We report a complete calculation of the quark and glue momenta and angular momenta in the proton. These include the quark contributions from both the connected and disconnected insertions. The quark disconnected insertion loops are computed with $Z_4
$ noise, and the signal-to-noise is improved with unbiased subtractions. The glue operator is comprised of gauge-field tensors constructed from the overlap operator. The calculation is carried out on a $16^3 times 24$ quenched lattice at $beta = 6.0$ for Wilson fermions with $kappa=0.154, 0.155$, and $0.1555$ which correspond to pion masses at $650, 538$, and $478$~MeV, respectively. The chirally extrapolated $u$ and $d$ quark momentum/angular momentum fraction is found to be $0.64(5)/0.70(5)$, the strange momentum/angular momentum fraction is $0.024(6)/0.023(7)$, and that of the glue is $0.33(6)/0.28(8)$. The previous study of quark spin on the same lattice revealed that it carries a fraction of $0.25(12)$ of proton spin. The orbital angular momenta of the quarks are then obtained from subtracting the spin from their corresponding angular momentum components. We find that the quark orbital angular momentum constitutes $0.47(13)$ of the proton spin with almost all of it coming from the disconnected insertions.
We report a complete calculation of the quark and glue momenta and angular momenta in the proton. These include the quark contributions from both the connected and disconnected insertions. The calculation is carried out on a $16^3 times 24$ quenched
lattice at $beta = 6.0$ and for Wilson fermions with $kappa = 0.154, 0.155,$ and 0.1555 which correspond to pion masses at 650, 538, and 478 MeV. The quark loops are calculated with $Z_4$ noise and signal-to-noise is improved further with unbiased subtractions. The glue operator is comprised of gauge-field tensors constructed from the overlap operator. The $u$ and $d$ quark momentum/angular momentum fraction is 0.66(5)/0.72(5), the strange momentum/angular momentum fraction is 0.024(6)/0.023(7), and that of the glue is 0.31(6)/0.25(8). The orbital angular momenta of the quarks are obtained from subtracting the angular momentum component from its corresponding spin. As a result, the quark orbital angular momentum constitutes 0.50(2) of the proton spin, with almost all it coming from the disconnected insertion. The quark spin carries a fraction 0.25(12) and glue carries a fraction 0.25(8) of the total proton spin.
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.
