Theoretical investigation of the molecular nature of $D_{s0}^*(2317)$ and $D_{s1}(2460)$ and the possibility of observing the $D\bar{D}K$ bound state $K_{c\bar{c}}(4180)$ in inclusive $e^+e^-\to c\bar{c}$ collisions

Searching for exotic multiquark states and elucidating their nature remains a central topic in understanding quantum chromodynamics--the underlying theory of the strong interaction. Two of the most studied such states are the charm-strange states $D_{s0}^*(2317)$ and $D_{s1}(2460)$. In this letter, we show for the first time that their prompt production yields in inclusive $e^+e^-\to c\bar{c}$ collisions near $\sqrt{s}=10.6$ GeV measured by the BABAR Collaboration, $Y(D_{s0}^*(2317))$ and $Y(D_{s1}(2460))$, in particular the ratio $R=Y(D_{s0}^*(2317))/Y(D_{s1}(2460))$, can be well explained in the molecular picture, which provide a highly nontrivial verification of their nature being $DK/D^*K$ molecules. On the contrary, treating them as pure $c\bar{s}$ $P-$wave states, the statistical model predicts a ratio $R$ smaller than unity, in contrast with the experimental central value, though in agreement with it considering its relatively large uncertainty. In addition, we predict the production yield of the $D\bar{D}K$ three-body bound state, $K_{c\bar{c}}(4180)$, in $e^+e^-\to c\bar{c}$ collisions and find that it is within the reach of the ongoing Belle II experiment. The present study demonstrates the feasibility of a novel method to unravel the nature of exotic hadrons and the potential of electron-positron collisions in this regard.