The perturbative QCD gradient flow to three loops

The gradient flow in qcd is treated perturbatively through next-to-next-to-leading order in the strong coupling constant. the evaluation of the relevant momentum and flow-time integrals is described, including various means of validation. for the vacuum expectation value of the action density, which turns out to be a useful quantity in lattice calculations, we find a very well-behaved perturbative series through nnlo. quark mass effects are taken into account through nlo. the theoretical uncertainty due to renormalization-scale variation is significantly reduced with respect to lo and nlo, as long as the flow time is smaller than about 0.1 fm.