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We investigate the functional renormalization group (FRG) flow of thetwo-particle vertex function of a model for X-ray absorption in metals.Concerning the appearance of logarithmic divergences, the model is prototypicalfor an important class of mostly zero- and one-dimensional systems whichincludes the Kondo model and the interacting one-dimensional Fermi gas. For ouranalysis we formulate the FRG in the framework of the real-timezero-temperature formalism, in which the model was studied before with aparquet-based approach. We establish that a reasonably crafted,purely-fermionic one-loop FRG approximation is fully equivalent on a detailedlevel to the leading-logarithmic parquet approximation. These two approximationschemes are thus found to just represent different perspectives on the sametechnical steps. This finding also reconfirms the traditional understanding ofthe capabilities of one-loop RG approximations for such models, which wasrecently put into question by an investigation of the X-ray-absorption modelwith multiloop FRG.

Parquet approximation and one-loop renormalization group: Equivalence on the leading-logarithmic level