New research has extended previous results on renormalizability in conformal field theories in curved spacetime, specifically applying them to conformal quantum gravity. This advance addresses a non-trivial issue in the quantization of classically conformal theories possessing gauge invariance, where the introduction of gauge-fixing and ghost terms breaks the original conformal symmetry. The formal proof of conformal invariant renormalizability in this situation, including interacting theories, was previously established in 1984 by one of the current authors.

The key to this demonstration lies in BRST symmetry. Thanks to it, the contributions from the gauge-fixing and ghost sectors to the conformal variation of the effective action cancel each other out. Although this cancellation does not hold in the finite part of the one-loop and higher-loop effective action, leading to the conformal anomaly, it is crucial for the renormalizability of the divergences. The new study applies these principles to conformal quantum gravity, including the case of conformal gravity coupled to other conformal matter fields.

The researchers have shown that, despite the increased complexity in gauge-fixing for Weyl-squared gravity, the proof of one-loop conformal invariant renormalization is feasible using BRST symmetry. As a preliminary step towards conformal quantum gravity, the work also presents a detailed general proof of conformal invariant one-loop divergences in the corresponding semiclassical theory. This result is fundamental for the development of a consistent theory of quantum gravity.