  We demonstrate how to devise a Matsubara-formalism-based one-loop
approximation to the flow of the functional renormalization group (FRG) that
reproduces identically the leading-logarithmic parquet approximation. This
construction of a controlled fermionic FRG approximation in a regime not
accessible by perturbation theory generalizes a previous study from the
real-time zero-temperature formalism to the Matsubara formalism and thus to the
de facto standard framework used for condensed-matter FRG studies. Our
investigation is based on a simple model for the absorption of x rays in
metals. It is a core part of our construction to exploit that in a suitable
leading-logarithmic approximation the values of the particle-hole
susceptibility on the real- and on the imaginary-frequency axes are identical.
