Universal leading quantum correction to the Newton potential

The derivation of effective quantum gravity corrections to Newton's potential is an important step in the whole effective quantum field theory approach. We hereby add new strong arguments in favor of omitting all the diagrams with internal lines of the massive sources, and we also recalculate the corrections to the Newtonian potential using functional methods in an arbitrary parametrization of the quantum fluctuations of the metric. The general proof of the gauge- and parametrization-independence within this approach is also explicitly given. On top of that, we argue that the universality of the result holds regardless of the details of the ultraviolet completion of quantum gravity theory. Indeed, it turns out that the logarithm quantum correction depends only on the low energy spectrum of the theory that is responsible for the analytic properties of loop's amplitudes.