$L_{\mu }-L_{\tau }$ gauge-boson production from lepton flavor violating $\tau$ decays at Belle II

Abstract

$L_{\mu }-L_{\tau }$ gauge boson ($Z^{\prime }$) with a mass in the MeV to GeV region can resolve not only the muon $g-2$ excess, but also the gap in the high-energy cosmic neutrino spectrum at IceCube. It was recently proposed that such a light gauge boson can be detected during the Belle II experiment with a luminosity of 50 ab${}^{-1}$ by the $e^{+}{e}^{-}\to \gamma +\text{slashed}E$ process through the kinetic mixing with the photon, where the missing energy $\text{slashed}E$ is from the $Z^{\prime }\to \bar{\nu }\nu$ decays. We study the phenomenological implications when a pair of singlet vector-like leptons carrying different $L_{\mu }-L_{\tau }$ charges are included, and a complex singlet scalar (${\varphi }_S$) is introduced to accomplish the spontaneous $U(1{)}_{L_{\mu }-L_{\tau }}$ symmetry breaking. It is found that the extension leads to several phenomena of interest, including (i) branching ratio (BR) for $h\to \mu \tau$ can be of the order of $10^{-3}$; (ii) ${\varphi }_S$-mediated muon $g-2$ can be of the order of $10\times 10^{-10}$; (iii) BR for $\tau \to \mu {\varphi }_S^{\ast }\to \mu Z^{\prime }{Z}^{\prime }$ can be $10^{-8}$, and (iv) kinetic mixing between the $Z^{\prime }$ boson and photon is sensitive to the relative heavy lepton masses. The predicted BRs for $\tau \to (3\mu +\text{slashed}E,5\mu$) through the leptonic $Z^{\prime }$ decays can reach a level of $10^{-9}$, in which the results fall within the sensitivity of the Belle II in the search for the rare tau decays.

Comments: 23 pages, 7 figures, 4 tables, version to appear in Phys. Rev. D