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Mathematical inverse problem for the world data inference of the parton distribution functions of the proton

This paper proposes a novel, model-bias-free methodology for extracting proton parton distribution functions by reformulating the global analysis of deep inelastic scattering data as a linear tensor reconstruction inverse problem, thereby replacing traditional parameter fitting with the direct solution of coupled linear integral equations derived from perturbative QCD.

Original authors: Henri Hänninen

Published 2026-09-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Henri Hänninen

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 of the visible universe, protons are not solid, indivisible spheres but rather bustling, chaotic clouds of smaller particles. These clouds are made of quarks and gluons, the fundamental building blocks of matter that are bound together by the strong nuclear force. Because these particles are so tightly confined, they can never be isolated and studied on their own; they only exist in groups. To understand how the universe works, from the collisions in massive particle accelerators to the earliest moments of the cosmos, physicists must know exactly how these internal particles are arranged. Specifically, they need to know the probability of finding a quark or a gluon at a certain speed and position within the proton. This probability map is called a parton distribution function. For decades, scientists have tried to draw this map by smashing particles together and measuring the debris, but they have always relied on a specific method that might be hiding the true shape of the proton.

A researcher at the University of Jyväskylä in Finland has proposed a radically different way to solve this puzzle. Instead of guessing the shape of the proton's internal map and then adjusting the guess until it fits the experimental data, this new approach treats the problem as a mathematical reconstruction, similar to how a doctor builds a three-dimensional image of a patient's body from a series of two-dimensional X-ray slices. In the traditional method, physicists assume the proton's internal structure follows a specific mathematical formula, like a curve with a few adjustable knobs. They turn these knobs until the curve matches the data. The new method, however, attempts to solve for the entire shape of the map directly from the data, without assuming any specific formula beforehand. This removes the risk of the scientists' own assumptions blinding them to unexpected features in the proton's structure.

The core of this work is a shift in perspective. The author shows that the complex equations describing how particles scatter inside a proton can be rewritten as a system of linear equations. In simpler terms, the data collected from experiments—such as the reduced cross sections, which measure the likelihood of a collision happening, and structure functions, which reveal details about the proton's internal layout—can be viewed as the result of a specific mathematical operation acting on the unknown map of the proton. By organizing all the available data from different types of particle collisions, including those involving electrons, neutrinos, and heavy particles like charm and bottom quarks, the author constructs a massive, unified system. This system links the experimental measurements directly to the unknown distribution of particles inside the proton through a set of integral equations.

The paper demonstrates that this system can be solved mathematically without fitting any parameters. The author reviews the most advanced theories of particle physics, known as perturbative quantum chromodynamics, and translates them into these linear equations. This includes not just the simplest approximations but also more complex, higher-order corrections that account for the subtle interactions between particles. The result is a framework where the proton's internal map is the only unknown variable. The data from the world's experiments acts as the constraints that force the solution to take a specific shape. The author also incorporates the rules that govern how the proton's structure changes as the energy of the collision changes, ensuring the solution is physically consistent.

While the paper does not present a final, completed map of the proton, it provides a proof of principle that such a map can be reconstructed without the bias of pre-set formulas. The author outlines a practical method to solve these equations using computer algorithms that break the continuous problem into a grid of points, a technique common in medical imaging. The work suggests that by using this direct reconstruction method, scientists could uncover features of the proton that have been missed by traditional fitting methods, such as unexpected behaviors at very low or very high energies. It opens a path to a more honest and direct understanding of the building blocks of matter, relying on the data itself to reveal the truth rather than on the assumptions of the observer.

Technical Summary: Mathematical Inverse Problem for the World Data Inference of the Parton Distribution Functions of the Proton

Problem Statement
The paper addresses the global analysis of Parton Distribution Functions (PDFs) for the proton, a central task in high-energy physics used to describe the internal structure of hadrons. Historically, this problem has been solved by fitting theoretical model parameters (ansatzes) to world data from Deeply Inelastic Scattering (DIS) experiments. The author identifies a fundamental limitation in this approach: the reliance on functional parametrizations (e.g., $xf(x) \approx Ax^\alpha(1-x)^\beta P(x)$ ) introduces model bias, potentially obscuring physical features not captured by the chosen ansatz.

The core problem is reformulated as a mathematical inverse problem: Can the PDFs be reconstructed directly from world data as smooth functions without assuming a specific functional form, analogous to how medical imaging reconstructs internal structures from projections without fitting parameters? The paper posits that the inference of PDFs is an indirect measurement problem where the unknowns are functions, not a finite set of parameters.

Methodology
The author constructs a framework that treats the global analysis of PDFs as a coupled system of linear integral equations. The methodology relies on three foundational postulates and two assumptions:

Postulate I: PDFs must be solved simultaneously as a unified tensor of functions.
Postulate II: PDFs are sufficiently smooth and integrable functions.
Postulate III: Perturbative QCD (pQCD) provides a physically accurate description of the observables.
Assumption (i): Non-perturbative degrees of freedom (masses, coupling constants, nuclear modifications) are known.
Assumption (ii): Quarks are treated as massless for the primary derivation, with heavy quark effects handled via specific schemes.

The technical approach involves the following steps:

Linearization: The author demonstrates that key DIS observables (reduced cross sections, structure functions, and sum rules) depend linearly on the PDFs. These observables are formalized as linear integral operators acting on a "parton distribution tensor" $f$ , which includes all quark flavors ( $u, d, s, c, b, t$ and their antiquarks) and the gluon.
Forward Operator Construction: The standard pQCD expressions for observables are rewritten as "forward operators." This involves a "second factorization" where the PDFs are separated from the hadronic tensor. The Wilson coefficient convolutions, which relate PDFs to observables, are identified as linear integral operators.
Coupled System Formulation: The problem is cast as a matrix-valued integral equation:
$\mathbf{O}_{WD} = \mathcal{G}_{lp \to l'X} f$
where $\mathbf{O}_{WD}$ represents the world data (cross sections, structure functions, sum rules) and $\mathcal{G}_{lp \to l'X}$ is the global data forward operator.
Order of Accuracy: The framework is explicitly developed at Leading Order (LO) and Next-to-Leading Order (NLO) in pQCD. At LO, the operators are primarily functional identity maps. At NLO, the operators become true linear integral equations of the first and second kind, involving Wilson coefficient convolutions and plus-distributions.
Constraints: The reconstruction is constrained by:
- Sum Rules: Integral constraints such as momentum conservation and flavor number conservation.
- DGLAP Evolution: The reconstructed PDFs must satisfy the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations, treated as a partial differential equation (PDE) constraint in the optimization problem.
Discretization: To solve the problem numerically, the continuous integral equations are discretized. The author proposes a "stacking" method to reduce the global problem to a finite-dimensional linear matrix-vector system ( $\mathbf{O} = \mathbf{A}\mathbf{f}$ ), solvable via standard regularization techniques (e.g., Tikhonov regularization).

Key Contributions

Formulation as an Inverse Problem: The paper provides the first explicit formulation of the global analysis of PDFs as a linear tensor reconstruction inverse problem, moving away from parameter fitting to functional reconstruction.
NLO Forward Operators: It systematically collects and rewrites the NLO accuracy results for a wide range of DIS processes (inclusive electromagnetic, neutral current, charged current, heavy flavor production, and neutrino DIS) into a unified linear integral operator formalism.
Identification of Mathematical Structure: The author identifies the Wilson coefficient convolutions as closely related to Wiener-Hopf integral equations. This suggests a path toward analytical insights and uniqueness proofs for the solution of the convolution equations.
Proof-of-Principle Methodology: A discretization-based algorithm is constructed to solve the coupled system of integral equations, including the handling of plus-distributions and flavor number schemes (ZF-VFNS).
Uncertainty Quantification: The paper outlines a method for estimating uncertainties using the "replica" approach (randomly sampling data points within experimental errors), similar to the NNPDF collaboration, but applied to the functional reconstruction rather than parameter fitting.

Results

The paper successfully derives the explicit form of the forward operators for all major DIS observables at LO and NLO.
It demonstrates that the global analysis problem can be written as a coupled system of linear integral equations where the unknowns are the PDFs themselves.
A "toy-model" discretized example is provided for inclusive DIS at NLO, showing how the continuous problem reduces to a linear algebra problem ( $\mathbf{O} = \mathbf{A}\mathbf{f}$ ).
The analysis suggests that with sufficient data (e.g., from HERA, NuTeV, etc.) and the inclusion of diverse observables (NC, CC, neutrino), the forward operator matrix can achieve full rank, theoretically allowing for a unique solution to the PDFs without model bias.
The computational complexity is estimated to be manageable (on the order of gigabytes for the operator matrix), comparable to existing global analysis efforts, provided the forward operators are pre-computed.

Significance and Claims
The paper claims to offer a "model-bias-free" alternative to traditional global analysis. By treating PDFs as unknown functions to be solved from data rather than parameters to be fitted, the methodology aims to:

Reveal physical features of the proton structure that might be missed by restrictive functional ansatzes.
Provide a rigorous mathematical framework for the existence and uniqueness of PDF solutions.
Align the methodology of high-energy physics with established inverse problem techniques used in fields like medical imaging (e.g., computed tomography).

The author is modest regarding the immediate practical application, noting that this is a "proof-of-principle" methodology. The paper does not claim to have solved the PDFs from real data yet; rather, it establishes the mathematical feasibility of doing so. It posits that future high-fidelity data from experiments like the Electron-Ion Collider (EIC) will be necessary to fully realize this approach. The work serves as a foundational step toward a new paradigm in QCD phenomenology where the inference of non-perturbative quantities is decoupled from specific model parametrizations.

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