Melting Hadrons, Boiling Quarks

EPJ manuscript No. (will be inserted by the editor) arXiv:1508.03260 13 Aug 2015 and PREPRINT CERN-PH-TH-2015-194

Johann Rafelski12 1 CERN-PH/TH, 1211 Geneva 23, Switzerland 2 Department of Physics, The University of Arizona Tucson, Arizona, 85721, USA

Submitted: August 11, 2015 / Print date: August 14, 2015

Abstract. In the context of the Hagedorn temperature half-centenary I describe our understanding of the hot phases of hadronic matter both below and above the Hagedorn temperature. The first part of the review addresses many frequently posed questions about properties of hadronic matter in different phases, phase transition and the exploration of quark-gluon plasma (QGP). The historical context of the discovery of QGP is shown and the role of strangeness and strange antibaryon signature of QGP illustrated. In the second part I discuss the corresponding theoretical ideas and show how experimental results can be used to describe the properties of QGP at hadronization. Finally in two appendices I present previously unpublished reports describing the early prediction of the different forms of hadron matter and of the formation of QGP in relativistic heavy ion collisions, including the initial prediction of strangeness and in particular strange antibaryon signature of QGP.

PACS. 24.10.Pa Thermal and statistical models – 25.75.-q Relativistic heavy-ion collisions – 21.65.Qr Quark matter – 12.38.Mh Quark-gluon plasma

1 Introduction A report on ‘Melting Hadrons, Boiling Quarks and TH ’ relates strongly to quantum chromodynamics (QCD), the The year 1964/65 saw the rise of several new ideas which theory of quarks and gluons, the building blocks of had- in the following 50 years shaped the discoveries in funda- rons, and its lattice numerical solutions; QCD is the quan- mental subatomic physics: tum (Q) theory of color-charged (C) quark and gluon dy- 1. The Hagedorn temperature TH ; later recognized as the namics (D); for numerical study the space-time continuum melting point of hadrons into is discretized on a ‘lattice’. 2. Quarks as building blocks of hadrons; and, Telling the story of how we learned that strong inter- 3. The Higgs particle and field escape from the Goldstone actions are a gauge theory involving two types of parti- theorem, allowing the understanding of weak interac- cles, quarks and gluons, and the working of the lattice tions, the source of inertial mass of the elementary par- numerical method would change entirely the contents of ticles. this article, and be beyond the expertise of the author. I The topic in this paper is Hagedorn temperature TH recommend instead the book by Weinberg [8], which also and the strong interaction phenomena near to TH . I present shows the historical path to QCD. The best sources of the an overview of 50 years of effort with emphasis on: QCD relation to the topic of this aritcle are: (a) the book a) The hot nuclear and hadronic matter; by Kohsuke Yagi and Tetsuo Hatsuda [9] as well as, (b) b) The critical behavior near TH ; the now 15 year old monograph by Letessier and the au- c) The quark-gluon plasma (QGP); thor [6]. We often refer to lattice-QCD method to present d) Relativistic heavy ion (RHI) collisions1; QCD properties of interest in this article. There are books e) The hadronization process of QGP; and many reviews on lattice implementation of gauge the- f) Abundant production of strangeness flavor. ories of interacting fields, also specific to hot-lattice-QCD This presentation connects and extends a recent retro- method at the time of writing I do not have a favorite to spective work, Ref. [1]: Melting Hadrons, Boiling Quarks; recommend. From Hagedorn temperature to ultra-relativistic heavy-ion collisions at CERN; with a tribute to Rolf Hagedorn. This Immediately in the following Subsection 1.1 the fa- report complements prior summaries of our work: 1986 [2], mous Why? is addressed. After that I turn to answering 1991 [3],1996 [4], 2000 [5], 2002 [6], 2008 [7]. the How? question in Subsection 1.2, and include a few reminiscences. I close this Introduction with Subsection 1 We refer to atomic nuclei which are heavier than the α- 1.3 where the organization and contents of this review particle as ‘heavy ions’. will be explained. 2 Johann Rafelski: Melting Hadrons, Boiling Quarks

1.1 What are the conceptual challenges of the into mass of matter. One can wonder aloud if this sheds QGP/RHI collisions research program? some light on the reverse process: Is it possible to convert matter into energy in the laboratory? Our conviction that we achieved in laboratory experi- The last two points show the potential of ‘applications’ ments the conditions required for melting (we can also of QGP physics to change both our understanding of, and say, dissolution) of hadrons into a soup of boiling quarks our place in the world. For the present we keep these ques- and gluons became firmer in the past 15-20 years. Now tions in mind. This review will address all the other chal- we can ask, what are the ‘applications’ of the quark-gluon lenges listed under points 1), 2), and 3) above; however, plasma physics? Here is a short wish list: see also thoughts along comparable foundational lines pre- 1) Nucleons dominate the mass of matter by a factor 1000. sented in Subsections 7.3 and 7.4. The mass of the three ‘elementary’ quarks found in nucleons is about 50 times smaller than the nucleon mass. Whatever compresses and keeps the quarks within the 1.2 From melting hadrons to boiling quarks nucleon volume is thus the source of nearly all of mass of matter. This clarifies that the Higgs field provides the With the hindsight of 50 years I believe that Hagedorn’s mass scale to all particles that we view today as elemen- effort to interpret particle multiplicity data has led to the tary. Therefore only a small %-sized fraction of the mass of recognition of the opportunity to study quark deconfine- matter originates directly in the Higgs field; see Section 7.1 ment at high temperature. This is the topic of the book [1] for further discussion. The question: What is mass? can be Melting Hadrons, Boiling Quarks; From Hagedorn temper- studied by melting hadrons into quarks in RHI collisions. ature to ultra-relativistic heavy-ion collisions at CERN; with a tribute to Rolf Hagedorn published at Springer 2) Quarks are kept inside hadrons by the ‘vacuum’ prop- Open, i.e. available for free on-line. This article should erties which abhor the color charge of quarks. This expla- be seen as a companion addressing more recent develop- nation of 1) means that there must be at least two dif- ments, and setting a contemporary context for this book. ferent forms of the modern æther that we call ‘vacuum’: How did we get here? There were two critical mile- the world around us, and the holes in it that are called stones: hadrons. The question: Can we form arbitrarily big holes filled with almost free quarks and gluons? was and remains I) The first milestone occurred in 1964–1965, when Hage- the existential issue for laboratory study of hot matter dorn, working to resolve discrepancies of the statistical made of quarks and gluons, the QGP. Aficionados of the particle production model with the pp reaction data, pro- lattice-QCD should take note that the presentation of two duced his “distinguishable particles” insight. Due to a phases of matter in numerical simulations does not answer twist of history, the initial research work was archived this question as the lattice method studies the entire Uni- without publication and has only become available to a verse, showing hadron properties at low temperature, and wider public recently; that is, 50 years later, see Chapter QGP properties at high temperature. 19 in [1] and Ref.[10]. Hagedorn went on to interpret the observation he made. Within a few months, in Fall 1964, 3) We all agree that QGP was the primordial Big-Bang he created the Statistical Bootstrap Model (SBM) [11], stuff that filled the Universe before ‘normal’ matter formed. showing how the large diversity of strongly interacting Thus any laboratory exploration of the QGP properties particles could arise; Steven Frautschi [12] coined in 1971 solidifies our models of the Big Bang and allows us to ask the name ‘Statistical Bootstrap Model’. these questions: What are the properties of the primordial matter content of the Universe? and How does ‘normal’ II) The second milestone occurred in the late 70s and early matter formation in early Universe work? 80s when we spearheaded the development of an experimental program to study ‘melted’ hadrons and the ‘boil- 4) What is flavor? In elementary particle collisions, we ing’ quark-gluon plasma phase of matter. The intense the- deal with a few, and in most cases only one, pair of newly oretical and experimental work on the thermal properties created 2nd, or 3rd flavor family of particles at a time. of strongly interacting matter, and the confirmation of a A new situation arises in the QGP formed in relativis- new quark-gluon plasma paradigm started in 1977 when tic heavy ion collisions. QGP includes a large number of the SBM mutated to become a model for melting nuclear particles from the second family: the strange quarks and matter. This development motivated the experimental ex- also, the yet heavier charmed quarks; and from the third ploration in the collisions of heavy nuclei at relativistic en- family at the LHC we expect an appreciable abundance ergies of the phases of matter in conditions close to those of bottom quarks. The novel ability to study a large num- last seen in the early Universe. I refer to Hagedorn’s ac- ber of these 2nd and 3rd generation particles offers a new count of these developments for further details Chapter opportunity to approach in an experiment the riddle of 25 loc.cit. and Ref.[13]. We return to this time period in flavor. Subsection 4.1. 5) In relativistic heavy ion collisions the kinetic energy of At the beginning of this new field of research in the late ions feeds the growth of quark population. These quarks 70s, quark confinement was a mystery for many of my col- ultimately turn into final state material particles. This leagues; gluons mediating the strong color force were nei- means that we study experimentally the mechanisms lead- ther discovered nor widely accepted, especially not among ing to the conversion of the colliding ion kinetic energy my nuclear physics peers, and QCD vacuum structure was Johann Rafelski: Melting Hadrons, Boiling Quarks 3

Fig. 1. Time line of the CERN-SPS RHI program: on the left axis we see year and ion beam available, S=Sulfur, Pb=lead, In=Indium) as a ‘function’ of the experimental code. The primary observables are indicated next to each square; arrows connecting the squares indicate that the prior equipment and group, both in updated format, continued. Source: CERN release February 2000 modified by the author. just coming out of kindergarten. The discussion of a new lisions were divided on many important questions. In re- phase of deconfined quark-gluon matter was therefore in gard to what happens in relativistic collision of nuclei the many eyes not consistent with established wisdom and situation was most articulate: a) One group believed that certainly too ambitious for the time. nuclei (baryons) pass through each other with a new phase Similarly, the special situation of Hagedorn deserves of matter formed in a somewhat heated projectile and/or remembering: early on Hagedorn’s research was under- target. This picture required detection systems of very mined by outright personal hostility; how could Hage- different character than the systems required by, both: b) dorn dare to introduce thermal physics into the field gov- those who believed that in RHI collisions energy would be erned by particles and fields? However, it is interesting consumed by a shock wave compression of nuclear mat- to also take note of the spirit of academic tolerance at ter crashing into the center of momentum frame; and c) CERN. Hagedorn advanced through the ranks along with a third group who argued that up to top CERN-SPS p his critics, and his presence attracted other like-minded ( (sNN ) 'O(20) GeV) collision energy a high temper- researchers, who were welcome in the CERN Theory Di- ature, relatively low baryon density quark matter fireball vision, creating a bridgehead towards the new field of RHI will be formed. The last case turned out to be closest to collisions when the opportunity appeared on the horizon. results obtained at SPS and at RHIC. In those days, the field of RHI collisions was in other From outside, we were ridiculed as being speculative; ways rocky terrain: from within we were in state of uncertainty about the fate 1) RHI collisions required the use of atomic nuclei at high- of colliding matter and the kinetic energy it carried, with est energy. This required cooperation between experimen- disagreements that ranged across theory vs. experiment, tal nuclear and particle physicists. Their culture, back- and particle vs. nuclear physics. In this situation, ‘QGP ground, and experience differed. A similar situation pre- formation in RHI collisions’ was a field of research that vailed within the domain of theoretical physics, where an could have easily fizzled out. However, in Europe there interdisciplinary program of research merging the three was CERN, and in the US there was strong institutional traditional physics domains had to form. While ideas of support. Early on it was realized that RHI collisions re- thermal and statistical physics needed to be applied, very quired large experiments uniting much more human ex- few subatomic physicists, who usually deal with individual pertise and manpower as compared to the prior nuclear particles, were prepared to deal with many body ques- and even some particle physics projects. Thus work had tions. There were also several practical issues: In which to be centralized in a few ‘pan-continental’ facilities. This (particle, nuclear, stat-phys) journal can one publish and meant that expertise from a few laboratories would need who could be the reviewers (other than direct competi- to be united in a third location where prior investments tors)? To whom to apply for funding? Which conference would help limit the preparation time and cost. to contribute to? These considerations meant that in Europe the QGP 2) The small group of scientists who practiced RHI col- formation in RHI collisions research program found its 4 Johann Rafelski: Melting Hadrons, Boiling Quarks

Fig. 2. The Brookhaven National Laboratory heavy ion accelerator complex. Creative Commons picture modified by the author. home at CERN. The CERN site benefited from being a ‘Relativistic Heavy-Ion Collider (RHIC), see Fig. 2, was multi-accelerator laboratory with a large pool of engineer- to be built. ing expertise and where some of the required experimental In a first step already existing tandems able to create equipment was already on the ground, grandfathered from low energy heavy ion beams were connected by a trans- prior particle physics efforts. fer line to the already existing AGS proton synchrotron The time line of the many CERN RHI experiments adapted to accelerate these ions. The AGS-ion system per- through the beginning of this millennium is shown in Fig. 1, formed experiments with fixed targets serving as a train- representation is based on a similar CERN documenta- ing ground for the next generation of experimentalists. tion from thee year 2000. The experiments WA85, NA35, During this time another transfer line was built connect- HELIOS-2, NA38 comprised a large instrumental compo- ing AGS to the ISABELLE project tunnel, in which the nent from prior particle physics detectors. Other experi- RHIC was installed. The initial RHIC experiments are ments and/or experimental components were contributed shown around their ring locations: STAR, PHENIX, PHO- by US and European laboratories. This includes the heavy BOS and BRAHMS, see also Subsection 4.2. The first data ion source, and its preaccelerator complex required for taking at RHIC began in Summer 2001. heavy ion insertion into CERN beam lines. The success of the SPS research program at CERN has When the CERN SPS program faded out early in this strongly supported the continuation of the RHI collision millennium, the resources were focused on the LHC-ion program. The Large Hadron Collider (LHC) was designed collider operations, and in the US, the ‘RHIC’ collider to accept highest energy counter propagating heavy ion came on-line. As this is written, the SPS fixed target pro- beams opening the study of a new domain of collision gram experiences a second life; the experiment NA61, built energy. LHC-ion operation allows us to exceed the top with large input from the NA49 equipment, is searching RHIC energy range by more than an order of magnitude. for the onset of QGP formation, see Subsection 6.3. In preparation for LHC-ion operations the SPS groups While the CERN program took off by the late 80s, the founded in the mid-90s a new collider collaboration, and US-Department of Energy decided to make a large invest- have built one of the four LHC experiments which is ded- ment that assured the formation of QGP in laboratory. icated to the study of RHI collisions. Two other experi- However, this meant that the US-QGP research program ments also participate in the LHC-ion research program suffered a delay, in part due to the need to transfer the which we will introduce in Subsection 6.2. scientific expertise to the newly designated research center To conclude: a new fundamental set of science argu- at the Brookhaven National Laboratory (BNL) where the ments, and deep-rooted institutional support carried the Johann Rafelski: Melting Hadrons, Boiling Quarks 5

field forward. CERN was in a unique position to embark these questions. Assuming that this effort is successful, I on RHI research having not only the accelerators, engi- propose in Section 7 the next generation of physics chal- neering expertise, and research equipment, but mainly due lenges. The topics discussed are very subjective; other au- to Hagedorn, also the scientific expertise on the ground, thors will certainly see other directions and propose other for more detail consult Ref.[1]. In the US a major new challenges of their interest. experimental facility, RHIC at BNL, was developed. With In Section 8 we deepen the discussion of the origins and the construction of LHC at CERN a new RHI collision the contents of the theoretical ideas that have led Hage- energy domain was opened. The experimental programs dorn to invent the theoretical foundations leading on to at SPS, RHIC and LHC-ion continue today. TH and melting hadrons. The technical discussion is brief and serves as an introduction to Appendix A. Section 8 ends with a discussion, Subsection 8.5, of how the present 1.3 Format of this review day lattice-QCD studies test and verify the theory of hot nuclear matter based on SBM. More than 35 years into the QGP endeavor I can say with conviction that the majority of nuclear and particle physi- Selected theoretical topics related to the study of QGP cists and the near totality of the large sub-group directly hadronization are introduced in the following: In Section 9 involved with the relativistic heavy ion collision research we describe the numerical analysis tool within the Statisti- agree that a new form of matter, the (deconfined) quark- cal Hadronization Model (SHM); that is, the SHARE suite gluon plasma phase has been discovered. The discovery of computer programs and its parameters. We introduce announced at CERN in the year 2000, see Subsection 4.2, practical items such as triggered centrality events and ra- has been confirmed both at RHIC and by the recent re- pidity volume dV/dy, resonance decays, particle number sults obtained at LHC. This review has, therefore, as its fluctuations, which all enter into the RHIC and LHC data primary objective, the presentation of the part of this an- analysis. nouncement that lives on, see Subsection 4.3, and how Section 10 presents the results of the SHM analysis more recent results are addressing these questions: What with emphasis put on bulk properties of the fireball; Sub- are the properties of hot hadron matter? How does it turn section 10.1, addresses SPS and RHIC prior to LHC, while into QGP, and how does QGP turn back into normal mat- in Subsection 10.2: it is shown how hadron production can ter? These are to be the topics addressed in the second be used to determine the properties of QGP and how the half of this review. threshold energy for QGP formation is determined. The There are literally thousands of research papers in this results of RHIC and LHC are compared and the univer- field today; thus this report cannot aim to be inclusive sality of QGP hadronization across a vast range of energy of all work in the field. We follow the example of John and fireball sizes described. Subsection 10.3 explains, in A. Wheeler. Addressing in his late age a large audience terms of evaluation by example of prior work, why the of physicists, he showed one transparency with one line, prior two subsections address solely the SHARE-model ˙ “What is the question?”In this spirit, this review begins results. In Subsection 10.4 the relevance of LHC results to with a series of questions, and answers, aiming to find QGP physics is described, and further lattice-QCD rela- the answer to: Which question is THE question today? tions to these results pointed out. A few issues we raise are truly fundamental present day challenges. Many provide an opportunity to recognize the The final Section 11 does not attempt a summary which state of the art, both in theory and experiment. Some in case of a review would mean presenting a review of a questions are historical in character and will kick off a review. Instead, a few characteristic objectives and results debate with other witnesses with a different set of personal of this review are highlighted. memories. An integral part of this review are two unpublished These introductory questions are grouped into three technical papers, one from 1980 (Appendix A) and an- separate sections: first come the theoretical concepts on other from 1983 (Appendix B). These two are just a tip the hadron side of hot hadronic matter, Section 2; next, of an iceberg; there are many other unpublished papers concepts on the quark side, Section 3; and third, the ex- by many authors hidden in conference volumes. There is perimental ‘side’ Section 4 about RHI collisions. Some of already a published work reprint volume [14] in which the the questions formulated in Sections 2, 3, and 4 intro- pivotal works describing QGP theoretical foundations are duce topics that this review addresses in later sections in reproduced; however, the much less accessible and often depth. The roles of strangeness enhancement and strange equally interesting unpublished work is at this juncture antibaryon signature of QGP are highlighted. in time practically out of sight. This was one of the rea- We follow this discussion by addressing the near future sons leading to the presentation of Ref.[1]. These two pa- of the QGP and RHI collision research in the context of pers were selected from this volume and are shown here this review centered around the strong interactions and unabridged. They best complement the contents of this re- hadron-quarks phase. In Section 5 I present several con- view, providing technical detail not presented in the body, ceptual RHI topics that both are under present active while also offering a historical perspective. Beside the key study, and which will help evaluate which direction the results and/or discussion they also show the rapid shift field will move on in the coming decade. Section 6 shows in the understanding that manifested itself within a short the current experimental research program that address span of two years. 6 Johann Rafelski: Melting Hadrons, Boiling Quarks

Appendix A presents “Extreme States of Nuclear Mat- ter” from the Workshop on Future Relativistic Heavy Ion Experiments held 7-10 October 1980. This small gather- ing is now considered to be the first of the “Quark Mat- ter” series i.e. QM80 conference. Most of this report is a summary of the theory of hot hadron gas based on Hage- dorn’s Statistical Bootstrap Model (SBM). The key new insight in this work was that in RHI collisions the production of particles rather than the compression of existent matter was the determining factor. The hadron gas phase study was complemented by a detailed QGP model presented as a large, hot, interacting quark-gluon bag. The phase boundary between these two phases characterized Fig. 3. Illustration of the Statistical Bootstrap Model idea: by Hagedorn temperature T was evaluated in quantita- H a volume comprising a gas of fireballs when compressed to tive manner. It was shown how the consideration of differ- natural volume is itself again a fireball. Drawing from Ref.[17] ent collision energies allows us to explore the phase bound- modified for this review. ary. This 1980 paper ends with the description of strangeness flavor as the observable of QGP. Strange antibaryons are introduced as a signature of quark-gluon plasma. It is convenient to introduce the mass spectrum ρ(m), Appendix B presents “Strangeness and Phase Changes where in Hot Hadronic Matter” from the Sixth High Energy Heavy Ion Study, Berkeley, 28 June – 1 July 1983. The meeting, ρ(m)dm = number of ‘i’ hadron states in {m, m + dm} , which had a strong QGP scientific component, played an (2) important role in the plans to develop a dedicated rela- Thus we have tivistic heavy ion collider (RHIC). In this lecture I sum- " # T 3/2 Z ∞ marize and update in qualitative terms the technical phase Z(T,V ) = exp V ρ(m)m3/2e−m/T dm . transition consideration seen in Appendix A, before turn- 2π 0 ing to the physics of strangeness in hot hadron and quark (3) matter. The process of strangeness production is presented On the other hand, a hadronic fireball comprising many as being a consequence of dynamical collision processes components seen on the left in Fig. 3, when compressed both among hadrons and in QGP, and the dominance of to its natural volume V → V0, is itself a highly excited gluon-fusion processes in QGP is described. The role of hadron, a resonance that we must include in Eq. (3). This strangeness in QGP search experiments is presented. For is what Hagedorn realized in 1964 [11]. This observation a more extensive historical recount see Ref.[15]. leads to an integral equation for ρ(m) when we close the ‘bootstrap’ loop that emerges. Frautschi [12] transcribed Hagedorn’s grand canonical 2 The Concepts: Theory Hadron Side formulation into microcanonical format. The microcanonical bootstrap equation reads in invariant Yellin [16] no- 2.1 What is the Statistical Bootstrap Model (SBM)? tation 2 P 2 2 Hτ(p ) = H δ0(p − m ) Considering that the interactions between hadronic parti- min in cles are well characterized by resonant scattering, see Sub- + P∞ 1 R δ4 p − Pn p Qn Hτ(p2)d4p , section 2.4, we can describe the gas of interacting hadrons n=2 n! i=1 i i=1 i i as a mix of all possible particles and their resonances ‘i’. (4) This motivates us to consider the case of a gas comprising where H is a universal constant assuring that Eq. (4) is several types of particles of mass mi, enclosed in a heat dimensionless; τ(p2) on the left-hand side of Eq. (4) is the √ µ bath at temperature T , where the individual populations fireball mass spectrum with the mass m = pµp which ‘i’ are unconstrained in their number, that is like photons we are seeking to model. The right-hand side of Eq. (4) in a black box adapting abundance to what is required for expresses that the fireball is either just one input particle the ambient T . The nonrelativistic limit of the partition of a given mass min, or else composed of several (two or function this gas takes the form 2 more) particles i having mass spectra τ(pi ), and 3/2 2 2 X T X 3/2 τ(m )dm ≡ ρ(m)dm, (5) ln Z = ln Z = V m e−mi/T , (1) i 2π i i i A solution to Eq. (4) has naturally an exponential form

−a where the momentum integral was carried out and the sum ρ(m) ∝ m exp(m/TH ). (6) ‘i’ includes all particles of diﬀering parity, spin, isospin, baryon number, strangeness etc. Since each state is counted, The appearance of the exponentially growing mass spec- there is no degeneracy factor. trum, Eq. (6), is a key SBM result. One of the important Johann Rafelski: Melting Hadrons, Boiling Quarks 7

Table 1. Thermodynamic quantities assuming exponential In view of the entries shown in Table 1 an important form of hadron mass spectrum with preexponential index a, further result can be obtained using these leading order Eq. (6); results from Ref.[25] terms for all cases of a considered: the speed of sound at which the small density perturbations propagate a P ε δε/ε 2 dP 1/2 C/∆T 2 C/∆T 3 C + C∆T c := ∝ ∆T → 0 . (7) s dε 1 C/∆T 3/2 C/∆T 5/2 C + C∆T 3/4 This universal for all a result is due to the exponential 2 1/2 2 3/2 C/∆T C/∆T C + C∆T mass spectrum of hadron matter studied here. cs → 0 1/2 3/2 1/4 at TH harbors an interesting new deﬁnition of the phase 2 C/∆T C/∆T C + C∆T boundary in the context of lattice-QCD. A nonzero but 2 5/2 C ln(T0/∆T ) C/∆T C small value cs should arise from the subleading terms con-

1/2 1/2 1/4 tributing to P and ε not shown in Table 1. The way sin- 3 P0 − C∆T C/∆T C/∆T 2 gular properties work, it could be that the cs = 0 point 1/2 7/2 P0 − C∆T ε0 C/∆T exists. The insight that the sound velocity vanishes at TH is known since 1978, see Ref.[25]. An ‘almost’ rediscovery 3/2 1/2 3/4 4 P0 − C∆T ε0 − C∆T C/∆T of this result is seen in Sections 3.5 and 8.7 of Ref.[26]. The above discussion shows both the ideas that led to the invention of SBM, and how SBM can evolve with consequences is that the number of different hadron states our understanding of the strongly interacting matter, be- grows so rapidly that practically every strongly interacting coming more adapted to the physical properties of the el- particle found in the fireball is distinguishable. Hagedorn ementary ‘input’ particles. Further potential refinements realized that the distinguishability of hadron states was an include introducing strange quark related scale into char- essential input in order to reconcile statistical hadron mul- acterization of the hadron volume, making baryons more tiplicities with experimental data. Despite his own initial compressible as compared to mesons. These improvements rejection of a draft paper, see Chapters 18 and 19 loc.cit., could generate a highly realistic shape of the mass spec- this insight was the birth of the theory of hot hadronic trum, connecting SBM more closely to the numerical study matter as it produced the next step, a model [13]. of QCD in lattice approach. We will return to SBM, and SBM solutions provide a wealth of information includ- the mass spectrum, and describe the method of finding a ing the magnitude of the power index a seen in Eq. (6). solution of Eq. (4) in Section 8. Frautschi, Hamer, Carlitz [12,18,19,20,21,22] studied solutions to Eq. (4) analytically and numerically and by 1975 drew important conclusions: 2.2 What is the Hagedorn temperature TH ?

– Fireballs would predominantly decay into two frag- Hagedorn temperature is the parameter entering the ex- ments, one heavy and one light. ponential mass spectrum Eq. (6). It is measured by fitting – By iterating their bootstrap equation with realistic to data the exponential shape of the hadron mass spec- input, they found numerically TH ≈ 140 MeV and trum. The experimental mass spectrum is discrete; hence a = 2.9 ± 0.1, which ruled out the previously adopted a smoothing procedure is often adopted to fit the shape approximate value [11,23] a = 5/2 . Eq. (8) to data. In technical detail one usually follows the – Each imposed conservation law implemented by fixing method of Hagedorn (see Chapter 20 loc.cit. and Ref.[23]), a quantum number, e.g., baryon number ρ(B, m), in applying a Gaussian distribution with a width of 200 MeV the mass spectrum, increases the value of a by 1/2. for all hadron mass states. However, the accessible experimental distribution allows fixing TH uniquely only if we Werner Nahm independently obtained a = 3 [24]. Further know the value of the preexponential power ‘a’. refinement was possible. In Appendix A, a SBM with com- The fit procedure is encumbered by the singularity for pressible finite-size hadrons is introduced where one must m → 0. Hagedorn proposed a regularized form of Eq. (6) consistently replace Eq. (A29) by Eq. (A30). This leads to a finite energy density already for a model which produces em/TH ρ(m) = c . (8) a = 3 with incompressible hadrons. 2 2 a/2 (m0 + m ) For any ρ(m) with a given value of a, Eq. (6), it is easy to understand the behavior near to TH . Inserting In fits to experimental data all three parameters TH , m0, c Eq. (6) into the relativistic form of Eq. (1), see Chapter must be varied and allowed to find their best value. In 1967 23 loc.cit., allows the evaluation near critical condition, Hagedorn fixed m0 = 0.5 GeV as he was working in the TH −T ≡ ∆T → 0 of the physical properties such as shown limit m > m0, and he also fixed a = 2.5 appropriate for in Table 1: pressure P , energy density ε, and other physical his initial SBM approach [23]. The introduction of a fitted properties, as example the mean relative fluctuations δε/ε value m0 is necessary to improve the characterization of of ε are shown, for a = 1/2, 2/2,..., 8/2. We see that, as the hadron mass spectrum for low values of m, especially T → TH (∆T → 0), the energy density diverges for a ≤ 3. when a range of possible values for a is considered. 8 Johann Rafelski: Melting Hadrons, Boiling Quarks

Table 2. Parameters of Eq. (8) ﬁtted for a prescribed preexponential power a. Results from Ref.[27].

a−1 12 a c[GeV ] m0[GeV] TH [MeV] TH [10 · K]

2.5 0.83479 0.6346 165.36 1.9189 3. 0.69885 0.66068 157.60 1.8289 3.5 0.58627 0.68006 150.55 1.7471 4. 0.49266 0.69512 144.11 1.6723 5. 0.34968 0.71738 132.79 1.5410 6. 0.24601 0.73668 123.41 1.4321

The fits to experimental mass spectrum shown in Ta- ble 2 are from 1994 [27] and thus include a smaller set of Fig. 4. The experimental mass spectrum (solid line), the hadron states than is available today. However, these re- fit (short dashed), compared to 1967 fit of Hagedorn (long sults are stable since the new hadronic states found are dashed): The case a = 3 is shown, for parameters see table at high mass. We see in Table 2 that as the preexpo- 2. Figure from Ref.[17] with results obtained in [28] modified nential power law a increases, the fitted value of TH de- for this review. creases. The value of c for a = 2.5 corresponds to c = 2.64×104 MeV3/2, in excellent agreement to the value obtained by Hagedorn in 1967. In Fig. 4 the case a = 3 is These results of Ref.[29] are seen in Fig. 5: the top illustrated and compared to the result of the 1967 fit by frame for mesons and the bottom frame for baryons. Two Hagedorn and the experimental smoothed spectrum. All different fits are shown characterized by a model param- fits for different a were found at nearly equal and convinc- eter R which, though different from H seen in Eq. (A15), ing confidence level as can be inferred from Fig. 4. plays a similar role. Thus the two results bracket the value of T from above (blue, T ' 162 MeV) and from below Even cursory inspection of Table 2 suggests that the H H (red, TH ' 145 MeV) in agreement with typical empirical value of TH that plays an important role in physics of RHI results seen in table 2. collisions depends on the understanding of the value of a. We further see in Fig. 5 that a noticeably different This is the reason that we discussed the different cases number of M > 2 GeV states can be expected depending in depth in previous subsection 2.1. The preexponential on the value of TH , even if the resonances for M < 1.7 GeV power value a = 2.5 in Eq. (8) corresponds to Hagedorn’s are equally well fitted in both cases. Thus it would seem original preferred value; the value a = 3 was adopted by that the value of TH can be fixed more precisely in the the mid-70s following extensive study of the SBM as de- future when more hadronic resonances are known. How- scribed. However, results seen in Table 1 and Appendix A ever, for M ' 3 GeV there are about 105 different meson imply a ≥ 7/2. or baryon states per GeV. This means that states of this This is so since for a < 7/2 we expect TH to be a maxi- mass are on average separated by 10 eV in energy. On the mum temperature, for which we see in Table 1 a divergence other hand, their natural width is at least 106 larger. Thus in energy density. Based on study of the statistical boot- there is little if any hope to experimentally resolve such strap model of nuclear matter with conserved baryon num- ‘Hagedorn’ states. Hence we cannot expect to determine, ber and compressible hadrons presented in Appendix A, I based on experimental mass spectrum, the value of TH believe that 3.5 ≤ a ≤ 4. A yet greater value a ≥ 4 should more precisely than it is already done today. However, emerge if in addition strangeness and charge are intro- there are other approaches to measure the value of TH . duced as a distinct conserved degree of freedom – in any For example, we address at the end of Subsection 3.3 why consistently formulated SBM with canonically conserved the behavior of lattice-QCD determined speed of sound quantum numbers one unique value of TH will emerge for suggests that TH ' 145 MeV. the mass spectrum, that is ρ(m, b, S, . . .) ∝ exp(m/TH ) To summarize, our current understanding is that Hage- for any value of b, S, Q, . . . i.e. the same TH for mesons dorn temperature has a value still needing an improved and baryons. Only the preexponential function can de- determination, pend on b, S, Q, . . . An example for this is provided by 12 the SBM model of Beitel, Gallmeister and Greiner [29]. 140 ≤ TH ≤ 155 MeV TH ' (1.7 ± 0.1) × 10 K. (9) Using a conserved discrete quantum numbers approach, explicit fits lead to the same (within 1 MeV) value of TH TH is the maximum temperature at which matter can ex- for mesons and baryons [29]. ist in its usual form. TH is not a maximum temperature Johann Rafelski: Melting Hadrons, Boiling Quarks 9

phase boundary could be a smooth line in the µB,T plane. Their qualitative remarks did not address a method to form, or to explore, the phase boundary connecting these limits. The understanding of high baryon density matter properties in the limit T → 0 is a separate vibrant research topic which will not be further discussed here [34,35,36, 37]. Our primary interest is the domain in which the eﬀects of temperature dominate, in this sense the limit of small µB T .

2.3 Are there several possible values of TH ?

The singularity of the SBM at TH is a unique singular point of the model. If and when within SBM we implement distinguishability of mesons from baryons, or/and of strange and nonstrange hadrons, all these families of particles would have a mass spectrum with a common value of TH . No matter how complex are the so-called SBM ‘input’ states, upon Laplace transform they lead to always one singular point, see Subsection 8.3. In subsequent projection of the generating SBM function onto individual families of hadrons one common exponential for all is found. On the other hand, it is evident from the formal- Fig. 5. Meson- (top) and baryon- (bottom) mass spectra ρ(M) ism that when extracting from the common expression the (particles per GeV): dashed line the experimental spectrum specific forms of the mass spectrum for different particle including discrete states. Two different fits are shown, see test. families, the preexponential function must vary from fam- Figure from Ref.[29] modified for this review ily to family. In concrete terms this means that we must fit the individual mass spectra with common TH but particle family dependent values of a and dimensioned parameter in the Universe. The value of TH which we evaluate in the c, m0 seen in table 2, or any other assumed preexponential study of hadron mass spectra is, as we return to discuss in function. Section 3.3, the melting point of hadrons dissolving into There are several recent phenomenological studies of the quark-gluon plasma (QGP), a liquid phase made of the hadron mass spectrum claiming to relate to SBM of Debye-screened color-ionic quarks and gluons. A further Hagedorn, and the approaches taken are often disappoint- heating of the quark-gluon plasma ‘liquid’ can and will ing. The frequently seen defects are: i) Assumption of continue. A similar transformation can occur already at a a = 2.5 along with the Hagedorn 1964-67 model, a value lower temperature at a finite baryon density. obsolete since 1971 when a = 3 and higher was recog- Indeed, there are two well studied ways to obtain de- nized; and ii) Choosing to change TH for different particle confinement: a) high temperature; and b) high baryon families, e.g. baryons and mesons or strange/nonstrange density. In both cases the trick is that the number of par- hadrons instead of modifying the preexponential function ticles per unit volume is increased. for different particle families. iii) A third technical prob- a) In absence of all matter (zero net baryon number cor- lem is that an integrated (‘accumulated’) mass spectrum responding to baryochemical potential µB → 0), in is considered, full thermal equilibrium temperature alone controls Z M the abundance of particles as we already saw in the R(M) = ρ(m)dm . (10) context of SBM. The result of importance to this re- 0 view is that confinement is shown to dissolve in the While the Hagedorn-type approach requires smoothing study of QCD by Polyakov [30], and this has been also of the spectrum, adopting an effective Gaussian width for argued early on and independently in the context of all hadrons, the integrated spectrum Eq. (10) allows one lattice-QCD [31]. to address directly the step function arising from integrat- b) At nuclear (baryon) densities an order of magnitude ing the discrete hadron mass spectrum, i.e. avoiding the greater than the prevailing nuclear density in large nu- Hagedorn smoothing. One could think that the Hagedorn clei, this transformation probably can occur near to, or smoothing process loses information that is now available even at, zero temperature; for further quantitative dis- in the new approach, Eq. (10). However, it also could be cussion see Appendix A. This is the context in which that a greater information loss comes from the consider- asymptotically free quark matter was proposed in the ation of the integrated ‘signal’. This situation is not un- context of neutron star physics [32]. common when considering any integrated signal function. Cabibbo and Parisi [33] were first to recognize that these The Krakow group Ref.[38,39] was first to consider the two distinct limits are smoothly connected and that the integrated mass spectrum Eq. (10). They also break the 10 Johann Rafelski: Melting Hadrons, Boiling Quarks large set of hadron resonances into different classes, e.g. 2.4 What is hadron resonance gas (HRG)? non-strange/strange hadrons, or mesons/baryons. How- ever, they chose same preexponential fit function and var- We are seeking a description of the phase of matter made ied TH between particle families. The fitted value of TH of individual hadrons. One would be tempted to think that was found to be strongly varying in dependence on sup- the SBM provides a valid framework. However, we already plementary hypotheses made about the procedure, with know from discussion above that the experimental reali- the value of TH changing by 100’s MeV, possibly showing ties limit the ability to fix the parameters of this model; the inconsistency of procedure aggravated by the loss of specifically, we do not know TH precisely. signal information. In the present day laboratory experiments one therefore approaches the situation differently. We employ all Ref.[40] fixes m0 = 0.5 GeV at a = 2.5, i.e. Hagedorn’s 1968 parameter choices. Applying the Krakow method ap- experimentally known hadrons as explicit partial fractions in the hadronic gas: this is what in general is called the proach, this fit produces with present day data TH = 174 MeV. We keep in mind that the assumed value of a hadron resonance gas (HRG), a gas represented by the is incompatible with SBM, while the assumption of a rel- non-averaged, discrete sum partial contributions, corresponding to the discrete format of ρ(m) as known empir- atively small m0 = 0.5 GeV is forcing a relatively large ically. value of TH , compare here also the dependence of TH on a seen in table 2. Another similar work is Ref.[41], which The emphasis here is on ‘resonances’ gas, reminding us seeing poor phenomenological results that emerge from an that all hadrons, stable and unstable, must be included. inconsistent application of Hagedorn SBM, criticizes un- In his writings Hagedorn went to great length to justify justly the current widely accepted Hagedorn approach and how the inclusion of unstable hadrons, i.e. resonances, ac- Hagedorn Temperature. For reasons already described, we counts for the dominant part of the interaction between do not share in any of the views presented in this work. all hadronic particles. His argument was based on work of Belenky (also spelled Belenkij) [45], but the intuitive However, we note two studies [42,43] of differentiated content is simple: if and when reaction cross sections are (meson vs. baryon) hadron mass spectrum done in the dominated by resonant scattering, we can view resonances way that we consider correct: using a common singularity, as being all the time present along with the scattering par- that is one and the same exponential T , but ‘family’ de- H ticles in order to characterize the state of the physical sys- pendent preexponential functions obtained in projection tem. This idea works well for strong interactions since the on the appropriate quantum number. It should be noted S-matrix of all reactions is pushed to its unitarity limit. that the hadronic volume V enters any reduction of the h To illustrate the situation, let us imagine a hadron mass spectrum by the projection method, see Appendix A, system at ‘low’ T ' T /5 and at zero baryon density; where volume effect for strangeness is shown. H this is in essence a gas made of the three types of pions, Biro and Peshier [42] search for TH within nonexten- π(+,−,0). In order to account for dominant interactions be- sive thermodynamics. They consider two different values tween pions we include their scattering resonances as in- of a for mesons and baryons (somewhat on the low side), dividual contributing fractions. Given that these particles and in their Fig. 2 the two fits show a common value of TH have considerably higher mass compared to that of two around 150–170 MeV. A very recent lattice motivated ef- pions, their number is relatively small. fort assumes differing shape of the preexponential function But as we warm up our hadron gas, for T > TH /5 res- for different families of particles [43], and uses a common, onance contribution becomes more noticeable and in turn but assumed, not fitted, value of TH . their scattering with pions requires inclusion of other res- Arguably, the most important recent step forward in onances and so on. As we reach, in the heat-up process regard to improving the Hagedorn mass spectrum analysis T . TH , Hagedorn’s distinguishable particle limit applies: is the realization first made by Majumder and Müller[44] very many different resonances are present such that this that one can infer important information about the hadron hot gas develops properties of classical numbered-ball sys- mass spectrum from lattice-QCD numerical results. How- tem, see Chapter 19 loc.cit.. ever, this first effort also assumed a = 2.5 without a good All heavy resonances ultimately decay, the process cre- reason. Moreover, use of asymptotic expansions of the ating pions observed experimentally. This yield is well Bessel functions introduced errors preventing a compar- ahead of what one would expect from a pure pion gas. ison of these results with those seen in table 2. Moreover, spectra of particles born in resonance decays To close let us emphasize that phenomenological ap- differ from what one could expect without resonances. As proach in which one forces same preexponential function a witness of the early Hagedorn work from before 1964, and fits different values of TH for different families of Maria Fidecaro of CERN told me recently, I paraphrase particles is at least within the SBM framework blatantly “when Hagedorn produced his first pion yields, there were wrong. A more general argument indicating that this is many too few, and with a wrong momentum spectrum”. always wrong could be also made: the only universal nat- As we know, Hagedorn did not let himself be discouraged ural constant governing phase boundary is the value of by this initial difficulty. TH , the preexponential function, which varies depending The introduction of HRG can be tested theoretically on how we split up the hadron particle family – projection by comparing HRG properties with lattice-QCD. In Fig. 6 of baryon number (meson, baryon), and strangeness, are we show the pressure presented in Ref.[68]. We indeed two natural choices. see a good agreement of lattice-QCD results obtained for Johann Rafelski: Melting Hadrons, Boiling Quarks 11

2.5 What does lattice-QCD tell us about HRG and about the emergence of equilibrium ?

The thermal pressure reported in Fig. 6 is the quantity least sensitive to missing high mass resonances which are nonrelativistic and thus contribute little to pressure. Thus the agreement we see in Fig. 6 is testing: a) the principles of Hagedorn’s HRG ideas; and b) consistency with the part of the hadron mass spectrum already known, see Fig. 4. A more thorough study is presented in Subsection 8.5, describing the compensating effect for pressure of finite hadron size and missing high mass states in HRG, which than produces good fit to energy density. Lattice-QCD results apply to a fully thermally equilibrated system filling all space-time. This in principle is true only in the early Universe. After hadrons are born at T . TH , the Universe cools in expansion and evolves, with the expansion time constant governed by the magnitude of the (applicable to this period) Hubble parameter; one finds [6,47] τq ∝ 25 · µs at TH , see also Subsection 7.4. the Fig. 6. Pressure P/T 4 of QCD matter evaluated in lattice value of τq is long on hadron scale. A full thermal equili- approach (includes 2+1 flavors and gluons) compared with bration of all HRG particle components can be expected their result for the HRG pressure, as function of T . The upper in the early Universe. limit of P/T 4 is the free Stephan-Boltzmann (SB) quark-gluon Considering the early Universe conditions, it is pos- pressure with three flavors of quarks in the relativistic limit sible and indeed necessary to interpret the lattice-QCD T strange quark mass. Figure from Ref. [68] mdified for this review results in terms of a coexistence era of hadrons and QGP. This picture is usually associated with a 1st order phase transition, see Kapusta and Csernai [48] where one finds T . TH with HRG, within the lattice-QCD uncertainties. separate spatial domains of quarks and hadrons. However, In this way we have ab-initio confirmation that Hagedorn’s as one can see modeling the more experimentally accessi- ideas of using particles and their resonances to describe a ble smooth transition of hydrogen gas to hydrogen plasma, strongly interacting hadron gas is correct, confirmed by this type of consideration applies in analogy also to any more fundamental theoretical ideas involving quarks, glu- smooth phase transformation. The difference is that for ons, QCD. smooth transformation, the coexistence means that the Results seen in Fig. 6 comparing pressure of lattice- mixing of the two phases is complete at microscopic level; QCD with HRG show that, as temperature decreases to- no domain formation occurs. However, the physical prop- wards and below TH , the color charge of quarks and glu- erties of the mixed system like in the 1st order transition ons literally freezes, and for T . TH the properties of case are obtained in a superposition of fractional gas com- strongly interacting matter should be fully characterized ponents. by a HRG. Quoting Redlich and Satz [46]: The recent analysis of lattice-QCD results of Biro and Jakovac [49] proceeds in terms of a perfect microscopic “The crucial question thus is, if the equation of mix of partons and hadrons. One should take note that state of hadronic matter introduced by Hagedorn as soon as QCD-partons appear, in such a picture color can describe the corresponding results obtained from deconfinement is present. In Figures 10 and 11 in [49] the QCD within lattice approach.” and they continue: appearance of partons for T > 140 MeV is noted. More- “There is a clear coincidence of the Hagedorn res- over, this model is able to describe precisely the interac- onance model results and the lattice data on the tion measure equation of states. All bulk thermodynamical ob- ε − 3P I ≡ (11) servables are very strongly changing with temper- m T 4 ature when approaching the deconfinement transi- as shown in Fig. 7. I is a dimensionless quantity that tion. This behavior is well understood in the Hage- m depends on the scale invariance violation in QCD. We dorn model as being due to the contribution of res- note the maximum value of I ' 4.2 in Fig. 7, a value onances. . . . resonances are indeed the essential de- m which reappears in the hadronization fit in Fig. 35, Sub- grees of freedom near deconfinement. Thus, on the section 10.2, where we see for a few classes of collisions thermodynamical level, modeling hadronic interac- the same value I ' 4.6 ± 0.2. tions by formation and excitation of resonances, as m Is this agreement between a hadronization fit, and lat- introduced by Hagedorn, is an excellent approxi- tice I an accident? The question is open since a priori mation of strong interactions.” m this agreement has to be considered allowing for the rapid dynamical evolution occurring in laboratory experiments, a situation vastly differing from the lattice simulation of 12 Johann Rafelski: Melting Hadrons, Boiling Quarks

2 Fig. 8. The square of speed of sound cs as function of temperature T , the relativistic limit is indicated by an arrow. Figure from Ref.[68] modified for this review Fig. 7. The Interaction measure (ε − 3P )/T 4 within mixed parton-hadron model, model fitted to match the lattice data of Ref.[68]. Figure from Ref.[49] modified for this review that the laboratory QGP cannot be close to the full chemical equilibrium; a kinetic computation will be needed to assess how the properties of parton-hadron phase evolve −22 static properties. The dynamical situation is also more given a characteristic lifespan of about τh ∝ 10 s. Such complex and one cannot expect that the matter content a study may be capable of justifying accurately specific of the fireball is a parton-hadron ideal mix. The rapid ex- hadronization models. pansion could and should mean that the parton system evolves without having time to enter equilibrium mixing with hadrons, this is normally called super-cooling in the 2.6 What does lattice-QCD tell us about TH ? context of a 1st order phase transition, but in context of a mix of partons and hadrons [49], these ideas should also We will see in Subsection 3.3 that we do have two dif- apply: as the parton phase evolves to lower temperature, ferent lattice results showing identical behavior at T ∈ the yet nonexistent hadrons will need to form. {150 ± 25} MeV. This suggests that it should be possible To be specific, consider a dense hadron phase created to obtain a narrow range of TH . Looking at Fig. 6, some in RHI collisions with a size R ∝ 5 fm and a T ' 400 h see TH at 140–145 MeV, others as high as 170 MeV. Such MeV, where the fit of Ref.[49] suggest small if any pres- disparity can arise when using eyesight to evaluate Fig. 6 ence of hadrons. Exploding into space this parton domain without applying a valid criterion. In fact such a criterion dilutes at, or even above, the speed of sound in thee trans- is available if we believe in exponential mass spectrum. verse direction and even faster into the longitudinal direc- When presenting critical properties of SBM table 1 tion. For relativistic√ matter the speed of sound Eq. (7) we reported that sound velocity Eq. (7) has the unique approaches cs = c/ 3, see Fig. 8 and only near to TH be- property c → 0 for T → T . What governs this result −22 s H comes small. Within time τh ∝ 10 s a volume dilution is solely the exponential mass spectrum, and this result by a factor 50 and more can be expected. holds in leading order irrespective of the value of the power It is likely that this expansion is too fast to allow index a. Thus a surprisingly simple SBM-related criterion hadron population to develop from the parton domain. for the value of TH is that there cs → 0. Moreover, cs is 2 What this means is that for both the lattice-QGP inter- available in lattice-QCD computation; Fig. 8 shows cs as preted as parton-hadron mix, and for a HRG formed in function of T , adapted from Ref.[68]. There is a noticeable laboratory, the reaction time is too short to allow develop- domain where cs is relatively small. ment of a multi-structure hadron abundance equilibrated In Fig. 8 the bands show the computational uncer- state, which one refers to as ‘chemical’ equilibrated hadron tainty. To understand better the value of TH we follow the gas, see here the early studies in Refs.[50,51,52]. 2 drop of cs when temperature increases, and when cs begins To conclude: lattice results allow various interpreta- to increase that is presumably, in the context of lattice- tions, and HRG is a consistent simple approximation for QCD, when the plasma material is mostly made of decon- T . 145 MeV. More complex models which include coexis- fined and progressively more mobile quarks and gluons. As tence of partons and hadrons manage a good fit to all lat- temperature rises further, we expect to reach the speed of 2 tice results, including the hard to get interaction measure sound limit of ultra relativistic matter cs → 1/3, indicated 2 Im. Such models in turn can be used in developing dynam- in Fig. 8 by an arrow. This upper limit, cs ≤ 1/3 arises ical model of the QGP fireball explosion. One can argue according to Eq. (7) as long as the constraint ε − 3P → 0 Johann Rafelski: Melting Hadrons, Boiling Quarks 13

from above at high T applies; that is Im > 0 and Im → 0 In the Koppe-Fermi-model, as of the instant of their at high T . formation, all hadrons are free-streaming. This is also Hage- The behavior of the lattice result-bands in Fig. 8 sug- dorn’s fireball pot with boiling matter. This reaction view gests hadron dominance below T = 125 MeV, and quark was formed before two different phases of hadronic mat- dominance above T = 150 MeV. This is a decisively more ter were recognized. With the introduction of a second narrow range compared to the wider one seen in the fit in primordial phase a new picture emerges: there are no had- which a mixed parton-hadron phase was used to describe rons to begin with. In this case in a first step quarks lattice results [49]; see discussion in Subsection 2.5. freeze into hadrons at or near TH , and in a second step 2 The shape of cs in Fig. 8 suggests that TH = 138 ± 12 at T < TH hadrons decouple into free-streaming parti- MeV. There are many ramifications of such a low value, cles. It is possible that TH is low enough so that when the as is discussed in the context of hadronization model in quark freezing into hadrons occurs, hadrons are immedi- the following Subsection 2.7. ately free-streaming; that is T ' TH , in which case one would expect abundances of observed individual particles to be constrained by the properties of QGP, and not of 2.7 What is the statistical hadronization model (SHM)? the HRG. On the other hand if in the QGP hadronization a dense The pivotal point leading on from last subsection is that phase of hadron matter should form, this will assure both chemical and thermal equilibrium of later free-streaming in view of Fig. 6 we can say that HRG for T . TH ' 150 MeV works well at a precision level that rivals the nu- hadrons as was clearly explained in 1985 [57]: “Why the merical precision of lattice-QCD results. This result jus- Hadronic Gas Description of Hadronic Reactions Works: tifies the method of data analysis that we call Statistical The Example of Strange Hadrons”. It is argued that the Hadronization Model (SHM). SHM was invented to char- way parton deconfinement manifests itself is to allow a acterize how a blob of primordial matter that we call QGP short lived small dynamical system to reach nearly full falls apart into individual hadrons. At zero baryon den- thermal and chemical equilibrium. sity this ‘hadronization’ process is expected to occur near The analysis of the experimental data within the SHM if not exactly at TH . The SHM relies on the hypothesis allows us to determine the degree of equilibration for dif- that a hot fireball made of building blocks of future had- ferent collision systems. The situation can be very different rons populates all available phase space cells proportional in pp and AA collisions and depend on both collision en- to their respective size, without regard to any additional ergy and the size A of atomic nuclei, and the related vari- interaction strength governing the process. able describing the variable classes, the participant num- The model is presented in depth in Section 9. Here we ber Npart, see Subsection 9.3. Study of strangeness which would like to place emphasis on the fact that the agree- is not present in initial RHI states allows us to address the ment of lattice-QCD results with the HRG provides today equilibration question in a quantitative way as was noted a firm theoretical foundation for the use of the SHM, and already 30 years ago [57]. We return to the SHM stran- it sets up the high degree of precision at which SHM can geness results in Section 10 demonstrating the absence of be trusted. chemical equilibrium in the final state, and the presence Many argue that Koppe [53], and later, independently, of (near) chemical equilibrium in QGP formed at LHC, Fermi [54] with improvements made by Pomeranchuk [55], see Fig. 36 and 37. invented SHM in its microcanonical format; this is the so One cannot say it strongly enough: the transient pres- called Fermi-model, and that Hagedorn [11,56] used these ence of the primordial phase of matter means that there ideas in computing within grand canonical formulation. are two different scenarios possible describing production However, in all these approaches the particles emitted of hadrons in RHI collisions: were not newly formed; they were seen as already being a) A dense fireball disintegrates into hadrons. There can the constituents of the fireball. Such models therefore are be two temporally separate physical phenomena: the re- what we today call freeze-out models. combinant-evaporative hadronization of the fireball made The difference between QGP hadronization and freeze- of quarks and gluons forming a HRG; this is followed by out models is that a priori we do not know if right at the freeze-out; that is, the beginning of free-streaming of the time of QGP hadronization particles will be born into a newly created particles. condition that allows free-streaming and thus evolve in b) The quark fireball expands significantly before convert- hadron form to the freeze-out condition. In a freeze-out ing into hadrons, reaching a low density before hadroni- model all particles that ultimately free-stream to a de- zation. As a result, some features of hadrons upon produc- tector are not emergent from a fireball but are already tion are already free-streaming: i) The hadronization tem- present. The fact that the freeze-out condition must be perature may be low enough to freeze-out particle abun- established in a study of particle interactions was in the dance (chemical freeze-out at hadronization), yet elastic early days of the Koppe-Fermi model of no relevance since scattering can still occur and as result momentum dis- the experimental outcome was governed by the phase space tribution will evolve (kinetic non-equilibrium at hadroni- and microcanonical constraints as Hagedorn explained in zation). ii) At a yet lower temperature domain, hadrons his very vivid account “The long way to the Statistical would be born truly free-streaming and both chemical and Bootstrap Model”, Chapter 17 loc.cit.. kinetic freeze-out conditions would be the same. This con- 14 Johann Rafelski: Melting Hadrons, Boiling Quarks dition has been proposed for SPS yields and spectra in the We do not know how the bulk energy density ε and pres- year 2000 by Torrieri [58], and named ‘single freeze-out’ sure P at hadronization after scaling with T 4 evolve in in a later study of RHIC results [59,60]. time to freeze-out point, and even more interesting is how Im Eq. (11) evolves. This can be a topic of future study. Once scattering processes came into discussion, the 2.8 Why value of TH matters to SHM analysis? concept of dynamical models of freeze-out of particles could be addressed. The review of Koch et.al. [2] comprises many What exactly happens in RHI collisions in regard to par- original research results and includes for the first time the ticle production depends to a large degree on the value of consideration of dynamical QGP fireball evolution into 2 free-streaming hadrons and an implementation of SHM in the chemical freeze-out temperature T ≤ TH . The value a format that we could today call SHM with sudden ha- of TH as determined from mass spectrum of hadrons depends on the value of the preexponential power index a, dronization. In parallel it was recognized that the experimentally observed particle abundances allow the determi- see table 2. The lower is TH , the lower the value of T must be. Since the value of T controls the density of particles, nation of physical properties of the source. This insight as seen in e.g. in Eq. (1), the less dense would be the HRG is introduced in Appendix B: in Fig. B3 we see how the ratio K+/K− allows the deduction of the baryochemical phase that can be formed. Therefore, the lower is TH the more likely that particles boiled off in the hadronization potential µB; this is stated explicitly in pertinent discus- process emerge without rescattering, at least without the sion. Moreover, in the following Fig. B4 the comparison is rescattering that changes one type of particle into another made between abundance of final state Λ/Λ particle ra- i.e. ‘chemical’ free streaming. In such a situation in chem- tio emerging from equilibrated HRG with abundance expected in direct evaporation of the quark-fireball an effect ical abundance analysis we expect to find T ' TH . The SHM analysis of particle production allows us to that we attribute today to chemical nonequilibrium with determine both the statistical parameters including the enhanced phase space abundance. value of T characterizing the hadron phase space, as well Discussion of how sudden the hadronization process is as the extensive (e.g. volume) and intensive (e.g. baryon reaches back to the 1986 microscopic model description density) physical properties of the fireball source. These of strange (antibaryon) formation by Koch, Müllerand govern the outcome of the experiment on the hadron side, the author [2] and the application of hadron afterburner. and thus can be measured employing experimental data Using these ideas in 1991, SHM model saw its first hum- on hadron production as we show in Section 10. ble application in the study of strange (anti)baryons [61]. The faster is the hadronization process, the more infor- Strange baryon and antibaryon abundances were inter- mation is retained about the QGP fireball in the hadronic preted assuming a fast hadronization of QGP – fast mean- populations we study. For this reason there is a long- ing that their relative yields are little changed in the fol- lasting discussion in regard to how fast or, one often says, lowing evolution. For the past 30 years the comparison sudden is the breakup of QGP into hadrons. Sudden ha- of data with the sudden hadronization concept has never dronization means that the time between QGP breakup led to an inconsistency. Several theoretical studies support and chemical freeze-out is short as compared to the time the sudden hadronization approach, a sample of works in- needed to change abundances of particles in scattering of cludes Refs.[62,63,64,65,66]. Till further notice we must hadrons. presume that the case has been made. Among the source (fireball) observables we note the Over the past 35 years a simple and naive thermal nearly conserved, in the hadronization process, entropy model of particle production has resurfaced multiple times, content, and the strangeness content, counted in terms reminiscent of the work of Hagedorn from the early-60s. of the emerging multiplicities of hadronic particles. The Hadron yields emerge from a fully equilibrated hadron fire- physical relevance of these quantities is that they origi- ball at a given T,V and to account for baryon content at nate, e.g. considering entropy or strangeness yield, at an low collision energies one adds µ . As Hagedorn found out, earlier fireball evolution stage as compared to the hadroni- B the price of simplicity is that the yields can differ from ex- zation process itself; since entropy can only increase, this periment by a factor two or more. His effort to resolve this provides a simple and transparent example how in hadron riddle gave us SBM. abundances which express total entropy content there can be memory of the the initial state dynamics. However, in the context of experimental results that Physical bulk properties such as the conserved (baryon need attention, one seeks to understand systematic be- number), and almost conserved (strangeness pair yields, havior across yields varying by many orders of magni- entropy yield) can be measured independent of how fast tude as parameters (collision energy, impact parameter) the hadronization process is, and independent of the com- of RHI collision change. So if a simple model practically plexity of the evolution during the eventual period in time ‘works’, for many the case is closed. However, one finds in while the fireball cools from TH to chemical freeze-out T . such a simple model study the value of chemical freeze- out T well above TH . This is so since in fitting abundant 2 We omit subscript for all different ‘temperatures’ under strange antibaryons there are two possible solutions: either consideration – other than TH – making the meaning clear in a T TH , or T . TH with chemical nonequilibrium. A the text contents. model with T TH for the price of getting strange an- Johann Rafelski: Melting Hadrons, Boiling Quarks 15

Fig. 9. T, µB diagram showing current lattice value of critical temperature Tc (bar on left [67,68], the SHM-SHARE results (full circles) [7,69,70,71,72,73,74,75] and results of other groups [76,77,78,79,80,81,82,83,84]. Figure from Ref.[69] modiﬁed for this review

tibaryons right creates other contradictions, one of which experimentally as large as possible p⊥ coverage in order is discussed in Subsection 10.4. to minimize extrapolation errors on particles yields con- How comparison of chemical freeze-out T with TH works sidered. is shown in Fig. 9. The bar near to the temperature axis Evaluation: In first step we evaluate, given an as- displays the range TH = 147±5 MeV [67,68]. The symbols sumed SHM parameter set, the phase space size for all show the results of hadronization analysis in the T –µB and every particle fraction that could be in principle mea- plane as compiled in Ref.[69] for results involving most sured, including resonances. This complete set is necessary (as available) central collisions and heaviest nuclei. The since the observed particle set includes particles arising solid circles are results obtained using the full SHM pa- from a sequel chain of resonance decays. These decays are rameter set [7,69,70,71,72,73,74,75]. The SHARE LHC implemented and we obtain the relative phase space size freeze-out temperature is clearly below the lattice criti- of all potential particle yields. cal temperature range. The results of other groups are Optional: Especially should hadronization T be at a obtained with simplified parameter sets: marked GSI [76, relatively large value, the primary particle populations can 77], Florence [78,79,80], THERMUS [81], STAR [82] and undergo modifications in subsequent scattering. However, ALICE [83,84]. These results show the chemical freeze-out since T < TH , a large T requires an even larger TH which temperature T in general well above the lattice TH . This shows importance of knowing TH . If T is large, a further means that these restricted SHM studies are incompatible evolution of hadrons can be treated with hadron ‘after- with lattice calculations, since chemical hadron decoupling burners’ taking the system from TH to T . Since in our should not occur inside the QGP domain. analysis the value of hadronization T is small, we do not address this stage further here; see however Refs.[85,86]. Iteration: The particle yields obtained from phase 2.9 How is SHM analysis of data performed? space evaluation represent the SHM parameter set assumed. A comparison of this predicted yield with observed Here the procedure steps are described which need tech- yields allows the formation of a value parameter such as nical implementation presented in Section 9. X χ2 = (theory − data)2/FWHM2, (12) Data: The experiment provides, within a well defined collision class, see Subsection 9.3, spectral yields of many i particles. For the SHM analysis we focus on integrated where FWHM is the error in the data, evaluated as ‘Full spectra, the particle number-yields. The reason that such Width at Half Maximum’ of the data set. In an iterative data are chosen for study is that particle yields are in- approach minimizing χ2 a best set of parameters is found. dependent of local matter velocity in the fireball which Constraints: There may be significant constraints; an imposes spectra deformation akin to the Doppler shift. example is the required balance ofs ¯ = s as strangeness is However, if the p⊥ coverage is not full, an extrapolation of produced in pairs and strangeness changing weak decays spectra needs to be made that introduces the same uncer- have no time to operate [87]. Such constraints can be im- tainty into the study. Therefore it is important to achieve plemented most effectively by constraints in the iteration 16 Johann Rafelski: Melting Hadrons, Boiling Quarks steps; the iterative steps do not need till the very end to conserve e.g. strangeness. Bulk properties: When our iteration has converged, we have obtained all primary particle yields; those that are measured, and all other that are, in essence, extrapo- lations from known to unknown. It is evident that we can use all these yields in order to compute the bulk properties of the fireball source, where the statement is exact for the conserved quantities such as net baryon number (baryons less antibaryons) and approximate for quantities where kinetic models show little modification of the value during hadronization. An example here is the number of strange quark pairs or entropy. Fig. 10. Illustration of the quark bag model colorless states: Discussion: The best fit is characterized by a value baryons qqq and mesons qq. The range of the quantum wave 2 function, typically χ , Eq. (12). Depending on the com- function of quarks, the hadronic radius is indicated as a (pink) plexity of the model, and the accuracy of the inherent cloud, the color electrical field lines connect individual quarks. physics picture, we can arrive at either a well converged fit, or at a poor one where χ2 normalized by degrees of freedom (dof) is significantly above unity. Since the ob- the other interactions the outcome of strong interactions jective of the SHM is the description of the data, for the is different. 2 case of a bad χ one must seek a more complex model. A clear statement is seen in the September 28, 1979 The question about analysis degeneracy also arises: are lecture by T.D. Lee [89] and the argument is also presented there two different SHM model variants that achieve in in T.D. Lee’s textbook [90]: at zero temperature quarks a systematic way as a function of reaction energy and/or can only appear within a bound state with other quarks as collision parameters always a success? Should degeneracy a result of transport properties of the vacuum state, and be suspected, one must attempt to break degeneracy by NOT as a consequence of the enslaving nature of inter- looking at specific experimental observables, as was ar- quark forces. However, indirectly QCD forces provide the gued in Ref.[88]. vacuum structure, hence quarks are enslaved by the same We perform SHM analysis of all ‘elementary’ hadrons QCD forces that also provide the quark-quark interaction. produced – that is we exclude composite light nuclei and Even so the conceptual difference is clear: We can liberate antinuclei that in their tiny abundances may have a dif- quarks by changing the nature of the vacuum, the modern ferent production history; we will allow the data to decide day æther, melting its confining structure. what are the necessary model characteristics. We find that The quark confinement paradigm is seen as an ex- for all the data we study and report on in Section 10, pression of the incompatibility of quark and gluon color- the result is strongly consistent with the parameter set electrical fields with the vacuum structure. This insight and values associated with chemical non-equilibrium. In was inherent in the work by Ken Wilson [91] which was the any case, we obtain a deeper look into the history of the backdrop against which an effective picture of hadronic expanding QGP fireball and QGP properties at chemi- structure, the ‘bag model’ was created in 1974/1975 [92, cal freeze-out temperature T < T and, we argue that H 93,94,95]. Each hadronic particle is a bubble [92]: Below QGP was formed. In a study of the bulk fireball proper- T , with their color field lines expelled from the vacuum, ties a precise description of all relevant particle yields is H quarks can only exist in colorless cluster states: baryons needed. Detailed results of SHM analysis are presented in qqq (and antibaryons qqq) and mesons qq as illustrated in Section 10. Fig. 10. These are bubbles with the electric field lines contained in a small space domain, and the color-magnetic (spin- 3 The Concepts: Theory Quark Side spin hyperfine) interactions contributing the details of the hadron spectrum [93]. This implementation of quark-con- 3.1 Are quarks and gluons ‘real’ particles? finement is the so-called (MIT) quark-bag model. By imposing boundary conditions between the two vacuua quark- The question to be addressed in our context is: How can hadron wave functions in a localized bound state were ob- quarks and gluons be real particles and yet we fail to pro- tained; for a succinct review see Johnson [94]. The later duce them? The fractional electrical charge of quarks is a developments which address the chiral symmetry are sum- strong characteristic feature and therefore the literature marized in 1982 by Thomas [95], completing the model. is full of false discoveries. Similarly, the understanding The quark-bag model works akin to the localization of and explanation of quark confinement has many twists quantum states in an infinite square-well potential. A new and turns, and some of the arguments though on first ingredient is that the domain occupied by quarks and/or sight contradictory are saying one and the same thing. their chromo-electrical fields has a higher energy density Our present understanding requires introduction of a new called bag constant B: the deconfined state is the state paradigm, a new conceptual context how in comparison to of higher energy compared to the conventional confining Johann Rafelski: Melting Hadrons, Boiling Quarks 17

more accessible model of what happens to light quarks in close vicinity of TH , where considerable clustering before and during hadronization must occur. The shape of the heavy quark potential, and thus the stability of ‘onium states can be studied as a function of quark separation, and of the temperature, in the framework of lattice-QCD, showing how the properties of the heavy quark potential change when deconfinement sets in for T > TH [100,101]. To conclude, quarks and gluons are real particles and Fig. 11. Illustration of the heavy quark Q = c, b and antiquark can, for example, roam freely above the vacuum melting ¯ Q =c, ¯ b connected by a color field string. As QQ separate, a point, i.e. above Hagedorn temperature TH . This under- pair of light quarks qq¯ caps the broken field-string ends. standing of confinement allows us to view the quark-gluon plasma as a domain in space in which confining vacuum structure is dissolved, and chromo-electric field lines can vacuum state. In our context an additional finding is im- exist. We will return to discuss further ramifications of the portant: even for small physical systems comprising three QCD vacuum structure in Subsections 7.2 and 7.3. quarks and/or quark-antiquark pairs once strangeness is correctly accounted for, only the volume energy density B without a “surface energy” is present. This was shown 3.2 Why do we care about lattice-QCD? by an unconstrained hadron spectrum model study [96, 97]. This result confirms the two vacuum state hypothesis The understanding of quark confinement as a confinement as the correct picture of quark confinement, with non- of the color-electrical field lines and characterization of analytical structure difference at T = 0 akin to what is hadrons as quark bags suggests as a further question: how expected in a phase transition situation. can there be around us, everywhere, a vacuum structure The reason that in the bag model the color-magnetic that expels color-electric field lines? Is there a lattice- hyperfine interaction dominates the color-electric interac- QCD based computation showing color field lines confine- tion is due to local color neutrality of hadrons made of ment? Unfortunately, there seems to be no answer avail- light quarks; the quark wave-function of all light quarks fill able. Lattice-QCD produces values of static observables, the entire bag volume in same way, hence if the global state and not interpretation of confinement in terms of moving is colorless so is the color charge density in the bag. How- quarks and dynamics of the color-electric field lines. ever, the situation changes when considering the heavy So why care about lattice-QCD? For the purpose of charm c, or bottom b, quarks and antiquarks. Their mass this article lattice-QCD upon convergence is the ultimate scale dominates, and their semi-relativistic wave functions authority, resolving in an unassailable way all questions are localized. The color field lines connecting the charges pertinent to the properties of interacting quarks and glu- are, however, confined. When we place heavy quarks rela- ons, described within the framework of QCD. The word tively far apart, the field lines are, according to the above, lattice reminds us how continuous space-time is repre- squeezed into a cigar-like shape, see top of Fig. 11. sented in a discrete numerical implementation on the most The field occupied volume grows linearly with the size powerful computers of the world. of the long axis of the cigar. Thus heavy quarks interact The reason that we trust lattice-QCD is that it is not when pulled apart by a nearly linear potential, but only a model but a solution of what we think is the founda- when the ambient temperature T < TH . One can expect tional characterization of the hadron world. Like in other that at some point the field line connection snaps, produc- theories, the parameters of the theory are the measured ing a quark-antiquark pair. This means that when we pull properties of observed particles. In case of QED we use on a heavy quark, a colorless heavy-meson escapes from the Coulomb force interaction strength at large distance, the colorless bound state, and another colorless heavy- α = e2/~c = 1/137. In QCD the magnitude of the strength 2 antimeson is also produced; this sequence is shown from of the interaction αs = g /~c is provided in terms of a top to bottom in Fig. 11. The field lines connecting the scale, typically a mass that the lattice approach captures quark to its color-charge source are called a ‘QCD string’. precisely; a value of αs at large distance cannot be mea- The energy per length of the string, the string tension, is sured given the confinement paradigm. nearly 1 GeV/fm. This value includes the modification of There are serious issues that have impacted the capa- the vacuum introduced by the color field lines. bility of the lattice-QCD in the past. One is the problem For T > TH the field lines can spread out and mix of Fermi-statistics which is not easily addressed by classi- with thermally produced light quarks. However, unlike cal computers. Another is that the properties we wanted light hadrons which melt at TH , the heavy QQ¯ mesons (of- to learn about depend in a decisive way on the inclusion ten referred as ‘onium states, like in charmonium cc¯) may of quark flavors, and require accurate value of the mass of remain bound, albeit with different strength for T > TH . the strange quark; the properties of QCD at finite T are Such heavy quark clustering in QGP has been of profound very finely tuned. Another complication is that in view interest: it impacts the pattern of production of heavy par- of today’s achievable lattice point and given the quark-, ticles in QGP hadronization [98,99]. Furthermore, this is a and related hadron-, scales, a lattice must be much more 18 Johann Rafelski: Melting Hadrons, Boiling Quarks

finely spaced than was believed necessary 30 years ago. Se- rious advances in numerical and theoretical methods were needed, see e.g. Refs.[9,26]. Lattice capability is limited by how finely spaced lattice points in terms of their separation must be so that over typical hadron volume sufficient number is found. Therefore, even the largest lattice implemented at present cannot ‘see’ any spatial structure that is larger than a few proton diameters, where for me: few=2. The rest of the Universe is, in the lattice approach, a periodic repetition of the same elementary cell. The reason that lattice at finite temperature cannot replace models in any foreseeable future is the time evolution: temperature and time are related in the theoretical formulation. Therefore considering hadrons in a heat bath we are restricted to consideration of a thermal equilibrium system. When we include temperature, nobody knows how to include time in lattice-QCD, let alone the question of time sequence that has not been so far implemented at T = 0. Thus all we can hope for in hot-lattice-QCD is Fig. 12. Illustration of quark-gluon plasma (QGP) compris- what we see in this article, possibly much refined in un- ing several red, green, blue (RGB colored) quarks and springy derstanding of internal structure, correlations, transport gluons in a modified vacuum state. coefficient evaluation, and achieved computational precision. After this description some may wonder why we should In a nutshell, QGP in the contemporary use of the lan- bother with lattice-QCD at all, given on one hand its limi- guage is an interacting localized assembly of quarks and tations in scope, and on another the enormous cost rivaling gluons at thermal (kinetic) and (close to) chemical (abun- the experimental effort in terms of manpower and com- dance) equilibrium. The word ‘plasma’ signals that free puter equipment. The answer is simple; lattice-QCD pro- color charges are allowed. Since the temperature is above vides what model builders need, a reference point where TH and thus above the scale of light quark u, d-mass, the models of reality meet with solutions of theory describing pressure exhibits the relativistic Stefan-Boltzmann for- the reality. mat, We have already by example shown how this works. 4 2 2 4 In the previous Section 2 we connected in several differ- ∗ 7 ∗ (πT ) ∗∗ (πT ) µ µ P = gB + gF 2 + gF 2 + 2 . (13) ent ways the value of TH to lattice results. It seems clear 8 90π 24π 48π that the interplay of lattice, with experimental data and The stars next to degeneracy g for Bosons B, and Fermions with models can fix TH with a sufficiently small error. A further similar situation is addressed in the following Sub- F, indicate that these quantities are to be modified by section 3.3 where we seek to interpret the lattice results the QCD interaction which affects this degeneracy signifi- on hot QCD and to understand the properties of the new cantly, and differently for B, F and also fort the two terms ∗ ∗∗ phase of matter, quark-gluon plasma. F vs. F. In Eq. (13) the traditional Stefan-Boltzmann T 4 terms, and the zero temperature limit quark-chemical potential µ4 term are well known and also easy to obtain by inte- 3.3 What is quark-gluon plasma? grating the Bose/Fermi gas expressions in the respective limit. The ideal (QCD interaction αs = 0) relativistic hot An artist’s view of Quark-Gluon Plasma (QGP), Fig. 12, quark gas at finite T including the T 2µ2 term in explicit shows several quarks and ‘springy’ gluons – in an image analytical form of the expression was for the first time pre- similar to Fig. 10. It is common to represent gluons by sented by Harrington and Yildiz in 1974 [102] in a work springs, a historical metaphor from times when we viewed which has the telling title “High-Density Phase Transi- gluons as creating a force that grew at a distance so as to tions in Gauge Theories”. be able to permanently keep quarks confined. Our views Regarding the degeneracy factors for the ideal gases: of confinement evolved, but springs remain in gluon illus- The Boson term generalizes the usual Stefan-Boltzmann trations. In principle these springs are also colored: there expression by an added factor 8c for color degeneracy of are 9 bi-color RGB combination, and excluding the ‘white’ gluons: case we have 8 bi-colors of gluons. As this is hard to illus- gB = 2s × 8c = 16. (14) trate, these springs are gray. The domain of space comprising quarks and gluons is colored to indicate that we The corresponding T 4 Fermi (quark) Stefan-Boltzmann expect this to be a much different space domain from the term differs by the well known factor 7/8 for each degree surroundings. of freedom. We count particles and antiparticles as degrees Johann Rafelski: Melting Hadrons, Boiling Quarks 19

are the same results as we see in Fig. 6 connecting at low T with HRG. These results of the Wuppertal-Budapest group at first contradicted the earlier and highly cited result of [104]. However, agreement between both lattice groups was restored by the revised results of Bazavov 2014 [HotQCD Collaboration] [105]. Let us also remember that the low value of TH we obtained at the end of Subsection 2.6 is due to the difference seen below in Fig. 13 between the results of 2009, and those reported a few years later, through 2014. This difference is highly relevant and shows that hadrons melt into quarks near to TH = 138 ± 12 MeV corresponding to 4 < a < 4.5 see table 2. As the results of lattice-QCD became reliable in the high T > 300 MeV domain a decade ago it became apparent that an accurate understanding of g∗ emerges [106] by taking O(αs) corrections literally and evaluating the behavior α (T ). A more modern study of the behavior Fig. 13. The number of degrees of freedom g∗ as function s of temperature T . The solid line includes the effect of QCD of thermal QCD and its comparison with lattice QCD is interactions as obtained within the framework of lattice QCD available [107,108,109]. The thermal QCD explains the by Borsanyi et.al. beginning in 2012 and published in 2014 (see difference between the asymptotic value g = 47.5 and lat- text). Horizontal dashed line: g = 47.5 for free quark-gluon gas tice results which we see in Fig. 13 to be significant at the highest T = 400 considered. In fact thermal quarks are never asymptotically free; asymptotic freedom for hot of freedom: QCD matter quarks suffers from logarithmic behavior. αs(T ) drops slowly and even at the thermal end of the gF = 2s × 2p × 3c × (2 + 1)f = 31.5, (15) standard model T → 150, 000 MeV the QCD interaction remains relevant and g∗/g ' 0.9. This of course is also where indices stand for: s=spin (=2), p-particle and an- true for very high density cold QCD matter, a small dis- tiparticle (=2), c=color (=3, or =8), f-flavor: 2 flavors appointment when considering the qualitative ideas seen q = u, d always satisfy mq T and one flavor (stran- in the work of Collins and Perry [32]. geness s) at phase boundary satisfies m T and turns s . We can conclude by looking at high T domains of all into a light flavor at high temperatures. To make sure this these results that the state of strongly interacting mat- situation is remembered we write (2 + 1) . f ter at T ' 4T is composed of the expected number of The analytical and relatively simple form of the first H nearly free quarks and gluons, and the count of these par- order in O(α ) thermal QCD perturbative correction re- s ticles in thermal-QCD and lattice-QCD agree. We can say sults are given in analytical format in the work of Chin in that QGP emerges to be the phase of strongly interacting 1979 [103], and result in the following degeneracy: matter which manifests its physical properties in terms of g ≡ g∗ + 7 g∗ = 2 × 8 1 − 15αs nearly free dynamics of practically massless gluons and ∗ B 8 F s c 4π quarks. The ‘practically massless’ is inserted also for glu- 7 50αs + 8 2s × 2p × 3c × (2 + 1)f 1 − 21π (16) ons as we must remember that in dense plasma matter all color charged particles including gluons acquire an effec- ∗∗ 2αs gF = 2s × 2p × 3c × (2 + 1)f 1 − π . tive in medium mass. It seems that today we are in control of the hot QCD αs is the QCD energy scale dependent coupling con- matter, but what properties characterize this QGP that stant. In the domain of T we consider αs ' 0.5, but it differ in a decisive way from more ‘normal’ hadron matter? is rapidly decreasing with T . To some extent this is why It seems that the safest approach in a theoretical review in Fig. 13, showing the lattice-QCD results for pressure, is to rely on theoretical insights. As the results of lattice- we see a relatively rapid rise of g∗ as a function of T , to- QCD demonstrate, the “quark-gluon plasma” is a phase wards the indicated limit g = 47.5 of a free gas, horizontal of matter comprising color charged particles (gluons and dashed line. In Fig. 13 three results also depict the path to quarks) that can move nearly freely so as to create ambient the current understanding of the value of TH . The initial pressure close to the Stefan-Boltzmann limit and whose results (triangles)were presented by Bazavov 2009 [104]; motion freezes into hadrons across a narrow temperature this work did not well describe the ‘low’ temperature do- domain characteristic of the Hagedorn temperature TH . main where the value of TH is determined. This is the The properties of QGP that we check for are thus: origin of the urban legend that TH ' 190 MeV, and a lot of confusion. 1. Kinetic equilibrium – allowing a meaningful definition The solid line in Fig. 6 shows Borsanyi et.al. 2012 [67] of temperature; results presented at the Quark Matter 2012 meeting, with 2. Dominance by effectively massless particles assuring later formal paper comprising the same results [68]; these that P ∝ T 4; 20 Johann Rafelski: Melting Hadrons, Boiling Quarks

3. Both quarks in their large number, and gluons, must – A theory of thermal quark matter is that of Harring- be present in conditions near chemical (yield) equilib- ton and Yidliz 1974 [102], but has no discussion of the rium with their color charge ‘open’ so that the count of role of gauge interaction in quantitative terms. This their number produces the correctly modified Stefan- paper is little known in the field of RHI collisions yet Boltzmann constant of QCD. it lays the foundation for the celebrated work by Linde on electroweak symmetry restoration in the early Uni- verse [116]. There is a remarkable bifurcation in the 3.4 How did the name QGP come into use? literature: those who study the hot Universe and its early stages use the same physics as those who explore In this article we use practically always the words Quark- the properties of hot quarks and gluons; yet the cross- Gluon Plasma and the acronym QGP to describe the phase citations between the two groups are sparse. of matter made of deconfined quarks and gluons interact- – Collins and Perry 1975 [32] in “Superdense Matter: ing according to (thermal) QCD and described in numeri- Neutrons or Asymptotically Free Quarks” propose that cal lattice simulations with ever increasing accuracy. How- high density nuclear matter turns into quark matter ever, even today there is a second equivalent name; the due to weakness of asymptotically free QCD. Com- series of conferences devoted to the study of quark-gluon pared to Harrington and Yidliz this is a step back to a plasma formation in laboratory calls itself “Quark Mat- zero-temperature environment, yet also a step forward ter”LéonVan˙ Hove (former scientific director general of as the argument that interaction could be sufficiently CERN) wrote a report in 1987 entitled “Theoretical pre- weak to view the dense matter as a Fermi gas of quarks diction of a new state of matter, the “quark-gluon plasma” is explicitly made. (also called “quark matter”)” [110] establishing the common meaning of these two terms. Following this there are a few, at times parallel develop- When using Quark Matter we can be misunderstood ments – but this is not the place to present a full history to refer to zero-temperature limit. That is why QGP seems of the field. However fragmentary, let me mention instead the preferred term. However, to begin, QGP actually meant those papers I remember best: something else. This is not unusual; quite often in physics in the naming of an important new insight older terms are – Freedman and McLerran 1976/77 [117] who address reused. This phenomenon reaches back to antiquity: the the thermodynamic potential of an interacting rela- early ancient Greek word ‘Chaos’ at first meant ‘empti- tivistic quark gas. ness’. The science of that day concluded that emptiness – Shuryak 1977/78 [118], writes about “Theory of Hadron would contain disorder, and the word mutated in its mean- Plasma” developing the properties of QGP in the frame- ing into the present day use. work of QCD. At first QGP denoted a parton gas in the context of – Kapusta 1978/79 [119] which work completes “Quan- pp collisions; Hagedorn attributes this to Bjorken 1969, tum Chromodynamics at High Temperature,”. but I could not find in the one paper Hagedorn cited the – Chin 1978 [103] synthesized all these results and was explicit mention of ‘QGP’ (see Chapter 25 in [1]). Shuryak the first to provide the full analytical first order αs in 1978 [111] used ‘QGP’ in his publication title addressing corrections as seen in Eq. (16). partons in pp collisions, thus using the language in the old However, in none of the early thermal QCD work is the fashion. acronym ‘QGP’, or spelled out ‘Quark-Gluon Plasma” in- Soon after “QGP” appears in another publication title, troduced. So where did Kalashnikov-Klimov [112] get the in July 1979 work by Kalashnikov and Klimov [112], now idea to use it? I can speculate that seeing the work by describing the strongly interacting quark-gluon thermal Shuryak on “Theory of Hadron Plasma” they borrowed equilibrium matter. This work did not invent what the the term from another Shuryak paper [111] where he used authors called QGP. They were, perhaps inadvertently, ‘QGP’ in his title addressing partons in pp collisions. In- connecting with the term used by others in another con- deed, in an aberration of credit Shuryak’s pp parton work text giving it the contemporary meaning. The results of is cited in AA QGP context, clearly in recognition of the Kalashnikov-Klimov agree with our Eq. (16) attributed to use of the QGP acronym in the title, while Shuryak’s ‘true’ a year earlier, July 1978, work of S.A. Chin [103] presented QGP paper, “Theory of Hadron Plasma” is often not cited under the title “Hot Quark Matter”. This work (despite in this context. In his 1980 review Shuryak [120] is almost the title) included hot gluons and their interaction with shifting to QGP nomenclature, addressing ‘QCD Plasma’ quarks and with themselves. and also uses in the text ‘Quark Plasma’, omitting to men- But QGP in its new meaning already had deeper roots. tion ‘gluons’ which are not established experimentally for Quark-star models [113] appear as soon as quarks are pro- a few more years. In this he echoes the approach of others posed; ‘after’ gluons join quarks [114], within a year in this period. – Peter Carruthers in 1973/74 [115] recognized that dense Having said all the above, it is clear that when ‘QGP’ quark matter would be a quite ‘bizarre’ plasma and he is mentioned as the theory of both hot quarks and hot explores its many body aspects. His paper has priority gluons, we should remember Kalashnikov-Klimov [112] for but is also hard to obtain, published in a new journal as I said, the probably inadvertent introduction of this that did not last. name into its contemporary use. Johann Rafelski: Melting Hadrons, Boiling Quarks 21

4 Quark-Gluon Plasma in Laboratory idea paper, a continuation of an earlier effort by Chapline and others in 1974 [122] where we read (abstract): 4.1 How did RHI collisions and QGP come together? It is suggested that very hot and dense nuclear matter may be formed in a transient state in “head-on” collisions of very energetic heavy ions with medium and heavy nuclei. A study of the particles emitted in these collisions should give clues as to the nature of dense hot nuclear matter. At the time of the initial Chapline effort in 1974 it was too early for a mention of quark matter and heavy ions in together. Indeed, at the Bear Mountain [123] workshop in Fall 1974 the physics of the forthcoming RHI collisions was discussed in a retreat motivated by Lee-Wick [124] matter, a proposed new state of nuclear matter. These authors claim: . . . the state . . . inside a very heavy nucleus can become the minimum-energy state, at least within the tree approximation; in such a state, the “effective” nucleon mass inside the nucleus may be much lower Fig. 14. Ink painting masterpiece 1986: “Nuclei as Heavy as than the normal value. Bulls, Through Collision Generate New States of Matter” by Li Keran, reproduced from open source works of T.D.Lee. In presenting this work, the preeminent theorists T.D. Lee and G.C. Wick extended an open invitation to explore in relativistic heavy ion collisions the new exotic state of The artistic representation of RHI collisions is seen dense nuclear matter. This work generated exciting scien- in Fig. 14 – two fighting bulls. The ink masterpiece was tific prospects for the BEVELAC accelerator complex at created in 1986 by Li Keran and has been around the field Berkley. We keep in mind that there is no mention of quark of heavy ions for the past 30 years, a symbol of nascent matter in any document related to BEVELAC [125], nor symbiosis between science and art, and also a symbol of at the Bear Mountain workshop [123]. However, the en- great friendship between T.D. Lee and Li Keran. The bulls suing experimental search for the Lee-Wick nuclear mat- are the heavy ions, and the art depicts the paradigm of ter generated the experimental expertise and equipment heavy-bull(ion) collisions. needed to plan and perform experiments in search of quark- So how did the heavy ions connect to QGP? In Oc- gluon plasma [126]. And, ultimately, T.D. Lee will turn to tober 1980 I remarked in the citation [20, Appendix A]: recognize QGP as the new form of hot nuclear matter re- “The possible formation of quark-gluon plasma in nuclear sulting, among other things, in the very beautiful painting collisions was first discussed quantitatively by S.A. Chin: by Li Keran, Fig. 14. Phys. Lett. B 78, 552 (1978); see also N. Cabibbo, G. Now back to the March 1978 Chapline-Kerman manu- Parisi: Phys. Lett. B 59, 67 (1975),” see Ref.[20, Appendix script: why was it never published? There are a few pos- A]. Let me refine this: sible answers: a) It is very qualitative; b) In the 5y run up period 1973–1978 the field of RHI collisions was dom- a) The pioneering insight of the work by Cabibbo and inated by other physics such as Lee-Wick. In fact at the Parisi [33] is: i) to recognize the need to modify SBM time quarks were not part of nuclear physics which ‘owned’ to include melting of hadrons and ii) in a qualitative the field of heavy ions. Judging by personal experience I drawing to recognize that both high temperature and am not really surprised that Chapline-Kerman work was baryon density allow a phase transformation process. not published. Planck was dead for 30 years3. It is regret- However, there is no mention direct or indirect in this table that once Chapline-Kerman ran into resistance they work about ‘bull’ collision. did not pursue the publication, or/and further develop- b) The paper by Chin [103], of July 1978, in its Ref. [7] ment of their idea; instead, grants the origin of the idea connecting RHI with QGP to Chapline and Kerman [121], an unpublished manu- a) A year later, Kerman (working with Chin who gave script entitled “On the possibility of making quark him the credit for the QGP-RHI connection idea in his matter in nuclear collisions” of March 1978. This pa- paper), presents strangelets [127], cold drops of quark per clearly states the connection of QGP and RHI matter containing a large strangeness content. collisions that Chin explores in a quantitative fashion b) And a few years later, Chapline [128] gives credit for recomputing the QCD thermodynamic potential, and the quark-matter connection to RHI collisions both

enclosing particles, quarks and gluons, in the bag-like 3 structure, see Section 3.1. Many credit Planck with fostering an atmosphere of open- ness and tolerance as a publisher; certainly he did not hesitate The preprint of Chapline and Kerman is available on- to take responsibility for printing Einstein miraculous 1905 pa- line at MIT [121]. It is a qualitative, mostly conceptual pers. 22 Johann Rafelski: Melting Hadrons, Boiling Quarks

to Chapline-Kerman [121] work, and the work of An- et.al. [129]. Hagedorn explains the time line of our and re- ishetty, Koehler, and McLerran of 1980 [129]. Anishetty lated work in his 1984 review [13]. His later point of view et.al. claim in their abstract is succinctly represented in a letter of September 1995, . . . two hot fireballs are formed. These fireballs Fig. 15, where he says4: “. . . can I hope to witness a proof would have rapidities close to the rapidities of of existence of QG plasma? I am in any case convinced of the original nuclei. We discuss the possible for- its existence, where else could the phase transition (which mation of hot, dense quark plasmas in the fire- with certainty is present) lead?. . . ”. balls. That Anishetty, Koehler, and McLerran view of RHI collision dynamics is in direct conflict with the effort of Hage- dorn to describe particle production in pp collisions which at the time was being adapted to the AA case and presented e.g. in the QM1-report [130]. Anishetty et.al. created the false paradigm that QGP was not produced centrally (as in center of momentum), a point that was corrected a few years later in 1982/83 in the renowned paper of J.D. Bjorken [131]. He obtained an analytical, one dimensional, solution of relativistic hydrodynamics that could be interpreted for the case of the RHI collision as description at asymptotically high energy of the collision events. If so, the RHI collision outcome Fig. 15. Hagedorn in September 1995 awaiting QGP discovery, would be a trail of energy connecting the two nuclei that see text. naturally qualifies to be the QGP. While this replaced the Anishetty, Koehler, and McLerran “cooking nuclei, nothing in-between” picture, this new asymptotic energy idea also distracted from the laboratory situation of the period which had to deal with realistic, rather than asymptotic 4.2 When and where was QGP discovered? collision energies. In that formative period I wrote papers which argued Both CERN and BNL have held press conferences describ- , that the hot, dense QGP fireball would be formed due ing their experimental work. In Fig. 16 a screen shot shows to hadron inelasticity stopping some or even all of nuclear how CERN advertised its position in February 2000 to a matter in the center of momentum frame (CM). However, wider public [134]. The document for scientists agreed to my referees literally said I was delusional. As history has by those representing the seven CERN experiments (see shown (compare Chapline and Kerman) referees are not the time line of CERN-SPS experiments in Fig. 1) pro- always useful. The long paper on the topic of forming QGP vided at the event read: at central rapidity was first published 20 years later in “The year 1994 marked the beginning of the CERN the memorial volume dedicated to my collaborator on this lead beam program. A beam of 33 TeV (or 160 GeV project, Michael Danos [132]. per nucleon) lead ions from the SPS now extends Here it is good to remember that the CERN-SPS dis- the CERN relativistic heavy ion program, started covery story relies on the formation of a baryon-rich QGP in the mid eighties, to the heaviest naturally occur- in the CM frame of reference i.e. at ‘central rapidity’. ring nuclei. A run with lead beam of 40 GeV per nu- RHIC is in transition domain in energy, and LHC energy cleon in fall of 1999 complemented the program to- scale, finally and 30 years later, is near to the Bjorken wards lower energies. Seven large experiments par- ‘scaling’ limit. The word scaling is used, as we should in ticipate in the lead beam program, measuring many a rather wide range of rapidity observe the same state of different aspects of lead-lead and lead-gold colli- hot QGP, a claim still awaiting an experiment. sion events: NA44, NA45/CERES, NA49, NA50, To close the topic, some regrets: an ‘idea’ paper equiv- NA52/NEWMASS, WA97/NA57, and WA98. . . . alent to Ref.[121] introducing the bootstrap model of hot Physicists have long thought that a new state of finite sized hadron matter and transformation into QGP matter could be reached if the short range repulsive in RHI collisions could have been written by Hagedorn and forces between nucleons could be overcome and if myself in late 1977. Hagedorn, however, desired a work- squeezed nucleons would merge into one another. ing model. After 10 months of telling the world about our Present theoretical ideas provide a more precise work, and much further effort in Summer 1978 we wrote picture for this new state of matter: it should be with I. Montvay a 99 page long paper [133], as well as a a quark-gluon plasma (QGP), in which quarks and few months later a much evolved shorter version[25]. Only in the Spring of 1980 was Hagedorn sure we un- 4 German Original: . . . werde ich noch den eindeutigen Nach- derstood the SBM and the hadron melting into QGP in weis der Existenz des QGP erleben? Ich bin sowieso davon RHI. Of course we were looking at central rapidity i.e. überzeugt denn wohin soll der Phasenübergang (den es doch CM system, quite different from the work of Anishetty sicher gibt) sonst führen? Johann Rafelski: Melting Hadrons, Boiling Quarks 23

ing that “piecing together even this indirect evidence of the quark-gluon plasma is a tour de force. The CERN teams have pressed their capabilities to the limit to extract these tantalizing glimpses into a new domain of matter.” Dr. Marburger was evidently expecting a better ‘direct evidence’ to ultimately emerge. Let us look at what this may be: The turn of BNL to announce its QGP arrived 5 years later. At the April 2005 meeting of the American Physical Society, held in Tampa, Florida a press conference took place on Monday, April 18, 9:00 local time. The public announcement of this event was made April 4, 2005: EVIDENCE FOR A NEW TYPE OF NUCLEAR MATTER At the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Lab (BNL), two beams of gold atoms are smashed together, the goal being to recreate the conditions thought to have prevailed in the universe only a few microseconds after the big bang, so that novel forms of nuclear Fig. 16. The press release text: “At a special seminar on matter can be studied. At this press conference, 10 February 2000, spokespersons from the experiments on RHIC scientists will sum up all they have learned CERN’s Heavy Ion program presented compelling evidence for from several years of observing the worlds most the existence of a new state of matter in which quarks, instead energetic collisions of atomic nuclei. The four ex- of being bound up into more complex particles such as protons perimental groups operating at RHIC will present and neutrons, are liberated to roam freely.” a consolidated, surprising, exciting new interpretation of their data. Speakers will include: Dennis Ko- var, Associate Director, Office of Nuclear Physics, gluons, the fundamental constituents of matter, are U.S. Department of Energy’s Office of Science; Sam no longer confined within the dimensions of the nu- Aronson, Associate Laboratory Director for High cleon, but free to move around over a volume in Energy and Nuclear Physics, Brookhaven National which a high enough temperature and/or density Laboratory. Also on hand to discuss RHIC results prevails. . . . (explicative in original:) A common and implications will be: Praveen Chaudhari, Di- assessment of the collected data leads us to rector, Brookhaven National Laboratory; represen- conclude that we now have compelling evi- tatives of the four experimental collaborations at dence that a new state of matter has indeed the Relativistic Heavy Ion Collider; and several the- been created, . . . . The new state of matter oretical physicists. found in heavy ion collisions at the SPS fea- The participants at the press conference each obtained a tures many of the characteristics of the theo- “Hunting for Quark-Gluon Plasma” report, of which the retically predicted quark-gluon plasma.. . . In cover in Fig. 17 shows the four BNL experiments operating spite of its many facets the resulting picture is sim- at the time: BRAHMS, PHOBOS, PHOENIX, and STAR, ple: the two colliding nuclei deposit energy into which reported on the QGP physical properties that have the reaction zone which materializes in the form been discovered in the first three years of RHIC oper- of quarks and gluons which strongly interact with ations. These four experimental reports were later pub- each other. This early, very dense state (energy lished in an issue of Nuclear Physics A [136,137,138,139]. density about 3–4 GeV/fm3, mean particle mo- The 10 year anniversary was relived at the 2015 RHIC menta corresponding to T ≈ 240 MeV) suppresses & AGS Users’ Meeting, June 9-12, which included a spe- the formation of charmonia, enhances strangeness cial celebration session “The Perfect Liquid at RHIC: 10 and begins to drive the expansion of the fireball.. . . ” Years of Discovery”. Berndt Mueller, the 2015 Brookhaven’ BNL presented the following comment [135] Associate Laboratory Director for Nuclear and Particle The CERN results are quite encouraging, says Tom Physics is quoted as follows [140]: Ludlam, Brookhaven’s Deputy Associate Director “RHIC lets us look back at matter as it existed for High-Energy and Nuclear Physics. “These re- throughout our universe at the dawn of time, be- sults set the stage for the definitive round of exper- fore QGP cooled and formed matter as we know it, iments at RHIC in which the quark-gluon plasma . . . The discovery of the perfect liquid was a turn- will be directly observed, opening up a vast land- ing point in physics, and now, 10 years later, RHIC scape for discovery regarding the nature and origins has revealed a wealth of information about this re- of matter.” markable substance, which we now know to be a Brookhaven’s Director John Marburger congratu- QGP, and is more capable than ever of measuring lated CERN scientists on their achievement, stat- its most subtle and fundamental properties.” 24 Johann Rafelski: Melting Hadrons, Boiling Quarks

Fig. 18. Multistrange (anti)baryons as signature of QGP, see text for further discussion

tions we need to address: What makes the interacting quark-gluon plasma such a nearly perfect liquid? How exactly does the transition to confined quarks work? Are there conditions under which the transition becomes discontinuous first-order phase transition? Today we are ready to address these Fig. 17. The cover of the BNL-73847-2005 Formal Report pre- questions. We are eagerly awaiting new results from pared by the Brookhaven National Laboratory, on occasion of the RHIC experimental program press conference April 2005. the upgraded STAR and PHOENIX experiments at The cover identified the four RHIC experiments. RHIC.

4.3 How did the SPS-QGP announcement An uninvolved scientist will ask: “Why is the flow prop- withstand the test of time? erty of QGP: a) Direct evidence of QGP and b) Worth full scientific attention 15 years after the new phase of matter It is impossible to present in extensive manner in this re- was announced for the first time?” Berndt Mueller answers view all the physics results that have driven the SPS an- for this article: nouncement, and I will not even venture into the grounds Nuclear matter at ‘room temperature’ is known to of the RHIC announcement. I will focus here instead on behave like a superfluid. When heated the nuclear what I consider my special expertise, the strangeness sig- fluid evaporates and turns into a dilute gas of nu- nature of QGP. The events accompanying the discovery cleons and, upon further heating, a gas of baryons and development of strangeness signature of QGP more and mesons (hadrons). But then something new than 30 years ago have been reported [15], and the first extensive literature mention of strangeness signature of happens; at TH hadrons melt and the gas turns back into a liquid. Not just any kind of liquid. At QGP from 1980 is found in Appendix A. RHIC we have shown that this is the most perfect So, what exactly is this signature? The situation is il- liquid ever observed in any laboratory experiment lustrated in Fig. 18 and described in detail in Appendix B. at any scale. The new phase of matter consisting In the center of the figure we see thermal QCD based of dissolved hadrons exhibits less resistance to flow strangeness production processes. This thermal produc- than any other substance known. The experiments tion dominates the production occurring in first collision at RHIC have a decade ago shown that the Uni- of the colliding nuclei. This is unlike heavier flavors where verse at its beginning was uniformly filled with a the mass threshold 2mQ T,Q = c, b. Strange quark new type of material, a super-liquid, which once pairs: s and antiquarkss ¯, are found produced in processes dominated by gluon fusion [141]. Processes based on light Universe cooled below TH evaporated into a gas of hadrons. quark collisions contribute fewer ss¯-pairs by nearly a factor 10 [142]. When T ≥ ms the chemical equilibrium abun- Detailed measurements over the past decade have dance of strangeness in QGP is similar in abundance to shown that this liquid is a quark-gluon plasma; i.e. the other light u and d quarks, Appendix A. matter in which quarks, antiquarks and gluons flow Even for the glue fusion processes enough lifespan of independently. There remain very important ques- QGP is needed to reach the large abundance of strange Johann Rafelski: Melting Hadrons, Boiling Quarks 25 quark pairs in chemical equilibrium. The lifespan of the QGP fireball increases as the collision volume increases and/or the energy increases. Since the gluon fusion GG → ss¯ dominates quark flavor conversion qq¯ → ss¯ the abundance of strangeness is signature of the formation of a thermal gluon medium. Of course we need to ask, how come there is a gluon medium at SPS energy scale? In the cascade evolution model one finds that gluons are in general the first to equilibrate in their number and momentum distribution. Equilibration means entropy S production, a topic of separate importance as S production is proceeding in temporal sequence other hadronic observables of QGP, and how entropy is produced remains today an unresolved question, see Subsection 5.2. The gluon based processes are driving the equilibra- 0 tion of quarks and antiquarks; first light q = u, d, next Fig. 19. Results obtained at the CERN-SPS Ω -spectrometer the slightly massive s and also some thermal evolution for Ξ/Λ-ratio in fixed target S-S and S-Pb at 200 A GeV/c; of charm is possible. Strangeness evolves along with the results from the compilation presented in Ref.[146] adapted light (u, u,¯ d, d¯) quarks and gluons G until the time of ha- for this report dronization, when these particles seed the formation of hadrons observed in the experiment. In QGP, s ands ¯ can Given this opinion of the ‘man in charge’, strange an- move freely and their large QGP abundance leads to un- tibaryons became the intellectual cornerstone of the ex- expectedly large yields of particles with a large s ands ¯ perimental strangeness program carried out at the CERN content [143,144], as is illustrated exterior of the QGP SPS, see Fig. 1. Thus it was no accident that SPS research domain in Fig. 18. program included as a large part the exploration of the A signature of anything requires a rather background predicted strange (anti)baryon enhancement. We see this free environment, and a good control of anything that is on left in Fig. 1 noting that ‘hadrons’ include of course there as no signature is background free. There are ways (multi)strange hadrons and strange antibaryons. two other than QGP to make strange antibaryons: In AA collisions at the CERN-SPS Ω0-spectrometer, I) Direct production of complex multistrange (anti)baryons the production of higher strangeness content baryons and is less probable for two reasons: antibaryons was compared to lower strangeness content ¯ ¯ 1. When new particles are produced in a color string particles, Ξ/Λ and Ξ/Λ. These early SPS experiments breaking process, strangeness is known to be produced published in 1997 clearly confirmed the QGP prediction less often by a factor 3 compared to lighter quarks. in a systematic fashion, as we see in the 1997 compilation 2. The generation of multistrange content requires mul- of the pertinent experimental WA85 and WA94 results by tiple such suppressed steps. Antinori [146], see Fig. 19. Given the systematic multiple observable 3s.d. agreement of experiment with the model Thus the conclusion is that with increasing strangeness predictions, I saw this result as first and clear experimental content the production by string processes of strange had- evidence of QGP obtained by the experiment-line WA85 rons is progressively more suppressed. and WA94 designed to discover QGP. II) Hadron-hadron collisions can redistribute strangeness In these experiments WA85 and WA94 (see Fig. 1) into multistrange hadrons. Detailed kinetic model study the sulfur ions (S) at 200 A GeV hit stationary labora- shows that the hadron-reaction based production of mul- tory targets, S, W (tungsten), respectively, with reference tistrange hadrons is rather slow and requires time that date from pp (AFS-ISR experiment at CERN) and p on S ¯ ¯ exceeds collision time of RHI collisions significantly. This shown for comparison. The Ξ/Λ and Ξ/Λ ratio enhance- means that both Ξ, Ξ and Ω, Ω are in their abundance sig- ment rises with the size of the reaction volume measured natures of QGP formation and hadronization, for further in terms of target A, and is larger for antimatter as com- details see Appendix B, Refs.[52] and [2]. pared to matter particles. Looking at Fig. 19, the effect LéonVan Hove, the former DG (1976-1980), charac- is systematic, showing the QGP predicted pattern [52,2], terized the strange antibaryon signature after hearing the see also Appendix A and B. reports [143,144] as follows [145]: The ‘enhancement’ results obtained by the same group now working in CERN North Area for the top SPS energy In the “Signals for Plasma” section: . . . implying Pb (lead) beam of 156 A GeV as published in 1999 by An- (production of) an abnormally large antihyperon dersen [NA57 Collaboration] [147] is shown in Fig. 20. On to antinucleon ratio when plasma hadronizes. The the right hadrons made only of quarks and antiquarks that qualitative nature of this prediction is attractive, are created in the collision are shown. On the left some of all the more so that no similar effect is expected in the hadron valence quarks from matter can be brought the absence of plasma formation. into the reaction volume. 26 Johann Rafelski: Melting Hadrons, Boiling Quarks

the results of Fig. 20 which show that the enhancement is volume independent are in Fig. 21 compressed to a relatively small domain on the right in both panels. The SPS- NA57 results in Fig. 21 are in agreement with the 1999 ‘high’ participant number results shown in Fig. 20. The rise of enhancement which we see in Fig. 21 as a function of the number of participants 2 < hNparti < 80 reflects on the rise of strangeness content in QGP to its chemical equilibrium abundance with an increase in volume and thus lifespan of QGP fireball. It is not surprising that the enhancement at SPS is larger than that seen at RHIC and LHC, considering that the reference yields play an important role in this comparison. Especially the high energy LHC pp reactions should begin to create space domains that resemble QGP and nearly achieve the degree of chemical strangeness equilibration that could erase the enhancement effect entirely. Fig. 20. Results obtained by the CERN-SPS NA57 experiment The study of the φ(ss¯) abundance and enhancement (former Ω0-spectrometer WA85 and WA94 team) for multi- corroborates these findings [148]. The importance in the strangeness enhancement at mid-rapidity |yCM| < 0.5 in fixed present context is that while φ(ss¯) by its strangeness con- target Pb-Pb collisions at 158 A GeV/c as a function of the nects to Ξ−(ssd), Ξ, φ(ss¯) is a net-strangeness free parti- mean number of participants hNparti, from Ref.[147]. cle. Therefore if it follows the pattern of enhancement established for Ξ, Ξ it fingerprints strangeness as being the quantity that causes the effect. For some of my colleagues, The enhancement in production of higher strangeness these year 2008 results were the decisive turnpoint to dif- content baryons and antibaryons in AA collisions increases ferentiate the strangeness effect from the effect associated with the particle strangeness content. To arrive at this re- with the source volume described in the closing discussion sult, the ‘raw’ AA yields are compared with reference pp-, of Appendix A. Those reading more contemporary litera- pA -reaction results and presented per number of ‘partici- ture should note that this volume source effect has been pants’ hNparti obtained from geometric models of reaction rediscovered three times since, and at some point in time based on energy and particle flows. We will discuss this in called “canonical suppression”. Subsection 9.3. The number of collision participants for The reader should also consult Subsection 10.1, where all data presented in Fig. 20 is large, greater than 100, a it is shown that QGP formation threshold for Pb–Pb col- point to remember in further discussion. lisions is found at about 1/4 of the 156 A GeV projectile We see that production of hadrons made entirely from energy, and that the properties of physical QGP fireball newly created quarks are up to 20 times more abundant formed at SPS are just the same, up to volume size, when in AA -reactions when compared to pA reference mea- SPS results are compared to RHIC, and with today data surement. This enhancement falls with decreasing strange- from LHC. Today, seen across energy, participant num- ness content and increasing contents of the valence quarks ber, and type of hadron considered, there cannot be any which are brought into collision. These reference results doubt that the source of enhancement is the mobility of at yield ratio ‘1’ provide the dominant error measure. The quarks in the fireball, with the specific strangeness content pattern of enhancement follows the QGP prediction and showing gluon based processes. is now at a level greater than 10 s.d.. There is no other Recall, in February 2000 in the snap of the QGP an- than QGP known explanation of these results. nouncement event, the highly influential Director of BNL, These discoveries are now all more than 15 years old. Jeff Marburger5 called these NA57 results and other CERN- They have been confirmed by further results obtained at ion experimental results, I paraphrase the earlier year 2000 SPS, at RHIC, and at the LHC. The present day exper- precise quote: “pieced together indirect glimpse of QGP”. imental summary is shown in the figure Fig. 21. We see Today I would respond to this assessment as follows: the results obtained by the collaborations:√ NA57 results seen in Fig. 20 and confirmed in past 15 SPS: NA57 for collision energy sNN = 17.2 GeV years of work, see Fig. 21 are a direct, full panoramic (lighter open symbols); √ sight of QGP, as good as one will ever obtain. There is RHIC: STAR for collision energy sNN = 200 GeV nothing more direct, spectacular, and convincing that we (darker open symbols); √ have seen as evidence of QGP formation in RHI collision LHC: Alice for collision energy sNN = 2760 GeV experiments. (filled symbols). These results span a range of collision energies that differ by a factor 160 and yet they are remarkably similar. 5 Jeff Marburger was a long term Presidential Science Advi- Comparing the results of Fig. 21 with those seen in sor, President of Stony Brook campus of the NY State Univer- Fig. 20 we note that hNparti is now on a logarithmic scale: sity System, Director of BNL. Johann Rafelski: Melting Hadrons, Boiling Quarks 27

− + − + Fig. 21. Enhancements of Ξ , Ξ , Ω + Ω in the rapidity range |yCM| < 0.5 as a function of the mean number of participants hNparti: LHC-ALICE: full symbols; RHIC-STAR and SPS-NA57: open symbols. The LHC reference data use interpolated in energy pp reference values. Results at the dashed line (at unity) indicate statistical and systematic uncertainties on the pp or pBe (at SPS) reference. Error bars on the data points represent the corresponding uncertainties for all the heavy-ion measurements. Results presented and compiled in Ref.[149].

5 The RHI physics questions of today The situation changes if we model this like a collision of the two bulls of Li Keran and T.D. Lee. Once some of 5.1 How is energy and matter stopped? the energy (and baryon number) of two nuclei has slowed down to rest in CM, the clocks of both ‘slowed’ bulls tick We arrange to collide at very high, relativistic energies, nearly at the same speed as the clock comoving with CM two nuclei such as lead (Pb) or gold (Au), having each frame – for the stopped energy and baryon number the about 12 fm diameter. In the rest frame of one of the lifespan of the fireball is again quite large. two nuclei we are looking at the other Lorentz-contracted But how do we stop the bulls or at least some of nucleus. The Lorentz contraction factor is large and thus their energy? The answer certainly depends on the energy what an observer traveling along with each nucleus sees regime. The lower is the energy of the bulls, the less we approaching is a thin, ultra dense matter pancake. As this need to worry; the pancakes are not thin and one can try pancake penetrates into the other nucleus, there are many to make parton-collision cascade to describe the physics reactions that occur, slowing down projectile matter. case, see e.g. Geiger-Sriwastava [150,154,155] for SPS en- For sufficiently high initial energy the collision occurs ergy range. The use of these methods for RHIC or even at the speed of light c despite the loss of motion energy. LHC energies looks less convincing [151]. Hence each observer comoving with with each of the nuclei records the interaction time τ that a pancake needs To put the problem in perspective, we need a way to to traverse the other nucleus. The geometric collision time concentrate entropy so that a thermal state can rapidly thus is cτ 0 = 12fm as measured by an observer comoving arise. Beginning with the work of Bjorken [131] a forma- with one of the nuclei. Thus if you are interested like An- tion time is introduced, which is more than an order of ishetty et.al. [129] in hot projectile and target nuclei there magnitude shorter compared to τ 0. It is hard to find tan- is no doubt this is one of the outcomes of the collision. gible experimental evidence which compels a choice such An observer in the center of momentum (CM) frame as 0.5fm/c, and theory models describing this stage are can determine the fly-by time that two nuclei need to pass not fully convincing. A model aims to explain how as a each other should they miss to hit: this is τ 0/γ, where γ is function of collision energy and centrality the easy to ob- the Lorentz-factor of each of the nuclei with respect to CM serve final entropy (hadron multiplicity) content arises. frame. This time is, in general, very short and even if nuclei For some related effort see review work of the Werner- were to touch in such short time very little could happen. group [152] and Iancu-Venugopalan [153]. 28 Johann Rafelski: Melting Hadrons, Boiling Quarks

To summarize, in the ‘low’ energy regime of SPS we ence of a strong field: the stronger is the field the greater can try to build a parton cascade model to capture the is the rate of field conversion into particles. One finds that essence of heavy-ion collision dynamics [154,155]. The un- when the field strength is such that it is able of accelerat- derstanding of the initial ‘formation’ of QGP as a function ing particles with a unit strength critical acceleration, the of collision energy and the understanding of the mecha- speed of field decay into particle pairs is such that a field nism that describe energy and baryon number stopping filled state makes no sense as it decays too fast [158]. For remains one of the fundamental challenges of the ongoing this reason there is an effective limit to the strength of the theoretical and experimental research program. field, and forces capable to accelerate particles at critical limit turn the field filled space into a gas of particles. The conversion of energy stored in fields into particles, 5.2 How and what happens, allowing QGP creation? often referred to, in the QCD context, as the breaking of color strings, must be an irreversible process. Yet the In the previous Subsection 5.1 we addressed the question textbook wisdom will assign to the time evolution pure how the energy and baryon number is extracted from fast quantum properties, and in consequence, while the com- moving nuclei. In this section the added challenge is, how plexity of the state evolves, it remains ‘unobserved’ and is the entropy produced that we find in the fireball? While thus a pure state with vanishing entropy content. Intu- in some solutions of the initial state formation in RHI itively, this makes little sense. Thus the riddle of entropy collisions these two topics are confounded, these are two production in RHI collisions which involve an encounter different issues: stopping precedes and is not the same as of two pure quantum states and turns rapidly into state of abundant entropy production. large entropy carried by many particles maybe related to For many the mechanism of fast, abundant entropy our poor formulation of quantum processes for unstable formation is associated with the breaking of color bonds, critical field filled states decaying into numerous pairs. the melting of vacuum structure, and the deconfinement of However, the situation may also call for a more fun- quarks and gluons. How exactly this should work has never damental revision of the laws of physics. The reason is been shown: Among the first to address a parton based en- that our understanding is based in experience, and we tropy production quantitatively within a kinetic collision really do not have much experience with critical accelera- model was Klaus Geiger [154,155] who built computer cas- tion conditions. We study usually acceleration phenomena cade models at parton level, and studied thermalization as which on microscopic scale, either are very small, or even a collision based process. zero in principle. However, in RHI collisions when we stop In order to understand the QGP formation process a partons in the central rapidity region we encounter the solution of this riddle is necessary. There is more to en- critical acceleration, an acceleration that in natural units tropy production: it controls the kinetic energy conversion is unity and which further signals a drastic change in the into material particles. The contemporary wisdom how to way fields and particles behave. The framework of physical describe the situation distinguishes several reaction steps laws which is based on present experience may not be suf- in RHI collision: ficiently complete to deal with this situation and we will 1) Formation of the primary fireball; a momentum need to increase the pool of our experience by performing equipartitioned partonic phase comprising in a limited many experiments involving critical forces. space-time domain, speaking in terms of orders of mag- To conclude: a) The measurement of entropy produc- nitude, almost the final state entropy; tion is relatively straightforward as all entropy produced 2) The cooking of the energy content of the hot matter at the end is found in newly produced particles; b) The fireball towards the particle yield (chemical) equilibrium QGP formation presents an efficient mechanism for the in a hot perturbative QGP phase; conversion of the kinetic energy of the colliding nuclei into 3) Expansion and evaporation cooling towards the tem- particles in a process that is not understood despite many perature phase boundary; years of effort; c) Exploration and understanding of the 4) Hadronization; that is, combination of effective and principles that lead to the abundant formation of entropy ¯ strongly interacting u, d, s, u,¯ d ands ¯ quarks and anti- in the process of QGP formation in RHI collisions harbors quarks into the final state hadrons, with the yield proba- potential opportunity to expand the horizons of knowl- bility weighted by accessible phase space. edge. It is the first step that harbors a mystery. The current textbook wisdom is that entropy production requires the immersion of the quantum system in a classical environment. Such an environment is not so read- 5.3 Nonequilibrium in fireball hadronization ily available for a RHI collision system that has a lifespan of below 10−22 s and a size less than 1/10,000 of atomic Heavy flavor production cross sections, in lowest order in 2 2 size. For a year 2011 review on entropy production dur- coupling constant, scale according to σ ∝ αs/m . Consid- ing the different stages of RHI collision see Ref.[156]. The ering a smaller (running) coupling, and a much larger mass search for a fast entropy generating mechanism continues, of e.g. heavy quarks c, b, we obtain a significant reduction see for example Ref.[157]. in the speed of thermal QGP production reactions. For So what could be a mechanism of rapid entropy forma- charm and bottom, contribution for thermal production tion? Consider the spontaneous pair production in pres- depends on the profile of temperature but is very likely Johann Rafelski: Melting Hadrons, Boiling Quarks 29 negligible, and for charm it is at the level of a few per- and, in the context of present discussion, also a good ap- cent. Conversely, light quarks equilibrate rather rapidly proximation at RHIC. compared to the even more strongly self coupled gluons Considering charm abundance, in QGP chemical equi- and in general can be assumed to follow and define QGP librium γQGP(t) → 1. However, we recall that charm froze matter properties. out shortly after first collisions. Therefore the value of QGP Heavy quark yields originate in the pre-thermal parton γc (t) in Eq. (19) is established by need to preserve the dynamics. However, heavy quarks may acquire through total charm pair number Nc = Const. The exponential elastic collisions a momentum distribution characteristic factor m/T changes from about 2 to 8 near to hadroni- of the medium, providing an image of the collective dy- zation. Thus for prescribed yields at LHC and RHIC it QGP namics of the dense quark matter flow. Moreover, the is likely that γc (t) > 1. More generally there is no- QGP question of yield evolution arises, in particular with re- body who disagrees with the need to have γc 6= 1. QGP gard to annihilation of heavy flavor in QGP evolution. γc = 1 is an accidental condition. We have established Our ‘boiling’ QGP fireball is not immersed in a bath. that charm, and for the very same reason, bottom fla- It is expanding or, rather, exploding into empty space at vor, cannot be expected to emerge in chemical equilibrium a high speed. This assures that the entropy S ∝ V is abundance at hadronization. not decreasing, but increasing, in consideration of inter- A QGP filled volume at high T cooks up a high content nal collisions which describe the bulk viscosity. The ther- of strangeness pairs, in essence as many as there are of mal energy content is not conserved since the sum of the each light flavor u, d; in plasma strangeness suppression kinetic energy of expanding motion, and thermal energy, disappears; the Wroblewski suppression factor [173] (see is conserved. Since the positive internal pressure of QGP also next subsection) is therefore close to unity. As plasma accelerates the expansion into empty space, an explosion, evolves and cools at some relatively low temperature the the thermal energy content decreases and the fireball cools yield of strangeness freezes-out, just like it did for charm rapidly. (and bottom) at higher value of T . In this dynamical evolution quark flavors undergo chem- In earlier discussion we have assumed that in QGP ical freeze-out. The heavier the quark, the earlier the abun- strangeness will follow the evolution in its pair abundance, dance freeze-out should occur. Charm is produced in the and always be in chemical equilibrium in the fireball. This first collisions in the formative stage of QGP. The cou- tacit assumption is not supported by kinetic theory for pling to thermal environment is weak. As ambient temper- T < Ts ' 180 MeV; however for such low value of T ature drops the charm quark phase space given its mass the systematic error of perturbative QCD is large, thus drops rapidly. The quantum Fermi phase space distribu- we really do not know where approximately strangeness tion which maximizes the entropy at fixed particle number pair yield freezes out. We must introduce a pair fugacity is [159] parameter aside of charm also for strangeness and we now QGP 1 have γ 6= 1. The phase space size of strangeness on n (t) = , (17) b,c,s F γ−1(t)e(E∓µ)/T (t) + 1 the hadron side is smaller so once strangeness emerges 6 one must expect that a relatively large value could be d NF g p = n ,E = p2 + m2 , measured. d3pd3x (2π)3 F So what about γu,d? If the evolution as a function of T where g is the statistical degeneracy, and the chemical of the fireball properties is smooth as lattice computation nonequilibrium fugacity (phase space occupancy) γ(t) is suggests, then the strongly coupled light quarks and glu- the same for particles and antiparticles while the chemical ons are defining the QGP properties and, remain in equi- potential µ describes particle-antiparticle asymmetry, and librium . . . really? The flaw in this argument is that only changes sign as indicated. Our µ is ‘relativistic’ chemical quarks define final hadrons. Thus gluons transform into potential. In the nonrelativistic limit µ ≡ m + µnr such quark pairs feeding additional mesons and baryons in that that m implicit in E cancels out for particle but turns way and helping preserve entropy content. Thus gluon to a 2m/T suppression for the antiparticles. Note that dissolution into additional hadrons assures that the light independent of the values of all parameters, nF ≤ 1 as quark phase space occupancies as measured in terms of ob- HG QGP required. served hadron abundances should show γu,d > γu,d > 1. The integral of the distribution Eq. (17) provides the The introduction hadron-side of phase space occupancy particle yield. When addressing SHARE phase space prop- γs [61] and later γu,d [75] into the study of hadron pro- erties in Subsection 9.4, we will inspect the more exact reduction in the statistical hadronization approach has been sult, here we consider the Boltzmann nonrelativistic limit challenged; There was no clear and obvious scientific case. suitable for heavy quarks (c, b) However if the arguments are driven by an intuitive ar- 3 gument that in RHI collisions at sufficiently high reac- gV T ±µ/T 2 m N = γe x K2(x) , x = (18) tion energy aside of thermal, also chemical equilibrium is 2π2 T reached. One of the objectives of this review is to explain why this intuition is wrong when QGP is formed. → gV (mT/2π)3/2γe−(m∓µ)/T (19) Note further that there is a difference between an as- T (t) is time dependent because the system cools. Let us sumption and the demonstration of a result. All know that look at the case µ = 0, appropriate for physics at LHC to make a proof one generally tries to show a contrary 30 Johann Rafelski: Melting Hadrons, Boiling Quarks behavior and arrives at a contradiction: in this case one The observation of strange hadrons involves the identifica- starts with γs,u,d 6= 1 and shows that results are right only tion of non-strange hadrons and thus a full characteriza- for γs,u,d → 1. However, we will see in Section 10 that re- tion of all particles emitted is possible. This in turn creates sults are right when γs,u,d 6= 1 and we show by example an opportunity to understand the properties of the QGP in Subsection 10.3 how the urban legend ‘chemical equi- at time of hadronization, see Subsection 10.1. librium works’ formed relying on a set of errors and/or omissions. Hard hadrons: jet quenching The question about chemical non/equilibrium condi- With increasing energy, like in pp, also in AA collisions tions has to be resolved so that consensus can emerge hard parton back-scattering must occur, with a rate de- about the properties of the hadronizing QGP drop, and scribed by the perturbative QCD [160,161]. Such hard the mechanisms and processes that govern the hadroni- partons are observed in back-to-back jets, that is two jet- zation process. like assembly of particles into which the hard parton hadronizes. These jets are created within the primordial medium. If geometrically such a pair is produced near to the edge of colliding matter, one of the jet-partons can escape 6 How is the experimental study of QGP and the balancing momentum of the immersed jet-parton continuing today? tells us how it travels across the entire nuclear domain, in essence traversing QGP that has evolved in the collision. Today RHI collisions and QGP is a research field that has The energy of such a parton can be partially or completely grown to be a large fraction of nuclear science research dissipated, ‘thermalized’ within the QGP distance trav- programs on several continents. A full account of meth- eled. Since at the production point a second high energy ods, ongoing experiments, scheduled runs, future plans quark (parton) was produced, we can deduce from the ‘jet’ including the development of new experimental facilities asymmetry that the dense matter we form in RHI colli- is a separate review that this author cannot write. The sions is very opaque, and with some effort we can quantify question how to balance presenting ‘nothing’, with ‘every- the strength of such an interaction. This establishes the thing’, is never satisfactorily soluble. The selection of the strength of interaction of a parton at given energy with following few topics is made in support of a few highlights the QGP medium. of greatest importance to this review. Direct Photons Hot electromagnetic charge plasma radiates both photons and virtual photons, dileptons [162,163]. The hotter 6.1 Short survey of recent QGP probes & results is the plasma, the greater is the radiation yield; thus we hope for a large early QGP stage contribution. Electro- A short list of contemporary QGP probes and results in- magnetic probes emerge from the reaction zone without cludes: noticeable loss. The yield is the integral over the history of QGP evolution, and the measured uncorrected yield is Strangeness and other soft hadrons polluted by contributions from the ensuing hadron decays. This cornerstone observable of QGP is a topic of per- On the other hand, at first glancce photons are the sonal expertise of the author and is addressed elsewhere ideal probes of the primordial QGP period if one can con- and at length in these pages. The following is a brief sum- trol the background photons from the decay of strongly mary: Strangeness, the lightest unstable quark flavor, ap- interacting particles such as π0 → γγ which in general pears in pp collisions with an abundance that is about a are dominant6. Recognition of the signal as direct QGP factor 2.5–3 below that of each light quark flavor; this is photon depends on a very precise understanding of the the mentioned ‘Wroblewski’ ratio [173]. It is natural to background. expect that in a larger physical AA collision system addi- At the highest collision energy the initial QGP tem- tional scattering opportunity among all particles creates perature increases and thus direct photons should be more a more democratic abundance with u, d, s quarks being abundant. In Fig. 22 we see the first still at the time of available in nearly equal abundance. However, this initial writing preliminary result from the Alice experiment at simple hypothesis, see Appendix A, needed to be refined LHC. The yield shown is ‘direct’ that is after the indirect with actual kinetic theory evaluation; see Appendix B, in photon part has been removed. The removal procedure consideration of the short time available and demonstra- appears reliable as for large p⊥ scaled pp yields match the tion that quark collisions were too slow [142] to achieve outcome. At small p , we see a very strong excess above this goal. ⊥ the scaled pp yields. The p⊥ is high enough to believe It was shown that the large abundance of strangeness that the origin are direct QGP photons, and not collec- depends on gluon reactions mechanism; thus the ‘gluon’ tive charge acceleration-radiation phenomena. particle component in quark gluon plasma is directly in- A virtual photon with q2 6= 0 is upon materializa- volved [141], see Appendix B. The high strangeness den- tion a dilepton e+e−, µ+µ− in the final state. The dilep- sity in QGP and ‘democratic’ abundance at nearly the same level also implies that the production of (anti)baryons 6 Note that γ when used as a symbol for photons is not to with multiple strangeness content is abundant, see Fig. 18, be confounded with parallel use of γ as a fugacity, meaning is which attracted experimental interest, see Subsection 4.3. always clear in the context. Johann Rafelski: Melting Hadrons, Boiling Quarks 31

Fig. 22. Direct photon LHC-AA yield; Adapted from: F. Antinori presentation July 2014. ton yields, compared to photons, are about a factor 1000 assumption that soft hadrons emerge from the hadroni- smaller; this creates measurement challenges e.g. for large zation fireball with transverse size as large as R ' 9 fm p⊥. Backgrounds from vector meson intermediate states for most central collisions. Aside of two particle correla- and decays are very large and difficult to control. Despite tion, more complex multi-particle correlations can be and many efforts to improve detection capabilities and the un- are explored – their non vanishing strength reminds us derstanding of the background, this author considers the that the QGP source can have color-charge confinement situation as fluid and inconclusive: dilepton radiance not related multi-particle effects that remain difficult to quan- directly attributed to hadrons is often reported and even tify. As an example of recent work on long range rapidity more often challenged. An observer view is presented in correlations see Ref.[171] Ref.[164]. Fluctuations J/Ψ(cc¯) yield modification Any physical system that on first sight appears homo- This is the other cornerstone observable often quoted geneous will under a magnifying glass show large fluctua- in the context of the early QGP search. The interest in tions; the color of the sky and for that matter of our planet the bound states of heavy charm quarks cc¯ and in partic- originate in how the atmospheric density fluctuations scat- ular J/Ψ is due to their yield evolution in the deconfined ter light. To see QGP fluctuation effects we need to study state as first proposed by Matsui and Satz [165] just when each individual event forming QGP apart from another. first result J/Ψ became available. Given that the varia- The SHARE suite of SHM programs also computes sta- tions in yield are subtle, and that there are many model tistical particle yield fluctuations, see Subsection 9.4. The interpretations of the effects based on different views of in- search is for large, nonstatistical fluctuations that would teraction of J/Ψ in the dense matter – both confined and signal competition between two different phases of mat- deconfined – this has been for a long time a livid topic ter, a phase transformation. This topic is attracting at- which is beyond the scope of this review [166]. tention [172]. To see the phase transformation in action Modern theory addresses both ‘melting’, and recom- smaller reaction systems may provide more opportunity. bination in QGP as processes that modify the final J/Ψ yield [98,167]. Recent results of the Alice collaboration [168] support, in my opinion, the notion of recombinant cc¯ for- 6.2 Survey of LHC-ion program July 2015 mation. Some features of these results allow suggesting that a yield equilibrium between melting and recombina- The Large Hadron Collider (LHC) in years of operation tion has been reached for more central collisions. This is sets aside 4 weeks of run time a year to the heavy ion beam clearly a research topic, not yet suitable for a review anal- experiments, typically AA (Pb-Pb) collisions but also p- ysis. Pb. The pp collision LHC run which lasts considerably longer addressing Higgs physics and beyond the standard Particle correlations and HBT model searches for new physics. This long run provides Measurement of two particle and in particular two heavy ion experimental groups an excellent opportunity pion and two kaon correlations allows within the frame- to obtain relevant data from the smallest collision system, work of geometric source interpretation the exploration of creating a precise baseline against which AA is evaluated. the three dimensional source size and the emission lifes- Furthermore, at the LHC energy, one can hope that in pan of the fireball. For a recent review and update of some measurable fraction of events conditions for QGP PHENIX-RHIC results see Ref.[169] and for ALICE-LHC could be met in select, triggered events (i.e. collision class see Ref.[170]. These reports are the basis for our tacit feature selected). 32 Johann Rafelski: Melting Hadrons, Boiling Quarks

When LHC reaches energy of 7 TeV+7 TeV for pro- collision energy. Therefore the chemical equilibration is tons, for Pb-Pb collisions this magnet setting√ will corre- achieved more rapidly at higher energy. It seems that just spond to a center-of-mass energy of up to sNN = 5.52 TeV about everyone agrees to this even though one can easily per nucleon pair in Pb-Pb collisions. However, due to argue the opposite, that more time is available at lower en- magnet training considerations the scheduled heavy√ ion ergy. In any case, this urban legend that energy and time run starting in mid-November 2015 should be at sNN = grow together is the main reason why QGP search exper- 5.125 TeV and the maximum energy achieved in the fol- iments started at the highest available accelerator energy. lowing year. The results we discuss in this review, see Sec- This said, the question about the threshold of QGP pro- tion 10, were obtained at a lower√ magnet setting in the duction as a function of energy is open. LHC run 1, corresponding to sNN = 2.76 TeV. Considered from a theoretical perspective one recog- Several experiments at LHC take AA collision data. nizes in a energy and A scan the opportunity to explore 1. The ALICE (A Large Ion Collider Experiment) was qualitative features of the QCD phase diagram in the conceived specifically for the exploration of the QGP T, µB plane. Of particular importance is the finding of formed in nucleus-nucleus collisions at the LHC. Within the critical point where at a finite value of µB the smooth the central rapidity domain 0.5 ≤ y ≤ 0.5, ALICE de- transformation between quark-hadron phases turns into tectors are capable of precise tracking and identifying an expected 1st order transition, see Ref.[174]. There are particles over a large range of momentum. This permits other structure features of quark matter that may become the study of the production of strangeness, charm and accessible, for a review see Ref.[37] and comments at the resonances, but also multi-particle correlations, such end of Subsection 2.2. as HBT and (moderate energy) jets. In addition, AL- At CERN the multipurpose NA61 experiment surveys ICE consists of a muon spectrometer allowing us to in its heavy-ion program tasks the domain in energy and study at forward rapidities heavy-flavor and quarko- collision system volume where threshold of deconfinement nium production. The detector system also has the is suspected in consideration of available data. This ex- ability to trigger on different aspects of collisions, to periment responds to the results of a study of head-on select events on-line based on the particle multiplic- Pb–Pb collisions as a function of energy at SPS did pro- ity, or the presence of rare probes such as (di-)muons, duce by 2010 tantalizing hints of an energy threshold to and the electromagnetic energy from high-momentum new phenomena [175,176,177,178]. electrons, photons and jets. There is a significant discontinuities as a function of 2. ATLAS (A Toroidal LHC ApparatuS) has made its collision energy in the K+/π+ particle yield ratio, see name by being first to see jet quenching. It has high Fig. 23 on left. Similarly, the inverse slope parameter of − p⊥ particle ID allowing the measurement of particle the m⊥ spectra of K , see Fig. 23 on right, also displays a spectra in a domain inaccessible to other LHC experi- √local maximum near to 30 A GeV, that is at 3.8+3.8 GeV, ments. sNN = 7.6 GeV collider energy collisions in both quanti- 3. The CMS (Compact Muon Solenoid) offers high rate ties. These behavior ‘thresholds’ are to some degree mir- and high resolution calorimetry, charged particle track- rored in the much smaller pp reaction system also shown ing and muon identification over a wide acceptance, al- in Fig. 23. These remarkable results are interpreted as the lowing detailed measurements of jets as well as heavy- onset of deconfinement as a function of collision energy. quark open and bound states. The large solid angle Turning to comparable effort at RHIC: in 2010 and coverage also provides unique opportunities in the study 2011, RHIC ran the first phase of a beam energy scan of global observables. program (RHIC-BES) to probe the nature of the phase The LHCb experiment has at present no footprint in the boundary between hadrons√ and QGP as a function of µB. study of AA collisions but has taken data in pA trial run. With beam energy settings sNN = 7.7, 11.5, 19.6, 27, 39 GeV, with 14.5 GeV included in year 2014, complementing the full energy of 200 GeV, and the run at 62.4 GeV, a 6.3 Energy and A scan relatively wide domain of µB can be probed, as the matter vs anti-matter excess increases when energy decreases. For The smaller the size of colliding nuclei, the shorter is the a report on these result see Refs. [179,180]. collision time. Thus in collisions of small sized objects such Among the first phase of the beam energy scan discov- as pp or light nuclei, one cannot presume, especially at a eries is the µB dependence of azimuthal asymmetry of flow relatively low collision energy, that primordial and yet not of matter, v2. Particle yield ratio fluctuations show sig- well understood processes (compare Subsection 5.2) will nificant deviation from Poisson expectation within HRG have time to generate the large amount of entropy leading model. This and other results make it plausible that√ QGP to QGP formation that would allow a statistical model to is formed down to the lowest RHIC beam energy of sNN = work well, and in particular would allow QGP formation. 7.7 GeV, corresponding to fixed target collision experi- This than suggests that one should explore dependence on ments at 32 A GeV. This is the collision energy where SPS reaction volume size, both in terms of collision centrality energy scan also found behavior characteristic of QGP, and a scan of projectile ion A. see Fig. 23. These interesting results motivate the second An important additional observation is that particle RHIC-BES phase after detector upgrades are completed production processes are more effective with increasing in 2018/19. Johann Rafelski: Melting Hadrons, Boiling Quarks 33

Fig. 23. The so called horn (left) and step (right) structures in energy dependence of the K+/π+ ratio, and the inverse − slope parameter of K m⊥ spectra, respectively. signal indicating threshold in strangeness to entropy yield in central Pb+Pb (Au+Au) collisions, from [177].

7 What are the Conceptual Challenges the localized, confined, light quarks governs the mass of of the QGP/RHI Collisions Program? matter. The ultimate word is, however, expected from an experiment. Most think that setting quarks free in a large In subsection 1.1 we have briefly addressed the Why? of vacuole created in RHI collision laboratory experiment is the RHI collision research program. Here we return to offering a decisive opportunity to test this understanding deepen some of the points raised, presenting a highly sub- of mass of matter. The same lattice-QCD that provided jective view of foundational opportunities that await us. the numerical evaluation of mass of matter, provides properties of the hot QGP Universe. 7.1 The origin of mass of matter Others go even further to argue nothing needs to be confirmed: given the QCD action, the computer provides Confining quarks to a domain in space means that the hadron spectrum and other static properties of hadron typical energy each of the light quarks will have inside structure. For a recent review of “Lattice results concern- a hadron is Eq ∝ 1/R mq, where R is the size of the ing low-energy particle physics,”see Ref.[183]. That is true: ‘hole’ in the vacuum – a vacuole. Imposing a sharp bound- the relatively good agreement of lattice-QCD theory with ary and forbidding a quark-leak results in a square-well- low-energy particle physics proves that QCD is the the- like relativistic Dirac quantum waves. This model allows ory of strong interactions. In fact, many textbooks argue quantification of Eq. One further argues that the size R that this has already been settled 20 years ago in accel- of the vacuole arises from the internal Fermi and Casimir erator experiments, so a counter question could be, why pressures balancing the outside vacuum which presses to bother to do lattice-QCD to prove QCD? One can present erase any vacuole comprising energy density that is higher. as example of a new insight the argument that the mass In a nutshell this is the math known from within the of matter is not due to the Higgs field [181,182]. context of quark-bag model [92,93,94], rounded off al- However, the mass argument is not entirely complete. lowing color-magnetic hyperfine structure splitting. This The vacuole size R directly relates to QCD vacuum prop- model explains how baryons and mesons have a mass much erties – in bag models we relate it to the bag constant B greater than the sum of quark masses. It is also easy describing the vacuum pressure acting on the vacuole. But to see that a larger vacuole with hot quarks and gluons is this hadron energy scale B1/4 ' 170 MeV fundamental? would provide a good starting point to develop a dynam- The understanding of the scale of the QCD vacuum struc- ical model of expanding QGP fireball formed in RHI col- ture has not been part of the present-day lattice-QCD. lisions. In lattice-QCD work one borrows the energy scale from The advent of lattice-QCD means we can address static an observable. In my opinion hadron vacuum scale is due time independent properties of strongly interacting parti- to the vacuum Higgs field, and thus the scale of hadron cles. A test of bag models ideas is the computation of the masses is after all due to Higgs field; it is just that the hadron mass spectrum and demonstration that the mass mechanism is not acting directly. of hadrons is not determined by the mass of quarks bound inside. Indeed, this has been shown [181,182]; the confin- Let me explain this point of view: By the way of top ing vacuum structure contributes as much as 96% of the interaction with Higgs there is a relation of the Higgs with mass of the matter, the Higgs field the remaining few-%. the QCD vacuum scale; Based on both bag model consideration and lattice- a) The intersection between QCD and the Higgs field is QCD we conclude that the quantum zero-point energy of provided by the top quark, given the remarkable value of 34 Johann Rafelski: Melting Hadrons, Boiling Quarks

the minimal coupling gt A few months earlier, in November, 1919 Einstein an- 2 nounced the contents of this address in a letter to Lorentz: mt t gt gt ≡ ' 1 , αh ≡ = 0.08 ' αs(mt) = 0.1 . (20) It would have been more correct if I had limited myself, hhi 4π in my earlier publications, to emphasizing only the non- Note that the same strength of interaction: top with glu- existence of an æther velocity, instead of arguing the total t non-existence of the æther . . . ons αs(mt), and with Higgs field fluctuations αh. b) The size of QCD vacuum fluctuations [184], has been estimated at 0.3 fm. This is large compared to the top −3 7.3 The quantum vacuum: Natural constants quark Compton wavelength λt = ~c/mt = 1.13 ×10 fm. This means that for the top-field the QCD vacuum looks In the quark–gluon plasma state of matter, we fuse and like a quasi-static mountainous random field driving large dissolve nucleons in the primordial æther state, different top-field fluctuations in the QCD vacuum. in its structure and properties from the æther of our ex- The possible relation of the QCD vacuum structure via perience. In Einstein’s writings quoted above the case of top quark with Higgs requires much more study, I hope transition between two coexistent æther states was not that this will keep some of us busy in coming years. foreseen, but properties such as the velocity of light were That something still needs improvement in our under- seen as being defined by the æther. One should thus ask: standing of strong interactions is in fact clear: Why i) all Is velocity of light the same out there (vacuole) as it is hadrons we know have qqq and qq¯ structure states, and around here? Such a question seems on first sight empty why ii) we do not observe internal excitations of quarks in as the velocity of light connects the definition of a unit of bags appearing as hadron resonances. These two questions length with the definition of a time increment. However, show that how we interpret QCD within the bag model is ifc ¯ in the vacuole is the same as c, it means that time incomplete. ‘advances’ at the same rate there as it does here. This I hope to have dented somewhat the belief that lattice- assumption is not necessary. QCD is capable of replacing the experimental study of Is it possible, both in practical and in principle terms, vacuum structure. In a nutshell, lattice does neither ex- using RHI collisions to answer if c¯ = c, where the bar plain scales of vacuum structure, nor it can address any indicates the property in the vacuole? dynamical phenomena, by necessity present in any labo- We can for example study the relation between energy ratory recreation of the early Universe QGP conditions. and momentum of photons produced in QGP, and the rate And, the QCD vacuum structure paradigm needs an ex- at which these processes occur. The photon emitted is de- perimental confirmation. fined by its wavelength k = 1/λ, the energy of the photon is ~ck. This energy is different in the vacuole from what 7.2 The quantum vacuum: Einstein’s æther we observe in the laboratory – energy conservation for the photon is not maintained since the translation symmetry The quantum vacuum state determines the prevailing form in time would need to be violated to make time tick differ- of the ‘fundamental’ physics laws. Within the standard ently in different vacuum states. However, global energy model, the nature of particles and their interactions is de- conservation is assured. Transition radiation, Cherenkov termined by the transport properties of the vacuum state. radiation are more mundane examples of what happens As just discussed above, the mass of matter is inherent when a superluminal photon enters a dielectric medium. in the scale of QCD, which itself relates in a way to be Thus we will need to differentiate with what would be studied in the future with the Higgs vacuum structure. called medium effect when considering photon propaga- The existence of a structured quantum vacuum as the tion across thec ¯ 6= c. boundary. That maybe difficult. carrier of the laws of physics was anticipated by Lorentz, Turning now to the rate of photon production in the and Einstein went further seeking to reconcile this with vacuole: we keep to gauge invariance, thus charge cannot the principles of relativity. What we call quantum vac- change between two quantum vacuum states. The way the uum, they called æther. The concluding paragraph from change from the vacuole to the normal vacuum rate will be a lecture by Albert Einstein is creating the philosophical looked at is that we assume the space size of the vacuole foundation of the quantum vacuum as carrier of laws of to be measured in units of length evaluated in the normal physics (translation by author) [185] vacuum. The rate of an electromagnetic process in mod- . . . space is endowed with physical qualities; in this ified vacuum should be, according to the Fermi golden rule proportional to ~¯α¯2 = e4~¯/(~¯2c¯2). This expression sense the æther exists. According to the general ¯ theory of relativity, space without æther is unthink- reminds that we also can have ~ 6= ~, but the result will able: without æther light could not only not prop- involve the product c~ only. The rate per unit volume and agate, but also there could be no measuring rods time of an electromagnetic process is in the vacuole with ¯ and clocks, resulting in nonexistence of space-time ∆t = ∆L/c¯ is distance as a physical concept. On the other hand, α¯2 1 1 this æther cannot be thought to possess properties W ∝ ¯ ∝ . (21) ~∆3L ∆t¯ ¯c¯ ∆4L characteristic of ponderable matter, such as having ~ parts trackable in time. Motion cannot be inherent The number of events we observe is ∆L4W. The produc- to the æther. tion of direct dileptons and direct photons is thus pre- Johann Rafelski: Melting Hadrons, Boiling Quarks 35 dicted to scale with (~¯c¯)−1 in a space-time volume deter- arise there must be a baryon number conservation violat- mined in our vacuum by for example the HBT method. ing process. This seems to be a requirement on fundamen- The above consideration cannot be applied to strong tal interactions which constrains most when and how one interactions since there is no meaning to αs in the nor- must look for the asymmetry formation. It would be hard mal vacuum, we always measureα ¯s. Similarly, the ther- to place this in the domain of physics today accessible to mal properties of the vacuole, in particular addressing the experiments as no such effect has come on the horizon. quark energy, are intrinsic properties. The direct connec- A variant model of asymmetry could be a primordial tion of intrinsic to external properties occurs by electro- acoustical chemical potential wave inducing an asymme- magnetic phenomena. The practical problem in using the try in the local distribution of quantum numbers. It has −1 rate of electromagnetic processes to compare in-out (~c) been established that at the QGP hadronization T = TH is that all production processes depend on scattering of temperature a chemical potential amplitude at the level of electrically charged quanta (quarks) in QGP, and that in 0.3 eV achieves the present day baryon to photon number turn depends on a high power of T . This means that small in our domain of the Universe [186]. Constrained by local, changes in ~c could be undetectable. However, it will be electrical charge neutrality, and B = L (local net baryon quite difficult to reconcile an order of magnitude ~c modi- density equal to local net lepton density), this chemical −9 fication by pushing T and HBT sizes. We hope to see such potential amplitude is about 10 fraction of TH . studies in the near future, where one triees to determine This insight sets the scale of energy we are looking for: for electromagnetic processes an in-medium strength of the absence in the SM of any force related to baryon num- α as this is how one would reinterpret vacuole modified ber and operating at the scale of eV is what allows us to physical natural constants. imagine local baryon number chemical fluctuation. This ‘random fluctuation’ resolution of the baryon asymmetry riddle implies that our matter domain in the Universe bor- 7.4 The primordial Quark Universe in Laboratory ders on an antimatter domain – however a chemical potential wave means that this boundary is where µ = 0 and Relativistic heavy ion (RHI) collisions recreate the ex- thus where no asymmetry is present; today presumably a treme temperature conditions prevailing in the early Uni- space domain void of any matter or antimatter. Therefore, verse: a) dominated by QGP; b) in the era of evolution be- a change from matter to antimatter across the boundary ginning at a few µs after the big-bang; c) lasting through is impossible to detect by astronomical observations – we the time when QGP froze into individual hadrons at about have to look for antimatter dust straying into local parti- 20-30 µs. We record especially at the LHC experiments the cle detectors. One of the declared objectives of the Alpha initial matter-antimatter symmetry a nearly net-baryon- ¯ Magnetic Spectrometer (AMS) experiment mounted on free (B = b − b → 0) primordial QGP. The early Universe the International Space Station (ISS) is the search for an- (but not the lab experiment) evolved through the7 matter- −9 tihellium, which is considered a characteristic signature of antimatter annihilation leaving behind the tiny 10 resid- antimatter lurking in space [187]. ual matter asymmetry fraction. We recall that the acoustical density oscillation of mat- The question in which era the present day net baryon ter is one of the results of the precision microwave back- number of the Universe originates remains unresolved. ground studies which explore the conditions in the Uni- Most believe that the net baryon asymmetry is not due to verse at temperatures near the scale of T = 0.25 eV where an initial condition. For baryon number to appear in the hydrogen recombines and photons begin to free-steam. Universe the three Sakharov condition have to be fulfilled: This is the begin of observational cosmology era. Another 1) In terms of its evolution, the Universe cannot be in the factor 30,000 into the primordial depth of the Universe full equilibrium stage; or else whatever created the asym- expansion, we reach the big-bang nuclear synthesis stage metry will also undo it. This requirement is generally un- occurring at the scale of T ' 10 keV. Abundance of he- derstood to mean that the asymmetry has to originate lium compared to hydrogen constrains significantly the in the period of a phase transformation, and the focus timescale of the Universe expansion and hence the present of attention has been on electro-weak symmetry restoring day photon to baryon ratio. A further factor 30,000 in- condition at a temperature scale 1000 × T . However the H crease of temperature is needed to reach the stage at which time available for the asymmetry to arise is in this condi- the hadronization of quark Universe occurs at Hagedorn tion on the scale of 10−8 s and not 10−5 s or longer if the temperature T . asymmetry is related to QGP evolution, hadronization, H and/or matter-antimatter annihilation period. We have focused here on conservation, or not, of baryon 2) During this period interactions must be able to differ- number in the Universe. But another topic of current in- entiate between matter and antimatter, or else how could terest is if the hot QGP fireball in its visible energy compo- the residual asymmetry be matter dominated? This asym- nent conserves energy; the blunt question to ask is: What metry requires CP-nonconservation, well known to be in- if the QGP radiates darkness, that is something we can- herent in the SM as a complex phase of the Kobayashi- not see?[188]. I will return under seperate cover to discuss Maskawa flavor mixing, the QGP in the early Universe, connecting these different 3) If true global excess of baryons over antibaryons is to stages. For a preliminary report see Ref.[189]. The understanding of the quark Universe deepens profoundly the 7 Here b, ¯b is not related to bottom quarks. reach of our understanding of our place in this world. 36 Johann Rafelski: Melting Hadrons, Boiling Quarks

8 Melting hadrons to properly represent the evolving scientific understanding, today’s perspective is different. In particular the in- Two paths towards the quark phase of matter started in sight that the appearance of a large number of different parallel in 1964-65, when on one hand quarks were intro- hadronic states allows to effectively side-step the quan- duced triggering the first quark matter paper [113], and tum physics nature of particles within statistical physics on another, Hagedorn recognized that the yields and spec- became essentially invisible in the ensuing work. Few sci- tra of hadrons were governed by new physics involving entists realize that this is a key property in the SBM, and TH and he proposed the SBM [11]. This briefly addresses the fundamental cause allowing the energy content to in- the events surrounding Hagedorn discovery and the result- crease without an increase in temperature. ing modern theory of hot hadronic matter. In the SBM model, a hadron exponential mass spectrum with the required ‘fine-tuned’ properties is a natural outcome. The absence of Hagedorn’s ‘Distinguishable Par- 8.1 The tale of distinguishable particles ticles’ preprint delayed the recognition of the importance of the invention of the SBM model. The SBM paper with- In early 1978 Rolf Hagedorn shared with me a copy of out its prequel looked like a mathematically esoteric work; his unpublished manuscript ‘Thermodynamics of Distin- the need for exponential mass spectrum was not immedi- guishable Particles: A Key to High-Energy Strong Interac- ately evident. tions?’, a preprint CERN-TH-483 dated 12 October 1964. Withdrawal of ‘Distinguishable Particles’ also removed He said there were two copies; I was looking at one; an- from view the fact that quantum physics in hot hadronic other was in the CERN archives. A quick glance sufficed matter loses its relevance, as not even Boltzmann’s 1/n! to reveal that this was, actually, the work proposing a factor was needed, the exponential mass spectrum effec- limiting temperature and the exponential mass spectrum. tively removes it. Normally, the greater the density of par- Hagedorn explained that upon discussions of the contents ticles, the greater the role of quantum physics. To the best of his paper with LéonVan Hove, he evaluated in greater of my knowledge the dense, strongly interacting hadronic detail the requirements for the hadron mass spectrum and gas is the only physical system where opposite happens. recognized a needed fine-tuning. Hagedorn concluded that Thus surfacing briefly in Hagedorn’s withdrawn ‘Thermo- his result was therefore too arbitrary to publish, and in dynamics of Distinguishable Particles’ paper, this original the CERN archives one finds Hagedorn commenting on finding faded from view. Hagedorn presented a new idea this shortcoming of the paper, see Chapter 18 in Ref.[1]. that has set up his SBM model, and for decades this new However, Hagedorn’s ‘Distinguishable Particles’ is a idea was to be hidden in archives. clear stepping stone on the road to modern understand- On the other hand, the Hagedorn limiting tempera- ing of strong interactions and particle production. The ture TH got off the ground. Within a span of only 90 days insights gained in this work allowed Hagedorn to rapidly between the withdrawal of his manuscript, and the date invent the Statistical Bootstrap Model (SBM). The SBM of his new CERN-TH preprint, Hagedorn formulated the paper ‘Statistical Thermodynamics of Strong Interactions SBM. Its salient feature is that the exponential mass spec- at High Energies’, preprint CERN-TH-520 dated 25 Jan- trum arises from the principle that hadrons are clusters 9 uary 1965, took more than a year to appear in press8 [11]. comprising lighter (already clustered) hadrons . The key The beginning of a new idea in physics often seems point of this second paper is a theoretical model based on to hang on a very fine thread: was anything lost when the very novel idea of hadrons made of other hadrons. Such ‘Thermodynamics of Distinguishable Particles’ remained a model bypasses the need to identify constituent content unpublished? And what would Hagedorn do after with- of all these particles. And, Hagedorn does not need to drawing his first limiting-temperature paper? My discus- make explicit the phenomenon of Hadron distinguishabil- sion of the matter with Hagedorn suggests that his vision ity that clearly was not easy to swallow just 30 years after at the time of how limiting temperature could be justi- quantum statistical distributions saw the light of day. fied evolved very rapidly. Presenting his more complete Clustering pions into new hadrons and then combin- insight was what interested Hagedorn and motivated his ing these new hadrons with pions, and with already pre- work. Therefore, he opted to work on the more complete formed clusters, and so on, turned out to be a challeng- theoretical model, and publish it, rather than to deal with ing but soluble mathematical exercise. The outcome was complications that pressing ‘Thermodynamics of Distin- that the number of states of a given mass was growing guishable Particles’ would generate. exponentially. Thus, in SBM, the exponential mass spec- While the withdrawal of the old, and the preparation trum required for the limiting temperature arose naturally of an entirely new paper seemed to be the right path ab-initio. Furthermore the model established a relation between the limiting temperature, the exponential mass 8 Publication was in Nuovo Cim. Suppl. 3, pp. 147–186 spectrum slope, and the pion mass, which provides the (1965) actually printed in April 1966. This is also one of the scale of energy in the model. last Journals one does not find on-line. Inspection of the journal reveals: a few ‘extreme’papers and reviews with thousands 9 In this paper as is common today we refer to all discovered of citations such as the Hagedorn work; and another extreme, hadron resonance states – Hagedorn’s clusters – as resonances, thousands of pages of Italian Physical Society bulletin news of and the undiscovered ‘heavy’ resonances are called Hagedorn long gone period. states Johann Rafelski: Melting Hadrons, Boiling Quarks 37

Models of clustering type are employed in other areas strong interactions; if introduced as new, independent par- of physics. An example is the use of the α-substructure ticles in a statistical model, they express the strong inter- in the description of nuclei structure: atomic nuclei are actions to which they owe their existence. To account in made of individual nucleons, yet improvement of the un- full for strong interactions effects we need all resonances; derstanding is achieved if we cluster 4 nucleons (two pro- that is, we need the complete mass spectrum ρ(m). tons and two neutrons) into an α-particle substructure. In order to obtain the mass spectrum ρ(m), we will The difference between the SBM and the nuclear α- implement in mathematical terms the self-consistent re- model is that the number of input building blocks in SBM quirement that a cluster is composed of clusters. This i.e. pions and more generally of all strongly interacting leads to the ‘bootstrap condition and/or bootstrap equa- clusters is not conserved but is the result of constraints tion’ for the mass spectrum ρ(m). The integral bootstrap such as available energy. As result one finds rapidly grow- equation (BE) can be solved analytically with the result ing with energy size of phase space with undetermined that the mass spectrum ρ(m) has to grow exponentially. number of particles. This in turn provides justification for Consequently, any thermodynamics employing this mass the use of the grand canonical statistical methods in the spectrum has a singular temperature TH generated by the description of particle physics phenomena at a time when asymptotic mass spectrum ρ(m) ∼ exp(m/T0). Today this only a few particles were observed. singular temperature is interpreted as the temperature where (for baryon chemical potential µB = 0) the phase conversion of hadron gas ←→ quark–gluon plasma occurs. 8.2 Roots and contents of the SBM 8.3 Implementation of the model The development of SBM in 1964/65 had a few preceding pivotal milestones, see Chapter 17 in Ref.[1] for a fully Let us look at a simple toy model proposed by Hagedorn referenced list. One should know seeing point 1. below to illustrate the Frautschi-Yellin reformulation [12,16] of that it was Heisenberg who hired Hagedorn as a postdoc in the original model which we find in comparable detail in June 1952 to work on cosmic emulsion ‘evaporation stars’, Appendix A, also shown in Subsection 2.1, Eq. (4). Like and soon after sent him on to join in 1954 the process of in SBM, in the toy model particle clusters are composed building CERN: of clusters; however we ignore kinetic energy. Thus

1. The realization and a first interpretation of many sec- ρ(m) =δ(m − m0)+ (22) ondaries produced in a single hadron–hadron collision ∞ n ! n X 1 Z X Y (Heisenberg 1936 [190]), δ m − m ρ(m )dm . n! i i i 2. The concept of the compound nucleus and its thermal n=2 i=1 i=1 behavior (1936–1937). 3. The construction of simple statistical/thermodynamical In words, the cluster with mass m is either the ‘input par- models for particle production in analogy to compound ticle’ with mass m0 or else it is composed of any number nuclei (1948–1950) (Koppe 1949 [53], Fermi 1950 [54]). of clusters of any masses mi such that Σmi = m. We Enter Hagedorn: Laplace-transform Eq. (22): 4. The inclusion of resonances to represent interactions Z recognized via phase shifts (Belenky 1956 [45]). ρ(m)e−βmdm =e−βm0 + (23) 5. The discovery of limited hp⊥i (1956). ∞ n 6. The discovery that fireballs exist and that a typical X 1 Y Z e−βmi ρ(m )dm . pp collision seems to produce just two of them, projec- n! i i tile and target fireball (1954–1958). n=2 i=1 7. The discovery that large-angle elastic cross-sections Define decrease exponentially with CM energy (1963). Z 8. The discovery of the parameter-free and numerically z(β) ≡ e−βm0 ,G(z) ≡ e−βmρ(m)dm . (24) correct description of this exponential decrease buried in Hagedorn’s archived Monte Carlo phase-space re- Thus Eq. (23) becomes G(z) = z + exp[G(z)] − G(z) − 1 sults obtained earlier at CERN (1963). or G(z) Hagedorn introduced a model based on an unlimited z = 2G(z) − e + 1 , (25) sequence of heavier and heavier bound and resonance states which provides implicitly the function G(z), the Laplace he called clusters10, each being a possible constituent of transform of the mass spectrum. a still heavier resonance, while at the same time being A graphic solution is obtained drawing z(G) in Fig. 24 itself composed of lighter ones. The pion is the lightest top frame a) and transiting in Fig. 24 from top a) to bot- ‘one-particle-cluster’. Hadron resonance states are due to tom frame b) by exchanging the axis. The parabola-like maximum of z(G) implies a square root singularity of G(z) 10 In the older literature Hagedorn and others initially called at z0, first remarked by Nahm [24] decaying clusters fireballs, this is another example of how a physics term is recycled in a new setting. zmax(G) ≡ z0 = ln 4−1 = 0.3863 ...,G0 = G(z0) = ln 2 . 38 Johann Rafelski: Melting Hadrons, Boiling Quarks

Upon inverse Laplace asymptotic transformation of the Bootstrap function G(z) one obtains

ρ(m) ∼ m−3em/TH , (26)

where in the present case (not universally):

m0 m0 TH = − = . (27) ln z0 0.95

Using the natural choice m0 = mπ we obtain:

TH (toy model) = 145 MeV . (28) The simple toy model already yields all essential features of SBM: the exponential mass spectrum with a = 3 and the right magnitude of TH .

8.4 Constituents of ﬁnite size

The original point-volume bootstrap model was adapted to be applicable to collisions of heavy ions where the reaction volume was relevant. This work began in 1977 and was in essence complete by 1980 [133,25,193], see the details presented Appendix A. The new physics is that cluster volumes are introduced. For overview of work that followed see for example Ref.[194]. However, ‘in principle’ we are today where the subject was when the initial model was completed in 1979. While many refinements were proposed, these were in physical terms of marginal impact. A new well posed question is how the van der Waals excluded volume extension of Hagedorn SBM connects to present day lattice-QCD [195], and we address this in the next Subsection. Before we discuss that, here follows one point of principle. Current work takes for granted the ability to work in a context similar to non-relativistic gas including relativistic phase space. This is not at all self-evident. To get there, Fig. 24. (a) z(G) according to Eq. (25). (b) Bootstrap func- see also Appendix A, we argued that particle rest-frame tion G(z), the graphical solution of Eq. (25). The dashed line volumes had to be proportional to particle masses. Follow- ing Touschek [196], we defined a ‘four-volume’; arbitrary represents the unphysical branch. The root singularity is at µ observer would attribute to each particle the 4-volume Vi ϕ0 = ln(4/e) = 0.3863 µ moving with particle four-velocity ui µ µ µ µ pi Vi = Viui , ui ≡ . (29) as also shown in Fig. 24. mi It is remarkable that the ‘Laplace-transformed BE’ The entire volume of all particles is comoving with the Eq. (25) is ‘universal’ in the sense that it is not restricted four-velocity of the entire particle assembly of mass m to the above toy model, but turns out to be the same in pµ all (non-cutoff) realistic SBM cases [12,16]. Moreover, it V µ = V uµ , uµ = ; (30) is independent of: m n µ X µ p µ p = pi , m = pµp – the number of space-time dimensions [191]; the ‘toy i=1 model’ extends this to the cae of ‘zero’-space dimen- We explored a simple additive model applicable when all sions, hadrons have the same energy density, see Appendix A – the number of ‘input particles’ (z becomes a sum over modified Hankel functions of input masses), V Vi – Abelian or non-Abelian symmetry constraints [192]. = = Const = 4B , (31) m mi Johann Rafelski: Melting Hadrons, Boiling Quarks 39 where the proportionality constant is written 4B in order to emphasize the similarity to MIT bags [92,93,94], which have the same mass–volume relation in absence of any other energy scale. However, in QCD two relevant scales enter in higher order: that of strange quark mass, and parameters characterizing the running of QCD parameters; the coupling constant αs and mass of strange quark ms are here relevant. Given that the assembly of particles of mass m oc- cupies a comoving volume V and the same applies to the constituent particles and their volumes one can henceforth ignore the Lorentz covariance challenges associated with introduction of particle proper volume. However, this has been shown only if all hadrons have the same energy density, nobody extended this argument to a more general case. In an independent consideration the energy spectrum of such SBM clusters we obtained and that of MIT bags was found to be the same [197,198,199]. This suggests these two models are two different views of the same object, a snapshot taken once from the hadron side, and another time from thequark side. MIT bags ‘consist of’ quarks and gluons, SBM clusters of hadrons. This leads on to a phase transition to connect these two aspects of the same, as is further developed in Appendix A.

8.5 Connection with lattice-QCD Fig. 25. Lattice-QCD [68] energy density ε/T 4 and Pressure 4 P/T for T < 155, µB = 0 MeV, in RHG in a model with ex- Today the transformation between hadrons and QGP is cluded volume parameter r = 0, 0.2,.0.3, 0.4 (dashed lines) and characterized within the lattice-QCD evaluation of the allowing for extension of the HRG with exponential mass spec- thermal properties of the quark-hadron Universe. In the trum (solid lines) for assumed TH =160 MeV. See text. After Ref.[195] context of our introductory remarks we have addressed the close relation of the HRG with the lattice results, see Subsection 2.4 and in particular Fig. 6. But what does change a = 5/2 → a = 3 with appropriate other changes this agreement between lattice-QCD and HRG have to but this does not resolve the above sharp cutoff matter. say about SBM of hadrons of finite size? That is an im- Note that the normalization parameter C in Eq. (32) is portant question, it decides also the fate of the 1979 effort the only free parameter and for C = 0 the complement described in Appendix A. states are excluded (dashed lines in Fig. 25), the model Shevchenko, Anchishkin, and Gorenstein [195] analyzed reverts to be HRG with finite size particle volume, but the lattice-QCD for the pressure and energy density at only for r 6= 0, for r = 0 we have point HRG. How this T < 155, µ = 0 MeV within the hadron resonance gas B model modifies energy density ε/T 4 and pressure P/T 4 is model allowing for effects of both the excluded volume and seen in Fig. 25. the undiscovered part of Hagedorn mass spectrum. That work is within a specific model of finite sized hadron gas: The authors conclude that lattice data exclude taking particles occupy a volume defined by v = 16πr3/3 where the two effects apart, i.e. consideration of each of these r is a parameter in range 0 < r < 0.4 fm, and it is the individually. This is so since for C = 0 the fit of pressure Fig. 6 favors finite hadron volume parameter r . 0.4 fm; density of particles that characterizes the size of excluded ∼ volume. however the best fit of energy density shows r = 0. When both: excluded volume r 6= 0, and heavy resonances C 6= 0 The shape of their exponentially extended mass spec- are considered simultaneously the model works better: the trum effect of finite volume and the possibly yet undiscovered θ(m − M0) ρ(m) = C em/TH , (32) high mass Hagedorn mass spectrum thus complement each (m2 + m2)a/2 0 other when considered simultaneously: the suppression ef- 4 4 where the authors assumed with Hagedorn TH = 160 MeV, fects for pressure P/T and energy density ε/T due to m0=0.5 GeV, and place the cutoff at M0 = 2 GeV. Es- the excluded volume effects and the enhancement due to pecially the assumed a = 5/2 is in conflict with prior art, the Hagedorn mass spectrum make the data fit marginally see Subsection 2.2, and the sharp cutoff leaves an unfilled better as we can see in Fig. 25. ‘hole’ in the intermediate mass domain. The authors re- In effect Ref.[195] tests in quantitative fashion the sen- port in a side remark that their results are insensitive to a sitivity of the lattice-QCD results to physics interpreta- 40 Johann Rafelski: Melting Hadrons, Boiling Quarks tion: it seems that even if and when the lattice results Table 3. Thermal parameters and their SHARE name. The should be a factor 5 more precise, the correlation between values are to be presented in units GeV and fm3, where appli- the contribution of undiscovered states and the van der cable. Waals effect will compensate within error margin. Symbol Parameter Parameter description As disappointing as these results may seem to some, it V norm absolute normalization in fm3 is a triumph for the physics developed in 1979. Namely, re- T temp chemical freeze-out temperature T sults of Ref.[195] also mean that upon a reasonable choice λq lamq light quark fugacity factor of the energy density in Hadrons 4B, the model presented λs lams strangeness fugacity factor in Appendix A will fit well the present day lattice-QCD γq gamq light quark phase space occupancy data T < 155, µB = 0 MeV, since it has both the correct γs gams strangeness phase space occupancy mass spectrum, and the correct van der Waals repulsion λ3 lmi3 I3 fugacity factor (Eq. (54)) effects due to finite hadron size. Therefore, this model is γ3 gam3 I3 phase space occupancy (Eq. (52)) bound to be accurate as a function of µB as well. λc lamc charm fugacity factor e.g. λc = 1 Reading in Appendix A it seems that the perturbative Nc+¯c Ncbc number of c +c ¯ quarks QCD phase has properties that do not match well to the Tc/T tc2t ratio of charm to the light quark ha- SBM, requiring a strong 1st order phase transition match- dronization temperature ing to SBM. This was a result obtained with a fixed value of αs in thermal-QCD. The results of lattice-QCD teach us that a more refined model with either running αs and/or thermal quarks and gluon masses [6,106] is needed. Con- sign of ‘a charge’. Thus it flips sign between particles and temporary investigation of the latest lattice-QCD results antiparticles. in such terms is promising [107] as we already mentioned The resultant shape of the Fermi-Dirac distribution in Subsection 3.4. is seen in Eq. (17). The occupancy γ is in Boltzmann approximation the ratio of produced particles to the number of particles expected in chemical equilibrium. Since there is one quark and one antiquark in each meson, yield is 9 Hadronization of QGP fireball 2 proportional to γq and accordingly the baryon yield to 3 In this section the method and implementation of the fire- γq . When necessary we will distinguish the flavor of the ball hadronization model will be presented allowing us to valance quark content q = u, d, s, . . . address the task defined in Subsection 5.3. This model The occupancy parameters describing the abundance was already described in Subsection 2.7 and thus we can of valance quarks counted in hadrons emerge in a complex proceed rapidly to develop the technical details. evolution process described in Subsection 5.3. In general, we expect a nonequilibrium value γi 6= 1. A much simplified argument to that used in Subsection 5.3 is to to 9.1 A large parameter set assume that we have a completely equilibrated QGP with all quantum charges zero (baryon number, etc) and thus, Our task is to describe precisely a multitude of hadrons in QGP all λi = 1, γi = 1. Just two parameters describe by a relatively small set of parameters. This then allows the QGP under these conditions: temperature T and vol- us to characterize the drop of QGP at the time of ha- ume V . dronization. In our view, the key objective is to charac- This state hadronizes preserving energy, and increasing terize the source of hadrons rather than to argue about or preserving entropy and essentially the number of pairs the meaning of parameter values in a religious fashion. of strange quarks. On the hadron side temperature T and For this procedure to succeed, it is necessary to allow for volume V would not suffice to satisfy these constraints, the greatest possible flexibility in the characterization of and thus we must at least introduce γs > 1. The value is in the particle phase space, consistent with conservation laws general above unity because near to chemical equilibrium and related physical constraints at the time of QGP ha- the QGP state contains a greater number of strange quark dronization. For example, the number yield of strange and pairs compared to the hadron phase space. light quark pairs has to be nearly preserved during QGP Table 3 presents the here relevant parameters which hadronization. Such an analysis of experimental hadron must be input with their guessed values or assumed con- yield results requires a significant bookkeeping and fitting ditions, in order to run the SHARE with CHARM pro- effort, in order to allow for resonances, particle widths, gram as input in file thermo.data. When and if we allow full decay trees and isospin multiplet sub-states. We use γs to account for excess of strangeness content, we must SHARE (Statistical HAdronization with REsonances), a also introduce γq to account for a similar excess of QGP data analysis program available in three evolution stages light quark content as already discussed in depth in Sub- for public use [200,201,202]. section 5.3. The important parameters of the SHM, which control In regard to the parameters γq, γs 6= 1 we note: the relative yields of particles, are the particle specific fu- (a) We do not know all hadronic particles, and the in- gacity factors λ ≡ eµ/T and the space occupancy factors complete hadron spectrum used in SHM can be to some γ. The fugacity is related to particle chemical potential degree absorbed into values of γs, γ˜q; µ = T ln λ. µ follows a conserved quantity and senses the (b) In our analysis of hadron production results we do not Johann Rafelski: Melting Hadrons, Boiling Quarks 41

fit spectra but yields of particles. This is so since the dy- y is recognized realizing that a fireball emitting particles namics of outflow of matter in an exploding fireball is hard will have some specific value of yf which we recognize dis- to control; integrated spectra (i.e., yields) are not affected playing particle yields as function of rapidity, integrated by collective flow of hot matter. with respect to p⊥. (c) γq, γs 6= 1 complement γc, γb to form a set of nonequi- The meaning of an analysis of particle data multiplic- librium parameters. ities dN/dyp is that we look at the particles that emerged Among the arguments advanced against use of chemi- from dV/dyp: the fireball incremental volume dV per unit cal nonequilibrium parameters is the urban legend that it of rapidity of emitted particles dyp, is hard, indeed impossible, to find in the enlarged parameter space a stable fit to the hadron yield data. A large set dN dV dL ∝ ≡ D (L) , (34) of parameters often allows spurious local minima which dy dy ⊥ dy cloud the physical minimum – when there are several fit minima, a random search can oscillate between such non- where D⊥ is the transverse surface at hadronization of the physics minima rendering the fit neither reproducible, nor fireball. Considering the case of sufficiently high energy physically relevant. where one expects that particle yields dN/dyp are flat as This problem is solved as follows using the SHARE function of rapidity, we can expect that D⊥(L) ' D⊥(L = suite of programs: we recall that SHARE allows us to use 0) and thus dL/dyp = Const, where L = 0 corresponds to any of the QGP bulk properties to constrain fits to par- the CM-location of the hadron-hadron collision. ticle yield. In extreme, one can reverse the process: given The quantity dL/dyp relates to the dynamics of each a prescribed fireball bulk properties one can fit a statisti- of the positions L from which measured particles emerge cal parameter set –provided that the information that is with a measured rapidity yp. Each such location has its introduced is sufficient. proper time τ which applies to both the dynamics of the To find a physics best fit, what a practitioner of SHARE longitudinal volume element dL and the dynamics of par- will do is to loosely constrain the physical bulk properties ticle production from this volume element. We thus can at hadronization. One speeds up considerably the conver- write gence by requiring that fits satisfy some ballpark value 3 dL dL/dτ f(y ) such as = 0.45 ± 0.15 GeV/fm . Once a good physics ≡ = L = Const. (35) minimum is obtained, a constraint can be removed. If the dyp dyp/dτ d(yth + yL)/dτ minimum is very sharp, one must repeat this process recursively; when imposing a value such as a favorite value In the last step we recognize the longitudinal dynamics of freeze-out T , the convergence improvement constraint introducing the local flow rapidity yL in the nominator has to be adjusted. where dL/dτ = f(yL), and in the denominator given the additivity of rapidity we can break up the particle rapidity into the longitudinal dynamics yL and the thermal com- 9.2 Rapidity density yields dN/dy ponent yth, describing the statistical thermal production of particles. We have so obtained In fitting the particle produced at RHIC and LHC energies we rarely have full information available about the dy dy f(y ) = R th + L . (36) yields. The detectors are typically designed to either cover L dτ dτ the center of momentum domain (central rapidity) or the forward/backward ‘projectile/target’ domains. Thus prac- Since we form dN/dy observing many particles emitted tically always – with the exception of results in SPS range forward (yth > 0) or backward (yth < 0) in rapidity with of energies – we do need to focus our analysis on particle respect to local rest frame, the statistical term averages yields emerging from a domain, typically characterized by out and thus we obtain as the requirement for a flat dN/dy rapidity y of a particle. that the local longitudinal flow satisfies As a reminder, the rapidity of a particle y replaces in set of kinematic variables the momentum component dL dy = f(y ) = R L , (37) parallel to the axis RHI motion. For a particle of mass m dτ L dτ with momentum vector p = pk +p⊥ split into components parallel and perpendicular to the axis RHI motion, the that is a linear relation relation is q L − L0 = R(yL − y0) . (38) 2 2 pk = E⊥ sinh y , E⊥ = m + p⊥ (33) It is tempting to view f(yL) ≡ dL/dτ = sinh yL as we which implies the useful relation E = E⊥ cosh y. would expect if L were a coordinate of a material particle. Rapidity is popular due to the additivity of the value The implicit system of equations allows usthen to deter- of y under a change of reference frame in k-direction char- mine the dependence of yL and thus L on τ and thus of acterized by the Lorentz transformation where cosh yL = time evolution in Eq. (34) and the relation of dV/dy with γL, sinhL = βLγL. In this restricted sense rapidity replaces geometric (HBT) volume, a connection that is at present velocity in the context of relativistic motion. The value of not understood. This will be a topic for further study. 42 Johann Rafelski: Melting Hadrons, Boiling Quarks

9.4 Particle yields and fluctuations For full and correct evaluation of the final hadron state in the LHC era, one has to calculate 1. Primary particle yields at chemical freeze-out, 2. Charm hadron decays for a given charm quark abundance, followed by 3. Decays of all hadron resonances. The point 2. is the new module that rounds of SHARE for LHC energies [202] p 2 2 Every hadron of species i with energy Ei = mi + pi Fig. 26. Illustration of relativistic heavy ion collision: two populates the energy states according to Fermi-Dirac or Lorentz-contracted nuclei impact with offset, with some of the Bose-Einstein distribution function: nucleons participating, and some remaining spectators, i.e. nu- 1 cleons that miss the other nucleus, based on Ref.[203] ni ≡ ni (Ei) = −1 , (39) Υi exp (Ei/T ) ± 1 where the upper sign corresponds to Fermions and the 9.3 Centrality classes lower one to Bosons. The fugacity Υi of the i-th hadron species is described and reduced to thee valence quark properties in Subsection 9.5 below. Then the hadron species When two atomic nuclei collide at relativistic speed, only i yield will correspond to the integral of the distribution matter in the collision path, see Fig. 26, participates in function (Eq.39) over the phase space multiplied by the the reaction. Two fraction of nuclei are shaved of and fly hadron spin degeneracy gi = (2Ji + 1) and volume V by along collisions axis – we call these nucleons spectators. Z d3p The sum of of the number of participants and specta- hNii ≡ hNi(mi, gi,V,T,Υi)i = giV 3 ni. (40) tors must be exactly the number of nucleons introduced (2π) into the reaction: for Pb-Pb this number is 2A = 416 or The fluctuation of the yield Eq. (40) is: perhaps better said, there are Nq = 1248 valance u, d- ∂hN i Z d3p quarks, and for Au-Au we have 2A = 358 or Nq = 1074. 2 i (∆Ni) = Υi = giV 3 ni (1 ∓ ni) . How many of these quarks actually have interacted in ∂Υi T,V (2π) each reaction is hard to know or directly measure. One (41) applies a ‘trigger’ to accept a class of collision events It is more practical for numerical computation to express which then is characterized in terms of some macroscopic the above yields and fluctuations as an expansion in mod- observable relating to nearly forward flying spectator. A ified Bessel functions numerical model connects the artificially created reaction 3 ∞ n−1 n giVT X (±1) Υ nmi classes with the mean number of participants Npart that hN i = i W , (42) i 2π2 n3 T contributed. For further details for the LHC work we refer n=1 to the recent ALICE review of their approach [204]. 3 ∞ n−1 n giVT X (±1) Υ 2+n−1 nmi In Fig. 27 we see how this works. All inelastic colli- (∆N )2 = i W , i 2π2 n3 n T sion classes are divided into groups related to how big a n=1 fraction of all inelastic events the trigger selects. So 0-5% (43) means that we are addressing the 5% most central colli- 2 W (x) ≡ x K2(x) . (44) sions, nearly head-on. How head-on this is we can see by considering the distribution in Npart one obtains in the These expansions can be calculated to any desired accu- Monte-Carlo Glauber model as shown in Fig. 27. racy; for Bosons convergence requires Υi exp(−mi/T ) < 1, How do we know that such classification, that is a char- otherwise the expansion makes no sense. For heavy (m acterization of events in terms of some forward observable T ) particles, such as charm hadrons, the Boltzmann dis- which is model-converted into participant distribution, is tribution is a good approximation, i.e., it is sufficient to meaningful? Experimental work provides direct confirma- evaluate the first term of the expansion in Eq. (42), which tion by connecting different observables [204]. I will in is indeed implemented in the CHARM module of SHARE the analysis of other experimental results evaluate specific to reduce computation time at no observable loss of pre- properties of the fireball of matter in terms of the number cision. of participants. Some of these properties turn out to be To evaluate the yield of hadron resonance with finite very flat across many of the collision classes as a function width Γi, one has to weigh the yield (Eq.40) by the resonance mass using the Breit-Wigner distribution: of Npart which entered into the discussion. This shows that the expected extensivity of the property holds: as more Z Γ hN (M, g ,T,V,Υ )i hN˜ Γ i = dM i i i i (45) participants participate the system expands accordingly. i 2π (M − m )2 + Γ 2/4 Moreover, this finding also validates the analysis method, i i a point which will be raised in due time. −→ hNii for Γi → 0. Johann Rafelski: Melting Hadrons, Boiling Quarks 43

Fig. 27. Distribution in Npart for each of experimental trigger classes called a-b% based on a MC Glauber model, data from Ref.[204]

For low energy states with a large width one has to use the 9.5 Hadron fugacity Υi and quark chemistry energy dependent resonance width, since an energy independent width implies a way too large probability of the The fugacity of hadron states defines the yields of different resonance being formed with unrealistically small mass. hadrons based on their quark content. It can be calculated The partial width of a decay channel i → j can be well from the individual constituent quark fugacities. In the i i i approximated by most general case, for a hadron consisting of Nu,Nd,Ns i and Nc up, down, strange and charm quarks respectively i i i i lij +1/2 and N ,N ,N and N anti-quarks, the fugacity can be m 2 u¯ d¯ s¯ c¯ Γ (M) = b Γ 1 − ij for M > m , expressed as i→j i→j i M ij N i N i N i N i (46) Υi =(λuγu) u (λdγd) d (λsγs) s (λcγc) c × (50) where bi→j is the decay channel branching ratio, mij is the i i i i Nu¯ N ¯ Ns¯ Nc¯ decay threshold (i.e., sum of the decay product masses) (λu¯γu¯) (λd¯γd¯) d (λs¯γs¯) (λc¯γc¯) , and lij is the angular momentum released in the decay. where γf is the phase space occupancy of flavor f and The total energy dependent width Γi(M) is obtained using λf is the fugacity factor of flavor f. Note that we allow the partial widths Eq. (46) for all decay channels of the for non-integer quark content to account for states like η resonance in question as meson, which is implemented as η = 0.55(uu¯+dd¯)+0.45ss¯ X in agreement with [205]. It can be shown that for quarks Γi(M) = Γi→j(M). (47) and anti-quarks of the same flavor j −1 γf = γf¯ and λf = λf¯ , (51) For a resonance with a finite width, we can then replace Eq. (45) by which reduces the number of variables necessary to evaluate the fugacity by half. ∞ Z It is a common practice to take advantage of the isospin X Γi→j(M) hNi(M, gi,T,V,Υi)i hN Γ i = dM , symmetry and to treat the two lightest quarks (q = u, d) i A (M − m )2 + Γ (M)2/4 j i i i using light quark and isospin phase space occupancy and mij fugacity factors which are obtained via a transformation (48) of parameters: where A is a normalization constant i r √ γu ∞ γ = γ γ , γ = , (52) Z q u d 3 X Γi→j(M) γd A = dM . (49) i (M − m )2 + Γ (M)2/4 j i i with straightforward backwards transformation mij γu = γqγ3, γd = γq/γ3, (53) Equation (48) is the form used in the program to evaluate hadron resonance yield, whenever calculation with finite and similarly for the fugacity factors width is required. Note that yield evaluation with finite r p λu width is implemented only for hadrons with no charm con- λq = λuλd, λ3 = , (54) λd stituent quark, zero width(Γci = 0) is used for all charm hadrons. λu = λqλ3, λd = λq/λ3. (55) 44 Johann Rafelski: Melting Hadrons, Boiling Quarks

Chemical potentials are closely related to fugacity; one is complemented by the unobserved isospin symmetric chan- can express an associated chemical potential µi for each nel + 0 + hadron species i via Λc → nK π (3.3 ± 1.0)%, (63) with the same branching ratio. Υ = eµi/T . (56) i The influence of resonance feed-down on fluctuations It is more common to express chemical potentials related is the following: to conserved quantum numbers of the system, such as h(∆N )2i = B (N −B )hN i + B2 h(∆N )2i. baryon number B, strangeness s, third component of isospin j→i j→i j→i j→i j j→i j I and charm c: (64) 3 The first term corresponds to the fluctuations of the mother particle j, which decays into particle i with branching ra- µB = 3T log λq , (57) tio Bj→i. Nj→i is the number of particles i produced in the µS = T log λq/λs , (58) P decay of i (inclusive production) so that i Bj→i = Nj→i. µI3 = T log λ3 , (59) For nearly all decays of almost all resonances Nj→i = 1, µC = T log λcλq . (60) however, there are significant exceptions to this, including the production of multiple π0, such as η → 3π0. The sec- Notice the inverse, compared to intuitive definition of µS, ond term in Eq. (64) corresponds to the fluctuation in the which has a historical origin and is a source of frequent yield of the mother particle (resonance). mistakes.

10 Hadrons from QGP: What do we learn? 9.6 Resonance decays A comparison of lattice results with freeze-out conditions The hadron yields observed include the post-hadronization were shown in Fig. 9. The band near to the tempera- decays of in general free streaming hadron states – only ture axis displays the lattice estimate for TH presented in a few are stable enough to reach detectors. In fact heav- Ref. [67], TH = 147 ± 5 MeV. As Fig. 9 demonstrates, ier resonances decay rapidly after the freeze-out and feed many SHM are in more or less severe conflict with this lighter resonances and ‘stable’ particle yields. The final value of TH . The model SHARE we detailed in previous stable particle yields are obtained by allowing all reso- Section 9 is, however, in excellent agreement. One of the nances to decay sequentially from the heaviest to the light- reasons to write this review is to highlight how the change est and thus correctly accounting for resonance cascades. in understanding of TH impacts the resultant choice that The observable yield of each hadron i including into emerges in terms of SHM applicability. the study the resonances populated by more massive res- The SHARE toolbox permits a complete analysis of onances, is then a combination of primary production and any sufficiently large family set of particle yields that is feed from resonance decays consistently presented in terms of a given reaction energy X and participant number class Npart. Especially as a func- hNii = hNiiprimary + Bj→ihNji, (61) tion of Npart this is not always the case, whence some in- j6=i terpolation of data is a part of the analysis. We do not discuss this practical issue further here. The material selected where Bj→i is the probability (branching ratio) that parti- for presentation is not comprehensive and it is only rep- cle j will decay into particle i. Applied recursively, Eq. (61) resentative of the work manifestly consistent with Fig. 9. generated the model result that corresponds to the ex- Another criterion that we use is to focus on parti- perimentally observable yields of all hadrons, ‘stable’ and cle yields only. Doing this, we need to mention upfront unstable resonances, which are often of interest. the work of Begun, Florkowski and Rybczynski [206,207] The SHARE program includes for non-charm hadrons which applies the same nonequilibrium methods in an am- all decay channels with branching ratio ≥ 10−2 in data ta- bitious effort to describe all LHC particle spectra and does bles. To attain the parallel level of precision for the higher this with good success. These results are directly relevant number of charm hadron decays (a few hundred(!) in some to our study of LHC data presenting complementing in- cases) with small branching ratios required to set the ac- formation that confirms our statistical parameter deter- ceptance for decay channels at a branching ratio ≥ 10−4. mination. Since charm hadrons in many cases decay into more than We will also show, by an example, some of the issues three particles, a more complex approach in implementing that have affected the SHM analysis carried out by an- them had to be used [202]. other group. There is still a lot of uncertainty regarding charm decay channels. Some of them are experimentally difficult to confirm, but required and had to be estimated based on 10.1 Hadron source bulk properties before LHC + symmetries. For example, a measured Λc decay channel Among the important features built into the SHARE pro- + 0 0 Λc → pK π (3.3 ± 1.0)%, (62) gram is the capability to fully describe the properties of Johann Rafelski: Melting Hadrons, Boiling Quarks 45 the fireball that produces the particles analyzed. This is not done in terms of produced particles: each carries away ‘content’, such as the energy of the fireball. We evaluate and sum all fractional contributions to the fireball bulk properties from the observed and, importantly, unobserved particles, predicted by the fit in their abundance. The energy content is only thermal, as we eliminate using yields the effect of expansion flow on the spectra, i.e. the dynamical collective flow energy of matter. Thus the energy content we compute is the ‘comoving’ total thermal energy. Given the large set of parameters that SHARE makes available we fit all particles well and thus the physical properties that we report are rather precise images of the observed particle yields. The question what the SHM parameters mean does not enter the discussion at all. If a measurement error has crept in then our results would look anomalous when inspected as a function of collision energy or collision centrality. The fit of SHM parameters then provides an extrapolation from the measured particle abundances to unmea- sured yields of all particles known and listed in SHARE tables. Most of these are of no great individual relevance, being too massive. The bulk properties we report here are, for the most part, defined by particles directly observed. We expect smooth√ lines describing the fireball properties as a function of sNN, the CM energy per pair of nucleons or/and as function of collision centrality class Nnpart√. Ap- pearance of discontinuous behavior as a function of sNN can indicate a change related to QGP formation. In Fig. 28 we see in the SPS and RHIC energy domain for most head-on collisions, from top to bottom, the pressure P , energy density ε, the entropy density σ, and the Fig. 28. Fireball bulk properties in the SPS and RHIC energy net baryon density ρB−B¯ . SPS and RHIC 4π data were used, for RHIC range also results obtained fitting dN/dy domain, see text. Update of results published in Ref.[7]. are shown by the dashed line, particles originating in a volume dV/dy, y ∈ {−0.5, 0.5}. Only for the baryon density can we recognize a serious difference; the baryon density gradual decline with increasing energy. But what makes in the central rapidity region seems to be a factor 5 below quarks stop just then? And why they decide to stop less average baryon density. Not shown is the change in the at higher energy, instead ‘shooting through’? Note that a fitted volume, which is the one changing quantity (aside possible argument that a decrease in baryon density is due of ρB−B¯ ). Volume grows to accommodate the rapid rise in to volume growth is not right considering that the ther- particle multiplicity with the available energy. mal energy density , and the entropy density σ remain Figure 28 shows exciting features worth further discus- constant above the threshold in collision energy. sion. There can be no doubt that over a relatively small I would argue that when first color bonds are melted, domain of collision energy – in laboratory frame, between gluons are stopped while quarks are more likely to run out. That would agree with our finding in context of stran- 20 and 40 A GeV√ (SPS projectile per nucleon energies) and in CM frame sNN ∈ {6.5, 7.5} GeV per nucleon pair geness production, see Appendix B, that despite similar – the properties of the fireball change entirely. Is this a looking matrix elements in perturbative QCD, gluons are signal of the onset of new physics? And if so why, is this much more effective in making things happen due to their happening at this energy? Though this experimental re- ‘high’adjoint representation color charge; the best analogy sult has been recognized for nearly 10 years now, Ref.[7] would be to say that gluons have double-color charge. The and private communication by M. Ga´zdzicki, I have no high gluon density at first manages to stop some quarks clear answer to offer to these simple questions. but the probability decreases with increasing energy. It We find a peak in the net baryon density, bottom frame is remarkable how fast the dimensionless ρB−B¯ /σ ≡ b/S of the Fig. 28. The K+/π+ peaking, Fig. 23, discussed in drops. This expresses the ability to stop quarks normalized to the ability to produce entropy. Subsection 6.3 seems to be related to the√ effect of baryon stopping, perhaps a rise as function of sNN in stopping Seeing all these results, one cannot but ask what the power at first, when color bonds are broken, and a more total abundance of strange quark pairs will do. Before the 46 Johann Rafelski: Melting Hadrons, Boiling Quarks

defined collision energy range. (b) Beyond this threshold in collision energy the hadronization proceeds more effectively into strange antibaryons. (c) The universality of hadronization source properties, such as energy density, or entropy density above the same energy threshold, suggest as explanation that a new phase of matter hadronizes. There is little doubt considering these cornerstone analysis results that at SPS at and above the projectile energy of 30 A GeV we produced a rapidly evaporating (hadronizing) drop of QGP. The analysis results we presented for the properties of the fireball leave very little space for other interpretation. The properties of the QGP fireball created in the energy range of 30–156 A GeV Pb–Pb collisions at CERN are just the same as those obtained for RHIC beam energy scan, see end of Subsection 6.3.

10.2 LHC SHM analysis √ We consider now LHC results obtained at sNN = 2760 GeV as a function of participant number Npart, Section 9.3 and compare with√ an earlier similar analysis of STAR results available at sNN = 62 GeV [72]. In comparison, there is a nearly a factor 50 difference in collision energy. The results presented here for LHC are from the ALICE experiment as analyzed in Refs.[69,70,71]. The ex- Fig. 29. Strangeness pair production s yield from SPS and √ perimental data inputs were discussed extensively in these RHIC as a function sNN: yield normalized by net baryon references, the data source includes Refs.[83,149,208,209]. abundance b in top frame, entropy S in middle frame. At The analysis of hadron production as a function of par- bottom the energy cost to produce strangeness. Total parti- ticipant number N at RHIC and LHC proceeds in es- cle yields, except for dN/dy results shown as dashed lines in part sentially the same way as already described. The results the RHIC energy range. Update of results published in Ref.[7] here presented were obtained without the contribution of charmed hadrons. Given the large set of available SHARE parameters all discussion of results seen in Fig. 29 it is wise to read the particles are described very well, a non-complete example conclusions in Appendix B where in 1983 the overall stran- of the data included is seen in Fig. 30. Note that the cen- geness yield enhancement alone was not predicted to be a tral rapidity yields are divided by Npart/2; that is they striking signature. In Fig. 29 ratios are shown, in the top are per nucleon pair as in pp collisions. This also means frame: the pair strangeness abundance s per net baryon that our fit spans a range of a yield of dN ' 10−4 for abundance b; per entropy in the middle frame; and in the Ω the most peripheral collisions to dNπ ' 2000 for the most bottom frame we see the energy cost in GeV to make a central collisions, thus more than 7 orders of magnitude strange quark pair, E/s, mind you that this energy is the alone of particles shown in Fig. 30. final state thermal fireball energy. About three orders of magnitude of the large range of We see in Fig. 29 that the s/b ration is smooth. This yields dN/dy that are fitted are absorbed into the rapidly means that strangeness production takes off where baryon changing volume dV/dy from which these particles emerge, √stopping takes off, being in the QGP attributed range of see Fig. 31. Note that this result is already reduced by the sNN faster than rise in entropy production. And, the en- factor Npart/2; thus this is volume per colliding nucleon ergy cost of a pair seems to be very low at high energy: pair. For RHIC we see that this is a rather constant value only 5–6 times the energy that the pair actually carries by to which the LHC results seem to converge for small value itself, and this factor reflects accurately on how abundant of Npart. However for large Npart at LHC the specific vol- strangeness is in comparison to all the other constituents ume keeps growing. Keep in mind that the interpretation of the QGP fireball. This by itself clearly indicates that the of dV/dy is difficult and a priori is not geometric, see Sub- yield converges to chemical QGP equilibrium. The clear section 9.2. break in the cost of making a strange quark pair near 30 The corresponding LHC and RHIC chemical freeze- GeV energy shows the threshold above which strangeness, out temperature T varies both at RHIC and LHC in the as compared to other components, becomes an equal fire- same fashion with larger values found for smaller hadro- ball partner. nization volumes. This is natural, as scattering length for Our analysis thus shows: (a) There is an onset of baryon decoupling must be larger than the size of the system and transparency and entropy production at a very narrowly thus the more dense hotter condition is possible for the Johann Rafelski: Melting Hadrons, Boiling Quarks 47

√ Fig. 32. The chemical freeze-out temperature T at sNN = 2.76 TeV, as a function of the number of participants Npart, lines guide the eye. Results adapted from Refs.[69,70,71].

Fig. 30. LHC experimental data measured by the ALICE √ experiment in Pb–Pb collisions at sNN = 2.76 TeV as function of centrality described by Npart, normalized by Npart/2. Results adapted from Refs.[69,70,71]

Fig. 33. Light quark γ and strange quark γ fugacities at √ q s sNN = 2.76 TeV as a function of the number of participants Npart, lines guide the eye. Results adapted from Refs.[69,70, √ Fig. 31. The source volume dV/dy at sNN = 2.76 TeV, 71]. normalized by number of nucleon pairs Npart/2, as a function of the number of participants N . For comparison, a similar √ part STAR sNN = 62 GeV data analysis is shown. Results adapted In Fig. 34 we see the physical properties of the fireball from Refs.[69,70,71] as obtained by the same procedure as discussed in Sub- section 10.1. With increasing participant number all these bulk properties decrease steadily. This is the most marked smaller fireball. One can also argue with the same out- difference to the RHIC results. We should here remember come that the rapid expansion of the larger fireball can that the hadronization volume at LHC given the greater lead to stronger supercooling of QGP which directly trans- total energy content of the fireball is much greater and forms into free-streaming hadrons. The possibility of di- thus the dynamics of fireball expansion should be differ- rect QGP hadronization is supported by the strong chem- ent. ical non-equilibrium with γq > 1, γs > 1 for all collision Results seen in Fig. 34 show a remarkable universality, centralities. These results are seen in Fig. 33 both when LHC is compared to RHIC, and as a func- 48 Johann Rafelski: Melting Hadrons, Boiling Quarks

Fig. 36. hsi+hs¯i strangeness density measured in the hadron phase (red squares) as a function of centrality described by Npart. The dashed (blue) line is a ﬁt with strangeness in the QGP phase, see text. Results adapted from Refs.[69,70,71]

rapidity is produced. For the more peripheral collisions the rise of the total strangeness yield is very rapid, as both the Fig. 34. From top to bottom as function of centrality de- size of the reaction volume and within the small fireball scribed by N : energy density ε, entropy density σ reduced part √ by a factor 10 to fit in figure, and 3P at s = 2.76 TeV. the approach to saturation of strangeness production in √ NN the larger QGP fireball combine. The dotted line are RHIC sNN = 62 GeV analysis results not showing the (larger) error band. Results adapted from Refs.[69, It is of considerable interest to understand the mag- 70,71]. nitude of strangeness QGP density at hadronization. We form a sum of all (strange) hadron multiplicities dNh/dy h weighting the sum with the strange content −3 ≤ ns ≥ 3 of any hadron h and include hidden strangeness, to obtain the result shown in Fig. 36. Within the error bar the result is a constant; strange quarks and antiquarks in the fireball are 20% more dense than are nucleons bound in nuclei. However, is this hsi+hs¯i strangeness density shown by error bars in Fig. 36 a density related to QGP? To give this result a quantitative QGP meaning we evaluate QGP phase strangeness density at a given T , see Eq. (42)

3 ∞ gT X n Fig. 35. Hadronization universality: the interaction measure s(m ,T ; γQGP) = − (−γQGP) × (65) (ε−3P )/T 4 evaluated at hadronization condition of the hadron s s 2π2 s √ n=1 ﬁreball created in s = 2.76 TeV Pb–Pb collisions as a NN 2 function of centrality described by N . Results adapted from 1 nms nms part K2 , Refs.[69,70,71] n3 T T

QGP where ms is the (thermal) strange quark mass, γs is the phase space occupancy: here the superscript QGP helps tion of centrality; variation as a function of Npart is much to distinguish from that measured in the hadronization smaller than that seen in particle yields in Fig. 30 (keep analysis as γs, used without a superscript. The degeneracy in mind that these results are divided by N /2). The part is g = 12 = 2spin3color2p where the last factor accounts for universality of the Hadronization condition is even more the presence of both quarks and antiquarks. 4 pronounced when we study, see Fig. 35, (ε − 3P )/T , the In central LHC collisions, the large volume (longer interaction measure Im Eq. (11) (compare Subsection 2.5, lifespan) also means that strangeness approaches satu- Fig. 7). rated yield in the QGP. In peripheral collisions, the short We observe that the lattice-QCD maximum from Fig. 7 lifespan of the fireball may not be sufficient to reach chem- (ε − 3P )/T 4 falls right into the uncertainty band of this ical equilibrium. Therefore we introduce a centrality de- result. Only for γ ' 1.6 and γ ' 2 a high value for I QGP q s m pendent strangeness phase space occupancy γs (Npart) shown in Fig. 35 can be obtained. The equilibrium hadron which is to be used in Eq. (65). gas results are about a factor 3 smaller in the relevant QGP A model of the centrality dependence of γs (Npart) is domain of temperature. not an important consideration, as the yield for Npart > 30 Turning our attention now to strangeness: In the most is nearly constant. The value of strangeness density re- central 5% Pb–Pb collisions at the LHC2760, a total of quires ms = 299 MeV in a QGP fireball at hadronization. dNss¯/dy ' 600 strange and anti-strange quarks per unit of For ms ' 140 MeV (mass at a scale of µ ' 2πT ' Johann Rafelski: Melting Hadrons, Boiling Quarks 49

Their lattice references are [15, from the year 2001] [212] and [16, from the year 1999] [213]. The two references disagree in regard to value of TH , verbatim: (1999)“If the quark mass dependence does not change drastically closer to the chiral limit the current data suggest Tc ' (170–190) MeV for 2-flavor QCD in the chiral limit. In fact, this estimate also holds for 3-flavor QCD.”. (2001) “The 3-flavor theory, on the other hand, leads to consistently smaller values of the critical temperature,. . . 3 flavor QCD: Tc = (154±8) MeV” While the authors of Ref.[211] were clearly encouraged by the 1999 side remark in Ref.[213] about 3 flavors, they also cite in the same breath the correction [212] which Fig. 37. Ratio of strangeness pair yield to entropy s/S as func- renders their RHIC SHM fit invalid: for a lattice result tion of collision centrality described by N . Results adapted part T = 154±8 MeV chemical freeze-out at T ' 174±7 MeV from Refs.[69,70,71] H seems inconsistent since T < TH strictly. When reading Ref.[211] in Spring 2002 I further spot- ted that it is technically wanting. Namely, the experimen- 0.9 GeV). γQGP ' 0.77 is found. The higher value of m s final s tal Ξ/Ξ ratio used in the paper predicts a value µ = makes more sense in view of the need to account for the Ξ µ − 2µ = 18.8 MeV while the paper determines from thermal effects. Thus we conclude that for N > 30 B S part this ratio a value µ = µ − 2µ = 9.75 MeV. In conclu- the fireball contains QGP chemical equilibrium strange- Ξ B S sion: the cornerstone manuscript of the GSI group is at ness abundance, with strangeness thermal mass m = s the time of publication inconsistent with the lattice used 299 MeV [71]. as justification showing chemical freeze-out T > TH by The ratio of strangeness to entropy is easily recog- 20 MeV, and its computational part contains a technical nized to be, for QGP, a measure of the relative number mistake. But, this paper had a ‘good’ confidence level. of strange to all particles - adding a factor ' 4 describing The key argument of the paper is that χ2/dof ' 1. the amount of entropy that each particle carries. Thus a However, χ2 depends in that case on large error bars in QGP source will weigh in with ratio s/S ' 0.03 [210]. This the initial 130 GeV RHIC results. Trusting χ2 alone is not is about factor 1.4 larger than one computes for hadron appropriate to judge a fit result11. A way to say this is to phase at the same T , and this factor describes the stran- argue that a fit must be ‘confirmed’ by theory, and indeed geness enhancement effect in abundance which was pre- that is what Ref.[211] claimed, citing Ref.[212] which how- dicted to be that small, see Appendix B. However, if a ever, provided a result in direct disagreement. QGP fireball was formed we do expect a rather constant Thus we can conclude that Ref.[211] at time of publica- s/S as a function of Npart. tion had already proved itself wrong. And while ‘humanum errare est’, students lack the experience to capture theirs effectively. Today this work is cited more than 500 times 10.3 Earlier work – meaning that despite the obvious errors and omissions it has entered into the contemporary knowledge base. Its results confuse the uninitiated deeply. These results could Results of SHM that provide freeze-out T well above TH seen only be erased by a direct withdrawal note by the authors. in Fig. 9 should today be considered obsolete. As an example let us enlarge here on the results of Ref.[211] which would be marked in Fig. 9 GSI-RHIC at√T ' 174±7, µB ' 10.4 Evaluation of LHC SHM fit results 46 ± 5 MeV corresponding to a fit of sNN = 130 GeV RHIC results (but the point is not shown above the upper The chemical non-equilibrium SHM describes very well all T margin). This reference assumes full chemical equilib- available LHC-2760 hadron production data obtained in rium. They draw attention to agreements with other re- a wide range of centralities Npart, measured in the CM sults and expectations, both in their conclusions, as well within the rapidity interval −0.5 ≤ 0 ≤ 0.5. A value of as in the body of their text, verbatim: freeze-out temperature that is clearly below the range for

“The chemical freeze-out temperature Tf ' 168 ± 11 Hagedorn explained the abuse of χ2 as follows: he carried 2.4 MeV found from a thermal analysis of experi- an elephant and mouse transparency set, showing how both mental data in PbPb collisions at SPS is remark- transparencies are fitted by a third one comprising a partial ably consistent within error with the critical tem- picture of something. Both mouse and elephant fitted the some- perature Tc ' 170 ± 8 MeV obtained from lat- thing very well. In order to distinguish mouse from elephant tice Monte Carlo simulations of QCD at vanishing external scientific understanding; in his example, the scale, was baryon density [15] and [16]” required. 50 Johann Rafelski: Melting Hadrons, Boiling Quarks

dronization at LHC, RHIC and SPS: the bulk properties of the fireball that we determine are all very similar to each other. This can be seen by comparing RHIC-SPS results presented in Fig. 28 with those shown in Fig. 34 for LHC-RHIC. This universality includes the strangeness content of the fireball. The LHC particle multiplicity data has relatively small errors, allowing establishment of relatively precise results. The strong nonequilibrium result γs → 2 seen in Fig. 33 allows the description of the large abundance of multi-strange hadrons despite the relatively small value of freeze-out temperature. The value of light quark fugacity, γq → 1.6 allows a match in the high entropy content of the QGP fireball with the γq enhanced phase space of hadrons, especially mesons. As noted above, this effect naturally provides the correct p/π ratio at small T . There are two noticeable differences that appear in comparing RHIC62 to LHC2760 results; in Fig. 31 we see as a function of centrality the specific volume parameter (dV/dy)/Npart. The noticeable difference is that at RHIC this value is essentially constant, while at LHC there is clearly a visible increase. One can associate this with a corresponding increase in entropy per participant, implying that a novel component in entropy production must have opened up in the LHC energy regime. This additional Fig. 38. Data: most central spectra of pions, kaons and entropy production also explains why at LHC the maxi- protons from ALICE experiment [83,84] as a function of p⊥. mum value of specific strangeness pair yield per entropy is Lines: Top the nonequilibrium model for parameters of this re- smaller when compared to RHIC62 for most central colli- view; bottom the outcome with equilibrium constraint. Figure sions, see Fig. 37. adapted from Ref.[206] The results found in the LHC-SHM analysis characterize a fireball that has properties which can be directly compared with results of lattice-QCD, and which have not TH reported in Fig. 9 for lattice-QCD, arises only when ac- been as yet reported; thus this analysis offers a predic- cepting a full chemical nonequilibrium outcome. Chemical tion which can be used to verify the consistency of SHM nonequilibrium is expected for the hadron phase space if results with lattice. For example, note the dimensionless QGP fireball was in chemical equilibrium. In that sense, ratio of the number of strange quark pairs with entropy theory supports the finding, and this result also has a very 2 s/S → 0.03. Since strange quarks have a mass scale, this good χ /ndf < 1 for all collision centralities. ratio can be expected to be a function of temperature The value of the ratio p/π|experiment= 0.046±0.003 [208, in lattice-QCD evaluation. The interesting question is, at 83] is a LHC result that any model of particle production what T will lattice obtain this strangeness hadronization in RHI collisions must agree with. The value p/π ' 0.05 condition s/S = 0.03? is a natural outcome of the chemical non-equilibrium fit Further, there is a variation as a function of central- with γq ' 1.6. This result was predicted in Ref.[214]: ity seen in Fig. 37; s/S decreases with decreasing Npart. p/π|prediction = 0.047 ± 0.002 for the hadronization pres- 3 Seeing that freeze-out T increases, compare Fig. 32, we ob- sure seen at RHIC and SPS P = 82 ± 5 MeV/fm . Chemi- tain the prediction that as freeze-out T increases, s/S de- cal equilibrium model predicts and fits a very much larger creases. On first sight this is counterintuitive as we would result. This is the so called proton anomaly, there is no think that at higher T there is more strangeness. This is anomaly if one does not dogmatically prescribe chemical an interesting behavior that may provide an opportunity equilibrium conditions. to better understand the relation of the freeze-out analysis A recent study of the proton spectra within the freeze- with lattice-QCD results. out model developed in Krakow confirms the chemical nonequilibrium [206]: In Fig. 38 we show a comparison between a spectral fit of pions, kaons and protons within the equilibrium and nonequilibrium approaches. These re- 11 Comments and Conclusions sults show the strong overprediction of soft protons and some overprediction of kaons that one finds in the equilib- To best of my knowledge nobody has attempted a syn- rium model. The chemical nonequilibrium model provides thesis of the theory of hot hadronic matter, lattice-QCD an excellent description of this key data. results, and the statistical hadronization model. This is Further evidence for the chemical non-equilibrium out- done here against the backdrop of the rich volume of soft come of SHM analysis arises from the universality of ha- hadron production results that have emerged in the past Johann Rafelski: Melting Hadrons, Boiling Quarks 51

20 years in the field of RHI collisions, covering the en- This immediately takes us to the question that ex- tire range of SPS,√ RHIC and now LHC energy, ranging a perts in the field will pose: is in this synthesis anything factor 1000 in sNN. scientifically new? The answer is yes. The list is actually This is for sure not the ultimate word since we expect quite long, and the advice is: read, and fill the gaps where new and important experimental results in the coming few the developments stop. Let me point here to one result, years: LHC-ion operations will reach near maximum en- the new item which is really not all that new: 1978 [25], ergy by the end of 2015; further decisive energy increases Hagedorn and I discovered that hadronic matter with ex- could take another lifespan. On the other energy range ponential mass spectrum at the point of singularity TH has end, we are reaching out to the domain where we expect a universally vanishing speed of sound cs → 0 in leading the QGP formation energy threshold, both at RHIC-BES, order. and at SPS-NA61. The future will show if new experimen- Universally means that this is true for all functions tal facilities today in construction and/or advanced plan- that I have tried, including in particular a variation on pre- ning will come online within this decade and join in the exponential singularity index a. In ‘leading order’ means study of QGP formation threshold. Such plans have been that the most singular parts of both energy density and made both at GSI and at DUBNA laboratories. pressure are considered. This result is found in a report In order for this report to be also a readable RHI colli- that was submitted to a Bormio Winter Meeting proceed- sion introduction, I provided pages distributed across the ings, Chapter 23 in Ref.[1] which today is archived elec- manuscript, suitable both for students starting in the field, tronically at CERN [25]. and readers from other areas of science who are interested The reason that this important result cs → 0 was not in the topic. I realize that most of technical material is not preprinted is that Hagedorn and I were working on a larger accessible to these two groups, but it is better to build a manuscript from which this Bormio text was extracted. bridge of understanding than to do nothing. Moreover, We did not want to preclude the publication of the large some historical considerations may be welcome in these paper. However, at the same time we were developing a circles. new field of physics. The main manuscript was never to I did set many of the insights into their historical con- be finished and submitted, but the field of physics took text which I have witnessed personally. In my eyes under- off. Thus the Bormio report is all that remains in public standing the history, how topics came to be looked at the view. way they are, helps both the present generation to learn This result, cs|T →TH → 0 has other remarkable conse- what we know, and the future generation in resolving the quences: Sound velocity that goes to zero at the critical misunderstandings that block progress. boundary implies that the matter is sticky there, when For this reason I felt that many insights that I needed pressed from inside many unusual things can happen, one to develop could be presented here equally well in the for- being filamenting break-up we call sudden hadronization mat of work done many years ago. Therefore, in the fol- – the amazing thing is that while I was writing this, Gior- lowing appendices there are two unpublished reports from gio Torrieri was just reminding me about this insight he conference proceedings, long gone from library shelves, had shared before with me. Could cs|T →TH → 0 actually which I think provide quite appropriate material for this be the cause why the SHM study of the fireball properties report. Perhaps I should have abridged this bonus mate- obtains such clean set of results? rial to omit a few obsolete developments, and/or to avoid Any universal hot hadron matter critical property can duplication across these 80+ pages. However, any contem- be tested with lattice-QCD and the results available do porary change will modify in a damaging way the histor- show a range where cs|T →TH ' 0. Thus, the value of ical context of these presentations. TH may become available as the point of a minimum sound There was a very special reason to prepare this report velocity. This criterion comes with the Hagedorn exponen- now. I took up this task after I finished editing a book tial mass spectrum ‘attached’: the result is valid if and to honor 50 years of Hagedorn remarkable achievements, when there is an exponential growth in hadron mass spec- Ref.[1]. Given the historical context and the target of in- trum. terest in the book being also a person, I could not inject In this text I also answer simple questions which turn there all the results that the reader sees in these pages. out to have complicated answers. For example, what is, The overlap between this work and Ref.[1] is small, mostly and when was, QGP discovered? It turns out that QGP when I describe how the field developed historically before as a phrase meant something else initially, and all kinds of 1985. variants such as: hot quark matter; hadron plasma, were This background material as presented here is more in use. This makes literature search difficult. extensive compared to Ref.[1] as I can go into the detail I also tracked the reporting and interviews from the without concern about the contents balance of an edited time when CERN decided in February 2000 to step for- book. For this reason a reader of Ref.[1] should look at this ward with its announcement of thee QGP discovery. I text as an extension, and conversely, the reader of this re- learned that the then director of BNL was highly skep- view should also obtain Ref.[1] (which is freely available tical of the CERN results. And I was shocked to learn on-line, published in open format by the publisher) in or- that one of the two authors of the CERN scientific con- der to access some of the hard to get references used in sensus report declared a few months down the road that this volume. he was mistaken. 52 Johann Rafelski: Melting Hadrons, Boiling Quarks

Seeing these initial doubts, and being expert on stran- 5. J. Letessier and J. Rafelski: “Observing quark gluon geness I thought I ought to take a late deep look at how plasma with strange hadrons,” Int. J. Mod. Phys. E 9, the signature of CERN February 2000 announcement held (2000) 107 [nucl-th/0003014]. up in past 15 years. I am happy to tell that it is doing very 6. J. Letessier and J. Rafelski, Hadrons and quark-gluon well; the case of QGP at CERN-SPS in terms of strange plasma, Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. antibaryon signature is very convincing. I hope the reader 18 (2002) pp 1–397, Cambridge University Press (Cam- will join me in this evaluation, seeing the results shown. bridge, 2002) These are not comprehensive (my apologies) but sufficient 7. J. Letessier and J. Rafelski, “Hadron production and phase to make the point. changes in relativistic heavy ion collisions,” Eur. Phys. J. I have spent a lot of time, ink, and paper, to explain A 35, (2008) 221 [nucl-th/0504028]. here why RHI collisions and QGP physics is, was, and 8. S. Weinberg, The Quantum Theory of Fields, Volume 1 remains a frontier of our understanding of physics. It is and 2, Cambridge University Press (Cambridge 1995 and true that for the trees we sometimes lose the view of the 1996) forest. Thus at a few opportunities in this report I went 9. Kohsuke Yagi and Tetsuo Hatsuda, Quark-Gluon Plasma: From Big Bang to Little Bang Cambridge University Press out of the trees to tell how the forest looks today, after (Cambridge 2008 35 year of healthy growth. While some will see my com- 10. J. Rafelski “Birth of the Hagedorn temperature,” CERN ments as speculative, others may choose to work out the Courier, 54 (2014) 10:57 http://cerncourier.com/cws/ consequences, both in theory and experiment. article/cern/59346 11. R. Hagedorn, “Statistical thermodynamics of strong in- Commendation: I am deeply indebted to Rolf Hagedorn of teractions at high-energies,” Nuovo Cim. Suppl. 3, (1965) CERN-TH whose continued mentoring nearly 4 decades ago 147. provided much of the guidance and motivation in my long 12. S. C. Frautschi, “Statistical bootstrap model of hadrons,” pursuit of strangeness in quark–gluon plasma and hadroni- Phys. Rev. D 3, 2821 (1971). zation mechanisms. Rolf Hagedorn was the scientist whose 13. R. Hagedorn, “How We Got to QCD Matter From the dedicated, determined personal commitment formed the deep Hadron Side by Trial and Error,” Lect. Notes Phys. 221, roots of this novel area of physics. In 1964/65 Hagedorn pro- (1985) 53; and preprint CERN-TH-3918/84; reprinted in posed the Hagedorn Temperature TH and the Statistical Boot- Chapter 25 of Ref.[1]. strap Model (SBM). These novel ideas opened up the physics 14. J. Kapusta, B. Muller and J. Rafelski, Quark-Gluon of hot hadronic matter to at first, theoretical and later, exper- Plasma: Theoretical Foundations: An annotated reprint imental study, in relativistic heavy ion collision experiments. collection (Elsevier, Amsterdam, 2003) pp 1-817 Acknowledgments: I thank (alphabetically) Michael Da nos15. J. Rafelski, “Strangeness Enhancement: Challenges and (deceased), Wojciech Florkowski, Marek Ga´zdzicki, Rolf Hage- Successes,” Eur. Phys. J. ST 155, (2008) 139. dorn (deceased), Peter Koch, Inga Kuznetsova, Jean Letessier, 16. J. Yellin, “An explicit solution of the statistical bootstrap,” Berndt Muller, Emanuele Quercigh, Krzysztof Redlich, Helmut Nucl. Phys. B 52 (1973) 583. Satz, and Giorgio Torrieri, who have all contributed in an es- 17. T. E. O. Ericson and J. Rafelski, “The tale of the Hagedorn sential way to further my understanding of relativistic heavy temperature,” CERN Cour. 43N7 (2003) 30. ion collisions, hot hadronic matter, statistical hadronization, 18. C. J. Hamer and S. C. Frautschi, “Determination of and the strangeness signature of QGP. I thank Tamas Biro asymptotic parameters in the statistical bootstrap model,” for critical comments of a draft manuscript helping to greatly Phys. Rev. D 4 (1971) 2125 improve the contents; and Victoria Grossack for her kind as- 19. S. C. Frautschi and C. J. Hamer, “Effective temperature of sistance with the manuscript presentation. resonance decay in the statistical bootstrap model,” Nuovo I thank the CERN-TH for hospitality in Summer 2015 Cim. A 13 (1973) 645. while this project was created and completed. This work has 20. C. J. Hamer, “Explicit solution of the statistical boot- been in part supported by the US Department of Energy, Of- strap model via laplace transforms,” Phys. Rev. D 8, 3558 fice of Science, Office of Nuclear Physics under award number (1973). DE-FG02-04ER41318 . 21. C. J. Hamer, “Single Cluster Formation in the Statistical Bootstrap Model,” Phys. Rev. D 9 (1974) 2512. 22. R. D. 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A Extreme States of Nuclear Matter – 1980 and restrict the discussion to the interesting phenomenological consequences. Thus the theoretical part of this re- Johann Rafelski1 port will be devoted mainly to the strongly interacting phase of hadronic gas. We will also describe some experi- From: Invited lecture at Quark Matter 1: mental consequences for relativistic nuclear collisions such Workshop on Future Relativistic Heavy Ion Experiments as particle temperatures, i.e., mean transverse momenta heldGSI, Darmstadt, Germany, 7-10 October 1980; and entropy. printed in: GSI81-6 Orange Report, pp. 282-324, As we will deal with relativistic particles throughout R. Bock and R. Stock, editors. this work, a suitable generalization of standard thermodynamics is necessary, and we follow the way described ABSTRACT: The theory of hot nuclear fireballs consisting of by Touschek [A.3]. Not only is it the most elegant, but all possible finite-size hadronic constituents in chemical and it is also by simple physical arguments the only physi- thermal equilibrium is presented. As a complement of this cal generalization of the concepts of thermodynamics to hadronic gas phase characterized by maximal temperature and relativistic particle kinematics. Our notation is such that energy density, the quark bag description of the hadronic fire- ~ = c = k = 1. The inverse temperature β and volume V ball is considered. Preliminary calculations of temperatures are generalized to become four-vectors: and mean transverse momenta of particles emitted in high E −→ pµ = (p0, p) = muµ , u uµ = 1 , multiplicity relativistic nuclear collisions together with some µ considerations on the observability of quark matter are offered. 1 1 −→ βµ = (β0, β) = vµ , v vµ = 1 , (A1) T T µ µ 0 µ µ V −→ V = (V , V ) = V w , wµw = 1 , A.1 Overview where uµ, vµ, and wµ are the four-velocities of the to- I wish to describe, as derived from known traits of strong tal mass, the thermometer, and the volume, respectively. interactions, the likely thermodynamic properties of hadro- Usually, huµi = vµ = wµ. nic matter in two different phases: the hadronic gas con- We will often work in the frame in which all velocities sisting of strongly interacting but individual baryons and have a timelike component only. In that case we shall often mesons, and the dissolved phase of a relatively weakly drop the Lorentz index µ, as we shall do for the arguments interacting quark-gluon plasma. The equations of state of V = Vµ, β = βµ of different functions. the hadronic gas can be used to derive the particle temper- The attentive reader may already be wondering how atures and mean transverse momenta in relativistic heavy the approach outlined here can be reconciled with the ion collisons, while those of the quark-gluon plasma are concept of quark confinement. We will now therefore ex- more difficult to observe experimentally. They may lead plain why the occurrence of the high temperature phase to recognizable effects for strange particle yields. Clearly, of hadronic matter – the quark-gluon plasma – is still con- the ultimate aim is to understand the behavior of hadronic sistent with our incapability to liberate quarks in high en- matter in the region of the phase transition from gas to ergy collisions. It is thus important to realize that the cur- plasma and to find characteristic features which will allow rently accepted theory of hadronic structure and interac- its experimental observation. More work is still needed to tions, quantum chromodynamics [A.4], supplemented with reach this goal. This report is an account of my long and its phenomenological extension, the MIT bag model [A.5], fruitful collaboration with R. Hagedorn [A.1]. allows the formation of large space domains filled with (al- The theoretical techniques required for the description most) free quarks. Such a state is expected to be unstable of the two phases are quite different: in the case of hadro- and to decay again into individual hadrons, following its nic gas, a strongly attractive interaction has to be ac- free expansion. The mechanism of quark confinement re- counted for, which leads to the formation of the numerous quires that all quarks recombine to form hadrons again. hadronic resonances – which are in fact bound states of Thus the quark-gluon plasma may be only a transitory several (anti) quarks. if this is really the case, then our in- form of hadronic matter formed under special conditions tuition demands that at sufficiently high particle (baryon) and therefore quite difficult to detect experimentally. density the individuality of such a bound state will be lost. We will recall now the relevant postulates and results In relativistic physics in particular, meson production at that characterize the current understanding of strong in- high temperatures might already lead to such a transition teractions in quantum chromodynamics (QCD). The most at moderate baryon density. As is currently believed, the important postulate is that the proper vacuum state in quark–quark interaction is of moderate strength, allowing QCD is not the (trivial) perturbative state that we (naively) a perturbative treatment of the quark-gluon plasma as rel- imagine to exist everywhere and which is little changed ativistic Fermi and Bose gases. As this is a very well stud- when the interactions are turned on/off. In QCD, the true ied technique to be found in several reviews [A.2], we shall vacuum state is believed to a have a complicated structure present the relevant results for the relativistic Fermi gas which originates in the glue (‘photon’) sector of the theory. The perturbative vacuum is an excited state with an 1 Present address: Depaartment of Physics, The University energy density B above the true vacuum. It is to be found of Arizona, Tucson, AZ 85721, USA inside hadrons where perturbative quanta of the theory, Johann Rafelski: Melting Hadrons, Boiling Quarks 59 in particular quarks, can therefore exist. The occurrence force us to formulate the theory of an extended, though of the true vacuum state is intimately connected to the otherwise ideal multicomponent gas. glue–glue interaction. Unlike QED, these massless quanta It is a straightforward exercise, carried through in the of QCD, also carry a charge – color – that is responsible beginning of the next section, to reduce the grand par- for the quark–quark interaction. tition function Z to an expression in terms of the mass spectrum τ(m2, b). In principle, an experimental form of In the above discussion, the confinement of quarks is τ(m2, b) could then be used as an input. However, the a natural feature of the hypothetical structure of the true more natural way is to introduce the statistical bootstrap vacuum. If it is, for example, a color superconductor, then model [A.7], which will provide us with a theoretical τ an isolated charge cannot occur. Another way to look at that is consistent with assumptions and approximations this is to realize that a single colored object would, ac- made in determining Z. cording to Gauss’ theorem, have an electric field that can In the statistical bootstrap, the essential step consists only end on other color charges. In the region penetrated in the realization that a composite state of many quark by this field, the true vacuum is displaced, thus effectively bags is in itself an ‘elementary’ bag [A.1,10]. This leads raising the mass of a quasi-isolated quark by the amount directly to a nonlinear integral equation for τ. The ideas BV . field of the statistical bootstrap have found a very success- Another feature of the true vacuum is that it exercises ful application in the description of hadronic reactions a pressure on the surface of the region of the perturba- [A.11] over the past decade. The present work is an exten- tive vacuum to which quarks are confined. Indeed, this is sion [A.1,9,12] and application [A.1,13] of this method to just the idea of the original MIT bag model [A.6]. The the case of a system containing any number of finite size Fermi pressure of almost massless light quarks is in equi- hadronic clusters with their baryon numbers adding up librium with the vacuum pressure B. When many quarks to some fixed number. Among the most successful pre- are combined to form a giant quark bag, then their prop- dictions of the statistical bootstrap, we record here the erties inside can be obtained using standard methods of derivation of the limiting hadronic temperature and the many-body theory [A.2]. In particular, this also allows the exponential growth of the mass spectrum. inclusion of the effect of internal excitation through a fi- We see that the theoretical description of the two hadro- nite temperature and through a change in the chemical nic phases – the individual hadron gas and the quark- composition. gluon plasma – is consistent with observations and with A further effect that must be taken into consideration the present knowledge of elementary particles. What re- is the quark–quark interaction. We shall use here the first mains is the study of the possible phase transition between order contribution in the QCD running coupling constant those phases as well as its observation. Unfortunately, we 2 2 2 can argue that in the study of temperatures and mean αs(q ) = g /4π. However, as αs(q ) increases when the average momentum exchanged between quarks decreases, transverse momenta of pions and nucleons produced in this approach will have only limited validity at relatively nuclear collisions, practically all information about the low densities and/or temperatures. The collective screen- hot and dense phase of the collision is lost, as most of ing effects in the plasma are of comparable order of mag- the emitted particles originate in the cooler and more di- nitude and should reduce the importance of perturbative lute hadronic gas phase of matter. In order to obtain re- contributions as they seem to reduce the strength of the liable information on quark matter, we must presumably quark–quark interaction. perform more specific experiments. We will briefly point out that the presence of numerous s quarks in the quark From this general description of the hadronic plasma, plasma suggest, as a characteristic experiment, the obser- it is immediately apparent that, at a certain value of tem- vation Λ hyperons. perature and baryon number density, the plasma must We close this report by showing that, in nuclear colli- disintegrate into individual hadrons. Clearly, to treat this sions, unlike pp reactions, we can use equilibrium thermo- process and the ensuing further nucleonisation by pertur- dynamics in a large volume to compute the yield of strange bative QCD methods is impossible. It is necessary to find and anti-strange particles. The latter, e.g., Λ, might be sig- a semi-phenomenological method for the treatment of the nificantly different from what one expects in pp collisions thermodynamic system consisting of a gas of quark bags. and give a hint about the properties of the quark-gluon The hadronic gas phase is characterized by those reactions phase. between individual hadrons that lead to the formation of new particles (quark bags) only. Thus one may view [A.7, 8,9] the hadronic gas phase as being an assembly of many A.2 Thermodynamics of the Gas Phase and the SBM different hadronic resonances, their number in the interval (m2, m2 + dm2) being given by the mass spectrum Given the grand partition function Z(β, V, λ) of a many- τ(m2, b)dm2. Here the baryon number b is the only dis- body system, all thermodynamic quantities can be deter- crete quantum number to be considered at present. All mined by differentiation of ln Z with respect to its argu- bag–bag interaction is contained in the mutual transmu- ments. Here, λ is the fugacity introduced to conserve a tations from one state to another. Thus the gas phase has discrete quantum number, here the baryon number. The the characteristic of an infinite component ideal gas phase conservation of strangeness can be carried through in a of extended objects. The quark bags having a finite size similar fashion leading then to a further argument λs of 60 Johann Rafelski: Melting Hadrons, Boiling Quarks

Z. Whenever necessary, we will consider Z to be implicitly The density of states in Eq. (A4) implies that the cre- dependent on λs. ation and absorption of particles in kinetic and chemi- The grand partition function is a Laplace transform of cal equilibrium is limited only by four-momentum and the level density σ(p, V, b), where pµ is the four-momentum baryon number conservation. These processes represent and b the baryon number of the many-body system en- the strong hadronic interactions which are dominated by closed in the volume V : particle productions. τ(m2, b) contains all participating el- ∞ Z ementary particles and their resonances. Some remaining X µ Z(β, V, λ) = λb σ(p, V, b)e−βµp d4p . (A2) interaction is here neglected or, as we do not use the com- b=−∞ plete experimental τ, it may be considered as being taken care of by a suitable choice of τ. The short range repulsive We recognize the usual relations for the thermodynamic forces are taken into account by the introduction of the expectation values of the baryon number, proper volume V of hadronic clusters. ∂ One more remark concerning the available volume ∆ is hbi = λ ln Z(β, V, λ) , (A3a) ∂λ in order here. If V were considered to be given and an independent thermodynamic quantity, then in Eq. (A4), a fur- and the energy–momentum four-vector, ther built-in restriction limits the sum over N to a certain ∂ Nmax, such that the available volume ∆ in Eq. (A5) re- hpµi = − ln Z(β, V, λ) , (A3b) ∂βµ mains positive. However, this more conventional assumption of V as the independent variable would significantly which follow from the definition in Eq. (A2). obscure our mathematical formalism. It is important to re- The theoretical problem is to determine σ(p, V, b) in alize that we are free to select the available volume ∆ as terms of known quantities. Let us suppose that the phys- the independent thermodynamic variable and to consider ical states of the hadronic gas phase can be considered as V as a thermodynamic expectation value to be computed being built up from an arbitrary number of massive ob- from Eq. (A5): jects, henceforth called clusters, characterized by a mass 2 2 2 µ µ µ µ spectrum τ(m , b), where τ(m , b)dm is the number of V −→ hV i = ∆ + Vc (β, ∆, λ) . (A7) different elementary objects (existing in nature) in the 2 2 2 µ mass interval (m , m +dm ) and having the baryon num- Here hVc i is the average sum of proper volumes of all ber b. As particle creation must be permitted, the number hadronic clusters contained in the system considered. As N of constituents is arbitrary, but constrained by four- already discussed, the standard quark bag leads to the momentum conservation and baryon conservation. Neglect- proportionality between the cluster volume and hadron ing quantum statistics (it can be shown that, for T & mass. Similar arguments within the bootstrap model [A.9], 40 MeV, Boltzmann statistics is sufficient), we have as for example discussed in the preceding lecture by R. Hagedorn [A.10], also lead to ∞ Z N X 1 4 X σ(p, V, b) = δ p − pi × µ N! µ p (β, ∆, λ) N=0 i=1 hVc i = , (A8) N 4B X X δ b − b × k i (A4) where 4B is the (at this point arbitrary) energy density of {bi} i=1 isolated hadrons in the quark bag model [A.5]. N µ Y 2∆µp Since our hadrons are under pressure from neighbors i τ(p2, b )d4p . (2π)3 i i i in hadronic matter, we have in principle to take instead of i=1 4B the energy density of a quark bag exposed to a pressure The sum over all allowed partitions of b into different bi P [see Eq. (A54) below] is included and ∆ is the volume available for the motion of the constituents, which differs from V if the different εbag = 4B + 3P. clusters carry their proper volume V : ci Combining Eqs. (A7)–(A9), we find, with ε(β, ∆, λ) = N hpµi/hV µi = hEi/hV i, that µ µ X µ ∆ = V − Vci . (A5) ∆ ε(β, ∆, λ) i=1 = 1 − . (A9) The phase space volume used in Eq. (A4) is best explained hV (β, ∆, λ)i 4B + 3P (β, ∆, λ) by considering what happens for one particle of mass m 0 As we shall see, the pressure P in the hadronic matter in the rest frame of ∆ and β : µ µ never rises above ' 0.4B, see Fig. A5a below, and argu- Z µ Z 3 √ 2∆µp d pi 2 2 ments following Eq. (A60). Consequently, the inclusion of d4p i e−β·pδ (p2 − m2) =∆ e−β0 p +m i (2π)3 0 i 0 (2π)3 P above – the compression of free hadrons by the hadronic matter by about 10% – may be omitted for now from fur- T m2 ther discussion. However, we note that both ε and P will = ∆ K (m/T ) . 0 2π2 2 be computed as ln Z becomes available, whence Eq. (A9) (A6) is an implicit equation for ∆/hV i. Johann Rafelski: Melting Hadrons, Boiling Quarks 61

It is important to record that the expression in Eq. (A9) by some expression containing τ linearly and by taking can approach zero only when the energy density of the into account the volume condition expressed in Eqs. (A7) hadronic gas approaches that of matter consisting of one and (A8). big quark bag: ε → 4B, P → 0. Thus the density of states We cannot simply put V = Vc and ∆ = 0, because in Eq. (A4), together with the choice of ∆ as a thermody- now, when each cluster carries its own dynamically de- namic variable, is a consistent physical choice only up to termined volume, ∆ loses its original meaning and must this point. Beyond we assume that a description in terms be redefined more precisely. Therefore, in Eq. (A4), we of interacting quarks and gluons is the proper physical de- tentatively replace scription. Bearing all these remarks in mind, we now consider the available volume ∆ as a thermodynamic variable 2Vc(m, b) · p σ(p, V , b) −→ τ(p2, b) which by definition is positive. Inspecting Eq. (A4) again, c (2π)3 we recognize that the level density of the extended objects in volume hV i can be interpreted for the time being as the 2m2 = τ(p2, b) , level density of point particles in a fictitious volume ∆ : (2π)34B (A14) σ(p, V, b) = σpt(p, ∆, b) , (A10) 2∆ · pi 2 2Vc(mi, bi) · pi 2 τ(pi , bi) −→ τ(pi , bi) whence this is also true for the grand canonical partition (2π)3 (2π)3 function in Eq. (A2): 2 2mi 2 = τ(p , bi) . Z(β, V, λ) = Zpt(β, ∆, λ) . (A11) (2π)34B i Combining Eqs. (A2) and (A4), we also find the important Next we argue that the explicit factors m2 and m2 arise relation i from the dynamics and therefore must be absorbed into ∞ Z µ 2 2 2 2 X 2∆µp µ τ(pi , bi) as dimensionless factors mi /m0. Thus, ln Z (β, ∆, λ) = λb τ(p2, b)e−βµp d4p . pt (2π)3 b=−∞ 2m2 σ(p, V , b) −→ 0 τ(p2, b) = Hτ(p2, b) , (A12) c (2π)34B This result can only be derived when the sum over N in Eq. (A4) extends to infinity, thus as long as ∆/hV i in 2 2∆ · pi 2m Eq. (A9) remains positive. τ(p2, b ) −→ 0 τ(p2, b ) = Hτ(p2, b ) , (2π)3 i i (2π)34B i i i i In order to continue with our description of hadronic (A15) matter, we must now determine a suitable mass spectrum with τ to be inserted into Eq. (A4). For this we now introduce 2 2m0 the statistical bootstrap model. The basic idea is rather H := 3 , old, but has undergone some development more recently (2π) 4B making it clearer, more consistent, and perhaps more con- where either H or m0 may be taken as a new free pa- vincing. The details may be found in [A.9] and the refer- rameter of the model, to be fixed later. (If m0 is taken, ences therein. Here a simplified naive presentation is given. then it should be of the order of the ‘elementary masses’ We note, however, that our present interpretation is non- appearing in the system, e.g., somwhere between mπ and trivially different from that in [A.9]. mN in a model using pions and nucleons as elementary The basic postulate of statistical bootstrap is that the input.) Finally, if clusters consist of clusters which consist mass spectrum τ(m2, b) containing all the ‘particles’, i.e., of clusters, and so on, this should end at some ‘elemen- elementary, bound states, and resonances (clusters), is tary’ particles (where what we consider as elementary is generated by the same interactions which we see at work fixed by convention). Inserting Eq. (A15) into Eq. (A4), if we consider our thermodynamical system. Therefore, if the bootstrap equation (BE) then reads we were to compress this system until it reaches its natural 2 2 2 volume Vc(m, b), then it would itself be almost a cluster Hτ(p , b) = Hgbδ0(p − mb ) + (A16) appearing in the mass spectrum τ(m2, b). Since σ(p, ∆, b) ∞ N N and τ(p2, b) are both densities of states (with respect to X 1 Z X X X + δ4 p − p δ b − b × the different parameters d4p and dm2), we postulate that N! i k i N=2 i=1 {bi} i=1

2 σ(p, ∆, b) =b const. × τ(p , b) , (A13) N hV i −→ Vc(m,b) Y 2 4 ∆→0 × Hτ(pi , bi)d pi . i=1 where =b means ‘corresponds to’ (in some way to be spec- ified). As σ(p, ∆, b) is [see Eq. (A4)] the sum over N of Clearly, the bootstrap equation (A16) has not been de- N-fold convolutions of τ, the above ‘bootstrap postulate’ rived. We have made it more or less plausible and state will yield a highly nonlinear integral equation for τ. The bootstrap postulate (A13) requires that τ should 2 Here is the essential difference with [A.9], where another obey the equation resulting from replacing σ in Eq. (A4) choice was made. 62 Johann Rafelski: Melting Hadrons, Boiling Quarks it as a postulate. For more motivation, see [A.9]. In other words, the bootstrap equation means that the cluster with mass pp2 and baryon number b is either elementary (mass mb, spin isospin multiplicity gb), or it is composed of any number N ≥ 2 of subclusters having the same internal composite structure described by this equation. The bar over mb indicates that one has to take the mass which the ‘elementary particle’ will have effectively when present in a large cluster, e.g., in nuclear matter, m = m − hEbindi, and mN ≈ 925 MeV. That this must be so becomes obvious if one imagines Eq. (A16) solved by iteration (the iteration solution exists and is the physical solution). Then Hτ(p2, b) becomes in the end a complicated function of 2 p , b, all mb, and all gb. In other words, in the end a single cluster consists of the ‘elementary particles’. As these are all bound into the cluster, their mass m should be the effective mass, not the free mass m. This way we may include a small correction for the long-range attractive meson exchange by choosing mN = m − 15 MeV. Let us make a brief excursion to the bag model at this point. There the mass of a hadron is computed from the assumption of an isolated particle (= bag) with its size Fig. A1. Bootstrap function G(ϕ). The dashed line represents the unphysical branch. The root singularity is at ϕ = and mass being determined from the equilibrium between 0 ln(4/e) = 0.3863 the vacuum pressure B and the internal Fermi pressure of the (valence) quarks. In a hadron gas, this is not true as a finite pressure is exerted on hadrons in matter. After a Once the set of input particles {mb, gb} is given, ϕ(β, λ) short calculation, we find the pressure dependence of the is a known function, while Φ(β, λ) is unknown. Applying bag model hadronic mass: the double Laplace transformation to the BE, we obtain " # M(P ) 1 + 3P/4B 3 P 2 Φ(β, λ) = ϕ(β, λ) + exp Φ(β, λ) − Φ(β, λ) − 1 . (A19) = = 1 + + ··· . M(0) (1 + P/B)3/4 32 B This implicit equation for Φ in terms of ϕ can be solved (A17) without regard for the actual β, λ dependence. Writing We have already noted that the pressure never exceeds 0.4B in the hadronic gas phase, see Fig. A5a below, and G(ϕ) := Φ(β, λ) , ϕ = 2G − eG + 1 , (A20) arguments following Eq. (A60). Hence we see that the increase in mass of constituents (quark bags) in the hadronic we can draw the curve ϕ(G) and then invert it graphi- gas never exceeds 1.5% and is at most comparable with cally (see Fig. A1) to obtain G(ϕ) = Φ(β, λ). G(ϕ) has a the 15 MeV binding in m. In general, P is about 0.1B and square root singularity at ϕ = ϕ0 = ln(4/e) = 0.3863. Be- the pressure effect may be neglected. yond this value, G(ϕ) becomes complex. Apart from this Thus we can consider the ‘input’ first term in Eq. (A16) graphical solution, other forms of solution are known: as being fixed by pions, nucleons, and whenever necessary ∞ ∞ by the usual strange members of meson and baryon mul- X X integral G(ϕ) = s ϕn = w (ϕ − ϕ)n/2 =representation . tiplets. Furthermore, we note that the bootstrap equation n n 0 n=1 n=0 (A16) makes use of practically all the same approxima- (A21) tions as our description of the level density in Eq. (A4). The expansion in terms of (ϕ −ϕ)n/2 has been used in our Thus the solution of Eq. (A16) is particularly suitable for 0 numerical work (12 terms yield a solution within computer our use. accuracy) and the integral representation will be published We solve the BE by the same double Laplace transfor- elsewhere3. Henceforth, we consider Φ(β, λ) = G(ϕ) to mation which we used before Eq. (A2). We define be a known function of ϕ(β, λ). Consequently, τ(m2, b) ∞ is also in principle known. From the singularity at ϕ = Z µ 2 −βµp X b 2 2 4 ϕ , it follows [A.1] that τ(m , b) grows, for m m b, ϕ(β, λ) := e λ Hgbδ0(p − m )d p 0 N b exponentially ∼ m−3 exp(m/T ). In some weaker form, b=−∞ 0 ∞ this has been known for a long time [A.7,15,16]. X b = 2πHT λ gbmbK1(mb/T ) , 3 Extensive discussion of the analytical properties of the b=−∞ bootstrap function was publisched in: R. Hagedorn and Z ∞ J. Rafelski: Analytic Structure and Explicit Solution of an Im- µ X Φ(β, λ) := e−βµp λbHτ(p2, b)d4p . (A18) portant Implicit Equation, Commun. Math. Phys. 83, (1982) b=−∞ 563 Johann Rafelski: Melting Hadrons, Boiling Quarks 63

A.3 The Hot Hadronic Gas Considering Z as a function of the chemical potential, viz.,

µβ The definition of Φ(β, λ) in Eq. (A18) in terms of the mass Z(β, V, λ) = Z(β, V, e ) = Z˜(β, V, µ) = Z˜pt(β, ∆, µ) , spectrum allows us to write a very simple expression for (A32) ln Z in the gas phase (passing now to the rest frame of the we find gas): ∂ ∂ ˜ ln Z = ln Zpt(β, ∆, µ) = −E + µhbi , (A33) 2∆ ∂ ∂β ∂β ln Z(β, V, λ) = ln Z (β, ∆, λ) = − Φ(β, λ) . µ,∆ pt (2π)3H ∂β (A22) with E being the total energy. From Eqs. (A31) and (A33), We recall that Eqs. (A9) and (A19) define (implicitly) the we find the ‘first law’ of thermodynamics to be quantities ∆ and Φ in terms of the physical variables V , ∂ β, and λ. E = −P hV i + T (P hV i) + µhbi . (A34a) Let us now introduce the energy density εpt of the ∂T hypothetical pointlike particles as Now quite generally, ∂ 2 ∂2 ε (β, λ) = − ln Z (β, ∆, λ) = Φ(β, λ) , E = −P hV i + TS + µhbi , (A34b) pt ∆∂β pt (2π)3H ∂β2 (A23) so that which will turn out to be quite helpful as it is independent ∂ h i of ∆. The proper energy density is S = P (β, ∆, µ)hV (β, ∆, µ)i . (A35) ∂T µ,∆ 1 ∂ ∆ ε(β, λ) = − ln Z = ε , (A24) hV i ∂β hV i pt Equations (A25) and (A33) now allow us to write while the pressure follows from ∂ E − µb S = (P hV i) = ln Z˜ (T, ∆, µ) + . (A36) ∂T pt T P (β, λ)hV i = T ln Z(β, V, λ) = T ln Zpt(β, ∆, λ) , (A25) The entropy density in terms of the already defined quan- ∆ 2T ∂ ∆ P (β, λ) = − Φ(β, λ) =: P . tities is therefore hV i (2π)3H ∂β hV i pt S P + ε − µν (A26) S = = . (A37) Similarly, for the baryon number density, we find hV i T hbi ∆ We shall now take a brief look at the quantities P , ε, ν, ν(β, λ) = =: ν (β, λ) , (A27) hV i hV i pt ∆/hV i. They can be written in terms of ∂Φ(β, λ)/∂β and its derivatives. We note that [see Eq. (A20)] with ∂ ∂G(ϕ) ∂ϕ 1 ∂ 2 ∂ ∂ Φ(β, λ) = , (A38) ν (β, λ) = λ ln Z = − λ Φ(β, λ) . ∂β ∂ϕ ∂β pt ∆ ∂λ pt (2π)3H ∂λ ∂β (A28) −1/2 and that ∂G/∂ϕ ∼ (ϕ0 −ϕ) near to ϕ = ϕ0 = ln(4/e) From Eqs. (A23)–(A23), the crucial role played by the (see Fig. A1). Hence at ϕ = ϕ0, we find a singularity in factor ∆/hV i becomes apparent. We note that it is quite the point particle quantities εpt, νpt, and Ppt. This implies straightforward to insert Eqs. (A24) and (A25) into Eq. (A9) that all hadrons have coalesced into one large cluster. In- and solve the resulting quadratic equation to obtain ∆/hV i deed, from Eqs. (A24), (A26), (A27), and (A29), we find as an explicit function of εpt and Ppt. First we record the limit P B : ε −→ 4B , ∆ ε(β, λ) ε (β, λ)−1 P −→ 0 , (A39) = 1 − = 1 + pt , (A29) hV i 4B 4B ∆/hV i −→ 0 . while the correct expression is We can easily verify that this is correct by establishing the average number of clusters present in the hadronic s 2 gas. This is done by introducing an artificial fugacity ξN ∆ 1 εpt 2B 4B 1 εpt 2B = − − + + − − . in Eq. (A4) in the sum over N, where N is the number of hV i 2 6P 3P 3P 2 6P 3P pt pt pt pt pt clusters. Denoting by Z(ξ) the associated grand canonical (A30) partition functions in Eq. (A22), we find The last of the important thermodynamic quantities is the entropy S. By differentiating Eq. (A25), we find ∂ ξ 2∆ ∂ hNi = ξ ln Zpt(β, ∆, λ; ξ) = − 3 Φ(β, λ) , ∂ ∂ ∂ ∂ξ ξ=1 (2π) H ∂β ln Z = βP hV i = P hV i − T (P hV i) . (A31) ∂β ∂β ∂T (A40) 64 Johann Rafelski: Melting Hadrons, Boiling Quarks

value of T0 seemed necessary to yield a maximal average decay temperature of the order of 145 MeV, as required by [A.17]. (However, a new value of the bag constant then induces a change [A.1] to a lower value of T0 = 180 MeV.) Here we use −2 H = 0.724 GeV ,T0 = 0.19 GeV , 4 (A43b) m0 = 0.398 GeV [when B = (145 MeV) ] ,

where the value of m0 lies as expected between mπ and 1/2 mN [(mπmN) = 0.36 GeV]. The critical curve limits the hadron gas phase. By approaching it, all hadrons dissolve into a giant cluster, which is not in our opinion a hadron solid [A.14]. We would prefer to identify it with a quark-gluon plasma. Indeed, as the energy density along the critical curve is constant (= 4B), the critical curve can be attained and, if the energy density becomes > 4B, we enter into a region which cannot be described without making assumptions about the inner structure and dynamics of the ‘elementary particles’ {mb, gb} – here pions and nucleons – entering into the input function ϕ(β, λ). Considering pions and nucleons Fig. A2. The critical curve corresponding to ϕ(T, µ) = ϕ0 in as quark-gluon bags leads naturally to this interpretation. the µ, T plane. Beyond it, the usual hadronic world ceases to exist. In the shaded region, our theory is not valid, because we neglected Bose-Einstein and Fermi-Dirac statistics A.4 The Quark–Gluon Phase

We now turn to the discussion of the region of the strongly which leads to the useful relation interacting matter in which the energy density would be P hV i = hNiT. (A41) equal to or higher than 4B. As a basic postulate, we will assume that it consists of – relatively weakly – interact- Thus as P hV i → 0, so must hNi, the number of clusters, ing quarks. To begin with, only u and d flavors will be for finite T . We record the astonishing fact that the hadron considered as they can easily be copiously produced at gas phase obeys an ‘ideal’ gas equation, although of course T & 50 MeV. Again the aim is to derive the grand parti- hNi is not constant as for a real ideal gas but a function tion function Z. This is a standard exercise. For the mass- of the thermodynamic variables. less quark Fermi gas up to first order in the interaction The boundary given by [A.1,2,12], the result is

" 2 ϕ(β, λ) = ϕ = ln(4/e) (A42) 8V 2αs 1 4 π 2 0 ln Z (β, λ) = 1 − ln λ + ln λ + q 6π2β3 π 4 q 2 q thus deﬁnes a critical curve in the β, λ plane. Its position # depends, of course, on the actually given form of ϕ(β, λ), 50 α 7π4 + 1 − s , i.e., on the set of ‘input’ particles {mb, gb} assumed and 21 π 60 the value of the constant H in Eq. (A15). In the case of three elementary pions π+, π0, and π− and four elemen- (A44) tary nucleons (spin ⊗ isospin) and four antinucleons, we valid in the limit mq < T ln λq. have from Eq. (A18) Here g = (2s + 1)(2I + 1)C = 12 counts the number of the components of the quark gas, and λq is the fugacity related to the quark number. As each quark has baryon ϕ(β, λ) = 2πHT [3mπK1(mπ/T )+ number 1/3, we ﬁnd (A43a) 1 3 µ/T +4 λ + λ mNK1(mN/T ) , λq = λ = e , (A45) and the condition (A42), written in terms of T and µ = where as before λ allows for conservation of the baryon T ln λ, yields the curve shown in Fig. A2, i.e., the ‘criti- number. Consequently, cal curve’. For µ = 0, the curve ends at T = T0, where 3µq = µ . (A46) T0, the ‘limiting temperature of hadronic matter’, is the same as that appearing in the mass spectrum [A.7,9,15, The glue contribution is 2 −3 16] τ(m , b) ∼ m exp(m/T0) (for b bmN). 2 The value of the constant H in Eq. (A15) has been cho- 8π −3 15 αs ln Zg(β, λ) = V β 1 − . (A47) sen [A.13] to yield T0 = 190 MeV. This apparently large 45 4 π Johann Rafelski: Melting Hadrons, Boiling Quarks 65

We notice the two relevant differences with the photon e.g., quark mass mq or the quantity Λ which enters into 2 gas: the running coupling constant αs(q ). Therefore the relativistic relation between the energy density and pressure, – The occurence of the factor eight associated with the viz., ε − B = 3(P + B), is preserved, which leads to number of gluons. – The glue–glue interaction as gluons carry color charge. 1 P = (ε − 4B) , (A54) Finally, let us introduce the vacuum term, which accounts 3 for the fact that the perturbative vacuum is an excited state of the ‘true’ vacuum which has been renormalized to a relation we have used occasionally before [see Eq. (A9)]. have a vanishing thermodynamic potential, Ω = −β−1 ln Z. From Eq. (A54), it follows that, when the pressure van- Hence in the perturbative vacuum, ishes, the energy density is 4B, independent of the values of µ and T which fix the line P = 0. This behavior is con- ln Zvac = −βBV. (A48) sistent with the hadronic gas phase. This may be used as a reason to choose the parameters of both phases in such This leads to the required positive energy density B within a way that the two lines P = 0 coincide. We will return the volume occupied by the colored quarks and gluons and to this point again below. For P > 0, we have ε > 4B. to a negative pressure on the surface of this region. At this Recall that, in the hadronic gas, we had 0 < ε < 4B. stage, this term is entirely phenomenological, as discussed Thus, above the critical curve of the µ, T plane, we have above. The equations of state for the quark-gluon plasma the quark-gluon plasma exposed to an external force. are easily obtained by differentiating In order to obtain an idea of the form of the P = 0 critical curve in the µ, T plane for the quark-gluon plasma, ln Z = ln Zq + ln Zg + ln Zvac (A49) we rewrite Eq. (A52) using Eqs. (A45) and (A46) for P = 0: with respect to β, λ, and V . The baryon number density, energy, and pressure are respectively: 1 − 2α /π 2 B = s µ2 + (3πT )2 162π2 1 ∂ 2T 3 2α 1 π2 ν = λ ln Z = 1 − s ln3 λ + ln λ , (A55) 2 4 T 4π2 5 α 15 α V ∂λ π π 3 9 + 12 1 − s + 8 1 − s . (A50) 45 3 π 4 π

1 ∂ ε = − ln Z = B Here, the last term is the glue pressure contribution. (If V ∂β the true vacuum structure is determined by the glue– glue interaction, then this term could be modified signifi- " 2 6 4 2αs 1 4 π 2 cantly.) We find that the greatest lower bound on temper- + 2 T 1 − 4 ln λ + 2 ln λ π π 4 · 3 2 · 3 ature Tq at µ = 0 is about

4 # 2 1/4 50 αs 7π 8π 15 αs T ∼ B ≈ 145–190 MeV . (A56) + 1 − + T 4 1 − , (A51) q 21 π 60 15 4 π This result can be considered to be correct to within 20%. Its order of magnitude is as expected. Taking Eq. (A55) 1/4 ∂ as it is, we find for αs = 1/2, Tq = 0.88B . Omitting the P = T ln Z = −B 1/4 ∂V gluon contribution to the pressure, we find Tq = 0.9B . " It is quite likely that, with the proper treatment of the 2T 4 2α 1 π2 + 1 − s ln4 λ + ln2 λ glue field and the plasma corrections, and with larger 2 4 2 1/4 π π 4 · 3 2 · 3 B ∼ 190 MeV, the desired value of Tq = T0 corresponding to the statistical bootstrap choice will follow. 4 # 2 50 αs 7π 8π 15 αs Furthermore, allowing some reasonable T, µ dependence + 1 − + T 4 1 − . (A52) 21 π 60 45 4 π of αs, we can then easily obtain an agreement between the critical curves. Let us first note that, for T µ and P = 0, the baryon However, it is not necessary for the two critical curves chemical potential tends to to coincide, even though this would be preferable. As the quark plasma is the phase into which individual hadrons 2 1/4 dissolve, it is sufficient if the quark plasma pressure van- 1/4 2π µB = 3µq −→ 3B = 1010 MeV , ishes within the boundary set for non-vanishing positive (1 − 2α /π) s pressure of the hadronic gas. It is quite satisfactory for the 1/4 [αs = 1/2 , B = 145 MeV] , theoretical development that this is the case. In Fig. A3a, (A53) a qualitative picture of the two P = 0 lines is shown in the which assures us that interacting cold quark matter is an µ, T plane. Along the dotted straight line at constant tem- excited state of nuclear matter. We have assumed that, perature, we show in Fig. A3b the pressure as a function except for T , there is no relevant dimensional parameter, of the volume (a P,V diagram). The volume is obtained 66 Johann Rafelski: Melting Hadrons, Boiling Quarks

Fig. A3. a) The critical curves (P = 0) of the two models in the T, µ plane (qualitatively). The region below the full line is described by the statistical bootstrap model and the region above the broken line by the quark-gluon plasma. The critical curves can be made to coincide. b) P,V diagram (qualitative) of the phase transition (hadron gas to quark-gluon plasma) along the broken line T = const. of Fig. A3a. The coexistence region is found from the usual Maxwell construction (the shaded areas being equal) by inverting the baryon density at constant fixed baryon The qualitative discussion presented above can be eas- number: ily supplemented with quantitative results. But first we hbi turn our attention to the modifications forced onto this V = . (A57) ν simple picture by the experimental circumstances in high energy nuclear collisions. The behavior of P (V,T = const.) for the hadronic gas phase is as described before in the statistical bootstrap model. For large volumes, we see that P falls with ris- A.5 Nuclear collisions and inclusive particle spectra ing V . However, when hadrons get close to each other so that they form larger and larger lumps, the pressure We assume that in relativistic collisions triggered to small drops rapidly to zero. The hadronic gas becomes a state of impact parameters by high multiplicities and absence of few composite clusters (internally already consisting of the projectile fragments [A.18], a hot central fireball of hadro- quark plasma). The second branch of the P (V,T = const.) nic matter can be produced. We are aware of the whole line meets the first one at a certain volume V = Vm. problematic connected with such an idealization. A proper The phase transition occurs for T = const. in Fig. A3b treatment should include collective motions and distribution of collective velocities, local temperatures, and so on at a vapor pressure Pv obtained from the conventional Maxwell construction: the shaded regions in Fig. A3b are [A.19], as explained in the lecture by R. Hagedorn [A.10]. Triggering for high multiplicities hopefully eliminates some equal. Between the volumes V1 and V2, matter coexists in the two phases with the relative fractions being deter- of the complications. In nearly symmetric collisions (pro- mined by the magnitude of the actual volume. This leads jectile and target nuclei are similar), we can argue that to the occurrence of a third region, viz., the coexistence the numbers of participants in the center of mass of the region of matter, in addition to the pure quark and hadron fireball originating in the projectile or target are the same. Therefore, it is irrelevant how many nucleons do form the domains. For V < V1, corresponding to ν > ν1 ∼ 1/V1, all matter has gone into the quark plasma phase. fireball – and the above symmetry argument leads, in a The dotted line in Fig. A3b encloses (qualitatively) the straightforward way, to a formula for the center of mass domain in which the coexistence between the two phases energy per participating nucleon: of hadronic matter seems possible. We further note that, s at low temperatures T ≤ 50 MeV, the plasma and hadro- Ec.m. Ek,lab/A U := = mN 1 + , (A58) nic gas critical curves meet each other in Fig. A3a. This A 2mN is just the domain where, at present, our description of the hadronic gas fails, while the quark-gluon plasma also where Ek,lab/A is the projectile kinetic energy per nucleon begins to suffer from infrared difficulties. Both approaches in the laboratory frame. While the fireball changes its have a very limited validity in this domain. baryon density and chemical composition (π+p ↔ ∆, etc.) Johann Rafelski: Melting Hadrons, Boiling Quarks 67

Fig. A4. a) The critical curve of hadron matter (bootstrap), together with some ‘cooling curves’ in the T, µ plane. While the system cools down along these lines, it emits particles. When all particles have become free, it comes to rest on some point on these curves (‘freeze out’). In the shaded region, our approach may be invalid b) The critical curve of hadron matter (bootstrap), together with some ‘cooling curves’ [same energy as in Fig. A4a] in the variables T and ν/ν0 = ratio of baryon number density to normal nuclear baryon number density. In the shaded region, our approach may be invalid during its lifetime through a change in temperature and composition changes more rapidly than the tempera- chemical potential, the conservation of energy and baryon ture. number assures us that U in Eq. (A58) remains constant, 2. The baryon density in Fig. A4b is of the order of 1–1.5 assuming that the influence on U of pre-equilibrium emis- of normal nuclear density. This is a consequence of the sion of hadrons from the fireball is negligible. As U is the choice of B1/4 = 145 MeV. Were B three times as large, total energy per baryon available, we can, supposing that i.e., B1/4 = 190 MeV, which is so far not excluded, then kinetic and chemical equilibrium have been reached, set it the baryon densities in this figure would triple to 3– equal to the ratio of thermodynamic expectation values of 5ν0. Furthermore, we observe that, along the critical the total energy and baryon number: curve of the hadronic gas, the baryon density falls with rising temperature. This is easily understood as, at hEi E(β, λ) higher temperature, more volume is taken up by the U = = . (A59) hbi ν(β, λ) numerous mesons. 3. Inspecting Fig. A4b, we see that, at given U, the tem- Thus we see that, through Eq. (A59), the experimental peratures at the critical curve and those at about ν0/2 value of U in Eq. (A58) fixes a relation between allow- differ little (10%) for low U, but more significantly for able values of β, λ : the available excitation energy defines large U. Thus, highly excited fireballs cool down more the temperature and the chemical composition of hadro- before dissociation (‘freeze out’). As particles are emit- nic fireballs. In Fig. A4a and Fig. A4b, these paths are ted all the time while the fireball cools down along the shown for a choice of kinetic energies Ek,lab/A in the µ, T lines of Fig. A4a and Fig. A4b, they carry kinetic en- plane and in the ν, T plane, respectively. In both cases, ergies related to various different temperatures. The only the hadronic gas domain is shown. inclusive single particle momentum distribution will We wish to note several features of the curves shown yield only averages along these cooling lines. in Fig. A4a and Fig. A4b that will be relevant in later considerations: Another remark which does not follow from the curves shown is: 1. Beginning at the critical curve, the chemical potential first drops rapidly when T decreases and then rises 4. Below about 1.8 GeV, an important portion of the to- slowly as T decreases further (Fig. A4a). This corre- tal energy is in the collective (hydrodynamical) motion sponds to a monotonically falling baryon density with of hadronic matter, hence the cooling curves at con- decreasing temperature (Fig. A4b, but implies that, in stant excitation energy do not properly describe the the initial expansion phase of the fireball, the chemical evolution of the fireball. 68 Johann Rafelski: Melting Hadrons, Boiling Quarks

Fig. A5. a) P,V diagram of ‘cooling curves’ belonging to diﬀerent kinetic laboratory energies per nucleon: (1) 1.8 GeV, (2) 3.965 GeV, (3) 5.914 GeV. In the history of a collision, the system comes down the quark lines and jumps somewhere over to the hadron curves (Maxwell). Broken lines show the diverging pressure of pointlike bootstrap hadrons b)The total speciﬁc entropy per baryon in the hadronic gas phase. Same energies per nucleon as in Fig. A5a, and a fourth value 1.07 GeV

Calculations of this kind can also be carried out for the meet each other at P = 0, since both have the same en- quark plasma. They are, at present, uncertain due to the ergy density ε = 4B and therefore V (P = 0) ∼ 1/ν = 1/4 unknown values of αs and B . Fortunately, there is one U/ε = U/4B. However, what we cannot see by inspecting particular property of the equation of state of the quark- Fig. A5a and Fig. A5b is that there will be a discontinu- gluon plasma that we can easily exploit. ity in the variables µ and T at this point, except if pa- Combining Eq. (A54) with Eq. (A59), we obtain rameters are chosen so that the critical curves of the two phases coincide. Indeed, near to P = 0, the results shown 1 P = (Uν − 4B) . (A60) in Fig. A5a should be replaced by points obtained from the 3 Maxwell construction. The pressure in a nuclear collision will never fall to zero. It will correspond to the momentary Thus, for a given U (the available energy per baryon in vapor pressure of the order of 0.2B as the phase change a heavy ion collision), Eq. (A60) describes the pressure– occurs. volume (∼ 1/ν) relation. By choosing to measure P in units of B and ν in units of normal nuclear density ν0 = A further aspect of the equations of state for the hadro- 0.14/fm3, we find nic gas is also illustrated in Fig. A5a. Had we ignored the finite size of hadrons (one of the van der Waals effects) P 4 U ν = γ − 1 , (A61) in the hadron gas phase then, as shown by the dash- B 3 mN ν0 dotted lines, the phase change could never occur because the point particle pressure would diverge where the quark with pressure vanishes. In our opinion, one cannot say it often m ν enough: inclusion of the finite hadronic size and of the γ := N 0 = 0.56 , for : B1/4 = 145 MeV , ν = 0.14/fm3 . 4B 0 finite temperature when considering the phase transition to quark plasma lowers the relevant baryon density (from Here, γ is the ratio of the energy density of normal nu- 8–14ν0 for cold point-nucleon matter) to 1–5ν0 (depend- clei (εN = mNν0) and of quark matter or of a quark bag ing on the choice of B) in 2–5 GeV/A nuclear collisions (εq = 4B). In Fig. A5a, this relation is shown for three [A.20]. projectile energies: Ek,lab/A = 1.80 GeV, 3.965 GeV, and 5.914 GeV, corresponding to U = 1.314 GeV, 1.656 GeV, The physical picture underlying our discussion is an and 1.913 GeV, respectively. We observe that, even at the explosion of the fireball into vacuum with little energy lowest energy shown, the quark pressure is zero near the being converted into collective motion, e.g., hydrodynam- baryon density corresponding to 1.3 normal nuclear den- ical flow, or being taken away by fast pre-hadronization sity, given the current value of B. particle emission. Thus the conserved internal excitation Before discussing this point further, we note that the energy can only be shifted between thermal (kinetic) and hadronic gas branches of the curves in Fig. A5a and Fig. A5b chemical excitations of matter. ‘Cooling’ thus really means show a quite similar behavior to that shown at constant that, during the explosion, the thermal energy is mostly temperature in Fig. A3b. Remarkably, the two branches convered into chemical energy, e.g., pions are produced. Johann Rafelski: Melting Hadrons, Boiling Quarks 69

While it is at present hard to judge the precise amount The chemical potential acts only for nucleons. In the case of expected deviation from the cooling curves shown in of pions, it has to be dropped from the above expression. Fig. A2, it is possible to show that they are entirely in- For the mean temperature, we thus find consistent with the notion of reversible adiabatic, i.e., en- Z tropy conserving, expansion. As the expansion proceeds RescRemisT dT along U = const. lines, we can compute the entropy per hT i = c , (A64) participating baryon using Eqs. (A36) and (A37), and we Z RescRemisdT find a significant growth of total entropy. As shown in c Fig. A5b, the entropy rises initially in the dense phase of the matter by as much as 50–100% due to the pion where the subscript c on the integral indicates here a line production and resonance decay. Amusingly enough, as integral along that particular cooling curve in Fig. A4a the newly produced entropy is carried mostly by pions, and Fig. A4b which belongs to the energy per baryon fixed one will find that the entropy carried by protons remains by the experimentalist. constant. With this remarkable behavior of the entropy, In practice, the temperature is most reliably measured we are in a certain sense, victims of our elaborate theory. through the measurement of mean transverse momenta of Had we used, e.g., an ideal gas of Fermi nucleons, then the particles. It may be more practical therefore to cal- the expansion would seem to be entropy conserving, as culate the average transverse momentum of the emitted pion production and other chemistry were forgotten. Our particles. In principle, to obtain this result we have to per- fireballs have no tendency to expand reversibly and adi- form a similar averaging to the one above. For the average abatically, as many reaction channels are open. A more transverse momentum at given T, µ, we find [A.8] complete discussion of the entropy puzzle can be found in Z √ − p2+m2−µ)/T 3 [A.1]. p⊥e d p Inspecting Fig. A4a and Fig. A4b again, it seems that hp⊥(m, T, µ)ip = Z √ a possible test of the equations of state for the hadro- e− p2+m2−µ)/T d3p nic gas consists in measuring the temperature in the hot (A65)

fireball zone, and doing this as a function of the nuclear p m µ/T collision energy. The plausible assumption made is that πmT/2 K 5 T e = 2 . the fireball follows the ‘cooling’ lines shown in Fig. A4a m µ/T K2 T e and Fig. A4b until final dissociation into hadrons. This presupposes that the surface emission of hadrons during The average over the cooling curve is then the expansion of the fireball does not significantly alter the Z r available energy per baryon. This is more likely true for ∆ 3/2 πm m µ/T T K 5 e dT 2 sufficiently large fireballs. For small ones, pion emission c Vex 2 T hp⊥(m, T, µ)ip c = Z . by the surface may influence the energy balance. As the ∆ m µ/T fireball expands, the temperature falls and the chemical TK2 e dT c Vex T composition changes. The hadronic clusters dissociate and (A66) more and more hadrons are to be found in the ‘elementary’ We did verify numerically that the order of averages does form of a nucleon or a pion. Their kinetic energies are not matter: reminiscent of the temperature found at each phase of the expansion. p⊥(m, hT ic, µ) p ≈ hp⊥(m, T, µ)ip c , (A67) To compute the experimentally observable final temperature [A.1,13], we shall argue that a time average must which shows that the mean transverse momentum is also be performed along the cooling curves. Not knowing the the simplest (and safest) method of determining the aver- reaction mechanisms too well, we assume that the tem- age temperature (indeed better than fitting ad hoc expo- perature decreases approximately linearly with the time nential type functions to p⊥ distributions). in the significant expansion phase. We further have to al- In the presented calculations, we chose the bag con- low that a fraction of particles emitted can be reabsorbed stant B = (145 MeV)4, but we now believe that a larger in the hadronic cluster. This is a geometric problem and, B should be used. As a consequence of our choice and the ex in a first approximation, the ratio of the available volume measured pion temperature of hT iπ = 140 MeV at high- ∆ to the external volume Vex is the probability that an est ISR energies, we have to choose the constant H such emitted particle not be reabsorbed, i.e., that it can escape: that T0 = 190 MeV [see Eq. (A43b)]. The average temperature, as a function of the range ∆ ε(β, λ) Resc = = 1 − . (A62) of integration over T , reaches different limiting values for Vex 4B different particles. The limiting value obtained thus is the The relative emission rate is just the integrated momen- observable ‘average temperature’ of the debris of the in- tum spectrum teraction, while the initial temperature Tcr at given Ek,lab (full line in Fig. A6) is difficult to observe. When integrat- Z 3 √ 2 d p 2 2 m T R = e− p +m /T +µ/T = K (m/T ) eµ/T . ing along the cooling line as in Eq. (A64), we can easily, at emis (2π)3 2π2 2 each point, determine the average hadronic cluster mass. (A63) The integration for protons is interrupted (protons are 70 Johann Rafelski: Melting Hadrons, Boiling Quarks

Fig. A6. Mean temperatures for nucleons and pions together with the critical temperature belonging to the point where the ‘cooling curves’ start oﬀ the critical curve (see Fig. A4a). The Fig. A7. Mean transverse momenta of nucleons and pions mean temperatures are obtained by integrating along the cool- found by integrating along the ‘cooling curves’. ing curves. Note that TN is always greater than Tπ .

claim that most properties of inclusive spectra are reminis- ‘frozen out’) when the average cluster mass is about half cent of the equations of state of the hadronic gas phase and the nucleon isobar mass. We have also considered baryon that the memory of the initial dense state is lost during density dependent freeze-out, but such a procedure de- the expansion of the fireballs as the hadronic gas rescat- pends strongly on the unreliable value of B. ters many times while it evolves into the final kinetic and Our choice of the freeze-out condition was made in chemical equilibrium state. such a way that the nucleon temperature at E /A = k,lab In order to observe properties of quark-gluon plasma, 1.8 GeV is about 120 MeV. The model dependence of our we must design a thermometer, an isolated degree of free- freeze-out introduces an uncertainty of several MeV in the dom weakly coupled to the hadronic matter. Nature has, average temperature. In Fig. A6, the pion and nucleon av- in principle (but not in practice) provided several such erage temperatures are shown as a function of the heavy thermometers: leptons and heavy flavors of quarks. We ion kinetic energy. Two effects contributed to the differ- would like to point here to a particular phenomenon per- ence between the π and N temperatures: haps quite uniquely characteristic of quark matter. First 1. The particular shape of the cooling curves (Fig. A4a). we note that, at a given temperature, the quark-gluon The chemical potential drops rapidly from the criti- plasma will contain an equal number of strange (s) quarks cal curve, thereby damping relative baryon emission at and antistrange (s) quarks, naturally assuming that the lower T . Pions, which do not feel the baryon chemical hadronic collision time is much too short to allow for light potential, continue being created also at lower temper- flavor weak interaction conversion to strangeness. Thus, atures. assuming equilibrium in the quark plasma, we find the 2. The freeze-out of baryons occurs earlier than the freeze- density of the strange quarks to be (two spins and three out of pions. colors) A third effect has been so far omitted – the emission of Z 3 √ 2 pions from two-body decay of long-lived resonances [A.1] s s d p − p2+m2/T T ms = = 6 e s = 3 K (m /T ) , would lead to an effective temperature which is lower in V V (2π)3 π2 2 s nuclear collisions. (A68) In Fig. A7, we show the dependence of the average neglecting for the time being the perturbative corrections transverse momenta of pions and nucleons on the kinetic and, of course, ignoring weak decays. As the mass ms of energy of the heavy ion projectiles. the strange quarks in the perturbative vacuum is believed to be of the order of 280–300 MeV, the assumption of equilibrium for ms/T ∼ 2 may indeed be correct. In Eq. (A68), A.6 Strangeness in Heavy Ion Collisions we were able to use the Boltzmann distribution again, as the density of strangeness is relatively low. Similarly, there From the averaging process described here, we have learned is a certain light antiquark density (q stands for either u that the temperatures and transverse momenta of parti- or d): cles originating in the hot fireballs are more reminiscent of the entire history of the fireball expansion than of the initial hot compressed state, perhaps present in the form q Z d3p 6 = 6 e−|p|/T −µq /T = e−µq /T T 3 , (A69) of quark matter. We may generalize this result and then V (2π)3 π2 Johann Rafelski: Melting Hadrons, Boiling Quarks 71

where the quark chemical potential is µq = µ/3, as given where the one-particle function Z1 for a particle of mass by Eq. (A46). This exponent suppresses the qq pair pro- ms is given in Eq. (A16). To include both particles and an- duction. tiparticles as two thermodynamically independent phases What we intend to show is that there are many more in Eq. (A71), the sum over s in Eq. (A71) must include s quarks than antiquarks of each light flavor. Indeed, them both. As the quantum numbers of particles (p) and antiparticles (a) must always be present with exactly the 2 s 1 ms ms µ/3T same total number, not each term in Eq. (A71) can con- = K2 e . (A70) q 2 T T tribute. Only when n = k/2 = number of particles = number of antiparticles is exactly fulfilled do we have a physical 2 The function x K2(x) is, for example, tabulated in [A.21]. state. Hence, For x = ms/T between 1.5 and 2, it varies between 1.3 n n and 1. Thus, we almost always have more s than q quarks 1 2n X s X Zpair = Z p Zsa . (A72) and, in many cases of interest, s/q ∼ 5. As µ → 0, there 2n (2n)! n 1 1 are about as many u and q quarks as there are s quarks. sp sa When the quark matter dissociates into hadrons, some n of the numerous s may, instead of being bound in a qs We now introduce the fugacity factor f to be able to kaon, enter into a q q s antibaryon and, in particular4, a Λ count the number of strange pairs present. Allowing an or Σ0. The probability for this process seems to be compa- arbitrary number of pairs to be produced, we obtain rable to the similar one for the production of antinucleons ∞ f n n n by the antiquarks present in the plasma. What is particu- X X sp X sa Zs(β, V ; f) = Z1 Z1 larly noteworthy about the s-carrying antibaryons is that n!n! n=0 sp sa (A73) they can conventionally only be produced in direct pair p production reactions. Up to about Ek,lab/A = 3.5 GeV, = I0( 4y) , this process is very strongly suppressed by energy–momentum conservation because, for free pp collisions, the thresh- where I0 is the modified Bessel function and old is at about 7 GeV. We would thus like to argue that 0 s a study of the Λ and Σ in nuclear collisions for 2 < X p X sa y = f Z1 Z1 . (A74) E /A < 4 GeV could shed light on the early stages k,lab sp sa of the nuclear collisions in which quark matter may be formed. We have to maintain the difference between the particles Let us mention here another effect of importance in (p) and antiparticles (a), as in nuclear collisions the sym- this context: the production rate of a pair of particles metry is broken by the presence of baryons and there is an with a conserved quantum number like strangeness will associated need for a baryon fugacity (chemical potential usually be suppressed by the Boltzmann factor e−2m/T , µ) that controls the baryon number. We obtain rather than a factor e−m/T as is the case in thermome- chanical equilibrium (see, for example, the addendum in p,a X sp,a Z1 := Z1 [A.8]). As relativistic nuclear collisions are just on the bor- sp,a derline between those two limiting cases, it is important VT 3 n o when considering the yield of strange particles to under- = 2W (x ) + 2e±µ/T W (x ) + 3W (x ) , stand the transition between them. We will now show how 2π2 K Λ Σ one can describe these different cases in a unified statisti- (A75) cal description [A.22]. for particles (+µ) and antiparticles (−µ), where W (x) = 2 As we have already implicitly discussed [see Eq. (A12)], x K2(x), xi = mi/T , and all kaons and hyperons are the logarithm of the grand partition function Z is a sum counted. In the quark phase, we have over all different particle configurations, e.g., expressed 3 with the help of the mass spectrum. Hence, we can now p,a VT h ±µ/3T i Z1,q = 6 e W (xs) , (A76) concentrate in particular on that part of ln Z which is 2π2 exclusively associated with the strangeness. with T xs = ms ∼ 280 MeV. We note in passing that the As the temperatures of interest to us and which allow baryon chemical potential cancels out in y of Eq. (A74) appreciable strangeness production are at the same time when Eq. (A76) is inserted in the quark phase [compare high enough to prevent the strange particles from being with Eq. (A68)]. thermodynamically degenerate, we can restrict ourselves By differentiating ln Zs of Eq. (A73) with respect to again to the discussion of Boltzmann statistics only. f, we find the strangeness number present at given T and The contribution to Z of a state with k strange parti- V : cles is k √ 1 X s ∂ I1( 4y)√ Zk = Z (T,V ) , (A71) hni = f ln Z = √ y . (A77) k! I s s s ∂f f=1 I0( 4y)

4 Σ0 decays into Λ by emitting a photon and is always For large y, that is, at given T for large volume V , we √ −m/T counted within the Λ abundance. ﬁnd hnis = y ∼ e , as expected. For small y, we ﬁnd 72 Johann Rafelski: Melting Hadrons, Boiling Quarks

larger hadronic clusters, while in the second domain, individual hadrons dissolve into one large cluster consisting of hadronic constituents, viz., the quark-gluon plasma. In order to obtain a theoretical description of both phases, we have used some ‘common’ knowledge and plausible interpretations of currently available experimental observations. In particular, in the case of hadronic gas, we have completely abandoned a more conventional La- grangian approach in favour of a semi-phenomenological statistical bootstrap model of hadronic matter that incor- porates those properties of hadronic interaction that are, in our opinion, most important in nuclear collisions. In particular, the attractive interactions are included through the rich, exponentially growing hadronic mass spectrum τ(m2, b), while the introduction of the finite volume of each hadron is responsible for an effective short- range repulsion. Aside from these manifestations of strong interactions, we only satisfy the usual conservation laws of energy, momentum, and baryon number. We neglect quantum statistics since quantitative study has revealed Fig. A8. The quenching factor for strangeness production as 3 that this is allowed above T ≈ 50 MeV. But we allow a function of the active volume V/Vh, where Vh = 4π/3 fm particle production, which introduces a quantum physical aspect into the otherwise ‘classical’ theory of Boltzmann particles. hni = y ∼ e−2m/T . In Fig. A8, we show the dependence of s Our approach leads us to the equations of state of the quenching factor I1/I0 = η as a function of the volume 3 hadronic matter which reflect what we have included in V measured in units of Vh = 4π/3 fm for a typical set of parameters: T = 150, µ = 550 MeV (hadronic gas phase). our considerations. It is the quantitative nature of our work that allows a detailed comparison with experiment. The following observations follow from inspection of Fig. A8: This work has just begun and it is too early to say if the 1. The strangeness yield is a qualitative measure of the features of strong interactions that we have chosen to in- hadronic volume in thermodynamic equilibrium. clude in our considerations are the most relevant ones. It 2. Total strangeness yield is not an indicator of the phase is important to observe that the currently predicted pion p transition to quark plasma, as the enhancement ( ηq/η and nucleon mean transverse momenta and temperatures = 1.25) in yield can be reinterpreted as being due to a show the required substantial rise (see Fig. A7) as re- change in hadronic volume. quired by the experimental results available at Ek,lab/A = 3. We can expect that, in nuclear collisions, the active 2 GeV (BEVALAC, see [A.18]) and at 1000 GeV (ISR, see volume will be sufficiently large to allow the strange- [A.17]). Further comparisons involving, in particular, par- ness yield to correspond to that of ‘infinite’ volume for ticle multiplicities and strangeness production are under reactions triggered on ‘central collisions’. Hence, e.g., consideration. Λ production rate will significantly exceed that found We also mention the internal theoretical consistency of in pp collisions. our two-fold approach. With the proper interpretation, the statistical bootstrap leads us, in a straightforward fashion, Our conclusions about the significance of Λ as an indicator to the postulate of a phase transition to the quark-gluon of the phase transition to quark plasma remain valid as plasma. This second phase is treated by a quite different the production of Λ in the hadronic gas phase will only be method. In addition to the standard Lagrangian quantum possible in the very first stages of the nuclear collisions, if field theory of weakly interacting particles at finite tem- sufficient center of mass energy is available. perature and density, we also introduce the phenomenological vacuum pressure and energy density B. Perhaps the most interesting aspect of our work is the A.7 Summary realization that the transition to quark matter will occur at much lower baryon density for highly excited hadro- Our aim has been to obtain a description of hadronic mat- nic matter than for matter in the ground state (T = 0). ter valid for high internal excitations. By postulating the The precise baryon density of the phase transition depends kinetic and chemical equilibrium, we have been able to somewhat on the bag constant, but we estimate it to be at develop a thermodynamic description valid for high tem- about 2–4ν0 at T = 150 MeV. The detailed study of the peratures and different chemical compositions. In our work different aspects of this phase transition, as well as of pos- we have found two physically different domains: firstly, the sible characteristic signatures of quark matter, must still hadronic gas phase, in which individual hadrons can ex- be carried out. We have given here only a very preliminary ist as separate entities, but are sometimes combined into report on the status of our present understanding. Johann Rafelski: Melting Hadrons, Boiling Quarks 73

We believe that the occurrence of the quark plasma [A.11] H. Grote, R. Hagedorn, J. Ranft: Atlas of Particle Pro- phase is observable and we have proposed therefore a mea- duction Spectra, CERN 1970 surement of the Λ¯/p relative yield between 2 and 10 GeV/N [A.12] J. Rafelski, H.Th. Elze, R. Hagedorn: CERN preprint kinetic energies. In the quark plasma phase, we expect a TH 2912 (August 1980) in Proceedings of the Fifth significant enhancement of Λ¯ production which will most European Symposium on Nucleon–Antinucleon Interac- likely be visible in the Λ¯/p relative rate. tions, Bressanone, Italy, June 1980, ed. by M. Cresti, CLEUP publishers, Padova, p. 357 [A.13] R. Hagedorn, J. Rafelski: Phys. Lett. B 97, (1980) 136 Appendix A 1980 Acknowledgements [A.14] F. Karsch, H. Satz: Phys. Lett. 21, (1980) 1168. The ‘solidification’ of hadronic matter, a purely geometric Many fruitful discussions with the GSI/LBL Relativistic effect, discussed in this reference should not be confused with the transition to the quark plasma phase discussed Heavy Ion group stimulated the ideas presented here. I here would like to thank R. Bock and R. Stock for their hospi- [A.15] S. Frautschi: Phys. Rev. D 3, (1971) 2821 tality at GSI during this workshop. As emphasized before, [A.16] R. Hagedorn, I. Montvay: Nucl. Phys. B 59, (1973) 45 this work was performed in collaboration with R. Hage- [A.17] G. Giacomelli, M. Jacob: Phys. Rep. 55, (1979) 1 dorn. [A.18] A. Sandoval, R. Stock, H.E. Stelzer, R.E. Renfordt, J.W. Harris, J. Brannigan, J.B. Geaga, L.J. Rosenberg, L.S. Schroeder, K.L. Wolf:Phys. Rev. Lett. 45, (1980) References 874 [A.19] R. Hagedorn, J. Ranft: Suppl. Nuovo Cimento 6, (1968) [A.1] R. Hagedorn, J. Rafelski: Manuscript in preparation for 169; Physics Reports5 W.D. Myers: Nucl. Phys. A 296, (1978) 177; [A.2] The many-body theory for QCD at finite temperatures J. Gosset, J.L. Kapusta, G.D. Westfall: Phys. Rev. C has been discussed in: 18, (1978) 844; B.A. Freedman, L.D. McLerran: Phys. Rev. D 16, R. Malfliet: Phys. Rev. Lett. 44, (1980) 864 (1977) 1169; [A.20] The possible formation of quark-gluon plasma in nu- P.D. Morley, M.B. Kislinger: Phys. Rep. 51, (1979) 63; clear collisions was first discussed quantitatively by J.I. Kapusta: Nucl. Phys. B 148, (1979 461); S.A. Chin: Phys. Lett. B 78, (1978 552); see also E.V. Shuryak: Phys. Lett. B 81, 65 (1979); ibid. Phys. N. Cabibbo, G. Parisi: Phys. Lett. B 59, (1975) 67 Rep. 61, (1980) 71; [A.21] M. Abramowitz, I.A. Stegun (Eds.): Handbook of Math- O.K. Kalashnikov, V.V. Klimov: Phys. Lett. B 88, ematical Functions, NBS (1964). (1979) 328 [A.22] J. Rafelski, M. Danos: Phys. Lett. B 97, (1980) 279 [A.3] B. Touschek: Nuovo Cimento B 58, (1968) 295 [A.4] W. Marciano, H. Pagels: Phys. Rep. 36, (1978) 137 [A.5] K. Johnson: The MIT bag model. Acta Phys. Polon. B 6, (1975) 865 [A.6] A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, V.F. Weisskopf: Phys. Rev. D 9, (1974) 3471 [A.7] R. Hagedorn: Suppl. Nuovo Cimento 3, (1965) 147 [A.8] R. Hagedorn: In Lectures on the Thermodynamics of Strong Interactions, CERN Yellow Report 71-12 (1971) [A.9] R. Hagedorn, I. Montvay, J. Rafelski: Lecture at Erice Workshop, Hadronic Matter at Extreme Energy Den- sity, ed. by N. Cabibbo and L. Sertorio, Plenum Press (New York 1980), p. 49 [A.10] R. Hagedorn: How to Deal with Relativistic Heavy Ion Collisions, Invited lecture at Quark Matter 1: Workshop on Future Relativistic Heavy Ion Experiments at the Gesellschaft für Schwerionenforschung (GSI), Darm- stadt, Germany, 7-10 October 1980; circulated in the GSI81-6 Orange Report, pp. 282-324, R. Bock and R. Stock, editors., reprinted in Chapter 26 of: J. Rafelski, (ed.) Melting Hadrons, Boiling Quarks: From Hagedorn temperature to ultra-relativistic heavy-ion collisions at CERN; with a tribute to Rolf Hagedorn (Springer, Hei- delberg 2015) pp 1-444

5 This manuscript was never completed – see also: J. Rafel- ski, (ed.), Melting Hadrons, Boiling Quarks: From Hagedorn temperature to ultra-relativistic heavy-ion collisions at CERN; with a tribute to Rolf Hagedorn (Springer, Heidelberg 2015) pp 1-444 74 Johann Rafelski: Melting Hadrons, Boiling Quarks

B Strangeness and Phase Changes in Hot i.e., nonzero temperature in reaching the transition to the Hadronic Matter – 1983 quark-gluon plasma: to obtain a high particle density, instead of only compressing the matter (which as it turns Johann Rafelski1 out is quite difficult), we also heat it up; many pions are generated in a collision, allowing the transition to occur From: Sixth High Energy Heavy Ion Study2, at moderate, even vanishing baryon density [B.3]. Printed in: LBL-16281 pp. 489-510; Consider, as an illustration of what is happening, the Also: report number UC-34C; DOE CONF-830675; p, V diagram shown in Fig. B1. Here we distinguish three Also: preprint CERN-TH-3685 available at domains. The hadronic gas region is approximately a Boltz- https://cds.cern.ch/record/147343/files/198311019.pdf mann gas where the pressure rises with reduction of the volume. When the internal excitation rises, the individual hadrons begin to cluster. This reduces the increase in the Boltzmann pressure, since a smaller number of particles ABSTRACT: Two phases of hot hadronic matter are de- exercises a smaller pressure. In a complete description of scribed with emphasis put on their distinction. Here the role the different phases, we have to allow for a coexistence of of strange particles as a characteristic observable of the quark- hadrons with the plasma state in the sense that the in- gluon plasma phase is particularly explored. ternal degrees of freedom of each cluster, i.e., quarks and gluons, contribute to the total pressure even before the dissolution of individual hadrons. This does indeed become B.1 Phase transition or perhaps transformation: necessary when the clustering overtakes the compressive Hadronic gas and the quark-gluon plasma effects and the hadronic gas pressure falls to zero as V reaches the proper volume of hadronic matter. At this I explore here consequences of the hypothesis that the point the pressure rises again very quickly, since in the energy available in the collision of two relativistic heavy absence of individual hadrons, we now compress only the nuclei, at least in part of the system, is equally divided hadronic constituents. By performing the Maxwell con- among the accessible degrees of freedom. This means that struction between volumes V1 and V2, we can in part ac- there exists a domain in space in which, in a suitable count for the complex process of hadronic compressibility Lorentz frame, the energy of the longitudinal motion has alluded to above. been largely transformed to transverse degrees of freedom. As this discussion shows, and detailed investigations The physical variables characterizing such a ‘fireball’ are confirm [B.4], we cannot escape the conjecture of a first energy density, baryon number density, and total volume. order phase transition in our approach. This conjecture of The basic question concerns the internal structure of the [B.1]item (g) has been criticized, and only more recent lat- fireball. It can consist either of individual hadrons, or in- tice gauge theory calculations have led to the widespread stead, of quarks and gluons in a new physical phase, the acceptance of this phenomenon, provided that an inter- plasma, in which they are deconfined and can move freely nal SU(3) (color) symmetry is used – SU(2) internal sym- over the volume of the fireball. It appears that the phase metry leads to a second order phase transition [B.1]item transition from the hadronic gas phase to the quark-gluon (i). It is difficult to assess how such hypothetical changes plasma is controlled mainly by the energy density of the in actual internal particle symmetry would influence phe- fireball. Several estimates [B.1] lead to 0.6–1 GeV/fm3 for nomenological descriptions based on an observed picture the critical energy density, to be compared with nuclear of nature. For example, it is difficult to argue that, were matter 0.16 GeV/fm3. the color symmetry SU(2) and not SU(3), we would still We first recall that the unhandy extensive variables, observe the resonance dominance of hadronic spectra and viz., energy, baryon number, etc., are replaced by intensive quantities. To wit, the temperature T is a measure of energy per degree of freedom; the baryon chemical potential µ controls the mean baryon density. The statistical quantities such as entropy (= measure of the number of available states), pressure, heat capacity, etc., will also be functions of T and µ, and will have to be determined. The theoretical techniques required for the description of the two quite different phases, viz., the hadronic gas and the quark-gluon plasma, must allow for the formulation of numerous hadronic resonances on the one side [B.2], which then at sufficiently high energy density dissolve into the state consisting of their constituents. At this point, we must appreciate the importance and help by a finite,

1 Present address: Depaartment of Physics, The University Fig. B1. p–V diagram for the gas–plasma ﬁrst order tran- of Arizona, Tucson, AZ 85721, USA sition, with the dotted curve indicating a model-dependent, 2 Invited lecture, Berkeley, 28 June – 1 July 1983 unstable domain between overheated and undercooled phases. Johann Rafelski: Melting Hadrons, Boiling Quarks 75

Table 1. Phase transition of hot hadronic matter in theoretical physics

Object −→ Observational hypothesis −→ Theoretical consequence Nature −→ Internal SU(3) symmetry −→ First order phase transition (on a lattice)

Nature −→ Bootstrap =b resonance −→ First order phase transition dominance of hadronic in a phenomenological interactions bootstrap approach ? −→ Internal SU(2) symmetry −→ Second order phase transition (on a lattice) could therefore use the bootstrap model. All present un- made in the absence of experimental guidance. A careful derstanding of phases of hadronic matter is based on ap- study of the hadronization process most certainly remains proximate models, which requires that Table 1 be read to be performed. from left to right. In closing this section, let me emphasize that the ques- I believe that the description of hadrons in terms of tion whether the transition hadronic gas ←→ quark-gluon bound quark states on the one hand, and the statisti- plasma is a phase transition (i.e., discontinuous) or contin- cal bootstrap for hadrons on the other hand, have many uous phase transformation will probably only be answered common properties and are quite complementary. Both in actual experimental work; as all theoretical approaches the statistical bootstrap and the bag model of quarks are suffer from approximations unknown in their effect. For based on quite equivalent phenomenological observations. example, in lattice gauge computer calculations, we es- While it would be most interesting to derive the phe- tablish the properties of the lattice and not those of the nomenological models quantitatively from the accepted continuous space in which we live. fundamental basis – the Lagrangian quantum field the- The remainder of this report is therefore devoted to ory of a non-Abelian SU(3) ‘glue’ gauge field coupled to the study of strange particles in different nuclear phases colored quarks – we will have to content ourselves in this and their relevance to the observation of the quark-gluon report with a qualitative understanding only. Already this plasma. will allow us to study the properties of hadronic matter in both aggregate states: the hadronic gas and the state in which individual hadrons have dissolved into the plasma B.2 Strange particles in hot nuclear gas consisting of quarks and of the gauge field quanta, the gluons. My intention in this section is to establish quantitatively It is interesting to follow the path taken by an isolated the different channels in which the strangeness, however quark-gluon plasma fireball in the µ, T plane, or equiv- created in nuclear collisions, will be found. In our follow- alently in the ν, T plane. Several cases are depicted in ing analysis (see Ref.[B.8]) a tacit assumption is made Fig. B2. In the Big Bang expansion, the cooling shown by that the hadronic gas phase is practically a superposition the dashed line occurs in a Universe in which most of the of an infinity of different hadronic gases, and all informa- energy is in the radiation. Hence, the baryon density ν is tion about the interaction is hidden in the mass spectrum quite small. In normal stellar collapse leading to cold neutron stars, we follow the dash-dotted line parallel to the ν axis. The compression is accompanied by little heating. In contrast, in nuclear collisions, almost the entire ν, T plane can be explored by varying the parameters of the colliding nuclei. We show an example by the full line, and we show only the path corresponding to the cooling of the plasma, i.e., the part of the time evolution after the termination of the nuclear collision, assuming a plasma formation. The figure reflects the circumstance that, in the beginning of the cooling phase, i.e., for 1–1.5×10−23 s, the cooling happens almost exclusively by the mechanism of pion radiation [B.5]. In typical circumstances, about half of the available energy has been radiated away before the expansion, which brings the surface temperature close to the temperature of the transition to the hadronic phase. Hence a possible, perhaps even likely, scenario is that in which the freezing out and the expansion happen simul- Fig. B2. Paths taken in the ν, T plane by different physical taneously. These highly speculative remarks are obviously events. 76 Johann Rafelski: Melting Hadrons, Boiling Quarks

τ(m2, b) which describes the number of hadrons of baryon dition, we get3 number b in a mass interval dm2 and volume V ∼ m. 1/2 When considering strangeness-carrying particles, all we W (x ) + λ−1W (x ) + 3W (x ) K B Λ Σ then need to include is the influence of the non-strange λs = λq ≡ λqF, hadrons on the baryon chemical potential established by W (xK) + λB W (xΛ) + 3W (xΣ) the non-strange particles. (B.2.5) The total partition function is approximately multi- a result contrary to intuition: λs 6= 1 for a gas with to- plicative in these degrees of freedom: tal hsi = 0. We notice a strong dependence of F on the −1 baryon number. For large µ, the term with λB will tend to zero and the term with λ will dominate the expression ln Z = ln Znon-strange + ln Zstrange . (B.2.1) B for λs and F . As a consequence, the particles with fugacity λs and strangeness S = −1 (note that by convention For our purposes, i.e., in order to determine the parti- strange quarks s carry S = −1, while strange antiquarkss ¯ cle abundances, it is sufficient to list the strange particles carry S = 1) are suppressed by a factor F which is always separately, and we find smaller than unity. Conversely, the production of particles which carry the strangeness S = +1 will be favored strange n −1 −1 −1 ln Z (T, V, λs, λq) = C 2W (xK)(λsλq + λs λq) by F . This is a consequence of the presence of nuclear matter: for µ = 0, we find F = 1. o + 2W (x ) + 3W (x ) (λ λ2 + λ−1λ−2) , In nuclear collisions, the mutual chemical equilibrium, Λ Σ s q s q that is, a proper distribution of strangeness among the (B.2.2) strange hadrons, will most likely be achieved. By studying where the relative yields, we can exploit this fact and eliminate m 2 m W (x ) = i K i . (B.2.3) the absolute normalization C [see Eq. (B.2.2)] from our i T 2 T considerations. We recall that the value of C is uncertain for several reasons: We have C = VT 3/2π2 for a fully equilibrated state. How- ever, strangeness-creating (x → s +s ¯) processes in hot i V is unknown. hadronic gas may be too slow (see below) and the to- ii C is strongly (t, r)-dependent, through the space-time tal abundance of strange particles may fall short of this dependence of T . 3 2 value of C expected in absolute strangeness chemical equi- iii Most importantly, the value C = VT /2π assumes librium. On the other hand, strangeness exchange cross- absolute chemical equilibrium, which is not achieved sections are very large (e.g., the K−p cross-section is ∼ owing to the shortness of the collision. 100 mb in the momentum range of interest), and there- Indeed, we have [see Eq. (B.4.3) for in plasma strangeness fore any momentarily available strangeness will always be formation and further details and solutions] distributed among all particles in Eq. (B.2.2) according to 1/3 2 the values of the fugacities λq = λ and λs. Hence we dC C(t) B = A 1 − , (B.2.6) can speak of a relative strangeness chemical equilibrium. dt H C(∞)2 We neglected to write down quantum statistics corrections as well as the multistrange particles Ξ and Ω−, and the time constant for strangeness production in nu- as our considerations remain valid in this simple approxi- clear matter can be estimated to be [B.7] mation [B.6]. Interactions are effectively included through −21 explicit reference to the baryon number content of the τH = C(∞)/2AH ∼ 10 s . strange particles, as just discussed. Non-strange hadrons influence the strange faction by establishing the value of Thus C does not reach C(∞) in plasmaless nuclear col- λq at the given temperature and baryon density. lisions. If the plasma state is formed, then the relevant The fugacities λs and λq as introduced here control the C > C(∞) (since strangeness yield in plasma is above strangeness and the baryon number, respectively. While λs strangeness yield in hadron gas (see below). counts the strange quark content, the up and down quark Now, why should we expect relative strangeness equi- content is counted by λ = λ1/3. librium to be reached faster than absolute strangeness q B equilibrium [B.8]? Consider the strangeness exchange in- Using the partition function Eq. (B.2.2), we calculate teraction for given µ, T , and V the mean strangeness by evaluating K−p −→ Λπ0 (B.2.7)

∂ strange hns − ns¯i = λs ln Z (T, V, λs, λq) , (B.2.4) ∂λs which is the diﬀerence between strange and antistrange components. This expression must be equal to zero due to the fact that the strangeness is a conserved quantum 3 Notation has been changed γ → F in order to avoid con- number with respect to strong interactions. From this confusion with phase space occupancy γ. Johann Rafelski: Melting Hadrons, Boiling Quarks 77 which has a cross-section of about 10 mb at low energies, while the ss¯ ‘strangeness creating’ associate production

pp −→ pΛK+ (B.2.8)

has a cross-section of less than 0.06 mb, i.e., 150 times smaller. Since the latter reaction is somewhat disfavored by phase space, consider further the reaction −2 Fig. B3. The ratio hnK+ i/hnK− i ≡ F as a function of the πp −→ YK (B.2.9) baryon chemical potential µ, for T = 100, (20), 160 MeV. The lines cross where µ = mY −mK; mY is the mean hyperon mass.

From the above equations, we can derive several very instructive conclusions. In Fig. B3 we show the ratio

−2 where Y is any hyperon (strange baryon). This has a cross- hnK+ i/hnK− i = F section of less than 1 mb, still 10 times weaker than one of the s-exchange channels in Eq. (B.2.7). Consequently, I as a function of the baryon chemical potential µ for sev- expect the relative strangeness equilibration time to be eral temperatures that can be expected and which are about ten times shorter than the absolute strangeness seen experimentally. We see that this particular ratio is equilibration time, namely 10−23 s, in hadronic matter a good measure of the baryon chemical potential in the of about twice nuclear density. hadronic gas phase, provided that the temperatures are We now compute the relative strangeness abundances approximately known. The mechanism for this process is expected from nuclear collisions. Using Eq. (B.2.5), we find as follows: the strangeness exchange reaction of Eq. (B.2.7) from Eq. (B.2.2) the grand canonical partition sum for tilts to the left (K−) or to the right (abundance F ∼ K+), zero average strangeness: depending on the value of the baryon chemical potential. In Fig. B4 the long dashed line shows the upper limit h strange −1 for the abundance of Λ as measured in terms of Λ abun- ln Z0 C 2W (xK) F λK + F λK dances. Clearly visible is the substantial relative suppres- −1 −1 + 2W (xΛ) F λBλΛ + F λB λΛ (B.2.10) sion of Λ, in part caused by the baryon chemical potential i factor of Eq. (B.2.14), but also by the strangeness chem- + 6W (x )F λ λ + F −1λ−1λ , Σ B Σ B Σ istry (factor F 2), as in K+K− above. Indeed, the actual relative number of Λ will be even smaller, since Λ are where, in order to distinguish different hadrons, dummy in relative chemical equilibrium and Λ in hadron gas are fugacities λi, i = K, K, Λ, Λ, Σ, Σ have been written. The not: the reaction K+p → Λπ0, analogue to Eq. (B.2.7), strange particle multiplicities then follow from will be suppressed by low p abundance. Also indicated in Fig. B4 by shading is a rough estimate for the produc- ∂ Λ strange tion in the plasma phase, which suggests that anomalous hnii = λi ln Z0 . (B.2.11) ∂λi λi=1 Λ abundance may be an interesting feature of highly energetic nuclear collisions [B.16], for further discussion see Explicitly, we find (notice that the power of F follows the Section B.5 below. s-quark content):

∓ hn ± i = CF W (x ) , (B.2.12) K K B.3 Quark-Gluon Plasma

+1 +µB/T hn 0 i = CF W (x 0 )e , (B.2.13) Λ/Σ Λ/Σ From the study of hadronic spectra, as well as from hadron– −1 −µB/T hadron and hadron–lepton interactions, there has emerged hn 0 i = CF W (x 0 )e . (B.2.14) Λ/Σ Λ/Σ convincing evidence for the description of hadronic struc- In Eq. (B.2.14) we have indicated that the multiplicity ture in terms of quarks [B.9]. For many purposes it is of antihyperons can only be built up if antibaryons are entirely satisfactory to consider baryons as bound states present according to their (small) phase space. This still of three fractionally charged particles, while mesons are seems an unlikely proposition, and the statistical approach quark–antiquark bound states. The Lagrangian of quarks may be viewed as providing an upper limit on their mul- and gluons is very similar to that of electrons and pho- tiplicity. tons, except for the required summations over flavour and 78 Johann Rafelski: Melting Hadrons, Boiling Quarks color: As the u and d quarks are almost massless inside a bag, they can be produced in pairs, and at moderate in- 1 µν L = ψ F · (p − gA) − m ψ − Fµν F . (B.3.1) ternal excitations, i.e., temperatures, many qq¯ pairs will 4 be present. Similarly, ss¯ pairs will also be produced. We The flavour-dependent masses m of the quarks are small. will return to this point at length below. Furthermore, For u, d flavours, one estimates mu,d ∼ 5–20 MeV. The real gluons can be excited and will be included here in strange quark mass is usually chosen at about 150 MeV our considerations. [B.10]. The essential new feature of QCD, not easily visible Thus, what we are considering here is a large quark bag in Eq. (B.3.1), is the non-linearity of the field strength F with substantial, equilibrated internal excitation, in which in terms of the potentials A. This leads to an attractive the interactions can be handled (hopefully) perturbatively. glue–glue interaction in select channels and, as is believed, In the large volume limit, which as can be shown is valid requires an improved (non-perturbative) vacuum state in for baryon number b & 10, we simply have for the light which this interaction is partially diagonalized, providing quarks the partition function of a Fermi gas which, for for a possible perturbative approach. practically massless u and d quarks can be given analyti- The energy density of the perturbative vacuum state, cally (see ref.[B.1]item (b) and [B.12]), even including the 2 defined with respect to the true vacuum state, is by defi- effects of interactions through first order in αs = g /4π : nition a positive quantity, denoted by B. This notion has been introduced originally in the MIT bag model [B.11], gV 2α 1 π2 ln Z (β, µ) = β−3 1 − s (µβ)4 + (µβ)2 logically, e.g., from a fit to the hadronic spectrum [B.11], q 6π2 π 4 2 which gives 50 α 7π4 + 1 − s . 4 B = (140–210) MeV = (50–250) MeV/fm3 . (B.3.2) 21 π 60 (B.3.3) The central assumption of the quark bag approach is that, Similarly, the glue is a Bose gas: inside a hadron where quarks are found, the true vacuum 8π2 15 α structure is displaced or destroyed. One can turn this point ln Z (β, λ) = V β−3 1 − s , (B.3.4) around: quarks can only propagate in domains of space g 45 4 π in which the true vacuum is absent. This statement is a reformulation of the quark confinement problem. Now the while the term associated with the difference to the true remaining difficult problem is to show the incompatibility vacuum, the bag term, is of quarks with the true vacuum structure. Examples of such behavior in ordinary physics are easily found; e.g., ln Zbag = −BV β . (B.3.5) a light wave is reflected from a mirror surface, magnetic field lines are expelled from superconductors, etc. In this It leads to the required positive energy density B within picture of hadronic structure and quark confinement, all the volume occupied by the colored quarks and gluons and colorless assemblies of quarks, antiquarks, and gluons can to a negative pressure on the surface of this region. At this form stationary states, called a quark bag. In particular, stage, this term is entirely phenomenological, as discussed all higher combinations of the three-quark baryons (qqq) above. The equations of state for the quark-gluon plasma and quark–antiquark mesons (qq¯) form a permitted state. are easily obtained by differentiating

ln Z = ln Zq + ln Zg + ln Zvac , (B.3.6)

with respect to β, µ, and V . An assembly of quarks in a bag will assume a geometric shape and size such as to make the total energy E(V, b, S) as small as possible at fixed given baryon number and fixed total entropy S. Instead of just considering one bag we may, in order to be able to use the methods of statistical physics, use the microcanonical ensemble. We find from the first law of thermodynamics, viz.,

dE = −P dV + T dS + µdb , (B.3.7)

that ∂E(V, b, S) P = − . (B.3.8) ∂V We observe that the stable configuration of a single bag, Fig. B4. Relative abundance of Λ/Λ. The actual yield from viz., ∂E/∂V = 0, corresponds to the configuration with the hadronic gas limit may still be 10–100 times smaller than vanishing pressure P in the microcanonical ensemble. Rather the statistical value shown. than work in the microcanonical ensemble with fixed b and Johann Rafelski: Melting Hadrons, Boiling Quarks 79

S, we exploit the advantages of the grand canonical ensemble and consider P as a function of µ and T : ∂ P = − T ln Z(µ, T, V ) , (B.3.9) ∂V with the result 1 P = (ε − 4B) , (B.3.10) 3 where ε is the energy density: 6 2α 1 µ4 1 µ2 ε = 1 − s + (πT )2 π2 π 4 3 2 3 50 α 7 + 1 − s (πT )4 21 π 60

15 α 8 + 1 − s (πT )4 + B . (B.3.11) 4 π 15π2 Fig. B5. Lowest order QCD diagrams for ss¯ production: In Eq. (B.3.10), we have used the relativistic relation be- a,b,c) gg → ss¯, and d) qq¯ → ss¯. tween the quark and gluon energy density and pressure: 1 1 P = ε ,P = ε . (B.3.12) q 3 q g 3 g B.4 Strange Quarks in Plasma From Eq. (B.3.10), it follows that, when the pressure vanishes in a static configuration, the energy density is 4B, In lowest order in perturbative QCD, ss¯ quark pairs can independently of the values of µ and T which fix the line be created by gluon fusion processes, Fig. B5a,b,c; and by P = 0. We note that, in both quarks and gluons, the inter- annihilation of light quark-antiquark pairs, see Fig. B5d. action conspires to reduce the effective available number The averaged total cross-sections for these processes were calculated by Brian Combridge [B.13]. of degrees of freedom. At αs = 0, µ = 0, we find the handy relation Given the averaged cross-sections, it is easy to calcu- 4 late the rate of events per unit time, summed over all final T GeV and averaged over initial states: εq + εg = . (B.3.13) 160 MeV fm3 Z Z 3 3 dN X d k1d k2 It is important to appreciate how much entropy must be = d3x ρ (k , x)ρ (k , x) dt (2π)3|k |(2π)3|k | i,1 1 i,2 2 created to reach the plasma state. From Eq. (B.3.6), we i 1 2 find for the entropy density S and the baryon density ν: Z ∞ 2 µ 2 × ds δs − (k + k ) k k σ(s) . (B.4.1) 2 2αs µ 14 50 αs 3 1 2 1 2µ S = 1 − πT + 1 − (πT ) 4M 2 π π 3 15π 21 π 32 15 α The factor k1 · k2/|k1||k2| is the relative velocity for mass- + 1 − s (πT )3 , 45π 4 π less gluons or light quarks, and we have introduced a (B.3.14) dummy integration over s in order to facilitate the calculations. The phase space densities ρi(k, x) can be approximated by assuming the x-independence of temperature 2 2α µ3 µ T (x) and the chemical potential µ(x), in the so-called lo- ν = 1 − s + (πT )2 , (B.3.15) 3π2 π 3 3 cal statistical equilibrium. Since ρ then only depends on the absolute value of k in the rest frame of the equilibrated which leads for µ/3 = µq < πT to the following expres- plasma, we can easily carry out the relevant integrals and sions for the entropy per baryon [including the gluonic obtain for the dominant process of the gluon fusion re- entropy second T 3 term in Eq. (B.3.14)]: action Fig. B5a,b,c the invariant rate per unit time and S 37 T T ∼µ volume [B.14]: ≈ π2 ←→q 25 ! (B.3.16) ν 15 µ q d4N 7α2 51 T A = ≈ A = s MT 3e−2M/T 1 + + ··· , As this simple estimate shows, plasma events are extremely d3xdt g 6π2 14 M entropy-rich, i.e., they contain very high particle multiplic- (B.4.2) ity. In order to estimate the particle multiplicity, one may where M is the strange quark mass4. simply divide the total entropy created in the collision by the entropy per particle for massless black body radiation, 4 In Eq. (B.4.2) a factor 2 was included to reduce the in- which is S/n = 4. This suggests that, at T ∼ µq, there are variant rate A, see Erratum: “Strangeness Production in the roughly six pions per baryon. Quark-Gluon Plasma” Johann Rafelski and Berndt Müller 80 Johann Rafelski: Melting Hadrons, Boiling Quarks

The abundance of ss¯ pairs cannot grow forever. At some point the ss¯ annihilation reaction will restrict the strange quark population. It is important to appreciate that the ss¯ pair annihilations may not proceed via the two-gluon channel, but instead occasionally through γG (photon-Gluon) final states [B.15]. The noteworthy feature of such a reaction is the production of relatively high energy γ’s at an energy of about 700–900 MeV (T = 160 MeV) stimulated by coherent glue emission. These γ’s will leave the plasma without further interactions and provide an independent confirmation of the s-abundance in the plasma. The loss term of the strangeness population is proportional to the square of the density ns of strange and antistrange quarks. With ns(∞) being the saturation density at large times, the following differential equation deter- Fig. B6. Time evolution of the strange quark to baryon mines n as a function of time [B.2]item (b) s number abundance in the plasma for various temperatures ( ) T ∼ µq = µ/3. M = 150 MeV, αs = 0.6 . dn n (t) 2 s ≈ A 1 − s . (B.4.3) dt ns(∞) There are two elements in point (1) above: firstly, stran- Thus we find geness in the quark-gluon phase is practically as abundant as the anti-light quarks u = d =q ¯, since both phase spaces tanh(t/2τ) + ns(0) ns(∞) ns(∞) have similar suppression factors: for u, d it is the baryon ns(t) = ns(∞) , τ = . 1 + ns(0) tanh(t/2τ) 2A chemical potential, for s, s¯ the mass (M ≈ µq): ns(∞) (B.4.4) s s Z d3p 1 where = = 6 √ , (B.4.7a) n (∞) V V (2π)3 p2+M 2/T τ = s . (B.4.5) e + 1 2A q Z d3p 1 = 6 . (B.4.7b) The relaxation time τ of the strange quark density in V (2π)3 e|p|/T +µq /T + 1 Eq. (B.4.5) is obtained using the saturated phase space in Eq. (B.4.5). We have [B.14] Note that the chemical potential of quarks suppresses the q¯ density. This phenomenon reflects on the chemical equi- π 1/2 9M 1/2 99 T −1 librium between qq¯ and the presence of a light quark den- τ ≈ τ = T −3/2eM/T 1 + + ··· . g 2 sity associated with the net baryon number. Secondly, 2 7αs 56 M (B.4.6) strangeness in the plasma phase is more abundant than For αs ∼ 0.6 and M ∼ T , we find from Eq. (B.4.6) that in the hadronic gas phase (even if the latter phase space τ ∼ 4×10−23 s. τ falls off rapidly with increasing temper- is saturated) when compared at the same temperature and ature. Figure B6 shows the approach of ns(t), normalized baryon chemical potential in the phase transition region. with baryon density, to the fully saturated phase space The rationale for the comparison at fixed thermodynamic as a function of time. For M . T = 160 MeV, the satu- variables, rather than at fixed values of microcanonical ration requires 4 × 10−23 s, while for T = 200 MeV, we variables such as energy density and baryon density, is need 2 × 10−23 s, corresponding to the anticipated life- outlined in the next section. I record here only that the time of the plasma. But it is important to observe that, abundance of strangeness in the plasma is well above that even at T = 120 MeV, the phase space is half-saturated in the hadronic gas phase space (by factors 1–6) and the in 2 × 10−23 s, a point to which we will return below. two become equal only when the baryon chemical poten- Another remarkable fact is the high abundance of stran- tial µ is so large that abundant production of hyperons geness relative to baryon number seen in Fig. B6 – here, becomes possible. This requires a hadronic phase at an 3 baryon number was computed assuming T ∼ µq = µ/3 energy density of 5–10 GeV/fm . [see Eq. (B.3.15)]. These two facts, namely: 1. high relative strangeness abundance in plasma, B.5 How to Discover the Quark–Gluon Plasma 2. practical saturation of available phase space, have led me to suggest the observation of strangeness as Here only the role of the strange particles in the antici- a possible signal of quark-gluon plasma [B.16]. pated discovery will be discussed. My intention is to show that, under different possible transition scenarios, charac- Phys. Rev. Lett. 56, 2334 (1986). This factor did not carry teristic anomalous strange particle patterns emerge. Ex- through to any of the following results. However, additional amples presented are intended to provide some guidance definition factors ‘2’ show up below in Eqs.B.4.4, B.4.5. to future experiments and are not presented here in order Johann Rafelski: Melting Hadrons, Boiling Quarks 81 to imply any particular preference for a reaction channel. gas phase, i.e., roughly ten times more strangeness than I begin with a discussion of the observable quantities. otherwise expected. For similar reasons, i.e., in view of the The temperature and chemical potential associated with rather long strangeness production time constants in the the hot and dense phase of nuclear collision can be con- hadronic gas phase, strangeness abundance survives prac- nected with the observed particle spectra, and, as dis- tically unscathed in this final part of the hadronization as cussed here, particle abundances. The last grand canoni- well. Facit: cal variable – the volume – can be estimated from particle if a phase transition to the plasma state has oc- interferences. Thus, it is possible to use these measured curred, then on return to the hadron phase, there variables, even if their precise values are dependent on will be most likely significantly more strange parti- a particular interpretational model, to uncover possible cles around than there would be (at this T and µ) rapid changes in a particular observable. In other words, if the hadron gas phase had never been left. instead of considering a particular√ particle multiplicity as a function of the collision energy s, I would consider it In my opinion, the simplest observable proportional to the as a function of, e.g., mean transverse momentum hp⊥i, strange particle multiplicity is the rate of V-events from which is a continuous function of the temperature (which the decay of strange baryons (e.g., Λ) and mesons (e.g., is in turn continuous across any phase transition bound- Ks) into two charged particles. Observations of this rate ary). require a visual detector, e.g., a streamer chamber. To To avoid possible misunderstanding of what I want to estimate the multiplicity of V-events, I reduce the total say, here I consider the (difficult) observation of the width strangeness created in the collision by a factor 1/3 to select of the K+ two-particle correlation function in momentum only neutral hadrons and another factor 1/2 for charged space as a function of the average K+ transverse momen- decay channels. We thus have √ + tum obtained at given s. Most of K would originate 1 hsi + hsi hbi from the plasma region, which, when it is created, is rela- hnVi ≈ hbi ∼ , (B.5.1) tively small, leading to a comparatively large width. (Here 6 hbi 15 I have assumed a first order phase transition with substan- where I have taken hsi/hbi ∼ 0.2 (see Fig. B6). Thus for tial increase in volume as matter changes from plasma events with a large baryon number participation, we can to gas.) If, however, the plasma state were not formed, expect to have several V’s per collision, which is 100–1000 K+ originating from the entire hot hadronic gas domain times above current observation for Ar-KCl collision at would contribute a relatively large volume which would be 1.8 GeV/Nuc kinetic energy [B.17]. seen; thus the width of the two-particle correlation func- Due to the highs ¯ abundance, we may further expect tion would be small. Thus, a first order phase transition an enrichment of strange antibaryon abundances [B.16]. I implies a jump in the K+ correlation width as a function of would like to emphasize heres ¯s¯q¯ states (anticascades) increasing hp i + , as determined in the same experiment, √ ⊥ K created by the accidental coagulation of twos ¯ quarks varying s. helped by a gluon → q¯ reaction. Ultimately, thes ¯s¯q¯ states From this example emerges the general strategy of becomes ¯q¯q¯, either through ans ¯ exchange reaction in the my approach: search for possible discontinuities in observ- gas phase or via a weak interaction much, much later. ables derived from discontinuous quantities (such as vol- However, half of thes ¯q¯q¯ states are then visible as Λ de- ume, particle abundances, etc.) as a function of quantities cays in a visual detector. This anomaly in the apparent measured experimentally and related to thermodynamic Λ abundance is further enhanced by relating it to the de- variables always continuous at the phase transition: tem- creased abundance of antiprotons, as described above. perature, chemical potentials, and pressure. This strategy, Unexpected behavior of the plasma–gas phase transi- of course, can only be followed if, as stated in the first tion can greatly influence the channels in which strange- sentence of this report, approximate local thermodynamic ness is found. For example, in an extremely particle-dense equilibrium is also established. plasma, the produced ss¯ pairs may stay near to each other Strangeness seems to be particularly useful for plasma – if a transition occurs without any dilution of the den- diagnosis, because its characteristic time for chemical equi- sity, then I would expect a large abundance of φ(1020) ss¯ libration is of the same order of magnitude as the expected mesons, easily detected through their partial decay mode lifetime of the plasma: τ ∼ 1–3 × 10−23 s. This means (1/4%) to a µ+µ− pair. that we are dominantly creating strangeness in the zone Contrary behavior will be recorded if the plasma is where the plasma reaches its hottest stage – freezing over cool at the phase transition, and the transition proceeds the abundance somewhat as the plasma cools down. How- slowly – major coagulation of strange quarks can then ever, the essential effect is that the strangeness abundance be expected with the formation of sss ands ¯s¯s¯ baryons in the plasma is greater, by a factor of about 30, than and in general (s)3n clusters. Carrying this even further, that expected in the hadronic gas phase at the same val- supercooled plasma may become ‘strange’ nuclear (quark) ues of µ, T . Before carrying this further, let us note that, matter [B.18]. Again, visual detectors will be extremely in order for strangeness to disappear partially during the successful here, showing substantial decay cascades of the phase transition, we must have a slow evolution, with time same heavy fragment. constants of ∼ 10−22 s. But even so, we would end up In closing this discussion, I would like to give warning with strangeness-saturated phase space in the hadronic about the pions. From the equations of state of the plasma, 82 Johann Rafelski: Melting Hadrons, Boiling Quarks we have deduced in Sect. B.3 a very high specific entropy in the GSI81-6 Orange Report, pp. 282-324, R. Bock per baryon. This entropy can only increase in the phase and R. Stock, editors., reprinted in Chapter 26 of: transition and it leads to very high pion multiplicity in nu- J. Rafelski, (ed.) Melting Hadrons, Boiling Quarks: clear collisions, probably created through pion radiation From Hagedorn temperature to ultra-relativistic heavy- from the plasma [B.5] and sequential decays. Hence by re- ion collisions at CERN; with a tribute to Rolf Hagedorn lating anything to the pion multiplicity, e.g., considering (Springer, Heidelberg 2015) pp 1-444 K/π ratios, we dilute the signal from the plasma. Further- [B.3] R. Hagedorn, J. Rafelski: Phys. Lett. B 97, (1980) 136; more, pions are not at all characteristic for the plasma; see also [B.1]item (g) they are simply indicating high entropy created in the [B.4] (a) R. Hagedorn: On a Possible Phase Transition Be- collision. However, we note that the K/π ratio can show tween Hadron Matter and Quark-Gluon Matter [CERN preprint TH 3392 (1982)], Chapter 24 in J. Rafelski, substantial deviations from values known in pp collisions (ed.) Melting Hadrons, Boiling Quarks: From Hagedorn – but the interpretations of this phenomenon will be dif- temperature to ultra-relativistic heavy-ion collisions at ficult. CERN; with a tribute to Rolf Hagedorn (Springer, Hei- It is important to appreciate that the experiments dis- delberg 2015) pp 1-444; see also: R. Hagedorn, Z. Phys. cussed above would certainly be quite complementary to C 17, (1983 265) for a later version of this manuscript the measurements utilizing electromagnetically interact- (b) R. Hagedorn, I. Montvay, J. Rafelski: Hadronic Mat- ing probes, e.g., dileptons, direct photons. Strangeness- ter at Extreme Energy Density, ed. by N. Cabibbo, based measurements have the advantage that they have Plenum Press, NY (1980); and J. Rafelski and R. Hage- much higher counting rates than those recording electro- dorn: Thermodynamics of hot nuclear matter in the sta- magnetic particles. tistical bootstrap model – 1978, Chapter 23 in J. Rafel- ski (ed.) loc. cit. [B.5] J. Rafelski, M. Danos: Pion radiation by hot quark- Appendix B 1983 Acknowledgements gluon plasma, CERN preprint TH 3607 (1983); M. Danos, J. Rafelski: Phys. Rev. D 27, (1983) 671 I would like to thank R. Hagedorn, B. Müller,and P. Koch [B.6] P. Koch: Diploma thesis, UniversitätFrankfurt (1983) for fruitful and stimulating discussions, and R. Hagedorn [B.7] A.Z. Mekjian: Nucl. Phys. B 384, (1982) 492 for a thorough criticism of this manuscript. Thie work was [B.8] P. Koch, J. Rafelski, W. Greiner: Phys. Lett. B 123, in part supported by Deutsche Dorschungsgemeinschaft. (1983) 151 [B.9] See, e.g., S. Gasiorowicz, J.L. Rosner: Hadron spectra and quarks, Am. J. Phys. 49, (1981) 954 [B.10] (a) P. Langacker, H. Pagels: Phys. Rev. D 19, (1979) References 2070 and references therein; (b) S. Narison, N. Paver, E. de Rafael, D. Treleani: Nucl. [B.1] An incomplete list of quark-gluon plasma papers in- Phys. B 212, (1983) 365 cludes: [B.11] (a) A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, V.F. (a) B.A. Freedman, L.C. McLerran: Phys. Rev. D 16, Weisskopf: Phys. Rev. D 9, (1974) 3471; (1977) 1169; (b) K. Johnson: Acta Phys. Polon. B 6, (1975) 865; (b) S.A. Chin: Phys. Lett. B 78, 552 (1978); (c) T. De Grand, R.L. Jaffe, K. Johnson, J. Kiskis: (c) P.D. Morley, M.B. Kislinger: Phys. Rep. 51, (1979) Phys. Rev. D 12, (1975) 2060 63; [B.12] H.Th. Elze, W. Greiner, J. Rafelski: J. Phys. G 6, (d) J.I. Kapusta: Nucl. Phys. B 148, (1978) 461; (1980) L149; J. Rafelski, H.Th. Elze, R. Hagedorn: Hot (e) O.K. Kalashnikov, V.V. Kilmov: Phys. Lett. B 88, hadronic and quark matter in p annihilation on nuclei, (1979 328); CERN preprint TH 2912 (1980), in Proceedings of Fifth (f) E.V. Shuryak: Phys. Lett. B 81, 65 (1979); also European Symposium on Nucleon–Antinucleon Interac- Phys. Rep. 61, (1980) 71; tions, Bressanone, 1980, CLEUP, Padua (1980) (g) J. Rafelski, R. Hagedorn: From hadron gas to quark [B.13] B.L. Combridge: Nucl. Phys. B 151, (1979) 429 matter II. In Thermodynamics of Quarks and Hadrons, [B.14] J. Rafelski, B. Müller:Phys. Rev. Lett. 48, (1982) 1066 ed. by H. Satz, North Holland, Amsterdam (1981); [B.15] G. Staadt: Diploma thesis, Universität Frankfurt (h) J. Rafelski, M. Danos: Perspectives in high en- (1983), [later published in: Phys.Rev. D 33, (1986) 66] ergy nuclear collisions, NBS-IR-83-2725; US Depart- [B.16] J. Rafelski: Extreme states of nuclear matter, Invited ment of Commerce, National Technical Information lecture at Quark Matter 1: Workshop on Future Rela- Service (NTIS) Accession Number PB83-223982 (1983) tivistic Heavy Ion Experiments at the Gesellschaft für (i) H. Satz: Phys. Rep. 88, (1982) 321 Schwerionenforschung (GSI), Darmstadt, Germany, 7- [B.2] These ideas originate in Hagedorn’s statistical boot- 10 October 1980; circulated in the GSI81-6 Orange strap theory: Report, pp. 282-324, R. Bock and R. Stock, editors., (a) R. Hagedorn: Suppl. Nuovo Cimento 3, 147 (1964); reprinted in Chapter 27 of: J. Rafelski, (ed.) loc.cit. Nuovo Cimento 6, (1968) 311; [B.17] J.W. Harris, A. Sandoval, R. Stock, et al.: Phys. Rev. (b) R. Hagedorn: How to Deal with Relativistic Heavy Lett. 47, (1981) 229 Ion Collisions Invited lecture at Quark Matter 1: Work- [B.18] S.A. Chin, A.K. Kerman: Phys. Rev. Lett. 43, (1979) shop on Future Relativistic Heavy Ion Experiments 1292 at the Gesellschaft für Schwerionenforschung (GSI), Darmstadt, Germany, 7-10 October 1980; circulated