Dynamics of the quasielastic ${}^{16}\mathrm{O}(e,e^{\prime }p)$ reaction at $Q^2\approx 0.8{(\mathrm{GeV}∕c)}^2$

The physics program in Hall A at Jefferson Lab commenced in the summer of 1997 with a detailed investigation of the ${}^{16}\mathrm{O}(e,e^{\prime }p)$ reaction in quasielastic, constant $(q,\omega )$ kinematics at $Q^2\approx 0.8{(\mathrm{GeV}∕c)}^2$, $q\approx 1\mathrm{GeV}∕c$, and $\omega \approx 445\mathrm{MeV}$. Use of a self-calibrating, self-normalizing, thin-film waterfall target enabled a systematically rigorous measurement. Five-fold differential cross-section data for the removal of protons from the $1p$-shell have been obtained for $0<p_{\mathrm{miss}}<350\mathrm{MeV}∕c$. Six-fold differential cross-section data for $0<E_{\mathrm{miss}}<120\mathrm{MeV}$ were obtained for $0<p_{\mathrm{miss}}<340\mathrm{MeV}∕c$. These results have been used to extract the $A_{LT}$ asymmetry and the $R_L$, $R_T$, $R_{LT}$, and $R_{L+TT}$ effective response functions over a large range of $E_{\mathrm{miss}}$ and $p_{\mathrm{miss}}$. Detailed comparisons of the $1p$-shell data with Relativistic Distorted-Wave Impulse Approximation (RDWIA), Relativistic Optical-Model Eikonal Approximation (ROMEA), and Relativistic Multiple-Scattering Glauber Approximation (RMSGA) calculations indicate that two-body currents stemming from meson-exchange currents (MEC) and isobar currents (IC) are not needed to explain the data at this $Q^2$. Further, dynamical relativistic effects are strongly indicated by the observed structure in $A_{LT}$ at $p_{\mathrm{miss}}\approx 300\mathrm{MeV}∕c$. For $25<E_{\mathrm{miss}}<50\mathrm{MeV}$ and $p_{\mathrm{miss}}\approx 50\mathrm{MeV}∕c$, proton knockout from the $1s_{1∕2}$-state dominates, and ROMEA calculations do an excellent job of explaining the data. However, as $p_{\mathrm{miss}}$ increases, the single-particle behavior of the reaction is increasingly hidden by more complicated processes, and for $280<p_{\mathrm{miss}}<340\mathrm{MeV}∕c$, ROMEA calculations together with two-body currents stemming from MEC and IC account for the shape and transverse nature of the data, but only about half the magnitude of the measured cross section. For $50<E_{\mathrm{miss}}<120\mathrm{MeV}$ and $145<p_{\mathrm{miss}}<340\mathrm{MeV}∕c$, $(e,e^{\prime }pN)$ calculations which include the contributions of central and tensor correlations (two-nucleon correlations) together with MEC and IC (two-nucleon currents) account for only about half of the measured cross section. The kinematic consistency of the $1p$-shell normalization factors extracted from these data with respect to all available ${}^{16}\mathrm{O}(e,e^{\prime }p)$ data is also examined in detail. Finally, the $Q^2$-dependence of the normalization factors is discussed.

Figure 1

Longitudinal-transverse interference effective responses $R_{LT}$ as a function of $p_{\mathrm{miss}}$ for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$. The open and filled circles were extracted from QE data obtained by Chinitz et al. at $Q^2=0.3{(\mathrm{GeV}∕c)}^2$ and Spaltro et al. at $Q^2=0.2{(\mathrm{GeV}∕c)}^2$, respectively. The dashed $[Q^2=0.3{(\mathrm{GeV}∕c)}^2]$ and solid $[Q^2=0.2{(\mathrm{GeV}∕c)}^2]$ curves are modern RDWIA calculations. Overall, agreement is good, and improves with increasing $Q^2$.

Figure 2

The experimental infrastructure in Hall A at Jefferson Lab at the time of this experiment. The electron beam passed through a beam-current monitor (BCM) and beam-position monitors (BPMs) before striking a waterfall target located in the scattering chamber. Scattered electrons were detected in the ${\mathrm{HRS}}_e$, while knocked-out protons were detected in the ${\mathrm{HRS}}_h$. Noninteracting electrons were dumped. The spectrometers could be rotated about the central pivot.

Figure 3

Yield spectrum obtained at $E_{\text{beam}}=0.843\mathrm{GeV}$ and ${\theta }_{pq}=+8°$, corresponding to $p_{\mathrm{miss}}=148\mathrm{MeV}∕c$. Pion rejection has been performed, and all timing corrections have been applied. The top panel shows a scatterplot of $p_{\mathrm{miss}}$ versus $E_{\mathrm{miss}}$. The dark vertical bands project into the peaks located at 12.1 and $18.3\mathrm{MeV}$ in the bottom panel. These peaks correspond to protons knocked-out of the $1p_{1∕2}$- and $1p_{3∕2}$-states of ${}^{16}\mathrm{O}$, respectively. The $E_{\mathrm{miss}}$ resolution was roughly $0.9\mathrm{MeV}$ FWHM, which did not allow for separation of the $2s_{1∕2}1d_{5∕2}$-doublet located at $E_{\mathrm{miss}}=17.4\mathrm{MeV}$ from the $1p_{3∕2}$-state at $18.3\mathrm{MeV}$. The bump located at roughly $23\mathrm{MeV}$ is a negative-parity doublet which was not investigated. The insert shows the corresponding optimized coincidence TOF peak which has a FWHM of $1.8\mathrm{ns}$. The signal-to-noise ratio was about 8:1 in these kinematics.

Figure 4

Measured ${}^1\mathrm{H}(e,e)$ cross-section data normalized to the absolute predictions of a parametrization at similiar $Q^2$. Statistical error bars are shown. The data shown were taken over the course of a three-month run period. The different data points for each $E_{\text{beam}}$ represent different ${\mathrm{HRS}}_h$ angular settings.

Figure 5

Cross-section data for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ measured in parallel kinematics at three different beam energies as a function of the separation lever arm $v_T∕v_L$. The data points correspond to beam energies of 2.442, 1.643, and $0.843\mathrm{GeV}$ from left to right. The effective response functions $R_L$ (offset) and $R_T$ (slope) have been extracted from the fitted line. The uncertainties shown are statistical only. The extremely linear behavior of the data (which persists even after corrections for Coulomb distortion are applied) indicates that the statistical uncertainties were dominant (see Sec. 3E for a complete discussion).

Figure 6

$R_{LT}$ for ${\theta }_{pq}=\pm 8°(p_{\mathrm{miss}}=145\mathrm{MeV}∕c)$ as a function of $E_{\mathrm{miss}}$ for $E_{\text{beam}}=1.643\mathrm{GeV}$ and $2.442\mathrm{GeV}$. Statistical uncertainties only are shown. The statistical agreement over a broad range of $E_{\mathrm{miss}}$ emphasizes the systematic precision of the measurement (see Sec. 3E for a complete discussion). Note that the averages of these $R_{LT}$ values are presented as the $p_{\mathrm{miss}}=145\mathrm{MeV}∕c$ data in Fig. 22.

Figure 7

Comparisons between RPWIA and ${\mathrm{\text{RDWIA}}}_{(U^{\mathrm{opt}}=0)}$ calculations for the removal of protons from the $1p_{1∕2}$-state of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. In the upper panels, the solid curves represent the reduced cross section for both the ${\mathrm{\text{RDWIA}}}_{(U^{\mathrm{opt}}=0)}$ and the RPWIA calculations (see the text for details). The dashed curves correspond to the momentum distributions. In the lower panels, ${\mathrm{\text{RDWIA}}}_{(U^{\mathrm{opt}}=0)}∕$RPWIA reduced cross-section ratios are shown. Agreement to much better than 1% is obtained for both kinematics over the entire $p_{\mathrm{miss}}$ range.

Figure 8

Comparison baseline RDWIA calculations by the Madrid Group and Kelly (LEA) for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. For the purposes of this comparison, the input into both calculations was identical (see Table 5). Overall agreement is very good, and agreement is excellent for $-250<p_{\mathrm{miss}}<250\mathrm{MeV}∕c$.

Figure 9

Comparison baseline RDWIA calculations for the $A_{LT}$ asymmetry by the Madrid Group and Kelly (LEA) for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. For the purposes of this comparison, the input into both calculations was identical (see Table 5). Overall agreement is excellent over the entire $p_{\mathrm{miss}}$ range.

Figure 10

RDWIA calculations compared to “bare” (no MEC nor IC) RMSGA calculations by the Ghent Group for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. Both calculations employ the input presented in Table 6. Apart from the treatment of FSI, all ingredients are identical.

Figure 11

Momentum distributions for the HS, NLSH, and NLSH-P models. There is only a slight difference between HS and NLSH—for the $1p_{3∕2}$-state, HS is broader spatially and thus drops off faster with increasing $p_{\mathrm{miss}}$. On the other hand, NLSH-P differs appreciably from both HS and NLSH, and is clearly distinguishable for $p_{\mathrm{miss}}>250\mathrm{MeV}∕c$ for both the $1p_{1∕2}$- and $1p_{3∕2}$-states. Note that both the NLSH and NLSH-P wave functions predict binding energies, single-particle energies, and a charge radius for ${}^{16}\mathrm{O}$ which are all in good agreement with the data.

Figure 12

Left-right asymmetry $A_{LT}$ together with RDWIA calculations for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. Uncertainties are statistical. Note that the solid curves shown here are identical to those shown in Figs. 13 and 15.

Figure 13

Left-right asymmetry $A_{LT}$ together with RDWIA calculations for the removal of protons from the $1p_{1∕2}$-state of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. Uncertainties are statistical. The solid curves in all three panels are the same and are identical to those shown for the removal of protons from the $1p_{1∕2}$-state of ${}^{16}\mathrm{O}$ in Figs. 12 and 15.

Figure 14

Measured cross-section data for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ as compared to relativistic calculations at $E_{\text{beam}}=2.442\mathrm{GeV}$. Uncertainties are statistical and, on average, there is an additional $\pm 5.6\%$ systematic uncertainty associated with the data. The solid line is the RDWIA calculation, while the dashed and dashed-dotted lines are, respectively, the bare ROMEA and RMSGA calculations.

Figure 15

Left-right asymmetry $A_{LT}$ together with relativistic calculations of the $A_{LT}$ asymmetry for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. Uncertainties are statistical. The solid line is the RDWIA calculation, while the dashed and dashed-dotted lines are, respectively, the bare ROMEA and RMSGA calculations. Note that the solid curves shown here are identical to those shown in Figs. 12 and 13.

Figure 16

Data from this work together with relativistic calculations for the $R_{L+TT}$, $R_{LT}$, and $R_T$ effective response functions for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$. Uncertainties are statistical. The solid line is the RDWIA calculation, while the dashed and dashed-dotted lines are, respectively, the bare ROMEA and RMSGA calculations.

Figure 17

Measured cross-section data for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ together with calculations by the Ghent Group at $E_{\text{beam}}=2.442\mathrm{GeV}$. Uncertainties are statistical and, on average, there is an additional $\pm 5.6\%$ systematic uncertainty. The curves labeled “bare” are identical to those shown in Fig. 14.

Figure 18

Left-right asymmetry $A_{LT}$ together with calculations by the Ghent Group of the $A_{LT}$ asymmetry for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. Error bars are statistical. The curves labeled “bare” are identical to those shown in Fig. 15.

Figure 19

Data from this work together with ROMEA calculations by the Ghent Group for the $R_{L+TT}$, $R_{LT}$, and $R_T$ effective response functions for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$. Uncertainties are statistical. The curves labeled “bare” are identical to those shown in Fig. 16.

Figure 20

Data from this work together with RMSGA calculation by the Ghent Group for the $R_{L+TT}$, $R_{LT}$, and $R_T$ effective response functions for the removal of protons from the $1p$-shell of ${}^{16}\mathrm{O}$ as a function of $p_{\mathrm{miss}}$. Uncertainties are statistical. The curves labeled “bare” are identical to those shown in Fig. 16.

Figure 21

Data from this work together with ROMEA calculations by the Ghent Group for the $E_{\mathrm{miss}}$-dependence of the cross section obtained at $E_{\text{beam}}=2.442\mathrm{GeV}$. The data are the averaged cross section measured on either side of $\mathbf{q}$ at each ${\theta }_{pq}$. Normalization factors of 0.6, 0.7, and 1.0 have been used for the $1p_{1∕2}$-, $1p_{3∕2}$-, and $1s_{1∕2}$-states, respectively. Uncertainties are statistical and, on average, there is an additional $\pm 5.9\%$ systematic uncertainty associated with the data. Also shown are calculations by the Ghent Group for the $(e,e^{\prime }pN)$ contribution.

Figure 22

Data from this work together with ROMEA calculations by the Ghent Group for the $E_{\mathrm{miss}}$-dependence of the $R_{L+TT}$, $R_{LT}$, and $R_T$ effective response functions. Uncertainities are statistical. Also shown is the $(e,e^{\prime }pN)$ contribution.

Figure 23

(Color online) Calculations by the Ghent Group of the contribution to the differential ${}^{16}\mathrm{O}(e,e^{\prime }p)$ cross section from two-nucleon knockout as a function of $E_{\mathrm{miss}}$ and ${\theta }_p$ for $E_{\text{beam}}=2.442\mathrm{GeV}$. The upper-left panel shows the contribution of central correlations. The upper-right panel shows the combined contribution of central and tensor correlations. The lower-left panel shows the combined contribution of central and tensor correlations (two-nucleon correlations) together with MEC and IC (two-body currents). The relationship between the various kinematic quantities is shown in the bottom-right panel.

Figure 24

Fits to various ${}^{16}\mathrm{O}(e,e^{\prime }p)$ data sets based on the HS bound-nucleon wave function and the EDAD1 optical potential. See Table 9 for the key to the dataset labels. Open points and solid lines pertain to the $1p_{1∕2}$-state, while points and dashed lines pertain to the $1p_{3∕2}$-state. The dashed-dotted lines include the contributions of the positive parity $2s_{1∕2}1d_{5∕2}$-doublet to the $1p_{3∕2}$-state. Panel (d) shows the data from this work.

Figure 25

Data from this work together with calculations by the Ghent Group for the $E_{\mathrm{miss}}$-dependence of the cross-section data obtained in dip-region kinematics for $E_{\text{beam}}=1.643\mathrm{GeV}$. Uncertainties are statistical and, on average, there is an additional $\pm 5.9\%$ systematic uncertainty associated with the data.