Glue Spin and Helicity in the Proton from Lattice QCD

We report the first lattice QCD calculation of the glue spin in the nucleon. The lattice calculation is carried out with valence overlap fermions on $2+1$ flavor domain-wall fermion gauge configurations on four lattice spacings and four volumes including an ensemble with physical values for the quark masses. The glue spin $S_G$ in the Coulomb gauge in the modified minimal subtraction ($\overline{\mathrm{MS}}$) scheme is obtained with one-loop perturbative matching. We find the results fairly insensitive to lattice spacing and quark masses. We also find that the proton momentum dependence of $S_G$ in the range $0\le |\overrightarrow{p}|<1.5\mathrm{GeV}$ is very mild, and we determine it in the large-momentum limit to be $S_G=0.251(47)(16)$ at the physical pion mass in the $\overline{\mathrm{MS}}$ scheme at ${\mu }^2=10{\mathrm{GeV}}^2$. If the matching procedure in large-momentum effective theory is neglected, $S_G$ is equal to the glue helicity measured in high-energy scattering experiments.

Figure 1

The ratio $R(t_f,t)$ as a function of the source-sink separation $t_f$ and the current time slice $t$, for the bare glue spin matrix element in the proton $S_G^{\text{bare}}$ is plotted at the unitary point on the 24I ensemble. The gray band shows the result extrapolated to infinite separation, which corresponds to the prediction of $S_G$. The excited-state contamination is small when the source-sink separation is larger than 1 fm. The inset shows that the prediction of $\sum _{t}R(t_f,t)$ from the two-state fit (the band) in Eq. (8) agrees well with the data points.

Figure 2

The HYP-smearing steps dependence of the bare glue spin $S_G$ at the unitary point on the ensemble 24I, for $p=0$ (red squares), $p=0.47\mathrm{GeV}$ (blue dots), and $p=0.94\mathrm{GeV}$ (black triangles). The values of $S_G$ are unchanged after two or three steps of smearing.

Figure 3

The valence pion mass dependence of $S_G$ at ${\mu }^2=10{\mathrm{GeV}}^2$, in the rest frame of the proton. These dependencies are fairly mild and can be well described with a linear fit. The gray band shows the result based on the global fit with the empirical form in Eq. (11).

Figure 4

The results extrapolated to the physical pion mass as a function of the absolute value of $\overrightarrow{p}=(0,0,p_3)$, on all the five ensembles. All the results have been converted to $\overline{\mathrm{MS}}$ at ${\mu }^2=10{\mathrm{GeV}}^2$. The data on several ensembles are shifted horizontally to enhance the legibility. The green band shows the frame dependence of the global fit [with the empirical form in Eq. (11)] of the results.