⚛️ phenomenology

Parton distribution functions and theory parameters: an NNPDF perspective

This paper presents the NNPDF collaboration's perspective on the current status and challenges of determining Parton Distribution Functions, emphasizing their critical role in extracting fundamental Standard Model parameters and Beyond the SM Wilson coefficients through both stand-alone and joint analyses.

Original authors: Richard D. Ball, Tommaso Giani, Felix Hekhorn, Jaco ter Hoeve, Tanjona R. Rabemananjara, Juan Rojo, Roy Stegeman, Maria Ubiali

Published 2026-03-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Richard D. Ball, Tommaso Giani, Felix Hekhorn, Jaco ter Hoeve, Tanjona R. Rabemananjara, Juan Rojo, Roy Stegeman, Maria Ubiali

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master chef trying to recreate a famous, complex dish (like a perfect soufflé) based on a recipe you found in a dusty cookbook. But here's the catch: the cookbook (the Proton) is written in a language you don't fully understand, and the ingredients inside it (Partons) are constantly shifting and changing.

To cook the dish perfectly, you need to know exactly how much of each ingredient is in the bowl at any given moment. These ingredient amounts are called Parton Distribution Functions (PDFs).

However, there's a problem. The recipe also depends on other variables you don't know for sure, like the exact temperature of the oven (Strong Coupling Constant, $\alpha_s$ ) or the weight of a specific spice (Top Quark Mass, $m_t$ ). If you guess the oven temperature wrong, your measurement of the ingredients will be wrong. If you guess the ingredient amounts wrong, your calculation of the oven temperature will be wrong.

This paper is about a team of scientists (the NNPDF collaboration) who are trying to solve this "chicken-and-egg" problem. They want to figure out the ingredients and the cooking conditions at the exact same time, without messing up the results.

Here is a breakdown of their work using simple analogies:

1. The Problem: The "Blind Taste Test"

For a long time, scientists tried to measure the ingredients (PDFs) first, assuming they knew the oven temperature (Standard Model parameters). Then, they would use those ingredients to measure the temperature.

The Flaw: If the oven is actually hotter than you thought, you might think you have less flour than you do, just to make the math work. This creates a "bias."
The New Approach: The NNPDF team says, "Let's stop guessing. Let's measure the ingredients and the oven temperature simultaneously." They treat everything as a giant, interconnected puzzle where every piece affects every other piece.

2. The Toolkit: How They Do It

The team uses three main "tools" (methods) to solve this puzzle:

The "Monte Carlo" Simulator (Correlated Replica Method):
Imagine you have 1,000 different versions of the same recipe book, each with slightly different random scribbles (representing experimental errors). You try to cook the dish using all 1,000 books. By seeing how the results vary across all 1,000 attempts, you can figure out exactly how much the "oven temperature" and the "flour amount" are dancing together.
The "Mathematical Shortcut" (Theory Covariance Method):
Instead of running 1,000 simulations, this method uses a clever math trick. It asks: "If I nudge the oven temperature by a tiny bit, how much does the recipe change?" It calculates the answer instantly, allowing them to find the perfect balance between ingredients and temperature without doing all the heavy lifting.
The "Smart AI" (SIMUnet):
This is a neural network (a type of AI) that has been upgraded. Usually, the AI just learns the ingredients. Now, they added a "control panel" to the AI. They can turn knobs on this panel to adjust the "oven temperature" or "spice weight" while the AI is still learning the ingredients. The AI learns both at the same time, finding the perfect combination that fits all the data.

3. The "Stress Test": The "Fake Data" Check

How do they know their new method isn't broken? They use a Closure Test.

The Analogy: Imagine the scientists create a "perfect" fake recipe book where they know the exact truth (e.g., exactly 200g of flour and 350°F oven). They then feed this fake book into their new method.
The Result: If their method is good, it should look at the fake book and say, "Ah, it's 200g and 350°F!"
The Discovery: In their tests, they found that if they forced the AI to only look at "positive" numbers (you can't have negative flour!), it got slightly confused and gave a wrong answer. By relaxing that rule, they fixed the error. This proved that their "Stress Test" was vital for catching hidden mistakes.

4. The "New Physics" Trap: Searching for Ghosts

The biggest challenge is looking for New Physics (Beyond the Standard Model). Imagine you are looking for a ghost in your kitchen.

The Trap: If you don't know your ingredients well enough, you might think a shadow is a ghost, when it's actually just a pile of flour you didn't measure right.
The Solution: The paper shows that if you try to find a "Ghost" (New Physics) using a recipe book that was biased by the ghost's presence, you will never find it! The ghost will just hide inside the "flour measurement."
The Fix: To find the ghost, you need to measure the ingredients while looking for the ghost. The paper shows that by using data from different sources (like a "Forward Physics Facility" or an "Electron Ion Collider"—think of them as different angles of light in the kitchen), they can separate the "flour" from the "ghost."

5. Why This Matters

The Large Hadron Collider (LHC) is like a giant, high-speed collision of two protons. To understand what happens when they smash together, we need to know exactly what was inside them before they hit.

If we get the ingredients (PDFs) wrong, we might miss a new particle (like the Higgs boson or something even bigger).
If we get the cooking conditions ( $\alpha_s$ , $m_t$ ) wrong, we might think we found new physics when we actually just made a math error.

In Summary:
This paper is a guide on how to stop guessing. The NNPDF team has built a sophisticated system that measures the "ingredients" of the proton and the "rules of the universe" at the exact same time. They use AI, advanced math, and rigorous "fake data" tests to ensure that when they claim to have found a new particle or a new law of physics, they haven't just been fooled by a bad measurement of flour.

They are essentially saying: "Don't just read the recipe; understand the whole kitchen, the oven, and the chef, all at once."

1. Problem Statement

The determination of Parton Distribution Functions (PDFs) and Standard Model (SM) or Beyond the Standard Model (BSM) parameters are deeply intertwined.

Interdependence: Precision predictions for Large Hadron Collider (LHC) observables rely on PDFs, while the extraction of fundamental parameters (e.g., strong coupling $\alpha_s(m_Z)$ , top mass $m_t$ , $W$ -mass $m_W$ , and SMEFT Wilson coefficients) depends on these same PDFs.
Bias Risk: If PDFs are fitted assuming fixed SM parameters, and those parameters are later extracted using the resulting PDFs, systematic biases can occur. Conversely, if BSM effects distort high-energy data tails, fitting PDFs without accounting for BSM physics can lead to "contaminated" PDFs that absorb new physics signals, masking them in future searches.
Current Limitations: Traditional approaches often separate data into disjoint sets (one for PDFs, one for parameters) or fix parameters to external values. This is suboptimal for parameters like $\alpha_s$ and $m_t$ , which are strongly correlated with PDF shapes (e.g., the gluon distribution), and for BSM searches where high- $x$ regions are poorly constrained.

2. Methodology

The paper outlines the NNPDF framework's approach to simultaneous extraction of PDFs and theory parameters using three primary methodologies:

A. Theoretical Framework

NNPDF4.0: Uses neural networks parametrized via gradient descent and Monte Carlo (MC) importance sampling to propagate uncertainties.
Higher Order Precision: Incorporates Next-to-Next-to-Next-to-Leading Order (N3LO) QCD corrections and Next-to-Leading Order (NLO) QED corrections (including photon PDFs).
Uncertainty Handling: Utilizes a Theory Covariance Matrix formalism to include Missing Higher Order Uncertainties (MHOU) and experimental correlations.

B. Simultaneous Fitting Strategies

The authors propose three distinct methods to handle the joint optimization of continuous PDF functions and discrete/continuous parameters:

Correlated Replica Method (CRM):
- Generates $N_{rep}$ data replicas including experimental and theory uncertainties.
- Performs fits for a grid of parameter values (e.g., different $\alpha_s$ ).
- Constructs a joint probability distribution in the $(\text{PDF}, \text{Parameter})$ space to extract the most likely parameter value and correlations.
Theory Covariance Method (TCM):
- Treats the target parameter (e.g., $\alpha_s$ or $m_t$ ) as a nuisance parameter $\lambda$ .
- Assumes Gaussian priors and likelihoods to analytically derive the posterior distribution of $\lambda$ .
- The shift in theory predictions is linearized ( $T \to T + \lambda \beta$ ), allowing the parameter to be determined alongside the PDFs without retraining the neural network for every parameter value.
SIMUnet (Simultaneous Network):
- An extension of the NNPDF architecture where Wilson coefficients (SMEFT parameters) are added as trainable parameters on the edges of the final output layer.
- Allows for the simultaneous minimization of the $\chi^2$ with respect to both PDF parameters and BSM coefficients.

C. Validation: Closure Tests

To ensure statistical robustness, the authors employ closure tests:

Synthetic data is generated from a known "truth" (specific PDFs and parameter values).
The fitting methodology is applied to this synthetic data.
Metric: The Normalized Bias ( $R_{bv}$ ) and Root-Mean-Square Bias are calculated to verify if the extracted uncertainties are faithful (Gaussian) and if the central values recover the input truth.
Key Finding: The paper highlights that enforcing positivity constraints (physical requirement that PDFs $>0$ ) can break Gaussian assumptions, introducing bias in parameter extraction unless carefully managed (e.g., by releasing constraints during the closure test phase).

3. Key Contributions and Results

A. Simultaneous Determination of $\alpha_s(m_Z)$

Result: Using NNPDF4.0 data and TCM/CRM, the authors extract $\alpha_s(m_Z) = 0.1194^{+0.0007}_{-0.0014}$ .
Improvement: This determination benefits from the inclusion of N3LO QCD, NLO QED, and a photon PDF.
Validation: Closure tests revealed a bias when positivity constraints were active. By correcting for this, the method achieves consistency with the input truth.
Impact: Including $\alpha_s$ uncertainty in a correlated manner with PDFs leads to a more robust gluon-gluon luminosity uncertainty estimate.

B. Simultaneous Determination of $m_t$ and $\alpha_s$

Method: Applied TCM to jointly fit $m_t$ and $\alpha_s$ with PDFs.
Result: The top mass is determined most precisely using the invariant mass ( $m_{t\bar{t}}$ ) distribution, followed by transverse momentum ( $p_T$ ) and rapidity ( $y_t$ ).
Correlation: The study finds limited correlation between $m_t$ and $\alpha_s$ in the joint fit, suggesting that simultaneous extraction does not significantly degrade the precision of either parameter compared to fixed-PDF extractions.

C. Simultaneous Determination of PDFs and SMEFT Wilson Coefficients

Challenge: BSM effects in high-mass tails (e.g., Drell-Yan, $t\bar{t}$ ) can be absorbed into PDFs (specifically large- $x$ gluons and antiquarks), leading to false constraints on BSM physics.
Findings:
- Top Sector: Simultaneous fits of top-sector SMEFT operators and PDFs show that while Wilson coefficient bounds remain stable, the large- $x$ gluon PDF becomes "softer."
- Drell-Yan: Universal BSM operators (e.g., $W'$ , $Z'$ ) can be mimicked by softer large- $x$ PDFs. If not accounted for, this erases the BSM signal in theoretical predictions.
Solution: The paper demonstrates that joint fits including low-energy data (e.g., from the Electron-Ion Collider or Forward Physics Facility) alongside high-energy LHC data can break this degeneracy, preventing BSM signals from being absorbed into PDFs.

4. Significance and Future Outlook

Precision Era: As the LHC moves toward the High-Luminosity (HL-LHC) era, PDF uncertainties are becoming the dominant systematic in many precision measurements. Simultaneous fitting is no longer optional but necessary for sub-percent precision.
Methodological Robustness: The paper establishes that closure tests are critical for validating simultaneous fits, particularly regarding the handling of positivity constraints and non-Gaussian uncertainties.
BSM Sensitivity: The work provides a framework to distinguish between PDF variations and genuine BSM signals, preventing the "hiding" of new physics in global fits.
Future Directions:
- Extending joint fits to heavy quark masses ( $m_c, m_b$ ) and electroweak parameters ( $m_W, \sin^2\theta_W$ ).
- Incorporating data from future colliders (EIC, FPF, LHeC, FCC) to constrain large- $x$ PDFs and BSM effects simultaneously.
- Moving to unbinned or detector-level observables to maximize sensitivity.

In summary, this paper presents a comprehensive NNPDF perspective on the necessity and methodology of jointly determining PDFs and theory parameters. It validates these methods through rigorous closure tests and demonstrates their critical role in achieving precision SM measurements and maintaining sensitivity to BSM physics.