Experimental study of isovector spin sum rules

We present the Bjorken integral extracted from Jefferson Lab experiment EG1b for $0.05<Q^2<2.92{\mathrm{GeV}}^2$. The integral is fit to extract the twist-4 element $f_2^{p-n}$ which appears to be relatively large and negative. Systematic studies of this higher twist analysis establish its legitimacy at $Q^2$ around $1{\mathrm{GeV}}^2$. We also performed an isospin decomposition of the generalized forward spin polarizability ${\gamma }_0$. Although its isovector part provides a reliable test of the calculation techniques of chiral perturbation theory, our data disagree with the calculations.

Figure 1

The Bjorken integral ${\Gamma }_1^{p-n}(Q^2)$. The solid blue circles give the results from this work with the horizontal band giving the systematic uncertainties. Other symbols show the data from experiments E143 [3] (open diamonds), E155 [5] (open star), HERMES [7] (open triangles), and JLab [8] (open circles and open squares). For those, the error bars represent the quadratic sum of the statistic and systematic uncertainties. The gray band represents the leading-twist NNLO pQCD calculation. The curves correspond to $\chi PT$ calculations ([35, 36]) and phenomenological models ([33, 34]).

Figure 2

World data on the Bjorken integral, including the elastic contribution. The error bars represent the quadratic sum of the statistic and point-to-point uncorrelated systematic uncertainties for the JLab data, and the quadratic sum of the statistic and full systematic uncertainties for the rest of the data. The continuous line is our three parameter fit in the $Q^2$ range from 0.66 to $10{\mathrm{GeV}}^2$.

Figure 3

$f_2^{p-n}(Q^2=1{\mathrm{GeV}}^2)$ for the fits performed over the $0.66<Q^2<10{\mathrm{GeV}}^2$ range for this study and Ref. 8 (JLab A/B). Calculations 40, 42, 43, 50, 51 are shown by the bands. Sum rule (1) refers to Ref 50 and (2) to Ref 40.

Figure 4

The isovector ${\gamma }_0^p-{\gamma }_0^n$ (top) and isoscalar ${\gamma }_0^p+{\gamma }_0^n$ (bottom) generalized forward spin polarizabilities together with $\chi PT$-based calculations and the results from the MAID model. The proton and neutron data are, respectively, from CLAS [21] and Hall A [24]