Precision Measurement of the Proton and Deuteron Spin Structure Functions g2 and Asymmetries A2

We have measured the spin structure functions g2p and g2d and the virtual photon asymmetries A2p and A2d over the kinematic range 0.02<x<0.8 and 0.7<Q^2<20 GeV^2 by scattering 29.1 and 32.3 GeV longitudinally polarized electrons from transversely polarized NH3 and 6LiD targets. Our measured g2 approximately follows the twist-2 Wandzura-Wilczek calculation. The twist-3 reduced matrix elements d2p and d2n are less than two standard deviations from zero. The data are inconsistent with the Burkhardt-Cottingham sum rule if there is no pathological behavior as x->0. The Efremov-Leader-Teryaev integral is consistent with zero within our measured kinematic range. The absolute value of A2 is significantly smaller than the sqrt[R(1+A1)/2] limit.

Measurements of the longitudinal and transverse spin-dependent structure functions g 1 (x, Q 2 ) and g 2 (x, Q 2 ) for deep-inelastic lepton-nucleon scattering have become an increasingly important tool in unraveling the complex structures of the proton and neutron. Of particular interest are the integrals Γ p 1 (Q 2 ) = 1 0 g p 1 (x, Q 2 )dx for the proton and Γ n 1 (Q 2 ) = 1 0 g n 1 (x, Q 2 )dx for the neutron. Ellis and Jaffe [1] have made sum rule predictions for each integral under the assumptions of SU(3) flavor symmetry and an unpolarized strange sea. Previous measurements of g p 1 [2][3][4][5] have found Γ p 1 to be below the Ellis-Jaffe predictions: this has been interpreted to mean that the strange sea and/or the gluons may be significantly polarized, and that the net quark helicity content of the nucleon may be smaller than expected. A fundamental sum rule originally derived from current algebra by Bjorken [6] predicts the difference Γ p 1 (Q 2 ) − Γ n 1 (Q 2 ). Measurements of Γ n 1 from 3 He [7] and deuterium [8] targets combined with the most recent proton data [5] are in agreement with this sum rule prediction when QCD corrections [9] are included.
In this Letter we report new measurements of g p 1 at moderate Q 2 that have considerably smaller errors than previous electron scattering experiments in the same Q 2 range (SLAC E80 [2], SLAC E130 [3]) and muon scattering experiments at higher Q 2 (EMC [4] and SMC [5]). The present experiment, E143, used the SLAC polarized electron beam with energies E of 9.7, 16.2, and 29.1 GeV scattering from polarized proton and deuteron targets in End Station A (ESA) to measure g p 1 , g p 2 , g d 1 , and g d 2 . This Letter reports only the g p 1 results at E = 29.1 GeV, covering 1.3 < Q 2 < 10 (GeV/c) 2 and 0.029 < x < 0.8.
The longitudinally polarized electron beam was produced by photoemission from a strained-lattice GaAs crystal illuminated by a flash-lamp-pumped Ti-sapphire laser operated at 850 nm [10]. Beam pulses were typically 2 µsec long, contained 2-4×10 9 electrons, and were delivered at a rate of 120 Hz. The helicity was selected randomly on a pulse-topulse basis to minimize instrumental asymmetries. The longitudinal beam polarization P b was measured in ESA using Møller scattering from thin ferromagnetic foils (49% Fe, 49% Co, 2% Va) magnetized by a Helmholtz coil. Results from two detectors (one to detect just one of the final-state electrons, the other to detect both in coincidence) agreed within errors. The high statistics coincidence measurements had smaller total errors since they were essentially free of background, and were used for the final values of P b . Corrections were made for electronics dead time, geometric acceptance, radiative losses in the foil, and a small contribution from the atomic motion of the target electrons (< 1%). Tests including reversing the direction of the foil polarization P f and varying the beam current over a wide range indicated no systematic bias. Values of P b using six foils of varying thickness agreed to better than 1%. P b was observed to vary weakly from 0.83 to 0.86 with the continuously monitored photocathode quantum efficiency (QE). An absolute error of ± 0.02 was assigned to P b , dominated by the uncertainty in P f (measured using an induction coil technique) and the observed spread in the daily P b measurements versus the QE fit.
The beam current was measured for each beam pulse by two independent toroid systems with an uncertainty of < 1%. A steering feedback system kept the average angle and position of the beam at the polarized target essentially constant. Asymmetries induced by changes in beam parameters correlated with helicity were found to be negligible.
The polarized target assembly contained a permeable target cell filled with granules of 15 NH 3 (99.7% isotopic purity) and immersed in a vessel filled with liquid He, maintained at 1 K using a high-power evaporation refrigerator. A superconducting Helmholtz coil provided a uniform field of 4.8 T. The ammonia granules were pre-irradiated [11] with 30 to 350 MeV electron beams to create a dilute assembly of paramagnetic atoms. During the experiment, they were exposed to 138 GHz microwaves to drive the hyperfine transition which aligns the nucleon spins. This technique of dynamic nuclear polarization produced proton polarizations of 65 to 80% in 10 to 20 minutes. The polarization then slowly decreased due to radiation damage: after eight to twelve hours of exposure to the incident electron beam the polarization had dropped to 50 to 55%. Most of the radiation damage was repaired by annealing the target at about 80 K. The electron beam was rastered over the 4.9 cm 2 front surface of the target to uniformly distribute beam heating and radiation damage. After typically ten anneal cycles, the average polarization began to decline and the material was replaced.
The target polarization direction was usually reversed after each anneal by adjusting the microwave frequency. Also, the direction of the magnetic field was reversed several times during the experiment. Approximately equal amounts of data were taken in each of the four polarization/field direction combinations, and the measured asymmetries were consistent for the four data samples. The target polarization P t was measured using a series LCR resonant circuit and Q-meter detector [12]. The inductance was supplied by an NMR coil embedded in the ammonia granules, calibrated by measuring the thermal-equilibrium (TE) signal near 1.6 K with beam and microwaves off. The total relative systematic error on P t was 2.5%, dominated by the observed 2.2% rms spread in the TE measurements.
Scattered electrons with energy E between 6 and 25 GeV were detected in two independent magnetic spectrometers [13] (first used in experiment E142 [7]) positioned at angles of 4.5 • and 7 • with respect to the incident beam. Electrons were distinguished from a background of pions in each spectrometer using two threshold gasČerenkov counters and a 24-radiation-length shower-counter array composed of 200 lead-glass blocks. Seven planes of plastic scintillator hodoscopes were used to measure particle momenta and scattering angles.
The experimental asymmetries A and A ⊥ were determined from where the target polarization is parallel (transverse) to the beam direction for A (A ⊥ ); N − and N + are the number of scattered electrons per incident charge for negative and positive beam helicity, respectively; C N is a correction factor for the polarized nitrogen nuclei; f is the dilution factor representing the fraction of measured events originating from polarizable hydrogen within the target; and A RC is the radiative correction.
The dilution factor f varied with x between 0.13 and 0.17; it was determined from the number of measured counts expected from each component of the 15 NH 3 target, which contained about 13% free protons, 65% 15 N, 10% 4 He, 6% Al, 5% Cu, and 1% Ti by weight.
The relative systematic error in f ranged from 2.2 to 2.6%, as determined from uncertainties in the target composition and uncertainties in the expected ratios of cross sections from different nuclei.
The dead-time corrected rates N − and N + were adjusted for contributions from secondary sources (such as e + /e − pair production from photons) measured by reversing the spectrometer polarity. These processes showed no measurable asymmetry, and the corrections to the rates were 10% at the lowest x bin, decreasing rapidly at higher x. The factor C N varied from 0.97 to 0.98, depending on target polarization, and was determined from measured 15 N polarizations and a shell-model calculation to determine the contribution of the unpaired p-shell proton.
The internal radiative corrections for both A and A ⊥ were evaluated using the formulae of Kukhto and Shumeiko [14]. The cross section components of the asymmetry were "externally radiated" according to Tsai [15] to form the "fully radiated" asymmetry corrections tio of virtual photon total absorption cross sections R = σ L /σ T we used a global fit [16]. The ratio g 1 /F 1 is related to the virtual photon longitudinal asymmetry A 1 = (g 1 /F 1 )−γ 2 (g 2 /F 1 ),

The approximation
The values of g p 1 /F p 1 from this experiment at E = 29.1 GeV are listed in Table I and are displayed in Fig. 1 along with results of previous experiments. Data from the two spectrometers are consistent in the overlap region 0.07 < x < 0.55, and therefore have been averaged together. Since the two data sets differ by about a factor of 2 in average Q 2 , the comparison indicates no strong Q 2 -dependence for g p 1 /F p 1 . The systematic errors include contributions from P b , P t , f , and A RC discussed above, as well as 3 to 5% in d arising from the uncertainty in R.
It can be seen in Fig. 1 that both the previous SLAC data [3] and the higher Q 2 SMC data [5] (< Q 2 >= 10 (GeV/c) 2 ) are in agreement with the data of this experiment, indicating that to a good approximation, g 1 /F 1 is independent of Q 2 over the (x, Q 2 ) range where this ratio has been measured. The SLAC E130 data are plotted assuming A ⊥ = 0 (the experiment measured A only), and the SMC data are plotted assuming g 1 /F 1 ≈ A 1 , which is a good approximation at their beam energy of 190 GeV.
Values of xg p 1 at the average Q 2 = 3 (GeV/c) 2 of this experiment are shown in Fig. 2.
The evaluation at constant Q 2 is model-dependent, and we made the assumption that g p 1 /F p 1 depends only on x [17]. For F p we used the NMC fit [18] to F p 2 and the SLAC fit [16] to R. Using the SLAC global fit [19] to F p 2 gives similar results. The systematic errors on g p 1 include an x-dependent error on the ratio F p 1 /d which varies from 2.5% in the mid-x range to 4% at low x and 10% at high x. The integral of g p 1 over the measured range 0.029 < x < 0.8 is proportional to the area under the data points in Fig. 2, yielding 0.8 0.029 g p 1 (x)dx = 0.120 ± 0.004 ± 0.008, where the first error is statistical, and the second is systematic. We note that the integral is decreased by 0.006 if we assume that both A 1 and A 2 are independent of Q 2 instead of assuming that g 1 /F 1 is independent of Q 2 .
An extrapolation from x = 0.8 to x = 1 was done assuming g 1 is proportional to (1 − x) 3 at high x; this yields 1 0.8 g p 1 (x)dx = 0.001 ± 0.001. The extrapolation to x = 0 is more model dependent, and could be large if g p 1 were to increase strongly at low x. We studied the x-dependence of g p 1 using our data combined with SMC and EMC data. We observe that, consistent with Regge theory [20], all data for x < x max = 0.1 are well-fit (χ 2 /d.f.=0.9) by a constant value of g p 1 = 0.29 ± 0.02 at Q 2 = 3 (GeV/c) 2 (see curve on Fig. 2). We use this value to estimate 0.029 0 g p 1 (x)dx = 0.008 ± 0.001 ± 0.005. The systematic error was estimated by varying x max for the fit from 0.03 (for which only SMC and EMC data contribute) to 0.12 (for which the present data dominate). Given the two assumptions that g 1 /F 1 depends only on x and that g p 1 is constant at low x, we obtain the total integral Γ p 1 = 0.129 ± 0.004 ± 0.009. This is in good agreement with the value from SMC [5] asymmetries, Γ p 1 = 0.122±0.011±0.011, obtained at Q 2 = 3 (GeV/c) 2 assuming g 1 /F 1 ≈ A 1 is independent of Q 2 . Our result is more than two standard deviations below the Ellis-Jaffe sum rule prediction of 0.160 ± 0.006, evaluated using the QCD corrections of Ref. [21] with α s = 0.35 ± 0.05 at Q 2 = 3 (GeV/c) 2 [22].
We can use the quark-parton model and the SU(3) coupling constants F + D = 1.2573 ± 0.0028 and F/D = 0.575 ± 0.016 [23] to extract the total quark contribution to the proton helicity ∆q = ∆u + ∆d + ∆s = 0.29 ± 0.10, which is small compared to the Ellis-Jaffe prediction 3F − D ≈ 0.58 for ∆s = 0. Using our value of ∆q along with the definition 3F − D = ∆q − 3∆s, we find the strange quark helicity contribution ∆s = −0.10 ± 0.04, which is negative and inconsistent with zero.
For Q 2 = 3 (GeV/c) 2 and for three flavors, the Bjorken sum rule prediction with third order QCD corrections [9] is Γ p−n  [8] gives a result that is also in agreement with the prediction, within larger errors. More data, including the deuteron data from this experiment, will be useful in improving the accuracy with which the Bjorken sum rule can be tested, and in learning about the xand Q 2 -dependence of the nucleon spin structure functions.
We wish to acknowledge the SLAC staff for making this experiment successful; H. Wiedemann and the SSRL, and the staffs of the Bates Linear Accelerator, CEBAF, and the Saskatoon Accelerator Laboratory for providing electron beams to pre-irradiate the target mate- at low x to 10 (GeV/c) 2 at high x. Also shown are data from SLAC E130 [3] and SMC [5].