Assessing the Impact of Fitting Methodology at aN$^3$LO with FPPDF: an Open Source Tool for Extracting Parton Distribution Functions in the Hessian Approach

The paper introduces FPPDF, a new open-source tool that implements the MSHT collaboration's polynomial parameterization and Hessian error approach using NNPDF's theoretical and experimental libraries, ultimately demonstrating that the impact of moving to aN$^3$LO perturbative orders is largely independent of the chosen PDF parameterization methodology.

The Gist

Imagine you are trying to create the ultimate recipe for a world-famous sourdough bread. To get it perfect, you need to know exactly how much flour, water, and salt is inside every single loaf.

In the world of particle physics, scientists are trying to do something similar, but instead of bread, they are studying the "ingredients" inside a proton (the tiny building blocks of atoms). These ingredients are called Parton Distribution Functions (PDFs).

Here is a breakdown of the paper using that analogy:

1. The Problem: Different Chefs, Different Recipes

Right now, there are several "Master Chefs" (scientific collaborations like NNPDF, MSHT, and CT) all trying to write the definitive recipe for the proton.

The problem is that they use different cooking methods:

• Chef A (NNPDF) uses a high-tech, AI-driven robot (Neural Networks) to guess the recipe. It’s incredibly flexible but can be a bit unpredictable.
• Chef B (MSHT/CT) uses a traditional, strict mathematical formula (Polynomials). It’s more rigid but very structured.

Because they use different methods, they end up with slightly different recipes. This is a problem because when we use these "recipes" to predict what will happen in massive machines like the Large Hadron Collider, the tiny differences in the recipes lead to different predictions about how the universe works.

2. The Solution: The "FPPDF" Kitchen Tool

The authors of this paper have created a new, open-source tool called FPPDF.

Think of FPPDF as a universal kitchen workstation. It allows a scientist to take the high-quality ingredients used by Chef A (the data and theory) but use the traditional cooking method of Chef B.

By doing this, they can perform a "fair fight." They can ask: "If we use the exact same ingredients, but just change the cooking method, how much does the final bread actually change?"

3. The Experiment: Upgrading the Oven (aN3LO)

The scientists didn't just want to test the method; they wanted to test it at the highest possible precision. In physics, precision is like the temperature of your oven.

• NNLO is like a standard oven.
• aN3LO is like a super-advanced, hyper-precise molecular oven.

They used their new tool to see how the "recipes" changed when they moved from the standard oven to the super-precise oven, comparing the results between the AI-robot method and the traditional formula method.

4. The Discovery: It’s the Ingredients, Not the Chef!

After running all the tests, they found something very important: The "flavor" of the proton is mostly determined by the ingredients (the physics), not the chef (the math method).

Specifically, they found that:

• When they upgraded to the "super-precise oven" (aN3LO), both the AI-chef and the Traditional-chef saw the same changes in the recipe.
• While the two methods gave slightly different "uncertainty" levels (how much they trust their own recipe), the core "taste" of the proton remained consistent.

Why does this matter?

In plain English: This paper gives scientists a way to double-check their work. It proves that the big changes we see in our understanding of particles aren't just "math glitches" caused by using different formulas—they are real, physical changes caused by the deeper laws of nature.

By releasing FPPDF as an open-source tool, they’ve handed a "universal kitchen" to every physicist in the world, allowing everyone to test their recipes and ensure we are all reading the same cookbook for the universe.

Technical Summary

Technical Summary: Assessing the Impact of Fitting Methodology at aN3LO with FPPDF

1. Problem Statement

As precision physics at the Large Hadron Collider (LHC) moves toward the percent level, Parton Distribution Functions (PDFs) have become a dominant source of theoretical uncertainty. While different global fitting groups (CT, MSHT, and NNPDF) aim for the same goal, non-negligible tensions have emerged between their results regarding central values and uncertainty estimates.

These discrepancies are difficult to disentangle because they arise from three distinct sources:

1. Data/Theory Inputs: Differences in datasets used and theoretical treatments (e.g., heavy quark schemes).
2. Perturbative Order: The transition from Next-to-Next-to-Leading Order (NNLO) to approximate Next-to-Next-to-Next-to-Leading Order (aN3LO).
3. Fitting Methodology: The mathematical approach to parameterizing the PDFs (Neural Networks vs. Fixed Polynomials) and the method of error propagation (Monte Carlo replicas vs. the Hessian approach).

Previous studies could not easily isolate the impact of the fitting methodology itself because they lacked a tool that could use the exact same data and theory settings while switching only the parameterization and error treatment.

2. Methodology

The authors introduce FPPDF, a new open-source public code designed to bridge this gap.

• The FPPDF Framework: It implements a fixed polynomial parameterization (using a Chebyshev polynomial basis, similar to the MSHT methodology) and the Hessian approach to error propagation. Crucially, it utilizes the public NNPDF libraries for all data and theory computations. This allows for a "like-for-like" comparison where the only variable changed is the fitting methodology.
• Comparative Setup: The authors perform global fits at both NNLO and aN3LO accuracy. They compare the FPPDF results (Hessian/Polynomial) against the established NNPDF results (Monte Carlo/Neural Network).
• Consistency: Both methodologies use identical input datasets, theory grids for cross-sections and evolution, and treatments of Missing Higher Order Uncertainties (MHOU) via the theory covariance matrix approach.

3. Key Contributions

• Software Release: The release of FPPDF, an open-source tool that enables the community to perform benchmark studies to standardize PDF determinations.
• Methodological Disentanglement: The paper provides a systematic way to separate "genuine" physics effects (from higher-order QCD corrections) from "methodological" effects (from the choice of mathematical parameterization).
• aN3LO Benchmark: It provides a new family of Hessian PDF sets at NNLO, NNLO+MHOU, and aN3LO accuracy to complement existing NNPDF sets.

4. Results

• Stability of Perturbative Effects: The most significant finding is that the impact of moving from NNLO to aN3LO (and the inclusion of MHOU) is insensitive to the fitting methodology. Both the Hessian/Polynomial and NN/Monte Carlo approaches show the same qualitative trends: a suppression of the gluon-gluon luminosity at small/intermediate $m_X$ and an enhancement at large $m_X$. This confirms that these are genuine physical effects of N3LO corrections.
• $\chi^2$ Trends: The fit quality trends are consistent across both methods, showing that the inclusion of MHOU generally improves $\chi^2$ for most processes, except for top-pair production.
• PDF-Level Differences:
  • Gluon/Singlet: The central values are highly compatible (within $1\sigma$) across the $x$ range.
  • Charm: The charm PDFs are compatible in central value, though uncertainty sizes differ.
  • Valence Quarks: A non-negligible difference was observed in the valence distribution ($V$) at medium-$x$, likely due to how different methodologies implement sum rules.
• Uncertainty Sizes: The methodologies produce different uncertainty profiles. The Hessian approach (with dynamic tolerance) tends to be more conservative in the data and low-$x$ regions, whereas the NNPDF (Monte Carlo) approach tends to produce larger uncertainties in the high-$x$ extrapolation region.
• Cross Sections: Benchmark predictions for Higgs ($\sigma_{ggH}$) and $W/Z$ boson production confirm that the relative impact of N3LO corrections is consistent regardless of the parameterization used.

5. Significance

This work is highly significant for the high-precision era of particle physics. By proving that the shifts caused by aN3LO corrections are robust and not artifacts of the fitting method, the authors provide confidence in the use of higher-order QCD calculations for LHC predictions. Furthermore, the release of FPPDF provides the community with a standardized tool to investigate the "tensions" between PDF sets, moving the field toward a more unified and understood understanding of proton structure.