  We study the Fermi-edge singularity, describing the response of a degenerate
electron system to optical excitation, in the framework of the functional
renormalization group (fRG). Results for the (interband) particle-hole
susceptibility from various implementations of fRG (one- and two-
particle-irreducible, multi-channel Hubbard-Stratonovich, flowing
susceptibility) are compared to the summation of all leading logarithmic (log)
diagrams, achieved by a (first-order) solution of the parquet equations. For
the (zero-dimensional) special case of the X-ray-edge singularity, we show that
the leading log formula can be analytically reproduced in a consistent way from
a truncated, one-loop fRG flow. However, reviewing the underlying diagrammatic
structure, we show that this derivation relies on fortuitous partial
cancellations special to the form of and accuracy applied to the X-ray-edge
singularity and does not generalize.
