Closed-channel parameters of Feshbach resonances

In the work of Phys. Rev. A 100, 042710 (2019), it was argued that the general description of two-body Feshbach resonances requires "closed-channel parameters" related to the bare bound state causing the resonance. The present work shows that once the resonance theory is renormalised (i.e. its parameters are expressed in terms of physical observables) the dependence of two-body observables on the closed-channel parameters disappears when the inter-channel coupling is localised in a region where the wave functions are energy independent. In such a regime, the resulting quantum defect theory is surprisingly independent of the bare bound state energy. This result highlights a qualitative difference between resonances in this regime, such as magnetic resonances in ultracold atoms, and other kinds of resonances, such as hadron resonances. Although the closed-channel parameters of magnetic resonances do not affect two-body observables, it is suggested that they could be evidenced experimentally from short-distance probing and three-body observables.