Corrections of the GR eikonal limit by a class of renormalizable gravity models

In the present work, the scattering between a light scalar particle $φ$ and a heavy scalar $σ$ in the eikonal limit is considered, for gravity scenarios containing higher order derivatives, such as the ones studied in \cite{stelle1}-\cite{modesto4}. It is suggested that if one of the new gravity scales introduced in the higher order action is smaller than the Planck mass, for instance of the order of $M_{GUT}\sim 10^{15}$ GeV, the functional form of the GR eikonal formulas appears changed by a factor. However, in this situation, the conditions for the eikonal approximation to hold has to be revised, this issue is analyzed in the text. The statements of the present work should be taken with a grain of salt, as the Schwarzschild radius for these polynomials theories is not yet established. The results presented here, in our opinion, are in agreement with the suppression of corrections to GR pointed out in \cite{brandhuber}, \cite{fradkin} and \cite{deser} for the Stelle gravity. The next to leading order approximation and part of the seagull diagrams are estimated. Different to the GR case, this order generically is non vanishing. An explicit regularization scheme is presented, based on Riesz and Hadamard procedures. The need of a regularization is partially expected, as the inclusion of small energy fluctuations may spoil the eikonal approximation.