We study the phenomenology of the simplest renormalisable model that, at low energy, leads to the effective field theory of the Standard Model extended with right-handed neutrinos (nu SMEFT). Our aim is twofold. First, to contextualise new collider signatures in models with sterile neutrinos so far studied only using the bottom-up approach. And second and more important, to provide a thorough example of one-loop matching in the diagrammatic approach, of which other matching techniques and automatic tools can benefit for cross-checks. As byproducts of this work, we provide for the first time: (i) a complete off-shell basis for the nu SMEFT and explicit relations between operators linked by equations of motion; (ii) a complete basis for the low-energy effective field theory (nu LEFT) and the tree-level matching onto the nu SMEFT; (iii) partial one-loop anomalous dimensions in the nu LEFT. This way, our work comprises a new step forward towards the systematisation of one-loop computations in effective field theories, especially if the SM neutrinos are Dirac.
