An approach to quantum gravity from 4-$\epsilon$ dimension

A calculational scheme of quantum-gravitational effects on the physical quantities is proposed. The calculations are performed in 4-$\epsilon$ dimension with $1/N$-expansion scheme, where the Einstein gravity is renormalizable and it has an ultraviolet fixed-point within the 1/N-expansion. In order to perform a consistent perturbation in $4-\epsilon$ dimension, spin-3/2 fields should be adopted as the N matter-fields whose loop-corrections are included in the effective action. After calculating the physical quantities at $4-\epsilon$ dimension, the four-dimensional aspects of them can be seen by taking the limit of $\epsilon=0$. In taking this limit, any higher derivative terms are not introduced as the counter terms since no divergence appears at $\epsilon=0$ in our scheme. According to this approach, we have examined the effective potential of a scalar field to see the possibility of the spontaneous symmetry breaking due to the gravitational loop corrections.