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We discuss a list of possible light gauge boson interpretations for thelong-standing experimental anomaly in $(g-2)_\mu$ and also recent anomalousexcess in $K_L \rightarrow \pi^0 + \text{(invisible)}$ events at the J-PARCKOTO experiment. We consider two models: $i$) $L_\mu - L_\tau$ gauge boson withheavy 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 gaugeboson has mass close to the neutral pion in order to satisfy the Grossman-Nirbound, the models successfully explain the anomalies simultaneously whilesatisfying all known experimental constraints. We extensively provide thefuture prospect of suggested models.

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