Dilatonic gravity near two dimensions and asymptotic freedom of the gravitational coupling constant

Two models of dilatonic gravity are investigated: (i) dilaton-Yang-Mills gravity and (ii) higher-derivative dilatonic gravity. Both are renormalizable in 2 + ε dimenions and have a smooth limit for ε → 0. The corresponding one-loop effective actions and beta-functions are found. Both theories are shown to possess a non-trivial ultraviolet fixed point — for all dilatonic couplings — in which the gravitational constant is asymptotically free (at ε → 0). It is shown that in the regime of asymptotic freedom the matter central charge can be significantly increased by two different mechanisms — as compared with pure dilatonic gravity, where n < 24.