I am currently working on a modified gravity theory which has non-minimal coupling between Ricci scalar and Maxwell term. The precise action is
$$\int d^4x\sqrt{-g} \left(R + \alpha R^2 + (1 + \beta R) F^{\mu \nu} F_{\mu \nu}\right)$$
I want to conformally transform this action to an Einstein frame where I expect there to be a linear Ricci scalar (R), a massive scalar ($\phi$), and maybe a term of the from $f(\phi)F^{\mu \nu} F_{\mu \nu}$. (Might be wrong, I also have a suspicion if it is even possible to decouple the Ricci scalar and Maxwell term by performing some transformation)
I tried to first convert this f (R, $F^{\mu \nu} F_{\mu \nu}$) theory to a scalar-tensor theory by introducing an auxiliary field $\Phi$ as we do for normal f(R) theory. After which it's easy to perform a conformal transformation to convert it to the Einstein frame. But my calculations were not fruitful. Any help and input is highly appreciated.
