  We investigate the pairing physics of a three-component spin-orbit coupled
Fermi gas in two spatial dimensions. The three atomic hyperfine states of the
system are coupled by the recently realized synthetic spin-orbit coupling
(SOC), which mixes different hyperfine states into helicity branches in a
momentum-dependent manner. As a consequence, the interplay of spin-orbit
coupling and the hyperfine-state dependent interactions leads to the emergence
of Fulde-Ferrell (FF) pairing states with finite center-of-mass momenta even in
the absence of the Fermi-surface asymmetry that is usually mandatory to
stabilize an SOC-induced FF state. We show that, for different combinations of
spin-dependent interactions, the ground state of the system can either be the
conventional Bardeen-Cooper-Schrieffer pairing state with zero center-of-mass
momentum or be the FF pairing states. Of particular interest here is the
existence of a three-component FF pairing state in which every two out of the
three components form FF pairing. We map out the phase diagram of the system
and characterize the properties of the three-component FF state, such as the
order parameters, the gapless contours and the momentum distributions. Based on
these results, we discuss possible experimental detection schemes for the
interesting pairing states in the system.
