  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.
