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Landau Fermi liquid theory is a fixed point theory of metals that includesthe forward scattering amplitudes as exact marginal couplings. However, thefixed point theory that only includes the strict forward scatterings isnon-local in real space. In this paper, we revisit the Fermi liquid theoryusing the field-theoretic functional renormalization group formalism and showhow the scale invariant fixed point emerges as a local theory, which includesnot only the forward scatterings but also non-forward scatterings with smallbut non-zero momentum transfers. In the low-energy limit, the non-forwardscattering amplitude takes a scale invariant form. If the bare coupling isattractive beyond a critical strength, the coupling function exhibits arun-away flow drived by non-forward scattering amplitudes, signifying potentialinstabilities in particle-hole channels. The pairing interaction also obeys ascaling relation if the center of mass momentum of Cooper pairs is comparablewith energy. The coupling functions fully capture the universal low-energydynamics of the collective modes and instabilities of Fermi liquids. Thedivergence of the cocupling function in the particle-hole channel beyond acritical interaction suggests an instability toward an ordered phase with amomentum 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 strongerattractive interaction. In the particle-particle channel, the coupling functionreveals the dynamics of the unstable mode associated with the BCS instability.When an unstable normal metal evolves into the superconducting state, thereexists a period in which a superconducting state with spatially non-uniformphase appears due to the presence of unstable Cooperon modes with non-zeromomenta.

Fermi liquids beyond the forward scattering limit: the role of non-forward scatterings for scale invariance and instabilities