Scattering amplitudes in dimensionless quadratic gravity coupled to QED

Abstract

We study ultra-Planckian $2\to2$ scattering in an Abelian gauge theory coupled to agravity, the scale-free and renormalizable realization of quadratic quantum gravity. Focusing on charged fermions and scalars interacting with the photon and the higher-derivative graviton, we present compact analytic expressions for the unpolarized squared matrix elements for a broad set of tree-level processes, including photon--photon, fermion--fermion, fermion--photon, scalar--fermion, scalar--photon, scalar--scalar, and annihilation channels. In contrast to purely graviton-mediated analyses, we retain systematically the photon--graviton interference contributions and verify explicitly the independence of the results on the gravitational gauge-fixing parameter. The amplitudes display characteristic forward/backward enhancements associated with small momentum transfer, amplified by the $1/p^{4}$ graviton propagator, while their high-energy scaling reflects the underlying dimensionless gravitational couplings. Moreover, for all channels analyzed the corresponding differential cross sections exhibit the universal ultra-Planckian scaling $dσ/dΩ\propto 1/s$, where $s$ is the Mandelstam invariant (the squared center-of-momentum energy). Our results furnish a unified amplitude-level description of how higher-derivative gravity reshapes familiar QED scattering at ultra-Planckian energies and provide analytic building blocks for further studies of IR definitions and UV consistency in agravity with matter.

Figures (8)

• Figure 1: Tree-level diagrams contributing to $e^-e^- \to e^-e^-$. Straight, wavy, and curly lines denote, respectively, the electron, photon, and graviton propagators.
• Figure 2: Tree-level graviton-exchange diagrams for $\gamma\gamma \to \gamma\gamma$.
• Figure 3: Tree-level diagrams for $e^-e^+ \to \gamma\gamma$, including the standard QED contributions and the additional graviton-exchange diagram.
• Figure 4: Tree-level diagrams for Compton scattering $e^-\gamma \to e^-\gamma$, obtained from Fig. \ref{['fig03']} by crossing.
• Figure 5: Tree-level diagrams contributing to $\phi\,e^- \to \phi\,e^-$ (charged scalar--electron scattering), including photon exchange and graviton exchange. Straight, wavy, and curly lines denote, respectively, electrons, photons, and gravitons, while dashed lines denote the charged scalar field.