Quark contribution to the proton spin from 2 +1 +1 -flavor lattice QCD

We present the first chiral-continuum extrapolated up, down, and strange quark spin contribution to the proton spin using lattice QCD. For the connected contributions, we use 11 ensembles of 2 +1 +1 -flavor of highly improved staggered quarks (HISQ) generated by the MILC Collaboration. They cover four lattice spacings, a ≈{0.15 ,0.12 ,0.09 ,0.06 } fm , and three pion masses, M_π≈{315 ,220 ,135 } MeV , of which two are at the physical pion mass. The disconnected strange calculations are done on seven of these ensembles, covering the four lattice spacings but only one with the physical pion mass. The disconnected light quark calculation was done on six ensembles at two values of M_π≈{315 ,220 } MeV . High-statistics estimates on each ensemble for all three quantities allow us to quantify systematic uncertainties and perform a simultaneous chiral-continuum extrapolation in the lattice spacing and the light-quark mass. Our final results are Δ u ≡⟨1 ⟩_{Δ u⁺}=0.777 (25 )(30 ) , Δ d ≡⟨1 ⟩_{Δ d⁺}=-0.438 (18 )(30 ) , and Δ s ≡⟨1 ⟩_{Δ s⁺}=-0.053 (8 ) , adding up to a total quark contribution to proton spin of ∑_{q =u ,d ,s}(1/2 Δ q )=0.143 (31 )(36 ) . The second error is the systematic uncertainty associated with the chiral-continuum extrapolation. These results are obtained without model assumptions and are in good agreement with the recent COMPASS analysis 0.13 <1/2 Δ Σ <0.18 and with the Δ q obtained from various global analyses of polarized beam or target data.