  If neutrinos are Dirac particles and, as suggested by the so far null LHC
results, any new physics lies at energies well above the electroweak scale, the
Standard Model effective field theory has to be extended with operators
involving the right-handed neutrinos. In this paper, we study this effective
field theory and set constraints on the different dimension-six interactions.
To that aim, we use LHC searches for associated production of light (and tau)
leptons with missing energy, monojet searches, as well as pion and tau decays.
Our bounds are generally above the TeV for order one couplings. One particular
exception is given by operators involving top quarks. These provide new signals
in top decays not yet studied at colliders. Thus, we also design an LHC
analysis to explore these signatures in the $t\overline{t}$ production. Our
results are also valid if the right-handed neutrinos are Majorana and
long-lived.
