Initial conditions from the shadowed Glauber model for Pb+Pb at √s = 2.76 TeV NN Snigdha Ghosh,∗ Sushant K. Singh,† Sandeep Chatterjee,‡ Jane Alam, and Sourav Sarkar Theoretical Physics Division, Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata, 700064, India Abstract We study the initial conditions for Pb+Pb collisions at √sNN = 2.76 TeV using the two component Monte-Carlo Glauber modelwith shadowingofthenucleonsintheinteriorbytheleadingones. Themodelparametersarefixedbycomparingtothe multiplicitydataofp+PbandPb+Pbat √sNN=5.02 and2.76 TeVrespectively. Wethencomputethecentrality dependence 6 of the eccentricities upto the fourth order as well as their event by event distributions. The inclusion of shadowing brings the 1 Monte-Carlo Glauber model predictions in agreement with data as well as with results from other dynamical models of initial 0 conditions based on gluon saturation at high energy nuclear collisions. Further, we find that the shadowed Glauber model 2 provides the desired relative magnitude between the ellipticity and triangularity of the initial energy distribution required to y explain the data on the even and odd flow harmonics v2 and v3 respectively at theLHC. a M I. INTRODUCTION physics like IP-Glasma [9] and EKRT [12] are in agree- 3 ment with data. ] One of the most important ingredients in understand- MCGMs provide a simple and intuitive description of h ingtheevolutionofmatterformedinheavyioncollisions the IC and hence there have been considerable efforts to -t (HIC)isitsinitialcondition(IC).Currentlytherearesev- address the above issues within the geometric approach cl eralmodelsofICavailablewithvaryingdegreesofsuccess of the MCGM [10, 16]. Recently, we have shown that u inexplainingthedata[1–12]. TheMonte-CarlobasedIC the inclusion of shadowing effect due to the leading nu- n modelsgeneratethe eventbyevent(E/E)fluctuationsin cleonsonthoselocateddeepinsideprovidesasimpleand [ observableswhichcanbe comparedtothosemeasuredin physical picture that brings the predictions of the shad- 2 experiments. Most of these models share the first step- owedMCGM(shMCGM)inagreementwiththatofdata v sampling the positions of the constituent nucleons of the as well as dynamical models like IP-Glasma at the top 1 two colliding nuclei from their nuclear density distribu- RHIC energy for Au+Au as well as U+U collisions [17]. 7 tion which is usually taken to be a Woods-Saxon pro- In this paper we present the results of the shMCGM 39 file [13]. In the second step, they all differ in the energy for Pb+Pb collisions at √sNN = 2.76 TeV and compare deposition scheme corresponding to a specific configura- withIP-Glasmamodelpredictions[9]aswellasthe LHC 0 tion of the nucleon positions. This finally results in dif- data [18–21]. . 1 ferentpredictionsofcentralitydependenceofmultiplicity, 0 eccentricitiesandtheir eventby eventdistributions. The In the next section II, we provide the details of our 6 shMCGM and the values of the parameters of the model largest source of uncertainties on the extracted values of 1 estimated by comparing with data. In section III we the medium properties obtained by comparing the pre- : present the results obtained in the shMCGM. We pro- v dictions of the theoretical models to data is known to i vide estimates for the centrality dependence of various X stem from the choice of the IC [14]. eccentricities and their E/E distributions. We find good Monte-CarloGlaubermodels(MCGMs)havebeenrea- r quantitative agreement with data as well as IP-Glasma a sonably successful in describing the qualitative features results. Finally, in section IV we summarise. of various observables [6, 7]. The energy deposition schemeislargelygeometricalwiththeonlydynamicalin- put being a constantnucleon-nucleon cross-sectionσ . NN The recent data on v dNch correlation for the top 2 − dη ZDC events in U+U collisions at √sNN = 193 GeV II. THE MODEL and Au+Au interactions at √sNN = 200 GeV could not be reproduced within the ambit of the standard MCGM [11, 15]. The MCGM predictions are also in dis- ThedetailsoftheshMCGMaregiveninRef.[17]. Here agreement with the E/E distribution of the second flow we summarise its main features. The shMCGM is an harmonic for Pb+Pb collisions at √s = 2.76 TeV. extension of the two component MCGM. In the latter, NN However, dynamical models based on gluon saturation the energy deposited at (x,y) on the plane transverse to the beam axis (which is along the z axis) is assumed to be a linear superposition of two terms- N (x,y) part andN (x,y)whereN countsthe numberofpartic- coll part ∗ [email protected] ipantnucleonsandNcoll isthenumberofpossiblebinary † [email protected] collisions between them. The total charged multiplicity ‡ [email protected] dN /dη is also assumed to havea similar linear relation ch 1 with the total N and N part coll MCGM ǫ(x,y)=ǫ [(1 f)N (x,y)+fN (x,y)] (1) 0 part coll shMCGM − dNch = 1−f Npartw +fNcollw (2) η) 10-2 CMS dη (cid:18) 2 (cid:19) Xi i Xi i N/dch where the wisare sampled from a negative binomial dis- P(d 10-4 tribution (P (w,n ,k)) with variance 1 [22] NBD 0 ∼ k P (w,n ,k)= Γ(k+w) nw0kk (3) 10-6 (a) NBD 0 Γ(k)Γ(w+1)(n +k)w+k 0 50 100 150 200 250 0 dN /dη ǫ andn aretheoverallnormalizationparametersforthe ch 0 0 energy deposited and multiplicity produced respectively. f is usually called the hardness factor which is fixed by comparingwithdata. The criterionfor abinary collision MCGM betweennucleonifromnucleusAat xiA,yAi andnucleon art 10 shMACLIGCME j from nucleus B at xjB,yBj is g(cid:0)iven by(cid:1)rAijB ≤ σNπN )/Np 8 whererij isthesquar(cid:16)eddista(cid:17)nceinthetransverseplane ηd betweenAtBhe two nucleons /ch N d 6 rij = xi xj 2+ yi yj 2 (4) 2 ( AB A− B A− B (b) (cid:16) (cid:17) (cid:16) (cid:17) 4 Allnucleonsthatsufferatleastasinglebinarycollisionis 0 100 200 300 400 treated as a participant. Thus in the standard two com- ponent MCGM approachall the participants are treated Npart democraticallyirrespectiveoftheirpositionsalongz-axis. IntheshMCGMweintroducetheeffectofshadowingdue to leading participants on the others through the follow- FIG. 1. (Color online) Top: The probability distribution of ing ansatz dNch/dη for p+Pb at √sNN = 5.02 TeV compared between S(n,λ)=e−nλ (5) data [23], MCGM andshMCGM. Bottom: Thecentrality de- pendence of dNch/dη for Pb+Pb at √sNN = 2.76 TeV com- whereS(n,λ)istheshadowingeffectonaparticipantdue pared between data [18], MCGM and shMCGM. to n other nucleons from the same nucleus which are in frontandshadowit. Thusalltheparticipantsarenomore including NBD fluctuation [22] with k 1. In Fig. 1 treatedonequal footing- the leading nucleons contribute ∼ (b) we display the centrality dependence of the charged toenergydepositionmorethanthoselocateddeepinside. particle multiplicity in Pb+Pbcollisions at √s =2.76 Thusoverallwehavethefollowingfourparametersinthe NN TeVandcontrasttheresultswiththeavailabledata[18]. shMCGM-n , k, f andλwhichareconstrainedby data. 0 It is clear that this plot does not discriminate between MCGM and shMCGM as both the models describe the III. RESULTS data well for suitably adjusted values of the model pa- rameters. Thustheshadowparameterλcannotbe fixed fromthisplot. WehavefixedλfromtheE/Edistribution We will now present the results of the shMCGM. In plotofscaledv assuminglinearhydrodynamicresponse, Fig. 1 (a) we have plotted the probability distribution of 2 i.e. v ε . The values of the parameters so obtained is charged particle multiplicity observed in p+Pb collision 2 ∝ 2 tabulatedinTableI.Apartfromthe parameterslistedin at5.02TeVfor η <2.5[23]. Wehavecomparedthedata with resultsfrom| |both MCGMas wellas shMCGM.The Table I, we need the nucleon-nucleon cross-section σNN whichis takenhereas64mb(the correspondingvalue at nucleons are sampled from a Woods-Saxon distribution RHICis42mb). Wetakeσ =0.5fm,whichisusedinthe of the Pb nuclear density. The Woods-Saxonparameters Gaussian ansatz below to smear the energy ǫ deposited of the Pb nucleus are: the radius R = 6.7 fm obtained i by a participant located at (x ,y ), fromtheparametrisationR=1.12A0.33 0.86A−0.33and i i − tshcreipstuirofnacoefdtihffeuslioonngδt=ail0i.n54tfhme d[2i4st].ribWuetiofinndongloyodafdteer- ǫi(x,y)= 2πǫ0σ2e−(x−xi)22σ+2(y−yi)2 (6) 2 We notice that in both MCGM as well as shMCGM, 0.7 n nearly doubles from √s = 200 GeV to 2.76 TeV. MCGM 0 NN Onthe other hand, f drops by 20%as we go from RHIC 0.6 shMCGM toLHCinMCGMwhileitstaysalmostconstantinshM- IP-Glasma 0.5 CGM. However, the shadow parameter λ drops by 25% at LHC compared to highest RHIC energy. This sug- > 0.4 2 gests that λ increases with decrease in √sNN and hence ε< 0.3 the distinction between the two Glauber approaches will becomeevenmoresignificantatlowerenergies(e.g. GSI- 0.2 FAIR energies). Thus,havingfixedtheparametersoftheGlaubermod- 0.1 (a) els, we now look at the model predictions for other ob- 0 10 20 30 40 50 servables and compare them with data as well as results from other dynamical models of IC like IP-Glasma. We Centrality (%) first study the centrality dependence of the mean eccen- 0.4 tricityharmonicsεn oftheinitialenergydepositedinthe MCGM overlapregion shMCGM IP-Glasma rneinφ 0.3 ε eiΨn = h i (7) n rn h i >3 ε where n = 2,3,... The coordinate system is chosen such < 0.2 that ~r =0. ... representsaveragingoverthetransverse h i h i plane with the initial energy deposited on the transverse plane ǫ(x,y) as the weight function. In Fig. 2, we have 0.1 (b) plotted the ensemble average of ε , ε and ε vs their 2 3 4 centrality which is determined from the final state (FS) 0 10 20 30 40 50 charged hadron multiplicity. We have shown the results Centrality (%) for MCGM, shMCGM and IP-Glasma. All the models show a rising trend for ε2 with centrality which is also 0.5 expected from geometrical arguments- events with lower MCGM shMCGM multiplicity occur with larger impact parameter which 0.4 IP-Glasma results in larger ellipticity of the overlap region. For all centralities, ε in shMCGM is enhanced as compared to 2 MCGM and is also in good agreement with IP-Glasma > 0.3 4 results. The higher value of ε2 in shMCGM as compared ε< to MCGM was also found at RHIC energy and is a typ- 0.2 ical effect due to nucleon shadowing [17]. The ends of the minor axis of the collision zone have on an average 0.1 (c) largernumber ofparticipantsas comparedto the ends of themajoraxisofthe overlapregionwhichcallsforlarger 0 10 20 30 40 50 shadowingeffectattheendsoftheminoraxis. Thiseffec- Centrality (%) tively reduces the minor axis more than the major axis, thus increasing the ε of the overlapregion. 2 The higher harmonics arise due to granularity in the FIG.2. (Coloronline)Thecentralitydependenceofeccentric- IC which is controlled by N : smaller N corre- part part ities compared between IP-Glasma, MCGM and shMCGM. sponds to larger granularity [25, 26]. This explains the rising trend in ε and ε with centrality. We note that 3 4 both MCGM and shMCGM yield almost similar values Model n0 f k λ for these higher eccentricities even though in shMCGM MCGM 4.05 0.11 1 - the effective number of participants is smaller than in shMCGM 4.05 0.32 1 0.08 MCGM due to the shadowing effect. This is so because the shadowing effect systematically weakens only those TABLEI.ThevaluesoftheparametersoftheGlaubermodels sources which have other sources in front. Thus, weak- used in this work. ening of these sources in the bulk do not increase the 3 1.4 MCGM MCGM 3 1.2 shMCGM ATLAS 0. shMCGM 2>) 1 >) 1 ε3 ε2 < < 0.5)/( 0.8 εP(/2 0.1 0-5% > 0.6 2 2 ε < 0.4 (a) ( 0.01 0.2 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 35 40 ε /<ε > 2 2 Centrality (%) MCGM FIG.3. (Color online)Thecentralitydependenceoftheratio ATLAS of r.m.s. ε2 to ε3 compared between MCGM and shMCGM. shMCGM AlsoshownisthebandasproposedinRef.[27]thatisrequired ) 1 IP-Glasma > to explain the correlation of v2 v3 in data assuming linear 2 − ε hydrodynamicresponse. < /2 ε P( 0.1 10-15% granularity in the transverse plane. (b) The correlation between the even-odd harmonics 0.01 largely stem from the ε2 ε3 correlation of the initial 0 0.5 1 1.5 2 2.5 3 − state (IS). Starting from the observed correlation in the ε /<ε > dataofv v atthe LHC,anallowedbandforthe ratio 2 2 2 3 − ofr.m.svaluesofε toε wasobtainedinRef.[27]within 2 3 therealmoflinearresponse. InFig.3wehaveshownthis MCGM band. We also show the values obtained for the same ATLAS quantity in MCGM and shMCGM. The enhancement of shMCGM ε2 in shMCGM as compared to MCGM as noted earlier ) 1 IP-Glasma > in Fig. 2 also helps here- it pushes the prediction for the 2 ε ratio of r.m.s. of ε to ε into the band that is favored < 2 3 /2 by data unlike the case of MCGM which underpredicts ε P( 0.1 20-25% as compared to the band. We now turn our attention from the mean geometric (c) propertiesintheICtotheirfluctuations. Wefirstanalyse 0.01 the E/E distributions of the εn scaled by their ensemble 0 0.5 1 1.5 2 2.5 3 average values and compare with that of IP-Glasma [9] ε /<ε > as well as ATLAS data of v [19, 20]. As long as the 2 2 n hydrodynamic response is linear (v = k ǫ where k is n n n n a constant), we expect the E/E distributions of ε / ǫ n n h i to be a good representative of vn/hvni. In Figs. 4, 5 FIG. 4. (Color online) The E/E distribution of ε2 compared and 6 we have plotted the E/E distribution plots for ε2, between data [19, 20], IP-Glasma, MCGM and shMCGM. ε and ε for the following centrality classes: (0 5)%, 3 4 − (10 15)%and(20 25)%. Overall,thereisgoodquan- − − titativeagreementbetweenshMCGManddataaswellas predictions. IP-Glasma. It is well known that the standard MCGM producesabroaderE/Edistributionascomparedtodata For further peripheral centralities, the agreement in aswellasIP-Glasmaresults[12,26]. Howeverasalready case of the E/E distributions of ε worsen. On the other 2 argued earlier in Ref. [17], the shadowing effect shadows hand, the distributions of the higher harmonics continue the participants as well as their E/E fluctuations in po- to be in good agreement. Recently, this has been shown sition which eventually results in narrowerE/E distribu- as evidenceofcubic hydrodynamicresponsetoellipticity tion that is in good agreementwith data and IP-Glasma in peripheral collisions [28]. A modified predictor of v , 2 4 MCGM MCGM ATLAS ATLAS shMCGM shMCGM ) 1 IP-Glasma ) 1 IP-Glasma > > 3 4 ε ε < < /3 /4 ε ε P( 0.1 0-5% P( 0.1 0-5% (a) (a) 0.01 0.01 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 ε /<ε > ε /<ε > 3 3 4 4 MCGM MCGM ATLAS ATLAS shMCGM shMCGM ) 1 IP-Glasma ) 1 IP-Glasma > > 3 4 ε ε < < /3 /4 ε ε P( 0.1 10-15% P( 0.1 10-15% (b) (b) 0.01 0.01 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 ε /<ε > ε /<ε > 3 3 4 4 MCGM MCGM ATLAS ATLAS shMCGM shMCGM ) 1 IP-Glasma ) 1 IP-Glasma > > 3 4 ε ε < < /3 /4 ε ε P( 0.1 20-25% P( 0.1 20-25% (c) (c) 0.01 0.01 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 ε /<ε > ε /<ε > 3 3 4 4 FIG. 5. (Color online) The E/E distribution of ε3 compared FIG. 6. (Color online) The E/E distribution of ε4 compared between data [19, 20], IP-Glasma, MCGM and shMCGM. between data [19, 20], IP-Glasma, MCGM and shMCGM. in terms of ε was shown to do a much better job in 1,2 the EKRTmodel for suchperipheralbins [12] whereit is for the centrality bins of (40 45)% and (50 55)%. − − defined as: We indeed find better agreementbetween data of v2 and shMCGM prediction for ε than ε . rei2φ 1,2 2 ε1,2eiΨ1,2 = h i (8) The standard deviation is a good measure of fluctua- r h i tion about the mean value. ATLAS data is available on In Fig. 7 we have plotted the E/E distribution for ε the scaled variance of the flow coefficients v which can 1,2 n andcomparedwiththe caseofε as wellasv fromdata becomparedtothatofinitialeccentricityassuminglinear 2 2 5 1 shMCGM , m = 1 MCGM shMCGM , m = 2 shMCGM 0.8 ) ATLAS ATLAS >2 1 > m, ε2 0.6 ε < ε/<m,2 0.1 40-45% σε()/2 0.4 ( P 0.2 (a) (a) 0.01 0 0 0.5 1 1.5 2 2.5 3 0 100 200 300 400 ε /<ε > N m,2 m,2 part 1 shMCGM , m = 1 MCGM shMCGM , m = 2 shMCGM 0.8 ) ATLAS ATLAS >2 1 > m, ε3 0.6 ε < ε/<m,2 0.1 50-55% σε()/3 0.4 ( P 0.2 (b) (b) 0.01 0 0 0.5 1 1.5 2 2.5 3 0 100 200 300 400 ε /<ε > N m,2 m,2 part 1 MCGM FIG. 7. (Color online) The comparison between E/E dis- shMCGM tribution of ε1,2 and ε2 obtained in shMCGM with data of 0.8 ATLAS v2 [19, 20]. > 4 0.6 ε < )/ response[20]. Thesequantitieshavebeenrecentlyshown ε4 0.4 ( to agree well with MCGM computations [29]. In Fig. 8, σ wehaveplottedthe centralitydependence ofσ(ε )/ ε 0.2 n n in MCGM and shMCGM. We find almost equally ghoodi (c) 0 agreement between data and the different Glauber ap- 0 100 200 300 400 proaches. Twoparticlecumulantsε 2 andfourparticle n cumulants εn 4 are other quantities{o}f interest that are Npart { } often measured in experiments and compared to theory, ε 2 = ε2 1/2 (9) n{ } h ni ε 4 = 2 ε2 2 ε4 1/4 (10) FIG. 8. (Color online) The centrality dependence of nor- n{ } h ni −h ni malised standarddeviationoftheeccentricities comparedbe- Fromthesecumulantsit(cid:0)ispossibletod(cid:1)efinethefollowing tween data [20], MCGM and shMCGM. scaled moments which within linear response will have one to one correspondence between the initial state(IS) and the final state (FS) ε 2 2 ε 4 2 data [21], MCGM and shMCGM. Thus it is clear from n n F(εn)=sεn{2}2+−εn{4}2 (11) Figs. 3, 4,8and9thatthe relativemagnitudes ofε2−ε3 { } { } and E/E distribution plots of scaled ε can discriminate 2 We have plotted the centrality dependence of F (ε ) clearly between MCGM and shMCGM. The rest of the n in Fig. 9 and find good qualitative agreement between observables have weaker discriminatory power. 6 heavy ion collisions. This model provides the central- 1.6 MCGM ity dependence of various ensemble average observables shMCGM like charged particle multiplicity, anisotropies of the ini- 1.2 ATLAS Cum. tialenergydepositedetc. TheMCGMgenerateseventby ATLAS EbyE eventdistributionsofthesequantities. Theaboveobserv- ) ε2 0.8 ables including their event by event distribution can be ( F measuredinexperimentsandcomparedtosuchmodelsof initial condition. Overall, Monte-Carlo Glauber models 0.4 manage to provide a good qualitative description of the (a) data. However,suchgeometricalmodelscannotdescribe 0 the recent data on v dN /dη correlation from U+U 2 ch 0 100 200 300 400 − at √s = 193 GeV and event by event flow data for NN Npart Pb+Pb at √sNN =2.76 TeV. This has called for a lot of ongoingefforttoaddresssuchissueswithinthegeometric approachofthe Glauber model. We haveearliersuccess- 1.6 MCGM fully addressed these issues at the top RHIC energy by shMCGM introducing the effect of shadowing due to leading nu- 1.2 ATLAS cleons on the other participants located in the bulk [17]. Here we have extended our study to Pb+Pbcollisions at ε)3 0.8 √sNN =2.76TeV.Thisideaofeclipseofnucleonsinpres- F( enceofothernucleonsinsideanucleushasbeenproposed about sixty years back [30]. However, the significance of 0.4 this effect in the phenomenology of heavy ion collisions has been hardly explored before. The current as well as (b) 0 our earlier work [17] suggest that the nucleon shadowing 0 100 200 300 400 plays an important role in the ICs of HICs. Npart We find that for all centralities ε2 is enhanced while ε and ε do not change much due to the inclusion of 3 4 shadowing. This brings the predictions of the shadowed 1.6 Glauber model for the mean values as well as event by MCGM shMCGM eventdistributionsoftheeccentricitiesinagreementwith 1.2 ATLAS IP-Glasma results. The relative magnitudes of ε2 ε3 − and event by event distributions of ε clearly demon- 2 ε)4 0.8 stratethesuperiorperformanceoftheshadowedGlauber ( model as compared to its conventional version, MCGM. F The shadow parameter λ drops by 25% as we go from 0.4 topRHICtoLHCenergies. Thisim∼pliesthatatloweren- ergies, the effect of shadowing in Glauber models will be (c) 0 even more important. Currently for low energies where 0 100 200 300 400 the fireball is expected to carry a non-zero net baryon N number, dynamical models are yet to be formulated and part predictionsmade. Ontheotherhand,withintheambitof the shadowedMonte-CarloGlauber model, it is straight- forward to make predictions at all energies as long as FIG. 9. 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