Moments of Nucleon's Parton Distribution for the Sea and Valence Quarks from Lattice QCD

TL;DR

The paper computes the first two moments of nucleon parton distributions, including sea-quark (disconnected) contributions, on a quenched $16^3 imes24$ lattice with Wilson fermions. It develops a stochastic complex $Z_2$-noise framework with unbiased subtraction to access all-to-all propagators and employs symmetry-based noise reduction, multiple nucleon sources, and careful renormalization to obtain $0.027 \\pm 0.006$ for $ig\<xig e_{s+ar{s}}$ at $\,\mu=2$ GeV and a ratio $ig\<xig e_{s+ar{s}} / ig\<x e_{u+ar{u}}$ (DI) of $0.88 \\pm 0.07$. The DI signals for light and strange quarks are strong for the first moment, while the second moments are statistically consistent with zero; CI results align with prior lattice studies. A notable finding is that the DI strange-to-light momentum ratio is larger than phenomenological fits, suggesting distinct connected and disconnected sea contributions and motivating future dynamical (2+1 flavor) lattice analyses to clarify the sea-quark structure and Gottfried-sum-type effects. Overall, the work demonstrates a feasible lattice-QCD route to sea-quark moments and highlights the need for dynamical fermions to reduce systematic uncertainties.

Abstract

We extend the study of lowest moments, $<x>$ and $<x^2>$, of the parton distribution function of the nucleon to include those of the sea quarks; this entails a disconnected insertion calculation in lattice QCD. This is carried out on a $16^3 \times 24$ quenched lattice with Wilson fermion. The quark loops are calculated with $Z_2$ noise vectors and unbiased subtractions, and multiple nucleon sources are employed to reduce the statistical errors. We obtain 5$σ$ signals for $<x>$ for the $u,d,$ and $s$ quarks, but $<x^2>$ is consistent with zero within errors. We provide results for both the connected and disconnected insertions. The perturbatively renormalized $<x>$ for the strange quark at $μ= 2$ GeV is $<x>_{s+\bar{s}} = 0.027 \pm 0.006$ which is consistent with the experimental result. The ratio of $<x>$ for $s$ vs. $u/d$ in the disconnected insertion with quark loops is calculated to be $0.88 \pm 0.07$. This is about twice as large as the phenomenologically fitted $\displaystyle\frac{< x>_{s+\bar{s}}}{< x>_{\bar{u}}+< x>_{\bar{d}}}$ from experiments where $\bar{u}$ and $\bar{d}$ include both the connected and disconnected insertion parts. We discuss the source and implication of this difference.

Figures (19)

• Figure 1: Quark line diagrams of the three-point function in the Euclidean path integral formalism. (a) Connected insertion and (b) disconnected insertion.
• Figure 2: Plaquette terms (a) for the operator ${\cal O}_{\mu\mu}$ when $\kappa^3 D^3$ term is considered and (b) for ${\cal O}_{4ii}$ when $\kappa^2 D^2$ term is considered.
• Figure 3: Errors of the noise estimation plotted against the number of configurations for different sets of noise vectors for the loop part of the current, ${\cal O}_{4i}$ at $\kappa_s = 0.154$ nd insertion time, $t_1 = 14$ (a) without subtraction and (b) with four subtraction terms.
• Figure 4: Errors of the noise estimation plotted against the number of noise vectors for different sets of configurations for the loop part of the current ${\cal O}_{4i}$ at $\kappa_s = 0.154$ and insertion time, $t_1 = 14$ (a) without subtraction and (b) with four subtraction terms.
• Figure 5: The ratio (D.I.) of the three-point to two-point functions at $\kappa_v=\kappa_s=0.154$ for the ${\cal O}_{4i}$ operator is plotted against the nucleon sink time ($t_2$) by using four different methods: (a) summation of insertion time from 5 to 20, (b) summation of insertion time from source to sink time, (c) method used in R. Lewis et al.wilcox3, and (d) summation of insertion time from (source time $+$ 1) to (sink time + 1).