f (R ,Rμν 2) at one loop

We compute the one-loop divergences in a theory of gravity with a Lagrangian of the general form f (R ,R_{μ ν}R^{μ ν}), 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.