Observation of large missing-momentum $(e,e^{\prime }p)$ cross-section scaling and the onset of correlated-pair dominance in nuclei

We report the measurement of $x_B$ scaling in $(e,e^{\prime }p)$ cross-section ratios off nuclei relative to deuterium at large missing momentum of $350\le p_{\mathrm{miss}}\le 600 \mathrm{MeV}/c$. The observed scaling extends over a kinematic range of $0.7\le x_B\le 1.8$, which is significantly wider than $1.4\le x_B\le 1.8$ previously observed for inclusive $(e,e^{\prime })$ cross-section ratios. The $x_B$-integrated cross-section ratios become constant (i.e., scale) beginning at $p_{\mathrm{miss}}\approx k_F$, the nuclear Fermi momentum. Comparing with theoretical calculations we find good agreement with generalized contact formalism calculations for high missing momentum ($>375 \mathrm{MeV}/c$), suggesting the observed scaling results from interacting with nucleons in short-range correlated (SRC) pairs. For low missing momenta, mean-field calculations show good agreement with the data for $p_{\mathrm{miss}}<k_F$, and suggest a potential non-negligible contribution to the measured cross-section ratios from scattering off single, uncorrelated, nucleons up to $p_{\mathrm{miss}}\approx 350 \mathrm{MeV}/c$. Therefore, SRCs become dominant in nuclei at $p_{\mathrm{miss}}\approx 350 \mathrm{MeV}/c$, well above the nuclear Fermi Surface of $k_F\approx 250 \mathrm{MeV}/c$.

Figure 1

The intrinsic width of the ${}^{12}\mathrm{C}$ missing mass ($M_{\mathrm{miss}}$) distribution, plotted vs $x_B$. Black points show the data. The red curve and uncertainty band shows an SRC based generalized contact formalism (GCF) calculation [19, 30]. The two main model parameters of the calculation, namely SRC pair c.m. momentum distribution width ${\sigma }_{c.m.}$ and the residual $A-2$ system excitation energy $E^{*}$, are fit to the data. Data error bars and calculation error band show the total uncertainty (statistical $+$ systematical) at the $1\sigma$ or $68\%$ confidence level. Inset: The resulting confidence intervals of the correlation between the fitted values of ${\sigma }_{\mathrm{c}.\mathrm{m}.}$ and $E^{*}$. The inner region (red) shows the $1\sigma$ ($68\%$) confidence region with each region increasing the confidence by an additional $1\sigma$. The observed agreement between the data and GCF calculation, and the agreement of the fitted model parameters with previous extractions, show the measured $(e,e^{\prime }p)$ events are consistent with resulting from the hard breakup of SRC pairs.

Figure 2

Measured $(e,e^{\prime }p)$ per nucleon cross-section ratios for $350\le p_{\mathrm{miss}}\le 600 \mathrm{MeV}/c$ for carbon, aluminum, iron, and lead relative to deuterium as function of $x_B$. Open squares are the inclusive $(e,e^{\prime })$ per nucleon cross section ratios of Ref. [26]. The horizontal lines show the average $(e,e^{\prime })$ cross section ratio for $1.45\le x_B\le 1.8$ [26]. Error bars show the data uncertainty (statistical plus point-to-point systematical) at the $1\sigma$ or $68\%$ confidence level. Overall $(e,e^{\prime }p)$ systematic uncertainties of 10% (C) to 15% (Pb) are not shown.

Figure 3

The per-nucleon cross-section ratios for carbon relative to deuterium as a function of $x_B$ (left) and $p_{\mathrm{miss}}$ (right). In the left panel full symbols with different colors indicate different lower limits of the $p_{\mathrm{miss}}$ integration. The upper $p_{\mathrm{miss}}$ limit is fixed at 600 $\mathrm{MeV}/c$. The colored bands mark the statistical plus point-to-point systematical uncertainty of the data at the $\pm 1\sigma$ or $68\%$ confidence level. Open black squares show the inclusive $(e,e^{\prime })$ per nucleon cross-section ratios of Ref. [26]. The horizontal line shows the average $(e,e^{\prime })$ cross section ratio for $1.45\le x_B\le 1.8$ [26]. In the right panel the cross-section ratios are integrated over $0.7\le x_B\le 1.8$. Filled circles show the data. The brown line shows calculated cross sections for scattering off SRC nucleons in carbon using the GCF model while the other lines show calculations for one-body mean-field nucleons obtained using QMC (teal), IPSM (black), and Skryme (azure) models. Data error bars and the widths of the bands around the calculation lines show their uncertainties (statistical plus point-to-point systematical) at the $1\sigma$ or $68\%$ confidence level. In both panels overall data systematic data uncertainties of 10–15 % are not shown.

Figure 4

Same as Fig. 3, for the cross-section ratio of aluminum, iron, and lead relative to deuterium and shown as a function of $x_B$ (left) and $p_{\mathrm{miss}}$ (right).