Renormalization of Yukawa model with sterile scalar in curved space-time

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.