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 $5\sigma$ 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 $\mu =2\mathrm{GeV}$ is $⟨x{⟩}_{s+\overline{s}}=0.027\pm 0.006$ 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 $\frac{⟨x{⟩}_{s+\overline{s}}}{⟨x{⟩}_{\overline{u}}+⟨x{⟩}_{\overline{d}}}$ from experiments where $\overline{u}$ and $\overline{d}$ include both the connected and disconnected insertion parts. We discuss the source and implication of this difference.

• Received 11 November 2008

DOI:https://doi.org/10.1103/PhysRevD.79.094502