Lattice results for the longitudinal spin structure and color forces on quarks in a nucleon

Using lattice QCD, we calculate the twist-2 contribution a2 to the third Mellin moment of the spin structure functions g1 and g2 in the nucleon. In addition we evaluate the twist-3 contribution d2. Our computations make use of Nf = 2 + 1 gauge field ensembles generated by the Coordinated Lattice Simulations (CLS) effort. Neglecting quark-line disconnected contributions we obtain as our best estimates a (p) 2 = 0.066(30), d (p) 2 = 0.0107(76) and a (n) 2 = 0.007(11), d (n) 2 = −0.0006(77) for the proton and the neutron, respectively, where we use the normalizations given in Eqs. (58) and (59). The d2 results can be interpreted as corresponding to a transverse color Lorentz force on a quark in a transversely polarized proton of size F (u) = 116(69) MeV/fm and F (d) = −36(73) MeV/fm for u and d quarks, respectively. The error estimates quoted include statistical and systematic uncertainties added in quadrature.