  We show that in two dimensions (2D) a systematic expansion of the self-energy
and the effective interaction of the dilute electron gas in powers of the
two-body T-matrix T_0 can be generated from the exact hierarchy of functional
renormalization group equations for the one-particle irreducible vertices using
the chemical potential as flow parameter. Due to the interference of
particle-particle and particle-hole channels at order T_0^2, in 2D the ladder
approximation for the self-energy is not reliable beyond the leading order in
T_0. We also discuss two-body scattering in vacuum in arbitrary dimensions from
the renormalization group point of view and argue that the singular interaction
proposed by Anderson [Phys. Rev. Lett. 65, 2306 (1990)] cannot be ruled out on
the basis of the ladder approximation.
