Inelastic electron-nucleus scattering and scaling at high inelasticity

Inelastic electron-nucleus scattering and scaling at high inelasticity M. B. Barbaro,1J. A. Caballero,2T. W. Donnelly,3and C. Maieron2,4 1Dipartimento di Fisica Teorica, Università di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy 2Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, Apartado Postal 1065, E-41080 Sevilla, Spain 3Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4INFN, Sezione di Catania, Via S. Sofia 64, 95123 Catania, Italy (Received 24 November 2003; published 24 March 2004) Highly inelastic electron scattering is analyzed within the context of the unified relativistic approach previously considered in the case of quasielastic kinematics. Inelastic relativistic Fermi gas modeling that includes the complete inelastic spectrum—resonant, nonresonant, and deep inelastic scattering—is elaborated and compared with experimental data. A phenomenological extension of the model based on direct fits to data is also introduced. Within both models, cross sections and response functions are evaluated and binding energy effects are analyzed. Finally, an investigation of the second-kind scaling behavior is also presented. DOI: 10.1103/PhysRevC.69.035502 PACS number(s): 25.30.Fj, 24.10.Jv, 13.60.Hb I. INTRODUCTION In this work we consider highly inelastic electron scattering and compare its analysis with the case of quasielastic (QE)electron scattering. The latter is dominated by the process where the exchanged virtual photon interacts with a nucleon in the nuclear ground state and ejects that nucleon, thereby forming a nuclear particle-hole excitation. Corrections to this dominant process involve going beyond the impulse approximation to account for two-body currents, finalstate interactions, and nuclear correlations. Although these contributions are known not to be entirely negligible [1–7] this simple process accounts for the basic feature seen in the vicinity of elastic scattering from a nucleon at rest, namely, the QE peak. Models such as those discussed below take into account the fact that the nucleons in the nucleus are moving and are bound and thereby produce a broad peak in the inelastic spectrum. In the present work our goal is to extend the analysis, still maintaining the same basic features of the relativistic modeling used for the QE region, and now focus on what we call highly inelastic scattering, or for brevity, simply the inelastic region. This includes everything that goes beyond the QE process: that is, whereas the QE process assumes elastic scattering from the nucleons, the inelastic process will assume inelastic e-Nscattering. For relatively low final-state invariant masses one lies in the region of resonance excitation and two cases of this sort have been explored in recent work [8,9]. In the present study these ideas are generalized to include the complete inelastic spectrum, both resonant and nonresonant, including deep inelastic scattering (DIS), within the context of the unified relativistic approach used in our previous work. Thus, in the present work our goal is to begin by exploring extensions of the relativistic Fermi gas (RFG)model [6,10,11]to an inelastic version of this approach. While this bears some connection with traditional convolution models for the high-energy response of nuclei (see, for example, Refs. [12–15]) it is not the same in that, albeit within a model, it correctly incorporates a specific relativistic nuclear spectral function into the problem, whereas some other approaches make additional assumptions and use only the integral of the spectral function, namely, the nuclear momentum distribution or make nonrelativistic approximations when dealing with the spectral function. This distinction can be seen quite clearly in studies of first- and second-kind scaling [11,16–20]and will not be elaborated here. Once the inelastic RFG modeling is in hand, it becomes clear that it might be useful to explore a phenomenological extension of this model, namely, what we call the extended relativistic Fermi gas (ERFG). In this approach we take the result of doing the correct integral over the nuclear spectral function (i.e., not the full integral, which is the momentum distribution, as alluded to above)directly from fits made previously to the data [21]. We shall see that this has a significant impact on the nuclear responses at high inelasticity.An issue which will also become clear later is that the story is not yet complete: in addition to the modeling done in the present work, where the focus is placed on incorporating inelastic effects at high energies, there are still other contributions that must be added. Specifically, in recent work [22] on 2p-2hmeson-exchange current effects it is seen that a significant incoherent contribution must be added to those explored here. Given that the work on 2p-2heffects is, as yet, incomplete—correlation contributions are presently being included—it is premature to make too much of comparisons with experimental data, and, as we remark later in the appropriate places, the final understanding of how all of the various reaction mechanisms enter, while becoming clearer is not yet achieved. The paper is organized as follows: in Sec. II we recall the general formalism for inelastic electron-nucleus scattering; in Sec. III we derive the expressions for the inelastic hadronic tensor in three different models: the pure RFG model [Sec. III A], the RFG including the effects of binding energy [Sec. III B]and the ERFG [Sec. III C]; in Sec. IV we present numerical results for cross sections [Sec. IV A], response functions [Sec. IV B], and scaling functions [Sec. IV C]and finally, in Sec. V we draw our conclusions. PHYSICAL REVIEW C 69, 035502 (2004) 0556-2813/2004/69(3)/035502(16)/$22.50 ©2004 The American Physical Society69 035502-1 II. INELASTIC ELECTRON-NUCLEUS SCATTERING: GENERAL FORMALISM With the goal outlined above in mind, we start by rewriting the general expressions that apply in both elastic and inelastic regimes. The general formalism describing inclusive electron-nucleus scattering processes is widely available [23–25]; here we simply focus on those aspects that are of special relevance for the discussion that follows. We follow the conventions and metric of Ref. [26]and use capital letters to refer to four-vectors. The incident and scattered electron four-momenta are denoted by Ki ␮ =共␧i,ki兲and Kf ␮ =共␧f,kf兲. The hadronic variables, PA ␮ =共MA,0兲and PB ␮ =共EB,pB兲represent the four momenta of the target and residual nucleus, respectively. The four-momentum transfer is given by Q ␮ =共 ␻ ,q兲(we assume the Born approximation, i.e., only one virtual photon exchanged in the process). Following standard procedures the differential cross section may be written d ␴ d⍀fd␧f=2 ␣ 2 Q4 ␧f ␧i ␩ ␮ ␯ W ␮ ␯ ,共1兲 where ␣ is the fine structure constant, ␩ ␮ ␯ is the leptonic tensor that can be evaluated directly using trace techniques 关27兴, and W ␮ ␯ is the hadronic tensor containing all of the nuclear structure and dynamics information. Assuming that the final state can be described in terms of a recoiling nuclear state 兩 ␺ B典plus a 共highly兲inelastic state 兩⌽X典, its general expression is given by W ␮ ␯ =兺 A兺 B兺 X具 ␺ B,⌽X兩J ˆ ␮ 共q兲兩 ␺ A典*具 ␺ B,⌽X兩J ˆ ␯ 共q兲兩 ␺ A典 ⫻ ␳ 共EB兲dEB ␳ 共EX兲dEX ␦ 共␧i−␧f+EA−EB−EX兲, 共2兲 where 兺 ¯ A共兺B兺X兲indicates the appropriate average 共sum兲 over initial 共final兲states. Here J ˆ ␮ 共q兲is the Fourier transform of the nuclear current operator evaluated, 兩 ␺ A典and 兩 ␺ B,⌽X典 represent the initial and final states, respectively, and the distribution functions ␳ 共EB兲and ␳ 共EX兲are introduced to account for the energy-momentum dispersion relation of the final nuclear 共B兲and hadronic 共X兲systems. In this work we assume that the inelasticity of the process is totally accounted for by the final state ⌽X; hence for the energy distribution function of the residual nuclear system we use ␳ 共EB兲= ␦ 共EB−E ¯ B兲, where E ¯ B=冑pB 2+共MB *兲2. Note that W ␮ ␯ in Eq. 共2兲is meant to be evaluated at pB+pX=q=ki−kf. The nuclear tensor can equivalently be expressed as an integral in the 共E,p兲plane, with −p=pBthe three-momentum of the recoiling daughter nucleus and E⬅冑p2+共MB *兲2 −冑p2+共MB 0兲2the excitation energy of the residual nucleus (see Ref. [19]). The domain of integration is the kinematically allowed region max关E共0兲,0兴艋E艋E共 ␲ 兲,共3兲 where E共 ␪ 兲=MA 0+ ␻ −冑共MB 0兲2+p2−冑WX 2+q2+p2+2pq cos ␪ 共4兲 with ␪ the angle between pand q, and where WXis the invariant mass of the final state. In the MB 0→⬁limit the above expression becomes E⬁共 ␪ 兲=mN+ ␻ ˜ −冑WX 2+q2+p2+2pq cos ␪ ,共5兲 where mNis the nucleon mass, ␻ ˜ ⬅ ␻ −ESand ES=MB 0+mN −MA 0is the separation energy. The upper curve E共 ␲ 兲crosses the paxis at p−=−yXand p+=YX, where yX=1 2W2关共MA 0+ ␻ 兲冑共W−MB 0兲2−WX 2 ⫻冑共W+MB 0兲2−WX 2−2q⌳X兴共6兲 and YX=1 2W2关共MA 0+ ␻ 兲冑共W−MB 0兲2−WX 2 ⫻冑共W+MB 0兲2−WX 2+2q⌳X兴,共7兲 and where W=冑共MA 0+ ␻ 兲2−q2and ⌳X=1 2关W2+共MB 0兲2−WX 2兴. 共8兲 The variable yXis the generalization of the usual y-scaling variable to the inelastic process where a resonance Xis produced. In the limit MB 0→⬁it reads yX,⬁=冑共 ␻ ˜ +mN兲2−WX 2−q.共9兲 Note that the allowed region decreases with WXand collapses to a point when −yX=YX, which implies W=MB 0+WX or, in the MB 0→⬁limit, 共yX,⬁兲min=−q, corresponding to 共WX兲max= ␻ ˜ +mN. Summarizing, for fixed four-momentum transfer, the resonant mass is limited to the range mN+m ␲ 艋WX艋mN+ ␻ −ES.共10兲 III. THE RELATIVISTIC FERMI GAS MODEL In this section we proceed by evaluating the hadronic nuclear tensor assuming the impulse approximation and by working within the framework of the RFG model. In this case, the virtual photon is absorbed by an on-shell nucleon described by a Dirac spinor u共h,sh兲, with energy E ¯ h =冑h2+mN 2. Integrating over the momenta in the Fermi sea, the following expression for the inelastic hadronic tensor results: BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-2 W ␮ ␯ 共q, ␻ 兲=3N 4 ␲ pF 3 冕 FdhmN E ¯ h 冕 dEX ␦ 共 ␻ +E ¯ h−EX兲 ⫻1 2兺 sh兺 Xi ␳ 共EXi兲关⌽ ¯ XiJ ˆ ␮ u共h,sh兲兴*关⌽ ¯ XiJ ˆ ␯ u共h,sh兲兴, 共11兲 where Nis the number of nucleons 共protons or neutrons兲and 兰Fdh⬅兰dh ␪ 共pF−h兲,pFbeing the Fermi momentum. The symbol 兰dEXstands for the integral over the energy of the inelastic final state, while 兺Xiindicates in general the sum/ integral over all the internal quantum numbers of all possible inelastic final states ⌽Xi, having total energy EXand total momentum pX, fixed by momentum conservation to be pX =h+q. The hadronic tensor in Eq. (11)for inelastic processes can be also written in the form Winel ␮ ␯ 共q, ␻ 兲=3N 4 ␲ pF 3 冕 dEX 冕 FdhmN E ¯ h winel ␮ ␯ 共H,Q,EX兲 ⫻ ␦ 共 ␻ +E ¯ h−EX兲,共12兲 where H ␮ =共E ¯ h,h兲and we have introduced the inelastic single-nucleon tensor winel ␮ ␯ 共H,Q,EX兲=1 2兺 sh兺 Xi ␳ 共EXi兲关⌽ ¯ XiJ ˆ ␮ u共h,sh兲兴* ⫻关⌽ ¯ XiJ ˆ ␯ u共h,sh兲兴.共13兲 Note that the above single-nucleon tensor has dimensions of E−1. As will be shown later, this is in contrast with our past work on QE and N→⌬scattering where the single-nucleon tensors were defined to be dimensionless. Next we choose to express the inelastic hadronic tensor in Eq. (12)in terms of the invariant mass WX, Winel ␮ ␯ 共q, ␻ 兲=3N 4 ␲ pF 3 冕 dWX 冕 FdhmNWX E ¯ hEX winel ␮ ␯ 共H,Q,EX兲 ⫻ ␦ 共 ␻ +E ¯ h−EX兲共14兲 with EX=冑pX 2+WX 2. The energy integral can be performed by exploiting the ␦ function, yielding Winel ␮ ␯ 共q, ␻ 兲=3N 4 ␲ pF 3 冕 FdhmN E ¯ h winel ␮ ␯ 共H,Q, ␻ +E ¯ h兲.共15兲 In the case of DIS on a single nucleon, the inelastic tensor simply reduces to the single-nucleon tensor winel ␮ ␯ . Before entering into a detailed analysis of the inelastic nuclear tensor, it is interesting to notice how the usual expressions for the QE and N→⌬hadronic tensors are recovered from the general result given in Eq. (11). First, in the case of QE scattering, the nuclear final state is simply a particle-hole excitation, hence, in the RFG model, ⌽Xdescribes an on-shell nucleon, namely, ⌽X=冑mN/E ¯ pu共p,sp兲. The energy distribution function is simply ␳ 共EX兲= ␦ 共EX −E ¯ p兲and the sum over the final states reduces to a sum over spin projections, 兺Xi=兺sp. The QE hadronic tensor then reads WQE ␮ ␯ 共q, ␻ 兲=3N 4 ␲ pF 3 冕 FdhmN 2 E ¯ hE ¯ p wQE ␮ ␯ 共H,Q兲 ␦ 共 ␻ +E ¯ h−E ¯ p兲, 共16兲 where wQE ␮ ␯ is the usual dimensionless QE single-nucleon tensor wQE ␮ ␯ =1 2兺 sh兺 sp 关u ¯ 共p,sp兲J ˆ ␮ u共h,sh兲兴*关u ¯ 共p,sp兲J ˆ ␯ u共h,sh兲兴. 共17兲 In the case of the transition N→⌬, the final state ⌽X, within the context of the RFG model, is an on-shell ⌬, namely, ⌽X=冑m⌬/E ¯ ⌬u⌬共p,s⌬兲, with on-shell energy E ¯ ⌬=冑p2+m⌬ 2. The energy distribution function in this case is ␳ 共EX兲 = ␦ 共EX−E ¯ ⌬兲and 兺Xi=兺s⌬. The N→⌬hadronic tensor that results is W⌬ ␮ ␯ 共q, ␻ 兲=3N 4 ␲ pF 3 冕 FdhmN 2 E ¯ hE ¯ ⌬ w⌬ ␮ ␯ 共H,Q兲 ␦ 共 ␻ +E ¯ h−E ¯ ⌬兲 共18兲 with w⌬ ␮ ␯ the dimensionless nucleon-⌬tensor w⌬ ␮ ␯ =m⌬ 2mN兺 sh兺 s⌬ 关u ¯ ⌬共p,sp兲J ˆ ␮ u共h,sh兲兴*关u ¯ ⌬共p,sp兲J ˆ ␯ u共h,sh兲兴. 共19兲 As expected, these expressions for the dimensionless singlenucleon tensors coincide with the ones introduced in Ref. 关8,28兴. Likewise for the Roper resonance the expressions obtained in Ref. 关9兴are recovered. A. The RFG inelastic nuclear tensor and response functions In this section we evaluate the inelastic nuclear tensor in the RFG framework. For convenience, as usual we first define the dimensionless variables ␬ ␮ =共␭, ␬ 兲= 冉 ␻ 2mN,q 2mN 冊 , ␶ = ␬ 2−␭2, ␩ F=pF mN, ⑀ F=冑1+ ␩ F 2, 共20兲 ␩ ␮ =共 ⑀ ¯ , ␩ 兲= 冉 E ¯ h mN,h mN 冊 , ␮ X=WX mN, ⑀ X=冑 ␮ X 2+共 ␩ +2 ␬ 兲2, in terms of which the hadronic tensor in Eq. (14)reads INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004) 035502-3 Winel ␮ ␯ 共 ␬ ,␭兲=3N 4 ␲ ␩ F 3 冕 d ␮ X 冕 d ␩ ␮ X ⑀ ¯ ⑀ Xwinel ␮ ␯ 共 ␩ , ␮ X; ␬ ,␭兲 ⫻ ␦ 共2␭+ ⑀ ¯ − ⑀ X兲 ␪ 共 ␩ F− ␩ 兲.共21兲 Before presenting the explicit results for the RFG response functions, let us discuss an important ingredient of the calculation, the single-nucleon inelastic hadronic tensor winel ␮ ␯ . For unpolarized scattering, the latter can be parametrized in terms of two structure functions, w1and w2, according to winel ␮ ␯ =−w1 冉 g ␮ ␯ + ␬ ␮ ␬ ␯ ␶ 冊 +w2共 ␩ ␮ + ␬ ␮ ␳ 兲共 ␩ ␯ + ␬ ␯ ␳ 兲. 共22兲 For on-shell nucleons, the structure functions w1and w2de- pend on two variables, the four-momentum transfer Q2and the invariant mass WXof the final state reached by the nucleon, or, equivalently, the single-nucleon Bjorken variable x=兩Q2兩 2H·Q=兩Q2兩 WX 2−mN 2−Q2= ␶ ␩ · ␬ .共23兲 In our formalism it is convenient to introduce the inelasticity parameter [8,29] ␳ ⬅1+ 1 4 ␶ 共 ␮ X 2−1兲,共24兲 the value unity corresponding to elastic scattering. Note that ␳ is simply linked to the Bjorken scaling variable of the on-shell nucleon moving inside the target nucleus by the relation ␳ =1/x, thus in the following we will use ␳ as argument of the structure functions w1,w2. In presenting our results we will also use the “laboratory” Bjorken variable xL=兩Q2兩 2mN ␻ = ␶ ␭,共25兲 corresponding to a single nucleon at rest in the laboratory frame. Let us now return to the inelastic nuclear tensor of Eq. (21): after performing the polar angular integration by means of the energy-conserving ␦ function one gets Winel ␮ ␯ 共 ␬ ,␭兲=3N ␶ 2 ␩ F 3 ␬ 冕 0 2 ␲ d⌽ 2 ␲ 冕 ␳ 1共 ␬ ,␭兲 ␳ 2共 ␬ ,␭兲d ␳ 冕 ⑀ 0共 ␳ 兲 ⑀ Fd ⑀ ¯ ⫻winel ␮ ␯ 共 ⑀ ¯ , ␪ 0, ␳ ; ␬ ,␭兲,共26兲 where cos ␪ 0=1 ␬ ␩ 共␭ ⑀ ¯ − ␶ ␳ 兲.共27兲 The condition 兩cos ␪ 0兩艋1 fixes the integration limits over ⑀ ¯ : ⑀ ¯ 艌 ⑀ 0共 ␳ 兲⬅ ␬ 冑1 ␶ + ␳ 2−␭ ␳ .共28兲 Moreover, by requiring that ⑀ 0共 ␳ 兲艋 ⑀ Fand that the resonance mass is above the pion-production threshold (i.e., ␮ X 艌 ␮ thresh⬅1+ ␮ ␲ )the following region is obtained for the integration over ␳ : 关 ␳ 1共 ␬ ,␭兲, ␳ 2共 ␬ ,␭兲兴 = 冋 max 再 ␭ ⑀ F− ␬ ␩ F ␶ , ␳ thresh 冎 ,␭ ⑀ F+ ␬ ␩ F ␶ 册 共29兲 with ␳ thresh =1+ ␮ ␲ 共 ␮ ␲ +2兲 4 ␶ .共30兲 Note that the upper integration limit ␳ 2共 ␬ ,␭兲always lies below the cutoff corresponding to Eq. 共10兲. Indeed using Eqs. 共10兲and 共24兲and keeping in mind that the 共negative兲separation energy of the RFG is ES RFG=−TF⬅mN共1− ⑀ F兲, the maximum ␳ allowed by Eq. 共10兲reads ␳ max RFG =1+ 1 4 ␶ 关共2␭+ ⑀ F兲2−1兴= ␳ 2共 ␬ ,␭兲+1 4 ␶ 共2 ␬ − ␩ F兲2, 共31兲 ␳ 2therefore resulting in the more stringent integration limit. Now by writing the single-nucleon inelastic tensor in terms of structure functions w1and w2as in Eq. (22)and choosing the zdirection along q, the integration over ⌽and ⑀ ¯ can be performed analytically (see Appendix A)and the hadronic inelastic tensor can be expressed in the general form Winel ␮ ␯ 共 ␬ ,␭兲=3N ␶ 2 ␩ F 3 ␬ ␰ F 冕 ␳ 1共 ␬ ,␭兲 ␳ 2共 ␬ ,␭兲d ␳ 共1− ␺ X 2兲 ␪ 共1− ␺ X 2兲 ⫻U ␮ ␯ 共 ␬ , ␶ , ␳ 兲,共32兲 where ␰ F= ⑀ F−1 is the Fermi kinetic energy and the inelastic scaling variable ␺ X⬅sgn共␭− ␶ ␳ 兲冑 ⑀ 0共 ␳ 兲−1 ⑀ F−1 共33兲 has been defined. For each value of ␳ 共and hence ␮ X兲a “peak” can thus be identified, corresponding to the region −1艋 ␺ X艋1, centered at ␺ X=0, ␭P= ␶ P ␳ =1 2 ␳ 共冑1+4 ␬ P 2 ␳ 2−1兲, ␬ P=冑 ␶ P共1+ ␶ P ␳ 2兲,共34兲 whose width ⌬␭ =1 2关冑共2 ␬ + ␩ F兲2+ ␮ X 2−冑共2 ␬ − ␩ F兲2+ ␮ X 2兴⯝ 2 ␬ ␩ F 冑4 ␬ 2+ ␮ X 2 共35兲 is a function that grows with ␬ and decreases with ␮ X. BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-4 The general expression for the tensor U ␮ ␯ is derived in Appendix A. Here we only report the longitudinal and transverse components UL=U00 = ␬ 2 ␶ 关共1+ ␶ ␳ 2兲w2共 ␶ , ␳ 兲−w1共 ␶ , ␳ 兲 +w2共 ␶ , ␳ 兲D共 ␬ , ␶ , ␳ 兲兴,共36兲 UT=U11 +U22 =2w1共 ␶ , ␳ 兲+w2共 ␶ , ␳ 兲D共 ␬ , ␶ , ␳ 兲,共37兲 which are linked to the longitudinal and transverse response functions by the following relations: Rinel L,T共 ␬ , ␶ 兲=3N ␶ 2 ␩ F 3 ␬ ␰ F 冕 −1 ␺ X max共 ␬ ,␭兲d ␺ X冏 ⳵␳ ⳵ ␺ X冏共1− ␺ X 2兲 ⫻UL,T„ ␬ , ␶ , ␳ 共 ␺ X兲…,共38兲 with ␺ X max共 ␬ ,␭兲= min 冦 1, 冢 ␬ 冑1 ␶ + ␳ thresh 2−␭ ␳ thresh −1 ␰ F 冣 1/2 冧 共39兲 and ⳵␳ ⳵ ␺ X=−冑2 ␰ F ␬ 共1+ ␰ F ␺ X 2兲−␭ ␺ X冑2 ␰ F 冉 1+1 2 ␰ F ␺ X 2 冊 ␶ 冑1+1 2 ␰ F ␺ X 2 . 共40兲 In Eqs. 共36兲and 共37兲the function D共 ␬ , ␶ , ␳ 兲=1 ⑀ F− ⑀ 0共 ␳ 兲 冕 ⑀ 0共 ␳ 兲 ⑀ Fd ⑀ ¯ 冕 0 2 ␲ d⌽ 2 ␲ 共 ␩ ⫻ ␬ ˆ兲2 = ␶ ␬ 2 再 1 3关 ⑀ F 2+ ⑀ F ⑀ 0共 ␳ 兲+ ⑀ 0共 ␳ 兲2兴 +␭关 ⑀ F+ ⑀ 0共 ␳ 兲兴 +␭2 冎 −共1+ ␶ 兲+共 ␳ −1兲 ⫻ ␶ ␬ 2兵␭关 ⑀ F+ ⑀ 0共 ␳ 兲兴 − ␶ 共 ␳ +1兲其 = ␰ F共1− ␺ X 2兲 冋 1+ ␰ F ␺ X 2−␭ ␬␺ X冑 ␰ F共2+ ␰ F ␺ X 2兲 + ␶ 3 ␬ 2 ␰ F共1− ␺ X 2兲 册 共41兲 arises from the Fermi motion and goes to zero as ␰ F→0; being proportional to ␰ F⬵ ␩ F 2/2Ⰶ1, this provides relatively moderate corrections to the rest of the contributions in Eqs. 共36兲and 共37兲. The value ␳ =1 corresponds to QE kinematics: in this case the well-known expressions for the QE responses are recovered. The “total” observables are then obtained by adding the usual RFG QE response to the inelastic results: Rtot L,T=RQE L,T+Rinel L,T.共42兲 In the deep inelastic regime it is customary to deal with nuclear structure functions W1,2 Aand/or F1,2 A. These can be expressed in terms of the longitudinal and transverse response functions through the following relations: W1 A=1 2RT,共43兲 W2 A= 冉 ␶ ␬ 2 冊 2RL+1 2 ␶ ␬ 2RT共44兲 and F1 A=mNW1 A,共45兲 F2 A=2mN␭W2 A.共46兲 B. Effects of binding energy In the study of superscaling for inclusive QE electron scattering from nuclei, an appropriate scaling variable ␺ ⬘ was introduced by including a small energy shift to have the QE peak occur at the place where the scaling variable is zero. A detailed study of the sensitivity of the scaling function to variations of the Fermi momentum and energy shift was presented in Refs. [16,18,20]. Here we extend this analysis to the inelastic region. In principle, the introduction of an energy shift ␻ shift in the formalism is straightforward and the calculation of the inelastic responses proceeds as in the ␻ shift=0 case. However, as will be made clear in the following, some complications arise. First, due to the general form assumed for the single-nucleon inelastic hadronic tensor, a certain asymmetry appears between the energy shift effects in the longitudinal and transverse responses. These shift effects are larger in the longitudinal response. Notice that this asymmetry already enters at the level of the QE nuclear responses. Second, there exists an ambiguity in the definition of the variable which should be used as the Bjorken x-scaling variable corresponding to the moving nucleon. The effects of the inclusion of an energy shift on the inelastic nuclear hadronic tensor have been studied in the literature, with particular emphasis on the structure function F2 and the European Muon Collaboration (EMC)effect at large values of x, in the context of so called “binding models” (see, for example, Refs. [30,31]and the general reviews [12,13]). The approach we follow here is the self-consistent generalization of previous works on the RFG. It is formally similar to the binding model approach, where in general the on-shell energy of the initial nucleon is modified by subtracting a constant term which effectively accounts for the nucleon separation energy and for the possibility that the residual nuclear system is left in an (highly)excited state. However, since the existing models either focus only on EMC ratios and/or use more realistic, although generally nonrelativistic wave functions, a precise quantitative comparison with those models is not possible. As we will discuss in the Results INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004) 035502-5 section, our calculations still miss some ingredients, coming from meson-exchange currents, and this makes a detailed quantitative comparison with experimental data premature; it is clear that, when this comparison will be made in the future, a more in-depth study of binding effects will also be needed. In the RFG the energy shift is usually introduced by modifying the argument of the ␦ function appearing in the general expression of the inelastic hadronic tensor in Eq. (12), according to ␻ +E ¯ h−EX→ ␻ ⬘+E ¯ h−EX, where ␻ ⬘= ␻ − ␻ shift. Then, introducing the invariant mass WX ⬘2⬅EX 2−pX 2 =共 ␻ ⬘+E ¯ h兲2−pX 2, we can write the inelastic hadronic tensor in the form Winel ␮ ␯ 共 ␬ ,␭兲=3N 4 ␲ ␩ F 3 冕 d ␮ X ⬘ 冕 d ␩ ␮ X ⬘ ⑀ ¯ ⑀ Xwinel ␮ ␯ 共 ␩ , ␮ X ⬘; ␬ ,␭兲 ⫻ ␦ 共2␭⬘+ ⑀ ¯ − ⑀ X兲 ␪ 共 ␩ F− ␩ 兲,共47兲 where ␮ X ⬘=WX ⬘/mNand ␭⬘= ␻ ⬘/共2mN兲. As in the unshifted analysis, the ␦ function can be used to perform the polar angular integration, leading to the result Winel ␮ ␯ 共 ␬ ,␭兲=3N ␶ ⬘ 2 ␩ F 3 ␬ 冕 ␳ 1 ⬘共 ␬ ,␭⬘兲 ␳ 2 ⬘共 ␬ ,␭⬘兲d ␳ ⬘ 冕 0 2 ␲ ⫻d⌽ 2 ␲ 冕 ⑀ 0 ⬘共 ␳ ⬘兲 ⑀ Fd ⑀ ¯ winel ␮ ␯ 共 ⑀ ¯ , ␳ ⬘; ␬ ,␭兲,共48兲 where the variable ␳ ⬘is defined as ␳ ⬘⬅2H·Q⬘ 兩Q⬘2兩= 冋 1+ 1 4 ␶ ⬘共 ␮ X ⬘2−1兲 册 共49兲 and ␶ ⬘⬅ ␬ 2−␭⬘2. The inclusion of the energy shift modifies the integration limits over ⑀ ¯ in the following way: ⑀ ¯ 艌 ⑀ 0 ⬘共 ␳ ⬘兲⬅ ␬ 冑1 ␶ ⬘+ ␳ ⬘2−␭⬘ ␳ ⬘.共50兲 Correspondingly, the region for the integration over ␳ ⬘is given by 关 ␳ 1 ⬘共 ␬ ,␭⬘兲, ␳ 2 ⬘共 ␬ ,␭⬘兲兴 = 冋 max 再 ␭⬘ ⑀ F− ␬ ␩ F ␶ ⬘, 1+ ␮ ␲ 4 ␶ ⬘共2+ ␮ ␲ 兲 冎 ,␭⬘ ⑀ F+ ␬ ␩ F ␶ ⬘ 册 . 共51兲 The definition of the inelastic scaling variable becomes now ␺ X ⬘2⬅ ⑀ 0 ⬘共 ␳ ⬘兲−1 ⑀ F−1 共52兲 and the inelastic longitudinal and transverse response functions, calculated as Rinel L=Winel 00 and Rinel T=Winel 11 +Winel 22 , have the following general forms: Rinel L,T共 ␬ , ␶ 兲=3N ␶ ⬘ 2 ␩ F 3 ␬ ␰ F 冕 ␳ 1 ⬘共 ␬ ,␭⬘兲 ␳ 2 ⬘共 ␬ ,␭⬘兲d ␳ ⬘共1− ␺ X ⬘2兲UL,T共 ␬ , ␶ , ␳ ⬘兲. 共53兲 In order to evaluate the longitudinal and transverse nuclear functions UL,T共 ␬ , ␶ , ␳ ⬘兲one needs to assume a specific form for the inelastic single-nucleon tensor winel ␮ ␯ 共 ⑀ ¯ , ␳ ⬘; ␬ ,␭兲.Itis important to remark that there exists some ambiguity in the choice made here: for instance, several alternatives involving different expressions containing the four-momenta Q ␮ and/or Q⬘ ␮ are possible, and these can lead to different results. For example, in Ref. 关30兴, the modified four momentum transfer Q⬘ ␮ is used, although then a prescription must be used in order to recover the gauge invariance of the nuclear hadronic tensor, which is lost by making this choice. In Appendix B we present the specific expressions of the inelastic and QE responses obtained for a given selection of the singlenucleon tensors accounting for the energy shift. Apart from the specific form of the tensor w ␮ ␯ , the choice of the arguments of the single-nucleon inelastic structure functions, w1,w2, also presents some ambiguities. In fact, the available parametrizations for w1,w2that we employ in our calculations are given for free, on-shell, nucleons, while the inclusion of the energy shift effectively introduces some “offshellness” of the initial nucleon, by altering the energy balance at the vertex where it couples to the exchanged virtual photon. In this case the bound-nucleon Bjorken variable is not uniquely determined by the final-state invariant mass and, since no theoretically derived prescriptions exist, one has to make some assumptions. As shown in Appendix B, the inelasticity parameter selected in this work, ␳ ˜ , corresponds to the one given as 共2mN ␻ ˜ 兲/兩Q2兩= ␳ ⬘共 ␶ ⬘/ ␶ 兲, where ␻ ˜ is the energy transferred to the nucleon in the system in which the nucleon is at rest. This means that in our numerical calculations, for a given set of values of ␻ ,Q2, and ␳ ⬘we employ free-nucleon structure functions taken at fourmomentum Q2and Bjorken variable 1/ ␳ ˜ . C. Extended relativistic fermi gas As discussed in previous works [8,16,18,20], for a fixed value of the invariant mass ␮ X, the RFG yields a scaling function f共 ␺ X ⬘兲=fL共 ␺ X ⬘兲=fT共 ␺ X ⬘兲=3 4共1− ␺ X ⬘2兲 ␪ 共1− ␺ X ⬘2兲共54兲 which, as a function of the appropriate scaling variable ␺ X ⬘,is the same for all values of ␮ X.1 In Ref. [18]the behavior of the longitudinal scaling function was studied for the existing world data in the QE region. This study showed that to a good approximation fL共 ␺ ⬘兲superscales, that is, it does not show any significant dependence on the momentum transfer ␬ (scaling of the first kind) and is approximately the same for all nuclear species (scal- 1The function in Eq. (54)differs from the one used in previous work [20]by a multiplicative function 2 ␰ F/ ␩ F 2关1+1 2 ␰ F共1+ ␺ X ⬘2兲兴.We have checked that this is numerically unimportant for all of the kinematical conditions considered here. BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-6 ing of the second kind). An expression for a phenomenological longitudinal scaling function, funiv共 ␺ ⬘兲, was obtained by fitting the data [21]. Based on these results, we now make the following hypothesis: we assume that this funiv共 ␺ ⬘兲, derived from the data, provides a good description of f共 ␺ X ⬘兲=fL共 ␺ X ⬘兲 =fT共 ␺ X ⬘兲,(“scaling of the zeroth kind”)as it implicitly contains the initial-state physics, and thus we make, for any ␮ X, the following substitution: 3 4共1− ␺ X ⬘2兲 ␪ 共1− ␺ X ⬘2兲→fERFG共 ␺ X ⬘兲=funiv共 ␺ X ⬘兲.共55兲 To be more specific, we calculate the response functions as RQE L,T=N ␩ F 3 ␬ mN ␰ Ffmodel共 ␺ ⬘兲UQE L,T,共56兲 Rinel L,T共 ␬ , ␶ 兲=N ␩ F 3 ␬ ␰ F 冕 ␮ thresh 1+2␭− ⑀ Sd ␮ X ␮ Xfmodel共 ␺ X ⬘兲UL,T,共57兲 where ⑀ S=ES/mNis the dimensionless separation energy and fmodel共 ␺ X ⬘兲= 再 3 4共1− ␺ X ⬘2兲 ␪ 共1− ␺ X ⬘2兲model = RFG funiv共 ␺ X ⬘兲model = ERFG. 共58兲 The functions fRFG and fERFG are shown in Fig. 1 as functions of ␺ X ⬘, while the functions UQE L,Tand UL,Tin Eqs. 共56兲 and 共57兲are given in Appendix B. IV. RESULTS In this section we present our results for cross sections and response and structure functions. In computing the inelastic hadronic tensor of Eq. (47), we employ phenomenological fits of the single-nucleon inelastic structure functions. The latter are measured in DIS experiments and a variety of parametrizations for w1and w2can be found in the literature [15,32–36], including some variations arising from the different assumptions made for how to extract the neutron structure functions from deuteron data. Unless stated otherwise, in the following we adopt the Bodek et al. fit of Refs. [15,32,33], which describes both the deep inelastic and resonance regions. For the QE contributions, we employ the form factor parametrization of Ref. [37]. The sensitivity of the results to the different parametrization choices will be discussed later. Additionally, for the Fermi momentum and the energy shift we will employ the values obtained in Ref. [20], namely, kF=220 MeV/c, ␻ shift=20 MeV for carbon, kF =236 MeV/c, ␻ shift=18 MeV for aluminum, kF =241 MeV/c, ␻ shift=23 MeV for iron, and kF=245 MeV/c, ␻ shift=25 MeV for gold. A. Cross sections In this section we present our results for the cross sections in the RFG and ERFG models and compare them with the available experimental data [38–41]. FIG. 2. Inclusive cross section for electron scattering from carbon at Einc=500 MeV and ␪ e=60° vs the energy transfer. The calculation includes an energy shift ␻ shift=20 MeV and the separate QE and inelastic contributions to the cross section are shown. Data are from Ref. [41]. FIG. 3. As for Fig. 2, but at Einc=2.020 GeV and scattering angle ␪ e=15° (a)and ␪ e=20° (b). Data are from Ref. [39]. FIG. 1. Scaling function fmodel共 ␺ X ⬘兲of Eq. (58)for the RFG and ERFG models. INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004) 035502-7 In Fig. 2 we show the inclusive cross section for a 12C target at Ee=500 MeV and ␪ e=60°. We separate the QE from the inelastic contribution. We notice that the shifted RFG model (solid line)yields roughly the right position and height of the QE peak, but fails to reproduce the tails of the peak, giving in particular an unobserved dip at ␻ ⯝200 MeV. On the other hand the ERFG model (dotted line), while reproducing the data in the tails better, significantly underestimates the cross section at the peak. This is related to the fact that, as shown in Fig. 1, the peak of the ERFG universal function fERFG is lower than the corresponding RFG value. Due to the larger extension of fERFG over ␺ X ⬘ the normalization of the two functions is the same, namely, 兰fRFGd ␺ X ⬘=兰fERFGd ␺ X ⬘=1. One might then naively expect the integral in Eq. (57)which yields the inelastic response functions to be the same in the two models. However, a closer inspection shows that this is not the case because the integration limits and/or the weighting provided by UL,Tare such that the ERFG integral does not “saturate” as does the RFG one. Figures 3–5 correspond to different kinematical conditions, namely, Ee=2.020 and 3.595 GeV (SLAC)and Ee =4.045 GeV (JLab)and various scattering angles. Concerning Figs. 3(a)and 3(b), a similar trend persists, with the FIG. 4. As for Fig. 2, but at Ee=3.595 GeV and scattering angle ␪ e=16° (a), 20° (b), 25° (c), and 30° (d). Data are from Ref. [39]. FIG. 5. As for Fig. 2, but at Einc=4.045 GeV and scattering angle ␪ e=15° (a), 30° (b), 45° (c), and 74° (d). Data are from Ref. [38]. BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-8 ERFG model significantly underestimating the data in the region of the QE peak, whereas the RFG is closer to the data (particularly for ␪ e=20°), although it leaves no room for other contributions to be added. Note also that for this scattering angle the inelastic channel starts to be sizable. Examining Figs. 4 we remark that the QE peak, which is more clearly separated from the inelastic region in Fig. 4(a), is again well reproduced in the low- ␻ tail by the ERFG, while its maximum agrees better with the RFG. On the other hand the inelastic cross section is in all cases underestimated by the ERFG, while the RFG alone would roughly account for what is observed. Similar comments apply to Fig. 5(a), corresponding to higher energy and low scattering angle. For higher angles [Figs. 5(b)–5(d)] the data lie roughly in between the predictions of the ERFG (smaller)and RFG (larger)models, the former again reproducing the low- ␻ behavior better. As a general result we observe that as the scattering angle increases the range of validity of the ERFG also increases. Finally, in Figs. 6 and 7 we consider slightly different kinematical conditions, corresponding to fixed values of 兩Q2兩 in the range 2–10 共GeV/c兲2, and various electron energies (in the range 8–25 GeV)and angles 共12° –22°兲. Theoretical results for 56Fe are shown as functions of the “laboratory” Bjorken variable xL. The data corresponding to a fixed Q2are taken at different values of Eeand ␪ e. For 兩Q2兩=2 and 10 共GeV/c兲2(Fig. 6)the various data fit reasonably well on one plot, whereas for 兩Q2兩=5 共GeV/c兲2(Fig. 7), for clarity we have separated the data into three sets as indicated in the figure caption. We notice that at large xL共艌0.6兲the data are closer to the ERFG predictions, at low xL共0.1–0.3兲they are closer to the RFG calculation and for 0.3艋xL艋0.6 they lie in between the two models. This general trend seems to be respected for all values of Q2(at least where data are available). We have also analyzed the effect introduced by different electromagnetic form factor parametrizations ([42–44]) and verified that it can produce a ±3 % uncertainty at the QE peak, but does not change the general agreement/ disagreement of the models with the data. Moreover, it should be remarked that, at the energies considered in this section, the contribution from the resonance region to the inelastic part of the cross section is quite important and thus a comparison with results obtained by using purely DIS parametrizations [34,36]of the single-nucleon structure functions is not appropriate. At the highest 兩Q2兩values considered here [Figs. 6(b)and 7], the use of different parametrizations [34,36]does not produce significant variations in the results. An important comment, already anticipated in the introduction, is in order. The RFG and ERFG models considered FIG. 6. Differential cross section d ␴ /d ␻ d⍀efor electron scattering on iron, shown as a function of xLat fixed 兩Q2兩. Panel (a): 兩Q2兩=2共GeV/c兲2, the experimental points, from right to left, are taken at 关Ee共GeV兲, ␪ e共deg兲兴=共8,11.8兲,共8,12.4兲,共8,13.6兲,共9.7, 11.8兲,共12,11.8兲,共15,11.8兲. Panel (b):兩Q2兩=10 共GeV/c兲2, the experimental points, from right to left, are taken at 关Ee共GeV兲, ␪ e共deg兲兴⫽(15,16.0),(15,17.1),(17,15.0),(17,16.9),(21, 14.1),(24,14.1). Data are from Ref. [40]. FIG. 7. As for Fig. 6, but at 兩Q2兩=5共GeV/c兲2. The two data in the upper panel, from right to left, correspond to Ee=8 GeV and ␪ e=22° and to Ee=9.7 GeV and ␪ e=19.7°; the data in the middle panel have fixed Ee=12 GeV and, from right to left, ␪ e =12.8,13.3,14.2,15.8,20.6°; the data in the lower panel, from right to left, are taken at 共Ee=15 GeV, ␪ e=13.2°兲,共Ee=17 GeV, ␪ e =13.5°兲,共Ee=24.5 GeV ␪ e=11.1°兲. Data are from Ref. [40]. INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004) 035502-9 ␺ ⬘2=1 ␰ F 冉 ␬ 冑1 ␶ ⬘+1−␭⬘−1 冊 共B13兲 and the electromagnetic structure functions by w1,QE共 ␶ 兲= ␶ GM 2共 ␶ 兲, w2,QE共 ␶ 兲=GE 2共 ␶ 兲+ ␶ GM 2共 ␶ 兲 1+ ␶ with GE,Mthe proton or neutron Sachs electromagnetic form factors. Finally, the “total” response functions are evaluated by adding the above QE responses to the inelastic ones, i.e., Rtot L,T=RQE L,T+Rinel L,T. [1]W. M. Alberico, T. W. Donnelly, and A. Molinari, Nucl. Phys. A512, 541 (1990). [2]W. M. Alberico, M. B. Barbaro, A. De Pace, T. W. Donnelly, and A. Molinari, Nucl. Phys. A563, 605 (1993). [3]M. B. Barbaro, A. De Pace, T. W. Donnelly, and A. Molinari, Nucl. Phys. A596, 553 (1996). [4]J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, and A. Molinari, Nucl. Phys. A643, 349 (1998). [5]J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, and A. Molinari, Nucl. Phys. A697, 388 (2002). [6]J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, and A. Molinari, Phys. Rep. 368, 317 (2002). [7]J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, and A. Molinari, Nucl. Phys. A723, 181 (2003). [8]J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, and A. Molinari, Nucl. Phys. A657, 161 (1999). [9]L. Alvarez-Ruso, M. B. Barbaro, T. W. Donnelly, and A. Molinari, Nucl. Phys. A724, 157 (2003). [10]B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16,1(1986). [11]W. M. Alberico, A. Molinari, T. W. Donnelly, E. L. Kronenberg, and J. W. Van Orden, Phys. Rev. C 38, 1801 (1988). [12]R. P. Bickerstaff and A. W. Thomas, J. Phys. G 15, 1523 (1989). [13]P. R. Norton, Rep. Prog. Phys. 66, 1253 (2003). [14]S. Simula, Few-Body Syst., Suppl. 8, 423 (1995). [15]A. Bodek and J. L. Ritchie, Phys. Rev. D 23, 1070 (1981);24, 1400 (1981). [16]T. W. Donnelly and I. Sick, Phys. Rev. Lett. 82, 3212 (1999). [17]M. B. Barbaro, R. Cenni, A. De Pace, T. W. Donnelly, and A. Molinari, Nucl. Phys. A643, 137 (1998). [18]T. W. Donnelly and I. Sick, Phys. Rev. C 60, 065502 (1999). [19]D. B. Day, J. S. McCarthy, T. W. Donnelly, and I. Sick, Annu. Rev. Nucl. Part. Sci. 40, 357 (1990). [20]C. Maieron, T. W. Donnelly, and I. Sick, Phys. Rev. C 65, 025502 (2002). [21]J. Jourdan, Nucl. Phys. A603,117(1996). [22]A. De Pace, M. Nardi, W. M. Alberico, T. W. Donnelly, and A. Molinari, Nucl. Phys. A726, 303 (2003). [23]T. De Forest and J. D. Walecka, Adv. Phys. 15,1(1966). [24]S. Boffi, C. Giusti, and F. D. Pacati, Phys. Rep. 226,1(1993). [25]T. W. Donnelly, in Nuclear/Nucleonic Structure and Inclusive Electron Scattering, Proceedings of the International School of Physics “Enrico Fermi,” Course CLIII, edited by A. Molinari and L. Riccati (IOS, Amsterdam, 2003), p. 183. [26]J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1965). [27]T. W. Donnelly and A. S. Raskin, Ann. Phys. (N.Y.)169, 247 (1986). [28]T. W. Donnelly, M. J. Musolf, W. M. Alberico, M. B. Barbaro, A. De Pace, and A. Molinari, Nucl. Phys. A541, 525 (1992). [29]L. Alvarez-Ruso, M. B. Barbaro, T. W. Donnelly, and A. Molinari, Phys. Lett. B 497, 214 (2001). [30]O. Benhar, V. R. Pandharipande, and I. Sick, Phys. Lett. B 410,79(1997). [31]C. Ciofi Degli Atti and S. Liuti, Phys. Lett. B 225, 215 (1989). [32]A. Bodek et al., Phys. Rev. D 20, 1471 (1979). [33]A. Stein et al., Phys. Rev. D 12, 1884 (1975). [34]P. Amaudruz et al., New Muon Collaboration, Phys. Lett. B 295, 159 (1992). [35]L. W. Whitlow, S. Rock, A. Bodek, E. M. Riordan, and S. Dasu, Phys. Lett. B 250, 193 (1990). [36]A. D. Martin, R. G. Roberts, W. J. Stirling, and R. S. Thorne, Eur. Phys. J. C 4, 463 (1998); see also http://durpdg.dur.ac.uk/ hepdata/mrs.html [37]G. Hohler, E. Pietarinen, I. Sabba Stefanescu, F. Borkowski, G. G. Simon, V. H. Walther, and R. D. Wendling, Nucl. Phys. B114, 505 (1976). [38]J. Arrington et al., Phys. Rev. Lett. 82, 2056 (1999). [39]D. B. Day et al., Phys. Rev. C 48, 1849 (1993). [40]R. G. Arnold et al., Phys. Rev. Lett. 52, 727 (1984). [41]R. R. Whitney, I. Sick, J. R. Ficenec, R. D. Kephart, and W. P. Trower, Phys. Rev. C 9, 2230 (1974). [42]S. Galster, H. Klein, J. Moritz, K. H. Schmidt, D. Wegener, and J. Bleckwenn, Nucl. Phys. B32, 221 (1971). [43]F. Iachello, A. D. Jackson, and A. Lande, Phys. Lett. B 43, 191 (1973). [44]P. E. Bosted, Phys. Rev. C 51, 409 (1995). BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-16