Analysis of polarized 16 O ( → e , e ′ → p ) observables within the relativistic distorted wave impulse approximation

Analysis of polarized 16O„e ᠬ ,e⬘p ᠬ …observables within the relativistic distorted wave impulse approximation M. C. Martínez,1J. R. Vignote,2J. A. Caballero,1T. W. Donnelly,3E. Moya de Guerra,4and J. M. Udías2 1Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apartado Postal 1065, E-41080 Sevilla, Spain 2Departamento de Física Atómica, Molecular y Nuclear, Universidad Complutense de Madrid, E-28040 Madrid, Spain 3Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain (Received 22 September 2003; published 12 March 2004) Recoil nucleon transferred polarization observables in coincidence quasielastic electron scattering are studied within the relativistic distorted wave impulse approximation. Results for response functions and polarization asymmetries are discussed for proton knockout from p1/2,p3/2, and s1/2 shells in 16O. The impact of spinor distortion is examined by comparing the fully relativistic calculation with results obtained by projecting out the negative-energy components. In particular, a careful analysis of effects linked to the description of the bound and scattered relativistic nucleon wave functions is presented. The high sensitivity of some polarization observables to the dynamical enhancement of the lower components, already shown within the relativistic plane wave impulse approximation, is proven to be maintained in the relativistic distorted wave approach. Semirelativistic approaches based on the effective momentum approximation are also studied. Finally, comparison with experimental data and a brief analysis of effects linked to medium modified form factors is presented. DOI: 10.1103/PhysRevC.69.034604 PACS number(s): 25.30.Rw, 14.20.Gk, 24.10.Jv, 24.30.Gd I. INTRODUCTION A very topical issue in nuclear physics at present is the search for evidence of possible modification of the nucleon form factors inside the nuclear medium. A number of double polarized 共e ជ ,e⬘p ជ 兲experiments have been proposed or carried out recently to measure polarization transfer asymmetries, motivated by the hope that such observables may provide valuable information that can shed some light on this issue. Importantly, transferred polarization observables have been identified as being ideally suited for such studies: they are believed to be the least sensitive to most standard nuclear structure uncertainties and accordingly to provide the best opportunities for studying the nucleon form factors in the nuclear medium. Polarization transfer data have been reported recently for the case of 16O共e ជ ,e⬘p ជ 兲15N in Ref. [1]and for 4He共e ជ ,e⬘p ជ 兲3H in Refs. [2,3]. Although the experimental uncertainties in both cases make it difficult to draw unambiguous conclusions on the nucleon form factors inside the nuclei, the data in Ref. [3]do seem to favor such a possibility. Specifically, this means that comparisons of measured polarization asymmetries with those computed using the best currently available nuclear models for the states and operators involved in the coincidence reaction in fact show disagreements, and that these can be removed by modifying the nucleon form factors in a reasonable way. Of course, what constitutes the “best currently available nuclear models” must be judged carefully. In particular, the kinematic regime where the measurements have been undertaken is at relatively high energy—to make the reaction sufficiently impulsive to be at all interpreted as a simple singlenucleon knockout reaction—and it is clear that relativistic effects in wave functions and operators are essential. So, for instance, the data in Ref. [2]disagree significantly with the standard nonrelativistic calculations; however, this cannot be taken as evidence for nucleon modifications, since one finds that the results are (not unexpectedly)much more in accord with a fully relativistic approach. Also recent data on induced polarization in 12C[4]strongly support an analysis based on the fully relativistic formalism [5]. These results are not surprising since spin and relativity are intrinsically related, and hence one may a priori consider the relativistic formalism to be better suited to describe polarization observables. Indeed, most electron scattering experiments performed in the last decade have involved energies and momenta high enough to invalidate the nonrelativistic approximations assumed within the standard nonrelativistic distorted wave impulse approximation (DWIA), i.e., bound and scattered wave functions given as solutions of the Schrödinger equation, and one-body current operator resulting from a nonrelativistic reduction. In the relativistic distorted wave impulse approximation (RDWIA), nucleon wave functions are described by solutions of the Dirac equation with scalar and vector 共S-V兲 potentials, and the relativistic free nucleon current operator is used. Relativistic effects can be classified into two basic categories according to their origin, namely, kinematical and dynamical effects. The former are due to the truncation of the current operator within the nonrelativistic approach, the latter, dynamical effects, come from the difference between the relativistic and nonrelativistic wave functions. Here one may distinguish a dynamical depression of the upper component of the scattered nucleon wave function in the nuclear interior (Darwin term)and a dynamical enhancement of the lower components, mainly that corresponding to the bound nucleon wave function. So far, RDWIA calculations for cross sections and rePHYSICAL REVIEW C 69, 034604 (2004) 0556-2813/2004/69(3)/034604(15)/$22.50 ©2004 The American Physical Society69 034604-1 sponse functions at low and high missing momenta [6–10] have clearly improved the comparison with experimental data over the previous nonrelativistic approaches. Moreover, RDWIA also predicts larger spectroscopic factors which are more in accord with theoretical calculations which incorporate correlations [6,10]. Concerning the current operators, in some recent studies [11–15]new so-called ⬙semirelativistic⬙approaches have been introduced to describe 共e,e⬘p兲reactions. Here the semirelativistic current operators are obtained by expanding only in missing momentum over the nucleon mass while treating the transferred energy and momentum exactly. This new approach has been proven to retain important aspects of relativity, and hence its predictions, compared with the standard DWIA, agree much better with the RDWIA calculations. Concerning dynamical effects, the enhancement of the lower components of bound Dirac spinors [9,10](not present in the semirelativistic approaches)has been shown to play a crucial role in the description of the interference RTL response and left-right asymmetry ATL. Meson exchange currents and the ⌬-isobar contribution have recently been analyzed in Refs. [16,17]within the semirelativistic approach, also showing very significant effects, particularly due to ⌬,at large missing momentum p艌300 MeV/c. In this paper we focus on the analysis of polarized A共e ជ ,e⬘p ជ 兲B observables within the framework of the RDWIA. Our aim is to study the role played by both kinematical and dynamical relativistic effects in a consistent description of the polarized responses and asymmetries. This work extends the previous analyses presented in Refs. [18,19]within the plane wave approach, now including a realistic description of the final-state interactions (FSI) through relativistic optical potentials. The magnitude of relativistic effects on various transfer polarization observables is carefully examined, disentangling the role played by the various ingredients that enter in the fully relativistic formalism. In particular, we extend the study of Ref. [18]where within relativistic plane wave impulse approximation (RPWIA)we demonstrated the importance of the negativeenergy components of the relativistic bound nucleon in the description of the polarized responses and transferred polarization asymmetries. The RDWIA analysis performed here allows one to examine also the dynamical enhancement of the lower components in the scattered Dirac wave functions and moreover, makes it possible to carry out meaningful comparisons with measured observables. Returning to the issue of potential medium modifications of the nucleon form factors, the current study has the following goal: we wish to explore a selected set of model “variations on a theme” of the type discussed above. In all cases we choose only modeling, that is, within the context of the general relativistic approach being adopted, consistent with what we know about initialand final-state wave functions and one-body electromagnetic operators. Since equally acceptable relativistic potentials exist when obtaining the states and since alternative descriptions of the current operators are likewise acceptable, it is impossible at present to define what is “the best” model. Our goal is to explore these acceptable models and where the resulting polarization observables differ with the choice of model to ascribe these variations to a (minimal)theoretical uncertainty. Needless to say, all of this is within the general context of relativistic mean-field modeling and so the resulting uncertainties are minimal in the sense that effects that go beyond the scope of the modeling might increase the uncertainties. In the final analysis, only if medium modification effects are larger than the uncertainties we find here, and only if the uncertainties that arise from ingredients not in the present model can ultimately be shown to be small, will a convincing case be made for the necessity of having such medium modification effects. The paper is organized as follows. In Sec. II we briefly introduce the general formalism for A共e ជ ,e⬘p ជ 兲B reactions focusing on the relativistic distorted wave impulse approximation. Within this context, we also introduce the projected approach, the effective momentum approximation (EMA-noSV)and the use of semirelativistic current operators. By comparing them one may get a clear image of the importance of relativity in these processes. In Sec. III we present and discuss the results, paying special attention to the polarized responses and transferred polarization asymmetries. Finally, in Sec. IV we summarize our conclusions. II. DESCRIPTION OF A„e ᠬ ,e⬘p ᠬ …BREACTIONS A. General formalism: RDWIA In this section we briefly review the general formalism needed to describe coincidence 共e ជ ,e⬘p ជ 兲reactions. We consider plane waves for the incoming and outgoing electron (treated in the extreme relativistic limit)and the Born approximation (one virtual photon exchanged). When the incoming electron is polarized and the final nucleon polarization is measured, the differential cross section can be written as [20–24] d ␴ d␧ed⍀ed⍀F= ␴ 0 2关1+P· ␴ +h共A+P⬘· ␴ 兲兴,共1兲 where the variables 兵␧e,⍀e其refer to the scattered electron and ⍀Fto the ejected nucleon. The term ␴ 0is the unpolarized cross section, his the incident electron helicity, Adenotes the electron analyzing power, and P共P⬘兲represents the induced 共transferred兲polarization. Note that both Pand P⬘ depend on the outgoing nucleon polarization, but P⬘only becomes accessible when the incoming electron beam is polarized. The cross section in Eq. 共1兲can also be written in terms of nuclear responses as follows: d ␴ d␧ed⍀ed⍀F=K ␴ Mfrec −1 兵vL共RL+Rn LS ˆn兲+vT共RT+Rn TS ˆn兲 +vTL关共RTL +Rn TLS ˆn兲cos ␾ +共Rl TLS ˆ l +Rs TLS ˆs兲sin ␾ 兴+vTT关共RTT +Rn TTS ˆn兲cos 2 ␾ +共Rl TTS ˆ l+Rs TTS ˆs兲sin 2 ␾ 兴+h兵vTL⬘关共Rl TL⬘S ˆl +Rs TL⬘S ˆs兲cos ␾ +共RTL⬘+Rn TL⬘S ˆn兲sin ␾ 兴 +vT⬘关Rl T⬘S ˆl+Rs T⬘S ˆs兴其其,共2兲 where ␾ is the azimuthal angle that determines the outgoing M. C. MARTÍNEZ et al. PHYSICAL REVIEW C 69, 034604 (2004) 034604-2 nucleon momentum. The term Kis a kinematical factor given by K=pFMNMB/MA, with pFthe outgoing nucleon momentum, MNthe nucleon mass, and MB共MA兲the mass of the residual nucleus 共target兲, respectively. The Mott cross section is represented by ␴ M,frec is the recoil factor given by frec=1+共 ␻ pF−qEFcos ␪ F兲/MApF, where EFis the outgoing nucleon energy and ␪ Fis the angle between pFand the transferred momentum, and the vK,K=L,T,…are the standard electron scattering kinematical factors 共see Refs. 关24,25兴兲. The indices l,s,nrefer as usual to the directions selected to specify the recoil nucleon polarization: l共parallel to the momentum pF兲,n共perpendicular to the plane containing pFand the transfer momentum q兲, and s共determined by n⫻l兲. From this large number of possible response functions some selection can be made to limit the focus: 共i兲Assuming coplanar kinematics, i.e., ␾ =0° ,180°, from the total set of 18 responses in Eq. 共2兲 only 12 survive. 共ii兲From these twelve responses, the four transferred polarization ones Rl,s K⬘only contribute when the electron is polarized, while the four induced polarization ones Rn Konly enter when FSI are taken into account. Following the analysis presented in Ref. [18], in this work we limit our attention to those observables that survive in the plane wave limit, i.e., transferred polarization responses Rl TL⬘,Rl T⬘,Rs TL⬘,Rs T⬘and transferred asymmetries Pl ⬘,Ps ⬘.A detailed study of the induced polarization observables within RDWIA has been presented in Ref. [5]. The response functions in Eq. (2)are constructed directly by taking the appropriate components of the hadronic tensor W ␮ ␯ which, within the RDWIA, comes from bilinear combinations of the nucleon current matrix elements JN ␮ 共 ␻ ,q兲= 冕 dp⌿ ¯ F共p+q兲J ˆN ␮ ⌿B共p兲,共3兲 where ⌿Band ⌿Fare relativistic wave functions describing the initial bound and final outgoing nucleons, respectively, and J ˆN ␮ is the relativistic one-body current operator. The bound wave function ⌿Bis a four-spinor with well-defined parity and angular momentum quantum numbers ␬ b, ␮ b, obtained within the framework of the relativistic independent particle shell model. The mean field in the Dirac equation is determined through a Hartree procedure from a phenomenological relativistic Lagrangian with scalar 共S兲and vector 共V兲 terms. It may be written ⌿B共p兲=⌿ ␬ b ␮ b共p兲=1 共2 ␲ 兲3/2 冕 dre−ip·r⌿ ␬ b ␮ b共r兲 =共−i兲ᐉb 冢 g ␬ b共p兲 S ␬ bf ␬ b共p兲 ␴ ·p p 冣 ⌽ ␬ b ␮ b共p ˆ兲共4兲 with ⌽ ␬ b ␮ b共p ˆ兲the usual spinor harmonics. The wave function for the ejected proton ⌿Fis a scattering solution of a Diraclike equation, which includes S-Vglobal optical potentials obtained by fitting elastic proton scattering data. This wave function, obtained as a partial wave expansion, is given in momentum space by ⌿F共p兲=4 ␲ 冑EF+MN 2EF ⫻兺 ␬␮ me−i ␦ ␬ *iᐉ具ᐉm1 2sF兩j ␮ 典Yᐉ m*共p ˆF兲⌿ ␬ ␮ 共p兲,共5兲 where ⌿ ␬ ␮ 共p兲are four-spinors of the same form as in Eq. 共4兲, but the phase shifts and radial functions are complex because of the complex optical potential involved. Finally, for the nucleon current operator we consider the two choices denoted as CC1 and CC2 [26] J ˆCC1 ␮ =共F1+F2兲 ␥ ␮ −F2 2MN共P ¯ +PF兲 ␮ ,共6兲 J ˆCC2 ␮ =F1 ␥ ␮ +iF2 2MN ␴ ␮ ␯ Q ␯ ,共7兲 where F1and F2are the Dirac and Pauli nucleon form factors related to the electric and magnetic Sachs form factors in the usual form. The variable P ¯ ␮ in Eq. 共6兲is the fourmomentum of the initial nucleon for on-shell kinematics, i.e., P ¯ ␮ =共E ¯ ,p兲共E ¯ =冑p2+MN 2and p=pF−q兲. B. Dynamical effects: projected approach and effective momentum approximation In recent years a considerable effort has been devoted to the analysis of quasielastic 共e,e⬘p兲reactions using a fully relativistic formalism. Within this framework, particular emphasis has been placed on comparison between relativistic and nonrelativistic approaches, trying to identify and disentangle clearly the ingredients which lead to different results in the two types of calculations. In some recent works [27], relativistic effects have been analyzed by comparing directly results obtained from a standard nonrelativistic DWIA code (DWEEPY)with those provided by a relativistic calculation. These investigations were aimed at providing systematic and precise information on the magnitude of the effects introduced by relativity when compared with the standard nonrelativistic description based on DWEEPY. The latter was widely used in the 1980’s to analyze low-energy experimental data. However, although interesting, this study did not allow one to identify clearly the role played by the various ingredients entering into the relativistic formalism. Note that apart from the four-spinor versus two-spinor structure involved in relativistic and nonrelativistic calculations, respectively, also the potentials used in the Dirac and Schrödinger equations for the bound and scattered nucleon are different. Moreover, the nonrelativistic current operator results from an expansion in a basis of free nucleon plane waves and a Pauli reduction with the operator expanded in powers of p/MN,q/MN, and/or ␻ /MN,pbeing the missing momentum, qand ␻ the transfer momentum and energy, respectively. In this work we focus on the separate analysis of the various ingredients that enter in the general formalism, and evaluate their impact on the transferred polarization observables. Hence, in order to minimize the mismatch coming from the different assumptions involved in relativistic and nonrelativistic approaches, ANALYSIS OF POLARIZED 16O共e ជ ,e⬘p ជ 兲OBSERVABLES …PHYSICAL REVIEW C 69, 034604 (2004) 034604-3 all of the results presented in this work have been evaluated using the same potentials and code. Dynamical effects arise from the differences between relativistic and nonrelativistic potentials and wave functions. A detailed study on this subject has been already presented in Refs. [9,10,28], so here we simply summarize the basic concepts needed for later discussion of the results. As is well known, interacting Dirac wave functions have a nonzero overlap with the Dirac sea [29]. The presence of the S-V potentials leads to a significant dynamical enhancement of the lower components of the Dirac solution at the nuclear interior. This fact is clearly illustrated by realizing that for a general solution of the Dirac equation with scalar and vector potentials, its upper and lower components are related by ⌿down = ␴ ·p E+MN+S−V⌿up 共8兲 with S⬍0 and V⬎0. Note that these lower components are enhanced with respect to the ones corresponding to free positive energy spinors where S=V=0. This effect has been referred to as dynamical enhancement of the lower components, and more recently as spinor distortion 关30兴. The analysis of these dynamical effects can be done by constructing properly normalized four-spinor wave functions where the negative-energy components have been projected out. Thus, instead of the fully relativistic expression given in Eq. (3), the nucleon current is evaluated as JN ␮ 共+,+兲共 ␻ ,q兲= 冕 dp⌿ ¯ F 共+兲共p+q兲J ˆN ␮ ⌿B 共+兲共p兲,共9兲 where ⌿B 共+兲共p兲,关⌿F 共+兲共p兲兴 is the positive-energy projection of ⌿B共p兲,关⌿F共p兲兴, i.e., ⌿B 共+兲共p兲=⌳共+兲共p兲⌿B共p兲 ⌿F 共+兲共p+q兲=⌳共+兲共p+q兲⌿F共p+q兲,共10兲 where ⌳共+兲共p兲=共MN+P ¯ 兲/2MNis the positive-energy projector. Then the effects due to the dynamical enhancement of the lower components show up clearly by comparing the results obtained using the fully relativistic amplitude given in Eq. 共3兲with those evaluated by using Eq. 共9兲. Note that the relationship between lower and upper components in the projected wave functions is similar to that corresponding to free nucleon wave functions, but with the positive-energy projectors depending explicitly on the integration variable p. An additional approach, referred to as asymptotic projection, consists of introducing the asymptotic values of the momenta into the positive-energy projectors acting on the bound and scattered wave functions. This asymptotic projection is very similar (although it is not completely equivalent)to the EMA-noSV introduced originally by Kelly [30]. Within the EMA-noSV approach, the four spinors used have the same upper components as those of the Dirac equation solutions, but the lower components are obtained by enforcing the “free” relationship between upper and lower components and using the asymptotic momenta at the nucleon vertex. Note that these wave functions also lack the dynamical enhancement of the lower components. Finally, one also has the dynamical quenching of the upper component of the Dirac wave function in the nuclear interior compared with the nonrelativistic solution. This effect, associated with the Darwin term, is implicitly included in all calculations presented in this work. Hence the differences between the EMA-noSV approach (or equivalently the asymptotic projection)and the fully relativistic calculation can be solely ascribed to the negative-energy components. C. Kinematical effects: semi-relativistic reductions Another ingredient which leads to differences between the relativistic and nonrelativistic approaches concerns the specific form of the current operator used to evaluate Eq. (3). Instead of the fully relativistic operator considered in RDWIA, truncated expressions up to first or higher orders in p/MN, ␻ /MN, and/or q/MNare employed in standard nonrelativistic DWIA calculations. These effects, here referred to as kinematical relativistic effects [9,10,19], include not only the relativistic kinematics of the nucleon energies and momenta [16,31](which must be accounted for in order to describe properly the form of the momentum distribution), but also the effects linked to the use of the relativistic nucleon current operator. Improved nonrelativistic expansions of the nucleon current operator, denoted as semirelativistic approaches, which contain important aspects of relativity, have been derived recently and are available in the literature [12–15]. In this paper we investigate the kinematical effects associated with these expansions in polarized 共e ជ ,e⬘p ជ 兲observables. To this end we have also incorporated the semirelativistic expressions in the relativistic code, so that a direct comparison between the fully relativistic calculation and the semirelativistic approach becomes more meaningful because the effects due to the choice of wave functions and/or potentials are minimized. To make the analysis clearer, in what follows we explain in some detail the procedure used to get the semirelativistic results. In the case in which spinor distortion is neglected and asymptotic momenta are used, the relativistic 共4⫻4兲 current matrix element can be recast in an equivalent form that involves an effective 共2⫻2兲current operator J ¯ eff ␮ that occurs between the upper two component spin 1 2spinors. The 共2⫻2兲operator J ¯ eff ␮ is obtained without any approximation concerning nonrelativistic reductions; it corresponds to an exact expression for the on-shell electromagnetic current operator [15]. This means that the results obtained using J ¯ eff ␮ between bispinors corresponding to the upper components of the relativistic wave functions should coincide exactly with those obtained using the original relativistic 共4⫻4兲electromagnetic current operator within the EMA-noSV approach M. C. MARTÍNEZ et al. PHYSICAL REVIEW C 69, 034604 (2004) 034604-4 [30]. Finally, a comparison between these results and those provided by making use of the semirelativistic expressions for the operator, leads to direct information on the magnitude associated with the kinematical relativistic effects. It is important to point out that the semirelativistic reduction is done in the context of the effective momentum approximation, i.e., using asymptotic momenta. The semirelativistic expression of the electromagnetic current operator relies on the direct Pauli reduction method, by expanding only in the missing momentum (p)over the nucleon mass. The transfer energy and momentum are treated exactly. Up to first-order in p/MN, the following results for the electromagnetic current operators are obtained: J ¯ 0= ␬ 冑 ␶ GE+i 冑1+ ␶ 冉 GM−GE 2 冊 共 ␬ ⫻ ␩ 兲· ␴ ,共11兲 J ¯ =1 冑1+ ␶ 再 iGM共 ␴ ⫻ ␬ 兲+ 冉 GE+ ␶ 2GM 冊 ␩ +GE ␬ −GM 2共1+ ␶ 兲共 ␬ · ␩ 兲 ␬ −iGE 2共1+ ␶ 兲共 ␴ ⫻ ␬ 兲 ␬ · ␩ −i ␶ ⫻ 冉 GM−GE 2 冊 共 ␴ ⫻ ␩ 兲+i共GM−GE兲 2共1+ ␶ 兲共 ␬ ⫻ ␩ 兲 ␴ · ␬ 冎 , 共12兲 where we have introduced the usual dimensionless variables: ␶ =兩Q2兩/4MN 2, ␬ =q/2MNand ␩ =p/MN. Obviously, when computing response functions, evaluated by taking bilinear combinations of the electromagnetic current matrix elements, terms of order ␩ 2should be dismissed. As shown, the spin-orbit part of the charge and the relativistic correction to the transverse current, the first-order convective spin-orbit term, are included in Eqs. (11)and (12). Although the above expressions have been already presented in the literature [12–15,31], in most of these previous works the analysis of the observables has been performed adopting additional approximations on the vector current, namely, J ¯ is simply taken as the standard nonrelativistic reduction except for a global kinematical factor 共1+ ␶ 兲−1/2 that includes relativistic corrections coming from the Dirac spinors (see Refs. [12–15]for details). Here we evaluate the recoil nucleon polarized observables by making use of the full semirelativistic currents in Eqs. (11)and (12)taken between the upper components of the original relativistic wave functions. III. RESULTS AND DISCUSSION In this section we analyze the recoil nucleon transferred polarization observables for proton knockout from 16O. Although we focus on results for the 1p1/2 shell, similar conclusions are reached for the 1p3/2 and 1s1/2 shells unless otherwise specified. Results are computed for both CC1 and CC2 choices of the current operator in Eqs. (6)and (7), and the Coulomb gauge is assumed. A detailed study on gauge ambiguities in RPWIA has been presented in Ref. [18]showing that the Coulomb and Landau gauges lead to very similar results, differing significantly from the ones corresponding to the Weyl gauge. These results are proven to persist within the relativistic distorted approach. The bound nucleon wave function is obtained using the parameters of the set NLSH [32]. Results computed with other parameterizations are found to be similar and do not change the general conclusions. For the outgoing nucleon wave function, we use the energy-dependent, A-independent potential derived by Clark et al. for 16O(EDAIO)[33]which describes fairly well the existing elastic proton-16O scattering data. Although our main interest in this work concerns the effects introduced by dynamical and kinematical relativistic effects, a brief study of the sensitivity of the polarized observables to the description of final-state interactions is also presented. Hence in following section, results evaluated with different relativistic optical potentials are shown and compared. Finally, the Coulomb distortion of the electron wave functions is accounted for by using the effective momentum approximation with the nuclear Coulomb potential equal to 3.5 MeV (see Refs. [6,7] for details). All the results shown throughout this work correspond to the nucleon form factor parametrization of Gari and Krumplemann [34]. A. Final-State Interactions: relativistic optical potentials We start our discussion with the analysis of the longitudinal and sideways transferred polarization asymmetries and their dependence on FSI. In Fig. 1, Pl ⬘and Ps ⬘are presented as functions of the missing momentum p. The kinematics are chosen with 共q, ␻ 兲constant, q=1 GeV/cand ␻ =439 MeV, yielding 兩Q2兩=0.8 共GeV/c兲2. This roughly corresponds to the experimental conditions of experiments E89-003 and E89033 performed at JLab [35–37]. Left panels correspond to the p1/2 shell and right panels to p3/2. In each case, RDWIA results obtained with the EDAIO optical potential parametrization [33]are compared with the RPWIA results. Plane wave calculations after projecting out the negative-energy components of the bound nucleon wave function, denoted as PWIA, are also shown. Note that PWIA polarization transfer asymmetries coincides with what one would obtain using free Dirac spinors wave functions for both nucleons in Eq. (3). The electron beam energy has been fixed to ␧beam =2.445 GeV which corresponds to an electron scattering angle ␪ e=23.4° (forward scattering). First note the difference between the RPWIA calculations (dot-dashed lines)and the RDWIA results (solid lines). For low missing momentum values pⱗ200 MeV/c, the effects of FSI do not modify substantially the behavior of the polarization asymmetries, particularly for Pl ⬘. However, in the case of Ps ⬘, the difference is of the order of 20–25% for p ⯝100 MeV/cwhich corresponds to the momentum where the responses reach their maxima for the p1/2 shell. Similar comments also apply to the results obtained for the p3/2 and s1/2 shells, although in these cases a smaller effect of FSI is observed for Ps ⬘. It is important to point out that FSI lead to a significant reduction of the individual response functions: ~50–60%共Rl TL⬘兲and ~25% (Rs TL⬘and Rl T⬘ )at p ⯝100 MeV/c. The response Rs T⬘is very small and its contribution to the transferred polarization is hardly visible. ANALYSIS OF POLARIZED 16O共e ជ ,e⬘p ជ 兲OBSERVABLES …PHYSICAL REVIEW C 69, 034604 (2004) 034604-5 Hence, the results in Fig. 1 clearly indicate that for low p values, FSI effects are partially canceled when constructing the transferred polarization asymmetries. Note also that, for these low-pvalues, the PWIA approach is more in accord with the RDWIA. This means that in RPWIA the role of dynamical relativity stands out more clearly. For high missing momentum, pⲏ200 MeV/c, FSI strongly modify the behavior of the polarizations, which is in accord with the peculiar sensitivity to the interaction presented by each response function. When comparing RDWIA with RPWIA we see that the main effect is a global displacement to lower momenta of the polarization profiles. Let us recall that the oscillatory behavior shown by Pl ⬘and Ps ⬘ within RPWIA is a direct consequence of the dynamical enhancement of the lower components in the bound Dirac wave functions [18]; thus disappearing within PWIA. The oscillations are also present in the relativistic distorted wave calculations, although being very different from the RPWIA results with the maxima and minima located at different p values. Let us note that the oscillatory behavior of the polarization asymmetries persists even when nonrelativistic distorted wave approaches are assumed (see Refs. [17,27,38]). This outcome emerges due to the fact that both FSI and dynamical relativistic effects cause a breakdown of factorization. A study of the latter is presently in progress and the results will be presented in a forthcoming publication [39]. Let us next focus on the analysis of the uncertainties introduced by different relativistic optical potentials. In Fig. 2 we present the transferred ratios Pl ⬘and Ps ⬘for the p1/2 shell evaluated using three different relativistic optical potential parametrizations: EDAIO, EDAD1, and EDAD2 [33]. Results with EDAD3 parameterization are practically identical to those obtained with EDAD1 and therefore have not been plotted. The left panels refer to calculations involving the CC1 current operator and right panels to CC2. As pointed out in previous papers [17,21,27,30], transferred polarization FIG. 1. Transferred polarization asymmetries for the p1/2 (left panels)and p3/2 (right panels) shells in 共q, ␻ 兲-constant kinematics (see text). Top and bottom panels correspond to the longitudinal and sideways components, respectively. RPWIA results (dot-dashed lines)are compared with RDWIA calculations using EDAIO (solid lines), and with the PWIA (dotted line)(see text for details). All calculations correspond to the CC2 current operator. FIG. 2. Transferred polarization asymmetries for the p1/2 shell in 共q, ␻ 兲-constant kinematics. Top and bottom panels correspond to the longitudinal and sideways components, respectively. Right panels refer to results obtained with the CC2 current operator and left ones to the CC1 current. RDWIA calculations using EDAIO (solid lines), EDAD1 (dot-dashed lines)and EDAD2 (dotted lines)optical potential parameterizations are compared. M. C. MARTÍNEZ et al. PHYSICAL REVIEW C 69, 034604 (2004) 034604-6 asymmetries are expected to be relatively insensitive to the choice of optical potential at low missing momenta. This can be seen in Fig. 2, at least up to p=150 MeV/cwhich is where the cross section reaches its maximum value [37]. This trend is also followed in the other two shells, p3/2 and s1/2. However, as shown in Fig. 2, Pl ⬘exhibits a strong dependence on the optical potential parametrization, resulting in important differences for larger values of the missing momentum: ⬃20% (CC1)and ⬃40% (CC2)for p ⯝250 MeV/c. Note that in this kinematical region the cross section [37]has already decreased by almost two orders of magnitude with regards to the maximum, making measurements of transferred polarization responses very difficult. This result contrasts with nonrelativistic and semirelativistic approaches where the effects introduced by different nonrelativistic optical potentials are small [17]. Note also that the current operator choice, CC1 versus CC2, gives rise to very significant differences in Pl ⬘within this pregion, being of the same order as those introduced by the optical potentials. Only for high pvalues, pⲏ350 MeV/c, is the uncertainty associated with FSI larger than that due to the choice of current operator. In the case of the sideways polarization Ps ⬘, in general less dependence on the interaction model as well as on the current is seen, which is more in accord with nonrelativistic analyses. Finally, note that for very high momentum values pⲏ400 MeV/c,Pl ⬘and Ps ⬘evaluated with the EDAIO potential deviate from the results corresponding to the EDAD1 and EDAD2 parameterizations. To end with this discussion, we conclude that both transferred polarization asymmetries at moderate pvalues 共p ⯝100 MeV/c兲are independent of the optical potential choice. Increasing pfrom here, each optical potential starts to follow a different curve especially in the case of Pl ⬘. For very high p共pⲏ350 MeV/c兲, both transferred polarizations present large sensitivity to the choice of optical potential. However, caution should be placed on drawing general conclusions from the results given here in this kinematical region because other ingredients beyond the impulse approximation, such as meson exchange currents (MEC),⌬-isobar, short-range correlations, etc., may also play a crucial role. B. Dynamical relativistic effects This section, which constitutes the main focus of the present work, is devoted to the analysis of dynamical relativistic effects for nucleon polarized observables within the framework of the RDWIA. With this aim we present in Fig. 3 the longitudinal and sideways transferred polarization asymmetries for the three shells involved in 16O: p1/2,p3/2, and s1/2. All of the results have been obtained using the EDAIO optical potential parametrization [33], and the choice of kinematics is the same as in the previous figures. To make explicit the effects introduced by spinor distortion, in each graph we compare the fully relativistic calculations (solid lines)using both current operators, CC1 (thin lines)and CC2 (thick lines), with the results after projecting out the negative-energy components [see Eqs. (9)and (10)] (dashed lines). Finally we also present for reference the results corresponding to the EMA-noSV approach evaluated with the CC2 current operator (dot-dashed line). Within EMA-noSV, FIG. 3. Same observables as in Fig. 1. Right panels correspond to Ps ⬘and left ones to Pl ⬘.On top, middle, and bottom panels, results for the 1p1/2,1p3/2, and 1s1/2 shells are plotted, respectively. In each graph, RDWIA calculations evaluated with EDAIO (solid line)are compared with positive-energy projection results (dashed line) and EMA-noSV approach (dot-dashed line). Thick lines correspond to the CC2 current operator and thin lines to CC1. ANALYSIS OF POLARIZED 16O共e ជ ,e⬘p ជ 兲OBSERVABLES …PHYSICAL REVIEW C 69, 034604 (2004) 034604-7 the results provided by the two current operators are very similar, differing only due to the off-shell kinematical quantities involved in the operator [19,28]. A detailed analysis of the transferred polarizations within the relativistic plane wave approach was presented in Ref. [18]. In said reference, it is shown that the dynamical enhancement of the lower components in the bound nucleon wave function leads to strong oscillations in Pl,s ⬘for high missing momentum values, p艌300 MeV/c. This behavior disappears after projecting out the negative-energy components. From the results shown in Fig. 3, it is clear that, within the relativistic distorted wave approximation, the oscillatory behavior in the polarization asymmetries persists even after projecting the bound and scattered proton wave functions over positive-energy states. The same comment applies to the EMA-noSV approach. On the contrary, this last fact is not applicable to the behavior shown by the left-right asymmetry ATL [9,10], defined as the difference of unpolarized cross sections evaluated at ␾ =0° and ␾ =180° divided by their sum. These results are connected with the interplay between polarization degrees of freedom and dynamical relativistic effects. Whereas in RPWIA, projecting out the negativeenergy components of the bound nucleon wave function leads to factorization, hence destroying the oscillatory behavior in Pl,s ⬘, in RDWIA factorization breaks down even after projection over positive-energy components. From inspection of Fig. 3, and in accord with previous results for unpolarized observables [9,10,28]and polarized ones in RPWIA [18], we note that dynamical relativistic effects are maximized for the CC1 current operator. This applies to both polarization ratios and the three shells considered. Particularly noteworthy is the behavior displayed by Pl ⬘ even at intermediate pvalues in the case of the fully relativistic CC1 calculation. This result deviates significantly from the others, modifying even the global shape of the observable. This contrasts with the situation for Ps ⬘where, apart from the specific discrepancies introduced by relativity, the five calculations follow the same general oscillatory pattern. Hence it would be interesting to investigate further this intermediate pregion where new high quality data on Pl ⬘could make it possible to constrain the theoretical choices for current operator. As shown in Refs. [9,28], the contribution from the negative-energy components to the current are of the same order as the positive-energy ones with the CC2 operator, whereas with the CC1 choice the negative-energy terms may become much larger. This explains the much wider spread shown by the CC1 results, particularly the large effects introduced by the dynamical enhancement of the lower components in Pl ⬘. As we will show later, this emerges from the polarized responses that enter in the longitudinal polarization in contrast with the sideways case. Note also that the CC1 projected calculations get closer to the CC2 ones and to the EMA-noSV approach. This may indicate that the CC1 current emphasizes the role played by the lower components in the wave functions, agreeing with the findings for unpolarized responses [10]. Precise comparisons with data would yield definite conclusions on the reliability of the various approximations. Finally, it is also interesting to compare the effects arising from dynamical relativity with those due to FSI models. As shown in Figs. 1–3, Pl ⬘presents the strongest sensitivity to both kinds of effects for intermediate pvalues, 200艋p 艋350 MeV/c. This can make it difficult to isolate the role played by each ingredient when compared with data; however, note that the important deviation between the results obtained with the two currents tends to persist, no matter which optical potential is used. Hence, precise measurements of Pl ⬘in this pregion, in conjunction with Ps ⬘data, may give us important clues to constrain final-state interactions and the choice of current operator. To complete the analysis of dynamical relativistic effects, we focus on the four separate responses that contribute when the polarization of the outgoing nucleon is measured and the electron beam is polarized: Rl T⬘,Rl TL⬘,Rs T⬘, and Rs TL⬘ (Rn TL⬘ does not enter for coplanar kinematics). Results are shown in Fig. 4 for proton knockout in 16O from the p1/2 shell. Let us recall that Coulomb distortion of the electron waves breaks the simplicity of Eq. (2), leading to responses which also depend on the electron kinematic variables. However, the effective momentum approximation for the electrons adopted in this work makes Eq. (2)reliable when analyzing the response functions. For 16O we have proven [10]that Coulomb distortion effects, and consequently the dependence of the responses with ␪ e, are very small. As a general rule we observe that Rs T⬘and Rl TL⬘show the highest sensitivity to relativistic dynamics, while the uncertainties in Rl T⬘and Rs TL⬘are much smaller. This coincides with the analysis already performed in RPWIA [18]and, although not shown here for simplicity, applies also to the p3/2 and s1/2 shells. In addition, Gordon ambiguities are also significantly enhanced for Rs T⬘and Rl TL⬘. Finally, note that the largest spread due to relativistic dynamical effects arises for the CC1 current operator, which is in accord with RPWIA results [18], and can be traced back to the strong influence of the negative-energy projections of the wave functions in this case. Let us study in more detail each individual response. As shown in Fig. 4, the contributions of Rl T⬘and Rs TL⬘are rather similar, and moreover, the EMA-noSV predictions almost coincide (evaluated at the maxima)with the fully relativistic calculations, the largest difference being of the order of 3.6% for the CC1 current in Rs TL⬘. Positive-energy projected results also follow the RDWIA curves closely, although sizeable differences are observed for the CC1 current, particularly in the case of Rl T⬘ (~11% at the maximum). Concerning Rl TL⬘, we observe that the projected calculations differ substantially from the RDWIA results, especially for the CC1 current operator. This resembles the large relativistic dynamical effects shown by this response in RPWIA [18]. On the contrary, it is interesting to note that different choices of the current operator within RDWIA lead to very similar results, which is somewhat opposed to the situation observed in the plane wave limit [18]. Finally, the EMAnoSV approach provides a description of Rl TL⬘that basically coincides with the two RDWIA calculations, the largest difference being observed at very low-pvalues. In fact, this result is proven to be valid only at q=1 GeV/c, where the effective momentum approach (EMA)applied to the bound M. C. MARTÍNEZ et al. PHYSICAL REVIEW C 69, 034604 (2004) 034604-8 wave function, leads to effects which cancel almost exactly those coming from the ejected nucleon. For lower values of q this cancellation does not occur, and so an important discrepancy between the EMA-noSV prediction and the RDWIA calculations emerges. The smallest Rs T⬘response presents a large dependence on the current operator choice. This applies to the full RDWIA calculation as well as to the positive-energy projected approach. Note, however, that the difference between RDWIA and projected results is tiny, almost negligible for the CC2 current. Contrary to Rl TL⬘case, the EMA-noSV approach for Rs T⬘deviates significantly from the fully relativistic and projected results, the uncertainty spread (significantly enhanced for the CC1 current)being even larger than that obtained in RPWIA [18]. We should also recall that Rs T⬘is strongly affected by the choice of the optical potential (results corresponding to the parameterizations EDAIO and EDAD2 are very different from those for EDAD1 and EDAD3). Although not shown in the figure, it is also important to point out that at low q共q艋350 MeV/c兲, the projection over positive-energies in the bound nucleon wave function clearly dominates, while at higher q, the reverse occurs. This result contrasts with the behavior seen for the unpolarized observables and also with the other three polarized responses, where for high enough transfer momentum projecting out the negative-energy components in the ejected nucleon wave function is proven not to alter the fully relativistic predictions. The behaviors presented by the four polarized responses, their relative contributions and their sensitivity to dynamical relativistic effects give us important clues to understand the results obtained for the longitudinal and sideways transferred polarization asymmetries. The large effects introduced by relativity in Pl ⬘, particularly when comparing full relativistic and projected calculations for CC1, can be traced back to the similar contributions given by the two responses Rl T⬘and Rl TL⬘that enter in Pl ⬘. Although relativistic dynamics affect Rl TL⬘more, their effect on Rl T⬘is also sizeable. The case of Ps ⬘ is clearly different. Here the two polarized responses involved contribute very differently, Rs T⬘being much smaller (more than one order of magnitude). Therefore, the asymmetry Ps ⬘is almost given uniquely by Rs TL⬘, whose uncertainty due to dynamical relativistic effects presents the lowest spread. Although results for p3/2 and s1/2 show basically similar behavior to those of the p1/2 shell, off-shell and dynamical relativity play a less significant role for the p3/2 shell in Rs T⬘ and Rl TL⬘. As already mentioned, in RDWIA spinor distortion affects both the bound and ejected nucleon wave functions. Hence in what follows, we analyze the role of dynamical relativity, isolating the spinor distortion contribution in each nucleon wave function separately. We show results for the ratios Pl,s ⬘ and the left-right asymmetry ATL, focusing on the CC2 current, which minimizes dynamical effects, and the p1/2 shell. Results for p3/2 and s1/2 follow the same general trends, but with a significant reduction of the effects due to relativistic dynamics. In Fig. 5 we show the observables for three values of the momentum transfer q. In each case, quasiperpendicular kinematics (q, ␻ constant)have been selected, and RDWIA and projected calculations are compared. Within the projected results, we distinguish the EMA-noSV approach, where negative-energy components of the bound and scattered nucleon wave functions have been projected out, from the results where the projection over positive-energy components affects only one of the nucleon wave functions: bound (referred to as EMAb)and ejected (EMAf). From inspection of Fig. 5, a clear difference emerges in the behavior observed for ATL and the polarized ratios Pl,s ⬘. FIG. 4. Transferred polarized responses for the 1p1/2 shell. Same kinematics as in preceding figures, and the labeling as in Fig. 2. ANALYSIS OF POLARIZED 16O共e ជ ,e⬘p ជ 兲OBSERVABLES …PHYSICAL REVIEW C 69, 034604 (2004) 034604-9