We report on the progress of our study on the color-electric correlation functions under gradient flow on the lattice. This calculation is the first step of our long-term project to estimate a series of important transport coefficients, of which the heavy quark momentum diffusion coefficient is our first attempt, as it can be extracted from a color-electric correlation function that has been calculated non-perturbatively using noise reduction technique in the quenched approximation . By comparing the flowed correlation function with those obtained by other signal-improving techniques, for instance the multi-level algorithm used in , one can gain insight into the applicability of the gradient flow and the renormalization of the correlation functions.
We start with quenched, isotropic lattices. Currently we have finished measuring the color-electric correlation functions on 4 different lattices with $\beta$-values corresponding to a temperature of $T\approx 1.5T_c$. We perform a continuum extrapolation on the flowed correlators at fixed physical flow times followed by an extrapolation of the continuum estimate back to zero flow time. The next step is to extend the study to different temperatures and to extend the study to dynamical QCD.