$B^{+}\to K^{+}\nu \bar{\nu }$ Excess and DM semi-annihilation

In 2023, Belle II collaboration announced the observarion of the $B^{+}\to K^{+}\nu \bar{\nu }$ decay channel for the first time. This decay channel provides a clean signal with high precision in theoretical calculation. However, we encounter $2.8\sigma$ deviation from the Standard Model (SM) prediction. To resolve this excess, we study scalar dark matter (DM) model with local discrete $Z_3$ symmetry. Assuming dark $U(1{)}_X\equiv U(1{)}_{L_{\mu }-L_{\tau }}$ symmetry, this $U(1{)}_{L_{\mu }-L_{\tau }}$ symmetry is spontaneously broken into local discrete $Z_3$ by non-zero vacuum expectation value of dark Higgs boson. Considering dark Higgs mass is $2$GeV, we can explain the recent $\mathrm{B}\mathrm{r}(B^{+}\to K^{+}\nu \bar{\nu })$ excess reported from Belle II collaboration and relic abundance at the same time.