Measurements of the Q 2 -dependence of the Proton and Deuteron Spin Structure Functions G the E143 Collaboration ?

The ratio g1=F1 has been measured over the range 0:03 < x < 0:6 and 0:3 < Q2 < 10 (GeV/c)2 using deep-inelastic scattering of polarized electrons from polarized protons and deuterons. We nd g1=F1 to be consistent with no Q2-dependence at xed x in the deep-inelastic region Q2 > 1 (GeV/c)2. A trend is observed for g1=F1 to decrease at lower Q2. Fits to world data with and without a possible Q2-dependence in g1=F1 are in agreement with the Bjorken sum rule, but q is substantially less than the quark-parton model expectation. The longitudinal spin-dependent structure function g1(x;Q2) for deep-inelastic leptonnucleon scattering has become increasingly important in unraveling the quark and gluon spin structure of the proton and neutron. The g1 structure function depends both on x, the fractional momentum carried by the struck parton, and on Q2, the fourmomentum transfer squared of the virtual photons used as a probe of nucleon structure. Of particular interest are the xed-Q2 integrals p1(Q2) = R 1 0 gp 1(x;Q2)dx for the proton and n1 (Q2) = R 1 0 gn 1 (x;Q2)dx for the neutron. These integrals are directly related to the net quark helicity q in the nucleon. Measurements of p1 [1{5], d1 [6{7], and n1 [8] have found q 0:3; signi cantly less than a prediction [9] that q = 0:58 assuming zero net strange quark helicity and SU(3) avor symmetry in the baryon octet. A fundamental sum rule originally derived from current algebra by Bjorken [10] predicts the di erence p1(Q2) n1 (Q2). Recent measurements are in agreement with this sum rule prediction when perturbative QCD (pQCD) corrections [11] are included. There are two main reasons for measuring g1 over a wide range of x and Q2. The rst is that experiments make measurements at xed beam energies rather than at xed Q2. To evaluate rst moment integrals of g1(x;Q2) at constant Q2 [typically between 2 and 10 (GeV/c)2], extrapolations are needed. Data at low x are at lower Q2 than desired [as low as 1 (GeV/c)2], while data at high x are at higher Q2 [up to 80 (GeV/c)2]. Data at multiple beam energies allow for a measurement of the kinematic dependence of g1, rather than relying on model-dependent extrapolations for the moment determinations. The second motivation is that the kinematic dependence of g1 can be used to obtain the underlying nucleon polarized quark and gluon distribution functions. According to 2 the GLAP equations [12], g1 is expected to evolve logarithmically with Q2, increasing with Q2 at low x, and decreasing with Q2 at high x. A similar Q2-dependence has been observed in the spin-averaged structure functions F1(x;Q2) and F2(x;Q2). For reference, in changing Q2 from 2 to 10 (GeV/c)2, F1 decreases by 40% for x 0:5, but increases by the same amount for x 0:035 [13,14]. Since the GLAP equations are similar for F1 and g1, the Q2 dependence of g1 is expected to be similar to that of F1, but the precise behavior is sensitive to the underlying spin-dependent quark and gluon distribution functions. Fits to polarized quark and gluon distribution functions have been made [15{19] using leading-order (LO) GLAP equations and data for g1(x;Q2). Because of the limitedQ2 range and statistical precision of the data, constraints from QCD counting rules and Regge theory on the x-dependence have generally been imposed. Recently, ts have also been made [20,21] using next-to-leading-order (NLO) GLAP equations [17]. The results indicate that NLO ts are more sensitive to the strength of the polarized gluon distribution function G(x;Q2) than LO ts. The theoretical interpretation of g1 at low Q2 is complicated by higher twist contributions not embodied in the GLAP equations. These terms are expected to be proportional to C(x)=Q2, D(x)=Q4, etc., where C(x) and D(x) are unknown functions. Higher twist contributions to the rst moments p1 and n1 have been estimated to be only a few percent [22] for Q2 > 3 (GeV/c)2, but very little is known about their strength as a function of x. In this Letter we study the Q2 dependence of g1 by supplementing our previously published results for gp 1 [5], gd 1 [7], and gp 2 and gd 2 [24] measured at average incident electron beam energy E of 29.1 GeV with data for gp 1 and gd 1 at beam energies of 9.7 and 16.2 GeV. Data at all energies were taken at scattering angles of 4:5 and 7 . The ratio of polarized to unpolarized structure functions was determined from measured longitudinal asymmetries Ak using g1=F1 = Ak=d + (g2=F1)[(2Mx)=(2E )] ; (1) where d = [(1 )(2 y)]=fy[1+ R(x;Q2)]g, y = =E, = E E0, E0 is the scattered electron energy, 1 = 1 + 2[1 + 2] tan2( =2), 2 = Q2= 2, is the electron scattering angle, M is the nucleon mass, and R(x;Q2) = [F2(x;Q2)(1 + 2)]=[2xF1(x;Q2)] 1 is typically 0.2 for the kinematics of this experiment [14]. For the contribution of the 3 transverse spin structure function g2 we used the twist-two model of Wandzura and Wilczeck (gWW 2 ) [23]g2(x;Q2) = g1(x;Q2) + Z 1 x g1( ;Q2)d = ; (2) evaluated with g1 based on a global t to the virtual photon asymmetry A1 (see ts V, Table I). The g1 and g2 structure functions are related to A1 (which is bounded by jA1j < 1) by A1 = (g1=F1) 2(g2=F1). The gWW 2 model is in good agreement with our g2 data at E = 29 GeV [24], the only energy at which both Ak and the transverse asymmetry A? were measured. Using other reasonable models for g2 (such as g2 = 0) has relatively little impact on the results for g1 due to the factor 2Mx=(2E ) in Eq. 1. The data analysis was essentially identical to that reported for the 29 GeV data [5,7], with Ak calculated from the di erence over the sum of rates for scattering longitudinally polarized electrons with spin either parallel or anti-parallel to polarized protons or deuterons in a cryogenic ammonia target. The most important corrections made were for the beam polarization (typically 0:85 0:02), target polarization (typically 0:65 0:017 for NH3, 0:25 0:011 for ND3), fraction of polarizable nucleons (0.12 to 0.17 for NH3, 0.22 to 0.24 for ND3), and for contributions from polarized nitrogen atoms. Radiative corrections were calculated [25] using iterated global ts to all data (see ts V in Table I). The data at 29 GeV used here di er slightly from our previously published results [5,7] due to the new radiative corrections, the inclusion of more data runs, and improvedmeasurements of the polarization of the target and beam. Data in the it resonance region de ned by missing mass W < 1:8 GeV were not included in the present analysis, but those for Q2 below the traditional deep-inelastic cuto of Q2 = 1 (GeV/c)2 were kept. The results for gp 1=F p 1 and gd 1=F d 1 are shown in Figs. 1 and 2, respectively, at eight values of x, and are listed in Table II. We display the ratio g1=F1 since it is closer to our measured asymmetries than g1 alone, and because g1 and F1 are expected theoretically to have a similarQ2 dependence, so that di erences are emphasized in the ratio. Data from other experiments [1{4,6] are plotted using published longitudinal asymmetries Ak and the same model for R(x;Q2) [14] and g2 [23] as for the present data. Improved radiative corrections have been applied to the E80 [1] and E130 [2] results. Only statistical errors have been plotted. For the present experiment, most systematic errors (beam polarization, target polarization, fraction of polarizable nucleons in the target) for a 4 given target are common to all data and correspond to an overall normalization error of about 5% for the proton data and 6% for the deuteron data. The remaining systematic errors (radiative corrections, model uncertainties for R(x;Q2), resolution corrections) vary smoothly with x in a locally correlated fashion, ranging from a few percent for moderate x bins, up to 15% for the highest and lowest x bins at E = 29 GeV. For all data, the statistical errors dominate over the point-to-point systematic error. The most striking feature of the data is that g1=F1 is approximately independent of Q2 at xed x, although there is a noticeable trend for the ratio to decrease for Q2 < 1 (GeV/c)2. To quantify the possible signi cance of this trend, we made two ts to the data. The rst t is motivated by possible di erences in the twist-4 contributions to g1 and F1. We t the data in each x bin with the form g1=F1 = a(1+C=Q2). The results for the C coe cients are shown in Fig. 3 for all Q2 [Q2 > 0:3 (GeV/c)2] (circles) and for Q2 > 1 (GeV/c)2 (squares). The coe cients indicate signi cantly negative values for C at intermediate values of x for the ts over all Q2. The errors are much larger when data with Q2 < 1 (GeV/c)2 are excluded, and the resulting coe cients are consistent with no Q2-dependence to g1=F1 (C = 0). There is no evidence for a signi cant x-dependence to C. Another t to the data in each x bin used the form g1=F1 = a[1 + C ln(1=Q2)], motivated by looking for di erences in the logarithmic evolution of g1 and F1. Again, the C coe cients tend to be less than zero when no Q2 cut is applied. The present data do not have su cient precision to distinguish between a logarithmic and power law Q2 dependence, but can rule out large di erences between the Q2-dependence of g1 and F1, especially for Q2 > 1 (GeV/c)2. Shown in Figs. 1 and 2 as the dot-dashed curves are the low-Q2 predictions from a representative global NLO pQCD t [20] to all proton and deuteron data excluding those at the 9.7 GeV and 16.2 GeV beam energies of this experiment. This group [20] nds considerably less Q2 dependence to g1=F1 when a minimal polarized gluon strength is used than when a maximal strength is chosen. Another group has made NLO pQCD ts to proton, deuteron, and neutron data using di erent constraints on the underlying parton distribution functions [21], examining the sensitivity to SU(3) symmetry breaking in the baryon decays. The results for their standard set are shown as the dotted curves in Figs. 1 and 2. Both [20] and [21] predict that gp 1=F p 1 increases with Q2 in the moderate x range (0:03 < x < 0:3), i