Scattering hypervolume of fermions in two dimensions

We define the three-body scattering hypervolume D-F for identical spin-polarized fermions in two dimensions by considering the wave function of three such fermions colliding at zero energy and zero orbital angular momentum. We derive the asymptotic expansions of such a wave function when three fermions are far apart or one pair and the third fermion are far apart, and D-F appears in the coefficients of such expansions. For weak-interaction potentials, we derive an approximate formula of D-F by using the Born expansion. We then study the shift of energy of three such fermions in a large periodic area due to D-F. This shift is proportional to D-F times the square of the area of the triangle formed by the momenta of the fermions. We also calculate the shifts of energy and of pressure of spin-polarized two-dimensional Fermi gases due to a nonzero D-F and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases in two dimensions.