Nonperturbative aspects of Euclidean Yang-Mills theories in linear covariant gauges: Nielsen identities and a BRST-invariant two-point correlation function

In order to construct a gauge-invariant two-point function in a Yang-Mills theory, we propose the use of the all-order gauge-invariant transverse configurations $A^h$. Such configurations can be obtained through the minimization of the functional $A_{\min }^2$ along the gauge orbit within the BRST-invariant formulation of the Gribov-Zwanziger framework recently put forward in [1,2] for the class of the linear covariant gauges. This correlator turns out to provide a characterization of nonperturbative aspects of the theory in a BRST-invariant and gauge-parameter-independent way. In particular, it turns out that the poles of $⟨A_{\mu }^h(k)A_{\nu }^h(-k)⟩$ are the same as those of the transverse part of the gluon propagator, which are also formally shown to be independent of the gauge parameter $\alpha$ entering the gauge condition through the Nielsen identities. The latter follow from the new exact BRST-invariant formulation introduced before. Moreover, the correlator $⟨A_{\mu }^h(k)A_{\nu }^h(-k)⟩$ enables us to attach a BRST-invariant meaning to the possible positivity violation of the corresponding temporal Schwinger correlator, giving thus for the first time a consistent, gauge parameter independent, setup to adopt the positivity violation of $⟨A_{\mu }^h(k)A_{\nu }^h(-k)⟩$ as a signature for gluon confinement. Finally, in the context of gauge theories supplemented with a fundamental Higgs field, we use $⟨A_{\mu }^h(k)A_{\nu }^h(-k)⟩$ to probe the pole structure of the massive gauge boson in a gauge-invariant fashion.