  We consider the resonant Fermi gas, that is, two-component fermions in three
dimensions interacting by a short-range potential of large scattering length.
We introduce a quantity, the three-body contact, that determines several
observables. Within the zero-range model, the number of nearby fermion
triplets, the large-momentum tail of the center-of-mass momentum distribution
of nearby fermion pairs, as well as the large-momentum tail of the two-particle
momentum distribution, are expressed in terms of the three-body contact. For a
small finite interaction range, the formation rate of deeply bound dimers by
three-body recombination, as well as the three-body contribution to the
finite-range correction to the energy, are expressed in terms of the three-body
contact and of a three-body parameter. This three-body parameter, which
vanishes in the zero-range limit, is defined through the asymptotic behavior of
the zero-energy scattering state at distances intermediate between the range
and the two-body scattering length. In general, the three-body contact has
different contributions labeled by spin and angular momentum indices, and the
three-body parameter can depend on those indices. We also include the
generalization to unequal masses for $\uparrow$ and $\downarrow$ particles.
With respect to the relation between three-body loss rate and number of nearby
triplets stated in [Petrov, Salomon and Shlyapnikov, PRL 93, 090404 (2004)],
the present work adds a derivation, expresses the proportionality factor in
terms of the three-body parameter, and includes the general case where there
are several contributions to the three-body contact and several three-body
parameters.
