Seesaw right-handed neutrino as the sterile neutrino for LSND

We show that a double seesaw framework for neutrino masses with $\mu -\tau$ exchange symmetry can lead to one of the right-handed seesaw partners of the light neutrinos being massless. This can play the role of a light sterile neutrino, giving a $3+1$ model that explains the Liquid Scintillator Neutrino Detector (LSND) results. We get a very economical scheme, which makes it possible to predict the full $4\times 4$ neutrino mass matrix if $CP$ is conserved. Once $CP$ violation is included, the effect of the LSND mass range sterile neutrino is to eliminate the lower bound on neutrinoless double beta decay rate which exists for the three neutrino case with inverted mass hierarchy. The same strategy can also be used to generate a natural $3+2$ model for LSND, which is also equally predictive for the $CP$ conserving case in the limit of exact $\mu -\tau$ symmetry.