Lattice QCD calculations of the quark and gluon contributions to the proton spin

Abstract: A review of the calculations of the proton's spin using lattice QCD is presented. Results for the three contributions, the quark contribution $\sum_{q=u,d,s,c} (\frac{1}{2} {\Delta q})$, the total angular momentum of the quarks $J_q$ and of the gluons $J_g$, and the orbital angular momentum of the quarks are discussed. The best measured is the the quark contribution $\sum_{q=u,d,s,c} (\frac{1}{2} {\Delta q})$, and its analysis is used to discuss the relative merits of calculations by the PNDME, ETMC and $\chi$QCD collaborations and the level of control over systematic errors achieved in each. The result by the PNMDE collaboration, $\sum_{q=u,d,s} \left[ \frac{1}{2} {\Delta q} \right] = 0.143(31)(36) $, is consistent with the COMPASS analysis $0.13 < \frac{1}{2} \Delta \Sigma < 0.18$. Results for $J_q$ and $J_g$ by the ETMC collaborations are also consistent with phenomenology. Lastly, I review first results from the LHPC collaboration for the calculation of the orbital angular momentum of the quarks. With much larger computing resources anticipated over the next five years, high precision results for all three will become available and provide a detailed description of their relative contributions to the nucleon spin.