one could choose ${\mathcal{O}}=|{\mathcal{A}}_l|$. However, unitarity relates ${\mathcal{A}}_l=1/(k\cot\delta_l-{\mathrm{i}}k)$ to the phase shift $\delta_l$. This constraint dominates when $\delta_l$ is between about $\pi/4$ and $3\pi/4$ – which affects much of the $\mathrm{NN}$ S-wave phase shifts. Even outside this interval, the additional contribution to eq.  is not sensitive to dynamics. In addition, analyticity dictates that phase shifts approach zero like $k^{2l+1}$ for $k\to0$ in the $l$th partial wave. Since both numerator and denominator in eq.  are then zero, ${\mathcal{O}}=\delta_l$ is dominated by numerical uncertainties as $k\to0$. This may not be a problem if the region in which the slopes are determined is far away, but only a closer inspection could tell if that holds. Likewise, one eliminates phase-space factors in decay constants, production cross sections, etc.

A sensible choice for single-channel scattering appears thus to be ${\mathcal{O}}=k^{2l+1}\cot\delta_l$: It is only constrained to be real below the first inelasticity, and imaginary parts are usually small above it. Indeed, the S-wave example above kept track of the imaginary part by plotting $$\left|1-\frac{k\cot\delta_0(\Lambda_2)}{k\cot\delta_0(\Lambda_1)}\right|\;\;.$$ While factors of $k$ formally cancel, one computes ${\mathcal{A}}$ ($k\cot\delta_0$), so that numerics is more benign.

[**Partial-Wave Mixing**]{} In the $\mathrm{NN}$ system, two partial waves with total angular momentum $J$ mix. The 