  We investigate forces induced by the exchange of two light neutrinos between
Standard Model (SM) fermions in the presence of effective operators
parametrising physics beyond the SM. We first set up a general framework in
which we derive the long-range potential mediated by weakly interacting
neutrinos in the SM, retaining both spin-independent and spin-dependent terms.
We then derive neutrino-mediated potentials when there are vector, scalar and
tensor non-standard interactions present as well as an exotic neutrino magnetic
moment. Examining the phenomenology of such long-range potentials in atomic
scale laboratory experiments, we derive upper bounds on the Wilson coefficients
of the effective operators and compare these to those from processes such as
charged lepton flavour violation.
