4D QUANTUM N-DILATON GRAVITY AND ONE-LOOP DIVERGENCE OF EFFECTIVE ACTION ON CONSTANT DILATON

We consider 4D quantum gravity with N-dilatons with the most general c ouplings. In particular, on constant dilaton and arbitrary metric back ground, we determine the structure of the divergent terms. We derive t he constraint between the couplings necessary to cancel the coefficien t of the square of the Weyl tensor. Next we show tbe N dependence of a non-renormalizable divergent term, and and that it cannot be cancelle d in the case N greater than or equal to 1 with any fine-tuning of the couplings.