  We formulate an Effective Field Theory (EFT) for Non Standard neutrino
Interactions (NSI) in elastic scattering with light quarks, leptons, gluons and
photons, including all possible operators of dimension 5, 6 and 7. We provide
the expressions for the cross sections in coherent neutrino-nucleus scattering
and in deep inelastic scattering. Assuming single operator dominance we
constrain the respective Wilson coefficient using the measurements by the
COHERENT and CHARM collaborations. We also point out the constraining power of
future elastic neutrino-nucleus scattering experiments. Finally, we explore the
implications of the bounds for SMEFT operators above the electroweak breaking
scale.
