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We utilize the lepton number violation signal process $p\, e^- \to \tau^+jjj$ to search for heavy Majorana neutrinos at future proton-electroncolliders. The LHeC (FCC-eh) is considered to run with an electron beam energyof 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 applydetector configurations and simulate signal and related standard modelbackground events for both hadronic $\tau_h$ and leptonic $\tau_\ell$ finalstates, $\ell$ being a muon. After preselection, multivariate analyses areperformed to reject the background. The strategy to reconstruct the heavyneutrino mass is developed and distributions of reconstructed mass arepresented. 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 3000GeV are predicted. At the 2-$\sigma$ significance, the best discoverysensitivity 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 similarto those for the hadronic $\tau_h$ final state for most of the parameter spaceinvestigated. We also derive the limits on mixing parameters from electroweakprecision data (EWPD) and DELPHI experiment. Assuming $|V_{\tau N}|^2 =|V_{eN}|^2 = |V_{\ell N}|^2$, sensitivity bounds from the LHeC and FCC-ehexperiments are found to be stronger than those from EWPD when $m_N \lesssim900$ 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.

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