Quark spins and anomalous Ward identity

We calculate the intrinsic quark spin contribution to the total proton spin using overlap valence quarks on three ensembles of $2+1$-flavor RBC/UKQCD domain-wall configurations with different lattice spacings. The lowest pion mass of the ensembles is around 171 MeV, which is close to the physical point. With overlap fermions and a topological charge derived from the overlap operator, we verify the anomalous Ward identity between nucleon states with momentum transfer. Both the connected and the disconnected insertions of the axial-vector current are calculated. For the disconnected-insertion part, the cluster-decomposition error reduction technique is utilized for the lattice with the largest volume and the error can be reduced by 10%--40%. Nonperturbative renormalization is carried out and the final results are all reported in the $\overline{\mathrm{MS}}$ scheme at 2 GeV. We determine the total quark spin contribution to the nucleon spin to be $\mathrm{\ensuremath{\Delta}}\mathrm{\ensuremath{\Sigma}}=0.405(25)(37)$, which is consistent with the recent global fitting result of experimental data. The isovector axial coupling we obtain in this study is ${g}_{A}^{3}=1.254(16)(30)$, which agrees well with the experimental value of 1.2723(23).