Gluon Propagation in Curved Space

In quantum chromodynamics (QCD), the vacuum structures are taken to be formed by nonperturbative interactions between gluons. The contributions of gluonic interactions can be parametrized by gluon condensates and especially the dimension-2 condensate $\langle A_{\mu}^{2} \rangle$ can be used to generalize the $\theta$ vacuum. In the generalized picture, each gauge fixed slice forms a curved surface around a hadron and other strongly interacting objects, and the propagation of a gluon has to be considered on this curved surface. The gluon propagator turns out to be massive due to the curvature of the gauge fixed surface implementing the possibility that the massless gauge bosons might provide the majority of the hadronic mass and even behave as dark matter in cosmological scales.