We sketch an approximate method to quantify the number of correlated pairs in
any nucleus $A$. It is based on counting independent-particle model (IPM)
nucleon-nucleon pairs in a relative $S$-state with no radial excitation. We
show that IPM pairs with those quantum numbers are most prone to short-range
correlations and are at the origin of the high-momentum tail of the nuclear
momentum distributions. Our method allows to compute the $a_2$ ratios extracted
from inclusive electron scattering. Furthermore, our results reproduce the
observed linear correlation between the number of correlated pairs and the
magnitude of the EMC effect. We show that the width of the pair center-of-mass
distribution in exclusive two-nucleon knockout yields information on the
quantum numbers of the pairs.Comment: 4 pages, 2 figures, 1 table. Based on a talk given at INPC 2013
  (Firenze, June 2-7 2013). Minor changes in the text. Accepted for publication
  in EPJ Web of conference

Cosyn, Wim

Ryckebusch, Jan

Vanhalst, Maarten

EPJ Web of Conferences

English

arXiv

We sketch an approximate method to quantify the number of correlated pairs in any nucleus A. It is based on counting independent-particle model (IPM) nucleon-nucleon pairs in a relative S-state with no radial excitation. We show that IPM pairs with those quantum numbers are most prone to short-range correlations and are at the origin of the high-momentum tail of the nuclear momentum distributions. Our method allows to compute the a2 ratios extracted from inclusive electron scattering. Furthermore, our results reproduce the observed linear correlation between the number of correlated pairs and the magnitude of the EMC effect. We show that the width of the pair center-ofmass distribution in exclusive two-nucleon knockout yields information on the quantum numbers of the pairs

Ghent University Academic Bibliography

Mass dependence of short- range correlations in nuclei and theEMC effectWim Cosyn1,a, Maarten Vanhalst1, and Jan Ryckebusch11Department of Physics and Astronomy,Ghent University, Proeftuinstraat 86, B-9000 Gent, BelgiumAbstract. We sketch an approximate method to quantify the number of correlated pairsin any nucleus A. It is based on counting independent-particle model (IPM) nucleon-nucleon pairs in a relative S -state with no radial excitation. We show that IPM pairs withthose quantum numbers are most prone to short-range correlations and are at the originof the high-momentum tail of the nuclear momentum distributions. Our method allowsto compute the a2 ratios extracted from inclusive electron scattering. Furthermore, ourresults reproduce the observed linear correlation between the number of correlated pairsand the magnitude of the EMC effect. We show that the width of the pair center-of-mass distribution in exclusive two-nucleon knockout yields information on the quantumnumbers of the pairs.One of the striking features of the nucleon-nucleon potential is the appearance of a huge repulsivecore in the potential for inter-nucleon distances of . 1 fm. This is a reflection of the finite sizeof the nucleon. The hard core introduces short-range correlations (SRC) in the nuclear quantummany-body system. These cause high-density and high-momentum fluctuations that are reflected inthe fat tails of the one-body momentum distribution. Recent theoretical and experimental effortshave identified proton-neutron pairs as the dominant contribution to SRC, an effect which can beattributed to the tensor part of the nuclear force (for recent reviews see [1, 2]). Somewhat surprisingly,a very nice linear correlation was also observed between the magnitude of the EMC (European MuonCollaboration) effect for a nucleus A and the ratios of inclusive electron scattering from nucleus Arelative to the deuteron (taken as a measure for the amount of correlated pairs in a nucleus) [3, 4]. Thismight indicate that both phenomena share a common origin, being caused by high-density and/or high-virtuality fluctuations in the nucleus. Here, high-virtuality refers to high-momentum bound nucleons,consequently far off the mass shell.In Refs. [5–7], we have introduced an approximate method to quantify the number of correlatedpairs in any nucleus. We start from the well-known approach where a symmetrical correlation operatorĜ is used to transform a Slater determinant of independent particle model (IPM) single-particle wavefunctions ΨIPMA into a realistic wave function ΨA [8]:| ΨA〉 = 1√〈 ΨIPMA | Ĝ†Ĝ | ΨIPMA 〉Ĝ | ΨIPMA 〉 . (1)ae-mail: wim.cosyn@ugent.beDOI: 10.1051/C© Owned by the authors, published by EDP Sciences, 2014,/02022 (2014)20166epjconfEPJ Web of Conferences46 6020 22This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20146602022The operator Ĝ is complicated but as far as the SRC are concerned, it is dominated by the central,tensor and spin-isospin correlations [9, 10]. We then exploit the short-ranged nature of the dominantcontributions in the following way. We start from a harmonic oscillator (HO) basis for the Slater deter-minant ΨIPMA . This does not impose restrictions on the choice of IPM model as any wave function canbe expanded in the HO basis. Next, for the two-particle wave function, we perform a transformationto relative and center-of-mass (c.o.m.) coordinates using Talmi-Moshinsky brackets [11]. This allowsus to identify the contributions of all relative (nl) and c.o.m. (NL) HO quantum numbers to the two-particle wave functions [5, 6]. As the typical central and tensor correlation operators in Eq. (1) projecton pairs with a substantial close-proximity (r ≈ 0) probability, we identify the IPM (n = 0, l = 0)nucleon pairs as those parts of ΨIPMA prone to SRC. The validity of this assumption is nicely illustratedin Fig. 1, where we show the different (n, l) contributions to the two-body momentum distribution in56Fe as computed in a lowest order cluster approximation of Eq. (1). One clearly observes that theSRC generated high-momentum tail in the momentum distribution is dominated by the contributionfrom correlation operators acting on IPM pairs with (n = 0, l = 0) relative quantum numbers. Wewish to stress that the high-momentum tails in the momentum distributions can have very differentquantum numbers than (n = 0, l = 0) through the operation of the correlation operator G. The mostobvious example is the dynamical generation of the deuteron D-wave (l = 2) as the tensor part of Ĝacts on the “IPM” S-wave (l = 0) component of the wave function.10−210−11001011021031040 0.1 0.2 0.3 0.4 0.5 0.6 0.7n2n+l2(k12)[GeV−3]k12 [GeV]2n+l = 02n+l = 12n+l = 22n+l = 32n+l = 42n+l = 5allIPMFigure 1: (Color online) Momentum dependence of the relative two-body momentum distribution n2(solid black curve) in function of the relative momentum k12 of the two nucleons, as computed in thelowest-order cluster expansion for 56Fe. The black dashed curve is the IPM prediction. The otherlines show the different contributions n2n+l2 to n2 from the correlation operators acting on two-nucleonstates with various nl.The mass dependence of nuclear SRC has experimentally been quantified in inclusive electronscattering (for an overview of the world data see [12]). The ratio of the inclusive cross section atmoderate Q2 on a nucleus A to the deuteron shows a scaling plateau at values of Bjorken xB of1.5 . xB . 2. The value of the measured ratio rescaled by 2A (a quantity denoted by a2(A)) can beinterpreted as a measure of the amount of correlated pairs in A relative to the deuteron. We compare inFig. 2a the experimental values of these a2 in superratios to 4He with our calculations. The kinematicsin the inclusive experiments probe initial nucleon momenta in the range of 300-500 MeV, where thetensor correlation function forms the dominant contribution to the high-momentum tail. Thereby, inour calculations we compute the a2(A) by counting the close-proximity (n = 0, l = 0) np pairs in aEPJ Web of Conferences02022-p.2spin triplet (S = 1) state. One can observe that the predictions are in acceptable agreement with theslope and magnitude of the measured A-dependence data for light and medium nuclei, but tend tooverestimate the measured values at high A. No corrections accounting for the c.o.m. motion or final-state interactions of the pairs in the nucleus A were applied to the calculations in Fig. 2a, providing apossible explanation for this overestimation. It is also obvious that the scaling with A of a2(A) is a lotsofter than one would expect from the naive counting of all np pairs, also depicted in Fig. 2a. 1 10 100 10  100a2(A)/ a2( 4He) mass number An=0,l=0all n and all lratio of JLab data(a) 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 2  3  4  5  6  7  8  9  10-d R/ dx B(2/A) (Npn + Npp + Nnn)4He9Be12C27Al40Ca56Fe108Ag197Au(b)Figure 2: (Color online) (a) Mass dependence of the a2 relative to 4He. The black crosses are thepredictions given that all relative quantum numbers equally contribute. For the red squares only pnpairs in a triplet spin state and with n = 0, l = 0 are counted. The data are from Ref. [12]. The curvesare guide to the eyes. (b) The magnitude of the EMC effect versus the computed number of correlatedNN pairs per nucleon. The EMC data are from the analysis presented in Ref. [12].Given the experimentally established linear correlation between the size of the EMC effect andthe amount of correlated pairs in a nucleus, we want to check if this correlation is also present whenwe compare our model calculations to EMC data. In Fig. 2b, we display the magnitude of the EMCeffect - quantified by means of the slope −dR/dxB of the EMC ratio for Bjorken 0.35 < xB < 0.7 [13]- versus our predictions for the probability for NN SRC per nucleon relative to the deuteron. Here,contrary to the a2 ratios in Fig. 2a, we compare to the combined number of pp, np and nn pairs. Thisis motivated by the partonic nature of the deep-inelastic scattering process (all pairs can contributeequally), whereas the inclusive scattering is dominated by the tensor-correlated triplet-spin proton-neutron pairs. We stress that the numbers which one finds on the x axis of Fig. 2a are the results ofparameter-free calculations. A nice linear relationship can be observed between the quantity whichwe propose as a per nucleon measure for the magnitude of the SRC and the magnitude of the EMCeffect. If the observed trends apply to the whole nuclear mass range, we could predict the size of theEMC effect for any nucleus A.So far, in Fig. 2, we have compared our model calculations to quantities extracted from inclusiveelectron scattering experiments. To study the precise nature of nuclear SRC, the exclusive A(e, e′NN)reaction provides more opportunities than inclusive measurements, but is of course more demand-ing to measure. Similarly to the scaling with the distorted one-body momentum distribution for thequasi-elastic A(e, e′N) cross section, it can be shown that under kinematic conditions probing corre-lated nucleon-nucleon pairs, the A(e, e′NN) cross section scales with the distorted c.o.m. momentumdistribution of close-proximity pairs. When comparing the width of the computed c.o.m. momentumdistribution restricted to the (n = 0, l = 0) pairs, with the width of the distribution for all pairs, oneINPC 201302022-p.3Table 1: The width of the 12C c.m. distribution for pp pairs with relative orbital momentum l.l = 0 l = 1 l = 2 all lσ [MeV] 154 135 121 140observes a significant difference between the two. The numbers for the widths of the c.o.m. distri-butions for several relative l quantum numbers in 12C are listed in Table 1. The width of the c.o.m.distribution for close-proximity l = 0 pairs is 14 MeV wider than the one for the distribution whichdoes not impose restrictions on the l.To conclude, we have shown that the amount of correlated pairs in any nucleus can be approxi-mately quantified by counting the number of relative (n = 0, l = 0) pairs in the IPM wave function.Our calculations for the a2 ratios are consistent with the data for light and medium nuclei, but tendto overestimate these at high A. We observe a nice correlation between the computed number ofcorrelated pairs and the measured magnitude of the EMC effect. We have shown that the width ofthe c.o.m. momentum distribution differs significantly between all possible and correlated nucleon-nucleon pairs, the latter being 14 MeV wider in 12C. This is a quantity that can be accessed in exclusiveA(e, e′NN) reactions at moderate Q2.AcknowledgementsThis work is supported by the Research Foundation Flanders (FWO-Flanders) and by the Interuniversity At-traction Poles Programme initiated by the Belgian Science Policy Office. The computational resources (StevinSupercomputer Infrastructure) and services used in this work were provided by Ghent University, the HerculesFoundation and the Flemish Government – department EWI.References[1] J. Arrington, D. Higinbotham, G. Rosner, M. Sargsian, Prog. Part. Nucl. Phys. 67, 898 (2012)[2] L. Frankfurt, M. Sargsian, M. Strikman, Int. J. Mod. Phys. A23, 2991 (2008)[3] L.B. Weinstein, E. Piasetzky, D.W. Higinbotham, J. Gomez, O. Hen, R. Shneor, Phys. Rev. Lett.106, 052301 (2011)[4] O. Hen, D. Higinbotham, G. Miller, E. Piasetzky, L. Weinstein, Int. J. Mod. Phys. E. 22, 1330017(2013)[5] M. Vanhalst, W. Cosyn, J. Ryckebusch, Phys. Rev. C84, 031302 (2011)[6] M. Vanhalst, J. Ryckebusch, W. Cosyn, Phys.Rev. C86, 044619 (2012)[7] M. Vanhalst, J. Ryckebusch, W. Cosyn (2012), 1210.6175[8] S.C. Pieper, R.B. Wiringa, V. Pandharipande, Phys. Rev. C46, 1741 (1992)[9] S. Janssen, J. Ryckebusch, W. Van Nespen, D. Debruyne, Nucl. Phys. A 672, 285 (2000)[10] J. Ryckebusch et al., Nucl. Phys. A 624, 581 (1997)[11] M. Moshinsky, Y. Smirnov, The harmonic oscillator in modern physics (Harwood AcademicPublishers,Amsterdam, 1996)[12] O. Hen, E. Piasetzky, L. Weinstein, Phys. Rev. C85, 047301 (2012)[13] J. Seely, A. Daniel, D. Gaskell, J. Arrington, N. Fomin et al., Phys.Rev.Lett. 103, 202301 (2009)EPJ Web of Conferences02022-p.4

Mass dependence of nuclear short- range correlations and the EMC effect

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