KINEMATIC COVERAGE

We propose a measurement of inclusive electron scattering from hydrogen and light nuclei. Scattered electrons will be measured in the HMS and SOS spectrometers, which will run independently. The majority of the data will be taken in the HMS, while the SOS will be used to make measurements of electrons from background (charge symmetric) processes and to take additional data at the largest Q² values. All data will be taken at the highest beam energy available (6 GeV assumed for the proposed kinematics). We will take data at 5 angles, over a range of scattered electron energies covering 0.3 < x < 1.0. Data will be taken on hydrogen, deuterium, ³He, ⁴He, and aluminum (for subtraction of the target endcap contributions). This measurement uses the standard Hall C spectrometers and detector packages, the standard hydrogen and deuterium cryotargets, and the ³He and ⁴He targets that were used in recent pion and kaon electroproduction experiments in Hall C.

  

Figure 4: Overview of the proposed kinematics. The dark lines indicate the coverage for hydrogen (limited at high-x values by the rapidly falling cross section) and the grey lines indicate the additional coverage for the nuclear targets. The dashed lines correspond to W²=2.0 and 4.0 (GeV)².

Figure 4 shows the proposed kinematic coverage at 6 GeV ($\theta \leq 60^\circ$) as a function of x and Q². The data above Q² = 4.0(GeV/c)² and at W² > 4.0 (GeV)² (to the left of the dashed line) are in the standardly defined DIS region. For x<0.65, we will have DIS data for EMC ratios, neutron extraction, and tests of models of nuclear effects in deuterium and helium. In the DIS region, we see scaling of the structure function in x, but also in the Nachtmann variable, $\xi = 1 / (1 + \sqrt{1+4m^2x^2/Q^2} )$. $\xi$ can be thought of as a modification to x, taking into account target mass effects. For very large Q², $\xi \rightarrow x$, and so in the DIS limit, the structure function will scale in $\xi$, and $F_2(\xi)$ will be related to the quark momentum distribution in the target, as was the case for x. However, the scaling violations at finite Q² will be smaller when the data is examined in terms of $\xi$ rather than x.

  

Figure 5: Structure function for Iron as a function of $\xi$. The data are taken at fixed scattering angle, and the quoted Q² is the value for x=1. The arrows indicate the value of $\xi$ corresponding to the quasielastic peak for each setting.

  

Figure 6: Structure function for Iron at fixed $\xi$ values as a function of Q². Inner errors are statistical, outer errors are statistical and systematic added in quadrature.

While the data at higher x are below the typical cut for DIS scattering, we believe that the scaling of the structure function will continue. Inclusive measurements designed to probe x>1 [22,23] saw that scaling in scattering from nuclei occurred at kinematics far from the DIS region. Figure 5 shows the structure function for iron plotted against $\xi$. In iron, the smearing caused by the Fermi motion causes resonance structure and even the quasielastic peak disappear at high Q². Once the resonance structure has been washed out, we observe scaling at all $\xi$, both in the resonance region and even when the data is almost entirely dominated by quasielastic scattering. Figure 6 shows the structure function for iron as a function of Q² for several values of $\xi$. The structure function above Q²=2-3 (GeV/c)² is constant to better than 10-20%. Scaling violations resulting from QCD evolution would be expected to cause variations of roughly 10% for large values of x. The largest remaining scaling violations occur at the top of the quasielastic peak (Q² corresponding to x=1 for the fixed $\xi$ value). Both the QCD scaling violations and the violations coming from the QE peak will be reduced at values of x somewhat lower than 1. They will also decrease as Q² increases and the quasielastic contribution becomes a smaller fraction of the total cross section.

Figure 7 shows the structure function for deuterium, as a function of $\xi$. In this case, the quasielastic peak is clearly visible in the structure function, as is the resonance structure at lower Q², and the scaling that was observed in iron breaks down. As Q² increases the peaks move to higher $\xi$, but fall in strength in such a way as to roughly follow the curve in the scaling region. However, for $Q^2 \mathrel{\raise.3ex\hbox{$\gt$}\mkern-14mu \lower0.6ex\hbox{$\sim$}}3$ (GeV/c)² the resonance structure is washed out and even the $\Delta$ resonance is no longer visible. Because deuterium has the lowest Fermi momentum, $\xi$-scaling should break down sooner (at higher W and Q²) in deuterium than in any other nuclear target. The success of $\xi$-scaling in deuterium at extremely low values of W and relatively low momentum transfers leads us to believe that the scaling observed in the DIS region should extend to W² = 2.0 (GeV)² or below for the larger Q² values of this measurement.

  

Figure 7: Structure function for Deuterium as a function of $\xi$. The data are taken at fixed scattering angle, and the quoted Q² is the value for x=1. The arrows indicate the value of $\xi$ corresponding to the quasielastic peak for each setting.

In the Bjorken limit, the parton model predicts that the structure function will scale, and that the scaling curve is directly related to the quark distributions. At finite $\nu$ and Q², in the `DIS' region, scaling is observed, and it is therefore assumed that the structure function is sensitive to the quark structure of the target. It is not clear that this assumption must be correct, but the success of the scaling is taken as a strong indication that it is true. In addition, the quantitative observation of scaling is enough to make some connection between the structure function measured at finite Q² and in the Bjorken limit. If scaling is perfect, than the finite Q² structure function is equal to the high Q² structure function, even if one cannot explicitly show that it must be directly related to the quark distributions. While scaling is not perfect at finite Q², the connection to the high-Q² structure can be made as long as the scaling violations are well understood. Quantitative measurements of the deviation from scaling can be used to determine how precisely the data will match the value that would be measured in the scaling region. If these deviations are small, or are largely independent of the target nucleus, then data taken at lower W² can also be used for measurements of nuclear effects.

  

Figure 8: Hall C measurements of the resonance region structure function at low Q², and a fit to the combined data set.

Finally, one way to interpret the success of $\xi$-scaling in the resonance region is local duality. In 1970, Bloom and Gilman observed [24,25] that the electroproduction of resonances in inclusive e-p scattering was closely related to the scaling limit in DIS scattering. Recent measurements at Jefferson Lab [26,27,28] have examined duality in the proton more carefully. Figure 8 shows the JLab data (plus two SLAC data sets at higher Q²) along with a global fit to the data. For each fixed Q² data set, the resonance region structure function agrees with the fit globally (when averaged over the entire resonance region) and locally when averaged over any prominent resonance. This agreement extends down to Q² = 1.0(GeV/c)² without significant violations. In a nucleus, the Fermi motion of the nucleons averages over the resonances, and so rather than seeing local agreement between the DIS and resonance data, we see scaling at all values of $\xi$. In recent years, several people have begun to look into the theoretical basis for local duality. With a better understanding of the underlying cause of the observed duality, it may be possible to make a rigorous statement on the precision of $\xi$-scaling in nuclei. In the meantime, we can use the precision measurements of duality in the proton, along with our quantitative measurements of $\xi$-scaling in nuclei, to set an upper limit on possible scaling violations as we move from the DIS region into the resonance region. These tests will determine how far in $\xi$ we can extend these measurements while still maintaining the quantitative connection to the quark distributions of the nucleus.