Precise determination of the deuteron spin structure at low to moderate $Q^2$ with CLAS and extraction of the neutron contribution

We present the final results for the deuteron spin structure functions obtained from the full data set collected in 2000–2001 with Jefferson Lab's continuous electron beam accelerator facility (CEBAF) using the CEBAF large acceptance spectrometer (CLAS). Polarized electrons with energies of 1.6, 2.5, 4.2, and 5.8 GeV were scattered from deuteron (${}^{15}{\mathrm{ND}}_3$) targets, dynamically polarized along the beam direction, and detected with CLAS. From the measured double-spin asymmetry, the virtual photon absorption asymmetry $A_1^d$ and the polarized structure function $g_1^d$ were extracted over a wide kinematic range ($0.05{\mathrm{GeV}}^2<Q^2<5{\mathrm{GeV}}^2$ and $0.9\text{GeV}<W<3\text{GeV}$). We use an unfolding procedure and a parametrization of the corresponding proton results to extract from these data the polarized structure functions $A_1^n$ and $g_1^n$ of the (bound) neutron, which are so far unknown in the resonance region, $W<2$ GeV. We compare our final results, including several moments of the deuteron and neutron spin structure functions, with various theoretical models and expectations, as well as parametrizations of the world data. The unprecedented precision and dense kinematic coverage of these data can aid in future extractions of polarized parton distributions, tests of perturbative QCD predictions for the quark polarization at large $x$, a better understanding of quark-hadron duality, and more precise values for higher-twist matrix elements in the framework of the operator product expansion.

Figure 1

Kinematic coverage in $Q^2$ vs $x$ for each of the four main electron beam energy groupings used in the EG1b experiment. The solid and dotted lines denote the $W=1.08$-GeV and $W=2.00$-GeV thresholds, respectively. The coverages for proton (${\mathrm{NH}}_3$) and deuterium (${\mathrm{ND}}_3$) targets were nearly identical.

Figure 2

Distribution of quasielastic $d(e,e^{\prime }p)$ events versus the angle $\varphi$ between the azimuth of the scattered electron and the azimuth of the observed proton. The background due to nitrogen, liquid ${}^4\mathrm{He}$, and various foils is strongly suppressed by the cuts described in the text, leading to a relatively clean signal from the deuteron component (solid line) of the target. A final cut is applied from $\varphi ={177}^{\circ }$ to $\varphi ={183}^{\circ }$.

Figure 3

Missing mass $W$ before (red open circles) and after (blue solid circles) the kinematic corrections for the 4.238+ data set for ${\mathrm{NH}}_3$ (top) and ${\mathrm{ND}}_3$ (bottom) targets. The corrections decreased the distribution width and centered the mean value of the (quasi-)elastic peak on the nucleon mass.

Figure 4

Dilution factors as a function of $W$, shown for four different combinations of beam energies and torus polarity [1.6+ (a), $2.5-$ (b), $4.2-$ (c), and $5.7-$ (d)]. The results from our standard method (using cross-section models; see text) are shown as blue lines, while the results from the data-based method (see text) are shown as the red data points.

Figure 5

$P_b{P}_t$ values for the 2.5-GeV inbending data sets. The plot shows the resulting $P_b{P}_t$ values extracted independently for each $Q^2$ bin with available data. The results from the exclusive (blue solid symbols) and the inclusive (red open symbols) methods are shown. The corresponding constants fit to the data are also shown as lines: The solid blue line is for the exclusive and the dashed red line is for the inclusive method.

Figure 6

Representative results for the fully corrected double-spin asymmetry $A_{\left|\right|}$ versus final state invariant mass $W$ for three different $Q^2$ bins and beam energies. The red solid line represents our model parametrization of $A_{\left|\right|}$ (see Sec. 5d). The dashed blue lines represent the model including radiative effects. The difference between those lines corresponds to the magnitude of radiative corrections applied. The error bars reflect statistical uncertainties while the shaded bands at the bottom of each plot represent the total systematic uncertainties.

Figure 7

Representative values for the double-spin asymmetry $A_1+\eta A_2$ versus final-state invariant mass $W$. The top panel is for 0.16 ${\mathrm{GeV}}^2\le Q^2\le 0.19{\mathrm{GeV}}^2$ (1.6-GeV data) and the bottom panel for 0.45 ${\mathrm{GeV}}^2\le Q^2\le 0.54{\mathrm{GeV}}^2$ (2.5-GeV data). The red solid line represents our model parametrization of $A_1+\eta A_2$ (see Sec. 5d). The shaded band at the bottom (green) is the total systematic uncertainty. The individual contributions are offset vertically, from top to bottom: pion and pair symmetric contamination (−0.4; barely visible); dilution factor (−0.6); $P_b{P}_t$ (−0.8); models plus radiative corrections (−1.0); and polarized background (−1.2).

Figure 8

Same as Fig. 7, except for two higher beam energies. The top panel is for 0.64 ${\mathrm{GeV}}^2\le Q^2\le 0.77{\mathrm{GeV}}^2$ (4.2-GeV data) and the bottom panel for 1.1 ${\mathrm{GeV}}^2\le Q^2\le 1.3{\mathrm{GeV}}^2$ (5.7-GeV data).

Figure 9

Virtual photon asymmetry $A_1$ for the deuteron versus $W$ for a few $Q^2$ bins: 0.16 ${\mathrm{GeV}}^2\le Q^2\le 0.19{\mathrm{GeV}}^2$ (top), 0.45 ${\mathrm{GeV}}^2\le Q^2\le 0.54{\mathrm{GeV}}^2$ (middle), and 1.1 ${\mathrm{GeV}}^2\le Q^2\le 1.3{\mathrm{GeV}}^2$ (bottom). The statistical uncertainties are indicated by error bars, while the total systematic uncertainties are indicated by the shaded band at the bottom. Again, the individual contributions are shown separately as offset bands: pion and pair symmetric contamination (−0.4); dilution factor (−0.6); ${\mathrm{P}}_b{\mathrm{P}}_t$ (−0.8); models plus radiative corrections (−1.0); and polarized background (−1.2).

Figure 10

$A_1$ for the deuteron versus $W$ for our 14 lowest $Q^2$ bins. Total systematic uncertainties are shown as a shaded area at the bottom of each plot. Our parametrized model is also shown as a red line on each plot. Only the data points with ${\sigma }_{\mathrm{stat}}<0.3$ and ${\sigma }_{\mathrm{sys}}<0.2$ are plotted. In addition, we also show data from SLAC E143 [15, 83] (open magenta squares).

Figure 11

Continuation of Fig. 10 for the remaining $Q^2$ bins. In addition to our data and the SLAC data (see above), we also show the data from the Jefferson Lab RSS experiment [14, 84] (blue open circles).

Figure 12

$A_1^d$ versus $x$ in the DIS region ($Q^2>1{\mathrm{GeV}}^2$ and $W>2$ GeV) from EG1b and several other experiments: EG1-dvcs at Jefferson Lab [21], SMC at CERN [85], E143 and E155 at SLAC [15, 17, 83], and HERMES at DESY [18]. Statistical uncertainties are indicated by error bars, and EG1b systematic uncertainties are shown by the shaded band at the bottom. Various theoretical predictions and parametrizations are shown as lines and shaded band and are discussed in the text.

Figure 13

The ratio $g_1/F_1$ for the deuteron versus ${\mathrm{Q}}^2$ and for various bins in the Bjorken variable $x$, together with our model shown as the red line on each plot. All data are corrected by our model to center them on the middle of each $x$ bin. The shaded area at the bottom of each plot represents the systematic uncertainty. Published world data are shown as open magenta squares (E143 [15, 83]) and open blue triangles (EG1-dvcs [21]). Arrows on the $x$ axis indicate the limit $W=2$ GeV. The horizontal arrows on the right-hand side of the right panel indicate the results for $g_1/F_1$ of a recent NLO analysis of world data [87] for our bin centers and $Q^2=5{\mathrm{GeV}}^2$.

Figure 14

The product $x{g}_1$ versus $x$ for all $Q^2$ bins, together with our model (red lines). The shaded area at the bottom of each plot represents the systematic uncertainty. The corresponding DIS parametrization for $Q^2=10{\mathrm{GeV}}^2$ is also shown (blue dashed lines). World data are shown for HERMES [18] (red circles), SLAC E143 [15, 83] (open magenta squares), SLAC E155 [17] (magenta inverted triangles), RSS [14, 84] (blue circles), and EG1-dvcs [21] (cyan triangles).

Figure 15

${\mathrm{\Gamma }}_1$ for the deuteron versus $Q^2$ from our data only (open blue circles) and from data plus model (solid blue circles), including the extrapolation to the unmeasured kinematics. The left-hand side shows the full $Q^2$ range (leaving out our data for $Q^2<0.3{\mathrm{GeV}}^2$, to avoid clutter) and the right-hand side focuses on the small-$Q^2$ region. The systematic uncertainty is shown at the bottom of the plot, for data only (light beige shaded area in the foreground) and for combined data and model (blue shade in the background). Corresponding results from SLAC E143 [15, 83], HERMES [18], and EG1-dvcs [21] are shown, as well as several predictions (explained in the text).

Figure 16

Higher moments of $g_1^d$ extracted from the EG1b data versus $Q^2$. The third moment for the deuteron, ${\mathrm{\Gamma }}_1^3$, is shown on the left, and the fifth moment, ${\mathrm{\Gamma }}_1^5$, on the right. The open squares were calculated with no model contribution, while the solid squares include model input for the kinematic regions with no available data. The total systematic uncertainty is shown by the blue (experimental only) and black (experimental plus extrapolation) shaded areas.

Figure 17

Forward spin polarizability (${\gamma }_0$) for the deuteron versus $Q^2$. The open squares represent the result using only data and the solid black circles are data plus model results. The shaded area close to the $x$ axis is the total systematic uncertainty (blue for experimental only and black with extrapolation uncertainty included). Our model is shown as a red solid line. Our results are compared to three $\chi \mathrm{PT}$ calculations (see text) and the MAID parametrization [102] for single-pion production.

Figure 18

Our results for the spin structure function $x{g}_1$ of the bound neutron, extracted in the impulse approximation framework of Ref. [27] versus the Bjorken variable $x$ for all (combined) $Q^2$ bins (solid circles). Our model is shown as red lines on each plot, and the asymptotic form of $g_1\left(x\right)$ in the DIS region is shown as dashed blue lines. The shaded area at the bottom of each plot represents the systematic uncertainty. Additional data from other experiments are shown as well: E154 [95] (magenta inverted triangles), HERMES [18, 97] (red circles), and E142 [94] (brown triangles).

Figure 19

$A_1$ for the bound neutron, extracted from our results for $g_1^n$ (see Fig. 18), versus $W$ for our combined $Q^2$ bins. Systematic uncertainties are shown as the shaded area at the bottom of each plot. Our parametrized model is also shown as a red line on each plot. Only the data points with ${\sigma }_{\mathrm{stat}}<0.6$ and ${\sigma }_{\mathrm{sys}}<0.2$ are plotted. The cyan diamonds indicate data from measurements on ${}^3\mathrm{He}$ [12, 88].

Figure 20

${\mathrm{\Gamma }}_1$ for the neutron versus $Q^2$ from data only (open blue circles) and data plus model (solid blue circles), including the extrapolation to the unmeasured kinematics. Also shown are phenomenological calculations from Pasechnik et al. [100] (lower light blue line) and Burkert and Ioffe [99] (upper magenta line), together with the $\chi \mathrm{PT}$ results from Lensky et al. [101] (wider dark green band) and Bernard et al. [38] (thin gray band). The GDH slope (black solid line) and pQCD prediction (black dotted line) are also shown. The right-hand side plot is a magnification of the low $Q^2$ region (which is omitted from the left-hand side). Systematic uncertainties of our data are shown as shaded areas at the bottom of the plot. Results from other experiments are also shown, with statistical and systematic uncertainties (added in quadrature) reflected in their total error bars.

Figure 21

Forward spin polarizability ${\gamma }_0$ for the neutron versus $Q^2$. The open squares represent the result using only data and the solid black circles are data plus model results. The shaded area close to the $x$ axis is the total systematic uncertainty (blue for experimental data only and black including the extrapolation). Our model is also shown as a red solid line. Our results are compared to three $\chi \mathrm{PT}$ calculations (see text) and to the ${}^3\mathrm{He}$ data from Hall A [11].