Extraction of Pion Unpolarized Quark and Gluon Generalized Parton Distributions using Deep Neural-Networks
We present a deep neural-network (DNN) extraction of the pion unpolarized quark and gluon generalized parton distributions (GPDs) using the corresponding parton distribution functions (PDFs) from the JAM21 and xFitter analysis, together with experimental measurements of the pion electromagnetic form factor (EMFF) and lattice quantum chromodynamics (QCD) results. The GPDs are parameterized using a physics-informed neural-network (PINN) that incorporates the known PDF behavior, an exponential momentum-transfer dependence, and a trainable neural network (NN) component. The network parameters are determined by minimizing a $\chi^2$-based loss function. For the valence-quark GPDs, the loss function includes contributions from the EMFF, squared EMFF, charge-normalization constraints, and regularization terms. For the gluon GPDs, it incorporates constraints from the gluon gravitational form factors together with regularization. This framework enables a flexible, nonparametric extraction while preserving the essential theoretical and phenomenological constraints. By employing the full ensemble of available PDF replicas, we quantify the uncertainties of the extracted GPDs over a broad kinematic range in the longitudinal momentum fraction and momentum transfer, with the uncertainty bands corresponding to the $1\sigma$ confidence interval. The extracted valence-quark GPDs are found to be in good agreement with available lattice-QCD calculations. Our study demonstrates that DNN-based methods provide a flexible and robust framework for extracting pion GPDs and probing the multidimensional internal structure of the pion, offering a promising avenue for future investigations of hadron tomography.