---
abstract: 'We analyze data for $^{12}$C$(e,e^\prime p)$ with $Q^2 < 2$ (GeV/$c$)$^2$ using the relativistic distorted-wave impulse approximation (RDWIA) based upon Dirac-Hartree wave functions. The $1p$ normalization extracted from data for $Q^2 > 0.6$ (GeV/$c$)$^2$ is approximately 0.87, independent of $Q^2$, which is consistent with the predicted depletion of the $1p_{3/2}$ orbital by short-range correlations. The total $1p$ and $1s$ strength for $E_m < 80$ MeV approaches 100% of IPSM, consistent with a continuum contribution for $30< E_m < 80$ MeV of about 12% of IPSM. Similarly, a scale factor of 1.12 brings RDWIA calculations into good agreement with $^{12}$C$(e,e^\prime p)$ data for transparency. We also analyzed low $Q^2$ data from which a recent NRDWIA analysis suggested that spectroscopic factors might depend strongly upon the resolution of the probe. We find that the momentum distributions for their empirical Woods-Saxon wave functions fit to low $Q^2$ data for parallel kinematics are too narrow to reproduce data for quasiperpendicular kinematics, especially for larger $Q^2$, and are partly responsible for reducing fitted normalization factors. Although the RDWIA normalization factors for $Q^2 < 0.2$ (GeV/$c$)$^2$ are also smaller than obtained for $Q^2 > 0.6$ (GeV/$c$)$^2$, the effect is smaller and we argue that it should be attributed to the effective single-nucleon current operator instead of to spectroscopic factors which are probe-independent properties of nuclear structure. However, remediation of the failure of RDWIA calculations to reproduce low $Q^2$ data for parallel kinematics will require a more sophisticated modification of the current than a simple multiplicative factor.'
author:
- 'James J. Kelly'
date: 'January 28, 2005'
title: 'RDWIA analysis of $^{12}$C$(e,e^\prime p)$ for $Q^2 < 2$ (GeV/$c$)$^2$'
---

Introduction
============

It is generally believed that single-nucleon electromagnetic knockout provides unambiguous measurements of the momentum distributions and spectroscopic factors for single-hole states near the Fermi surface. Reviews of analyses based upon the nonrelativistic distorted-wave impulse approximation (NRDWIA) can be found in Refs. [@Lapikas93; @Kelly96; @Boffi96], which show that the momentum distributions are described well by mean-field calculations, such as Skyrme-Hartree-Fock, while spectroscopic factors for low-lying state are reduced relative to the independent particle shell model (IPSM) by an average factor of about $65\%$ over a broad range of $A$. Part of the depletion of IPSM strength can be attributed to short-range correlations that shift approximately $15\%$ of the hole strength to very large missing energies, beyond 100 MeV. Recently, a direct measurement of the spectral function using $^{12}$C$(e,e^\prime p)$ in parallel kinematics [@Rohe04] observed approximately 0.6 protons in a region with $p_m \gtrsim 240$ MeV/$c$ and $E_m \gtrsim 50$ MeV attributable to single-nucleon knockout from correlated clusters. This finding is consistent with the $16\%$ IPSM predicted by Frick [[*et al.* ]{}]{}[@Frick04] using the self-consistent Green’s function (SCGF) method [@Dickhoff04]. Similar estimates of the depletion of hole states are available from the correlated basis function (CBF) theory [@Benhar90] and other methods also. Furthermore, SCGF calculations [@Muther94; @Muther95a] show that the momentum distribution for low-lying quasihole states is hardly affected by short-range correlations, remaining very similar to mean-field calculations, while the momentum distributions for the large-$E_m$ continuum are substantially broadened.

If the occupancy of IPSM orbitals is predicted to be approximately $85\%$ by theories that reproduce the correlated strength at large $(E_m,p_m)$, why is only $65\%$ observed by $(e,e^\prime p)$ experiments? Coupling to collective modes is expected to produce significant fragmentation of the valence quasi-hole strength spread over a range of perhaps 10 MeV, but the extended RPA calculations presently available do not yet reproduce those fragmentation patterns well [@Dickhoff04]. Thus, it is possible that many small fragments are missed experimentally even if they lie within the experimental acceptance. Alternatively, problems in the reaction model may lead to systematic underestimation of spectroscopic factors. Most of the data contributing to the aforementioned estimate of $65\%$ IPSM were limited to $T_p < 100$ MeV and were analyzed using NRDWIA calculations based upon empirical Woods-Saxon wave functions. More recent relativistic analyses (RDWIA) typically produce larger spectroscopic factors [@Udias95] and describe interference response functions more accurately [@Kelly99b; @Udias99; @Udias00]. The effects due to dynamical enhancement of Dirac spinors by the nuclear mean field are actually stronger at low $Q^2$ than at higher energies [@Kelly05a].

While spectroscopic factors are of obvious importance to theories of nuclear structure, they also affect analyses of nuclear transparency that look for the onset of color transparency [@Dutta03; @Garrow02]. Those experiments typically compare the $(e,e^\prime p)$ yield integrated over wide but finite ranges of missing momentum, $p_m$, and missing energy, $E_m$, with calculations based upon a model spectral function. This spectral function is usually based upon the IPSM with a correction factor for the depletion by correlations of the single-hole strength within the experimental acceptance. Therefore, the accuracy of the depletion factor affects the accuracy of transparency measurements.

Recently Lapikás [[*et al.* ]{}]{}[@Lapikas00] analyzed the data for $^{12}$C$(e,e^\prime p)$ with $Q^2 < 0.4$ (GeV/$c$)$^2$ using NRDWIA calculations with Woods-Saxon bound-state wave functions fit to the NIKHEF data for $T_p = 70$ MeV in parallel kinematics [@vdSteenhoven88a] and concluded that the normalization factors relative to IPSM were $N_{1p} = 0.56 \pm 0.02$ and $N_{1s} = 0.59 \pm 0.04$ for $E_m < 80$ MeV. They then argued that the transparency results based upon an estimated depletion factor of $f = 0.9$ for the single-hole spectral function with $E_m < 80$ MeV should be increased by a factor of approximately 1.25 using appropriately weighted attenuation factors for the $1p$ and $1s$ shells. Finally, they argued that the normalization factors for $^{12}$C vary strongly with $Q^2$ from the low values fitted to the NIKHEF data to values approaching the IPSM limit for $Q^2 > 2$ (GeV/$c$)$^2$. Frankfurt [[*et al.* ]{}]{}[@Frankfurt01] speculated that quenching of spectroscopic factors for low $Q^2$ might be caused by probing a dressed quasiparticle using long wave lengths and that at higher resolution the effective current operator for the $(e,e^\prime p)$ reaction approaches more closely that for a free nucleon. However, although the $Q^2$ range was smaller, our recent relativistic analysis showed no significant slope in the normalization factors fit to $^{16}$O$(e,e^\prime p)$ data for $Q^2 < 0.8$ (GeV/$c$)$^2$ [@Fissum04]. Furthermore, we find that RDWIA calculations based upon the ordinary single-nucleon current with dynamical enhancement of lower components of Dirac spinors describe the left-right asymmetry for quasiperpendicular kinematics very well [@Kelly05a]; it is not obvious that one should describe quasiparticles for low $Q^2$ using Dirac spinors in a similar mean field.

In this paper we use relativistic distorted wave (RDWIA) calculations to analyze the data for $^{12}$C$(e,e^\prime p)$ with $Q^2 \lesssim 2$ (GeV/$c$)$^2$. In addition to the data considered by Lapikás [[*et al.* ]{}]{}[@Lapikas00], we include recent data from Dutta [[*et al.* ]{}]{}[@Dutta03] for $Q^2 = 0.6$, 1.2, and 1.8 (GeV/$c$)$^2$. Our RDWIA fits produce systematically larger normalization factors using Dirac-Hartree wave functions than using the Woods-Saxon wave functions that were fit to the NIKHEF data. The latter do not describe the momentum distributions for larger $Q^2$ well; nor do they describe the data for low $Q^2$ with quasiperpendicular kinematics as well as do Dirac-Hartree wave functions. Therefore, the fits to data for parallel kinematics with small ejectile energies appear to produce artifically narrow momentum distributions that result in anomalously low spectroscopic factors.

The model and fitting procedures are described in Sec. \[sec:model\]. Sec. \[sec:EMA\] presents an analysis based upon the effective momentum approximation (EMA) that permits direct comparison between nonrelativistic and relativistic bound-state wave functions. Sec. \[sec:RDWIA\] presents a more rigorous analysis, without using the EMA, of both spectroscopic factors and transparency. Our conclusions are summarized in Sec. \[sec:conclusions\].

Model and Data {#sec:model}
==============

Detailed descriptions of RDWIA for $(e,e^\prime p)$ reactions are been given in many recent papers. We will use the methods and terminology of Refs. [@Fissum04; @Kelly05a]. All calculations treat electron distortion in the $q_\text{eff}$ approximation and use the CC2 current operator with MMD form factors [@MMD] in Coulomb gauge. Optical potentials for Dirac phenomenology were taken from the global analysis by Cooper [[*et al.* ]{}]{}[@Cooper93]. We consider Dirac-Hartree wave functions from the original Horowitz and Serot (HS) analysis [@Horowitz86] and NLSH wave functions from Sharma [[*et al.* ]{}]{}[@Sharma93]. In addition we consider Woods-Saxon wave functions fit to $^{12}$C$(e,e^\prime p)$ data.

The data we consider are summarized in Table \[table:data\]. Reduced cross sections for the Tokyo, Saclay, and SLAC data were provided by Lapikás [@Lapikas00; @Lapikas04] and include small adjustments to a common convention for reduced cross section. In addition, radiative corrections were applied to the data from Ref. [@Bernheim82]. Only for the NIKHEF experiment [@vdSteenhoven88a] was it possible to resolve the lowest three $1p$ fragments. Although there are small differences between the missing momentum distributions for these three fragments, the ground state represents about $81\%$ and the next $1p_{3/2}$ fragment an additional $9\%$ of the total $1p$ strength for $E_m < 25$ MeV. Therefore, we represent the $1p$ strength using either a single Dirac-Hartree wave function for $1p_{3/2}$ or the NIKHEF fit to the ground state. That experiment was also able to resolve several weak positive-parity states with $E_m < 30$ MeV, but their strength is small enough to neglect in the present analysis [@vdSteenhoven88b]. For most other experiments the lowest missing energy bin contains a small contribution from the low-energy tail of the $1s$ shell. Furthermore, for all experiments the bin intended to emphasize the $1s$ shell inevitably contains contributions from a broad continuum that may include additional $1p$ strength. Therefore, each $E_m$ bin will be fit as an incoherent mixture of $1p$ and $1s$ contributions according to $$\label{eq:mixture}
\sigma_\text{red} = 
N_{1p} \sigma_\text{red}(1p_{3/2}) +  N_{1s} \sigma_\text{red}(1s_{1/2})$$ where the overlap function for each shell is assumed to be independent of missing energy and is normalized to full occupancy. In addition, for the JLab data sets we fit a normalization factor for the independent particle shell model (IPSM) to the inclusive data in the range $10 \leq E_m \leq 80$ MeV according to $$\label{eq:NIPSM}
\sigma_\text{red} = N_\text{IPSM} 
\left( \sigma_\text{red}(1p) +  \sigma_\text{red}(1s) \right)$$ Calculations will be shown for the entire $p_m$ range measured for each data set, but only data in the range $|p_m| \leq 200$ MeV$c$ will used to fit the normalization factors because larger $p_m$ may be susceptible to increasing corrections for effects not contained in the RDWIA model based upon single-nucleon current operators.

We also require a model of the $1s$ energy distribution to obtain the spectroscopic factor from the normalization $N_{1s}(E_\text{min},E_\text{max})$ fit to limited and variable intervals of $E_m$. One often employs a Lorentzian distribution $$\label{eq:L}
L_{1s}(E_m) = \frac{1}{2\pi} \frac{\Gamma(E_m)}
{(E_m-E_{1s})^2 + (\Gamma(E_m)/2)^2}$$ whose energy-dependent width is given by the Brown-Rho parametrization [@Brown81] $$\label{eq:BR}
\Gamma(E_m) = \frac{ a (E_m - E_F)^2 }{(E_m-E_F)^2+b}$$ where $E_F$ is the Fermi energy and $a$,$b$ are constants. For the present purposes it suffices to use the parameters $a=24$ MeV and $b=500$ MeV$^2$ originally proposed by Brown and Rho [@Brown81] with $E_F = 15.96$ MeV and $E_{1s} = 39$ MeV for $^{12}$C. Lapikás [[*et al.* ]{}]{}obtained slightly different parameters for $b$ and $E_{1s}$ by fitting the Tokyo data, but the effect upon $S_{1s}$ is insignificant. Thus, one would expect approximately $11\%$ of the $1s$ strength to be broadly distributed above 80 MeV missing energy, which is consistent with the depletion factor attributed to short-range correlations. Therefore, in the absence of background, the $1s$ spectroscopic factor deduced from a fitted normalization factor would be $$\label{eq:1s}
S_{1s} = 2 N_{1s}(E_\text{min},E_\text{max})
\frac{ \int_0^\infty L_{1s}(E_m) dE_m } 
{ \int_{E_\text{min}}^{E_\text{max}} L_{1s}(E_m) dE_m }$$

  --------------------------- --------- ------------------ ------------------ ------- --------------- --------------------------------------------------------
  data set                        $E_0$   $\Delta E_m(1p)$   $\Delta E_m(1s)$   $T_p$           $Q^2$ notes
                                    MeV                MeV                MeV     MeV   (GeV/$c$)$^2$ 
  Tokyo [@Nakamura76]               700               6-30              21-66     159            0.29 
                                    700                                 21-66     136                 
  Saclay [@Mougey76]                497              15-22              30-50      87            0.16 
  Saclay [@Bernheim82]              500              15-22                 NA      99            0.18 constant $(\omega,\bm{q})$
  Saclay [@Bernheim82]              500              15-22                 NA      99       0.09-0.32 parallel; $1s$ not available
  NIKHEF [@vdSteenhoven88a]     285-481              15-22              30-39      70       0.02-0.26 parallel
  SLAC [@Makins94]                 2015              15-25              30-80     600            1.11 $1s$ not radiatively unfolded
  JLab [@Dutta03]                  2445              15-25              30-50     350            0.64 inclusive bin with $10 \leq E_m \leq 80$ MeV available
                                   2445                                           700            1.28 
                                   3245                                           970            1.84 
  --------------------------- --------- ------------------ ------------------ ------- --------------- --------------------------------------------------------

  : Summary of data for $^{12}$C$(e,e^\prime p)$. $E_0$ is the beam energy and $T_p$ is the central proton kinetic energy. Quasiperpendicular kinematics with constant $(\omega,\bm{q})$ were used unless a $Q^2$ range is given. \[table:data\]

Effective momentum approximation {#sec:EMA}
================================

Figure \[fig:12Cpm\] compares momentum distributions for the relativistic HS and NLSH models with the Woods-Saxon models used by Lapikás [[*et al.* ]{}]{}[@Lapikas00] and by Dutta [[*et al.* ]{}]{}[@Dutta03]. The latter were based upon a NRDWIA analysis of the Saclay data for $T_p = 99$ MeV made by Bernheim [[*et al.* ]{}]{}[@Bernheim82]. The differences between the HS and NLSH wave functions are relatively small, but the NIKHEF $1p$ wave function has a much stronger peak and declines more rapidly with respect to $p_m$. Similarly, the NIKHEF $1s$ wave function starts higher and falls faster than the relativistic wave functions. Interestingly, the Saclay wave functions show similar behavior but the effects, compared with the NIKHEF wave functions, are smaller for the $1p$ and larger for the $1s$ wave function. The large spread among these momentum distributions will obviously be reflected in the corresponding DWIA calculations.

![(Color online) Momentum distribution for $^{12}$C$(e,e^\prime p)$. []{data-label="fig:12Cpm"}](pmiss.ps){width="3.0in"}

To evaluate the effect of variations of $\rho(p_m)$ upon DWIA calculations using both relativistic and nonrelativistic wave functions, it is simplest to employ the EMA-noSV approach. Several previous papers have shown that the differences between EMA and full RDWIA calculations are relatively small for cross sections. Although that approximation does not provide the best description of the left-right asymmetry, $A_{LT}$, it has the advantage that the nonrelativistic wave function can be used directly.

Figure \[fig:Dutta-EMAnoSV\] compares EMA calculations using NLSH, HS, and NIKHEF wave functions with the JLab data for $Q^2 = 0.6$, 1.2, and 1.8 (GeV/$c$)$^2$. For each kinematics the top set are semi-inclusive data for $E_m < 80$ MeV, the second set shows a bin $15 \leq E_m \leq 25$ MeV dominated by the $1p$ shell, and the bottom set shows a bin $30 \leq E_m \leq 50$ MeV dominated by the $1s$ shell. The curves were fit to data in the range $|p_m| \leq 200$ MeV/$c$ neglecting, at present, the mixing between the shells. We will later find that filling of the $1p$ minimum and broadening of the $1s$ distribution can be described by mixing. These calculations employ the EDAD1 optical potential; other choices of potentials from Cooper [[*et al.* ]{}]{}[@Cooper93] produce small variations in the normalization factors with little dependence upon the choice of overlap function and practically no discernible effect upon the curves. The results for the two relativistic wave functions, NLSH and HS, are very similar but the fitted $p_m$ distributions for the NIKHEF wave function tend to be too narrow, especially for the $1p$ shell. Consequently, the $1p$ normalization factors listed in Table \[table:normperp\_EMAnoSV\] are systematically smaller for the NIKHEF wave function than for the NLSH or HS wave functions. These normalization factors are correlated well with the peak of the momentum distribution, with the NLSH results largest and NIKHEF results smallest.

![EMA-noSV calculations for $^{12}$C$(e,e^\prime p)$ are compared with quasiperpendicular data for $0.6 < Q^2 < 1.8$ (GeV/$c$)$^2$ from JLab . In each panel the top set of curves shows the inclusive $10 \leq E_m \leq 80$ MeV bin scaled by a factor of 20, the middle set shows the $p$-shell bin with $15 \leq E_m \leq 25$ MeV, and the bottom set shows the $s$-shell bin with $30 \leq E_m \leq 50$ MeV scaled by a factor of 0.1. Solid, dashed, and dash-dotted curves are based upon NLSH, HS, and NIKHEF wave functions, respectively. []{data-label="fig:Dutta-EMAnoSV"}](emanosv_sigred1.ps){width="5.0in"}

Similar comparisons are shown in Figs. \[EMA:Saclay\_kin2\]-\[EMA:Tokyo\] for several low $Q^2$ experiments performed using quasiperpendicular kinematics. Except when noted otherwise, a normalization factor for each curve was obtained using a least-squares fit to the data for $|p_m| \leq 200$ MeV/$c$. Fig. \[EMA:Saclay\_kin2\] shows that the relativistic wave functions describe the data for the Saclay experiment with $T_p = 87$ MeV in quasiperpendicular kinematics better than the NIKHEF wave functions, which fall too rapidly with increasing $p_m$. The results shown in Fig. \[EMA:Tokyo\] for the Tokyo experiment with $Q^2 = 0.29$ (GeV/$c$)$^2$ are somewhat ambiguous: although the NLSH and HS wave functions give better fits to for $p_m \gtrsim 60$ MeV/$c$, particularly for the ground-state, none of the calculations can describe the enhancement at lower $p_m$ seen for the ground state or the dip for the $s$-state. The low $p_m$ bulge for the ground-state data seems somewhat implausible and it is likely that there were difficulties in cleanly separating the $1p$ and $1s$ contributions to the low $p_m$ data. Therefore, the $1p$ normalization for this set was fit to $50 \leq p_m \leq 200$ MeV/$c$. Similar calculations are also compared with the SLAC data for $Q^2 = 1.1$ (GeV/$c$)$^2$ in Fig. \[EMA:SLAC\]. We again find that the $p_m$ distribution for the NIKHEF wave function is too narrow to fit the $1p$ data well. However, the $1s$ data remain well above all of the calculations for large $p_m$. Lapikás [[*et al.* ]{}]{}[@Lapikas00] attributed this problem to $1p$ contamination due to inadequate radiative unfolding.

![EMA-noSV calculations for $^{12}$C$(e,e^\prime p)$ are compared with quasiperpendicular data for $T_p = 87$ MeV from Saclay . Solid, dashed, and dash-dotted curves are based upon NLSH, HS, and NIKHEF wave functions, respectively. []{data-label="EMA:Saclay_kin2"}](emanosv_saclay_kin2.ps){width="3.0in"}

![EMA-noSV calculations for $^{12}$C$(e,e^\prime p)$ are compared with quasiperpendicular data for $T_p = 99$ MeV from Saclay . Solid, dashed, and dash-dotted curves are based upon NLSH, HS, and NIKHEF wave functions, respectively. []{data-label="EMA:Saclay_kin1"}](emanosv_saclay_kin1.ps){width="3.0in"}

![EMA-noSV calculations for $^{12}$C$(e,e^\prime p)$ are compared with quasiperpendicular data for $Q^2 = 0.29$ (GeV/$c$)$^2$ from Tokyo . []{data-label="EMA:Tokyo"}](emanosv_tokyo_sigred1.ps){width="2.5in"}

![EMA-noSV calculations for $^{12}$C$(e,e^\prime p)$ are compared with quasiperpendicular data for $Q^2 = 1.1$ (GeV/$c$)$^2$ from SLAC . Solid, dashed, and dash-dotted curves are based upon NLSH, HS, and NIKHEF wave functions, respectively. []{data-label="EMA:SLAC"}](emanosv_slac.ps){width="2.5in"}

The normalization factors obtained for single-component EMA-noSV fits are listed in Table \[table:normperp\_EMAnoSV\]; improved two-component RDWIA fits will be presented in the next section. The Tokyo result is omitted because the $1s$ admixture is appreciable. We find that these factors are correlated well with the peak value of the $\rho(p_m)$, such that the NIKHEF wave functions produce the smallest and the NLSH wave functions the largest normalization factors. Over a wide range of $Q^2$, including both early data at low $Q^2$ and recent data for $Q^2 \sim 1$ (GeV/$c$)$^2$, the relativistic wave functions fit the $p_m$ distributions relatively well while the NIKHEF wave functions are systematically narrower than the data. Thus, the data for quasiperpendicular kinematics consistently prefer the broader $p_m$ distributions and larger normalization factors obtained with Dirac-Hartree wave functions.

  ------------ --------------- ------- ------- ------- -------- ------- ------- --------
               $Q^2$           $E_m$                                            
  experiment   (GeV/$c$)$^2$   MeV     NLSH    HS      NIKHEF   NLSH    HS      NIKHEF
  Saclay       0.16            15-22   0.637   0.599   0.483    0.572   0.542   0.444
  Saclay       0.18            15-22   0.757   0.705   0.554    0.684   0.640   0.510
  JLab         0.6             15-25   0.934   0.878   0.697    0.909   0.856   0.682
  SLAC         1.1             15-25   0.852   0.791   0.613    0.887   0.822   0.633
  JLab         1.2             15-25   0.904   0.849   0.671    0.950   0.890   0.699
  JLab         1.8             15-25   0.901   0.854   0.689    0.886   0.841   0.679
  ------------ --------------- ------- ------- ------- -------- ------- ------- --------

  : Normalization factors for $^{12}$C$(e,e^\prime p)(1p)^{-1}$ fit to quasiperpendicular data for the $1p$ bin using several overlap functions in the EMAnoSV approximation. Possible $1s$ contributions were neglected. The first set uses the EDAD1 and the second the EDAIC optical potential. \[table:normperp\_EMAnoSV\]

The data for parallel kinematics, on the other hand, present special problems. We omit the Mainz data [@Blomqvist95] because there appear to be significant normalization problems for that experiment [@Lapikas00]. The NIKHEF and Saclay data for parallel kinematics are compared with EMA-noSV calculations for NLSH, HS, and NIKHEF wave functions in Figs. \[fig:NIKHEF-para\] and \[fig:Saclay-para\]. The results for both experiments are similar but none of these calculations reproduce the data for the $1p$ shell. Due to obvious the difficulties for $p_m < 0$, the normalization factors listed in Table \[table:normpara\_EMAnoSV\] were fit to $0 < p_m < 200$ MeV/$c$. These factors are also correlated with the peak values of $\rho(p_m)$ but for the relativistic wave functions are somewhat lower than were obtained using quasiperpendicular kinematics. On the other hand, these fits underestimate the cross sections for $p_m <0$ and larger $Q^2$.

  experiment   $T_p$   NLSH    HS      NIKHEF
  ------------ ------- ------- ------- --------
  NIKHEF       70      0.635   0.616   0.541
  Saclay       99      0.685   0.636   0.505

  : Normalization factors for $^{12}$C$(e,e^\prime p)(1p)^{-1}$ fit to data for parallel kinematics using several overlap functions in the EMAnoSV approximation. These results were extracted for the $15 \leq E_m \leq 22$ MeV bin using NLSH wave functions and the EDAD1 optical potential. \[table:normpara\_EMAnoSV\]

In order to improve standard NRDWIA calculations for parallel kinematics, van der Steenhoven [[*et al.* ]{}]{}[@vdSteenhoven88a; @Ireland94] resorted to adjusting the overlap function, the optical model, and the current operator simultaneously. The NIKHEF analysis used the nonrelativistic expansion of the current operator in powers of $p/m$ proposed by McVoy and van Hove [@McVoy]; we verified that at second order this expansion, denoted NR2, gives results that are practically indistinguishable from our EMA-noSV calculations. Their analysis also used a nonrelativistic optical potential produced by Comfort and Karp [@Comfort80], denoted CK, and a Perey factor [@Perey62] for the ejectile with $\beta=0.85$ fm. Figure \[fig:NR2-para\] compares several variations of the NR2 calculations that are similar to those of van der Steenhoven [*et al*]{}. The dashed curve uses this optical potential and current operator with the upper component of the NLSH wave function and fails to reproduce the data for parallel kinematics with $p_m <0$; the results are similar to those shown in Fig. \[fig:NIKHEF-para\] in the EMA approach. The narrower NIKHEF momentum distribution improves the fit to the data but appears to be shifted to the left — a shift to the right was accomplished by modifying the optical potential. This modified potential, denoted MCO in Ref. [@vdSteenhoven88a], was intended to simulate channel coupling in the final state but our more detailed analysis of channel coupling [@Kelly99a] gave much smaller effects for this reaction; hence, we regard this modification of the optical potential to be somewhat [*ad hoc*]{}. Finally, van der Steenhoven [[*et al.* ]{}]{}inserted an enhancement factor for transverse components of the current operator and adjusted that factor to fit the peak for negative $p_m$. Applying this factor without further adjustment does provide a reasonably accurate fit to the reduced cross section for parallel kinematics.

We have not attempted to tune this fit because we doubt that it is really possible to distinguish between variations of the overlap function, the current operator, and the final-state interactions using these data. Nor does it appear possible to fit the low $Q^2$ data for both parallel and quasiperpendicular kinematics simultaneously using unique overlap functions and global optical potentials fit to proton elastic scattering. The relativistic wave functions provide goods fits to the $p_m$ dependence of the data for quasiperpendicular kinematics even if the normalization factors for $Q^2 \ge 0.6$ (GeV/$c$)$^2$ are significantly larger than those for low $Q^2$. On the other hand, the normalization factors for the two Saclay experiments with quasiperpendicular kinematics do not agree well with each other either. Therefore, the failure of EMA calculations with NLSH wave functions to reproduce the $p_m$ dependence of data for parallel kinemtics with $T_p \lesssim 100$ MeV seems to be a more serious problem than the variation of normalization factors with $Q^2$. Recognizing that off-shell ambiguities are more serious for parallel kinematics because $x=Q^2/2m \omega$ varies over a wide range and expecting the reaction mechanism to be more reliable for large $Q^2$, we believe that the reaction mechanism for parallel kinematics with $T_p \lesssim 100$ MeV is not sufficiently reliable to justify fitting the bound-state wave function and that it is likely that in such analyses variations of the overlap function would tend to compensate for deficiencies in the reaction model.

If one wants to fit the momentum distribution for a discrete single-hole state to $(e,e^\prime p)$ data, it is probably best to use quasifree kinematics where $(\omega,\bm{q})$ is constant and $Q^2$ is as large as possible. The reliability of the reaction mechanism should improve as $Q^2$ increases and the use of quasifree kinematics should minimize distortion of $\rho(p_m)$ due to variation of current operator off shell. It would then be of interest to test the reaction mechanism by comparison with data for parallel kinematics with large $T_p$. Ideally such tests should have sufficient resolution in $E_m$ to isolate discrete states without the complications of incoherent mixtures. Unfortunately, no high-resolution $(e,e^\prime p)$ data for parallel kinematics with $T_p > 200$ MeV and sufficient coverage in $p_m$ are available.

![EMA-noSV calculations for $^{12}$C$(e,e^\prime p)$ are compared with data from from NIKHEF using parallel kinematics for $T_p = 70$ MeV/$c$. []{data-label="fig:NIKHEF-para"}](emanosv_nikhef.ps){width="3.0in"}

![EMA-noSV calculations for $^{12}$C$(e,e^\prime p)$ are compared with data from from Saclay using parallel kinematics for $T_p = 99$ MeV. Solid, dashed, and dash-dotted curves are based upon NLSH, HS, and NIKHEF wave functions, respectively. []{data-label="fig:Saclay-para"}](emanosv_saclay_para.ps){width="3.0in"}

![(Color online) NR2 calculations for $^{12}$C$(e,e^\prime p)$ are compared with data from from NIKHEF using parallel kinematics for $T_p = 70$ MeV/$c$. WS refers to the NIKHEF Woods-Saxon wave function and $\eta>0$ shows the effect of enhancing transverse components of the current opertor. []{data-label="fig:NR2-para"}](nr2_nikhef.ps){width="3.0in"}

RDWIA analysis {#sec:RDWIA}
==============

Normalization factors {#sec:norm}
---------------------

### JLab {#sec:norm-JLab}

Full RDWIA fits to the JLab data [@Dutta03] are shown in Fig. \[fig:Dutta\_nlsh\] using NLSH wave functions, EDAD1 optical potentials, and the CC2 current operator in Coulomb gauge. For the inclusive bin, these fits are slightly better than the corresponding EMA-noSV fits shown in Fig. \[fig:Dutta-EMAnoSV\] because full RDWIA calculations reproduce the left-right asymmetry better than the EMA-noSV approximation [@Kelly05a]. These fits for the $1p$ and $1s$ bins are significantly better than single-component fits because neither bin represents a pure single-particle configuration. The $1s$ contribution to the $1p$ bin fills in the minimum and rounds the peaks. Similarly the $1p$ contribution to the $1s$ bin broadens the momentum distribution — the reduced $1s$ component provides a better fit for small $p_m$ while the $1p$ component improves the fit to larger $p_m$. However, for $p_m > 200$ MeV/$c$ the data for both the $s$-shell and inclusive bins remain significantly above these two-component fits. Similar data for $^{16}$O$(e,e^\prime p)$ [@Liyanage01; @Fissum04] covering a broad range of missing energy show that for $p_m \gtrsim 150$ MeV/$c$ there is much more strength at large $E_m$ than can be attributed to single-nucleon knockout from the $s$-shell. The continuum fraction increases with $p_m$, which may be simulated in the two-component model by a $1p$ admixture. Assuming that the $1s$ quasihole momentum distribution is similar to the Dirac-Hartree wave function, it would appear that the continuum contribution to $E_m > 30$ MeV has a broader momentum distribution than the $1p_{3/2}$ wave function used in these two-component fits. Although part of this may be contributed by $1p$ fragmentation, much of the fitted $1p$ strength is probably a surrogate for multinucleon continuum. Recent SCGF estimates suggest that approximately $68\%$ of the correlated continuum is in partial waves with $\ell \leq 1$ [@Muther04]. No doubt higher partial waves begin to contribute to the continuum as $p_m$ increases and it is likely that the $\ell=1$ continuum is also broader than the $1p_{3/2}$ quasihole momentum distribution.

![ Contributions of $1p_{3/2}$ and $1s_{1/2}$ fitted to JLab data for $^{12}$C$(e,e^\prime p)$. In each panel the top set of curves shows the inclusive $10 \leq E_m \leq 80$ MeV bin scaled by a factor of 20, the middle set shows the $p$-shell bin with $15 \leq E_m \leq 25$ MeV, and the bottom set shows the $s$-shell bin with $30 \leq E_m \leq 50$ MeV scaled by a factor of 0.1. The solid curves show the total fit, the dashed curves show the $1p$ contributions to the inclusive and $p$-shell bins, and the dash-dotted curves show the $1s$ contributions to the inclusive and $s$-shell bins. In addition, scale-factor fits for the IPSM are shown as dotted curves for the inclusive bin but are practically indistinguishable from the solid curves. All calculations use NLSH wave functions and EDAD1 optical potentials in full RDWIA.[]{data-label="fig:Dutta_nlsh"}](Dutta_nlsh.ps){width="5in"}

RDWIA fits to the JLab data using NLSH, HS, or NIKHEF wave functions are compared in Fig. \[fig:Dutta\_fits\]. The two Dirac-Hartree wave functions provide very similar fits with a slight preference for NLSH. Mixing between $1p$ and $1s$ contributions improves fits to the $1s$ bin using NIKHEF wave functions by flattening the small $p_m$ and lifting the large $p_m$ regions. However, two-component fits using NIKHEF wave functions were impossible for the $1p$ bin because the $1s$ normalizations were negative; hence, the $1p$ NIKHEF fits shown here simplify scale RDWIA calculations for $1p_{3/2}$. Furthermore, the $1p$ NIKHEF fits are even worse and their normalization factors are even smaller in RDWIA than in EMA-noSV: with EDAD1 those normalization factors are 0.66, 0.63, and 0.66 for RDWIA compared with 0.70, 0.67, and 0.69 for EMA-noSV at the three $Q^2$ settings. This reduction occurs because the RDWIA calculations for inherently nonrelativistic Woods-Saxon wave functions were made by replacing the Perey factor with the Darwin factor obtained using the spin-orbit potential; this procedure is explained in Ref. [@Kelly05a]. However, because the spin-orbit potential used by van der Steenhoven [[*et al.* ]{}]{}[@vdSteenhoven88a] is stronger than that for NLSH or HS Dirac-Hartree wave functions, the smaller Darwin factor leads to a narrower momentum distribution for RDWIA than for EMA-noSV and, consequently, smaller normalization factors. The fits for the $1p$ bin overshoot the peaks and then fall too rapidly as $p_m$ increases and these problems carry over, in slightly diluted form, to the inclusive bin. Therefore, we reject those parametrizations of the $1p_{3/2}$ and $1s_{1/2}$ overlap functions.

![RDWIA fits to the JLab data for JLab data for $^{12}$C$(e,e^\prime p)$ are compared for NLSH (solid), HS (dashed), and NIKHEF (dash-dot) wave functions. In each panel the top set of curves shows the inclusive $10 \leq E_m \leq 80$ MeV bin scaled by a factor of 20, the middle set shows the $p$-shell bin with $15 \leq E_m \leq 25$ MeV, and the bottom set shows the $s$-shell bin with $30 \leq E_m \leq 50$ MeV scaled by a factor of 0.1.[]{data-label="fig:Dutta_fits"}](Dutta_fits.ps){width="5in"}

RDWIA normalization factors fit to the JLab reduced cross section data for $^{12}$C$(e,e^\prime p)$ in three bins of missing energy are collected in Table \[table:Dutta\_norm\]. These results were obtained using EDAD1 optical potentials but the corresponding results for EDAD2, EDAD3, EDAIC, or EEI optical potentials differ by less than $\pm 5\%$ with systematics similar to Table \[table:normperp\_EMAnoSV\]. The differences between NLSH and HS results are small for $1p_{3/2}$ and modest for $1s_{1/2}$. Most of the discussion will be based upon the NLSH results because they provide slightly better fits to the data.

  ------------ ------------- ------------- ------------- ------------- ------- ------------- ------------- ------------- -------
                                                                                                                         
  state        $E_m$ bin     $Q^2 = 0.6$   $Q^2 = 1.2$   $Q^2 = 1.8$   ave     $Q^2 = 0.6$   $Q^2 = 1.2$   $Q^2 = 1.8$   ave
  $1p_{3/2}$   15 – 25 MeV   0.885         0.868         0.847         0.87    0.852         0.836         0.830         0.84
               30 – 50 MeV   0.060         0.078         0.077         0.072   0.063         0.058         0.062         0.061
               10 – 80 MeV   1.021         1.001         0.977         1.00    1.008         0.985         0.973         0.99
  $1s_{1/2}$   15 – 25 MeV   0.133         0.102         0.198         0.14    0.082         0.050         0.110         0.08
               30 – 50 MeV   0.669         0.616         0.667         0.65    0.628         0.575         0.620         0.61
               10 – 80 MeV   0.953         0.887         1.009         0.95    0.818         0.757         0.845         0.81
  IPSM         10 – 80 MeV   0.981         0.991         0.984         0.985   0.949         0.914         0.940         0.934
  ------------ ------------- ------------- ------------- ------------- ------- ------------- ------------- ------------- -------

  : Normalization factors for $^{12}$C$(e,e^\prime p)$ fit to JLab data for three bins of missing energy and three values of $Q^2$ in (GeV/$c$)$^2$. The columns labeled [*ave*]{} is averaged with respect to $Q^2$. The first set uses NLSH and the second HS wave functions. Both sets use EDAD1 optical potentials. \[table:Dutta\_norm\]

A rather large fraction, approximately $87\%$, of the IPSM $1p$ strength is found in the lowest $E_m$ bin. If we assume that fragmentation of the $1p$-strength by collective modes is largely confined to excitation energies below 10 MeV, the $87\%$ we find for the $1p$-shell is consistent with SCGF estimates. The additional $1p$ strength fit to the reduced cross section for larger $E_m$ probably represents continuum contributions with $\ell > 0$ that should not be added to the valence $1p$ strength.

The $1s$ contribution to the $p$-shell bin is more than predicted by Eqs. (\[eq:L\]-\[eq:BR\]), but could arise either from resolution, discrete $\frac{1}{2}^+$ states, or a stronger low-$E_m$ tail. Approximately $65\%$ of the $1s$ IPSM strength appears in the $30 < E_m < 50$ MeV bin and corresponds to a net occupancy of $94\%$ when scaled according to Eq. (\[eq:1s\]) to account for the spreading width. However, this is probably an overestimate of the $1s$ spectroscopic factor because one would expect approximately $10\%$ of the $1s$ strength to lie beyond 80 MeV missing energy; hence, if we scale $N_{1s}$ for the NLSH fit to the inclusive bin we would obtain more than $100\%$. Although the overestimation appears to be smaller for the HS fits, neither accounts for continuum contributions from rescattering or two-body currents that could artificially enhance $N_{1s}$. The difficulties in reproducing the cross sections for $p_m > 200$ MeV/$c$ in the $E_m > 30$ MeV region also suggest additional continuum contributions with broader momentum distribution that are not described by these two-component fits. Therefore, it is reasonable to assume that the depletion of the $1s$ shell by short-range correlations is similar to that for the $1p$ shell and to attribute the additional strength in the inclusive cross section to continuum processes that are not related directly to the single-hole spectral function.

The final line of Table \[table:Dutta\_norm\] lists IPSM scale factors fit to the $E_m < 80$ MeV data according to Eq. (\[eq:NIPSM\]); these results are in reasonable agreement with the weighted average of the $1p$ and $1s$ normalization factors. The corresponding fits are shown in Fig. \[fig:Dutta\_nlsh\] as dotted curves that are practically indistinguishable from the two-component fits to the inclusive data. Therefore, we find approximately $98.5\%$ of the IPSM strength in $E_m < 80$ MeV, which is considerably more than the $85\%$ one expects after accounting for short-range correlations. It is likely that much of the extra strength observed for $E_m > 30$ MeV arises from processes other than single-nucleon knockout from the $1s$-shell, such that the total single-hole strength for $E_m < 80$ MeV is overestimated by this analysis.

Under these conditions, we consider $N_{1p}(15,25)=0.87$ to be the most reliable estimate of the depletion of IPSM orbitals in $^{12}$C by short-range correlations. This estimate is consistent with the direct measurements of correlated continuum made by Rohe [[*et al.* ]{}]{}[@Rohe04] using parallel kinematics and the analysis of those data by Müther and Sick [@Muther04] based upon SCGF. The extra yield observed by Dutta [[*et al.* ]{}]{}[@Dutta03] in quasiperpendicular kinematics should then be attributed to two-body currents and rescattering processes in which some of the flux absorbed by the optical potential is distributed to final states that remain within the experimental acceptance. Several studies suggest that rescattering contributions to the continuum are minimized for parallel kinematics [@Sick97; @Rohe04; @Barbieri04]. These background contributions affect the interpretation of nuclear transparency measurements, which will be considered in Sec. \[sec:transparency\].

For the sake of completeness, we also show in Fig. \[fig:alt\_fit\] the left-right asymmetry in reduced cross section $$\label{eq:alt}
a_{LT} = 
\frac{\sigma_\text{red}(\phi=0) - \sigma_\text{red}(\phi=\pi)}
     {\sigma_\text{red}(\phi=0) + \sigma_\text{red}(\phi=\pi)}$$ where the azimuthal angle $\phi=0$ corresponds to an ejectile momentum in the electron scattering plane between the beam direction and the momentum transfer. The sensitivity of this quantity to spinor distortion was studied in some detailed in Ref. [@Kelly05a] for pure single-particle configurations. Figure \[fig:alt\_fit\] shows that the $1p_{3/2}$ admixture for the $s$-shell bin provides slightly better fits to the data for $p_m \lesssim 300$ MeV/$c$. That effect is small for the $p$-shell bin because the $1p$ admixture is small and is also small for the inclusive bin because the asymmetries for the two components are similar, depending more upon the Dirac potentials than upon the details of the bound-state wave functions. Two-component fits based upon EMA-noSV calculations demonstrate that $a_{LT}\approx 0$ without spinor distortion because the characteristic left-right asymmetry for electron scattering by a moving free proton is removed by using the reduced cross section. The remaining asymmetry due to dynamical enhancement of lower components of Dirac spinors is described very well by RDWIA calculations. As argued in Ref. [@Kelly05a], the flattening of the $p_m$ distribution for $a_{LT}$ in the $s$-shell and inclusive bins for increasing $Q^2$ is probably caused by continuum contributions that need not share the characteristic asymmetry of single-nucleon knockout.

![RDWIA calculations for the left-right asymmetry in reduced cross section for quasiperpendicular kinematics with $0.6 \leq Q^2 \leq 1.8$ (GeV/$c$)$^2$. NLSH wave functions, EDAD1 optical potentials, and the CC2 current operator were used. The dashed curves show single-component calculations while results for two-component $1p+1s$ fits are shown as solid curves. The dash-dotted curves show two-component fits in EMA-noSV.[]{data-label="fig:alt_fit"}](alt_fit.ps){width="5in"}

### Previous experiments {#sec:RDWIA-lowQ2}

The RDWIA analysis of the JLab data presented in the preceding section provides a satisfying degree of internal consistency over a broad range of $Q^2$. Unfortunately, similar analyses for older data sets present a variety of consistency problems that cannot be resolved by simply by choosing the best overlap functions or optical potentials. Therefore, in this section we consider those data in chronological order using only a single set of options, namely NLSH and EDAD1, in the RDWIA framework.

The fits shown in Fig. \[fit:Tokyo\] appear to describe the Tokyo data [@Nakamura76] very well, but the $1s$ contribution to the $p$-shell bin is implausibly large. With NLSH wave functions and EDAD1 optical potential, we find $(N_{1p},N_{1s}) = (0.56,0.42)$ for $6 \leq E_m \leq 30$ MeV and $(N_{1p},N_{1s}) = (0.08,0.69)$ for $21 \leq E_m \leq 66$ MeV. Apparently there was an experimental problem in defining the $p$-shell or there is an unidentified problem in the reaction mechanism that enhances the cross section for small $p_m$. Because this is the only quasiperpendicular data set with severe problems in the $p$-shell bin for small $p_m$, we believe that energy resolution is a more likely explanation than reaction mechanism. The single-component fit to $p_m > 50$ MeV shown in Fig. \[EMA:Tokyo\] corresponds to $N_{1p} = 0.77$, which is more consistent with the other quasiperpendicular data for this bin. The $1p$ contribution to the $1s$ bin, on the other hand, is fairly small. Scaling the $1s$ fit for $21 \leq E_m \leq 66$ MeV according to Eq. (\[eq:1s\]) gives a net occupancy of $80\%$, which is reasonable but smaller than the corresponding JLab result. Perhaps the continuum contamination was smaller for the Tokyo experiment.

![RDWIA fit to Tokyo data using NLSH wave functions. Solid curves show the fits while dashed and dash-dotted curves show $1p$ and $1s$ contributions. []{data-label="fit:Tokyo"}](tokyo_fit.ps){width="2.5in"}

RDWIA provides no qualitative improvement over EMA-noSV fits to NIKHEF or Saclay data for parallel kinematics. As discussed before, we believe the severe variation of Bjorken $x$ in parallel kinematics distorts momentum distributions fit to $(e,e^\prime p)$ data for low $Q^2$ using familiar off-shell extrapolations of the single-nucleon current operator. Therefore, we do not consider those data further. Conversely, single-component RDWIA fits to the Saclay data for quasiperpendicular kinematics are similar to those shown in Figs. \[EMA:Saclay\_kin2\]-\[EMA:Saclay\_kin1\]. There is no evidence for $1s$ contamination of the $15 \leq E_m \leq 25$ MeV bin and the $30 \leq E_m \leq 50$ MeV bin is too narrow and the range of $p_m$ is too small to find significant $1p$ strength there. Furthermore, mixed fits sometimes produce negative contamination factors with large uncertainties; hence, only single-component fits are considered for the Saclay quasiperpendicular data. We find $1p$ normalization factors of 0.63 for Ref. [@Mougey76] or 0.75 for Ref. [@Bernheim82] that are somewhat smaller than the JLab results but are not in especially good agreement with each other either. Similarly, $N_{1s}=0.46$ for 30-50 MeV corresponds to an occupancy of $67\%$, which is also somewhat less than the JLab result. Given that we believe the reaction mechanism is more reliable at higher $Q^2$ and that there are unresolved problems in reproducing data in parallel kinematics at the same $Q^2$, we are not especially alarmed by a 10-20% discrepancy between $Q^2 < 0.2$ and $Q^2 > 0.6$ (GeV/$c$)$^2$.

Finally, the statistical quality of the $1p$ SLAC data for $Q^2 = 1.1$ (GeV/$c$)$^2$ do not support an extraction of a $1s$ admixture either. Nor did we attempt to fit the $1p$ contribution to the $30 \leq E_m \leq 80$ MeV bin because the enhancement for large positive $p_m$ is probably an artifact of inadequate radiative unfolding. Single-component fits using RDWIA calculations are very similar to the EMA fits shown in Fig. \[EMA:SLAC\] and the figures are omitted. The SLAC $1p$ normalization factor for the $p$-shell bin is 0.86 for NLSH and EDAD1, which is perfectly consistent with the JLab results. However, $N_{1s} = 1.2$ for 30-80 MeV is much larger than the JLab result and scaled up to unrestricted $E_m$ would represent 140% of IPSM, suggesting a rather substantial underlying continuum.

### $Q^2$ dependence of spectroscopic factors {#sec:Q2}

Lapikás [[*et al.* ]{}]{}[@Lapikas00] proposed that the $Q^2$ variation seen in their analysis of the spectroscopic factors for $^{12}$C could be explained in terms of the resolution with which the electromagnetic knockout reaction probes the structure of a quasiparticle, such that the true spectroscopic factor would be observed in the limit $Q^2 \longrightarrow \infty$. Although this model was expressed in terms of a scale-dependent renormalization of the spectroscopic factor, it would be more appropriate to attribute such an effect, if present, to the single-nucleon current operator instead of to the spectroscopic factor. The overlap function for single-nucleon knockout takes the form $$\langle \Psi^{(A-1)}_\alpha | \Psi^{(A)}_0 \rangle = 
\sum_\beta c_{\alpha\beta} \phi_\beta(\bm{r})$$ where $\Psi$ is a many-body state of either the $A$ or $A-1$ nucleus, $\alpha$ is a state label with 0 being the target ground state, $c_{\alpha\beta}$ is a parentage coefficient, and $\phi_\beta$ is a local overlap function that is expected to resemble an orbital for the mean field. If the expansion basis is orthnormal, the spectroscopic factors become $$S_\beta = \sum_\alpha \left| c_{\alpha\beta} \right|^2$$ The parentage expansion depends upon the nuclear structure and is independent of the probe; hence, the spectroscopic factors should not depend upon $Q^2$. However, the assumption that the electromagnetic current operator is adequately approximated by the free nucleon current with the energy transfer placed on shell and with free form factors is questionable. Such approximations violate current conservation, Lorentz covariance, and unitarity. Furthermore, the current in a many-body system becomes nonlocal when expressed in terms of single-nucleon degrees of freedom; such nonlocality could contribute to an apparent scale dependence. Nevertheless, one must resort to simplified models of the current operator because there is no practical method for performing rigorous calculations for nucleon knockout from nuclei with $A>2$. Therefore, $Q^2$ dependent modifications of the off-shell single-nucleon current operator should not be surprising.

The recent JLab data for $^{12}$C show no evidence for systematic variations of the current operator within the range $0.6 \leq Q^2 \leq 1.8$ (GeV/$c$)$^2$. Although consistency among previous experiments is not entirely satisfactory, our RDWIA analysis shows much less variation with $Q^2$ than suggested by Lapikás [[*et al.* ]{}]{}Because the continuum contamination for $E_m > 30$ MeV has not been modeled accurately, we consider $N_{1p}$ for the $p$-shell bin to be the most reliable gauge of possible $Q^2$ dependencies. Comparing the average of the two Saclay results for quasiperpendicular kinematics with the average of the JLab results for NLSH and EDAD1, we see an increase of $0.87/0.69 = 1.26$ compared with a factor of about 1.6 for $S_{1p}+S_{1s}$ from Ref. [@Lapikas00]. There may still be a small effect, which we would attribute to the current instead of spectroscopic factors, but until the failure of RDWIA to reproduce low $Q^2$ data for parallel kinemtics is understood we are reluctant to interpret this as a simple scale factor.

Transparency {#sec:transparency}
------------

The experimental definition of nuclear transparency $$\label{eq:transp}
{\cal T}_\text{exp} = \frac{ \int_V d^3p_m dE_m d\Omega_e \; 
\sigma_{e,e^\prime p}}
{f \sum_\alpha \int_V d^3p_m dE_m d\Omega_e \; K
\rho_\alpha(p_m) S_\alpha(E_m) \sigma_{ep}}$$ compares the measured semi-inclusive differential cross section for proton knockout with a PWIA calculation based upon factorization of a model spectral function and the free [*e-p*]{} cross section. The phase-space volume is indicated by $V$ and we leave possible correction factors for nonuniform acceptance implicit, assuming that they are applied in both numerator and denominator consistently. Here $K$ is a kinematic factor, $\rho_\alpha(p_m)$ is the momentum distribution for orbital $\alpha$, $S_\alpha(E_m)$ is the energy distribution for single-nucleon removal from orbital $\alpha$, and $f$ is a correction factor describing the depletion of the single-hole spectral function by correlations that shift strength to large $E_m$. Dutta [[*et al.* ]{}]{}assumed that $f$ is independent of $\alpha$ and obtained an estimate of 0.9 for $^{12}$C from [@Makins94] that is consistent with the $1p$ normalization factor we extracted in the preceding section.

Similarly, a theoretical definition of nuclear transparency appropriate to quasiperpendicular kinematics takes the form $$\label{eq:Tperp}
{\cal T}_\bot =
\frac{ \sum_\alpha \int dp_m p_m \; \sigma_\text{RDWIA}(p_m,E_\alpha)}
{\sum_\alpha \int dp_m p_m \; \sigma_\text{RPWIA}(p_m,E_\alpha)}$$ where the numerator integrates the RDWIA and the denominator the RPWIA cross section for the same model of the spectral function. For simplicity we neglect the spread of electron kinematics and assume quasifree kinematics for the ejectile, neglecting the small variation of transparency with nucleon kinetic energy within the acceptance. Similarly, we also neglect variations over the spreading widths of nuclear orbitals and assume that the bound-nucleon kinematics can be approximated by the IPSM. A more rigorous calculation would use Monte Carlo integration over a realistic model of the experimental acceptance, but the dependence of nuclear transparency upon energy is very mild and the differences between transparencies for parallel and quasiperpendicular kinematics are small [@Kelly96b].

Note that one should not include the correlation factor $f$ in the theoretical definition of transparency because any strength that is removed from the numerator by correlations is also removed from the denominator by using the same spectral function for both. This factor is appropriate for Eq. (\[eq:transp\]) because the numerator integrates experimental yield while the denominator integrates a theoretical calculation based upon a model spectral function. When the denominator uses the IPSM one requires a theoretical estimate for the fraction of the the spectral strength that is shifted out of the experimental range of $E_m$. Unfortunately, a recent comparison of the data with RDWIA and Glauber calculations includes the correlation factor in both theoretical and experimental definitions of transparency [@Lava04], which we believe is incorrect.

RDWIA calculations using NLSH wave functions and several optical potentials from Dirac phenomenology [@Cooper93] are compared in Fig. \[fig:transparency\] with transparency data for $^{12}$C$(e,e^\prime p)$ from MIT [@Garino92], SLAC [@Makins94; @O'Neill95], and JLab [@Abbott98; @Dutta03]. The present results are very similar to those we previously obtained [@Kelly96b] using the EMA-noSV approximation and a definition of transparency that replaces the differential cross sections in Eqs. (\[eq:transp\],\[eq:Tperp\]) with the corresponding reduced cross sections. Neither change is more than a couple percent. Our results are also similar to the RDWIA results of Lava [[*et al.* ]{}]{}[@Lava04] after removal of the correlation factor. Also note that Lava [[*et al.* ]{}]{}used a transparency definition based upon reduced cross section. However, Meucci [@Meucci02a] obtained somewhat larger transparency factors using an EMA-SV calculation based upon differential cross section. The latter are the only RDWIA calculations, of which we are aware, that slightly overestimate the transparency for $^{12}$C$(e,e^\prime p)$, but the origin of their enhancement is not known.

![ RDWIA calculations using several optical potentials are compared to transparency data for $^{12}$C$(e,e^\prime p)$. The solid curve has been multiplied by 1.12 to provide a better fit to the data from MIT (open circle), SLAC (open square) and JLab (filled circles. The SLAC datum includes systematic uncertainties, but other error bars are statistical only.[]{data-label="fig:transparency"}](transparency.ps){width="4in"}

The variation among $A$-independent global optical potentials is small, but the $A$-dependent potential fit to data for $^{12}$C$(p,p)$ provides the best description of the energy dependence for $T_p \lesssim 400$ MeV. However, all of these calculations systematically underestimate the experimental transparency. Multiplying the EDAIC results by 1.12 provides a reasonable description of the data over this range of ejectile energy. Interestingly, this factor is very close to the inverse of the correlation correction applied by Abbott [[*et al.* ]{}]{}[@Abbott98]. If we assume that their correction factor is a reasonable estimate of the fraction of the single-nucleon yield shifted by correlations to large $E_m$, this comparison suggests that the experimental yield is approximately $12\%$ larger than single-nucleon knockout. This is consistent with the fact that the RDWIA normalization factors sum to the full IPSM strength without depletion. Therefore, either the RDWIA is approximately $12\%$ too absorptive or there is a significant continuum arising from multinucleon knockout that contributes approximately $12\%$ of the strength for $E_m < 80$ MeV. The rather strong $^{16}$O$(e,e^\prime p)$ continuum seen in Fig. 21 of Ref. [@Fissum04] suggests that the underlying continuum is probably responsible for the large normalization factors and transparency values for $^{12}$C$(e,e^\prime p)$.

The consistency between the fitted $1p$ normalization factors for $Q^2 > 0.6$ (GeV/$c$)$^2$ and the predicted IPSM depletion factor supports the accuracy of $(e,e^\prime p)$ attenuation factors calculated using RDWIA with global optical potentials. The observation that the yield for $30 \leq E_m \leq 80$ MeV is considerably larger than can be attributed to $1s$ knockout demonstrates that there is significant background in the numerator that is not described by the spectral function used in the denominator of Eq. (\[eq:transp\]). The contribution to this background from multinucleon currents artificially increases the measured transparency insofar as that quantity is intended to represent the loss of flux from single-nucleon knockout channel. However, the contribution to this background from rescattering processes that simply redistribute that flux within the experimental acceptance, instead of removing it entirely, represents a limitation of the distorted wave approximation which does not account for where flux “absorbed” by the optical potential finally appears [@Kelly96b]. The background fraction probably increases with $A$, but a more detailed model of the continuum is needed to distinguish between the processes that increase the experimental transparency with respect to RDWIA.

Summary and Conclusions {#sec:conclusions}
=======================

We have used RDWIA calculations based upon Dirac-Hartree bound-state wave functions to analyze data for $^{12}$C$(e,e^\prime p)$ for $E_m < 80$ MeV. Good fits are obtained for the recent JLab data [@Dutta03] for $0.6 \leq Q^2 \leq 1.8$ (GeV/$c$)$^2$, with a slight preference for NLSH over HS wave functions and practically no variation with any of the optical potentials provided by Cooper [[*et al.* ]{}]{}[@Cooper93]. We find that the $p$-shell bin contains approximately 87% of the IPSM strength independent of $Q^2$ over that range. Although the $s$-shell bin appears to carry almost 100% of IPSM, independent of $Q^2$, there is evidence in the momentum distribution for a significant continuum that artificially increases the $1s$ normalization factor for two-component fits mixing $1p_{3/2}$ and $1s_{1/2}$ contributions. Therefore, we consider the $1p$ contribution to the $15 \leq E_m \leq 25$ MeV bin to be the most reliable estimate of the depletion of IPSM orbitals by short-range correlations; the occupancy of 87% is consistent with estimates based upon the self-consistent Green’s function method and with recent direct measurements of the correlated continuum using parallel kinematics [@Rohe04; @Muther04].

We have also analyzed the same low $Q^2$ data sets used by Lapikas [[*et al.* ]{}]{}[@Lapikas00] to study the $Q^2$ dependence of normalization factors for $^{12}$C$(e,e^\prime p)$. We find that the RDWIA calculations for Dirac-Hartree wave functions reproduce the low $Q^2$ data for quasiperpendicular kinematics well, but with somewhat smaller normalization factors. For example, the $1p$ normalization factors for two Saclay experiments were 0.63 and 0.75 using NLSH wave functions and EDAD1 optical potentials, compared with 0.87 for the JLab data. However, the same model fails to reproduce low $Q^2$ data for parallel kinematics and remediation of this problem will require more than a multiplicative factor. Similar problems in previous NRDWIA analyses prompted van der Steenhoven [[*et al.* ]{}]{}[@vdSteenhoven88a] to adjust the Woods-Saxon wave function, the optical potential, and an empirical enhancement of the transverse components of the current simultaneously. Although a better fit was achieved, this wave function does not reproduce the data for quasiperpendicular kinematics for either early low $Q^2$ experiments or the more recent high $Q^2$ experiments. The momentum distributions for the fitted Woods-Saxon wave functions are too narrow, which artificially reduces the normalization factors fit to data. Thus, Lapikas [[*et al.* ]{}]{}[@Lapikas00] obtained a $1p$ normalization factor of only 0.56 for $Q^2 < 0.2$ (GeV/$c$)$^2$ and suggested that spectroscopic factors might depend strongly upon the resolution of the probe. We argue that spectroscopic factors are properties of nuclear structure that are independent of probe, but that the effective current operator for single-nucleon knockout may include additional $Q^2$ dependencies beyond those in familiar off-shell extrapolations of the free nucleon current operator. On the other hand, the additional $Q^2$ dependence is probably smaller than their estimate. Further investigation is needed to explain the discrepancy between parallel and quasiperpendicular kinematics at low $Q^2$ — we believe that it is unwise to fit the momentum distribution to data with a strong variation of $x=Q^2/2m\omega$ in the absence of a more fundamental theory of the effective current operator.

Finally, we found that RDWIA calculations of nuclear transparency need to be multiplied by approximately 1.12 to reproduce data for $^{12}$C and attribute much of this enhanced transparency to contributions to the continuum for $30 \leq E_m \leq 80$ MeV that are not directly related to single-nucleon knockout, such as multinucleon electromagnetic currents. However, a more detailed model of the continuum is need to distinguish between contributions due to multinucleon currents and rescattering processes.

We thank Dr. Dutta for data tables and Dr. Udías for tables of the NLSH and HS wave functions. We especially thank Dr. Lapikás for the parameters of the NIKHEF wave functions and data tables for the Tokyo, Saclay, and SLAC experiments. The support of the U.S. National Science Foundation under grant PHY-0140010 is gratefully acknowledged.
