Universality of Gluon Saturation from Physics-Informed Neural Networks

TL;DR

The paper tackles the universality of the gluon saturation picture by extracting a momentum-space dipole amplitude $N(k,Y)$ through a Physics-Informed Neural Network (PINN) constrained by the BK evolution in the diffusion regime and anchored to inclusive $F_2$ data. Using a two-phase Teacher–Student training, the framework yields a universal $N(k,Y)$ without fixing initial conditions and then makes zero-parameter predictions for exclusive $J/\psi$ photoproduction, including the $t$-slope, in strong agreement with HERA data. The approach provides uncertainty estimates via bootstrap aggregation and extracts physical parameters such as the saturation exponent $\lambda_s \approx 0.239 \pm 0.010$ and an effective proton radius $R_p \approx 5.46$ GeV$^{-1}$, supporting the universality of the saturation scale. This work demonstrates that physics-informed deep learning can unify perturbative evolution with non-perturbative gluon structure and outlines a path toward more precise EIC-era analyses with explicit uncertainty quantification and beyond-diffusion corrections.

Abstract

The universality of the color dipole amplitude is a cornerstone of high-energy Quantum Chromodynamics (QCD). However, standard phenomenological approaches typically rely on rigid parametric ansatzes and often require ad-hoc geometric adjustments to reconcile inclusive and diffractive measurements. To resolve this tension, we introduce Physics-Informed Neural Networks (PINNs) employing a ``Teacher--Student'' strategy. The rigorous momentum-space Balitsky-Kovchegov evolution dynamics act as the ``Teacher,'' constraining the solution manifold, while the network ``Student'' is refined against inclusive HERA $F_2$ data. This approach extracts a model-independent dipole amplitude without assuming initial states. Strikingly, we demonstrate that this amplitude -- without parameter retuning or geometric rescaling -- successfully predicts exclusive $J/ψ$ photoproduction cross-sections. This zero-parameter prediction rigorously confirms the universality of the gluon saturation scale and establishes PINNs as a transformative paradigm for uncovering non-perturbative QCD structures.

Figures (9)

• Figure 1: Workflow of the BK-constrained PINN framework. The training process integrates physical constraints (BK equation) and experimental data (HERA $F_2$) through a composite loss function. The optimized network yields the momentum-space dipole amplitude $N(k,Y)$, which is then transformed to coordinate space to predict exclusive vector meson production cross-sections.
• Figure 2: Evolution of the extracted momentum-space dipole amplitude $N(k, Y)$ with rapidity $Y$. The curves show the solution at $Y=0.5, 1.5, 2.5, 3.5$ (from left to right). The wavefront propagation towards higher $k^2$ illustrates the dynamic generation of the saturation scale $Q_s(Y)$, an emergent property of the PINN solution.
• Figure 3: Zero-parameter predictions for exclusive $J/\psi$ photoproduction. Top: Comparison with ZEUS data at $W=90$ GeV ZEUS:2004yeh. Bottom: Comparison with H1 data at $W=100$ GeV H1:2005dtp. The solid and dashed lines represent predictions using Gaus-LC and Boosted Gaussian wavefunctions, respectively. The agreement validates the universality of the PINN-extracted amplitude.
• Figure S1: Schematic of the BK-PINN architecture. The network maps kinematic inputs $(L, Y)$ to the momentum-space dipole amplitude $N$. The training is governed by two loss components: the physics-based $\mathcal{L}_{\text{PDE}}$ derived from the BK equation via automatic differentiation, and the data-driven $\mathcal{L}_{\text{Data}}$ obtained by directly integrating the amplitude with photon wavefunctions in momentum space to match HERA $F_2$ data.
• Figure S2: Training dynamics showing the total loss, PDE loss (Teacher), and Data loss (Student) for both training and testing sets.