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

• URL: http://arxiv.org/abs/2501.05194v1
• Date: Thu, 09 Jan 2025 12:33:43 GMT
• Title: Three-body scattering hypervolume of two-component fermions in three dimensions
• Authors: Jiansen Zhang, Zipeng Wang, Shina Tan,
• Abstract summary: We study the zero-energy collision of three fermions, two of which are in the spin-down ($downarrow$) state. When the interactions are weak, we calculate $D$ approximately using the Born expansion. We also analyze the energy shift of three such fermions in a large periodic cube due to $D$ and generalize this result to the many-fermion system.
• Score: 5.735035463793008
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• Abstract: 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 appears in the coefficients of such expansions at the order of $B^{-5}$, where $B$ is the hyperradius of the triangle formed by the three fermions. When the interactions are weak, we calculate $D$ approximately using the Born expansion. We also analyze the energy shift of three such 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 rate in two-component Fermi gases, and we calculate the three-body recombination rate in terms of $D$ and the density and temperature of the gas.

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