$f(R, R_{mu u}^2)$ at one loop

We compute the one-loop divergences in a theory of gravity with Lagrangian of the general form $f(R,R_{mu u}R^{mu u})$, on an Einstein background. We also establish that the one-loop effective action is invariant under a duality that consists of changing certain parameters in the relation between the metric and the quantum fluctuation field. Finally, we discuss the unimodular version of such a theory and establish its equivalence at one-loop order with the general case.