  We apply the finite-temperature renormalization-group (RG) to a model based
on an effective action with a short-range repulsive interaction and a rotation
invariant Fermi surface. The basic quantities of Fermi liquid theory, the
Landau function and the scattering vertex, are calculated as fixed points of
the RG flow in terms of the effective action's interaction function. The
classic derivations of Fermi liquid theory, which apply the Bethe-Salpeter
equation and amount to summing direct particle-hole ladder diagrams, neglect
the zero-angle singularity in the exchange particle-hole loop. As a
consequence, the antisymmetry of the forward scattering vertex is not
guaranteed and the amplitude sum rule must be imposed by hand on the components
of the Landau function. We show that the strong interference of the direct and
exchange processes of particle-hole scattering near zero angle invalidates the
ladder approximation in this region, resulting in temperature-dependent
narrow-angle anomalies in the Landau function and scattering vertex. In this RG
approach the Pauli principle is automatically satisfied. The consequences of
the RG corrections on Fermi liquid theory are discussed. In particular, we show
that the amplitude sum rule is not valid.
