A Monte Carlo generator of nucleon configurations in complex nuclei including Nucleon-Nucleon correlations
aa r X i v : . [ nu c l - t h ] S e p A Monte Carlo generator of nucleon configurations in complex nuclei includingNucleon-Nucleon correlations
M. Alvioli a , H. J. Drescher b , M. Strikman a a
104 Davey Lab, The Pennsylvania State University, University Park, PA 16803, USA b Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universit¨at Routh-Moufang-Str. 1, 60438 Frankfurt am Main,Germany
Abstract
We developed a Monte Carlo event generator for production of nucleon configurations in complex nuclei consistentlyincluding effects of Nucleon-Nucleon (NN) correlations. Our approach is based on the Metropolis search for configurationssatisfying essential constraints imposed by short- and long-range NN correlations, guided by the findings of realisticcalculations of one- and two-body densities for medium-heavy nuclei. The produced event generator can be used forMonte Carlo (MC) studies of pA and AA collisions. We perform several tests of consistency of the code and comparisonwith previous models, in the case of high energy proton-nucleus scattering on an event-by-event basis, using nucleusconfigurations produced by our code and Glauber multiple scattering theory both for the uncorrelated and the correlatedconfigurations; fluctuations of the average number of collisions are shown to be affected considerably by the introductionof NN correlations in the target nucleus. We also use the generator to estimate maximal possible gluon nuclear shadowingin a simple geometric model.
1. Introduction
The structure of nuclei has been described for a longtime by independent particle models in which the nu-cleus is treated as a collection of fermions freely movingabout, this picture being suggested by the fact that nu-clear matter is a dilute system. Nevertheless the shortrange structure of nuclei cannot be accurately described bythis simplified picture, and essential aspects of nuclei suchas high momentum component of the nuclear wave func-tion are completely missed by the non-interacting model;configurations of nucleons at short separations are knownas short range correlations (SRC)[1, 2]. NN correlationshave been recently unambiguously observed[3] in a seriesof dedicated experiments, in which high-momentum nu-cleons were knocked out by high energy probes and theircorrelated partner were observed with opposite momen-tum; the probability for a nucleon to belong to a SRC pairwas measured to be about 20% in C and 25% in in heavynuclei; see Ref.[4] for a review. A theoretical descriptionof SRC can be made in a satisfactory way using nonrela-tivistic Hamiltonians containing two- and three-body real-istic potentials. It is currently possible to solve exactly theSchr¨odinger equation for light nuclei while, for A >
12, ap-proximate methods must be considered, which have beenshown to produce a reasonable approximation to the basicground state quantities such as one- and two-body densi-ties and momentum distributions. These studies demon-strated that NN correlations produce non-negligible ef-fects in a variety of processes including those which arenot specially probing SRC. For example, significant ef- fects are found for high energy total nucleon-nucleus crosssections[5], and it has thus been shown that they must betaken into account in the theoretical calculation of suchprocesses. In this Letter we are addressing the challengingproblem of including NN correlations in the descriptionof high energy proton-nucleus and nucleus-nucleus colli-sions. It was shown in Ref.[6] that using correlated two-body densities for the analysis of fluctuations of the meannumber of participating nucleons in proton-nucleus colli-sions significantly modifies the results, especially the tailsof the distribution, and it was pointed out the importanceof implementing NN correlations in the event generatorfor nucleon configurations in a way consistent with thesingle nucleon nuclear density and use it for description ofthe heavy ion collisions on an event-by-event basis. Sim-ulations of such processes are commonly performed usingconfigurations of nucleons as an input, consisting of nu-cleons spatial locations and isospins, generated assumingthe independent particle model; however, inclusion of NNcorrelations can provide a more realistic description of ini-tial states of nuclei and appears to be feasible with a goodaccuracy.The Letter is organized as follows; in section II, de-tails of NN correlations are discussed and the methodadopted for generation of configurations is outlined; wedescribe the generation of nuclear configurations withina MC approach including central two-body correlationswhich are consistent with realistic calculations of one- andtwo-body density distributions. In section III we introducethe probability distribution functions P N for the N -th or- Preprint submitted to Physics Letters October 28, 2018 er, N = 1 , ..., A , interaction of a projectile nucleon with anucleus A , and we present results for fluctuations. In sec-tion IV we use our event generator to calculate maximalpossible shadowing for gluons based on simple geometricconsiderations.
2. NN correlations
The aim of the present work is to develop a MC proce-dure for generating spatial nucleon configurations to beused as an input for simulations of collisions involvingheavy nuclei which treats NN correlations in a realisticway. The commonly used approach to this problem is tak-ing a given number of nucleons out of a Woods-Saxon den-sity distribution, which describes the probability densityfunction of a nucleon to be located at a distance r fromthe center of the system. As a result, the A nucleons arepositioned independently from each other, such that alltwo-particle (and higher) correlations are completely ig-nored; this is justified by the assumption that inclusivequantities are not very much dependent on these correla-tions. Alternatively one puts nucleons in consequently oneafter another imposing the condition that the distance be-tween nucleons should be larger than some minimal one.This procedure, however, results in a wrong single nucleondistribution of the nucleons and must be improved.We argue that when considering event-by-event observ-ables, NN correlations in the nuclear wave function are rel-evant. This extends observation of[6] that NN correlationssignificantly modify the variance of the distribution overthe number of collisions. In the present work we analyzeseveral different approximations for correlations. A fullimplementation of realistic short range correlations[7, 8]in the wave function Ψ of a complex nucleus can be madeusing an independent particle model wave function Φ anda proper correlation operator ˆ F = Q Ai Figure 1: The quantity of Eq.(6), calculated for O . Dashedhistogram : step function (excluded volume) correlation; solid his-togram : Gaussian correlation; solid curve : the realistic calculationof Ref.[7]. infinite nuclear matter and which is a good approximationin finite nuclei, far enough from the edge of the nucleus),the pair distribution function C ( r ) can be obtained fromthe correlated ρ (2) C ( r ) and uncorrelated ρ (2) U ( r ) two-bodyradial densities: C ( r ) = 1 − ρ (2) C ( r ) / ρ (2) U ( r ) . (6)The quantities defined in Eq.(5) and Eq.(6) were calcu-lated using the MC code using several approximations forthe correlation function f ( r ) (see Eq.(1)). They are shownin Fig.1; the results of Ref.[7] are also shown. Our proce-dure was to start with a central correlation function f ( r )taken as a step function, i.e. vanishing wave function for r < f m ; we then used a Gaussian f ( r ) = 1 − exp ( − β r )in order to match better the realistic result. It is knownthat the realistic central correlations has a small overshoot-ing over unity, and it is in general different from our Gaus-sian correlation; however, in realistic calculations of finitenuclei[7, 12] the peak in the radial two-body density ismainly due to the existence of state-dependent, mainlytensor correlations, and three-body diagrams. For thisreason we prefer not to introduce an unrealistic centralcorrelation to reproduce such a peak and use a parameter β = 0 . f m − in the Gaussian correlation which gives ahealing distance similar to the one exhibited by the real-istic approach. This amounts to calculate the wave func-tion with a central correlation operator; the fully realisticimplementation of the operator of Eq.(1), involving spinand isospin degrees of freedom, extremely computation-ally intensive, is thus beyond the aim of the present con-tribution and will be performed elsewhere. In the presentwork we randomly assigned isospin degrees of freedom forA nucleons, so that each generated configuration consistsof A quads of numbers, containing three spatial coordi- P N ( b ) P bg bg P P B&T Uncorr. Gaussian b [fm] O Figure 2: The probabilities P N , N = 1 , , 3, as a function of theimpact parameter b (see text), calculated for √ s = 42 GeV ( toppanels ) and for √ s = 5500 GeV (LHC energies; lower panels ) on the O nucleus. Squares : analytic results obtained with the approxima-tion of Eq.(12), the Glauber approach with zero-range interaction ofBertocchi and Treleani, Ref.[15]; dashes : uncorrelated result; solidlines : Gaussian correlations. nates and one, randomly assigned, isospin variable; thecorrelated nuclear configurations are downloadable fromthe PSU physics department web server at the address[13]. 3. Probability distribution functions for hadron-nucleus collisions In order to illustrate the present implementation of cor-relations, we have focused on the O , Ca and P b nuclei.For a given configuration of nucleons, the probability ofinteraction with the i -th nucleon for an incoming projectilewith impact parameter b is given by P ( b , b i ) = 1 − [1 − Γ( b − b i )] , (7)the corresponding probability of no interaction being 1 − P ( b , b i ). The Γ function in Eq.(7) for high energy incidentnucleons can be parameterized asΓ( s ) = σ totNN πB e − s / B , (8)where σ totNN is the total cross section of NN scattering andthe t dependence of the cross section dσ/dt ∝ exp ( Bt ),neglecting small corrections due to the real part of theamplitude. Let us define the probability of interactionwith N nucleons as a function of the impact parameter as P N ( b ) = N X i ,...,i N P ( b , b i ) · . . . · P ( b , b i N ) A − N Y j = i ,...,i N [1 − P ( b , b j )] . (9)3 s = 5500 GeVP P P P P N ( b ) b [fm] Pb P Figure 3: The probabilities P N , N = 1 , , , , 12, as a function ofthe impact parameter b , calculated for √ s = 5500 GeV , correspond-ing LHC energies, on the P b nucleus, using (gaussian) correlatedconfigurations. The inelastic cross section due to collisions with N nucle-ons is then given by σ inN = Z d b P N ( b ) , (10)and the total inelastic cross section is σ inNA = P Ai σ ini ,and it can also be calculated as σ inNA = σ totNA − σ elNA − σ qeNA , (11) σ elNA and σ qeNA being the elastic and quasi-elastic cross sec-tion, respectively; this calculation is being performed withrealistic wave functions (Ref.[14]) and a comparison withthe present approach will be presented elsewhere.We can evaluate the probabilities given by Eq.(9) to anyorder in N ≤ A as a function of b = | b | . The results for A = 16 , 208 are shown in Figs.2, 3. The number of con-figurations used in all the calculations is large enough toproduce negligible statistical errors in the results, typicallyover 100 thousands configurations. All the results havebeen obtained using parameters of the NN amplitude cor-responding to √ s = 42 GeV /c ( σ tot = 41 . mb , B = 12 . GeV − ), except the lower panel of Fig.2 for O , where √ s = 5500 GeV /c ( σ tot = 94 . mb , B = 17 . GeV − ) ischosen, corresponding to LHC energies, to be comparedwith the top panel of the same figure, and in Fig.3 for P b where √ s = 5500 GeV /c as well.We have checked the independent particle model re-sults with the analytic approximation of Ref.[15], wherethe probability distributions are given by: P N ( b ) = A !( A − N )! N ! (cid:0) σ in T ( b ) (cid:1) N (cid:2) − σ in T ( b ) (cid:3) ( A − N ) , (12)where T ( b ) = R ∞−∞ dz ρ ( b , z ), which neglects the finiteradius of NN interaction. The comparison is made for the D ( b ) B&T Gaussian Uncorrelated Eqs. (13) (14) Ca b [fm] Pb O Figure 4: The dispersion D ( b ), as a function of the impact parameter b , defined in Eq.(18) calculated for O , Ca and P b nuclei, andfor a 920 GeV incident nucleon. Dashed curves correspond to uncor-related configurations; solid lines to the Gaussian correlations. Theresults obtained using eikonal model expression of B&T (Ref.[15]) forthe zero-range NN interactions are given by dotted lines; for largeA they coincide with the uncorrelated MC calculations. Symbols :direct calculation using Eqs.(13), (14), from Ref.[6], with C ( r ) ex-tracted from the MC configurations and shown in Fig.1. O nucleus in Fig.2, showing a more narrow distributionif the approximation of Eq.(12) is used; the same holdstrue for the other considered nuclei. The quantities h N i and h N ( N − i , can be written as follows[6]: h N i = Z d r ρ ( r ) dσ in d b ( b − b ) , (13) h N ( N − i = Z d r d r ρ (2) ( r , r ) dσ in d b ( b − b ) dσ in d b ( b − b ) (14)which can be calculated in the present framework using dσ in d b ( b − b i ) = 1 − (1 − Γ( b − b i )) . (15)Alternatively we can calculate h N i , h N ( N − i usingEqs.(13), (14) as h N i = X N N P N ( b ) , (16) h N ( N − i = X N (cid:0) N − N (cid:1) P N ( b ) . (17)4 .0 0.2 0.4 0.6 0.8 1.0012012 Gaussian P ( x ) x Dipole Pb b=0 fm b=5 fm b=6 fm Figure 5: The probability density P ( x ) for the quantity x definedin Eq.(24) calculated as a function of the impact parameter b , for m g = 0 . GeV and for P b . We have used the dipole ( top ) andGaussian ( left ) gluon density distributions of Ref.[17] and correlatedconfigurations. As a result we can write the variance of the mean numberof collisions as a function of the impact parameter b , asfollows D ( b ) = h N i − [ h N i ] h N i = P N N P N ( b ) − [ P N N P N ( b )] P N N P N ( b ) . (18)In order to check the accuracy of our method, D ( b ) wascalculated both using Eqs.(13), (14) and Eq.(18) using an-alytical one-body densities and C ( r ) extracted from theMC calculation; we then used Eqs.(16), (17), calculatedwith the P N ( b ) functions obtained with the MC. The re-sults are compared in Fig.4 for O , Ca and P b ; asmall discrepancy is exhibited, between the calculationwith C ( r ) (shown with symbols) and the correspondingGaussian correlated result, but the overall agreement issatisfactory. The small difference between the results ofthe two methods, especially for lead, is to be ascribed tothe fact that the quantity C ( r ) has been extracted fromthe MC configurations using Eq.(6), assuming the function C ( r , r ) to be a function of the relative distance only,and not to depend on the particular region of the nucleusconsidered, while surface effects should be present on thelevel of few %. We have compared the performance of theGlauber type interaction with the approach of Ref.[16], Pb R g ( b ) b [ fm ] Gaussian Dipole O Figure 6: The ratio R g ( b ) defined in Eq.(25) for O and P b ,calculated using the gaussian ( dots ) and dipole ( solid ) gluon densitydistributions of Ref.[17] and correlated configurations as a functionof the impact parameter b , plotted for m g = 0 . GeV and for thosevalues of b for which there is enough statistics for the normalization R dxP ( x ) not to deviate from unity for more than 5%. where the probability of interaction in Eq.(9) was takenas a step function, which vanishes for nucleons sitting out-side the cylinder centered at the given | b | from the centerof the nucleus. We took for the cylinder the value whichgives an area corresponding to the one given by σ in = Z d b σ in ( b ) = Z d b h − (1 − Γ( b − b i )) i (19)where Γ is calculated with the Glauber parameters we haveused in the present work. The results, as compared withour Glauber approach showed to differ substantially fromboth the uncorrelated and correlated MC results. 4. Lower limit on the parton nuclear shadowing In this section we present application of our MC code tothe study of maximal possible nuclear shadowing in nuclei.We use the well known observation that shadowing in thescattering off the deuteron cannot reduce the cross sectionto the value smaller than cross section of scattering offone nucleon. Basically it is due to one nucleon screeninganother one but not itself.Similarly, it is natural to expect that in any dynam-ics of parton interactions the gluon (quark) density at agiven impact parameter in a particular configuration can-not be less than the maximum of individual transversegluon densities, g N ( x, ρ ) in the nucleons at given b. Here g N ( x, ρ ) = g N ( x ) F g ( ρ ) is the generalized diagonal gluondensity in the nucleon. g A ( x, b ) conf ≥ max i =1 ,A ( g iN ( x, r t − b )) , (20)5 .4 0.6 0.8 1.0 1.20.20.40.6 Pb R g ( b = ) m [ GeV ] Gaussian Dipole O Figure 7: The same quantity of Fig. 6 but plotted at b = 0 and as afunction of m g , which is a parameter in Eq.(25). leading to g A ( x, b ) min ≥ h g A ( x, b ) conf i , (21)In the limit of A → ∞ this leads to g A ( x, b ) ≥ g N ( ρ = 0) . (22)The onset of the limiting behavior depends on the trans-verse shape of the gluon GPD. For the same g N ( x ) a shapemore peaked at small ρ will asymptotically lead to largervalue of g A ( x, b ) min though the approach to the asymp-totic value will require larger values of A as chances thatthere is a nucleon with ρ small enough that g N ( x, ρ ) is closeto g N ( x, 0) is smaller in this case. In Ref.[17], two param-eterizations for the two-gluon form factor were discussed,fitted to the J/ψ photoproduction data ([18]); they weretaken in the forms of an exponential and a dipole. The cor-responding transverse spatial distributions of gluon GPDsare thus F g ( ρ ) = 12 πB g e − ρ / (2 B g ) ,F g ( ρ ) = m g π m g ρ K ( m g ρ ) , (23)where K is the modified Bessel function, m g = 0 . GeV for x ∼ − and B g = 3 . /m g . Using the configurationsdescribed in the previous sections, we have calculated forgiven impact parameter b the maximum transverse gluondensity normalized to its peak value, i.e. the maximumvalue of the probability density P ( x, b ) of the quantity x = F g ( ρ j ) /F g (0) , (24)as a function of ρ j = b − b j , with b j the j -th nucleon trans-verse coordinate; here P ( x, b ) is normalized according to < x > A m =0.6 m =1.1 Gaussian Dipole Figure 8: The quantity < x > as a function of A , calculated withthe densities of Eqs.(23) and large values of A . The values of m g aregiven in GeV . R dxP ( b, x ) = 1, for given b . The results of the calculationfor P b are shown in Fig.5. It can be seen that the twodensities of Eq.(23) produce very similar results.We next apply these results to determine minimal valueof the gluon shadowing at given b defined as the ratio oflower limit on g A ( x, b ) and its value in the impulse approx-imation, g A ( x, b ) = g N ( x ) T A ( b ); to this end, we take theratio R g ( b ) = g N ( x, ρ = 0) R dx x P ( x, b ) g N ( x ) T A ( b )= F g (0) R dx x P ( x, b ) T A ( b ) . (25)Results are presented for the two used models of the gluonGPDs of Eq.(23) in Fig.6 and Fig.7. For A → ∞ the lim-its using the two different fits differ by F (1) g (0) /F (2) g (0) =2 / . 24 = 0 . 62; however, this limit is reached at extremelylarge A, as shown in Fig.8. This is due to a rathersmall radius of the transverse gluon density r trg ≤ . f m and low nuclear density, leading to a small probabil-ity for more than three nucleons to significantly screeneach other up to very large A. To investigate the onset ofasymptotic, we evaluated the A dependence of the quan-tity h x i = R dxP ( x ) at zero impact parameter for the twoconsidered gluon GPDs for two values of m g correspond-ing to x ∼ . 01 and 0 . GeV corresponding to x ∼ − .We modeled nuclei with very large A by generating ran-dom nucleons with constant density ρ = 0 . f m − ina cylinder centered at b = 0, with height 2 R A , where R A = 1 . A / ; the results, shown in Fig.8, are indepen-dent from the radius of the cylinder, provided it is largerthan about 0 . f m . It can be seen that, the GPDs beingvery peaked at ρ = 0, the increase of h x i is very slow andthe maximum value is not reached even with A as large as60 . Onset of asymptotic is somewhat faster for smaller m g due to a smoother behavior of GP Ds at ρ ≃ 5. Conclusions We developed a MC event generator for nucleon configu-rations in nuclei which correctly reproduces single nucleondensities and central NN correlations in nuclei. The gener-ator can be used in modeling a wide range of processes. Inparticular, it would be interesting to explore how it wouldmodify effects of fluctuations of the number of woundednucleons in the heavy ion collisions. 6. Acknowledgments We thank G. Baym, C. Ciofi degli Atti and L. Frankfurtfor useful discussions. This work is supported by DOEgrant under contract DE-FG02-93ER40771 References [1] D. Higinbotham, E. Piasetzky and M. Strikman, CERN Cour. (2009) 22[2] R. Subedi et al. , Science (2008) 1476[3] R. Shneor et al. [Jefferson Lab Hall A Collaboration], Phys.Rev. Lett. (2007) 072501; K. S. Egiyan at al. (CLAS), Phys.Rev. Lett. (2006) 082501; A. Tang et al., Phys. Rev. Lett. (2003) 042301[4] L. Frankfurt, M. 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