Direct Deep Neural-network Extraction of Generalized Parton Distributions

TL;DR

This work establishes a flexible, scalable, and model-independent strategy for extracting multidimensional hadronic structure from current and future DVCS data and other GPD-related processes.

Abstract

We present a machine-learning method for the nonparametric extraction of generalized parton distributions (GPDs) from Compton form factors (CFFs) constrained by experimental data. The method addresses the longstanding inverse problem posed by the principal-value (PV) linear integral transform with a singular kernel that relates the charge-conjugation-even (C-even) quark GPD $H^{(+)}$ to the real part of the deeply virtual Compton scattering (DVCS) amplitude. Our approach constructs a differentiable representation of the Quantum Chromodynamics (QCD) PV kernel and embeds it as a fixed, physics-preserving layer inside a neural network that parameterizes the GPD $H^{(+)}(x,ξ,t,Q^{2})$ itself. The model enforces exact oddness in $x$, implements endpoint suppression, and includes curvature-based regularization that stabilizes the inversion in kinematically ill-conditioned regions. A Monte Carlo ensemble of CFFs, obtained from a global neural-network fit to unpolarized DVCS measurements with propagated experimental uncertainties, serves as input to a replica ensemble of GPD networks, yielding a fully probabilistic extraction of $H^{(+)}$ over the phase space. We demonstrate the method using a global determination of $\mathrm{Re}\,\mathcal{H}$ for Jefferson Lab measurements, and present a direct neural-network reconstruction of three-dimensional GPD surfaces $H^{(+)}(x_{0},ξ,t,Q_{0}^{2})$ obtained from experimental CFF inputs. This work establishes a flexible, scalable, and model-independent strategy for extracting multidimensional hadronic structure from current and future DVCS data and other GPD-related processes.

Figures (4)

• Figure 1: Reconstructed one-dimensional slice of $H^{(+)}(x \mid \xi_0, t_0, Q_0^2)$ at the median kinematic point of the data, corresponding to $\xi_0=0.25$, $t_0=-0.750$ GeV$^2$, and $Q_0^2=3.75$ GeV$^2$ showing a comparison between the true analytic function and the reconstructed GPD. The solid line represents the 1 replica instance and the dashed line represent the true function.
• Figure 2: Reconstructed one-dimensional slice of $H^{(+)}(x \mid \xi_0, t_0, Q_0^2)$ at the median kinematic point of the data, corresponding to $\xi_0=0.257$, $t_0=-0.742$ GeV$^2$, and $Q_0^2=4.81$ GeV$^2$. The solid line represents the ensemble mean and the shaded region the $68\%$ credible band obtained from the replica distribution. The vertical dashed lines mark $x=\pm\xi_0$.
• Figure 3: Three-dimensional GPD surface $H^{(+)}(x_0,\xi,t,Q_0^2)$ evaluated at fixed $x_0=0.257$ and $Q_0^2=4.81~\mathrm{GeV}^2$. The colored sheet shows the ensemble mean over $N_{\rm rep}$ replicas, while the two translucent sheets above and below indicate the $\pm1\sigma$ uncertainty band. The plane $\xi=x_0$ separates the DGLAP ($\xi<x_0$) and ERBL ($\xi>x_0$) domains.
• Figure 4: Comparison of reconstructed surfaces $H^{(+)}(x_0,\xi,t,Q_0^2)$ at fixed $x_0=0.257$ and three values of the virtuality, $Q_0^2 = 1,\,4,\,8\ \mathrm{GeV}^2$. All surfaces are ensemble means plotted on the same $(\xi,t,z)$ axes. Differences across the three scales reflect the smooth $Q^2$ dependence inferred from the experimental CFFs and the learned representation. No uncertainty bands are shown for clarity.