In conformal gravity theory, the action is given by
$$L=\int \sqrt{-g}C^{abcd} C_{abcd} d^4x=\int \sqrt{-g}(R^{ab}R_{ab}- \frac{1}{3}R^2)d^4 x.$$
However, the variation of the first term $\int \sqrt{-g}R^{ab}R_{ab}d^4 x$ according to the article "Alternatives to dark matter and Dark energy" contains a Ricci scalar term. I have no ideal how the Ricci scalar term is from.
