Weakly interacting $K\to \pi X^0$ emission with $m_{X^0}\cong m_{{\pi }^0}$ is out of sight of the current $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ study, but it can be sensed by the $K_L\to {\pi }^0\nu \overline{\nu }$ search. This evades the usual Grossman-Nir bound of $\mathcal{B}(K_L\to {\pi }^0\nu \overline{\nu })<1.4\times {10}^{-9}$; thus, the KOTO experiment is already starting to probe new physics. An intriguing possibility is the $Z^{\prime }$ gauge boson of a weak leptonic force that couples to $L_{\mu }-L_{\tau }$ (the difference between the muon and tauon numbers), which may explain the long-standing “muon $g-2$” anomaly, but is constrained by ${\nu }_{\mu }N\to {\nu }_{\mu }N{\mu }^{+}{\mu }^{-}$ scattering to $m_{Z^{\prime }}\lesssim 400\mathrm{MeV}$. An explicit model for $K\to \pi Z^{\prime }$ is given, which illustrates the link between rare kaon and $B\to K{\mu }^{+}{\mu }^{-}$, $K^{(*)}\nu \overline{\nu }$ decays. Complementary to these searches and future lepton experiments, the LHC might discover the scalar boson $\varphi$ responsible for light $m_{Z^{\prime }}$ generation via $\varphi \to Z^{\prime }{Z}^{\prime }\to 2({\mu }^{+}{\mu }^{-})$.

• Received 16 December 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.171802