Using the diagrammatic approach, here we study how spin-orbit coupling (SOC) affects the fermion-dimer and dimer-dimer scattering lengths in the Born approximation, and we benchmark their accuracy with the higher-order approximations. We consider both isotropic and Rashba couplings in three dimensions and show that the Born approximation gives accurate results in the $1/\left(m\alpha a_s\right)\ll -1$ limit, where $m$ is the mass of the fermions, $\alpha$ is the strength of the SOC, and $a_s$ is the $s$-wave scattering length between fermions. This is because the higher-loop contributions form a perturbative series in the $1/\left(m\alpha a_s\right)<0$ region that is controlled by the smallness of the residue $Z$ of the dimer propagator. In sharp contrast, since $Z$ grows with the square root of the binding energy of the dimer in the $1/\left(m\alpha a_s\right)>0$ region, all of the higher-loop contributions are of similar order.

中文翻译：

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自旋轨道耦合费米气体中的原子二聚体和二聚体二聚体散射

使用图解方法，在这里我们研究了自旋轨道耦合（SOC）如何在Born近似中影响费米-二聚体和二聚体-二聚体的散射长度，并使用高阶近似对它们的准确性进行基准测试。我们在三个维度上同时考虑了各向同性和Rashba耦合，并证明了Born逼近可以得出精确的结果。$\mathrm{1个}/（米\alpha {\mathrm{一种}}_s）\ll -\mathrm{1个}$ 极限，在哪里 $米$ 是费米子的质量 $\alpha$ 是SOC的强度，并且 ${\mathrm{一种}}_s$ 是个 $s$费米子之间的波散射长度。这是因为较高的回路贡献在系统中形成一个扰动级数。$\mathrm{1个}/（米\alpha {\mathrm{一种}}_s）<0$ 残留物细小程度控制的区域 $ž$二聚体繁殖体。与之形成鲜明对比的是，$ž$ 与二聚体结合能的平方根一起增长 $\mathrm{1个}/（米\alpha {\mathrm{一种}}_s）>0$ 在所有区域中，所有较高回路的贡献都具有相似的顺序。