Precision Measurement of the Neutron Twist-3 Matrix Element $d_2^n$: Probing Color Forces

Double-spin asymmetries and absolute cross sections were measured at large Bjorken $x(0.25\le x\le 0.90)$, in both the deep-inelastic and resonance regions, by scattering longitudinally polarized electrons at beam energies of 4.7 and 5.9 GeV from a transversely and longitudinally polarized ${}^3\mathrm{He}$ target. In this dedicated experiment, the spin structure function $g_2^{{}^3\mathrm{He}}$ was determined with precision at large $x$, and the neutron twist-3 matrix element $d_2^n$ was measured at $⟨Q^2⟩$ of 3.21 and $4.32{\mathrm{GeV}}^2/c^2$, with an absolute precision of about $10^{-5}$. Our results are found to be in agreement with lattice QCD calculations and resolve the disagreement found with previous data at $⟨Q^2⟩=5{\mathrm{GeV}}^2/c^2$. Combining $d_2^n$ and a newly extracted twist-4 matrix element $f_2^n$, the average neutron color electric and magnetic forces were extracted and found to be of opposite sign and about $30\mathrm{MeV}/\mathrm{fm}$ in magnitude.

Figure 1

$x^2$-weighted $g_2^{{}^3\mathrm{He}}$ plotted against $x$ for data from E99-117 [24], E97-103 [47], E142 [48], E154 [49], and E01-012 [50] with $Q^2>1{\mathrm{GeV}}^2/c^2$. (a) All error bars on the world data represent statistical and systematic uncertainties added in quadrature. (b) The error bars on our data are statistical only. The top (red) and bottom (blue) bands represent the systematic uncertainty associated with the $E=4.7$ and 5.9 GeV data sets, respectively. The gray band shows the $g_2^{\mathrm{WW},{}^3\mathrm{He}}$ coverage from several global analyses [46, 51, 52, 53, 54, 55].

Figure 2

$d_2^n$ data plotted against $Q^2$ for data with $⟨Q^2⟩\ge 1{\mathrm{GeV}}^2/c^2$. The error bars on the world data from E01-012 [50], E155x [23], $\mathrm{E}99\text{−}117+\mathrm{E}155\mathrm{x}$ [24], and RSS [62] represent the in quadrature sum of their statistical and systematic uncertainties. Our results are displayed with and without the low-$x$ contribution added, and are offset in $Q^2$ for clarity. The inset figure zooms in around our results, with the shaded boxes representing our systematic uncertainty.