Quark-mass effects in gradient-flow observables through next-to-next-to-leading order in QCD

We provide results for the vacuum expectation values of the flowed action density, the quark condensate, and the quark kinetic operator in the gradient-flow formalism. We work in $N_{\text{F}}$-flavor QCD, keeping the heaviest quark massive and all others massless. The vacuum expectation values of these operators are calculated numerically through next-to-next-to-leading order QCD, providing important input for the extraction of fundamental QCD parameters from lattice calculations. While the focus is on charm- and bottom-quark mass effects, we provide the results in a form that is independent of the specific quark mass.