Search for heavy Majorana neutrinos in the $\tau$ final state at proton-electron colliders

We utilize the lepton number violation signal process $p\, e^- \to \tau^+ jjj$ to search for heavy Majorana neutrinos at future proton-electron colliders. The LHeC (FCC-eh) is considered to run with an electron beam energy of 60 GeV, a proton beam energy of 7 (50) TeV and an integrated luminosity of 1 (3) ab$^{-1}$, and the electron beam is considered to be unpolarized. We apply detector configurations and simulate signal and related standard model background events for both hadronic $\tau_h$ and leptonic $\tau_\ell$ final states, $\ell$ being a muon. After preselection, multivariate analyses are performed to reject the background. The strategy to reconstruct the heavy neutrino mass is developed and distributions of reconstructed mass are presented. Discovery sensitivities on parameter $|V_{\tau N}|^2 |V_{eN}|^2 / ( |V_{\tau N}|^2 + |V_{eN}|^2 )$ for the heavy neutrino mass between 10 and 3000 GeV are predicted. At the 2-$\sigma$ significance, the best discovery sensitivity is $\sim 1.2 \times10^{-5} \,\,(5.0 \times 10^{-6})$ at the LHeC (FCC-eh) when $m_N \sim 100$ GeV for the hadronic $\tau_h$ final state. Sensitivities for the leptonic $\tau_\ell$ final state are found to be similar to those for the hadronic $\tau_h$ final state for most of the parameter space investigated. We also derive the limits on mixing parameters from electroweak precision data (EWPD) and DELPHI experiment. Assuming $|V_{\tau N}|^2 = |V_{eN}|^2 = |V_{\ell N}|^2$, sensitivity bounds from the LHeC and FCC-eh experiments are found to be stronger than those from EWPD when $m_N \lesssim 900$ GeV, and also stronger than those from DELPHI when $m_N \gtrsim 70$ GeV. Constraints are also interpreted and compared in the $|V_{\tau N}|^2$ vs. $|V_{e N}|^2$ plane.