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

Abstract

Describing the proton structure function $F_2$ in the non-perturbative and transition regimes of quantum chromodynamics (QCD) remains a significant theoretical challenge. In this work, we introduce a Physics-Guided Neural Network (PGNN) that integrates Holographic QCD with deep learning. By embedding the five-dimensional ${\text{AdS}}_5$ Dirac equation and the string diffusion kernel directly into the computational graph, the network is strictly constrained to the physical proton mass ($M_p\equiv 0.938\text{ GeV}$). Applying this framework to high-precision SLAC deep inelastic scattering data yields a global fit of ${\chi }^2/\text{d.o.f.}\simeq 0.91$. Rather than relying on predetermined empirical forms, the network dynamically extracts the transition between the $s$-channel bulk fermion mechanism (hadronic resonance excitations) and the $t$-channel holographic Pomeron exchange (diffractive background), identifying a kinematic crossover near $x\approx 0.19$. Furthermore, the optimization naturally recovers a Pomeron intercept of ${\alpha }_0\approx 1.0786$ and generates higher-twist scale-breaking effects through the evolution of resonance mass spectra. This demonstrates that embedding analytical differential equations into neural networks provides an interpretable, data-driven approach for phenomenological studies of strongly coupled systems.

Comments: 11 pages, 6 figures