Quark-hadron duality in spin structure functions $g_1^p$ and $g_1^d$

New measurements of the spin structure functions of the proton and deuteron $g_1^p(x,Q^2)$ and $g_1^d(x,Q^2)$ in the nucleon resonance region are compared with extrapolations of target-mass-corrected next-to-leading-order (NLO) QCD fits to higher energy data. Averaged over the entire resonance region ($W<2$ GeV), the data and QCD fits are in good agreement in both magnitude and $Q^2$ dependence for $Q^2>1.7$ GeV${}^2/c^2$. This “global” duality appears to result from cancellations among the prominent “local” resonance regions: in particular strong ${\sigma }_{3/2}$ contributions in the $\Delta (1232)$ region appear to be compensated by strong ${\sigma }_{1/2}$ contributions in the resonance region centered on 1.5 GeV. These results are encouraging for the extension of NLO QCD fits to lower $W$ and $Q^2$ than have been used previously.

Figure 1

Present data for proton $g_1^p(x,Q^2)$ (left panels) and deuteron $g_1^d(x,Q^2)$ (defined to be per nucleon, right panels) at four representative values of $Q^2$. The errors include statistical and systematic contributions added in quadrature. The three arrows on each plot indicate the central kinematic position of the three prominent resonance regions at $W=1.7$, 1.5, and 1.23 GeV from left to right. The hatched band represents the range of $g_1$ predicted by modern NLO PDF fits (GRSV [17] and AAC [16]) to high-energy data, evolved to the $Q^2$ of our data and corrected for TM as described in the text.

Figure 2

The $Q^2$ dependence of $Q^2{g}_1(x,Q^2)$, averaged over a region in $x$ corresponding to $1.08<W<2$ GeV (solid circles) for (a) the proton and (b) the deuteron. The inner error bars reflect only statistical contributions; the outer error bars include statistical and systematic components added in quadrature. The open circles represent our data after adding the contribution from $\mathit{ep}$ elastic ($\mathit{ed}$ quasi-elastic) scattering at $x=1$ for the proton (deuteron). For clarity these results are slightly displaced in $Q^2$, and the error bars include only statistical contributions. The hatched bands represent the range of the averages calculated from extrapolated NLO DIS fits, as in Fig. 1 (see text for details).

Figure 3

The $Q^2$ dependence of $Q^2{g}_1(x,Q^2)$ for the proton, averaged over various regions of $x$. At each $Q^2$, the $x$ range over which $g_1$ was averaged is determined by the corresponding range in $W$ as indicated in each panel (see text). Symbols and curves are as in Fig. 2.