Generalized Parton Distributions from Symbolic Regression

The tool of symbolic regression (SR) is shown to be extremely helpful in the mission of understanding the internal structure of hadrons, including extracting information about the spatial distributions of quarks and gluons. We utilize a highly customizable SR package PySR to model a Generalized Parton Distribution Function (GPD) as a function of kinematic variables, specifically the flavor isovector combination Hu−d(x, t, ξ) at ξ = 0 . These PySR models were trained on GPD results provided by both Lattice QCD and phenomenological sources GGL, GK, and VGG. SR produces simple analytic closed form expressions that optimize the fit quality and simplicity. SR is used to obtain 1000 fits to these results, and their x-dependence is found in 3 clusters of solutions. Because SR allows for improved extrapolation capabilities beyond the range of LQCD results, we were able to single out a pathway for a quantitative description of the spatial distributions of quarks and gluons on the transverse plane, extracting quantitative trends of their radii in different ranges of the momentum fraction x. These trends emerge in the absence of any known priors, using solely results from LQCD on GPDs and nucleon form factors. In addition to PySR penalizing models with higher complexity and mean-squared error, we implement schemes that test specific physics hypotheses, including factorized x and t dependence and Regge behavior in PySR GPDs. Knowing the precise behavior of the GPDs, and their uncertainties in a wide range in x and t, crucially impacts our ability to concretely and quantitatively predict hadronic spatial distributions and their derived quantities.