  An investigation of relatively light (GeV-scale), long-lived right-handed
neutrinos is performed within minimal left-right symmetric models using the
neutrino-extended Standard Model Effective Field Theory framework. Light
sterile neutrinos can be produced through rare decays of kaons, $D$-mesons, and
$B$-mesons at the Large Hadron Collider (LHC) and the Long-Baseline Neutrino
Facility (LBNF) of Fermilab. Their decays could result in displaced vertices,
which can be reconstructed. By performing Monte-Carlo simulations, we assess
the sensitivities of the future LHC far-detector experiments ANUBIS, CODEX-b,
FACET, FASER(2), MoEDAL-MAPP1(2), MATHUSLA, the recently approved beam-dump
experiment SHiP, and the upcoming neutrino experiment DUNE at the LBNF, to the
right-handed gauge-boson mass $M_{W_R}$ as functions of neutrino masses. We
find that DUNE and SHiP could be sensitive to right-handed gauge-boson masses
up to $\sim 25$ TeV. We compare this reach to indirect searches such as
neutrinoless double beta decay, finding that displaced-vertex searches are very
competitive.
