⚛️ phenomenology

Analytical solution for QCD $\otimes$ QED evolution

This paper presents an analytical solution for the DGLAP evolution of both polarized and unpolarized parton distributions incorporating mixed-order QCD and QED corrections, offering improved computational efficiency and precision for theoretical predictions in phenomenological applications.

Original authors: Daniel de Florian, Lucas Palma Conte

Published 2026-02-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniel de Florian, Lucas Palma Conte

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 trying to predict exactly how a complex machine, like a proton (a tiny building block of matter), behaves when you smash it together at high speeds. To do this, physicists use a set of rules called Parton Distribution Functions (PDFs). Think of these PDFs as a "recipe" or a "map" that tells you how much of the proton is made of quarks, gluons, and even photons (particles of light) at any given moment.

However, this map isn't static. As you change the energy of the collision (the "zoom level" of your microscope), the recipe changes. The particles inside the proton interact, split, and merge. This changing map is governed by a complex set of mathematical rules known as the DGLAP equations.

For decades, physicists have been incredibly good at calculating these changes using the rules of QCD (the theory of the strong nuclear force, which holds the proton together). They have reached a level of precision where they can predict things to the third or fourth decimal place.

The Problem: The "Ghost" in the Machine
Recently, as our predictions have become so precise, a tiny, previously ignored factor has started to matter: QED (Quantum Electrodynamics, the theory of light and electricity).

Think of it like this: For years, you were baking a cake using a perfect recipe for flour and sugar (QCD). But now, you realize that the humidity in the room (QED) is actually affecting how the cake rises. If you ignore the humidity, your cake might look okay, but if you want to win a gold-medal baking competition, you need to account for that tiny bit of moisture.

In the proton, this "humidity" is the electromagnetic interaction. It's weak compared to the strong force, but at the high levels of precision we are aiming for today, it creates a "mixed" effect. The strong force and the electromagnetic force are interacting in a way that previous recipes didn't fully capture.

The Solution: A New, Smarter Calculator
The authors of this paper, Daniel de Florian and Lucas Palma Conte, have created a new, analytical solution to update these recipes.

Here is how they did it, using some creative analogies:

The "Abelianization" Algorithm (The Magic Translator):
They needed to figure out how the strong force and light force mix. Instead of starting from scratch, they used a clever trick called "Abelianization." Imagine you have a complex, tangled knot of yarn (the QCD rules). You know exactly how to untangle it. This algorithm acts like a translator that says, "Okay, if we take this tangled knot and apply a simple rule of light to it, here is exactly how the new knot looks." This allowed them to quickly generate the new "mixed-order" rules without doing millions of hours of new calculations.
Two Ways to Solve the Puzzle:
To update the map (the PDFs) using these new rules, they proposed two different methods:
- Method 1: The U-Matrix (The Modular Lego Set): This method takes the existing, well-known blocks for QCD and QED and snaps a new, special "mixed" block in between them. It's efficient because it reuses the old, trusted parts of the machine.
- Method 2: The Magnus Expansion (The Swiss Army Knife): This is a more mathematical, "all-in-one" approach that tries to solve the whole equation in one giant, elegant formula. It's beautiful and precise, but like a Swiss Army knife with 50 tools, it's a bit harder to carry around and use for everyday tasks.
They found that both methods gave almost the same result, but the "Lego" method (U-Matrix) is easier for other scientists to use in their own work.
The Result: A Sharper Lens:
When they applied these new rules to the "recipe" for the proton, they found something interesting:
- For most particles, the change was tiny (like a whisper).
- However, for the photon (the particle of light) inside the proton, the change was significant—up to 10% in some cases!
- This is huge because if you are trying to predict a process where light is created (like in electron-proton collisions), ignoring this "mixed" effect would be like trying to navigate a ship without accounting for the wind.

Why Does This Matter?
This paper is like upgrading the GPS in a self-driving car. For a long time, the GPS was good enough to get you to the grocery store. But now, we are trying to drive to the moon. We need to account for every tiny gravitational pull and atmospheric shift.

By providing an exact, analytical solution for how these mixed forces work, the authors have:

Speeded up calculations: Computers can now run these simulations much faster.
Increased precision: Future experiments at particle colliders (like the Large Hadron Collider) will be able to test theories with much higher accuracy.
Clarified the "Photon" role: They showed that the photon inside a proton is more important than we thought, which could change how we interpret future data.

In short, they took a messy, complex problem of two forces fighting and mixing, and they wrote down a clean, exact formula that allows scientists to see the proton with a much sharper, more accurate lens.

1. Problem Statement

The precision of theoretical calculations for partonic cross-sections has reached the Next-to-Next-to-Next-to-Leading Order (N3LO) in Quantum Chromodynamics (QCD). However, to match this precision, previously neglected effects, specifically Quantum Electrodynamics (QED) corrections and their mixed-order QCD ⊗ QED contributions, must be incorporated into Parton Distribution Functions (PDFs) and polarized PDFs (pPDFs).

The core challenges addressed in this work are:

Lack of Analytical Solutions: While numerical solutions for DGLAP evolution with QED corrections exist, fully analytical solutions for the mixed-order $O(\alpha_S \alpha)$ evolution are scarce, particularly for the polarized case.
Computational Efficiency: Numerical integration of evolution equations is computationally expensive, especially for global fits requiring repeated evaluations.
Photon PDF Uncertainty: The photon PDF is a crucial component in QED-corrected analyses but relies heavily on phenomenological models. A rigorous analytical framework is needed to reduce these uncertainties.
Polarized Observables: There is a need to understand how mixed-order corrections affect polarized structure functions (specifically $g_1$ ) and the associated Wilson coefficients.

2. Methodology

The authors employ a rigorous analytical approach working in Mellin $N$ -space, where convolutions in $x$ -space become simple products.

A. Theoretical Framework

Splitting Kernels: The authors compute the Altarelli-Parisi (AP) splitting functions at mixed-order $O(\alpha_S \alpha)$ $O (α_{S} α)$ . They utilize the "Abelianization" algorithm (previously developed for unpolarized cases) to derive polarized kernels from known NLO QCD results.
- This includes calculating photon-initiated processes ( $\gamma \to q, g$ ) and quark/gluon initiated processes with photon emission.
- The treatment of $\gamma_5$ follows the HVBM scheme, and calculations are performed in the $\overline{\text{MS}}$ scheme.
Basis Decomposition: The evolution equations are decomposed into Non-Singlet (NS) and Singlet (S) sectors.
- NS Sector: Decoupled equations for valence quarks and specific flavor combinations ( $\Delta_{uc}, \Delta_{ds}$ , etc.).
- Singlet Sector: A coupled matrix system involving $\Delta_{UD}$ , $\Sigma$ (flavor singlet), gluon ( $g$ ), and photon ( $\gamma$ ) distributions.

B. Solving the DGLAP Equations

The paper proposes and compares two distinct analytical methods to solve the evolution equations in Mellin space:

Extended U-Matrix Method:
- An extension of the conventional U-matrix approach.
- The total evolution operator is factorized as $E = E_{\text{QCD}} \cdot E_{\text{QED}} \cdot E_{\text{MIX}}$ .
- $E_{\text{MIX}}$ is a new operator accounting for mixed-order corrections and coupling evolution.
- To handle the non-commutativity of the QCD and QED kernel matrices, the method introduces a series of correction operators ( $A_i$ ) derived iteratively. The dominant contribution is resummed to provide a practical solution.
- Advantage: Allows for the reuse of existing, well-tested QCD and QED evolution codes.
Magnus Expansion Method:
- Based on the Magnus expansion for linear ordinary differential equations, providing a solution in a closed exponential form: $E(t) = \exp(\Omega(t))$ .
- $\Omega(t)$ is expanded in terms of integrals of the kernel matrix and its commutators.
- The authors compute terms up to the necessary order, including the double integral of the commutator $[P(1,0), P(0,1)]$ .
- Disadvantage: While elegant, higher-order terms become analytically intractable, and it does not easily reuse legacy QCD codes.

C. Coupling Evolution

The authors solve the Renormalization Group Equations (RGE) for the strong coupling ( $\alpha_S$ ) and electroweak coupling ( $\alpha$ ) at mixed-order. They propose two approaches to integrate the mixed coupling term $\int \alpha_S(t) \alpha(t) dt$ , demonstrating that using Leading Order (LO) approximations for the couplings within the integral is sufficient for the desired accuracy.

D. Structure Function $g_1$

The polarized structure function $g_1$ is computed including mixed-order Wilson coefficients ( $\Delta C^{(1,1)}$ ).
The "Abelianization" technique is applied to known NNLO QCD Wilson coefficients to derive the mixed-order terms.
The authors separate the impact of the mixed-order PDF evolution from the Wilson coefficients.

3. Key Contributions

First Analytical Polarized Solution: Provides the first exact analytical solution for the DGLAP equations including mixed-order QCD $\otimes$ QED corrections for polarized parton distributions.
Explicit Kernel Derivation: Derives explicit expressions for the mixed-order polarized splitting functions ( $\Delta P^{(1,1)}$ ) for all relevant channels ( $q\gamma, g\gamma, \gamma\gamma, qg, \gamma g, gg, qg$ ).
Methodological Comparison: Systematically compares the U-matrix and Magnus expansion methods. They find that both yield consistent results for mixed-order corrections, with differences arising mainly from the treatment of higher-order pure QCD terms.
Wilson Coefficients: Calculates the mixed-order Wilson coefficients for the polarized structure function $g_1$ , separating quark, gluon, and photon contributions.

4. Results

Coupling Evolution: The mixed-order corrections to the running couplings are found to be of order $O(10^{-4})$ , confirming that higher-order terms beyond the mixed order are negligible for current phenomenological needs.
PDF Corrections:
- For most polarized PDFs (valence, sea, gluon), the relative mixed-order corrections are small, approximately $O(10^{-4})$ .
- Photon PDF: The corrections are significantly larger, reaching the percent level ( $\sim 1\%$ ) and up to $\sim 12\%$ at high $x$ . This highlights the critical importance of mixed-order evolution for processes sensitive to the photon PDF (e.g., prompt photon production).
- Both the U-matrix and Magnus methods predict identical corrections for non-singlet distributions and very similar results for singlet distributions.
Structure Function $g_1$ :
- The relative corrections to $g_1$ are $O(10^{-4})$ at $x \approx 0.1$ and grow to $O(10^{-3})$ as $x \to 1$ .
- The dominant contribution to the correction comes from the mixed-order Wilson coefficients (specifically the quark channel), rather than the PDF evolution itself, except at very low $x$ .
- Including mixed-order corrections reduces the theoretical scale uncertainty of the cross-section, though the uncertainty remains larger than the correction itself.

5. Significance

Phenomenological Precision: This work provides the necessary theoretical tools to include QED and mixed-order effects in global PDF fits with the same precision as modern QCD-only fits. This is essential for future high-luminosity colliders (e.g., EIC, LHeC, FCC).
Computational Efficiency: The analytical nature of the solution (especially the U-matrix approach) allows for rapid evaluation of PDFs, making it feasible to include these corrections in complex global fitting frameworks where numerical integration would be too slow.
Photon PDF Constraints: By providing a rigorous framework for the photon PDF evolution, the paper supports the LuxQED approach and reduces reliance on phenomenological models for the photon density.
Benchmarking: The comparison between the U-matrix and Magnus methods serves as a valuable benchmark for future developments in solving evolution equations, validating the Magnus expansion as a viable alternative for mixed-order problems despite its complexity.

In conclusion, this paper bridges a critical gap in high-precision QCD phenomenology by delivering a complete, analytical, and efficient framework for the mixed-order evolution of both polarized and unpolarized parton distributions.