Fast Dynamical Evolution of Hadron Resonance Gas via Hagedorn States M. Beitel,1 C. Greiner,1 and H. Stoecker1,2,3 1Institut fu¨r Theoretische Physik, Goethe-Universita¨t Frankfurt am Main, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany 2GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH, Planckstraße 1, D-64291 Darmstadt, Germany 3Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany Hagedorn states are the key to understand how all hadrons observed in high energy heavy ion collisions seem to reach thermal equilibrium so quickly. An assembly of Hagedorn states is formed in elementary hadronic or heavy ion collisions at hadronization. Microscopic simulations within the transport model UrQMD allow to study the time evolution of such a pure non-equilibrated Hagedorn state gas towards a thermally equilibrated Hadron Resonance Gas by using dynamics, 6 which unlike strings, fully respect detailed balance. Propagation, repopulation, rescatterings and 1 decaysofHagedorn statesprovidetheyieldsofall hadronsuptoamassofm=2.5GeV. Ratios of 0 feed down corrected hadron multiplicities are compared to corresponding experimental data from 2 the ALICE collaboration at LHC. The quick thermalization within t=1 2fm/c of the emerging − n Hadron Resonance Gas exposes Hagedorn states as a tool to understand hadronization. a J 1 Before the emergence of QCD as the theory of strong perons. Via a coupled set of rate equations,one for each 1 interactions physicists already developed several models species,thechemicalequilibrationtimesforp,K,Λareof andideastodescribeobservablesinconnectionwithhigh order t∼1−2fm/c. The inclusion of Hagedorn spectra ] energetic particle collisions of several types. A promi- inthepartitionfunctionsoftheHRGprovidesalowering h p nentphenomenologicalmodelistheStatisticalBootstrap of the speed of sound, cs, at the phase transition and a - Model (SBM) emerging from first applications of statis- significant decrease of the shear viscosity to entropy ra- p ticalmeans[1–3]. Especiallyin(ultra-)relativisticheavy tio η/s [11, 12]. This results are in good agreementwith e h ioncollisionsthermalmodels[4,5]seemtoshowanexcel- corresponding lattice calculations [13–15]. The general [ lentdescriptionofvarioushadronicparticlemultiplicities impact of HS on the occurrence of various phases from by choosing a temperature, volume and chemical poten- HRG to deconfined partonic matter was also studied in 1 v tials. Inthispaperweprovideamicroscopicexplanation various MIT bag model descriptions [16–22]. There the 4 for the validity of the thermal model and the very fast Hagedorn spectrum ρ(m) with 7 equilibration at hadronization. In our approach the sys- 4 m tem temperature equals the Hagedorn temperature. 2 ρ(m)=f(m)exp (1) 0 The Hagedorn temperature TH is the highest tempera- (cid:18)TH(cid:19) . ture thatsystemsofhadronicparticleswithanexponen- 1 is applied, where the pre-function f(m) mimics the tially growing spectrum of (mass) states can have, be- 0 low-mass part of the Hagedorn spectrum. yond which the partition function diverges [1]. Beyond 6 1 TH deconfinement will start, and, depending on the un- To describe the hadronization of jets in e+-e− an- : dersaturation of quarks in the matter [6] a Yang Mills v nihilation events the concept of color strings [23] was i plasma or a 2+1 flavour Quark Gluon Plasma (QGP) X applied. Here the basic idea is that partons tend to will form. T in the present approach is identified with H cluster in color singlet states from the very beginning r the critical temperature T . The Hagedorn states (HS) a c of the generated event. These clusters decay to smaller within the SBM are the presently not yet discovered clusters until some cut-off scale is reached and hadrons heavy “missing hadron states” which can be attributed are formed [24, 25]. In the framework of RQMD to the exponential part of the Hagedorn spectrum and multi-particle collisions and decays were considered as which are most abundant in the vicinity of T . In [7] H particle clusters for which separable interactions have HS being createdin multiparticle collisions areproposed to exist [26]. A statistical approach [27] considers the to serve as a tool for a microscopic description of the hadronization of quark matter droplets to hadrons phase transition from HRG to QGP. The HS can have within the microcanonical ensemble. According to a anyquantumnumbercombinationcompatiblewiththeir n-body phase space these quark matter droplets decay mass. This property of HS was applied in [8–10] in or- via Markov chains into various hadron configurations. der to understand why (multi-) strange (anti-) hyperons A further statistical treatment of HS was performed in attheRelativisticHeavyIonCollider(RHIC)chemically Ref. [28] using a simplistic description of the Hagedorn equilibratemuchfasterthanthetypicallifetimeofafire- spectrum. The authors regarded one single massive ball(∼10fm/c). InthevicinityofT themostabundant H (m∼100GeV) initial resonance which consecutively mesons, i.e. pions and kaons, do ’cluster’ to Hagedorn cascadesdownviadecaychainsuntilonlystablehadrons states which in turn can decay into several kinds of hy- as protons, neutrons, pions etc. are left. The various 2 terms like ’clusters’, ’quark matter droplets’,’massive cascade-type transport model which is based on a resonances’ or ’bags’ could be identified with possible geometrical interpretation of cross sections as it is now Hagedorn states. implemented in UrQMD. The total decay width is expressed in terms of HS creation cross section σ which Following our recent approach [29] we here imple- are in the range Γ ≈ 0.4 − 3.5GeV. Various results ment for the first time Hagedorn states in microscopical can be found in [29]. There are some light HS whose dynamical box simulations: In contrast to a non- total decay width exceeds the mass. Their total yield covariant bootstrap equation [30, 31] we have employed islessthan15%,sowedecidedtoignorethissmalleffect. a covariant one The HS with mass m, quantum numbers C~, total m m−m1 R3 decay width Γ and the various branching ratios BR τ (m)= dm dm τ (m )m (2) C~ C~ 3πm Z 1Z 2 C~1 1 1 have , in the present work, been implemented fully into C~X1,C~2m01 m02 the UrQMD model. The evolution from nonequilibrated ×τ (m )m p (m,m ,m )δ(3) , initial HS gas through detailed balance between HS cre- C~2 2 2 cm 1 2 C~,C~1+C~2 ations and HS decays to equilibrated HRG is simulated in a 10 ∗ 10 ∗ 10fm3 cubic box with reflecting walls. where τ (m) denotes the mass density of Hagedorn C~ Each simulation is done at an energy density between stateswithchargeC~ =(B,S,Q)andmassm,whereB is ǫ = 0.3−2.0GeV/fm3 in steps of ∆ǫ = 0.1GeV/fm3. the baryonnumber, S the strangenessandQ the electric The quantum numbers of all initial heavy HS in the box charge. The terms τ andτ standfor spectra of both C~1 C~2 are assumed to have C~ = (0,0,0) to simulate an un- constituents which make up spherical HS with radius R charged gas. The time evolution will therefore produce whosedensityisdescribedbyτ (m). Intherestframeof C~ all charges C~ only by the decays of HS into charged createdHS,p denotesthemomentaofthedecayprod- cm hadrons and lighter HS, as is the case in ultrarelativistic ucts and ensures strict energy-momentum conservation. heavy ion collisions at RHIC and at the LHC, where all Exact charge conservation is applied too. This highly the net charges at midrapidity are close to zero, e. g. a non-linear integral equation of Volterra type is solved net baryon density of ρ ≈ 0 has been measured. Thus B numerically. The initial input forτ are spectralfunc- C~1,2 the initial ensemble of (heavy) HS creates dynamically tionsofthehadronictransportmodelUrQMD(Ultrarel- all kinds of (light) hadrons until chemical equilibrium ativisc Quantum Molecular Dynamics) [32] consisting of between HS and hadrons is achieved. As a more 55 baryons and 32 mesons. Inserting the hadronic spec- conventionalalternate conceptual point of view consider tral functions into the r.h.s. of Eq. 2 results in first HS the following picture of hadronization : In an (ultra-) consisting of two hadrons only. In the subsequent steps, relativistic collision of two heavy ions a large QGP these created HS serve as constituents of next heavier drop is being created. This drop expands, cools and HS, which now may consist of one HS and one hadron decays into smaller droplets/HS close to the transition or of two lighter HS. This means that every Hagedorn temperature T ≈ T . The HS propagate, collide with H c spectrum onthe l.h.s. sooneror later will appearas con- each other and with hadrons, until they decay. Among stituent on the r.h.s. of Eq. 2 to create next heavy HS. thedecayproductsoftheHSatfirsthadronswillappear The upper Hagedorn spectrum mass m is increased by quasi instantly. The hadrons and the HS may create ∆m = 0.01GeV until a final value of m = 8.6GeV is new HS or hadrons, which then can go on to collide reacheddue to computationallimitations. Numericalre- elastically or inelastically with each other. As long sults of Eq. 2 are provided in [29]. We find that the as the system stays at a high temperature T ≈ T H Hagedorn temperature rises when R gets smaller and is the dynamical interplay between hadrons and HS will very weakly dependent on charges C~. In addition with drive both into thermal and chemical equilibrium. To theHagedornspectrumwewereabletoderiveanexpres- examine this process we distribute initial HS uniformly sion for HS total decay width Γ in momentum and configuration space and demand: m m−m1 σ N0 N0 ΓC~ (m)= 2π2τC~ (m)C~X1,C~2mZ01dm1mZ02dm2τC~1(m1)τC~2(m2) E =Xi=1Ei, ~p=Xi=1p~i =0 (4) ×pcm(m,m1,m2)2δC(~3,)C~1+C~2. (3) Tlahstesiunnittiialltn=um2b0efrmo/fcHaSndisaslletretsoulNts0a=re2a0v0e.raEgaecdhorvuenr To compute Γ we applied the principle of detailed 1200 runs. C~ balance between binary collisions to create HS and Fig. 1 shows the final mass distribution of all HS which their decays into two particles, i. e. 2 → 1 and arises when a system with exponential growth of mass 1 → 2 only. The limitation to 2 ↔ 1 processes is states is subjected to the Boltzmann distribution. The necessary when implementing HS into a customary equilibrated mass distribution of HS shown in Fig. 1 re- 3 1 *ε=0.3 ε=1.0 ε=2.0 N /N 12 ε=0.5 ε=1.4 HS init. 0.9 N0=200 EHS/Etot ε=0.8 ε=1.7 N0=200 0.8 HS init. HS tot 10 t=20 fm/c 0.7 t=5 fm/c MHS/Mtot 8 n 0.6 M o d cti 0.5 N/ 6 ra 0.4 d f 0.3 4 0.2 2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 3 4 5 6 7 8 9 ε [GeV/fm3] M [GeV] *ε/[GeV/fm3] FIG. 1: Mass distribution of HS in thermal equilibrium 0.3 2(t.0=G2e0Vf/mfm/c3). for energy densities in the range ǫ = 0.3 − 0.25 tHN=0S2= 0i2n f0imt0./c Λnp ΩΣΞ0-- TΚπH00 0.2 V] sτu∼ltsexfrpo(mm/tThe)caonndvotlhuetiBonoltozfmtahnenHdiastgreidbourtnionspfe(cEtr)um∼ Ge 0.15 exp(−E/T). HObserve that for higher energy densities T [ 0.1 more HS with higher masses are formed. The impact of HS on the system’s total particle number, energy and 0.05 mass after t = 5fm/c is shown in Fig. 2. At the largest energy density, ǫ = 2.0GeV/fm3, already every fourth 0 0.3 0.8 1.3 1.8 particle is a HS and more than ∼ 60% of the total en- ergy and ∼ 70% of the total mass are occur by HS. All ε [GeV/fm3] results are in full accordance to the SBM: HS appear most abundantly at T , which is reached when ǫ → ∞. H FIG.2: Fraction(top)oftotalmultiplicities,energyandmass Wealsoobservethatwithincreasingenergydensitymore occupied by HS in thermal equilibrium (t 5fm/c) as func- and more of the available total energy is converted into tion of energy density. Boltzmann slopes≥(temperatures) of (massive) HS rather than being distributed over the ki- hadrons (bottom) as function of energy density in thermal netic degrees of freedom. The latter result is backed by equilibrium (t=20fm/c). Red solid line denotes the Hage- the dependence of hadrons’ Boltzmann slopes T on ǫ as dorn temperature TH. showninthe lowerpartofFig.2. Increasingenergyden- sityǫcausesthetemperatureT ofallhadronstoconverge totheHagedorntemperatureTH. Thisresultcontradicts their decay chains as discussed in [29]. The very fast the usualHRG thermodynamics, where the temperature thermalization in our simulations of t ≤ 2fm/c is now grows with the energy density beyond any limit. Our obtainedfromtheinitialdecayingHSinthesystem. The result manifests the SBM statement upper figure shows that the number of HS drops down within t = 1fm/c and then saturates. The emerging lim T =TH. (5) hadrons and hadronic resonances are build up during ǫ→∞ by these decays and by the subsequent regenerations of Note that decay chains of a single massive HS already HS on such short time scale (lower figure). The sum of show slopes with a Hagedorn temperature [29], - a con- the yield hadrons stemming from feed down of the HS sistent picture. and of hadronic resonances (shown in the lower figure) Observe the very fast thermalization time accounts for the total stable particle yields in the upper t = 1 − 2fm/c, where hadrons, hadron resonances figure. Within t =1fm/c more than a half of the initial and HS interact rapidly, changing energy and quantum HS has decayedinto hadronsreachinga stationaryvalue numbers. Fig. 3 shows the time evolution of pions, at t = 1 − 2fm/c and a further moderate saturation. kaons, protons and lambdas as ’direct’ decay hadrons The very fast chemical equilibration occurs by means and feed down of hadron resonances and of HS. The of decays, recreation and rescatterings of HS in such later feed down corresponds to (potential) hadronic a dense environment. This is in contrast to standard particles ’stored’ in the existing HS as calculated via hadronic transport approaches of [33, 34]. 4 60 N0=200 π+/10 Λ 60 N0=200 π+/10 Λ HS init. Κ+/2 HS/5 HS init. Κ+/2 HS/5 50 ε=1.0 GeV/fm3 p 50 t=5 fm/c p ultiplicity 3400 multiplicity 234000 m 20 10 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0 5 10 15 20 ε [GeV/fm3] t [fm/c] FIG. 4: Total multiplicity dependence on energy density of 60 π+ /10 Κ+ /2 Λ π+,K+,pandΛ0 closetochemicalequilibrium(t=5fm/c). π+had./10 HpS. Λhad. In case of HS dynamical multiplicities were considered. 50 Κ+HS. /2 phad. HSH/5S. had. HS. 40 p-p Pb-Pb 0.3 0.8 1.0 2.0 y cit HS init. ε=1.0 GeV/fm3 K−/π− 0.123(14) 0.149(16) 0.192 0.197 0.193 0.185 ultipli 30 N0=200 Λp¯//ππ−− 00..003523((46)) 00..003465((55)) 00..000175 00..002429 00..002542 00..002690 m 20 Λ/p¯ 0.608(88) 0.78(12) 0.475 0.456 0.469 0.499 Ξ−/π− 103 3.000(1) 5.000(6) 1.565 6.492 5.769 7.106 10 Ω−/π−∗103 - 0.87(17) 0.137 0.815 0.823 0.994 ∗ 0 TABLE I: Comparison of particle multiplicity ratios from 0 1 2 3 4 5 theory vs. p-p at √sNN = 0.9TeV [35] and Pb-Pb at t [fm/c] √sNN =2.76TeV[36–38],bothfromALICEatLHC.Calcu- lated values are listed for some energy densities in the range ǫ=0.3 2.0GeV/fm3. Numbersinbracketsdenotetheerror FIG. 3: Time evolution of total (feed down corrected) mul- − tiplicities (top) of π+, K+, p and Λ0 at energy density in thelast digits of theexperimentalmultiplicity ratios. The ǫ = 1.0GeV/fm3. Time evolution (bottom) of first 5 fm/c statistical error is less than 25% for strange baryons. ofdirectmultiplicitiesplusfeeddowncontributionsfromres- onance decay (had.) and from HS decays (HS.) for same hadrons as mentioned above. In case of HS direct multiplici- thermally equilibrated. ties were considered. In summary, it was shown that a system of HS, e. g. as emerging of large QGP drops created in heavy Fig. 4 shows that in chemical equilibrium total mul- ion collisions gives a new insight how hadronization tiplicities rise nearly linear with energy density in the can take place. Starting in non-equilibrium dynamical evolving system originating from the initial assembly decay and (re-) creation of HS provide on a very short of HS. The yields are determined predominantely by time scale of t = 1 − 2fm/c all hadrons of the HRG the particle’s masses. HS exhibit the steepest slope as advocated over the years in [4, 5]. Potential decays as demanded by SBM. Tab. I confronts the calculated of HS might also explain the finding of e− − e+ [39] hadron multiplicity ratios at different energy densities and p − p¯ [40] within a thermal model analysis. The with corresponding experimental values as obtained by implementation in microscopic transport models of the ALICE collaboration at the LHC. To demonstrate full heavy ion collisions sets a new venue at future that a real thermal (chemical+kinetic) equilibrium is investigations, also for finite net baryon densities at reached we compare in Fig. 5 simulated hadron multi- NICA facilities in Dubna and CBM experiments at plicities at ǫ=1.0GeV/fm3 with multiplicities provided the FAIR facility which is build adjecent to the GSI by a standard HRG thermal model. A temperature of in Darmstadt. Multibaryonic and multistrange HS T ∼ 0.154GeV is assumed in the thermal model and can serve as an energy reservoir for production of rare for ǫ = 1.0GeV/fm3 in the full evolution as depicted in hadronic particles. 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