Individual lepton flavors and the total lepton number are strictly conserved quantities in the standard model (SM). However, neutrino oscillations evidence lepton flavor violation in the neutral lepton sector, suggesting the need to consider SM extensions capable of accounting for massive light neutrinos and lepton mixing. The extensions considering sterile right-handed neutrinos, with Majorana mass terms, lead to Majorana massive states (being their own anti-particles) which predict the occurrence of total lepton number violation (LNV). In turn, the observation of LNV would be a clear signal of new physics, and of the existence of Majorana fermions.
The effective field theory extending the standard model with right-handed neutrinos ($\nu$SMEFT) parameterizes new high-scale weakly coupled physics in a model independent manner. We consider massive Majorana neutrinos coupled to ordinary matter by dimension-6 effective operators, focusing on a simplified scenario with only one right-handed neutrino added, which provides us with a manageable parameter space to probe. Here we present the prospects of the future LHeC electron-proton collider to discover or constrain the $\nu$SMEFT interactions, with a realistic analysis of the well known lepton-trijet signals, both for the lepton flavor violating $p ~ e^{-} \rightarrow \mu^{-} + 3 \mathrm{j}$ (LFV) and the lepton number violating $p ~ e^{-} \rightarrow \mu^{+} + 3 \mathrm{j}$ (LNV) channels, for HNLs masses in the electroweak scale range: $100 ~\rm GeV \leq m_N \leq 500 ~\rm GeV$.