  We study the renormalizable quantum gravity formulated as a perturbed theory
from conformal field theory (CFT) on the basis of conformal gravity in four
dimensions. The conformal mode in the metric field is managed
non-perturbatively without introducing its own coupling constant so that
conformal symmetry becomes exact quantum mechanically as a part of
diffeomorphism invariance. The traceless tensor mode is handled in the
perturbation with a dimensionless coupling constant indicating asymptotic
freedom, which measures a degree of deviation from CFT. There are no massive
ghosts because they are not gauge invariant in this formulation. Higher order
renormalization is carried out using dimensional regularization, in which the
Wess-Zumino integrability condition is applied to reduce indefiniteness
existing in higher-derivative actions. The effective action of quantum gravity
improved by renormalization group is obtained. We then make clear that
conformal anomalies are indispensable quantities to preserve diffeomorphism
invariance. Anomalous scaling dimensions of the cosmological constant and the
Planck mass are calculated. The effective cosmological constant is obtained in
the large number limit of matter fields.
