Probe the Mixing Parameter $|V_{\tau N}|^2$ for Heavy Neutrinos

Because of the difficulty in detecting final state taus, the mixing parameter $|V_{\tau N}|^2$ for heavy neutrino $N$ is not well studied at current experiments, compared with other mixing parameters $|V_{e N}|^2$ and $|V_{\mu N}|^2$. In this paper, we focus on a challenging scenario where $N$ mixes with active neutrino of tau flavour only, i.e. $|V_{\tau N}|^2 \neq 0$ and $|V_{e N}|^2 = |V_{\mu N}|^2 = 0$. We derive current constraints on $|V_{\tau N}|^2$ from the rare $Z$-boson decay and electroweak precision data (EWPD). To forecast the future limits, we also investigate the signal $p p \to \tau^{\pm} \tau^{\pm} j j$ via a Majorana heavy neutrino at future proton-proton colliders. To suppress the background, both taus are required to decay leptonically into muons, leading to the final state containing two same sign muons, at least two jets plus moderate missing energy. The signal and relevant background processes are simulated at the HL-LHC and SppC/FCC-hh with center-of-mass energy of 14 TeV and 100 TeV. The preselection and multivariate analyses based on machine-learning are performed to reduce background. Limits on $|V_{\tau N}|^2$ are shown for heavy neutrino mass in the range 10-1000 GeV based on measurements from the rare $Z$-boson decay and EWPD, and searches at the HL-LHC and SppC/FCC-hh with integrated luminosities of 3 and 20 ab$^{-1}$.