Differentiating Dimension-6 and Dimension-8 Effects in $\nu$SMEFT at the HL-LHC

We study dimension-eight effects in the Standard Model Effective Field Theory extended by right-handed neutrinos ($\nu$SMEFT). Using the Hilbert series formalism, we derive the complete basis of dimension-eight operators and confirm agreement with existing classifications, providing a systematic framework beyond the conventional dimension-six truncation. We analyse the collider phenomenology of the representative operator $\mathcal{O}_{N^{2}q^{2}B}^{(1,2)}$ at the High-Luminosity LHC. The resulting signatures involve pair production of right-handed neutrinos in association with jets, followed by decays into electron-jet final states with potentially displaced vertices. Since similar final states are generated by leading dimension-six operators, we explicitly address whether dimension-eight contributions can be experimentally distinguished from dimension-six effects. Using a Boosted Decision Tree analysis based on kinematic observables, we show that the dimension-eight signal can be reliably separated from each relevant dimension-six hypothesis. Our results demonstrate that dimension-eight operators in the $\nu$SMEFT can give rise to experimentally resolvable signatures and should be included in collider EFT interpretations.