arXiv:1009.5666v4 [hep-ph] 18 Jan 2011 hs fdueimat deuterium of those 0 in fnce with nuclei of tions not and medium. effect nuclear density the local and of a property light [8–11] is bulk on it in a that data found suggest high-precision be [7] Recent nuclei can Comprehensive effect therein. EMC references medium. the nuclear of nucleon the bound reviews to the due of structure modification and and distributions both energy) (momentum include binding effects to effect. need structure EMC explanations nuclear the proposed of general, explanation There In accepted [1–7]. nuclei generally of no range wide is a for measured been has GeV DS rs etoso uliwith nuclei of sections cross (DIS) xlr h eainhpbtende nlsi n large- and will inelastic x paper deep This between relationship nucleus. the the explore in (SRC) correlations range eimfor terium ur itiuin.Ieatcsatrn at scattering Inelastic distributions. quark o n eim hw lta hnpotda func- a as plotted when of plateau tion a shows helium) and bon where h rs eto ai o w ieetnce ( nuclei different two for ratio section cross The n ie nyi antd.Terto(nteplateau the ( (in inclusive ratio per-nucleon The the of magnitude. region) in only differ mo- and high nucleon for the nuclei mentum, that different indicates of plateau distributions The momentum 14]. Lab- [13, Jefferson at oratory subsequently and [12] SLAC at observed iin,epcal h ormmnu n nrytrans- energy con- fers, and kinematic four-momentum specific the selecting especially By ditions, nuclei. studying for ula hredsrbto.De nlsi cteigat scattering inelastic the Deep Q measure to distribution. used charge been nuclear has scattering Elastic nucleus. . B 35 2 h e-ulo lcrnieatcsatrn rs sec- cross scattering inelastic electron per-nucleon The h e-ulo lcrnde nlsi scattering inelastic deep electron per-nucleon The nlsv lcrnscattering, electron Inclusive nlsi scattering. inelastic > 2 ≤ Q and m GeV 2 2 x x and B B p stencenms)i estv otenuclear the to sensitive is mass) nucleon the is x ≤ Q ( ..Weinstein, L.B. ≥ i.e., B nlsi cteig(I)a intermediate at (DIS) scattering inelastic orlto SC cl atrotie rmeeto incl electron from obtained factor scale (SRC) Correlation etosand sections hnmnlgclrltosi sue oetatteratio the extract to used is relationship phenomenological ASnmes 13.60.Hb,21.30.-x numbers: a PACS effect nucleus. EMC the the in both nucleons momentum) because (high is virtuality correlation observed the that 2 ω 2 0 p > n 0 and , n a ou ndffrn set fthe of aspects different on focus can one , . > hspprsosqatttvl httemgiueo h EM the of magnitude the that quantitatively shows paper This thresh .Ti ffc,kona h M effect, EMC the as known effect, This 7. 1 ti needn of independent is it 1 . . ssniiet ulo-ulo short nucleon-nucleon to sensitive is 5 GeV 4 A 3 0 = hmsJffro ainlAclrtrFclt,NwotNew Newport Facility, Accelerator National Jefferson Thomas . ≥ 35 Q 2 . F r rae hntoeo deu- of those than greater are 3 7 e/,aesmlri shape in similar are GeV/c, 275 2 ≤ 2 n ≥ hr ag orltosadteECEffect EMC 1, the and Correlations Range Short n large and /F x ∗ GeV 2 2 B p .Piasetzky, E. h ai ftefe eto ofe rtnsrcuefunct structure proton free to neutron free the of ratio the , A ≤ ( 1 ,e e, A l oiinUiest,Nrok igna23529 Virginia Norfolk, University, Dominion Old 0 2 ≥ n moderate and , . 2 ′ x ( 7 ,e e, ,i aubetool valuable a is ), x e vvUiest,TlAi 97,Israel 69978, Aviv Tel University, Aviv Tel B r mle than smaller are 3 B .Ti a first was This ). x ′ 1 , rs sections cross ) B . 5 2 = ≤ ..Higinbotham, D.W. Q Dtd uut2,2018) 22, August (Dated: Q e.g. 2 x 2 B / > x 2 car- , B ≤ mω x 0 , 1 B . 2. 4 . , , 35 ≤ ntergo 0 region cross the in scattering nucleus electron of inelastic sections deep per-nucleon the nucleus otevleo h rs eto ai at ratio section cross the of value the to esrdi e.[] safnto of function a as [3], Ref. in measured For together. points data the of all lower 3 that or uncertainties raise normalization merely overall by unaffected is 7 esrda 3 at measured [7] R cl atro htncesrltv odeuterium. to Higinbotham by relative suggested nucleus was that idea of This factor nucleus scale in SRC effect EMC the of fac- scale “SRC of the ratio region the plateau call mo- tor”. Fermi the will in we the Thus, sections to cross relative nuclei. are heavy high in mentum and momen- low mass of where center tum, low and with momentum nucleons of relative pairs high between occur cen- SRC two-nucleon to [21–24]. Correlations (SRC) 20] Range due Short entirely nucleon-nucleon [19, tensor almost and leptonic tral be and to experiments 18] knockout [17, hadronic in 16]. [15, nuclei find two to those probabilities the in of nucleons ratio momentum the high then is nuclei two for x Q n 0GeV 10 and 5 w esrmnsfor measurements two ffcso h nisaoigrgo at The region anti-shadowing I. possible the Table account into See of take effects to two. meant the not of are uncertainties average weighted the use h em oinrgo at region motion Fermi the orce e-ulo ai fteicuie( inclusive the of ratio per-nucleon corrected interest. of region rs etoso nucleus on sections cross x He, B 2 B ecaatrz h tegho h M ffc for effect EMC the of strength the characterize We hspprwl hwqatttvl httemagnitude the that quantitatively show will paper This recently shown were nucleons high-momentum These h R cl atr eemndfo h isospin- the from determined factors scale SRC The ( ≤ ≤ i.e., 4 3 sv cteigat scattering usive 0 He, ftedueo otefree the to deuteron the of 0 . .Gomez, J. . ,i ieryrltdt h hr Range Short the to related linearly is 7, A 8 h eut r vrgsoe l measured all over averages are results The 68. Q 9 olwn e.[]a h lp ftertoof ratio the of slope the as [7] Ref. following dSCaedmntdb h high the by dominated are SRC nd 2 eand Be ffc esrdi lcrndeep electron in measured effect C 2 = . 35 2 , o larger for n 0GeV 10 and 5 3 ≤ ≤ 12 A ,Vrii 23606 Virginia s, .Hen, O. Q x euetepbihdsoe from slopes published the use we C B eaiet deuterium, to relative 2 4 eand He x ≤ ≤ B GeV 6 0 os especulate We ions. ≥ . 2 x x A .Ti lp sproportional is slope This 7. B .Teobserved The 1. B n .Shneor R. and A .Terslsfo the from results The ). 12 > and 2 pn slnal eae othe to related linearly is 2 r ossetadwe and consistent are C for eas tteratios, the fit also We . 0 . arcross pair 5etnigit the into extending 75 3 x He, B < x x tal. et B a B 0 x 2 . 2 dR ( and 6 ≈ A/ ≈ o 0 for EMC 0 [25]. 3 0 . e = He) 5and 15 . ,but 5, . Q 36 ,e e, /dx 2 ≤ = ′ ) , 2

dREMC /dx dREMC /dx dREMC /dx This implies that both stem from the same underlying nu- Nucleus (Ref. [7]) (Ref. [3]) (combined) clear physics, such as high local density or large nucleon Deuteron 0 virtuality (v = P 2 −m2 where P is the four momentum). 3He −0.070 ± 0.029 −0.070 ± 0.029 4 This striking correlation means that we can predict He −0.199 ± 0.029 −0.191 ± 0.061 −0.197 ± 0.026 9Be −0.271 ± 0.029 −0.207 ± 0.037 −0.243 ± 0.023 the SRC scale factors for a wide range of nuclei from 12C −0.280 ± 0.029 −0.318 ± 0.040 −0.292 ± 0.023 Be to Au using the linear relationship from Eq. 1 and 9 27Al −0.325 ± 0.034 −0.325 ± 0.034 the measured EMC slopes (see Table II). Note that Be 40Ca −0.350 ± 0.047 −0.350 ± 0.047 is a particularly interesting nucleus because of its clus- 56Fe −0.388 ± 0.032 −0.388 ± 0.032 ter structure and because its EMC slope is much larger 108Ag −0.496 ± 0.051 −0.496 ± 0.051 than that expected from a simple dependence on aver- 197Au −0.409 ± 0.039 −0.409 ± 0.039 age nuclear density [7]. The EMC slopes and hence the predicted SRC scale factors may saturate for heavy nu- TABLE I. The measured EMC slopes dREMC /dx for 0.35 ≤ clei but better data are needed to establish the exact A ≤ xB 0.7. dependence.

2 2 (3/A)(σA(Q , xB )/σ3He(Q , xB ) are listed in Table II us- χ2 / ndf 0.7688 / 3 ing data from [14]. We used the ratio of deuterium to 3He /dx 56Fe 0.4 a -0.07879 ± 0.006376 determined in Ref. [14] primarily from the calculated ra- EMC 12 tio of their momentum distributions above the scaling C

threshold (p =0.275±0.025 GeV/c). We combined -dR thresh 4He the statistical and systematic uncertainties in quadrature 0.2 to give the total uncertainties shown in the table. The SRC scale factors for nucleus A relative to deuterium, 3He a2(A/d), are calculated from the second column. d The value of the SRC scale factors was shown to be 0.0 Q2 independent for 1.5 ≤ Q2 ≤ 2.5 GeV2 [13] and more recently for 1.5 ≤ Q2 ≤ 5 GeV2 [26]. Similarly, the EMC effect was shown to be Q2 independent for SLAC, BCDMS and NMC data for 2 ≤ Q2 ≤ 40 GeV2 [3]. This 0 2 4 6 Q2-independence allows us to compare these quantities a2(A/d) in their different measured ranges. FIG. 1. The EMC slopes versus the SRC scale factors. The Measured Measured Predicted 3 uncertainties include both statistical and systematic errors Nucleus a2(A/ He) a2(A/d) a2(A/d) added in quadrature. The fit parameter is the intercept of Deuteron 0.508 ± 0.025 1 the line and also the negative of the slope of the line. 3He 1 1.97 ± 0.10 4He 1.93 ± 0.14 3.80 ± 0.34 12C 2.41 ± 0.17 4.75 ± 0.41 56Fe 2.83 ± 0.18 5.58 ± 0.45 This correlation between the EMC slopes and the SRC 9 scale factors also allows us to extract significant infor- Be 4.08 ± 0.60 27 mation about the deuteron itself. Due to the lack of Al 5.13 ± 0.55 40Ca 5.44 ± 0.70 a free neutron target, the EMC measurements used the 108Ag 7.29 ± 0.83 deuteron as an approximation to a free proton and neu- 197Au 6.19 ± 0.65 tron system and measured the ratio of inclusive DIS on nuclei to that of the deuteron. This seems like a reason- TABLE II. The SRC scale factors for nucleus A with respect able approximation since the deuteron is loosely bound 3 to He and to deuterium. The third column is calculated from (≈ 2 MeV) and the average distance between the nu- the second. The resulting uncertainties are slightly overesti- 3 cleons is large (≈ 2 fm). But the deuteron is not a free mated since the uncertainty in the d/ He ratio of about 5% is system; the pion tensor force binds the two nucleons even added to all of the other ratios. The predicted values (fourth column) are calculated from the values in Table I and Eq. 1. if weakly. To quantify the effects of the binding of nucleons in deuterium, we define the In-Medium Correction (IMC) Fig. 1 shows the EMC slopes versus the SRC scale effect as the ratio of the DIS cross section per nucleon factors. The two values are strongly linearly correlated, bound in a nucleus relative to the free (unbound) pn pair cross section (as opposed to the EMC effect which uses − dREMC/dx = (a2(A/d) − 1) × (0.079 ± 0.006) . (1) the ratio to deuterium). 3

The deuteron IMC effect can be extracted from the reaches its minimum of 0.97 at xB ≈ 0.5 and increases data in Fig. 1. If the IMC effect and the SRC scale rapidly, crossing 1 at xB ≈ 0.7. factor both stem from the same cause, then the IMC If the structure function F2 is proportional to the DIS effect and the SRC scale factor will both vanish at the cross section (i.e., if the ratio of the longitudinal to trans- same point. The value a2(A/d) = 0 is the limit of free verse cross sections is the same for n,p and d [see dis- nucleons with no SRC. Extrapolating the best fit line in cussion in [8]]), then the free neutron structure func- n 2 Fig. 1 to a2(A/d) = 0 gives an intercept of dREMC /dx = tion, F2 (xB,Q ), can also be deduced from the measured −0.079 ± 0.006. The difference between this value and deuteron and proton structure functions: the deuteron EMC slope of 0 is the deuteron IMC slope: d 2 p 2 n 2 2F2 (xB ,Q ) − [1 − a(xB − b)]F2 (xB,Q ) F (xB ,Q )= 2 [1 − a(x − b)] dRIMC (d) B =0.079 ± 0.006 . (2) (5) dx which leads to

d 2 This slope is the same size as the EMC slope measured F2 (xB,Q ) 3 n 2 2 p 2 − [1 − a(xB − b)] for the ratio of He to deuterium [7]. It is slightly smaller F2 (xB,Q ) F2 (xB,Q ) p 2 = . (6) than the deuterium IMC slope of ≈ 0.10 derived in [3] F2 (xB ,Q ) [1 − a(xB − b)] assuming that the EMC effect is proportional to the av- erage nuclear density and the slope of 0.098 deduced by This is only valid for 0.35 ≤ xB ≤ 0.7. n p Frankfurt and Strikman based on the relative virtuality Fig. 2 shows the ratio of F2 /F2 extracted in this work 2 2 of nucleons in iron and deuterium [16] and the iron EMC using the IMC-based correction and the Q = 12 GeV d p d p slope [3]. ratio F2 /F2 from Ref. [28]. Note that the ratio F2 /F2 2 2 2 The IMC effect for nucleus A is then just the differ- is Q -independent from 6 ≤ Q ≤ 20 GeV for 0.4 ≤ ence between the measured EMC effect and the value xB ≤ 0.7 [28]. The dominant uncertainty in this ex- p d dREMC /dx = −0.079 ± 0.006. Thus traction is the uncertainty in the measured F2 /F2 . The IMC-based correction increases the extracted free neu-

tron structure function (relative to that extracted using dRIMC (A) dREMC (A) = +0.079±0.006 . (3) the deuteron momentum density [28]) by an amount that dx dx n meas increases with xB . Thus, the IMC-based F2 strongly fa- vors model-based extractions of F n that include nucleon This is true when the slopes are small compared to one. 2 modification in the deuteron [29]. The free neutron cross section can be obtained from The IMC-based F n appears to be constant or slightly the measured deuteron and proton cross sections using 2 increasing in the range from 0.6 ≤ xB ≤ 0.7. The the observed phenomenological relationship presented in p d/u ratio is simply related to the ratio of F n/F in Fig. 1 to determine the nuclear corrections. Since the 2 2 the deep inelastic limit, x2 ≪ Q2/4m2 [28], d/u = EMC effect is linear for 0.3 ≤ xB ≤ 0.7, we have n p n p (4F2 /F2 − 1)/(4 − F2 /Fn ). While it is quite hazardous σd to extrapolate from our limited xB range all the way to =1 − a(xB − b) for 0.3 ≤ xB ≤ 0.7, (4) σp + σn xB = 1, these results appear to disfavor models of the proton with d/u ratios of 0 at xB = 1 (see [29] and ref- where σd and σp are the measured DIS cross sections erences therein). for the deuteron and free proton, σn is the free neu- By using the deuteron IMC slope, these results take tron DIS cross section that we want to extract, a = into account both the nuclear corrections as well as any |dRIMC (d)/dx| = 0.079 ± 0.006 and b = 0.31 ± 0.04 possible changes to the internal structure of the neutron is the average value of xB where the EMC ratio is in the deuteron. Note that this assumes either that the 2 unity (i.e., where the per-nucleon cross sections are equal EMC and F2 data are taken at the same Q or that they 2 2 2 σA(xB)/A = σd(xB )/2) as determined in Refs. [3, 7] and are Q -independent for 6 ≤ Q ≤ 12 GeV . The fact that taking into account the quoted normalization uncertain- the measured EMC ratios for nuclei with A ≥ 3 decrease ties. linearly with increasing xB for 0.35 ≤ xB ≤ 0.7 indicates Our results imply that σd/(σp + σn) decreases linearly that Fermi smearing is not significant in this range. from 1 to 0.97 over the range 0.3 ≤ xB ≤ 0.7. (More We now speculate as to the physical reason for the precisely, that it decreases by 0.031 ± 0.004 where the EMC-SRC relation presented above. Assuming that the uncertainty is due to the fit uncertainties in Eq. 3.) This IMC/EMC effect is due to a difference in the quark dis- depletion (see Eq. 4) is similar in size to the depletion cal- tributions in bound and free nucleons, these differences culated by Melnitchouk using the weak binding approx- could occur predominantly in either mean field nucleons imation smearing function with target mass corrections or in nucleons affected by SRC. and an off-shell correction [27]. However, the distribution According to Ref. [30], the IMC/EMC effect is mainly in xB is very different. Melnitchouk’s calculated ratio associated with nucleons at high virtuality. These nucle- 4

Entries 0 Mean x 0 Mean y 0 RMS x 0 RMS y 0

p Gross, Jerry Miller, and Wally Melnitchouk. This 2 work was supported by the U.S. Department of En- / F n

2 0.6 ergy, the U.S. National Science Foundation, the Israel F Science Foundation, and the US-Israeli Bi-National Sci- ence Foundation. Jefferson Science Associates operates IMC Corrected (This Work) the Thomas Jefferson National Accelerator Facility under 0.5 DOE contract DE-AC05-06OR23177.

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