Light gauge boson interpretation for $(g-2{)}_{\mu }$ and the $K_L\to {\pi }^0+\text{(invisible)}$ anomaly at the J-PARC KOTO experiment

We discuss a list of possible light gauge boson interpretations for the long-standing experimental anomaly in $(g-2{)}_{\mu }$ and also recent anomalous excess in $K_L\to {\pi }^0+\text{(invisible)}$ events at the J-PARC KOTO experiment. We consider two models: $i$) $L_{\mu }-L_{\tau }$ gauge boson with heavy vector-like quarks and $ii$) $(L_{\mu }-L_{\tau })+ϵ(B_3-L_{\tau })$ gauge boson in the presence of right-handed neutrinos. When the light gauge boson has mass close to the neutral pion in order to satisfy the Grossman-Nir bound, the models successfully explain the anomalies simultaneously while satisfying all known experimental constraints. We extensively provide the future prospect of suggested models.