a gauge-fixing parameter $α$, the SU(2) gluon propagator is well described by a simple quenched mass-mixing formula. The gluon mass parameter appears to be independent of $α$ for sufficiently large $α$.