Semirelativistic meson-exchange currents in (e,e) and (e,e' p) reactions

Semirelativistic meson-exchange currents in „e,e⬘…and „e,e⬘p…reactions J. E. Amaro,1M. B. Barbaro,2,3 J. A. Caballero,3and F. Kazemi Tabatabaei1 1Departamento de Fı ´sica Moderna, Universidad de Granada, E-18071 Granada, Spain 2Dipartimento di Fisica Teorica, Universita `di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy 3Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear Universidad de Sevilla, Apartado Postal 1065, E-41080 Sevilla, Spain 共Received 3 February 2003; published 21 July 2003兲 Electron-induced one-nucleon knockout observables are computed for a moderate to high momentum transfer, making use of semirelativistic expressions for the one-body and two-body meson-exchange current matrix elements. Emphasis is laid on the semirelativistic form of the ⌬-isobar exchange current and several prescriptions for the dynamical-equivalent form of the ⌬propagator are analyzed. To this end, the inclusive transverse response function, evaluated within the context of the semi-relativistic approach and using different prescriptions for the ⌬propagator, is compared with the fully relativistic calculation performed within the scheme of the relativistic Fermi gas model. It is found that the best approximation corresponds to the one using the traditional static ⌬propagator. These semirelativistic approaches, which contain important aspects of relativity, are implemented in a distorted-wave analysis of quasielastic (e,e⬘p) reactions. Final state interactions are incorporated through a phenomenological optical potential model and relativistic kinematics is assumed when calculating the energy of the ejected nucleon. The results indicate that the meson-exchange currents may modify substantially the TL asymmetry for high missing momentum. DOI: 10.1103/PhysRevC.68.014604 PACS number共s兲: 25.30.Fj, 14.20.Gk, 24.10.Jv, 24.10.Eq I. INTRODUCTION Electron-induced one-nucleon emission reactions near the quasielastic peak are clearly dominated by one-body 共OB兲 dynamics, since such knematics roughly corresponds to the virtual photon interacting directly with a bound nucleon 关1–3兴. Under these conditions it has been possible to analyze a considerable amount of data for a variety of nuclei and to study many aspects of the reaction mechanism. Most of these studies have been based on the standard distorted-wave impulse approximation 共DWIA兲, where the OB current matrix elements are computed using single-particle wave functions obtained as solutions of the Schro ¨dinger equation with phenomenological potentials. Recently 关4–7兴, efforts have been made to develop a fully relativistic description of the process in order to describe consistently the high momentum and energy transfer regions 关8兴. This constitutes the basis of the relativistic DWIA 共RDWIA兲. Within the scheme of nonrelativistic approaches, the role played by correlations beyond the mean field and their impact on the overlap functions and spectroscopic factors have been investigated in several works. Many-body calculations 关9,10兴including short-range correlations 共SRC兲, of central type only, have shown a small effect over the quasihole overlap functions and, accordingly, over the (e,e⬘p) cross section. In Ref. 关11兴the SRC were shown not to modify substantially the mean-field (e,e⬘p) results at high momentum and low excitation energy. In recent works, a significant effort has been made to improve the analysis by including the tensor and spin-isospin channels 关12,13兴as well as longrange correlations 关14,15兴. It is important to point out that the extraction of spectroscopic factors from the analysis of (e,e⬘p) experiments is still not free from ambiguities. This would require an accurate knowledge of the reaction mechanism. In this sense, a RDWIA calculation, compared to DWIA, gives rise to a different quenching of the (e,e⬘p) cross section, hence leading to different spectroscopic factors 关4兴共larger for RDWIA兲. The two-body meson-exchange currents 共MEC兲may also play a significant role in the description of electron scattering observables, since they are connected to the nuclear correlations by the continuity equation. In fact, the interplay between correlations and MEC in (e,e⬘p) reactions is far from trivial; rigorously, the concept of overlap function is not enough to describe the process in the presence of two-body current operators and, therefore, the effect of correlations in the presence of MEC may be not simply parametrizable by a spectroscopic factor. This idea is supported by calculations of the inclusive transverse response for nuclear matter within the correlated basis function perturbation theory 关16兴, which have shown a significant effect due to MEC, contrary to the much smaller effect usually found for uncorrelated calculations 关17兴. Before a complete calculation of the exclusive response functions including MEC within a sophisticated correlated model of the reaction be attempted, it is necessary to calibrate different approaches to the calculation in uncorrelated models, as a first step beyond the DWIA. Thus, in this work we restrict our attention to the meanfield 共uncorrelated兲model and evaluate the role played by the MEC in quasielastic inclusive and exclusive electron scattering reactions. In the case of exclusive (e,e⬘p) processes, a theoretical evaluation of MEC, including the usual pion in flight P, contact C, and Delta isobar ⌬, has been presented in Refs. 关18–20兴. In Ref. 关20兴substantial differences were found with respect to the results of Ref. 关19兴, particularly for the interference TL response. The analysis in Ref. 关19兴, which treats final state interactions 共FSI兲through a real potential, was extended to higher momentum transfer in Refs. 关21,22兴where some relativistic corrections were also included in the OB current operator. Instead, the standard nonrelativistic MEC operators 关23,24兴were used except for a modified dynamical ⌬propagator that depends on the invari- PHYSICAL REVIEW C 68, 014604 共2003兲 0556-2813/2003/68共1兲/014604共12兲/$20.00 ©2003 The American Physical Society68 014604-1 ant energy 冑 sof the ⌬. More recently, the theoretical model of Ref. 关18兴has been refined in Ref. 关25兴modifying also the ⌬propagator in order to include the ⌬-invariant mass. The use of a proper dynamical propagator in the ⌬current has been discussed at length, concerning mainly the case of twoparticle emission reactions 关26–28兴. One of the central questions is that how to determine 冑 sin a way such that it could be easily implemented in the existing models for (e,e⬘) and (e,e⬘p) reactions. In general, the appropriate choice of 冑 swill depend on the specific dynamical model for the reaction mechanism. Hence, a proper description of the reaction dynamics through an adequate choice of the currents and kinematics is necessary. For high momentum transfer, relativistic effects play a significant role and therefore relativity should enter in the description of kinematics and/or current matrix elements. Within a fully relativistic model, MEC effects in (e,e⬘p) have been computed recently in Ref. 关29兴, where only the contact 共or ‘‘seagull’’兲current has been considered. Due to the complexity of the fully relativistic two-body currents, particularly the ⌬one, in this work we start by developing reasonable semirelativistic expressions for all the two-body currents (P,C, and ⌬) that can be easily implemented in the existing nonrelativistic descriptions of the reaction mechanism. These semirelativistic currents retain important aspects of relativity and hence they may be used to describe properly recent experiments performed at high momentum and energy transfers. In this paper we discuss the one-particle emission sector, introducing a recently developed semirelativistic 共SR兲approximation for the three MEC operators which, in conjunction with the SR form of the OB current derived in Ref. 关30兴 共see Ref. 关31兴for a recent review on these expansions兲, makes it possible to evaluate the quasielastic (e,e⬘) and (e,e⬘p) observables. The SR-OB current was applied in inclusive (e,e⬘)关32兴and for coincidence (e,e⬘p)关6,10,33– 35兴reactions. The SR expressions for the MEC operators were obtained in Ref. 关36兴in the case of the pionic and contact currents. In Refs. 关31,37兴, a comparison between the SR-MEC predictions for the inclusive transverse response and the exact relativistic Fermi gas 共RFG兲results is presented. One of the goals of this work is to derive a semirelativistic expression for the ⌬-exchange current and compare its prediction for the inclusive transverse response function with the exact relativistic result obtained within the RFG model of Ref. 关38兴. This comparison, carried out in the inclusive channel 共Sec. II兲, makes it possible to test the reliability of the different prescriptions of the dynamical ⌬propagator to be used in exclusive (e,e⬘p) processes 共Sec. III兲. Once this is settled, the impact of the relativistic MEC over the exclusive response functions is evaluated for the moderate to high momentum transfer. Finally, in Sec. IV we draw our conclusions. II. INCLUSIVE TRANSVERSE RESPONSE The impact of meson-exchange currents on inclusive electron scattering has been traditionally investigated in a nonrelativistic framework and relativistic corrections have been implemented making use of a power expansion in the momenta of the nucleons and of the exchanged photon 关39–41兴. In Refs. 关31,38兴a semirelativistic analysis of the MEC (P, C, and ⌬) currents, represented in Fig. 1, has been performed in the context of the RFG with the aim of finding simple prescriptions for implementing relativistic effects into nonrelativistic calculations. Within the RFG model, it is possible to perform a fully relativistic calculation of the inclusive response functions, using exact relativistic kinematics, currents, and propagators. Hence, the inclusive transverse response function of nuclear matter can be used to test the quality of the semirelativistic MEC, to be applied in the following section to semiinclusive processes. In this section, we focus on the ⌬current, represented by Figs. 1共d兲–1共g兲, laying emphasis on the dynamical treatment of the ⌬propagator, an aspect of the problem which was neglected in the SR model of Ref. 关38兴. Restricting our attention to the case of one-particle emission processes, the contribution of the ⌬current to the transverse nuclear response function is obtained as the interference between the ⌬ and the OB currents. In the Fermi gas model, it reads R⌬-OB T共q, ␻ 兲⫽3Z 8 ␲ kF 32Re 冕 h⬍kF d3h ␦ 共Ep⫺Eh⫺ ␻ 兲 ⫻NphTr关jT OB共p,h兲*•jT ⌬共p,h兲兴,共1兲 FIG. 1. Feynman diagrams contributing to the two-body current with one-pion exchange. Contact C关共a兲,共b兲兴, pionic P共c兲, and isobar ⌬共d兲–共g兲 are included in this work. AMARO, BARBARO, CABALLERO, AND KAZEMI TABATABAEI PHYSICAL REVIEW C 68, 014604 共2003兲 014604-2 where p⫽h⫹qis the momentum of the ejected particle, jTis the transverse 共perpendicular to q) component of j, and kFis the Fermi momentum. In the RFG model Ek⫽ 冑 k2⫹mN 2, whereas in the nonrelativistic 共NR兲Fermi gas Ek⫽mN⫹ ⑀ k ⫽mN⫹K2/(2mN). Moreover, the factor Nph , arising from the spinor normalization, is mN 2/(EpEh) for RFG and 1 in a NR model. Finally, jT a(p,h)共which implicitly include spin and isospin indices兲are the OB or ⌬particle-hole currents 关31兴. The nonrelativistic two-body ⌬current used in this paper has been obtained from the relativistic one in the limit of small momenta by following the procedure sketched in Ref. 关38兴; it can be written in momentum space as jnr ⌬共p1 ⬘,p2 ⬘,p1,p2兲 ⫽1 9 G1 2mN f ␲ N⌬ m ␲ f m ␲ 兵 G⌬共P1⫹Q兲q ⫻关⫺k2⫻ ␴ (1)⫹2ik2兴关2 ␶ 3 (2)⫺i共 ␶ (1)⫻ ␶ (2)兲z兴 ⫹G⌬共P1 ⬘⫺Q兲q⫻关k2⫻ ␴ (1)⫹2ik2兴 ⫻关2 ␶ 3 (2)⫹i共 ␶ (1)⫻ ␶ (2)兲z兴 其 k2• ␴ (2) k2 2⫹m ␲ 2⫹共1↔2兲,共2兲 where Pi,Pi ⬘are the four-momenta of the initial and final nucleons defined in Fig. 1, ki⫽pi ⬘⫺piis the momentum transferred to the ith nucleon, and G⌬(P) is the nonrelativistic 共dynamical兲⌬propagator, which will be later discussed in detail. We use the following values for the coupling constants: G1⫽4.2, f2/4 ␲ ⫽0.079, and f ␲ N⌬⫽2.24. In Ref. 关38兴a more general relativistic current was considered, containing additional couplings G2,G3and dependence on the so-called off-shell parameters z1,z2, and z3. Therein, it was shown that the terms G2and G3have a weak impact on the transverse response for qbelow 1 GeV and that this response displays a small dependence upon the offshell parameters compared with its sensitivity to 冑 s, which will be discussed at the end of this section. Therefore, in this work we use the Peccei Lagrangian, which neglects G2and G3and corresponds to zi⫽⫺1/4. The MEC particle-hole matrix elements are obtained by summing over all the single-particle states occupied in the ground-state Slater determinant 关31,38兴. In the case of nuclear matter only the exchange term, represented by the many-body diagrams of Fig. 2, survives due to spin and isospin symmetries. Moreover, it can be proved that the exchange ⌬terms represented by Figs. 2共f兲and 2共g兲are also zero in nuclear matter. In the case of finite nuclei, there exists however a contribution coming from the direct ⌬term, which, although small, has been included in the calculations presented in following section. Using Eq. 共2兲the particle-hole matrix element of the nonrelativistic ⌬current in nuclear matter results jnr ⌬共p,h兲⫽⫺i ␦ tpth ␶ h 4 9 G1 2mN f ␲ N⌬ m ␲ f m ␲ 冕 d3k 共2 ␲ 兲3 ␪ 共kF⫺k兲 ⫻ 兵 G⌬共k⫹q兲B共h⫺k兲spsh ⫹G⌬共k⫺q兲B共p⫺k兲spsh 其 ,共3兲 where ␶ h⫽1ifhis a proton and ⫺1 if neutron, and we have defined the function B共p⫺k兲ss⬘⫽关 ␴ ss⬘•共p⫺k兲兴共p⫺k兲⫹共p⫺k兲2 ␴ ss⬘ 共p⫺k兲2⫹m ␲ 2.共4兲 If the kdependence of the ⌬propagator is neglected, then the integral in Eq. 共3兲can be solved analytically1共see Ref. 关17兴for its explicit form兲. The relativistic expression for the currents can be found in Ref. 关38兴and yields results which differ significantly from the nonrelativistic ones even if the momentum transfer is as low as 500 MeV/c. This difference is due in part to the relativistic kinematics 共RK兲, which can be easily implemented in the NR model by the replacement ␻ → ␻ (1⫹ ␻ /2mN)关32兴, with the exception of the electromagnetic form factors, that 1Note that in this section we are not including ␲ N form factors in order to obtain an analytical result for numerical convenience. In the following section, they will be properly included. FIG. 2. Exchange MEC diagrams contributing to the one-particle emission responses computed in this work. In the vertex-type isobar diagrams 关共d兲,共e兲兴, different prescriptions for the dynamical ⌬propagator are discussed in the text. SEMIRELATIVISTIC MESON-EXCHANGE CURRENTS IN . . . PHYSICAL REVIEW C 68, 014604 共2003兲 014604-3 must be computed for the unshifted value of ␻ . Moreover, the NR results can be brought closer to the relativistic ones if the SR form of the current operators jT,SR a共p,h兲⬅1 冑 1⫹ ␶ jT,nr a共p,h兲共5兲 is employed for a⫽OB,C,P,⌬, with ␶ ⫽ 兩 Q2 兩 /4mN 2. Equation 共5兲has been obtained 共see Refs. 关31,36,38兴for details兲by a direct Pauli reduction, expanding in powers of h/mNto first order the OB current 关30兴and to leading order the MEC of pionic and contact types 关36,37兴. In this paper, we use a similar ‘‘factorized’’ expression for the SR ⌬current, which was already proposed in Ref. 关38兴. The ␶ -dependent factor in Eq. 共5兲arises from the spinology and produces a reduction of the responses. Let us now discuss the ⌬propagator G⌬, appearing in the ⌬current, Eqs. 共2兲and 共3兲. In a fully relativistic theory, G⌬ is given by the Rarita-Schwinger tensor 关38兴. The possible decay of the isobar into an N ␲ state is accounted for by the substitution of the ⌬mass m⌬→m⌬⫺(i/2)⌫(P2) in the denominator of the ⌬propagator, where the function ⌫(P2)is the width of the resonance 共see Refs. 关42,43兴兲. The nonrelativistic version of this propagator is defined by the positive energy sector for spatial indices and small momenta 关38兴 Gij ⌬共P兲⯝P”⫹m⌬ P2⫺m⌬ 2 冉 ␦ ij⫹1 3 ␥ i ␥ j 冊 →G⌬共P兲 冉 2 3 ␦ ij⫺i 3 ⑀ ijk ␴ k 冊 .共6兲 Note that Figs. 1共d兲–1共g兲correspond to different momenta in the propagators, namely, G⌬(P1⫹Q) and G⌬(P1 ⬘⫺Q): these, referred to as ⌬excitation (G⌬ I) and ⌬deexcitation (G⌬ II) in Refs. 关21,25兴, denote the propagator for a ⌬created after and before photo-absorption, respectively. The static limit approximation to G⌬involves taking P”⫹m⌬ P2⫺m⌬ 2⯝mN ␥ 0⫹m⌬ P2⫺m⌬ 2→mN⫹m⌬ mN 2⫺m⌬ 2⫽1 mN⫺m⌬ ,共7兲 where we have used P2⯝mN 2for P⫽P1⫹Qor P⫽P1 ⬘⫺Q and q, ␻ small, since P1and P1 ⬘are the four-momenta of the initial and final nucleons. In this case, the two propagators are equal, i.e., G⌬(P1⫹Q)⫽G⌬(P1 ⬘⫺Q)⫽(mN⫺m⌬)⫺1 and can be factorized out in Eq. 共2兲, thus yielding the traditional form of the nonrelativistic current 关17,23兴. The effects introduced by relativity can be appreciated in Fig. 3. Here, we show the inclusive transverse response of 40Ca for two values of the momentum transfer, q ⫽0.5 GeV/c共top panel兲and q⫽1 GeV/c共bottom panel兲. We use the Fermi momentum kF⫽237 MeV/cand the Galster parametrization of the nucleon form factors 关44兴. For the ⌬current, we use the electric form factor of the proton. For comparison with the nonrelativistic result we use strong form factors F ␲ NN⫽F ␲ N⌬⫽1. The fully relativistic calculation performed within the RFG model of Ref. 关38兴共thick solid line兲is compared with various nonrelativistic approaches: the traditional NR model, the NR model but including RK, and the results corresponding to the semirelativistic currents, using in addition relativistic kinematics 共denoted as SR0兲. Apart from the RFG, where the Rarita-Schwinger propagator is used including the ⌬width, the other results have been evaluated employing the static limit approximation for the ⌬ propagator, for which 冑 s⫽mN, and therefore ⌫⌬(s)⫽0, since sis below the threshold mN⫹m ␲ . Note the crucial role played by the relativistic kinematics and, moreover, how the SR approach, compared to the RK case, improves the agreement with the RFG results. Next, we study the effect of including a dynamical ⌬ propagator within the SR model. This can be done by using different approximations for the propagators G⌬(K⫹Q) and G⌬(K⫺Q) appearing in the nonrelativistic current, Eq. 共3兲, under the requirement that the new dynamical propagator be independent of k: only in this case it can be easily introduced into nonrelativistic calculations. The results for the ⌬-OB transverse response computed within the SR model, and using different prescriptions for the ⌬propagator, are displayed in Fig. 4. As in the previous figure and for comparison, we present the fully relativistic results 共RFG兲and the SR approach using the static limit for the ⌬propagator 共SR0兲. In all the cases, relativistic kinematics is assumed. The various approaches considered to describe the dynamical ⌬propagator are listed below and discussed in some detail. FIG. 3. Contribution of ⌬to the transverse response function for q⫽500 and 1000 MeV/c. The results corresponding to the RFG are compared with the NR Fermi gas, including RK, and using the SR0 approach for the electromagnetic currents. AMARO, BARBARO, CABALLERO, AND KAZEMI TABATABAEI PHYSICAL REVIEW C 68, 014604 共2003兲 014604-4 共1兲The propagator is written, as suggested in Refs. 关26,28兴,as G⌬共P兲⫽1 冑 s⫺m⌬⫹i 2⌫⌬共s兲 ,共8兲 where s⫽P2and 冑 sis the available energy in the ⌬rest system. The approximate values of sare obtained by neglecting the momentum kand kinetic energy ⑀ kcompared with q and the nucleon mass mN, i.e., sI⬅共K⫹Q兲2⫽共Ek⫹ ␻ 兲2⫺共k⫹q兲2⯝共mN⫹ ␻ 兲2⫺q2, 共9兲 sII⬅共K⫺Q兲2⯝共mN⫺ ␻ 兲2⫺q2.共10兲 As we observe in Fig. 4, this prescription 共denoted as SR1兲 produces a hardening of the response function, which is more pronounced for high q, and underestimates the exact result at the peak. 共2兲This second prescription is obtained, as suggested in Refs. 关25,28兴, by using the static approximation in the deexcitation diagram, Fig. 2共e兲, i.e., sII⯝mN 2, and a dynamical propagator in the ⌬excitation diagram, Fig. 2共d兲. In this case, we use the same value sIas in Eq. 共9兲. As shown in Fig. 4, this prescription 共labeled as SR2兲still produces a hardening of the response, but now the response function is larger than the relativistic one. We also note that the SR2 curve crosses the static SR0 result precisely at the position of the quasielastic peak 共QEP兲. In fact, this is determined by (mN ⫹ ␻ )2⫺q2⫽mN 2, hence Eq. 共9兲gives the static result, 冑 sI ⫽mN. 共3兲From Eq. 共6兲we may write P”⫹m⌬ P2⫺m⌬ 2⯝E ␥ 0⫺q• ␥ ⫹m⌬ 冑 s⫹m⌬ 1 冑 s⫺m⌬ 共11兲 for P⫽K⫹Q. The result given by Eq. 共8兲is reobtained when the first term in the right hand side of Eq. 共11兲is close to one. Note that for qlarge, 冑 scan be significantly different from E, while one cannot neglect the term q• ␥ . This would give an additional spin-dependent term and the propagator would get tangled. Instead, a possible way to proceed is by taking partially into account the qdependence in the numerator in the way: G⌬共K⫹Q兲⯝mN⫹ ␻ ⫹q⫹m⌬ 冑 sI⫹m⌬ 1 冑 sI⫺m⌬ ,共12兲 G⌬共K⫺Q兲⯝mN⫹ ␻ ⫺q⫹m⌬ 冑 sII⫹m⌬ 1 冑 sII⫺m⌬ ,共13兲 where sI,II are given in Eqs. 共9兲and 共10兲. This procedure allows us to write down an approximate diagonal expression for the ⌬propagator, and the corresponding results are denoted in Fig. 4 as SR3. The reduction of the response function is shown not to be as large as in the SR1 case. Summarizing, from Fig. 4 we may conclude that none of the above approximations proposed for the dynamical ⌬ propagator is entirely satisfactory. In fact, we observe that the best result, compared with the fully relativistic RFG one, corresponds to the static limit approximation. To complete this discussion, we note that the role of the width ⌫in the SR models of Fig. 4 is irrelevant, since the approximate values taken for 冑 sare far below the pole for these kinematics. In fact, for the prescription SR1 we have, from Eq. 共9兲, ␻ ⫽ 冑 sI⫹q2⫺mN.共14兲 This means that for q⫽1 GeV/cand 冑 sI⫽m⌬, we get ␻ ⬃649 MeV, which is well above the allowed region of the QEP, while the threshold 冑 sI⫽mN⫹m ␲ is reached for ␻ ⬃531 MeV, below which the ⌬width is zero. Hence, only the small tail of the ⌬close to the threshold is being considered in this region of energy. Actually in the relativistic model, things are totally different because the pole is reached inside the allowed energy region. Indeed, the delicate energy balance in the denominator makes the value of the inner momentum kto play a role. Instead of Eq. 共14兲, a better approximation to the exact relation is represented by ␻ ⫽ 冑 sI⫹共q⫹k兲2⫺mN.共15兲 In this case, the pole is first reached at FIG. 4. Contribution of ⌬to the transverse response function, using different prescriptions for the ⌬propagator, discussed in the text. SEMIRELATIVISTIC MESON-EXCHANGE CURRENTS IN . . . PHYSICAL REVIEW C 68, 014604 共2003兲 014604-5 ␻ ⫽ 冑 m⌬ 2⫹共q⫺kF兲2⫺mN,共16兲 which gives ␻ ⬃511 MeV, close to the peak for q ⫽1 GeV/c. Thus, we have considered two new prescriptions, denoted as SR4 and SR5. To help the reader they are shown in a separate figure 共Fig. 5兲, where they are again compared with the fully relativistic calculation 共RFG兲and the static limit approach 共SR0兲. These new prescriptions are based on the value assigned to 冑 s, such that the position of the pole be closer to the relativistic case. 共4兲More precisely, one considers 冑 sI⫽ 冑 sNN⫺mN,sNN being the invariant energy of the two outgoing nucleons, suggested in Refs. 关26,28兴for two-nucleon knockout, and apparently applied also to one-nucleon emission in Ref. 关25兴. The meaning of sNN for this case is doubtful, since there is only one particle in the final state. However, from the ⌬-excitation diagram, Fig. 2共d兲, we see that k,hare two entering momenta and p,kthe two exiting ones. Hence, the prescription SR4 in Fig. 5 is based on the approximation sNN⯝共P⫹K兲2⫽共Eh⫹Ek⫹ ␻ 兲2⫺共h⫹k⫹q兲2 ⯝共2mN⫹ ␻ 兲2⫺q2.共17兲 In this case, the pole for q⫽1 GeV/cis reached at ␻ ⯝513 MeV, inside the QE region. However, from the results in Fig. 5, this prescription appears to be less satisfactory than the previous ones. 共5兲Finally, the prescription SR5 is obtained by exploiting Eq. 共16兲that gives the right position at which the pole is first reached and by defining sI⫽共mN⫹ ␻ 兲2⫺共q⫺kF兲2,共18兲 sII⫽共mN⫺ ␻ 兲2⫺共q⫺kF兲2.共19兲 The results show that for q⫽1 GeV/cthe pole is reached at the right position close to the peak of the response. The change of sign of the response is a consequence of the change of sign of the ⌬propagator when one crosses the pole. From Fig. 5 we observe that it is not possible to recover the exact relativistic result by using a ⌬propagator independent of k, even if it reaches the pole 共in absence, of course, of the ⌬width兲inside the allowed region. In fact, the present singularity can only be smoothed, even in the presence of the ⌬width, after an integration over kis performed, as in the relativistic model. In general, we conclude that, if a dynamical propagator has to be used, like in the semirelativistic approaches discussed here, a smoothed form not hitting the pole is needed in order not to deviate too much from the exact result for the response function. Among the prescriptions analyzed in this section, the static form of the ⌬propagator produces, in spite of its simplicity, the best agreement with the RFG results in the case of one-particle knockout sector once the relativistic kinematics and the semirelativistic corrections of the currents are used. Therefore, a procedure to dynamize the ⌬propagator, as the ones presented here, and its use within the context of a nonrelativistic or a semirelativistic distorted-wave analysis of quasielastic (e,e⬘) and (e,e⬘p) reactions appears not to give better results than the static approximation. Obviously, this conclusion does not affect the validity of the approximations discussed in Ref. 关28兴for the two-particle emission channel, where the kinematical conditions are completely different. In the following section, we apply the present semirelativistic model of MEC to the exclusive (e,e⬘p) response functions of nuclei. III. EXCLUSIVE „e,e⬘p…OBSERVABLES In this section, we compute the quasielastic (e,e⬘p) response functions for intermediate to high momentum transfer values within the context of the SR approach, introduced in the preceding section. As shown in the case of quasielastic inclusive (e,e⬘) responses 关30,31,37,38兴, accounting for relativistic effects requires at least to treat properly the kinematics and relativistic factors in the currents. Referring to the ⌬current, we make use of the static limit approach for the ⌬ propagator, as this gives rise to the best agreement with the fully relativistic calculation in the inclusive channel, as shown in the preceding section. As a complete relativistic analysis of all the MEC (P,C, and ⌬) currents in (e,e⬘p) processes is still lacking, the use of dynamized ⌬propagators within existing NR or SR descriptions of the reaction mechanism does not appear to be well founded and, moreover, these may lead to large discrepancies with the exact calculation. The general formalism for (e,e⬘p) reactions has been presented in detail in Refs. 关2–4,45兴and we refer to them for FIG. 5. Contribution of ⌬to the transverse response function using different prescriptions for the ⌬propagator is discussed in the text. AMARO, BARBARO, CABALLERO, AND KAZEMI TABATABAEI PHYSICAL REVIEW C 68, 014604 共2003兲 014604-6 specifications on the kinematics. Assuming plane waves for the incoming and outgoing electron 共treated in the extreme relativistic limit兲and parity conservation, the exclusive cross section can be written in the form d5 ␴ d ⑀ ⬘d⍀e ⬘d⍀p⬘ ⫽K ␴ M共vLWL⫹vTWT⫹vTLWTLcos ␾ ⬘ ⫹vTTWTTcos2 ␾ ⬘兲,共20兲 where ⑀ ⬘and ⍀e ⬘are the energy and solid angle corresponding to the scattered electron and ⍀p⬘⫽( ␪ ⬘, ␾ ⬘) is the solid angle for the ejected proton with four-momentum P⬘ ␮ ⫽(E⬘,p⬘). In Eq. 共20兲K⫽2p⬘mN/(2 ␲ ប)3and ␴ Mis the Mott cross section. Finally, v ␣ are the electron kinematical factors given in Refs. 关32,45兴. The labels Land Trefer to the longitudinal and transverse projections of the current matrix elements with respect to the virtual photon direction, respectively. The hadronic content of the problem enters via the response functions W ␣ , which are obtained by taking the appropriate components of the hadronic tensor W ␮ ␯ ⫽1 K兺 msM ␣ 具 p⬘ms,⌽ ␣ (A⫺1) 兩 J ˆ ␮ 共Q兲 兩 ⌽0 (A) 典 * ⫻ 具 p⬘ms,⌽ ␣ (A⫺1) 兩 J ˆ ␯ 共Q兲 兩 ⌽0 (A) 典 ,共21兲 where a sum over undetected final polarization states is performed. In Eq. 共21兲,J ˆ ␮ (Q) is the nuclear current operator and we assume the initial state 兩 ⌽0 A 典 to correspond to a spinzero nuclear target in its ground state with energy E0 (A). The final state 兩 p⬘ms,⌽ ␣ (A⫺1) 典 is assumed to behave asymptotically as a knockout nucleon with momentum p⬘and spin quantum number msand a residual nucleus left in a bound state, i.e. 兩 ⌽ ␣ (A⫺1) 典 ⫽ 兩 J ␣ ,M ␣ 典 , with energy E ␣ (A⫺1) . In the present distorted-wave analysis of (e,e⬘p) reactions, the outgoing nucleon state is described by a wave function solution of the Schro ¨dinger equation with an optical potential Vopt fitted to elastic nucleon-nucleus scattering data. The matrix elements of the hadronic tensor are computed by performing a multipole expansion of both the distorted nucleon wave and the current operators in terms of the usual Coulomb CJ, electric EJ, and magnetic MJmultipoles 共see Refs. 关10,32,33兴for details on the model and Refs. 关17,24,32兴for explicit expressions of the multipole matrix elements of the currents兲. The electromagnetic form factors used in the contact and pionic MEC are given in Ref. 关24兴, while for ⌬we use the same form factor as the electromagnetic proton form factor. In this section, we include monopole form factors in the MEC F ␲ NN(K)⫽F ␲ N⌬(K)⫽(⌳2⫺m ␲ 2)/(⌳2⫺K2) with ⌳ ⫽1300 MeV. For low qthese form factors are small 关24兴, but for high values of q共close to 1 GeV, as here兲they can reduce the MEC contribution of about 25%. Note that the ⌬-current contributions can also be sensitive to the ⌬-nucleus potential in the medium 共see Ref. 关21兴兲. Indeed, in-medium modifications of the ⌬and ␲ propagators have been studied in inclusive (e,e⬘) scattering in the region of the ⌬peak and for pion electroproduction 关46兴, but not, to our knowledge, for the MEC in the QEP. A rough estimate of these effects can be performed by adding a constant ⌬-nucleus optical potential V⌬⫽⫺30⫺40iMeV 关47兴in the denominator of the static ⌬propagator: since V⌬turns out to affect our results very little, we have accordingly neglected it. To illustrate the role of the relativistic corrections we display in Fig. 6 the four unpolarized exclusive response functions for proton knockout from the 1p1/2 shell in 16O leading to the residual nucleus 15N. The kinematics selected corresponds to (q, ␻ )-constant kinematics 共sometimes also referred to as quasiperpendicular kinematics兲and the values q⫽460 MeV/cand q⫽995 MeV/chave been chosen. In each case, the selected value of the transferred energy ␻ almost corresponds to the quasielastic peak value, i.e., ␻ ⫽100 MeV and 439 MeV, respectively. FSI are taken into account for q⫽460 MeV/cthrough the optical potential of Comfort and Karp 关48兴, which is appropriate for proton kinetic energies below 183 MeV. In the case of q ⫽995 MeV/c, as the proton kinetic energy is ⬃430 MeV, we use instead a Schr ¨odinger-equivalent form of the relativistic global optical potential of Ref. 关49兴. Thus, the nonrelativistic wave functions correspond to the upper components of the relativistic ones, containing the Darwin term. Results obtained within this approach were compared with a fully relativistic calculation in the impulse approximation 关6兴, i.e., without including the effects of MEC. In each panel of Fig. 6 we compare the results corresponding to the traditional nonrelativistic model with NR kinematics, including RK, and finally, the SR approach discussed in the preceding section. All the calculations include MEC. From these results it emerges that, particularly for high q, the relativistic kinematics plays a crucial role in describing properly the form of the momentum distribution. Note that the allowable missing momentum values are determined by the relation 兩 p⬘⫺q 兩 ⭐p⭐p⬘⫹q, with p⬘fixed by energy conservation. Assuming nonrelativistic kinematics, the value of p⬘is given by p⬘⫽ 冑 2mN共 ⑀ h⫹ ␻ 兲,共22兲 while for relativistic kinematics it results p⬘⫽ 冑 共mN⫹ ⑀ h⫹ ␻ 兲2⫺mN 2⫽ 冑 2mN共 ⑀ h⫹ ␻ 兲 冉 1⫹ ⑀ h⫹ ␻ 2mN 冊 . 共23兲 In Eqs. 共22兲and 共23兲, the energy ⑀ hrepresents the 共negative兲 energy of the bound nucleon. Hence in the case of RK we solve the Schro ¨dinger equation with equivalent energy ( ⑀ h ⫹ ␻ )关1⫹( ⑀ h⫹ ␻ )/2mN兴for the ejected proton instead of the NR energy ⑀ h⫹ ␻ . Once relativistic kinematics is selected, the use of SR currents produces an enhancement of the Land TL responses and a reduction of the Tand TT ones, whose magnitude increases with q. These effects are connected with the factors ␬ / 冑 ␶ ⬎1 and 冑 ␶ / ␬ ⯝1/(1⫹ ␶ ) that enter in the SR expres- SEMIRELATIVISTIC MESON-EXCHANGE CURRENTS IN . . . PHYSICAL REVIEW C 68, 014604 共2003兲 014604-7 sions of the longitudinal and transverse currents, respectively 关31,37兴. It is important to point out the particularly large relativistic enhancement observed for the interference TL response. This is connected to the spin-orbit term 关32兴con- tained in the SR charge density, but neglected within the NR approach. As shown in Ref. 关32兴, the interference between the spin-orbit term and the magnetization current gives rise to a contribution in WTL which is of the same order of magnitude as the interference between the charge density and convection current. Thus the presence of the spin-orbit term in the density is essential to describe properly the response WTL, even for moderate qvalues. Analogous results hold for a proton knockout from the p3/2 shell in 16O: the role of relativity in each response is similar to the one observed for the p1/2 case and, again, the most sensitive response to relativistic effects is WTL. However, the relativistic enhancement for the p3/2 orbit is somewhat smaller than the one presented for p1/2 , particularly for large q. The separate contributions of the MEC is presented in Figs. 7 and 8 for p1/2 and p3/2 shells, respectively. Kinematics is as in the previous figure and we do not show results for the pure longitudinal response as it is not affected by MEC within the present approaches. We compare the OB responses 共dotted line兲with the results obtained when including the contact Ccurrent 共dashed line兲, the contact and pionin-flight (C⫹P) currents 共dot-dashed line兲, and the contact, pion-in-flight, and ⌬(C⫹P⫹⌬) currents 共solid line兲. Let us analyze each response separately. In the case of WT, the contact current produces an increase which is partially canceled by a small reduction introduced by the pionic current. Note that the role of the Cand Pcurrents is negligible for large q. The ⌬current gives rise to an additional reduction whose relative magnitude is rather similar for both shells. The total effect of C⫹P⫹⌬MEC is a reduction of the T response, which is slightly more important for high q. Although a great caution should be taken in extending the conclusions drawn from the analysis of exclusive reactions to inclusive ones or vice versa, it is illustrative to discuss the results relative to quasielastic (e,e⬘p) reactions in connection with the quasielastic (e,e⬘) ones. Note that, apart from the potentials usually considered in both types of reactions, the inclusive responses are calculated by integrating the exclusive ones over the ejected nucleon variables, and summing over all occupied hole states, i.e., including also the contribution given by the 1s1/2 shell. In Refs. 关31,37,38兴we evaluated the role of MEC for the inclusive Tresponse in a relativistic Fermi gas model. The contact Ccontribution was FIG. 6. Exclusive response functions for proton knockout from the 1p1/2 shell in 16O. Two (q, ␻ )-constant kinematical situations are selected. The traditional NR results are compared with the responses obtained including RK and using the SR form of the currents. All the curves include MEC. AMARO, BARBARO, CABALLERO, AND KAZEMI TABATABAEI PHYSICAL REVIEW C 68, 014604 共2003兲 014604-8 found to be larger than the pion-in-flight term P, a dominance increasing with q. The total (C⫹P) contribution presents an oscillatory behavior with respect to the transferred energy ␻ , having a node close to the QEP value, particularly for high q. For q⫽1 GeV/c, this node is reached for ␻ ⬃435 MeV, which is very close to the value ␻ ⫽439 MeV selected in the (e,e⬘p) calculations 共right hand panels in Figs. 7 and 8兲. On the contrary, in the case q⫽500 MeV/c, the contribution of the C⫹Pcurrents in the inclusive Tresponse presents a node for ␻ ⬃150 MeV and is positive and non-negligible for the value ␻ ⫽100 MeV selected here in the case of the (e,e⬘p) process. A similar discussion can also FIG. 7. Separate contribution of each one of the MEC to the exclusive response functions. SR results are shown for knockout from the 1p1/2 shell of 16O for the same kinematics as in Fig. 6, including the OB current only, and in addition the contact C, pionic P, and ⌬ currents in the calculation. FIG. 8. Same as Fig. 7 for knockout from the 1p3/2 shell of 16O. SEMIRELATIVISTIC MESON-EXCHANGE CURRENTS IN . . . PHYSICAL REVIEW C 68, 014604 共2003兲 014604-9