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Multichannel quantum defect theory (MQDT) provides a powerful toolkit fordescribing and understanding collisions of cold alkali atoms. Various MQDTapproximations differ primarily in how they characterize the so-calledshort-ranged $K$-matrix, ${\mathbf K}_{\text{sr}}$, which encapsulates theshort-ranged, high-energy physics into a handful of low-energy parameters thatexhibit simple and smooth dependence on energy and field. Here, we comparethree different methods for computing ${\mathbf K}_{\text{sr}}$ for homonuclearcollisions of alkali atoms, from lithium to cesium. The MQDT calculations arebenchmarked against numerically converged coupled-channels calculations thatuse a log-derivative propagator out to the asymptotic region. We study how wellthese approximations reproduce positions of $s$-wave magnetic Feshbachresonances, comparing to experiment where possible, and identify thelimitations of various approximations.

Homonuclear ultracold elastic $s$-wave collisions of alkali atoms via multichannel quantum defect theory