Comment on "Do near-threshold molecular states mix with neighboring $\bar QQ$ states?"

I comment on a paper by Christoph Hanhart and Alexey Nefediev, published in Phys. Rev. D 106, 114003 (2022). The authors discuss the interpretation of mesons close to their lowest decay threshold and present a mechanism for the formation of molecular states. The proposed formalism is then applied to the axial-vector mesons $D_{s1}(2536)$ and $D_{s1}(2460)$, presenting two scenarios for the lighter meson, namely a $D^\star K$ molecule or a compact $c\bar{s}$ state. The authors argue that the latter hypothesis requires a fine-tuning of the mixing angle between the $J^{PC}=1^{++}$ and $J^{PC}=1^{+-}$ $C$-parity eigenstates. In this Comment I show that no such fine-tuning is needed, as demonstrated in an article published in Phys. Rev. D 84, 094020 (2011), where a unitarized quark model was applied to the two $C$-parity eigenstates, coupled to several two-meson channels including $D^\star K$. The coupled-channel dynamics naturally leads to a mixing angle very close to the required one. Moreover, I argue that the $D_1(2420)$ and $D_1(2430)$ axial-vectors, not considered by the authors, as well as a lattice simulation in Phys. Rev. D 90, 034510 (2014), also not mentioned by the authors, do not lend support to a molecular interpretation of the $D_{s1}(2460)$. I conclude with some more general remarks about mesons coupling to $S$-wave thresholds.