⚛️ phenomenology

Integrating Heisenberg uncertainty and maximum entropy principles verses statistical approach to nucleon structure

This paper determines nucleon parton distribution parameters using two distinct approaches—one combining Heisenberg uncertainty and maximum entropy principles with sum rules, and another employing a statistical thermodynamic model—and demonstrates that both methods yield valence quark and gluon densities consistent with existing parametrizations for calculating structure functions.

Original authors: A. Mirjalili, S. Khosravi Kia, S. Atashbar Tehrani

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: A. Mirjalili, S. Khosravi Kia, S. Atashbar Tehrani

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

Deep inside every atom lies a proton, a tiny but mighty core that gives matter its substance. For decades, physicists have tried to map the invisible landscape within this core, asking a simple but difficult question: how is the momentum of the proton shared among the smaller particles that make it up? These particles, called quarks and gluons, are not static bricks but a seething, dynamic fluid. To understand the proton, scientists must determine the "parton distribution functions," which are essentially maps showing how likely it is to find a specific particle carrying a specific fraction of the proton's total speed. Traditionally, creating these maps has relied heavily on fitting mathematical curves to vast amounts of experimental data from particle colliders. However, this new study attempts a different path, one that relies less on matching data points and more on fundamental laws of nature to predict how these particles should behave before any experiment is even run.

The researchers approached this problem by treating the proton not just as a collection of particles, but as a system governed by two powerful, universal principles. The first is the Heisenberg uncertainty principle, a rule of quantum mechanics stating that we cannot know both the exact position and the exact speed of a particle at the same time; the more precisely we know one, the less we know the other. The second is the principle of maximum entropy, a concept from thermodynamics which suggests that a system in equilibrium will naturally settle into the state of greatest disorder or randomness, given the constraints it faces. By combining these two ideas with the strict rules that the proton must contain exactly three valence quarks and that the total momentum of all its parts must add up to the whole, the team set out to calculate the internal structure of the proton from first principles.

In their first method, the team treated the proton as a confined space where the uncertainty in a particle's location limits how its momentum can be distributed. They imagined the proton's interior as a cylinder to simplify the geometry of the problem, allowing them to calculate the uncertainty in position based on the proton's known size. This uncertainty in space translates directly into an uncertainty in momentum. By applying this logic to the up and down quarks that make up the proton, they derived a set of equations that described how these particles are likely to be distributed. They then used the maximum entropy principle to find the specific arrangement that maximizes disorder while still obeying the rules of particle count and momentum conservation. This approach allowed them to determine the unknown parameters of the quark distributions without needing to look at experimental data first. The results, when evolved to higher energy levels using standard physics equations, matched well with existing models and experimental measurements of the proton's structure, particularly for the ratio of down to up quarks.

Recognizing that this first method had limitations—specifically, it could not easily account for the gluons, the particles that carry the force holding the quarks together—the researchers developed a second, more comprehensive approach. In this version, they abandoned the geometric constraints of the first method and instead viewed the proton as a statistical system in thermal equilibrium, similar to a gas of particles. In this picture, the proton is defined by thermodynamic variables: a temperature, a volume, and chemical potentials that act like energy levels for the different types of particles. Using the same maximum entropy principle and the rules for particle count and momentum, they solved for these thermodynamic variables. This allowed them to generate a complete set of distributions for up quarks, down quarks, anti-quarks, and crucially, the gluons. This statistical model provided a full picture of the proton's interior, including the previously elusive gluon density.

The findings from this second approach proved to be even more robust. When the researchers evolved their calculated distributions to the high energy scales used in modern experiments, the resulting maps of the proton's interior aligned closely with the most sophisticated models used by the global physics community. They were able to calculate the proton's structure function, a key quantity measured in experiments where neutrinos smash into protons, and their predictions matched the available experimental data with high precision. Furthermore, the ratio of down to up quarks derived from this statistical model showed a remarkable independence from the energy scale, a feature that theoretical physics predicts should exist but is often difficult to reproduce in other models. The study suggests that by treating the proton as a statistical system governed by entropy and uncertainty, it is possible to reconstruct its internal structure with a high degree of accuracy, offering a powerful alternative to purely data-driven methods.

Ultimately, this work demonstrates that the fundamental laws of thermodynamics and quantum mechanics contain enough information to reveal the hidden architecture of the proton. While the first method provided a solid foundation by linking spatial uncertainty to momentum, the second method, by embracing a full statistical description, succeeded in capturing the role of gluons and providing a more complete and reliable map of the nucleon. The researchers conclude that these principles offer a viable, theory-driven path to understanding the subatomic world, one that complements the massive experimental efforts currently underway. By showing that the proton's internal structure can be deduced from the principles of maximum disorder and quantum limits, the study opens a new window into the nature of matter, suggesting that the chaotic dance of particles inside the proton follows a predictable, statistical order.

Technical Summary: Integrating Heisenberg Uncertainty and Maximum Entropy Principles vs. Statistical Approach to Nucleon Structure

Problem Statement
The determination of parton distribution functions (PDFs) and nucleon structure functions remains a central challenge in Quantum Chromodynamics (QCD). Conventional methods rely heavily on fitting parameterized functions to experimental data, often neglecting the non-perturbative quantum correlations between initial and final states. While statistical approaches exist (e.g., assuming Fermi-Dirac or Bose-Einstein distributions), they often require extensive parameterization or fail to determine gluon densities without additional constraints. This paper addresses the problem of determining unknown parameters of valence quark and gluon densities at an initial low energy scale ( $Q_0$ ) using theoretical constraints rather than direct experimental fitting, aiming to provide a non-perturbative input for evolution equations.

Methodology
The authors propose and compare two distinct theoretical approaches to extract parton densities:

First Approach (Hybrid Uncertainty-Entropy):
- Assumptions: Valence densities are parameterized in the form $f(x) = Ax^B(1-x)^C$ .
- Constraints: The unknown parameters are determined by simultaneously satisfying:
  - Quark Number Sum Rules: Integrating valence densities to yield 2 up-quarks and 1 down-quark.
  - Momentum Sum Rule: Integrating momentum fractions to equal 1 (assuming only valence quarks at $Q_0$ ).
  - Heisenberg Uncertainty Principle: Applied to estimate the spatial and momentum uncertainties ( $\sigma_X \sigma_P \geq \hbar/2$ ). The authors map the proton's spatial uncertainty to a cylindrical geometry to simplify phase space calculations, deriving uncertainties for the Bjorken- $x$ variable ( $\sigma_x$ ).
  - Maximum Entropy Principle: The entropy function is maximized with respect to the distribution parameters to find the most probable distribution given the constraints.
- Evolution: The resulting initial densities are evolved to higher energy scales ( $Q^2 = 10, 20, 40 \text{ GeV}^2$ ) using DGLAP evolution equations at Leading Order (LO) and Next-to-Leading Order (NLO).
Second Approach (Pure Statistical Thermodynamics):
- Assumptions: The nucleon is modeled as a thermal equilibrium system of free partons (quarks, anti-quarks, and gluons) within a volume $V$ at temperature $T$ .
- Distributions: Partons are assigned statistical distributions: Fermi-Dirac for quarks/anti-quarks and Bose-Einstein for gluons. These depend on thermodynamic parameters: Temperature ( $T$ ), Volume ( $V$ ), and chemical potentials ( $\mu_u, \mu_d$ ).
- Constraints: Unknown parameters are determined using:
  - Quark Number and Momentum Sum Rules.
  - Maximum Entropy Principle: Optimized with respect to the chemical potentials.
  - Note: The Heisenberg uncertainty principle is not used in this approach.
- Evolution: Similar to the first approach, the derived statistical densities are evolved using DGLAP equations to calculate structure functions.

Key Contributions

Dual Methodology: The paper introduces a comparative framework where a hybrid quantum-statistical method (Approach 1) is contrasted with a full thermodynamic statistical model (Approach 2).
Gluon Density Extraction: A significant contribution of the second approach is the successful determination of the gluon density function. The first approach, relying on valence-only sum rules at $Q_0$ , could not determine gluon densities, leading to a higher number of free parameters. The statistical approach resolves this by including gluons in the thermodynamic ensemble.
Parameter Reduction: The statistical approach reduces the number of unknown parameters to four ( $T, V, \mu_u, \mu_d$ ) while simultaneously determining valence and gluon distributions, whereas the first approach required six parameters for valence densities alone.
Structure Function Calculation: Both approaches are used to calculate the nucleon structure function $xF_3$ (relevant for neutrino scattering) and the ratio of valence densities ( $d_v/u_v$ ) at various energy scales.

Results

Valence Densities: Both approaches yield valence quark densities that, when evolved to $Q^2 = 10 \text{ GeV}^2$ , show satisfactory agreement with established parametrization models (CT10 and MSTW08). The NLO results generally show better agreement with these models than LO results.
Structure Function $xF_3$ : The calculated $xF_3$ structure functions at $Q^2 \approx 20 \text{ GeV}^2$ align well with experimental data from CCFR and NuTeV, as well as with CT10 and MSTW08 models. The NLO results from both approaches demonstrate good consistency with available data.
Valence Ratio ( $d_v/u_v$ ): The ratio of valence densities exhibits minimal dependence on the energy scale, a theoretical expectation that is confirmed by the results. The second approach (statistical) shows particularly strong consistency with experimental data (WA21, CDHS, HERMES) in the $0.05 < x < 0.5$ interval.
Comparison of Approaches: The authors find that the second (statistical) approach yields more reliable results. This is attributed to its ability to include gluon densities (crucial for NLO accuracy) and its use of a complete statistical framework without relying on the geometric interpretation of the Heisenberg uncertainty principle, which the authors note is an approximation.

Significance and Claims
The paper claims that determining PDF parameters using fundamental principles (Maximum Entropy, Sum Rules, and in the first case, Uncertainty) rather than purely empirical fitting offers a valuable theoretical alternative.

Theoretical Insight: The work suggests that statistical-thermodynamic principles are well-realized in experimental data, particularly in the intermediate- $x$ region.
Methodological Advantage: The second approach is presented as superior because it utilizes the full statistical features of the parton system, extracts gluon densities (essential for NLO calculations), and requires fewer free parameters to achieve comparable or better agreement with data.
Limitations: The authors modestly acknowledge limitations, including the neglect of quark masses and inter-parton interactions, and the use of a simplified non-perturbative input. They note that the momentum uncertainty increases slightly if the confinement potential is not comprehensively modeled.
Future Directions: The paper suggests that these principles could be extended to polarized PDFs, Generalized Parton Distributions (GPDFs), and Transverse Momentum Dependent PDFs (TMDPDFs), and that future work could involve investigating the $F_2$ structure function and refining constraints in asymptotic regions ( $x \to 0$ and $x \to 1$ ).

The authors conclude that while their models are approximations, the results are "totally acceptable" and provide a robust method for understanding primary aspects of nucleon structure without resorting to extensive experimental fitting for initial parameters.

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