4D Quantum N-Dilaton Gravity and One-Loop Divergence of Effective Action on Constant Dilaton

We consider 4D quantum garvity with N-dilatons with the most general couplings. In particular, on constant dilaton and arbitrary metric background, we determine the structure of the divergent terms. We derive the constraint between the couplings necessary to cancel the coefficient of the square of the Weyl tensor. Next we show the N dependence of a non-renormalizable divergent term, and find that it cannot be cancelled in the case N ≥ 1 with any fine-tuning of the couplings.