reproduce the scattering lengths in each channel we ensure that $\mathbf{Y}$ contains only off-diagonal terms, $Y_{11}=Y_{22}=0$ and $Y_{12}=Y_{21}=\sqrt{y}$. Under these assumptions we can obtain an analytic formula for the $S$ matrix using formulas -. The off-diagonal element $S_{1,2}$, which is of particular importance here as it describes the reaction process, is given by

$$\left|S_{1,2}\right|^2=\frac{4yC_1^{-2}C_2^{-2}}{1+2y\left(C_1^{-2}C_2^{-2}-\tan\lambda_1 \tan\lambda_2\right)+y^2\left(C_1^{-2}+\tan\lambda_1\right)\left(C_2^{-2}+\tan\lambda_2\right)}$$

For $y\ll 1$ this reduces to $$\left|S_{1,2}\right|^2=4y C_1 ^{-2}(E)C_2 ^{-2}(E).$$ The scaling of the loss rate is thus given by the product of $C^{-2}$ functions and the $y$ parameter.

To better understand the meaning of $y$ parameter, one can define a short-range $S$ matrix as $\mathbf{S}_{\rm sh}=(1-i\mathbf{Y})(1+i\mathbf{Y})^{-1}$ in analogy to Eq. , obtaining $S_{11}=\frac{1-y}{1+y}$ and $S_{12}=\frac{2i\sqrt{y}}{1+y}$. This defines the short range reaction probability $P^{\mathrm}{re} = \left|S_{12}\right|^2 = 4 y/(1+y)^2$. $y$ is thus a parameter describing the short-range reactivity and fulfills $0\leq y\leq 1$. The same intuition can be gained by using WKB to find the wave function at short range, as discussed in Section \[apprwkb\].

A convenient way to describe the scattering process is to use energy-dependent complex scattering length, which can be defined as [@Hutson2007; @Idziaszek2010] $$\begin{aligned}
\label{alm}
\tilde{a}_{p\ell m}(E) & = \tilde{\alpha}_{p\ell m}(E) - i \tilde{\beta}_{p\ell m}(E) =
\frac{1}{i k} \frac{1-S_{p\ell m,p\ell m}}{1+S_{p\ell 