Structure functions for light nuclei

We discuss the nuclear EMC effect with particular emphasis on recent data for light nuclei including ${}^2\mathrm{H}$, ${}^3\mathrm{He}$, ${}^4\mathrm{He}$, ${}^9\mathrm{Be}$, ${}^{12}\mathrm{C}$, and ${}^{14}\mathrm{N}$. In order to verify the consistency of available data, we calculate the ${\chi }^2$ deviation between different data sets. We find a good agreement between the results from the NMC, SLAC E139, and HERMES experiments. However, our analysis indicates an overall normalization offset of about $2\%$ in the data from the recent JLab E03-103 experiment with respect to previous data for nuclei heavier than ${}^3\mathrm{He}$. We also discuss the extraction of the neutron/proton structure function ratio $F_2^n/F_2^p$ from the nuclear ratios ${}^3\mathrm{He}$/${}^2\mathrm{H}$ and ${}^2\mathrm{H}$/${}^1\mathrm{H}$. Our analysis shows that the E03-103 data on ${}^3\mathrm{He}$/${}^2\mathrm{H}$ require a renormalization of about $3\%$ in order to be consistent with the $F_2^n/F_2^p$ ratio obtained from the NMC experiment. After such a renormalization, the ${}^3\mathrm{He}$ data from the E03-103 and HERMES experiments are in a good agreement. Finally, we present a detailed comparison between data and model calculations, which include a description of the nuclear binding, Fermi motion, and off-shell corrections to the structure functions of bound proton and neutron, as well as the nuclear pion and shadowing corrections. Overall, a good agreement with the available data for all nuclei is obtained.

Figure 1

The $\mathcal{R}$ ratios of ${}^{14}\mathrm{N}$, ${}^{12}\mathrm{C}$, ${}^9\mathrm{Be}$, ${}^4\mathrm{He}$ with respect to deuterium and the ratio ${}^9\mathrm{Be}$/${}^{12}\mathrm{C}$ as measured by the NMC [4, 5, 6] (full squares), SLAC E139 [7] (full circles), JLab E03-103 [12] (open circles), and HERMES [8] (full triangles) experiments. Statistical and systematic uncertainties are added in quadrature, while the normalization uncertainty is not shown. For the E139 and E03-103 experiments the ratio ${}^9\mathrm{Be}$/${}^{12}\mathrm{C}$ is calculated as a double ratio of data on ${}^9\mathrm{Be}$/${}^2\mathrm{H}$ and ${}^{12}\mathrm{C}$/${}^2\mathrm{H}$.

Figure 2

Ratios of cross sections for positron scattering off ${}^{14}\mathrm{N}$ and ${}^{84}\mathrm{Kr}$ nuclear targets with respect to a deuterium target. Data from the HERMES experiment [8] (full triangles) are compared with predictions of Ref. [17] calculated for the same kinematics (solid lines). Statistical and systematic uncertainties are added in quadrature, while the normalization uncertainty is not shown. The data from the NMC experiment for ${}^{12}\mathrm{C}$ target and the results of Ref. [17] (dash-dotted line) are also shown in the top panel to illustrate the effect of the $Q^2$ dependence at low $x$ values (see text).

Figure 3

Data on the $\mathcal{R}$ ratios of ${}^{12}\mathrm{C}$, ${}^9\mathrm{Be}$, and ${}^4\mathrm{He}$ with respect to deuterium compared with predictions of the model of Ref. [17] for the same kinematics. A common normalization factor of $0.98$ has been applied to all data points of Ref. [12], and statistical and systematic uncertainties are added in quadrature. The result of a calculation in impulse approximation with no off-shell correction is also shown as dashed-dotted line for comparison.

Figure 4

The results of the extraction of the ratio $F_2^n/F_2^p$ from data on the ${}^2\mathrm{H}$/${}^1\mathrm{H}$ cross-section ratio by the NMC experiment [32] (green squares) and the data on the ${}^3\mathrm{He}$/${}^2\mathrm{H}$ cross-section ratio by JLab E03-103 experiment [12] (red circles). The full symbols are obtained with the model of nuclear effects of Ref. [17], while the open symbols show the points obtained with no nuclear corrections. The solid curve is the result of a calculation according to Eq. (8) with the NNLO PDFs and the HT corrections of Refs. [15, 25, 33] for the $Q^2$ corresponding to the NMC kinematics. The dashed line is a $Q^2$-independent model used in Ref. [7].

Figure 5

The ratios $R_2$ and $R_3$ calculated at the values of $x$ and $Q^2$ of Ref. [12] for $x>0.3$ and at fixed $Q^2=3$ ${\mathrm{GeV}}^2$ for $x<0.3$.

Figure 6

Ratio of cross sections for electron scattering off ${}^3\mathrm{He}$ with respect to a deuterium target, corrected for the proton excess. Data from the HERMES experiment [8] (triangles) and the JLab E03-103 experiment [12] (open circles) are compared with the E03-103 raw data renormalized by a factor $1.03$ and corrected for the proton excess according to Eq. (5) and using the proton and neutron structure functions of Refs. [15, 33] (full circles). Our predictions calculated for the same kinematics with the complete model described in this article (solid line) and in the impulse approximation (dashed line) are also shown for comparison. See text for details.

Figure 7

The isoscalar correction for ${}^3\mathrm{He}$ calculated by Eq. (5) using different models of $F_2^n/F_2^p$.