Search for heavy Majorana neutrinos at electron-proton colliders

We develop the search strategy for a heavy Majorana neutrino via the lepton number violation signal process $p\, e^- \to \mu^+ jjj$ at future electron-proton colliders. The signal and dominant standard model background events are generated with the fast detector simulation. We apply the pre-selection criteria and perform the multi-variate analysis based on machine-learning to reject the background. Distributions of representative kinematic observables are presented for both signal and background processes and effects on final limits are compared by inputting two different set of observables when performing multi-variate analysis. The 2- and 5-$\sigma$ limits on the mixing parameter $|V_{\ell N}|^2$ are predicted for the heavy neutrino mass $m_N$ in the range of 10$-$1000 GeV. At the LHeC (FCC-eh) with an electron beam energy of 60 GeV, a proton beam energy of 7 (50) TeV and an integrated luminosity of 1 (3) ab$^{-1}$, the mixing parameter $|V_{\ell N}|^2$ can be constrained to be below $\sim 3.0~(1.0) \times 10^{-6}$ for $m_N$ around $\mathcal{O}(100)$ GeV at 2-$\sigma$ level. The limits are much stronger than the current experiment limits at the LHC for $m_N$ above 30 GeV. The positron signal final state and the effect of long-lived cases of heavy neutrinos are also checked and commented.