Experimental Determination of the Evolution of the Bjorken Integral at Low $Q^2$

We extract the Bjorken integral ${\Gamma }_1^{p\mathrm{\text{−}}n}$ in the range $0.17<Q^2<1.10{\mathrm{G}\mathrm{e}\mathrm{V}}^2$ from inclusive scattering of polarized electrons by polarized protons, deuterons, and ${}^3\mathrm{H}\mathrm{e}$, for the region in which the integral is dominated by nucleon resonances. These data bridge the domains of the hadronic and partonic descriptions of the nucleon. In combination with earlier measurements at higher $Q^2$, we extract the nonsinglet twist-4 matrix element $f_2$.

Figure 1

Inelastic contribution to the Bjorken sum. The triangles (squares) represent the results with the neutron extracted from ${}^3\mathrm{H}\mathrm{e}$ (deuteron) data, with the horizontal bands covering the $0.17–0.99{\mathrm{G}\mathrm{e}\mathrm{V}}^2$ and $0.34–1.1{\mathrm{G}\mathrm{e}\mathrm{V}}^2$ ranges the corresponding systematic uncertainties. The E143 data are shown for comparison. The gray band represents the leading-twist (LT) contribution calculated to third order in ${\alpha }_s$. The curves correspond to theoretical calculations (see text).

Figure 2

Total moment ${\Gamma }_1^{p\mathrm{\text{−}}n}$, including the inelastic contribution from Fig. 1 together with the elastic. The data extracted from ${}^3\mathrm{H}\mathrm{e}$ (deuterium) together with proton data are indicated by the triangles (squares). The leading-twist (LT) contribution is given by the gray band. The point to point correlated uncertainty for the data extracted from ${}^3\mathrm{H}\mathrm{e}$ and proton is shown by the horizontal band. The error bars on the symbols represent the uncorrelated uncertainty. The data from other experiments are assumed to be uncorrelated. The fits to the total moment are indicated by the solid curves.