From Ji to Jaffe-Manohar orbital angular momentum in lattice QCD using a direct derivative method

A lattice QCD approach to quark orbital angular momentum in the proton based on generalized transverse momentum-dependent parton distributions (GTMDs) is enhanced methodologically by incorporating a direct derivative technique. This improvement removes a significant numerical bias that had been seen to afflict results of a previous study. In particular, the value obtained for Ji quark orbital angular momentum is reconciled with the one obtained independently via Ji’s sum rule, validating the GMTD approach. Since GTMDs simultaneously contain information about the quark impact parameter and transverse momentum, they permit a direct evaluation of the cross product of the latter. They are defined through proton matrix elements of a quark bilocal operator containing a Wilson line; the choice in Wilson line path allows one to continuously interpolate from Ji to Jaffe-Manohar quark orbital angular momentum. The latter is seen to be significantly enhanced in magnitude compared to Ji quark orbital angular momentum, confirming previous results.

Figure 1

Path of the gauge connection $U$ [cf. (3)] in the correlator (2).

Figure 2

Ji quark orbital angular momentum, i.e., the $\eta =0$ limit, for the three values of $\widehat{\zeta }$ probed, with the average plotted at $\widehat{\zeta }=\infty$ (open square). The filled diamond represents the value extracted at the same pion mass in the $\overline{\mathrm{MS}}$ scheme at the scale ${\mu }^2=4{\mathrm{GeV}}^2$ via Ji’s sum rule [7]. The isovector $u-d$ quark combination was evaluated. The shown uncertainties are statistical jackknife errors.

Figure 3

Quark orbital angular momentum as a function of staple length parameter $\eta$, normalized to the magnitude of Ji quark orbital angular momentum, i.e., the result obtained at $\eta =0$. This quantity is even under $\eta \to -\eta$, corresponding to time reversal. Accordingly, the $|\eta |\to \infty$ extrapolated values are obtained by averaging the $\eta >0$ and $\eta <0$ plateaus, which are determined by fitting to the $|\eta ||v|/a=7,\dots ,9$ range. The isovector $u-d$ quark combination is shown, with the three panels corresponding to the three available values of $\widehat{\zeta }$. The shown uncertainties are statistical jackknife errors.

Figure 4

Integrated torque accumulated by a quark struck in a deep inelastic scattering process along its trajectory exiting the proton, as a function of Collins-Soper parameter $\widehat{\zeta }$. The data pertain to the isovector $u-d$ quark channel and are normalized to the magnitude of the $\eta =0$ Ji orbital angular momentum. An ad hoc extrapolation to the $\widehat{\zeta }\to \infty$ limit is also exhibited. The shown uncertainties are statistical jackknife errors.

Figure 5

Flavor-separated quark orbital angular momentum as a function of staple length $\eta$, analogous to Fig. 3, at fixed $\widehat{\zeta }=0.315$. Results are displayed for $d$ quarks and for two $u$ quarks (i.e., the $u$-quark data for $L_3/n$ have been multiplied by 2 to compensate for $n=2$), as well as for the isoscalar total quark orbital angular momentum, i.e., the sum of the “$d$” and “$2u$” data. All results are still normalized by the magnitude of isovector Ji orbital angular momentum (thus, at $\eta =0$, the $2u$ and $d$ data differ by unity). The shown uncertainties are statistical jackknife errors.

Figure 6

Flavor-separated integrated torque accumulated by a quark struck in a deep inelastic scattering process, as a function of $\widehat{\zeta }$, analogous to Fig. 4, together with ad hoc extrapolations to infinite $\widehat{\zeta }$. Data for $d$ quarks, for two $u$ quarks (i.e., the $u$-quark data for ${\tau }_3$ have been multiplied by 2 to compensate for $n=2$ in the $u$-quark case), and their sum, the isoscalar $u+d$ quark combination, are shown. Data are normalized by the magnitude of isovector Ji quark orbital angular momentum, as in previous figures. For better visibility, isoscalar data are slightly displaced horizontally. The shown uncertainties are statistical jackknife errors.