Inelastic electron-nucleus scattering and scaling at high inelasticity

Inelas ic elec on-nucleus sca e ing and scaling a high inelas ici y M. B. Ba ba o,1J. A. Caballe o,2T. W. Donnelly,3and C. Maie on2,4 1Dipa imen o di Fisica Teo ica, Uni e si à di To ino and INFN, Sezione di To ino, Via P. Giu ia 1, 10125 To ino, I aly 2Depa amen o de Física A ómica, Molecula y Nuclea , Uni e sidad de Se illa, Apa ado Pos al 1065, E-41080 Se illa, Spain 3Cen e o Theo e ical Physics, Labo a o y o Nuclea Science and Depa men o Physics, Massachuse s Ins i u e o Technology, Camb idge, Massachuse s 02139, USA 4INFN, Sezione di Ca ania, Via S. So ia 64, 95123 Ca ania, I aly (Recei ed 24 No embe 2003; published 24 Ma ch 2004) Highly inelas ic elec on sca e ing is analyzed wi hin he con ex o he uni ied ela i is ic app oach p e i- ously conside ed in he case o quasielas ic kinema ics. Inelas ic ela i is ic Fe mi gas modeling ha includes he comple e inelas ic spec um— esonan , non esonan , and deep inelas ic sca e ing—is elabo a ed and com- pa ed wi h expe imen al da a. A phenomenological ex ension o he model based on di ec i s o da a is also in oduced. Wi hin bo h models, c oss sec ions and esponse unc ions a e e alua ed and binding ene gy e ec s a e analyzed. Finally, an in es iga ion o he second-kind scaling beha io is also p esen ed. DOI: 10.1103/PhysRe C.69.035502 PACS numbe (s): 25.30.Fj, 24.10.J , 13.60.Hb I. INTRODUCTION In his wo k we conside highly inelas ic elec on sca e - ing and compa e i s analysis wi h he case o quasielas ic (QE)elec on sca e ing. The la e is domina ed by he p o- cess whe e he exchanged i ual pho on in e ac s wi h a nucleon in he nuclea g ound s a e and ejec s ha nucleon, he eby o ming a nuclea pa icle-hole exci a ion. Co ec- ions o his dominan p ocess in ol e going beyond he im- pulse app oxima ion o accoun o wo-body cu en s, inal- s a e in e ac ions, and nuclea co ela ions. Al hough hese con ibu ions a e known no o be en i ely negligible [1–7] his simple p ocess accoun s o he basic ea u e seen in he icini y o elas ic sca e ing om a nucleon a es , namely, he QE peak. Models such as hose discussed below ake in o accoun he ac ha he nucleons in he nucleus a e mo ing and a e bound and he eby p oduce a b oad peak in he in- elas ic spec um. In he p esen wo k ou goal is o ex end he analysis, s ill main aining he same basic ea u es o he ela i is ic modeling used o he QE egion, and now ocus on wha we call highly inelas ic sca e ing, o o b e i y, simply he inelas ic egion. This includes e e y hing ha goes beyond he QE p ocess: ha is, whe eas he QE p ocess assumes elas ic sca e ing om he nucleons, he inelas ic p ocess will assume inelas ic e-Nsca e ing. Fo ela i ely low inal-s a e in a ian masses one lies in he egion o eso- nance exci a ion and wo cases o his so ha e been ex- plo ed in ecen wo k [8,9]. In he p esen s udy hese ideas a e gene alized o include he comple e inelas ic spec um, bo h esonan and non esonan , including deep inelas ic sca - e ing (DIS), wi hin he con ex o he uni ied ela i is ic app oach used in ou p e ious wo k. Thus, in he p esen wo k ou goal is o begin by explo - ing ex ensions o he ela i is ic Fe mi gas (RFG)model [6,10,11] o an inelas ic e sion o his app oach. While his bea s some connec ion wi h adi ional con olu ion models o he high-ene gy esponse o nuclei (see, o example, Re s. [12–15]) i is no he same in ha , albei wi hin a model, i co ec ly inco po a es a speci ic ela i is ic nuclea spec al unc ion in o he p oblem, whe eas some o he ap- p oaches make addi ional assump ions and use only he in e- g al o he spec al unc ion, namely, he nuclea momen um dis ibu ion o make non ela i is ic app oxima ions when dealing wi h he spec al unc ion. This dis inc ion can be seen qui e clea ly in s udies o i s - and second-kind scaling [11,16–20]and will no be elabo a ed he e. Once he inelas ic RFG modeling is in hand, i becomes clea ha i migh be use ul o explo e a phenom- enological ex ension o his model, namely, wha we call he ex ended ela i is ic Fe mi gas (ERFG). In his app oach we ake he esul o doing he co ec in eg al o e he nuclea spec al unc ion (i.e., no he ull in eg al, which is he mo- men um dis ibu ion, as alluded o abo e)di ec ly om i s made p e iously o he da a [21]. We shall see ha his has a signi ican impac on he nuclea esponses a high inelas ic- i y.An issue which will also become clea la e is ha he s o y is no ye comple e: in addi ion o he modeling done in he p esen wo k, whe e he ocus is placed on inco po a ing inelas ic e ec s a high ene gies, he e a e s ill o he con i- bu ions ha mus be added. Speci ically, in ecen wo k [22] on 2p-2hmeson-exchange cu en e ec s i is seen ha a signi ican incohe en con ibu ion mus be added o hose explo ed he e. Gi en ha he wo k on 2p-2he ec s is, as ye , incomple e—co ela ion con ibu ions a e p esen ly be- ing included—i is p ema u e o make oo much o compa i- sons wi h expe imen al da a, and, as we ema k la e in he app op ia e places, he inal unde s anding o how all o he a ious eac ion mechanisms en e , while becoming clea e is no ye achie ed. The pape is o ganized as ollows: in Sec. II we ecall he gene al o malism o inelas ic elec on-nucleus sca e ing; in Sec. III we de i e he exp essions o he inelas ic had- onic enso in h ee di e en models: he pu e RFG model [Sec. III A], he RFG including he e ec s o binding ene gy [Sec. III B]and he ERFG [Sec. III C]; in Sec. IV we p esen nume ical esul s o c oss sec ions [Sec. IV A], esponse unc ions [Sec. IV B], and scaling unc ions [Sec. IV C]and inally, in Sec. V we d aw ou conclusions. PHYSICAL REVIEW C 69, 035502 (2004) 0556-2813/2004/69(3)/035502(16)/$22.50 ©2004 The Ame ican Physical Socie y69 035502-1 II. INELASTIC ELECTRON-NUCLEUS SCATTERING: GENERAL FORMALISM Wi h he goal ou lined abo e in mind, we s a by ew i - ing he gene al exp essions ha apply in bo h elas ic and inelas ic egimes. The gene al o malism desc ibing inclu- si e elec on-nucleus sca e ing p ocesses is widely a ailable [23–25]; he e we simply ocus on hose aspec s ha a e o special ele ance o he discussion ha ollows. We ollow he con en ions and me ic o Re . [26]and use capi al le - e s o e e o ou - ec o s. The inciden and sca e ed elec- on ou -momen a a e deno ed by Ki ␮ =共␧i,ki兲and K ␮ =共␧ ,k 兲. The had onic a iables, PA ␮ =共MA,0兲and PB ␮ =共EB,pB兲 ep esen he ou momen a o he a ge and e- sidual nucleus, espec i ely. The ou -momen um ans e is gi en by Q ␮ =共 ␻ ,q兲(we assume he Bo n app oxima ion, i.e., only one i ual pho on exchanged in he p ocess). Following s anda d p ocedu es he di e en ial c oss sec- ion may be w i en d ␴ d⍀ d␧ =2 ␣ 2 Q4 ␧ ␧i ␩ ␮ ␯ W ␮ ␯ ,共1兲 whe e ␣ is he ine s uc u e cons an , ␩ ␮ ␯ is he lep onic enso ha can be e alua ed di ec ly using ace echniques 关27兴, and W ␮ ␯ is he had onic enso con aining all o he nuclea s uc u e and dynamics in o ma ion. Assuming ha he inal s a e can be desc ibed in e ms o a ecoiling nuclea s a e 兩 ␺ B典plus a 共highly兲inelas ic s a e 兩⌽X典, i s gene al ex- p ession is gi en 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−␧ +EA−EB−EX兲, 共2兲 whe e 兺 ¯ A共兺B兺X兲indica es he app op ia e a e age 共sum兲 o e ini ial 共 inal兲s a es. He e J ˆ ␮ 共q兲is he Fou ie ans o m o he nuclea cu en ope a o e alua ed, 兩 ␺ A典and 兩 ␺ B,⌽X典 ep esen he ini ial and inal s a es, espec i ely, and he dis ibu ion unc ions ␳ 共EB兲and ␳ 共EX兲a e in oduced o ac- coun o he ene gy-momen um dispe sion ela ion o he inal nuclea 共B兲and had onic 共X兲sys ems. In his wo k we assume ha he inelas ici y o he p ocess is o ally ac- coun ed o by he inal s a e ⌽X; hence o he ene gy dis- ibu ion unc ion o he esidual nuclea sys em we use ␳ 共EB兲= ␦ 共EB−E ¯ B兲, whe e E ¯ B=冑pB 2+共MB *兲2. No e ha W ␮ ␯ in Eq. 共2兲is mean o be e alua ed a pB+pX=q=ki−k . The nuclea enso can equi alen ly be exp essed as an in eg al in he 共E,p兲plane, wi h −p=pB he h ee-momen um o he ecoiling daugh e nucleus and E⬅冑p2+共MB *兲2 −冑p2+共MB 0兲2 he exci a ion ene gy o he esidual nucleus (see Re . [19]). The domain o in eg a ion is he kinema i- cally allowed egion max关E共0兲,0兴艋E艋E共 ␲ 兲,共3兲 whe e E共 ␪ 兲=MA 0+ ␻ −冑共MB 0兲2+p2−冑WX 2+q2+p2+2pq cos ␪ 共4兲 wi h ␪ he angle be ween pand q, and whe e WXis he in a ian mass o he inal s a e. In he MB 0→⬁limi he abo e exp ession becomes E⬁共 ␪ 兲=mN+ ␻ ˜ −冑WX 2+q2+p2+2pq cos ␪ ,共5兲 whe e mNis he nucleon mass, ␻ ˜ ⬅ ␻ −ESand ES=MB 0+mN −MA 0is he sepa a ion ene gy. The uppe cu e E共 ␲ 兲c osses he paxis a p−=−yXand p+=YX, whe e 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 whe e 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 a iable yXis he gene aliza ion o he usual y-scaling a iable o he inelas ic p ocess whe e a esonance Xis p o- duced. In he limi MB 0→⬁i eads yX,⬁=冑共 ␻ ˜ +mN兲2−WX 2−q.共9兲 No e ha he allowed egion dec eases wi h WXand col- lapses o a poin when −yX=YX, which implies W=MB 0+WX o , in he MB 0→⬁limi , 共yX,⬁兲min=−q, co esponding o 共WX兲max= ␻ ˜ +mN. Summa izing, o ixed ou -momen um ans e , he esonan mass is limi ed o he ange mN+m ␲ 艋WX艋mN+ ␻ −ES.共10兲 III. THE RELATIVISTIC FERMI GAS MODEL In his sec ion we p oceed by e alua ing he had onic nuclea enso assuming he impulse app oxima ion and by wo king wi hin he amewo k o he RFG model. In his case, he i ual pho on is abso bed by an on-shell nucleon desc ibed by a Di ac spino u共h,sh兲, wi h ene gy E ¯ h =冑h2+mN 2. In eg a ing o e he momen a in he Fe mi sea, he ollowing exp ession o he inelas ic had onic enso e- sul s: 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 whe e Nis he numbe o nucleons 共p o ons o neu ons兲and 兰Fdh⬅兰dh ␪ 共pF−h兲,pFbeing he Fe mi momen um. The symbol 兰dEXs ands o he in eg al o e he ene gy o he inelas ic inal s a e, while 兺Xiindica es in gene al he sum/ in eg al o e all he in e nal quan um numbe s o all possible inelas ic inal s a es ⌽Xi, ha ing o al ene gy EXand o al momen um pX, ixed by momen um conse a ion o be pX =h+q. The had onic enso in Eq. (11) o inelas ic p ocesses can be also w i en in he o m 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 whe e H ␮ =共E ¯ h,h兲and we ha e in oduced he inelas ic single-nucleon enso 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 No e ha he abo e single-nucleon enso has dimensions o E−1. As will be shown la e , his is in con as wi h ou pas wo k on QE and N→⌬sca e ing whe e he single-nucleon enso s we e de ined o be dimensionless. Nex we choose o exp ess he inelas ic had onic enso in Eq. (12)in e ms o he in a ian 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 wi h EX=冑pX 2+WX 2. The ene gy in eg al can be pe o med by exploi ing he ␦ unc ion, yielding Winel ␮ ␯ 共q, ␻ 兲=3N 4 ␲ pF 3 冕 FdhmN E ¯ h winel ␮ ␯ 共H,Q, ␻ +E ¯ h兲.共15兲 In he case o DIS on a single nucleon, he inelas ic enso simply educes o he single-nucleon enso winel ␮ ␯ . Be o e en e ing in o a de ailed analysis o he inelas ic nuclea enso , i is in e es ing o no ice how he usual ex- p essions o he QE and N→⌬had onic enso s a e eco - e ed om he gene al esul gi en in Eq. (11). Fi s , in he case o QE sca e ing, he nuclea inal s a e is simply a pa icle-hole exci a ion, hence, in he RFG model, ⌽Xde- sc ibes an on-shell nucleon, namely, ⌽X=冑mN/E ¯ pu共p,sp兲. The ene gy dis ibu ion unc ion is simply ␳ 共EX兲= ␦ 共EX −E ¯ p兲and he sum o e he inal s a es educes o a sum o e spin p ojec ions, 兺Xi=兺sp. The QE had onic enso hen eads 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 whe e wQE ␮ ␯ is he usual dimensionless QE single-nucleon en- so 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 he case o he ansi ion N→⌬, he inal s a e ⌽X, wi hin he con ex o he RFG model, is an on-shell ⌬, namely, ⌽X=冑m⌬/E ¯ ⌬u⌬共p,s⌬兲, wi h on-shell ene gy E ¯ ⌬=冑p2+m⌬ 2. The ene gy dis ibu ion unc ion in his case is ␳ 共EX兲 = ␦ 共EX−E ¯ ⌬兲and 兺Xi=兺s⌬. The N→⌬had onic enso ha esul s 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 wi h w⌬ ␮ ␯ he dimensionless nucleon-⌬ enso 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 expec ed, hese exp essions o he dimensionless single- nucleon enso s coincide wi h he ones in oduced in Re . 关8,28兴. Likewise o he Rope esonance he exp essions ob- ained in Re . 关9兴a e eco e ed. A. The RFG inelas ic nuclea enso and esponse unc ions In his sec ion we e alua e he inelas ic nuclea enso in he RFG amewo k. Fo con enience, as usual we i s de- ine he dimensionless a iables ␬ ␮ =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 e ms o which he had onic enso in Eq. (14) eads 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 Be o e p esen ing he explici esul s o he RFG e- sponse unc ions, le us discuss an impo an ing edien o he calcula ion, he single-nucleon inelas ic had onic enso winel ␮ ␯ . Fo unpola ized sca e ing, he la e can be pa am- e ized in e ms o wo s uc u e unc ions, w1and w2, ac- co ding o winel ␮ ␯ =−w1 冉 g ␮ ␯ + ␬ ␮ ␬ ␯ ␶ 冊 +w2共 ␩ ␮ + ␬ ␮ ␳ 兲共 ␩ ␯ + ␬ ␯ ␳ 兲. 共22兲 Fo on-shell nucleons, he s uc u e unc ions w1and w2de- pend on wo a iables, he ou -momen um ans e Q2and he in a ian mass WXo he inal s a e eached by he nucleon, o , equi alen ly, he single-nucleon Bjo ken a i- able x=兩Q2兩 2H·Q=兩Q2兩 WX 2−mN 2−Q2= ␶ ␩ · ␬ .共23兲 In ou o malism i is con enien o in oduce he inelas- ici y pa ame e [8,29] ␳ ⬅1+ 1 4 ␶ 共 ␮ X 2−1兲,共24兲 he alue uni y co esponding o elas ic sca e ing. No e ha ␳ is simply linked o he Bjo ken scaling a iable o he on-shell nucleon mo ing inside he a ge nucleus by he ela ion ␳ =1/x, hus in he ollowing we will use ␳ as a gu- men o he s uc u e unc ions w1,w2. In p esen ing ou esul s we will also use he “labo a o y” Bjo ken a iable xL=兩Q2兩 2mN ␻ = ␶ ␭,共25兲 co esponding o a single nucleon a es in he labo a o y ame. Le us now e u n o he inelas ic nuclea enso o Eq. (21): a e pe o ming he pola angula in eg a ion by means o he ene gy-conse ing ␦ unc ion one ge s 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 whe e cos ␪ 0=1 ␬ ␩ 共␭ ⑀ ¯ − ␶ ␳ 兲.共27兲 The condi ion 兩cos ␪ 0兩艋1 ixes he in eg a ion limi s o e ⑀ ¯ : ⑀ ¯ 艌 ⑀ 0共 ␳ 兲⬅ ␬ 冑1 ␶ + ␳ 2−␭ ␳ .共28兲 Mo eo e , by equi ing ha ⑀ 0共 ␳ 兲艋 ⑀ Fand ha he eso- nance mass is abo e he pion-p oduc ion h eshold (i.e., ␮ X 艌 ␮ h esh⬅1+ ␮ ␲ ) he ollowing egion is ob ained o he in eg a ion o e ␳ : 关 ␳ 1共 ␬ ,␭兲, ␳ 2共 ␬ ,␭兲兴 = 冋 max 再 ␭ ⑀ F− ␬ ␩ F ␶ , ␳ h esh 冎 ,␭ ⑀ F+ ␬ ␩ F ␶ 册 共29兲 wi h ␳ h esh =1+ ␮ ␲ 共 ␮ ␲ +2兲 4 ␶ .共30兲 No e ha he uppe in eg a ion limi ␳ 2共 ␬ ,␭兲always lies be- low he cu o co esponding o Eq. 共10兲. Indeed using Eqs. 共10兲and 共24兲and keeping in mind ha he 共nega i e兲sepa- a ion ene gy o he RFG is ES RFG=−TF⬅mN共1− ⑀ F兲, he maximum ␳ allowed by Eq. 共10兲 eads ␳ 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 ␳ 2 he e o e esul ing in he mo e s ingen in eg a ion limi . Now by w i ing he single-nucleon inelas ic enso in e ms o s uc u e unc ions w1and w2as in Eq. (22)and choosing he zdi ec ion along q, he in eg a ion o e ⌽and ⑀ ¯ can be pe o med analy ically (see Appendix A)and he had onic inelas ic enso can be exp essed in he gene al o m 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 whe e ␰ F= ⑀ F−1 is he Fe mi kine ic ene gy and he inelas ic scaling a iable ␺ X⬅sgn共␭− ␶ ␳ 兲冑 ⑀ 0共 ␳ 兲−1 ⑀ F−1 共33兲 has been de ined. Fo each alue o ␳ 共and hence ␮ X兲a “peak” can hus be iden i ied, co esponding o he egion −1艋 ␺ X艋1, cen e ed a ␺ 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 wid h ⌬␭ =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 unc ion ha g ows wi h ␬ and dec eases wi h ␮ X. BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-4 The gene al exp ession o he enso U ␮ ␯ is de i ed in Appendix A. He e we only epo he longi udinal and ans- e se componen s 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 a e linked o he longi udinal and ans e se esponse unc ions by he ollowing ela ions: 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 wi h ␺ X max共 ␬ ,␭兲= min 冦 1, 冢 ␬ 冑1 ␶ + ␳ h esh 2−␭ ␳ h esh −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 he unc ion 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 a ises om he Fe mi mo ion and goes o ze o as ␰ F→0; being p opo ional o ␰ F⬵ ␩ F 2/2Ⰶ1, his p o ides ela i ely mode a e co ec ions o he es o he con ibu ions in Eqs. 共36兲and 共37兲. The alue ␳ =1 co esponds o QE kinema ics: in his case he well-known exp essions o he QE esponses a e eco - e ed. The “ o al” obse ables a e hen ob ained by adding he usual RFG QE esponse o he inelas ic esul s: R o L,T=RQE L,T+Rinel L,T.共42兲 In he deep inelas ic egime i is cus oma y o deal wi h nuclea s uc u e unc ions W1,2 Aand/o F1,2 A. These can be exp essed in e ms o he longi udinal and ans e se e- sponse unc ions h ough he ollowing ela ions: 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. E ec s o binding ene gy In he s udy o supe scaling o inclusi e QE elec on sca e ing om nuclei, an app op ia e scaling a iable ␺ ⬘ was in oduced by including a small ene gy shi o ha e he QE peak occu a he place whe e he scaling a iable is ze o. A de ailed s udy o he sensi i i y o he scaling unc ion o a ia ions o he Fe mi momen um and ene gy shi was p e- sen ed in Re s. [16,18,20]. He e we ex end his analysis o he inelas ic egion. In p inciple, he in oduc ion o an en- e gy shi ␻ shi in he o malism is s aigh o wa d and he calcula ion o he inelas ic esponses p oceeds as in he ␻ shi =0 case. Howe e , as will be made clea in he ollow- ing, some complica ions a ise. Fi s , due o he gene al o m assumed o he single-nucleon inelas ic had onic enso , a ce ain asymme y appea s be ween he ene gy shi e ec s in he longi udinal and ans e se esponses. These shi e - ec s a e la ge in he longi udinal esponse. No ice ha his asymme y al eady en e s a he le el o he QE nuclea e- sponses. Second, he e exis s an ambigui y in he de ini ion o he a iable which should be used as he Bjo ken x-scaling a iable co esponding o he mo ing nucleon. The e ec s o he inclusion o an ene gy shi on he in- elas ic nuclea had onic enso ha e been s udied in he li - e a u e, wi h pa icula emphasis on he s uc u e unc ion F2 and he Eu opean Muon Collabo a ion (EMC)e ec a la ge alues o x, in he con ex o so called “binding models” (see, o example, Re s. [30,31]and he gene al e iews [12,13]). The app oach we ollow he e is he sel -consis en gene ali- za ion o p e ious wo ks on he RFG. I is o mally simila o he binding model app oach, whe e in gene al he on-shell ene gy o he ini ial nucleon is modi ied by sub ac ing a cons an e m which e ec i ely accoun s o he nucleon sepa a ion ene gy and o he possibili y ha he esidual nuclea sys em is le in an (highly)exci ed s a e. Howe e , since he exis ing models ei he ocus only on EMC a ios and/o use mo e ealis ic, al hough gene ally non ela i is ic wa e unc ions, a p ecise quan i a i e compa ison wi h hose models is no possible. As we will discuss in he Resul s INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004) 035502-5 sec ion, ou calcula ions s ill miss some ing edien s, coming om meson-exchange cu en s, and his makes a de ailed quan i a i e compa ison wi h expe imen al da a p ema u e; i is clea ha , when his compa ison will be made in he u- u e, a mo e in-dep h s udy o binding e ec s will also be needed. In he RFG he ene gy shi is usually in oduced by modi ying he a gumen o he ␦ unc ion appea ing in he gene al exp ession o he inelas ic had onic enso in Eq. (12), acco ding o ␻ +E ¯ h−EX→ ␻ ⬘+E ¯ h−EX, whe e ␻ ⬘= ␻ − ␻ shi . Then, in oducing he in a ian mass WX ⬘2⬅EX 2−pX 2 =共 ␻ ⬘+E ¯ h兲2−pX 2, we can w i e he inelas ic had onic enso in he o m 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 whe e ␮ X ⬘=WX ⬘/mNand ␭⬘= ␻ ⬘/共2mN兲. As in he unshi ed analysis, he ␦ unc ion can be used o pe o m he pola angula in eg a ion, leading o he esul 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 whe e he a iable ␳ ⬘is de ined 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 o he ene gy shi modi ies he in eg a ion limi s o e ⑀ ¯ in he ollowing way: ⑀ ¯ 艌 ⑀ 0 ⬘共 ␳ ⬘兲⬅ ␬ 冑1 ␶ ⬘+ ␳ ⬘2−␭⬘ ␳ ⬘.共50兲 Co espondingly, he egion o he in eg a ion o e ␳ ⬘is gi en 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 de ini ion o he inelas ic scaling a iable becomes now ␺ X ⬘2⬅ ⑀ 0 ⬘共 ␳ ⬘兲−1 ⑀ F−1 共52兲 and he inelas ic longi udinal and ans e se esponse unc- ions, calcula ed as Rinel L=Winel 00 and Rinel T=Winel 11 +Winel 22 , ha e he ollowing gene al o ms: 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 o de o e alua e he longi udinal and ans e se nuclea unc ions UL,T共 ␬ , ␶ , ␳ ⬘兲one needs o assume a speci ic o m o he inelas ic single-nucleon enso winel ␮ ␯ 共 ⑀ ¯ , ␳ ⬘; ␬ ,␭兲.I is impo an o ema k ha he e exis s some ambigui y in he choice made he e: o ins ance, se e al al e na i es in ol ing di e en exp essions con aining he ou -momen a Q ␮ and/o Q⬘ ␮ a e possible, and hese can lead o di e en esul s. Fo example, in Re . 关30兴, he modi ied ou momen um ans e Q⬘ ␮ is used, al hough hen a p esc ip ion mus be used in o de o eco e he gauge in a iance o he nuclea had onic enso , which is los by making his choice. In Appendix B we p esen he speci ic exp essions o he inelas ic and QE esponses ob ained o a gi en selec ion o he single- nucleon enso s accoun ing o he ene gy shi . Apa om he speci ic o m o he enso w ␮ ␯ , he choice o he a gu- men s o he single-nucleon inelas ic s uc u e unc ions, w1,w2, also p esen s some ambigui ies. In ac , he a ailable pa ame iza ions o w1,w2 ha we employ in ou calcula- ions a e gi en o ee, on-shell, nucleons, while he inclu- sion o he ene gy shi e ec i ely in oduces some “o - shellness” o he ini ial nucleon, by al e ing he ene gy balance a he e ex whe e i couples o he exchanged i - ual pho on. In his case he bound-nucleon Bjo ken a iable is no uniquely de e mined by he inal-s a e in a ian mass and, since no heo e ically de i ed p esc ip ions exis , one has o make some assump ions. As shown in Appendix B, he inelas ici y pa ame e selec ed in his wo k, ␳ ˜ , co e- sponds o he one gi en as 共2mN ␻ ˜ 兲/兩Q2兩= ␳ ⬘共 ␶ ⬘/ ␶ 兲, whe e ␻ ˜ is he ene gy ans e ed o he nucleon in he sys em in which he nucleon is a es . This means ha in ou nume i- cal calcula ions, o a gi en se o alues o ␻ ,Q2, and ␳ ⬘we employ ee-nucleon s uc u e unc ions aken a ou - momen um Q2and Bjo ken a iable 1/ ␳ ˜ . C. Ex ended ela i is ic e mi gas As discussed in p e ious wo ks [8,16,18,20], o a ixed alue o he in a ian mass ␮ X, he RFG yields a scaling unc ion 共 ␺ X ⬘兲= L共 ␺ X ⬘兲= T共 ␺ X ⬘兲=3 4共1− ␺ X ⬘2兲 ␪ 共1− ␺ X ⬘2兲共54兲 which, as a unc ion o he app op ia e scaling a iable ␺ X ⬘,is he same o all alues o ␮ X.1 In Re . [18] he beha io o he longi udinal scaling unc- ion was s udied o he exis ing wo ld da a in he QE egion. This s udy showed ha o a good app oxima ion L共 ␺ ⬘兲su- pe scales, ha is, i does no show any signi ican depen- dence on he momen um ans e ␬ (scaling o he i s kind) and is app oxima ely he same o all nuclea species (scal- 1The unc ion in Eq. (54)di e s om he one used in p e ious wo k [20]by a mul iplica i e unc ion 2 ␰ F/ ␩ F 2关1+1 2 ␰ F共1+ ␺ X ⬘2兲兴.We ha e checked ha his is nume ically unimpo an o all o he kinema ical condi ions conside ed he e. BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-6 ing o he second kind). An exp ession o a phenomenologi- cal longi udinal scaling unc ion, uni 共 ␺ ⬘兲, was ob ained by i ing he da a [21]. Based on hese esul s, we now make he ollowing hypo hesis: we assume ha his uni 共 ␺ ⬘兲, de i ed om he da a, p o ides a good desc ip ion o 共 ␺ X ⬘兲= L共 ␺ X ⬘兲 = T共 ␺ X ⬘兲,(“scaling o he ze o h kind”)as i implici ly con- ains he ini ial-s a e physics, and hus we make, o any ␮ X, he ollowing subs i u ion: 3 4共1− ␺ X ⬘2兲 ␪ 共1− ␺ X ⬘2兲→ ERFG共 ␺ X ⬘兲= uni 共 ␺ X ⬘兲.共55兲 To be mo e speci ic, we calcula e he esponse unc ions as RQE L,T=N ␩ F 3 ␬ mN ␰ F model共 ␺ ⬘兲UQE L,T,共56兲 Rinel L,T共 ␬ , ␶ 兲=N ␩ F 3 ␬ ␰ F 冕 ␮ h esh 1+2␭− ⑀ Sd ␮ X ␮ X model共 ␺ X ⬘兲UL,T,共57兲 whe e ⑀ S=ES/mNis he dimensionless sepa a ion ene gy and model共 ␺ X ⬘兲= 再 3 4共1− ␺ X ⬘2兲 ␪ 共1− ␺ X ⬘2兲model = RFG uni 共 ␺ X ⬘兲model = ERFG. 共58兲 The unc ions RFG and ERFG a e shown in Fig. 1 as unc- ions o ␺ X ⬘, while he unc ions UQE L,Tand UL,Tin Eqs. 共56兲 and 共57兲a e gi en in Appendix B. IV. RESULTS In his sec ion we p esen ou esul s o c oss sec ions and esponse and s uc u e unc ions. In compu ing he in- elas ic had onic enso o Eq. (47), we employ phenomeno- logical i s o he single-nucleon inelas ic s uc u e unc ions. The la e a e measu ed in DIS expe imen s and a a ie y o pa ame iza ions o w1and w2can be ound in he li e a u e [15,32–36], including some a ia ions a ising om he di - e en assump ions made o how o ex ac he neu on s uc u e unc ions om deu e on da a. Unless s a ed o he - wise, in he ollowing we adop he Bodek e al. i o Re s. [15,32,33], which desc ibes bo h he deep inelas ic and eso- nance egions. Fo he QE con ibu ions, we employ he o m ac o pa ame iza ion o Re . [37]. The sensi i i y o he esul s o he di e en pa ame iza ion choices will be dis- cussed la e . Addi ionally, o he Fe mi momen um and he ene gy shi we will employ he alues ob ained in Re . [20], namely, kF=220 MeV/c, ␻ shi =20 MeV o ca bon, kF =236 MeV/c, ␻ shi =18 MeV o aluminum, kF =241 MeV/c, ␻ shi =23 MeV o i on, and kF=245 MeV/c, ␻ shi =25 MeV o gold. A. C oss sec ions In his sec ion we p esen ou esul s o he c oss sec ions in he RFG and ERFG models and compa e hem wi h he a ailable expe imen al da a [38–41]. FIG. 2. Inclusi e c oss sec ion o elec on sca e ing om ca - bon a Einc=500 MeV and ␪ e=60° s he ene gy ans e . The cal- cula ion includes an ene gy shi ␻ shi =20 MeV and he sepa a e QE and inelas ic con ibu ions o he c oss sec ion a e shown. Da a a e om Re . [41]. FIG. 3. As o Fig. 2, bu a Einc=2.020 GeV and sca e ing angle ␪ e=15° (a)and ␪ e=20° (b). Da a a e om Re . [39]. FIG. 1. Scaling unc ion model共 ␺ X ⬘兲o Eq. (58) o he RFG and ERFG models. INELASTIC ELECTRON-NUCLEUS SCATTERING AND…PHYSICAL REVIEW C 69, 035502 (2004) 035502-7 In Fig. 2 we show he inclusi e c oss sec ion o a 12C a ge a Ee=500 MeV and ␪ e=60°. We sepa a e he QE om he inelas ic con ibu ion. We no ice ha he shi ed RFG model (solid line)yields oughly he igh posi ion and heigh o he QE peak, bu ails o ep oduce he ails o he peak, gi ing in pa icula an unobse ed dip a ␻ ⯝200 MeV. On he o he hand he ERFG model (do ed line), while ep oducing he da a in he ails be e , signi i- can ly unde es ima es he c oss sec ion a he peak. This is ela ed o he ac ha , as shown in Fig. 1, he peak o he ERFG uni e sal unc ion ERFG is lowe han he co espond- ing RFG alue. Due o he la ge ex ension o ERFG o e ␺ X ⬘ he no maliza ion o he wo unc ions is he same, namely, 兰 RFGd ␺ X ⬘=兰 ERFGd ␺ X ⬘=1. One migh hen nai ely expec he in eg al in Eq. (57)which yields he inelas ic esponse unc- ions o be he same in he wo models. Howe e , a close inspec ion shows ha his is no he case because he in e- g a ion limi s and/o he weigh ing p o ided by UL,Ta e such ha he ERFG in eg al does no “sa u a e” as does he RFG one. Figu es 3–5 co espond o di e en kinema ical condi- ions, namely, Ee=2.020 and 3.595 GeV (SLAC)and Ee =4.045 GeV (JLab)and a ious sca e ing angles. Conce n- ing Figs. 3(a)and 3(b), a simila end pe sis s, wi h he FIG. 4. As o Fig. 2, bu a Ee=3.595 GeV and sca e ing angle ␪ e=16° (a), 20° (b), 25° (c), and 30° (d). Da a a e om Re . [39]. FIG. 5. As o Fig. 2, bu a Einc=4.045 GeV and sca e ing angle ␪ e=15° (a), 30° (b), 45° (c), and 74° (d). Da a a e om Re . [38]. BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-8 ERFG model signi ican ly unde es ima ing he da a in he egion o he QE peak, whe eas he RFG is close o he da a (pa icula ly o ␪ e=20°), al hough i lea es no oom o o he con ibu ions o be added. No e also ha o his sca - e ing angle he inelas ic channel s a s o be sizable. Examining Figs. 4 we ema k ha he QE peak, which is mo e clea ly sepa a ed om he inelas ic egion in Fig. 4(a), is again well ep oduced in he low- ␻ ail by he ERFG, while i s maximum ag ees be e wi h he RFG. On he o he hand he inelas ic c oss sec ion is in all cases unde es ima ed by he ERFG, while he RFG alone would oughly accoun o wha is obse ed. Simila commen s apply o Fig. 5(a), co esponding o highe ene gy and low sca e ing angle. Fo highe angles [Figs. 5(b)–5(d)] he da a lie oughly in be ween he p edic- ions o he ERFG (smalle )and RFG (la ge )models, he o me again ep oducing he low- ␻ beha io be e . As a gene al esul we obse e ha as he sca e ing angle in- c eases he ange o alidi y o he ERFG also inc eases. Finally, in Figs. 6 and 7 we conside sligh ly di e en kinema ical condi ions, co esponding o ixed alues o 兩Q2兩 in he ange 2–10 共GeV/c兲2, and a ious elec on ene gies (in he ange 8–25 GeV)and angles 共12° –22°兲. Theo e ical esul s o 56Fe a e shown as unc ions o he “labo a o y” Bjo ken a iable xL. The da a co esponding o a ixed Q2a e aken a di e en alues o Eeand ␪ e. Fo 兩Q2兩=2 and 10 共GeV/c兲2(Fig. 6) he a ious da a i easonably well on one plo , whe eas o 兩Q2兩=5 共GeV/c兲2(Fig. 7), o cla i y we ha e sepa a ed he da a in o h ee se s as indica ed in he igu e cap ion. We no ice ha a la ge xL共艌0.6兲 he da a a e close o he ERFG p edic ions, a low xL共0.1–0.3兲 hey a e close o he RFG calcula ion and o 0.3艋xL艋0.6 hey lie in be ween he wo models. This gene al end seems o be espec ed o all alues o Q2(a leas whe e da a a e a ail- able). We ha e also analyzed he e ec in oduced by di e en elec omagne ic o m ac o pa ame iza ions ([42–44]) and e i ied ha i can p oduce a ±3 % unce ain y a he QE peak, bu does no change he gene al ag eemen / disag eemen o he models wi h he da a. Mo eo e , i should be ema ked ha , a he ene gies conside ed in his sec ion, he con ibu ion om he esonance egion o he inelas ic pa o he c oss sec ion is qui e impo an and hus a compa ison wi h esul s ob ained by using pu ely DIS pa- ame iza ions [34,36]o he single-nucleon s uc u e unc- ions is no app op ia e. A he highes 兩Q2兩 alues conside ed he e [Figs. 6(b)and 7], he use o di e en pa ame iza ions [34,36]does no p oduce signi ican a ia ions in he esul s. An impo an commen , al eady an icipa ed in he in o- duc ion, is in o de . The RFG and ERFG models conside ed FIG. 6. Di e en ial c oss sec ion d ␴ /d ␻ d⍀e o elec on sca - e ing on i on, shown as a unc ion o xLa ixed 兩Q2兩. Panel (a): 兩Q2兩=2共GeV/c兲2, he expe imen al poin s, om igh o le , a e aken a 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, he ex- pe imen al poin s, om igh o le , a e aken a 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). Da a a e om Re . [40]. FIG. 7. As o Fig. 6, bu a 兩Q2兩=5共GeV/c兲2. The wo da a in he uppe panel, om igh o le , co espond o Ee=8 GeV and ␪ e=22° and o Ee=9.7 GeV and ␪ e=19.7°; he da a in he middle panel ha e ixed Ee=12 GeV and, om igh o le , ␪ e =12.8,13.3,14.2,15.8,20.6°; he da a in he lowe panel, om igh o le , a e aken a 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. Da a a e om Re . [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 he elec omagne ic s uc u e unc ions by w1,QE共 ␶ 兲= ␶ GM 2共 ␶ 兲, w2,QE共 ␶ 兲=GE 2共 ␶ 兲+ ␶ GM 2共 ␶ 兲 1+ ␶ wi h GE,M he p o on o neu on Sachs elec omagne ic o m ac o s. Finally, he “ o al” esponse unc ions a e e alua ed by adding he abo e QE esponses o he inelas ic ones, i.e., R o L,T=RQE L,T+Rinel L,T. [1]W. M. Albe ico, T. W. Donnelly, and A. Molina i, Nucl. Phys. A512, 541 (1990). [2]W. M. Albe ico, M. B. Ba ba o, A. De Pace, T. W. Donnelly, and A. Molina i, Nucl. Phys. A563, 605 (1993). [3]M. B. Ba ba o, A. De Pace, T. W. Donnelly, and A. Molina i, Nucl. Phys. A596, 553 (1996). [4]J. E. Ama o, M. B. Ba ba o, J. A. Caballe o, T. W. Donnelly, and A. Molina i, Nucl. Phys. A643, 349 (1998). [5]J. E. Ama o, M. B. Ba ba o, J. A. Caballe o, T. W. Donnelly, and A. Molina i, Nucl. Phys. A697, 388 (2002). [6]J. E. Ama o, M. B. Ba ba o, J. A. Caballe o, T. W. Donnelly, and A. Molina i, Phys. Rep. 368, 317 (2002). [7]J. E. Ama o, M. B. Ba ba o, J. A. Caballe o, T. W. Donnelly, and A. Molina i, Nucl. Phys. A723, 181 (2003). [8]J. E. Ama o, M. B. Ba ba o, J. A. Caballe o, T. W. Donnelly, and A. Molina i, Nucl. Phys. A657, 161 (1999). [9]L. Al a ez-Ruso, M. B. Ba ba o, T. W. Donnelly, and A. Mo- lina i, Nucl. Phys. A724, 157 (2003). [10]B. D. Se o and J. D. Walecka, Ad . Nucl. Phys. 16,1(1986). [11]W. M. Albe ico, A. Molina i, T. W. Donnelly, E. L. K onen- be g, and J. W. Van O den, Phys. Re . C 38, 1801 (1988). [12]R. P. Bicke s a and A. W. Thomas, J. Phys. G 15, 1523 (1989). [13]P. R. No on, Rep. P og. Phys. 66, 1253 (2003). [14]S. Simula, Few-Body Sys ., Suppl. 8, 423 (1995). [15]A. Bodek and J. L. Ri chie, Phys. Re . D 23, 1070 (1981);24, 1400 (1981). [16]T. W. Donnelly and I. Sick, Phys. Re . Le . 82, 3212 (1999). [17]M. B. Ba ba o, R. Cenni, A. De Pace, T. W. Donnelly, and A. Molina i, Nucl. Phys. A643, 137 (1998). [18]T. W. Donnelly and I. Sick, Phys. Re . C 60, 065502 (1999). [19]D. B. Day, J. S. McCa hy, T. W. Donnelly, and I. Sick, Annu. Re . Nucl. Pa . Sci. 40, 357 (1990). [20]C. Maie on, T. W. Donnelly, and I. Sick, Phys. Re . C 65, 025502 (2002). [21]J. Jou dan, Nucl. Phys. A603,117(1996). [22]A. De Pace, M. Na di, W. M. Albe ico, T. W. Donnelly, and A. Molina i, Nucl. Phys. A726, 303 (2003). [23]T. De Fo es and J. D. Walecka, Ad . Phys. 15,1(1966). [24]S. Bo i, C. Gius i, and F. D. Paca i, Phys. Rep. 226,1(1993). [25]T. W. Donnelly, in Nuclea /Nucleonic S uc u e and Inclusi e Elec on Sca e ing, P oceedings o he In e na ional School o Physics “En ico Fe mi,” Cou se CLIII, edi ed by A. Molina i and L. Ricca i (IOS, Ams e dam, 2003), p. 183. [26]J. D. Bjo ken and S. D. D ell, Rela i is ic Quan um Mechanics (McG aw-Hill, New Yo k, 1965). [27]T. W. Donnelly and A. S. Raskin, Ann. Phys. (N.Y.)169, 247 (1986). [28]T. W. Donnelly, M. J. Musol , W. M. Albe ico, M. B. Ba ba o, A. De Pace, and A. Molina i, Nucl. Phys. A541, 525 (1992). [29]L. Al a ez-Ruso, M. B. Ba ba o, T. W. Donnelly, and A. Mo- lina i, Phys. Le . B 497, 214 (2001). [30]O. Benha , V. R. Pandha ipande, and I. Sick, Phys. Le . B 410,79(1997). [31]C. Cio i Degli A i and S. Liu i, Phys. Le . B 225, 215 (1989). [32]A. Bodek e al., Phys. Re . D 20, 1471 (1979). [33]A. S ein e al., Phys. Re . D 12, 1884 (1975). [34]P. Amaud uz e al., New Muon Collabo a ion, Phys. Le . B 295, 159 (1992). [35]L. W. Whi low, S. Rock, A. Bodek, E. M. Rio dan, and S. Dasu, Phys. Le . B 250, 193 (1990). [36]A. D. Ma in, R. G. Robe s, W. J. S i ling, and R. S. Tho ne, Eu . Phys. J. C 4, 463 (1998); see also h p://du pdg.du .ac.uk/ hepda a/m s.h ml [37]G. Hohle , E. Pie a inen, I. Sabba S e anescu, F. Bo kowski, G. G. Simon, V. H. Wal he , and R. D. Wendling, Nucl. Phys. B114, 505 (1976). [38]J. A ing on e al., Phys. Re . Le . 82, 2056 (1999). [39]D. B. Day e al., Phys. Re . C 48, 1849 (1993). [40]R. G. A nold e al., Phys. Re . Le . 52, 727 (1984). [41]R. R. Whi ney, I. Sick, J. R. Ficenec, R. D. Kepha , and W. P. T owe , Phys. Re . C 9, 2230 (1974). [42]S. Gals e , H. Klein, J. Mo i z, K. H. Schmid , D. Wegene , and J. Bleckwenn, Nucl. Phys. B32, 221 (1971). [43]F. Iachello, A. D. Jackson, and A. Lande, Phys. Le . B 43, 191 (1973). [44]P. E. Bos ed, Phys. Re . C 51, 409 (1995). BARBARO, CABALLERO, DONNELLY, AND MAIERON PHYSICAL REVIEW C 69, 035502 (2004) 035502-16