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

TL;DR

This work studies high-energy, small-angle scattering of a light scalar off a heavy scalar in renormalizable gravity models with higher-derivative terms, focusing on how the GR eikonal phase is modified. By deriving the graviton propagator for polynomial actions, computing the tree-level amplitude, and performing a full eikonal resummation, the authors obtain a phase $\chi_0(b^{\perp})$ that encodes the modifications via mass scales $m_i$ and is expressed through Bessel/Hankel functions; regularization schemes (Hadamard/Riesz) are extended to handle NE terms, with the Fradkin kernel framework guiding the analysis. The results show that if a new gravity scale is below the Planck mass, leading eikonal corrections are a multiplicative factor to GR and the eikonal validity becomes more stringent; if all new scales exceed Planck, the leading eikonal result reduces to GR with subleading, often exponentially suppressed, corrections, while NE and seagull contributions remain nonzero and require regularization. The Stelle gravity case illustrates that leading eikonal behavior is effectively GR-like after field redefinitions, aligning with prior literature, though poles can introduce nontrivial corrections via $H_0$ and $Y_0$ functions. Overall, the paper provides a coherent framework for evaluating eikonal corrections in renormalizable gravity and highlights the dependence on the mass-scale structure of the theory and the need for careful regularization of higher-order terms.

Abstract

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.