We provide the results for important observables in the gradient-flow formalism including effects from a massive quark.

Here we provide crucial input for the determination of parton density function on the lattice using the gradient flow formalism, see this paper.

This is a demonstration of how the gradient flow allows one to circumvent the fundamental obstacles that prohibit the determination of parton density functions (PDFs) from first principle lattice calculations. It evaluates the PDFs for the pion and shows that they agree with phenomenological extractions.

Using the small-flow-time expansion in the gradient-flow formalism, we use the PCAC relation to obtain the strange and charm quark mass by combining the perturbative results for the matching coefficients of the quark currents with lattice data.

We claim that calculating vacuum expectation values of flowed operators on the lattice and in perturbation theory allows one to derive the quark masses in the MS scheme. Here we provide the required perturbative results.