Future collider sensitivities to $\nu$SMEFT interactions

The discovery of neutrino oscillations and masses provides strong motivation to extend the Standard Model by including right-handed neutrinos, which lead to heavy neutrino states that could exist at the electroweak scale. These states may also be influenced by new high-scale, weakly interacting physics. Incorporating right-handed neutrinos into an effective field theory framework -- the $\nu$SMEFT -- offers a systematic approach to study the phenomenology of heavy neutrinos in current and upcoming experiments. In this work, we present the first prospective 95\% exclusion plots achievable at a future lepton collider operating at a center-of-mass energy of $\sqrt{s}=0.5 ~\rm{TeV}$ for what we term the agnostic $\nu$SMEFT scenario. This study focuses on the high-mass regime where the heavy neutrino $N$ decays promptly into leptons and jets. Specifically, we analyze the processes $e^+e^- \to \nu N \to \nu \mu^{-} \mu^{+} \nu$ and $e^+e^- \to \nu N \to \nu \mu^{-} \mathrm{j} \mathrm{j}$, deriving the exclusion regions in the $\frac{\alpha}{\Lambda^2}$ vs. $m_N$ parameter space. When compared to prospective limits for the LHeC, we find that the semi-leptonic process with final jets in a lepton collider offers the greatest sensitivity, even with a straightforward cut-based analysis. The expected bounds are as stringent as those considered in recent studies for the low-mass regime where the $N$ may be long-lived and detectable via displaced decay searches, both at the LHC and future colliders.