  Landau Fermi liquid theory is a fixed point theory of metals that includes
the forward scattering amplitudes as exact marginal couplings. However, the
fixed point theory that only includes the strict forward scatterings is
non-local in real space. In this paper, we revisit the Fermi liquid theory
using the field-theoretic functional renormalization group formalism and show
how the scale invariant fixed point emerges as a local theory, which includes
not only the forward scatterings but also non-forward scatterings with small
but non-zero momentum transfers. In the low-energy limit, the non-forward
scattering amplitude takes a scale invariant form. If the bare coupling is
attractive beyond a critical strength, the coupling function exhibits a
run-away flow drived by non-forward scattering amplitudes, signifying potential
instabilities in particle-hole channels. The pairing interaction also obeys a
scaling relation if the center of mass momentum of Cooper pairs is comparable
with energy. The coupling functions fully capture the universal low-energy
dynamics of the collective modes and instabilities of Fermi liquids. The
divergence of the cocupling function in the particle-hole channel beyond a
critical interaction suggests an instability toward an ordered phase with a
momentum that depends on the interaction strength. At the critical interaction,
the instability corresponds to the uniform Pomeranchuk or Stoner instability,
but the momentum of the leading instability becomes non-zero for stronger
attractive interaction. In the particle-particle channel, the coupling function
reveals the dynamics of the unstable mode associated with the BCS instability.
When an unstable normal metal evolves into the superconducting state, there
exists a period in which a superconducting state with spatially non-uniform
phase appears due to the presence of unstable Cooperon modes with non-zero
momenta.
