⚛️ phenomenology

Non-singlet coefficient functions for charged-current deep-inelastic scattering to the third order in QCD

This paper presents compact approximate expressions for the third-order QCD coefficient functions of the structure functions $F_2$ , $F_L$ , and $F_3$ in charged-current deep-inelastic scattering, thereby completing the description of unpolarized inclusive $W^{\pm}$ exchange processes and providing a unified collection of lower-order contributions for phenomenological analyses.

Original authors: J. Davies (Liverpool U., Dept. Math.), A. Vogt (Liverpool U., Dept. Math.), S. Moch (Hamburg U., Inst. Theor. Phys. II), J. A. M. Vermaseren (Nikhef, Amsterdam)

Published 2026-06-30

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Original authors: J. Davies (Liverpool U., Dept. Math.), A. Vogt (Liverpool U., Dept. Math.), S. Moch (Hamburg U., Inst. Theor. Phys. II), J. A. M. Vermaseren (Nikhef, Amsterdam)

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 the universe is built out of tiny, invisible Lego bricks called quarks. These bricks are glued together inside protons and neutrons (the building blocks of atoms) by a super-strong "glue" known as the strong force.

To figure out exactly how these bricks are arranged and how they move, scientists smash high-energy particles (like neutrinos) into protons. This is called Deep-Inelastic Scattering (DIS). It's like throwing a tennis ball at a cloud of dust to see how the dust swirls; by watching how the ball bounces off, you can map out the dust cloud.

In this specific experiment, the scientists are using neutrinos (ghost-like particles) that interact via the weak force (specifically, by swapping a particle called a $W$ boson). This is the "charged-current" interaction.

The Problem: The "Recipe" is Too Complicated

When scientists analyze the data from these collisions, they need a mathematical "recipe" to translate what they see into a picture of the quarks inside the proton. This recipe is called a coefficient function.

For a long time, scientists had the recipe for the first two steps of the calculation (the "first-order" and "second-order" approximations). But nature is messy. To get a truly precise picture, you need to calculate the third step (the "third-order" correction). Until now, this third step for neutrino collisions was missing a crucial piece of the puzzle.

The Solution: A New, Ultra-Precise Recipe

The authors of this paper (Davies, Moch, Vermaseren, and Vogt) have finally calculated these missing third-order coefficient functions.

Think of it like this:

The Old Recipe: Good enough to bake a decent cake, but the frosting might be a little lumpy.
The New Recipe: Now they have the exact measurements to make the frosting perfectly smooth. They have calculated the math for three different types of "flavors" of data ( $F_2$ , $F_L$ , and $F_3$ ) that come out of these neutrino collisions.

The "Approximate" Shortcut

The exact math for this third step is so incredibly complex that it would fill thousands of pages of equations. It's like trying to describe a mountain range by listing the height of every single grain of sand.

To make this useful for other scientists, the authors created compact approximate expressions.

The Analogy: Imagine you need to describe a mountain to a pilot. Instead of listing the height of every rock, you give them a smooth, curved line that is 99.9% accurate.
The Result: The authors' new formulas are like that smooth line. They are accurate enough for real-world experiments (phenomenological analyses) but much easier to use than the raw, messy math. They are accurate to within 0.1% for most of the data, and even at the very edges of the data (where the numbers get tiny), they are still within 1% to 3% accuracy.

Why Does This Matter?

The paper mentions two main reasons why this "perfect recipe" is important:

Understanding the Proton: It helps us understand the internal structure of protons and nuclei with unprecedented precision.
Measuring the "Weak Mixing Angle": The paper specifically notes that these calculations are needed to determine a fundamental number in physics called $\sin^2 \theta_W$ (the weak mixing angle). This is a parameter that defines how the weak force behaves. By using their new, precise formulas, scientists can measure this angle more accurately using data from neutrino experiments.

The "Small X" Surprise

The paper also looked at what happens when the particles involved have very low momentum (a region called "small $x$ ").

The Metaphor: Imagine a crowd of people. Usually, if you look at the whole crowd, the behavior is predictable. But if you zoom in on a tiny, specific corner of the crowd, things might get chaotic.
The Finding: The authors found that at these very low levels, the math gets very "spiky" and rises sharply. They discovered that to predict this behavior correctly, you need to include specific "logarithmic" terms in your math (think of these as special ingredients that only appear when the crowd gets very dense). Without these specific ingredients, the recipe fails in that corner.

Summary

In short, this paper is a mathematical upgrade. The authors have finished the calculation for the "third layer" of complexity in how neutrinos interact with protons. They turned a messy, impossible-to-read equation into a clean, usable tool that allows physicists to measure the fundamental properties of our universe with much greater precision. They didn't invent a new machine or discover a new particle; they simply perfected the calculator used to interpret the data from existing machines.

Technical Summary: Non-singlet coefficient functions for charged-current deep-inelastic scattering to the third order in QCD

Problem Statement
This work addresses the calculation of the non-singlet coefficient functions for the structure functions $F_2$ , $F_L$ , and $F_3$ in $\nu-\bar{\nu}$ charged-current deep-inelastic scattering (DIS) within massless perturbative Quantum Chromodynamics (QCD). While the inclusive charged-current DIS process is a primary source for determining nucleon parton structure and Standard Model parameters (such as the strong coupling $\alpha_s$ and the weak mixing angle $\sin^2 \theta_W$ ), the theoretical description at the third order (N $^3$ LO) in the strong coupling constant was incomplete. Specifically, the third-order contributions for the $\nu-\bar{\nu}$ combinations had not been fully established in a form suitable for phenomenological analysis.

Methodology
The authors calculate the coefficient functions in the $\overline{\text{MS}}$ scheme with the standard choice of renormalization and factorization scales $\mu_F = \mu_R = Q$ . The perturbative expansion is organized in powers of $a_s \equiv \alpha_s(Q^2)/(4\pi)$ .

The methodology involves two distinct approaches for different orders and flavor combinations:

Exact and Parametrized Results: For the $\nu+\bar{\nu}$ combinations, the authors present results that are "approximate but sufficiently accurate" for phenomenology. These expressions are derived by fitting coefficients to exact results at $x \ge 10^{-6}$ .
Difference Calculations: For the $\nu-\bar{\nu}$ combinations, the authors utilize the observation that the difference between $\nu+\bar{\nu}$ and $\nu-\bar{\nu}$ coefficient functions is suppressed by two powers of $(1-x)$ at large $x$ . Consequently, they calculate the differences $\delta C_a$ (defined as $C_{\nu p + \bar{\nu} p} - C_{\nu p - \bar{\nu} p}$ ) rather than the full functions.
Flavor Treatment: The calculations distinguish between the flavor non-singlet contributions. Notably, the flavor structure $fl_{02}$ (which appears at three loops) does not contribute to the flavor asymmetries probed in the $\nu-\bar{\nu}$ combinations and is therefore excluded from the difference calculations for $F_3$ .
Validation: The accuracy of the parametrized expressions is verified against typical quark distributions, with deviations generally kept below 0.1% for $x \ge 10^{-6}$ . Accuracy degrades at very small $x$ but remains within 1% at $x=10^{-8}$ and 3% at $x=10^{-10}$ .

Key Contributions and Results
The primary contribution of this paper is the completion of the N $^3$ LO description of unpolarized inclusive $W^\pm$ -exchange DIS by providing the third-order non-singlet coefficient functions for both $\nu+\bar{\nu}$ and $\nu-\bar{\nu}$ cases.

$\nu+\bar{\nu}$ Coefficient Functions: The paper provides compact, approximate expressions for the third-order contributions to $c^{(3)}_{2,+}$ , $c^{(3)}_{L,+}$ , and $c^{(3)}_{3,+}$ . These are presented in terms of logarithmic variables ( $L_0 = \ln x$ , $L_1 = \ln(1-x)$ ) and plus-distributions ( $D_k$ ). The expressions include terms dependent on the number of light flavors ( $n_f$ ) and $n_f^2$ .
$\nu-\bar{\nu}$ Coefficient Functions: The authors present new third-order results for the differences $\delta c^{(3)}_2$ , $\delta c^{(3)}_L$ , and $\delta c^{(3)}_3$ . These results supersede previous approximate expressions based on Mellin moments. The new parametrizations are valid for the full range of $x$ relevant to phenomenology.
Small- $x$ Behavior: The analysis of the third-order functions reveals a sharp rise at small $x$ . For $F_2$ and $F_3$ , this rise becomes significant below $x < 10^{-5}$ , while for $F_L$ , it occurs below $x \lesssim 10^{-4}$ . The authors note that the $fl_{02}$ flavor structure, which dominates $c^{(3)}_{a,+}$ at small $x$ , is absent in the $\nu-\bar{\nu}$ differences.
Mellin Moments: The paper provides truncated numerical values for the second moments of $\delta c^{(3)}_2$ and $\delta c^{(3)}_L$ for $n_f$ light flavors. These moments are critical for QCD corrections to the Paschos-Wolfenstein relation used in determining $\sin^2 \theta_W$ .

Significance and Claims
The authors state that these results "complete the description of unpolarized inclusive $W^\pm$ -exchange DIS to this order of massless perturbative QCD." The paper emphasizes that the provided expressions are "sufficiently accurate for phenomenological analyses," allowing for the inclusion of N $^3$ LO corrections in global fits and precision measurements.

The work is modest in its scope, focusing strictly on the massless perturbative QCD calculation of coefficient functions. It does not propose new experimental facilities or applications beyond the stated utility for future facilities like the LHeC and for high-energy cosmic neutrino scattering. The authors note that while the exact analytical expressions for the third-order terms are available in other references (specifically Ref. [25]), the compact parametrizations provided here are designed for immediate practical use in data analysis. The availability of Fortran subroutines via the arXiv source is highlighted as a resource for the community.