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Short Range Correlations and the EMC Effect L.B. Weinstein,1,∗ E. Piasetzky,2 D.W. Higinbotham,3 J. Gomez,3 O. Hen,2 and R. Shneor2 1Old Dominion University, Norfolk, Virginia 23529 2Tel Aviv University, Tel Aviv 69978, Israel 3Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 (Dated: January 19, 2011) ThispapershowsquantitativelythatthemagnitudeoftheEMCeffectmeasuredinelectrondeep inelasticscattering(DIS)atintermediatexB,0.35≤xB ≤0.7,islinearlyrelatedtotheShortRange Correlation (SRC)scale factorobtained from electron inclusivescattering atxB ≥1. Theobserved phenomenological relationship is used to extract the ratio of the deuteron to the free pn pair cross sections and F2n/F2p, the ratio of the free neutron to free proton structure functions. We speculate that the observed correlation is because both the EMC effect and SRC are dominated by the high 1 virtuality (high momentum) nucleonsin thenucleus. 1 0 2 PACSnumbers: 13.60.Hb,21.30.-x n a ′ Inclusiveelectronscattering,A(e,e),isavaluabletool fortwonucleiisthentheratiooftheprobabilitiestofind J for studying nuclei. By selecting specific kinematic con- high momentum nucleons in those two nuclei [15, 16]. 8 ditions,especiallythefour-momentumandenergytrans- These high-momentum nucleons were shown recently 1 fers, Q2 and ω, one can focus on different aspects of the in hadronic [17, 18] and leptonic [19, 20] two-nucleon ] nucleus. Elastic scatteringhas been used to measure the knockout experiments to be almost entirely due to cen- h nuclear charge distribution. Deep inelastic scattering at tralandtensornucleon-nucleonShortRangeCorrelations p - Q2 > 2 GeV2, and 0.35 ≤ xB ≤ 0.7 (xB = Q2/2mω, (SRC)[21–24]. SRCoccurbetweenpairsofnucleonswith p where m is the nucleon mass) is sensitive to the nuclear highrelativemomentumandlowcenterofmass momen- e quark distributions. Inelastic scattering at Q2 > 1.4 tum, where low and high are relative to the Fermi mo- h [ GeV2 and xB >1.5 is sensitive to nucleon-nucleonshort mentum in heavy nuclei. Thus, we will call the ratio of range correlations (SRC) in the nucleus. This paper will cross sections in the plateau region the “SRC scale fac- 4 exploretherelationshipbetweendeepinelasticandlarge- tor”. v 6 xB inelastic scattering. Thispaperwillshowquantitativelythatthemagnitude 6 The per-nucleon electron deep inelastic scattering of the EMC effect in nucleus A is linearly related to the 6 (DIS)crosssectionsofnucleiwithA≥3aresmallerthan SRC scale factor of that nucleus relative to deuterium. 5 those of deuterium at Q2 ≥ 2 GeV2, and moderate x , This idea was suggested by Higinbotham et al. [25]. . B 9 0.35 ≤ xB ≤ 0.7. This effect, known as the EMC effect, We characterize the strength of the EMC effect for 0 hasbeenmeasuredforawiderangeofnuclei[1–7]. There nucleus A following Ref. [7] as the slope of the ratio of 0 is no generally accepted explanation of the EMC effect. the per-nucleon deep inelastic electron scattering cross 1 : In general, proposed explanations need to include both sections of nucleus A relative to deuterium, dREMC/dx, v nuclear structure effects (momentum distributions and inthe region0.35≤x ≤0.7. Thisslope isproportional i B X binding energy) and modification of the bound nucleon to the value of the cross section ratio at x ≈ 0.5, but r structure due to the nuclear medium. Comprehensive is unaffected by overall normalization uncertainties that a reviews of the EMC effect can be found in [8–11] and merely raise or lowerall of the data points together. For references therein. Recent high-precision data on light 3He, 4He, 9Be and 12C we use the published slopes from nuclei [7] suggest that it is a local density effect and not [7]measuredat3≤Q2 ≤6GeV2. We alsofittheratios, a bulk property of the nuclear medium. measured in Ref. [3], as a function of x for 0.36 ≤ B The per-nucleonelectroninelastic scatteringcrosssec- xB ≤ 0.68. The results are averages over all measured tions of nuclei with A≥3 are greater than those of deu- Q2 (i.e., Q2 = 2,5 and 10 GeV2 for xB < 0.6 and Q2 = terium for Q2 > 1.4 GeV2 and large xB, 1.5 ≤ xB ≤ 2. 5 and 10 GeV2 for larger xB). The results from the The cross section ratio for two different nuclei (e.g., car- twomeasurementsfor4Heand12Careconsistentandwe bonandhelium)showsaplateauwhenplottedasafunc- use the weighted average of the two. See Table I. The tion of x (i.e., it is independent of x ). This was first uncertaintiesarenotmeanttotakeintoaccountpossible B B observedatSLAC[12]andsubsequentlyatJeffersonLab- effects of the anti-shadowing region at xB ≈ 0.15 and oratory [13, 14]. The plateau indicates that the nucleon the FermimotionregionatxB >0.75extendinginto the momentum distributions of different nuclei for high mo- region of interest. mentum,p≥p =0.275GeV/c,aresimilarinshape The SRC scale factors determined from the isospin- thresh ′ and differ only in magnitude. The ratio (in the plateau corrected per-nucleon ratio of the inclusive (e,e) region) of the per-nucleon inclusive (e,e′) cross sections cross sections on nucleus A and 3He, a (A/3He) = 2 2 dREMC/dx dREMC/dx dREMC/dx Thisimpliesthatbothstemfromthesameunderlyingnu- Nucleus (Ref. [7]) (Ref. [3]) (combined) clear physics, such as high local density or large nucleon Deuteron 0 virtuality(v =P2−m2 whereP isthefourmomentum). 3He −0.070±0.029 −0.070±0.029 4He −0.199±0.029 −0.191±0.061 −0.197±0.026 This striking correlation means that we can predict 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 27Al −0.325±0.034 −0.325±0.034 the measured EMC slopes (see Table II). Note that 9Be 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. (3/A)(σA(Q2,xB)/σ3He(Q2,xB)arelistedinTableIIus- dx χχ22 // nnddff 00..77668888 // 33 ingdatafrom[14]. Weusedtheratioofdeuteriumto3He / 56Fe determined in Ref. [14] primarily fromthe calculated ra- MC0.4 aa -- 00 .. 0077887799 ±± 00..000066337766 E tio of their momentum distributions above the scaling R 12C d threshold(p =0.275±0.025GeV/c). Wecombined thresh - 4He thestatisticalandsystematicuncertaintiesinquadrature 0.2 to give the total uncertainties shown in the table. The SRC scale factors for nucleus A relative to deuterium, 3He a (A/d), are calculated from the second column. 2 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 a (A/d) Q2-independence allows us to compare these quantities 2 in their different measured ranges. FIG. 1. The EMC slopes versus the SRC scale factors. The Measured Measured Predicted uncertainties include both statistical and systematic errors Nucleus a2(A/3He) a2(A/d) a2(A/d) added in quadrature. The fit parameter is the intercept of Deuteron 0.508±0.025 1 theline and also the negative of theslope of theline. 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 ThiscorrelationbetweentheEMCslopesandtheSRC scale factors also allows us to extract significant infor- 9Be 4.08±0.60 mation about the deuteron itself. Due to the lack of 27Al 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.TheSRCscale factors for nucleus Awith respect able approximation since the deuteron is loosely bound to3Heandtodeuterium. Thethirdcolumniscalculatedfrom (≈ 2 MeV) and the average distance between the nu- the second. The resulting uncertainties are slightly overesti- cleons is large (≈ 2 fm). But the deuteron is not a free matedsincetheuncertaintyinthed/3Heratioofabout5%is system;thepiontensorforcebindsthetwonucleonseven addedtoall oftheotherratios. Thepredicted values(fourth if weakly. column)arecalculatedfrom thevaluesinTableIandEq.1. 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, boundinanucleusrelativetothefree(unbound)pnpair cross section (as opposed to the EMC effect which uses −dR /dx=(a (A/d)−1)×(0.079±0.006) . (1) the ratio to deuterium). EMC 2 3 The deuteron IMC effect can be extracted from the reaches its minimum of 0.97 at x ≈ 0.5 and increases B data in Fig. 1. If the IMC effect and the SRC scale rapidly, crossing 1 at x ≈0.7. B factor both stem from the same cause, then the IMC If the structure function F is proportionalto the DIS 2 effect and the SRC scale factor will both vanish at the crosssection(i.e.,iftheratioofthelongitudinaltotrans- same point. The value a (A/d) = 0 is the limit of free verse cross sections is the same for n,p and d [see dis- 2 nucleons with no SRC. Extrapolating the best fit line in cussion in [8]]), then the free neutron structure func- Fig.1toa2(A/d)=0givesaninterceptofdREMC/dx= tion,F2n(xB,Q2),canalsobededucedfromthemeasured −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: 2Fd(x ,Q2)−[1−a(x −b)]Fp(x ,Q2) (cid:12)(cid:12)(cid:12)dRIMC(d)(cid:12)(cid:12)(cid:12)=0.079±0.006 . (2) F2n(xB,Q2)= 2 B [1−a(xBB−b)] 2 B (cid:12) dx (cid:12) (5) (cid:12) (cid:12) which leads to Tfohritshselorpateioisotfh3eHseatmoedesuizteeraiusmth[e7]E.MItCissslloigphetlmyesamsualrleedr F2n(xB,Q2) = 2FF22pd((xxBB,,QQ22)) −[1−a(xB −b)] . (6) than the deuterium IMC slope of ≈ 0.10 derived in [3] Fp(x ,Q2) [1−a(x −b)] 2 B B assuming that the EMC effect is proportional to the average nuclear density and the slope of 0.098 deduced by This is only valid for 0.35≤xB ≤0.7. Frankfurt and Strikman based on the relative virtuality Fig.2showstheratioofF2n/F2p extractedinthis work ofnucleonsinironanddeuterium[16]andthe ironEMC using the IMC-based correction and the Q2 = 12 GeV2 slope [3]. ratio F2d/F2p from Ref. [28]. Note that the ratio F2d/F2p The IMC effect for nucleus A is then just the differ- is Q2-independent from 6 ≤ Q2 ≤ 20 GeV2 for 0.4 ≤ ence between the measured EMC effect and the value xB ≤ 0.7 [28]. The dominant uncertainty in this ex- dREMC/dx=−0.079±0.006. Thus traction is the uncertainty in the measured F2p/F2d. The IMC-based correction increases the extracted free neu- (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)dR (A)(cid:12) (cid:12)dR (A)(cid:12) tron structure function (relative to that extracted using (cid:12)(cid:12) IMdxC (cid:12)(cid:12)=(cid:12)(cid:12) EMdxC (cid:12)(cid:12) +0.079±0.006 . (3) thedeuteronmomentumdensity[28])byanamountthat (cid:12) (cid:12) (cid:12) (cid:12)meas increaseswith xB. Thus, the IMC-basedF2n strongly fa- vorsmodel-basedextractions of Fn 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 Fn appears to be constant or slightly the measured deuteron and proton cross sections using 2 increasing in the range from 0.6 ≤ x ≤ 0.7. The B the observedphenomenologicalrelationshippresented in d/u ratio is simply related to the ratio of Fn/Fp 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 (4Fn/Fp−1)/(4−Fn/Fp). While it is quite hazardous 2 2 2 n σd to extrapolate from our limited xB range all the way to =1−a(x −b) for 0.3≤x ≤0.7, (4) σp+σn B B xB = 1, these results appear to disfavor models of the proton with d/u ratios of 0 at x = 1 (see [29] and ref- B where σ and σ are the measured DIS cross sections erences therein). d p for the deuteron and free proton, σ is the free neu- By using the deuteron IMC slope, these results take n tron DIS cross section that we want to extract, a = into account both the nuclear corrections as well as any |dR (d)/dx| = 0.079 ± 0.006 and b = 0.31 ± 0.04 possible changes to the internal structure of the neutron IMC is the average value of x where the EMC ratio is in the deuteron. Note that this assumes either that the B unity(i.e.,wheretheper-nucleoncrosssectionsareequal EMCandF dataaretakenatthe sameQ2 orthatthey 2 σ (x )/A=σ (x )/2)as determinedin Refs.[3,7] and areQ2-independentfor6≤Q2 ≤12GeV2. Thefactthat A B d B taking into account the quoted normalization uncertain- the measuredEMC ratiosfor nucleiwith A≥3 decrease ties. linearlywithincreasingx for0.35≤x ≤0.7indicates B B Our results imply that σ /(σ +σ ) decreaseslinearly that Fermi smearing is not significant in this range. d p n from 1 to 0.97 over the range 0.3 ≤ x ≤ 0.7. (More We now speculate as to the physical reason for the B precisely, that it decreases by 0.031±0.004 where the EMC-SRC relation presented above. Assuming that the uncertaintyisduetothefituncertaintiesinEq. 3.) This IMC/EMC effect is due to a difference in the quark dis- depletion(seeEq. 4)issimilarinsizetothedepletioncal- 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. andanoff-shellcorrection[27]. However,thedistribution According to Ref. 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