Proton's isovector PDF with updated analysis of large-momentum lattice data

Technical Summary: Proton's Isovector PDF with Updated Analysis of Large-Momentum Lattice Data

Problem Statement
The proton's unpolarized isovector parton distribution function (PDF), $u(x) - d(x)$, is a well-constrained quantity from decades of high-energy experimental data. However, recent first-principles calculations using Lattice Quantum Chromodynamics (QCD) within the Large Momentum Effective Theory (LaMET) framework have yielded results that exhibit significant variations and often deviate from phenomenological determinations. These discrepancies are attributed to two primary sources:

1. Lattice Artifacts: Including excited-state contaminations, unphysical pion masses, and discretization effects. These are particularly severe in PDF calculations compared to first-moment calculations due to smaller energy gaps at large momenta and suppressed signal-to-noise ratios.
2. Theoretical Limitations: Previous calculations often predated recent theoretical advancements in large-momentum expansion, lacking sophisticated treatments such as hybrid renormalization, large-distance extrapolations, and higher-order matching (NNLO) with log resummations.

Methodology
The authors reanalyze four existing lattice QCD datasets (MSULat20, ETMC20, LPC18, and CLQCD24) covering various lattice spacings, pion masses, and hadron momenta ($P_z \sim 1.85 - 2.27$ GeV). The analysis employs a state-of-the-art theoretical framework to mitigate systematic errors:

• Renormalization: The authors utilize hybrid renormalization to handle ultraviolet (UV) divergences. This scheme combines a ratio method for short distances ($|z| < z_s$) to remove discretization effects and an MS-scheme-like subtraction for long distances ($|z| > z_s$), avoiding uncontrolled infrared effects.
• Self-Renormalization and Matching: The conversion factor to the $\overline{\text{MS}}$ scheme is determined via self-renormalization, fitting the lattice data to perturbative coefficients calculated at Next-to-Next-to-Leading Order (NNLO). This is supplemented by DGLAP log resummation (RGR), leading renormalon resummation (LRR), and threshold log resummation (TR) to ensure perturbative convergence and handle renormalon ambiguities.
• Mitigation of Lattice Artifacts: To address distortions in the global behavior of the PDF (specifically the first moment $\langle x \rangle$), the authors enforce a constraint based on phenomenological data. They introduce an empirical correction factor, $e^{izP_z \delta y}$, to the renormalized matrix elements. The parameter $\delta y$ is fitted to the short-distance Operator Product Expansion (OPE) to align the first moment with the CT18 phenomenological value ($\langle x \rangle = 0.1562$ at $\mu=2$ GeV).
• Large-Distance Extrapolation: Since lattice data at large separations ($z$) suffer from poor signal-to-noise ratios, the authors perform an asymptotic analysis. They fit the lattice data to Next-to-Leading Asymptotic (NLA) forms derived from confinement physics before performing the Fourier Transform (FT) to momentum space.
• Matching: The resulting quasi-PDFs are matched to the physical PDFs using the LaMET expansion formula with NNLO accuracy and the aforementioned resummations.

Key Contributions

• Unified Framework: The application of a consistent, modern theoretical pipeline (hybrid renormalization, NNLO matching, and resummations) to multiple disparate lattice datasets.
• Empirical Artifact Mitigation: The introduction of a specific ansatz to correct the first moment, effectively filtering out lattice artifacts that previously caused deviations from experimental values.
• Systematic Error Reduction: A rigorous treatment of uncertainties, including statistical fluctuations, renormalization parameters, Fourier transform extrapolations, and perturbative scale variations.

Results

• Consistency: After applying the updated analysis, the spread among the PDFs derived from the four different lattice datasets is substantially reduced.
• Agreement with Phenomenology: The reanalyzed PDFs, particularly the MSULat20 dataset extrapolated to the continuum and physical pion mass limits, are consistent with global phenomenological fittings (MMHT2014 and CT18) within approximately $1\sigma$ in the accessible moderate-$x$ region.
• Residual Deviations: The remaining datasets (ETMC20, LPC18, CLQCD24) show broadly compatible $x$-dependence with small deviations, likely attributable to residual discretization effects or unphysical pion masses, which are expected to diminish with further lattice improvements.

Significance and Claims
The paper claims that the previous disagreements in lattice QCD PDF calculations were largely due to analysis methodologies and unmitigated lattice artifacts rather than a fundamental failure of the LaMET approach. By employing advanced theoretical tools and empirical constraints, the authors demonstrate that:

1. Large momentum expansion is capable of accurately predicting the $x$-dependence of PDFs when ideal lattice data and modern analysis techniques are combined.
2. Hadron momenta of order $P_z \sim 2$ GeV may already be sufficient for precision studies when paired with these updated techniques.
3. This work represents a critical step toward precision LaMET calculations, providing essential inputs for future high-energy experiments at the Large Hadron Collider (LHC), Jefferson Laboratory (JLab), and Electron-Ion Colliders (EICs).

The authors remain modest, noting that while the results are compelling, further lattice data is required to fully confirm these expectations and reduce residual systematic uncertainties.