Correlations and Fluctuations in the Initial State of high energy Heavy Ion Collisions
aa r X i v : . [ nu c l - t h ] A ug Correlations and Fluctuations in the Initial State of high energy Heavy Ion Collisions
Christoffer Flensburg Dept. of Astronomy and Theoretical Physics, Lund university, Sweden (Dated: July 31, 2018)Initial states of high energy heavy ion collisions are studied using a dipole model through the
DIPSY event generator that dynamically includes saturation together with the fluctuations andcorrelations of the BFKL cascade. The eccentricities ε , , , are calculated at RHIC and LHC.Predictions are made for correlations and fluctuations in rapidity of the eccentricities, and conven-tional theoretical approximations are tested. A large set of initial state Au-Au, Cu-Au and Pb-Pbcollisions have been generated and are published online. I. INTRODUCTION
The study of azimuthal correlations and their originplays a big role in high energy heavy ion collisions. Oftenthe azimuthal correlations are explained by an anisotropyin the transverse plane of the state just after collision,that propagates to the final state through collective ef-fects [1].Traditionally these studies have been made with av-erage transverse shapes as function of impact parameter b , but lately more and more studies have incorporatedevent-by-event fluctuations of the interaction shape [2, 3].The first sign was that fluctuations are necessary to ex-plain the large elliptic flow in central Cu-Cu collisionsseen at PHOBOS [4], but recently also fluctuation drivenobservables such as triangular flow [5–10] and directedflow [11, 12] have been studied. Some of the approachesto model these fluctuations are:The colour glass condensate model [13], for exampleimplemented in the KLN model [14], calculates quanti-ties from an average transverse gluon density, evolvedthrough rapidity. This approach does not include theevent-by-event and region-by-region fluctuations and cor-relations in BFKL.NeXus [15] is an initial state event generator based ona Gribov-Regge model of hadronic collisions, and is nowcombined with the hydrodynamical model SPheRIO [16]to make the full final state event generator NeXSPhe-RIO [17]. UrQMD, Ultra-relativistic Quantum Molecu-lar Dynamics [6, 18] uses flux tube excitation and frag-mentation to generate initial states. Another model isAMPT [19, 20], a multi-phase transport model, usingHIJING [21] for initial states, where nucleon-nucleon col-lisions are found from an eikonal formalism.NeXus, UrQMD and HIJING all use binary collisionbetween the nucleons, and thus do not take saturation inthe cascade into account properly.In this letter the dipole model introduced in [22–25],with the Monte Carlo implementation DIPSY , will be usedto describe the states at time t = 0. This dipole model isbased on BFKL in impact parameter space, and includesall the fluctuations and correlations from a partonic cas-cade while still taking saturation into account in bothevolution and interaction.The model is briefly introduced in section II and results of common quantities such as eccentricities ε , , , followin section III. In section IV, the effects of saturation onthe BFKL correlations and fluctuations are studied, andsection V ends the paper with concluding remarks. II. THE MODEL
The model and its Monte Carlo implementation
DIPSY are described in detail in [22]. Below follows a shortsummary of the key components, and why it is well suitedfor the analysis of azimuthal eccentricities in high energyheavy ion collisions.
DIPSY is an event generator using colour dipoles intransverse space. The two starting dipole states, repre-senting the colliding particles, evolve in rapidity towardseach other to meet at the interaction rapidity y . Theinteracting dipoles in the virtual cascade are traced backtowards their valence parents, separating the real gluons,referred to as backbone gluons, from virtual fluctuations.The evolution and interaction amplitudes are basedon leading logarithm BFKL, and is corrected for severalnonleading effects: • Running α s . The scale is set by the dipole sizes. • Energy conservation. The full 4-momentum andrecoils of each parton are tracked, and p + and p − ordering is required, simulating non-singular termsin the splitting function and “energy scale terms”. • Confinement is included through a gluon mass, sup-pressing large dipole emissions. • Saturation. Unitarisation provide saturation in theinteraction, and the “dipole swing” allows for merg-ing of gluon chains in the cascade.The running α s and the lightcone momentum orderingmakes the model take the essential parts of non-leadinglogarithmic effects into account, while still including thefluctuations from a BFKL description. Multiple interac-tions together with the dipole swing gives a descriptionof merging and splitting gluon chains at any rapidity,independently of where the interaction rapidity y is set.A proton in DIPSY starts out as a triangle of dipoles,that then evolves in rapidity to form more complicatedstates before collision. A heavy ion is here described < ε n2 > / N part n = 1n = 2n = 3n = 4 FIG. 1: (Colour online) The eccentricities at Au-Au collisionsfor √ s NN = 200 GeV as function of impact parameter b . Theerror bars show σ ε n = p h ε n i − h ε n i . by a Wood–Saxon distribution of nucleons with a hardcore [26, 27], where each nucleon is described by a tri-angle just like a proton. This set of triangles in impactparameter space is then evolved as one state giving a webof gluon ladders and loops connecting the nucleons.The backbone gluons describe the t = 0 state, whichwill be used to study properties such as eccentricitiesin this letter. The gluonic states can also be usedas initial condition in a final state evolution model.For this purpose, a large set of collisions of severalreactions has been generated and published online athttp://home.thep.lu.se/ christof/DIPSYEvents/. III. RESULTSA. ε n at RHIC and LHC The calculation of ε n in this letter will follow themethod in [5] with the formula ε n = p h r cos( nφ ) i + h r sin( nφ ) i h r i . (1)Here the averages are over all real gluons in a rapidityslice η ∈ [ − , r and φ for the gluons are deter-mined with respect to the center of gravity.The eccentricities will be shown as function of the num-ber of participants. The spectators can be identified in DIPSY as the nucleons that have no interacting emissions.Also the formula ε n = p h r n cos( nφ ) i + h r n sin( nφ ) i h r n i (2)is in use [6, 28] for the eccentricities. The difference be-tween the two definition is essentially only a constant < ε n2 > / N part n = 1n = 2n = 3n = 4 FIG. 2: (Colour online) The eccentricities at Pb-Pb collisionsfor √ s NN = 2 .
76 TeV as function of impact parameter b . Theerror bars show σ ε n = p h ε n i − h ε n i . factor ( n + 2) / DIPSY data. ε however, is calculated according to [11, 12] from ε = p h r cos( φ ) i + h r sin( φ ) i h r i . (3)For a closer comparison to observables, p h ε n i will beshown rather than h ε n i [28]. This adds about 15%, 7%,13% and 10% to ε , , , respectively.The eccentricities for Au-Au collisions at √ s NN =200 GeV as function of impact parameter are shown infig. 1, and the corresponding plot for Pb-Pb at √ s NN =2 .
76 TeV is shown in fig. 2. The error bars show thefluctuations σ ε n = p h ε n i − h ε n i and, assuming not toolarge fluctuations in the final state evolution, the Au-Auratio σ ε / p h ε i agrees with data for σ v /v from PHO-BOS [29] and STAR [30] for all centralities.The angle φ n between the participant plane of the n :thmoment and the event plane can be seen in fig. 3. As seenin [6], φ is strongly correlated to the event plane angleΨ b at medium impact parameters where the geometriceffect is strong, and significantly less correlated at centralevents where the ellipticity is more fluctuation driven. φ is independent of impact parameter, and φ only weaklycorrelated. IV. CORRELATIONS AND FLUCTUATIONS
With the dynamics from the saturated BFKL cascade,
DIPSY is expected to describe correlations and fluctua-tions between different rapidity slices within one event.One such observable is the ellipticity in the forwardregion as function of the ellipticity in the backward re-gion, seen in figure 4. The ellipticity is seen to be more φ n - Ψ b | n = 1n = 2 ( N part > 350)n = 2 ( 250 > N part > 100)n = 3n = 4 FIG. 3: (Colour online) The angle between the event planeand the participant plane for N part >
100 Au-Au collisionsat √ s NN = 200 GeV. The normalisation is such that uncor-related angles yield a constant 1.
0 0.2 0.4 0.6 0.8 1 ε for η in (-3,-1) 0 0.2 0.4 0.6 0.8 1 ε f o r η i n ( , )
0 0.2 0.4 0.6 0.8 1 ε for η in (-3,-1) 0 0.2 0.4 0.6 0.8 1 ε f o r η i n ( , ) FIG. 4: (Colour online) The correlation between ε n in theforward region and backward region at N part >
100 Au-Aucollisions for √ s NN = 200 GeV. n = 2 (left) and n = 3 (right). correlated over rapidity than the triangularity. The cor-relation between ε n in η ∈ (1 ,
3) and in η ∈ ( − , −
1) canbe quantified with the correlation coefficient ρ ε F n ,ε B n = h ε F n ε B n i − h ε F n ih ε B n i p h ( ε F n ) i − h ε F n i p h ( ε B n ) i − h ε B n i (4)where the index F and B implies the eccentricity in theforward, η ∈ (1 , η ∈ ( − , − ρ n is 1 if the eccentricities are perfectly correlated, and0 if they are completely uncorrelated. At N part > ρ ε F n ,ε B n , or ρ n for short, is 0.49, 0.81, 0.57,0.58 for n = 1 , , , ε is morecorrelated over rapidity as its origin is a systematic effect.The same is seen at LHC Pb-Pb, where ρ , , , are 0.62,0.88, 0.71, 0.69 respectively.Looking at the angle between the participant plane inthe forward and backward region in fig. 5, it is seen thatthe strong correlation in the ellipticity is coming from themid-centralities, where the systemtic effect is strongest.Experimental verification of these correlations in theangle and magnitude of the corresponding anisotropic φ n,(-3,-1) - φ n,(1,3) | n = 1n = 2 (N part > 350)n = 2 (250 > N part > 100)n = 3n = 4 FIG. 5: (Colour online) The angle between φ n in the forwardregion and backward region at N part >
100 Au-Au collisionsfor √ s NN = 200 GeV. The normalisation is such that uncor-related angles yield a constant 1. corr FIG. 6: (Colour online) The number of correlated gluons n corr , and the correlated gluons typical transverse range, inpseudorapidity (2 ,
3) due to a trigger gluon in pseudorapidity( − , −
2) in b = 0 Au-Au with √ s NN = 200 GeV. R is thedistance from the center of the collision. flow observables would reinforce the picture of flow com-ing from systematic effects and fluctuations in the initialtransverse geometry.A more direct measure of the positional correlationbetween rapidities is to measure the transverse density ofgluons in the forward rapidity slice triggered by a gluonin the backward rapidity slice. An extra n corr gluonsare seen on top of the background in the correspondingtransverse position in the forward slice, due to the gluonchain from the trigger gluon that may pass through alsothe opposite rapidity bin. This correlation is analogous toa flux tube in other models. Fig. 6 shows central ( b = 0)Au-Au collisions with trigger particles at a distance R from the center of the collision.It is seen that in the center of the collision, the mostdense region, fewer gluons are correlated to the triggergluon. This is a sign of saturation, as the trigger gluonchain does not propagate unhindered to the other ra-pidity slice, but has a large probability of merging withother chains before that, giving fewer correlated gluons inthe other slice. In the peripheral region of the collisions, n corr ≈ .
4, which is similar to pp collisions in DIPSY .This shows that the approximation of binary collisions,where flux tubes propagate independently through ra-pidity, breaks down in the dense region of a heavy ioncollisions.The shorter range of the correlation in the center ofthe collision is again due to the saturated environmentpreferring smaller dipoles through the dipole swing.
V. CONCLUSIONS
A new method to generate t = 0 states of high en-ergy heavy ion collisions has been introduced with DIPSY .The model has been tuned to pp and γ ∗ p minimum biasevents, and no new parameters have been introduced forheavy ions. DIPSY is based on BFKL and includes allmergings and splittings of gluon chains, describing allfluctuations and correlations in a saturated environment. t = 0 events for Au-Au and Cu-Au at RHIC, and Pb-Pb at LHC are generated and avaliable online, see sec. II.Eccentricities ε , , , and their angles φ , , , havebeen studied and give results similar to other mod-els, with fluctuation-driven quantities generally slightlylarger. Predictions were made for correlations between eccentricities in the forward and backward regions, whichputs the concept of azimuthal flow from initial state ge-ometry to the test. v for mid-centrality classes are ex-pected to be significantly more correlated, both in ampli-tude and orientation, over rapidity than other momentsas well as the elliptic flow at central collisions.Correlations over long range in rapidity are found be-tween the transverse gluon distributions, as is expectedfrom a flux tube approach. Studying head on Au-Au col-lisions at RHIC energy, the correlation is weaker by afactor 2 in the center compared to the peripherial regionof the collision. The weaker correlation is caused by gluonchain mergings, as a gluon chain (or correspondingly aflux tube) passing the trigger rapidity slice, does not nec-cessarily imply that the same chain (flux tube) pass theother rapidity slice. Also the range of the transverse cor-relation is shortened a factor 2, due to the smaller averagedipole size in a saturated environment, corresponding toa larger saturation scale Q s . VI. ACKNOWLEDGEMENTS
First I wish to thank G¨osta Gustafson and LeifL¨onnblad, without whom this work could not have beendone. The work depends on the model for nucleon po-sitions which Andras Ster helped us include in
DIPSY . Iam also grateful for valuable discussion with Peter Chris-tiansen, Jamie Nagle and Jean-Yves Ollitrault. [1] J.-Y. Ollitrault, Phys. Rev.
D46 , 229 (1992).[2] B. Alver, B. Back, M. Baker, M. Ballintijn, D. Barton,et al., Phys.Rev.
C77 , 014906 (2008), 0711.3724.[3] R. S. Bhalerao, M. Luzum, and J.-Y. Ollitrault (2011),1107.5485.[4] B. Alver et al. (PHOBOS), Phys. Rev. Lett. , 242302(2007), nucl-ex/0610037.[5] B. Alver and G. Roland, Phys. Rev. C81 , 054905 (2010),1003.0194.[6] H. Petersen, G.-Y. Qin, S. A. Bass, and B. Muller, Phys.Rev.
C82 , 041901 (2010), 1008.0625.[7] B. Schenke, S. Jeon, and C. Gale, Phys. Rev. Lett. ,042301 (2011), 1009.3244.[8] B. Alver et al. (PHOBOS), Phys. Rev.
C81 , 024904(2010), 0812.1172.[9] B. Alver et al. (PHOBOS), Phys. Rev. Lett. , 062301(2010), 0903.2811.[10] B. I. Abelev et al. (STAR) (2008), 0806.0513.[11] D. Teaney and L. Yan, Phys. Rev.
C83 , 064904 (2011),1010.1876.[12] F. G. Gardim, F. Grassi, Y. Hama, M. Luzum, and J.-Y.Ollitrault, Phys. Rev.
C83 , 064901 (2011), 1103.4605.[13] E. Iancu and R. Venugopalan (2003), hep-ph/0303204.[14] H.-J. Drescher and Y. Nara, Phys.Rev.
C75 , 034905(2007), nucl-th/0611017.[15] H. J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog,and K. Werner, Phys. Rept. , 93 (2001), hep-ph/0007198. [16] C. E. Aguiar, Y. Hama, T. Kodama, and T. Osada, Nucl.Phys.
A698 , 639 (2002), hep-ph/0106266.[17] J. Takahashi et al., Phys. Rev. Lett. , 242301 (2009),0902.4870.[18] H. Petersen, C. Greiner, V. Bhattacharya, and S. A. Bass(2011), 1105.0340.[19] Z.-W. Lin, C. M. Ko, B.-A. Li, B. Zhang, and S. Pal,Phys. Rev.
C72 , 064901 (2005), nucl-th/0411110.[20] J. Xu and C. M. Ko (2011), 1103.5187.[21] M. Gyulassy and X.-N. Wang, Comput. Phys. Commun. , 307 (1994), nucl-th/9502021.[22] C. Flensburg, G. Gustafson, and L. L¨onnblad (2011),1103.4321.[23] E. Avsar, G. Gustafson, and L. L¨onnblad, JHEP , 062(2005), hep-ph/0503181.[24] E. Avsar, G. Gustafson, and L. L¨onnblad, JHEP , 012(2007), hep-ph/0610157.[25] C. Flensburg, G. Gustafson, and L. L¨onnblad, Eur. Phys.J. C60 , 233 (2009), 0807.0325.[26] M. Rybczynski and W. Broniowski (2010), 1012.5607.[27] H. De Vries, C. De Jager, and C. De Vries, Atom.DataNucl.Data Tabl. , 495 (1987).[28] B. H. Alver, C. Gombeaud, M. Luzum, and J.-Y. Olli-trault, Phys. Rev. C82 , 034913 (2010), 1007.5469.[29] B. Alver et al. (PHOBOS Collaboration), Phys.Rev.Lett. , 142301 (2010), nucl-ex/0702036.[30] P. Sorensen (STAR Collaboration), J.Phys.G