  The failure of Landau-Fermi liquid theory is often considered a telltale sign
of universal, scale-invariant behavior in the emergent field theory of
interacting fermions. Nevertheless, there exist borderline cases where weak
scale invariance coupled with particle-hole asymmetry can coexist with the
Landau quasiparticle paradigm. In this letter, I show explicitly that a
Landau-Fermi liquid can exist for weak power-law scaling of the retarded
Green's function. Such an exotic variant of the traditional Fermi liquid,
although exhibiting a finite quasiparticle weight and large quasiparticle
lifetime, is shown to always be incompatible with Luttinger's theorem for any
non-trivial scaling. This result yields evidence for a Fermi liquid-like ground
state in the high-field, underdoped pseudogap phase of the high-temperature
cuprate superconductors.
