the open channels at the same energy $\Delta{\epsilon}^*_\ell$ above the channel threshold, as it is illustrated in Fig. \[fig:transmission\_decouple\_sp\]. Fig. \[fig:transmission\_decouple\_sp\](a) shows the bosonic case, with $\ell=0$ and $\Delta{\epsilon}^{*}_0\approx 0.69$, where the transmission coefficients ${T_{0}^{(0)}},{T_{1}^{(0)}},{T_{2}^{(0)}}$ versus the scaled $s$-wave scattering length ${\bar{a}_{0}}$ are shown for one, two and three open channels, respectively. The symmetric line shape centered around the non-interacting case ${\bar{a}_{0}}=0$ is clearly seen. Transmissions for $\ell=1$ and ${\epsilon}^{*}_1\approx 0.31$, are shown in panel (b) as a function of the scaled $p$-wave scattering volume $({\bar{a}_{1}})^3$. In both panels the respective scaled scattering length are considered as external parameters which is in the [$\ell$-SPWA]{} equivalent to a change of the $C_{10}$ parameter.

### The CIR limit {#sss:cir_limit}

As already emphasized above a CIR occurs in the vicinity of a free-space resonance, i.e. in parameter regions $|\bar{a}_{\ell}|\gg1$. For this free-space unitarity regime the scattering amplitude matrix in the [$\ell$-SPWA]{} is simply given by $$\boldsymbol f_{\infty}^{(\ell)}=-\frac{{\mathbf{F}_{\ell\ell}^{\mathbf{o}}}}{{\mathfrak{U}_{\ell\ell}}+{\operatorname{Tr}[{\mathbf{F}_{\ell\ell}^{\mathbf{o}}}]}},
  \label{eq:scattering_amplitude_unitarity}$$ from which we readily derive the corresponding transmission coefficients $$T_{\infty,n}^{(\ell)}=\frac{{\mathfrak{U}_{\ell\ell}}^2-{\operatorname{Tr}[{\mathbf{F}_{\ell\ell}^{\mathbf{o}}}]}({\operatorname{Tr}[{\mathbf{F}_{\ell\ell}^{\mathbf{o}}}]}-({\mathbf{F}_{\ell\ell}^{\mathbf{o}}})_{nn})}{{\mathfrak{U}_{\ell\ell}}^2-{\operatorname{Tr}[{\mathbf{F}_{\ell\ell}^{\mathbf{o}}}]}^2}.
\label{eq:transmission_unitarity_temp}$$ This equation gives the values in the wings of large scattering length of the transmission coefficients of Figs. \[fig:transmission\_t123\_boson\_fermion\] and \[fig:transmission\_decouple\_sp\]. Furthermore, its general energy dependence is show in panels (a) and (b) of 