  We explore the classical and quantum properties of a sterile scalar field
coupled to $N$ copies of Dirac fermions in an external gravitational field. We
find that the self-interaction scalar potential of a model that is consistent
at the quantum level, includes odd (first and third) powers of a scalar. In
particular, one has to consider, besides the standard non-minimal coupling of
the form $\xi \varphi^2 R$, the new type of non-minimal coupling of the form
$f\varphi R$ with new non-minimal parameter $f$. We study the one-loop
renormalization of such a theory including renormalization of the new
non-minimal coupling. Also, we calculate the one-loop effective potential using
the renormalization group and show how the renormalization group analysis
should be extended compared to the standard expression which was derived in
1980-ies. This conclusion is supported by the direct calculation of effective
potential using normal coordinates and covariant cut-off regularization. The
important features of the classical theory with a sterile scalar are related to
the presence of the qualitatively new terms in the induced action of gravity,
coming from the odd terms. We show that this new feature of the theory may have
phenomenologically relevant consequences, both in the low-energy gravitational
physics and at the high energies, corresponding to inflation.
