On the definition and interpretation of a static quark anti-quark potential in the colour-adjoint channel

We study possibilities to define a static quark anti-quark pair in a colour-adjoint orientation based on Wilson loops with generator insertions, using both lattice QCD and leading order perturbation theory in various gauges. Non-perturbatively, the only way to obtain non-zero results while maintaining positivity of the Hamiltonian is by some form of temporal gauge. In this case the correlator is equivalent to a gauge invariant correlation function of a static quark anti-quark pair and a static adjoint quark, the resulting three-point potential is attractive. Saturating open colour indices with colour magnetic fields instead also leads to a gauge invariant correlator. However this object is found to couple to the singlet sector only. There appears to be no lattice observable that reproduces the repulsive adjoint potential predicted by perturbation theory in Lorenz or Coulomb gauges.