A model for energy, pressure and flow velocity distributions at the beginning
of relativistic heavy ion collisions is presented, which can be used as initial
condition for hydrodynamical calculations. The results show that QGP forms a
tilted disk, such that the direction of the largest pressure gradient stays in
the reaction plane, but deviates from both the beam and the usual transverse
flow directions. Such initial condition may lead to the creation of "antiflow"
or "third flow component".Comment: Talk given at the New Trends in High-Energy Physics, Yalta (Crimea),
  Ukraine, May 27 - June 4, 2000. 8 pages, 2 figures. The misprint t_0 instead
  of x_0 has been corrected in several place

Csernai, L. P.

Magas, V. K.

Strottman, D.

English

arXiv

A model for energy, pressure and flow velocity distributions at the beginning of relativistic heavy ion collisions is presented, which can be used as initial condition for hydrodynamical calculations. The results show that QGP forms a tilted disk, such that the direction of the largest pressure gradient stays in the reaction plane, but deviates from both the beam and the usual transverse flow directions. Such initial condition may lead to the creation of antiflow or third flow component

Csernai, L.

Magas, V.

UNT Digital Library

Approved forpublic release;distribution is unlimited.Title: The Source of Elliptic Flow and Initial Conditions forHydrodynamical CalculationsAuthor(s): D~iel D. Strottman, LANLL.P. CsernaiV.K. MagasSubmitted to: Iproceedings of Crimea 2000Los AlamosNATIONAL LABORATORYLos Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by the University of California for the U.S.Department of Energy under contract W-7405- ENG-36. By acceptance of this article, the publisher recognizes that the US. Governmentretains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or to allow others to do so, for U.S.Government purposes. Los Alamos National Laboratory requests that the publisher identify this article as work performed under theauspices of the U.S. Department of Energy. Los Alamos National Laboratory strongly supports academic freedom and a researcher’s right topublish; as an institution, however, the Laboratory does not endorse the viewpoint of a publication or guarantee its technical correctness.Form 836 (10/96)DISCLAIMERThis repoti was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.DISCLAIMERPortions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.1 .The source of elliptic flow and initial conditions for hydrodynamical calculationsV.K. Magasl, L.P. Csernai, 12 and Daniel D. Strottman 3lSection for Theoretical and Computational Physics, Department of PhysicsUniversity of Bergen, Allegaten 55, 5007 Bergen, Norway2 KFKI Research Institute for Particle and Nuclear Physics~p~~j~.~~P. O. Box 49, 1525 Budapest, Hungary3 Theory Division, Los Alamos National Laboratory,DEC 13 211(18LOS Akunos, NM, 87454, USA m-hA model for energy, pressure and flow velocity distributions at the beginning of relativistic heavy ion collisionsis presented, which can be used as initial condition for hydrodynamical calculations. The results show that QGPforms a tilted disk, such that the direction of the largest pressure gradient stays in the reaction plane, but deviatesfrom both the beam and the usual transverse flow directions. Such initial condition may lead to the creation of“ antiflow” or “third flow component” [10].Introduction. Fluid dynamical models are widely used to describe ultra-relativistic heavy ioncollisions. Their advantage is that one can vary flexibly the Equation of State (EoS) of the matter andtest its consequences on the reaction dynamics and outcome. In energetic collisions of large heavy ions,especially if Quark-Gluon Plasma ( QGP) is formed in the collision, one-fluid dynamics is a valid andgood description for the intermediate stages of the reaction. Here, interactions are strong and frequent,so that other models, (e.g. transport models, string models, etc., assuming binary collisions, with freepropagation of constituents between collisions) have limited validity. On the other hand, the initialand final, Freeze-Out (FO), stages of the reaction are outside the domain of applicability of the fluiddynamical model.In conclusion, the realistic, and detailed description of an energetic heavy ion reaction requires a MultiModule Model, where the different stages of the reaction are each described with suitable theoreticalapproaches. It is important that these Modules are coupled to each other correctly: on the interface,which is a 3 dimensional hyper-surface in space-time with normal dd, all conservation laws should besatisfied (e.g. [T@VdaV] = O), and entropy should not decrease, [S~doP] > 0. These matching conditionswere worked out and studied for the matching at FO in detail in refs. [1].The final FO stages of the reaction, after hadronization, can be described well with kinetic modelswhere the matter is already dilute.The initial stages are more problematic. Frequently two or three fluid models are used to remedy thedifficulties, and to model the process of QGP formation and thermalization. [2, 3, 4] Here, the problem istransferred to the determination of drag-, friction- and transfer- terms among the fluid components, anda new problem is int reduced with the (unjustified) use of EoS in each component in a nonequilibratedsit uations, where EoS does not exist. Strictly speaking this approach can only be justified for mixturesof noninteract ing ideal gas components. Similarly, the use of transport theoretical approaches assumingdilute gases with binary interactions is questionable, as due to the extreme Lorentz contraction, inthe C.M. frame, enormous particle and energy densities, with the immediate formation of perturbativevacuum should be handled. Even in most parton cascade models these initial stages of the dynamics arejust assumed in form of some initial condition, with little justification behind.Our goal in the present work is to construct a model, based on the recent experiences gained in stringMonte Carlo models and in parton cascades. One important conclusion of heavy ion research in the lastdecade is that standard ‘hadronic’ string models fail to describe heavy ion experiments.All string models had to introduce new, energetic objects: string ropes [5, 8], quark clusters [6], fusedstrings [7], in order to describe the abundant formation of massive particles like strange antibaryons., .Based on this, we describe the initial moments of the reaction in the framework of classical (or coherent)Yang-Mills theory, following ref. [9] assuming larger field strength (string tension) than in ordinaryhadron-hadron collisions. In addition we now satisfy all conservation laws exactly, while in ref. [9] infiniteprojectile energy was assumed, and so, overall energy and momentum conservation was irrelevant. We donot solve simultaneously the kinetic problem leading to parton equilibration, but assume that the arisingfriction is such that the heavy ion system will be an overdamped oscillator, i.e. yo-yoing of the two heavyions will not occur. This assumption is based on recent string and parton cascade results.Formulation of model. Our basic idea is to generalize the model developed in [9], for collisions oftwo heavy ions and improve it by strictly satisfying conservation laws. First of all, we would create agrid in [z, y] plane (Z – is the beam axes, [z, a] – is reaction plane). We will describe the nucleus-nucleuscollision in terms of steak-by-streak collisions, corresponding to the same transverse coordinates, {x;, yj }.We assume that baryon recoil for both target and projectile arise from the acceleration of partons inan effective field ~~p, produced in the interaction. Of course, the physical picture behind this modelshould be based on chromoelectric flux tube or string models, but for our purpose we consider F~U as aneffective abelian field. Phenomenological parameters describing this field must be fixed from comparisonwith experimental data.Let describe the streak-streak collision.~P~n~=O, i=l,2, (2)‘in; is the baryon current of ith nucleus (we are working in the Center of Rapidity Frame (CRF), whichis the same for all streaks. The concept of using target and projectile reference frames has no advantageany more). We will use the parameterization:n? = piu~ , U! = (Cosh yz, sinh yi) . (3)T’~V is a energy-momentum flux tensor. It consists of five parts, corresponding to both nuclei and freefield energy (also divided into two parts) and one defines the QGP perturbative vacuum.(4)B – is the bag constant, the equation of state is F?i = c~ei, where ei and Pi are energy density andpressure of QGP.In complete analogy to electro-magnetic field(5)Oi = 83A; – dOA~ , (6)In our case the string tensions, ai, will have the same absolute value a and opposite sign (in completeanalogy to the usual string with two ends moving in opposite directions), and ai will be constant in thespace-time region after string creation and before string decay.To get the analytic solutions of the above equations, we use light cone variables(z)t) + (x+,x-), z+=t+.z .Following [9], we insist that el, gl, PI, A! are functions of z- only andIn terms of light cone variables:●yin? = ni,~ = pi(u~ + U~) = pie ,(9)e2, y2, p2, A; depend on Z+ only.(lo)where(~:z’)=(hi~~h$:~’2yi)+TF’)hi+ = (1+ c~)ei, hi– = (1 – c~)ei .(11)(12)(13)(14)At the time of first touch of two streaks, t = O, there is no string tension. We assume that strings arecreated, i.e. the sting tension achieves the value a at time t = to,corresponding to complete penetrationof streaks through each other.Conservation laws — String rope creation. In light cone variables eq. (2) may be rewritten as6Ln; + ~+n~ = O . (15)So, we have a sum of two terms, depending on different independent variables, and the solution can befound in the following way.d_n~ = a, i?+n~ = –a ,(16)n; = m- + (nl)o, n; = —a%+ + (nz)o .Since both n; and n; are positive (and also more or less symmetric) we can conclude that for our caseU=o.Finallyn~ = ple–gl = poego , n; = p2eg2 = poevo , (17)PI = ~oevo+w > p2 = poe~O–~2, (18)Let us come back to the energy-momentum tensor T~v. Based on eqs. (6, 7, 8) and taking intoaccount the Feynman gauge, dOA~ — ~3Ag = O, we can findAs mentioned before, after string creation, i.e. t > to,and before string decay we choose the stringtensions in the form:q=-q=o ’>o. (20)To satisfy the above choice and the Feynman gauge condition we take the vector potentials in the followingform:At = O, A; = –’ox– ,A; = –’ax+, A; = O .(21)In our calculations we used the parameterization:()2~.A ~ POV”GL ,m(22)where the typical values of A are around 0.05 — 0.06. Notice, that there is only one free parameter inparameterization (22). The typical values of o are 8 – 15 GeV/fm for co = 100 GeV per nucleon.The problem with eq. (1) is that we do not know what the really conserved quantities are. Using thedefinition of F’flV, eq. (6), we can rewrite eq. (1) asThe solution for n; and n;, eq. (17), shows as that second and fourth terms vanish. So, we can definenew energy-moment urn tensor ~~v, such that~pm” = () , (24)T~v = ~yip” +T~, = ~ (TF - A~nf +~P”A~n,a) + B#” (25)‘i ‘iUsing the exact definition of A: – eqs. (21) – we obtainrjjw = (h1+e2~l + 502 hl– + 4cn–nf )( h2+e2~’ +02 + 2crx+n~ h2– – 2ox+n;hl– + 2ox-nf hl+e–2y1 +02 – 2ox–n; + h2_ – 4ox~n; h2+e –aJ2 + 5.2 )Now the new conserved quantities areQO = J fOOdV = ~ I ?iOOdV ,i Q(26)(27)(28)Based on conservation of Qo, Q3 we can calculate rapidity, energy and baryon densities at the momentt = to, when the string with tension o is created. These new quantities are used as initial conditions forour differential eqs. (1, 2).(*)= (3&-pOE{:+C#%”8 S07flC#1+2’2)-POeO: +C##) ‘2’)(*) = (%)& -POEi:+C~)%-8S:flC3, (z2+2z’)-Poso:+c3,% - ’30)Here the ei is energy per nucleon. Now the proper baryon density can be found, p, (to) = PO&,&i(to)_—‘i=&– ‘“For x* > to we should solve eqs. (24), with boundary conditions+–nl (x = to)= poe=~o n~(x+ = to) = poe+yohl+(z- = to) = el(to)(l + c~) h2+(x+ = to) = el(to)(l + c;)?Jl(z- = to) = !/I(to)(31)3&!(fc+ = to) = 3/2(~o)CTl (x– = to)= –(7 02(Z+= to)= C7Let us present the complete analytical solution in the following formd (i+lzyi _ _J + —~(–) _ : + ~(-)i+’wo)bi ~ )(1 %’;’0)-+ ,“~+12vie2(to) (I+ c~)ehi+ = e(-)(-(-)2+ 1w(to) 1 –@ = pOevOet-)i+19i,wherexl =x–, X2=X+, i,j= 1,2 , i#j, and using the not ationsbi = CKLj+ 2apoe~0 ,dz = Ci – 2apOeV0e(-)’+’2V’(tO) ,~i = ez(to)(l + ~)e(–) ~+12W(to)aj )al = c1 + 2apOey0—2apOe90e2y1tto) , az = cz + 2apOey0Ci = ~((1 + c~)ez(to) – eO)/tO .Then the trajectories of nucleons (or cell elements) for both nuclei‘- dx e2Yl (2) =Xf(z; ) = q + Jtorxi — t,)>Ti– 2apOev0e–2v2(t0) ,are given by:‘+ ~x e–2Y2(*) =x;(q) = q + ~.20- *(x – to) +(*+ e-2y2(’0))2-2* [(l-=) ’”:’Y-l]~(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)where X. = z(O), for nucleon or cell element in the position z = z(O) at the time t = O.Recreation of the matter. As we may see from the trajectories, eqs, (40, 41), nucleons (or celldomains) will keep going in the initial direction up to the time t = ti,turn)then they will turn and gobackwards until the two streaks again penetrate through each other and new oscillation will start. Sucha motion is analogous to the “ Yo-Yo” motion in the string models. Of course, it is difficult to believethat such a process would really happen in heavy ion collisions, because of string decays, string-stringinteractions, interaction between streaks and other reasons, which are quite difficult to take into account.To be realistic we should stop the motion described by eqs. (40, 41) at some moment before the projectileand target cross again.We assume that the final result of collisions of two streaks after stopping the string’s expansionand after its decay, is one streak with homogeneous energy density distribution, ef, and baryon chargedistribution, nf, moving like one object with rapidity yf. We assume that this is due to string-stringinteractions and string decays. As it was mentioned above the typical values of the string tension, a,are of the order of 10 GeV/fm, and these may be treated as several parallel strings. The string-stringinteraction will produce a kind of “string rope” between our two streaks, which is responsible for finalenergy density and baryon charge homogeneous distributions. Now it is worth to mention that decay ofour “string rope” does not allow charges to remain at the ends of the final streak, as it would be if weassume full transparency.The homogeneous distributions are the simplest assumptions, which may be modified based on ex-perimental data. Its advantage is a simple expression for ef, nj, gf.The final energy density and rapidity, ef and yf, may be determined from conservation laws.(~2(1 + d) + 2C3 + JM2(1 + c;) + 2C:)2 + 4cf(M2 - 1)cosh2 Vf =2(1 + C;)(M2 – 1)>where we neglected B A lf next to Q. and introduced the notation M = (12+ 11)/(12– 11),&– BAlf‘f= ((1+ c:)cosh2 gf - C;) ~ it “(42)(43)figure=endtrajectory.ps ,height=9.OcmFigure 1: The typical trajectory of the ends of two initial streaks, corresponding to numbers of nucleons,nl and n2. Stars denote the stopping and turning points, where Yi = yf. From tO to the turning pointsstreak ends keep going in their initial direction according to eqs. (40, 41). Later the final streak startsto move like one object with rapidity, yf (42) in CRF.The typical trajectory of the streak ends is presented in Fig. 1. From to they move according to eqs.(40, 41) until they reach the rapidity yi = yf. Later the final string starts to move like one object withrapidity yf.The turning points can be found from the condition:Yi=yf,which gives for ith nucleus (~1 = x–, X2 = Z+)[q, t~~~ = to + 7~ 1 —* + e(–)i+12y2(to) ~2$ + e(–)i+12Vf.1(44)(45)Initial conditions for hydrodynamical calculations. In this section we present the results ofour calculations. We are interested in the shape of QGP formed, when string expansions stop and theirmatter is locally equilibrated. This will be the initial state for further hydrod ynamical calculations. Wemay see in Figs. 2, that QGP forms a tilted disk for b # O. So, the direction of fastest expansion, thesame as largest pressure gradient, will be in the reaction plane, but will deviate from both the beam axisand the usual transverse flow direction. So, the new flow component, called “ antiflow” or “third flowcomponent” , will appear in addition to the usual transverse flow component in the reaction plane. Withincreasing beam energy the usual transverse flow is getting weaker, while this new flow component isstrengthened. The mutual effect of the usual directed transverse flow and this new “ antiflow” or “thirdflow component” lead to an enhanced emission in the reaction plane. This was actually observed andwidely studies earlier and referred to as “elliptic flow”.Conclusions. Based on earlier Coherent Yang-Mills field theoretical models, and introducing effectiveparameters based on Monte-Carlo string cascade and parton cascade model results, a simplified model isintroduced to describe the pre fluid dynamical stages of heavy ion collisions at the highest SPS energiesand above. The model predicts limited transparency for massive heavy ions.figure=evo1055. ps,height=ll .OcmFigure 2: The Au+Au collisions, E. = 100 GeV/nucl, b = 0.5(R1 + R2), A = 0.055 (parameter Aintroduced in (22)), y = O (ZX plane through the centers of nuclei). We would like to notice that finalshape of QGP volume is a tilted disk % 45°, and the direction of the fastest expansion will deviate fromboth the beam axis and the usual transverse flow direction, and might be a reason for the third flowcomponent, as argued in [10].Contrary to earlier expectations, — based on standard string tensions of 1 GeV/fm which lead to theBjorken model type of initial state, — effective string tensions are introduced for collisions of massiveheavy ions, as a consequence of collective effects related to QGP formation. These collective effectsin central and semi central collisions lead to an effective string tension of the order of 10 GeV/frn andconsequently cause much less transparency than earlier estimates. The resulting initial locally equilibratedstate of matter in semi central collisions takes a rather unusual form, which can be then identified bythe asymmetry of the caused collective flow. Our prediction is that this special initial state may be thecause of the recently predicted “ antiflow” or “third flow component”.Detailed fluid dynamical calculations as well as flow experiments at semi central impact parametersfor massive heavy ions are needed at SPS and RHIC energies to connect the predicted special initial statewith observable.References[1][2][3][4][5][6][7][8][9][10]V.K. Magas, Cs. Anderlik, L.P. Csernai, F. Grassi, W. Greiner, Y. Hama, T. Kodama, 2s. L4z&- andH. Stoker, (nucl-th/9808024), Phys. Rev. C59 (1999) 388; (nucl-th/9806004), Phys. Rev. C59(1999) 3309; (nucl-th/9903045), HIP 9 (1999) 193; (nucl-th/9905054), Phys. Lett. B 459 (1999)33.A.A. Amsden et al., Phys. Rev. C17 (1978) 2080.L.P. Csernai et al., Phys. Rev. C26 (1982) 149.J. Brachmann, S. Soff, A. Dumitru, H. Stoker, J.A. Maruhn, W. Greiner, D.H. Rischke, L. Bravina,Phys. Rev. C61 (2000) 024909.T.S. Biro, H.B. Nielsen, J. Knoll, Nucl. Phys. B245 (1984) 449.K. Wegner, J. Aichelin, Phys. Rev. Lett. 76 (1996) 1027.N.S. Amelin, M.A. Braun, C. Pajares, Phys. Lett. B306 (1993) 312, Z. Phys. C63 (1994) 507.H. Sorge, Phys. Rev. C52 (1995) 3291.M. Gyulassy, L.P. Csernai, ~UC~. Phys. A460 (1986) 723.L.P. Csernai, D. Rohrich, nucl-th/9908034, Phys. Lett. B458 (1999) 454.

The source of elliptic flow and initial conditions for hydrodynamical calculations

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The source of elliptic flow and initial conditions for hydrodynamical calculations

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