  We consider the neutrino self-energy in a background composed of a scalar
particle and a fermion using a simple model for the coupling of the form
$\lambda\bar f_R\nu_L\phi$. The results are useful in the context of Dark
Matter-neutrino interaction models in which the scalar and/or fermion
constitute the dark-matter. The corresponding formulas for models in which the
scalar particle couples to two neutrinos via a coupling of the form
$\lambda^{(\nu\nu\phi)}\bar\nu^c_R\nu_L\phi$ are then obtained as a special
case. In the presence of these interactions there can be new contributions to
the neutrino effective potential and index of refraction in the context of
neutrino collective oscillations in a supernova and in the Early Universe hot
plasma before neutrino decoupling. The formulas obtained here can be used to
estimate those effects and/or put limits on the model parameters based on the
contribution that such interactions can have in those contexts. A particular
feature of the results is that the contribution to the neutrino self-energy or
effective potential in a neutrino background due to the $\nu\nu\phi$ coupling
is proportional to the antineutrino-neutrino asymmetry $(n_{\bar\nu} -
n_{\nu})$, in contrast to the standard $Z$ contribution which is proportional
to $(n_{\nu} - n_{\bar\nu})$. Therefore the two contributions tend to cancel.
If the cancellation is significant, it is conceivable that the $O(1/m^4_\phi)$
terms give the dominant contribution. Alternatively, a limit can be set by
requiring that the contribution of the $\nu\nu\phi$ interaction to the neutrino
effective potential does not cancel the standard contribution in an appreciable
way.
