Universal and nonuniversal renormalizations in Fermi liquids

We discuss an interplay between the Fermi-liquid (FL) theory and diagrammatic perturbative approach to interacting Fermi systems. In the FL theory for Galilean-invariant systems, mass renormalization m*/m comes exclusively from fermions at the Fermi surface. We show that in a diagrammatic perturbation theory the same result for m*/m comes from fermions both at and away from the Fermi surface. The equivalence of the FL and perturbative approaches is based on a particular relation between self-energy contributions from high- and low-energy fermions. We argue that care has to be exercised in the renormalization-group approach to a FL in order not to miss the high- energy contribution to m*/m. As particular examples, we discuss m*/m and the quasiparticle residue Z for two-dimensional and three-dimensional systems with both SU(2) and SU(N) symmetries, and with a short-range interaction. We derive an expression for the anisotropic part of the Fermi-liquid vertex in the large-N limit of the SU(N) case.