Meson-exchange currents in (e,e'p) recoil polarization observables

Meson-exchange currents in „e,e⬘p…recoil polarization observables F. Kazemi Tabatabaei,1J. E. Amaro,1and J. A. Caballero2 1Departamento de Fı ´sica Moderna, Universidad de Granada, Granada 18071, Spain 2Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Apartado Postal 1065, Sevilla 41080, Spain 共Received 23 May 2003; published 22 September 2003兲 A study of the effects of meson-exchange currents and isobar configurations in A(e ជ ,e⬘p ជ )Breactions is presented. We use a distorted wave impulse approximation model where final-state interactions are treated through a phenomenological optical potential. The model includes relativistic corrections in the kinematics and in the electromagnetic one- and two-body currents. The full set of polarized response functions is analyzed, as well as the transferred polarization asymmetry. Results are presented for proton knockout from closed-shell nuclei for moderate to high momentum transfer. DOI: 10.1103/PhysRevC.68.034611 PACS number共s兲: 25.30.Fj, 24.10.Eq, 24.70.⫹s, 24.10.Jv I. INTRODUCTION For the past decades coincidence (e,e⬘p) reactions on complex nuclei have provided precise information on bound nucleon properties, which have made it possible to test carefully the validity of present nuclear models 关1–4兴. Although the analysis of these processes, making use of different distorted wave approaches and coupled-channel models, has been extremely useful, there are still uncertainties associated to the various ingredients that enter in the description of the reaction mechanism: treatment of final-state interactions 共FSI兲, nuclear correlations, off-shell effects, Coulomb distortion of the electrons, relativistic degrees of freedom, mesonexchange currents 共MEC兲, etc. All of these ingredients affect the evaluation of the differential cross section and hence lead to ambiguities in the extraction of the spectroscopic factors. The origin of this uncertainty is directly connected with the complexity of the dynamics of the reaction and the different approaches to handle it, which produce different cross sections. It is clear that a reliable determination of spectroscopic factors requires an accurate description of the reaction mechanism. Important efforts in this direction have been made in recent works 关5–9兴. The measurement of the separate nuclear response functions and asymmetries imposes additional restrictions over the theory. The exclusive response functions, which include different components of the hadronic tensor taken along the longitudinal 共L兲or transverse 共T兲directions with respect to the momentum transfer q, may present very different sensitivities to the different aspects of the reaction. In this sense, it is interesting to point out that MEC are shown to contribute mainly to the transverse components 关10–12兴, while relativistic degrees of freedom play a crucial role in the interference TL response 关8,13兴. Thus, a joint analysis of cross sections and response functions, comparing the experimental data with the theoretical predictions, can provide very relevant and complementary information on the reaction mechanism. Separate response functions and the TL asymmetry have been measured for 16O(e,e⬘p) at moderate 关14,15兴and high 关16兴qvalues. The asymmetry ATL , obtained from the difference of cross sections measured at opposite azimuthal angles 共with respect to q) divided by the sum, is particularly relevant because it does not depend on the spectroscopic factors. For high missing momentum p⭓300 MeV/c,ATL presents an oscillatory structure that has been shown to be consistent with predictions of ‘‘dynamical’’ relativistic calculations 关13,17–19兴. The advent of longitudinally polarized beams 关20兴and recoil polarization measurements 关21,22兴has importantly enlarged the number of observables which can be accessible with this type of experiment, a fact that is welcome to challenge the theory strongly. In recent experiments carried out at MIT-Bates and Jefferson Lab, the induced (P) and transferred (P⬘) polarization asymmetries were measured for complex nuclei 12C关21兴and 16O关22兴, respectively. In both cases (q, ␻ )-constant kinematics has been selected with q⬇760 MeV/c, ␻ ⬇290 MeV at MIT-Bates and q ⬇1000 MeV/c, ␻ ⬇450 MeV at TJlab. Since the transfer momentum values are high enough, relativistic degrees of freedom should be incorporated in a consistent description of these reactions. After the pioneering work in Refs. 关23,24兴,a detailed study on the induced normal polarization Pnhas been presented in Refs. 关25,26兴within the framework of the relativistic distorted wave impulse approximation 共RDWIA兲. A comparison with nonrelativistic analyses was also discussed. The sensitivity of polarized observables to channel coupling in final-state interactions was analyzed in Refs. 关3,27兴, while in Ref. 关17兴the study was focused on the effects of spinor distortion over the transfer polarization ratio Px ⬘/Pz ⬘. In Ref. 关28兴the whole 18 recoil nucleon polarized responses were computed from intermediate to high momentum transfer in the Dirac eikonal formalism. A comparison between the predictions of the Glauber and eikonal models for Pnwas presented in Ref. 关29兴with the aim of bridge the gap between the low- and high-energy description of FSI. More recently a theoretical study of kinematical and dynamical relativistic effects over polarized response functions and polarization asymmetries has been performed in Refs. 关30,31兴within the relativistic plane-wave impulse approximation 共RPWIA兲. A general analysis of all the polarized observables within the RDWIA is at present in progress 关32兴. Our main aim in this work is to explore in depth the role played by the two-body currents in recoil nucleon polarization observables. Some previous analyses on this subject PHYSICAL REVIEW C 68, 034611 共2003兲 0556-2813/2003/68共3兲/034611共21兲/$20.00 ©2003 The American Physical Society68 034611-1 have been done by the Pavia group 关33,34兴and the Gent group 关35,36兴. The calculation of MEC in Refs. 关33,34兴 makes use of an effective one-body operator leading to results which, in the unpolarized case, differ significantly from those obtained with other approaches that describe properly the two-body currents 关37,38兴. Recently the unpolarized model of Ref. 关34兴has been improved in Ref. 关39兴, but differences with other MEC calculations still persist 关40兴.In Ref. 关36兴the induced and transferred polarization asymmetries Pnand Pl ⬘and Pt ⬘were evaluated for different kinematical situations. The model considered did not rely on any empirical input with respect to the FSI, describing the bound and scattering states as the solutions of the Schro ¨dinger equation with a mean field potential obtained from a Hartree- Fock calculation. MEC were included based on the formalism developed in Ref. 关37兴which also differs from the MEC analysis performed in Refs. 关38,40兴. In addition, in Ref. 关36兴 results for high momentum transfer 共up to q⫽1 GeV/c) were evaluated including relativistic corrections into the onebody current operator obtained through the Foldy- Wouthuysen method. In this work we extend the DWIA⫹MEC model developed for unpolarized reactions in Refs. 关38,40兴in order to describe the spin observables in (e ជ ,e⬘p ជ ) processes from closed-shell nuclei. This model takes care of relativistic degrees of freedom by making use of semirelativistic 共SR兲operators for the one-body 共OB兲current 关13,41–43兴as well as for the two-body MEC 关40,44–46兴. The SR currents are obtained by a direct Pauli reduction of the corresponding relativistic operators by expanding only in missing momentum over the nucleon mass while treating the transferred energy and momentum exactly. Relativistic kinematics for the ejected nucleon is assumed throughout this work. Finally FSI are incorporated through a phenomenological optical potential which, for high momentum transfer, is taken as the Schro ¨dinger-equivalent form of a S-V Dirac optical potential. The goal of this work is to use the SR approach to evaluate the importance of MEC effects upon the spin observables and their dependence on the FSI for intermediate to high momentum transfer. As a complete relativistic distorted wave analysis of MEC in (e,e⬘p) processes is still lacking— the only study in this direction has been performed taking into account only the contact current 关47兴—the use of the SR model becomes, as a starting point, a convenient way of implementing relativistic effects in existing nonrelativistic descriptions of the reaction mechanism in order to explore the high momentum region. The paper is organized as follows: in Sec. II we outline the DWIA formalism describing in detail the multipole expansion done for the separate response functions. In Sec. III we present our results for the polarized response functions and transferred polarization asymmetries for selected kinematics near the quasielastic peak. Finally our conclusions are drawn in Sec. IV. II. DWIA MODEL OF „e ᠬ ,e⬘p ᠬ … A. Cross section and response functions The general formalism for coincidence electron scattering on nuclei involving polarization degrees of freedom has been presented in detail in Refs. 关4,23,24,48兴. In this section we simply provide the basic description of our DWIA model focusing on the development of the multipole expansion used to compute the response functions. For this end we follow closely the multipole formalism developed in Ref. 关49兴for polarized nuclei. We consider the process depicted in Fig. 1, in which an incident electron with four-momentum Ke ␮ ⫽( ⑀ e,ke) and helicity hinteracts with a nucleus A, scatters through an angle ␪ eto four-momentum K⬘e ␮ ⫽( ⑀ ⬘,ke ⬘), and is detected in coincidence with a nucleon with momentum p⬘and energy E⬘. The four-momentum transferred to the nucleus is Q ␮ ⫽Ke ␮ ⫺K⬘e ␮ ⫽( ␻ ,q), verifying Q2⫽ ␻ 2⫺q2⬍0. The polarization of the final nucleon is measured along an arbitrary direction defined by the unitary vector s. Assuming plane waves for the electrons and neglecting the nuclear recoil, the cross section can be written in the extreme relativistic limit meⰆ ⑀ eas 关48兴 d ␴ d ⑀ e ⬘d⍀e ⬘dp ˆ⬘ ⫽⌺⫹h⌬,共1兲 where a separation has been made into terms involving polarized and unpolarized incident electrons. Using the general properties of the leptonic tensor it can be shown that both terms, ⌺and ⌬, have the following decompositions: ⌺⫽K ␴ M共vLRL⫹vTRT⫹vTLRTL⫹vTTRTT兲,共2兲 ⌬⫽K ␴ M共vTL⬘RTL⬘ ⫹vT⬘RT⬘兲,共3兲 ␴ Mis the Mott cross section, the factor K⬅mNp⬘/(2 ␲ ប)3, FIG. 1. Kinematics for the (e,e⬘p) reaction. The (x,y,z) coordinate system is referred to the scattering plane with the zaxis lying along the direction of the momentum transfer q. The barycentric system (l,t,n) is referred to the reaction plane: llies along the direction of the ejected nucleon p⬘, the direction nis defined by q⫻p⬘, and t⫽n⫻l. TABATABAEI, AMARO, AND CABALLERO PHYSICAL REVIEW C 68, 034611 共2003兲 034611-2 with mNthe nucleon mass, and the v ␣ coefficients are the usual electron kinematical factors.1 The hadronic dynamics of the process is contained in the exclusive response functions RK, which are given as RL⫽W00,RT⫽Wxx⫹Wyy,共4兲 RTL⫽ 冑 2共W0x⫹Wx0兲,RTT⫽Wyy⫺Wxx,共5兲 RT⬘ ⫽i共Wxy⫺Wyx兲,RTL⬘ ⫽i 冑 2共W0y⫹Wy0兲,共6兲 with W ␮ ␯ the hadronic tensor W ␮ ␯ ⫽1 K兺 MB 具 p⬘s,B 兩 J ˆ ␮ 共q兲 兩 A 典 * 具 p⬘s,B 兩 J ˆ ␯ 共q兲 兩 A 典 共7兲 constructed from the matrix elements of the electromagnetic nuclear current operator J ˆ ␮ (q) between the ground state of the target nucleus 兩 A 典 共assumed to have zero total angular momentum兲and the final hadronic states 兩 p⬘s,B 典 . In what follows we assume the residual nucleus to be left in a bound state, hence its wave function can be written down in the form 兩 B 典 ⫽ 兩 JBMB 典 with JBas the total angular momentum. The state 兩 p⬘s 典 represents the asymptotic distorted wave function of the ejected nucleon polarized along an arbitrary s direction, determined by the angles ( ␪ s, ␾ s) referred to the xyz coordinate system of Fig. 1. It is given by 兩 p⬘s 典 ⫽兺 ␯ ⫽⫺1/2 1/2 D ␯ 1/2 (1/2)共 ␪ s, ␾ s,0兲 兩 p⬘ ␯ 典 ,共8兲 where 兩 p⬘ ␯ 典 is referred to the system with the quantization axis along qand the arguments of the rotation matrices are the Euler angles that specify the sdirection. Isolating the explicit dependences on the azimuthal angle of the ejected nucleon ␾ ⬘⫽ ␾ , the hadronic responses can be expressed in the form RL⫽WL,共9兲 RT⫽WT,共10兲 RTL⫽cos ␾ WTL⫹sin ␾ W ˜ TL,共11兲 RTT⫽cos2 ␾ WTT⫹sin2 ␾ W ˜ TT,共12兲 RT⬘ ⫽W ˜ T⬘,共13兲 RTL⬘ ⫽cos ␾ W ˜ TL⬘ ⫹sin ␾ WTL⬘,共14兲 where the functions WKand W ˜ Kare totally specified by four kinematical variables, for instance 兵 E, ␻ ,q, ␪ ⬘ 其 , and the polarization direction 兵 ␪ s,⌬ ␾ ⫽ ␾ ⫺ ␾ s 其 . The responses with and without tilde refer to their dependence on the spin vector s. As shown below, W ˜ Kare purely spin vector, while WK present also a spin-scalar dependence, so only the latter survive when the polarization of the ejected nucleon is not measured. In the case of (e ជ ,e⬘N ជ ) processes, the hadronic response functions are usually given by referring the recoil nucleon polarization vector sto the barycentric system defined by the axes 共see Fig. 1兲:l共along the p⬘direction兲,n共normal direction to the plane defined by qand p⬘, i.e., along q⫻p⬘), and t共determined by n⫻l).2It can be shown 共see Refs. 关23,24兴 for details兲that a total of 18 response functions enter in the analysis of (e ជ ,e⬘N ជ ) reactions. These are given by the decomposition WK⫽1 2Wunpol K⫹Wn Ksn,K⫽L,T,TL,TT,TL⬘,共15兲 W ˜ K⫽Wl Ksl⫹Wt Kst,K⫽TL,TT,T⬘,TL⬘,共16兲 where, as mentioned above, only the Wunpol Kresponses survive within the unpolarized case. Moreover, Wunpol TL⬘共referred to as fifth response兲enters only when the polarization of the incident electron is measured. Owing to the above decomposition, the response functions 共9兲–共14兲can be expressed in the form RK⫽Runpol K/2 ⫹RK•s, and similarly the cross section 共1兲–共3兲can be written as a sum of unpolarized and spin-vector dependent terms d ␴ d ⑀ e ⬘d⍀e ⬘dp ˆ⬘ ⫽1 2⌺unpol⫹⌺•s⫹h 冉 1 2⌬unpol⫹⌬•s 冊 共17兲 ⫽1 2⌺unpol关1⫹P•s⫹h共A⫹P⬘•s兲兴, 共18兲 where the usual polarization asymmetries have been introduced 关3兴: P⫽⌺/共1 2⌺unpol兲共induced polarization兲,共19兲 P⬘⫽⌬/共1 2⌺unpol兲共transferred polarization兲,共20兲 A⫽⌬unpol /⌺unpol 共electron analyzing power兲.共21兲 1In this work we consider the kinematical factors similar to those expressions presented in Ref. 关48兴. Note that these factors differ from the ones of Ref. 关24兴in a global sign for vTL ,vTL⬘, and vTT , and an additional 1/ 冑 2 factor in the case of the interference TL coefficients. 2Note that this notation does not coincide with Refs. 关30,31兴 where the tdirection is denoted as s共sideways兲. MESON EXCHANGE CURRENTS IN (e,e⬘p) RECOIL... PHYSICAL REVIEW C 68, 034611 共2003兲 034611-3 B. Multipole analysis of response functions In this section we present the multipole expansion of the response functions to be used in our DWIA model. The final expressions, where the sums over third components of angular momenta have been performed analytically, are convenient in the present work since the computational time can be considerably reduced, especially the calculation concerning the MEC. Note that the number of multipoles needed to get convergence increases with q, ␻ , and up to ⬃36 multipoles are needed for q⫽1 GeV/c. The expansion is performed following the formalism developed in Ref. 关49兴for exclusive reactions from polarized nuclei. A basic difference between the present work and that of Ref. 关49兴lies on the sums performed over the third components which are different when initial- and/or final-state polarizations are considered. Here we simply present the final expressions, referring to Ref. 关49兴 for details on the expansion method and to the Appendix for an outline on the procedure used to perform the sum over third components in the present case. In order to compute the hadronic tensor in our DWIA model we first perform a multipole expansion of the ejected nucleon wave function in partial waves. The final hadronic states may then be written as 兩 p⬘ ␯ ,B 典 ⫽兺 lMjmpJfMf ilYlM *共p ˆ⬘兲 具 1 2 ␯ lM 兩 jmp 典 ⫻ 具 jmpJBMB 兩 JfMf 典 兩 共lj兲JB,JfMf 典 ,共22兲 where the partial waves (lj) are coupled to the angular momentum JBof the residual nucleus to give a total angular momentum Jfin the final states 兩 f 典 ⫽ 兩 (lj)JB,JfMf 典 . The electromagnetic charge and transverse current operators are expanded as sums involving Coulomb 共C兲, electric 共E兲, and magnetic 共M兲tensor operators, ␳ ˆ共q兲⫽ 冑 4 ␲ 兺 J⫽0 ⬁ iJ关J兴M ˆJ0共q兲,共23兲 J ˆm⫽⫺ 冑 2 ␲ 兺 J⫽1 ⬁ iJ关J兴关T ˆJm el 共q兲⫹mT ˆJm mag共q兲兴,m⫽⫾1, 共24兲 where, as usual, we assume the transfer momentum qalong the zdirection and J ˆmare the spherical components of the current operator J ˆ. We use the bracket symbol 关J兴 ⫽ 冑 2J⫹1 for angular momenta. Inserting Eqs. 共8兲and 共22兲– 共24兲into the hadronic tensor 共7兲, the following expansion for the responses WKand W ˜ Kis obtained: WK⫽1 2Wunpol K⫹2 ␲ P1 1共cos ␪ s兲sin共⌬ ␾ 兲W ˜ 11 K共25兲 for K⫽L,T,TL,TT,TL⬘, and W ˜ K⫽2 ␲ ␣ K关P1 0共cos ␪ s兲W10 K⫹P1 1共cos ␪ s兲cos共⌬ ␾ 兲W11 K兴 共26兲 for K⫽TL,TT,T⬘,TL⬘, with the coefficient ␣ K⫽⫺1 for TL,TT and ␣ K⫽1 for T⬘,TL⬘, and PJ M(cos ␪ s) the Legendre functions. The five response functions Wunpol Kare the only ones that survive when summing over final spins ⫾s, in which case the 1/2 factor cancels and the unpolarized cross section is recovered. The spin dependence is determined from the 13 reduced response functions W1M K(M⫽0,1) and W ˜ 11 Kintroduced above. Explicit expressions for these reduced responses can be written in terms of the reduced matrix elements of the current multipole operators C ␴ ⫽ 具 共lj兲JB,J 储 M ˆJ 储 0 典 ,共27兲 E ␴ ⫽ 具 共lj兲JB,J 储 T ˆJ el 储 0 典 ,共28兲 M ␴ ⫽ 具 共lj兲JB,J 储 iT ˆJ mag 储 0 典 ,共29兲 where we have defined a multiple index ␴ ⫽(l,j,J) corresponding to the quantum numbers of the final states. Note that the initial state 兩 A 典 ⫽ 兩 0 典 has total angular momentum equal to zero, so Jf⫽J. The response functions involve quadratic products of these multipole matrix elements which can be decomposed into their real (R ␴ ⬘ ␴ K) and imaginary (I ␴ ⬘ ␴ K) parts: C ␴ ⬘ *C ␴ ⫽R ␴ ⬘ ␴ L⫹iI ␴ ⬘ ␴ L,共30兲 E ␴ ⬘ *E ␴ ⫹M ␴ ⬘ *M ␴ ⫽R ␴ ⬘ ␴ T1⫹iI ␴ ⬘ ␴ T1,共31兲 E ␴ ⬘ *M ␴ ⫺M ␴ ⬘ *E ␴ ⫽R ␴ ⬘ ␴ T2⫹iI ␴ ⬘ ␴ T2,共32兲 C ␴ ⬘ *E ␴ ⫽R ␴ ⬘ ␴ TL1⫹iI ␴ ⬘ ␴ TL1,共33兲 C ␴ ⬘ *M ␴ ⫽R ␴ ⬘ ␴ TL2⫹iI ␴ ⬘ ␴ TL2,共34兲 E ␴ ⬘ *E ␴ ⫺M ␴ ⬘ *M ␴ ⫽R ␴ ⬘ ␴ TT1⫹iI ␴ ⬘ ␴ TT1,共35兲 E ␴ ⬘ *M ␴ ⫹M ␴ ⬘ *E ␴ ⫽R ␴ ⬘ ␴ TT2⫹iI ␴ ⬘ ␴ TT2.共36兲 Expressions for the unpolarized response functions Wunpol Kin terms of Eqs. 共30兲–共36兲are given in Ref. 关41兴, while the recoil nucleon polarized responses can be written as W ˜ 11 L⫽1 K兺 J⬘Lh ˜ 1J⬘L0 1共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 000 冊 ␰ J⬘⫺l⬘,J⫺l ⫹I ␴ ⬘ ␴ L,共37兲 TABATABAEI, AMARO, AND CABALLERO PHYSICAL REVIEW C 68, 034611 共2003兲 034611-4 W ˜ 11 T⫽⫺ 1 K兺 J⬘LPJ⬘⫹L ⫹h ˜ 1J⬘L0 1共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 1⫺10 冊 共 ␰ J⬘⫺l⬘,J⫺l ⫹I ␴ ⬘ ␴ T1⫹ ␰ J⬘⫺l⬘,J⫺l ⫺I ␴ ⬘ ␴ T2兲, 共38兲 W ˜ 11 TL⫽⫺ 1 K2 冑 2兺 J⬘L 共⫺1兲J⬘⫹Lh ˜ 1J⬘L1 1共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 01⫺1 冊 共 ␰ J⬘⫺l⬘,J⫺l ⫹I ␴ ⬘ ␴ TL1⫺ ␰ J⬘⫺l⬘,J⫺l ⫺I ␴ ⬘ ␴ TL2兲, 共39兲 W ˜ 11 TT⫽⫺ 1 K兺 J⬘L 共⫺1兲J⬘⫹Lh ˜ 1J⬘L2 1共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 11⫺2 冊 共 ␰ J⬘⫺l⬘,J⫺l ⫹I ␴ ⬘ ␴ TT1⫺ ␰ J⬘⫺l⬘,J⫺l ⫺I ␴ ⬘ ␴ TT2兲, 共40兲 W ˜ 11 TL⬘ ⫽⫺ 1 K2 冑 2兺 J⬘L 共⫺1兲J⬘⫹Lh ˜ 1J⬘L1 1共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 01⫺1 冊 共 ␰ J⬘⫺l⬘,J⫺l ⫹R ␴ ⬘ ␴ TL1⫺ ␰ J⬘⫺l⬘,J⫺l ⫺R ␴ ⬘ ␴ TL2兲, 共41兲 W1M TL ⫽⫺ 1 K2 冑 2兺 J⬘L 共⫺1兲J⬘⫹Lh1J⬘L1 M共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 01⫺1 冊 共 ␰ J⬘⫺l⬘,J⫺l ⫹I ␴ ⬘ ␴ TL1⫺ ␰ J⬘⫺l⬘,J⫺l ⫺I ␴ ⬘ ␴ TL2兲, 共42兲 W1M TT ⫽⫺ 1 K兺 J⬘L 共⫺1兲J⬘⫹Lh1J⬘L2 M共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 11⫺2 冊 共 ␰ J⬘⫺l⬘,J⫺l ⫹I ␴ ⬘ ␴ TT1⫺ ␰ J⬘⫺l⬘,J⫺l ⫺I ␴ ⬘ ␴ TT2兲, 共43兲 W1M T⬘ ⫽⫺ 1 K兺 J⬘LPJ⬘⫹L ⫺h1J⬘L0 M共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 1⫺10 冊 共 ␰ J⬘⫺l⬘,J⫺l ⫹R ␴ ⬘ ␴ T1⫹ ␰ J⬘⫺l⬘,J⫺l ⫺R ␴ ⬘ ␴ T2兲, 共44兲 W1M TL⬘ ⫽⫺ 1 K2 冑 2兺 J⬘L 共⫺1兲J⬘⫹Lh1J⬘L1 M共 ␪ ⬘兲兺 ␴ ⬘ ␴ Pl⫹l⬘⫹J⬘ ⫹⌽ ␴ ⬘ ␴ ⫻ 冉 J⬘JL 01⫺1 冊 共 ␰ J⬘⫺l⬘,J⫺l ⫹R ␴ ⬘ ␴ TL1⫺ ␰ J⬘⫺l⬘,J⫺l ⫺R ␴ ⬘ ␴ TL2兲, 共45兲 where we use the parity functions Pn ⫾⫽关1⫾共⫺1兲n兴/2, ␰ J⬘J ⫹⬅共⫺1兲(J⬘⫺J)/2PJ⬘⫹J ⫹, ␰ J⬘J ⫺⬅共⫺1兲(J⬘⫺J⫹1)/2PJ⬘⫹J ⫺,共46兲 and the angular dependence of the above responses is determined by the functions hJJ⬘LM M( ␪ ⬘) and h ˜ JJ⬘LM M( ␪ ⬘), defined through the coupling of two spherical harmonics 关see Eq. 共A21兲兴. Re关YJ共s兲YJ⬘共p ˆ⬘兲兴LM ⫽cos M ␾ ⬘兺 M⫽0 J hJJ⬘LM M共 ␪ ⬘兲PJ M共cos ␪ s兲cos共M⌬ ␾ 兲 ⫹sinM ␾ ⬘兺 M⫽0 J h ˜ JJ⬘LM M共 ␪ ⬘兲PJ M共cos ␪ s兲sin共M⌬ ␾ 兲. 共47兲 Finally, the coefficients ⌽ ␴ ⬘ ␴ are derived in the Appendix and are given by Eq. 共A19兲selecting J⫽1. Although the above expressions correspond formally to those denoted as WJM K(⫺)and W ˜ JM K(⫺)in Ref. 关49兴for polarized nuclei and J ⫽1, it is important to point out that the coefficients ⌽ ␴ ⬘ ␴ contain the whole information on the polarization distribution of the particles. Hence the significance of ⌽ ␴ ⬘ ␴ is clearly different when polarization degrees of freedom are considered for the ejected nucleon 共present work兲or the target nucleus 关49兴. The nuclear structure information in Eqs. 共37兲–共45兲is contained in the quadratic forms 共30兲–共36兲of the C,E, and Mmultipoles.3Thus the present expansion can be applied to any nuclear model of the reaction as far as it provides multipole matrix elements 共27兲–共29兲for high enough angular momenta ␴ ⫽(l,j,J). Note that only the responses involving the real parts R ␴ ⬘ ␴ Ksurvive when FSI are neglected since in this case all the C,E, and Mmultipoles are strictly real functions. Therefore those responses which depend on the imaginary parts are expected to be particularly sensitive to the description of FSI. Writing down explicitly the Legendre polynomials involved in the multipole expansion 共25兲and 共26兲, and com- 3Note that there is a typo in Eqs. 共40兲–共51兲of Ref. 关49兴: the order of Jand J⬘in the 3-jshould be reversed. This error has been corrected in Eqs. 共37兲–共45兲in the present paper. MESON EXCHANGE CURRENTS IN (e,e⬘p) RECOIL... PHYSICAL REVIEW C 68, 034611 共2003兲 034611-5 paring with the general expression 共15兲and 共16兲, we get the following relation between both sets of response functions: Wn K⫽⫺2 ␲ W ˜ 11 K,K⫽L,T,TL,TT,TL⬘,共48兲 Wl K⫽2 ␲ ␣ K共W11 Ksin ␪ ⬘⫹W10 Kcos ␪ ⬘兲,K⫽TL,TT,TL⬘,T⬘, 共49兲 Wt K⫽2 ␲ ␣ K共W11 Kcos ␪ ⬘⫺W10 Ksin ␪ ⬘兲,K⫽TL,TT,TL⬘,T⬘, 共50兲 with ␣ Kas introduced in Eq. 共26兲. C. Electromagnetic operators and PWIA In this work we evaluate the exclusive polarized responses using a SR model for describing the electromagnetic OB and two-body MEC current operators. The OB current has been obtained by a direct Pauli reduction of the fully relativistic operator in powers only of the initial nucleon momentum over the nucleon mass p/mN. The dependence on the transfer and final momenta, which can be large 关42–45兴, is treated exactly. The SR-OB current in momentum space can be written as J0共p⬘,p兲⫽ ␳ c⫹i ␳ so共cos ␾ ␴ x⫺sin ␾ ␴ y兲 ␹ ,共51兲 Jx共p⬘,p兲⫽iJm ␴ y⫹Jc ␹ cos ␾ ,共52兲 Jy共p⬘,p兲⫽⫺iJm ␴ x⫹Jc ␹ sin ␾ ,共53兲 where ␹ ⫽(p/mN)sin ␪ and ( ␪ , ␾ ) are the angles determining the direction of the initial momentum pin the (x,y,z) coordinate system. The charge ( ␳ c), spin-orbit ( ␳ so), magnetization (Jm), and convection (Jc) terms shown above include relativistic corrections and are given by the following expressions ␳ c⫽ ␬ 冑 ␶ GE, ␳ so⫽ ␬ 2GM⫺GE 2 冑 1⫹ ␶ ,共54兲 Jm⫽ 冑 ␶ GM,Jc⫽ 冑 ␶ ␬ GE,共55兲 where ␬ ⫽q/2mN, ␶ ⫽ 兩 Q2 兩 /4mN 2, and GEand GMare the electric and magnetic nucleon form factors for which we take the Galster parametrization 关50兴. The two-body MEC operators of pionic 共P兲, seagull or contact 共S兲, and ⌬-isobar kinds, displayed in the Feynman diagrams of Fig. 2, have been also obtained by making use of a SR approach leading to simple prescriptions that include relativistic corrections through a multiplicative factor 共see Refs. 关40,45,46兴for details on the SR expansion method兲 JSR MEC⫽1 冑 1⫹ ␶ JNR MEC ,共56兲 where JNR MEC is the traditional nonrelativistic MEC operator. The expressions for the reduced matrix elements of the OB and MEC multipole operators 共27兲–共29兲in the shell model are given in Refs. 关10,11,43兴except for the relativistic correction factors appearing within the SR operators. Because of the somewhat complex structure displayed by these multipoles it is not possible to predict the relative importance of each contribution separately without explicit numerical evaluation, even in the case of the OB current. Although in this work we perform a DWIA analysis of the response functions, we may take advantage of the significant simplifications introduced within the PWIA, where analytical expressions for the response functions can be obtained 关30,31兴. First, for intermediate to high values of q, the PWIA approach is expected to provide reasonable results, thus the analytical PWIA expressions allow us to estimate the contributions of the different pieces of the currents to the polarized response functions. Second, since the PWIA results should be recovered using the present multipole expansion in the limit of no FSI, the comparison between our calculation and the analytical PWIA responses makes it possible to fix the number of multipoles needed to get convergence. Hence, within PWIA, the matrix element of the OB current is written as 具 p⬘s,B 兩 J ␮ 共q兲 兩 A 典 ⫽兺 ␤ ⬘ ␤ D ␤ ⬘1/2 *共s兲J ␮ 共p⬘,p兲 ␤ ⬘ ␤ 具 B 兩 ap, ␤ 兩 A 典 , 共57兲 where ap, ␤ is the annihilation operator corresponding to a particle with momentum pand spin projection ␤ referred to the quantization axis. Inserting this expression into the hadronic tensor 共7兲, and following the procedure described in Ref. 关43兴we obtain W ␮ ␯ ⫽1 2mNp⬘w ␮ ␯ 共p⬘,p,s兲MS共p兲,共58兲 where we have defined the polarized single-nucleon tensor w ␮ ␯ 共p⬘,p,s兲 ⫽兺 ␣␣ ⬘ ␤ ⬘ D ␤ ⬘1/2 *共s兲J ␯ 共p⬘,p兲 ␤ ⬘ ␣ J ␮ 共p⬘,p兲 ␣ ⬘ ␣ *D ␣ ⬘1/2共s兲. 共59兲 FIG. 2. MEC diagrams contributing to the two-body current of this work: seagull 共a兲, pion-in-flight 共b兲, and ⌬共c,d兲currents are considered including relativistic corrections. TABATABAEI, AMARO, AND CABALLERO PHYSICAL REVIEW C 68, 034611 共2003兲 034611-6 In the case of interest here, a closed-shell nucleus, the scalar momentum distribution MS(p) for nucleon knockout from a shell nlj is given by MS共p兲⫽2j⫹1 4 ␲ R ˜ 2共p兲,共60兲 with R ˜ (p) the radial wave function of the hole in momentum space. Using the current matrix elements 共51兲–共53兲one can compute in the factorized approximation 共58兲the response functions 共4兲–共6兲. From these results the PWIA expressions for the reduced response functions can be identified. Expressions for the unpolarized responses in PWIA were given in Refs. 关43,49兴. In the case of the polarized responses, from the total of 18, only five survive in PWIA. These are given by W11 T⬘ ⫽mNp⬘ 4 ␲ 2JcJm ␹ MS共p兲,共61兲 W10 T⬘ ⫽mNp⬘ 4 ␲ 2Jm 2MS共p兲,共62兲 W ˜ 11 TL⬘ ⫽mNp⬘ 4 ␲ 2 冑 2共 ␳ cJm⫺ ␳ soJc ␹ 2兲MS共p兲,共63兲 W11 TL⬘ ⫽mNp⬘ 4 ␲ 2 冑 2 ␳ cJmMS共p兲,共64兲 W10 TL⬘ ⫽mNp⬘ 4 ␲ 2 冑 2 ␳ soJm ␹ MS共p兲,共65兲 where we have used the factors introduced in Eqs. 共54兲and 共55兲. Note that all the L-, T-, TL-, and TT-type polarized responses are zero in this approximation. III. RESULTS In this section we present results for selected recoil nucleon polarization observables corresponding to proton knockout from the p1/2 and p3/2 shells in 16O. In particular, we restrict ourselves to the analysis of all the polarized response functions, including the fifth response W0 TL⬘ ⫽Wunpol TL⬘/2 that does not depend on the nucleon polarization and only enters when the initial electron beam is polarized, and the transferred polarization asymmetries Pl,t,n ⬘. The study of cross sections and induced polarizations will be presented in a forthcoming publication 关51兴. Two different kinematical situations corresponding to (q, ␻ )-constant kinematics 共also referred to as quasiperpendicular kinematics兲have been selected: 共i兲q⫽460 MeV/c, ␻ ⫽100 MeV and 共ii兲q ⫽1 GeV/c, ␻ ⫽450 MeV. In both cases the value of the transfer energy ␻ corresponds almost to the quasielastic peak. In this work our main interest is focused on the role of the two-body MEC operators upon the recoil nucleon polarization observables, trying to identify kinematical conditions for which these effects can be important; however, a brief excursion on the FSI effects is also presented. All the calculations have been done within the formalism described in the preceding section, i.e., semirelativistic expressions for the one- and two-body current operators and a multipole expansion method have been used. The number of multipoles needed has been fixed by comparing the DWIA results, in the particular case of no FSI, with the exact factorized PWIA responses 共61兲–共65兲. Convergence in the multipole analysis is obtained with Jmax⫽30 for q⫽460 MeV/cand Jmax⫽35 for q⫽1000 MeV/c. Finally, in all the results which follow, the kinematics of the ejected nucleon is treated exactly. A. Polarized response functions Here we analyze the 13 responses defined in Eqs. 共48兲– 共50兲which arise from the ejected nucleon polarization, plus the ‘‘fifth’’ response function W0 TL⬘. Results are displayed in Figs. 3–12. A similar analysis for the unpolarized responses L,T,TL, and TT has been performed recently in Ref. 关40兴. FIG. 3. Induced polarized response functions (L-, T-, TL-, and TT-type responses兲for proton knockout from the 1p1/2 shell in 16O, with momentum transfer q⫽460 MeV/cand energy transfer ␻ ⫽100 MeV. Solid lines are the DWIA results using the optical potential of Comfort and Karp; dashed lines have been computed with the Schwandt optical potential. MESON EXCHANGE CURRENTS IN (e,e⬘p) RECOIL... PHYSICAL REVIEW C 68, 034611 共2003兲 034611-7 1. Effects of FSI We start our discussion with the effects of FSI on the polarized responses. A study of the dependence of the unpolarized responses on the particular FSI model was already presented in Ref. 关38兴. In Fig. 3 we show the eight induced polarized responses for proton knockout from the 1p1/2 shell in 16O as a function of the missing momentum p. Kinematics corresponds to q⫽460 MeV/cand ␻ ⫽100 MeV. The five transferred polarized responses plus the fifth one (T⬘and TL⬘types兲are displayed in Fig. 4. Similar results are obtained for the 1p3/2 shell and thus they are not shown here. In all of these results we use bound wave functions obtained as solutions of the Schro ¨dinger equation with a Woods-Saxon potential, with parameters taken from Ref. 关42兴. For the final states we use solutions for two different optical potentials. Solid lines correspond to calculations performed with the Comfort and Karp potential 关52兴, which was originally fitted to elastic proton scattering from 12C for energies below 183 MeV. We have extended it to 16O by introducing a dependence A1/3 in the radius parameters. The results shown with dashed lines have been computed with the Schwandt potential 关53兴, which also has been extrapolated here for 16O since it was originally fitted to higher mass nuclei. The induced polarized L,T,TL,TT and the fifth response functions, which are zero in the absence of FSI, are expected to be highly sensitive to the details of the particular optical potential considered and, in particular, to the spin-orbit term in the potential. In this sense note the significant difference introduced by both potentials in the case of the polarized TL and TT responses 共Fig. 3兲, while the FSI discrepancy gets smaller for the fifth response function 共Fig. 4兲and is considerably reduced for Wn Land Wn T. The five transferred polarized responses which survive in PWIA 共61兲–共65兲depend less on the details of the potential, being mostly affected by the central imaginary part of it. As known, these responses enter in the case in which also the initial electron is polarized, and they contribute to the transferred nucleon polarization asymmetry. We observe 共Fig. 4兲 that both potentials lead to close results, differing by less than 10% for the dominant responses Wn TL⬘,Wl T⬘, and Wl TL⬘, while the largest differences are shown for Wt T⬘, which is however very small. Similar results are found for the p3/2 shell. The sensitivity shown by some polarized responses to the details of the potential makes these observables of special interest to disentangle between the different models of FSI that can fit reasonably well the unpolarized cross sections. FIG. 4. The same as Fig. 3 for the fifth and transferred polarized response functions (T⬘- and TL⬘-type responses兲. With dotted lines we show also for reference the PWIA results. FIG. 5. Induced polarized response functions for proton knockout from the 1p1/2 shell in 16O, with momentum transfer q ⫽460 MeV/cand energy transfer ␻ ⫽100 MeV. Dotted lines are the DWIA results using only the OB current operator; dashed lines include in addition the seagull current 共OB⫹S兲; dot-dashed lines include the seagull plus pionic currents 共OB⫹S⫹P兲; and finally solid lines represent the total result 共OB⫹MEC兲including also the ⌬current. TABATABAEI, AMARO, AND CABALLERO PHYSICAL REVIEW C 68, 034611 共2003兲 034611-8 2. Effects of MEC The impact of MEC on the recoil nucleon polarized responses is shown in Figs. 5–12. In each panel we compare the distorted wave responses evaluated by using the OB current only 共dotted line兲with the results obtained when including also the two-body MEC operators considered in Fig. 2, namely, the seagull or contact (OB⫹S) current 共dashed lines兲, the contact, and pion-in-flight (OB⫹S⫹P) currents 共dot-dashed lines兲and, finally, including also the ⌬current 共solid lines兲, denoted as 共OB⫹MEC兲. Results in Figs. 5–8 correspond to kinematics q⫽460 MeV/c, ␻ ⫽100 MeV 共kinematics I兲, whereas in Figs. 9–12 we present the responses evaluated at q⫽1 GeV/c, ␻ ⫽450 MeV 共kinematics II兲. For both kinematics proton knockout from the p1/2 共Figs. 5, 6 and 9, 10兲and p3/2 共Figs. 7, 8 and 11, 12兲have been considered. Let us discuss first the results for kinematics I 共Figs. 5–8兲. Here we observe that the global sign of the polarized T,TL, and TT responses changes when comparing the p1/2 共Fig. 5兲 and p3/2 共Fig. 7兲shells. The same occurs for kinematics II. Concerning MEC effects, the various polarized responses display different sensitivities to the two-body component of the nuclear current. Apart from the pure longitudinal response Wn L, which shows no dependence on MEC because the ‘‘semirelativistic’’ MEC expressions only include the leading transverse components, the role of MEC on Wn Tis shown to be similar to the one found for the unpolarized T response in Ref. 关40兴: the enhancement 共in absolute value兲 produced by the S current is partially canceled by the reduction introduced by the P current; the ⌬current gives rise to an additional reduction, leading to a global decrease of the Wn Tresponse ⬃10% at the maximum; this effect being similar for both shells 共Figs. 5 and 7兲. Larger MEC effects are found for some of the induced polarized TL responses, particularly for Wt TL where the ⌬ current produces a very significant modification of the response, changing even its shape in the region close to p ⬃100 MeV/c. Note that, although the global effect introduced by the ⌬current in this response is similar for both shells, in the case of p1/2 there is a large increase, whereas for p3/2 the response is significantly reduced in absolute value. It is interesting to point out that the ⌬current also plays the most important role for the Wn TL response, this being clearly shown in the case of the p3/2 shell. The role of MEC on the three polarized TT responses shows a very different behavior for the two shells considered. In the case of p1/2 共Fig. 5兲, the global effect of MEC is a very significant reduction of the responses, particularly for Wt TT (⬃20%) and Wn TT (⬃30%), being the separate contributions of the S, P, and ⌬currents of rather similar importance. Note that the contributions introduced by the S and P currents have opposite signs for the p1/2 and p3/2 shells. As a consequence, for the p3/2 shell 共Fig. 7兲the large enhancement 共in absolute value兲produced by the S current is almost canceled exactly by the contributions of the P and ⌬cur- FIG. 6. Fifth and transferred polarized response functions for 16O. Kinematics corresponds to momentum transfer q ⫽460 MeV/cand energy transfer ␻ ⫽100 MeV. The meaning of the lines is the same as in Fig. 5. FIG. 7. The same as Fig. 5 for proton knockout from the 1p3/2 shell. MESON EXCHANGE CURRENTS IN (e,e⬘p) RECOIL... PHYSICAL REVIEW C 68, 034611 共2003兲 034611-9 clearly less important for these missing momentum values. 共2兲Ryckebusch et al. 关36兴do not present the separate l,t,nresponse functions. In general they find small MEC effects, as we do, in the transferred polarizations for low p ⬍300 MeV/c. These effects being larger as qand pincrease. Comparing specifically our results to theirs for kinematics II, we observe that the OB results clearly differ due to the different treatment of FSI, while somewhat larger and qualitatively different MEC effects are found in this work. Be that as it may, since the different treatment of FSI and of the current operators in Refs. 关33,36兴and in the present work produces discrepancies already at the impulse approximation, it is hard to draw general conclusions on MEC effects beyond the fact that in Ref. 关33兴MEC lead to excessively large contributions compared with ours, while their small size in Ref. 关36兴is in accord with our calculation. IV. SUMMARY AND CONCLUSIONS In this paper we have presented a distorted wave model of (e ជ ,e⬘p ជ ) reactions which goes beyond the impulse approximation with the inclusion of two-body meson-exchange currents. Relativistic kinematics to relate the energy and momentum of the ejected proton is used, and the currents are derived through an expansion in powers of the missing momentum over the nucleon mass. Explicit expressions of the polarized response functions in a general multipole expansion method are given. Results for the responses and trans- FIG. 16. The same as Fig. 15 for q⫽1000 MeV/cand ␻ ⫽450 MeV. TABATABAEI, AMARO, AND CABALLERO PHYSICAL REVIEW C 68, 034611 共2003兲 034611-16 ferred polarization asymmetries have been obtained for proton knockout from the different shells in 16O for quasiperpendicular kinematics with the transfer momentum fixed to q⫽460 and 1000 MeV/c. FSI have been considered in each case by using different optical potentials. One of our primary goals has been to estimate, within our present model, the validity of the impulse approximation by analyzing the effect of MEC on the different recoil nucleon polarized observables. Thus we compare the standard DWIA results, obtained using only the OB current, with the ‘‘full’’ calculation which includes the MEC. We have also explored the role played by the particular description of the FSI, hence we compare the results obtained by using different optical potentials which have been widely considered in the literature: Schwandt 关53兴and Comfort and Karp 关52兴parametri- zations. For higher energy we have used instead the Schro ¨dinger-equivalent form of a Dirac optical potential. From our present studies we may summarize and conclude the following. 共1兲The induced T,TL, and TT polarized responses are particularly sensitive to the details of the optical potential, allowing them, especially the TT ones, to constrain the theoretical model for FSI. The transferred polarized responses (T⬘and TL⬘), which survive in PWIA, show a much less sensitivity to the interaction. 共2兲In general, MEC effects over the transferred T⬘and TL⬘polarized responses for q⫽460 MeV/cand moderate missing momentum (p⬍300 MeV/c) are rather small (⬍5%), and tend to increase as qgoes higher, being of the FIG. 17. The same as Figs. 15 and 16 for the normal polarization transfer component Pn ⬘and ␾ ⫽90°. FIG. 18. Transferred polarization asymmetries Pl ⬘and Pt ⬘, and their quotient Pt ⬘/Pl ⬘for proton knockout from the three shells in 16O for q⫽1000 MeV/cand ␻ ⫽450 MeV. The incident electron energy is ⑀ e⫽2450 MeV and the proton azimuthal angle ␾ ⫽180°. The meaning of the lines is the following: dotted are the DWIA calculation with OB current only; solid are the total OB⫹MEC result; dot-dashed are also the OB ⫹MEC result but without the spin-orbit term of the optical potential. These three curves have been obtained using the electromagnetic nucleon form factors of Galster. Finally, dashed lines are the total OB⫹MEC result using instead the form factors of Gari- Krumplemann. MESON EXCHANGE CURRENTS IN (e,e⬘p) RECOIL... PHYSICAL REVIEW C 68, 034611 共2003兲 034611-17 order of a ⬃10% reduction 共due mainly to the ⌬current兲in the particular case of Wl T⬘and q⫽1 GeV/c. The role of MEC gets clearly more important for the induced T,TL, and TT polarized responses. Emphasis should be placed on Wt TT and Wn TT which are reduced at the maximum by ⬃20% and ⬃30%, respectively, for q⫽460 MeV/cand for the p1/2 shell; note however that these effects are negligible in the case of p3/2 . For q⫽1 GeV/cthe role of MEC diminishes. 共3兲FSI give rise to an important deviation of the transferred polarization asymmetries Pl ⬘and Pt ⬘with respect to the PWIA results, showing a very pronounced oscillatory behavior that starts for p⭓200 MeV/c. This behavior does not appear in the component Pn ⬘. The uncertainties introduced by the optical potentials are rather small for the missing momentum region analyzed. 共4兲MEC effects on Pt ⬘and Pl ⬘are negligible for q ⫽460 MeV/cand increase for q⫽1 GeV/c, especially for p⬎200 MeV/c. The role of MEC on Pn ⬘is clearly more important. Finally we are confident that the significant sensitivity shown by some polarized observables to MEC, particularly to the ⌬current, will be maintained within the scheme of a fully relativistic calculation which takes care of relativistic ingredients, such as the dynamical enhancement of lower components, not included in the present model. Work along this line is in progress. ACKNOWLEDGMENTS This work was partially supported by funds provided by DGI 共Spain兲and FEDER funds, under Contract Nos. BFM2002-03218, BFM2002-03315, and FPA2002-04181- C04-04, and by the Junta de Andalucı ´a. APPENDIX: SUM OVER THIRD COMPONENTS AND REDUCED RESPONSE FUNCTIONS In this appendix we perform the sum over third components in the multipole expansion of response functions and give their explicit expressions in terms of the reduced matrix elements of the multipole operators. First, we write the response functions 共4兲–共6兲in terms of the spherical components of the current matrix elements J⫾1⬅ 具 p⬘s,B 兩 J ˆ⫾1 兩 A 典 using the hadronic tensor 共7兲 RL⫽1 K兺 ␳ * ␳ ,共A1兲 RT⫽1 K兺共 兩 J⫺1 兩 2⫹ 兩 J⫹1 兩 2兲,共A2兲 RTL⫽⫺21 KRe兺 ␳ *共J⫹1⫺J⫺1兲,共A3兲 RTT⫽1 K兺共J⫺1 *J⫹1⫹J⫹1 *J⫺1兲,共A4兲 RTL⬘ ⫽⫺21 KRe兺 ␳ *共J⫹1⫹J⫺1兲,共A5兲 RT⬘ ⫽1 K兺共 兩 J⫹1 兩 2⫺ 兩 J⫺1 兩 2兲.共A6兲 Inserting the multipole expansion for the charge and current components as given in Eqs. 共23兲and 共24兲we find that each response can be written as a sum of terms of the type BJ⬘J m⬘m⬅1 K兺 MB 具 p⬘s,JBMB 兩 T ˆJ⬘m⬘ ⬘共q兲 兩 A 典 * ⫻ 具 p⬘s,JBMB 兩 T ˆJm共q兲 兩 A 典 ,共A7兲 where T ˆJ⬘m⬘ ⬘and T ˆJm represent in general the Coulomb, electric, or magnetic multipole operators. Introducing now the multipole expansion 共22兲corresponding to the final state, which is polarized along an arbitrary direction s共8兲,weget BJ⬘J m⬘m⫽1 K兺 MB兺 ␯ ⬘ ␯ D ␯ ⬘1/2 (1/2) 共s兲D ␯ 1/2 (1/2)共s兲*兺 l⬘M⬘j⬘mp ⬘ il⬘Yl⬘M⬘ *共p ˆ⬘兲 ⫻ 具 1 2 ␯ ⬘l⬘M⬘ 兩 j⬘mp ⬘ 典具 j⬘mp ⬘JBMB 兩 J⬘m⬘ 典 ⫻ 具 共l⬘j⬘兲JB,J⬘m⬘ 兩 TJ⬘m⬘ ⬘ 兩 A 典 *兺 lMjmp i⫺lYlM共p ˆ⬘兲 ⫻ 具 1 2 ␯ lM 兩 jmp 典具 jmpJBMB 兩 Jm 典具 共lj兲JB,Jm 兩 TJm 兩 A 典 , 共A8兲 where we have used Jf⫽Jand Mf⫽m, since the initial nucleus has total angular momentum equal zero. Using the Wigner-Eckart theorem for the matrix elements of tensor operators between states of definite angular momenta 具 共lj兲JB,Jm 兩 TJm 兩 0 典 ⫽1 关J兴 具 共lj兲JB,J 储 TJ 储 0 典 共A9兲 and reducing the products of two rotation matrices and two spherical harmonics to linear combinations of spherical harmonics D ␯ ⬘1/2 (1/2) 共s ˆ兲D ␯ 1/2 (1/2)共s ˆ兲*⫽ 冑 4 ␲ 兺 JM 共⫺1兲1/2⫹ ␯ ⫹JfJ (1/2) ⫻ 冉 1 2 1 2J ⫺ ␯␯ ⬘M 冊 YJM共s ˆ兲, 共A10兲 TABATABAEI, AMARO, AND CABALLERO PHYSICAL REVIEW C 68, 034611 共2003兲 034611-18 Yl⬘M⬘ *共p ˆ⬘兲YlM共p ˆ⬘兲⫽兺 J⬘M⬘ 共⫺1兲M关l兴关l⬘兴关J⬘兴 冑 4 ␲ 冉 ll ⬘J⬘ ⫺MM ⬘M⬘ 冊冉 ll ⬘J⬘ 00 0 冊 YJ⬘M⬘共p ˆ⬘兲,共A11兲 with fJ (1/2)⫽1/ 冑 2 the Fano tensor for spin-1/2 polarization, we obtain BJ⬘J m⬘m⫽兺 MB兺 ␯␯ ⬘兺 JM 兺 lMjmp兺 l⬘M⬘j⬘mp ⬘兺 J⬘M⬘ il⬘⫺l共⫺1兲(1/2)⫹ ␯ ⫹JfJ (1/2) 冉 1 2 1 2J ⫺ ␯␯ ⬘M 冊 YJM共s兲共⫺1兲M关l兴关l⬘兴关J⬘兴 ⫻ 冉 ll ⬘J⬘ ⫺MM ⬘M⬘ 冊冉 ll ⬘J⬘ 00 0 冊 YJ⬘M⬘共p ˆ⬘兲共⫺1兲l⬘⫺(1/2)⫺mp ⬘ 关j⬘兴 冉 1 2l⬘j⬘ ␯ ⬘M⬘⫺mp ⬘ 冊 共⫺1兲JB⫺j⬘⫺m⬘ ⫻ 冉 j⬘JBJ⬘ mp ⬘MB⫺m⬘ 冊 共⫺1兲l⫺(1/2)⫺mp关j兴 冉 1 2lj ␯ M⫺mp 冊 共⫺1兲JB⫺j⫺m 冉 jJ BJ mpMB⫺m 冊 ⫻ 具 共l⬘j⬘兲JB,J⬘ 储 TJ⬘ ⬘ 储 0 典 * 具 共lj兲JB,J 储 TJ 储 0 典 ,共A12兲 where we have transformed the Clebsch-Jordan to 3-Jcoefficients. Next we perform the sums over third components of angular momenta in the above expression. Note that the total phase inside the sum can be simplified to phase⫽共⫺1兲共1/2兲⫹mp共⫺1兲J⫹l⫹l⬘共⫺1兲j⫺j⬘.共A13兲 Therefore the following coefficient appears S⬅兺 MB兺 ␯␯ ⬘兺 MM⬘兺 mpmp ⬘ 共⫺1兲(1/2)⫹mp⫹J⫹l⫹l⬘⫹j⫺j⬘ 冉 1 2 1 2J ⫺ ␯␯ ⬘M 冊 冉 ll ⬘J⬘ ⫺MM ⬘M⬘ 冊 冉 1 2l⬘j⬘ ␯ ⬘M⬘⫺mp ⬘ 冊 ⫻ 冉 j⬘JBJ⬘ mp ⬘MB⫺m⬘ 冊 冉 1 2lj ␯ M⫺mp 冊 冉 jJ BJ mpMB⫺m 冊 .共A14兲 We first perform the sum over ␯ , ␯ ⬘,M,M⬘by using a 9-jcoefficient 兺 ␯␯ ⬘兺 MM⬘ 冉 1 2 1 2J ⫺ ␯␯ ⬘M 冊 冉 ll ⬘J⬘ ⫺MM ⬘M⬘ 冊 冉 1 2lj ␯ M⫺mp 冊 冉 1 2l⬘j⬘ ␯ ⬘M⬘⫺mp ⬘ 冊 ⫽共⫺1兲(1/2)⫹l⫹j兺 LM 关L兴2 冉 JJ ⬘L MM ⬘M 冊 冉 Lj j ⬘ Mm p⫺mp ⬘ 冊 冦 JJ ⬘L 1 2lj 1 2l⬘j⬘ 冧 .共A15兲 Next we compute the sum over mp,mp ⬘,MBusing a 6-jcoefficient 兺 mpmp ⬘MB 共⫺1兲mp⫹1/2 冉 Lj j ⬘ Mm p⫺mp ⬘ 冊 冉 jJ BJ mpMB⫺m 冊 冉 j⬘JBJ⬘ mp ⬘MB⫺m⬘ 冊 ⫽共⫺1兲(1/2)⫹m⬘⫹j⫹j⬘⫹JB 冉 LJ ⬘J M⫺m⬘m 冊 再 LJ ⬘J JBjj ⬘ 冎 .共A16兲 MESON EXCHANGE CURRENTS IN (e,e⬘p) RECOIL... PHYSICAL REVIEW C 68, 034611 共2003兲 034611-19 Then the Scoefficient 共A15兲results S⫽兺 LM 关L兴2共⫺1兲J⫹l⬘⫹j⫹JB⫹m⬘ 冉 LJ ⬘J M⫺m⬘m 冊 ⫻ 冉 JJ ⬘L MM ⬘M 冊 冦 JJ ⬘L 1 2lj 1 2l⬘j⬘ 冧 再 LJ ⬘J JBjj ⬘ 冎 . 共A17兲 To finish we insert result 共A18兲into Eq. 共A13兲, and define indices ␴ , ␴ ⬘corresponding to the quantum numbers of the final states ␴ ⫽共l,j,J兲, ␴ ⬘⫽共l⬘,j⬘,J⬘兲,共A18兲 and a coupling coefficient ⌽ ␴ ⬘ ␴ 共J,J⬘,L兲⫽ 冑 2关l兴关l⬘兴关j兴关j⬘兴关J兴关J⬘兴关J⬘兴关L兴 ⫻共⫺1兲l⫹j⫹JB⫹L⫹J⫹J⬘ 冉 ll ⬘J⬘ 00 0 冊 ⫻ 再 LJ ⬘J JBjj ⬘ 冎 冦 JJ ⬘L 1 2lj 1 2l⬘j⬘ 冧 . 共A19兲 The final expression for Bis BJ⬘J m⬘m⫽1 2兺 lj 兺 l⬘j⬘兺 JJ⬘LM il⬘⫺l共⫺1兲m ⫻ 冉 JJ ⬘L m⫺m⬘M 冊 ⌽ ␴ ⬘ ␴ 共J,J⬘,L兲 ⫻关YJ共s ˆ兲YJ⬘共p ˆ⬘兲兴L,⫺M 具 共l⬘j⬘兲JB,J⬘ 储 TJ⬘ ⬘ 储 0 典 * ⫻ 具 共lj兲JB,J 储 TJ 储 0 典 ,共A20兲 where the coupling between two spherical harmonics has been used 关YJ共s ˆ兲YJ⬘共p ˆ⬘兲兴LM ⫽共⫺1兲J⫺J⬘⫹M兺 MM⬘ 关L兴 ⫻ 冉 JJ ⬘L MM ⬘⫺M 冊 YJM共s ˆ兲YJ⬘M⬘共p ˆ⬘兲. 共A21兲 Although the result given in Eq. 共A21兲is formally identical, with the exception of the factor 1/2, to the one obtained in Ref. 关49兴for the case of polarized nuclei, there exists a basic difference concerning the polarization coefficient ⌽ ␴ ⬘ ␴ (J,J⬘,L), which contains all the information on the polarization properties of the particles in the initial and/or final state. 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