Studying Short-Range Correlations with Real Photon Beams at GlueX
O. Hen, M. Patsyuk, E. Piasetzky, A. Schmidt, A. Somov, H. Szumila-Vance, L. B. Weinstein, D. Dutta, H. Gao, M. Amaryan, A. Ashkenazi, A. Beck, V. Berdnikov, T. Black, W. J. Briscoe, T. Britton, W. Brooks, R. Cruz-Torres, M. M. Dalton, A. Denniston, A. Deur, H. Egiyan, C. Fanelli, S. Fegan, S. Furletov, L. Gan, L. Guo, F. Hauenstein, H. Haykobyan, D. W. Higinbotham, D. G. Ireland, G. Johansson, M. Kamel, I. Korover, S. Kuleshov, D. Lawrence, K. Livingston, D. Mack, H. Marukyan, M. McCaughan, B. Mckinnon, S. Meytal-Beck, F. Nerline, D. Nguyen, A. Papadopoulou, P. Pauli, R. Pedroni, L. Pentchev, J. R. Pybus, S. Ratliff, D. Romanov, C. Romera, C. Salgado, B. Schmookler, E. P. Segarra, E. Seroka, P. Sharp, S. Somov, I. I. Strakovsky, S. Taylor, A. Thiel, B. Zihlmann
SStudying Short-Range Correlations with Real Photon Beamsat GlueX
H. MarukyanA. I. Alikhanian National Science Laboratory(Yerevan Physics Institute), 0036 Yerevan, ArmeniaM. Patsyuk (Spokesperson)Joint Institute for Nuclear Research, Dubna, RussiaH. Gao (Spokesperson)Duke University, Durham, North Carolina 27708, USAM. Kamel, L. GuoFlorida International University, Miami, Florida 33199, USAD. G. Ireland, K. Livingston, B. Mckinnon, P. Pauli, A. ThielUniversity of Glasgow, Glasgow G12 8QQ, United KingdomF. NerlingGSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH,D-64291 Darmstadt, GermanyT. Britton, M. M. Dalton, A. Deur, H. Egiyan, S. Furletov, D.W. Higinbotham,D. Lawrence, D. Mack, M. McCaughan, L. Pentchev, A. Somov (Spokesperson),H. Szumila-Vance (Spokesperson), S. Taylor, B. ZihlmannThomas Jefferson National Accelerator Facility,Newport News, Virginia 23606, USAA. Ashkenazi, R. Cruz-Torres, A. Denniston, C. Fanelli, O. Hen (Spokesperson ∗ ),D. Nguyen, A. Papadopoulou, J. R. Pybus, E.P. SegarraMassachusetts Institute of Technology, Cambridge, Massachusetts 02139, USAD. Dutta (Spokesperson)Mississippi State University, Mississippi State, Mississippi 29762, USA a r X i v : . [ nu c l - e x ] O c t . Berdnikov, D. Romanov, S. SomovNational Research Nuclear University Moscow Engineering Physics Institute,Moscow 115409, RussiaC. SalgadoNorfolk State University, Norfolk, Virginia 23504, USAR. PedroniNorth Carolina A&T State University, Greensboro, North Carolina 27411, USAT. Black, L. GanUniversity of North Carolina at Wilmington,Wilmington, North Carolina 28403, USAA. Beck, I. Korover, and S. Maytal-BeckNuclear Research Center Negev, Beer-Sheva 84190, IsraelM. Amaryan, F. Hauenstein, L.B. Weinstein (Spokesperson)Old Dominion University, Norfolk, Virginia 23529, USAW. Brooks, H. Hakobyan, S. Kuleshov, C. RomeroUniversidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V Valpara´ıso, ChileB. SchmooklerState University of New York at Stony Brook, New York, 11794G. Johansson, E. Piasetzky (Spokesperson)Tel-Aviv University, Tel Aviv 69978, IsraelW. J. Briscoe, S. Fegan, S. Ratliff, A. Schmidt (Spokesperson), E. Seroka,P. Sharp, I. I. StrakovskyThe George Washington University, Washington, D.C. 20052, USA Theory SupportM. Strikman Pennsylvania State University, State College, PA.G.A. Miller University of Washington, Seattle, WA.A.B. Larionov Frankfurt Institute for Advanced Studies (FIAS), and National ResearchCentre “Kurchatov Institute”, Moscow, Russia.M. Sargsian Florida International University, Miami, FL.L. Frankfurt Tel-Aviv University, Tel Aviv, Israel. ∗ Contact Person: [email protected] bstract The past few years has seen tremendous progress in our understanding of short-range cor-related (SRC) pairing of nucleons within nuclei, much of it coming from electron scatteringexperiments leading to the break-up of an SRC pair. The interpretation of these experimentsrests on assumptions about the mechanism of the reaction. These assumptions can be directlytested by studying SRC pairs using alternate probes, such as real photons. We propose a 30-day experiment using the Hall D photon beam, nuclear targets, and the GlueX detector in itsstandard configuration to study short-range correlations with photon-induced reactions. Severaldifferent reaction channels are possible, and we project sensitivity in most channels to equal orexceed the 6 GeV-era SRC experiments from Halls A and B. The proposed experiment willtherefore decisively test the phenomena of np dominance, the short-distance NN interaction,and reaction theory, while also providing new insight into bound nucleon structure and the onsetof color transparency. Since the 1950s much effort has been devoted to understanding the detailed characteristics andorigin of the nucleon-nucleon interaction, and how it forms atomic nuclei. The development ofmodern superconducting accelerators—with high energy, high intensity and high duty factor—hasenabled scattering experiments that resolve the structure and dynamics of both individual nucleonsand nucleon pairs in nuclei, allowing significant progress. While many breakthroughs have beenmade, much still remains to be understood.Early measurements of inclusive ( e, e (cid:48) ) and single proton knockout ( e, e (cid:48) p ) processes helped es-tablish the shell-structure of nuclei and probe properties of nucleons bound in different nuclearshells [1–3]. While lending credence to many shell model predictions, these measurements alsorevealed that in nuclei ranging from lithium to lead, proton-knockout cross-sections are only 60%–70% of the mean-field one-body-based theoretical expectation [4,5], highlighting the need to considerhigher order two-body effects that go beyond the traditional mean-field approximation.Electron-scattering experiments, analyzed within a high-resolution theoretical framework, sug-gest that about 20% of the nucleons in nuclei are part of strongly interacting close-proximity nu-cleon pairs, with large relative ( k rel > k F ≈
250 MeV/c) and small center-of-mass (C.M.) momenta( k CM < k F ). These are referred to as short-range correlated (SRC) pairs [6–9]. Nucleons that are3art of SRC pairs are absent in the one-body shell-model description of the data. Their formationcan therefore explain some of the discrepancy between the measured and calculated single-protonknockout cross-sections [5,10] and have wide-spread implications for different phenomena in nuclear-,particle- and astro-physics.A large part of our understanding of SRC pairs comes from measurements of exclusive two-nucleon knockout reactions ( e, e (cid:48) N N ). In these experiments, a high-energy electron scatters off thenucleus, leading to the knockout of a nucleon with large missing momentum that is balanced by theemission of a single recoil nucleon, leaving the residual A − /c (just above k F ), neutron-proton ( np ) SRC pairs predominate over proton-proton ( pp ) and neutron-neutron ( nn )pairs by a factor of about 20, in both light and heavy nuclei [13,16,17]. This phenomenon, commonlyreferred to as “ np -dominance,” [12–17, 22, 23], is driven by the tensor nature of the N N interactionin the quoted momentum range [24–26] and indicates that increasing the fraction of neutrons in anucleus increases the fraction of protons that are part of SRC pairs [12, 27, 28].These experimental findings inspired a broad complementary program of theoretical and phe-nomenological studies of SRCs and their impact on various nuclear phenomena, including the in-ternal structure of nucleons bound in nuclei [11, 29–32], neutrinoless double beta decay matrixelements [33–39], nuclear charge radii [40], and the nuclear symmetry energy and neutron star prop-erties [9, 41, 42].However, almost all of our understand of SRC comes from electron scattering, with only asingle proton scattering C( p, ppn ) measurement [23]. Thus, the interpretation of these experimentalresults relies on an assumed electron interaction mechanism at large momentum transfers (detailed insection 2 below). Different assumptions could lead to different interpretations. It is crucial to studySRCs with different, non-electron, probes, in order to validate the reaction mechanism assumptionsand the connection between the experimental results and their interpretation in terms of SRC pairs.This proposal describes an experiment to study SRCs using photo-production reactions with realphotons on nuclear targets to determine the probe-dependence of SRC measurements. The new datawill complement the above-mentioned electron scattering studies and yield stringent constraints onpossible reaction mechanisms that could complicate the interpretation of the data. Photo-nuclearreactions have significantly different sensitivity to meson exchange currents, are dominated by thetransverse response function with backward emitted recoil nucleons (as oppose to both longitudinaland transverse response functions and forward emitted recoil nucleons in x B > Q , | t | , | u | > ) are indeed understood to the expected level, we should be able to confirm the ob-served neutron-proton pair dominance and the A -dependence of SRC-pair abundances through thisexperiment. We note that nuclear targets have been considered as part of the future plan for theGlueX Detector, as laid out by the recent GlueX collaboration white paper [43].We propose a 30-day measurement using the real photon beam in Hall D, three nuclear targets( d , He, and C), and the GlueX detector in its standard configuration. The main goal of theexperiment will be to study short-range correlations using photon-induced reactions. The experimentcan additionally provide information on bound nucleon structure (discussed in section 3) and color-transparency (discussed in section 4). 4
Recent progress in the quantitative study of SRCs
The study of short-range correlations is a broad subject. It covers a large body of experimentaland theoretical work, as well as phenomenological studies of the implications of SRCs for variousphenomena in nuclear, particle and astro-physics. The discussion below is focused primarily onrecent experimental activities co-led by the spokespersons, and theoretical developments that aremost relevant for the objectives of the current proposal. A full discussion of SRC physics is availablein a recent RMP review [6], as well as in a theory-oriented review [7].
A A–2 e – e – Np (0, m A ) ( q, ω )( p i , ϵ i )( p CM ,m A - ϵ A–2 ) ( p N ≡ p i +q, p N +m N ) ( p recoil , p recoil +m N ) ( - p CM ,E A-2 ≡ p CM +(m A-2 +E*) ) SRCSRC
Figure 1: Diagrammatic representation and kinematics of the triple-coincidence A ( e, e (cid:48) Np )reaction within the SRC breakup model. Dashed red lines represent off-shell particles. Openovals represent un-detected systems. Solid black lines represent detected particles. The mo-mentum and energy of the particles are also indicated. Previous studies of SRCs have used measurements of Quasi-Elastic (QE) electron scattering atlarge momentum-transfer, see Fig. 1. Within the single-photon exchange approximation, electronsscatter from the nucleus by transferring a virtual photon carrying momentum (cid:126)q and energy ω . Inthe one-body view of QE scattering, the virtual photon is absorbed by a single off-shell nucleon withinitial energy (cid:15) i and momentum (cid:126)p i . If the nucleon does not re-interact as it leaves the nucleus, it willemerge with momentum (cid:126)p N = (cid:126)p i + (cid:126)q and energy E N = (cid:112) p N + m N . Thus, we can approximate theinitial momentum and energy of that nucleon using the measured missing momentum, (cid:126)p i ≈ (cid:126)p miss ≡ (cid:126)p N − (cid:126)q , and missing energy, (cid:15) i ≈ m N − (cid:15) miss ≡ (cid:15) N − ω . When (cid:126)p miss > k F , the knockout nucleonis expected to be part of an SRC pair [6–9, 14, 16, 22]. The knockout of one nucleon from the pairshould therefore be accompanied by the simultaneous emission of the second (recoil) nucleon withmomentum (cid:126)p recoil ≈ − (cid:126)p miss . At the relevant high- Q of our measurements ( > . . /c ), thedifferential A ( e, e (cid:48) p ) cross-sections can be approximately factorized as [3, 44]: d σd Ω k (cid:48) d(cid:15) k (cid:48) d Ω N d(cid:15) N = p N (cid:15) N · σ ep · S ( p i , (cid:15) i ) , (1)where k (cid:48) = ( k (cid:48) , (cid:15) k (cid:48) ) is the final electron four-momentum, σ ep is the off-shell electron-nucleon cross-section [44], and S ( p i , (cid:15) i ) is the nuclear spectral function that defines the probability for finding anucleon in the nucleus with momentum p i and energy (cid:15) i . Different models of the N N interactioncan produce different spectral functions that lead to different cross-sections. Therefore, exclusivenucleon knockout cross-sections analyzed with this method are sensitive to the
N N interaction.In the case of two-nucleon knockout reactions, the cross-section can be factorized in a similarmanner to Eq. 1 by replacing the single-nucleon spectral function with the two-nucleon decay function D A ( p i , p recoil , (cid:15) recoil ) [9,22,45]. The latter represents the probability for a hard knockout of a nucleonwith initial momentum (cid:126)p i , followed by the emission of a recoil nucleon with momentum (cid:126)p recoil . (cid:15) recoil is the energy of the A − A − p miss final states that are not due to the knockout of nucleons from SRC pairs, thus breakingthe factorization shown in Eq. 1. To address this, the measurements discussed here are carriedout at anti-parallel kinematics with p miss ≥
300 MeV /c , Q ≡ q − ω ≥ . /c ) , and x B ≡ Q / m N ω ≥ .
2, where such non-QE reaction mechanisms were shown to be suppressed [6–9,46,47].For completeness, we note that from a theoretical standpoint, the reaction diagram shown inFig. 1 can be viewed as a ‘high-resolution’ starting point for a unitary-transformed calculation [48].Such calculations would soften the input
N N interactions and turn the electron scattering operatorsfrom one-body to many-body. This ‘unitary-freedom’ does not impact cross-section calculations butdoes make the extracted properties of the nuclear ground-state wave-function (e.g. the spectral func-tion) depend on the assumed interaction operator. This discussion focuses on the high-resolutionelectron interaction model of Fig. 1, as it constitutes the simplest reaction picture that is consistentwith both the measured observables [6–9] and various reaction and ground-state ab-initio calcula-tions [49].
N N
Interaction at the Generalized Contact Formal-ism
Precision SRC studies are only feasible if one has the ability to quantitatively relate experimentalobservables to theoretical calculations, ideally ones starting from the fundamental
N N interactionand accounting for all relevant reaction mechanisms. This is a challenging endeavor, as un-factorizedab-initio calculations of high- Q nucleon knockout cross-sections are currently unfeasible for A >
N N interaction as input [17, 51]. This is done byproviding a factorized model of the short-distance / high-momentum part of the many-body nuclearwave function leveraging the separation between the energy scales of the A − Q scattering reactions such as in Fig. 1 addsa third energy scale of the virtual photon (high-energy) that justifies the factorized approximationof Eq. 1.The GCF provides a consistent model for nuclear two-body momentum distribution at high-momenta and at short-distance, as well as for two-body continuum states of the nuclear spectraland decay functions. Recent studies of the GCF: • Demonstrated its ability to reproduce many-body ab-initio calculated nucleon momentumdistributions in nuclei from He to Ca, above k F , to ≈
10% accuracy [50]; • Extracted consistent SRC abundances (i.e., nuclear contacts) from ab-initio calculations oftwo-nucleon distributions in both coordinate and momentum space and from experimentaldata [50]; and • Derived a new factorized expression for the nuclear correlation function with implicationsfor calculations of double beta decay matrix elements [33] and demonstrated its relation tosingle-nucleon charge distribution measurements [40].The main application of the GCF germane to this proposal is the modeling of the nuclear spectraland decay functions [51], allowing calculations of nucleon knockout cross-sections. For example, usingEq. 1 and the reaction model of Fig. 1, the A ( e, e (cid:48) N N ) cross-section can be expressed within the6CF as [17]: d σdQ dx B dφ k d (cid:126)p CM d Ω recoil = K · σ eN · n ( (cid:126)p CM ) · (cid:34)(cid:88) α C α · | ˜ ϕ α ( | (cid:126)p CM − (cid:126)p recoil | ) | (cid:35) , (2)where subscripts ‘ N ’ and ‘recoil’ stand for the leading and recoil nucleon respectively, K is a kine-matic term, (detailed in Ref. [17]), σ eN is the off-shell electron-nucleon cross-section, and α representsthe spin and isospin quantum numbers of SRC pairs. ˜ ϕ α , n ( (cid:126)p CM ), and C α respectively describe therelative motion, CM motion, and abundances of SRC pairs with quantum numbers α . The functions˜ ϕ α are universal SRC pair relative momentum distributions, obtained by solving the zero-energytwo-body Schr¨odinger equation of an N N pair in quantum state α using an input N N potentialmodel. n ( (cid:126)p CM ) is the SRC pair CM momentum distribution, given by a three-dimensional Gaussianwith width of 150 ±
20 MeV /c [18, 53, 54]. C α are the nuclear contact terms that determine therelative abundance of SRC pairs in quantum state α . These are obtained through the analysis ofab-initio many-body calculations of two-nucleon densities [50, 52, 55]. C o un t s C ( e, e p ) DataAV18N2LO (1.0 fm)0 . . . . . . . p miss [GeV /c ]10 − C o un t s C ( e, e pp ) C o un t s C ( e, e p ) 02040 C o un t s < p miss < /c C ( e, e pp )0100200 C o un t s C o un t s < p miss < /c C o un t s C o un t s < p miss < /c m N − (cid:15) [GeV]050 C o un t s DataAV18N2LO (1.0 fm) m N − (cid:15) [GeV] 01020 C o un t s < p miss < /c Figure 2: Left panel: the p miss dependence of the C( e, e (cid:48) p ) (top) and C( e, e (cid:48) pp ) (bottom)event yields. Points show the measured data. Bands show the GCF calculations using theN2LO(1.0fm) (blue) and AV18 (black) interactions. Right panel: the (cid:15) miss dependence of the C( e, e (cid:48) p ) (left column) and C( e, e (cid:48) pp ) (right column) event yields in four different rangesof p miss . The purple arrow indicates the expected (cid:15) miss for standing SRC pair breakup with amissing-momentum that is equal to the mean value of the data. Ref. [17] shows the first comparisons between the prediction of Eq. 2 and measured A ( e, e (cid:48) N p )cross-section ratios. In Figs. 2 and 3 we showcase our most recent results, where Eq. 2 is used to cal-culate the individual ( e, e (cid:48) p ) and ( e, e (cid:48) pp ) cross-sections in the kinematics of our SRC measurements.The calculation was done using two N N interaction models to obtain ˜ ϕ α : the phenomenologicalAV18 [56], and Chiral EFT-based local N2LO(1.0 fm) [57]. Nuclear contacts C α and width ofthe CM momentum distribution were obtained from theoretical calculations [50, 52–55] and nucleartransparency and single-charge exchange reaction effects were accounted for as detailed in the on-line supplementary materials of Ref. [17], using the calculations of Ref. [46]. The model systematic7ncertainty is determined from the uncertainties in the GCF input parameters and reaction effectscorrection factors.The left panel of Fig. 2 shows the p miss dependence of the measured and GCF-calculated C( e, e (cid:48) pp ) and C( e, e (cid:48) p ) event yields for the two interactions. The AV18 interaction is observedto describe both ( e, e (cid:48) p ) and ( e, e (cid:48) pp ) data over the entire measured p miss range. The N2LO(1.0 fm)interaction agrees with the data up to its cutoff and, as expected, decreases exponentially above it.The right panel of Fig. 2 shows the (cid:15) miss - p miss correlation for the C( e, e (cid:48) pp ) and C( e, e (cid:48) p )reactions. The average value of m N − (cid:15) is observed to increase with p miss , peaking at the expectedvalue for the breakup of a standing SRC pair (indicated by the purple arrows) for both reactions.The GCF calculations follow the same trend. However, the AV18 interaction agrees with the dataover the entire (cid:15) miss - p miss range, while the chiral interactions under predict at the highest p miss . scalar limit N LO (1.0 fm)AV18
Nuclear Structure Experiment pp S R C F r a c t i o n p rel [GeV /c ] A) ( e , e ' pp ) / ( e , e ' p ) SRC Break-up Data p miss [GeV /c ] B) Data
AV4'
Figure 3: A: the pp pair fraction in C as predicted by GCF using AV18, AV4’, and ChiralN2LO(1.0 fm) interactions. B: the ratio of C( e, e (cid:48) pp ) to C( e, e (cid:48) p ) event yields for data (redpoints) and GCF (bands), including all experimental effects. Both the AV18 and N2LO(1.0 fm)interactions are consistent with data, and show an increase from a tensor-dominated regimeat p miss = 0 . /c to scalar spin-independent regime approaching p miss = 1 GeV /c . TheAV4’ interaction, which has no tensor component, leads to predictions that are inconsistentwith data. Fig. 3 considers the C( e, e (cid:48) pp ) / C( e, e (cid:48) p ) yield ratio, a measure of the impact of the tensorforce in the N N interaction. In this figure, the AV18 and the chiral N2LO(1.0 fm) interactions arecompared to the AV4 (cid:48) interaction, which does not include a tensor force. The right panel showsthe data yield ratio as well as the GCF-calculated yield ratio. Both the data, and the calculationswith the AV18 and N2LO(1.0 fm) interactions show the pp fraction increasing with p miss , consistentwith a transition from tensor- to scalar-dominated regions of the interaction [16]. By contrast, thecalculation with the AV4 (cid:48) interaction over-predicts the fraction of pp pairs observed in the data.The left panel shows the fraction of pp pairs in C as predicted by the GCF formalism asa function of p rel. ≡ | (cid:126)p miss − (cid:126)p recoil | . The AV18 and N2LO(1.0 fm) interactions approach limitpredicted by a purely spin-independent interaction. The AV4 (cid:48) interaction, without a tensor force,predicts a pp fraction above this scalar limit.We note that our confidence in these results is also supported by the fact that the GCF-basedcalculations describe well numerous other measured kinematical distributions not shown here dueto a lack of space. Thus, the results presented here showcase the use of high- Q electron scatteringdata to quantitatively study the nuclear interaction at very large momenta.It is interesting to note that for the AV18 interaction, we observe good agreement with the dataup to 1 GeV /c , which corresponds to SRC configurations with nucleons separated by a distancesmaller than their radii [58]. As discussed below, previous studies indicated that in such extremeconditions the internal quark-gluon structure of SRC nucleons can well be modified as compared with8hat of free nucleons [6, 11, 59–61]. The ability of the AV18-based GCF calculation to reproduce ourdata over the entire measured (cid:15) miss - p miss range suggests that such modifications do not significantlyimpact the effective modeling of the nuclear interaction, offering support for using point-like nucleonsas effective degrees of freedom for modeling of nuclear systems up to very high densities. The above-mentioned results constitute some of the most advanced, ongoing (i.e. unpublished),analysis that utilizes the scale-separated GCF to calculate factorized nucleon-knockout cross-sectionsusing different models of the
N N interaction. These studies are made possible by the vast progressmade in the study of SRCs using hard knockout reactions over the last decade. Below, we reviewkey published results from initial measurements of nuclei from He to
Pb. np -SRC dominance and the tensor interaction First measurements of exclusive SRC pair breakup reactions focused primarily on probing the isospinstructure of SRC pairs. These experiments were initially done at BNL using hadronic (proton) probeson C, and continued at JLab with leptonic (electron) probes on He, C, Al, Fe and
Pb.Focusing on a missing momentum range of 300–600 MeV /c , comparisons of the measured A ( e, e (cid:48) p )and A ( e, e (cid:48) pN ) cross-section indicated that the full single-proton knockout cross-section is exhaustedby the two-nucleon knockout cross-sections, i.e., the data were consistent with every ( e, e (cid:48) p ) eventhaving the correlated emission of a recoil nucleon [14, 16, 17, 22]. A common interpretation of theseresults is that the nucleon momentum distribution above k F is dominated by nucleons that aremembers of SRC pairs. While the current analysis uses the SCX calculations ofRef. [31] and the formalism detailed in the SupplementalMaterial [48], other calculations for these corrections canbe applied in the future. See Supplemental Material [48]for details on the numerical evaluation of Eq. (2) and itsuncertainty.These SCX-corrected pp=pn ratios agree within uncer-tainty with the ratios previously extracted from A ð e; e pp Þ and A ð e; e p Þ events [3], which assumed that all high-missing momentum nucleons belong to SRC pairs. Inaddition, the SCX-corrected pp=np ratio is in betteragreement with the GCF contacts fitted here but is notinconsistent with those determined in Ref. [28]. This is asignificant achievement of the GCF calculations that opensthe way for detailed data-theory comparisons. This will bepossible using future higher statistics data that will allowfiner binning in both recoil and missing momenta.The pp=np ratios measured directly in this work aresomewhat lower than both previous indirect measurementson nuclei from C to Pb [3], and previous direct measure-ments on C [20]. This is due to the more sophisticatedSCX calculations used in this work [31] compared to theprevious ones [57]. This is consistent with the lower valuesof the pp to np contact extracted from GCF calculations fitto these data mentioned above.To conclude, we report the first measurements of highmomentum-transfer hard exclusive np and pp SRC pairknockout reactions off symmetric ( C) and medium and heavy neutron-rich nuclei ( Al, Fe, and
Pb). We findthat the reduced cross-section ratio for proton-proton toproton-neutron knockout equals ∼ , consistent withprevious measurements off symmetric nuclei. Usingmodel-dependent SCX corrections, we also extracted therelative abundance of pp - to pn -SRC pairs in the measurednuclei. As expected, these corrections reduce the pp -to- np ratios to about 3%, so that the measured reduced cross-section ratios are an upper limit on the relative SRC pairsabundance ratios.The data also show good agreement with GCF calcu-lations using phenomenological as well as local and non-local chiral NN interactions, allowing for a higher precisiondetermination of nuclear contact ratios and a study of theirscale and scheme dependence. While the contact-term ratiosextracted for phenomenological and local-chiral interactionsare consistent with each other, they are larger than thoseobtained for the nonlocal chiral interaction examined here.Forthcoming data with improved statistics will allow map-ping the missing and recoil momentum dependence of themeasured ratios. This will facilitate detailed studies of theorigin, implications, and significance of such differences.Previous work [3] measured A ð e; e p Þ and A ð e; e pp Þ events and derived the relative probabilities of np and pp pairs assuming that all high-missing momentum A ð e; e p Þ events were due to scattering from SRC pairs. The agree-ment between the pp=np ratios directly measured here andthose of the previous indirect measurement, as well as withthe factorized GCF calculations, strengthens the np -pairdominance theory and also lends credence to the previousassumption that almost all high-initial-momentum protonsbelong to SRC pairs in nuclei from C to Pb.We acknowledge the efforts of the staff of theAccelerator and Physics Divisions at Jefferson Lab thatmade this experiment possible. We are also grateful formany fruitful discussions with L. L. Frankfurt, M.Strikman, J. Ryckebusch, W. Cosyn, M. Sargsyan, andC. Ciofi degli Atti. The analysis presented here was carriedout as part of the Jefferson Lab Hall B Data-Miningproject supported by the U.S. Department of Energy(DOE). The research was supported also by the NationalScience Foundation, the Pazy Foundation, the IsraelScience Foundation, the Chilean Comisin Nacional deInvestigacin Cientfica y Tecnolgica, the French CentreNational de la Recherche Scientifique and Commissariata l ’ Energie Atomique the French-American CulturalExchange, the Italian Istituto Nazionale di FisicaNucleare, the National Research Foundation of Korea,and the UK ’ s Science and Technology Facilities Council.Jefferson Science Associates operates the ThomasJefferson National Accelerator Facility for the DOE,Office of Science, Office of Nuclear Physics underContract No. DE-AC05-06OR23177. The raw data fromthis experiment are archived in Jefferson Lab ’ s massstorage silo. A pp / np r a t i o s [ % ] C Al Fe Pb
GCF
AV18 loc
N2LO non-loc
N3LO
JLab Hall A, directJLab CLAS, indirect(SCX corrected)This Work
FIG. 3. Extracted ratios of pp - to np -SRC pairs plotted versusatomic weight A . The filled green circles show the ratios of pp - to np -SRC pairs extracted from ð e; e pp Þ = ð e; e pn Þ cross-sectionratios corrected for SCX using Eq. (2). The shaded regions markthe 68% and 95% confidence limits on the extraction due touncertainties in the measured cross-section ratios and SCXcorrection factors (see Supplemental Material [48] for details).The magenta triangle shows the carbon data of Ref. [20], whichwere also corrected for SCX. The open black squares show theindirect extraction of Ref. [3]. The uncertainties on both previousextractions mark the 68% (i.e., σ ) confidence limits. Thehorizontal dashed lines show the C GCF-calculated contactratios for different NN potentials using contact values fitteddirectly to the measured cross-section ratios. See text for details.
PHYSICAL REVIEW LETTERS
Figure 4: np -SRC dominance in nuclei from C to
Pb extracted from A ( e, e (cid:48) Np ) and A ( e, e (cid:48) p ) measurements [13, 14, 17], compared with GCF calculations [17]. Furthermore, the measured A ( e, e (cid:48) pn ) and A ( e, e (cid:48) np ) cross-sections were found to be significantlyhigher than the A ( e, e (cid:48) pp ) cross-section. This finding, consistently observed in all measured nuclei,was interpreted as evidence for np -SRC pairs being about 20 × more abundant than pp -SRC pairs(Fig. 4). From a theoretical standpoint, this np -SRC predominance was interpreted as resulting fromthe dominance of the tensor part of the N N interaction at the probed sub-fm distances [6, 7, 24–26](see Fig. 5.It should be pointed out that, on average, the tensor part of the
N N interaction is long-rangedand small compared to the dominant scalar part. However, studies of the deuteron suggest that9 ucleon [9] and Urbana-IX three-nucleon [10] interactions(
AV18 = UIX ). The high accuracy of the VMC wave func-tions is well documented (see Refs. [11,12] and referencestherein), as is the quality of the
AV18 = UIX
Hamiltonian inquantitatively accounting for a wide variety of light nu-clei properties, such as elastic and inelastic electromag-netic form factors [13], and low-energy capture re-actions [14]. However, it is important to stress that thelarge effect of tensor correlations on two-nucleon mo-mentum distributions and the resulting isospin depen-dence of the latter remain valid, even if one uses a semi- realistic Hamiltonian model. This will be shown explicitlybelow.The double Fourier transform in Eq. (1) is computed byMonte Carlo (MC) integration. A standard Metropoliswalk, guided by j JM J ! r ; r ; r ; . . . ; r A " j , is used to sam-ple configurations [12]. For each configuration a two-dimensional grid of Gauss-Legendre points, x i and X j , isused to compute the Fourier transform. Instead of justmoving the position ( r and R ) away from a fixed position ( r and R ), both positions are moved sym-metrically away from r and R , so Eq. (1) becomes ! TM T ! q ; Q " A ! A $ " ! J % " X M J Z d r d r d r & & & d r A d x d X y JM J ! r % x = ; R % X = ; r ; . . . ; r A " e $ i q & x e $ i Q & X P TM T ! " JM J ! r $ x = ; R $ X = ; r ; . . . ; r A " : (3)Here the polar angles of the x and X grids are also sampledby MC integration, with one sample per pair. This proce-dure is similar to that adopted most recently in studies ofthe He ! e; e p " d and He ! ~e; e ~p " H reactions [15] and hasthe advantage of very substantially reducing the statisticalerrors originating from the rapidly oscillating nature of theintegrand for large values of q and Q . Indeed, earliercalculations of nucleon and cluster momentum distribu-tions in few-nucleon systems, which were carried out bydirect MC integration over all coordinates, were very noisyfor momenta beyond $ , even when the random walkconsisted of a very large number of configurations [2].The present method is, however, computationally inten-sive, because complete Gaussian integrations have to beperformed for each of the configurations sampled in therandom walk. The large range of values of x and X requiredto obtain converged results, especially for He , requirefairly large numbers of points; we used grids of up to 96and 80 points for x and X , respectively. We also sum overall pairs instead of just pair 12.The np and pp momentum distributions in He , He , Li , and Be nuclei are shown in Fig. 1 as functions of therelative momentum q at fixed total pair momentum Q ,corresponding to nucleons moving back to back. The sta-tistical errors due to the Monte Carlo integration are dis-played only for the pp pairs; they are negligibly small forthe np pairs. The striking features seen in all cases are(i) the momentum distribution of np pairs is much largerthan that of pp pairs for relative momenta in the range : – : $ , and (ii) for the helium and lithium isotopesthe node in the pp momentum distribution is absent in the np one, which instead exhibits a change of slope at acharacteristic value of p ’ : $ . The nodal structureis much less prominent in Be . At small values of q theratios of np to pp momentum distributions are closer tothose of np to pp pair numbers, which in He , He , Li ,and Be are, respectively, 2, 4, 3, and = . Note that the np momentum distribution is given by the linear combination ! TM T % ! TM T , while the pp momentum distributioncorresponds to ! TM T . The wave functions utilized in thepresent study are eigenstates of total isospin ( = for He ,and 0 for He , Li , and Be ), so the small effects of isospin-symmetry-breaking interactions are ignored. As a result, in He , Li , and Be the ! TM T is independent of the isospinprojection and, in particular, the pp and T np mo-mentum distributions are the same.The excess strength in the np momentum distribution isdue to the strong correlations induced by tensor compo-nents in the underlying NN potential. For Q , the pairand residual ( A $ ) system are in a relative S wave. In He and He with uncorrelated wave functions, = of the np pairs are in deuteronlike T; S ; states, while the pp , nn , and remaining = of np pairs are in T; S ; (quasibound) states. When multibody tensor correlationsare taken into account, 10%–15% of the T; S ; pairsare spin flipped to T; S ; pairs, but the number of -1 )10 -1 ρ NN ( q , Q = ) (f m ) He He Li Be FIG. 1 (color online). The np (lines) and pp (symbols) mo-mentum distributions in various nuclei as functions of therelative momentum q at vanishing total pair momentum Q . PRL week ending30 MARCH 2007
R. Weiss et al. / Physics Letters B 780 (2018) 211–215
Table 1
The nuclear contacts for a variety of nuclei. The contacts are extracted by fitting the asymptotic expressions of Eq. (4) to the
VMC two-body densities in momentum (k) and coordinate (r) space separately. For He and C the contacts extracted from electron scattering data are also shown. The nuclear contacts are divided by A/2 and multiplied by to give the percent of nucleons above k F in the different SRC channels.A k-space r-space C s = pn C s = pn C s = nn C s = pp C s = pn C s = pn C s = nn C s = pp He 12.3 ± ± ± ± ± ± ± Li 10.5 ± ± ± ± ± Li 10.6 ± ± ± ± ± ± ± ± Be 13.2 ± ± ± ± ± Be 12.3 ± ± ± ± ± ± ± ± B 11.7 ± ± ± ± ± C 16.8 ± ± ± ± ± ± ± O 11.4 ± ± Ca 11.6 ± ± Fig. 1.
The ratio of proton–proton to proton–neutron SRC pairs in He as a function of the pair momentum extracted from He(e,e’pN) measurements [47].
The colored lines show the equivalent ab-initio two-body momentum densities ratio integrated over the c.m. momentum from to K max that varies from zero to infinity [16]. The solid (dashed) black line is the contact theory prediction calculated using the contacts extracted in momentum (coordinate) space (Eq. (7)). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) consider pairs with high relative momentum k , and low c.m. mo-mentum K . The cut on K reduces the contributions from mean field nucleons significantly, and identifies SRC pairs with lower rel-ative momentum. It should be noted that in the limit of heavy nuclei the contribution of uncorrelated nucleon pairs with low c.m. momentum could increase.These two approaches can be demonstrated by comparing the two-body density calculations to data. Fig. 1 shows the calculated and measured proton–proton ( pp ) to proton–neutron ( pn ) pairs density ratio in He as a function of their relative momentum. The experimental data are obtained from recent electron induced two-nucleon knockout measurements performed in kinematics domi-nated by breakup of SRC pairs [47].
The calculated pair density ratio is shown as a function of the relative pair momentum and is given by: ! K max d K F pp ( k , K )/ ! K max d K F np ( k , K ) , where K max varies from zero to infinity. As can be seen, as long as the maximal c.m. momentum is small, i.e. K max < . − ∼ k F , the calcu-lated ratio describes well the experimental data for k > k F . This demonstrates the above second approach.
These results are inline with those of Ref. [49]. On the other hand, demonstrating the first approach, if we concentrate on very high relative momentum, i.e. k > − , we can see that the ratios are largely insensitive to the value of K max . Equipped with these observations, we are now in position to utilize the two-body densities to extract the values of the nuclear contacts. Extracting the nuclear contacts – As explained above, we con-sider four main nuclear contacts: singlet ℓ = pn , pp , and nn , and triplet pn deuteron channel. For symmetric nuclei, spin-zero pp and nn pairs are identical, leaving three nuclear contacts: C s = nn , C s = pn , and C s = pn . Isospin symmetry can be used to relate the various s = leaving two independent contacts: spin-singlet and spin-triplet. For what follows, we do not impose isospin symmetry in order to study its manifestation in the case of SRC pairs.We will extract the values of the contacts for nuclei up to Ca in three different methods. In the first two methods we use the available two-body densities [16], in momentum space and coor-dinate space, separately. In the third method, we use experimental data. The results are summarized in Table 1, where one can see a good agreement between all three methods.In the first (second) method, the values are extracted by fit-ting the factorized two-body momentum (coordinate) space ex-pressions of Eq. (4) to the equivalent two-body density obtained from many-body
VMC calculations [16].
The s = pp and nn con-tacts are obtained by fitting the VMC pp and nn two-body density respectively. The s = s = pn contacts are obtained from simultaneously fitting the spin–isospin ST = pn two-body den-sity and the total pn two-body density. In view of the discussion above, the fitting range was fm − to fm − , as F NN ( k ) is dom-inated by SRC pairs only for k > fm − . In coordinate space the fitting was done in the range from to The determi-nation of the uncertainties is described in [57]. As VMC coordinate space distributions are not available for the different spin–isospin states, we assumed isospin symmetry (i.e. equal s = for the symmetric nuclei. The universal functions ϕ α ij were calculated using the AV18 potential [57].Experimentally, the nuclear contacts can be evaluated using the measured pp -to- pn SRC pairs ratio discussed above,
SRC pp SRC pn ( k ) , and the high-momentum scaling factor, a ( A / d ) . The latter is ex-tracted from large momentum transfer inclusive electron scattering cross-section ratios and determines the relative number of high-momentum ( k > k F ) nucleons in a nucleus, A , relative to deu-terium [9,40–43], assuming the effects of Final-State
Interactions and other reaction channels are suppressed in the kinematics of these measurements due to the large momentum transfer and the use of cross-section ratios, see Ref. [8,55] and references therein for
Figure 5: Left: calculated pp (points) and np (lines) stationary pair momentum densities inlight nuclei [24]. Right: measured and calculated He pp/np pair density ratios as a functionof the pair relative momentum [50]. its second order effect, viewed as a two-pion exchange term, becomes important in the momentumrange where the scalar force approaches zero ( ≈ . pp -SRC pairs with much larger missing momentum. Fig. 5shows the measured increase in the fraction of pp -SRC pairs [16], which is overall consistent withtheoretical expectation based on calculations of two-nucleon momentum distributions [55] and theirGCF representation [50]. The large error bars of the He data makes it hard to draw any conclusivequantitative conclusions on the
N N interaction beyond the tensor limit. However, as shown above,the combination of improved data, and recent theoretical developments (such as the GCF), allowsaddressing such issues.
Measurements of exclusive two-nucleon knockout reactions allow us to probe the detailed charac-teristics of SRC pairs, going beyond their isospin structure. One such property of interest is theC.M. motion of SRC pairs. It is a measure of the interaction of the pair with the ‘mean-field’ po-tential created by the residual A − σ CM .Fig. 6 shows new results from the extraction of the σ CM for pp-SRC pairs from an analysisof A ( e, e (cid:48) pp ) data, led by graduate student E. Cohen and the spokespersons [18]. The extractedC.M. momentum distribution for the measured nuclei was observed to be consistent with a Gaussiandistribution in each direction, as expected. The extracted values of σ CM were observed to varybetween 140 and 160 MeV /c , and are consistent with a constant within experimental uncertainties.Comparisons with theory predictions show good agreement with either a simple Fermi-gas modelprediction (where the N N pairs are formed from two randomly chosen nucleons, each followinga Fermi-Gas momentum distribution with k F = 250 MeV /c ) or more realistic mean-field calcula-tions [53, 54]. Interestingly, the data seem to be higher than the mean-field predictions that assumeall N N pairs can form SRC pairs, but lower than the most restrictive S calculation (i.e., assum-ing only mean-field pp pairs in a relative S state can form pp -SRC pairs). This indicates someselectivity in the SRC pair formation process and was suggested to provide insight to their quantumnumbers [18, 54, 62]. 10 ossible NN pairs from shell-model orbits, while Ref. [35]considers both all pairs, and nucleons in a relative S state(i.e., nodeless s -wave with spin 0) [64,65]. The simplisticFermi-gas prediction samples two random nucleons from aFermi sea with k F from [63].The agreement of the data with calculations supports thetheoretical picture of SRC pair formation from temporalfluctuations of mean-field nucleons [15]. The experimen-tally extracted widths are consistent with the Fermi-Gasprediction and are higher than the full mean-field calcu-lations that consider formation from all possible pairs. Thedata are lower than the S calculation that assumesrestrictive conditions on the mean-field nucleons that formSRC pairs [35].We note that the SRC-pair c.m. momentum distributionsextracted from experiment differ from those extracteddirectly from ab initio calculations of the two-nucleonmomentum distribution. The latter are formed by summingover all two-nucleon combinations in the nucleus andtherefore include contributions from non-SRC pairs. Seediscussion in Ref. [34].In conclusion, we report the extraction of the width of thec.m. momentum distribution, σ c : m : , for pp -SRC pairs from A ð e; e pp Þ measurements in C, Al, Fe, and Pb. The newdata are consistent with previous measurements of thewidth of the c.m. momentum distribution for both pp and pn pairs in C. σ c : m : increases very slowly and mighteven saturate from C to Pb, supporting the claim that finalstate interactions are negligible between the two outgoingnucleons and the residual A − nucleus. The comparisonwith theoretical models supports the claim that SRC pairsare formed from mean-field pairs in specific quantumstates. However, improved measurements and calculationsare required to determine the exact states. The raw data from this experiment are archived inJefferson Labs mass storage silo [66].We acknowledge the efforts of the staff of theAccelerator and Physics Divisions at Jefferson Lab thatmade this experiment possible. We are also grateful formany fruitful discussions with L. L. Frankfurt, M.Strikman, J. Ryckebusch, W. Cosyn, M. Sargsyan, andC. Ciofi degli Atti. The analysis presented here was carriedout as part of the Jefferson Lab Hall B data-mining projectsupported by the U.S. Department of Energy (DOE). Theresearch was supported also by the National ScienceFoundation, the Israel Science Foundation, the ChileanComisin Nacional de Investigacin Cientfica y Tecnolgica,the French Centre National de la Recherche Scientifiqueand Commissariat a lEnergie Atomique the French-American Cultural Exchange, the Italian IstitutoNazionale di Fisica Nucleare, the National ResearchFoundation of Korea, and the UKs Science andTechnology Facilities Council. Jefferson ScienceAssociates operates the Thomas Jefferson NationalAccelerator Facility for the DOE, Office of Science,Office of Nuclear Physics under Contract No. DE-AC05-06OR23177. E. O. Cohen would like to acknowledge theAzrieli Foundation. * Corresponding [email protected] † On sabbatical leave from Nuclear Research Centre Negev,Beer-Sheva, Israel. ‡ Present address: Idaho State University, Pocatello, Idaho83209, USA.[1] L. L. Frankfurt, M. I. Strikman, D. B. Day, and M. Sargsyan,Phys. Rev. C , 2451 (1993).[2] K. Egiyan et al. (CLAS Collaboration), Phys. Rev. C ,014313 (2003).[3] K. Egiyan et al. (CLAS Collaboration), Phys. Rev. Lett. ,082501 (2006).[4] N. Fomin et al. , Phys. Rev. Lett. , 092502 (2012).[5] A. Tang et al. , Phys. Rev. Lett. , 042301 (2003).[6] E. Piasetzky, M. Sargsian, L. Frankfurt, M. Strikman, andJ. W. Watson, Phys. Rev. Lett. , 162504 (2006).[7] R. Shneor et al. , Phys. Rev. Lett. , 072501 (2007).[8] R. Subedi et al. , Science , 1476 (2008).[9] I. Korover, N. Muangma, O. Hen et al. , Phys. Rev. Lett. ,022501 (2014).[10] O. Hen et al. (CLAS Collaboration), Science , 614(2014).[11] J. Ryckebusch, W. Cosyn, and M. Vanhalst, J. Phys. G ,055104 (2015).[12] R. B. Wiringa, R. Schiavilla, S. C. Pieper, and J. Carlson,Phys. Rev. C , 024305 (2014).[13] D. Lonardoni, A. Lovato, S. C. Pieper, and R. B. Wiringa,Phys. Rev. C , 024326 (2017).[14] C. C. degli Atti and S. Simula, Phys. Rev. C , 1689(1996). A [ M e V / c ] c . m . σ He C Al Fe Pb
This WorkBNL (p,2pn)Hall-A (e,e'pp)Hall-A (e,e'pn) Ciofi and SimulaColle et al. (All Pairs) pairs) S Colle et al. (Fermi-Gas (All Pairs)
FIG. 3. The nuclear mass dependence of the one-dimensionalwidth of the c.m. momentum distribution. The data pointsobtained in this work (red full circles) are compared to previousmeasurements (blue full squares and triangles) [5,7,9] andtheoretical calculations by Ciofi and Simula (open stars) [14],Colle et al. , considering all mean-field nucleon pairs (dashed line)and only S pairs (solid line) [35] and a Fermi-gas prediction[63] considering all possible nucleon pairs. See text for details. PHYSICAL REVIEW LETTERS
Figure 6: Width of pp -SRC pairs C.M. momentum distribution, extracted from A ( e, e (cid:48) pp ) data(red circles) [18], compared with previous extractions (blue points). The width is extractedassuming a 3D gaussian for the C.M. distribution, defined by its width, σ CM . The lines andstars show mean-field theory predictions [53, 54]. A ( e, e N ) cross-section ratios were corrected forradiative e↵ects [21] in the same way as was donein [10, 22, 23]. The radiative correction to the trans-parency ratio was found to be ⇠ T N ( A ) = exp A ( e, e N ) PWIA A ( e, e N ) . (1)In the commonly used factorized approximation for large- Q reactions, PWIA is given by (see [24]): PWIA ( e, e N ) = K N tar · eN · I S A ( E, p i ) dEd p i , (2)in which N tar is the number of relevant nucleons in thetarget nucleus (i.e. number of protons for ( e, e p ) andneutrons for ( e, e n )), K = | ~p N | · E N is a kinematical fac-tor, eN is the o↵-shell electron-nucleon elementary crosssection, S A ( E, p i ) is the nuclear spectral function, whichis the probability for finding a nucleon in the nucleus withmomentum p i and separation energy E . S A ( E, p i ) is nor-malized as R S A ( E, p i ) dEd p i ⌘ N tar . The spectralfunction in Eq. 2 is integrated over the experimental ac-ceptance.If the two nuclei, nucleus with A nucleons and C,are measured in the same kinematics, then their trans-parency ratio is given by: T N ( A ) T N ( C ) = exp A ( e, e N ) exp C ( e, e N ) · H S C ( E, p i ) dEd p i H S A ( E, p i ) dEd p i , (3)in which the spectral functions for A and C are integratedover the same kinematical regions.For the MF kinematics, Eq. 3 can be expressed as: T MFN ( A ) T MFN ( C ) = R k n C ( p i ) dp i R k n A ( p i ) dp i · exp A ( e, e N ) exp C ( e, e N ) , (4)where exp A ( e, e N ) / exp C ( e, e N ) is the measured nu-cleon knockout cross-section ratio discussed above andthe first term is the ratio of integrals over the mean-field part of the nuclear momentum density, which, dueto the large missing-energy cut, replaces the integralsover the mean-field spectral functions. The nuclear mo-mentum density is defined as n A ( p i ) ⌘ R S A ( E, p i ) dE FIG. 2: The estimated transparency ratios for MF and SRCkinematics, both for protons and neutrons, together with apower law fit to a weighted average (grey line), as describedin the text. For Fe and Pb nuclei, also shown are results basedon three Glauber Calculations: [18] dotted line, [19] dashedline, and [20, 30] solid line. and was calculated following [18]. The integral calcula-tions in Eq. 4 were done using three di↵erent models forthe mean-field momentum distribution: Ciofi and Sim-ula [25], Woods-Saxon [26], and Serot-Walecka [27] with k , the upper limit of the MF momentum range, chosento be the average between 300 MeV/ c and the Fermi sealevel, k F = 221, 260, 260, and 260 (280) MeV/ c for C,Al, Fe, Pb, respectively, for protons (neutrons) [28]. Weassigned the half di↵erence between the two extreme val-ues obtained by considering the di↵erent values of k andthe di↵erent models as a corresponding systematic un-certainty. The values of the latter are 4.9% (3.8%), 4.2%(5.7%), and 4.3% (4.5%) for protons (neutrons) for theAl/C, Fe/C, and Pb/C ratios, respectively. The resultsof this calculation are consistent with those previouslyobtained by Hartree-Fock-Slater wave functions [18].The transparency ratios in SRC kinematics were ex-tracted following [10] as: T SRCN ( A ) T SRCN ( C ) = 1 a ( A/ C) · exp A ( e, e N ) /A exp C ( e, e N ) / , (5)where a ( A/C ) is the relative number of 2 N -SRC pairsper nucleon in nuclei A and C. These ratios were adaptedfrom [29] and are based on a compilation of world datafor the ( e, e ) cross-section ratio at large Q and x B > . . c forboth proton and neutron, and for each of the three nu- ↵ which is brought in a continuum state at the co-ordinate ~r is modeled by, P ↵ ( )CX ( ~r ) = 1 exp[ CX ( s ) Z + z d z ⇢ ↵ ( z )] . (1)The z -axis is chosen along the direction of propagation ofthe nucleon with initial quantum numbers ↵ . The quan-tum numbers of the correlated partner in the SRC pairare denoted with . The ⇢ ↵ is the density of the resid-ual nucleus available for SCX reactions. Obviously, for anejected proton (neutron) only the neutron (proton) den-sity of the residual nucleus a↵ects SCX reactions. CX ( s )in Eq. (1), with s the total c.m. energy squared of thetwo nucleons involved in the SCX [26], can be extractedfrom elastic proton-neutron scattering data [27]. Cross-section Model:
As outlined in Refs. [23, 28], inthe spectator approximation it is possible to factorizethe A ( e, e pN ) cross section in kinematics probing short-range correlated pairs asd [ A ( e, e pN )]d ⌦ e d ~P d ~k = K epN epN ( ~k ) F pN ( D )A ( ~P ) , (2)where ⌦ e is the solid angle of the scattered electron,and ~k and ~P are the relative and c.m. momenta ofthe nucleon pair that absorbed the virtual-photon. The K epN is a kinematic factor and epN ( ~k ) is the crosssection for virtual-photon absorption on a correlated pNpair. The F pN ( D )A ( ~P ) is the distorted two-body c.m.momentum distribution of the correlated pN pair. In thelimit of vanishing FSIs, it is the conditional c.m. momen-tum distribution of a pN pair with relative S n =0 quan-tum numbers. Distortions of F pN ( D )A ( ~P ) due to FSI arecalculated in the RMSGA. The factorized cross-sectionexpression of Eq. (2) hinges on the validity of the zero-range approximation (ZRA), which amounts to puttingthe relative pair coordinate ~r to zero. The ZRA worksas a projection operator for selecting the very short-rangecomponents of the IPM relative pair wave functions.The probability for charge-exchange reactions in pNknockout is calculated on an event per event basis, us-ing the SRC pair probability density F pN ( D )A ( ~R ) in theZRA corrected for FSI. With the aid of the factorizedcross-section expression of Eq. (2), the phase-space inte-grated A ( e, e pN ) to C( e, e pN ) cross-section ratios canbe approximately expressed as integrals over distorted
10 100 A A ( e , e pp ) c r o ss - s e c t i o n r a t i o s ZRAZRA-RMSGA
FIG. 2: (color online). The mass dependence of the A ( e, e pp ) / C( e, e pp ) cross-section ratios. The points showthe measured, uncorrected, cross section ratios. The lowerorange band and upper grey line denote ZRA reaction-modelcalculations for C, Al, Fe, and
Pb based on Eq. (3)with and without FSI corrections respectively. The width ofthe ZRA-RMSGA band reflects the maximum possible e↵ectof SCX. c.m. momentum distributions, [ A ( e, e pN )] [ C( e, e pN )] ⇡ R d ⌦ e d ~k K epN epN ( ~k ) R d ~P F pN ( D )A ( ~P ) R d ⌦ e d ~k K epN epN ( ~k ) R d ~P F pN ( D )C ( ~P )= R d ~P F pN ( D )A ( ~P ) R d ~P F pN ( D )C ( ~P ) . (3)In the absence of FSI, the integrated c.m. momentumdistributions R d ~P F pN ( D )A ( ~P ) equal the total num-ber of SRC-prone pN pairs in the nucleus A . Hence,the cross section ratios of Eq. (3) provide access to therelative number of SRC pN-pairs up to corrections stem-ming from FSI. We have evaluated the ratios of the dis-torted c.m. momentum distributions of Eq. (3) overthe phase space covered in the experiment. Given thealmost 4 ⇡ phase space and the high computational re-quirement of multidimensional FSI calculations, we usean importance-sampling approach. The major e↵ect onthe c.m. momentum distribution F pN ( D )A ( ~P ) when in-cluding FSIs is an overall attenuation, the shape is almostuna↵ected [23]. Motivated by this, we used the c.m. mo-mentum distributions without FSI as sampling distribu-tion for the importance sampling in the FSI calculations.When convergence is reached, the computed impact ofFSI is extrapolated to the whole phase space. Results:
Figure 2 shows the measured uncorrected [ A ( e,e pp ) ] [ C( e,e pp )] cross-section ratios compared with the ZRA Figure 7: Nucleon transparency ratios for nuclei relative to C, extracted from single-nucleonknockout measurements (left) [20], and calculations of the two-nucleon knockout reaction [62]using Glauber theory (right).
The results presented above in sections 2.1 and 2.2 require corrections for reaction effects suchas final-state interactions (FSI) and singe-charge exchange (SCX). Therefore, understanding theimpact of such reaction mechanism effects on hard electron QE scattering cross-sections is crucialfor the interpretation of measurements in general, and specifically their relation to ground-stateproperties of nuclei. In high- Q reactions, one may use the Generalized Eikonal approximationwithin a Glauber-framework to perform quantitative estimations of reaction effects such as FSIand SCX. However, additional experimental verification of this approach in the kinematics of ourmeasurements are needed. Several measurements of the nuclear transparency of proton knockout in( e, e (cid:48) p ) and ( e, e (cid:48) pp ) reactions in SRC kinematics were compared them with theoretical calculations11sing the Glauber approximation [19,62] (Fig. 7, right). The experimentally extracted transparencyratios showed good agreement with Glauber calculations. Recently, this work was extended tomeasurements of neutron knockout ( e, e (cid:48) n ) reactions in both SRC and Mean-Field kinematics [20](Fig. 7 top panel). The extracted transparency for both proton and neutron knockout in mean-fieldand SRC kinematics were observed to agree with each other and with Glauber calculations. Thecombined nuclear mass dependence of the data is consistent with power-law scaling of A α with α = − . ± . The predominance of np -SRC pairs leads to interesting phenomena in asymmetric nuclei. WithoutSRC pairs, neutrons in neutron-rich nuclei should have a higher Fermi momentum and thus ahigher average momentum and kinetic energy than the minority protons. However, since the high-momentum tail of the momentum distribution is dominated by np -pairs, there should be equalnumbers of protons and neutrons above k F . Therefore, the excess neutrons in a neutron-rich nucleusshould either increase the fraction of correlated protons or occupy low-momentum states. In eithercase, the fraction of high-momentum protons should be larger than that of neutrons [13, 27, 28]. common angular region for detecting both protons and neutrons, correcting for the neutron and proton detection efficiencies, and accounting for the different momentum resolutions. The latter was a significant challenge as, being neutral particles, neutrons do not interact in the tracking system. Thus, their momentum must be inferred from the time difference between the electron scattering in the target and the neutron detection in the calorimeter, leading to about 10 - 15% momentum resolution for the detected neutrons [25]. Proton momenta were determined to an uncertainty of about 1% from the curvature of their trajectories in the CLAS magnetic field. We accounted for this momentum resolution difference by: (1) selecting the desired A(e,e’p) events in high- and low-momentum kinematics, (2) “smearing” the proton momentum for each event using the measured neutron momentum resolution, and (3) using unsmeared and smeared A(e,e’p) event samples to study bin migration effects and optimize the event selection criteria. This allowed us to produce a smeared event sample with as many of the ‘original’ A(e,e’p) events as possible (i.e. high selection efficiency), and as few other events as possible (i.e. high purity). We used the smeared proton momenta in the final selection of A(e,e’p) events for consistency with the A(e,e’n) analysis. The final event selection criteria are detailed below. Low-initial-momentum events are characterized by low missing energy and low missing momentum ( E miss < *$++ = *$++ < 250 MeV/c) [25]. Because the neutron resolution was not good enough to select these events directly, we developed a set of alternative constraints to select the same events by using the detected electron momentum and the knocked-out nucleon angle, which were unaffected by the neutron momentum resolution [25]. We optimized these constraints using the unsmeared and smeared protons so that the final event sample contains about 90% of the desired sample with about 15% contamination, resulting in about 5% more events in our sample. We assumed cross section corrections of about 5%, which caused a less than 1% correction to ratios between different nuclei. We assumed systematic uncertainties equal to the corrections, see Supplementary Material for details. Similarly, we selected the high-initial-momentum events in two steps. We first selected QE events with a leading nucleon by cutting on the energy and momentum transfer and requiring that the outgoing nucleon be emitted with most of the transferred momentum in the general direction of the momentum transfer. We then selected high-initial-momentum events by requiring large missing momentum ( *$++ > 300
MeV/c). These selection criteria ensured that the electron interacted with a single high-initial-momentum proton or neutron in the nucleus [3, 27]. We then optimized the nucleon-momentum dependent cuts using the smeared and unsmeared protons. This gave us a final event sample with 85% of the desired sample and about 15% contamination, resulting in about the same number of events. We corrected for this when extracting the cross-section ratios. The procedures for the extraction of cross-section ratios and their associated systematic uncertainties from the measured event yields are detailed in the supplementary materials [25]. To verify the neutron detection efficiency, detector acceptance corrections, and event selection method, we extracted the carbon neutron-to-proton reduced cross-section ratio for both high and low initial nucleon momenta: [ C(e,e’n)/ ] / [ C(e,e’p)/ ] (i.e. measured cross-sections scaled by the known elastic electron-proton and electron-neutron cross-sections). Figure 2 shows these two measured cross-section ratios are consistent with unity, as expected for a symmetric nucleus. This shows that in both high- and low-initial-momentum kinematics we have restricted the reaction mechanisms to primarily QE scattering and have correctly accounted for the various detector-related effects. For the other measured nuclei, the (e,e’n)/(e,e’p) low-momentum reduced cross-section ratios grow approximately as N/Z, as expected from simple nucleon counting. However, the (e,e’n)/(e,e’p) high-momentum ratios are consistent with unity for all measured nuclei, see Fig. 2. The struck nucleons could reinteract as they emerged from the nucleus. Such an effect would cause the number of detected outgoing nucleons to decrease and modify the angles and momenta of the knocked-out nucleons. The decrease in the measured cross section would be nearly identical for protons and neutrons and thus, any effects should cancel when forming cross-section ratios [25]. Since rescattering changes the event kinematics, some of the events with high measured *$++ could have Fig 2 | The relative abundances of high- and low-initial-momentum neutrons and protons. [A(e,e’n)/ ] / [A(e,e’p)/ ] reduced cross-section ratio for low-momentum (green circles) and high-momentum (purple triangles) events. The initial nucleon momenta corresponding to each type of event are illustrated in the inset. The lines show the simple N/Z expectation for low-momentum nucleons and the np-dominance expectation (i.e., ratio = 1) for high-momentum nucleons. The inner error bars are statistical while the outer ones include both statistical and systematic uncertainties [25]. originated from electron scattering from a low-initial-momentum nucleon, which then rescattered, increasing *$++ . If the high-initial-momentum (high- *$++ ) nucleons were caused by electron scattering from the more-numerous low-initial-momentum nucleons followed by nucleon rescattering, then the high-momentum (e,e’n)/(e,e’p) ratio would show the same N/Z dependence as the low-momentum ratio. As the high-momentum (e,e’n)/(e,e’p) ratio is independent of A, nucleon-rescattering effects must be small. Thus, these data indicate that there are equal numbers of high-initial-momentum protons and neutrons in asymmetric nuclei, even though these nuclei contain up to 50% more neutrons than protons. This observation is consistent with high-momentum nucleons belonging primarily to np-SRC pairs, even in neutron rich nuclei [20]. This number equality implies a greater fraction of high-initial-momentum protons. For example, if 20% [1] of the 208 nucleons in lead-208 are at high-initial-momentum, then these consist of 21 protons and 21 neutrons. This corresponds to a high-momentum proton fraction of 21/82 ~ 25% and a corresponding neutron fraction of only 21/126 ~ 17%. In order to quantify the relative fraction of high-momentum protons and neutrons in the different nuclei with minimal experimental and theoretical uncertainties, we extracted the double ratio of (e,e’N) high-initial-momentum to low-initial-momentum events for nucleus A relative to carbon for both protons and neutrons. We find that the fraction of high-initial-momentum protons increases by about 50% from carbon to lead (see Fig. 3). Moreover, the corresponding fraction of high-initial-momentum neutrons seems to decrease by about 10%. Nucleon-rescattering should increase in larger nuclei and should affect protons and neutrons equally. Since, unlike the protons, the neutron ratio decreases with A, this also rules out significant nucleon rescattering effects. To determine whether the observed high-initial-momentum proton fraction is large enough to make the average kinetic energy of protons larger than that of neutrons in heavy neutron-rich nuclei, we compared it to a simple phenomenological (i.e. experiment-based) np-dominance model [3, 20] that uses the mean-field momentum distributions from one of three different models at low momentum ( k < k F ) and a deuteron-like high-momentum tail, scaled by the measured fraction of high-momentum nucleons in nuclei [25]. The model predictions for the relative fractions of high-momentum nucleons agree with the data, increasing our confidence in the model, see Fig. 3. This model also predicts a neutron-to-proton average kinetic energy ratio (
O. Hen et al., Rev. Mod. Phys. , 045002 (2017). 2. R. Subedi et al., Science , 1476 (2008). (cid:1)
Fig 3 |
Relative high-momentum fractions for neutrons and protons.
Red circles: The double ratio of the number of (e,e’p) high-momentum proton events to low-momentum proton events for nucleus A relative to carbon. The inner error bars are statistical while the outer ones include both statistical and systematic uncertainties. Red bands: the prediction of the phenomenological np-dominance model [25]. Blue squares and bands: the same for neutron events. The inset demonstrates how adding neutrons increases the fraction of protons in the high momentum tail. The red line at N / Z and the blue line at 1 are drawn to guide the eye. Figure 8: Nucleon knockout studies of heavy nuclei [12]. Left: Extracted ratio of protonto neutron knockout from above and below the nuclear Fermi momentum. Right: Extractedfraction of high-momentum ( k > k F ) protons and neutrons in nuclei relative to C, comparedwith SRC model predictions (shaded squares).
In a paper recently published in Nature [12], we reported the first simultaneous measurement ofhard QE electron scattering off protons and neutrons (i.e., A ( e, e (cid:48) p ) and A ( e, e (cid:48) n ) reactions) in C, Al, Fe, and
Pb. The simultaneous measurement of both proton and neutron knockout alloweda direct comparison of their properties with minimal assumptions. The measurement was made intwo different kinematical settings, one corresponding to electron scattering primarily off nucleonsfrom an SRC pair ( p miss > k F ), the other from nucleons in the nuclear mean field ( p miss < k F ). Usingthese event samples, the reduced cross-section ratios: [ A ( e, e (cid:48) n ) /σ en ] / [ A ( e, e (cid:48) p ) /σ ep ] (i.e., measuredcross-sections divided by the known elementary electron-proton, σ ep , and electron-neutron, σ en ,cross-sections) were extracted for each kinematical setting. The results shown in Fig. 8 (left) indicatethat the n/p mean-field reduced cross-section ratios grow approximately as N/Z for all nuclei, asexpected from simple nucleon counting. However, the SRC ratios in all nuclei are consistent withunity, consistent with np -SRC dominance of the high-momentum tail.To quantify the pairing mechanism leading to constant n/p ratios for SRC nucleons, we also ex-tracted the relative fraction of high-missing-momentum to low-missing-momentum events in neutron-rich nuclei relative to C, see Fig. 5 (right). This extraction was done separately for protons andneutrons, and shows that the neutron SRC probabilities are independent of the nuclear neutron12xcess (i.e. they saturate) while the corresponding proton probabilities grow linearly with
N/Z .This observation indicates that in neutron-rich nuclei, the outer excess neutrons form SRC pairswith protons from the inner ‘core’ of the nucleus.
Figure 9: The reaction mechanisms for electron-induced two nucleon knockout. The virtualphoton can be absorbed on one nucleon of an SRC pair, leading to the emission of both nucleons(SRC). The virtual photon can excite a nucleon to a ∆, which deexcite by exchanging a pion,resulting in the emission of two nucleons (IC). The virtual photon can be absorbed on a pion-in-flight (MEC). The virtual photon can be absorbed on one nucleon of an SRC pair whichrescatters from the other nucleon in the pair (FSI (left)). The virtual photon can be absorbedon an uncorrelated nucleon which rescatters from another nucleon (FSI (right)).
The results described above are almost all derived from electron scattering measurements, withonly a single proton scattering C( p, ppn ) measurement [23]. Thus, the interpretation of these exper-imental results relies on an assumed electron interaction mechanism at large momentum transfers.There are a number of different electron-scattering reaction mechanisms that can lead to two-nucleonemission (see Fig. 9). While the experiments described above have been performed at kinematicswhere many of these effects have been minimized, there are still interpretational uncertainties dueto these other possible reaction mechanisms. These reaction mechanisms are not present or are verydifferent for proton scattering.Photon scattering will also proceed through very different reaction mechanisms. Instead ofquasielastic nucleon knockout, the primary photo-induced reaction studied here will be γn → pπ − ,with a second nucleon (the correlated partner nucleon) emitted backward (see Fig. 10. For thisreaction, the IC and MEC reaction mechanisms will be absent or significantly different. In addition,because the correlated partner nucleon will be emitted backwards, the effects of Final State Interac-tions (FSI) will also be quite different. It is much more difficult to produce backward nucleons thatforward ones.Thus photonuclear measurements of SRCs will provide a crucial reaction mechanism check forSRC studies. 13 A–2 p (0, m A ) ( p CM ,m A - ϵ A–2 ) N ( p recoil , p recoil +m N ) ( p i , ϵ i )( q, ω =q ) ( - p CM ,E A-2 ≡ p CM +(m A-2 +E*) ) SRCSRC π ( p π , ϵ π ≡ p π +m π ) ( p N ≡ p i +q–p π , p N +m N ) Figure 10: Diagrammatic representation and kinematics of the triple-coincidence A ( γ, πNp )reaction, one of the main channels of interest for SRC breakup by a real photon beam. As inFig. 1, dashed red lines represent off-shell particles. Open ovals represent un-detected systems.Solid black lines represent detected particles. The momentum and energy of the particles arealso indicated. The relative abundance of SRC pairs in nuclei can be extracted from measurements of inclusive( e, e (cid:48) ) cross-section ratios for different nuclei at high- Q , x B > Q , these cross-section ratios scale as a function of x B starting approximately at x B ≥ . He to
Au.The latter is the slope of the deviation from unity of the isoscalar DIS cross-section ratio for nucleirelative to deuterium in the range 0 . ≤ x B ≤ .
7. The EMC effect is commonly interpreted asevidence for modification of the partonic structure function of bound nucleons [6, 60, 61].The observation of a correlation between the strength of the EMC effect and the SRC scalingcoefficients in nuclei generated new interest in the EMC effect (see e.g. CERN Courier cover paperfrom May 2013; ‘Deep in the nucleus: a puzzle revisited’ [32]) and gave new insight into its possibleorigin. Several models have been proposed by us and others that attempt to explain the underlyingdynamics that drive the EMC effect and its correlation with SRC pair abundances; see a recentreview in Ref. [6].In a data-mining analysis recently published in Nature [11], led by graduate student B. Schmook-ler and the spokespersons, a high-precision measurement of both the SRC scaling coefficients andthe EMC effect was performed for C, Al, Fe and
Pb (see Fig. 11). The new data were usedto examine the finer aspects of the EMC-SRC correlation. Specifically, we examined whether theEMC data can indeed be explained by assuming the nuclear structure function can be factorizedinto a collection of un-modified mean-field nucleons and modified SRC pairs: F A = ( Z − n A SRC ) F p + ( N − n A SRC ) F n + n A SRC (cid:0) F p ∗ + F n ∗ (cid:1) , (3)where n A SRC is the number of np -SRC pairs, F N ( x B ) are the free nucleon (proton and neutron)structure functions, and F N ∗ ( x B ) are the average modified nucleon structure functions in SRCpairs. n A SRC is taken from experiment (i.e. from ( e, e (cid:48) ) scaling ratios at x B > . F N ∗ ( x B ), is expected to be universal (i.e., independent of thesurrounding nuclear environment). 14 ETTER RESEARCH
We also constrained the internal structure of the free neutron using the extracted universal modification function and we concluded that in neutron-rich nuclei the average proton structure modification will be larger than that of the average neutron.We analysed experimental data taken using CLAS (CEBAF Large Acceptance Spectrometer) at the Thomas Jefferson National Accelerator Facility (Jefferson Laboratory). In our experiment, a 5.01-GeV electron beam impinged upon a dual target system with a liquid deuterium target cell followed by a foil of either C, Al, Fe or Pb. The scattered electrons were detected in CLAS over a wide range of angles and energies, which enabled the extraction of both quasi-elastic and DIS reaction cross-section ratios over a wide kinematical region (see Supplementary Information section I).The electron scattered from the target by exchanging a single virtual photon with momentum q and energy ν , giving a four-momentum trans-fer of Q = | q | – ν . We used these variables to calculate the invariant mass of the nucleon plus virtual photon, W = ( m + ν ) − | q | (where m is the nucleon mass), and the Bjorken scaling variable x B = Q /(2 m ν ).We extracted cross-section ratios from the measured event yields by correcting for effects of the experimental conditions, acceptance and momentum reconstruction, as well as reaction effects and bin-centring effects (see Supplementary Information section I). To our knowledge, this was the first precision measurement of inclusive quasi-elastic scat-tering for SRCs in both Al and Pb, as well as the first measurement of the EMC effect on Pb. For other measured nuclei our data are consistent with previous measurements, but with reduced uncertainties.The DIS cross-section on a nucleon can be expressed as a function of a single structure function, F ( x B , Q ). In the parton model, x B represents the fraction of the nucleon momentum carried by the struck quark. F ( x B , Q ) describes the momentum distribution of the quarks in the nucleon, and the ratio / / / F x Q A F x Q [ ( , ) ] [ ( , ) 2]
2A B 2 2d B 2 describes the relative quark momentum distributions in a nucleus A with mass number A and deuterium (d). For brevity, we often omit explicit reference to x B and Q —that is, we write / F F
2A 2d —with the understanding that the structure functions are being compared at iden-tical x B and Q values. Because the DIS cross-section is proportional to F , experimentally the cross-section ratio of two nuclei is assumed to equal their structure-function ratio . The magnitude of the EMC effect is defined by the slope of either the cross-section ratios or the structure-function ratios for 0.3 ≤ x B ≤ IV and V). Similarly, the relative probability for a nucleon to belong to an SRC pair is interpreted as equal to a , which denotes the average ratio of the inclusive quasi-elastic electron scattering cross-section per nucleon of nucleus A to that of deuterium at momentum transfer Q > GeV and 1.45 ≤ x B ≤ III).Other nuclear effects are expected to be negligible. The contribu-tion of three-nucleon SRCs should be an order of magnitude smaller than the SRC-pair contributions. The contributions of two-body cur-rents (called ‘higher-twist effects’ in DIS) should also be small (see Supplementary Information section
VIII).Figure x B . The red lines are fits to the data that are used to determine the EMC-effect slopes or SRC scaling coefficients (see Extended Data Tables
1, 2). Typical 1 σ cross-section-ratio normalization uncertainties of 1%–2% directly contribute to the uncertainty in the SRC scaling coefficients but introduce negligible uncertainty in the EMC slope. None of the ratios presented has isoscalar corrections (cross-section corrections for une-qual numbers of protons and neutrons), in contrast to much published data. We did not apply such corrections for two reasons: (1) to focus on asymmetric nuclei and (2) because isoscalar corrections are model- dependent and differ among experiments (see Extended Data Fig. Q > GeV and W > GeV, which is just above the resonance region and higher than the W > GeV cut used in previous Jefferson Laboratory measurements . The extracted EMC slopes are insensitive to variations in these cuts over Q and W ranges of 1.5 − GeV and 1.8 − GeV, respectively (see Supplementary Information Table a ), we model the modification of the nuclear structure function, F , as entirely caused by the modification of np SRC pairs. F is therefore decomposed into contributions from unmodified mean-field protons and neutrons (the first and second terms in equation (1)) and np SRC pairs with modified structure func-tions (third term): = − + − + += + + ∆ + ∆ ∗ ∗ F Z n F N n F n F FZF NF n F F ( ) ( ) ( )( ) (1) p n p np n p n
2A SRCA 2 SRCA 2 SRCA 2 22 2 SRCA 2 2
This workRef. x B (cid:109) C Al Fe Fe Pb eg fh Ref. C Al Pb x B a bc d ( (cid:86) A / A ) / ( (cid:86) d / ) ( (cid:86) A / A ) / ( (cid:86) d / ) Fig. 1 | DIS and quasi-elastic ( e , e ′ ) cross-section ratios. a – d , Ratio of the per-nucleon electron scattering cross-section of nucleus A (A = C ( a ), Al ( b ), Fe ( c ) and Pb ( d )) to that of deuterium for DIS kinematics (0.2 ≤ x B ≤ W ≥ GeV). The solid points show the data obtained in this work, the open squares show
SLAC (Stanford Linear Accelerator Center) data and the open triangles show Jefferson Laboratory data . The red lines show a linear fit. e , f , Corresponding ratios for quasi-elastic kinematics (0.8 ≤ x B ≤ . The red lines show a constant fit. The error bars shown include both statistical and point-to-point systematic uncertainties, both at the 1 σ or 68% confidence level. The data do not include isoscalar corrections. Figure 11: High-precision measurements of the EMC effect (left) and SRC scaling (right) ledby the spokespersons [11].
Figure 12 shows the measured structure function ratios of nuclei relative to deuterium (left panel),and the extracted modification function of SRC pairs, using ∆ F N = F N ∗ − F N (right panel). Ascan be seen, while the nuclear structure functions vary significantly between different nuclei, theextracted SRC pair modification function is universal for all nuclei. LETTERRESEARCH where n SRCA is the number of np SRC pairs in nucleus A,
F x Q ( , ) p and F x Q ( , ) n are the free-proton and free-neutron structure func-tions, ∗ F x Q ( , ) p and ∗ F x Q ( , ) n are the average modified structure functions for protons and neutrons in SRC pairs and ∆ = − ∗ F F F n n n (and similarly for ∆ F p ). ∗ F p and ∗ F n are assumed to be the same for all nuclei. In this simple model, nucleon-motion effects , which are also dominated by SRC pairs owing to their high relative momentum, are folded into ∆ F p and ∆ F n .This model resembles that used in ref. . However, that work focused on light nuclei and did not determine the shape of the modification function. Similar ideas using factorization were discussed in ref. , such as a model-dependent ansatz for the modified structure functions, which was shown to be able to describe the EMC data . The analysis presented here, to our knowledge, is the first data-driven determination of the modified structure functions for nuclei from He to Pb.Because there are no model-independent measurements of F n , we apply equation (1) to the deuteron, re writing F n as − − ∆ + ∆ F F n F F ( ) p p n
2d 2 SRCd 2 2 . We then rearrange equation (1) to get: ∆ + ∆ = − − − / − n F FF Z N NA a N ( ) ( )( 2) (2) p n FF FF SRCd 2 22d 2 p where / F F p has been previously extracted and a is the measured per-nucleon cross-section ratio shown by the red lines in Fig. a approximately equals the per-nucleon SRC-pair density ratio between nucleus A and deuterium : / / / n A n ( ) ( 2) SRCA SRCd .Because ∆ + ∆ F F p n is assumed to be nucleus-independent, our model predicts that the left-hand side of equation (2) should be a uni-versal function (that is, the same for all nuclei). This requires that the nucleus-dependent quantities on the right-hand side of equation (2) combine to give a nucleus-independent result.T h i s i s t e s t e d i n F i g . / / /
F x Q A F x Q [ ( , ) ] [ ( , ) 2]
2A B 2 2d B 2 , the per-nucleon structure-function ratio of different nuclei relative to deuterium, without isoscalar cor-rections. The approximately linear deviation from unity for 0.3 ≤ x B ≤ np SRC pairs, ∆ + ∆ / n F F F ( ) p n SRCd 2 2 2d , extracted using the right-hand side of equation (2).The EMC slope for all measured nuclei increases monotonically with A whereas the slope of the SRC-modified structure function is con-stant within uncertainties; see Fig.
2. Even He, which has a markedly different structure-function ratio owing to its very large proton-to-neutron ratio of 2, has a remarkably similar modified structure function to the other nuclei, with the same slope. Thus, we conclude that the magnitude of the EMC effect in different nuclei can be described by the abundance of np SRC pairs and that the proposed SRC-pair modification function is in fact universal. This universality appears to hold even beyond x B = and by measuring semi-inclusive DIS off the deuteron, tagged by the detec-tion of a high-momentum backward-recoiling proton or neutron, that will enable direct quantification of the relationship between the momen-tum and the structure-function modification of bound nucleons .The universal SRC-pair modification function can also be used to extract the free neutron-to-proton structure-function ratio, / F F n p , by applying equation (1) to the deuteron and using the measured proton and deuteron structure functions (see Extended Data Fig. F n can be used to apply self-consistent isoscalar corrections to the EMC effect data (see Supplementary Information equation (5)).To further test the SRC-driven EMC model, we consider the isopho-bic nature of SRC pairs (that is, np dominance), which leads to an approximately constant probability for a neutron to belong to an SRC pair in medium-to-heavy nuclei, while the proton probability increases as N / Z . If the EMC effect is indeed driven by high-momen-tum SRCs, then in neutron-rich nuclei both the neutron EMC effect and the SRC probability should saturate, whereas for protons both should grow with nuclear mass and neutron excess. This is done by a ( F / A ) / ( F / ) A d x B Ref. Ref. This work –0.0500.050.2 0.4 0.6 0.8Median normalization uncertainty b n S R C ( Δ F + Δ F ) / F d p n d x B A Fig. 2 | Universality of SRC-pair quark distributions. a , b , The EMC effect for different nuclei, as observed in ratios of / / / F A F ( ) ( 2)
2A 2d as a function of x B ( a ) and the modification of SRC pairs, as described by equation (2) ( b ). Different colours correspond to different nuclei, as indicated by the colour scale on the right. The open circles show SLAC data and the open squares show Jefferson Laboratory data . The nucleus- independent (universal) behaviour of the SRC modification, as predicted by the SRC-driven EMC model, is clearly observed. The error bars show both statistical and point-to-point systematic uncertainties, both at the 1 σ or 68% confidence level, and the grey bands show the median normalization uncertainty at the 1 σ or 68% confidence level. The data do not include isoscalar corrections. A E M C s l o p e Ref. This work dA F / F Universal function
Fig. 3 | EMC and universal modification function slopes.
The slopes of the EMC effect for different nuclei from Fig.
2a (blue) and of the universal function from Fig.
2b (red). The error bars shown include the fit uncertainties at the 1 σ or 68% confidence level. Figure 12: Left: measured structure function ratio for nuclei relative to deuterium (withoutmodel-dependent iso-scalar corrections). Right: the extracted universal modification functionof nucleons in SRC pairs [11].
To gain further insight to the modification of nucleons bound in SRC pairs, we propose to measurethe variation in the Branching Ratios (BRs) for hard photonuclear reactions off free (/quasifree) vs.deeply bound nucleons in the deuteron, He and C. Changes in the measured BRs, which maydepend on the momentum transfer, scattering angle and nuclear transparency, will shed new lighton the mechanisms of quark-gluon nucleon structure modification in nuclei. The detailed descriptionof this novel observable follows.The proton (or neutron) is a complex system that can be described in QCD at any given momentas a superposition of different Fock states: | proton (cid:105) = α PLC | PLC (cid:105) + α qg | q + g (cid:105) + α qq ¯ q | qq ¯ q (cid:105) + α qπ | qπ (cid:105) + . . . (4)15 able 1: List of possible exclusive photonuclear reactions off protons and neutrons that arewithin the detection capabilities of the GlueX spectrometer. Note that neutron reactions areonly possible using nuclear targets (deuteron and heavier). Proton Reactions Neutron Reactions γ + p → π + p γ + n → π − + pγ + p → π − + ∆ ++ γ + n → π − + ∆ + γ + p → ρ + p γ + n → ρ − + pγ + p → K + + Λ γ + n → K + Λ γ + p → K + + Σ γ + n → K + Σ γ + p → ω + p — γ + p → φ + p —where the different brackets represent states of the proton (or neutron) with the corresponding α representing the amplitude of each state. By definition all weights must sum to 1. The minimalstate of the nucleon includes only the 3 valence quarks and is assumed to be small in size and witha reduced strong interaction. Such a state is referred to as a Point-Like Configuration (PLC). Theother states include more complex configuration involving additional gluons, quark-antiquark pairs,pions etc. These states are all components of the wave function of the nucleon.
The modified structure of a proton (or neutron) bound in a nucleus can then be represented bya different, decomposition into the same Fock states: | proton ∗ (cid:105) = α ∗ PLC | PLC (cid:105) + α ∗ qg | q + g (cid:105) + α ∗ qq ¯ q | qq ¯ q (cid:105) + α ∗ qπ | qπ (cid:105) + . . . (5)where the difference between a free and bound proton is depicted by the difference between the α and α ∗ coefficients in Eqs. 4 and 5. An example of such an effect can be found in the ‘Point LineConfigurations Suppression’ model of Frankfurt and Strikman [67] or the ‘Blob-Like ConfigurationsEnhancement’ model of Frank, Jennings, and Miller [68] that propose a possible explanation to theEMC effect in which the PLC part of the bound nucleon is different than in a free one.We stress that the Fock space description of bound nucleons is somewhat more complex asnucleons bound in nuclei span various states: e.g. mean-field vs. SRC nucleons, high vs. low localdensity etc., allowing the α ∗ coefficients to possibly depend on the detailed nuclear state of thebound nucleon. For example, in the PLC suppression model [67], | α ∗ PLC /α PLC | − γp → π − ∆ ++ ; ρ p ). We propose to usethe unique capability of GlueX to measure simultaneously the BRs of many decay channels of anexcited nucleon following the absorption of a real photon at high momentum transfer (large t). Bymeasuring these BR for nucleons in a range of nuclei from deuterium through lead we will be able tosee differences in the Fock state decomposition, and hence the structure, of bound and free nucleons.For a free proton GlueX will measure the branching ratio (BR) for many reactions, including γp → π − ∆ ++ , ρ p , K + Λ, K + Σ , and others. By measuring these reactions, and the neutron equivalents(listed in Table 1), on deuteron and nuclear targets, we can extract the BRs for scattering off free(/quasi-free) vs. deeply bound nucleons. As each reaction is sensitive to a different combination ofFock states, modifying their contribution to the bound proton will modify their BRs.Current theoretical models do not allow us either to predict the exact change in BRs as a function16f the bound nucleon structure, or to translate the observable BRs to the modified α ∗ coefficients.However, this is a novel observable that allows us to observe or exclude deviations, and to studytheir dependence on the nucleon momentum and ‘hardness’ of the reaction. Any such observationwill therefore serve as clear and direct evidence for changes in bound nucleon structure.On average, differences between a bound and a free nucleon are expected to be small. However,we propose to select specific kinematics, focusing on deeply bound nucleons, that could enhance theeffect: e.g., selecting hard process with large s , t , and u is expected to emphasize the contribution ofthe PLC component. Alternatively, detecting the decay products along with a high-momentum recoilnucleon (which favors scattering from a nucleon from an SRC pair) should also significantly amplifythe medium effect to the level observed for the EMC effect at x B ∼ . . In addition to SRCs and bound nucleon structure, the data to be collected in this experiment will beused to study hadron color transparency (CT). At high energies, the phenomena of CT arises fromthe fact that exclusive processes on a nucleus at high momentum-transfer preferentially select thecolor singlet, small transverse size configuration, which then moves with high momentum throughthe nucleus. The interactions between the small transverse size configuration and other nucleonsare strongly suppressed because the gluon emission amplitudes arising from different quarks cancel.This suppression of the interactions is one of the essential ingredients needed to account for Bjorkenscaling in deep-inelastic scattering at small x B [45].CT at high energies was directly observed in the diffractive dissociation of 500 GeV /c pions intodijets when coherently scattering from carbon and platinum targets. The per-nucleon cross-sectionfor dijet production is parametrized as s = s A a , and the experiment found a = 1 . ± .
08 [69],consistent with CT predictions of a = 1 .
54 [70]. These results confirm the predicted strong increaseof the cross-section with A , and the dependence of the cross-section on the transverse momentum ofeach jet with respect to the beam axis ( k t ) indicates the preferential selection of the small transversesize configurations in the projectile. Such experiments have unambiguously established the presenceof small-size q ¯ q Fock components in light mesons and show that at transverse separations, d ∼ . q ¯ q -dipole-nucleon interactions. Thus, color transparency is wellestablished at high energies and low x B . However, these high-energy experiments do not provideany information about the appropriate energy regime for the onset of CT.At intermediate energies, in addition to the preferential selection of the small-size configuration,the expansion of the interacting small-size configuration is also very important. At these energies,the expansion distance scales are not large enough for the small-size configuration to escape withoutinteraction which, suppresses the color transparency effect [71–74]. The interplay between the selec-tion of the small transverse size and its subsequent expansion determine the scale of the momentumand energy transfers required for the onset of CT. As mentioned, a major difference between photo-induced and electron-induced reactions is that, in the former, much greater energy is transferredrelative to momentum, which can help disentangle the roles of freezing and squeezing.The first attempt to measure the onset of CT at intermediate energies used the large-angle A ( p, p ) reaction at the Brookhaven National Lab (BNL) [75–78]. In these experiments, large-angle17 p and quasielastic ( p, p ) scattering were simultaneously measured in hydrogen and several nucleartargets, at incident proton momenta of 6–12 GeV /c . The nuclear transparency was extracted fromthe ratio of the quasielastic cross-section from a nuclear target to the free pp elastic cross-section.The transparency was found to increase as predicted by CT, but only between 6–9.5 GeV /c ; thetransparency was found to decrease between 9.5 and 14.4 GeV /c . This decrease cannot be explainedby models incorporating CT effects alone. Though not fully understood to date, this behavioris commonly attributed to a lack of understanding of the fundamental two-body reactions, whichlimits one’s ability to relate the s , t scales for the onset of squeezing in different reactions. Thissituation raises doubts about our ability to study CT effects using such proton-induced QE scatteringreactions.In contrast to hadronic probes, weaker electromagnetic probes sample the complete nuclear vol-ume. The fundamental electron-proton scattering cross-section is smoothly varying and is accuratelyknown over a wide kinematic range. Detailed knowledge of the nucleon energy and momentum dis-tributions inside a variety of nuclei have been extracted from extensive measurements in low-energyelectron scattering experiments. Therefore, the ( e, e (cid:48) p ) reaction is simpler to understand than the( p, p ) reaction, an advantage immediately recognized following the BNL ( p, p ) experiments. Anumber of A ( e, e (cid:48) p ) experiments have been carried out over the years, first at SLAC [79, 80] andlater at JLab [81, 82] for a range of light and heavy nuclei. In high Q quasielastic ( e, e (cid:48) p ) scatteringfrom nuclei, the electron scatters preferably from a single proton, which need not be stationary dueto Fermi motion [83]. In the plane wave impulse approximation (PWIA), the proton is ejected with-out final state interactions with the residual A − A ( e, e (cid:48) p ) cross-sectionwould be reduced compared to the PWIA prediction in the presence of final state interactions, wherethe proton can scatter both elastically and inelastically from the surrounding nucleons as it exitsthe nucleus. The deviations from the simple PWIA expectation is used as a measure of the nucleartransparency. In the limit of complete color transparency, the final state interactions would vanishand the nuclear transparency would approach unity. In the conventional nuclear physics picture, oneexpects the nuclear transparency to show the same energy dependence as the energy dependence ofthe N N cross-section. Other effects such as short-range correlations and the density dependence ofthe
N N cross-section will affect the absolute magnitude of the nuclear transparency but have littleinfluence on the energy- (or Q -) dependence of the transparency. Thus, the onset of CT is expectedto be manifested as a rise in the nuclear transparency as a function of increasing Q .The existing world data rule out any onset of CT effects larger than 7% over the Q range of 2.0–8.1 (GeV /c ) with a confidence level of at least 90%. As mentioned earlier, the onset of CT dependsboth on momentum and energy transfers, which affect the squeezing and freezing respectively. Since A ( e, e (cid:48) p ) scattering measurements are carried out at x B = 1 kinematics, they are characterized bylower energy transfers as compared to the momentum transfer (e.g. 4.2 GeV for Q = 8 GeV ).Existing data seem to suggest that a Q of 8 (GeV /c ) with 4.2 GeV energy transfer is not enoughto overcome the expansion of the small transverse size objects selected in the hard ep scatteringprocess (i.e. freezing requirements are not met). A recent Hall C 12 GeV experiment, currentlyunder analysis, will extend these studies to Q ∼
16 GeV [84]. Although, no unambiguous evidencefor CT has been observed so far for nucleons from either A ( e, e (cid:48) p ) or A ( p, p ) reactions, it is expectedto be more probable to reach the CT regime at low energy for the interaction/production of mesonsthan for baryons, since only two quarks must come close together and a since a quark-antiquarkpair is more likely to form a small size object [85]. Indeed, pion production measurements at JLabreported evidence for the onset of CT [86] in the process e + A → e + p + A ∗ . The pion-nucleartransparency was calculated as the ratio of pion electroproduction cross-section from the nucleartarget to that from the deuteron. As proposed here, the use of the deuteron instead of the protonhelped reduce the uncertainty due to the unknown elementary pion electroproduction cross-sectionoff a free neutron and to uncertainties in the Fermi smearing corrections. The measured pion nucleartransparency shows a steady rise with increasing pion momentum for the A > p π is consistent with the rise in transparency predicted by various CT18alculations [87–89]. Although, all the calculations use an effective interaction based on the quantumdiffusion model [71] to incorporate the CT effect, the underlying conventional nuclear physics iscalculated very differently. The results of the pion electroproduction experiment demonstrate thatboth the energy and A dependence of the nuclear transparency show a significant deviation fromthe expectations of conventional nuclear physics and are consistent with calculations that includeCT. The results indicate that the energy scale for the onset of CT in mesons is ∼ q ¯ q ) pairs under well-controlled kinematical conditions.Soon after the observation of the onset of CT in pion electroproduction, results from a study of ρ -meson production from nuclei at JLab also indicated an early onset of CT in mesons [90]. Previous ρ production experiments had shown that nuclear transparency also depends on the coherence length, l c , which is the length scale over which the q ¯ q states of mass M q ¯ q can propagate. Therefore, tounambiguously identify the CT signal, one should keep l c fixed while measuring the Q dependenceof the nuclear transparency. The CLAS collaboration at JLab measured the nuclear transparencyfor incoherent exclusive ρ electroproduction off carbon and iron relative to deuterium [90] using a5 GeV electron beam. An increase of the transparency with Q for both C and Fe, was observedindicating the onset of CT phenomenon. The rise in transparency was found to be consistent withpredictions of CT by models [91,92] which had successfully described the increase in transparency forpion electroproduction. Therefore, the π and ρ electroproduction data also demonstrate an onset ofCT in the few GeV energy range as shown in Figure 13. Both of these experiments will be extendedto higher energies in future 12 GeV experiments [84, 93].In the case of large momentum transfer exclusive photoinduced reactions, while the predictedeffects are larger (Fig. ?? ) they were not studied in much detail. JLab experiment E94-104 searchedfor CT using the reaction γ + n → π − + p [94]. The experiment used an untagged mixed electron andphoton bremsstrahlung beam incident on a He target and the Hall-A high-resolution spectrometersto measure π − and p produced in the reaction. The momentum transfer was reconstructed assumingscattering off a mean-field neutron in He leaving the residual system in the ground state of He.Nuclear transparency was measured as a ratio of the pion photoproduction cross-section from Heto that of H. Figure 14 shows the extracted transparency as a function of the momentum transfer, | t | , for center-of-mass scattering angles of 70 ◦ and 90 ◦ . The results were compared to Glaubercalculations with and without CT effects. As can be seen, the measurement did not have therequired statistical and systematical accuracy to discriminate between the two calculations over themeasured | t | range. We propose, using the advantages of the GlueX spectrometer and the upgradedCEBAF 12 GeV electron beam, to add many more reaction channels, extend the measured | t | rangeup to 10–12 GeV , and add heavier nuclei. While we will keep comparable uncertainties, for heaviernuclei and larger momentum transfers the expected effects are considerably larger, which significantlyincreases the discovery potential.It should be pointed out that the rate of expansion/contraction of configurations involved in theinteraction with nucleons is the same for the different reactions. Hence, in light of the successfuldescription of CT for mesons, reliable estimate of space-time evolution effects were performed forother reactions with the conclusion that in the proposed kinematics for GlueX, CT is not washedout by the expansion/contraction effects due to the high photon energies used in the experiment. The kinematical distributions and expected event rates were simulated for the pion-proton photo-production reaction off a neutron bound in a nucleus, A ( γ, π − p ), using a dedicated Monte-Carlo eventgenerator. In this section, we present the simulation method and show the resulting kinematicaldistributions. 19 (GeV Q2 4 D T C ) (GeV Q2 4 T Al ) (GeV Q2 4 D T Cu ) (GeV Q2 40.20.30.4 Au Q (GeV ) N u c l e a r T r a n s p a re n c y C Fe GKM ModelGKM Model (CT) Q (GeV ) FMS Model (CT)FMS Model
Figure 13: The two JLab experiments which show conclusive evidence for the onset of CTin meson electroproduction at intermediate energies. (left panel)
Nuclear transparency vs Q for C, Al, Cu and
Au in the ( e, e (cid:48) π + ) reaction. The inner error bars are thestatistical uncertainties and the outer error bars are the statistical and point-to-point sys-tematic uncertainties added in quadrature. The solid circles (blue) are the high- (cid:15) (virtualphoton polarization) points, while the solid squares (red) are the low- (cid:15) points. The dashedand solid lines (red) are Glauber calculations from [87], with and without CT, respectively.Similarly, the dot-short dash and dot-long dash lines (blue) are Glauber calculations with andwithout CT from [88]. The dotted and dot-dot-dashed lines (green) are microscopic+ BUUtransport calculations from [89], with and without CT, respectively. (right panel) Nucleartransparency as a function of Q in the ( e, e (cid:48) ρ ) reaction. The curves are predictions of theFMS [91] (red) and GKM [92] (green) models with (dashed-dotted and dashed curves, respec-tively) and without (dotted and solid curves, respectively) CT. Both models include the pionabsorption effect when the ρ meson decays inside the nucleus. The inner error bars are thestatistical uncertainties and the outer ones are the statistical and the point-to-point systematicuncertainties added in quadrature. igure 14: The measured transparency for the reaction He( γ, π − p ) for two c.m. scatteringangles (70 ◦ -right, 90 ◦ -left). The measurements are compared to Glauber calculations withand without CT. See Ref. [91] for details. The simulation uses an incoming photon with energy sampled from the tagged photon spectraobtained from the standard GlueX simulation software (Fig. 15). The momentum distribution ofthe nucleons in the nucleus has two components: a mean-field region that spans low momentum (upto k F ) and account for 80% of the nucleons and an SRC region that spans high momentum (from k F and up) and account for 20% of the nucleons. The SRC-pairs are modeled using a three-dimensionalGaussian center of mass momentum distribution with width (sigma) of 140 MeV /c [15, 16, 23, 53, 54]In the case of the deuteron, the AV18 momentum distribution was used.The cross-section for the γ + n → π − + p reaction was calculated based on the experimental datafor 90 ◦ scattering in the c.m. with s > .
25 GeV assuming factorization of the s and c.m. angledependence, i.e., dσdt | θ c.m. = ( C × s − ) × f ( θ c.m. ), where C is a free fit parameter and f was extractedfrom the SLAC data assuming f (90 ◦ ) = 1 [95], see Fig. 16.The scattering was performed in the c.m. frame of the bound nucleon and gamma beam forscattering angles of 40 ◦ –140 ◦ . Hard reaction kinematics where enforced by requiring | t, u | > .We note that the rate for lower momentum transfers is very high and within the GlueX acceptance.Figures 17 and 18 show the kinematical distributions for the final state particles respectively forinteractions of the gamma with a mean-field nucleon and an SRC nucleon. For the case of SRCpair breakup, the distribution of the correlated recoil proton is shown in Fig. 19. As mentioned,the backward peak of the recoil proton is due to the s − weighting of the cross-section that prefersinteractions with forward going nucleons which, in the case of SRCs, enforce the recoil nucleon tobe emitted in a backward direction. We note that the resulting kinematical distributions are notvery different from those obtained for scattering off stationary nucleons, which is what GlueX wasdesigned to do.The simulation results were compared to a simple, back-of-the-envelope calculation for the reac-tion cross-section. This calculation is explained in Appendix A. The back-of-the-envelope result iswithin 20% of the simulation, giving us confidence in the validity of our simulation.21 igure 15: The energy distribution for the incoming photon beam hitting the GlueX targetassuming a 5 mm diameter collimator. The distribution is normalized to a flux of 2 . × photons/s in the beam energy range 7.5 GeV < E beam < s dependence of the photonuclear cross-section at 90 ◦ in the c.m. (left) and itsdependence on the c.m. angle (right). We extract the cross-section by fitting the s dependenceat high- s , after the low- s oscillations appear to be over. Figures were adapted from [95]. igure 17: Kinematical distributions for the final state particles of the γ + n → π − + p reactionfor the MF regime ( P miss < .
25 GeV /c ). igure 18: Same as Fig. 17 for the SRC regime ( P miss > .
25 GeV /c and q recoil < ◦ ).Figure 19: The angular distribution of the recoil nucleon when scattering off an SRC pair inthe nucleus. .2 Optimization of the Tagged Gamma Energy Range The Hall-D beam allows for a broad distribution of tagged photons on the target (Fig. 15). Due tothe coincidental rate limitation of the Hall-D tagger we cannot consider the full gamma spectrumand should focus on a given energy range. Fig. 20 shows the correlation of | t | and the beam-energy.As we are largely interested in large | t | reactions, we choose to focus the tagger at the coherent peak,with gamma energies of 8 GeV < E beam < Figure 20: The momentum transfer | t | as a function of the photon beam energies for tworegimes of event selection: MF on the left and SRC on the right. The center of mass angle isin the range between 50 ◦ and 130 ◦ . The number of photons in the coherent peak is regulated by the size of the collimator: a widercollimator ensures more coherent photons to hit the target. A collimator diameter of 5 mm wasfound to be optimal, as it allows measuring all the coherent photons with minimal ‘background’from low energy photons. Smaller collimator will reduce the high-energy gamma flux and largercollimator will increase the low energy background (leading to larger EM and neutron backgrounds)without improving the high-energy gamma flux.The factors limiting the beam luminosity are the coincidental rate in the tagger and the electro-magnetic background level in the GlueX spectrometer. The tagger coincidental rate for a photonflux on the target of 2 × photons/s and RF time of 4 ns is expected to be about 18%.To be conservative, the rate calculations presented below (and kinematical distribution presentedabove) are done for photon beam energies in the 8 GeV < E beam < The efficiencies for the reconstruction of final state particles (i.e. meson-baryon pair and, in the caseof SRC breakup, also the recoil proton) were simulated using the Geant-based GlueX simulationchain for the event generator described in the previous section. Fig. 21 shows the simulated detectionefficiency for each particle separately. The average efficiency for the simultaneous reconstruction ofthe proton and a pion was found to equal 64%.Based on the current GlueX data reconstruction efficiencies, we expect that more complex finalstates will have varying detection efficiencies reaching down to 30% for rho mesons. The totaldetection efficiency for the reaction γ + n → ρ − + p is therefore assumed to be 0 . × . ◦ in25 igure 21: The reconstruction efficiency for the charged tracks coming from the reaction γn → π − p . On the left panel the low momentum band is due to recoil protons from SRC pairbreakup. the lab and can extend up to 160 ◦ when the target is placed downstream. Fig. 23 shows the vertexreconstruction resolution, showing we can separate solid target foils with a distance of ∼ C. The rate calculations were done for the γ + n → π − + p and γ + n → ρ − + p reactions, usingthe simulation presented in section 5.1. We choose these two reactions as they have the smallestand largest cross-sections respectively, of the reactions listed in table 1 which makes them a goodrepresentative of the various expected rates.We assume a total of 30 beam days with a photon flux of 2 × s − (compared to the nominalGlueX photon flux of 10 photons/s) and four targets: D, He, and C. Based on the acceptancesimulations presented above, we assume 80% detection efficiency for each of the leading baryon andmeson and 65% for the recoil nucleon. We assume the nominal nuclear attenuation effect reducesthe total cross-section as A − / . Table 2 lists the parameters for the chosen targets. The factorslimiting the event rates are the following: • GlueX detector capabilities (maximum possible gamma flux on target) • Electromagnetic background in the GlueX spectrometer • Tagger coincidental rate • Neutron backgroundPresently, GlueX is operating with a 30 cm long liquid hydrogen (LH) target (3.4% radiation length, X ). For the nominal beam flux on the target of 10 photons/s the electromagnetic backgroundis reaching its upper limit. In order to comply with the electromagnetic background limits, weassume the carbon target thickness to be 7% X (note that unlike for hydrogen, for nuclei there are2 nucleons for each electron in the target. Therefore, for EM background estimations, 7% X onnuclei is equivalent to 3 . X on hydrogen). The radiation lengths for liquid hydrogen, deuterium,and helium are similar. The use of nuclear targets will increase the slow neutron background that caninduce some damage to GlueX detector components such as the SiPMs. Table 2 shows an estimateof the neutron background, done in collaboration with Hall D staff, based on JLAB-TN-11-005.GlueX was designed to handle one year of LH running with a photon flux of 10 photons/s. Whilethe proposed gamma flux for this measurement is smaller by a factor of 5, detailed estimations by26 igure 22: The detection efficiency for recoiling protons in GlueX as a function of the recoilangle and momentum for 3 different vertex locations.Figure 23: The vertex reconstruction resolution for recoiling protons at various recoil angles. able 2: Parameters for the proposed targets. The current GlueX liquid hydrogen target (LH)is shown for comparison. (*) The neutron flux for the LH target is taken under assumptionof the nominal flux of 10 photons/s in the coherent peak. Target Thickness[cm] / % X Atoms/cm for the given targetthickness EM bkg. rel.to GlueX Neutron bkg.rel. to GlueXD 30 / 4.1 1 . × He 30 / 4 5 . × C 1.9 / 7 1 . × . × Table 3: Event rates estimation. See text for details.
Target γ + n → π − p γ + n → ρ − p PACDaysMF SRC MF SRCD 13,600 750 57,000 3,000 5 He 16,000 840 68,000 3,500 10 C 8,900 2,800 37,000 11,000 12Calibration, commissioning, and overhead: 3
Total PAC Days: Hewill increase the neutron background by a factor of 4–5 (depending on the exact location). Thisimplies that the neutron background for He target in our running conditions will be similar to theGlueX design specifications. The estimated neutron backgrounds for all targets are shown in table 2,and are calculated from the Radcon He estimate and their reported A -dependence. For reference,the table includes the GlueX LH target under nominal running conditions, i.e. 10 photons/s.The background rates for the other targets take into account the five-fold reduced gamma flux forthe proposed experiment. While this background estimation procedure takes into account the maindifferences coming from nuclear targets relative to LH, the deuteron backgrounds could be somewhathigher. Given the very short deuteron beam time (table 3) this should not be an issue with regardto integrated damage.Table 3 lists the expected events rate and beam time for each target for | t, u | > for mean-field and SRC events separately. For Deuterium, as we use the AV18 distribution, the distinctionis based on the initial momentum of the nucleon that the gamma interacted with (above or below250 MeV /c ). Figure 24 shows the expected count rate of various | t | bins for Deuterium and C formean-field events. The statistics for | t | < is rich, allowing to map the transition betweendifferent transparency regimes. Other nuclei have the same | t | dependence and the expected countrate per bin can be scaled based on the total number of events listed in Table 3. We have chosento distribute the beam time between the different targets so as to obtain comparable discriminatingpower for transparency studies and scaling of SRC pairs; the larger nuclei have larger predictedeffects and therefore fewer statistics are needed to observe them at similar levels of significance.Fig. 25 shows the expected results for the color transparency for the γ + n → π − + p reaction.Other reactions from table 1 will have comparable or better discriminating power. We note thatby taking ratios for nuclei relative to deuterium we minimize many of the theoretical systematical28 igure 24: The expected count rate for 10 days running as a function of | t | for Deuterium(left) and C (right) targets in mean-field kinematics for two different reactions. He Glauber + CTGlauber T -t (GeV ) He Glauber + CTGlauber T Θ c.m. (deg) C Glauber + CTGlauber T -t (GeV ) C Glauber + CTGlauber T Θ c.m. (deg) Figure 25: Expected uncertainties (statistical + systematical) for the measurement of the γ + n → π − + p reaction off He (upper row), and C (lower row). p miss [GeV /c ] pp p a i r s / np p a i r s GCF+AV18CLAS-6This proposal C( γ , ρ pp)/C( γ , ρ + pp) Figure 26: The expected precision of this proposal for testing np -dominance via the ρ pro-duction channel, compared with electron scattering data from CLAS-6. For SRC studies, while the rates are modest, they are in fact comparable and even higher perreaction as compared to the 6 GeV measurements done in Hall A and B. Therefore, the cross-section ratios for scattering off nuclei relative to deuterium, in SRC kinematics, will allow us to testthe observed np -dominance and extract the relative number of SRC pairs in the measured nucleiwith < ρ -production, the highest-rate channel, compared with the preliminaryanalysis of CLAS data (shown previously in Fig. 3). The precision will be sufficient to confirm both np -dominance and explore the transition to the repulsive core. These data will therefore improve ourunderstanding of SRCs and help to reduce interpretation uncertainty in a unique way un-matchedby any other measurement that can be performed at JLab. The goals of our proposed measurement run complementary to those of several other approved JLab12 GeV experiments, which will be described in this section.Semi-inclusive and exclusive measurements of short-range correlations are the focus of severalrecent and upcoming experiments, which are complimentary to our proposed SRC program. TheHall A experiment E12-14-011 [96] ran in 2018, measuring asymmetries between the mirror nuclei H and He to study the isospin-dependent effects in SRCs [21], along with an inclusive counterpart,E12-11-112 [97]. The Hall C experiment E12-17-005 [98] will also look for isospin-dependent effectsthrough measurements on Ca, Ca, and Fe. The Hall B run group proposal E12-17-006 includesmeasurements of short range correlations in a wide range of targets [99].The interpretation of the above experiments depends on the general framework for our under-standing of SRCs, FSIs, and reaction mechanisms in electron-induced pair break-up, which ourproposed measurement will attempt to validate with the new method of photo-induced reactions.Understanding of the modification of nucleon structure within the nuclear medium is a salienttopic in nuclear physics under active investigation in several JLab experiments. Our proposal focuses30n a completely novel observable, branching ratio modification, which would complement traditional“EMC” electron-scattering measurements. Among the upcoming electron scattering measurements,we highlight E12-11-003A [100] and E12-11-107 [101], which will test the role that highly-virtualnucleons play in the EMC effect by looking at recoil-tagged F structure functions of bound nucleonsin deuterium.Color transparency for mesons, hints of which were seen in the JLab 6 GeV program [86], will bestudied at high- Q in two 12 GeV electron scattering experiments: E12-06-106 [93] (upcoming) andE12-06-107 [84] (under analysis). While these experiments are complementary to our proposed CTmeasurement, we point out that our measurement is sensitive to CT for baryons as well. The largenumber of different final states we consider will allow us to map out the spin and isospin dependenceof CT.E12-06-107 will also measure color transparency in for protons, which complements our baryonicCT measurements. A significant difference between this measurement and ours comes from theuse of photon-induced reactions, which will generally have larger energy transfer (and thus greater“freezing” of PLCs) than the majority of the kinematic space probed by E12-06-107.As discussed in section 4, the theoretical framework of PLCs suggests that both color trans-parency and the EMC effect have a common origin. Both these experiments and our proposedmeasurement will have complimentary roles in constraining that framework and in understandingthe origin of the EMC effect. We propose a 30-day measurement using the real photon beam in Hall D, d , He and C targets,with the GlueX detector in its standard configuration, with the goal of studying short-range correla-tions, transparency and bound nucleon structure in nuclei. The use of a real photon beam as a probeprovides an outstanding handle on reaction mechanism effects in SRC pair breakup, which comple-ment the successful electron-scattering SRC program in Halls A and B. We project count rates thatexceed those of the 6 GeV-era SRC experiments, allowing definitive measurements of SRC properties,the short-distance
N N interaction, in-medium modification, and nuclear transparency.
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To validate our simulation, we performed a back-of-the-envelope calculation of the rate of the γ + n → π − + p reaction. The differential cross-section for the reaction can be approximated using thefollowing: dσdt = C · f ( s ) · f (cos θ cm ) ≈ . · nb · GeV · s − · (1 − cos θ cm ) − (1 + cos θ cm ) − , (6)where the cross-section is in units of nb/GeV , and θ cm is the polar scattering angle in the centerof mass system.The polar scattering angle in the center of mass system θ cm can be approximated ascos θ cm = t − m π + 2 k i (cid:113) k f + m π k i k f , (7)where k i , and k f are the center of mass momenta of incoming and outgoing particles: k i = s − m p √ s (8) k f = (cid:114) ( s − ( m p − m π ) ) · ( s − ( m p + m π ) )4 s . (9)Equation (7) shows that for θ cm > ◦ , | t | > . Mandelstam variables are interrelatedas s + t + u = m , which means that for E γ = 9 GeV we have | t | > and | u | > for almost the whole range of θ cm between 40 ◦ and 140 ◦ .The cross-section (6) in terms of cos θ cm : dσ = C · s − · f (cos θ cm ) dt = C · s − · f (cos θ cm ) d cos θ cm , (10)where C = 1 . · · k i k f .The total cross-section for E γ = 9 GeV and θ cm in the range between 40 ◦ and 140 ◦ is then about2 . N = σ nucl · F · T · t · (cid:15) = 880 , (11)where F = 2 · photons/s - photon flux on target, T = 1 . · atoms/cm - target density for C, t = 24 hours · (cid:15) = 0 .
64 - detection efficiency, and σ nucl = σ · A/ · A − / = 5 . C. This is consistent within 20% with the simulation results (740events/day for the12