  Motivated by recent experimental advances in ultracold atomic gases placed in
cavities, we study the influence of the atom-cavity coupling on the Fermi gases
trapped in optical lattices. By adiabatic elimination of the cavity photon
field, the atom-cavity coupling gives rise to effective long-range
interactions. It results in a variety of two-body scattering processes, during
which the atomic pairs can acquire an additional center-of-mass momentum. This
reveals the possibility of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids
in which the atomic pairing momentum is nonzero. By inspecting the phase
diagram at the mean-field level, we confirm that the FFLO superfluid phase
coexists with the zero-momentum pairing, and is the ground state that hosts the
lowest energy. Furthermore, the order parameter characterizing the
nonzero-momentum pairing does not vanish as long as the cavity-induced
interaction is present.
