  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 $K_L \rightarrow \pi^0 + \text{(invisible)}$ events at the J-PARC
KOTO experiment. We consider two models: $i$) $L_\mu - L_\tau$ gauge boson with
heavy vector-like quarks and $ii$) $(L_\mu - L_\tau) + \epsilon (B_3 - L_\tau)$
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.
