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We study the beta functions for four-dimensional conformal gravity using twodifferent parametrizations of metric fluctuation, linear split and exponentialparametrization. We find that after imposing the traceless conditions, the betafunctions are the same in four dimensions though the dependence on thedimensions are quite different. This indicates the universality of theseresults. We also examine the beta functions in general quadratic theory withthe Einstein and cosmological terms for exponential parametrization, and findthat it leads to results for beta functions of dimensionful couplings differentfrom linear split, though the fact that there exists nontrivial fixed pointremains the same and the fixed points also remain the same.

Ultraviolet Fixed Points in Conformal Gravity and General Quadratic Theories