In a recent paper by Gao et al. [Phys. Rev. D 105, 094002 (2022).], $S$-wave $\pi \pi$ scattering phase shifts obtained in a lattice-QCD calculation are analyzed using dispersive $S$-matrix methods. We question the reliability of the conclusion from this analysis that, for a pion mass of 391 MeV, the lattice phases favor the presence of both a $\sigma$-meson bound state and a nearby virtual state. Our main criticism concerns the neglect of the $S$-wave $K\overline{K}$ channel, which was considered alongside additional $s\overline{s}$ interpolating fields in the lattice computation used by the authors of Gao et al. and also in typical coupled-channel models. As an illustration, some results from such a recent model are presented as well. Concluding remarks concern possible improvements of the analysis in Gao et al. as well as further model tests.

• Received 2 September 2022
• Accepted 28 November 2022
• Corrected 4 April 2023

DOI:https://doi.org/10.1103/PhysRevD.107.058501

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Published by the American Physical Society