Physics-Informed Global Extraction of the Universal Small-$x$ Dipole Amplitude

Abstract

We extract the universal small-$x$ dipole scattering amplitude $N(r,x_B)$ from a global analysis based on a physics-informed neural network (PINN), without imposing a priori MV-type parametrization of the initial condition. The network provides a smooth and differentiable surrogate for $N(r,x_B)$, whose rapidity dependence is constrained by the collinearly improved Balitsky--Kovchegov evolution equation, while its functional form is simultaneously constrained by Deep Inelastic Scattering (DIS) data for the reduced total and charm cross sections, exclusive $J/\psi$ photoproduction measurements, and a positivity requirement for the momentum-space dipole amplitude. The resulting single universal amplitude consistently describes all fitted observables within a unified framework, alleviating the long-standing tension between total and charm channels encountered in conventional small-$x$ fits based on rigid parametric ans\"atze. Within the fitted kinematic domain, the best extracted PINN solution yields a smooth, non-negative momentum-space dipole over the full transverse-momentum range examined. Our results provide a robust and well-behaved input for Color Glass Condensate phenomenology across a broad class of high-energy processes.