Constraining secret gauge interactions of neutrinos by meson decays

Secret coupling of neutrinos to a new light vector boson, $Z'$, with a mass smaller than 100 MeV is motivated within a myriad of scenarios which are designed to explain various anomalies in particle physics and cosmology. Due to the longitudinal component of the massive vector boson, the rates of three-body decay of charged mesons ($M$) such as the pion and the kaon to the light lepton plus neutrino and $Z'$ ($M \to l \nu Z'$) are enhanced by a factor of $(m_M/m_{Z'})^2$. On the other hand, the standard two body decay $M \to l \nu$ is suppressed by a factor of $(m_l/m_M)^2$ due to chirality. We show that in the case of ($M \to e \nu Z^\prime$), the enhancement of $m_M^4/m_e^2 m_{Z^\prime}^2\sim 10^8-10^{10}$ relative to two-body decay ($M \to e \nu$) enables us to probe very small values of gauge coupling for $\nu_e$. The strongest bound comes from the $R_K\equiv Br(K \to e +\nu)/Br(K \to \mu +\nu)$ measurement in the NA62 experiment. The bound can be significantly improved by customized searches for signals of three-body charged meson decay into the positron plus missing energy in the NA62 and/or PIENU data.