A complex logistic equation for universal energy evolution in hadronic elastic scattering

Abstract

We introduce a universal evolution equation for elastic scattering of hadrons, derived from Regge Field Theory (RFT) and solved in closed analytical form. The equation has a complex logistic structure and evolves initial amplitude profiles from existing models at a fixed energy, reproducing both differential cross sections and integrated quantities over a broad energy range. It admits a unique solution for each initial condition and rigorously satisfies unitarity, the Froissart--Martin bound, and dispersion relations. The dynamics are governed by two physically meaningful parameters: the effective Pomeron mass $\epsilon_{\mathcal{P}}$ and the nonlinear coupling $\lambda$, both fitted at a single energy. By decoupling the nonperturbative input from the universal energy evolution, the framework enables model-independent extrapolations and provides a minimal predictive alternative to eikonal resummation. Moreover, the structure of the equation -- featuring rapidity evolution, saturation, and impact-parameter dependence -- shares qualitative features with nonlinear QCD equations at small-$x$, such as BK and JIMWLK, suggesting a possible bridge between Regge-based and QCD-based approaches to high-energy scattering.