The gradient flow at higher orders in perturbation theory

The gradient or Wilson flow has proven to be a useful tool in lattice QCD calculations in the last years. It also offers promising possibilities of cross-fertilization of lattice and perturbative calculations. Up to now, perturbative calculations in the gradient-flow formalism have mostly been carried out to next-to-leading order. Experience shows that the step to next-to-next-to-leading order (NNLO) typically leads to a significant improvement of the perturbative accuracy. We apply well-known techniques for multi-loop calculations to the gradient-flow formalism in order to systematically compute higher orders. After describing our general setup, we show NNLO results of central observables like the gluon action density, which may offer a possibility to extract ${\alpha }_s$ from lattice results. We also present a lattice formulation of the energy-momentum tensor in the gradient-flow formalism through NNLO.