Physical Running in Conformal Gravity and Higher Derivative Scalars

We compute the physical running of a general higher derivative scalar coupled to a nondynamical metric and of higher derivative Weyl invariant gravity with a dynamical metric in four dimensions. In both cases, we find that the physical running differs from the $\mu$-running of dimensional regularization because of infrared divergences which are present in amplitudes also at large momenta, differently from what happens in standard two derivative theories. We use the higher derivative scalar as a toy-model to elaborate on the properties of the conformal limit in relation to the trace anomaly. The physical running of higher derivative Weyl gravity, while different from the $\mu$-running, remains asymptotically free, suggesting that the model is a viable completion of Einstein's gravity, at least from the point of view of its renormalization group properties.