Precision measurements of of the proton and the deuteron with 6 GeV electrons
The inclusive polarized structure functions of the proton and deuteron, and , were measured with high statistical precision using polarized 6 GeV electrons incident on a polarized ammonia target in Hall B at Jefferson Laboratory. Electrons scattered at lab angles between 18 and 45 degrees were detected using the CEBAF Large Acceptance Spectrometer (CLAS). For the usual DIS kinematics, GeV and the final-state invariant mass GeV, the ratio of polarized to unpolarized structure functions is found to be nearly independent of at fixed . Significant resonant structure is apparent at values of up to 2.3 GeV. In the framework of perturbative QCD, the high- results can be used to better constrain the polarization of quarks and gluons in the nucleon, as well as high-twist contributions.
pacs:13.60.Hb, 13.88.+e , 14.20.Dh
Current address:]Universidad Técnica Federico Santa María, Casilla 110-V Valparaíso, Chile Current address:]Old Dominion University, Norfolk, Virginia 23529 Current address:]Christopher Newport University, Newport News, Virginia 23606 Current address:]University of Kentucky, Lexington, KY 40506 Current address:]Los Alamos National Laboratory, Los Alamos, NM 87544 USA Current address:]Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 Current address:]University of Glasgow, Glasgow G12 8QQ, United Kingdom Current address:]Institut de Physique Nucléaire ORSAY, Orsay, France
The CLAS Collaboration
A full understanding of the spin structure of the nucleon, including the determination of the polarized gluon density in particular, is a major goal of hadronic physics. Deep-inelastic lepton scattering from nucleons has proved over the years to be the cleanest tool to study the short-distance structure of the nucleon. The pioneering experiments at SLAC, followed by several generations of experiments at FermiLab, SLAC, CERN, DESY, and elsewhere, have made great strides in determining the spin-averaged parton densities of the quarks and gluons in the proton and neutron. Starting in the 1970’s, experiments using polarized targets have been making steady progress in determining the spin-dependent longitudinal parton densities, although over a more restricted range in momentum fraction and momentum transfer squared due to the lower luminosity available with polarized targets. See Ref. reviewarticle () for a recent review. Initial studies from SLAC and CERN, borne out with increased precision by subsequent experiments at SLAC-E143 (); E155 (), CERN-COMPASS (), DESY-HERMES (), and JLab reviewarticle (); Eg1b (), showed that the up and down quarks sum to only a small fraction of total spin of the nucleon in the framework of perturbative QCD. This implied that the net contribution of polarized gluons, strange quarks, and parton angular momentum must be substantial. A large effort is ongoing to study these contributions.
For gluon spin, two approaches are being followed. The first is to try to isolate specific processes in which a polarized gluon is involved at leading order, for example, photon-gluon fusion leading to a pair of charmed quarks (COMPASS) COMPASS (), or quark-gluon scattering leading to a high energy photon (RHIC-spin) RHIC (). The interpretation of these interactions is complicated due both to background events (other tree-level processes that can lead to the same final state) and higher order QCD corrections reviewarticle ().
The second approach examines the dependence of the spin structure function . Perturbative QCD allows a simple expression of in terms of the quark, antiquark, and gluon distributions , and , which evolve in leading order according to the DGLAP equations DGLAP ():
where is the strong coupling factor, and are the Wilson coefficients, is the number of quark flavors, and indicates a convolution integral.
In the spin-averaged case, pQCD evolution is the benchmark approach to which reaction-specific determinations of the gluon density, , are compared. This is possible due to the high precision of measurements of the spin-averaged structure function over many decades in both and (needed because the logarithmic evolution due to gluon radiation is very slow). In the polarized case, the kinematic range of present precision data is considerably more limited. Nonetheless, the data are of sufficient quality to obtain a very good description of the valence up and down quark densities, and rough indications of the gluon and sea quark densities reviewarticle ().
In practice, fits to data should include the effects of both kinematic and dynamic higher twist, especially in the pre-asymptotic regime of this experiment. We can decompose into a leading twist and a higher twist component lss-2007 () :
The leading twist term can be expressed as a sum of the well-known pQCD term reviewarticle (), the calculable kinematic target mass corrections, and higher order terms:
The dynamic higher twist contribution represents the parton-parton correlations in the target nucleon and is given by:
The dynamic higher twist corrections are non-perturbative in nature and cannot presently be calculated in a model-independent way. However, they can be extracted directly from the experimental data, which has been done recently with CLAS and COMPASS data by the LSS group lss-2007 ().
The goal of the present experiment is to provide much higher statistical precision in measurements of and than was previously available in the kinematic range accessible with 6 GeV electrons at Jefferson Lab. This precision is needed to distinguish between power-law higher twist and logarithmic gluon radiation in the polarized parton densities in the nucleon. Much of the present data also lie in the pre-asymptotic region, and exhibit significant resonance structure, which needs to be taken into account by QCD fits including the higher twist terms. These data will be particularly valuable when combined with planned data with 11 GeV electrons.
Ii The Experiment
ii.1 Experimental Context
This experiment has been carried out at Jefferson Lab, using longitudinally polarized electrons incident on longitudinally polarized protons and deuterons in solid ammonia targets, and the CEBAF Large Acceptance Spectrometer (CLAS) CLAS (). Data from several previous experiments with solid polarized targets have been analyzed and published eg1a (); Eg1b (), and some are still in the analysis stage.
The present experiment used 6 GeV electrons and relatively larger scattering angles to focus on the Deep Inelastic region (DIS). The main focus of this experiment was on semi-inclusive DIS (SIDIS) and deeply virtual Compton scattering (DVCS), both of which required the detection of photons at small angles. For this reason, an Inner Calorimeter (IC) was installed, which blocked scattered electrons below about 18 degrees. The IC is not used in the present inclusive analysis.
ii.2 Experimental Overview
The experiment, named “eg1-dvcs”, used 6 GeV longitudinally polarized electrons from CEBAF at Jefferson Lab impinging on a longitudinally polarized solid ammonia target of 0.025 radiation length, immersed in liquid helium Keith (). Inclusive scattered electrons were detected in CLAS CLAS (). The typical beam current was 7 nA, which, when integrated over the 6 months of data taking, resulted in approximately electrons traversing the target. The beam polarization, as periodically measured using Møller scattering in an upstream polarimeter, averaged 85% for the first three quarters of the experiment. A lower polarization of about 75% was delivered during the remaining time.
About 70% of the running time was on polarized protons (NH target), 20% on polarized deuterons (ND target), 10% on a reference unpolarized carbon target, and 1% on an empty cell. The 1.5-cm-diameter cells typically contained 1 g/cm of material immersed in a 2-cm-long liquid helium bath. The target insert housed 2 such cells, which were called the ”top”, and the ”bottom” cells. In order to minimize loss of polarization from beam heating, the sub-millimeter-diameter beam was uniformly rastered over the front face of the target with a period of a few seconds.
The beam position, averaged over a few minutes or longer, was kept stable at the 0.1 mm level, using feedback from a set of beam position monitors. A split solenoid superconducting magnet provided a highly uniform 5-T magnetic field near the target, which effectively extended about 20 cm up- and downstream from the target center.
The CLAS, comprised of six azimuthally symmetric detector arrays, uses a toroidal field to bend charged particles. Particle momenta and scattering angles were measured with a drift chamber (DC) tracking system with a relative accuracy of 0.3% to 2% in momentum, and about 3 mr in angle. Electrons were separated from a significantly larger flux of charged pions using segmented gas Cherenkov detectors (pion threshold 2.6 GeV) and an electromagnetic calorimeter. A hardware trigger system made the first selection, rejecting about 90% of pions while keeping close to 99% of electrons, in order to not overwhelm the data acquisition system. The hardware Cherenkov and calorimeter thresholds were adjusted to give a trigger rate of about 3000 Hz, and a “dead time” in which no trigger could be accepted of about 10%. An additional unbiased trigger was prescaled by a large factor in order to measure the efficiency of the main electron trigger.
The data taking was divided into three parts: Part A in early 2009 used NH as the target, centered at 58 cm upstream of CLAS center ( cm); Part B (mid 2009) also used NH, this time at cm; and Part C (September 2009) mostly used ND as the target, again with cm. Each part had slightly different primary beam energies (between 5.7 and 6.0 GeV, with several days at 4.8 GeV at the end of Part A). The CLAS torus polarity was set to bend electrons inwards for almost all of parts A and B, and about two thirds of Part C. The field strength was two thirds of the maximum available (2250 A), resulting in better acceptance for low momentum charged particles, but somewhat worse momentum resolution.
Iii Data Analysis
For the present analysis, the only particles of interest were scattered electrons. The spin structure functions were determined from the difference in rates in a particular and bin for beam and target polarizations aligned or anti-aligned. A large background of scattering from unpolarized (or slightly polarized) material in the target was taken into account using a detailed model of the target composition, combined with a fit to world cross section data.
iii.1 Data processing
The raw data consisted of approximately 50,000 files, each about 2 GB in size and corresponding to a few minutes of data taking. The files were grouped into runs consisting of about 80 files, on average, all with the same experimental configuration. Each file was processed with the standard CLAS analysis package InclNote ().
The processed data were corrected using an improved method of tracking particles through the target magnetic field. The standard method in the CLAS reconstruction package swims particles backwards to a plane perpendicular to the appropriate sector. A more accurate method is to swim back to the and coordinates specified by the raster magnet. This method used the track coordinates at the first layer of Drift Chambers and a fit to a large ensemble of simulated forward-swimming particles. This ensured that the track would intersect the beam line for a given raster magnet setting. The net result was about a factor of two improved angular resolution, as verified by the width of the beam energy, , reconstructed from the electron and proton scattering angles in elastic scattering.
iii.2.1 Standard Calibrations
Standard calibration procedures were done for each of the subsystems in CLAS. These included: drift chamber (DC) alignment using straight-through tracks; DC timing alignments; gas Cherenkov detector (CC) pulse height alignment using single photo-electron peaks ; time-of-flight scintillator (SC) timing corrections; and electromagnetic calorimeter (EC) pulse height corrections using cosmic rays. Timing and position resolutions for each of the systems, after calibration, were similar to those obtained during other electron running periods CLAS (). Calibrations were made frequently enough to ensure very good stability for these detector systems.
iii.2.2 Raster Corrections
An additional calibration specific to experiments with polarized targets was done. This was to calibrate the beam position at the target, which depended on steering in the accelerator, as well as the strength of the magnetic field used to raster the beam. The and magnet field (in units of ADC counts) was recorded for each event trigger. By minimizing the width of the reconstructed target position along the beam line (), the relation between magnet current and beam position was determined, as well as the beam position relative to the CLAS center for no raster magnetic field. The fit also determined the target center along the beam line relative to the center of CLAS: -57.95 cm for Part A, and -67.95 cm for parts B and C of the run.
iii.2.3 Magnet Angle Correction
Another calibration specific to this experiment was the determination of the orientation of the target solenoid with respect to the beam line. The primary method used was to make the opening angle of reconstructed electron-positron pairs (from photon conversions) as close to zero as possible, independent of azimuthal angle of the pair relative to the beam line axis. The result of the study was an approximately 3 mr tilt, resulting in a significant polar deflection of charged particles, on top of the azimuthal rotation characteristic of a solenoidal field. An analysis using the over-constrained kinematics of exclusive elastic scattering confirmed the 3 mr tilt.
iii.3 Event selection
In this experiment, the events of interest are those with at least one well-identified electron originating from the target. The two detectors used for particle ID were the threshold gas Cherenkov detector (CC, pion threshold of 2.6 GeV) and the lead-scintillator electromagnetic calorimeter (EC), with a nominal sampling fraction of 0.30. The first level of selection was in the electronic trigger, which required about 1 photo-electron (p.e.) in the CC and an energy deposition of 0.5 GeV in the EC. In software, an electron was identified by first requiring that a time-based drift-chamber (DC) track have negative charge, use at least five of the six super-layers in the tracking system, and have a signal in each of the CC, EC, and scintillator time-of-flight counter system (SC). A further requirement was that the CLAS sector number for each the subsystems (DC, CC, SC, and EC) be matched. We then required eight additional particle ID cuts:
the number of photo-electrons (p.e.) in the CC must be greater than 2 p.e.
The energy in the EC divided by momentum (with 0.12 GeV offset) must be greater than , where is the peak of the distribution (from a first pass through the data), averaged over each run, for each sector separately. This procedure was performed to take out slow drifts in the EC gain, which were especially evident in rare cases when the high voltage on the PMTs had been off for a significant period of time. The offset of 0.12 GeV was obtained from a fit to data with GeV, and can be attributed to the energy losses in the detectors in front of the EC, as well as the minimum ADC signal size used in this analysis. The distribution of EC energy divided by momentum (with 0.12 GeV offset) plotted vs the number of photo-electrons in the Cherenkov detector can be seen in Fig. 2.
The target vertex position along the beam line () must be within 3 cm of the polarized target center.
The difference between the electron’s polar angle at the first drift chamber layer and the azimuthal component of its momentum at the same location must be zero within . Particles with higher are more likely to have scattered from materials that were not part of the target and so can have inaccurate kinematics.
The Cherenkov mirror number must be aligned with the value expected from the track trajectory (as defined by the nearby SC paddle number) within 2 SC paddles. This method uses the feature that the SC paddles and CC mirrors are close together so that their correlation is relatively insensitive to the track trajectories.
The Cherenkov signal time must agree with the EC signal time within 4 nsec.
The electron scattering angle must be less than 40 degrees for Part A, and less than 45 degrees for parts B and C, in order to eliminate large angle particles that could have passed through the significant amounts of the heat shield and super-insulation in the target.
The electrons’ trajectories must not pass too close to the lead shielding around the IC (or the IC support stand). The reason for this cut is that electrons passing through short lengths of the shielding could survive, but with strongly modified angles and momenta from multiple scattering and Bremsstrahlung, thus resulting in an incorrect determination of their original kinematic values.
iii.4 Quality Checks
iii.4.1 Beam Scraping
Careful on-line monitoring ensured that very few data were taken with the beam scraping on the relatively thick and unpolarized target side walls. An off-line check revealed a few files with this problem, which were removed from further analysis.
iii.4.2 Rate Stability
In order to avoid significant corrections to the measured asymmetries, and also to obtain reliable dilution factors from the comparison of ammonia and carbon target rates, we removed data files where an obvious drop in efficiency occurred. The most common reasons for the loss of efficiency were that parts of the DC or EC tripped off.
iii.4.3 Asymmetry Stability
The study of the average inclusive electron double-spin asymmetry proved to be a very valuable quality check. This asymmetry is zero for carbon targets, and generally at least 3-sigma difference from zero for polarized target runs. The sign of the asymmetry indicated whether the overall beam and target polarized signs were correct in the database. Figure 3 shows the final results for the double spin asymmetry as a function of run number. In Part C, the NH and ND targets could also be distinguished due to the much smaller asymmetries for ND compared to NH. After all corrections, the polarized target runs all show a positive asymmetry, while the carbon runs are all consistent with zero.
iii.4.4 Electron detection efficiency
The electron detection efficiency was determined by comparing the rates from the carbon target to those expected, based on fits to world data Adep (). Radiative effects and pair-symmetric backgrounds were taken into account. The typical efficiency was 75%, except for Sector 5, where it was only 40%, due to a leak in the Cherenkov detector. The efficiency was found to decrease with increasing beam current, at a relative rate of about 1%/nA. This decrease is attributed to higher drift-chamber occupancy at higher beam current, resulting in lower tracking efficiency.
iii.4.5 Beam Energy
The beam energy was determined using measurements from the other two experimental halls at JLab, which accurately measure the deflection of the beam in dipole magnets. These results were confirmed using the over-constrained kinematics of exclusive elastic scattering. The results are , , and GeV for Parts A, B and C, respectively. The estimated uncertainty is 5 MeV.
iii.5 Double-spin asymmetries
The double-spin asymmetry was formed in each two-dimensional physics bin using:
for NH and:
for ND, where () are the number of counts in the parallel (anti-parallel) beam helicity bins, is the ratio of incident beam charges for the two helicities, is the bin-averaged dilution factor, defined as the ratio of events from polarized protons or deuterons in the NH or ND target to the total number of events, is the pair-symmetric correction, accounts for polarized nitrogen in the NH and ND targets, accounts for the polarized NH mixed into the ND target, is the beam polarization, is the target polarization, is the measured value of the proton asymmetry, is the deuteron asymmetry, is a radiative dilution factor, and is an additive radiative correction. Each of the factors in Equations 5 and 6 is discussed below. Numerical values can be found in the tables of final results in the Appendix, and are also stored in the CLAS Physics Database clasdb ().
iii.6 Raw Asymmetry
Data processing consisted of determining the number of electron events passing the particle ID cuts mentioned above, for each run, in bins of momentum , polar scattering angle , and helicity (beam and target polarizations either aligned or anti-aligned). The main reason for choosing bins in at the data processing stage, was to allow flexibility later on to slightly vary the beam energy from the nominal value.
Counts were summed for each target and each run period: Part A, 5.9 GeV; Part B, in-bending; Part B, out-bending; Part C, in-bending; and Part C, out-bending.
The bin sizes were 0.04 GeV in momentum and 0.2 degree in , chosen to be small enough to allow re-distribution into physics bins of and ,
where is the invariant mass of the final state.
Here, is the virtual photon energy, equal to the difference in the incident and scattered electron energies: ,
is the four-momentum transfer, and
is the Bjorken scaling variable. During the re-distribution process, the average values of all relevant physics quantities were calculated.
iii.7 Correction Factor
The ratio of incident charges for the two beam helicity states takes into account any difference in the integrated incident current for one state compared to the other. Since the beam helicity was reversed at 30 Hz, and came in alternating helicity buckets, the difference came about only due to helicity-dependent beam current differences, which were kept to less than 1 part in 1000 on a few minute scale by a feedback system, and less than 1 part in 10000, averaged over a month-long time scale. Therefore, effectively = 1.
iii.8 Target and Beam Polarization
For this inclusive analysis, only the product of target polarization () and beam polarization () is important, but not the individual values. For the NH targets, the individual measurements of each quantity (NMR for and Møller scattering for ) had relative systematic uncertainties of over 4%. We therefore used elastic scattering, for which the uncertainty on the measured asymmetry is of order 1% statistical and systematic. Another advantage is that the same runs were used as for the inclusive analysis, so any run-dependent biases would tend to cancel.
We selected electrons using similar particle ID cuts as those used for the inclusive analysis. We selected protons using a nsec cut on the difference in predicted and measured times between the electron and proton, determined with the SC system. The cuts used to select elastic events were:
The missing energy must be less than 120 MeV
The missing longitudinal (transverse) momentum relative to the incident beam direction must be less than 120 MeV (80 MeV).
The beam energy reconstructed from electron and proton scattering angles only must agree with the nominal beam energy to 70 MeV.
The last cut was especially powerful in reducing the background from quasi-elastic events from nitrogen and helium to about 3%. The dilution-corrected double-spin asymmetry was then formed for each of seven bins, and was extracted from the ratio to the theoretically well-known asymmetry for elastic scattering. The values were then averaged over , which ranged from about 2 to 7 GeV. No dependence on was observed. The results are consistent with the product of the individual measurements of and .
|Part A (5.9 GeV)||NH|
|Part B (in-bending)||NH|
|Part B (out-bending)||NH|
|Part C (in-bending)||NH|
|Part C (out-bending)||NH|
|Part C (in-bending)||ND|
|Part C (out-bending)||ND|
For the ND target, the kinematic region where there is the best sensitivity to quasi-elastic scattering was heavily contaminated by the polarization of the 10% NH in the target. On the other hand, the direct deuteron polarization measurements using NMR are more accurate than for the proton, thanks to the “double peak” fitting method, which removes the sensitivity to the hard-to-measure thermal equilibrium signal DULYA ().
We therefore used the count-weighted product of target polarization from NMR and beam polarization from Møller measurements. We estimated the systematic uncertainty of 3% in and 4% in .
Using these values as a constraint, we then fit the coincidence data in the region of small missing momentum to obtain for the protons in the ND target. The results for all targets are given for each set of running conditions in Table 1.
iii.9 Dilution Factor
The dilution factor, , is defined as the fraction of inclusive scattering events originating from polarized hydrogen or deuterium, as compared to the total number of scattering events from all materials in the target. The electron beam in our experiment passed through: helium (He), Kapton (K), ammonia (NH/ND), and aluminum (Al). A schematic diagram of the three target cells is shown in Figure 4.
If we define as the electron scattering rate from a particular target material, we can write for the proton:
The event rate for each material is proportional to the product of the areal density and the inclusive DIS cross section :
where is the volume density and is the length of each material. The constant of proportionality depends on detector acceptance, but since the target materials do not change with time, this constant is the same for the numerator and denominator, and it cancels out of the ratio. The dilution factor for the NH target is then (we replace the symbol NH by a symbol ) :
where is the length of the He bath and is the effective ammonia length. The radiated cross sections are a function of the length of the material in units of radiation length and are obtained by modeling the available world data, and the areal density for each material was measured in the lab or obtained from the literature. The average areal densities of each of the materials are listed in Table 2.
As shown by Eq. 12, to accurately determine the dilution factor we need to know the total length of the He bath () in the target and the packing fraction or ‘length’ () of the NH beads in the target cup.
iii.9.1 Length of the He bath ()
The distance between the aluminum windows that contain the liquid He was measured at room temperature to be 2.3 0.3 cm. The large uncertainty arises because the windows are very thin and hence can bow inwards or outwards, depending on pressure differences.We therefore relied primarily on data taken with the empty target and no helium bath, where two peaks in the vertex- distribution could be clearly seen, corresponding to the windows. The peak separation from these measurements is 2.0 0.1 cm.
A check was made by extracting from measurements of the empty and carbon targets, with and without helium present, and assuming a helium density of 0.145 g/cm.The average value of from a set of ten such measurements was cm.
iii.9.2 Ammonia length (the packing fraction)
The other quantity extracted is the packing fraction of the ammonia beads in the Kapton target cup. This gives us the apparent length of the ammonia target as if it were packed into a solid piece as opposed to crushed beads.
The method used to determine was to fit for the value that best described the measured ratios of electron rates from the carbon and ammonia targets. The predictions used fits to world data taking into account radiative effects using the Mo-Tsai formalism motsai (). The dependence of external radiative corrections on were taken into account in the fits. For each carbon target run, the run on ammonia closest in time was chosen to obtain a value of in order to minimize the effects of slow drifts in average detection efficiency. A summary of the results is presented in Table 3.
|Part||Target Cup||Packing fraction|
iii.9.3 Nh Contamination of the ND Target
The manufacturer’s specification for the deuterium gas used to make the ammonia beads had hydrogen contamination. A check was made using elastic data, and the actual contamination was, surprisingly, found to be an order of magnitude larger. After standard elastic exclusivity cuts, the events from hydrogen, deuterium, and heavier nuclei could clearly be distinguished by plotting the event rates as a function of missing transverse momentum, , relative to the momentum transfer vector . In the case of a free proton, the distribution is a delta-function widened by our experimental resolution. In the case of the deuteron, it is considerably wider due to the average 50 MeV Fermi motion of the proton in the deuteron. In the case of heavier nuclei such as carbon and nitrogen, the peak is another factor of four wider, because the typical Fermi momentum is on the order of 200 MeV. These features are illustrated in Figure 5.
The distribution of for the carbon target is very wide, and that for the NH target has a very narrow peak, sitting on top of a wide distribution with the same shape as for the carbon target. The ND target spectrum has the expected medium-width peak from free deuterons in the target, sitting on top of a nuclear background from nitrogen. Unfortunately, a very narrow peak of the same width as seen in the NH target is also clearly visible. Using data from the carbon and NH targets as a guide, we performed fits to the three components visible in the ND spectra to obtain the relative fraction of free protons and deuterons. The result of the study was that of the ND effective target length was NH, for the in-bending runs, and for the outbending runs. No time-dependence to the contamination was observed within each of these run periods.
iii.9.4 Dilution factor results
Having determined the areal density of each component of each target, we used the method outlined above to calculate the corresponding dilution factor as a function of and . The results for NH, using a typical value cm, are shown in Fig. 6 as a function of for four values of .
Clear resonance structure can be seen corresponding to the three well-established resonance regions near , 1.5, and 1.7 GeV, arising primarily from the -dependence of the ratio of proton to neutron cross sections. The dilution factor for the deuteron shows much weaker structure, because the deuteron and the materials both contain almost equal numbers of proton and neutrons.
The correction factor primarily takes into account the background contribution from electrons produced in decays. The correction also accounts for a smaller background, from mesons misidentified as electrons.
Dalitz decay of the () or Bethe-Heitler conversions of one of the photons from “normal” decay () both produce electron-positron pairs. Since there are equal numbers of electrons and positrons produced from decays, the rate and asymmetry of positrons can be used to correct the electron sample. We therefore define . To compare the electron/positron rates within the same acceptance, with both particles in-bending or both particles out-bending, we use our data taken with opposite torus fields. The results, binned in momentum and polar angle (, ), are shown in Figure 7, for the case where both electron and positron are in-bending (red), and where both are out-bending (green). The average of the two was used to evaluate . It can be seen that the ratios are negligibly small for GeV, independent of , but rise to very large values below GeV. We therefore used the cut GeV, corresponding to , in this analysis.
The ratios of are compared with simulations based on the known rates of pion yields with 6 GeV electron beams Wiser (). The black curves are from decays, while the blue curves are from mis-identified . The simulation accounts quite well for the magnitude and kinematic dependence of the observed ratios. The enhancement seen above 2.65 GeV is a consequence of this value being the threshold for charged pions to produce light in the gas Cherenkov counter.
The functional form of the correction (basically a dilution factor), arises from the assumption that the double-spin asymmetry for inclusive pion production is zero. Our measurements of positron asymmetries were found to be consistent with this assumption, when the positron asymmetry is at least one order of magnitude smaller than the electron asymmetry.
iii.11 Polarized Nitrogen Contribution
The term in Eq. 5 accounts for the polarized nitrogen contribution to the measured double spin asymmetry. From the definition of the raw asymmetry in terms of the physics asymmetries of each of the polarizable nuclei in a target, it is straightforward to show that
where is the number of nuclei of a given species, is the cross section per nucleus, is the double-spin asymmetry (hence is the cross section difference), and is the polarization of a given material. For each of these four variables, the subscript is for nitrogen-14, and refers to either the proton (for the NH target) or deuteron (for the ND target). The first term in Eq. 13 is 1/3, by definition, for ammonia.
In the nuclear shell model, the spin-1 nitrogen-14 nucleus can be considered as a spinless carbon nucleus surrounded by an extra proton and neutron, each in a orbital state Rondon (). After doing the spin projections, it turns out that the proton and neutron are each twice as likely to have their spin anti-aligned with the nitrogen spin, as having it aligned. It then follows that the second factor in Eq. 13 can be evaluated using
where the subscript refers to the neutron, and we have neglected the small d-state correction and used the relation . Inserting this into the second term of Eq. 13, we obtain a constant value of for the ND target and for the NH target. The uncertainty of 0.08 comes from an evaluation SMCNMR () of a range of more sophisticated treatments of the nitrogen wave function than the simple shell model.
The third factor, the ratio of nitrogen to proton (deuteron) polarizations, can be evaluated using Equal Spin Temperature (EST) theory EST (); SMCNMR (). This gives for the average value of this experiment. Experimental measurements from the SMC SMCNMR () are consistent with this result, although about 10% to 15% higher. We therefore used . The EST theory predicts , essentially independent of . An experimental study at SLAC (E143 experiment, unpublished) yielded a much lower value of . We therefore used an average: .
Combining all these results together yields:
To evaluate , the ratio of deuteron to proton cross section per nucleus, we used recent fits to world data christy (); bosted (). To evaluate the ratio of double-spin asymmetries we used an empirical fit to previous world data Eg1bLONGPAPER ().
The values of vary between -0.007 at low and -0.013 at high , with little dependence on .
iii.12 Correction for NH in ND
The term in Eq. 6 accounts for the polarized NH contribution to the measured double spin asymmetry in the nominal ND target. As discussed above, the ND target contained an approximately 10.5% (by weight) admixture of NH (or equivalent), and the protons in this material were polarized. In the following treatment, we consider NH as a background, and we define as
where the variables have the same meaning as in Eq. 13. The derivation of Eq. 17 is valid only if the dilution factor in Eq. 6 is defined using the number of polarizable nucleons in deuterium only (not including the free protons), in the numerator of the ratio.
From the discussion above, the ratio of proton to deuteron nuclei is () for the in-bending (out-bending) portion of Part C. The ratio of proton to deuteron polarizations is () for the in-bending (out-bending) portion of Part C. Numerically, we then obtain for the average values:
iii.13 Radiative Corrections
In our analysis we approximate the scattering process as a one-photon exchange process, also called Born scattering. The higher order processes contributing to the total measured cross sections and asymmetries are taken into account by calculating radiative corrections. The radiative corrections can be broken into two kinds: internal and external. The internal processes occur within the field of the scattering nucleus and consist of the vertex corrections [which effectively account for the running of the fine coupling “constant” ], as well as the emission of hard photons from the incident or scattered electron. External radiation occurs when a Bremsstrahlung photon is emitted from the incident electron prior to scattering (from a different nucleus from which the hard scattering takes place), or a hard photon is radiated by the scattered electron. The probability of emitting a hard photon is approximately given by , where is the material thickness in radiation lengths, and is the photon energy. An important consideration is that an electron is de-polarized by the emission of Bremsstrahlung photons. As a rough guide at JLab energies, internal radiation is equivalent to external radiation when is on the order a few percent. The main difference between the two is that the electron angles are essentially unchanged in external radiation (characteristic angle ), whereas significant changes in the electron scattering at the vertex can occasionally occur in the internal radiation process. For our ammonia targets, the values of relevant for external radiation are about 1.2% for the incident electron and 4% for the scattered electron.
The radiative corrections require the evaluation of both polarized and unpolarized components for Born, internally radiated, and fully radiated cross sections and asymmetries. Polarization-dependent internal radiative cross sections were calculated using the formalism developed by Kuchto and Shumeiko kuchto (); E143 (). External radiation was taken into account by convoluting internal radiative corrections with the spectrum of incident electron energies, rather than a single monochromatic value, according to the formalism of Mo and Tsai motsai (). The external radiation from the scattered electron was similarly taken into account.
The calculations were done with the computer code RCSLACPOL, developed for the E143 experiment at SLAC in the early 1990’s E143 (). The code requires input models for inelastic electron scattering as well as elastic (proton target) or and quasi-elastic scattering. For spin-averaged inelastic cross sections, we used the recent fits to world data of Christy and Bosted christy () for the proton and Bosted and Christy bosted () for the deuteron. Spin-dependent inelastic cross sections were obtained using a recent fit to JLab data nguler ().
In our correction scheme, the radiative corrections are broken into an additive correction , and a radiative dilution factor, . The factor is nothing more than (), where is the fraction of events that have radiated down into a given bin from the elastic scattering process (or quasi-elastic process for the deuteron target). The factor accounts for all other radiative processes. The radiatively corrected asymmetry is then given by
The statistical uncertainty on the corrected asymmetry is given by
The corrections were found to be very small for , the cut used in this analysis, corresponding to typical changes in of less than 1%. The largest effects are in the resonance region, where the asymmetry is changing rapidly with . The error bars with radiative corrections applied are larger than without corrections, with the biggest increase at the largest of a given bin (corresponding to large values of ). The effect of radiative corrections for the deuteron is even smaller than for the proton.
Iv Results and Comparison to Theory
iv.1 Results for
In the one-photon-exchange (Born) approximation, the cross section for inclusive electron scattering with beam and target spin parallel () or anti-parallel () can be expressed in terms of the four structure functions and , all of which can depend on and Eg1bLONGPAPER ():
where the Mott cross section
We can now define the double spin asymmetry as
The measured asymmetry contains contributions from both the and structure functions. After some algebra, the equations can be rewritten as:
where the depolarization factor in this case is given by E143 ()
with and . The ratio of longitudinal to transverse virtual photon absorption cross section is:
where . The correction factor due to , is described in the section below.
iv.1.1 Depolarization Factor and R
This depolarization factor is a function of (Eq. 27), which we obtained from a fit to world data christy (). The fit describes the range GeV and GeV which covers the area of interest for this study. Typical values of vary between 0.1 and 0.3 for both the proton and the deuteron, with a typical uncertainty of 0.03.
The correction factor is given by:
To gauge the rough order of magnitude of the correction, we note that for , , GeV, then
Since is smaller in magnitude than , the overall correction differs by only a few percent from unity, as illustrated in Fig. 8.
The values of used in the analysis were taken from a fit to world data nguler (). In order to estimate the systematic uncertainty on the correction, we also used the assumption that there are no deviations from the twist-two model of Wandzura and Wilczeck () WW (), in which
In this relation, the magnitude of cancels, and the result depends only on the -dependence at higher . The corrections using only the Wandzura and Wilczeck contribution are shown as the dashed curves in Fig. 8. The differences from the world fit of Ref. nguler () are quite small, except for at low .
iv.1.3 Combining Data Sets
Physics quantities that should depend only on [or equivalently ] were first calculated for each beam energy and torus polarity. These physics quantities were then combined, weighted by their statistical uncertainties. The proton results used only Parts A and B, due to the very small amount of proton data in Part C. The deuteron data are from Part C only. Data from each of the individual run periods were compared with the corresponding averages, and found to be consistent within overall normalization uncertainties (dominated by the uncertainty in ).
iv.2 Systematic Uncertainties
In this section we first summarize the systematic uncertainties on arising from each of the terms in Eqs. 5 and 6. We then discuss the systematic uncertainties from three other additional sources, which were assumed to be negligible in these equations.
iv.2.1 Beam charge ratio
The uncertainty in the ratio of incident beam charge for positive and negative helicities (relative to the target polarization direction) was much less than 0.0001 on a few-minute time scale. This negligibly small value was achieved by three methods: 1) frequent reversal of the half-wave plate in the CEBAF injector; 2) keeping the charge asymmetry less than 0.1% using an on-line feedback system; and 3) generating beam helicity buckets in pairs.
iv.2.2 Dilution factor
The dilution factor is one of the two most important sources of systematic uncertainties in the determination of . There are several factors which contribute here:
The ammonia length has an estimated relative uncertainty of 3% for most of the experiment, based on the spread in the individual determinations for pairs of carbon and ammonia runs, combined with an overall uncertainty of 1% in the carbon target areal density.
The length of the He bath, (which determines how much helium there is in the target), has an estimated uncertainty of 0.2 cm, based on inconsistencies between the determination of 2.0 cm from the empty target runs, and 2.3 cm for a direct measurement in the lab when the target was at room temperature.
The areal density of the aluminum He containment windows has an estimated uncertainty of 0.005 g/cm (a relative uncertainty of 10%).
The areal density of the target Kapton windows has an estimated uncertainty of 0.005 g/cm (a relative uncertainty of 5%)
Approximately 3% uncertainties in the density of ammonia and helium at 1 K.
Folding together these uncertainties for our particular target (i.e. about 70% ammonia by areal density, 30% other nuclei), the result is a 1.5% relative uncertainty in , with no significant dependence for GeV.
Although all of the above contributions vanish in the limit of a pure ammonia target, there is still the overall scale uncertainty in the ratios of spin-averaged inclusive cross sections and . Based on the fluctuations between various experiments fit in Refs. bosted (); christy (); Adep (), we estimate the uncertainties be 1.5% in both cases. Combining this with the target-parameter-dependent uncertainty of 1.5%, the total relative uncertainty in is 2.3% for both proton and deuteron.
iv.2.3 Product of beam and target polarization
For the proton, the relative uncertainty on has a statistical component of 1% (averaged over Parts A and B). We estimate a further systematic uncertainty of 1% coming from the choice of cuts and the model for . The net systematic uncertainty on for the proton is therefore 1.4%.
For the deuteron, we estimate an overall normalization uncertainty of 3% to 5% in the target polarization values extracted from the NMR signals by the double-peak method. The systematic uncertainty on the beam polarization from the Møller methods is estimated to be 4%, based on comparison with more accurate measurements made in Halls A and C over a period of many years. The net result is an overall relative systematic uncertainty of 7% in for the deuteron.
iv.2.4 Uncertainty associated with pair-symmetric and mis-identified pion correction
The correction term, ,is negligibly small for GeV (), but rises to values as large as at (), the lowest value of used in the analysis. Based on the disagreement between our two measurements (both particles in-bending, or both out-bending), we assigned a systematic uncertainty of 30% on , corresponding to a systematic uncertainty of up to 3% on at the highest values of .
We assumed that the pair-symmetric asymmetry was zero, however, at low , there are some indications that the scaled contribution could be as large as 1% for GeV for the proton, and as much as 3% for the deuteron. We therefore assigned a relative systematic uncertainty of 1% (3%) to for the proton (deuteron) for GeV.
iv.2.5 Uncertainty associated with radiative corrections
The application of radiative corrections make typically less than 1% changes to . To study the systematic uncertainty, radiative corrections were calculated with several alternate cross section and asymmetry models. No significant changes were observed, at the 0.5% level.
iv.2.6 Uncertainty on
The uncertainty on the value of was estimated to be 0.003 (0.014) for the proton (deuteron), independent of , based on the uncertainties in modeling nitrogen-14, as discussed in Sec. III.11.
iv.2.7 Uncertainty on
The uncertainty on the value of was estimated to be 0.025 (0.038) for the in-bending (out-bending) deuteron asymmetries, independent of . The corresponding relative uncertainty in the deuteron asymmetries and structure functions is given by for the in-bending runs, and for the out-bending runs, where the ratio varies between 2 and 3 over the range of the experiment.
iv.2.8 Detection Efficiency
In Eqs. 5 and 6, we assumed that the detector efficiency was the same for target and beam polarization aligned or anti-aligned. A correction should be made if the overall particle rate is higher for one state than the other, resulting in a rate-dependence to the detection efficiency. We estimated that if the total particle rate were entirely good electrons, the measured asymmetry would be 7% lower than if the detector efficiency were not rate-dependent. Fortunately, the total particle rate was dominated () by photo-produced pions, rather than scattered electrons.
From measurements at SLAC E155pion (), we know that the raw asymmetry in pion production is an order-of-magnitude smaller than for electron scattering. Therefore the correction would be on the order of 0.7%. Taking into account that approximately the same correction applies to elastic scattering (from which we determine ), the net effect on the final electron asymmetry is reduced even further. We therefore made no correction for rate-dependent detection efficiency, and assign an overall systematic uncertainty of 0.7%.
iv.2.9 Parity-violating background
The raw asymmetry arising due to parity-violating inelastic scattering from any of the nucleons in the target is given to a good approximation by , independent of (from the Particle Data Book pdg ()). Since does not depend on target polarization (unlike the double spin asymmetry ), the contribution to cancels, by definition, for equal running times with the target polarization aligned (denoted by ) or anti-aligned () with the beam direction. For unequal running times, the contribution from is reduced by a factor . Averaged over the entire experiment, for NH and for ND. Since the measured raw double-spin raw asymmetry is approximately given by for our average virtual photon energy GeV, the relative parity-violating contribution was less than 0.1% in all bins, and was neglected.
iv.2.10 Summary of systematic uncertainties on
With all of the above uncertainties added in quadrature, the overall relative systematic uncertainty on has an -independent value of 2.8% for the proton and 8.1% for the deuteron. The proton uncertainty is dominated by the uncertainty in , while the deuteron is dominated by the uncertainty in . The only systematic uncertainty that depends strongly on kinematic values for the proton is the pair-symmetric correction, which increases the overall uncertainty to 4.1% (8.6%) for the proton (deuteron) at . For the deuteron, the contribution varies significantly with kinematics also. A summary of systematic uncertainty factors contributing to is presented in Table 4.
iv.3 Systematic Uncertainty on
Eq. 25 shows that the systematic uncertainty on has two additional sources compared to : the uncertainty on (dominated by the uncertainty in ), and the uncertainty on the correction. Since the knowledge of and may improve in the future, we list the values we used in the final results table.
iv.3.1 Uncertainty on
The dominant uncertainty in comes from the uncertainty in (since ). Fortunately, relatively recent precision data from JLab have reduced the overall uncertainty in in the kinematic region of the present experiment to approximately 0.03 (see Ref. christy ()). Since for the present experiment, the uncertainty in introduces a relative uncertainty of 1% to 3% in . Another source of uncertainty in comes from the estimated systematic uncertainties in the beam energy of 10 MeV, the electron momentum of about , and the scattering angle of about 0.5 mr. Taken together, these result in a relative uncertainty in of 0.5% to 1%, with the largest uncertainty at large .
iv.3.2 Uncertainty on the correction
The values of used in the analysis were taken from a fit to world data nguler (). In order to estimate the systematic uncertainty on the correction, we also used the assumption that there are no deviations from the twist-two model of Wandzura and Wilczeck () WW (). We estimated the systematic uncertainty on the correction to be about 3% for both targets.
iv.4 Physics Results
The essential physics results from this analysis are the ratios for the proton and deuteron, from which can be obtained by using fits of world data of . These results can be examined as a function of to look for resonance structure, and as a function of to study quark and gluon spin distributions, taking into account higher-twist contributions.
iv.4.1 as a function of
The results for as a function of are shown for the proton in nine bins of in Fig. 9.
The systematic uncertainty bands are dominated by an overall normalization uncertainty common to all points. The results are in reasonable agreement with the published results from the Eg1b experiment Eg1b (), but have typically five times higher statistical precision, and a factor of two better systematic precision. The Eg1b results are on average a few percent lower than the present results, which is well within the overall systematic uncertainty of the two experiments (about 3% for this experiment and 7% for Eg1b).
The bands at the bottom of each panel represent the total systematic uncertainty (point-to-point as well as overall normalization uncertainties combined). Note the offset from 0 in the vertical axis of most of the panels.
These higher precision results for clearly shows some resonant-like structure for values of as high as 2.4 GeV. To see the higher- structures more clearly, we plot in Fig. 10. In this plot, pairs of adjacent bins have been averaged together.
The downward trend seen in the plot, near GeV can be attributed to transitions to the spin-3/2 . The dip can be understood by realizing that is closely related to
where () represent transitions to final state helicity 1/2(3/2). For a pure transition, .
The strong peak near 1.5 GeV seen in both the and the plots is dominated by transitions to two nucleon resonances, the and states. The spin-1/2 resonance has only helicity = 1/2 transition amplitudes contributing, while the spin-3/2 state has contributions from both helicity = 1/2 and helicity = 3/2 transition amplitudes, and therefore can exhibit a range of asymmetry values, from for dominance to for dominance. As was found in the analysis of single pion () and single eta () exclusive cross section measurements from other CLAS experiments azn1 (); azn2 (), the helicity structure of this state changes rapidly from at GeV to at GeV. In the -range of this experiment both states have to create the strong peak at 1.5 GeV.
The peak near 1.7 GeV has a similar origin. While more resonances contribute in this mass range, the state was found to be by far the most dominant one for a proton target. The helicity structure is similar to the discussed above, and shows a dominant helicity 3/2 transition at GeV, which changes to helicity 1/2 dominance for GeV, resulting in asymmetry , growing with . All of these features have been seen before, at somewhat lower Eg1b (); RSS (). What is interesting is that all three primary resonance regions are visible for values as high as 4 GeV.
What is new in the present results is a confirmation of an apparent dip near GeV, with a possible peak near the GeV (sometimes known as the fourth resonance region). Hints of these structures were seen in the Eg1b results Eg1b (). The strength of the dip near GeV seems to decrease with increasing . This could indicate significant contributions from resonances with dominant helicity = 3/2 transitions. There is a number of states in this mass range that could contribute to the observed structure.
To investigate this possibility further, we are in the process of analyzing exclusive single pion () and eta () electroproduction from the same experiment. The analysis of pion production on neutrons may provide information on the isospin of the resonance contribution.
The results for for the deuteron are shown in Fig. 11. The comparison to the world data fit (dominated by Eg1b results Eg1b ()) is reasonably good. The structure seen in the proton results is not as pronounced, largely due to the factor of six to eight larger error bars, resulting from 1/3 of the integrated luminosity, a factor of almost three lower , and the NH contamination of the target.
iv.4.2 Results for as a function of
The results for as a function of are shown for the proton in nine bins of in Fig. 12. The systematic uncertainty bands are dominated by an overall normalization uncertainty common to all points. Small bin-centering corrections have been applied to the data: typically these are only significant in the highest and lowest bins in each panel.
The bands at the bottom of each panel represent the total systematic uncertainty (with point-to-point and overall normalization uncertainties combined). The dashed red and solid blue curves are representative pQCD calculation from the LSS group with two models for gluon polarization LSS ().
The results for the deuteron are shown in Fig. 13. These data are in good agreement with Eg1b Eg1b () SLAC E143 (); E155 (), COMPASS COMPASS (), and HERMES HERMES (). The data are also reasonably consistent with the fit used for radiative and other corrections (black curves) as well as the two pQCD calculations from LSS LSS (). As for the proton case, the deuteron data show less dependence than either model, and are completely consistent with no -dependence at all for GeV.
The data are consistent with previous CLAS data with 6 GeV electrons Eg1b () (blue points), and higher beam energy data from SLAC E143 (); E155 (), shown as the green points. In the region of overlap, the present data have much higher statistical and systematic precision, allowing for a detailed study of the dependence of at fixed .
The blue and red curves are pQCD calculations from the LSS group LSS () with positive (blue curves) and negative (red curves). In each case, higher twist coefficients were fit to give the best agreement with the data available in 2007. The difference in overall magnitude between the curves is of the same order or larger than our experimental error bars. However, the -dependence is generally larger than in the data, and the magnitude of the curves is above the data at low and below at higher . It appears that the curves with negative agree better with the flat -dependence of our data than the positive curves. Any such conclusions must be treated with great caution, because the data clearly show oscillations around a smooth curve, with a dip seen just below the values corresponding to GeV, shown as the arrows in the figure panels.
We have made the most precise measurements to date of and , in the kinematic range available with a 6 GeV lepton beam. Examined as a function of , the results show resonant structure in the three well-known resonance regions (, 1.5, and 1.7 GeV), as well as a possible peak near 2.2 GeV, with a dip near 1.9 GeV. This is the first time some of these resonant structures are clearly seen at values well above 1 GeV.
Examined in the variables , the most striking result is the almost complete lack of any significant -dependence in the ratio for GeV and . On the other hand, significant oscillations around a smooth curve are observed at values of corresponding to GeV.
A new global pQCD fit that includes our new data should be able to significantly improve the determination of higher twist corrections, and start to shed light on . Such a fit will need to take into account the effects of nucleon resonance structure below of approximately 2.4 GeV. Options include simply placing a cut GeV, using higher-twist forms that take nucleon resonance structure into accounts, or smoothing over resonance structure in some fashion, such as by invoking quark-hadron duality in the spin structure functions Eg1bduality ().
The availability of an 11 GeV electron beam at Jefferson Lab, coupled with a new CLAS12 detector, will soon permit measurements of at higher values of with good statistical precision. These data will be crucial for advancing our knowledge of the spin structure of the nucleon.
We would like to acknowledge the outstanding efforts of the staff of the Accelerator and the Physics Divisions at Jefferson Lab that made this experiment possible. This work was supported in part by the U.S. Department of Energy and the National Science Foundation, the Italian Instituto Nazionale di Fisica Nucleare, the French Centre National de la Recherche Scientifique, the French Commissariat à l’Energie Atomique,the United Kingdom’s Science and Technology Facilities Council, the Chilean CONICYT, and the National Research Foundation of Korea. The Southeastern Universities Research Association (SURA) operates the Thomas Jefferson National Accelerator Facility for the United States Department of Energy under contract DE-AC05-84ER-40150.
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Appendix A Tables of results