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 ($\downarrow$) state and one of which is in the spin-up ($\uparrow$) state. Assuming that the two-body and the 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-$\uparrow$ (spin-$\downarrow$) 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 $B^{-5}$, where $B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}$ 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.