Low-density expansion for the two-dimensional electron gas

We show that in two dimensions (2D) a systematic expansion of the self-energy and the effective interaction of a dilute electron gas in powers of the two-body T matrix T₀ can be generated from an exact hierarchy of the functional renormalization group equations for one-particle irreducible vertices using the chemical potential as flow parameter. Due to the interference of particle-particle and particle-hole channels at order T₀², in 2D the ladder approximation for the self-energy is not reliable beyond the leading order in T₀. 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.