Table Of Content

Observation of Nuclear Scaling in the A(e, e′) Reaction at x >1 B 1 1 11 36 27 31 11 K.Sh. Egiyan, N.Dashyan, M. Sargsian, S. Stepanyan, L.B. Weinstein, G. Adams, P. Ambrozewicz, E. Anciant,6 M. Anghinolfi,16 B. Asavapibhop,23 G. Asryan,1 G. Audit,6 T. Auger,6 H. Avakian,36 H. Bagdasaryan,27 J.P. Ball,2 S. Barrow,12 M. Battaglieri,16 K. Beard,20 I. Bedlinski,19 M. Bektasoglu,26 M. Bellis,31 N. Benmouna,13 B.L. Berman,13 N. Bianchi,15 A.S. Biselli,4 S. Boiarinov,19 B.E. Bonner,32 S. Bouchigny,17 R. Bradford,4 D. Branford,10 W.J. Briscoe,13 W.K. Brooks,36 V.D. Burkert,36 C. Butuceanu,40 J.R. Calarco,24 D.S. Carman,4 B. Carnahan,5 C. Cetina,13 L. Ciciani,27 P.L. Cole,35 A. Coleman,40 D. Cords,36 3 0 P. Corvisiero,16 D. Crabb,39 H. Crannell,5 J.P. Cummings,31 E. DeSanctis,15 R. DeVita,16 P.V. Degtyarenko,36 0 R. Demirchyan,1 H. Denizli,29 L. Dennis,12 K.V. Dharmawardane,27 K.S. Dhuga,13 C. Djalali,34 G.E. Dodge,27 2 D. Doughty,7 P. Dragovitsch,12 M. Dugger,2 S. Dytman,29 O.P. Dzyubak,34 M. Eckhause,40 H. Egiyan,36 n L. Elouadrhiri,36 A. Empl,31 P. Eugenio,12 R. Fatemi,39 R.J. Feuerbach,4 J. Ficenec,38 T.A. Forest,27 H. Funsten,40 a M. Gai,8 G. Gavalian,24 S. Gilad,22 G.P. Gilfoyle,33 K.L. Giovanetti,20 P. Girard,34 C.I.O. Gordon,14 K. Griffioen,40 J M. Guidal,17 M. Guillo,34 L. Guo,36 V. Gyurjyan,36 C. Hadjidakis,17 R.S. Hakobyan,5 J. Hardie,7 D. Heddle,7 4 P. Heimberg,13 F.W. Hersman,24 K. Hicks,26 R.S. Hicks,23 M. Holtrop,24 J. Hu,31 C.E. Hyde-Wright,27 Y. Ilieva,13 1 M.M. Ito,36 D. Jenkins,38 K. Joo,36 J.H. Kelley,9 M. Khandaker,25 D.H. Kim,21 K.Y. Kim,29 K. Kim,21 M.S. Kim,21 1 W. Kim,21 A. Klein,27 F.J. Klein,36 A. Klimenko,27 M. Klusman,31 M. Kossov,19 L.H. Kramer,11 Y. Kuang,40 v S.E. Kuhn,27 J. Kuhn,4 J. Lachniet,4 J.M. Laget,6 D. Lawrence,23 Ji Li,31 K. Lukashin,36 J.J. Manak,36 8 C. Marchand,6 L.C. Maximon,13 S. McAleer,12 J. McCarthy,39 J.W.C. McNabb,4 B.A. Mecking,36 0 0 S. Mehrabyan,29 J.J. Melone,14 M.D. Mestayer,36 C.A. Meyer,4 K. Mikhailov,19 R. Minehart,39 M. Mirazita,15 1 R. Miskimen,23 L. Morand,6 S.A. Morrow,6 M.U. Mozer,26 V. Muccifora,15 J. Mueller,29 L.Y. Murphy,13 0 G.S. Mutchler,32 J. Napolitano,31 R. Nasseripour,11 S.O. Nelson,9 S. Niccolai,13 G. Niculescu,26 I. Niculescu,20 3 B.B. Niczyporuk,36 R.A. Niyazov,27 M. Nozar,36 G.V. O’Rielly,13 A.K. Opper,26 M. Osipenko,16 K. Park,21 0 / E. Pasyuk,2 G. Peterson,23 S.A. Philips,13 N. Pivnyuk,19 D. Pocanic,39 O. Pogorelko,19 E. Polli,15 x S. Pozdniakov,19 B.M. Preedom,34 J.W. Price,3 Y. Prok,39 D. Protopopescu,14 L.M. Qin,27 B.A. Raue,11 e - G. Riccardi,12 G. Ricco,16 M. Ripani,16 B.G. Ritchie,2 F. Ronchetti,15 P. Rossi,15 D. Rowntree,22 P.D. Rubin,33 cl F. Sabati´e,6 K. Sabourov,9 C. Salgado,25 J.P. Santoro,38 V. Sapunenko,16 R.A. Schumacher,4 V.S. Serov,19 u A. Shafi,13 Y.G. Sharabian,1 J. Shaw,23 S. Simionatto,13 A.V. Skabelin,22 E.S. Smith,36 L.C. Smith,39 n D.I. Sober,5 M. Spraker,9 A. Stavinsky,19 P. Stoler,31 I. Strakovsky,13 S. Strauch,13 M. Strikman,28 : v M. Taiuti,16 S. Taylor,32 D.J. Tedeschi,34 U. Thoma,36 R. Thompson,29 L. Todor,4 C. Tur,34 M. Ungaro,31 Xi M.F. Vineyard,37 A.V. Vlassov,19 K. Wang,39 A. Weisberg,26 H. Weller,9 D.P. Weygand,36 C.S. Whisnant,34 E. Wolin,36 M.H. Wood,34 L. Yanik,13 A. Yegneswaran,36 J. Yun,27 B. Zhang,22 J. Zhao,22 and Z. Zhou22 r a (The CLAS Collaboration) 1 Yerevan Physics Institute, Yerevan 375036 , Armenia 2 Arizona State University, Tempe, Arizona 85287-1504 3 University of California at Los Angeles, Los Angeles, California 90095-1547 4 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 5 Catholic University of America, Washington, D.C. 20064 6 CEA-Saclay, Service de Physique Nucl´eaire, F91191 Gif-sur-Yvette, Cedex, France 7 Christopher Newport University, Newport News, Virginia 23606 8 University of Connecticut, Storrs, Connecticut 06269 9 Duke University, Durham, North Carolina 27708-0305 10 Edinburgh University, Edinburgh EH9 3JZ, United Kingdom 11 Florida International University, Miami, Florida 33199 12 Florida State University, Tallahassee, Florida 32306 13 The George Washington University, Washington, DC 20052 14 University of Glasgow, Glasgow G12 8QQ, United Kingdom 15 INFN, Laboratori Nazionali di Frascati, Frascati, Italy 16 INFN, Sezione di Genova, 16146 Genova, Italy 17 Institut de Physique Nucleaire ORSAY, Orsay, France 18 Institute fu¨r Strahlen und Kernphysik, Universit¨at Bonn, Germany 19 Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia 20 James Madison University, Harrisonburg, Virginia 22807 21 Kungpook National University, Taegu 702-701, South Korea 2 22 Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 23 University of Massachusetts, Amherst, Massachusetts 01003 24 University of New Hampshire, Durham, New Hampshire 03824-3568 25 Norfolk State University, Norfolk, Virginia 23504 26 Ohio University, Athens, Ohio 45701 27 Old Dominion University, Norfolk, Virginia 23529 28Pennsylvania State University, State Collage, Pennsylvania 16802 29 University of Pittsburgh, Pittsburgh, Pennsylvania 15260 30 Universita’ di ROMA III, 00146 Roma, Italy 31 Rensselaer Polytechnic Institute, Troy, New York 12180-3590 32 Rice University, Houston, Texas 77005-1892 33 University of Richmond, Richmond, Virginia 23173 34 University of South Carolina, Columbia, South Carolina 29208 35 University of Texas at El Paso, El Paso, Texas 79968 36 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 37Union College, Schenectady, NY 12308 38 Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435 39 University of Virginia, Charlottesville, Virginia 22901 40 College of William and Mary, Williamsburg, Virginia 23187-8795 (Dated: February 8, 2008) The ratios of inclusive electron scattering cross sections of 4He, 12C, and 56Fe to 3He have been measuredforthefirsttime. ItisshownthattheseratiosareindependentofxB atQ2 >1.4(GeV/c)2 forxB >1.5wheretheinclusivecrosssectiondependsprimarilyonthehigh-momentumcomponents ofthenuclearwavefunction. Theobservedscalingshowsthatthemomentumdistributionsathigh- momentahavethesameshapeforallnucleianddifferonlybyascalefactor. Theobservedonsetof thescalingatQ2 >1.4andxB >1.5isconsistentwiththekinematicalexpectationthattwonucleon short range correlations (SRC) are dominate the nuclear wave function at pm & 300 MeV/c. The values of these ratios in the scaling region can be related to the relative probabilities of SRC in nuclei with A≥3. Our data demonstrate that for nuclei with A≥12 these probabilities are 5-5.5 times larger than in deuterium,while for 4Heit is larger by a factor of about 3.5. PACSnumbers: PACS:13.60.Le,13.40.Gp,14.20.Gk I. INTRODUCTION and meson exchange currents (at high Q2) can be sig- nificantly reduced, which corresponds to studying the Due to the strong interaction and short distances be- low-energy-loss side of the quasi-elastic peak. In Eq. 1 tween the nucleons in nuclei, there is a significant prob- Q2 is the four-momentum squared of the virtual photon ability for nucleon wave functions to overlap, resulting (Q2 = −qµqµ >0), ν is the energy transfer, xB is the inshortrangenucleon-nucleoncorrelations(SRC)in nu- Bjorken scaling variable, and M is the nucleon mass, clei [1]. Investigation of SRC is important for at least Many previous analyses of data in this kinematic re- two reasons. First, because of the short range nature gionconcentrateonusingy-scalingtostudynucleonmo- ofthesecorrelations,they shouldcontributesignificantly mentum distributions (see e.g. Refs. [2], [3]). While to the high-momentum component of the nuclear wave this technique provides some information about the nu- function. Second, scattering from nucleons in SRC will clear wave function, it does not measure the probability provideuniquedataonthemodificationofdeeplybound of finding SRC in nuclei. nucleons, which is extremely important for a complete Meanwhile, the data at x >1 can be used to mea- B understanding of nucleon structure in general. sure the probability of finding SRC in nuclei. There are High-energy inclusive electron scattering from nuclei, theoretical predictions that at momenta higher than the A(e,e′), is one of the simplest ways to investigate SRC. Fermi momentum, the nucleon momentum distributions In particular, it is probably the best way to measure in light and heavy nuclei are similar (see e.g. Ref. [4]). the probabilities of SRC in nuclei. The main problem This implies that they originate predominantly from the in these studies is selecting the electron-SRC scattering interaction between two nearby nucleons, i.e. due to events from the orders-of-magnitude larger background SRC. If the A(e,e′) cross section depends primarily on of inelastic and/or quasi-elastic interaction of electrons the nuclear wave function and the shape of this wave with the uncorrelated low-momentum nucleons. function at high-momentum is really universal, then in By measuring cross sections at this high-momentum region the ratio of weighted (e,e′) cross sections for different nuclei [34] should scale, i.e. Q2 x = >1, (1) they should be independent of electron scattering vari- B 2Mν ables (Q2 and x ), with the magnitude of the scaling B contributions from inelastic electron-nucleon scattering factor being proportional to the relative probability of 3 SRC in the two nuclei [5, 6]. p are known.) From Eq.(2) one obtains: A InRef.[6]thiswascheckedbyanalyzingexistingSLAC A(e,e′) data for deuterium [7, 8, 9] and heavier nuclei ∆M2−Q2+ Q2 M − M2 +p~ 2 [10]. They found an indication of scaling at Q2 >1 and m x (cid:16) A q A−1 m (cid:17) N B x ≥1.5. However,since the data for deuterium and the B −2~q·~p −2M M2 +p~ 2 = 0, (3) heavy nuclei were collected in different experiments at m Aq A−1 m similarQ2 butatdifferentelectronscatteringanglesand incident electron energies, to find the ratios at the same where ∆M2 = MA2 +MA2−1−m2N and p~m = ~pf −~q = values of (xB,Q2), a complicated fitting and interpola- −p~A−1 is the recoil momentum involved in the reaction tion procedure was applied [6] to the data. The main (sometimes referred to as the ‘missing momentum’ in problem was that the cross sections varied very strongly (e,e′p) reactions). Eq. (3) defines a simple relationship with angle, incident energy, and Q2. To simplify the in- between |p~mmin| and xB at fixed Q2. This relation for 2 terpolation, the electron-deuteron cross section was first deuterium at various values of Q is shown in Fig. 2a. dividedby the theoreticalcalculationwithin the impulse Fig. 2b shows the same relationshipfor various nucleiat approximation. Therefore,thedataarenotpurelyexper- Q2 = 2 (GeV/c)2. Note that this relationship is differ- imental, since they include the theoretical calculations, ent for the different nuclei, due primarily to differences and the ratios may have been affected by the fitting and in the mass of the recoil A−1 system. This minimum interpolation procedures. recoilmomentum is one of the possible definitions of the Inthiswork,theyieldsofthereactionA(e,e′)for3He, scaling variable y. 4He, 12C, and 56Fe targets are measured in the same kinematical conditions, and the ratios A(e,e′)/3He(e,e′) 1 are obtained for 1< x < 2 and Q2 >0.65 (GeV/c)2. c a) 4.0 B v/ 0.8 3.0 Furthermore, using the scaling behavior of these ratios, Ge 2.5 hthaevereblaeteinveexptrroabcatbedil.ityofNN SRCforthevariousnuclei |, min 0.6 12..50 m 0.4 1.0 P | 0.5 0.2 ↑ II. KINEMATICS AND PREDICTIONS Q2 0 b) c In order to suppress the background from quasi- v/ 0.8 elastic interactions of electrons with the uncorrelated e G ltohwe-kminoemmeanttiucmvanruiacbleloenssx(BseaenFdigQ.21.a),wefurtherrestrict |, min 0.6 3DHe m 0.4 4He / / |P A>12 e e 0.2 e e 0 q p q i 00..22 00..44 00..66 00..88 11 11..22 11..44 11..66 11..88 22 pi pf -pi pf xB SRC p SRC AA A-1 i A-2 FIG.2: Theminimumrecoil momentumas afunction of xB. a)FordeuteriumatseveralQ2(in(GeV/c)2);b)Fordifferent a) b) nuclei at Q2 = 2.0 (GeV/c)2. Horizontal lines at 250 MeV/c indicatetheFermimomentumtypicaloftheuncorrelatedmo- FIG. 1: Two mechanisms of A(e,e′) scattering. a) single tion of nucleons in nuclei. nucleon model; b) Short Range Correlation model. OnecanseefromFig.2thatforanynucleusAandfixed For quasi-elastic A(e,e′) scattering, xB, Q2, and the Q2, we canfind the valuexoB suchthat atxB > xoB the minimum A − 1 recoil momentum contributing to the magnitude of the minimum recoil momentum, |p~ min|, m reaction are related by energy and momentum conserva- contributing to the reaction, exceeds the average Fermi tion: momentum in nucleus A. It shouldbe pointed outthat the initial momentumof 2 2 2 (q+pA−pA−1) =pf =mN, (2) the struck nucleon p~i is equal to p~m only in the simplest model where the virtual photon is absorbed on one nu- where q, pA, pA−1, and pf are the four-momenta of the cleonandthatnucleonleavesthenucleuswithoutfurther virtualphoton,targetnucleus,residualA−1system,and interactions (the Plane Wave Impulse Approximation). knocked-out nucleon respectively (note that only q and In reality, the (e,e′) reaction effectively integrates over 4 many values of p ≥ pmin. In addition, this simple re- The ratios in Fig. 3 show a nice plateau starting from m m lation between recoil momentum and initial momentum x >1.5 for both nuclei and all Q2. The experimentally B is modified by FinalState Interactions(FSI) andthe ex- obtainedratiointhe scalingregioncanbe usedto deter- citation energy of the residual nucleus. These make it mine the relative probability of finding correlated NN difficult to determine the nuclear wave function directly pairs in different nuclei. However one needs to take into from (e,e′) cross sections. However, for our purposes, account two main factors: first the final state interac- it is sufficient to know that when the minimum recoil tions of a nucleon with the residual system, and second momentum contributing to the reaction is much larger the NN pair center-of-mass motion. thanthe Fermimomentum, the initialmomentum ofthe In the SRC model, FSI do not destroy the scaling be- struck nucleon will also be larger. havior of the ratio, R. Indeed, in the light-cone approxi- Letusnowconsidervariouspredictionsoftheratiosof mationof the SRC model, if the invariantmass ofthe fi- weighted (e,e′) cross sections for different nuclei. In the nalNN systemissufficientlylarge, (q+m )2−m > D D mechanism for inclusive (e,e′) scattering at xB >1 with 50-100 MeV, then the scattering ampplitude will depend virtual photon absorption on a single nucleon and the mainly on the light-cone fraction of the interacting nu- A−1 system recoiling intact without FSI (see Fig. 1a), cleon’smomentumα=(E−p )/M,andhasonlyaweak z the minimum recoil momentum for different nuclei at dependence on the conjugated variables E +p and p fixedQ2 differsandthisdifferenceincreaseswithxB (see [6,17,18]. Asaresult,theclosureapproximatiozncanbet Fig.2). Therefore,the crosssectionratiobetweendiffer- applied in the light-cone reference frame, allowing us to ent nuclei will increase with xB and will not scale. sum over all final states and use the fact that this sum In the Short Range Correlations model of Frankfurt is normalized to unity. After using the closure approx- and Strikman [1] (see Fig. 1b) the high-momentum part imation the inclusive cross section will depend on the of the nuclear momentum distribution is due to corre- light-cone momentum distribution of the nucleon in the lated nucleon pairs. This means that when the electron nucleus,integratedoverthetransversemomentumofthe scatters from a high-momentum nucleon in the nucleus, nucleon, ρ (α) [5]. Thus, within the light cone descrip- A we can consider this scattering as an electron-deuterium tion the Eq.(4) measures the ratio of ρ (α) for nuclei A interaction with the spectator A−2 system almost at A1 and A2 in the high-momentum range of the target rest. Therefore,accordingtoFig. 2a,startingfromsome nucleon. threshold x0 for fixed Q2 the cross section ratio B In the lab frame description (in the virtual nucleon RAA21(Q2,xB)= σσAA21((QQ22,,xxBB))//AA21, (4) abmpeepnartopaap,claihen)d,dhfFooSrwIleasvrhegore,ultvdhaelbueecslcoasolucfrueilnatateeprdparceotxixnpimglicanittuliyocnle(oscenaenmen.oogt-. where σ and σ are the inclusive electron scattering Ref. [17]). Within the SRC model at high recoil mo- A1 A2 cross sections from nuclei with atomic numbers A1 and menta, FSI are dominated by the rescattering of the A2 respectively, will scale (will be constant). Scaling re- knocked-out nucleon with the correlated nucleon in the sultsfromthe dominanceofSRCinthe high-momentum SRC [6, 17]. Therefore, FSI will be localized in SRC, component of the nuclear wave function, and it should and will cancel in the ratio R. As a result, Eq.(4) at be observed, for example, for the cross section ratios of xB >x0B couldberelatedtotheratioofhigh-momrntum heavy nuclei to light nuclei such as 3He. part of nucleon-momentum distributions in A1 and A2 Fig. 3a shows R3CHe, for A1 = 12C and A2 = 3He, as nuclei [17]. a function of x for Q2 from 1.5 to 2.5 (GeV/c)2 calcu- Having an underlying model of the nuclear spectral B latedintheSRCmodel[11]. TheratioforA1 = 56Feand functions,onecanrelatethemeasuredratiosinEq.(4)to A2 = 3HeisshowninFig. 3b. Thecalculationsusedthe theSRCpropertiesofthe nuclearwavefunction. Within Faddeevwavefunctionfor3HecalculatedusingtheBonn the spectral function model [1], in which correlated nu- NN potential [12]. The momentum distributions for cleon pair is assumed at rest with the nucleon momen- heavier nuclei have been modeled through a two compo- tum distribution in pair identical to that in deuteron, nentofmomentumdistributionusingmeanfielddistribu- the ratio in Eq. (4) could be directly related to the per- tions for small nucleon momenta and using the deuteron nucleon SRC probability in nucleus A relative to deu- momentumdistributionforp>250MeV/c,scaledbyfac- terium, a2(A). In models of the nuclear spectral func- tora2(A),per-nucleonprobabilityofNN SRCinnucleus tion[19]inwhichtwo-nucleoncorrelationsaremovingin A, estimated from Ref. [6]. The mean field momentum themeanfieldofthespectatorA−2system,theanalysis distributionsusedtheHarmonicOscillatorwavefunction ofEq. (4)willyieldslightlysmallervaluesfora2(A). Cal- for 12C and the quasi-particle Lagrange Method of [13] culationsbyCiofidegliAtti[20]indicatethatthismotion for 56Fe. For the description of the eN interaction, the doesnotaffectthescalingbutcandecreasetheextracted inelastic form factor parameterization of Ref. [14] and a2(A) for 56Fe by up to 20%. Howeverit is importantto the dipole elastic form-factors have been used. These emphasize that since both approximations predict sim- calculations are in reasonable agreement with existing ilar (light cone) momentum distributions, both models A(e,e′)X experimental data from SLAC [15] and from leadtoasimilarratioofthelight-conespectralfunctions Jefferson Lab Hall C [16]. and the overall probability of high-momentum nucleons 5 remains practically the same. et al. [21] use the nuclear spectral function in the lab One can summarize the predictions of the SRC model system and calculate the FSI using a correlatedGlauber fortheratiosoftheinclusivecrosssectionsfromdifferent approximation (CGA), in which the initial momenta of nuclei as follows (see Fig. 3): the re-scattered nucleons are neglected. In this model the cross section at x >1 originates mainly from FSI 2 B • Scaling (xB independence) is expected for Q ≥1 andthereforethe crosssectionratioswillnotscale. This (GeV/c)2 andx0B ≤xB <2wherex0B isthethresh- modelpredictsthattheseratiosalsodependonQ2,since old for high recoil momentum. itincludesanoticeablereductionofFSIinordertoagree • No scaling is expected for Q2 < 1 (GeV/c)2. with the data at Q2 ≥2 (GeV/c)2. Benhar et al. at- tribute this reduction in FSI to color transparency ef- • For x ≤x0 the ratios should have a minimum at fects [35]. The requirement of large color transparency B B x =1andshouldgrowwithx sinceheavynuclei effects also results in a strong A dependence of the ratio B B have a broader momentum distribution than light since the amount of the FSI suppression depends on the nuclei for p<0.3 GeV/c. number of nucleons participating in the rescattering. ThemainpredictionsoftheCGAmodelforthenuclear • The onset of scalingdepends onQ2; x0 should de- cross section ratios are as follows: B crease with increasing Q2. • No scaling is predicted for Q2 ≥1 (GeV/c)2 and • Inthescalingregime,theratiosshouldbeindepen- x <2. B 2 dent of Q . • The nuclear ratios should vary with Q2. • Inthescalingregimetheratiosshoulddependonly • The ratios should depend on A. weakly on A for A≥ 10. This reflects nuclear sat- uration. • ThemodelisnotapplicablebelowQ2=1(GeV/c)2. • Ratiosinthescalingregimeareproportionaltothe Thus, measuring the ratios of inclusive (e,e′) scatter- ratios of the two-nucleon SRC probabilities in the ing at x >1 and Q2 >1 (GeV/c)2 will yield important B two nuclei. information about the reaction dynamics. If scaling is observed, then the dominance of the SRC in the nuclear 3 wavefunctionis manifestedandthe measuredratioswill containinformationabouttheprobabilityoftwo-nucleon 2.5 short range correlations in nuclei. e3 Q2=2.5 H 2 s2 Q2=1.5 1 1.5 /2 III. EXPERIMENT 1 C 1 s3 0.5 a) In this paper we present the first experimental studies of ratios of normalized, and acceptance- and radiative- 0 corrected, inclusive yields of electrons scattered from 2.5 4He, 12C, 56Fe, and 3He measured under identical kine- 3 He 2 Q2=2.5 matical conditions. s6 Q2=1.5 The measurements were performed with the CEBAF 5 /61.5 LargeAcceptanceSpectrometer(CLAS)inHallBatthe 5 e Thomas Jefferson National Accelerator Facility. This is F 1 s3 the first CLAS experiment with nuclear targets. Elec- 0.5 b) tronswith2.261and4.461GeVenergiesincidenton3He, 4 12 56 He, C, and Fe targets have been used. We used he- 0 00..88 11 11..22 11..44 11..66 11..88 22 lium liquefiedin cylindricaltargetcells 1 cmin diameter x and4cm long,positionedonthe beamapproximatelyin B the center of CLAS. The solid targets were thin foils of FIG. 3: SRC Model predictions for the normalized inclusive 12C (1mm) and56Fe (0.15mm) positioned1.5cmdown- cross section ratio as a function of xB for several values of stream of the exit window of the liquid target. Data on Q2 (in (GeV/c)2). Note the scaling behavior predicted for solidtargetshavebeentakenwithanemptyliquidtarget xB >1.4. a) 12C to 3He, b) 56Fe to3He. cell. The CLAS detector [24] consists of six sectors, each functioning as an independent magnetic spectrometer. Another possible mechanism for inclusive (e,e′) scat- Six superconducting coils generate a toroidal magnetic tering at x >1 is virtual photon absorption on a single field primarily in the azimuthal direction. Each sector B nucleon followed by NN rescattering [21, 22]. Benhar is instrumented with multi-wire drift chambers [25] and 6 0.6 time-of-flight scintillator counters [26] that coverthe an- gular range from 8◦ to 143◦, and, in the forward region 0.5 (8◦ <θ <45◦),withgas-filledthresholdCherenkovcoun- 0.4 ters (CC) [27] and lead-scintillator sandwich-type elec- c tromagnetic calorimeters (EC) [28]. Azimuthal coverage c0.3 A for CLAS is limited only by the magnetic coils, and is 0.2 approximately90%atlargepolaranglesand50%atfor- ward angles. The CLAS was triggered on scattered elec- 0.1 Q2= 1.55 Q2= 1.85 tronsbyaCC-ECcoincidenceat2.2GeVandbytheEC 0 alone with a ≈1 GeV electron threshold at 4.4 GeV. 0.5 Forouranalysis,electronsareselectedinthekinemati- calregionQ2 >0.65(GeV/c)2 andxB >1wherethecon- 0.4 tribution from the high-momentum components of the c nuclear wave function should be enhanced. We also re- Ac0.3 0.2 1 0.1 Q2= 2.15 Q2= 2.45 0.8 0 0.6 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 C E X X R 0.4 B B 0.2 FIG.5: TheacceptancecorrectionfactorsasafunctionofxB. • 3He, ◦ 12C. Q2 are in (GeV/c)2. 0 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 p ,GeV/c e FIG. 4: The ratio (REC) of energy deposited in the CLAS B. Acceptance Corrections electromagnetic calorimeter (EC) to the electron momentum pe asafunction ofpe at beamenergy 4.461 GeV.Thelineat REC ≈0.25islocated3standarddeviationsbelowthemean, We used the Monte Carlo techniques to determine the as determined by measurements at several values of pe. This electron acceptance correction factors. Two iterations cut was used to identify electrons. were done to optimize the cross section model for this purpose. In the first iteration we generated events using the SRC model [11] and determined the CLAS detec- quire that the energy transfer, ν, should be >300 MeV tor response using the GEANT-based CLAS simulation (the characteristicmissingenergyforSRC is∼260MeV program,taking into account “bad” or “dead” hardware [1]). In this region one expects that inclusive A(e,e′) channels in various components of CLAS, as well as re- scattering will proceedthroughthe interactionof the in- alistic position resolution for the CLAS drift chambers. coming electron with a correlated nucleon in a SRC. We then used the CLAS data analysis package to recon- structthese eventsusing the same electronidentification criteriathatwasappliedtotherealdata. Theacceptance A. Electron Identification correctionfactorswerefoundastheratiosofthenumber of reconstructed and simulated events in each kinematic Electrons were selected in the fiducial region of the bin. Then the acceptance corrections were applied to CLAS sectors. The fiducial region is a region of az- the data event-by-event, i.e. each event was weighted imuthal angle, for a given momentum and polar angle, by the acceptance factor obtained for the corresponding 2 where the electrondetection efficiency is constant. Then (∆x ,∆Q ) kinematic bin and the cross sections were B a cut on the ratio of the energy deposited in the EC to calculated as a function of x and Q2. For the second B themeasuredelectronmomentump (R )wasusedfor iteration the obtained cross sections were fitted and the e EC final selection. In Fig. 4 R vs. p for the 56Fe target fit-functions were used to generate a new set of data, EC e at 4.4 GeV is shown. The lines shows the applied cut at and the process was repeated. Fig. 5 shows the electron R ≈0.25whichislocated3standarddeviationsbelow acceptance factors after the second iteration for liquid EC the meanas determinedby measurementsatseveralval- (3He) and solid (12C) targets. We used the difference ues of p . A Monte Carlo simulation showed that these between the iterations as the uncertainty in the accep- e cuts reduce the A(e,e′) yield by less then 0.5%. tancecorrectionfactor. Notethatthe acceptanceforthe We estimated the π− contamination in the electron carbontargetis smallerthanforthe heliumtarget. This sampleforawideangularrangeusingthephoto-electron is due to the closer location of the solid targets to the distributionsintheCLASCherenkovcounters. Wefound CLAS coils, which limit azimuthal angular coverage of that for x >1 this is negligible. the detectors. B 7 4 3 3.5 2.5 Q2<1.4 3 Q2<1.4 2 2.5 2 3 4 3 C1 He 2 He He1.5 R R 1.5 1 1 0.5 0.5 0 0 3.5 2.5 Q2>1.4 3 Q2>1.4 2 2.5 2 3 4 3 1 e 2 e e1.5 C H H H R R 1.5 1 1 0.5 0.5 0 0 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 X X B B FIG.6: R3CHe,theper-nucleonyieldratios for 12Cto3He. a) FIG. 7: The same as Fig. 6 for 4He. ◦ 0.65 <Q2 < 0.85; (cid:3) 0.9 <Q2 < 1.1; △ 1.15 <Q2 < 1.35 (GeV/c)2, all at incident energy 2.261 GeV. b) ◦ 1.4 <Q2 < 1.6 (GeV/c)2 and at incident energy 2.261 4 GeV; (cid:3) 1.4<Q2 <2.0 ; △ 2.0<Q2<2.6 (GeV/c)2, both at 3.5 incident energy 4.461 GeV. Statistical errors are shown only. 3 Q2<1.4 2.5 3 Fe He 2 R C. Radiative Corrections 1.5 1 Thecrosssectionratioswerecorrectedforradiativeef- 0.5 fects. The radiative correction for each target as a func- 0 tion of Q2 and x was calculated as the ratio 3.5 B 3 Q2>1.4 2 dσrad(xB,Q2) 2.5 C (x ,Q )= , (5) rad B dσnorad(xB,Q2) Fe He3 2 R 1.5 where dσrad(x ,Q2) and dσnorad(x ,Q2) are the radia- B B 1 tively corrected and uncorrected theoretical cross sec- 0.5 tions, respectively. The cross sections have been calcu- 0 lated using [11]. 1 1.2 1.4 1.6 1.8 2 X B IV. RESULTS FIG. 8: The same as Fig. 6 for 56Fe. We constructed ratios of normalized, acceptance- and radiative-corrected inclusive electron yields on nuclei 4He, 12C and 56Fe divided the yield on 3He in the range whereNe andNT arethenumberofelectronsandtarget nuclei,respectively,Acc is the acceptancecorrectionfac- of kinematics 1< x <2. Assuming that electron detec- B tor,and∆Q2 and∆x arethebinsizesinQ2 andinx , tion efficiency from different targets is the same, these B B respectively. Since electron detection efficiency in CLAS ratios,weightedby atomicnumber, areequivalentto the is expected to be > 96%, we compare obtained yields ratios of cross sections in Eq. 4. The normalized yields for each x and Q2 bin have with radiatedcross sections calculatedby Ref. [11] code. B Within systematic uncertainies (see below) the satisfac- been calculated as: tory agreement has been found between our results and dY Ne′ 1 the calculationsfromRef. [11]that weretuned onSLAC = · (6) dQ2dx ∆Q2∆x N N Acc [15] and Jefferson Lab Hall C [16] data. B B e T 8 The ratios R3AHe, also corrected for radiative effects, Q2 1.55 1.85 2.15 2.45 are defined as: δRA 7.1 5.8 4.9 5.1 δRHe4 0.7 0.7 0.7 0.7 3YA CA RA (x ) = Rad, (7) 3He B AY3HeCR3Hade TABLE I: Systematic uncertainties δRA and δRHe4 for the where Y is the normalizedyield in a given (xB, Q2) bin, rtaivteioly,oifnn%or.mQal2iziend(GinecVlu/sciv)2e,yaineldds∆RQ3C2H,F=ee±a0n.d15R(34GHHeeeVr/ecs)p2e.c- and C is the radiative correction factor from Eq. 5 Rad for each nucleus. Fig. 6 shows these ratios for 12C at several values of Q2. Figs. 7 and 8 show these ratios for 4He and 56Fe, B. Probabilities of Two-Nucleon Short respectively. These data have the following important Range Correlations in Nuclei characteristics: • ThereisaclearQ2 evolutionoftheshapeofratios: Our data are clearly consistent with the predictions of the NN SRC model. The obtained ratios, RA , for 3He – At low Q2 (Q2 <1.4(GeV/c)2), RA (x )in- 1.4<Q2 <2.6 (GeV/c)2 region are shown in Table II as creases with x in the entire 1<x3H<e 2Brange a function of x . Fig. 9 shows these ratios for the 12C B B B (see Figs. 6a – 8a). and 56Fe targets together with the SRC calculation re- sults usingofRef. [11]whichusedthe estimatedscaling – At high Q2 (Q2 ≥1.4 (GeV/c)2) RA (x ) is independent of x for x > x0 ≈3H1.e5. B(See factors, a2(A), (per-nucleon probabilitiy of NN SRC in B B B nucleus A) from Ref. [6]. Good agreement between our Figs. 6b – 8b.) data and calculation is seen. Note that one of the goals of the present paper is to determine these factors more • The value of RA (x ) in the scaling regime is in- 3He B precisely (see below). dependent of Q2. • The value of RA (x ) in the scaling regime for 4 3He B A>10suggestsaweakdependenceontargetmass. 3.5 3 2.5 2 3 A. Systematic Errors C1 He 2 4 R → 1.5 ↑ 1 1 Thesystematicerrorsinthismeasurementaredifferent 0.5 a) for different targets and include uncertainties in: 0 3.5 • fiducial cut applied: ≈ 1%; 3 2.5 • radiative correction factors: ≈ 2%; 4 3 Fe He 2 → R • target densities and thicknesses: ≈ 0.5% and 1.0% 1.5 ↑1 for solid targets; 0.5% and 3.5% for liquid targets, 1 respectively. 0.5 b) 0 1 1.2 1.4 1.6 1.8 2 • acceptance correction factors (Q2 dependent): be- X tween 2.2 and 4.0% for solid targets and between B 1.8 and 4.3% for liquid targets. FIG. 9: R3AHe(xB) as a function of xB for 1.4<Q2 <2.6 (GeV/c)2, statistical errors are shown only. Curves are SRC Some of systematic uncertainties will cancel out in the model predictions for different Q2 in the range 1.4 (GeV/c)2 yield ratios. For the 3He/4He ratio, all uncertainties ex- (curve 1) to 2.6 (GeV/c)2 (curve 4), respectively, for a) 12C, cept those on the beam current and the target density b) 56Fe. divideout,givingatotalsystematicuncertaintyof0.7%. For the solid-target to 3He ratios, only the electron de- tectionefficiencycancels. Thequadraticsumoftheother uncertainties is between 5% and 7%, depending on Q2. Experimentaldatainthescalingregioncanbeusedto The systematic uncertainties onthe ratios for all targets estimate the relative probabilities of NN SRC in nuclei and Q2 are presented in Table I. comparedto3He. AccordingtoRef.[1]theratioofthese 9 e3 4 XB 4He 12C 56Fe A H3.5 , r3 3 01..9055 ±± 00..0055 00..8768 ±± 00..000044 00..7772 ±± 00..000033 00..8702 ±± 00..000044 ARHe2.5 1.15 ± 0.05 0.94 ± 0.006 0.96 ± 0.006 0.94 ± 0.007 2 1.25 ± 0.05 1.19 ± 0.012 1.33 ± 0.012 1.33 ± 0.015 1.5 1.35 ± 0.05 1.41 ± 0.021 1.77 ± 0.025 1.81 ± 0.030 1 1.45 ± 0.05 1.58 ± 0.033 2.12 ± 0.044 2.17 ± 0.055 0.5 a) 1.55 ± 0.05 1.71 ± 0.049 2.12 ± 0.059 2.64 ± 0.087 0 7 1.65 ± 0.05 1.70 ± 0.063 2.29 ± 0.085 2.40 ± 0.109 6 1.75 ± 0.05 1.85 ± 0.089 2.32 ± 0.110 2.45 ± 0.139 5 1.85 ± 0.05 1.65 ± 0.100 2.21 ± 0.128 2.70 ± 0.190 ) A 4 1.95 ± 0.05 1.71 ± 0.124 2.17 ± 0.157 2.57 ± 0.227 ( 2 a 3 2 TABLE II: The ratios, R3AHe, measured in 1.4<Q2 <2.6 (GeV/c)2 interval. Errors are statistical only. 1 b) 0 0 10 20 30 40 50 60 10 A 9 FIG.10: a) RA3He (◦) andr3AHe (•)versusA. b)a2(A)versus 8 A. ◦ - a (A) is obtaned from Eq.(10) using theexperimental 2 value of a2(3) Ref. [6]; (cid:3) - a2(A) is obtaned using the theo- 7 retical value of a (3). For errors shown see caption of Table 2 III.△ - data from Ref. [6]. 6 A) 5 a(2 4 probabilities is proportional to: 3 2 (2σ +σ )σ rA = p n A , (8) 3He (Zσp+Nσn)σ3He 1 0 where σA and σ3He are the A(e,e′) and 3He(e,e′) in- 1.2 1.4 1.6 Q1.28, (Ge2V/c)22.2 2.4 2.6 clusive cross sections respectively. σ and σ are the p n electron-proton and electron-neutron elastic scattering FIG. 11: Q2 - dependences of a (A) parameters obtained by cross sections respectively. Z and N are the number 2 of protons and neutrons in nucleus A. Using Eq. (4) multiplyingrA3He(=aa22((A3)))withthetheoreticalvaluesofa2(3). For errors shown see caption of Table III. N 4He, ◦ 12C and the ratio of Eq. (8) can be related to the experimentally measured ratios RA as • 56Fe. 3He A(2σ +σ ) rA =RA (x ,Q2)× p n (9) 3He 3He B 3(Zσp+Nσn) The per-nucleon SRC probability in nucleus A rela- tive to 3He is proportionalto rA3He ∼ a2(A)/a2(3), where To obtain the numerical values for rA3He we calculated a2(A) and a2(3) are the per-nucleon probability of SRC the second factor in Eq. (9) using parameterizations for relative to deuterium for nucleus A and 3He. As was the neutron and protonform factors [29, 30, 31, 32]. We discussed earlier, the direct relation of rA to the per- 3He foundtheaveragevaluesofthese factorstobe 1.14±0.02 nucleon probabilities of SRC has an uncertainty of up for 4He and 12C, and 1.18±0.02 for 56Fe. Note that to 20% due to pair center-of-mass motion. Within this these factors vary slowly over our Q2 range. For rA uncertainty, we will define the per-nucleon SRC proba- 3He calculations the experimental data were integrated over bilities of nuclei relative to deuterium as: Q2 >1.4 (GeV/c)2 and x >1.5 for each nucleus. The B ratioofintegratedyields,RA3He(xB),arepresentedinthe a2(A) = r3AHe·a2(3) (10) firstcolumnofTableIIIandinFig.10a(rectangles). The ratios rA3He are shown in the second column of Table III Twovaluesofa2(3)havebeenusedtocalculatea2(A). andbythecirclesinFig.10a. Onecanseethattheratios First is the experimentally obtained value from Ref. [6], r3AHe are 2.5 - 3.0 for 12C and 56Fe, and approximately a2(3)=1.7± 0.3, and the second the value from the cal- 1.95 for 4He. culation using the wave function for deuterium and 3He, 10 a2(3)=2±0.1. Similar results were obtained in Ref. [33]. 1. TheseratiosareindependentofxB (scale)forxB > 1.5 and Q2 >1.4 (GeV/c)2, i.e. for high recoil momentum. The ratios do not scale for Q2 <1.4 RA3He(xB) rA3He a2(A)exp a2(A)theor (GeV/c)2. 4He 1.72 ± 0.03 1.96 ± 0.05 3.39 ± 0.51 3.93 ± 0.24 2. These ratios in the scaling region are independent 12C 2.20 ± 0.04 2.51 ± 0.06 4.34 ± 0.66 5.02 ± 0.31 ofQ2,andapproximatelyindependentofAforA≥ 56Fe 2.54 ± 0.06 3.00 ± 0.08 5.21 ± 0.79 6.01 ± 0.38 12 TABLEIII:RA3He(xB)istheratioofnormalized (e,e′)yields 3. These features were predicted by the Short Range for nucleus A to 3He. r3AHe is the relative per-nucleon prob- Correlation model of inclusive A(e,e′) scattering ability of SRC for the two nuclei. Both values are obtained at large x , and consistent with the kinematical B from the scaling region (Q2 >1.4 (GeV/c)2 and xB >1.5). expectation that two nucleon short range correla- a (A)exp and a (A)theor are the a (A) parameters obtained 2 2 2 tions are dominating in the nuclear wave function by multiplying rA3He (=aa22((A3))) with the experimental and/or at pm & 300 MeV/c [6]. theoretical valuesof a2(3). TheRA3He(xB) ratios includesta- tisticalerrorsonly. ErrorsofrA3Heincludealsouncertaintiesin 4. Theobservedscalingshowsthatmomentumdistri- thesecondfactorinEq.(9). Errorsina (A)expanda (A)theor butions at high-momenta have the same shape for 2 2 alsoincludeuncertaintiesinthecorrespondingvaluesofa (3). all nuclei and differ only by a scale factor. 2 For systematic uncertainties see Table I. There is an overall theoretical uncertaintyof 20% in convertingthese ratios into 5. UsingtheSRCmodel,thevaluesoftheratiosinthe SRCprobabilities. scalingregionwereusedtoderivetherelativeprob- abilities of SRC in nuclei compared to deuterium. The per-nucleonprobabilityofShortRangeCorre- lationsinnucleirelativetodeuteriumis≈3.5times The per-nucleon probability of SRC for nucleus A rel- largerfor4Heand5.0-5.5timeslargerfor12Cand ative to deuterium is shown in the third and fourth 56Fe. columns of Table III and in Fig. 10b. The uncertain- ties inexistinga2(3)areincludedinthe errorsfora2(A). The results from Ref. [6] are shown as well. One can see that a2(A) changes significantly from A = 4 to A = 12 ACKNOWLEDGMENT but does not change significantly for A ≥ 12. There are approximately5.5timesasmuchSRCforA≥12thanfor deuterium,andapproximately3.5timesasmuchSRCfor WeacknowledgetheeffortsofthestaffoftheAccelerator 4He as for deuterium. These results are consistent with andPhysicsDivisions(especiallytheHallBtargetgroup) the analysis of the previous SLAC (e,e′) data [6]. They at JeffersonLab in their support of this experiment. We are also consistent with calculations of Ref. [33]. also acknowledge useful discussions with D. Day and Fig. 11 shows the measured Q2 dependence of the rel- E. Piasetzki. This work was supported in part by the ative SRC probability, a2(A), which appears to be Q2 U.S. Department of Energy, the National Science Foun- independent for all targets. dation, the French Commissariat a l’Energie Atomique, the FrenchCentreNationalde la RechercheScientifique, the ItalianIstituto NazionalediFisicaNucleare,andthe V. SUMMARY KoreaResearchFoundation. U.Thomaacknowledgesan “Emmy Noether” grand from the Deutsche Forschungs- The A(e,e′) inclusive electron scattering cross section gemeinschaft. TheSouteasternUniversitiesResearchAs- ratios of 4He, 12C, and 56Fe to 3He have been measured sociation (SURA) operates the Thomas Jefferson Na- for the first time under identical kinematical conditions. tional Acceleraror Facility for the United States Depart- It is shown that: ment of Energy under contract DE-AC05-84ER40150. [1] L.L. Frankfurt and M.I. Strikman, Phys. Rep. 76, 215 [6] L.L.Frankfurt,M.I.Strikman,D.B.Day,M.M.Sargsyan, (1981); , Phys.Rep. 160, 235 (1988). Phys. Rev.C 48, 2451 (1993). [2] T.W. Donnelly and I. Sick, Phys. Rev. C 60, 065502 [7] W.P. Schutzet al.,Phys. Rev.Lett. 38, 8259 (1977). (1999); [8] S. Rock et al., Phys.Rev.Lett. 49, 1139 (1982). [3] J. Arrington, et al.,Phys.Rev. Lett.82, 2056 (1999). [9] R.G. Arnold et al.,Phys. Rev.Lett. 61, 806 (1988). [4] C. Ciofi degli Atti,Phys.Rev. C 53, 1689, (1996). [10] D. Day et al., Phys.Rev.Lett. 59, 427 (1979). [5] L.FrankfurtandM.Strikman,inElectromagnetic Inter- [11] M. Sargsian, CLAS-NOTE90-007 (1990). actionswithNuclei,editedbyB.FroisandI.Sick(World [12] R. Machleidt, Phys.Rev.C 63, 024001 (2001). Scientific,Singapore, 1991). [13] M.V. Zverev and E.E. Saperstein, Yad. Fiz. 43, 304

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Observation of Nuclear Scaling in the $A(e,e^{\prime})$ Reaction at $x_B>$1 PDF

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Preview Observation of Nuclear Scaling in the $A(e,e^{\prime})$ Reaction at $x_B>$1

Observation of Nuclear Scaling in the A(e, e′) Reaction at x >1 B 1 1 11 36 27 31 11 K.Sh. Egiyan, N.Dashyan, M. Sargsian, S. Stepanyan, L.B. Weinstein, G. Adams, P. Ambrozewicz, E. Anciant,6 M. Anghinolfi,16 B. Asavapibhop,23 G. Asryan,1 G. Audit,6 T. Auger,6 H. Avakian,36 H. Bagdasaryan,27 J.P. Ball,2 S. Barrow,12 M. Battaglieri,16 K. Beard,20 I. Bedlinski,19 M. Bektasoglu,26 M. Bellis,31 N. Benmouna,13 B.L. Berman,13 N. Bianchi,15 A.S. Biselli,4 S. Boiarinov,19 B.E. Bonner,32 S. Bouchigny,17 R. Bradford,4 D. Branford,10 W.J. Briscoe,13 W.K. Brooks,36 V.D. Burkert,36 C. Butuceanu,40 J.R. Calarco,24 D.S. Carman,4 B. Carnahan,5 C. Cetina,13 L. Ciciani,27 P.L. Cole,35 A. Coleman,40 D. Cords,36 3 0 P. Corvisiero,16 D. Crabb,39 H. Crannell,5 J.P. Cummings,31 E. DeSanctis,15 R. DeVita,16 P.V. Degtyarenko,36 0 R. Demirchyan,1 H. Denizli,29 L. Dennis,12 K.V. Dharmawardane,27 K.S. Dhuga,13 C. Djalali,34 G.E. Dodge,27 2 D. Doughty,7 P. Dragovitsch,12 M. Dugger,2 S. Dytman,29 O.P. Dzyubak,34 M. Eckhause,40 H. Egiyan,36 n L. Elouadrhiri,36 A. Empl,31 P. Eugenio,12 R. Fatemi,39 R.J. Feuerbach,4 J. Ficenec,38 T.A. Forest,27 H. Funsten,40 a M. Gai,8 G. Gavalian,24 S. Gilad,22 G.P. Gilfoyle,33 K.L. Giovanetti,20 P. Girard,34 C.I.O. Gordon,14 K. Griffioen,40 J M. Guidal,17 M. Guillo,34 L. Guo,36 V. Gyurjyan,36 C. Hadjidakis,17 R.S. Hakobyan,5 J. Hardie,7 D. Heddle,7 4 P. Heimberg,13 F.W. Hersman,24 K. Hicks,26 R.S. Hicks,23 M. Holtrop,24 J. Hu,31 C.E. Hyde-Wright,27 Y. Ilieva,13 1 M.M. Ito,36 D. Jenkins,38 K. Joo,36 J.H. Kelley,9 M. Khandaker,25 D.H. Kim,21 K.Y. Kim,29 K. Kim,21 M.S. Kim,21 1 W. Kim,21 A. Klein,27 F.J. Klein,36 A. Klimenko,27 M. Klusman,31 M. Kossov,19 L.H. Kramer,11 Y. Kuang,40 v S.E. Kuhn,27 J. Kuhn,4 J. Lachniet,4 J.M. Laget,6 D. Lawrence,23 Ji Li,31 K. Lukashin,36 J.J. Manak,36 8 C. Marchand,6 L.C. Maximon,13 S. McAleer,12 J. McCarthy,39 J.W.C. McNabb,4 B.A. Mecking,36 0 0 S. Mehrabyan,29 J.J. Melone,14 M.D. Mestayer,36 C.A. Meyer,4 K. Mikhailov,19 R. Minehart,39 M. Mirazita,15 1 R. Miskimen,23 L. Morand,6 S.A. Morrow,6 M.U. Mozer,26 V. Muccifora,15 J. Mueller,29 L.Y. Murphy,13 0 G.S. Mutchler,32 J. Napolitano,31 R. Nasseripour,11 S.O. Nelson,9 S. Niccolai,13 G. Niculescu,26 I. Niculescu,20 3 B.B. Niczyporuk,36 R.A. Niyazov,27 M. Nozar,36 G.V. O’Rielly,13 A.K. Opper,26 M. Osipenko,16 K. Park,21 0 / E. Pasyuk,2 G. Peterson,23 S.A. Philips,13 N. Pivnyuk,19 D. Pocanic,39 O. Pogorelko,19 E. Polli,15 x S. Pozdniakov,19 B.M. Preedom,34 J.W. Price,3 Y. Prok,39 D. Protopopescu,14 L.M. Qin,27 B.A. Raue,11 e - G. Riccardi,12 G. Ricco,16 M. Ripani,16 B.G. Ritchie,2 F. Ronchetti,15 P. Rossi,15 D. Rowntree,22 P.D. Rubin,33 cl F. Sabati´e,6 K. Sabourov,9 C. Salgado,25 J.P. Santoro,38 V. Sapunenko,16 R.A. Schumacher,4 V.S. Serov,19 u A. Shafi,13 Y.G. Sharabian,1 J. Shaw,23 S. Simionatto,13 A.V. Skabelin,22 E.S. Smith,36 L.C. Smith,39 n D.I. Sober,5 M. Spraker,9 A. Stavinsky,19 P. Stoler,31 I. Strakovsky,13 S. Strauch,13 M. Strikman,28 : v M. Taiuti,16 S. Taylor,32 D.J. Tedeschi,34 U. Thoma,36 R. Thompson,29 L. Todor,4 C. Tur,34 M. Ungaro,31 Xi M.F. Vineyard,37 A.V. Vlassov,19 K. Wang,39 A. Weisberg,26 H. Weller,9 D.P. Weygand,36 C.S. Whisnant,34 E. Wolin,36 M.H. Wood,34 L. Yanik,13 A. Yegneswaran,36 J. Yun,27 B. Zhang,22 J. Zhao,22 and Z. Zhou22 r a (The CLAS Collaboration) 1 Yerevan Physics Institute, Yerevan 375036 , Armenia 2 Arizona State University, Tempe, Arizona 85287-1504 3 University of California at Los Angeles, Los Angeles, California 90095-1547 4 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 5 Catholic University of America, Washington, D.C. 20064 6 CEA-Saclay, Service de Physique Nucl´eaire, F91191 Gif-sur-Yvette, Cedex, France 7 Christopher Newport University, Newport News, Virginia 23606 8 University of Connecticut, Storrs, Connecticut 06269 9 Duke University, Durham, North Carolina 27708-0305 10 Edinburgh University, Edinburgh EH9 3JZ, United Kingdom 11 Florida International University, Miami, Florida 33199 12 Florida State University, Tallahassee, Florida 32306 13 The George Washington University, Washington, DC 20052 14 University of Glasgow, Glasgow G12 8QQ, United Kingdom 15 INFN, Laboratori Nazionali di Frascati, Frascati, Italy 16 INFN, Sezione di Genova, 16146 Genova, Italy 17 Institut de Physique Nucleaire ORSAY, Orsay, France 18 Institute fu¨r Strahlen und Kernphysik, Universit¨at Bonn, Germany 19 Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia 20 James Madison University, Harrisonburg, Virginia 22807 21 Kungpook National University, Taegu 702-701, South Korea 2 22 Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 23 University of Massachusetts, Amherst, Massachusetts 01003 24 University of New Hampshire, Durham, New Hampshire 03824-3568 25 Norfolk State University, Norfolk, Virginia 23504 26 Ohio University, Athens, Ohio 45701 27 Old Dominion University, Norfolk, Virginia 23529 28Pennsylvania State University, State Collage, Pennsylvania 16802 29 University of Pittsburgh, Pittsburgh, Pennsylvania 15260 30 Universita’ di ROMA III, 00146 Roma, Italy 31 Rensselaer Polytechnic Institute, Troy, New York 12180-3590 32 Rice University, Houston, Texas 77005-1892 33 University of Richmond, Richmond, Virginia 23173 34 University of South Carolina, Columbia, South Carolina 29208 35 University of Texas at El Paso, El Paso, Texas 79968 36 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 37Union College, Schenectady, NY 12308 38 Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435 39 University of Virginia, Charlottesville, Virginia 22901 40 College of William and Mary, Williamsburg, Virginia 23187-8795 (Dated: February 8, 2008) The ratios of inclusive electron scattering cross sections of 4He, 12C, and 56Fe to 3He have been measuredforthefirsttime. ItisshownthattheseratiosareindependentofxB atQ2 >1.4(GeV/c)2 forxB >1.5wheretheinclusivecrosssectiondependsprimarilyonthehigh-momentumcomponents ofthenuclearwavefunction. Theobservedscalingshowsthatthemomentumdistributionsathigh- momentahavethesameshapeforallnucleianddifferonlybyascalefactor. Theobservedonsetof thescalingatQ2 >1.4andxB >1.5isconsistentwiththekinematicalexpectationthattwonucleon short range correlations (SRC) are dominate the nuclear wave function at pm & 300 MeV/c. The values of these ratios in the scaling region can be related to the relative probabilities of SRC in nuclei with A≥3. Our data demonstrate that for nuclei with A≥12 these probabilities are 5-5.5 times larger than in deuterium,while for 4Heit is larger by a factor of about 3.5. PACSnumbers: PACS:13.60.Le,13.40.Gp,14.20.Gk I. INTRODUCTION and meson exchange currents (at high Q2) can be sig- nificantly reduced, which corresponds to studying the Due to the strong interaction and short distances be- low-energy-loss side of the quasi-elastic peak. In Eq. 1 tween the nucleons in nuclei, there is a significant prob- Q2 is the four-momentum squared of the virtual photon ability for nucleon wave functions to overlap, resulting (Q2 = −qµqµ >0), ν is the energy transfer, xB is the inshortrangenucleon-nucleoncorrelations(SRC)in nu- Bjorken scaling variable, and M is the nucleon mass, clei [1]. Investigation of SRC is important for at least Many previous analyses of data in this kinematic re- two reasons. First, because of the short range nature gionconcentrateonusingy-scalingtostudynucleonmo- ofthesecorrelations,they shouldcontributesignificantly mentum distributions (see e.g. Refs. [2], [3]). While to the high-momentum component of the nuclear wave this technique provides some information about the nu- function. Second, scattering from nucleons in SRC will clear wave function, it does not measure the probability provideuniquedataonthemodificationofdeeplybound of finding SRC in nuclei. nucleons, which is extremely important for a complete Meanwhile, the data at x >1 can be used to mea- B understanding of nucleon structure in general. sure the probability of finding SRC in nuclei. There are High-energy inclusive electron scattering from nuclei, theoretical predictions that at momenta higher than the A(e,e′), is one of the simplest ways to investigate SRC. Fermi momentum, the nucleon momentum distributions In particular, it is probably the best way to measure in light and heavy nuclei are similar (see e.g. Ref. [4]). the probabilities of SRC in nuclei. The main problem This implies that they originate predominantly from the in these studies is selecting the electron-SRC scattering interaction between two nearby nucleons, i.e. due to events from the orders-of-magnitude larger background SRC. If the A(e,e′) cross section depends primarily on of inelastic and/or quasi-elastic interaction of electrons the nuclear wave function and the shape of this wave with the uncorrelated low-momentum nucleons. function at high-momentum is really universal, then in By measuring cross sections at this high-momentum region the ratio of weighted (e,e′) cross sections for different nuclei [34] should scale, i.e. Q2 x = >1, (1) they should be independent of electron scattering vari- B 2Mν ables (Q2 and x ), with the magnitude of the scaling B contributions from inelastic electron-nucleon scattering factor being proportional to the relative probability of 3 SRC in the two nuclei [5, 6]. p are known.) From Eq.(2) one obtains: A InRef.[6]thiswascheckedbyanalyzingexistingSLAC A(e,e′) data for deuterium [7, 8, 9] and heavier nuclei ∆M2−Q2+ Q2 M − M2 +p~ 2 [10]. They found an indication of scaling at Q2 >1 and m x (cid:16) A q A−1 m (cid:17) N B x ≥1.5. However,since the data for deuterium and the B −2~q·~p −2M M2 +p~ 2 = 0, (3) heavy nuclei were collected in different experiments at m Aq A−1 m similarQ2 butatdifferentelectronscatteringanglesand incident electron energies, to find the ratios at the same where ∆M2 = MA2 +MA2−1−m2N and p~m = ~pf −~q = values of (xB,Q2), a complicated fitting and interpola- −p~A−1 is the recoil momentum involved in the reaction tion procedure was applied [6] to the data. The main (sometimes referred to as the ‘missing momentum’ in problem was that the cross sections varied very strongly (e,e′p) reactions). Eq. (3) defines a simple relationship with angle, incident energy, and Q2. To simplify the in- between |p~mmin| and xB at fixed Q2. This relation for 2 terpolation, the electron-deuteron cross section was first deuterium at various values of Q is shown in Fig. 2a. dividedby the theoreticalcalculationwithin the impulse Fig. 2b shows the same relationshipfor various nucleiat approximation. Therefore,thedataarenotpurelyexper- Q2 = 2 (GeV/c)2. Note that this relationship is differ- imental, since they include the theoretical calculations, ent for the different nuclei, due primarily to differences and the ratios may have been affected by the fitting and in the mass of the recoil A−1 system. This minimum interpolation procedures. recoilmomentum is one of the possible definitions of the Inthiswork,theyieldsofthereactionA(e,e′)for3He, scaling variable y. 4He, 12C, and 56Fe targets are measured in the same kinematical conditions, and the ratios A(e,e′)/3He(e,e′) 1 are obtained for 1< x < 2 and Q2 >0.65 (GeV/c)2. c a) 4.0 B v/ 0.8 3.0 Furthermore, using the scaling behavior of these ratios, Ge 2.5 hthaevereblaeteinveexptrroabcatbedil.ityofNN SRCforthevariousnuclei |, min 0.6 12..50 m 0.4 1.0 P | 0.5 0.2 ↑ II. KINEMATICS AND PREDICTIONS Q2 0 b) c In order to suppress the background from quasi- v/ 0.8 elastic interactions of electrons with the uncorrelated e G ltohwe-kminoemmeanttiucmvanruiacbleloenssx(BseaenFdigQ.21.a),wefurtherrestrict |, min 0.6 3DHe m 0.4 4He / / |P A>12 e e 0.2 e e 0 q p q i 00..22 00..44 00..66 00..88 11 11..22 11..44 11..66 11..88 22 pi pf -pi pf xB SRC p SRC AA A-1 i A-2 FIG.2: Theminimumrecoil momentumas afunction of xB. a)FordeuteriumatseveralQ2(in(GeV/c)2);b)Fordifferent a) b) nuclei at Q2 = 2.0 (GeV/c)2. Horizontal lines at 250 MeV/c indicatetheFermimomentumtypicaloftheuncorrelatedmo- FIG. 1: Two mechanisms of A(e,e′) scattering. a) single tion of nucleons in nuclei. nucleon model; b) Short Range Correlation model. OnecanseefromFig.2thatforanynucleusAandfixed For quasi-elastic A(e,e′) scattering, xB, Q2, and the Q2, we canfind the valuexoB suchthat atxB > xoB the minimum A − 1 recoil momentum contributing to the magnitude of the minimum recoil momentum, |p~ min|, m reaction are related by energy and momentum conserva- contributing to the reaction, exceeds the average Fermi tion: momentum in nucleus A. It shouldbe pointed outthat the initial momentumof 2 2 2 (q+pA−pA−1) =pf =mN, (2) the struck nucleon p~i is equal to p~m only in the simplest model where the virtual photon is absorbed on one nu- where q, pA, pA−1, and pf are the four-momenta of the cleonandthatnucleonleavesthenucleuswithoutfurther virtualphoton,targetnucleus,residualA−1system,and interactions (the Plane Wave Impulse Approximation). knocked-out nucleon respectively (note that only q and In reality, the (e,e′) reaction effectively integrates over 4 many values of p ≥ pmin. In addition, this simple re- The ratios in Fig. 3 show a nice plateau starting from m m lation between recoil momentum and initial momentum x >1.5 for both nuclei and all Q2. The experimentally B is modified by FinalState Interactions(FSI) andthe ex- obtainedratiointhe scalingregioncanbe usedto deter- citation energy of the residual nucleus. These make it mine the relative probability of finding correlated NN difficult to determine the nuclear wave function directly pairs in different nuclei. However one needs to take into from (e,e′) cross sections. However, for our purposes, account two main factors: first the final state interac- it is sufficient to know that when the minimum recoil tions of a nucleon with the residual system, and second momentum contributing to the reaction is much larger the NN pair center-of-mass motion. thanthe Fermimomentum, the initialmomentum ofthe In the SRC model, FSI do not destroy the scaling be- struck nucleon will also be larger. havior of the ratio, R. Indeed, in the light-cone approxi- Letusnowconsidervariouspredictionsoftheratiosof mationof the SRC model, if the invariantmass ofthe fi- weighted (e,e′) cross sections for different nuclei. In the nalNN systemissufficientlylarge, (q+m )2−m > D D mechanism for inclusive (e,e′) scattering at xB >1 with 50-100 MeV, then the scattering ampplitude will depend virtual photon absorption on a single nucleon and the mainly on the light-cone fraction of the interacting nu- A−1 system recoiling intact without FSI (see Fig. 1a), cleon’smomentumα=(E−p )/M,andhasonlyaweak z the minimum recoil momentum for different nuclei at dependence on the conjugated variables E +p and p fixedQ2 differsandthisdifferenceincreaseswithxB (see [6,17,18]. Asaresult,theclosureapproximatiozncanbet Fig.2). Therefore,the crosssectionratiobetweendiffer- applied in the light-cone reference frame, allowing us to ent nuclei will increase with xB and will not scale. sum over all final states and use the fact that this sum In the Short Range Correlations model of Frankfurt is normalized to unity. After using the closure approx- and Strikman [1] (see Fig. 1b) the high-momentum part imation the inclusive cross section will depend on the of the nuclear momentum distribution is due to corre- light-cone momentum distribution of the nucleon in the lated nucleon pairs. This means that when the electron nucleus,integratedoverthetransversemomentumofthe scatters from a high-momentum nucleon in the nucleus, nucleon, ρ (α) [5]. Thus, within the light cone descrip- A we can consider this scattering as an electron-deuterium tion the Eq.(4) measures the ratio of ρ (α) for nuclei A interaction with the spectator A−2 system almost at A1 and A2 in the high-momentum range of the target rest. Therefore,accordingtoFig. 2a,startingfromsome nucleon. threshold x0 for fixed Q2 the cross section ratio B In the lab frame description (in the virtual nucleon RAA21(Q2,xB)= σσAA21((QQ22,,xxBB))//AA21, (4) abmpeepnartopaap,claihen)d,dhfFooSrwIleasvrhegore,ultvdhaelbueecslcoasolucfrueilnatateeprdparceotxixnpimglicanittuliyocnle(oscenaenmen.oogt-. where σ and σ are the inclusive electron scattering Ref. [17]). Within the SRC model at high recoil mo- A1 A2 cross sections from nuclei with atomic numbers A1 and menta, FSI are dominated by the rescattering of the A2 respectively, will scale (will be constant). Scaling re- knocked-out nucleon with the correlated nucleon in the sultsfromthe dominanceofSRCinthe high-momentum SRC [6, 17]. Therefore, FSI will be localized in SRC, component of the nuclear wave function, and it should and will cancel in the ratio R. As a result, Eq.(4) at be observed, for example, for the cross section ratios of xB >x0B couldberelatedtotheratioofhigh-momrntum heavy nuclei to light nuclei such as 3He. part of nucleon-momentum distributions in A1 and A2 Fig. 3a shows R3CHe, for A1 = 12C and A2 = 3He, as nuclei [17]. a function of x for Q2 from 1.5 to 2.5 (GeV/c)2 calcu- Having an underlying model of the nuclear spectral B latedintheSRCmodel[11]. TheratioforA1 = 56Feand functions,onecanrelatethemeasuredratiosinEq.(4)to A2 = 3HeisshowninFig. 3b. Thecalculationsusedthe theSRCpropertiesofthe nuclearwavefunction. Within Faddeevwavefunctionfor3HecalculatedusingtheBonn the spectral function model [1], in which correlated nu- NN potential [12]. The momentum distributions for cleon pair is assumed at rest with the nucleon momen- heavier nuclei have been modeled through a two compo- tum distribution in pair identical to that in deuteron, nentofmomentumdistributionusingmeanfielddistribu- the ratio in Eq. (4) could be directly related to the per- tions for small nucleon momenta and using the deuteron nucleon SRC probability in nucleus A relative to deu- momentumdistributionforp>250MeV/c,scaledbyfac- terium, a2(A). In models of the nuclear spectral func- tora2(A),per-nucleonprobabilityofNN SRCinnucleus tion[19]inwhichtwo-nucleoncorrelationsaremovingin A, estimated from Ref. [6]. The mean field momentum themeanfieldofthespectatorA−2system,theanalysis distributionsusedtheHarmonicOscillatorwavefunction ofEq. (4)willyieldslightlysmallervaluesfora2(A). Cal- for 12C and the quasi-particle Lagrange Method of [13] culationsbyCiofidegliAtti[20]indicatethatthismotion for 56Fe. For the description of the eN interaction, the doesnotaffectthescalingbutcandecreasetheextracted inelastic form factor parameterization of Ref. [14] and a2(A) for 56Fe by up to 20%. Howeverit is importantto the dipole elastic form-factors have been used. These emphasize that since both approximations predict sim- calculations are in reasonable agreement with existing ilar (light cone) momentum distributions, both models A(e,e′)X experimental data from SLAC [15] and from leadtoasimilarratioofthelight-conespectralfunctions Jefferson Lab Hall C [16]. and the overall probability of high-momentum nucleons 5 remains practically the same. et al. [21] use the nuclear spectral function in the lab One can summarize the predictions of the SRC model system and calculate the FSI using a correlatedGlauber fortheratiosoftheinclusivecrosssectionsfromdifferent approximation (CGA), in which the initial momenta of nuclei as follows (see Fig. 3): the re-scattered nucleons are neglected. In this model the cross section at x >1 originates mainly from FSI 2 B • Scaling (xB independence) is expected for Q ≥1 andthereforethe crosssectionratioswillnotscale. This (GeV/c)2 andx0B ≤xB <2wherex0B isthethresh- modelpredictsthattheseratiosalsodependonQ2,since old for high recoil momentum. itincludesanoticeablereductionofFSIinordertoagree • No scaling is expected for Q2 < 1 (GeV/c)2. with the data at Q2 ≥2 (GeV/c)2. Benhar et al. at- tribute this reduction in FSI to color transparency ef- • For x ≤x0 the ratios should have a minimum at fects [35]. The requirement of large color transparency B B x =1andshouldgrowwithx sinceheavynuclei effects also results in a strong A dependence of the ratio B B have a broader momentum distribution than light since the amount of the FSI suppression depends on the nuclei for p<0.3 GeV/c. number of nucleons participating in the rescattering. ThemainpredictionsoftheCGAmodelforthenuclear • The onset of scalingdepends onQ2; x0 should de- cross section ratios are as follows: B crease with increasing Q2. • No scaling is predicted for Q2 ≥1 (GeV/c)2 and • Inthescalingregime,theratiosshouldbeindepen- x <2. B 2 dent of Q . • The nuclear ratios should vary with Q2. • Inthescalingregimetheratiosshoulddependonly • The ratios should depend on A. weakly on A for A≥ 10. This reflects nuclear sat- uration. • ThemodelisnotapplicablebelowQ2=1(GeV/c)2. • Ratiosinthescalingregimeareproportionaltothe Thus, measuring the ratios of inclusive (e,e′) scatter- ratios of the two-nucleon SRC probabilities in the ing at x >1 and Q2 >1 (GeV/c)2 will yield important B two nuclei. information about the reaction dynamics. If scaling is observed, then the dominance of the SRC in the nuclear 3 wavefunctionis manifestedandthe measuredratioswill containinformationabouttheprobabilityoftwo-nucleon 2.5 short range correlations in nuclei. e3 Q2=2.5 H 2 s2 Q2=1.5 1 1.5 /2 III. EXPERIMENT 1 C 1 s3 0.5 a) In this paper we present the first experimental studies of ratios of normalized, and acceptance- and radiative- 0 corrected, inclusive yields of electrons scattered from 2.5 4He, 12C, 56Fe, and 3He measured under identical kine- 3 He 2 Q2=2.5 matical conditions. s6 Q2=1.5 The measurements were performed with the CEBAF 5 /61.5 LargeAcceptanceSpectrometer(CLAS)inHallBatthe 5 e Thomas Jefferson National Accelerator Facility. This is F 1 s3 the first CLAS experiment with nuclear targets. Elec- 0.5 b) tronswith2.261and4.461GeVenergiesincidenton3He, 4 12 56 He, C, and Fe targets have been used. We used he- 0 00..88 11 11..22 11..44 11..66 11..88 22 lium liquefiedin cylindricaltargetcells 1 cmin diameter x and4cm long,positionedonthe beamapproximatelyin B the center of CLAS. The solid targets were thin foils of FIG. 3: SRC Model predictions for the normalized inclusive 12C (1mm) and56Fe (0.15mm) positioned1.5cmdown- cross section ratio as a function of xB for several values of stream of the exit window of the liquid target. Data on Q2 (in (GeV/c)2). Note the scaling behavior predicted for solidtargetshavebeentakenwithanemptyliquidtarget xB >1.4. a) 12C to 3He, b) 56Fe to3He. cell. The CLAS detector [24] consists of six sectors, each functioning as an independent magnetic spectrometer. Another possible mechanism for inclusive (e,e′) scat- Six superconducting coils generate a toroidal magnetic tering at x >1 is virtual photon absorption on a single field primarily in the azimuthal direction. Each sector B nucleon followed by NN rescattering [21, 22]. Benhar is instrumented with multi-wire drift chambers [25] and 6 0.6 time-of-flight scintillator counters [26] that coverthe an- gular range from 8◦ to 143◦, and, in the forward region 0.5 (8◦ <θ <45◦),withgas-filledthresholdCherenkovcoun- 0.4 ters (CC) [27] and lead-scintillator sandwich-type elec- c tromagnetic calorimeters (EC) [28]. Azimuthal coverage c0.3 A for CLAS is limited only by the magnetic coils, and is 0.2 approximately90%atlargepolaranglesand50%atfor- ward angles. The CLAS was triggered on scattered elec- 0.1 Q2= 1.55 Q2= 1.85 tronsbyaCC-ECcoincidenceat2.2GeVandbytheEC 0 alone with a ≈1 GeV electron threshold at 4.4 GeV. 0.5 Forouranalysis,electronsareselectedinthekinemati- calregionQ2 >0.65(GeV/c)2 andxB >1wherethecon- 0.4 tribution from the high-momentum components of the c nuclear wave function should be enhanced. We also re- Ac0.3 0.2 1 0.1 Q2= 2.15 Q2= 2.45 0.8 0 0.6 1 1.25 1.5 1.75 2 1 1.25 1.5 1.75 2 C E X X R 0.4 B B 0.2 FIG.5: TheacceptancecorrectionfactorsasafunctionofxB. • 3He, ◦ 12C. Q2 are in (GeV/c)2. 0 2.6 2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 p ,GeV/c e FIG. 4: The ratio (REC) of energy deposited in the CLAS B. Acceptance Corrections electromagnetic calorimeter (EC) to the electron momentum pe asafunction ofpe at beamenergy 4.461 GeV.Thelineat REC ≈0.25islocated3standarddeviationsbelowthemean, We used the Monte Carlo techniques to determine the as determined by measurements at several values of pe. This electron acceptance correction factors. Two iterations cut was used to identify electrons. were done to optimize the cross section model for this purpose. In the first iteration we generated events using the SRC model [11] and determined the CLAS detec- quire that the energy transfer, ν, should be >300 MeV tor response using the GEANT-based CLAS simulation (the characteristicmissingenergyforSRC is∼260MeV program,taking into account “bad” or “dead” hardware [1]). In this region one expects that inclusive A(e,e′) channels in various components of CLAS, as well as re- scattering will proceedthroughthe interactionof the in- alistic position resolution for the CLAS drift chambers. coming electron with a correlated nucleon in a SRC. We then used the CLAS data analysis package to recon- structthese eventsusing the same electronidentification criteriathatwasappliedtotherealdata. Theacceptance A. Electron Identification correctionfactorswerefoundastheratiosofthenumber of reconstructed and simulated events in each kinematic Electrons were selected in the fiducial region of the bin. Then the acceptance corrections were applied to CLAS sectors. The fiducial region is a region of az- the data event-by-event, i.e. each event was weighted imuthal angle, for a given momentum and polar angle, by the acceptance factor obtained for the corresponding 2 where the electrondetection efficiency is constant. Then (∆x ,∆Q ) kinematic bin and the cross sections were B a cut on the ratio of the energy deposited in the EC to calculated as a function of x and Q2. For the second B themeasuredelectronmomentump (R )wasusedfor iteration the obtained cross sections were fitted and the e EC final selection. In Fig. 4 R vs. p for the 56Fe target fit-functions were used to generate a new set of data, EC e at 4.4 GeV is shown. The lines shows the applied cut at and the process was repeated. Fig. 5 shows the electron R ≈0.25whichislocated3standarddeviationsbelow acceptance factors after the second iteration for liquid EC the meanas determinedby measurementsatseveralval- (3He) and solid (12C) targets. We used the difference ues of p . A Monte Carlo simulation showed that these between the iterations as the uncertainty in the accep- e cuts reduce the A(e,e′) yield by less then 0.5%. tancecorrectionfactor. Notethatthe acceptanceforthe We estimated the π− contamination in the electron carbontargetis smallerthanforthe heliumtarget. This sampleforawideangularrangeusingthephoto-electron is due to the closer location of the solid targets to the distributionsintheCLASCherenkovcounters. Wefound CLAS coils, which limit azimuthal angular coverage of that for x >1 this is negligible. the detectors. B 7 4 3 3.5 2.5 Q2<1.4 3 Q2<1.4 2 2.5 2 3 4 3 C1 He 2 He He1.5 R R 1.5 1 1 0.5 0.5 0 0 3.5 2.5 Q2>1.4 3 Q2>1.4 2 2.5 2 3 4 3 1 e 2 e e1.5 C H H H R R 1.5 1 1 0.5 0.5 0 0 1 1.2 1.4 1.6 1.8 2 1 1.2 1.4 1.6 1.8 2 X X B B FIG.6: R3CHe,theper-nucleonyieldratios for 12Cto3He. a) FIG. 7: The same as Fig. 6 for 4He. ◦ 0.65 <Q2 < 0.85; (cid:3) 0.9 <Q2 < 1.1; △ 1.15 <Q2 < 1.35 (GeV/c)2, all at incident energy 2.261 GeV. b) ◦ 1.4 <Q2 < 1.6 (GeV/c)2 and at incident energy 2.261 4 GeV; (cid:3) 1.4<Q2 <2.0 ; △ 2.0<Q2<2.6 (GeV/c)2, both at 3.5 incident energy 4.461 GeV. Statistical errors are shown only. 3 Q2<1.4 2.5 3 Fe He 2 R C. Radiative Corrections 1.5 1 Thecrosssectionratioswerecorrectedforradiativeef- 0.5 fects. The radiative correction for each target as a func- 0 tion of Q2 and x was calculated as the ratio 3.5 B 3 Q2>1.4 2 dσrad(xB,Q2) 2.5 C (x ,Q )= , (5) rad B dσnorad(xB,Q2) Fe He3 2 R 1.5 where dσrad(x ,Q2) and dσnorad(x ,Q2) are the radia- B B 1 tively corrected and uncorrected theoretical cross sec- 0.5 tions, respectively. The cross sections have been calcu- 0 lated using [11]. 1 1.2 1.4 1.6 1.8 2 X B IV. RESULTS FIG. 8: The same as Fig. 6 for 56Fe. We constructed ratios of normalized, acceptance- and radiative-corrected inclusive electron yields on nuclei 4He, 12C and 56Fe divided the yield on 3He in the range whereNe andNT arethenumberofelectronsandtarget nuclei,respectively,Acc is the acceptancecorrectionfac- of kinematics 1< x <2. Assuming that electron detec- B tor,and∆Q2 and∆x arethebinsizesinQ2 andinx , tion efficiency from different targets is the same, these B B respectively. Since electron detection efficiency in CLAS ratios,weightedby atomicnumber, areequivalentto the is expected to be > 96%, we compare obtained yields ratios of cross sections in Eq. 4. The normalized yields for each x and Q2 bin have with radiatedcross sections calculatedby Ref. [11] code. B Within systematic uncertainies (see below) the satisfac- been calculated as: tory agreement has been found between our results and dY Ne′ 1 the calculationsfromRef. [11]that weretuned onSLAC = · (6) dQ2dx ∆Q2∆x N N Acc [15] and Jefferson Lab Hall C [16] data. B B e T 8 The ratios R3AHe, also corrected for radiative effects, Q2 1.55 1.85 2.15 2.45 are defined as: δRA 7.1 5.8 4.9 5.1 δRHe4 0.7 0.7 0.7 0.7 3YA CA RA (x ) = Rad, (7) 3He B AY3HeCR3Hade TABLE I: Systematic uncertainties δRA and δRHe4 for the where Y is the normalizedyield in a given (xB, Q2) bin, rtaivteioly,oifnn%or.mQal2iziend(GinecVlu/sciv)2e,yaineldds∆RQ3C2H,F=ee±a0n.d15R(34GHHeeeVr/ecs)p2e.c- and C is the radiative correction factor from Eq. 5 Rad for each nucleus. Fig. 6 shows these ratios for 12C at several values of Q2. Figs. 7 and 8 show these ratios for 4He and 56Fe, B. Probabilities of Two-Nucleon Short respectively. These data have the following important Range Correlations in Nuclei characteristics: • ThereisaclearQ2 evolutionoftheshapeofratios: Our data are clearly consistent with the predictions of the NN SRC model. The obtained ratios, RA , for 3He – At low Q2 (Q2 <1.4(GeV/c)2), RA (x )in- 1.4<Q2 <2.6 (GeV/c)2 region are shown in Table II as creases with x in the entire 1<x3H<e 2Brange a function of x . Fig. 9 shows these ratios for the 12C B B B (see Figs. 6a – 8a). and 56Fe targets together with the SRC calculation re- sults usingofRef. [11]whichusedthe estimatedscaling – At high Q2 (Q2 ≥1.4 (GeV/c)2) RA (x ) is independent of x for x > x0 ≈3H1.e5. B(See factors, a2(A), (per-nucleon probabilitiy of NN SRC in B B B nucleus A) from Ref. [6]. Good agreement between our Figs. 6b – 8b.) data and calculation is seen. Note that one of the goals of the present paper is to determine these factors more • The value of RA (x ) in the scaling regime is in- 3He B precisely (see below). dependent of Q2. • The value of RA (x ) in the scaling regime for 4 3He B A>10suggestsaweakdependenceontargetmass. 3.5 3 2.5 2 3 A. Systematic Errors C1 He 2 4 R → 1.5 ↑ 1 1 Thesystematicerrorsinthismeasurementaredifferent 0.5 a) for different targets and include uncertainties in: 0 3.5 • fiducial cut applied: ≈ 1%; 3 2.5 • radiative correction factors: ≈ 2%; 4 3 Fe He 2 → R • target densities and thicknesses: ≈ 0.5% and 1.0% 1.5 ↑1 for solid targets; 0.5% and 3.5% for liquid targets, 1 respectively. 0.5 b) 0 1 1.2 1.4 1.6 1.8 2 • acceptance correction factors (Q2 dependent): be- X tween 2.2 and 4.0% for solid targets and between B 1.8 and 4.3% for liquid targets. FIG. 9: R3AHe(xB) as a function of xB for 1.4<Q2 <2.6 (GeV/c)2, statistical errors are shown only. Curves are SRC Some of systematic uncertainties will cancel out in the model predictions for different Q2 in the range 1.4 (GeV/c)2 yield ratios. For the 3He/4He ratio, all uncertainties ex- (curve 1) to 2.6 (GeV/c)2 (curve 4), respectively, for a) 12C, cept those on the beam current and the target density b) 56Fe. divideout,givingatotalsystematicuncertaintyof0.7%. For the solid-target to 3He ratios, only the electron de- tectionefficiencycancels. Thequadraticsumoftheother uncertainties is between 5% and 7%, depending on Q2. Experimentaldatainthescalingregioncanbeusedto The systematic uncertainties onthe ratios for all targets estimate the relative probabilities of NN SRC in nuclei and Q2 are presented in Table I. comparedto3He. AccordingtoRef.[1]theratioofthese 9 e3 4 XB 4He 12C 56Fe A H3.5 , r3 3 01..9055 ±± 00..0055 00..8768 ±± 00..000044 00..7772 ±± 00..000033 00..8702 ±± 00..000044 ARHe2.5 1.15 ± 0.05 0.94 ± 0.006 0.96 ± 0.006 0.94 ± 0.007 2 1.25 ± 0.05 1.19 ± 0.012 1.33 ± 0.012 1.33 ± 0.015 1.5 1.35 ± 0.05 1.41 ± 0.021 1.77 ± 0.025 1.81 ± 0.030 1 1.45 ± 0.05 1.58 ± 0.033 2.12 ± 0.044 2.17 ± 0.055 0.5 a) 1.55 ± 0.05 1.71 ± 0.049 2.12 ± 0.059 2.64 ± 0.087 0 7 1.65 ± 0.05 1.70 ± 0.063 2.29 ± 0.085 2.40 ± 0.109 6 1.75 ± 0.05 1.85 ± 0.089 2.32 ± 0.110 2.45 ± 0.139 5 1.85 ± 0.05 1.65 ± 0.100 2.21 ± 0.128 2.70 ± 0.190 ) A 4 1.95 ± 0.05 1.71 ± 0.124 2.17 ± 0.157 2.57 ± 0.227 ( 2 a 3 2 TABLE II: The ratios, R3AHe, measured in 1.4<Q2 <2.6 (GeV/c)2 interval. Errors are statistical only. 1 b) 0 0 10 20 30 40 50 60 10 A 9 FIG.10: a) RA3He (◦) andr3AHe (•)versusA. b)a2(A)versus 8 A. ◦ - a (A) is obtaned from Eq.(10) using theexperimental 2 value of a2(3) Ref. [6]; (cid:3) - a2(A) is obtaned using the theo- 7 retical value of a (3). For errors shown see caption of Table 2 III.△ - data from Ref. [6]. 6 A) 5 a(2 4 probabilities is proportional to: 3 2 (2σ +σ )σ rA = p n A , (8) 3He (Zσp+Nσn)σ3He 1 0 where σA and σ3He are the A(e,e′) and 3He(e,e′) in- 1.2 1.4 1.6 Q1.28, (Ge2V/c)22.2 2.4 2.6 clusive cross sections respectively. σ and σ are the p n electron-proton and electron-neutron elastic scattering FIG. 11: Q2 - dependences of a (A) parameters obtained by cross sections respectively. Z and N are the number 2 of protons and neutrons in nucleus A. Using Eq. (4) multiplyingrA3He(=aa22((A3)))withthetheoreticalvaluesofa2(3). For errors shown see caption of Table III. N 4He, ◦ 12C and the ratio of Eq. (8) can be related to the experimentally measured ratios RA as • 56Fe. 3He A(2σ +σ ) rA =RA (x ,Q2)× p n (9) 3He 3He B 3(Zσp+Nσn) The per-nucleon SRC probability in nucleus A rela- tive to 3He is proportionalto rA3He ∼ a2(A)/a2(3), where To obtain the numerical values for rA3He we calculated a2(A) and a2(3) are the per-nucleon probability of SRC the second factor in Eq. (9) using parameterizations for relative to deuterium for nucleus A and 3He. As was the neutron and protonform factors [29, 30, 31, 32]. We discussed earlier, the direct relation of rA to the per- 3He foundtheaveragevaluesofthese factorstobe 1.14±0.02 nucleon probabilities of SRC has an uncertainty of up for 4He and 12C, and 1.18±0.02 for 56Fe. Note that to 20% due to pair center-of-mass motion. Within this these factors vary slowly over our Q2 range. For rA uncertainty, we will define the per-nucleon SRC proba- 3He calculations the experimental data were integrated over bilities of nuclei relative to deuterium as: Q2 >1.4 (GeV/c)2 and x >1.5 for each nucleus. The B ratioofintegratedyields,RA3He(xB),arepresentedinthe a2(A) = r3AHe·a2(3) (10) firstcolumnofTableIIIandinFig.10a(rectangles). The ratios rA3He are shown in the second column of Table III Twovaluesofa2(3)havebeenusedtocalculatea2(A). andbythecirclesinFig.10a. Onecanseethattheratios First is the experimentally obtained value from Ref. [6], r3AHe are 2.5 - 3.0 for 12C and 56Fe, and approximately a2(3)=1.7± 0.3, and the second the value from the cal- 1.95 for 4He. culation using the wave function for deuterium and 3He, 10 a2(3)=2±0.1. Similar results were obtained in Ref. [33]. 1. TheseratiosareindependentofxB (scale)forxB > 1.5 and Q2 >1.4 (GeV/c)2, i.e. for high recoil momentum. The ratios do not scale for Q2 <1.4 RA3He(xB) rA3He a2(A)exp a2(A)theor (GeV/c)2. 4He 1.72 ± 0.03 1.96 ± 0.05 3.39 ± 0.51 3.93 ± 0.24 2. These ratios in the scaling region are independent 12C 2.20 ± 0.04 2.51 ± 0.06 4.34 ± 0.66 5.02 ± 0.31 ofQ2,andapproximatelyindependentofAforA≥ 56Fe 2.54 ± 0.06 3.00 ± 0.08 5.21 ± 0.79 6.01 ± 0.38 12 TABLEIII:RA3He(xB)istheratioofnormalized (e,e′)yields 3. These features were predicted by the Short Range for nucleus A to 3He. r3AHe is the relative per-nucleon prob- Correlation model of inclusive A(e,e′) scattering ability of SRC for the two nuclei. Both values are obtained at large x , and consistent with the kinematical B from the scaling region (Q2 >1.4 (GeV/c)2 and xB >1.5). expectation that two nucleon short range correla- a (A)exp and a (A)theor are the a (A) parameters obtained 2 2 2 tions are dominating in the nuclear wave function by multiplying rA3He (=aa22((A3))) with the experimental and/or at pm & 300 MeV/c [6]. theoretical valuesof a2(3). TheRA3He(xB) ratios includesta- tisticalerrorsonly. ErrorsofrA3Heincludealsouncertaintiesin 4. Theobservedscalingshowsthatmomentumdistri- thesecondfactorinEq.(9). Errorsina (A)expanda (A)theor butions at high-momenta have the same shape for 2 2 alsoincludeuncertaintiesinthecorrespondingvaluesofa (3). all nuclei and differ only by a scale factor. 2 For systematic uncertainties see Table I. There is an overall theoretical uncertaintyof 20% in convertingthese ratios into 5. UsingtheSRCmodel,thevaluesoftheratiosinthe SRCprobabilities. scalingregionwereusedtoderivetherelativeprob- abilities of SRC in nuclei compared to deuterium. The per-nucleonprobabilityofShortRangeCorre- lationsinnucleirelativetodeuteriumis≈3.5times The per-nucleon probability of SRC for nucleus A rel- largerfor4Heand5.0-5.5timeslargerfor12Cand ative to deuterium is shown in the third and fourth 56Fe. columns of Table III and in Fig. 10b. The uncertain- ties inexistinga2(3)areincludedinthe errorsfora2(A). The results from Ref. [6] are shown as well. One can see that a2(A) changes significantly from A = 4 to A = 12 ACKNOWLEDGMENT but does not change significantly for A ≥ 12. There are approximately5.5timesasmuchSRCforA≥12thanfor deuterium,andapproximately3.5timesasmuchSRCfor WeacknowledgetheeffortsofthestaffoftheAccelerator 4He as for deuterium. These results are consistent with andPhysicsDivisions(especiallytheHallBtargetgroup) the analysis of the previous SLAC (e,e′) data [6]. They at JeffersonLab in their support of this experiment. We are also consistent with calculations of Ref. [33]. also acknowledge useful discussions with D. Day and Fig. 11 shows the measured Q2 dependence of the rel- E. Piasetzki. This work was supported in part by the ative SRC probability, a2(A), which appears to be Q2 U.S. Department of Energy, the National Science Foun- independent for all targets. dation, the French Commissariat a l’Energie Atomique, the FrenchCentreNationalde la RechercheScientifique, the ItalianIstituto NazionalediFisicaNucleare,andthe V. SUMMARY KoreaResearchFoundation. U.Thomaacknowledgesan “Emmy Noether” grand from the Deutsche Forschungs- The A(e,e′) inclusive electron scattering cross section gemeinschaft. TheSouteasternUniversitiesResearchAs- ratios of 4He, 12C, and 56Fe to 3He have been measured sociation (SURA) operates the Thomas Jefferson Na- for the first time under identical kinematical conditions. tional Acceleraror Facility for the United States Depart- It is shown that: ment of Energy under contract DE-AC05-84ER40150. [1] L.L. Frankfurt and M.I. Strikman, Phys. Rep. 76, 215 [6] L.L.Frankfurt,M.I.Strikman,D.B.Day,M.M.Sargsyan, (1981); , Phys.Rep. 160, 235 (1988). Phys. Rev.C 48, 2451 (1993). [2] T.W. Donnelly and I. Sick, Phys. Rev. C 60, 065502 [7] W.P. Schutzet al.,Phys. Rev.Lett. 38, 8259 (1977). (1999); [8] S. Rock et al., Phys.Rev.Lett. 49, 1139 (1982). [3] J. Arrington, et al.,Phys.Rev. Lett.82, 2056 (1999). [9] R.G. Arnold et al.,Phys. Rev.Lett. 61, 806 (1988). [4] C. Ciofi degli Atti,Phys.Rev. C 53, 1689, (1996). [10] D. Day et al., Phys.Rev.Lett. 59, 427 (1979). [5] L.FrankfurtandM.Strikman,inElectromagnetic Inter- [11] M. Sargsian, CLAS-NOTE90-007 (1990). actionswithNuclei,editedbyB.FroisandI.Sick(World [12] R. Machleidt, Phys.Rev.C 63, 024001 (2001). Scientific,Singapore, 1991). [13] M.V. Zverev and E.E. Saperstein, Yad. Fiz. 43, 304

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