Quantum theory of 3+1 dimensional einstein gravity, dilatons and dark matter

We develop a quantum field theoretic formulation of four–dimensional Einstein gravity coupled to dilatons, aimed at providing a finite and unitary description of graviton and dilaton dynamics relevant to dark matter. The construction employs the Schwinger–Feynman variational principle together with the Lorentz–invariant convolution theory of ultrahyperfunctions, which enables a well-defined treatment of non-renormalizable interactions without invoking regularization or counterterms. Building on the Gupta–Feynman framework, and incorporating an additional constraint that restores unitarity, we obtain explicit expressions for the graviton self-energy and extend the analysis to include graviton–dilaton interactions. The resulting convolutions remain finite at every perturbative order, and the structure of the derived self-energies is exhibited analytically and through seven representative plots. We further show that, within this framework, the dilaton self-energy vanishes, highlighting structural differences between tensor and scalar sectors under ultrahyperfunctional convolution. The approach provides a consistent route toward a predictive quantum theory of gravity in the presence of dark-sector fields, demonstrating the utility of ultrahyperfunction techniques in resolving divergences endemic to conventional formulations.