Fantômas4QCD
Our team set out to create a new, flexible way to model parton distribution functions (PDFs) – the statistical profiles that show how a particle’s momentum is shared among its quarks and gluons.
We applied Bézier curves in our PDF modeling, giving our solution the clarity of a simple mathematical formula and the adaptability of a neural network – a unique combination not seen in previous models.
The Fantômas module we built quantifies uncertainties in the pion’s structure that earlier models could not measure, providing new insights for QCD analysis.
The xFitter program with the Fantômas module implemented is found here.
Bézier curves: From cars to physics
To better describe the internal structure of the pion (and other hadrons), we use Bézier curves, originally developed for car design.
Invented by Paul de Casteljau (Citroën, 1958) and Pierre Bézier (Renault, 1960), Bézier curves were first used to model smooth car body shapes using only a few control points. The Citroën DS (left) was one of the first cars designed using this technology, which later became foundational in computer-aided design (CAD).
Today, more than 60 years later, we apply the same mathematical framework in high-energy physics. Bézier curves allow us to construct smooth, continuous, and highly adaptable functions for modeling PDFs. Their interpretability and flexibility make them ideal for capturing model-dependent uncertainty in a physically meaningful way.
How we extract PDFs using Bézier curves
Since parton distribution functions (PDFs) are not directly observable, we must infer them from experimental data — specifically, measurements from fixed-target and collider experiments.
Our module was implemented into xFitter, an open-source framework for global QCD analyses. xFitter predicts theoretical models by using a gradient-descent algorithm (Minuit) to minimize the chi-squared (χ²) goodness-of-fit value to identify the best-fit Bézier parameterization for the given data.
By adjusting the number and location of control points, the module can scan a wide range of model possibilities, giving us a more complete picture of the theoretical uncertainties in PDF extraction.
Extracted PDFs from pion data
From the hundreds of solutions generated by the Fantômas module, we selected five good fits that captured the full range of observed features.
These were combined into a single model that incorporates both aleatoric (statistical) and epistemic (model-based) uncertainties — providing a more comprehensive understanding of the pion's structure.
The complete pion PDF set is located here.
Fifty Bézier curves fitted to randomly fluctuated data using fixed settings, illustrating aleatoric uncertainty via bootstrapping.
Fifty Bézier curves fitted to the same data using varied settings, illustrating epistemic uncertainty and model dependency.
Kinematic coverage of data points from various pion experiments.
Gluon PDFs showing the range of solutions obtained by varying the number, positions, and treatment of control points during fitting.
Quark sea PDFs showing the range of solutions obtained by varying the number, positions, and treatment of control points during fitting.
Gluon PDFs from selected fits that satisfy the imposed χ² condition.
Quark sea PDFs from selected fits that satisfy the imposed χ² condition.
Constituent pion quark (valence) PDFs from selected fits that satisfy the imposed χ² condition.
Combined pion PDFs showing contributions from both aleatoric and epistemic uncertainty.
Visualizing pion data
After identifying the best-fit models, we analyzed their statistical behavior and interrelationships.
Self-correlation heat map of the pion gluon PDF at fixed energy scale, plotted as a function of x on each axis.
Self-correlation heat map of the pion quark sea PDF at fixed energy scale, plotted as a function of x on each axis.
Self-correlation heat map of the pion valence PDF at fixed energy scale, plotted as a function of x on each axis.
Correlation heat map between the pion gluon and quark sea PDFs at a fixed energy scale, plotted as a function of x on each axis.
Correlation heat map between the pion gluon and valence PDFs at a fixed energy scale, plotted as a function of x on each axis.
Correlation heat map between the pion quark sea and valence PDFs at a fixed energy scale, plotted as a function of x on each axis.
Total momentum fractions carried by the gluon and quark sea in recent pion PDF studies. The FantoPDF combination highlights an anticorrelation between these two contributions.
Normalized distributions of momentum fractions from Monte Carlo replicas, comparing fits excluding the leading neutron dataset (transparent) and including it (opaque).