Abstract
We investigate the infrared limit of the quantum equation of motion of the gauge boson propagator in various gauges and models with a BRST symmetry. We find that the saturation of this equation at low momenta distinguishes between the Coulomb, Higgs and confining phase of the gauge theory. The Coulomb phase is characterized by a massless gauge boson. Physical states contribute to the saturation of the transverse equation of motion of the gauge boson at low momenta in the Higgs phase, while the saturation is entirely due to unphysical degrees of freedom in the confining phase. This corollary to the Kugo–Ojima confinement criterion in linear covariant gauges also is sufficient for confinement in general covariant gauges with BRST and anti-BRST symmetry, maximal Abelian gauges with an equivariant BRST symmetry, non-covariant Coulomb gauge and in the Gribov–Zwanziger theory.
Highlights
In their seminal work, [1], Kugo and Ojima develop the covariant operator formalism for gauge theories in linear covariant gauge
We do not show this, we find that BRST-exact states in principle can saturate the gluonic quantum equation of motion (QEoM) at vanishing momentum in various gauges with a BRST or equivariant BRST symmetry
The present article is organized as follows: In Sect. 2, we examine the gluonic QEoM in Abelian gauge theory in linear covariant gauges, and we review the arguments that distinguish between Coulomb and Higgs phases in this specific case
Summary
In their seminal work, [1], Kugo and Ojima develop the covariant operator formalism for gauge theories in linear covariant gauge. Proposition: In the confining phase of a gauge theory, unphysical states created by the color current jμa saturate the gluonic QEoM at vanishing momentum. One can discriminate between the Higgs and confinement phase in linear covariant gauge by whether or not physical states contribute to the matrix element of the (generalized) color current jμb at vanishing momentum. It is another matter to prove that the gluonic QEoM is saturated by unphysical states at vanishing momentum. Conventions and some technical details are deferred to three appendices
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