Scattering amplitudes in dimensionless quadratic gravity coupled to QED

We study ultra-Planckian $2\to 2$ 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 $1\mathrm{/}{p}^4$ 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 $d\sigma \mathrm{/}d\mathrm{\Omega }\propto 1\mathrm{/}s$, 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.