A Lattice Study of Quark and Glue Momenta and Angular Momenta in the Nucleon

TL;DR

This work presents a complete lattice QCD determination of the nucleon's momentum and angular momentum decomposition into quark (connected and disconnected) and gluon components, using an overlap-based construction for the gluon tensor to achieve clear signals. Form factors T1, T2, and T3 extracted from two- and three-point functions yield the quark and gluon momentum fractions and total angular momenta, with T2 contributions cancelling in the sum rules as required. After one-loop MSbar renormalization at 2 GeV, the results indicate quarks carry roughly 0.64 of the momentum (u+d) with 0.70 of the spin, strange quarks contribute ~0.024, and gluons ~0.33, while the quark orbital angular momentum is ~0.47 and the quark spin ~0.25, with the glue contributing ~0.28 to the proton spin. The study confirms that the total momentum and angular momentum fractions sum to unity within uncertainties and highlights a dominant role for disconnected insertions in the quark orbital angular momentum, motivating future dynamical (2+1 flavor) analyses at the physical pion mass and continuum limit.

Abstract

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 $β= 6.0$ for Wilson fermions with $κ=0.154, 0.155$, 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.

Figures (6)

• Figure 1: Quark line diagrams of the three-point function with current insertion in the Euclidean path integral formalism. (a) Connected insertions (CI), and (b) disconnected insertions (DI).
• Figure 2: CI plots at $\kappa = 0.1555$. (a) The sum of $T_1(q^2)$ and $T_2(q^2)$, extracted from Eqs. (\ref{['eq:three_pt_2_two_pt_rat_1']}) and (\ref{['eq:3_ind_eq_ratio']}) along with error bands from the dipole fit, is compared to $\left[T_1 + T_2\right](q^2)$ obtained from Eqs. (\ref{['eq:three_pt_2_two_pt_sp1_rat_1']}) and (\ref{['eq:three_pt_2_two_pt_sp2_rat_1']}) at comparable $q^2$ values for $u$ and $d$ quarks in the CI. (b) The sum of $u$ and $d$ quark contributions for $T_1(q^2)$ and $T_2(q^2)$. The red square at $q^2 = 0$ is $\left[T_1^u (0) + T_1^d (0)\right]$ which is obtained from forward matrix elements using Eq. (\ref{['eq:first_mom_rat_1']}). The black square at $q^2 = 0$ is $\left[T_2^u (0) + T_2^d (0)\right]$ which is obtained from dipole fit. To construct $J^{u+d}$ (CI), we add the values represented by the red and black squares.
• Figure 3: DI plots for $u, d$ at $\kappa_v = \kappa_{\hbox{\scriptsize loop}} = 0.1555$. (a) One of the ratios in Eq. (\ref{['eq:three_pt_2_two_pt_rat_2']}) plotted against the sink time, $t_2$. The term with form factor $T_3 (q^2)$ does not appear in this particular ratio. The slope is fitted to obtain $\left[a_1 T_1 (q^2) + a_2 T_2 (q^2)\right]^{u,d}$. (b) The ratio in Eq. (\ref{['eq:three_pt_2_two_pt_sp1_rat_2']}) plotted against the sink time, $t_2$. The slope is fitted to obtain $\left[T_1 + T_2\right]^{u,d} (q^2)$. (c) The sum of separately extracted $T_1(q^2)$ and $T_2(q^2)$ is compared with $\left[T_1+ T_2\right](q^2)$. $T_1(0)$ (red square) is from the forward matrix element. In order to construct $J$, the value represented by the red-square is used as $T_1(0)$. (d) Chiral extrapolation of $T_1(0)$ and $T_2(0)$ for the $u/d$ quark. The red and black squares in this figure represent chirally extrapolated values of $T_1(0)$ and $T_2(0)$, respectively. Please note that they are not renormalized in this figure.
• Figure 4: Plots for glue first moment: (a) ratio between three-point and two-point functions obtained by using Eq. (\ref{['eq:first_mom_rat_2']}) at $\kappa_v = 0.1555$, and (b) chiral extrapolation.
• Figure 5: Similar types of plots as in Fig. \ref{['fig:DI']} for the glue at $\kappa_v = 0.1555$.