CJ26 Global QCD Analysis with Large-$x$ Jefferson Lab 6 and 12 GeV Data

The CJ26 global QCD analysis presents a new set of NLO parton distribution functions by incorporating the complete suite of JLab 6 GeV and the first published 12 GeV data to uniquely disentangle higher-twist effects from off-shell nucleon corrections, thereby significantly reducing uncertainties in the large-$x$ $n/p$ structure function and $d/u$ valence quark ratios.

The Gist

Imagine the atom's nucleus as a bustling city, and the protons and neutrons inside it as the buildings. Inside those buildings live tiny, energetic workers called quarks. To understand how this city works, physicists need a map showing exactly where these workers are and how fast they are moving. This map is called a Parton Distribution Function (PDF).

For a long time, this map was very blurry at the "edge of town" (where quarks carry almost all the energy). This paper, CJ26, is like a team of cartographers who just finished a massive renovation of that map, specifically focusing on that blurry edge.

Here is how they did it, using simple analogies:

1. The New High-Resolution Camera (JLab Data)

Previously, the team had some old, grainy photos of the city edge. In this study, they added thousands of brand-new, ultra-high-definition photos taken by the Jefferson Lab (JLab).

• The 6 GeV and 12 GeV runs: Think of these as two different cameras. The 6 GeV camera took great pictures of the "middle" of the edge, while the new 12 GeV camera is powerful enough to see the very farthest, most distant corners of the city that were previously invisible.
• The Result: By combining these new photos with older ones, they created a map that is 30% to 50% more precise in those previously blurry areas.

2. Untangling a Messy Knot (The "Large-x" Problem)

In the physics world, "large-x" means a quark is carrying a huge chunk of the proton's energy. When you look at these high-energy quarks, the data gets messy because of two things happening at once:

• The "Off-Shell" Effect: Imagine a worker (a quark) inside a building (a proton) that is slightly squished because it's part of a larger structure (a nucleus). This squishing changes how the worker moves.
• The "Higher-Twist" Effect: Imagine the workers bumping into each other or the walls, creating extra noise and friction that isn't part of their normal movement.

In the past, it was hard to tell if a weird signal on the map was caused by the squished building or the bumping workers. They were tangled together like a knot.

• The Breakthrough: The new 12 GeV data acts like a magnifying glass. Because it looks at the data with more "leverage" (higher energy), the team could finally untangle the knot. They could separate the "squishing" effect from the "bumping" effect, allowing them to draw the map of the workers much more accurately.

3. Fixing the Deuteron Puzzle

To see the "down" quarks clearly, the team looked at deuterium (a nucleus made of one proton and one neutron). But looking at a pair is tricky because the two particles are holding hands and moving together.

• The Analogy: If you try to measure how fast one person in a dance pair is moving, you have to account for the fact that they are spinning around each other.
• The Fix: The paper introduces a new way to calculate this "dance." They found that by carefully accounting for how the two particles are bound together, they could determine the ratio of "down" to "up" quarks with much higher confidence.

4. The Importance of "Correlated Errors" (The Team Huddle)

When scientists take measurements, there are always small mistakes (uncertainties). Sometimes, these mistakes happen together across many measurements (like if a ruler was slightly bent, all measurements using that ruler would be off by the same amount).

• The Innovation: The team realized that for the new Jefferson Lab photos, these "bent ruler" errors were known and could be corrected. By treating these errors as a team huddle (correlated) rather than random noise, they improved the reliability of the whole map. They found that ignoring this "huddle" would have made the map look much less certain than it actually is.

5. The Final Map (The Results)

The result is the CJ26 map.

• What it shows: It gives a much clearer picture of how the "down" quarks behave compared to the "up" quarks at the very edge of the energy spectrum.
• Why it matters: This map is now the standard reference for anyone trying to understand the fundamental structure of matter. It helps other scientists predict what will happen in giant particle smashers (like the Large Hadron Collider) with greater accuracy.
• The "Tail" of the Map: The team found that the "tail" of the map (the very edge where quarks have almost all the energy) behaves differently than some older maps suggested. It's not as flat as some thought; it has a specific shape that depends on the complex interactions inside the nucleus.

Summary

Think of this paper as the release of a new, GPS-enabled atlas for the subatomic world. By using the best new cameras (JLab 12 GeV), learning how to untangle the traffic jams (separating off-shell and higher-twist effects), and correcting for the fact that the mapmakers sometimes made the same mistake twice (correlated errors), the team has produced the most accurate guide yet for the "edge of the universe" inside a proton.

Technical Summary

Technical Summary of CJ26: A Global QCD Analysis with Large-x Jefferson Lab 6 and 12 GeV Data

Problem Statement
Parton Distribution Functions (PDFs) are essential for connecting high-energy collision observables to perturbative QCD calculations. However, determining PDFs in the large momentum fraction region ($x \to 1$) remains challenging due to the rapid power-law suppression of parton densities and the scarcity of high-precision data in this kinematic regime. Furthermore, global analyses in this region must contend with significant non-perturbative effects, including target mass corrections (TMCs), higher-twist (HT) contributions, and nuclear binding effects when using deuteron targets. Previous analyses often lacked the kinematic leverage to disentangle these effects or relied on parametrizations that introduced theoretical biases. A specific tension exists between different global fits regarding the $d/u$ valence quark ratio and the $\bar{d}/\bar{u}$ sea quark asymmetry at large $x$, partly due to differing treatments of nuclear corrections and kinematic cuts.

Methodology
The paper presents CJ26, a new global QCD analysis at Next-to-Leading Order (NLO) that incorporates the complete suite of Jefferson Lab (JLab) 6 GeV and the first published JLab 12 GeV deep inelastic scattering (DIS) measurements. The analysis utilizes a dataset exceeding 5,000 points, including inclusive DIS on proton and deuteron targets, tagged DIS (BONuS), Drell-Yan processes, and vector boson production.

Key methodological components include:

• Formalism: The analysis employs the ACOT-$\chi$ heavy-quark scheme and includes TMCs calculated via the Georgi-Politzer momentum space approximation.
• Higher-Twist and Off-Shell Corrections: To address the interplay between power corrections and nuclear effects, the authors implement two distinct parametrizations for residual higher-twist terms: an additive form and a multiplicative form. These are fitted independently for protons and neutrons to allow for isospin dependence. The off-shell deformation of bound nucleons in the deuteron is modeled via a Taylor expansion of the nucleon structure function, parameterized by a flavor-independent off-shell function $\delta f(x)$.
• Parametrization: The PDF parametrization at the initial scale $Q_0^2 = 1.69 \text{ GeV}^2$ uses a standard five-parameter functional form. A specific modification allows the $d/u$ ratio to approach a finite limit as $x \to 1$ by mixing the $u_v$ distribution into the $d_v$ parametrization. The $\bar{d}-\bar{u}$ distribution is reparameterized to decouple normalization from shape parameters, stabilizing the uncertainty analysis.
• Uncertainty Treatment: The analysis explicitly incorporates correlated experimental systematic uncertainties where available (notably for SLAC and JLab data). Fit uncertainties are determined using the Hessian method with a local tolerance variant ($T^2 = 2.71$). The final uncertainty bands for extracted quantities are defined as the envelope of the results from the additive and multiplicative HT fits, treating the difference as an estimate of theoretical systematic uncertainty.

Key Contributions and Results

• Integration of JLab Data: For the first time in a global QCD analysis, the complete set of JLab 6 GeV and 12 GeV DIS data is included. The 12 GeV data provide a crucial increase in $Q^2$ leverage, particularly in the region $0.7 \lesssim x \lesssim 0.8$, allowing for a more effective separation of leading-twist dynamics from power corrections.
• Precision Improvements: The inclusion of JLab data and the improved treatment of correlated systematics lead to significant reductions in uncertainties. The uncertainty on the $n/p$ structure function ratio is reduced by 30–50%, and the uncertainty on the $d/u$ valence quark ratio is reduced by 5–10% in the large-$x$ region.
• Extraction of Physical Quantities:
  • $d/u$ Ratio: The CJ26 fit yields a $d/u$ ratio that is stable with respect to previous CJ analyses but with reduced uncertainty. The ratio approaches a finite limit as $x \to 1$, consistent with the specific parametrization choice.
  • Off-Shell Function ($\delta f$): The analysis extracts a non-zero off-shell function that is negative around $x \approx 0.4$ and becomes positive at larger $x$. The shape is robust up to $x \approx 0.7$, constrained by the new data, though the behavior at $x > 0.75$ remains an extrapolation.
  • Higher-Twist Terms: The analysis determines isospin-dependent higher-twist functions for protons and neutrons. The results show that the neutron HT correction is enhanced while the proton HT correction is reduced at large $x$, leading to a more pronounced tail in the $n/p$ ratio compared to previous fits.
• Systematic Uncertainty Analysis: The paper demonstrates that neglecting correlated systematic uncertainties leads to a misrepresentation of the pull distributions and an overestimation of experimental errors. Including these correlations is shown to be essential for the consistency of the global fit, particularly for SLAC and JLab 12 GeV data.
• Comparison with Other Fits: CJ26 results are compared with modern global analyses (CT18, NNPDF4.0, MSHT20, ABMP16). While generally consistent at intermediate $x$, CJ26 shows distinct behavior at large $x$ due to the inclusion of fixed-target data and explicit nuclear/power corrections. The $d/u$ ratio from CJ26 is compatible with JAM26 and PDF4LHC21 but differs from CT18 and ABMP16 in the very large-$x$ region, highlighting the sensitivity of this region to the treatment of nuclear corrections and parametrization flexibility.

Significance
The paper claims that CJ26 provides one of the most accurate and unbiased determinations of the $d/u$ ratio and the $n/p$ structure function ratio at large $x$ currently available. By simultaneously fitting higher-twist and nuclear corrections while propagating their associated systematic uncertainties, the analysis minimizes theoretical bias. The work highlights the critical role of experimental correlated systematic uncertainties in achieving high precision and validates the robustness of the CJ framework. Furthermore, the resulting PDFs and structure functions, provided in LHAPDF format, serve as a solid baseline for precision measurements of parity-violating processes at JLab and for background estimates in searches for physics beyond the Standard Model at the LHC, particularly where high-mass and large-rapidity observables probe the large-$x$ region. The authors emphasize that future data from dedicated large-$x$ measurements (e.g., BONuS12, JLab 22 GeV, EIC) will be necessary to further constrain the off-shell function and higher-twist effects in the extrapolation region.