m,p\ell m}}.\end{aligned}$$ The elastic and reactive rate constants for channel $p$ are defined as $$\begin{aligned}
\mathcal{K}_p^{\mathrm}{el}(E)=\sum_{\ell,m}{\mathcal{K}^{\mathrm}{el}_{p \ell m}(E)} & = g \frac{h}{2\mu k} \sum_{\ell,m}{\left| 1 - S_{p\ell m,p\ell m}(E) \right|^2}\,,   \label{Keldef} \\
\mathcal{K}_p^{\mathrm}{re}(E)=\sum_{\ell,m}{\mathcal{K}^{\mathrm}{re}_{p \ell m}(E)} & = g \frac{h}{2\mu k} \sum_{\ell,m}{\left(1- |S_{p\ell m,p\ell m}(E)|^2\right)}\,. \label{Klossdef}\end{aligned}$$ Here, $k^2 = 2 \mu E/\hbar^2$ with $E$ denoting the total energy and $g$ is a quantum statistical factor equal to $2$ in the case of identical bosons or fermions in the same internal states, for which only even or odd $\ell$ respectively can occur, or $1$ in other cases. Alternatively, using , we can write $$\begin{aligned}
\mathcal{K}^{\mathrm}{el}_{p \ell m}(E) & = 2 g \frac{h k}{\mu} |\tilde{a}_{p\ell m}(k)|^2 f_{p\ell m}(k) \,,   \label{Kel} \\
\mathcal{K}^{\mathrm}{re}_{p \ell m}(E) & = 2 g \frac{h}{\mu} \tilde{\beta}_{p\ell m}(k) f_{p\ell m}(k) \,, \label{Kloss}\end{aligned}$$ where $$\label{fellm}
f_{p\ell m}(k) = \frac{1}{1+k^2|\tilde{a}_{p\ell m}(k)|^2+2k \tilde{\beta}_{p\ell m}(k)}.$$ Parametrization using $f$ function can be useful, as near threshold we have $f\stackrel{k\to 0}{\rightarrow}1$.

Far from threshold exit channel
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We will now consider the case when the loss channel is strongly open, which means that $E^\infty _2$ is large and negative while we set $E^\infty_1$ to $0$. In this case one can apply the high energy limit for 