Quark orbital dynamics in the proton from lattice QCD: From Ji to Jaffe-Manohar orbital angular momentum

Given a Wigner distribution simultaneously characterizing quark transverse positions and momenta in a proton, one can directly evaluate their cross product, i.e., quark orbital angular momentum. The aforementioned distribution can be obtained by generalizing the proton matrix elements of quark bilocal operators which define transverse momentum-dependent parton distributions (TMDs); the transverse momentum information is supplemented with transverse position information by introducing an additional nonzero momentum transfer. A gauge connection between the quarks must be specified in the quark bilocal operators; the staple-shaped gauge link path used in TMD calculations yields the Jaffe-Manohar definition of orbital angular momentum, whereas a straight path yields the Ji definition. An exploratory lattice calculation, performed at the pion mass $m_{\pi }=518\mathrm{MeV}$, is presented which quasicontinuously interpolates between the two definitions and demonstrates that their difference can be clearly resolved. The resulting Ji orbital angular momentum is confronted with traditional evaluations based on Ji’s sum rule. Jaffe-Manohar orbital angular momentum is enhanced in magnitude compared to its Ji counterpart.

Figure 1

Path of the gauge connection $U$, cf. Eq. (4), in the correlator (3).

Figure 2

Quark orbital angular momentum in units of the number of valence quarks, as a function of the number of lattice spacings $d$ used to construct the derivative with respect to $z_T$ in Eqs. (14) and (16). Data are shown at a fixed $\widehat{\zeta }=0.39$ and for three different staple length parameters $\eta$; the isovector $u-d$ quark combination was evaluated. The shown uncertainties are statistical jackknife errors.

Figure 3

Ji quark orbital angular momentum, i.e., the $\eta =0$ limit, as a function of $\widehat{\zeta }$, with an ad hoc extrapolation to infinite $\widehat{\zeta }$ (open square). The filled diamond represents the value extracted on the same ensemble via Ji’s sum rule. The isovector $u-d$ quark combination was evaluated. The shown uncertainties are statistical jackknife errors.

Figure 4

Quark orbital angular momentum as a function of staple length parameter $\eta$, normalized to the magnitude of the $\eta =0$ Ji value. The quark orbital angular momentum is a time-reversal-even quantity, i.e., even under $\eta \to -\eta$. The extrapolated values are obtained by averaging over the $\eta >0$ and $\eta <0$ plateau values, which are each extracted by a fit to the $|\eta ||v|/a=7,\dots ,9$ range. The isovector $u-d$ quark combination was evaluated; the three panels display the data for the three available values of $\widehat{\zeta }$. The shown uncertainties are statistical jackknife errors.

Figure 5

Torque accumulated by the struck quark in a deep-inelastic scattering process (cf. main text), normalized to the magnitude of the $\eta =0$ Ji orbital angular momentum, as a function of $\widehat{\zeta }$, with an ad hoc extrapolation to infinite $\widehat{\zeta }$. The isovector $u-d$ quark combination was evaluated. The shown uncertainties are statistical jackknife errors.

Figure 6

Flavor-separated orbital angular momentum as a function of staple length parameter $\eta$, analogous to Fig. 4, at fixed $\widehat{\zeta }=0.39$. Shown are data for $d$ quarks, for two $u$ quarks (i.e., the $u$-quark data for $L_3^U/n$ have been multiplied by 2 to compensate for $n=2$ in the $u$-quark case), and the isoscalar total quark orbital angular momentum. The latter was obtained by adding the “$d$” and “$2u$” data. All data are still normalized by the magnitude of $u-d$ Ji orbital angular momentum, i.e., at $\eta =0$, the “$2u$” and “$d$” data differ by unity. The shown uncertainties are statistical jackknife errors.

Figure 7

Flavor-separated torque accumulated by the struck quark in a deep-inelastic scattering process, as a function of $\widehat{\zeta }$, analogous to Fig. 5, with ad hoc extrapolations to infinite $\widehat{\zeta }$. Shown are 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 the isoscalar combination. The latter was obtained by adding the “$d$” and “$2u$” data. All data are still normalized by the magnitude of $u-d$ Ji orbital angular momentum. Isoscalar data are slightly displaced horizontally for better visibility. The shown uncertainties are statistical jackknife errors.