  Non-Standard neutral current Interactions (NSIs) of neutrinos with matter can
alter the pattern of neutrino oscillation due to the coherent forward
scattering of neutrinos on the medium. This effect makes long-baseline neutrino
experiments such as NO$\nu$A and DUNE a sensitive probe of beyond standard
model (BSM) physics. We construct light mediator models that can give rise to
both lepton flavor conserving as well as Lepton Flavor Violating (LFV) neutral
current NSI. We outline the present phenomenological viability of these models
and future prospects to test them. We predict a lower bound on Br$(H\to \mu
\tau)$ in terms of the parameters that can be measured by DUNE and NO$\nu$A,
and show that the hint for $H\to \mu \tau$ in current LHC data can be
accommodated in our model. A large part of the parameter space of the model is
already constrained by the bounds on Br$(\tau \to Z^\prime \mu)$ and rare meson
decays and can in principle fully tested by improving these bounds.
