Constraining secret gauge interactions of neutrinos by meson decays

Secret coupling of neutrinos to a new light vector boson, $Z^{\mathrm{\prime }}$, 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 $Z^{\mathrm{\prime }}$ ($M\to l\nu Z^{\mathrm{\prime }}$) are enhanced by a factor of $(m_M\mathrm{/}{m}_{Z^{\mathrm{\prime }}}{)}^2$. On the other hand, the standard two body decay $M\to l\nu$ is suppressed by a factor of $(m_l\mathrm{/}{m}_M{)}^2$ due to chirality. We show that in the case of ($M\to e\nu Z^{\mathrm{\prime }}$), the enhancement of $m_M^4/m_e^2 m_{Z^\prime}^2\sim 10^8-10^{10}$relativetotwo-bodydecay($M \to e \nu$) enables us to probe very small values of gauge coupling for ${\nu }_e$. The strongest bound comes from the $R_K\equiv Br(K\to e+\nu )\mathrm{/}Br(K\to \mu +\nu )$ 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.