Light dark-matter window constrained by \boldmath$K^+\to\pi^+$$+$$\not{\!\!E}$

We explore the constraints on new physics from the recent NA62 observation of the kaon decay $K^+\to\pi^+$+$\not{\!\!E}$ with missing energy $\not{\!\!E}$ in the context of a dark-matter (DM) scenario recently used to accommodate the Belle II finding of an enhanced rate of the $b$-meson decay $B^+\to K^+$+$\not{\!\!E}$ compared to the standard-model expectation. Specifically, assuming that a light real scalar boson $\phi$ plays the role of DM and working in an effective field-theory framework, we study model independently the impact of operators involving $\phi$ and ordinary quarks on the aforementioned transitions over the kaon mode's kinematical mass region of $m_\phi < (m_K - m_\pi)/2 = 177$ MeV. Such a DM particle is subject to significant restrictions from the observed relic abundance and from DM direct-detection experiments incorporating the Migdal effect, as well as from indirect searches in cosmic microwave background data and collider experiments, except when its mass is between 110 and 146 MeV. We demonstrate that $K^+\to\pi^+\phi\phi$ can saturate the new-physics window in the NA62 result if $m_\phi$ lies in the 110-130 MeV portion of the range left by the DM constraints, thus providing a complementary constraint on this scenario. Improved data from future Belle II and NA62 measurements and DM quests can test it more stringently. In particular, expanding the NA62 signal window into the region that is now removed due to three-body decay background modes could further explore the remaining mass window for this type of invisible particle, $130 < m_\phi < 177$ MeV.