Determination of the proton spin structure functions for $0.05<Q^2<5{\mathrm{GeV}}^2$ using CLAS

We present the results of our final analysis of the full data set of $g_1^p\left(Q^2\right)$, the spin structure function of the proton, collected using CLAS at Jefferson Laboratory in 2000–2001. Polarized electrons with energies of 1.6, 2.5, 4.2, and 5.7 GeV were scattered from proton targets (${}^{15}\mathrm{NH}_3$ dynamically polarized along the beam direction) and detected with CLAS. From the measured double spin asymmetries, we extracted virtual photon asymmetries $A_1^p$ and $A_2^p$ and spin structure functions $g_1^p$ and $g_2^p$ over a wide kinematic range (0.05 ${\mathrm{GeV}}^2<Q^2<$ 5 ${\mathrm{GeV}}^2$ and 1.08 GeV $<W<$ 3 GeV) and calculated moments of $g_1^p$. We compare our final results with various theoretical models and expectations, as well as with parametrizations of the world data. Our data, with their precision and dense kinematic coverage, are able to constrain fits of polarized parton distributions, test pQCD predictions for quark polarizations at large $x$, offer a better understanding of quark-hadron duality, and provide more precise values of higher twist matrix elements in the framework of the operator product expansion.

Figure 1

An internal view of the target chamber, viewed from upstream, showing the orange transparent Kapton cylindrical LHe minicup into which the target stick was inserted. Note the metal “horn,” the source of microwave emission, on the left side.

Figure 2

The CLAS spectrometer. Different colors represent different components of the detector (from the central target outward): three layers of drift chambers (DCs) (blue) and the torus magnet (yellow), Cerenkov counters (CCs) (magenta), TOF counters (SCs) (red), and electromagnetic calorimeters (ECs) (green). The electron beam travels through the central axis from upper left to lower right.

Figure 3

Kinematic coverage in $Q^2$ vs $x$ for each of the 4 electron beam-energy groups in the EG1b experiment. The solid and dotted lines denote the $W=$ 1.08 and 2.00 GeV thresholds, respectively.

Figure 4

Helicity signal logic. The clock signal (top) provided a rising edge every 30 ns. The helicity bit train (middle) was a pseudorandom stream of opposite bit pairs. The logic analyzed each helicity bit into four categories (bottom): 1, negative first bit followed by its complement; 4, positive second bit preceded by its complement; 2, positive first bit followed by its complement; and 3, negative second bit preceded by its complement. Buckets without a complementary partner were removed from the analysis.

Figure 5

Raster pattern for a sample run, demonstrating some temporary settling of the target material. (The “crosshair” pattern is a nonphysical relic of the coordinate reconstruction.)

Figure 6

Cherenkov signal distributions before (red, solid line) and after (black, dotted line) requiring track matching.

Figure 7

Sample fiducial cuts for (a) inbending and (b) outbending electrons, shown in $\varphi$ vs $\theta$ for one CLAS sector.

Figure 8

The total energy $E{C}_{\mathrm{tot}}$ deposited in the EC vs the energy $E{C}_{\mathrm{in}}$ deposited in the inner (front) layer of the EC only for electron candidates. The plot shows a clear separation of electrons from light hadrons (bottom left corner). A cut on $E{C}_{\mathrm{in}}$ (shown by the vertical line) removes most of the hadron background.

Figure 9

Scatter plot of $E{C}_{\mathrm{tot}}/p$ vs CC signal, at $p<$ 3 GeV/$c$, after fiducial cuts. Only events to the right and above the straight lines are kept as inclusive electrons.

Figure 10

Pion contamination fraction (a) before and (b) after track-matching cuts for the 5.7-GeV beam energies, as a function of polar angle and momentum. The increase beyond $p=2.8$ GeV/$c$ indicates the threshold beyond which pions start to produce a signal in the CC.

Figure 11

Average positron asymmetries for the 5.7-GeV data set as a function of (a) momentum and (b) scattering angle $\theta$.

Figure 12

Ratios of $e^{+}/e^{-}$ as a function of electron momentum $p$, at various $\theta$ angles, for the (a) $2.561-$ and (b) 5.727+ data sets.

Figure 13

Kinematic cuts on (a) the difference between measured and expected momentum, (b) polar angle, and (c) azimuthal angle of elastic $ep$ events. Each of the distributions has the other two cuts applied.

Figure 14

Vertex $z$ positions for electrons after corrections for the raster. Secondary peaks correspond to target windows. A vertex cut of ($-58$ $<v_z<$ $-52$ cm) was applied as shown to select events from the target.

Figure 15

Missing energy and momentum distributions from elastic events in the 4.238-GeV inbending data set before (dashed red line) and after (black solid line) momentum corrections.

Figure 16

Measured energy mismatch $\mathrm{\Delta }{E}^{\prime }/E^{\prime }$ vs $\varphi$ for elastically scattered electrons (a) before and (b) after the kinematic corrections for the $1.723$-data set. After corrections, $\mathrm{\Delta }{E}^{\prime }/E^{\prime }$ is centered on zero for all azimuthal angles.

Figure 17

Elastic $W$ peaks (different colors/shades) for seven $\varphi$ bins spanning the detector acceptance, (a) before and (b) after kinematic corrections to the 2.286-GeV data set. The plots represent one sector and one polar-angle bin. The spurious $\varphi$ dependence of the elastic $W$ peak location is removed by these corrections.

Figure 18

Missing mass $W$ before (red, open circles) and after (blue, solid dots) the kinematic corrections for the 4.238+ data set. The corrections decrease the distribution width and center the mean on the 0.938-GeV proton mass, which is indicated by the vertical black dashed line.

Figure 19

Inclusive $W$ spectra normalized to the integrated Faraday cup charge for each target (ammonia, red circles; carbon, blue squares; and empty (MT), green triangles) in a selected $Q^2$ bin, at (a) the lower two beam energies and (b) the higher two beam energies.

Figure 20

Dilution factors (a) $F_{\mathrm{DF}}$ vs $W$ and (b) $F_{\mathrm{DF}}$ vs. $Q^2$, for the 5.7-GeV beam energy. The solid blue line shows the modeled dilution factor used in the analysis, and the dotted black line (most visible in plot (a) at low $W$) is a two-dimensional polynomial fit (in $Q^2$ and $W$) to the red points from the data-driven method. The difference between the solid blue and black dotted lines is an estimate of the model systematic uncertainty. Over much of the kinematics, $F_{\mathrm{DF}}$ is close to the naive ratio 3/18 of polarized to unpolarized nucleons in the target.

Figure 21

Background subtraction (using dilution factors) for inclusive inelastic $W$ spectra for a selected $Q^2$ bin at (a) 2.5-GeV and (b) 5.7-GeV beam energies. The blue circles are the inclusive counts for ammonia. The green squares show the subtracted background, as determined by the dilution factors, and the black triangles are the difference, which represents the free proton counts in the ammonia target.

Figure 22

Histogram of the azimuthal angular difference ${\varphi }_p-{\varphi }_e$ for elastic scattering events from the ${\mathrm{NH}}_3$ target (blue circles) overlaid with the scaled distribution from the carbon target (red triangles) for two different data sets.

Figure 23

Comparison of the elastic asymmetry $A_{\left|\right|}(W=M_p,Q^2)$ (solid lines) to the measured elastic asymmetries for all data sets, normalized by $P_b{P}_t$. (a) Inbending and (b) outbending sets are shown separately. Each line represents a specific beam energy, increasing in energy with descending order from the upper left. Each color and marker style (red circles, cyan squares, light green triangles, magenta inverted triangles, blue open circles, orange crosses, gray open triangles, dark green diamonds) represent a different beam energy (1.606, 1.723, 2.286, 2.561, 4.238, 5.615, 5.725, and 5.743 GeV, respectively).

Figure 24

Values of $A_{\parallel }$ (including radiative corrections) shown at beam energies of (a) 1.6, (b) 2.3, (c) 4.2, and (d) 5.7 GeV. The curves correspond to our model with (blue solid line) and without (red dotted line) radiative corrections, as discussed in the text.

Figure 25

Values of $A_{\parallel }/D$ vs $W$ for each beam energy, including systematic uncertainties. The green inverted triangles, blue triangles, red squares, and black circles correspond to data from approximate beam energies of 1.6, 2.5, 4.2, and 5.7 GeV, respectively.

Figure 26

Representative linear fit of $A_{\parallel }/D$ vs $\eta$ for one $W$, $Q^2$ bin (at $W=1.51$ GeV and $Q^2=0.5{\mathrm{GeV}}^2$). The three points were taken at three different beam energies (color and style coded as in Fig. 25). The $y$ intercept gives $A_1^p$ and the slope gives $A_2^p$.

Figure 27

$A_2^p$ vs $W$ extracted from the EG1b data (black filled circles) together with the RSS (blue open circles) [22] and Bates (purple inverted triangles) [44] data. The EG1b model (red solid line) is shown for comparison. The green band shows the systematic uncertainty.

Figure 28

$x{g}_2^p$ vs Bjorken $x$ for the proton (solid black circles), together with RSS data (blue open circles) [22] and E155x data [41] (diamonds). The red curve is our model for the $Q^2$ bin median (which differs significantly from the average $Q^2$ value for the other data sets).

Figure 29

Asymmetries $A_1^p$ vs $W$ for bins in $Q^2$. The solid black points are our data with statistical error bars. Open squares represent EG1a data [12], and the purple triangles are Bates data [44], visible on the left side of three of the four highest $Q^2$ plots shown. The red line shows our model of $A_1^p$ for comparison. The green bands show the systematic uncertainties.

Figure 30

Same as Fig. 29 but for the higher $Q^2$ bins. Additionally, here, blue hollow circles are RSS data [22] and open triangles are E143 data [37].

Figure 31

$A_1^p$ vs $x$ for DIS events, $W>2$ GeV, compared to world data. Curves and models are discussed in the text. The difference between EG1b data and higher energy data is discussed in the main text. The hatched region at the bottom represents the systematic uncertainty on the EG1b data.

Figure 32

Spin structure function $x{g}_1^p$ vs Bjorken $x$, for various bins in $Q^2$. Our data (black points) are plotted along with the world data at similar $Q^2$: from HERMES (red crosses) [35], E155 (diamonds) [40], E143 (hollow triangles) [37], RSS (blue circles) [22], and EG1a (hollow squares) [12]. The green band indicates total systematic uncertainties; the red solid line is our model for the median of each $Q^2$ bin, and the blue dashed line is the DIS model at $Q^2=$ 10 ${\mathrm{GeV}}^2$, included for reference.

Figure 33

Plots of $g_1^p/F_1^p$ vs $Q^2$ for different $x$ ranges of the combined EG1b data. The (red) line represents our model. The blue triangles correspond to the EG1-dvcs data [23], while magenta squares represent E143 data [37]. The downward-pointing black arrows indicate the upper limit of the resonance region at $=W=2$ GeV, while the red horizontal arrows indicate the results for $g_1^p/F_1^p$ of a recent analysis of world data for our bin centers and $Q^2=$ 5 ${\mathrm{GeV}}^2$.

Figure 34

The full $g_1^p$ data set from this experiment. For clarity, the $n\mathrm{th}$ $x$ distribution at fixed $Q^2$ is shifted upward by $1+n$.

Figure 35

${\mathrm{\Gamma }}_1^p$ vs $Q^2$ for EG1b data and selected world data. The right panel shows an expanded scale at small $Q^2$. The open circles represent our data, integrated over the measured region. The filled blue circles are the full integral from $x=0.001\to 1$, excluding the elastic region. The curves show phenomenological parametrizations by Burkert and Ioffe [122, 123] (magenta) and Pasechnik et al. [124] (cyan). The limiting cases of large $Q^2$ (“DIS limit”) and $Q^2\to 0$ (“GDH slope”) are also shown, as well as two bands showing $\chi \mathrm{PT}$ calculations (Lensky et al. [125] and Meissner et al. [126]). The green band at the bottom represents the total systematic uncertainty.

Figure 36

${\mathrm{\Gamma }}_3^p$ and ${\mathrm{\Gamma }}_5^p$ vs $Q^2$ for EG1b data. Solid (blue) circles are the total integral, whereas the open (blue) circles are the integral over measured data. The curve (red) is our model. The gold and gray bands at the bottom represent the systematic uncertainties on the data and the data $+$ model contributions, respectively.

Figure 37

World data on ${\mathrm{\Gamma }}_1^p\left(Q^2\right)$. The band (LT) is the pQCD leading-twist evolution. The error bars represent statistical (inner bars) and total (outer bars) uncertainties after applying the unbiased estimate procedure. The solid black line is a fit of the data starting at $Q_{min}^2=0.6{\mathrm{GeV}}^2$. The band labeled “elastic” shows the elastic contribution to ${\mathrm{\Gamma }}_1^p\left(Q^2\right)$ with its uncertainty.

Figure 38

The forward spin polarizability ${\gamma }_0^p$ vs $Q^2$. Open and closed circles represent the contribution to the integral from the data only and the data plus model, respectively (slightly offset horizontally for clarity). Our model is shown as a solid red line. Our results are compared to $\chi \mathrm{PT}$ calculations (as in Fig. 35), the MAID parametrization for single-pion production, and real photon data at $Q^2=0$ from MAMI [141, 142, 143].

Figure 39

The $Q^2$ dependence of $Q^2{g}_1^p(x,Q^2)$ averaged over a region in $x$ corresponding to $1.08<W<2$ GeV (solid circles) for the proton, with the green band showing systematic uncertainties. The open circles represent the data after adding the contribution from $ep$ elastic scattering. The shaded cyan band represents the range of the averages calculated from extrapolated NLO DIS fits.

Figure 40

Averages of $Q^2{g}_1^p(x,Q^2)$ vs $Q^2$ over limited spans in $x$ corresponding to prominent “resonance regions” as indicated by the ranges in $W$. Symbols are the same as in Fig. 39.