caballero

¹Instituto de Estructura de la Materia, CSIC Serrano 123, E-28006 Madrid, Spain
^aDepartamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid, Spain
^bDepartamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain
²Departamento de Física Moderna, Universidad de Granada, E-18071 Granada, Spain
³Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

$\textstyle \parbox{15truecm}{\footnotesize \parindent=0pt Predictions for elect... ... and kinematical relativistic effects which can be tested by experiment.\par }$

Most theoretical work on (e, e'p) has been carried out on the basis of nonrelativistic approximations to the nucleon current, namely the standard distorted wave impulse approximation (DWIA) [1]. Data analyses based on DWIA have met two major difficulties: a) The spectroscopic factors extracted from low-p_m data are too small compared with theoretical predictions. b) DWIA calculations compatible with the low-p_m data predict much smaller cross sections at high-p_m than those experimentally observed [2]. Although short-range correlations are expected to increase the high-momentum components, their effect is negligible [3] at the small missing energies of these high-p_m data.

In recent years the relativistic mean-field approximation has been successfully used for the analyses of both low-p_m [4,5,6] and high-p_m [7] data. In the relativistic distorted-wave impulse approximation (RDWIA), the nucleon current

J_N( $\displaystyle \omega$ , $\displaystyle \vec{q}\,$ ) = $\displaystyle \int$ d $\displaystyle \vec{p}\,$ $\displaystyle \bar{\psi}_{F}^{}$ ( $\displaystyle \vec{p}\,$ + $\displaystyle \vec{q}\,$ ) $\displaystyle \hat{J}^{\mu}_{N}$ ( $\displaystyle \omega$ , $\displaystyle \vec{q}\,$ ) $\displaystyle \psi_{B}^{}$ ( $\displaystyle \vec{p}\,$ )

(1)

We have recently studied [11,12] the effect on the individual response functions of the relativistic treatment of the nucleon current. Within the relativistic plane wave impulse approximation (RPWIA) we showed [11] that the TL response is very sensitive to the negative-energy components of the relativistic bound nucleon wave function. We have also shown [12] that, for the j = l $\pm$ 1/2 spin-orbit partners of a given shell, this sensitivity is much larger for the j = l - 1/2 than for the j = l + 1/2 case.

A certain degree of controversy surrounds the TL response measured in exclusive quasielastic electron scattering from the least bound protons in several nuclei (¹²C, ¹⁶O, ²⁰⁸Pb): in some cases [13] large deviations from standard DWIA calculations appear, while in others [4] the data are close to the calculations. New data on the R_TL response for proton knockout from the 1p_1/2 and 1p_3/2 orbits of ¹⁶O are available from Jefferson Laboratory (TJNAF) experiment 89-003 [14] at | Q²| $\cong$ 0.8 (GeV/c)². The purpose of this work is to show that there are important kinematical and dynamical relativistic effects for this case.

We can divide the differences between this fully relativistic approach and the standard nonrelativistic one into two categories: i) Effects due to the fully relativistic 4-vector current operator, compared to the nonrelativistic current operator which involves $\vec{p}\,$ /M expansions. We call these effects kinematical as they are independent of the dynamics introduced by the nuclear interaction. ii) Effects due to the differences between relativistic and nonrelativistic nucleon wave functions, which depend on the 4-spinor structure and importantly on the potentials used in the respective Dirac and Schrödinger equations. We call these effects dynamical.

One may identify two types of relativistic dynamical effects: ii - a) Effects coming from the difference between the upper components of $\psi_{F}^{}$ ( $\psi_{B}^{}$ ) and the solutions $\chi_{F}^{}$ ( $\chi_{B}^{}$ ) of the Schrödinger equation. Assuming equivalent central and spin-orbit potentials, this difference stems from the well-known Darwin term. The influence of this term on (e, e'p) observables has been demonstrated in several works [6,15]. It appears to be the main dynamical relativistic effect in the cross section in the low-p_m region [6], and is important for the correct determination of the spectroscopic factor from low-p_m data. Its omission reduces the spectroscopic factor by 15-20%. It is included in all calculations presented here. ii - b) The other dynamical effect is due to the negative-energy components of the relativistic $\psi_{B}^{}$ , $\psi_{F}^{}$ wave functions. Starting from Schrödinger-like solutions $\chi$ one may at best construct properly normalized four-spinors of the form

$\displaystyle \psi$ = $\displaystyle {\frac{1}{\sqrt{N}}}$ $\displaystyle \left(\vphantom{\chi(\vec{p}),\frac{\vec{\sigma}\cdot\vec{p}}{\bar{E}+M}\chi(\vec{p}) }\right.$ $\displaystyle \chi$ ( $\displaystyle \vec{p}\,$ ), $\displaystyle {\frac{\vec{\sigma}\cdot\vec{p}}{\bar{E}+M}}$ $\displaystyle \chi$ ( $\displaystyle \vec{p}\,$ ) $\displaystyle \left.\vphantom{\chi(\vec{p}),\frac{\vec{\sigma}\cdot\vec{p}}{\bar{E}+M}\chi(\vec{p}) }\right)$

(2)

**Figure 1:** Cross section in nb/(MeV sr) (left), R_TL in fm³ (middle) and A_TL (right) for proton knockout from ¹⁶O for the 1p_1/2 (upper panel) and 1p_3/2 (lower panel) orbits, versus missing momentum p_m in MeV/c. Results shown correspond to a fully relativistic calculation using the Coulomb gauge and the current operators: cc1 (solid line) and cc2 (dotted line). Also shown are the results after projecting the bound and scattered proton wave functions over positive-energy states (J_{+ +}, dashed line) and using the asymptotic momenta (J_as, long-dashed line). For the p_3/2 shell a small contribution was taken into account from the nearby 5/2⁺ and 1/2 + states known from the low-p_m data.
$\begin{figure} \begin{center} \mbox{\epsfig{file=fig1.eps,width=\textwidth}} \end{center}\end{figure}$

Fig. 1 shows the differential cross section, R_TL response and TL asymmetry (A_TL) for p_1/2 (top panels) and p_3/2 (bottom panels). R_TL and A_TL are obtained from the cross sections measured at $\phi_{F}^{}$ = 0^o and $\phi_{F}^{}$ = 180⁰ with the other variables ( $\omega$ , Q², E_m, p_m) held constant, where $\phi_{F}^{}$ is the azimuthal angle of the scattered proton (we follow the same convention as in [14]). Relativistic calculations using the cc1 and cc2 current operators are shown by solid and dotted lines, respectively. The Coulomb gauge has been used throughout this work. The role of the negative-energy components can be seen in Fig. 1 comparing the solid with the short-dashed lines. The dashed lines show the results obtained with the cc1 current operator when the negative-energy components are projected out. We call J_{+ +} the corresponding nucleon current, because $\psi_{F}^{}$ , $\psi_{B}^{}$ wave functions in eq. (1) are replaced by their positive energy projections [11] $\psi_{F}^{(+)}$ , $\psi_{B}^{(+)}$ , respectively. The difference between the solid and short dashed lines is due to the dynamical enhancement of the lower components. It is important to realize that the positive-energy projectors needed to compute J_{+ +} depend on the integration variable $\vec{p}\,$ . One may attempt to neglect this dependence by using projection operators corresponding to asymptotic values of the momenta, i.e. projectors acting on $\psi_{F}^{}$ and $\psi_{B}^{}$ respectively, with P_F = (E_F, $\vec{p}_{F}^{}$ ), P_F - $\bar{Q}^{\mu}_{}$ the asymptotic four-momentum of the outgoing and bound nucleon respectively. We refer to this approach as asymptotic projection (J_as). The results corresponding to this approximation are shown by long dashed lines in Fig. 1. They are obtained with the cc1 operator and are very similar for cc2.

**Table 1:** Spectroscopic factors obtained with the fully relativistic currents using cc1 and cc2 operators (second and third column), results obtained with the J_{+ +} and with J_as currents (see text) are presented in the fourth and fifth columns. For the p_3/2 shell we took into account the small contribution from the near lying 5/2⁺ and 1/2 + states known from the low-p_m data.
Shell	CC1	CC2	++ Proj.	As. Proj.
1p_1/2	0.72	0.77	0.83	0.77
1p_3/2	0.70	0.76	0.84	0.77

In Table 1 we show the spectroscopic factors needed by the different approximations. The spectroscopic factor simply scales down the curves for differential cross sections and R_TL while leaving A_TL unchanged. As seen in the left panel of Fig. 1, with these spectroscopic factors, the differential cross sections for | p_m| < 300 MeV/c are similar in the different approximations, but differ for | p_m| > 300 MeV/c where there is a substantial influence of negative energy components. Note that the two fully relativistic calculations, J_cc1 (solid line) and J_cc2 (dotted line), fit better the high-p_m data than the projected calculations. Note also that the cross sections obtained with positive-energy projected wave functions are more symmetrical around p_m = 0 than the RDWIA results. Therefore, the effect of removing the negative-energy components shows up more in R_TL and A_TL (see middle and right-hand panels of Fig. 1). Note that the dependence on the dynamical enhancement of the lower components is stronger for the p_1/2 R_TL response than for the p_3/2, a feature that was first seen in RPWIA [11] and that persists in RDWIA. The agreement with experimental R_TL data is generally better for the fully relativistic calculations, even in the high-p_m region where data are more precise, and for the p_1/2 shell where relativistic effects are more important. Particularly interesting is the oscillatory structure of the fully relativistic result for A_TL. This characteristic is preserved by the exact positive-energy projection method J_{+ +}, but not by J_as that severely modifies A_TL for both orbitals. We notice that the A_TL calculated with J_as are very similar to the ones obtained in [16], as the J_as calculation is similar to the EMA (noSV) of said reference. At low momentum this approach lies close to the fully relativistic ones and to the J_{+ +} ones, but beyond p_m $\simeq$ 200 MeV/c they are noticeable different. The oscillating trend of the A_TL calculated in RDWIA is confirmed by the data [14] and agrees qualitatively with previous calculations by Van Orden [14]. As it was the case for R_TL, the negative energy components clearly affect the A_TL of the j = l - 1/2 partner, largely improving the agreement with experiment. The experimental error bars for this observable at high-p_m should be reduced by a factor two or more to have more conclusive evidence. Anyway, for the p_3/2 shell the asymptotic calculation clearly fails to reproduced the experimental data, except for the point at around 150 MeV, where all the calculations predict very similar results.

**Figure 2:** R_TL (right panels) and A_TL asymmetries (left panels) for proton knockout from ¹⁶O for the 1p_1/2 (top panels) and 1p_3/2 (bottom panels) orbits. Results shown correspond to a fully relativistic calculation using the Coulomb gauge and the current operator cc2 (solid line), a calculation performed by projecting the bound and scattered proton wave functions over positive-energy states (dashed line) and two nonrelativistic calculations with (long-dashed) and without (dotted) the spin-orbit correction term in the charge density operator (see text for details).
$\begin{figure} \begin{center} {\epsfig{file=fig2.eps,width=0.85\textwidth}} \end{center}\end{figure}$

Other relativistic effects can be seen in Fig. 2, where we compare RDWIA results on A_TL (left panels) and R_TL (right panels) to nonrelativistic approaches at various levels. To minimize the differences we have used the cc2 current operator and nonrelativistic scattered wave functions obtained from Dirac-equivalent Schrödinger equations. This ensures that the nonrelativistic wave functions correspond to the upper components of the relativistic ones, containing in particular the Darwin term. For the nonrelativistic bound wave functions, we used the ones in [17]. In said reference new approximations to the on-shell relativistic one-body current operator were developed to take better account of relativistic kinematic effects in nonrelativistic calculations. In particular, the charge density contains a spin-orbit correction that affects R_TL [17]. In Fig. 2 we show by long dashed lines the results obtained with the ``relativized current'' and by dotted lines the results obtained with that relativized current when the spin-orbit correction term to the charge-density is neglected. One can see that the spin-orbit correction has a very large effect on R_TL and A_TL. Its omission causes large deviations from the relativized current results. Using the DWEEPY [18] code we have obtained for A_TL similar results to the dotted lines in Fig. 2.

The short dashed lines in Fig. 2 are results obtained with the relativistic current and 4-spinors constructed as in eq. (2) from the nonrelativistic bound and scattered wave functions. In this way, the relativistic kinematic is fully taken into account, only the dynamical enhancement of the lower components is missed. This is why these results (short-dashed lines) for A_TL and R_TL are much closer to the fully relativistic results shown by the solid lines. We see also that a large oscillation of A_TL can be recovered in the nonrelativistic approach. The much smaller oscillation of A_TL is a distinctive feature of the J_as results, and it seems to be ruled out by the experiment. We also note that, while the effect of the dynamical enhancement of the lower components is larger in p_1/2 than in p_3/2 shells, the effects of relativistic kinematics are of the same order in both shells. Everything put together, the data on A_TL and R_TL are a strong indication of the presence and crucial role played by dynamical effects of relativity affecting the lower components, in the description of electron-nucleus scattering reactions.

In conclusion, we have identified two types of relativistic effects on R_TL and A_TL. One is of kinematical origin, and has a large contribution from the spin-orbit correction to the charge density, and other is of dynamical origin. The latter is mainly due to the enhancement of the lower components and is stronger for p_1/2 than for the p_3/2 orbital. This is in addition to the dynamical effect on the upper component due to the Darwin term which is present in all the results given here, and that mostly affects the determination of spectroscopic factors [6]. It is encouraging that the data [14] agree so well with the predictions of the fully relativistic calculations and one anticipates being able to make even more stringent tests when a finer grid of high-precision data involving other nuclei become available in the range 200 $\le$ p_m $\le$ 400 MeV/c.

Acknowledgments

This work was partially supported under Contracts No. 940183 (NATO Collaborative Research grant), #DE-FC01-94ER40818 (cooperative agreement with the US department of Energy D.O.E.), PB/95-0123, PB/95-0533-A, PB/95-1204 (DGICYT, Spain), PB/96-0604 (DGES, Spain), PR156/97 (Complutense University, Spain) and by the Junta de Andalucía (Spain).

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