Constraining light dark matter and mediator with $B^{+}\to K^{+}\nu \bar{\nu }$ data

We study the decay of $B^{+}$ meson into $K^{+}$ plus a light mediator $\phi$, which subsequently decays into a dark matter pair, $\bar{\chi }\chi$. Integrating constraints from DM relic density, direct detection, collider data and $B$-physics, alongside the recently reported results form Belle II experiment, we analyze the couplings between the mediator, standard model fermions, and the dark matter particles. Our results indicate that if the decay process $\phi \to \bar{\chi }\chi$ is kinematically allowed, i.e. $m_{\phi }>2m_{\chi }$, then the mediator mass must be constrained within 0.35 GeV $\lesssim m_{\phi }\lesssim$ 3 GeV. Conversely, if $m_{\phi }<2m_{\chi }$, the mediator $m_{\phi }$ is long-lived relative to the detector size, and the only allowed decay channel is $\phi \to e^{+}{e}^{-}$.