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

We discuss a list of possible light gauge boson interpretations for the long-standing experimental anomaly in $(g-2)_μ$ and also recent anomalous excess in $K_L \rightarrow π^0 + \text{(invisible)}$ events at the J-PARC KOTO experiment. We consider two models: $i$) $L_μ- L_τ$ gauge boson with heavy vector-like quarks and $ii$) $(L_μ- L_τ) + ε(B_3 - L_τ)$ 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.