  We develop a pairing-field formalism for ab initio studies of
non-relativistic two-component fermions on a $(d\!+\!1)$-dimensional spacetime
lattice. More specifically, we focus on theories where the interaction between
the two components can be described by the exchange of a corresponding pairing
field. The introduction of a pairing field may indeed be convenient for studies
of, e.g., the finite-temperature phase structure and critical behavior of,
e.g., ultracold atomic Fermi gases. Moreover, such a formalism allows to
directly compute the momentum and frequency dependence of the pair propagator,
from which the pair-correlation function can be extracted. For a first
illustration of the application of our formalism, we compute the density
equation of state and the superfluid order parameter for a gas of unpolarized
fermions in $(0\!+\!1)$ dimensions by employing the complex Langevin approach
to surmount the sign problem.
