A Candidate for Renormalizable and Diffeomorphism Invariant 4D Quantum Theory of Gravity

We present evidence that there is a 4D model that satisfies the conditions of renormalizability and diffeomorphism invariance simultaneously at the 2-loop level. The traceless mode is treated perturbatively, while the conformal mode can be managed exactly. The two conditions constrain the theory strongly and determine the measure of the gravitational field uniquely. Quantum corrections of the cosmological constant are computed in part to 3-loop diagrams. The method to remove the negative-metric states is also discussed from the viewpoint of diffeomorphism invariance in analogy to the $R_{\xi }$ gauge in spontaneously broken gauge theory. The model may be a candidate for a continuum version of 4D simplicial quantum geometry realized in recent numerical simulations.