Running cosmological constant and running Newton constant in modified gravity theories

We discuss how to extract information about the cosmological constant from the Wheeler‐DeWitt equation, considered as an eigenvalue of a Sturm‐Liouville problem in a de Sitter and Anti‐de Sitter background. The equation is approximated to one loop with the help of a variational approach with Gaussian trial wave functionals. A canonical decomposition of modes is used to separate transverse‐traceless tensors (graviton) from ghosts and scalar. We show that no ghosts appear in the final evaluation of the cosmological constant. A zeta function regularization is used to handle with divergences. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation. We apply this procedure on the induced cosmological constant Λ and, as an alternative, on the Newton constant G. A brief discussion on the extension to a f(R) theory is considered.