Window observables for benchmarking parton distribution functions

TL;DR

The paper addresses the challenge of benchmarking PDFs extracted from lattice QCD against global phenomenology within limited $x$-ranges. It introduces window observables, including window moments $a_n(x_-,x_+)$ and Gaussian windows $g_n(x_-,x_+)$, defined over a constrained interval $[x_-,x_+]$ to enable cross-validation between SDF, LaMET, and global analyses via convolutions of the PDF $f(x,\mu^2)$. By leveraging the Ioffe-time distribution framework with representations in $\nu$ (Ioffe time) and $x$-space, the approach provides observables that remain well-behaved under inverse problems and extrapolations, and can be computed from both lattice ITD data and phenomenological PDFs. Demonstrations with synthetic JAM3D*/JAMDiFF data and HadStruc lattice data show that these window observables achieve higher precision benchmarks than pointwise comparisons, even when $\nu_{\max}$ or lattice discretization constraints limit full $x$-reconstructions. The proposed method offers a practical pathway to robustly validate and merge lattice QCD results with global PDF determinations, advancing the reliability of hadron structure predictions.

Abstract

Global analysis of collider and fixed-target experimental data and calculations from lattice quantum chromodynamics (QCD) are used to gain complementary information on the structure of hadrons. We propose novel ``window observables'' that allow for higher precision cross-validation between the different approaches, a critical step for studies that wish to combine the datasets. Global analyses are limited by the kinematic regions accessible to experiment, particularly in a range of Bjorken-$x$, and lattice QCD calculations also have limitations requiring extrapolations to obtain the parton distributions. We provide two different ``window observables'' that can be defined within a region of $x$ where extrapolations and interpolations in global analyses remain reliable and where lattice QCD results retain sensitivity and precision.

Figures (11)

• Figure 1: The GPR of synthetic data from JAM3D$\ast$. The GPR reconstruction (dashed line, lighter band) for the $CP$ even (blue) and $CP$ odd (orange) PDFs compared to the original input (solid line, darker band) for (Upper) the ITD and (Lower) the PDF.
• Figure 2: The window moment (upper) and Gaussian window (lower) with $x_-=0.1$ and $x_+=0.5$.
• Figure 3: (Upper) The real (squares) and imaginary (crosses) components of the lattice QCD data fit to a GPR. (Middle) The $CP$ even (left) and $CP$ odd (right) pseudo-PDF reconstructions. (Lower) The PDF at $\mu=2$ GeV resulting from LO matching and LL evolution.
• Figure 4: The window moment (upper) and Gaussian window (lower) with $x_-=0.1$ and $x_+=0.5$. The lattice QCD data for $q_-$ (squares) and $q_+$ (diamonds) are given with increasing $z/a$ in the range $[1,8]$ from left to right.
• Figure 5: The relative error in the reconstructions of the PDF (upper), Window moments (middle), and Gaussian windows (lower).