The strange and flavor-singlet axial form factors of the nucleon from lattice QCD

This work presents a comprehensive lattice-QCD determination of the flavor-singlet and strange axial form factors of the nucleon using $N_f = 2+1$ CLS gauge configurations with $O(a)$-improved Wilson fermions, providing a complete error budget for the extrapolations with respect to chiral symmetry, the continuum limit, and infinite volume, with a specific focus on the treatment of discontinuous contributions.

The Gist

Imagine the proton not as a solid marble, but as a bustling, chaotic city. In this city, there are three main types of inhabitants: Up-Quarks, Down-Quarks, and Strange-Quarks. These inhabitants are constantly moving, spinning, and interacting with one another, thereby generating the "spin" (or intrinsic angular momentum) that gives the proton its magnetic personality.

For a long time, scientists knew how to count the Up and Down inhabitants, as they are the most common and easiest to locate. The Strange-Quarks, however, are like shy ghosts; they rarely appear on the surface and are incredibly difficult to track. Furthermore, there is a "Singlet" channel, which is like trying to sum up the total spin of all the city's inhabitants, including the invisible background noise.

This article is a report by a team of scientists who used a massive digital simulation (called Lattice-QCD) to finally obtain a clear census of these hidden inhabitants and the total spin of the proton.

Here is a breakdown of their journey, using simple analogies:

1. The Challenge: The "Ghost" Problem

In the digital city, the Up and Down Quarks are like people walking on the street. You can see and count them easily. This is referred to as the "connected" contribution.

The Strange-Quarks, however, are like ghosts that only appear as fleeting shadows in the background. They do not walk on the street; they briefly pop in and out of the city's "vacuum." In physical terms, these are "disconnected contributions."

• The Problem: Since these ghosts are so weak and noisy, it is like trying to hear a whisper in a hurricane. The signal is buried under a mountain of static noise.
• The Article's Solution: The team developed a special "noise-reduction" strategy. Instead of listening directly to the whisper, they used a method called the Summation Method for the clear voices (Up/Down) and a Plateau Fit for the ghosts (Strange). This allowed them to separate the clear signal from the static noise and obtain a reliable count.

2. The Tools: Building a Digital City

To do this, the scientists did not build a real laboratory; they built a digital lattice (a Lattice) representing space and time.

• They created 14 different versions of this city, some with heavy "air" (heavy quarks) and some with light "air" (light quarks), as well as some with a coarse lattice and some with a fine lattice.
• By simulating the city at different scales and then mathematically "zooming out" to achieve the perfect, real-world size (the "continuum limit"), they could ensure that their results were not merely artifacts of their digital lattice.

3. The Discovery: What is the Proton Made Of?

Once they had cleaned up the noise and counted the inhabitants, they found two main things:

A. The Strange Contribution (The Ghosts)
They calculated the "Strange Axial Form Factor." Imagine this as a map showing how strongly the Strange-Quarks contribute to the proton's spin at different distances.

• The Result: The Strange-Quarks do contribute, but it is a small, negative amount. It is like a tiny group of ghosts spinning in the opposite direction of the main crowd, slightly canceling out the total spin.
• The Number: They found that the "charge" (the total contribution) of these strange ghosts is approximately -0.03.

B. The Singlet Contribution (The Total Spin)
This is the big picture: How much of the proton's spin comes from all the quarks (Up, Down, and Strange) combined?

• The Result: They found that the quarks themselves contribute about 35% of the proton's total spin.
• The Analogy: If the total spin of the proton is a cake, the quarks (Up, Down, and Strange) have baked only about one-third of it. The rest of the cake must consist of something else—likely the "glue" (gluons) holding the city together and the orbital motion of the inhabitants running around.

4. Why This Matters (According to the Article)

The article states that this work is crucial because:

• Completing the Puzzle: Previous studies could only clearly see the Up and Down inhabitants. This is the first time a team has successfully counted the Strange ghosts and the total spin together, with a complete "error budget" (a detailed accounting of how confident they are in their numbers).
• Neutrino Experiments: Understanding these hidden spins helps scientists predict how neutrinos (themselves tiny, ghostly particles) bounce off protons. This is critical for upcoming experiments like MicroBooNE and the P2 experiment, which require precise data to understand the universe.
• Dark Matter: Some theories about dark matter rely on knowing exactly how the proton's spin is structured. If the "strange" part is different than expected, it could change how we detect dark matter.

Summary

In short, this article is a masterclass in noise reduction. The scientists built a digital universe, developed clever tricks to filter out the static noise from "ghost" particles, and ultimately produced a clear, high-resolution map of how the Up-, Down-, and Strange-Quarks contribute to the proton's spin. They confirmed that quarks provide about 35% of the spin, with the remainder left for the "glue" and motion, and delivered the first precise map of the elusive Strange-Quark's role in this dance.

Technical Summary

Technical Summary: The Strange and Flavor-Singlet Axial Form Factors of the Nucleon from Lattice QCD

Problem Statement and Motivation
The axial form factor of the nucleon is central to understanding electroweak interactions, neutrino-nucleon scattering, and the intrinsic spin structure of the proton. In the $SU(3)$ flavor basis, the axial nucleon current decomposes into isovector ($u-d$), isoscalar octet ($u+d-2s$), and singlet components ($u+d+s$). While the isovector channel has seen significant progress in lattice QCD, the flavor-singlet and strange channels remain less explored, particularly regarding complete form factor calculations with controlled continuum extrapolations.

The singlet channel presents unique challenges: it involves quark-disconnected contributions that are computationally intensive and suffer from significant statistical noise, and it is subject to the axial anomaly, which introduces gluonic contributions and complicates renormalization. Existing studies of singlet and strange channels often lack controlled continuum limits, and previous lattice determinations of the isoscalar octet channel were only recently completed. The aim of this work is a comprehensive lattice QCD determination of the singlet axial form factor $G_A^{u+d+s}(Q^2)$ and the strange contribution $G_A^s(Q^2)$, thereby completing the flavor decomposition of the nucleon axial structure within a unified framework.

Methodology
The authors utilize a set of fourteen $N_f = 2+1$ CLS (Coordinated Lattice Simulations) gauge ensembles employing $O(a)$-improved Wilson fermions and tree-level-improved Lüscher-Weisz gauge actions. The ensembles span lattice spacings from $0.050$ fm to $0.086$ fm and pion masses from $130$ MeV to $350$ MeV.

• Correlation Functions: The calculation uses nucleon interpolating fields with Gaussian-smeared quark fields and APE-smeared gauge links. Both connected and disconnected three-point correlation functions are computed. The disconnected contributions, crucial for the strange and singlet components, are estimated using stochastic methods with variance reduction.
• Extraction Strategy:
  • Connected Contributions: The summation method is applied to suppress contamination from excited states. A linear fit is performed on the summed expression, followed by a $z$-expansion parametrization for the momentum transfer dependence.
  • Disconnected Contributions: Due to high statistical noise and the absence of excited state contamination within errors, plateau fits are used to extract the effective form factor for various source-sink separations.
  • Combination: The full form factor is constructed by combining the connected and disconnected contributions. Specifically, the strange form factor $G_A^s$ is extracted from the disconnected part, and the singlet form factor $G_A^{u+d+s}$ is constructed by combining the previously determined isoscalar octet result with $3G_A^s$.
• Renormalization: The renormalization of the singlet current is treated as a core component of this work. Mass-independent renormalization factors are computed on $N_f=3$ ensembles in the RI'-MOM scheme and converted to the $\overline{\text{MS}}$ scheme at 2 GeV. The mixing between flavor-singlet and octet currents under renormalization is accounted for, while non-diagonal coefficients induced by $O(a)$ effects are neglected as they are expected to be small.
• Extrapolation: The final results are obtained by extrapolating the $z$-expansion coefficients to the physical pion mass and the continuum limit using an ansatz linear in $M_\pi^2$ and $a^2$. Finite volume effects are included via a specific term in the axial charge. A model-averaging procedure based on the Akaike Information Criterion (AIC) is employed to combine results from different fit models and ensure a robust error budget.

Main Contributions

1. First Complete Error Budget: This work presents the first lattice QCD determination of the singlet axial form factor $G_A^{u+d+s}(Q^2)$ and the strange contribution $G_A^s(Q^2)$ with a complete error budget covering chiral, continuum, and infinite-volume limits.
2. Unified Framework: The study completes the flavor decomposition of the nucleon axial structure (isovector, octet, and singlet) within a single computational framework using the same gauge ensembles and analysis strategies.
3. Renormalization of the Singlet Current: The authors provide a dedicated calculation of the renormalization factors for the singlet axial current and address the complexities introduced by the axial anomaly and disconnected loops.
4. Treatment of Disconnected Contributions: The article describes a specific strategy for handling the noisy disconnected contributions, using plateau fits and window averaging to extract reliable signals for the strange and singlet components.

Results
The authors report the coefficients of the $z$-expansion for the strange and singlet form factors in the continuum limit at the physical pion mass.

• Strange Axial Charge: The strange axial charge is determined to be $g_A^s = -0.0325(78)$. This result is consistent with recent determinations by other collaborations (e.g., $\chi$QCD, PNDME, ETM).
• Singlet Axial Charge: The singlet axial charge is found to be $g_A^{u+d+s} = 0.355(43)$ in the $\overline{\text{MS}}$ scheme at 2 GeV.
• Momentum Dependence: The momentum dependence of both form factors is shown up to $Q^2 \approx 1$ GeV$^2$. The strange form factor proves to be small and negative, while the singlet form factor is positive and significantly larger, driven by the connected contributions.

Significance
The authors claim that their results provide a comprehensive lattice QCD determination of the nucleon axial structure across various flavor channels. The value $g_A^{u+d+s} \approx 0.355$ suggests that the intrinsic quark spin contributes approximately 35% to the proton spin, a figure consistent with historical deep inelastic scattering results (e.g., EMC) and recent COMPASS data, supplemented by octet estimates. The work highlights that the remaining portion of the proton spin must be provided by contributions from gluons and orbital angular momentum. By completing the flavor decomposition, this study provides essential input for understanding the proton spin problem and for interpreting data from current and future neutrino experiments (such as DUNE and Hyper-Kamiokande) as well as dark matter searches, where the strange and singlet axial form factors are critical inputs. The authors note that while their result is consistent with experimental constraints, the determination of the small strange contribution via $SU(3)$ symmetry breaking from hyperon decays remains too imprecise, underscoring the necessity of direct lattice calculations.