  Neutrino physics is advancing into a precision era with the construction of
new experiments, particularly in the few GeV energy range. Within this energy
range, neutrinos exhibit diverse interactions with nucleons and nuclei. This
study delves in particular into neutrino--nucleus quasi-elastic cross sections,
taking into account both standard and, for the first time, non-standard
interactions, all within the framework of effective field theory (EFT). The
main uncertainties in these cross sections stem from uncertainties in the
nucleon-level form factors, and from the approximations necessary to solve the
nuclear many-body problem. We explore how these uncertainties influence the
potential of neutrino experiments to probe new physics introduced by
left-handed, right-handed, scalar, pseudoscalar, and tensor interactions. For
some of these interactions the cross section is enhanced, making long-baseline
experiments an excellent place to search for them. Our results, including
tabulated cross sections for all interaction types and all neutrino flavors,
can serve as the foundation for such searches.
