  We study a model of two species of one-dimensional linearly dispersing
fermions interacting via an s-wave Feshbach resonance at zero temperature.
While this model is known to be integrable, it possesses novel features that
have not previously been investigated. Here, we present an exact solution based
on the coordinate Bethe Ansatz. In the limit of infinite resonance strength,
which we term the strongly interacting limit, the two species of fermions
behave as free Fermi gases. In the limit of infinitely weak resonance, or the
weakly interacting limit, the gases can be in different phases depending on the
detuning, the relative velocities of the particles, and the particle densities.
When the molecule moves faster or slower than both species of atoms, the atomic
velocities get renormalized and the atoms may even become non-chiral. On the
other hand, when the molecular velocity is between that of the atoms, the
system may behave like a weakly interacting Lieb-Liniger gas.
