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 $χ^2/\text{d.o.f.} \simeq 0.91$. 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 $α_0 \approx 1.0786$ 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.

Figures (6)

• Figure 1: The holographic dual-channel physical picture for deep inelastic scattering. (a) In the large-$x$ regime, the scattering is governed by the $s$-channel mechanism, where the virtual photon interacts with the ground-state bulk fermion field ($\Psi_0$) via an AdS overlap integral, exciting a discrete tower of hadronic resonance states ($\Psi_n$). This non-perturbative transition phenomenologically dualizes the large-$x$ behavior without invoking explicit valence quark distributions. (b) As $x$ decreases, the interaction transitions to a dipole-target formulation, dominated by the $t$-channel holographic Pomeron ($\mathbb{P}$) exchange, which smoothly captures the diffractive background at high energies.
• Figure 2: Schematic architecture of the PGNN proposed in this work. The kinematic inputs $(x, Q^2)$ are processed through a dual-track framework. The purely data-driven neural network (left track) dynamically extracts the mechanism weight $w(x)$ and a neural network residual $\Delta_{\mathrm{NN}}$. Concurrently, the physics-driven Holographic AdS/QCD module (right track) calculates the analytic structure functions $F_2^{\mathrm{Fermion}}$ and $F_2^{\mathrm{Pomeron}}$, strictly subjected to the rigid proton mass manifold constraint ($\mathcal{H}_{\mathrm{AdS}} \Psi = M_p^2 \Psi$). These outputs are synthesized in the physics-guided fusion layer. The overall loss function, incorporating the SLAC data fidelity and an $L_2$ regularization on the residual, updates solely the neural network weights via backpropagation, preserving the theoretical rigidity of the holographic background.
• Figure 3: The unpolarized proton structure function $F_2(x, Q^2)$ plotted as a function of $Q^2$ for various values of Bjorken $x$. The solid lines denote the theoretical outputs from the PGNN, while the markers represent the SLAC experimental data Whitlow:1990gkWhitlow:1991uw. The curves for different $x$ bins are scaled by constant factors for visual clarity.
• Figure 4: The neural-network-extracted mechanism weight $w(x)$ as a function of Bjorken $x$. The curve reveals a distinct transition from a Pomeron-dominated regime (where $w(x)$ is significantly suppressed to a finite minimum near the data limit $x \approx 0.08$) to a bulk-fermion-dominated regime ($w(x) \to 1$ at large $x$), with the crossover occurring near $x_c \approx 0.19$.
• Figure 5: The physical constraints and dynamical outputs of the PGNN holographic parameters. (a) The proton mass manifold constraint in the parameter space of the 5D fermion mass $M_5$ and the soft-wall scale $\kappa$. The red curve represents the loci where the ground state is rigidly fixed to $M_0 \equiv 0.938 \text{ GeV}$. (b) The dynamic evolution of the first and second excited resonance masses ($M_1, M_2$) as a function of the Bjorken scale $x$. While $M_0$ remains strictly constrained across the phase space, the higher states exhibit distinct scale-breaking behaviors optimized by the neural network.