  We study the quantum electrodynamics of Luttinger fermions with quadratic
band-crossing dispersion in three dimensions. The model can be viewed as the
low-energy effective theory of a putative $U(1)$ quantum spin liquid with
fermionic Luttinger spinons, or as an extension of the
Luttinger-Abrikosov-Beneslavskii (LAB) model that accounts for transverse gauge
fluctuations with finite photon velocity. Aided by a renormalization group
analysis below four dimensions, we elucidate the presence and stability of
quantum critical phenomena in this model. We find that the non-Fermi liquid LAB
phase is stable against gauge fluctuations, and can thus also be viewed as a
$U(1)$ spin liquid with gapless Luttinger spinons. We discover a multicritical
point with Lifshitz scaling that corresponds to a time-reversal
symmetry-breaking quantum phase transition from the LAB state to a chiral spin
liquid with spinon Landau levels and birefringent emergent photons. This
multicritical point is characterized by a finite fermion-photon coupling in the
infrared and can be viewed as a fermionic analog of the Rokhsar-Kivelson point
in three-dimensional quantum dimer models.
