Search of light sterile neutrinos from $W^\pm$ decays

We study the feasibility to observe sterile neutrinos at the LHC, with masses in the range \hbox{5~GeV~$< m_N < 20$~GeV}, using exclusive semileptonic modes involving pions $W\to \ell N \to \pi \ell\ell, 2\pi \ell\ell$ and $3\pi \ell\ell$. We thus cover a mass window for sterile neutrinos which is between what can be studied in meson factories and high energy colliders. We run simulations to study these exclusive events where pions should be distinguished from the background provided the neutrino decay exhibits a vertex displacement from its production point. In a previous work we have estimated the theoretical rates and here we analyze the observability of the processes at the LHC, given the fact that exclusive hadronic states may be difficult to identify. If a sample of $10^9$ $W$ bosons is available the end of the LHC Run~2, a sterile neutrino in the above mass range could be discovered, or at least the current bounds on the heavy-to-light lepton mixings could be improved by an order of magnitude, $|U_{\ell N}|^2\sim 2\times 10^{-6}$. Studying in addition equal sign and opposite sign dileptons, the Majorana or Dirac character of the sterile neutrino could be revealed.