## 2 Results
### Distorted Wave Born Approximation  
Having calculated the effective interaction potential, we now turn to the analysis of the scattering. At this point, an issue concerning the short-range effects arises. Namely, if the reactants form a collision complex, the short-range three-body physics becomes relevant and impacts the recombination process. Ergodic dynamics of the complex would lead to statistical distribution of the product states. The effective potential (3) is written in the rovibrational state basis and thus does not capture dynamics of the three-body complex. Furthermore, as demonstrated above, the coupling between different  
Figure 3: (**a**) Effective coupling coefficients \(C_{4}^{\rm eff}\) between the 5 th vibrational state and the ones right below it in units of the diagonal coupling constant. (**b**) Same, but for the most weakly bound state. Figure 2: (**a**) Diagonal effective potentials for the most weakly bound state (blue) and the 5 th one counting from the dissociation threshold (orange). (**b**) Couplings between the 5 th vibrational state and the ones right below it as a function of the atom–molecule distance. (**c**) Same couplings calculated for the most weakly bound state. The binding energies \(E_{v}\approx 2\cdot 10^{4}E^{\star}\) for \(v=5\) and \(0.16E^{\star}\) for \(v=1\). vibrational states of the molecular ion arising in the collision has rather long-range nature. It is then warranted to focus on the close-coupled system described by Equation (3), disregarding the possibility of the complex formation. A possible minimal extension of the current results would be to impose (partially) absorbing boundary conditions at short range in order to mimick the long-lived complex creation and treat the part of the flux reaching the inner region statistically [12; 35]. Rather than solving the scattering problem set by Equation (3) in full, as we are anyway working with an approximate model potential, we will now proceed by employing the distorted wave Born approximation (DWBA) to estimate the inelastic collision rates. DWBA is a method based on perturbation theory [26; 36], which utilizes the knowledge of the scattering solution to the elastic part of the problem and has been widely used to describe atom-molecule inelastic collisions [11; 37; 38]. This is particularly suitable for our problem since the diagonal potentials are analytically solvable [33]. Within the DWBA, the off-diagonal elements of the \(\mathbf{K}\) matrix in Equation (8) are given by  
\[K_{if}=-\pi\int dR\,u_{i}(R)u_{f}(R)V_{if}(R)\,, \tag{11}\]  
where the \(u_{i}\) functions denote the energy normalized scattering solution for the diagonal potential in channel \(i\) (in our case given solely by the \(1/R^{4}\) term) and \(V_{if}\) is the coupling matrix element between the initial and the final state given by Equation (4). The elastic scattering within the DWBA is not affected by the nondiagonal terms in the potential and can be described analytically by energy- and angular momentum-dependent phase shifts \(K_{li}=\tan\delta(\ell_{i},k_{i})\). Then the scattering matrix and all observable quantities can be calculated from Equation (8). Note that the matrix elements need to be calculated for all partial waves involved in the collision. Interestingly, assuming that the coupling between the channels behaves as \(1/R^{4}\), the matrix element (11) can be calculated analytically assuming the channel wavefunctions are given by spherical Bessel functions and for angular momenta of the two states fulfilling \(\ell+\ell^{\prime}\geq 1\). This provides the threshold law for the K-matrix element \(K\propto k_{i}^{\ell_{i}+1/2}\). In particular, for \(\ell_{i}=0\) one obtains \(K_{if}=2k_{i}^{1/2}k_{f}^{3/2}\ \,_{2}F_{1}((-1-\ell_{f})/2,(-1+\ell_{f})/2,1,k_{i}^{2}/k_{ f}^{2})/(\ell_{f}^{2}-1)\) where \(\ {}_{2}F_{1}\) is the hypergeometric function. In the limit of a far-from-threshold exit channel (large energy difference between the states compared to the initial kinetic energy) this quantity scales as \(k_{f}^{3/2}\).