Abstract
Using lattice QCD, we calculate the twist-2 contribution $a_2$ to the third Mellin moment of the spin structure functions $g_1$ and $g_2$ in the nucleon. In addition we evaluate the twist-3 contribution $d_2$. Our computations make use of $N_f=2+1$ gauge field ensembles generated by the Coordinated Lattice Simulations (CLS) effort. Neglecting quark-line disconnected contributions we obtain as our best estimates $a_2^{(p)}= 0.069(17)$, $d_2^{(p)}= 0.0105(68)$ and $a_2^{(n)}= 0.0068(88)$, $d_2^{(n)}= -0.0009(70)$ for the proton and the neutron, respectively, where we use the normalizations given in Eqs. (58) and (59). While the $a_2$ results have been converted to the $\overline{\mathrm{MS}}$ scheme using three-loop perturbation theory, the numbers for $d_2$ are given in the regularization independent momentum subtraction (RI$^\prime$-MOM) scheme, i.e., the conversion has been performed only in tree-level perturbation theory. The $d_2$ results can be interpreted as corresponding to a transverse color Lorentz force on a quark in a transversely polarized proton of size $F^{(u)} = 116(61)$ MeV/fm and $F^{(d)} = -38(66)$ MeV/fm for $u$ and $d$ quarks, respectively. The error estimates quoted include statistical and systematic uncertainties added in quadrature.
Highlights
For a number of reasons hadron spin structure has attracted intense interest for more than two decades with no sign of attenuation
The central goal of the JLAB and BNL programs is to better understand the structure of hadrons. This includes multiparton correlations, which are parametrized by higher-twist coefficients
Approaches based on so-called quasi- and pseudodistribution functions have gained prominence in lattice QCD calculations of hadron structure observables, due to their prospect of providing information that goes beyond the computation of Mellin moments of parton distribution functions, distribution amplitudes, etc., see Refs. [21–27] and references therein
Summary
For a number of reasons hadron spin structure has attracted intense interest for more than two decades with no sign of attenuation. This includes multiparton correlations, which are parametrized by higher-twist coefficients The most prominent such example is the matrix element d2, the third Mellin moment (x2) of the twist-3 contribution to the helicity structure function g2ðx; Q2Þ of deep-inelastic longitudinally polarized lepton-nucleon scattering. Several experiments have extracted estimates of dð2pÞ and dð2nÞ [12–19], where the superscript indicates proton or neutron, respectively These estimates are found to be quite small compared to various model predictions but compatible with the old lattice results (considering the large error bars), see, e.g., Fig. 2 of Ref. Approaches based on so-called quasi- and pseudodistribution functions have gained prominence in lattice QCD calculations of hadron structure observables, due to their prospect of providing information that goes beyond the computation of Mellin moments of (generalized) parton distribution functions, distribution amplitudes, etc., see Refs. Knowing d2 from a direct lattice calculation would obviously help to unravel the different contributions
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