The inclusive A(e,e′) cross section for x≃1 was measured on 2H, C, Fe, and Au for momentum transfers Q2 from 1 to 6.8 (GeV/c)2. The scaling behavior of the data was examined in the region of transition from y scaling to x scaling. Throughout this transitional region, the data exhibit ξ scaling, reminiscent of the Bloom-Gilman duality seen in free nucleon scattering

Anthony, P.

Arnold, R.G.

Arrington, J.

Beise, E.J.

Belz, J.E.

Bosted, P.E.

Bulten, H.-J.

Chapman, M.S.

Coulter, K.P.

Dietrich, F.

Ent, R.

Epstein, M.

Filippone, B.W.

Gao, H.

Gearhart, R.A.

Geesaman, D.F.

Hansen, J.-O.

Holt, R.J.

Jackson, H.E.

Jones, C.E.

Keppel, C.E.

Kinney, E.R.

Kuhn, S.

Lee, K.

Lorenzon, W.

Lung, A.

Makins, N.C.R.

Margaziotis, D.J.

McKeown, R.D.

Milner, R.G.

Mueller, B.

Napolitano, J.

Nelson, J.

O'Neill, T.G.

Papavassiliou, V.

Petratos, G.G.

Potterveld, D.H.

Rock, S.E.

Spengos, M.

Szalata, Z.M.

Tao, L.H.

van Bibber, K.

van den Brand, J.F.J.

White, J.L.

Winter, D.

Zeidman, B.

Physical Review C

English

Digital Repository at the University of Maryland

PHYSICAL REVIEW C MAY 1996VOLUME 53, NUMBER 5Inclusive electron scattering from nuclei atx.1J. Arrington,1 P. Anthony,2 R. G. Arnold,3 E. J. Beise,1,a J. E. Belz,1,b P. E. Bosted,3 H.-J. Bulten,4 M. S. Chapman,5K. P. Coulter,6,c F. Dietrich,2 R. Ent,5,d M. Epstein,7 B. W. Filippone,1 H. Gao,1,e R. A. Gearhart,8 D. F. Geesaman,6J.-O. Hansen,5 R. J. Holt,6 H. E. Jackson,6 C. E. Jones,4,f C. E. Keppel,3,g E. R. Kinney,9 S. Kuhn,10,h K. Lee,5W. Lorenzon,1,i A. Lung,3,j N. C. R. Makins,5 D. J. Margaziotis,7 R. D. McKeown,1 R. G. Milner,5 B. Mueller,1J. Napolitano,11 J. Nelson,5,k T. G. O’Neill,1,f V. Papavassiliou,6,l G. G. Petratos,8,m D. H. Potterveld,6 S. E. Rock,3M. Spengos,3 Z. M. Szalata,3 L. H. Tao,3 K. van Bibber,2 J. F. J. van den Brand,4 J. L. White,3 D. Winter,5,nand B. Zeidman61California Institute of Technology, Pasadena, California 911252Lawrence Livermore National Laboratory, Livermore, California 945503American University, Washington, D.C. 200164University of Wisconsin, Madison, Wisconsin 537065Massachusetts Institute of Technology, Cambridge, Massachusetts 021396Argonne National Laboratory, Argonne, Illinois 604397California State University, Los Angeles, California 900328Stanford Linear Accelerator Center, Stanford, California 943099University of Colorado, Boulder, Colorado 8030910Stanford University, Stanford, California 9430511Rensselaer Polytechnic Institute, Troy, New York 12180~Received 6 March 1995!The inclusiveA(e,e8) cross section forx.1 was measured on2H, C, Fe, and Au for momentum transfersQ2 from 1 to 6.8~GeV/c)2. The scaling behavior of the data was examined in the region of transition fromy scaling tox scaling. Throughout this transitional region, the data exhibitj scaling, reminiscent of theBloom-Gilman duality seen in free nucleon scattering.@S0556-2813~96!01705-0#PACS number~s!: 25.30.Fj, 13.60.Hbheepedfef-ftonngae-las--eferIn inclusive electron scattering, scaling functions are important in the study of constituent substructure and interations. Scaling is typically a sign that a simple reactiomechanism dominates the process, allowing one to extraPresent address: University of Maryland, College Park, Maryla20742.bPresent address: University of Colorado, Boulder, Colora80309.cPresent address: University of Michigan, Ann Arbor, Michiga48109.dPresent address: CEBAF, Newport News, Virginia 23606.ePresent address: University of Illinois at Urbana-Champaign, Ubana, Illinois 61801.fPresent address: Argonne National Laboratory, Argonne, Illino60439.gPresent address: Virginia Union University, Richmond, Virgini23220.hPresent address: Old Dominion University, Norfolk, Virginia23529.iPresent address: University of Pennsylvania, Philadelphia, Pesylvania 19104.jPresent address: California Institute of Technology, PasadeCalifornia 91125.kPresent address: SLAC, Stanford, California 94309.lPresent address: Illinois Institute of Technology, Chicago, Illino60616.mPresent address: Kent State University, Kent, Ohio 44242.nPresent address: Colombia University, New York, New Yor10027.530556-2813/96/53~5!/2248~4!/$10.00-c-nactinformation on structure in a model independent way. Tmost familiar scaling occurs in the limit of largen andQ2,wheren is the energy transfer and2Q25qm2 is the square ofthe four-momentum transfer. In this limit the nucleon deinelastic structure functionsMW1(n,Q2) and nW2(n,Q2)become functions only of the Bjorkenx5Q2/2Mn, whereM is the nucleon mass. In this limit,x can be interpreted asthe fraction of the nucleon’s longitudinal momentum carriby the struck quark, andMW1 and nW2 are related to thequark longitudinal momentum distribution. Violations oBjorken scaling in the free nucleon exist at lowQ2 due totarget mass and higher-twist effects. To correct for thefects of target mass at finiteQ2, the Nachtmann variablej52x/@11(114M2x2/Q2)1/2# has been used in place ox. This has been shown to be the correct variable in whichstudy the logarithmic QCD scaling violations in the nucleo@1#. A more recent work by Gurvitz proposes a new scalivariable that includes parton confinement effects@2#.A similar case exists for quasielastic scattering fromnucleon in a nucleus. At high momentum transfer, the ‘‘rduced’’ cross section was predicted@3#, and later observed@4# to exhibit scaling in the variabley(q,n) @in a simplerelativistic approximationy5(2Mn1n22q2)/2q, whereqis the momentum transfer#. In the simplest picture ofy scal-ing, the electron-nucleus cross section is divided by the etic nucleon cross section, leaving a universal functionF(y)which is independent ofQ2 in the plane wave impulse approximation. In the scaling limit,y can be interpreted as thnucleon’s initial momentum along the momentum transdirection, andF(y) is related to the nucleon’s momentumnddonr-isann-na,isk2248 © 1996 The American Physical Societygvcre-uietored.12c-thesedallec-tionata.delrededineltor-eltedncengia-ngs-dre-ec-s,sethe53 2249INCLUSIVE ELECTRON SCATTERING FROM NUCLEI ATx.1distribution in the nucleus. Thus,y plays a similar role fornucleons in a nucleus asx does for quarks in a nucleon.In the limit of high Q2, the scaling variablesx, y, andj are related. In the parton model,j replacesx as the scalingvariable when the target mass is not neglected. At larQ2, j can also be expressed as a function only of~with theleading scale-breaking termM2/Q2) @5#. There may also bea relationship between quasielastic and inelastic scatterinmore modest momentum transfers. In the case of the fnucleon, Bloom and Gilman@6# discovered that the reso-nance peaks in the structure function have the sameQ2 be-havior as the deep inelastic contribution when viewed asfunction of v8, a modified version of the Bjorken scalingvariable. It was later shown@7# that this connection betweenthe highQ2 structure function and the resonance form fators, called local duality, was expected from perturbatiQCD and should be valid for the nucleon elastic peak as was the resonance peaks if the structure function is analyzeterms ofj. When the structure function is viewed as a funtion of j, the elastic and resonance peaks have the saQ2 behavior as the deep inelastic structure function. Tmagnitude of the elastic and resonance peaks decreaseidly with Q2, but move to higherj, keeping a nearly con-stant strength with respect to the deep inelastic structfunction, which falls withj. Thus the strongQ2 dependenceof the higher twist effects~the elastic and resonance peaks! isremoved when the structure function is averaged overange inj. In the case of electron scattering from a nucleuthe Fermi motion can perform this ‘‘averaging’’ of the structure function. Thus, when examiningnW2A as a function ofj, scaling may be observed at lower momentum transfwherex scaling is not yet valid due to the quasielastic cotribution.Scaling in inclusive scattering from nuclei~He, C, Fe, andAu! was examined in a previous measurement@8, 5# forQ2 values of 0.3–3.1~GeV/c)2. For these values of momentum transfer, where the quasielastic contribution dominathe cross section, the data exhibity scaling for y,0(x.1). The positivey values represent the high energytransfer side of the quasielastic peak where they scalingbreaks down due to the increasing inelastic contributionhigherQ2. This same experiment examinedx andj scalingin the nucleus@5#. For low values ofx, the structure functionnW2 is a function only ofx, as predicted. For values ofxnear or above 1, scaling was not observed due to the conbution of quasielastic scattering. If one examines the strture function vs the Nachtmann variablej, a scaling behav-ior is suggested; at lowerj, the data are nearly independenof Q2, while at higherj, the data approach a universacurve. More recent data at higherQ2 show the same ap-proach to scaling for inclusive scattering from aluminum@9#.The beginning of scaling in this region suggests thatj scal-ing is not only applicable to deep inelastic scattering, butalso connected to quasielastic scattering andy scaling. Herewe examine scaling in the transition region from quasielasto deep inelastic scattering, to further study the connectbetweenj scaling andy scaling.The data presented here are from the NE18 experim@10, 11#, a coincidenceA(e,e8p) measurement performed inEnd Station A at the SLAC Nuclear Physics Facilitygeatreeac-eelld in-meherap-ureas,-rsn-tes-attri-c-tlisticonent~NPAS!. Electron singles were recorded as well as thelectron-proton coincidences and this data was analyzedextract the inclusive cross section. Scattering was measufrom cryogenic liquid1H and 2H targets and solid C, Fe, andAu targets with beam energies of 2.02, 3.19, 4.21, and 5GeV, at angles of 35.5°, 47.4°, 53.4°, and 56.6°, respetively @Q251, 3, 5, and 6.8~GeV/c)2#. The scattered elec-trons were detected in the SLAC 1.6 GeV/c spectrometer.The pion rate in the spectrometer was up to 500 timeselectron rate for runs on Au at the highestQ2. A CO2 gasČerenkov counter and lead glass shower counter were uto eliminate the pions. Tight cuts were used in the finanalysis, resulting in a pion rejection of 15 000 to 1, whimaintaining an electron efficiency of 90%.In order to extract the cross section, corrections for spetrometer acceptance, detector efficiencies, data acquisideadtimes, and radiative corrections were applied to the dThe acceptance was determined using a Monte Carlo moof the spectrometer, and deadtime corrections were measuon a run-by-run basis. Radiative corrections were appliusing an iterative procedure following the formulas of Steet al. @12#, which are based on the work of Mo and Tsai@13#and Tsai@14#. Radiative effects were calculated for a modcross section and the result was compared to the datadetermine a smooth correction to the model. The ‘‘corected’’ model was then used in place of the original modcross section, and the procedure repeated until the radiamodel was consistent with the data. The model dependeof the radiative correction procedure was tested by varyithe initial model cross section. We also compared the radtively corrected cross sections calculated from runs usitargets of different thickness. A final error of 3% was asigned to the radiative correction procedure.Extracting the structure functions from the measurecross section without performing a Rosenbluth separationquires a knowledge of the ratio of the absorption cross stions for longitudinal and transverse virtual photonR5sL /sT . However, the error in extractingnW2A due touncertainty inR is small for forward angles and forR,1.We have assumedR50.5/Q2 with an uncertainty of 50%,which is consistent with impulse approximation predictionas well as a recent measurement@9#. This leads to a worstcase contribution to the uncertainty innW2A of 6 3%. TheFIG. 1. F(y) vs y for iron for the present experiment and thprevious NE3 measurement. Errors in the new data~solid points!are dominated by a 4% systematic error, but are smaller thanpoints shown.du-on--n-le-e-t-gticn2250 53J. ARRINGTONet al.scaling functionF(y) was extracted from the measured crosection using the same method as Dayet al. @8# and Potter-veld @18#.The extracted scaling functionF(y) for iron is shown inFig. 1, along with the previous SLAC NE3 data@8#. Whilethe y,0 data exhibitedy scaling for the previous data, thescaling clearly breaks down at highQ2 for all y values mea-sured (y.280 MeV/c). The breakdown ofy scaling is dueto the transition from quasielastic scattering to inelastic sctering. To testy scaling in this region, one must calculatand subtract off inelastic contributions to the cross sectioThis introduces a model dependence and can only be dreliably when the inelastic contributions do not dominate tcross section. It is clear that in the case of inclusive scatting, the applicability ofy scaling is limited to lower momen-tum transfers, where quasielastic scattering dominatescross section.Figure 2 shows the measured structure function pnucleon for iron and carbon as a function ofx. Clearly thedata do not scale in this range but theQ2 dependence isdecreasing asQ2 increases. The structure function is nearidentical for all of the nuclear targets except deuteriumwhere the smaller Fermi momentum causes a peak instructure function nearx51. The larger Fermi momentum inthe heavier nuclei washes out the quasielastic peak, leadto a lower structure function nearx51. The difference be-tween carbon and iron decreases asx gets further from 1 andat higherQ2. Table I gives the ratio of the structure funcFIG. 2. nW2 /A vs x for iron ~solid points! and carbon~hollowpoints!.FIG. 3. nW2Fe vs j for the present experiment and the NEmeasurement.ssat-en.oneheer-theerly,theing-tions for different targets at eachQ2, for 0.95,x,1.05, andthe ratio of iron to deuterium forx51. TheQ2 behavior ofthe ratio to deuterium is consistent with the behavior founfor aluminum@15#.In Fig. 3,nW2 is plotted vsj and an approach to scalingis observed. The new data are all centered atx51, but moveto higherj asQ2 increases, lying on the universal curve. Tobetter understand the transition, we calculated the contribtions due to the different scattering processes using the cvolution model of Ji and Filippone@17# with a Woods-Saxonspectral function, dipole electric form factor, and the magnetic form factor of Gari and Kru¨mpelmann@16#. Figure 4shows the approach to scaling for fixedj along with ourcalculations showing the quasielastic and deep inelastic cotributions to the structure function. Also included is theNE11 data for aluminum@9#, which are in good agreementwhen the structure function is scaled by the number of nucons. As a function ofQ2, we see an increase in the structurfunction on the lowQ2 side of the quasielastic peak, andthen a decrease to the highQ2 value, where inelastic scatter-ing dominates. While the structure function is not independent ofQ2 for a fixedj, it shows lessQ2 dependence thanwhen viewed at a constantx. More importantly, the mea-sured structure function has relatively littleQ2 dependencein the region of transition from quasielastic to inelastic scatering, even though the quasielastic contribution is fallinrapidly with Q2. This is true for allj values measured, in-dicating a connection between the quasielastic and inelascross sections, reminiscent of local duality in the nucleon.To summarize, we have extracted the scaling functioF(y) and the structure functionW2 nearx51 for nucleiwith A ranging from 2 to 197 atQ2 values from 1 to 6.8~GeV/c)2. At the higherQ2 values,y scaling breaks down3FIG. 4. nW2Fe is plotted vsQ2 at j50.85. The lines are calcu-lations of the total~solid!, quasielastic~dashed!, and deep inelastic~dotted! contributions to the structure function.TABLE I. Ratio of structure functions (nW2 /A) for differenttargets nearx51.Q 2 @~GeV/c)2] C/Fe Fe/Au Fe/D1.0 1.1460.05 1.1160.053.0 1.1460.05 1.1960.05 0.4860.035.0 1.0760.05 1.16 0.05 0.6960.046.8 1.0560.05 0.96 0.05 0.9560.06ceY-n-o-an53 2251INCLUSIVE ELECTRON SCATTERING FROM NUCLEI ATx.1for all measured values ofy as deep inelastic scattering begins to dominate. When examiningnW2 , we do not yet seescaling inx, but we do begin to see scaling in the Nachmann scaling variablej. This suggests a connection betweequasielastic and inelastic scattering, similar to the caselocal duality in the nucleon.j scaling may prove to be auseful tool in understanding nuclear structure functions, bbetter coverage inj at the presentQ2 values, as well ashigherQ2 measurements~e.g., at CEBAF@19#! are needed tofully understand the scaling of the structure function, and trelation betweenj scaling andy scaling.E-t-nofutheThis work was supported in part by the National ScienFoundation under Grant Nos. PHY-9014406 and PH9114958 ~American!, PHY-9115574 ~Caltech!, PHY-9101404 ~CSLA!, PHY-9208119 ~RPI!, PHY-9019983~Wisconsin!, and by the Department of Energy under Cotract Nos. W-31-109-ENG-38 ~Argonne!, DE-FG02-86ER40269 ~Colorado!, W-2705-Eng-48 ~LLNL !, DE-AC02-76ER03069~MIT !, DE-AC03-76SF00515~SLAC!,DE-FG03-88ER40439~Stanford!. R.G.M. acknowledges thesupport of the Presidential Young Investigator Award Prgram at NSF. B.W.F. acknowledges the support of the SloFoundation.l-@1# H. Georgi and H. D. Politzer, Phys. Rev. D14, 1829~1976!.@2# S. A. Gurvitz, Phys. Rev. D52, 1433~1995!.@3# G. B. West, Phys. Rep.18, 263 ~1975!.@4# I. Sick, D. B. Day, and J. S. McCarthy, Phys. Rev. Lett.45,871 ~1980!.@5# B. W. Filipponeet al., Phys. Rev. C45, 1582~1992!.@6# E. Bloom and F. Gilman, Phys. Rev. D4, 2901~1971!.@7# A. DeRujula, H. Georgi, and H. D. Politzer, Ann. Phys.~N.Y.!103, 315 ~1977!. For a recent discussion of duality see C.Carlson and N. C. Mukhopadhyay, Phys. Rev. D41, 2343~1990!; 47, R1737~1993!.@8# D. B. Dayet al., Phys. Rev. Lett.59, 427 ~1987!.@9# P. E. Bostedet al., Phys. Rev. C46, 2505~1992!@10# T. G. O’Neill, Ph.D. thesis, California Institute of Technology.,1994; T. G. O’Neillet al., Phys. Lett. B351, 4775~1995!@11# N. C. R. Makinset al., Phys. Rev. Lett.72 1986 ~1994!.@12# S. Steinet al., Phys. Rev. D12, 1884~1975!.@13# L. W. Mo and Y. S. Tsai, Rev. Mod. Phys.41, 205 ~1969!.@14# Yung-Su Tsai, SLAC Report SLAC-PUB-848, 1971.@15# L. L. Frankfurt, M. I. Strikman, D. B. Day, and M. Sargsyan,Phys. Rev. C48, 2451~1993!.@16# M. F. Gari and W. Kru¨mpelmann, Z. Phys. A322, 689~1985!.@17# Xiangdong Ji and B. W. Filippone, Phys. Rev. C42, R2279~1990!.@18# D. H. Potterveld, Ph.D. thesis, California Institute of Technoogy, 1989.@19# D. B. Day and B. W. Filippone, CEBAF Proposal E89-008.

Inclusive electron scattering from nuclei at x≃1

The inclusive A(e,e\u27) cross section for x≃1 was measured on 2H, C, Fe, and Au for momentum transfers Q2 from 1 to 6.8 (GeV/c)2. The scaling behavior of the data was examined in the region of transition from y scaling to x scaling. Throughout this transitional region, the data exhibit ξ scaling, reminiscent of the Bloom-Gilman duality seen in free nucleon scattering

Arnold, R. G.

Beise, E. J.

Belz, J. E.

Bosted, P. E.

Bulten, H.- J.

Chapman, M. S.

Coulter, K. P.

Filippone, B. W.

Gearhart, R. A.

Geesaman, D. F.

Hansen, J.- O.

Holt, R. J.

Jackson, H. E.

Jones, C. E.

Keppel, C. E.

Kinney, E. R.

Malkins, N.C.R.

Margaziotis, D. J.

McKeown, R. D.

Milner, R. G.

O\u27Neill, T.G.

Petratos, G. G.

Potterveld, D. H.

Rock, S. E.

Szalata, Z. M.

Tao, L. H.

White, J. L.

Old Dominion University

Old Dominion University ODU Digital Commons Physics Faculty Publications Physics 1996 Inclusive Electron Scattering From Nuclei at x≃1 J. Arrington P. Anthony R. G. Arnold E. J. Beise J. E. Belz See next page for additional authors Follow this and additional works at: https://digitalcommons.odu.edu/physics_fac_pubs  Part of the Elementary Particles and Fields and String Theory Commons, Nuclear Commons, and the Quantum Physics Commons Original Publication Citation Arrington, J., Anthony, P. Arnold, R.G. ... White, J.L., Winter, D., Zeidman, B. (1996). Inclusive electron scattering from nuclei at x≃1. Physical Review C, 53(5), 2248-2251. https://doi.org/10.1103/PhysRevC.53.2248 This Article is brought to you for free and open access by the Physics at ODU Digital Commons. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact digitalcommons@odu.edu. Authors J. Arrington, P. Anthony, R. G. Arnold, E. J. Beise, J. E. Belz, P. E. Bosted, H.- J. Bulten, M. S. Chapman, K. P. Coulter, F. Dietrich, R. Ent, M. Epstein, B. W. Filippone, H. Gao, R. A. Gearhart, D. F. Geesaman, J.- O. Hansen, R. J. Holt, H. E. Jackson, C. E. Jones, C. E. Keppel, E. R. Kinney, S. Kuhn, K. Lee, W. Lorenzon, A. Lung, N.C.R. Malkins, D. J. Margaziotis, R. D. McKeown, R. G. Milner, B. Mueller, J. Napolitano, J. Nelson, T.G. O'Neill, V. Papavassiliou, G. G. Petratos, D. H. Potterveld, S. E. Rock, M. Spengos, Z. M. Szalata, L. H. Tao, K. van Bibber, J.F.J. van den Brand, J. L. White, D. Winter, and B. Zeidman This article is available at ODU Digital Commons: https://digitalcommons.odu.edu/physics_fac_pubs/509 Inclusive electron scattering from nuclei atx.1J. Arrington,1 P. Anthony,2 R. G. Arnold,3 E. J. Beise,1,a J. E. Belz,1,b P. E. Bosted,3 H.-J. Bulten,4 M. S. Chapman,5K. P. Coulter,6,c F. Dietrich,2 R. Ent,5,d M. Epstein,7 B. W. Filippone,1 H. Gao,1,e R. A. Gearhart,8 D. F. Geesaman,6J.-O. Hansen,5 R. J. Holt,6 H. E. Jackson,6 C. E. Jones,4,f C. E. Keppel,3,g E. R. Kinney,9 S. Kuhn,10,h K. Lee,5W. Lorenzon,1,i A. Lung,3,j N. C. R. Makins,5 D. J. Margaziotis,7 R. D. McKeown,1 R. G. Milner,5 B. Mueller,1J. Napolitano,11 J. Nelson,5,k T. G. O’Neill,1,f V. Papavassiliou,6,l G. G. Petratos,8,m D. H. Potterveld,6 S. E. Rock,3M. Spengos,3 Z. M. Szalata,3 L. H. Tao,3 K. van Bibber,2 J. F. J. van den Brand,4 J. L. White,3 D. Winter,5,nand B. Zeidman61California Institute of Technology, Pasadena, California 911252Lawrence Livermore National Laboratory, Livermore, California 945503American University, Washington, D.C. 200164University of Wisconsin, Madison, Wisconsin 537065Massachusetts Institute of Technology, Cambridge, Massachusetts 021396Argonne National Laboratory, Argonne, Illinois 604397California State University, Los Angeles, California 900328Stanford Linear Accelerator Center, Stanford, California 943099University of Colorado, Boulder, Colorado 8030910Stanford University, Stanford, California 9430511Rensselaer Polytechnic Institute, Troy, New York 12180~Received 6 March 1995!The inclusiveA(e,e8) cross section forx.1 was measured on2H, C, Fe, and Au for momentum transfersQ2 from 1 to 6.8~GeV/c)2. The scaling behavior of the data was examined in the region of transition fromy scaling tox scaling. Throughout this transitional region, the data exhibitj scaling, reminiscent of theBloom-Gilman duality seen in free nucleon scattering.@S0556-2813~96!01705-0#PACS number~s!: 25.30.Fj, 13.60.HbIn inclusive electron scattering, scaling functions are im-portant in the study of constituent substructure and interac-tions. Scaling is typically a sign that a simple reactionmechanism dominates the process, allowing one to extractinformation on structure in a model independent way. Themost familiar scaling occurs in the limit of largen andQ2,wheren is the energy transfer and2Q25qm2 is the square ofthe four-momentum transfer. In this limit the nucleon deepinelastic structure functionsMW1(n,Q2) and nW2(n,Q2)become functions only of the Bjorkenx5Q2/2Mn, whereM is the nucleon mass. In this limit,x can be interpreted asthe fraction of the nucleon’s longitudinal momentum carriedby the struck quark, andMW1 and nW2 are related to thequark longitudinal momentum distribution. Violations ofBjorken scaling in the free nucleon exist at lowQ2 due totarget mass and higher-twist effects. To correct for the ef-fects of target mass at finiteQ2, the Nachtmann variablej52x/@11(114M2x2/Q2)1/2# has been used in place ofx. This has been shown to be the correct variable in which tostudy the logarithmic QCD scaling violations in the nucleon@1#. A more recent work by Gurvitz proposes a new scalingvariable that includes parton confinement effects@2#.A similar case exists for quasielastic scattering from anucleon in a nucleus. At high momentum transfer, the ‘‘re-duced’’ cross section was predicted@3#, and later observed@4# to exhibit scaling in the variabley(q,n) @in a simplerelativistic approximationy5(2Mn1n22q2)/2q, whereqis the momentum transfer#. In the simplest picture ofy scal-ing, the electron-nucleus cross section is divided by the elas-tic nucleon cross section, leaving a universal functionF(y)which is independent ofQ2 in the plane wave impulse ap-proximation. In the scaling limit,y can be interpreted as thenucleon’s initial momentum along the momentum transferdirection, andF(y) is related to the nucleon’s momentumaPresent address: University of Maryland, College Park, Maryland20742.bPresent address: University of Colorado, Boulder, Colorado80309.cPresent address: University of Michigan, Ann Arbor, Michigan48109.dPresent address: CEBAF, Newport News, Virginia 23606.ePresent address: University of Illinois at Urbana-Champaign, Ur-bana, Illinois 61801.fPresent address: Argonne National Laboratory, Argonne, Illinois60439.gPresent address: Virginia Union University, Richmond, Virginia23220.hPresent address: Old Dominion University, Norfolk, Virginia23529.iPresent address: University of Pennsylvania, Philadelphia, Penn-sylvania 19104.jPresent address: California Institute of Technology, Pasadena,California 91125.kPresent address: SLAC, Stanford, California 94309.lPresent address: Illinois Institute of Technology, Chicago, Illinois60616.mPresent address: Kent State University, Kent, Ohio 44242.nPresent address: Colombia University, New York, New York10027.PHYSICAL REVIEW C MAY 1996VOLUME 53, NUMBER 5530556-2813/96/53~5!/2248~4!/$10.00 2248 © 1996 The American Physical Societydistribution in the nucleus. Thus,y plays a similar role fornucleons in a nucleus asx does for quarks in a nucleon.In the limit of high Q2, the scaling variablesx, y, andj are related. In the parton model,j replacesx as the scalingvariable when the target mass is not neglected. At largeQ2, j can also be expressed as a function only of~with theleading scale-breaking termM2/Q2) @5#. There may also bea relationship between quasielastic and inelastic scattering atmore modest momentum transfers. In the case of the freenucleon, Bloom and Gilman@6# discovered that the reso-nance peaks in the structure function have the sameQ2 be-havior as the deep inelastic contribution when viewed as afunction of v8, a modified version of the Bjorken scalingvariable. It was later shown@7# that this connection betweenthe highQ2 structure function and the resonance form fac-tors, called local duality, was expected from perturbativeQCD and should be valid for the nucleon elastic peak as wellas the resonance peaks if the structure function is analyzed interms ofj. When the structure function is viewed as a func-tion of j, the elastic and resonance peaks have the sameQ2 behavior as the deep inelastic structure function. Themagnitude of the elastic and resonance peaks decrease rap-idly with Q2, but move to higherj, keeping a nearly con-stant strength with respect to the deep inelastic structurefunction, which falls withj. Thus the strongQ2 dependenceof the higher twist effects~the elastic and resonance peaks! isremoved when the structure function is averaged over arange inj. In the case of electron scattering from a nucleus,the Fermi motion can perform this ‘‘averaging’’ of the struc-ture function. Thus, when examiningnW2A as a function ofj, scaling may be observed at lower momentum transferswherex scaling is not yet valid due to the quasielastic con-tribution.Scaling in inclusive scattering from nuclei~He, C, Fe, andAu! was examined in a previous measurement@8, 5# forQ2 values of 0.3–3.1~GeV/c)2. For these values of momen-tum transfer, where the quasielastic contribution dominatesthe cross section, the data exhibity scaling for y,0(x.1). The positivey values represent the high energy-transfer side of the quasielastic peak where they scalingbreaks down due to the increasing inelastic contribution athigherQ2. This same experiment examinedx andj scalingin the nucleus@5#. For low values ofx, the structure functionnW2 is a function only ofx, as predicted. For values ofxnear or above 1, scaling was not observed due to the contri-bution of quasielastic scattering. If one examines the struc-ture function vs the Nachtmann variablej, a scaling behav-ior is suggested; at lowerj, the data are nearly independentof Q2, while at higherj, the data approach a universalcurve. More recent data at higherQ2 show the same ap-proach to scaling for inclusive scattering from aluminum@9#.The beginning of scaling in this region suggests thatj scal-ing is not only applicable to deep inelastic scattering, but isalso connected to quasielastic scattering andy scaling. Herewe examine scaling in the transition region from quasielasticto deep inelastic scattering, to further study the connectionbetweenj scaling andy scaling.The data presented here are from the NE18 experiment@10, 11#, a coincidenceA(e,e8p) measurement performed inEnd Station A at the SLAC Nuclear Physics Facility~NPAS!. Electron singles were recorded as well as theelectron-proton coincidences and this data was analyzed toextract the inclusive cross section. Scattering was measuredfrom cryogenic liquid1H and 2H targets and solid C, Fe, andAu targets with beam energies of 2.02, 3.19, 4.21, and 5.12GeV, at angles of 35.5°, 47.4°, 53.4°, and 56.6°, respec-tively @Q251, 3, 5, and 6.8~GeV/c)2#. The scattered elec-trons were detected in the SLAC 1.6 GeV/cspectrometer.The pion rate in the spectrometer was up to 500 times theelectron rate for runs on Au at the highestQ2. A CO2 gasČerenkov counter and lead glass shower counter were usedto eliminate the pions. Tight cuts were used in the finalanalysis, resulting in a pion rejection of 15 000 to 1, whilemaintaining an electron efficiency of 90%.In order to extract the cross section, corrections for spec-trometer acceptance, detector efficiencies, data acquisitiondeadtimes, and radiative corrections were applied to the data.The acceptance was determined using a Monte Carlo modelof the spectrometer, and deadtime corrections were measuredon a run-by-run basis. Radiative corrections were appliedusing an iterative procedure following the formulas of Steinet al. @12#, which are based on the work of Mo and Tsai@13#and Tsai@14#. Radiative effects were calculated for a modelcross section and the result was compared to the data todetermine a smooth correction to the model. The ‘‘cor-rected’’ model was then used in place of the original modelcross section, and the procedure repeated until the radiatedmodel was consistent with the data. The model dependenceof the radiative correction procedure was tested by varyingthe initial model cross section. We also compared the radia-tively corrected cross sections calculated from runs usingtargets of different thickness. A final error of 3% was as-signed to the radiative correction procedure.Extracting the structure functions from the measuredcross section without performing a Rosenbluth separation re-quires a knowledge of the ratio of the absorption cross sec-tions for longitudinal and transverse virtual photons,R5sL /sT . However, the error in extractingnW2A due touncertainty inR is small for forward angles and forR,1.We have assumedR50.5/Q2 with an uncertainty of 50%,which is consistent with impulse approximation predictionsas well as a recent measurement@9#. This leads to a worstcase contribution to the uncertainty innW2A of 6 3%. TheFIG. 1. F(y) vs y for iron for the present experiment and theprevious NE3 measurement. Errors in the new data~solid points!are dominated by a 4% systematic error, but are smaller than thepoints shown.53 2249INCLUSIVE ELECTRON SCATTERING FROM NUCLEI ATx.1-0.3 -0.2 NE3 0 2=1.8 (GeV/c) 2 NE3 02=2.3 (GeV/c) 2 NE3 02=2.7 (GeV / c) 2 NE3 0 2=3.1 (GeV/c) 2 NE 18 0 2= 1.0 (GeV / c)2 NE18 0 2=3.0 (GeV /c) 2 NE18 0 2=5.0 (GeV/c) 2 NE18 0 2=6.8 (GeV /c) 2 -0.1 -0.0 0.1 y (GeV/c) 0.2 scaling functionF(y) was extracted from the measured crosssection using the same method as Dayet al. @8# and Potter-veld @18#.The extracted scaling functionF(y) for iron is shown inFig. 1, along with the previous SLAC NE3 data@8#. Whilethe y,0 data exhibitedy scaling for the previous data, thescaling clearly breaks down at highQ2 for all y values mea-sured (y.280 MeV/c). The breakdown ofy scaling is dueto the transition from quasielastic scattering to inelastic scat-tering. To testy scaling in this region, one must calculateand subtract off inelastic contributions to the cross section.This introduces a model dependence and can only be donereliably when the inelastic contributions do not dominate thecross section. It is clear that in the case of inclusive scatter-ing, the applicability ofy scaling is limited to lower momen-tum transfers, where quasielastic scattering dominates thecross section.Figure 2 shows the measured structure function pernucleon for iron and carbon as a function ofx. Clearly thedata do not scale in this range but theQ2 dependence isdecreasing asQ2 increases. The structure function is nearlyidentical for all of the nuclear targets except deuterium,where the smaller Fermi momentum causes a peak in thestructure function nearx51. The larger Fermi momentum inthe heavier nuclei washes out the quasielastic peak, leadingto a lower structure function nearx51. The difference be-tween carbon and iron decreases asx gets further from 1 andat higherQ2. Table I gives the ratio of the structure func-tions for different targets at eachQ2, for 0.95,x,1.05, andthe ratio of iron to deuterium forx51. TheQ2 behavior ofthe ratio to deuterium is consistent with the behavior foundfor aluminum@15#.In Fig. 3,nW2 is plotted vsj and an approach to scalingis observed. The new data are all centered atx51, but moveto higherj asQ2 increases, lying on the universal curve. Tobetter understand the transition, we calculated the contribu-tions due to the different scattering processes using the con-volution model of Ji and Filippone@17#with a Woods-Saxonspectral function, dipole electric form factor, and the mag-netic form factor of Gari and Kru¨mpelmann@16#. Figure 4shows the approach to scaling for fixedj along with ourcalculations showing the quasielastic and deep inelastic con-tributions to the structure function. Also included is theNE11 data for aluminum@9#, which are in good agreementwhen the structure function is scaled by the number of nucle-ons. As a function ofQ2, we see an increase in the structurefunction on the lowQ2 side of the quasielastic peak, andthen a decrease to the highQ2 value, where inelastic scatter-ing dominates. While the structure function is not indepen-dent ofQ2 for a fixedj, it shows lessQ2 dependence thanwhen viewed at a constantx. More importantly, the mea-sured structure function has relatively littleQ2 dependencein the region of transition from quasielastic to inelastic scat-tering, even though the quasielastic contribution is fallingrapidly with Q2. This is true for allj values measured, in-dicating a connection between the quasielastic and inelasticcross sections, reminiscent of local duality in the nucleon.To summarize, we have extracted the scaling functionF(y) and the structure functionW2 nearx51 for nucleiwith A ranging from 2 to 197 atQ2 values from 1 to 6.8~GeV/c)2. At the higherQ2 values,y scaling breaks downFIG. 2. nW2 /A vs x for iron ~solid points!and carbon~hollowpoints!.FIG. 3. nW2Fe vs j for the present experiment and the NE3measurement.FIG. 4. nW2Fe is plotted vsQ2 at j50.85. The lines are calcu-lations of the total~solid!, quasielastic~dashed!, and deep inelastic~dotted!contributions to the structure function.TABLE I. Ratio of structure functions (nW2 /A) for differenttargets nearx51.Q 2 @~GeV/c)2] C/Fe Fe/Au Fe/D1.0 1.1460.05 1.1160.053.0 1.1460.05 1.1960.05 0.4860.035.0 1.0760.05 1.1660.05 0.6960.046.8 1.0560.05 0.9660.05 0.9560.062250 53J. ARRINGTONet al.0.7 100 10-1 10-2 <( ?' 10-3 5-N ;:, 10-4 10-5 10-6 0.25 • • .. . .. 0.8 0.50 0.9 X 0.75 f 1.0 1.1 ,0-2 <( ',, ;,,;: ;:, 10-3 ~ 0 2 100 125 ◊ NE3 - ( • 0,85, fe X NE11 - t • 0.85, Al • NE18 - ( • 0.85, Fe Totol Structure Function Ouasielastic Contribution ' fnelostie Contribution ~ T \ ~ i l< •1 \ ' ' 1 \ . . . . . . . . . . . . . . . . . . . . . . . ' \ ' \ \ ' \ \ 3 4 5 6 7 8 02 [(GeV/c)2] for all measured values ofy as deep inelastic scattering be-gins to dominate. When examiningnW2 , we do not yet seescaling inx, but we do begin to see scaling in the Nacht-mann scaling variablej. This suggests a connection betweenquasielastic and inelastic scattering, similar to the case oflocal duality in the nucleon.j scaling may prove to be auseful tool in understanding nuclear structure functions, butbetter coverage inj at the presentQ2 values, as well ashigherQ2 measurements~e.g., at CEBAF@19#! are needed tofully understand the scaling of the structure function, and therelation betweenj scaling andy scaling.This work was supported in part by the National ScienceFoundation under Grant Nos. PHY-9014406 and PHY-9114958 ~American!, PHY-9115574 ~Caltech!, PHY-9101404 ~CSLA!, PHY-9208119 ~RPI!, PHY-9019983~Wisconsin!, and by the Department of Energy under Con-tract Nos. W-31-109-ENG-38 ~Argonne!, DE-FG02-86ER40269 ~Colorado!, W-2705-Eng-48 ~LLNL !, DE-AC02-76ER03069~MIT !, DE-AC03-76SF00515~SLAC!,DE-FG03-88ER40439~Stanford!. R.G.M. acknowledges thesupport of the Presidential Young Investigator Award Pro-gram at NSF. B.W.F. acknowledges the support of the SloanFoundation.@1# H. Georgi and H. D. Politzer, Phys. Rev. D14, 1829~1976!.@2# S. A. Gurvitz, Phys. Rev. D52, 1433~1995!.@3# G. B. West, Phys. Rep.18, 263 ~1975!.@4# I. Sick, D. B. Day, and J. S. McCarthy, Phys. Rev. Lett.45,871 ~1980!.@5# B. W. Filipponeet al., Phys. Rev. C45, 1582~1992!.@6# E. Bloom and F. Gilman, Phys. Rev. D4, 2901~1971!.@7# A. DeRujula, H. Georgi, and H. D. Politzer, Ann. Phys.~N.Y.!103, 315 ~1977!. For a recent discussion of duality see C. E.Carlson and N. C. Mukhopadhyay, Phys. Rev. D41, 2343~1990!; 47, R1737~1993!.@8# D. B. Dayet al., Phys. Rev. Lett.59, 427 ~1987!.@9# P. E. Bostedet al., Phys. Rev. C46, 2505~1992!@10# T. G. O’Neill, Ph.D. thesis, California Institute of Technology,1994; T. G. O’Neillet al., Phys. Lett. B351, 4775~1995!@11# N. C. R. Makinset al., Phys. Rev. Lett.72 1986 ~1994!.@12# S. Steinet al., Phys. Rev. D12, 1884~1975!.@13# L. W. Mo and Y. S. Tsai, Rev. Mod. Phys.41, 205 ~1969!.@14# Yung-Su Tsai, SLAC Report SLAC-PUB-848, 1971.@15# L. L. Frankfurt, M. I. Strikman, D. B. Day, and M. Sargsyan,Phys. Rev. C48, 2451~1993!.@16# M. F. Gari and W. Kru¨mpelmann, Z. Phys. A322, 689~1985!.@17# Xiangdong Ji and B. W. Filippone, Phys. Rev. C42, R2279~1990!.@18# D. H. Potterveld, Ph.D. thesis, California Institute of Technol-ogy, 1989.@19# D. B. Day and B. W. Filippone, CEBAF Proposal E89-008.53 2251INCLUSIVE ELECTRON SCATTERING FROM NUCLEI ATx.1

Inclusive Electron Scattering From Nuclei at x≃1

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Inclusive electron scattering from nuclei at x≃1

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