Probing Proton Structure via Physics-Guided Neural Networks in Holographic QCD

Abstract

Describing the proton structure function $F_2(x,Q^2)$ across the nonperturbative and transition regimes of QCD remains a major theoretical challenge. We construct a physics-guided phenomenological representation combining fermion-resonance and Pomeron-like AdS/QCD basis functions with a neural parametrization. The resonance sector is constrained by the AdS eigenvalue problem and physical proton mass, while the neural component describes effective kinematic mixing, Pomeron normalization, and a regularized log-space correction. For 187 selected SLAC measurements, a fixed 153/34 training--test split gives ${\chi }^2/N_{\mathrm{train}}=0.542$, ${\chi }^2/N_{\mathrm{test}}=1.990$, and ${\chi }^2/N_{\mathrm{all}}=0.805$. Conditional on the chosen holographic basis, the fitted channel fractions evolve continuously from Pomeron-like dominance at lower measured $x$ to fermion-resonance dominance at larger $x$. The mixing midpoint is at $x\simeq 0.187$, while equality of the final channel fractions occurs at $x\simeq 0.226$, defining a transition region $x\simeq 0.19$--$0.23$ rather than a unique boundary. Although ${\alpha }_0\simeq 1.08$ is compatible with the data, fixed-${\alpha }_0$ refits show that it is not separately identifiable from the flexible Pomeron normalization. The $x$-dependent holographic coordinates and effective excited scales should therefore be regarded as basis-dependent quantities rather than direct measurements of an $x$-dependent proton spectrum. A data-only neural baseline shows no predictive advantage for the holographic basis on the present dataset. The framework thus provides a basis-conditioned phenomenological decomposition rather than a model-independent extraction of proton dynamics.

arXiv comment17 pages, 6 figures, accepted by Chinese Physics C