  We study the ground-state properties of one-dimensional mixtures of bosonic
and fermionic atoms resonantly coupled to fermionic Feshbach molecules. When
the particle densities of fermionic atoms and Feshbach molecules differ, the
system undergoes various depletion transitions between binary and ternary
mixtures, as a function of the detuning parameter. However, when the particle
densities of fermionic atoms and Feshbach molecules are identical, the
molecular conversion and disassociation processes induce a gap in a sector of
low-energy excitations, and the remaining system can be described by a
two-component Tomonaga-Luttinger liquid. Using a bosonization scheme, we derive
the effective low-energy Hamiltonian for the system, which has a similar form
as that of the two-chain problem of coupled Tomonaga-Luttinger liquids. With
the help of improved perturbative renormalization group analysis of the latter
problem, we determine the ground-state phase diagram and find that it contains
a phase dominated by composite superfluid or pairing correlations between the
open and closed resonant channels.
