Moments of Nucleons Parton Distribution for the Sea and Valence Quarks from Lattice QCD

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 $mu = 2$ 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 $displaystylefrac{< x>_{s+bar{s}}}{< x>_{bar{u}}+< x>_{bar{d}}}$ 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.