Abstract
We present and discuss well known conditions for ultraviolet finiteness. The requirements for complete absence of ultraviolet divergences in quantum field theories and existence of a non-trivial fixed point for renormalization group flow in the ultraviolet regime are compared based on the example of a six-derivative quantum gravitational theory in \(d = 4\) spacetime dimensions. Here vanishing of beta functions is equivalent to the emergence of conformal symmetry on the quantum level. In this model, it is possible for the first time to have fully UV-finite quantum theory without adding matter or special symmetry, but by inclusion of additional terms cubic in curvatures. We discuss all necessary algebraic conditions for this to happen. Finally, we motivate the claim that actually asymptotic safety needs UV-finite models for providing explicit form of the ultraviolet limit of Wilsonian effective actions describing special situations at conformal fixed points.
