Three-body scattering hypervolume of two-component fermions in three dimensions

We study the zero-energy collision of three fermions, two of which are in the spin-down ($\ensuremath{\downarrow}$) state and one of which is in the spin-up ($\ensuremath{\uparrow}$) state. Assuming that the two-body and three-body interactions have a finite range, we find a parameter $D$ called the three-body scattering hypervolume. We study the three-body wave function asymptotically when three fermions are far apart or one spin-$\ensuremath{\uparrow}$ (spin-$\ensuremath{\downarrow}$) fermion and one pair, formed by the other two fermions, are far apart and derive three asymptotic expansions of the wave function. The three-body scattering hypervolume $D$ appears in the coefficients of such expansions at the order of ${B}^{\ensuremath{-}5}$, where $B={[({s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2})/2]}^{1/2}$ is the hyperradius of the triangle formed by the three fermions (we assume that the three fermions have the same mass), and ${s}_{1}, {s}_{2}, {s}_{3}$ are the sides of the triangle. We compute the $T$-matrix element for three such fermions colliding at low energy in terms of $D$ in the absence of two-body interactions. When the interactions are weak, we calculate $D$ approximately using the Born expansion. We also analyze the energy shift of three two-component fermions in a large periodic cube due to $D$ and generalize this result to the many-fermion system. $D$ also determines the three-body recombination rates in two-component Fermi gases, and we calculate the three-body recombination rates in terms of $D$ and the density and temperature of the gas.