Hadron yields in central nucleus-nucleus collisions, the statistical hadronization model and the QCD phase diagram
HHadron yields in central nucleus-nucleus collisions, thestatistical hadronization model and the QCD phasediagram
Anton Andronic
Institut f¨ur Kernphysik, Universit¨at M¨unster, 48149 M¨unster, Germany
Peter Braun-Munzinger
Research Division and EMMI, GSI Helmholtzzentrum f¨ur Schwerionenforschung,64291 Darmstadt, GermanyPhysikalisches Institut, Universit¨at Heidelberg, 69120 Heidelberg, GermanyInstitute of Particle Physics and Key Laboratory of Quark and Lepton Physics(MOE), Central China Normal University, Wuhan 430079, China
Krzysztof Redlich
University of Wroc(cid:32)law, Institute of Theoretical Physics, 50-204 Wroc(cid:32)law, Poland
Johanna Stachel
Physikalisches Institut, University of Heidelberg, 69120 Heidelberg, GermanyThe description of hadron production in relativistic heavy-ion collisionsin the statistical hadronization model is very good, over a broad range ofcollision energy. We outline this both for the light (u, d, s) and heavy(charm) quarks and discuss the connection it brings to the phase diagramof QCD.
1. Introduction
If one compresses or heats nuclear matter to higher densities and/or hightemperatures one expects [1, 2, 3, 4] that quarks are no longer confined butcan move over distances significantly larger than the size of the nucleon.Such a deconfined state of matter, the Quark-Gluon Plasma (QGP) [5],is likely to have existed in the Early Universe within the first (about 10)microseconds after its creation in the Big Bang [6] and is studied experi-mentally and theoretically via collisions of nuclei at high energies [7, 8]. One (1) a r X i v : . [ nu c l - t h ] J a n wroclaw2020 printed on January 15, 2021 stage in the complex dynamics of the system produced in heavy-ion colli-sions is that of the chemical freeze-out, at which the abundance of hadronspecies is fixed (frozen), addressed phenomenologically within the statisti-cal hadronization model (SHM) [9]. The value of the crossover temperature T c at vanishing µ B is currently calculated in Lattice QCD (LQCD) to be156.5 ± . ± . T c with increasing µ B as long as µ B (cid:46) Z can be very well approx-imated within the framework of the hadron resonance gas, as long as thetemperature stays below T c .The grand canonical partition function for specie (hadron) i is:ln Z i = V g i π (cid:90) ∞ ± p d p ln[1 ± exp( − ( E i − µ i ) /T )] (1)with + for fermions and − for bosons, where g i = (2 J i + 1) is the spindegeneracy factor, T is the temperature, E i = (cid:113) p + m i the total energy; µ i = µ B B i + µ I I i + µ S S i + µ C C i are the chemical potentials ensure conser-vation (on average) of baryon, isospin, strangeness and charmness quantumnumbers. Three initial conditions help fixing ( I i , µ S , µ C ): i) isospin stop-ping identical to baryon stopping: I tot / (cid:80) i n i I i = N totB / (cid:80) i n i B i , with I tot , N totB isospin and baryon numbers of the system (proportional to µ B / µ B reflecting baryon stopping in the collision); ii) vanishing net initialstrangeness: (cid:80) i n i S i = 0; iii) vanishing net initial charmness: (cid:80) i n i C i = 0.On needs for the calculations the knowledge of the complete hadron spec-trum and the default constitutes what is listed by PDG [15]; the presenceof resonances corresponds to attractive interactions among hadrons. Tradi-tionally, repulsive interactions are modelled with an ’excluded volume’ pre-scription [16]. For weak repulsion, implying excluded volume radii r ≤ . T and µ B are little affected. Strong repulsioncannot be modelled that way: significantly larger r values lead to, amongothers, unphysical (superluminous) equations of state, in contra-distinctionto results from LQCD. Other approaches, like temperature-dependent reso-nance widths [17] were recently proposed, but lack full consistency. A con-sistent approach is the implementation employing the S-matrix formulationof statistical mechanics with measured pion-nucleon interactions including, roclaw2020 printed on January 15, 2021 importantly, also non-resonant components [18]. In this approach, currentlyimplemented only for µ B (cid:39)
2. Statistical hadronization of light quarks
In practice, T CF , µ B , and V , the parameters at chemical freeze-out aredetermined from a fit to the experimental data. For the most-central (0-10%) Pb–Pb collisions at the LHC, the best description of the ALICE data(see [19] and ref. therein) on yields of particles in one unit of rapidityat midrapidity, is obtained with T CF = 156 . ± . µ B = 0 . ± . V = 4175 ±
380 fm (corresponding to a slice of one unit ofrapidity, centered at mid-rapidity) [18], shown in Fig. 1. The standarddeviations quoted here are exclusively due to experimental uncertaintiesand do not reflect the systematic uncertainties connected with the modelimplementation. - - - - - -
10 110 y / d N Y i e l d d =2.76 TeV, 0-10% centrality NN s Pb-Pb Data, ALICEStatistical HadronizationData, ALICEStatistical Hadronization + p - p + K - K s0 K f p p L L - X + X - W + W d dHe He H L H L He He D a t a / M ode l + p - p + K - K s0 K f p p L L - X + X - W + W d d He He H L H L He He Mass (GeV) - - - - - - -
10 110 Y i e l d pe r s p i n d . o .f. + p + K p f L - X - W d He H L He =2.76 TeV, 0-10% centrality NN s Pb-Pb
Statistical Hadronizationtotal (after decays)primordial (thermal)
Data, ALICEparticlesantiparticles
Fig. 1. Left: Hadron yields d N/ d y measured in central Pb–Pb collisions at the LHCand the best fit with SHM. The lower panel shows the ratio of data and model withuncertainties (statistical and systematic added in quadrature) of the data. Right:Mass dependence of hadron yields divided by the spin degeneracy factor (2 J + 1).For SHM, plotted are the “total” yields, including all contributions from high-mass resonances (for the Λ hyperon, the contribution from the electromagneticdecay Σ → Λ γ , which cannot be resolved experimentally, is also included), andthe (“primordial”) yields prior to strong and electromagnetic decays. Very good agreement is obtained between the measured particle yieldsand SHM over nine orders of magnitude in abundance values and encom- wroclaw2020 printed on January 15, 2021 passes strange and non-strange mesons, baryons including strange and multiply-strange hyperons as well as light nuclei and hypernuclei and their anti-particles. The initially-observed overprediction of data by the model forproton and antiproton yields (a deviation of 2.7 σ ) is entirely accounted forvia the S-matrix treatment of the interactions [18] included here (for con-sistency the excluded-volume correction is not applied anymore). It wasrecently shown that the addition (compared to what is listed by PDG [15])of about 500 new states predicted by LQCD and the quark model does leadto a deterioration of the fit, while no change is observed when the S-matrixtreatment is employed [20].The thermal origin of all particles including light nuclei and anti-nuclei isparticularly transparent when inspecting the dependence of their yields withparticle mass, shown in the right panel of Fig. 1. We note that the yieldsof the measured lightest mesons and baryons, ( π, K, p, Λ) are substantiallyincreased relative to their primordial thermal production by the resonancedecay contributions (for pions, e.g., the decay contribution amounts to 70%of the total yield). For the subset of light nuclei, the SHM predictionsare, however, not affected by resonance decays. For these nuclei, a smallvariation in temperature leads to a large variation of the yield, resulting ina relatively precise determination of the freeze-out temperature T nuclei =159 ± T CF extracted above.The rapidity densities of light (anti)-nuclei and hypernuclei were actu-ally predicted [21], based on the systematics of hadron production at lowerenergies. It is nevertheless remarkable that such loosely bound objects (thedeuteron binding energy is 2.2 MeV, much less than T CF ≈ T c ≈
157 MeV)are produced with temperatures very close to that of the phase boundaryat LHC energy, implying any further evolution of the fireball has to be closeto isentropic. The detailed production mechanism for loosely bound statesremains an open question (see recent review [22]). One possibility, con-sidered already long ago [23], is that such objects, at QGP hadronization,are produced as compact, colorless droplets of quark matter with quantumnumbers of the final state hadrons.The thermal nature of particle production in ultra-relativistic nuclearcollisions has been experimentally verified not only at LHC energy, butalso at the lower energies of the RHIC, SPS and AGS accelerators. Theessential difference is that, at these lower energies, the matter-antimattersymmetry observed at the LHC is lifted, implying non-vanishing values ofthe chemical potentials. Furthermore, in central collisions at energies below √ s NN ≈ roclaw2020 printed on January 15, 2021 (GeV) NN s ( M e V ) C F T Statistical Hadronization yields y /d N d yields p (GeV) NN s ( M e V ) B m parametrizations (GeV) NN s y i e l d r a t i o y / d N d - - - -
10 110 + p p/ - p /pd/p Points: DataLines: Statistical Hadronization (GeV) NN s y i e l d r a t i o y / d N d + p / + K - p / - K + p / L Fig. 2. Left: Energy dependence of chemical freeze-out parameters T CF and µ B .The results are obtained from the SHM analysis of hadron yields (at midrapidity,d N/ d y , and in full phase space, 4 π ) for central collisions at different energies.Right: Collision energy dependence of the relative abundance of several hadronspecies (the data are compiled in [24, 25]). for the conservation laws [26, 27]. Similar considerations apply for the de-scription of particle yields in peripheral nuclear and elementary collisions. Aconsequence of exact strangeness conservation is the suppression of strangeparticle yields when going from central to peripheral nucleus-nucleus colli-sions or from high multiplicity to low multiplicity events in proton-protonor proton-nucleus collisions [28, 29].While µ B decreases smoothly with increasing energy, the dependenceof T CF on energy exhibits a striking feature which is illustrated in Fig. 2: T CF increases with increasing energy (decreasing µ B ) from about 50 MeVto about 158 MeV, where it exhibits a saturation for √ s NN >
20 GeV. Theslight increase of this value compared to T CF = 156 . T CF observed in Fig. 2 lends support to the earlier proposal[30, 31, 32] that, at least at high energies, the chemical freeze-out temper-ature is very close to the QCD hadronization temperature [33], implying adirect connection between data from relativistic nuclear collisions and theQCD phase boundary. This is in accord with the earlier prediction, already wroclaw2020 printed on January 15, 2021 more than 50 years ago, by Hagedorn [34] that hadronic matter cannotbe heated beyond this limit. The parametrizations shown in Fig. 2 are: T CF = T limCF / (1 + exp(2 . − ln( √ s NN ) / . µ B = a/ (1 + 0 . √ s NN ),with √ s NN in GeV and the ’limiting temperature’ T limCF = 158 . ± . a = 1307 . K + /π + and Λ /π + ratios are naturally ex-plained [33] as the interplay between the energy dependence of T CF and µ B and the consequence of strangeness conservation. (MeV) B m ( M e V ) T NucleiQuark-Gluon MatterHadronic Matter CF T Points: Statistical Hadronization, c T Band: Lattice QCD,
Fig. 3. Phenomenological phase di-agram of strongly interacting matterconstructed from chemical freeze-outpoints for central collisions at differ-ent energies, extracted from experi-mental data sets in our own analy-sis (squares) and other similar analyses[35, 36, 37, 25] are compared to predic-tions from LQCD [10, 11] shown as aband. The inverted triangle marks thevalue for ground state nuclear matter(atomic nuclei).
Since the statistical hadronization analysis at each collision energy yieldsa pair of ( T CF , µ B ) values, these points can be used to construct a T vs. µ B diagram, shown in Fig. 3. Note that the points at low temperature seemto converge towards the value for ground state nuclear matter ( µ B = 931MeV). As argued in [38] this limit is not necessarily connected to a phasetransition. While the situation at low temperatures and collision energies iscomplex and at present cannot be investigated with first-principle calcula-tions, the high temperature, high collision energy limit allows a quantitativeinterpretation in terms of fundamental QCD predictions.
3. Statistical hadronization of charm quarks
An interesting question is whether the production of hadrons with heavyquarks can be described with similar statistical hadronization concepts. We roclaw2020 printed on January 15, 2021 note that the mass of the charm quark, m c (cid:39) . T c introduced above, such that ther-mal production of charm quarks is strongly Boltzmann suppressed, and thatat the LHC a copious production of charm quarks in relativistic nuclear col-lisions through hard scattering processes is expected. The produced charmquarks will, therefore, not resemble a chemical equilibrium population forthe temperature T . However, what is needed for the thermal descriptionproposed is that the heavy quarks produced in the collision reach a sufficientdegree of thermal equilibrium through scattering with the partons of the hotmedium. Indeed, the energy loss suffered by energetic heavy quarks in theQGP is indicative of their “strong coupling” with the medium, dominated bylight quarks and gluons. The measurements at the LHC [39, 40] and RHIC[41] of the energy loss and hydrodynamic flow of D mesons demonstrate thisquantitatively.Among the various suggested probes of deconfinement, charmonium (thebound states of c ¯ c ) plays a distinctive role. The J /ψ meson is the firsthadron for which a clear mechanism of suppression (melting) in the QGPwas proposed early on, based on the color analogue of Debye screening [42].A novel quarkonium production mechanism, based on statistical hadroniza-tion was proposed [43], based on thermalized charm quarks which are ”dis-tributed” into hadrons at the phase boundary, i.e. at chemical freeze-out,with thermal weights as discussed above for the light quarks, [44, 43, 45, 46].An alternative mechanism for the (re)combination of charm and anti-charmquarks into charmonium in a QGP [47] was proposed based on kinetic theory(for further developments see [48, 49]).In the SHM, the absence of chemical equilibrium for heavy quarks isaccounted for by introducing a fugacity g c . The parameter g c is obtainedfrom the balance equation [43] which accounts for the distribution of allinitially produced heavy quarks into hadrons at the phase boundary, witha thermal weight constrained by exact charm conservation. With the aboveapproach the knowledge of the heavy quark production cross section alongwith the thermal parameters obtained from the analysis of the yields ofhadrons composed of light quarks, see previous section, is sufficient to de-termine the yield of hadrons containing heavy quarks in ultra-relativisticnuclear collisions.In the SHM, the J /ψ nuclear modification factor R AA is obtained bycomputing the yields in AA collisions while the yields in proton-protoncollisions are taken from experimental data. The so determined R AA ispredicted to increase with increasing collision energy [55], implying reducedsuppression or even enhancement due to the rapid increase with energy ofthe charm production cross section. Clear evidence for such a pattern wasobtained with the first ALICE measurements at LHC energy [53]. Since wroclaw2020 printed on January 15, 2021 =0 h | h /d ch N d y J / AA R
8% syst.unc.) – <4.0, y =5.02 TeV (ALICE, 2.5< NN s
9% syst.unc.) – <2.2, y =0.2 TeV (PHENIX, 1.2< NN s Lines: Statistical Hadronization – = 0.344 y /d cc s d =0 h | h /d ch N d y J / AA R
7% syst.unc.) – |<0.9, y =5.02 TeV (ALICE, | NN s
12% syst.unc.) – |<0.35, y =0.2 TeV (PHENIX, | NN s
14% syst.unc.) – |<1.0, y =0.2 TeV (STAR, | NN s – = 0.532 y /d cc s d Fig. 4. The nuclear modification factor R AA for inclusive J /ψ production in depen-dence on the multiplicity density (at η =0) at forward rapidity (left panel) and atmidrapidity (right panel). The data are for Au–Au collisions from PHENIX (blue)[50, 51] and STAR (green) [52] at RHIC and for Pb–Pb collisions from the ALICEcollaboration (red) [53, 54] at the LHC. then a large number of additional data including detailed energy, rapidity,centrality and transverse momentum dependences of R AA for J /ψ as well ashydrodynamic flow [56] results have provided a firm basis for the statisticalhadronization scenario [44], with the biggest uncertainties still related to thenot yet measured value of the open charm cross section in Pb–Pb collisions.Current results on J /ψ production at midrapidity and forward rapidity asa function of the charged particle multiplicity and description within theSHM are summarized in Fig. 4. A dramatic increase of R AA with increasingcollision energy is clearly observed. Furthermore, the measurements at theLHC demonstrate [53, 54], that the increase is largely concentrated at J /ψ transverse momentum values less than the mass m J /ψ = 3 . R AA for J/ ψ is deeply connected toand provides unique evidence for the deconfinement of charm quarks [9, 43]in the hot medium. Acknowledgements
K.R. acknowledges partial support by the Ex-treme Matter Institute EMMI and the Polish National Science Center NCNunder Opus grant no.2018/31/B/ST2/01663. This work is part of and sup-ported by the DFG Collaborative Research Center ”SFB1225/ISOQUANT”. roclaw2020 printed on January 15, 2021 REFERENCES [1] N. Itoh, “Hydrostatic Equilibrium of Hypothetical Quark Stars,”
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