Interaction of Low - Energy Induced Gravity with Quantized Matter and Phase Transition Induced by Curvature

At high energy scale the only quantum effect of any asymptotic free and asymptotically conformal invariant GUT is the trace anomaly of the energy-momentum tensor. Anomaly generates the new degree of freedom, that is propagating conformal factor. At lower energies conformal factor starts to interact with scalar field because of the violation of conformal invariance. We estimate the effect of such an interaction and find the running of the nonminimal coupling from conformal value $\frac{1}{6}$ to $0$. Then we discuss the possibility of the first order phase transition induced by curvature in a region close to the stable fixed point and calculate the induced values of Newtonian and cosmological constants.