Lucas Kotz

Physics PhD · Data Scientist · Simulation and Modeling Specialist

I earned my PhD in Physics from Southern Methodist University, where I developed a strong foundation in data-driven analysis and predictive modeling. These skills have enabled me to tackle complex, data-intensive problems in areas like statistical modeling, uncertainty quantification, high-dimensional optimization, and data-driven theoretical analysis.

Learn about the particle that holds matter together within our universe: the pion. By learning about the internal structure of the pion at high energies, we can learn about the universe we interact with daily.

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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.

Visualizing pion data

After identifying the best-fit models, we analyzed their statistical behavior and interrelationships.

L2 Sensitivity

Understanding how each dataset influences a model is critical, especially when combining multiple datasets, where conflicting signals can obscure important patterns. That’s why it’s essential to evaluate dataset impact even after a model has been built.

Recently, a new technique called L2 sensitivity was developed to measure how much each dataset influences a model’s fit (link to study). In essence, it checks how much the model’s error (χ² goodness-of-fit) changes when a PDF parameter is adjusted by one standard deviation.

However, the standard L2 sensitivity method only works if the dataset is already part of the model’s analysis, which means it can’t directly evaluate brand-new or external data.

To solve that problem, I developed a more flexible procedure that applies L2 sensitivity to any dataset, even if it’s not part of the original model. I integrated this solution into the open-source xFitter tool and validated it with real data (as documented in my publication at arXiv).

Files related to the L2 sensitivity extraction method can be found here.

Baseline Comparison

I verified my method by comparing its results to an earlier study on how datasets affect proton PDFs. My results closely matched the prior study’s findings, confirming that my approach works reliably.

There were a few minor differences because I defined the chi-squared metric slightly differently and handled heavy-quark data in another way. These technical details only caused small changes in the L2 sensitivity results (as a function of the parton’s momentum fraction x).

Exploring New Datasets

After proving the method, I applied the L2 sensitivity method to new datasets whose influence hadn’t been examined before. Traditionally, to evaluate a new dataset’s influence, you’d have to spend weeks integrating it into the model’s codebase.

In contrast, my approach can assess a dataset’s impact in just a few hours and requires no changes to the core code. This makes it possible to quickly screen new data for its relevance (or redundancy), greatly speeding up the analysis pipeline.

Data Visualization

During my PhD, I relied heavily on data visualization to turn complex results into clear, accessible insights. I often analyzed parton distribution functions (PDFs) to predict measurable outcomes, and I had to create clear, informative charts both for technical reports and for communicating with non-specialist stakeholders.

Below are selected examples showcasing how I visualized data from my studies. Most of these figures were produced using Mathematica with the ManeParse package or CERN’s ROOT framework, widely used in particle physics. You can find more information about ManeParse here and ROOT here.

Gallery