Total Gluon Helicity Contribution to the Proton Spin from Lattice QCD

We report a state-of-the-art lattice QCD calculation of the total gluon helicity contribution to the proton spin, $\Delta G$. The calculation is done on ensembles with three different lattice spacings $a=\{0.08, 0.09, 0.11\}$ fm. By employing distillation and momentum smearing for proton external states, we extract the bare matrix elements of the topological current $K^\mu$ using 5-HYP smeared Coulomb gauge fixing configurations. Furthermore, we apply a non-perturbative $\mathrm{RI/MOM}$ renormalization scheme augmented by the Cluster Decomposition Error Reduction (CDER) technique to determine the renormalization constants of $K^\mu$. The results obtained from different components $K^{t,i}$ (with $i$ being the direction of proton momentum or polarization) are consistent with Lorentz covariance within uncertainties. After extrapolating to the continuum limit, $\Delta G$ is found to be $\Delta G = 0.231(17)^{\mathrm{sta.}}(44)^{\mathrm{sym.}}$ at the $\overline{\mathrm{MS}}$ scale ${\mu}^2=10\ \mathrm{GeV}^2$, which constitutes approximately $46(9)\%$ of the proton spin.