## Appendix A Derivation of Differential Scattering Cross Section  
For the derivation of the differential scattering cross section of Equation (2), we follow closely the derivation performed by [PERSON] and [PERSON] [PERSON] and [PERSON] (2018). We consider an encounter between two bodies \(A\) (free carrier) and \(B\) (exciton), which are in their ground state prior to the collision. The reduced mass of this system is \(M=\frac{m_{A}m_{B}}{m_{A}+m_{B}}\). Prior to the collision, the internal motions of each of the two bodies are given by their independent Hamiltonians  
\[H_{A}\chi_{A,n}(\mathbf{r}_{A})=E_{A,n}\chi_{A,n}(\mathbf{r}_{A}),\quad H_{B} \chi_{B,m}(\mathbf{r}_{B})=E_{B,m}\chi_{B,m}(\mathbf{r}_{B}), \tag{10}\]  
where \(\chi_{A,n}/\chi_{B,m}\) are the independent wave functions for the internal motion of the two bodies \(A/B\) in the states \(n/m\), respectively, and \(E_{A,n}/E_{B,m}\) their energies. In the absence of the interaction of the two bodies, their relative motion is given by  
\[\left[-\frac{\hbar^{2}}{2M}\
abla^{2}-\frac{1}{2}M\mathbf{v}^{2}\right]F_{11} (\mathbf{R})=0, \tag{11}\]  
with \(\mathbf{v}\) and \(\mathbf{R}\) the relative velocity and position of the two particles and \(F_{11}(\mathbf{R})\) the part of the wave function related to the relative motion of the particles in their ground state. The complete wave equation for this system is given by  
\[\left[-\frac{\hbar^{2}}{2M}\
abla^{2}-\frac{1}{2}M\mathbf{v}^{2}+H_{A}-E_{A, 1}+H_{B}-E_{B,1}+V(\mathbf{R},\mathbf{r}_{A},\mathbf{r}_{B})\right]\Psi( \mathbf{R},\mathbf{r}_{A},\mathbf{r}_{B})=0, \tag{12}\]  
where \(V(\mathbf{R},\mathbf{r}_{A},\mathbf{r}_{B})\) is the interaction potential between the two particles. The wave function can then be expanded in terms of the basis functions \(\chi_{A,n}/\chi_{B,m}\), reading  
\[\Psi(\mathbf{R},\mathbf{r}_{A},\mathbf{r}_{B})=\sum_{n,m}F_{nm}(\mathbf{R}) \chi_{A,n}(\mathbf{r}_{A})\chi_{B,m}(\mathbf{r}_{B}), \tag{13}\]  
meaning that \(F_{nm}(\mathbf{R})\) must obey  
\[\left[\
abla^{2}+k^{2}\right]F_{nm}(\mathbf{R})=\frac{2M}{\hbar^{2}}\int d \mathbf{r}_{A}d\mathbf{r}_{B}\,V(\mathbf{r}_{A},\mathbf{r}_{B},\mathbf{R}) \Psi(\mathbf{r}_{A},\mathbf{r}_{B},\mathbf{R})\chi_{An}^{*}(\mathbf{r}_{A}) \chi_{Bm}^{*}(\mathbf{r}_{B}), \tag{14}\]  
where  
\[\mathbf{k}^{2}=\frac{2M}{\hbar^{2}}\bigg{[}\frac{1}{2}M\mathbf{v}^{2}+E_{A,1} -E_{A,n}+E_{B,1}-E_{B,m}\bigg{]}. \tag{15}\]  
The relation in Equation (15) immediately leads to the thresholds for inelastic scattering, as \(k\) being real implies \(\frac{1}{2}M\mathbf{v}^{2}>E_{A,n}-E_{A,1}+E_{B,m}-E_{B,1}\). The solution to Equation (14) reads  
\[F_{nm}(\mathbf{R})=\frac{-iM}{2\hbar^{2}}\int d\mathbf{r}_{A}d\mathbf{r}_{B}d \mathbf{R}^{\prime}\,V\big{(}\mathbf{r}_{A},\mathbf{r}_{B},\mathbf{R}^{ \prime}\big{)}\Psi\big{(}\mathbf{R}^{\prime},\mathbf{r}_{A},\mathbf{r}_{B} \big{)}\chi_{A,n}^{*}(\mathbf{r}_{A})\chi_{B,m}^{*}(\mathbf{r}_{B})H_{0}^{ \mathrm{I}}\big{(}k\big{|}\mathbf{R}-\mathbf{R}^{\prime}\big{|}\big{)}, \tag{16}\]  
where \(H_{0}^{\mathrm{I}}(k\big{|}\mathbf{R}-\mathbf{R}^{\prime}\big{|})\) is the Hankel function of the first kind, which is the solution to Equation (11). This function satisfies the boundary condition that, for \(\mathbf{R}\gg\mathbf{R}^{\prime}\), the solution represents an outgoing circular wave. Following an identical procedure to that which is used to apply the Born approximation in a 3D system, the asymptotic regime as \(\mathbf{R}\to\infty\) for \(F_{nm}(\mathbf{R})\) reads  
\[F_{nm}(\mathbf{R})\to e^{i\mathbf{k}_{0}\cdot\mathbf{R}}+\frac{e^{i\mathbf{k} \cdot\mathbf{R}}}{\sqrt{\mathbf{R}}}f_{nm}(\theta), \tag{17}\]  
where it was assumed that the asymptotic form of the solution is the sum of an incoming plane wave and an outgoing circular wave.