Great progress was made after realizing that all infinities in quantum electrodynamics are related to two effects: the self-energy of the electron/positron, and vacuum polarization.
Indeed, one can show that in the diagrammatic expansion of the lattice self-energy, only local diagrams survive when one goes into the infinite coordination limit.
In addition to its interaction with other objects via the covariant derivative, the gauge field typically contributes energy in the form of a self-energy term.