Search for new light vector boson using $J/\Psi$ at BESIII and Belle II

We investigate various search strategies for light vector boson $X$ in $\mathcal{O}(10)~{\rm MeV}$ mass range using $J/\Psi$ associated channels at BESIII and Belle II: (i) $J/\Psi \to \eta_c X$ with $10^{10} J/\Psi$s at BESIII, (ii) $J/\Psi (\eta_c +X) +\ell \bar{\ell}$ production at Belle~II, and (iii) $J/\Psi +X$ with the displaced vertex in $X\to e^+e^-$ decay are analyzed and the future sensitivities at Belle II with 50 ${\rm ab}^{-1}$ luminosity are comprehensively studied. By requiring the displaced vertex to be within the beam pipe, the third method results in nearly background-free analysis, and the vector boson-electron coupling and the vector boson mass can be probed in the unprecedented range, $10^{-4}\leq |\varepsilon_e| \leq 10^{-3}$ and $9~{\rm MeV}\leq m_X\leq 100 {\rm MeV}$ with 50 ${\rm ab}^{-1}$ at Belle II. This covers the favored signal region of $^8{\rm Be}^*$ anomaly recently reported by Atomki experiment with $m_X \simeq 17~{\rm MeV}$.