标题： 两组分费米子在三维中的三体散射超体积 显示英文标题

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

摘要： 我们研究三个费米子的零能碰撞，其中两个处于自旋向下（$\downarrow$）状态，一个处于自旋向上（$\uparrow$）状态。 假设两体和三体相互作用具有有限范围，我们找到一个参数，$D$，称为三体散射超体积。 我们研究当三个费米子彼此相距很远，或者一个自旋-$\uparrow$（自旋-$\downarrow$）费米子和由其他两个费米子形成的对彼此相距很远时，三体波函数的渐近行为，并推导出波函数的三个渐近展开式。 三体散射超体积$D$出现在此类展开式的$B^{-5}$阶系数中，其中$B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}$是由三个费米子形成的三角形的超半径（我们假设三个费米子的质量相同），而$s_1,s_2,s_3$是三角形的边长。我们在没有两体相互作用的情况下，用$D$表示三个这样的费米子在低能下碰撞的$T$矩阵元。当相互作用较弱时，我们使用 Born 展开近似计算$D$。 我们还分析了由于$D$导致的三个两组分费米子在大周期立方体中的能量偏移，并将此结果推广到多费米子系统。 $D$也决定了两组分费米气体中的三体复合率，我们根据$D$以及气体的密度和温度计算了三体复合率。 显示英文摘要

摘要： 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.