Connection to Deep Inelastic Scattering (DIS)

The response of the nucleus in the range $x > 1$ is expected to be composed of both deep-inelastic scattering from quarks in the nucleus and elastic scattering from the bound nucleons (quasielastic scattering). For both the bound quark and bound nucleon cases it is the non-zero momentum of the bound nucleons that permits scattering into a kinematic region that is forbidden for the free nucleon. The scattering from quarks should exhibit scaling in the Bjorken $x$ variable (experimentally verified for $x < 1$), while the scattering from the nucleons exhibits $y$ scaling (discussed below). However the respective scaling functions for the two processes appear to be dramatically different. It is the inclusive structure functions (e.g. $\nu W_2^A$) that scale for the quark case while it is the cross section weighted by the elastic form factors [$G_E(Q^2)$ and $G_M(Q^2)$] that exhibits scaling for the nucleon case. In a simple impulse approximation (Quark-Parton model for quark scattering, quasielastic (QE) nucleon scattering for the nucleon scattering) the DIS scaling functions are related to the quark momentum distributions in the nucleus, while the quasielastic scaling function is related to the nucleon momentum distributions. It is the weighting by the elastic form factors, which fall with a high power of $Q^2$, that causes the quasielastic response to vanish in the limit of $Q^2 \rightarrow \infty$. In this limit the deep inelastic scattering from quarks should dominate the response even for $x > 1$. Thus the two types of scaling appear to be significantly different. A possible connection between the two has been suggested in several analyses of the previous data [1,2,3]. Here the nuclear structure function is taken versus the Nachtmann scaling variable $\xi$, and an interesting scaling (for all $\xi$) is suggested by the data [4] (Fig. 1). $\xi$ is a modified version of the deep inelastic scaling variable ( $\xi \rightarrow x$ as $Q^2 \rightarrow \infty$) that takes into account target mass effects and thus reduces scaling violations at finite $Q^2$ values. The $Q^2$ range of the previous SLAC data was too limited to draw firm conclusions about the nature of this scaling. One theoretical analysis [5] suggested that the observed scaling is accidental and would break down at larger $Q^2$. A more recent work [6] explains $\xi$-scaling as an approximation to scaling in $\xi_{QE}$, which is analogous to $\xi$ but describes scattering from quasifree nucleons in the nucleus. For both of these explanations, the scaling in $\xi$ is described as an approximation to scaling for quasielastic scattering, where scaling violations coming from the transformation from $y$ ($\xi_{QE}$) to $\xi$ are either small or cancelled by other contributions. However, in the kinematics covered by the previous JLab experiment with a 4 GeV beam, the scaling violations that come from the change of scaling variables are much larger than the observed scaling violations [4].

Figure 1: Structure function per nucleon for Fe vs. the Nachtmann scaling variable from Jefferson Lab E89-008. The $Q^2$ values are given for Bjorken $x=1$. Errors shown are statistical only.

The connection between the inelastic and quasielastic regions seems to be a consequence of duality, as observed first by Bloom and Gilman [7], and studied more precisely in recent Jefferson Lab experiments [8]. In the proton, it was observed that the resonance region structure function, averaged over the resonances, is identical to the DIS structure function. In the nucleus, the Fermi motion of the nucleons performs this averaging and duality yields true scaling, rather than scaling on average, in regions where the intrinsic averaging is sufficient. While this explains the scaling in the resonance region, it is not clear why the scaling works so well for $\xi > 1$, where at moderate $Q^2$ we are sensitive only to the quasielastic contributions, and where we average only over part of the quasielastic peak.

In addition to providing information about the scaling behavior at $x > 1$, these measurements provide the necessary data to perform precise moment analyses of nuclei. Current moment analyses are limited at moderate to high $Q^2$ values by the knowledge of the structure function at $x > 1$, especially for the higher moments. Combining this data with lower $x$ measurements from duality studies of hydrogen and deuterium [9,10] and other planned measurements of light nuclei [11] will allow a more precise determination of the first several moments of the nuclear structure function. A comparison of the moments of deuterium and hydrogen may allow a determination of the moments for the neutron without some of the theoretical ambiguities that arise when attempting to directly extract the neutron structure function from data on nuclei.

Exploring the transition from Quasielastic scaling (i.e. $y$-scaling) to DIS scaling ($x$-scaling) requires measurements at the highest possible $Q^2$. Measurements with a 6 GeV beam will significantly extend the accessible $Q^2$ range compared to what is possible with a 4 GeV beam. Comparisons of deuterium and heavy nuclei at $x > 1$ for high $Q^2$ allows one to study scattering from high momentum partons, as well as allowing searches for modifications of quark distributions due to the nuclear medium in a new kinematic regime.