  We study theoretically the properties of the interacting Dirac liquid, a
novel three-dimensional many-body system which was recently experimentally
realized and in which the electrons have a chiral linear relativistic
dispersion and a mutual Coulomb interaction. We find that the "intrinsic" Dirac
liquid, where the Fermi energy lies exactly at the nodes of the band
dispersion, displays unusual Fermi liquid properties similar to graphene,
whereas the "extrinsic" system with finite detuning or doping behaves as a
standard Landau Fermi liquid. We present analytical and numerical results for
the self-energy and spectral function based on both Hartree-Fock and the random
phase approximation (RPA) theories and compute the quasiparticle lifetime,
residue, and renormalized Fermi velocity of the extrinsic Dirac liquid. A full
numerical calculation of the extrinsic RPA spectral function indicates that the
Fermi liquid description breaks down for large-energy excitations. Furthermore,
we find an additional plasmaron quasiparticle sideband in the spectral function
which is discontinuous around the Fermi energy. Our predictions should be
observable in ARPES and STM measurements.
