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Eur. Phys. J. A (2015) 51: 114 DOI 10.1140/epja/i2015-15114-0

Melting hadrons, boiling quarks

Johann Rafelski 51 Eur. Phys. J. A (2015) : 114 THE EUROPEAN DOI 10.1140/epja/i2015-15114-0 PHYSICAL JOURNAL A Review

Melting hadrons, boiling quarks

Johann Rafelski1,2,a 1 CERN-PH/TH, 1211 Geneva 23, Switzerland 2 Department of Physics, The University of Arizona Tucson, Arizona, 85721, USA

Received: 12 August 2015 / Revised: 23 August 2015 Published online: 22 September 2015 c The Author(s) 2015. This article is published with open access at Springerlink.com Communicated by T.S. B´ır´o

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. The material of this review is complemented by two early and unpublished reports containing the prediction of the different forms of hadron matter, and of the formation of QGP in relativistic heavy ion collisions, including the discussion of strangeness, and in particular strange antibaryon signature of QGP.

1 Introduction This presentation connects and extends a recent retro- spective work, ref. [1]: Melting Hadrons, Boiling Quarks; The years 1964/65 saw the rise of several new ideas which From Hagedorn temperature to ultra-relativistic heavy-ion in the following 50 years shaped the discoveries in funda- collisions at CERN; with a tribute to Rolf Hagedorn. This mental subatomic physics: report complements prior summaries of our work: 1986 [2], 1991 [3],1996 [4], 2000 [5], 2002 [6], 2008 [7]. 1) The Hagedorn temperature TH; later recognized as the A report on “Melting Hadrons, Boiling Quarks and melting point of hadrons into TH” relates strongly to quantum chromodynamics (QCD), 2) Quarks as building blocks of hadrons; and the theory of quarks and gluons, the building blocks of 3) The Higgs particle and field escape from the Goldstone hadrons, and its lattice numerical solutions; QCD is the theorem, allowing the understanding of weak interac- quantum (Q) theory of color-charged (C) quark and gluon tions, the source of inertial mass of the elementary par- dynamics (D); for numerical study the space-time contin- ticles. uum is discretized on a “lattice”. Telling the story of how we learned that strong inter- The topic in this paper is Hagedorn temperature T H actions are a gauge theory involving two types of parti- and the strong interaction phenomena near to T .I H cles, quarks and gluons, and the working of the lattice present an overview of 50 years of effort with emphasis numerical method would entirely change the contents of on: this article, and be beyond the expertise of the author. I a) Hot nuclear and hadronic matter. recommend instead the book by Weinberg [8], which also shows the historical path to QCD. The best sources of b) Critical behavior near T . H the QCD relation to the topic of this article are: a) the c) Quark-gluon plasma (QGP). book by Kohsuke Yagi and Tetsuo Hatsuda [9] as well as, d) Relativistic heavy ion (RHI) collisions1. b) the now 15 year old monograph by Letessier and the au- e) The hadronization process of QGP. thor [6]. We often refer to lattice-QCD method to present f) Abundant production of strangeness flavor. QCD properties of interest in this article. There are books and many reviews on lattice implementation of gauge the- a e-mail: [email protected] ories of interacting fields, also specific to hot-lattice-QCD 1 We refer to atomic nuclei which are heavier than the α- method. At the time of writing I do not have a favorite to particle as “heavy ions”. recommend. Page 2 of 58 Eur. Phys. J. A (2015) 51: 114

Immediately in the following subsect. 1.1 the famous 5) In relativistic heavy ion collisions the kinetic en- Why? is addressed. After that I turn to answering the ergy of ions feeds the growth of quark population. These How? question in subsect. 1.2, and include a few reminis- quarks ultimately turn into final state material particles. cences about the accelerator race in subsect. 1.3. I close This means that we study experimentally the mechanisms this introduction with subsect. 1.4 where the organization leading to the conversion of the colliding ion kinetic en- and contents of this review will be explained. ergy into mass of matter. One can wonder aloud if this sheds some light on the reverse process: Is it possible to convert matter into energy in the laboratory? 1.1 What are the conceptual challenges of the The last two points show the potential of “applica- QGP/RHI collisions research program? tions” of QGP physics to change both our understanding of, and our place in the world. For the present we keep Our conviction that we achieved in laboratory experi- these questions in mind. This review will address all the ments the conditions required for melting (we can also other challenges listed under points 1), 2), and 3) above; say, dissolution) of hadrons into a soup of boiling quarks however, see also thoughts along comparable foundational and gluons became firmer in the past 15-20 years. Now we lines presented in subsects. 7.3 and 7.4. can ask, what are the “applications” of the quark-gluon plasma physics? Here is a short wish list: 1) Nucleons dominate the mass of matter by a factor 1.2 From melting hadrons to boiling quarks 1000. The mass of the three “elementary” quarks found in nucleons is about 50 times smaller than the nucleon With the hindsight of 50 years I believe that Hagedorn’s mass. Whatever compresses and keeps the quarks within effort to interpret particle multiplicity data has led to the nucleon volume is thus the source of nearly all of mass the recognition of the opportunity to study quark decon- of matter. This clarifies that the Higgs field provides the finement at high temperature. This is the topic of the mass scale to all particles that we view today as elemen- book [1] Melting Hadrons, Boiling Quarks; From Hage- tary. Therefore only a small %-sized fraction of the mass dorn temperature to ultra-relativistic heavy-ion collisions of matter originates directly in the Higgs field; see sect. 7.1 at CERN; with a tribute to Rolf Hagedorn published at for further discussion. The question: What is mass? can be Springer Open, i.e. available for free on-line. This article studied by melting hadrons into quarks in RHI collisions. should be seen as a companion addressing more recent de- 2) Quarks are kept inside hadrons by the “vacuum” velopments, and setting a contemporary context for this properties which abhor the color charge of quarks. This book. explanation of 1) means that there must be at least two How did we get here? There were two critical mile- different forms of the modern æther that we call “vac- stones: uum”: the world around us, and the holes in it that are I) The first milestone occurred in 1964–1965, when called hadrons. The question: Can we form arbitrarily big Hagedorn, working to resolve discrepancies of the statis- holes filled with almost free quarks and gluons? was and tical particle production model with the pp reaction data, remains the existential issue for laboratory study of hot produced his “distinguishable particles” insight. Due to matter made of quarks and gluons, the QGP. Aficionados a twist of history, the initial research work was archived of the lattice-QCD should take note that the presentation without publication and has only become available to a of two phases of matter in numerical simulations does not wider public recently; that is, 50 years later, see chapt. 19 answer this question as the lattice method studies the en- in [1] and ref. [10]. Hagedorn went on to interpret the ob- tire Universe, showing hadron properties at low tempera- servation he made. Within a few months, in Fall 1964, he ture, and QGP properties at high temperature. created the Statistical Bootstrap Model (SBM) [11], show- 3) We all agree that QGP was the primordial Big- ing how the large diversity of strongly interacting particles Bang stuff that filled the Universe before “normal” mat- could arise; Steven Frautschi [12] coined in 1971 the name ter formed. Thus any laboratory exploration of the QGP “Statistical Bootstrap Model”. properties solidifies our models of the Big Bang and allows II) The second milestone occurred in the late 70s and us to ask these questions: What are the properties of the early 80s when we spearheaded the development of an ex- primordial matter content of the Universe? and How does perimental program to study “melted” hadrons and the “normal” matter formation in early Universe work? “boiling” quark-gluon plasma phase of matter. The in- 4) What is flavor? In elementary particle collisions, we tense theoretical and experimental work on the thermal deal with a few, and in most cases only one, pair of newly properties of strongly interacting matter, and the confir- created 2nd, or 3rd flavor family of particles at a time. mation of a new quark-gluon plasma paradigm started A new situation arises in the QGP formed in relativistic in 1977 when the SBM mutated to become a model for heavy ion collisions. QGP includes a large number of par- melting nuclear matter. This development motivated the ticles from the second family: the strange quarks and also, experimental exploration in the collisions of heavy nuclei the yet heavier charmed quarks; and from the third family at relativistic energies of the phases of matter in condi- at the LHC we expect an appreciable abundance of bot- tions close to those last seen in the early Universe. I refer tom quarks. The novel ability to study a large number of to Hagedorn’s account of these developments for further these 2nd and 3rd generation particles offers a new oppor- details chapt. 25 loc. cit. and ref. [13]. We return to this tunity to approach in an experiment the riddle of flavor. time period in subsect. 4.1. Eur. Phys. J. A (2015) 51: 114 Page 3 of 58

At the beginning of this new field of research in the late formation in RHI collisions” was a field of research that 70s, quark confinement was a mystery for many of my col- could have easily fizzled out. However, in Europe there leagues; gluons mediating the strong color force were nei- was CERN, and in the US there was strong institutional ther discovered nor widely accepted, especially not among support. Early on it was realized that RHI collisions re- my nuclear physics peers, and QCD vacuum structure was quired large experiments uniting much more human ex- just finishing kindergarten. The discussion of a new phase pertise and manpower as compared to the prior nuclear of deconfined quark-gluon matter was therefore in many and even some particle physics projects. Thus work had eyes not consistent with established wisdom and certainly to be centralized in a few “pan-continental” facilities. This too ambitious for the time. meant that expertise from a few laboratories would need Similarly, the special situation of Hagedorn deserves to be united in a third location where prior investments remembering: early on Hagedorn’s research was under- would help limit the preparation time and cost. mined by outright personal hostility; how could Hagedorn dare to introduce thermal physics into the field governed by particles and fields? However, one should also take note 1.3 The accelerator race for quark matter of the spirit of academic tolerance at CERN. Hagedorn advanced through the ranks along with his critics, and These considerations meant that in Europe the QGP for- his presence attracted other like-minded researchers, who mation in RHI collisions research program found its home were welcome in the CERN Theory Division, creating a at CERN. The CERN site benefited from being a multi- bridgehead towards the new field of RHI collisions when accelerator laboratory with a large pool of engineering the opportunity appeared on the horizon. expertise and where some of the necessary experimental In those days, the field of RHI collisions was in other equipment already existed, thanks to prior related particle ways rocky terrain: physics efforts. The CERN program took off by the late 80s. The time 1) RHI collisions required the use of atomic nuclei line of the many CERN RHI experiments through the at highest energy. This required cooperation between ex- beginning of this millennium is shown in fig. 1; the rep- perimental nuclear and particle physicists. Their culture, resentation is based on a similar CERN document from background, and experience differed. A similar situation the year 2000. The experiments WA85, NA35, HELIOS- prevailed within the domain of theoretical physics, where 2, NA38 were built largely from instrumental components an interdisciplinary program of research merging the three from prior particle physics detectors. Other experiments traditional physics domains had to form. While ideas of and/or experimental components were contributed by US thermal and statistical physics needed to be applied, very and European laboratories. These include the heavy ion few subatomic physicists, who usually deal with individual source and its preaccelerator complex, required for heavy particles, were prepared to deal with many body ques- ion insertion into CERN beam lines. tions. There were also several practical issues: In which When the CERN SPS program faded out early in this (particle, nuclear, stat-phys) journal can one publish and millennium, the resources were focused on the LHC-ion who could be the reviewers (other than direct competi- collider operations, and in the US, the RHIC came on-line. tors)? To whom to apply for funding? Which conference As this is written, the SPS fixed target program experi- to contribute to? ences a second life; the experiment NA61, built with large 2) The small group of scientists who practiced RHI input from the NA49 equipment, is searching for the onset collisions were divided on many important questions. In of QGP formation, see subsect. 6.3. regard to what happens in relativistic collision of nuclei The success of the SPS research program at CERN has the situation was most articulate: a) One group believed strongly supported the continuation of the RHI collision that nuclei (baryons) pass through each other with a new program. The Large Hadron Collider (LHC) was designed phase of matter formed in a somewhat heated projectile to accept highest energy counter propagating heavy ion and/or target. This picture required detection systems beams opening the study of a new domain of collision of very different character than the systems required by, energy. LHC-ion operation allows us to exceed the top both: b) those who believed that in RHI collisions energy RHIC energy range by more than an order of magnitude. would be consumed by a shock wave compression of nu- In preparation for LHC-ion operations, in the mid-90s the clear matter crashing into the center of momentum frame; SPS groups founded a new collider collaboration, and have and c) a third group who argued that up to top CERN- built one of the four LHC experiments dedicated to the SPS ( (sNN) (20) GeV) collision energy a high tem- study of RHI collisions. Two other experiments also par- ≃O perature, relatively low baryon density quark matter fire- ticipate in the LHC-ion research program which we will  ball will be formed. The last case turned out to be closest introduce in subsect. 6.2. to results obtained at CERN-SPS and at Brookhaven Na- In parallel to CERN there was a decisive move in the tional Laboratory (BNL) RHI Collider (RHIC). same direction in the US. The roots of the US relativis- From outside, we were ridiculed as being speculative; tic heavy ion program predate the interest of CERN by from within we were in state of uncertainty about the fate nearly a decade. In 1975, the Berkeley SuperHILAC, a low of colliding matter and the kinetic energy it carried, with energy heavy ion accelerator was linked to the Bevatron, disagreements that ranged across theory vs. experiment, an antique particle accelerator at the time, yet capable of and particle vs. nuclear physics. In this situation, “QGP accelerating the injected ions to relativistic energies with Page 4 of 58 Eur. Phys. J. A (2015) 51: 114

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. the Lorentz factor above two. The system of accelerators modified to be a RHI Collider (RHIC), it was thought was called the Bevalac. It offered beams of ions which were that it could be completed within a few years, offering the used in study of properties of compressed nuclear matter, US a capability comparable to that expected by Pugh at conditions believed to be similar to those seen within col- CERN. lapsing neutron stars. This evaluation prompted a major investment decision As interest in the study of quark matter grew, by 1980 by the US Department of Energy to create a new relativis- the Bevalac scientists formulated the future Variable En- tic heavy ion research center at BNL shown in fig. 2, a plan ergy Nuclear Synchrotron (VENUS) heavy ion facility. that would be cementing the US leadership role in the field Representing the Heavy Ion Program at Berkeley How- of heavy ions. In a first step, already existing tandems able ell Pugh [14] opened in October 1980 the “Quark Matter to create low energy heavy ion beams were connected by 1” conference at GSI in Germany making this comment a transfer line to the already existing AGS proton syn- chrotron adapted to accelerate these ions. In this step a A A “20 GeV

Fig. 2. The Brookhaven National Laboratory heavy ion accelerator complex. Creative Commons picture modified by the author.

BNL, by the adaptation of ISABELLE design to fit RHIC CERN was in a unique position to embark on RHI research needs, and by typical funding constraints. As this work by having not only the accelerators, engineering expertise, progressed nobody rushed. I think this was so since at and research equipment, but mainly due to Hagedorn, also BNL the opinion prevailed that RHIC was invulnerable, a the scientific expertise on the ground, for more detail con- dream machine not to be beaten in the race to discover the sult ref. [1]. In the US a major new experimental facility, new phase of matter. Hereto I note that nobody back then RHIC at BNL, was developed. With the construction of could tell what the energy threshold for QGP formation LHC at CERN a new RHI collision energy domain was in the very heavy ion collisions would be. The theoreti- opened. The experimental programs at SPS, RHIC and cal presumption that this threshold was above the energy LHC-ion continue today. produced at SPS turned out to be false. Because data taking for the RHIC beam did not hap- 1.4 Format of this review pen until 2001, the priority in the field of heavy ions that the US pioneered in a decisive way at Berkeley in the More than 35 years into the QGP endeavor I can say with early 70s passed on to CERN where a large experimental conviction that the majority of nuclear and particle physi- program at SPS was developed, and as it is clear today, cists and the near totality of the large sub-group directly the energy threshold for QGP formation in Pb-Pb colli- involved with the relativistic heavy ion collision research sions was within SPS reach, see sect. 4.2. It is important agree that a new form of matter, the (deconfined) quark- to remember that CERN moved on to develop the rela- gluon plasma phase has been discovered. The discovery tivistic heavy ion research program under the leadership announced at CERN in the year 2000, see subsect. 4.2, of Herwig Schopper. Schopper, against great odds, bet his has been confirmed both at RHIC and by the recent re- reputation on Heavy Ions to become one of the pillars sults obtained at LHC. This review has, therefore, as its of CERN’s future. This decision was strongly supported primary objective, the presentation of the part of this an- by many national nuclear physics laboratories in Europe, nouncement that lives on, see subsect. 4.3, and how more where in my opinion the most important was the support recent results are addressing these questions: What are the offered by the GSI and the continued development of rela- properties of hot hadron matter? How does it turn into tivistic heavy ion physics by one of GSI directors, Rudolph QGP, and how does QGP turn back into normal matter? Bock. These are to be the topics addressed in the second half of To conclude the remarks about where we came from this review. and where we are now: a new fundamental set of science There are literally thousands of research papers in this arguments about the formation of quark-gluon plasma and field today; thus this report cannot aim to be inclusive deep-rooted institutional support carried the field forward. of all work in the field. We follow the example of John Page 6 of 58 Eur. Phys. J. A (2015) 51: 114

A. Wheeler. Addressing in his late age a large audience and fireball sizes described. Subsection 10.3 explains, in of physicists, he showed one transparency with one line, terms of evaluation by example of prior work, why the “What is the question?”. In this spirit, this review begins prior two subsections address solely the SHARE-model with a series of questions, and answers, aiming to find results. In subsect. 10.4 the relevance of LHC results to the answer to: Which question is THE question today? QGP physics is described, and further lattice-QCD rela- A few issues we raise are truly fundamental present day tions to these results pointed out. challenges. Many provide an opportunity to recognize the The final sect. 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 previously un- These introductory questions are grouped into three published technical papers which are for the first time separate sections: first come the theoretical concepts on in print as an addendum to this review, one from 1980 the hadron side of hot hadronic matter, sect. 2; next, con- (ref. [15]) and another from 1983 (ref. [16]). These two are cepts on the quark side, sect. 3; and third, the experi- just a tip of an iceberg; there are many other unpublished mental “side” sect. 4 about RHI collisions. Some of the papers by many authors hidden in conference volumes. questions formulated in sects. 2, 3, and 4 introduce topics There is already a published work reprint volume [17] in that this review addresses in later sections in depth. The which the pivotal works describing QGP theoretical foun- roles of strangeness enhancement and strange antibaryon dations are reproduced; however, the much less accessible signature of QGP are highlighted. and often equally interesting unpublished work is at this We follow this discussion by addressing the near future juncture in time practically out of sight. This was one of of the QGP and RHI collision research in the context of the reasons leading to the presentation of ref. [1]. These this review centered around the strong interactions and two papers were selected from this volume and are shown hadron-quarks phase. In sect. 5, I present several concep- here unabridged. They best complement the contents of tual RHI topics that both are under present active study, this review, providing technical detail not repeated here, and which will help determine which direction the field while also offering a historical perspective. Beside the key will move on in the coming decade. Section 6 shows the results and/or discussion they also show the rapid shift current experimental research program that address these in the understanding that manifested itself within a short questions. Assuming that this effort is successful, I pro- span of two years. pose in sect. 7 the next generation of physics challenges. Reference [15] presents Extreme States of Nuclear Mat- The topics discussed are very subjective; other authors ter,fromtheWorkshop on Future Relativistic Heavy Ion will certainly see other directions and propose other chal- Experiments held 7-10 October 1980. This small gather- lenges of their interest. ing convened by Rudolph Bock and Reinhard Stock is now In sect. 8 we deepen the discussion of the origins and considered to be the first of the “Quark Matter” series i.e. the contents of the theoretical ideas that have led Hage- QM80 conference. Most of this report is a summary of the dorn to invent the theoretical foundations leading on to theory of hot hadron gas based on Hagedorn’s Statisti- TH and melting hadrons. The technical discussion is brief cal Bootstrap Model (SBM). The key new insight in this and serves as an introduction to ref. [15] which is pub- work was that in RHI collisions the production of parti- lished for the first time as an addendum to this review. cles rather than the compression of existent matter was Section 8 ends with a discussion, subsect. 8.5, of how the the determining factor. The hadron gas phase study was present day lattice-QCD studies test and verify the theory complemented by a detailed QGP model presented as a of hot nuclear matter based on SBM. large, hot, interacting quark-gluon bag. The phase bound- Selected theoretical topics related to the study of QGP ary between these two phases characterized by Hagedorn hadronization are introduced in the following: In sect. 9 we temperature TH was evaluated in quantitative manner. It describe the numerical analysis tool within the Statistical was shown how the consideration of different collision en- Hadronization Model (SHM); that is, the SHARE suite ergies allows us to explore the phase boundary. This 1980 of computer programs and its parameters. We introduce paper ends with the description of strangeness flavor as the practical items such as triggered centrality events and ra- observable of QGP. Strange antibaryons are introduced as pidity volume dV/dy, resonance decays, particle number a signature of quark-gluon plasma. fluctuations, which all enter into the RHIC and LHC data Reference [16] presents Strangeness and Phase analysis. Changes in Hot Hadronic Matter,fromtheSixth High En- Section 10 presents the results of the SHM analysis ergy Heavy Ion Study, Berkeley, 28 June – 1 July 1983. with emphasis put on bulk properties of the fireball; sub- The meeting, which had a strong QGP scientific compo- sect. 10.1, addresses SPS and RHIC prior to LHC, while nent, played an important role in the plans to develop a in subsect. 10.2: it is shown how hadron production can dedicated relativistic heavy ion collider (RHIC). In this be used to determine the properties of QGP and how the lecture I summarize and update in qualitative terms the threshold energy for QGP formation is determined. The technical phase transition consideration seen in ref. [15], results of RHIC and LHC are compared and the univer- before turning to the physics of strangeness in hot hadron sality of QGP hadronization across a vast range of energy and quark matter. The process of strangeness production Eur. Phys. J. A (2015) 51: 114 Page 7 of 58

to its natural volume V V0, is itself a highly excited hadron, a resonance that→ we must include in eq. (3). This is what Hagedorn realized in 1964 [11]. This observation leads to an integral equation for ρ(m) when we close the “bootstrap” loop that emerges. Frautschi [12] transcribed Hagedorn’s grand canonical formulation into microcanonical format. The microcanon- ical bootstrap equation reads in invariant Yellin [19] no- tation Hτ(p2)=H δ (p2 m2 ) 0 − in min  Fig. 3. Illustration of the Statistical Bootstrap Model idea: ∞ n n a volume comprising a gas of fireballs when compressed to 1 4 2 4 + δ p pi Hτ(pi )d pi, natural volume is itself again a fireball. Drawing from ref. [20] n! − n=2  i=1 i=1 modified for this review.   (4) is presented as being a consequence of dynamical colli- where H is a universal constant assuring that eq. (4) is dimensionless; τ(p2) on the left-hand side of eq. (4) is the sion processes both among hadrons and in QGP, and the µ dominance of gluon-fusion processes in QGP is described. fireball mass spectrum with the mass m = √pµp which The role of strangeness in QGP search experiments is pre- we are seeking to model. The right-hand side of eq. (4) sented. For a more extensive historical recount see ref. [18]. expresses that the fireball is either just one input particle of a given mass min, or else composed of several (two or 2 more) particles i having mass spectra τ(pi ), and 2 The concepts: Theory hadron side τ(m2)dm2 ρ(m)dm. (5) ≡ 2.1 What is the Statistical Bootstrap Model (SBM)? A solution to eq. (4) has naturally an exponential form ρ(m) m−a exp(m/T ). (6) Considering that the interactions between hadronic parti- ∝ H cles are well characterized by resonant scattering, see sub- The appearance of the exponentially growing mass sect. 2.4, we can describe the gas of interacting hadrons spectrum, eq. (6), is a key SBM result. One of the impor- as a mix of all possible particles and their resonances “i”. tant consequences is that the number of different hadron This motivates us to consider the case of a gas comprising states grows so rapidly that practically every strongly in- several types of particles of mass m , enclosed in a heat i teracting particle found in the fireball is distinguishable. bath at temperature T , where the individual populations Hagedorn realized that the distinguishability of hadron “i” are unconstrained in their number, that is like photons states was an essential input in order to reconcile statisti- in a black box adapting abundance to what is required for cal hadron multiplicities with experimental data. Despite the ambient T . The non-relativistic limit of the partition his own initial rejection of a draft paper, see chapts. 18 function this gas takes the form and 19 loc. cit., this insight was the birth of the theory of hot hadronic matter as it produced the next step, a T 3/2 ln Z = ln Z = V m3/2e−mi/T , (1) model [13]. i 2π i i   i SBM solutions provide a wealth of information includ-   ing the magnitude of the power index a seen in eq. (6). where the momentum integral was carried out and the Frautschi, Hamer, Carlitz [12,21–25] studied solutions sum “i” includes all particles of differing parity, spin, to eq. (4) analytically and numerically and by 1975 drew isospin, baryon number, strangeness etc. Since each state important conclusions: is counted, there is no degeneracy factor. It is convenient to introduce the mass spectrum ρ(m), – Fireballs would predominantly decay into two frag- where ments, one heavy and one light. – By iterating their bootstrap equation with realistic ρ(m)dm =numberof“i” hadron states in m, m +dm . input, they found numerically TH 140 MeV and { } a =2.9 0.1, which ruled out the previously≈ adopted (2) ± Thus we have approximate value [11,26] a =5/2. – Each imposed conservation law implemented by fixing T 3/2 ∞ a quantum number, e.g., baryon number ρ(B,m), in Z(T,V ) = exp V ρ(m)m3/2e−m/T dm . 2π the mass spectrum, increases the value of a by 1/2.    0  (3) Werner Nahm independently obtained a = 3 [27]. Further On the other hand, a hadronic fireball comprising many refinement was possible. In ref. [15], a SBM with com- components seen on the left in fig. 3, when compressed pressible finite-size hadrons is introduced where one must Page 8 of 58 Eur. Phys. J. A (2015) 51: 114

Table 1. Thermodynamic quantities assuming exponential Table 2. Parameters of eq. (8) fitted for a prescribed pre- form of hadron mass spectrum with pre-exponential index exponential power a. Results from ref. [30]. a, eq. (6); results from ref. [28]. a−1 12 a c [GeV ] m0 [GeV] TH [MeV] TH [10 K] · a P ε fε/ε 2.5 0.83479 0.6346 165.36 1.9189 2 3 1/2 C/∆T C/∆T C + C∆T 3. 0.69885 0.66068 157.60 1.8289 3/2 5/2 3/4 1 C/∆T C/∆T C + C∆T 3.5 0.58627 0.68006 150.55 1.7471 2 1/2 3/2 C/∆T C/∆T C + C∆T 4. 0.49266 0.69512 144.11 1.6723 1/2 3/2 1/4 2 C/∆T C/∆T C + C∆T 5. 0.34968 0.71738 132.79 1.5410 5/2 C ln(T0/∆T ) C/∆T C 6. 0.24601 0.73668 123.41 1.4321 1/2 1/2 1/4 3 P0 C∆T C/∆T C/∆T − 1/2 7/2 P0 C∆T ε0 C/∆T − 2.2 What is the Hagedorn temperature TH? 3/2 1/2 3/4 4 P0 C∆T ε0 C∆T C/∆T − − Hagedorn temperature is the parameter entering the ex- ponential mass spectrum eq. (6). It is measured by fit- consistently replace eq. (29) by eq. (30). This leads to a ting to data the exponential shape of the hadron mass finite energy density already for a model which produces spectrum. The experimental mass spectrum is discrete; a = 3 with incompressible hadrons. hence a smoothing procedure is often adopted to fit the For any ρ(m) with a given value of a, eq. (6), it is easy shape eq. (8) to data. In technical detail one usually fol- to understand the behavior near to TH. Inserting eq. (6) lows the method of Hagedorn (see chapt. 20 loc. cit. and into the relativistic form of eq. (1), see chapt. 23 loc. cit., ref. [26]), applying a Gaussian distribution with a width of allows the evaluation near critical condition, T T 200 MeV for all hadron mass states. However, the acces- H − ≡ ΔT 0 of the physical properties such as shown in ta- sible experimental distribution allows fixing TH uniquely ble 1:→ pressure P , energy density ε, and other physical only if we know the value of the pre-exponential power properties, as example the mean relative fluctuations fε/ε “a”. of ε are shown, for a =1/2, 2/2,...,8/2. We see that, as The fit procedure is encumbered by the singularity for T TH (ΔT 0), the energy density diverges for a 3. m 0. Hagedorn proposed a regularized form of eq. (6) →In view of→ the entries shown in table 1 an important≤ → further result can be obtained using these leading order em/TH ρ(m)=c 2 2 a/2 . (8) terms for all cases of a considered: the speed of sound at (m0 + m ) which the small density perturbations propagate In fits to experimental data all three parameters TH, m0, c dP must be varied and allowed to find their best value. In 1967 c2 := ΔT 0. (7) s dε ∝ → Hagedorn fixed m0 =0.5GeV as he was working in the limit m>m0, and he also fixed a =2.5 appropriate for This universal for all a result is due to the exponential his initial SBM approach [26]. The introduction of a fitted mass spectrum of hadron matter studied here. c2 0 value m is necessary to improve the characterization of s → 0 at TH harbors an interesting new definition of the phase the hadron mass spectrum for low values of m, especially boundary in the context of lattice-QCD. A non-zero but when a range of possible values for a is considered. 2 small value cs should arise from the subleading terms con- The fits to experimental mass spectrum shown in ta- tributing to P and ε not shown in table 1. The way sin- ble 2 are from 1994 [30] and thus include a smaller set of 2 gular properties work, it could be that the cs = 0 point hadron states than is available today. However, these re- exists. The insight that the sound velocity vanishes at TH sults are stable since the new hadronic states found are at is known since 1978, see ref. [28]. An “almost” rediscovery high mass. We see in table 2 that as the pre-exponential of this result is seen in sects. 3.5 and 8.7 of ref. [29]. power law a increases, the fitted value of TH decreases. The The above discussion shows both the ideas that led to value of c for a =2.5 corresponds to c =2.64 104 MeV3/2, the invention of SBM, and how SBM can evolve with our in excellent agreement to the value obtained× by Hagedorn understanding of the strongly interacting matter, becom- in 1967. In fig. 4 the case a = 3 is illustrated and com- ing more adapted to the physical properties of the ele- pared to the result of the 1967 fit by Hagedorn and the mentary “input” particles. Further potential refinements experimental smoothed spectrum. All fits for different a include introducing strange quark related scale into char- were found at nearly equal and convincing confidence level acterization of the hadron volume, making baryons more as can be inferred from fig. 4. compressible as compared to mesons. These improvements Even cursory inspection of table 2 suggests that the could generate a highly realistic shape of the mass spec- value of TH that plays an important role in physics of RHI trum, connecting SBM more closely to the numerical collisions depends on the understanding of the value of a. study of QCD in lattice approach. We will return to SBM, This is the reason that we discussed the different cases in and the mass spectrum, and describe the method of find- depth in previous subsect. 2.1. The pre-exponential power ing a solution of eq. (4) in sect. 8. value a =2.5 in eq. (8) corresponds to Hagedorn’s original Eur. Phys. J. A (2015) 51: 114 Page 9 of 58

Fig. 4. The experimental mass spectrum (solid line), the fit (short dashed), compared to 1967 fit of Hagedorn (long dashed): The case a = 3 is shown, for parameters see table 2. Figure from ref. [20] with results obtained in [31] modified for this review. Fig. 5. Meson- (top) and baryon- (bottom) mass spectra ρ(M) (particles per GeV): dashed line the experimental spectrum including discrete states. Two different fits are shown, see test. preferred value; the value a = 3 was adopted by the mid- Figure from ref. [32] modified for this review. 70s following extensive study of the SBM as described. However, results seen in table 1 and ref. [15] imply a 7/2. ≥ that the value of TH can be fixed more precisely in the This is so since for a<7/2 we expect T to be a max- future when more hadronic resonances are known. How- H ever, for M 3 GeV there are about 105 different meson imum temperature, for which we see in table 1 a diver- ≃ gence in energy density. Based on study of the statistical or baryon states per GeV. This means that states of this mass are on average separated by 10 eV in energy. On bootstrap model of nuclear matter with conserved baryon 6 number and compressible hadrons presented in ref. [15], I the other hand, their natural width is at least 10 larger. believe that 3.5 a 4. A yet greater value a 4 should Thus there is little if any hope to experimentally resolve emerge if in addition≤ ≤ strangeness and charge≥ are intro- such “Hagedorn” states. Hence we cannot expect to deter- duced as a distinct conserved degree of freedom —in any mine, based on experimental mass spectrum, the value of consistently formulated SBM with canonically conserved TH more precisely than it is already done today. However, quantum numbers one unique value of T will emerge for there are other approaches to measure the value of TH. H For example, we address at the end of subsect. 3.3 why the mass spectrum, that is ρ(m,b,S,...) exp(m/TH) for any value of b,S,Q,..., i.e. the same T∝ for mesons the behavior of lattice-QCD determined speed of sound H suggests that T 145 MeV. and baryons. Only the pre-exponential function can de- H ≃ pend on b,S,Q,... An example for this is provided by To summarize, our current understanding is that Hage- the SBM model of Beitel, Gallmeister and Greiner [32]. dorn temperature has a value still needing an improved Using a conserved discrete quantum numbers approach, determination, explicit fits lead to the same (within 1MeV) value of T H 140 T 155 MeV T (1.7 0.1) 1012 K. (9) for mesons and baryons [32]. ≤ H ≤ H ≃ ± ×

These results of ref. [32] are seen in fig. 5: the top frame TH is the maximum temperature at which matter can exist for mesons and the bottom frame for baryons. Two differ- in its usual form. TH is not a maximum temperature in ent fits are shown characterized by a model parameter R the Universe. The value of TH which we evaluate in the which, though different from H seen in eq. (15) in ref. [15], study of hadron mass spectra is, as we return to discuss plays a similar role. Thus the two results bracket the value in sect. 3.3, the melting point of hadrons dissolving into of TH from above (blue, TH 162 MeV) and from below the quark-gluon plasma (QGP), a liquid phase made of ≃ (red, TH 145 MeV) in agreement with typical empirical Debye-screened color-ionic quarks and gluons. A further ≃ results seen in table 2. heating of the quark-gluon plasma “liquid” can and will We further see in fig. 5 that a noticeably different num- continue. A similar transformation can occur already at a ber of M>2GeV states can be expected depending on lower temperature at a finite baryon density. the value of TH, even if the resonances for M<1.7GeV Indeed, there are two well studied ways to obtain de- are equally well fitted in both cases. Thus it would seem confinement: a) high temperature; and b) high baryon Page 10 of 58 Eur. Phys. J. A (2015) 51: 114 density. In both cases the trick is that the number of par- for different particle families. iii) A third technical prob- ticles per unit volume is increased. lem is that an integrated (“accumulated”) mass spectrum is considered, a) In the absence of all matter (zero net baryon number corresponding to baryochemical potential o 0), B → M in full thermal equilibrium temperature alone controls R(M)= ρ(m)dm. (10) the abundance of particles as we already saw in the 0 context of SBM. The result of importance to this re- view is that confinement is shown to dissolve in the While the Hagedorn-type approach requires smoothing study of QCD by Polyakov [33], 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 [34]. 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, smoothing process loses information that is now available or even at, zero temperature; for further quantitative in the new approach, eq. (10). However, it also could be discussion see ref. [15]. 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 [35]. common when considering any integrated signal function. The Krakow group refs. [41,42] was first to consider the Cabibbo and Parisi [36] were first to recognize that these integrated mass spectrum eq. (10). They also break the two distinct limits are smoothly connected and that the large set of hadron resonances into different classes, e.g. phase boundary could be a smooth line in the , T plane. oB non-strange/strange hadrons, or mesons/baryons. How- Their qualitative remarks did not address a method to ever, they chose same pre-exponential fit function and var- form, or to explore, the phase boundary connecting these ied T between particle families. The fitted value of T limits. The understanding of high baryon density matter H H was found to be strongly varying in dependence on sup- properties in the limit T 0 is a separate vibrant research plementary hypotheses made about the procedure, with topic which will not be further→ discussed here [37–40]. Our the value of T changing by 100’s MeV, possibly showing primary interest is the domain in which the effects of tem- H the inconsistency of procedure aggravated by the loss of perature dominate, in this sense the limit of small o T . B ≪ signal information. Reference [43] fixes m0 =0.5GeV at a =2.5, i.e. 2.3 Are there several possible values of TH? Hagedorn’s 1968 parameter choices. Applying the Krakow method approach, this fit produces with present day data

The singularity of the SBM at TH is a unique singular TH = 174 MeV. We keep in mind that the assumed value point of the model. If and when within SBM we imple- of a is incompatible with SBM, while the assumption of a ment distinguishability of mesons from baryons, and/or of relatively small m0 =0.5 GeV is forcing a relatively large strange and non-strange hadrons, all these families of par- value of TH, compare here also the dependence of TH on ticles would have a mass spectrum with a common value of a seen in table 2. Another similar work is ref. [44], which TH. No matter how complex are the so-called SBM “input” seeing poor phenomenological results that emerge from states, upon Laplace transform they always lead to one an inconsistent application of Hagedorn SBM, criticizes singular point, see subsect. 8.3. In subsequent projection unjustly the current widely accepted Hagedorn approach of the generating SBM function onto individual families of and Hagedorn temperature. For reasons already described, hadrons one common exponential is found for all. On the we do not share in any of the views presented in this work. other hand, it is evident from the formalism that when ex- However, we note two studies [45,46] of differentiated tracting from the common expression the specific forms of (meson vs. baryon) hadron mass spectrum done in the way the mass spectrum for different particle families, the pre- that we consider correct: using a common singularity, that exponential function must vary from family to family. In is one and the same exponential TH, but “family” depen- concrete terms this means that we must fit the individual dent pre-exponential functions obtained in projection on mass spectra with common TH but particle family depen- the appropriate quantum number. It should be noted that dent values of a and dimensioned parameter c, m0 seen the hadronic volume Vh enters any reduction of the mass in table 2, or any other assumed pre-exponential function. spectrum by the projection method, see ref. [15], where There are several recent phenomenological studies of volume effect for strangeness is shown. the hadron mass spectrum claiming to relate to SBM of Biro and Peshier [45] search for TH within non- Hagedorn, and the approaches taken are often disappoint- extensive thermodynamics. They consider two different ing. The frequently seen defects are: i) Assumption of values of a for mesons and baryons (somewhat on the low a =2.5 along with the Hagedorn 1964-67 model, a value side), and in their fig. 2 the two fits show a common value obsolete since 1971 when a = 3 and higher was recog- of TH around 150–170 MeV. A very recent lattice moti- nized; and ii) Choosing to change TH for different particle vated effort assumes differing shape of the pre-exponential families, e.g. baryons and mesons or strange/non-strange function for different families of particles [46], and uses a hadrons instead of modifying the pre-exponential function common, but assumed, not fitted, value of TH. Eur. Phys. J. A (2015) 51: 114 Page 11 of 58

Arguably, the most important recent step forward in regard to improving the Hagedorn mass spectrum analysis is the realization first made by Majumder and M¨uller [47] that one can infer important information about the hadron mass spectrum from lattice-QCD numerical re- sults. However, this first effort also assumed a =2.5 with- out a good reason. Moreover, use of asymptotic expan- sions of the Bessel functions introduced errors, preventing a comparison of these results with those seen in table 2. To close let us emphasize that phenomenological ap- proach in which one forces same pre-exponential function and fits different values of TH for different families of parti- cles is at least within the SBM framework blatantly wrong. A more general argument indicating that this is always wrong could be also made: the only universal natural con- stant governing phase boundary is the value of TH, the pre- exponential function, which varies depending on how we split up the hadron particle family —projection of baryon number (meson, baryon), and strangeness, are two natural Fig. 6. Pressure P/T4 of QCD matter evaluated in lattice choices. approach (includes 2 + 1 flavors and gluons) compared with their result for the HRG pressure, as function of T . The upper limit of P/T4 is the free Stephan-Boltzmann (SB) quark-gluon pressure with three flavors of quarks in the relativistic limit 2.4 What is hadron resonance gas (HRG)? T strange quark mass. Figure from ref. [72] modified for this≫ review. We are seeking a description of the phase of matter made of individual hadrons. One would be tempted to think that the SBM provides a valid framework. However, we already But as we warm up our hadron gas, for T>TH/5 res- know from discussion above that the experimental reali- onance contribution becomes more noticeable and in turn ties limit the ability to fix the parameters of this model; their scattering with pions requires inclusion of other res- specifically, we do not know TH precisely. onances and so on. As we reach T  TH in the heat-up In the present day laboratory experiments one there- process, Hagedorn’s distinguishable particle limit applies: fore approaches the situation differently. We employ all very many different resonances are present such that this experimentally known hadrons as explicit partial fractions hot gas develops properties of classical numbered-ball sys- in the hadronic gas: this is what in general is called the tem, see chapt. 19 loc. cit. hadron resonance gas (HRG), a gas represented by the All heavy resonances ultimately decay, the process cre- non-averaged, discrete sum partial contributions, corre- ating pions observed experimentally. This yield is well sponding to the discrete format of ρ(m) as known empir- ahead of what one would expect from a pure pion gas. ically. Moreover, spectra of particles born in resonance decays The emphasis here is on “resonances” gas, reminding differ from what one could expect without resonances. As us that all hadrons, stable and unstable, must be included. a witness of the early Hagedorn work from before 1964, In his writings Hagedorn went to great length to justify Maria Fidecaro of CERN told me recently, I paraphrase how the inclusion of unstable hadrons, i.e. resonances, ac- “when Hagedorn produced his first pion yields, there were counts for the dominant part of the interaction between many too few, and with a wrong momentum spectrum”. all hadronic particles. His argument was based on work As we know, Hagedorn did not let himself be discouraged of Belenky (also spelled Belenkij) [48], but the intuitive by this initial difficulty. content is simple: if and when reaction cross sections are The introduction of HRG can be tested theoretically dominated by resonant scattering, we can view resonances by comparing HRG properties with lattice-QCD. In fig. 6 as being all the time present along with the scattering par- we show the pressure presented in ref. [72]. We indeed ticles in order to characterize the state of the physical sys- see a good agreement of lattice-QCD results obtained for tem. This idea works well for strong interactions since the T  TH with HRG, within the lattice-QCD uncertainties. S-matrix of all reactions is pushed to its unitarity limit. In this way we have ab initio confirmation that Hagedorn’s To illustrate the situation, let us imagine a hadron ideas of using particles and their resonances to describe a system at “low” T TH/5 and at zero baryon density; strongly interacting hadron gas is correct, confirmed by this is in essence a gas≃ made of the three types of pions, more fundamental theoretical ideas involving quarks, glu- r(+,−,0). In order to account for dominant interactions be- ons, QCD. tween pions we include their scattering resonances as in- Results seen in fig. 6 comparing pressure of lattice- dividual contributing fractions. Given that these particles QCD with HRG show that, as temperature decreases to- have considerably higher mass compared to that of two wards and below TH, the color charge of quarks and glu- pions, their number is relatively small. ons literally freezes, and for T  TH the properties of Page 12 of 58 Eur. Phys. J. A (2015) 51: 114 strongly interacting matter should be fully characterized by a HRG. Quoting Redlich and Satz [49]: “The crucial question thus is, if the equation of state of hadronic matter introduced by Hagedorn can describe the corresponding results obtained from QCD within lattice approach.” and they con- tinue: “There is a clear coincidence of the Hage- dorn resonance model results and the lattice data on the equation of states. All bulk thermodynam- ical observables are very strongly changing with temperature when approaching the deconfinement transition. This behavior is well understood in the Hagedorn model as being due to the contribution of resonances. . . . resonances are indeed the essen- tial degrees of freedom near deconfinement. Thus, on the thermodynamical level, modeling hadronic interactions by formation and excitation of reso- nances, as introduced by Hagedorn, is an excellent 4 Fig. 7. The Interaction measure (ε 3P )/T within mixed approximation of strong interactions.” parton-hadron model, model fitted to− match the lattice data of ref. [72]. Figure from ref. [52] modified for this review. 2.5 What does lattice-QCD tell us about HRG and about the emergence of equilibrium? The recent analysis of lattice-QCD results of Biro and The thermal pressure reported in fig. 6 is the quantity least Jakovac [52] proceeds in terms of a perfect microscopic mix sensitive to missing high mass resonances which are non- of partons and hadrons. One should take note that as soon relativistic and thus contribute little to pressure. Thus the as QCD-partons appear, in such a picture color deconfine- agreement we see in fig. 6 is testing: a) the principles of ment is present. In figs. 10 and 11 in [52] the appearance of Hagedorn’s HRG ideas; and b) consistency with the part partons for T>140 MeV is noted. Moreover, this model of the hadron mass spectrum already known, see fig. 4. A is able to describe precisely the interaction measure more thorough study is presented in subsect. 8.5, describ- ε 3P ing the compensating effect for pressure of finite hadron Im − , (11) ≡ T 4 size and missing high mass states in HRG, which than produces good fit to energy density. as shown in fig. 7. Im is a dimensionless quantity that Lattice-QCD results apply to a fully thermally equi- depends on the scale invariance violation in QCD. We note librated system filling all space-time. This in principle is the maximum value of Im 4.2 in fig. 7, a value which true only in the early Universe. After hadrons are born at reappears in the hadronization≃ fit in fig. 35, subsect. 10.2, T  TH, the Universe cools in expansion and evolves, with where we see for a few classes of collisions the same value the expansion time constant governed by the magnitude Im 4.6 0.2. of the (applicable to this period) Hubble parameter; one ≃Is this± agreement between a hadronization fit and lat- finds [6,50] τ 25 μ satT , see also subsect. 7.4. The tice I an accident? The question is open since a priori q ∝ · H m value of τq is long on hadron scale. A full thermal equili- this agreement has to be considered allowing for the rapid bration of all HRG particle components can be expected dynamical evolution occurring in laboratory experiments, in the early Universe. a situation differing vastly from the lattice simulation of Considering the early Universe conditions, it is pos- static properties. The dynamical situation is also more sible and indeed necessary to interpret the lattice-QCD complex and one cannot expect that the matter content results in terms of a coexistence era of hadrons and QGP. of the fireball is a parton-hadron ideal mix. The rapid ex- This picture is usually associated with a 1st order phase pansion could and should mean that the parton system transition, see Kapusta and Csernai [51] where one finds evolves without having time to enter equilibrium mixing separate spatial domains of quarks and hadrons. However, with hadrons, this is normally called super-cooling in the as one can see modeling the more experimentally accessi- context of a 1st order phase transition, but in context of ble smooth transition of hydrogen gas to hydrogen plasma, a mix of partons and hadrons [52], these ideas should also this type of consideration applies in analogy also to any apply: as the parton phase evolves to lower temperature, smooth phase transformation. The difference is that for the yet non-existent hadrons will need to form. smooth transformation, the coexistence means that the To be specific, consider a dense hadron phase cre- mixing of the two phases is complete at microscopic level; ated in RHI collisions with a size Rh 5 fm and a no domain formation occurs. However, the physical prop- T 400 MeV, where the fit of ref. [52] suggest∝ small if erties of the mixed system like in the 1st order transition any≃ presence of hadrons. Exploding into space this parton case are obtained in a superposition of fractional gas com- domain dilutes at, or even above, the speed of sound in the ponents. transverse direction and even faster into the longitudinal Eur. Phys. J. A (2015) 51: 114 Page 13 of 58

is solely the exponential mass spectrum, and this result holds in leading order irrespective of the value of the power index a. Thus a surprisingly simple SBM-related criterion for the value of TH is that there cs 0. Moreover, cs is available in lattice-QCD computation.→ Figure 8 shows 2 cs as function of T , adapted from ref. [72]. There is a noticeable domain where cs is relatively small. In fig. 8 the bands show the computational uncertainty. To understand better the value of TH we follow the drop of 2 cs when temperature increases, and when cs begins to in- crease that is presumably, in the context of lattice-QCD, when the plasma material is mostly made of deconfined and progressively more mobile quarks and gluons. As tem- perature rises further, we expect to reach the speed of 2 2 Fig. 8. The square of speed of sound cs as function of temper- sound limit of ultra relativistic matter cs 1/3, indicated 2→ ature T , the relativistic limit is indicated by an arrow. Figure in fig. 8 by an arrow. This upper limit, cs 1/3arisesac- from ref. [72] modified for this review. cording to eq. (7) as long as the constraint≤ ε 3P 0 − → from above at high T applies; that is Im > 0andIm 0 at high T . → direction. For relativistic matter the speed of sound eq. (7) The behavior of the lattice result-bands in fig. 8 sug- approaches cs = c/√3, see fig. 8 and becomes small only gests hadron dominance below T = 125 MeV, and quark −22 near to TH. Within time τh 10 s a volume dilution dominance above T = 150 MeV. This is a decisively more ∝ by a factor 50 and more can be expected. narrow range compared to the wider one seen in the fit in It is likely that this expansion is too fast to allow which a mixed parton-hadron phase was used to describe hadron population to develop from the parton domain. lattice results [52]; see discussion in subsect. 2.5. 2 What this means is that for both the lattice-QGP in- The shape of cs in fig. 8 suggests that TH = 138 terpreted as parton-hadron mix, and for a HRG formed 12 MeV. There are many ramifications of such a low value,± in laboratory, the reaction time is too short to allow de- as is discussed in the context of hadronization model in velopment of a multi-structure hadron abundance equi- the following subsect. 2.7. librated state, which one refers to as “chemical” equili- brated hadron gas, see here the early studies in refs. [53– 55]. 2.7 What is the statistical hadronization model (SHM)? To conclude: lattice results allow various interpreta- tions, and HRG is a consistent simple approximation for The pivotal point leading on from the last subsection is T  145 MeV. More complex models which include coexis- that in view of fig. 6 we can say that HRG for T  TH tence of partons and hadrons manage a good fit to all lat- 150 MeV works well at a precision level that rivals the nu-≃ tice results, including the hard to get interaction measure merical precision of lattice-QCD results. This result jus- Im. Such models in turn can be used in developing dynam- tifies the method of data analysis that we call Statistical ical model of the QGP fireball explosion. One can argue Hadronization Model (SHM). SHM was invented to char- that the laboratory QGP cannot be close to the full chem- acterize how a blob of primordial matter that we call QGP ical equilibrium; a kinetic computation will be needed to falls apart into individual hadrons. At zero baryon density assess how the properties of parton-hadron phase evolve this “hadronization” process is expected to occur near if −22 given a characteristic lifespan of about τh 10 s. Such not exactly at TH. The SHM relies on the hypothesis that a study may be capable of justifying accurately∝ specific a hot fireball made of building blocks of future hadrons hadronization models. populates all available phase space cells proportional to their respective size, without regard to any additional in- teraction strength governing the process. 2.6 What does lattice-QCD tell us about TH? The model is presented in depth in sect. 9. Here we would like to place emphasis on the fact that the agree- We will see in subsect. 3.3 that we do have two dif- ment of lattice-QCD results with the HRG provides today ferent lattice results showing identical behavior at T a firm theoretical foundation for the use of the SHM, and 150 25 MeV. This suggests that it should be possible∈ it sets up the high degree of precision at which SHM can { ± } to obtain a narrow range of TH. Looking at fig. 6, some be trusted. see TH at 140–145 MeV, others as high as 170 MeV. Such Many argue that Koppe [56,57], and later, indepen- disparity can arise when using eyesight to evaluate fig. 6 dently, Fermi [58] with improvements made by Pomer- without applying a valid criterion. In fact such a criterion anchuk [59], invented SHM in its microcanonical format; is available if we believe in exponential mass spectrum. this is the so called Fermi-model, and that Hagedorn [11, When presenting critical properties of SBM table 1 60] used these ideas in computing within grand canon- we reported that sound velocity eq. (7) has the unique ical formulation. However, in all these approaches the property c 0forT T . What governs this result particles emitted were not newly formed; they were seen s → → H Page 14 of 58 Eur. Phys. J. A (2015) 51: 114 as already being the constituents of the fireball. Such a) A dense fireball disintegrates into hadrons. There models therefore are what we today call freeze-out mod- can be two temporally separate physical phenomena: els. the recombinant-evaporative hadronization of the fireball The difference between QGP hadronization and freeze- made of quarks and gluons forming a HRG; this is followed out models is that a priori we do not know if right at the by freeze-out; that is, the beginning of free-streaming of time of QGP hadronization particles will be born into a the newly created particles. condition that allows free-streaming and thus evolve in b) The quark fireball expands significantly before con- hadron form to the freeze-out condition. In a freeze-out verting into hadrons, reaching a low density before ha- model all particles that ultimately free-stream to a de- dronization. As a result, some features of hadrons upon tector are not emergent from a fireball but are already production are already free-streaming: i) The hadroni- present. The fact that the freeze-out condition must be zation temperature may be low enough to freeze-out parti- established in a study of particle interactions was in the cle abundance (chemical freeze-out at hadronization), yet early days of the Koppe-Fermi model of no relevance since elastic scattering can still occur and as result momentum the experimental outcome was governed by the phase distribution will evolve (kinetic non-equilibrium at hadro- space and microcanonical constraints as Hagedorn ex- nization). ii) At a yet lower temperature domain, hadrons plained in his very vivid account “The long way to the would be born truly free-streaming and both chemical and Statistical Bootstrap Model”, chapt. 17 loc. cit. kinetic freeze-out conditions would be the same. This con- In the Koppe-Fermi-model, as of the instant of their dition has been proposed for SPS yields and spectra in the formation, all hadrons are free-streaming. This is also year 2000 by Torrieri [62], and named “single freeze-out” Hagedorn’s fireball pot with boiling matter. This reaction in a later study of RHIC results [63,64]. view was formed before two different phases of hadronic matter were recognized. With the introduction of a sec- ond primordial phase a new picture emerges: there are no 2.8 Why value of TH matters to SHM analysis? hadrons to begin with. In this case in a first step quarks freeze into hadrons at or near TH, and in a second step What exactly happens in RHI collisions in regard to par- at T

Among the source (fireball) observables we note the fireball at a given T , V and to account for baryon content nearly conserved, in the hadronization process, entropy at low collision energies one adds oB. As Hagedorn found content, and the strangeness content, counted in terms out, the price of simplicity is that the yields can differ from of the emerging multiplicities of hadronic particles. The experiment by a factor two or more. His effort to resolve physical relevance of these quantities is that they origi- this riddle gave us SBM. nate, e.g. considering entropy or strangeness yield, at an However, in the context of experimental results that earlier fireball evolution stage as compared to the hadroni- need attention, one seeks to understand systematic be- zation process itself; since entropy can only increase, this havior across yields varying by many orders of magni- provides a simple and transparent example how in hadron tude as parameters (collision energy, impact parameter) abundances which express total entropy content there can of RHI collision change. So if a simple model practically be memory of the initial state dynamics. “works”, for many the case is closed. However, one finds Physical bulk properties such as the conserved (baryon in such a simple model study the value of chemical freeze- number), and almost conserved (strangeness pair yields, out T well above TH. This is so since in fitting abundant entropy yield) can be measured independent of how fast strange antibaryons there are two possible solutions: either the hadronization process is, and independent of the com- a T TH,orT  TH with chemical non-equilibrium. A model≫ with T T for the price of getting strange an- plexity of the evolution during the eventual period in time ≫ H while the fireball cools from TH to chemical freeze-out T . tibaryons right creates other contradictions, one of which We do not know how the bulk energy density ε and pres- is discussed in subsect. 10.4. sure P at hadronization after scaling with T 4 evolve in How comparison of chemical freeze-out T with TH time to freeze-out point, and even more interesting is how works is shown in fig. 9. The bar near to the tempera- I eq. (11) evolves. This can be a topic of future study. ture axis displays the range TH = 147 5 MeV [71,72]. m ± Once scattering processes came into discussion, the The symbols show the results of hadronization analysis concept of dynamical models of freeze-out of particles in the T –oB plane as compiled in ref. [73] for results in- could be addressed. The review of Koch et al. [2] com- volving most (as available) central collisions and heav- prises many original research results and includes for the iest nuclei. The solid circles are results obtained using first time the consideration of dynamical QGP fireball evo- the full SHM parameter set [7,73–79]. The SHARE LHC lution into free-streaming hadrons and an implementation freeze-out temperature is clearly below the lattice criti- of SHM in a format that we could today call SHM with cal temperature range. The results of other groups are sudden hadronization. In parallel it was recognized that obtained with simplified parameter sets: marked GSI [80, the experimentally observed particle abundances allow the 81], Florence [82–84], THERMUS [85], STAR [86] and AL- determination of physical properties of the source. This in- ICE [87,88]. These results show the chemical freeze-out sight is introduced in ref. [16], fig. 3 where we see how the temperature T in general well above the lattice TH. This ratio K+/K− allows the evaluation of the baryochemical means that these restricted SHM studies are incompatible with lattice calculations, since chemical hadron decoupling potential oB; this is stated explicitly in pertinent discus- sion. Moreover, in the following fig. 4 the comparison is should not occur inside the QGP domain. made between abundance of final state N/N particle ra- tio emerging from equilibrated HRG with abundance ex- pected in direct evaporation of the quark-fireball an effect 2.9 How is SHM analysis of data performed? that we attribute today to chemical non-equilibrium with enhanced phase space abundance. Here the procedure steps are described which need tech- Discussion of how sudden the hadronization process is nical implementation presented in sect. 9. reaches back to the 1986 microscopic model description Data: The experiment provides, within a well defined of strange (antibaryon) formation by Koch, M¨uller and collision class, see subsect. 9.3, spectral yields of many par- the author [2] and the application of hadron afterburner. ticles. For the SHM analysis we focus on integrated spec- Using these ideas in 1991, SHM model saw its first hum- tra, the particle number-yields. The reason that such data ble application in the study of strange (anti)baryons [65]. are chosen for study is that particle yields are independent Strange baryon and antibaryon abundances were inter- of local matter velocity in the fireball which imposes spec- preted assuming a fast hadronization of QGP —fast mean- tra deformation akin to the Doppler shift. However, if the ing that their relative yields are little changed in the fol- p⊥ coverage is not full, an extrapolation of spectra needs lowing evolution. For the past 30 years the comparison to be made that introduces the same uncertainty into the of data with the sudden hadronization concept has never study. Therefore it is important to achieve experimentally led to an inconsistency. Several theoretical studies support as large as possible p⊥ coverage in order to minimize ex- the sudden hadronization approach, a sample of works in- trapolation errors on particles yields considered. cludes refs. [66–70]. Till further notice we must presume that the case has been made. Evaluation: In first step we evaluate, given an assumed Over the past 35 years a simple and naive ther- SHM parameter set, the phase space size for all and ev- mal model of particle production has resurfaced multiple ery particle fraction that could be in principle measured, times, reminiscent of the work of Hagedorn from the early including resonances. This complete set is necessary since 60s. Hadron yields emerge from a fully equilibrated hadron the observed particle set includes particles arising from Page 16 of 58 Eur. Phys. J. A (2015) 51: 114

Fig. 9. T , oB diagram showing current lattice value of critical temperature Tc (bar on left) [71,72], the SHM-SHARE results (full circles) [7,73–79] and results of other groups [80–88]. Figure from ref. [73] modified for this review.

a sequel chain of resonance decays. These decays are im- use all these yields in order to compute the bulk prop- plemented and we obtain the relative phase space size of erties of the fireball source, where the statement is exact all potential particle yields. for the conserved quantities such as net baryon number (baryons less antibaryons) and approximate for quantities Optional: Especially should hadronization T be at a where kinetic models show little modification of the value relatively large value, the primary particle populations can during hadronization. An example here is the number of undergo modifications in subsequent scattering. However, strange quark pairs or entropy. since T

Fig. 11. Illustration of the heavy quark Q = c, b and antiquark Fig. 10. Illustration of the quark bag model colorless states: Q =¯c, ¯b connected by a color field string. As QQ separate, a baryons qqq and mesons qq. The range of the quantum wave pair of light quarks qq¯ caps the broken field-string ends. function of quarks, the hadronic radius is indicated as a (pink) cloud, the color electrical field lines connect individual quarks. quark-hadron wave functions in a localized bound state were obtained; for a succinct review see Johnson [98]. 3 The concepts: Theory quark side The later developments which address the chiral symme- try are summarized in 1982 by Thomas [99], completing 3.1 Are quarks and gluons “real” particles? the model. The question to be addressed in our context is: How can The quark-bag model works akin to the localization of quarks and gluons be real particles and yet we fail to pro- quantum states in an infinite square-well potential. A new duce them? The fractional electrical charge of quarks is a ingredient is that the domain occupied by quarks and/or strong characteristic feature and therefore the literature is their chromo-electrical fields has a higher energy density called bag constant : the deconfined state is the state full of false discoveries. Similarly, the understanding and B explanation of quark confinement has many twists and of higher energy compared to the conventional confining turns, and some of the arguments though on first sight vacuum state. In our context an additional finding is im- contradictory are saying one and the same thing. Our portant: even for small physical systems comprising three present understanding requires the introduction of a new quarks and/or quark-antiquark pairs once strangeness is paradigm, a new conceptual context how in comparison to correctly accounted for, only the volume energy density without a “surface energy” is present. This was shown the other interactions the outcome of strong interactions B is different. by an unconstrained hadron spectrum model study [100, A clear statement is seen in the September 28, 1979 101]. This result confirms the two vacuum state hypothe- lecture by T.D. Lee [93] and the argument is also presented sis as the correct picture of quark confinement, with non- in T.D. Lee’s textbook [94]: at zero temperature quarks analytical structure difference at T = 0 akin to what is can only appear within a bound state with other quarks expected in a phase transition situation. as a result of transport properties of the vacuum state, The reason that in the bag model the color-magnetic and not as a consequence of the enslaving nature of inter- hyperfine interaction dominates the color-electric interac- quark forces. However, indirectly QCD forces provide the tion is due to local color neutrality of hadrons made of vacuum structure, hence quarks are enslaved by the same light quarks; the quark wave function of all light quarks fill QCD forces that also provide the quark-quark interaction. the entire bag volume in same way, hence if the global state Even so the conceptual difference is clear: we can liberate is colorless so is the color charge density in the bag. How- quarks by changing the nature of the vacuum, the modern ever, the situation changes when considering the heavy day æther, melting its confining structure. charm c, or bottom b, quarks and antiquarks. Their mass The quark confinement paradigm is seen as an ex- scale dominates, and their semi-relativistic wave functions pression of the incompatibility of quark and gluon color- are localized. The color field lines connecting the charges electrical fields with the vacuum structure. This insight are, however, confined. When we place heavy quarks rela- was inherent in the work by Ken Wilson [95] which was the tively far apart, the field lines are, according to the above, backdrop against which an effective picture of hadronic squeezed into a cigar-like shape, see top of fig. 11. structure, the “bag model” was created in 1974/1975 [96– The field occupied volume grows linearly with the size 99]. Each hadronic particle is a bubble [96]. Below TH, with of the long axis of the cigar. Thus heavy quarks interact their color field lines expelled from the vacuum, quarks when pulled apart by a nearly linear potential, but only can only exist in colorless cluster states: baryons qqq (and when the ambient temperature T

The energy per length of the string, the string tension, is precisely; a value of αs at large distance cannot be mea- nearly 1 GeV/fm. This value includes the modification of sured given the confinement paradigm. the vacuum introduced by the color field lines. There are serious issues that have impacted the capa- For T>TH the field lines can spread out and mix bility of the lattice-QCD in the past. One is the problem with thermally produced light quarks. However, unlike of Fermi-statistics which is not easily addressed by classi- light hadrons which melt at TH, the heavy QQ¯ mesons (of- cal computers. Another is that the properties we wanted ten referred as “onium states, like in charmonium cc¯)may to learn about depend in a decisive way on the inclusion remain bound, albeit with different strength for T>TH. of quark flavors, and require accurate value of the mass of Such heavy quark clustering in QGP has been of profound the strange quark; the properties of QCD at finite T are interest: it impacts the pattern of production of heavy par- very finely tuned. Another complication is that in view ticles in QGP hadronization [102,103]. Furthermore, this of today’s achievable lattice point and given the quark-, is a more accessible model of what happens to light quarks and related hadron-, scales, a lattice must be much more in close vicinity of TH, where considerable clustering be- finely spaced than was believed necessary 30 years ago. Se- fore and during hadronization must occur. rious advances in numerical and theoretical methods were The shape of the heavy quark potential, and thus the needed, see e.g. refs. [9,29]. stability of “onium states can be studied as a function of Lattice capability is limited by how finely spaced lat- quark separation, and of the temperature, in the frame- tice points in terms of their separation must be so that work of lattice-QCD, showing how the properties of the over typical hadron volume sufficient number is found. heavy quark potential change when deconfinement sets in Therefore, even the largest lattice implemented at present for T>TH [104,105]. cannot “see” any spatial structure that is larger than a few To conclude, quarks and gluons are real particles and proton diameters, where for me: few = 2. The rest of the can, for example, roam freely above the vacuum melting Universe is, in the lattice approach, a periodic repetition point, i.e. above Hagedorn temperature TH. This under- of the same elementary cell. standing of confinement allows us to view the quark-gluon The reason that lattice at finite temperature cannot plasma as a domain in space in which confining vacuum replace models in any foreseeable future is the time evolu- structure is dissolved, and chromo-electric field lines can tion: temperature and time are related in the theoretical exist. We will return to discuss further ramifications of the formulation. Therefore considering hadrons in a heat bath QCD vacuum structure in subsects. 7.2 and 7.3. 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 3.2 Why do we care about lattice-QCD? T = 0. Thus all we can hope for in hot-lattice-QCD is what we see in this article, possibly much refined in un- The understanding of quark confinement as a confinement derstanding of internal structure, correlations, transport of the color-electrical field lines and characterization of coefficient evaluation, and achieved computational preci- hadrons as quark bags suggests as a further question: how sion. can there be around us, everywhere, a vacuum structure After this description some may wonder why we should that expels color-electric field lines? Is there a lattice- bother with lattice-QCD at all, given on one hand its limi- QCD based computation showing color field lines confine- tations in scope, and on another the enormous cost rivaling ment? Unfortunately, there seems to be no answer avail- the experimental effort in terms of manpower and com- able. Lattice-QCD produces values of static observables, puter equipment. The answer is simple; lattice-QCD pro- and not interpretation of confinement in terms of moving vides what model builders need, a reference point where quarks and dynamics of the color-electric field lines. models of reality meet with solutions of theory describing So why care about lattice-QCD? For the purpose of the reality. this article lattice-QCD upon convergence is the ultimate We have already by example shown how this works. In authority, resolving in an unassailable way all questions the previous sect. 2 we connected in several different ways pertinent to the properties of interacting quarks and glu- the value of TH to lattice results. It seems clear that the in- ons, described within the framework of QCD. The word terplay of lattice, with experimental data and with models lattice reminds us how continuous space-time is repre- can fix TH with a sufficiently small error. A further similar sented in a discrete numerical implementation on the most situation is addressed in the following subsect. 3.3 where powerful computers of the world. we seek to interpret the lattice results on hot QCD and The reason that we trust lattice-QCD is that it is not to understand the properties of the new phase of matter, a model but a solution of what we think is the founda- quark-gluon plasma. tional characterization of the hadron world. Like in other theories, the parameters of the theory are the measured properties of observed particles. In case of QED we use 3.3 What is quark-gluon plasma? the Coulomb force interaction strength at large distance, α = e2/c =1/137. In QCD the magnitude of the strength An artist’s view of Quark-Gluon Plasma (QGP), fig. 12, 2 of the interaction αs = g /c is provided in terms of a shows several quarks and “springy” gluons —in an image scale, typically a mass that the lattice approach captures similar to fig. 10. It is common to represent gluons by Eur. Phys. J. A (2015) 51: 114 Page 19 of 58

Fig. 13. The number of degrees of freedom g∗ as function Fig. 12. Illustration of quark-gluon plasma (QGP) compris- of temperature T . The solid line includes the effect of QCD ing several red, green, blue (RGB colored) quarks and springy interactions as obtained within the framework of lattice QCD gluons in a modified vacuum state. by Borsanyi et al. beginning in 2012 and published in 2014 (see text). Horizontal dashed line: g =47.5 for free quark-gluon gas. springs, a historical metaphor from times when we viewed gluons as creating a force that grew at a distance so as to 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 combination, and excluding the “white” case gluons: we have 8 bi-colors of gluons. As this is hard to illustrate, gB =2s 8c =16. (14) × these springs are gray. The domain of space comprising The corresponding T 4 Fermi (quark) Stefan-Boltzmann quarks and gluons is colored to indicate that we expect term differs by the well known factor 7/8 for each degree this to be a much different space domain from the sur- of freedom. We count particles and antiparticles as degrees roundings. of freedom: In a nutshell, QGP in the contemporary use of the lan- guage is an interacting localized assembly of quarks and g =2 2 3 (2 + 1) =31.5, (15) gluons at thermal (kinetic) and (close to) chemical (abun- F s × p × c × f dance) equilibrium. The word “plasma” signals that free where indices stand for: s = spin (= 2), p-particle and color charges are allowed. Since the temperature is above antiparticle (= 2), c = color (= 3, or = 8), f-flavor: 2 TH and thus above the scale of light quark u, d-mass, the flavors q = u, d always satisfy m T and one flavor pressure exhibits the relativistic Stefan-Boltzmann for- q ≪ (strangeness s) at phase boundary satisfies ms  T and mat, turns into a light flavor at high temperatures. To make 4 2 2 4 sure this situation is remembered we write (2 + 1)f . ∗ 7 ∗ (πT) ∗∗ (πT) μ μ P = gB + gF +gF + . (13) The analytical and relatively simple form of the first 8 90π2 24π2 48π2     order in (α ) thermal QCD perturbative correction re- O s The stars next to degeneracy g for Bosons B, and Fermions sults are given in analytical format in the work of Chin in F, indicate that these quantities are to be modified by 1979 [107], and result in the following degeneracy: the QCD interaction which affects this degeneracy signifi- cantly, and differently for B, F and also fort the two terms 7 15αs g g∗ + g∗ =2 8 1 ∗F vs. ∗∗F. ∗ ≡ B 8 F s × c − 4π In eq. (13) the traditional Stefan-Boltzmann T 4 terms,   7 50αs and the zero temperature limit quark-chemical potential + 2 2 3 (2 + 1) 1 (16) 8 s × p × c × f − 21π μ4 term are well known and also easy to obtain by inte-   grating the Bose/Fermi gas expressions in the respective ∗∗ 2αs gF =2s 2p 3c (2 + 1)f 1 . limit. The ideal (QCD interaction αs = 0) relativistic hot × × × − π quark gas at finite T including the T 2μ2 term in explicit   analytical form of the expression was for the first time pre- αs is the QCD energy scale dependent coupling con- sented by Harrington and Yildiz in 1974 [106] in a work stant. In the domain of T we consider αs 0.5, but it which has the telling title “High-Density Phase Transi- is rapidly decreasing with T . To some extent≃ this is why tions in Gauge Theories”. in fig. 13, showing the lattice-QCD results for pressure, Page 20 of 58 Eur. Phys. J. A (2015) 51: 114

we see a relatively rapid rise of g∗ as a function of T , to- review is to rely on theoretical insights. As the results wards the indicated limit g =47.5 of a free gas, horizontal of lattice-QCD demonstrate, the “quark-gluon plasma” is dashed line. In fig. 13 three results also depict the path to a phase of matter comprising color charged particles (glu- the current understanding of the value of TH. The initial ons and quarks) that can move nearly freely so as to cre- results (triangles)were presented by Bazavov 2009 [108]; ate ambient pressure close to the Stefan-Boltzmann limit this work did not well describe the “low” temperature do- and whose motion freezes into hadrons across a narrow main where the value of TH is determined. This is the temperature domain characteristic of the Hagedorn tem- origin of the urban legend that TH 190 MeV, and a lot perature TH. The properties of QGP that we check for are of confusion. ≃ thus: The solid line in fig. 6 shows Borsanyi et al. 2012 [71] results presented at the Quark Matter 2012 meeting, with 1) Kinetic equilibrium —allowing a meaningful definition later formal paper comprising the same results [72]; these of temperature. are the same results as we see in fig. 6 connecting at low 2) Dominance by effectively massless particles assuring that P T 4. T with HRG. These results of the Wuppertal-Budapest ∝ group at first contradicted the earlier and highly cited 3) Both quarks in their large number, and gluons, must be result of [108]. However, agreement between both lattice present in conditions near chemical (yield) equilibrium groups was restored by the revised results of Bazavov 2014 with their color charge “open” so that the count of (HotQCD Collaboration) [109]. their number produces the correctly modified Stefan- Boltzmann constant of QCD. Let us also remember that the low value of TH we obtained at the end of subsect. 2.6 is due to the difference seen below in fig. 13 between the results of 2009, and those 3.4 How did the name QGP come into use? reported a few years later, through 2014. This difference is highly relevant and shows that hadrons melt into quarks In this article we use practically always the words Quark- near to T = 138 12 MeV corresponding to 4 300 MeV domain a decade ago, it became ap- numerical lattice simulations with ever increasing accu- parent that an accurate understanding of g∗ emerges [110] racy. However, even today there is a second equivalent by taking (α ) corrections literally and evaluating the O s name; the series of conferences devoted to the study of behavior αs(T ). A more modern study of the behavior quark-gluon plasma formation in laboratory calls itself of thermal QCD and its comparison with lattice QCD is “Quark Matter”. In 1987 L´eon Van Hove (former scientific available [111–113]. The thermal QCD explains the dif- director general of CERN) wrote a report entitled “The- ference between the asymptotic value g =47.5 and lat- oretical prediction of a new state of matter, the “quark- tice results which we see in fig. 13 to be significant at gluon plasma” (also called “quark matter”)” [114] estab- the highest T = 400 considered. In fact thermal quarks lishing the common meaning of these two terms. are never asymptotically free; asymptotic freedom for hot When using Quark Matter we can be misunderstood QCD matter quarks suffers from logarithmic behavior. to refer to zero-temperature limit. That is why QGP αs(T ) drops slowly and even at the thermal end of the seems the preferred term. However, to begin, QGP ac- standard model T 150, 000 MeV the QCD interaction tually meant something else. This is not unusual; quite remains relevant and→ g /g 0.9. This of course is also ∗ ≃ often in physics in the naming of an important new in- true for very high density cold QCD matter, a small dis- sight older terms are reused. This phenomenon reaches appointment when considering the qualitative ideas seen back to antiquity: the early ancient Greek word “Chaos” in the work of Collins and Perry [35]. at first meant “emptiness”. The science of that day con- We can conclude by looking at high T domains of all cluded that emptiness would contain disorder, and the these results that the state of strongly interacting mat- word mutated in its meaning into the present day use. ter at T 4TH is composed of the expected number of At first QGP denoted a parton gas in the context of pp ≃ nearly free quarks and gluons, and the count of these par- collisions; Hagedorn attributes this to Bjorken 1969, but ticles in thermal-QCD and lattice-QCD agree. We can say I could not find in the one paper Hagedorn cited the ex- that QGP emerges to be the phase of strongly interacting plicit mention of “QGP” (see chapt. 25 in [1]). Shuryak in matter which manifests its physical properties in terms of 1978 [115] used “QGP” in his publication title addressing nearly free dynamics of practically massless gluons and partons in pp collisions, thus using the language in the old quarks. The “practically massless” is inserted also for glu- fashion. ons as we must remember that in dense plasma matter all Soon after “QGP” appears in another publication title, color charged particles including gluons acquire an effec- in July 1979 work by Kalashnikov and Klimov [116], now tive in medium mass. describing the strongly interacting quark-gluon thermal It seems that today we are in control of the hot QCD equilibrium matter. This work did not invent what the matter, but what properties characterize this QGP that authors called QGP. They were, perhaps inadvertently, differ in a decisive way from more “normal” hadron mat- connecting with the term used by others in another con- ter? It seems that the safest approach in a theoretical text giving it the contemporary meaning. The results of Eur. Phys. J. A (2015) 51: 114 Page 21 of 58

Kalashnikov-Klimov agree with our eq. (16) attributed to a year earlier, July 1978, work of S.A. Chin [107] pre- sented under the title Hot Quark Matter. This work (de- spite the title) included hot gluons and their interaction with quarks and with themselves. But QGP in its new meaning already had deeper roots. Quark-star models [117] appear as soon as quarks are pro- posed; “after” gluons join quarks [118], within a year – Peter Carruthers in 1973/74 [119] recognized that dense quark matter would be a quite “bizarre” plasma and he explores its many body aspects. His paper has priority but is also hard to obtain, published in a new journal that did not last. – A theory of thermal quark matter is that of Harring- Fig. 14. Ink painting masterpiece 1986: Nuclei as Heavy as ton and Yidliz 1974 [106], but has no discussion of the Bulls, Through Collision Generate New States of Matter by Li role of gauge interaction in quantitative terms. This Keran, reproduced from open source works of T.D. Lee. paper is little known in the field of RHI collisions yet it lays the foundation for the celebrated work by Linde on electroweak symmetry restoration in the early Uni- use of the QGP acronym in the title, while Shuryak’s verse [120]. There is a remarkable bifurcation in the “true” QGP paper, Theory of Hadron Plasma is often not literature: those who study the hot Universe and its cited in this context. In his 1980 review Shuryak [124] is early stages use the same physics as those who explore almost shifting to QGP nomenclature, addressing “QCD the properties of hot quarks and gluons; yet the cross- Plasma” and also uses in the text “Quark Plasma”, omit- citations between the two groups are sparse. ting to mention “gluons” which are not established ex- – Collins and Perry 1975 [35] in Superdense Matter: Neu- perimentally for a few more years. In this he echoes the trons or Asymptotically Free Quarks propose that high approach of others in this period. density nuclear matter turns into quark matter due to Having said all the above, it is clear that when “QGP” weakness of asymptotically free QCD. Compared to is mentioned as the theory of both hot quarks and hot Harrington and Yidliz this is a step back to a zero- gluons, we should remember Kalashnikov-Klimov [116] for temperature environment, yet also a step forward as as I said, the probably inadvertent introduction of this the argument that interaction could be sufficiently name into its contemporary use. weak to view the dense matter as a Fermi gas of quarks is explicitly made. 4 Quark-gluon plasma in laboratory Following this there are a few, at times parallel develop- ments —but this is not the place to present a full history 4.1 How did RHI collisions and QGP come together? of the field. However fragmentary, let me mention instead those papers I remember best: The artistic representation of RHI collisions is seen in fig. 14 —two fighting bulls. The ink masterpiece was – Freedman and McLerran 1976/77 [121] who address created in 1986 by Li Keran and has been around the field the thermodynamic potential of an interacting rela- of heavy ions for the past 30 years, a symbol of nascent tivistic quark gas. symbiosis between science and art, and also a symbol of – Shuryak 1977/78 [122], writes about Theory of Hadron great friendship between T.D. Lee and Li Keran. The bulls Plasma developing the properties of QGP in the frame- are the heavy ions, and the art depicts the paradigm of work of QCD. heavy-bull(ion) collisions. – Kapusta 1978/79 [123] which work completes Quan- So how did the bulls aka heavy ions connect to QGP? tum Chromodynamics at High Temperature. In October 1980, I remarked in a citation3 “The possible – Chin 1978 [107] synthesized all these results and was formation of quark-gluon plasma in nuclear collisions was the first to provide the full analytical first order α s first discussed quantitatively by S.A. Chin: Phys. Lett. B corrections as seen in eq. (16). 78, 552 (1978); see also N. Cabibbo, G. Parisi: Phys. Lett. However, in none of the early thermal QCD work is the B 59, 67 (1975)”. Let me refine this: acronym “QGP”, or spelled out “Quark-Gluon Plasma” a) The pioneering insight of the work by Cabibbo and introduced. So where did Kalashnikov-Klimov [116] get Parisi [36] is: i) to recognize the need to modify SBM the idea to use it? I can speculate that seeing the work to include melting of hadrons and ii) in a qualitative by Shuryak on Theory of Hadron Plasma they borrowed the term from another Shuryak paper [115] where he used 3 The present day format requirement means that these “QGP” in his title addressing partons in pp collisions. In- words are now found in the text of ref. [15], end of 3rd para- deed, in an aberration of credit Shuryak’s pp parton work graph below eq. (61), so that each of the two references can be is cited in AA QGP context, clearly in recognition of the cited and hyperlinked as a separate citation item. Page 22 of 58 Eur. Phys. J. A (2015) 51: 114

drawing to recognize that both high temperature and the time quarks were not part of nuclear physics which baryon density allow a phase transformation process. “owned” the field of heavy ions. Judging by personal ex- However, there is no mention direct or indirect in this perience I am not really surprised that Chapline-Kerman work about “bull” collision. work was not published. Planck was dead for 30 years4.It b) The paper by Chin [107], of July 1978, in its ref. [7] is regrettable that once Chapline-Kerman ran into resis- grants the origin of the idea connecting RHI with QGP tance they did not pursue the publication, and/or further to Chapline and Kerman [125], an unpublished manu- development of their idea; instead, script entitled On the possibility of making quark mat- ter in nuclear collisions of March 1978. This paper a) A year later, Kerman (working with Chin who gave clearly states the connection of QGP and RHI col- him the credit for the QGP-RHI connection idea in his lisions that Chin explores in a quantitative fashion paper), presents strangelets [131], cold drops of quark recomputing the QCD thermodynamic potential, and matter containing a large strangeness content. enclosing particles, quarks and gluons, in the bag-like b) And a few years later, Chapline [132] gives credit for structure, see sect. 3.1. the quark-matter connection to RHI collisions both to Chapline-Kerman [125] work, and the work of An- The preprint of Chapline and Kerman is available on- ishetty, Koehler, and McLerran of 1980 [133]. An- line at MIT [125]. It is a qualitative, mostly conceptual ishetty et al. claim in their abstract idea paper, a continuation of an earlier effort by Chapline . . . two hot fireballs are formed. These fireballs and others in 1974 [126] where we read (abstract): would have rapidities close to the rapidities of the original nuclei. We discuss the possible for- It is suggested that very hot and dense nuclear mat- mation of hot, dense quark plasmas in the fire- ter may be formed in a transient state in “head-on” balls. collisions of very energetic heavy ions with medium and heavy nuclei. A study of the particles emitted That Anishetty, Koehler, and McLerran view of RHI colli- in these collisions should give clues as to the nature sion dynamics is in direct conflict with the effort of Hage- of dense hot nuclear matter. dorn to describe particle production in pp collisions which at the time was being adapted to the AA case and pre- At the time of the initial Chapline effort in 1974 it was sented e.g. in the QM1-report [134]. too early for a mention of quark matter and heavy ions Anishetty et al. created the false paradigm that QGP in together. Indeed, at the Bear Mountain [127] workshop was not produced centrally (as in center of momentum), in Fall 1974 the physics of the forthcoming RHI collisions a point that was corrected a few years later in 1982/83 was discussed in a retreat motivated by Lee-Wick [128] in the renowned paper of J.D. Bjorken [135]. He obtained matter, a proposed new state of nuclear matter. These an analytical, one dimensional, solution of relativistic hy- authors claim: drodynamics that could be interpreted for the case of the RHI collision as description at asymptotically high energy ... the state ... inside a very heavy nucleus can of the collision events. If so, the RHI collision outcome become the minimum-energy state, at least within would be a trail of energy connecting the two nuclei that the tree approximation; in such a state, the “effec- naturally qualifies to be the QGP. While this replaced the tive” nucleon mass inside the nucleus may be much Anishetty, Koehler, and McLerran “cooking nuclei, noth- lower than the normal value. ing in-between” picture, this new asymptotic energy idea also distracted from the laboratory situation of the period In presenting this work, the preeminent theorists T.D. which had to deal with realistic, rather than asymptotic Lee and G.C. Wick extended an open invitation to explore collision energies. in relativistic heavy ion collisions the new exotic state of In that formative period I wrote papers which argued dense nuclear matter. This work generated exciting scien- that the hot, dense QGP fireball would be formed due to tific prospects for the BEVELAC accelerator complex at hadron inelasticity stopping some or even all of nuclear Berkley. We keep in mind that there is no mention of quark matter in the center of momentum frame (CM). However, matter in any document related to BEVELAC [129], nor my referees literally said I was delusional. As history has at the Bear Mountain workshop [127]. However, the en- shown (compare Chapline and Kerman) referees are not suing experimental search for the Lee-Wick nuclear mat- always useful. The long paper on the topic of forming QGP ter generated the experimental expertise and equipment at central rapidity was first published 20 years later in needed to plan and perform experiments in search of the memorial volume dedicated to my collaborator on this quark-gluon plasma [130]. And, ultimately, T.D. Lee will project, Michael Danos [136]. turn to recognize QGP as the new form of hot nuclear Here it is good to remember that the CERN-SPS dis- matter resulting, among other things, in the very beauti- covery story relies on the formation of a baryon-rich QGP ful painting by Li Keran, fig. 14. in the CM frame of reference i.e. at “central rapidity”. Now back to the March 1978 Chapline-Kerman manu- script: why was it never published? There are a few pos- 4 Many credit Planck with fostering an atmosphere of open- sible answers: a) It is very qualitative; b) In the 5y run ness and tolerance as a publisher; certainly he did not hesitate up period 1973–1978 the field of RHI collisions was dom- to take responsibility for printing Einstein miraculous 1905 pa- inated by other physics such as Lee-Wick. In fact at pers. Eur. Phys. J. A (2015) 51: 114 Page 23 of 58

Fig. 15. Hagedorn in September 1995 awaiting QGP discovery, see text.

RHIC is in transition domain in energy, and LHC energy scale, finally and 30 years later, is near to the Bjorken “scaling” limit. The word scaling is used, as we should in a rather wide range of rapidity observe the same state of Fig. 16. The press release text: “At a special seminar on hot QGP, a claim still awaiting an experiment. 10 February 2000, spokespersons from the experiments on CERN’s Heavy Ion program presented compelling evidence for To close the topic, some regrets: an “idea” paper equiv- the existence of a new state of matter in which quarks, instead alent to ref. [125] introducing the bootstrap model of hot of being bound up into more complex particles such as protons finite sized hadron matter and transformation into QGP and neutrons, are liberated to roam freely.” in RHI collisions could have been written by Hagedorn and myself in late 1977. Hagedorn, however, desired a work- ing model. After 10 months of telling the world about our work, and much further effort in Summer 1978 we wrote the CERN relativistic heavy ion program, started with I. Montvay a 99 page long paper [137], as well as a in the mid eighties, to the heaviest naturally occur- few months later a much evolved shorter version [28]. ring nuclei. A run with lead beam of 40 GeV per nu- Only in the Spring of 1980 was Hagedorn sure we un- cleon in fall of 1999 complemented the program to- derstood the SBM and the hadron melting into QGP in wards lower energies. Seven large experiments par- RHI. Of course we were looking at central rapidity i.e. ticipate in the lead beam program, measuring many CM system, quite different from the work of Anishetty different aspects of lead-lead and lead-gold colli- et al. [133]. Hagedorn explains the time line of our and sion events: NA44, NA45/CERES, NA49, NA50, related work in his 1984 review [13]. His later point of NA52/NEWMASS, WA97/NA57, and WA98. . . . view is succinctly represented in a letter of September Physicists have long thought that a new state of 1995, fig. 15, where he says5: “... can I hope to witness matter could be reached if the short range repulsive a proof of existence of QG plasma? I am in any case con- forces between nucleons could be overcome and if vinced of its existence, where else could the phase transi- squeezed nucleons would merge into one another. tion (which with certainty is present) lead?. . . ”. Present theoretical ideas provide a more precise picture for this new state of matter: it should be a quark-gluon plasma (QGP), in which quarks and 4.2 When and where was QGP discovered? gluons, the fundamental constituents of matter, are no longer confined within the dimensions of the nu- Both CERN and BNL have held press conferences describ- cleon, but free to move around over a volume in ing their experimental work. In fig. 16 a screen shot shows which a high enough temperature and/or density how CERN advertised its position in February 2000 to a prevails. . . . (explicative in original:) A common wider public [138]. The document for scientists agreed to assessment of the collected data leads us to by those representing the seven CERN experiments (see conclude that we now have compelling evi- the time line of CERN-SPS experiments in fig. 1) provided dence that a new state of matter has indeed at the event read: been created, .... The new state of mat- ter found in heavy ion collisions at the SPS “The year 1994 marked the beginning of the CERN features many of the characteristics of the lead beam program. A beam of 33 TeV (or 160 GeV theoretically predicted quark-gluon plasma per nucleon) lead ions from the SPS now extends .... In spite of its many facets the resulting pic- 5 German original: . . . werde ich noch den eindeutigen Nach- ture is simple: the two colliding nuclei deposit en- weis der Existenz des QGP erleben? Ich bin sowieso davon ergy into the reaction zone which materializes in uberzeugt¨ denn wohin soll der Phasen¨ubergang (den es doch the form of quarks and gluons which strongly in- sicher gibt) sonst f¨uhren? teract with each other. This early, very dense state Page 24 of 58 Eur. Phys. J. A (2015) 51: 114

(energy density about 3–4 GeV/fm3, mean par- ticle momenta corresponding to T 240 MeV) suppresses the formation of charmonia,≈ enhances strangeness and begins to drive the expansion of thefireball....” BNL presented the following comment [139] The CERN results are quite encouraging, says Tom Ludlam, Brookhaven’s Deputy Associate Director for High-Energy and Nuclear Physics. “These re- sults set the stage for the definitive round of exper- iments at RHIC in which the quark-gluon plasma will be directly observed, opening up a vast land- scape for discovery regarding the nature and origins of matter.” Brookhaven’s Director John Marburger congratu- lated CERN scientists on their achievement, stat- ing that “piecing together even this indirect evi- dence 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 Fig. 17. The cover of the BNL-73847-2005 Formal Report pre- Physical Society, held in Tampa, Florida a press confer- pared by the Brookhaven National Laboratory, on occasion of the RHIC experimental program press conference April 2005. ence took place on Monday, April 18, 9:00 local time. The The cover identified the four RHIC experiments. public announcement of this event was made April 4, 2005:

at the time: BRAHMS, PHOBOS, PHENIX, and STAR, EVIDENCE FOR A NEW TYPE OF NUCLEAR which reported on the QGP physical properties that have MATTER At the Relativistic Heavy Ion Collider been discovered in the first three years of RHIC oper- (RHIC) at Brookhaven National Lab (BNL), two ations. These four experimental reports were later pub- beams of gold atoms are smashed together, the goal lished in an issue of Nuclear Physics A [140–143]. being to recreate the conditions thought to have The 10 year anniversary was relived at the 2015 RHIC prevailed in the universe only a few microseconds & AGS Users’ Meeting, June 9-12, which included a spe- after the big bang, so that novel forms of nuclear cial celebration session “The Perfect Liquid at RHIC: 10 matter can be studied. At this press conference, Years of Discovery”. Berndt M¨uller, the 2015 Brookhaven’ RHIC scientists will sum up all they have learned Associate Laboratory Director for Nuclear and Particle from several years of observing the worlds most Physics is quoted as follows [144]: energetic collisions of atomic nuclei. The four ex- perimental groups operating at RHIC will present “RHIC lets us look back at matter as it existed a consolidated, surprising, exciting new interpreta- throughout our universe at the dawn of time, be- tion of their data. Speakers will include: Dennis Ko- fore QGP cooled and formed matter as we know it, var, Associate Director, Office of Nuclear Physics, ... The discovery of the perfect liquid was a turn- U.S. Department of Energy’s Office of Science; Sam ing point in physics, and now, 10 years later, RHIC Aronson, Associate Laboratory Director for High has revealed a wealth of information about this re- Energy and Nuclear Physics, Brookhaven National markable substance, which we now know to be a Laboratory. Also on hand to discuss RHIC results QGP, and is more capable than ever of measuring and implications will be: Praveen Chaudhari, Di- its most subtle and fundamental properties.” rector, Brookhaven National Laboratory; represen- An uninvolved scientist will ask: “Why is the flow prop- tatives of the four experimental collaborations at erty of QGP: a) Direct evidence of QGP and b) Worth full the Relativistic Heavy Ion Collider; and several the- scientific attention 15 years after the new phase of matter oretical physicists. was announced for the first time?” Berndt M¨uller answers The participants at the press conference each obtained a for this article: “Hunting for Quark-Gluon Plasma” report, of which the Nuclear matter at “room temperature” is known to cover in fig. 17 shows the four BNL experiments operating behave like a superfluid. When heated the nuclear Eur. Phys. J. A (2015) 51: 114 Page 25 of 58

fluid evaporates and turns into a dilute gas of nu- cleons and, upon further heating, a gas of baryons and mesons (hadrons). But then something new happens; at TH hadrons melt and the gas turns back into a liquid. Not just any kind of liquid. At RHIC we have shown that this is the most perfect liquid ever observed in any laboratory experiment at any scale. The new phase of matter consisting of dissolved hadrons exhibits less resistance to flow than any other substance known. The experiments at RHIC have a decade ago shown that the Uni- verse at its beginning was uniformly filled with a new type of material, a super-liquid, which once Universe cooled below TH evaporated into a gas of hadrons. Detailed measurements over the past decade have shown that this liquid is a quark-gluon plasma; i.e. matter in which quarks, antiquarks and gluons flow Fig. 18. Multistrange (anti)baryons as signature of QGP, see independently. There remain very important ques- text for further discussion. tions we need to address: What makes the inter- acting quark-gluon plasma such a nearly perfect liquid? How exactly does the transition to confined QGP fireball increases as the collision volume increases quarks work? Are there conditions under which the and/or the energy increases. Since the gluon fusion GG transition becomes discontinuous first-order phase ss¯ dominates quark flavor conversion qq¯ ss¯ the abun-→ transition? Today we are ready to address these dance of strangeness is signature of the→ formation of a questions. We are eagerly awaiting new results from thermal gluon medium. the upgraded STAR and PHENIX experiments at Of course we need to ask, how come there is a gluon RHIC. 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. 4.3 How did the SPS-QGP announcement withstand Equilibration means entropy S production, a topic of sepa- the test of time? rate importance as S production is proceeding in temporal sequence other hadronic observables of QGP, and how en- It is impossible to present in extensive manner in this re- tropy is produced remains today an unresolved question, view all the physics results that have driven the SPS an- see subsect. 5.2. nouncement, and I will not even venture into the grounds The gluon based processes are driving the equilibra- of the RHIC announcement. I will focus here instead on tion of quarks and antiquarks; first light q = u, d, next what I consider my special expertise, the strangeness sig- the slightly massive s and also some thermal evolution nature of QGP. The events accompanying the discovery of charm is possible. Strangeness evolves along with the and development of strangeness signature of QGP more light (u, u,¯ d, d¯) quarks and gluons G until the time of ha- than 30 years ago have been reported [18], and the first dronization, when these particles seed the formation of extensive literature mention of strangeness signature of hadrons observed in the experiment. In QGP, s ands ¯ can QGP from 1980 is found in ref. [15]. move freely and their large QGP abundance leads to un- So, what exactly is this signature? The situation is expectedly large yields of particles with a large s ands ¯ illustrated in fig. 18 and described in detail in ref. [16]. content [147,148], as is illustrated exterior of the QGP In the center of the figure we see thermal QCD based domain in fig. 18. strangeness production processes. This thermal produc- A signature of anything requires a rather background tion dominates the production occurring in first collision free environment, and a good control of anything that is of the colliding nuclei. This is unlike heavier flavors where there as no signature is background free. There are ways the mass threshold 2mQ T , Q = c, b. Strange quark two other than QGP to make strange antibaryons: ≫ pairs: s and antiquarkss ¯, are found produced in processes I) Direct production of complex multistrange dominated by gluon fusion [145]. Processes based on light (anti)baryons is less probable for two reasons: quark collisions contribute fewer ss¯-pairs by nearly a fac- tor 10 [146]. When T ms the chemical equilibrium abun- dance of strangeness≥ in QGP is similar in abundance to 1) When new particles are produced in a color string the other light u and d quarks [15]. breaking process, strangeness is known to be produced Even for the gluon fusion processes enough lifespan of less often by a factor 3 compared to lighter quarks. QGP is needed to reach the large abundance of strange 2) The generation of multistrange content requires mul- quark pairs in chemical equilibrium. The lifespan of the tiple such suppressed steps. Page 26 of 58 Eur. Phys. J. A (2015) 51: 114

′ Fig. 19. Results obtained at the CERN-SPS Y -spectrometer for Z/N-ratio in fixed target S-S and S-Pb at 200 A GeV/c; Fig. 20. Results obtained by the CERN-SPS NA57 experiment ′ results from the compilation presented in ref. [150] adapted for (former Y -spectrometer WA85 and WA94 team) for multi- this report. strangeness enhancement at mid-rapidity yCM < 0.5infixed target Pb-Pb collisions at 158 A GeV/c as| a function| of the mean number of participants Npart , from ref. [151]. Thus the conclusion is that with increasing strangeness   content the production by string processes of strange had- rons is progressively more suppressed. evidence of QGP obtained by the experiment-line WA85 II) Hadron-hadron collisions can redistribute strange- and WA94 designed to discover QGP. ness into multistrange hadrons. Detailed kinetic model In these experiments WA85 and WA94 (see fig. 1) study shows that the hadron-reaction based production the sulfur ions (S) at 200 A GeV hit stationary labora- of multistrange hadrons is rather slow and requires time tory targets, S, W (tungsten), respectively, with reference that exceeds collision time of RHI collisions significantly. date from pp (AFS-ISR experiment at CERN) and p on S This means that both Z, Z and Y, Y are in their abun- shown for comparison. The Z/N and Z¯/N¯ ratio enhance- dance signatures of QGP formation and hadronization, for ment rises with the size of the reaction volume measured further details see refs. [16,55,2]. in terms of target A, and is larger for antimatter as com- L´eon Van Hove, the former DG (1976-1980), charac- pared to matter particles. Looking at fig. 19, the effect terized the strange antibaryon signature after hearing the is systematic, showing the QGP predicted pattern [15,16, reports [147,148] as follows [149]: 55,2]. In the “Signals for Plasma” section: ... implying The “enhancement” results obtained by the same (production of) an abnormally large antihyperon group now working in CERN North Area for the top to antinucleon ratio when plasma hadronizes. The SPS energy Pb (lead) beam of 156 A GeV as published qualitative nature of this prediction is attractive, in 1999 by Andersen (NA57 Collaboration) [151] is shown all the more so that no similar effect is expected in in fig. 20. On the right hadrons made only of quarks and the absence of plasma formation. antiquarks that are created in the collision are shown. On the left some of the hadron valence quarks from matter Given this opinion of the “man in charge”, strange an- can be brought into the reaction volume. tibaryons became the intellectual cornerstone of the ex- The enhancement in production of higher strangeness perimental strangeness program carried out at the CERN content baryons and antibaryons in AA collisions increases SPS, see fig. 1. Thus it was no accident that SPS research with the particle strangeness content. To arrive at this re- program included as a large part the exploration of the sult, the “raw” AA yields are compared with reference pp-, predicted strange (anti)baryon enhancement. We see this pA-reaction results and presented per number of “partici- on left in fig. 1 noting that “hadrons” include of course pants” Npart obtained from geometric models of reaction  (multi)strange hadrons and strange antibaryons. based on energy and particle flows. We will discuss this in In AA collisions at the CERN-SPS Y′-spectrometer, subsect. 9.3. The number of collision participants for all the production of higher strangeness content baryons and data presented in fig. 20 is large, greater than 100, a point antibaryons was compared to lower strangeness content to remember in further discussion. particles, Z/N and Z¯/N¯. These early SPS experiments We see that production of hadrons made entirely from published in 1997 clearly confirmed the QGP prediction newly created quarks are up to 20 times more abundant in a systematic fashion, as we see in the 1997 compilation in AA-reactions when compared to pA reference measure- of the pertinent experimental WA85 and WA94 results by ment. This enhancement falls with decreasing strangeness Antinori [150], see fig. 19. Given the systematic multiple content and increasing contents of the valence quarks observable 3 s.d. agreement of experiment with the model which are brought into collision. These reference results predictions, I saw this result as first and clear experimental at yield ratio “1” provide the dominant error measure. Eur. Phys. J. A (2015) 51: 114 Page 27 of 58

− + − + Fig. 21. Enhancements of Z , Z , Y + Y in the rapidity range yCM < 0.5 as a function of the mean number of participants | | Npart : LHC-ALICE: full symbols; RHIC-STAR and SPS-NA57: open symbols. The LHC reference data use interpolated in energypp 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. [153].

The pattern of enhancement follows the QGP prediction volume and thus lifespan of QGP fireball. It is not surpris- and is now at a level greater than 10 s.d.. There is no ing that the enhancement at SPS is larger than that seen known explanation of these results other than QGP. at RHIC and LHC, considering that the reference yields This is also the largest “medium effect” observed in RHI play an important role in this comparison. Especially the collision experiments. high energy LHC pp reactions should begin to create space These discoveries are now all more than 15 years old. domains that resemble QGP and nearly achieve the degree They have been confirmed by further results obtained at of chemical strangeness equilibration that could erase the SPS, at RHIC, and at the LHC. The present day exper- enhancement effect entirely. imental summary is shown in fig. 21. We see results ob- The study of the h(ss¯) abundance and enhancement tained by the collaborations: corroborates these findings [152]. The importance in the present context is that while h(ss¯) by its strangeness con- SPS: NA57 for collision energy √s =17.2GeV NN nects to −(ssd), , (ss¯) is a net-strangeness free par- (lighter open symbols). Z Z h ticle. Therefore if it follows the pattern of enhancement RHIC: STAR for collision energy √s = 200 GeV NN established for , this confirms strangeness as being (darker open symbols). Z Z the quantity that causes the effect. For some of my col- LHC: Alice for collision energy √s = 2760 GeV NN leagues, these year 2008 results were the decisive turning (filled symbols). point to differentiate the strangeness effect from the effect These results span a range of collision energies that differ associated with the source volume described in the closing by a factor 160 and yet they are remarkably similar. discussion of ref. [15]. Those reading more contemporary Comparing the results of fig. 21 with those seen literature should note that this volume source effect has in fig. 20 we note that Npart is now on a logarithmic been rediscovered three times since, and at some point in scale: the results of fig. 20 which show that the enhance- time was called “canonical suppression”. ment is volume independent are in fig. 21 compressed to The reader should also consult subsect. 10.1, where it is a relatively small domain on the right in both panels. The shown that QGP formation threshold for Pb–Pb collisions SPS-NA57 results in fig. 21 are in agreement with the 1999 is found at about 1/4 of the 156 A GeV projectile energy, “high” participant number results shown in fig. 20. and that the properties of physical QGP fireball formed The rise of enhancement which we see in fig. 21 as a at SPS are just the same, up to volume size, when SPS function of the number of participants 2 < Npart < 80 results are compared to RHIC, and with today data from reflects on the rise of strangeness content in QGP to LHC. Today, seen across energy, participant number, and its chemical equilibrium abundance with an increase in type of hadron considered, there cannot be any doubt that Page 28 of 58 Eur. Phys. J. A (2015) 51: 114 the source of enhancement is the mobility of quarks in to make parton-collision cascade to describe the physics the fireball, with the specific strangeness content showing case, see e.g. Geiger-Sriwastava [154,158,159] for SPS en- gluon based processes. ergy range. The use of these methods for RHIC or even Recall, in February 2000 in the snap of the QGP LHC energies looks less convincing [155]. announcement event, the highly influential Director of To put the problem in perspective, we need a way to BNL, Jeff Marburger6 called these NA57 results and other concentrate entropy so that a thermal state can rapidly CERN-ion experimental results, I paraphrase the earlier arise. Beginning with the work of Bjorken [135] a forma- year 2000 precise quote: “pieced together indirect glimpse tion time is introduced, which is more than an order of of QGP”. Today I would respond to this assessment as fol- magnitude shorter compared to τ 0. It is hard to find tan- lows: the NA57 results seen in fig. 20 and confirmed in past gible experimental evidence which compels a choice such 15 years of work, see fig. 21 are a direct, full panoramic as 0.5 fm/c, and theory models describing this stage are sight of QGP, as good as one will ever obtain. There is not fully convincing. A model aims to explain how as a nothing more direct, spectacular, and convincing that we function of collision energy and centrality the easy to ob- have seen as evidence of QGP formation in RHI collision serve final entropy (hadron multiplicity) content arises. experiments. For some related effort see review work of the Werner- group [156] and Iancu-Venugopalan [157]. To summarize, in the “low” energy regime of SPS we 5 The RHI physics questions of today can try to build a parton cascade model to capture the essence of heavy-ion collision dynamics [158,159]. The un- 5.1 How is energy and matter stopped? derstanding of the initial “formation” of QGP as a func- tion of collision energy and the understanding of the mech- We arrange to collide at very high, relativistic energies, anism that describe energy and baryon number stopping two nuclei such as lead (Pb) or gold (Au), having each remains one of the fundamental challenges of the ongoing about 12 fm diameter. In the rest frame of one of the two theoretical and experimental research program. nuclei we are looking at the other Lorentz-contracted nu- cleus. The Lorentz contraction factor is large and thus 5.2 How and what happens, allowing QGP creation? what an observer traveling along with each nucleus sees approaching is a thin, ultra dense matter pancake. As this In the previous subsect. 5.1 we addressed the question pancake penetrates into the other nucleus, there are many how the energy and baryon number is extracted from fast reactions that occur, slowing down projectile matter. moving nuclei. In this section the added challenge is, how For sufficiently high initial energy the collision occurs is the entropy produced that we find in the fireball? While at the speed of light c despite the loss of motion energy. in some solutions of the initial state formation in RHI Hence each observer comoving with each of the nuclei collisions these two topics are confounded, these are two records the interaction time τ that a pancake needs to different issues: stopping precedes and is not the same as traverse the other nucleus. The geometric collision time abundant entropy production. 0 thus is cτ = 12 fm as measured by an observer comoving For many the mechanism of fast, abundant entropy with one of the nuclei. Thus if you are interested like An- formation is associated with the breaking of color bonds, ishetty et al. [133] in hot projectile and target nuclei there the melting of vacuum structure, and the deconfinement of is no doubt this is one of the outcomes of the collision. quarks and gluons. How exactly this should work has never An observer in the center of momentum (CM) frame been shown: Among the first to address a parton based en- can determine the fly-by time that two nuclei need to pass tropy production quantitatively within a kinetic collision 0 each other should they miss to hit: this is τ /γ, where γ is model was Klaus Geiger [158,159] who built computer cas- the Lorentz-factor of each of the nuclei with respect to CM cade models at parton level, and studied thermalization as frame. This time is, in general, very short and even if nuclei a collision based process. were to touch in such short time very little could happen. In order to understand the QGP formation process a The situation changes if we model this like a collision of solution of this riddle is necessary. There is more to en- the two bulls of Li Keran and T.D. Lee. Once some of tropy production: it controls the kinetic energy conversion the energy (and baryon number) of two nuclei has slowed into material particles. The contemporary wisdom how to down to rest in CM, the clocks of both “slowed” bulls tick describe the situation distinguishes several reaction steps nearly at the same speed as the clock comoving with CM in RHI collision: frame —for the stopped energy and baryon number the lifespan of the fireball is again quite large. 1) Formation of the primary fireball; a momentum equi- But how do we stop the bulls or at least some of partitioned partonic phase comprising in a limited their energy? The answer certainly depends on the energy space-time domain, speaking in terms of orders of mag- regime. The lower is the energy of the bulls, the less we nitude, almost the final state entropy. need to worry; the pancakes are not thin and one can try 2) The cooking of the energy content of the hot matter fireball towards the particle yield (chemical) equilib- 6 Jeff Marburger was a long term Presidential Science Advi- rium in a hot perturbative QGP phase. sor, President of Stony Brook campus of the NY State Univer- 3) Expansion and evaporation cooling towards the tem- sity System, Director of BNL. perature phase boundary. Eur. Phys. J. A (2015) 51: 114 Page 29 of 58

4) Hadronization; that is, combination of effective and years of effort; c) Exploration and understanding of the strongly interacting u, d, s,¯u, d¯ ands ¯ quarks and principles that lead to the abundant formation of entropy anti-quarks into the final state hadrons, with the yield in the process of QGP formation in RHI collisions harbors probability weighted by accessible phase space. potential opportunity to expand the horizons of knowl- edge. It is the first step that harbors a mystery. The current textbook wisdom is that entropy produc- tion requires the immersion of the quantum system in a 5.3 Non-equilibrium in fireball hadronization classical environment. Such an environment is not so read- ily available for a RHI collision system that has a lifespan Heavy flavor production cross sections, in lowest order in −22 2 2 of below 10 s and a size less than 1/10000 of atomic coupling constant, scale according to σ αs/m . Consid- size. For a year 2011 review on entropy production dur- ering a smaller (running) coupling, and a∝ much larger mass ing the different stages of RHI collision see ref. [160]. The of e.g. heavy quarks c, b, we obtain a significant reduction search for a fast entropy generating mechanism continues, in the speed of thermal QGP production reactions. For see for example ref. [161]. charm and bottom, contribution for thermal production So what could be a mechanism of rapid entropy forma- depends on the profile of temperature but is very likely tion? Consider the spontaneous pair production in pres- negligible, and for charm it is at the level of a few per- ence of a strong field: the stronger is the field the greater cent. Conversely, light quarks equilibrate rather rapidly is the rate of field conversion into particles. One finds that compared to the even more strongly self coupled gluons when the field strength is such that it is capable of accel- and in general can be assumed to follow and define QGP erating particles with a unit strength critical acceleration, matter properties. the speed of field decay into particle pairs is such that a Heavy quark yields originate in the pre-thermal parton field filled state makes no sense as it decays too fast [162]. dynamics. However, heavy quarks may acquire through For this reason there is an effective limit to the strength of elastic collisions a momentum distribution characteristic the field, and forces capable to accelerate particles at crit- of the medium, providing an image of the collective dy- ical limit turn the field filled space into a gas of particles. namics of the dense quark matter flow. Moreover, the The conversion of energy stored in fields into particles, question of yield evolution arises, in particular with re- often referred to, in the QCD context, as the breaking gard to annihilation of heavy flavor in QGP evolution. of color strings, must be an irreversible process. Yet the Our “boiling” QGP fireball is not immersed in a bath. textbook wisdom will assign to the time evolution pure It is expanding or, rather, exploding into empty space at quantum properties, and in consequence, while the com- a high speed. This assures that the entropy S V is plexity of the state evolves, it remains “unobserved” and not decreasing, but increasing, in consideration of∝ inter- thus a pure state with vanishing entropy content. Intu- nal collisions which describe the bulk viscosity. The ther- itively, this makes little sense. Thus the riddle of entropy mal energy content is not conserved since the sum of the production in RHI collisions which involve an encounter kinetic energy of expanding motion, and thermal energy, of two pure quantum states and turns rapidly into state of is conserved. Since the positive internal pressure of QGP large entropy carried by many particles maybe related to accelerates the expansion into empty space, an explosion, our poor formulation of quantum processes for unstable the thermal energy content decreases and the fireball cools critical field filled states decaying into numerous pairs. rapidly. However, the situation may also call for a more fun- In this dynamical evolution quark flavors undergo damental revision of the laws of physics. The reason is chemical freeze-out. The heavier the quark, the earlier the that our understanding is based in experience, and we abundance freeze-out should occur. Charm is produced in really do not have much experience with critical accelera- the first collisions in the formative stage of QGP. The cou- tion conditions. When we study acceleration phenomena pling to thermal environment is weak. As ambient temper- on microscopic scale, usually these are very small, even in ature drops the charm quark phase space given its mass principle zero. However, in RHI collisions when we stop drops rapidly. The quantum Fermi phase space distribu- partons in the central rapidity region we encounter the tion which maximizes the entropy at fixed particle number critical acceleration, an acceleration that in natural units is [163,164] is unity and which further signals a drastic change in the 1 way fields and particles behave. The framework of physical nF(t)= , laws which is based on present experience may not be suf- γ−1(t)e(E∓µ)/T (t) +1 ficiently complete to deal with this situation and we will d6N g F = n ,E= p2 + m2 , (17) need to increase the pool of our experience by performing d3pd3x (2π)3 F many experiments involving critical forces.  To conclude: a) The measurement of entropy produc- where g is the statistical degeneracy, and the chemical tion is relatively straightforward as all entropy produced non-equilibrium fugacity (phase space occupancy) γ(t)is at the end is found in newly produced particles; b) The the same for particles and antiparticles while the chemical QGP formation presents an efficient mechanism for the potential μ describes particle-antiparticle asymmetry, and conversion of the kinetic energy of the colliding nuclei into changes sign as indicated. Our μ is “relativistic” chemical particles in a process that is not understood despite many potential. In the non-relativistic limit μ m + μ such ≡ nr Page 30 of 58 Eur. Phys. J. A (2015) 51: 114 that m implicit in E cancels out for particle but turns dissolution into additional hadrons assures that the light to a 2m/T suppression for the antiparticles. Note that quark phase space occupancies as measured in terms of ob- HG QGP independent of the values of all parameters, nF 1as served hadron abundances should show γ >γ > 1. ≤ u,d u,d required. The introduction hadron-side of phase space occu- The integral of the distribution eq. (17) provides the pancy γs [65] and later γu,d [79] into the study of hadron particle yield. When addressing SHARE phase space prop- production in the statistical hadronization approach has erties in subsect. 9.4, we will inspect the more exact result, been challenged. However there was no scientific case, here we consider the Boltzmann non-relativistic limit suit- challenges were driven solely by an intuitive argument that able for heavy quarks (c, b) in RHI collisions at sufficiently high reaction energy aside of thermal, also chemical equilibrium is reached. One of gV T 3 m ±µ/T 2 the objectives of this review is to explain why this intu- N = 2 γe x K2(x),x= (18) 2π T ition 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- → sumption and the demonstration of a result. All know that T (t) is time dependent because the system cools. Let us to make a proof one generally tries to show a contrary look at the case μ = 0, appropriate for physics at LHC behavior and arrives at a contradiction: in this case one and, in the context of present discussion, also a good ap- starts with γ = 1 and shows that results are right only proximation at RHIC. s,u,d for γ 1. However, we will see in sect. 10 that re- Considering charm abundance, in QGP chemical equi- s,u,d sults are right→ when γ = 1 and we show by example in librium γQGP(t) 1. However, we recall that charm froze s,u,d subsect. 10.3 how the urban legend “chemical equilibrium out shortly after→ first collisions. Therefore the value of works” formed relying on a set of errors and/or omissions. γQGP(t) in eq. (19) is established by need to preserve the c The question about chemical non-equilibrium condi- total charm pair number N = const. The exponential fac- c tions has to be resolved so that consensus can emerge tor m/T changes from about 2 to 8 near to hadronization. about the properties of the hadronizing QGP drop, and Thus for prescribed yields at LHC and RHIC it is likely QGP the mechanisms and processes that govern the hadroni- that γc (t) > 1. More generally there is nobody who QGP QGP zation process. disagrees with the need to have γc =1.γc =1is an accidental condition. We have established that charm, and for the very same reason, bottom flavor, cannot be expected to emerge in chemical equilibrium abundance at 6 How is the experimental study of QGP hadronization. continuing today? A QGP filled volume at high T cooks up a high content of strangeness pairs, in essence as many as there are of Today RHI collisions and QGP is a research field that has each light flavor u, d; in plasma strangeness suppression grown to be a large fraction of nuclear science research disappears; the Wroblewski suppression factor [178] (see programs on several continents. A full account of methods, also next subsection) is therefore close to unity. As plasma ongoing experiments, scheduled runs, future plans includ- evolves and cools at some relatively low temperature the ing the development of new experimental facilities is a sep- yield of strangeness freezes-out, just like it did for charm arate review that this author cannot write. The question (and bottom) at higher value of T . how to balance presenting “nothing”, with “everything”, In earlier discussion we have assumed that in QGP is never satisfactorily soluble. The selection of the follow- strangeness will follow the evolution in its pair abundance, ing few topics is made in support of a few highlights of and always be in chemical equilibrium in the fireball. This greatest importance to this review. tacit assumption is not supported by kinetic theory for T

Fig. 22. Direct photon LHC-AA yield; Adapted from: F. Antinori presentation July 2014. a more democratic abundance with u, d, s quarks being from the “jet” asymmetry that the dense matter we form available in nearly equal abundance. However, this initial in RHI collisions is very opaque, and with some effort we simple hypothesis, see ref. [15], needed to be refined with can quantify the strength of such an interaction. This es- actual kinetic theory evaluation; see ref. [16], in consider- tablishes the strength of interaction of a parton at given ation of the short time available and demonstration that energy with the QGP medium. quark collisions were too slow [146] to achieve this goal. It was shown that the large abundance of strangeness depends on gluon reactions mechanism; thus the “gluon” Direct photons particle component in quark gluon plasma is directly in- volved [145], see ref. [15]. The high strangeness density Hot electromagnetic charge plasma radiates both photons in QGP and “democratic” abundance at nearly the same and virtual photons, dileptons [167,168]. The hotter is the level also implies that the production of (anti)baryons plasma, the greater is the radiation yield; thus we hope with multiple strangeness content is abundant, see fig. 18, for a large early QGP stage contribution. Electromagnetic which attracted experimental interest, see subsect. 4.3. probes emerge from the reaction zone without noticeable The observation of strange hadrons involves the identi- loss. The yield is the integral over the history of QGP fication of non-strange hadrons and thus a full charac- evolution, and the measured uncorrected yield is polluted terization of all particles emitted is possible. This in turn by contributions from the ensuing hadron decays. creates an opportunity to understand the properties of the On the other hand, at first glance photons are the ideal QGP at time of hadronization, see subsect. 10.1. probes of the primordial QGP period if one can control the background photons from the decay of strongly interacting particles such as π0 γγ which in general are dominant7. Hard hadrons: jet quenching Recognition of the signal→ as direct QGP photon depends on a very precise understanding of the background. With increasing energy, like in pp, also in AA collisions At the highest collision energy the initial QGP temper- hard parton back-scattering must occur, with a rate de- ature increases and thus direct photons should be more scribed by the perturbative QCD [165,166]. Such hard abundant. In fig. 22 we see the first still at the time of partons are observed in back-to-back jets, that is two jet- writing preliminary result from the Alice experiment at like assembly of particles into which the hard parton ha- LHC. The yield shown is “direct”; that is, after the in- dronizes. These jets are created within the primordial me- direct photon part has been removed. The removal pro- dium. If geometrically such a pair is produced near to cedure appears reliable as for large p⊥ scaled pp yields the edge of colliding matter, one of the jet-partons can match the outcome. At small p⊥,weseeaverystrongex- escape and the balancing momentum of the immersed jet- cess above the scaled pp yields. The p⊥ is high enough to parton tells us how it travels across the entire nuclear believe that the origin are direct QGP photons, and not domain, in essence traversing QGP that has evolved in collective charge acceleration-radiation phenomena. the collision. The energy of such a parton can be partially or completely dissipated, “thermalized” within the QGP 7 Note that γ when used as a symbol for photons is not to distance traveled. Since at the production point a second be confounded with parallel use of γ as a fugacity, meaning is high energy quark (parton) was produced, we can deduce always clear in the context. Page 32 of 58 Eur. Phys. J. A (2015) 51: 114

A virtual photon with q2 = 0 is upon materializa- tions; the color of the sky and for that matter of our planet tion a dilepton e+e−, μ+μ− in the final state. The dilep- originate in how the atmospheric density fluctuations scat- ton yields, compared to photons, are about a factor 1000 ter light. To see QGP fluctuation effects we need to study smaller; this creates measurement challenges e.g. for large each individual event forming QGP apart from another. p⊥. Backgrounds from vector meson intermediate states The SHARE suite of SHM programs also computes sta- and decays are very large and difficult to control. Despite tistical particle yield fluctuations, see subsect. 9.4. The many efforts to improve detection capabilities and the un- search is for large, non-statistical fluctuations that would derstanding of the background, this author considers the signal competition between two different phases of mat- situation as fluid and inconclusive: dilepton radiance not ter, a phase transformation. This topic is attracting at- directly attributed to hadrons is often reported and even tention [177]. To see the phase transformation in action more often challenged. An observer view is presented in smaller reaction systems may provide more opportunity. ref. [169].

6.2 Survey of LHC-ion program July 2015 J/Ψ(cc¯) yield modification This is the other cornerstone observable often quoted in The Large Hadron Collider (LHC) in years of operation the context of the early QGP search. The interest in the sets aside 4 weeks of run time a year to the heavy ion beam bound states of heavy charm quarks cc¯ and in particu- experiments, typically AA (Pb-Pb) collisions but also p- lar J/Ψ is due to their yield evolution in the deconfined Pb. The pp collision LHC run which lasts considerably state as first proposed by Matsui and Satz [170] just when longer addresses Higgs physics and beyond the standard first result J/Ψ became available. Given that the varia- model searches for new physics. This long run provides tions in yield are subtle, and that there are many model heavy ion experimental groups an excellent opportunity interpretations of the effects based on different views of in- to obtain relevant data from the smallest collision system, teraction of J/Ψ in the dense matter —both confined and creating a precise baseline against which AA is evaluated. deconfined— this has been for a long time a livid topic Furthermore, at the LHC energy, one can hope that in which is beyond the scope of this review [171]. some measurable fraction of events conditions for QGP Modern theory addresses both “melting” and recom- could be met in select, triggered events (i.e. collision class bination in QGP as processes that modify the final J/Ψ feature selected). yield [102,172]. Recent results of the Alice collabora- When LHC reaches energy of 7 TeV + 7 TeV for pro- tion [173] support, in my opinion, the notion of recom- tons, for Pb-Pb collisions this magnet setting will cor- binant cc¯ formation. Some features of these results allow respond to a center-of-mass energy of up to √sNN = suggesting that a yield equilibrium between melting and 5.52 TeV per nucleon pair in Pb-Pb collisions. However, recombination has been reached for more central colli- due to magnet training considerations the scheduled heavy sions. This is clearly a research topic, not yet suitable for ion run starting in mid-November 2015 should be at a review analysis. √sNN =5.125 TeV and the maximum energy achieved in the following year. The results we discuss in this review, see sect. 10, were obtained at a lower magnet setting in Particle correlations and HBT the LHC run 1, corresponding to √sNN =2.76 TeV. Several experiments at LHC take AA collision data. Measurement of two particle and in particular two pion and two kaon correlations allows within the framework 1) The ALICE (A Large Ion Collider Experiment) was of geometric source interpretation the exploration of the conceived specifically for the exploration of the QGP three dimensional source size and the emission lifespan of formed in nucleus-nucleus collisions at the LHC. the fireball. For a recent review and update of PHENIX- Within the central rapidity domain 0.5 y 0.5, AL- RHIC results see ref. [174] and for ALICE-LHC see ICE detectors are capable of precise tracking≤ ≤ and iden- ref. [175]. These reports are the basis for our tacit assump- tifying particles over a large range of momentum. This tion that soft hadrons emerge from the hadronization fire- permits the study of the production of strangeness, ball with transverse size as large as R 9 fm for most cen- ≃ charm and resonances, but also multi-particle correla- tral collisions. Aside of two particle correlation, more com- tions, such as HBT and (moderate energy) jets. In ad- plex multi-particle correlations can be and are explored dition, ALICE consists of a muon spectrometer allow- —their non-vanishing strength reminds us that the QGP ing us to study at forward rapidities heavy-flavor and source can have color-charge confinement related multi- quarkonium production. The detector system also has particle effects that remain difficult to quantify. As an ex- the ability to trigger on different aspects of collisions, ample of recent work on long range rapidity correlations to select events on-line based on the particle multiplic- see ref. [176]. ity, or the presence of rare probes such as (di-)muons, and the electromagnetic energy from high-momentum Fluctuations electrons, photons and jets. 2) ATLAS (A Toroidal LHC ApparatuS) has made its Any physical system that at first sight appears homoge- name by being first to see jet quenching. It has high neous will under a magnifying glass show large fluctua- p⊥ particle ID allowing the measurement of particle Eur. Phys. J. A (2015) 51: 114 Page 33 of 58

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 [182].

spectra in a domain inaccessible to other LHC experi- Considered from a theoretical perspective one recog- ments. nizes in an energy and A scan the opportunity to explore 3) The CMS (Compact Muon Solenoid) offers high rate qualitative features of the QCD phase diagram in the T , and high resolution calorimetry, charged particle track- oB plane. Of particular importance is the finding of the ing and muon identification over a wide acceptance, al- critical point where at a finite value of oB the smooth lowing detailed measurements of jets as well as heavy- transformation between quark-hadron phases turns into quark open and bound states. The large solid an- an expected 1st order transition, see ref. [179]. There are gle coverage also provides unique opportunities in the other structure features of quark matter that may become study of global observables. accessible, for a review see ref. [40] and comments at the end of subsect. 2.2. The LHCb experiment has at present no footprint in the At CERN the multipurpose NA61 experiment surveys study of AA collisions but has taken data in pA trial run. in its heavy-ion program tasks the domain in energy and collision system volume where threshold of deconfinement is suspected in consideration of available data. This ex- 6.3 Energy and A scan periment responds to the results of a study of head-on Pb–Pb collisions as a function of energy at SPS did pro- The smaller the size of colliding nuclei, the shorter is the duce by 2010 tantalizing hints of an energy threshold to collision time. Thus in collisions of small sized objects such new phenomena [180–183]. as pp or light nuclei, one cannot presume, especially at a There are significant discontinuities as a function relatively low collision energy, that primordial and yet not of collision energy in the K+/π+ particle yield ratio, well understood processes (compare subsect. 5.2) will have see fig. 23 on left. Similarly, the inverse slope parameter of − time to generate the large amount of entropy leading to the m⊥ spectra of K , see fig. 23 on right, also displays a QGP formation that would allow a statistical model to local maximum near to 30 A GeV, that is at 3.8+3.8GeV, work well, and in particular would allow QGP formation. √sNN =7.6 GeV collider energy collisions in both quanti- This then suggests that one should explore dependence on ties. These behavior “thresholds” are to some degree mir- reaction volume size, both in terms of collision centrality rored in the much smaller pp reaction system also shown and a scan of projectile ion A. in fig. 23. These remarkable results are interpreted as the An important additional observation is that particle onset of deconfinement as a function of collision energy. production processes are more effective with increasing Turning to comparable efforts at RHIC: in 2010 and collision energy. Therefore the chemical equilibration is 2011, RHIC ran the first phase of a beam energy scan achieved more rapidly at higher energy. It seems that just program (RHIC-BES) to probe the nature of the phase about everyone agrees to this even though one can easily boundary between hadrons and QGP as a function of argue the opposite, that more time is available at lower en- oB. With beam energy settings √sNN =7.7, 11.5, 19.6, ergy. In any case, this urban legend that energy and time 27, 39 GeV, with 14.5 GeV included in year 2014, com- grow together is the main reason why QGP search exper- plementing the full energy of 200 GeV, and the run at iments started at the highest available accelerator energy. 62.4 GeV, a relatively wide domain of oB can be probed, This said, the question about the threshold of QGP pro- as the matter vs. anti-matter excess increases when energy duction as a function of energy is open. decreases. For a report on these result see refs. [184,185]. Page 34 of 58 Eur. Phys. J. A (2015) 51: 114

Among the first phase of the beam energy scan discov- vacuole created in RHI collision laboratory experiments is eries is the oB dependence of azimuthal asymmetry of flow offering a decisive opportunity to test this understanding of matter, v2. Particle yield ratio fluctuations show sig- of mass of matter. The same lattice-QCD that provided nificant deviation from Poisson expectation within HRG the numerical evaluation of mass of matter, provides prop- model. This and other results make it plausible that erties of the hot QGP Universe. QGP is formed down to the lowest RHIC beam energy Others go even further to argue nothing needs to be of √sNN =7.7 GeV, corresponding to fixed target colli- confirmed: given the QCD action, the computer provides sion experiments at 32 A GeV. This is the collision energy hadron spectrum and other static properties of hadron where SPS energy scan also found behavior characteris- structure. For a recent review of “Lattice results concern- tic of QGP, see fig. 23. These interesting results motivate ing low-energy particle physics,” see ref. [188]. That is the second RHIC-BES phase after detector upgrades are true: the relatively good agreement of lattice-QCD the- completed in 2018/19. ory with low-energy particle physics proves that QCD is the theory of strong interactions. In fact, many textbooks argue that this has already been settled 20 years ago in 7 What are the conceptual challenges of the accelerator experiments, so a counter question could be, why bother to do lattice-QCD to prove QCD? One can QGP/RHI collisions program? present as example of a new insight the argument that the mass of matter is not due to the Higgs field [186,187]. In subsect. 1.1 we have briefly addressed the Why? of the However, the mass argument is not entirely complete. RHI collision research program. Here we return to explore The vacuole size R directly relates to QCD vacuum prop- some of the points raised, presenting a highly subjective erties —in bag models we relate it to the bag constant view of foundational opportunities that await us. describing the vacuum pressure acting on the vacuole. ButB is this hadron energy scale 1/4 170 MeV fundamental? The understanding of the scaleB of≃ the QCD vacuum struc- 7.1 The origin of mass of matter ture has not been part of the present-day lattice-QCD. In lattice-QCD work one borrows the energy scale from Confining quarks to a domain in space means that the an observable. In my opinion hadron vacuum scale is due typical energy each of the light quarks will have inside to the vacuum Higgs field, and thus the scale of hadron ahadronisEq 1/R mq, where R is the size of masses is after all due to Higgs field; it is just that the ∝ ≫ the “hole” in the vacuum —a vacuole. Imposing a sharp mechanism is not acting directly. boundary and forbidding a quark-leak results in a square- Let me explain this point of view: By the way of top well-like relativistic Dirac quantum waves. This model al- interaction with Higgs there is a relation of the Higgs with lows quantification of Eq. One further argues that the the QCD vacuum scale. size R of the vacuole arises from the internal Fermi and a) The intersection between QCD and the Higgs field Casimir pressures balancing the outside vacuum which is provided by the top quark, given the remarkable value presses to erase any vacuole comprising energy density of the minimal coupling gt that is higher. 2 In a nutshell this is the math known from within mt gt g 1,αt =0.08 α (m )=0.1. the context of quark-bag model [96–98], rounded off al- t ≡ h ≃ h ≡ 4π ≃ s t lowing color-magnetic hyperfine structure splitting. This  (20) model explains how baryons and mesons have a mass much Note that the same strength of interaction: top with glu- t greater than the sum of quark masses. It is also easy ons αs(mt), and with Higgs field fluctuations αh. to see that a larger vacuole with hot quarks and gluons b) The size of QCD vacuum fluctuations has been es- would provide a good starting point to develop a dynam- timated at 0.3 fm [189]. This is large compared to the top −3 ical model of expanding QGP fireball formed in RHI col- quark Compton wavelength λt = c/mt =1.13 10 fm. lisions. This means that for the top-field the QCD vacuum× looks The advent of lattice-QCD means we can address like a quasi-static mountainous random field driving large static time independent properties of strongly interacting top-field fluctuations in the QCD vacuum. particles. A test of bag models ideas is the computation The possible relation of the QCD vacuum structure via of the hadron mass spectrum and demonstration that the top quark with Higgs requires much more study, I hope mass of hadrons is not determined by the mass of quarks that this will keep some of us busy in coming years. bound inside. Indeed, this has been shown [186,187]; the That something still needs improvement in our under- confining vacuum structure contributes as much as 96% standing of strong interactions is in fact clear: Why i) all of the mass of the matter, the Higgs field the remaining hadrons we know have qqq and qq¯ structure states, and few-%. why ii) we do not observe internal excitations of quarks in Based on both bag model consideration and lattice- bags appearing as hadron resonances. These two questions QCD we conclude that the quantum zero-point energy of show that how we interpret QCD within the bag model is the localized, confined, light quarks governs the mass of incomplete. matter. The ultimate word is, however, expected from an I hope to have dented somewhat the belief that lattice- experiment. Most think that setting quarks free in a large QCD is capable of replacing the experimental study of vac- Eur. Phys. J. A (2015) 51: 114 Page 35 of 58 uum structure. In a nutshell, lattice neither explains scales “advances” at the same rate there as it does here. This of vacuum structure, nor can it address any dynamical assumption is not necessary. phenomena, by necessity present in any laboratory recre- Is it possible, both in practical and in principle terms, ation of the early Universe QGP conditions. Finally, the using RHI collisions to answer if c¯ = c, where the bar QCD vacuum structure paradigm needs an experimental indicates the property in the vacuole? confirmation. We can for example study the relation between energy and momentum of photons produced in QGP, and the rate at which these processes occur. The photon emitted is de- 7.2 The quantum vacuum: Einstein’s æther fined by its wavelength k =1/λ, the energy of the photon is ck. This energy is different in the vacuole from what The quantum vacuum state determines the prevailing we observe in the laboratory —energy conservation for the form of the “fundamental” physics laws. Within the stan- photon is not maintained since the translation symmetry dard model, the nature of particles and their interactions in time would need to be violated to make time tick differ- is determined by the transport properties of the vacuum ently in different vacuum states. However, global energy state. As just discussed above, the mass of matter is in- conservation is assured. Transition radiation, Cherenkov herent in the scale of QCD, which itself relates in a way to radiation are more mundane examples of what happens be studied in the future with the Higgs vacuum structure. when a superluminal photon enters a dielectric medium. The existence of a structured quantum vacuum as the Thus we will need to differentiate with what would be carrier of the laws of physics was anticipated by Lorentz, called medium effect when considering photon propaga- and Einstein went further seeking to reconcile this with tion across thec ¯ = c. boundary. That may be difficult.  the principles of relativity. What we call quantum vac- Turning now to the rate of photon production in the uum, they called æther. The concluding paragraph from vacuole: we keep to gauge invariance, thus charge cannot a lecture by Albert Einstein is creating the philosophical change between two quantum vacuum states. The way the foundation of the quantum vacuum as carrier of laws of change from the vacuole to the normal vacuum rate will be physics (translation by author) [190] looked at is that we assume the space size of the vacuole to be measured in units of length evaluated in the normal ... space is endowed with physical qualities; in this vacuum. The rate of an electromagnetic process in mod- sense the æther exists. According to the general ified vacuum should be, according to the Fermi golden theory of relativity, space without æther is unthink- rule proportional to ¯α¯2 = e4¯/(¯2c¯2). This expression able: without æther light could not only not prop- reminds that we also can have ¯ = , but the result will agate, but also there could be no measuring rods involve the product c only. The rate per unit volume and and clocks, resulting in non-existence of space-time time of an electromagnetic process is in the vacuole with distance as a physical concept. On the other hand, Δt¯= ΔL/c¯ is this æther cannot be thought to possess properties characteristic of ponderable matter, such as having α¯2 1 1 parts trackable in time. Motion cannot be inherent ¯ . (21) to the æther. W∝ Δ3LΔt¯ ∝ ¯c¯ Δ4L

A few months earlier, in November, 1919 Einstein an- The number of events we observe is ΔL4 . The produc- nounced the contents of this address in a letter to Lorentz: tion of direct dileptons and direct photonsW is thus pre- It would have been more correct if I had limited myself, dicted to scale with (¯c¯)−1 in a space-time volume deter- in my earlier publications, to emphasizing only the non- mined in our vacuum by for example the HBT method. existence of an æther velocity, instead of arguing the total The above consideration cannot be applied to strong non-existence of the æther . . . interactions since there is no meaning to αs in the nor- mal vacuum; we always measureα ¯s. Similarly, the ther- mal properties of the vacuole, in particular addressing the 7.3 The quantum vacuum: Natural constants quark energy, are intrinsic properties. The direct connec- tion of intrinsic to external properties occurs by electro- In the quark–gluon plasma state of matter, we fuse and magnetic phenomena. The practical problem in using the dissolve nucleons in the primordial æther state, different rate of electromagnetic processes to compare in-out (c)−1 in its structure and properties from the æther of our ex- is that all production processes depend on scattering of perience. In Einstein’s writings quoted above the case of electrically charged quanta (quarks) in QGP, and that in transition between two coexistent æther states was not turn depends on a high power of T . This means that small foreseen, but properties such as the velocity of light were changes in c could be undetectable. However, it will be seen as being defined by the æther. One should thus ask: quite difficult to reconcile an order of magnitude c mod- Is velocity of light the same out there (vacuole) as it is ification by pushing T and HBT sizes. We hope to see around here? Such a question seems on first sight empty such studies in the near future, where one tries to deter- as the velocity of light connects the definition of a unit of mine for electromagnetic processes an in-medium strength length with the definition of a time increment. However, of α as this is how one would reinterpret vacuole modified ifc ¯ in the vacuole is the same as c, it means that time physical natural constants. Page 36 of 58 Eur. Phys. J. A (2015) 51: 114

7.4 The primordial quark universe in laboratory to imagine local baryon number chemical fluctuation. This “random fluctuation” resolution of the baryon asymmetry Relativistic heavy ion (RHI) collisions recreate the ex- riddle implies that our matter domain in the Universe bor- treme temperature conditions prevailing in the early Uni- ders on an antimatter domain —however a chemical po- verse: a) dominated by QGP; b) in the era of evolution be- tential wave means that this boundary is where μ =0and ginning at a few μs after the big-bang; c) lasting through thus where no asymmetry is present; today presumably a the time when QGP froze into individual hadrons at about space domain void of any matter or antimatter. Therefore, 20–30 μs. We record especially at the LHC experiments the a change from matter to antimatter across the boundary initial matter-antimatter symmetry a nearly net-baryon- is impossible to detect by astronomical observations— we free (B = b ¯b 0) primordial QGP8. The early Universe have to look for antimatter dust straying into local parti- (but not the− lab→ experiment) evolved through the matter- cle detectors. One of the declared objectives of the Alpha antimatter annihilation leaving behind the tiny 10−9 resid- Magnetic Spectrometer (AMS) experiment mounted on ual matter asymmetry fraction. the International Space Station (ISS) is the search for an- tihelium, which is considered a characteristic signature of The question in which era the present day net baryon antimatter lurking in space [192]. number of the Universe originates remains unresolved. We recall that the acoustical density oscillation of mat- Most believe that the net baryon asymmetry is not due to ter is one of the results of the precision microwave back- an initial condition. For baryon number to appear in the ground studies which explore the conditions in the Uni- Universe the three Sakharov conditions have to be fulfilled: verse at temperatures near the scale of T =0.25 eV where 1) In terms of its evolution, the Universe cannot be hydrogen recombines and photons begin to free-steam. in the full equilibrium stage; or else whatever created the This is the begin of observational cosmology era. Another asymmetry will also undo it. This requirement is generally factor 30000 into the primordial depth of the Universe understood to mean that the asymmetry has to originate expansion, we reach the big-bang nuclear synthesis stage in the period of a phase transformation, and the focus occurring at the scale of T 10 keV. Abundance of he- of attention has been on electro-weak symmetry restoring lium compared to hydrogen≃ constrains significantly the condition at a temperature scale 1000 TH. However the timescale of the Universe expansion and hence the present time available for the asymmetry to arise× is in this condi- −8 −5 day photon to baryon ratio. A further factor 30000 in- tion on the scale of 10 s and not 10 s or longer if the crease of temperature is needed to reach the stage at which asymmetry is related to QGP evolution, hadronization, the hadronization of quark Universe occurs at Hagedorn and/or matter-antimatter annihilation period. temperature TH. 2) During this period interactions must be able to dif- We have focused here on conservation, or not, of ferentiate between matter and antimatter, or else how baryon number in the Universe. But another topic of cur- could the residual asymmetry be matter dominated? This rent interest is if the hot QGP fireball in its visible energy asymmetry requires CP–non-conservation, well known component conserves energy; the blunt question to ask is: to be inherent in the SM as a complex phase of the What if the QGP radiates darkness, that is something we Kobayashi-Maskawa flavor mixing. cannot see? [193]. I will return under separate cover to 3) If true global excess of baryons over antibaryons is discuss the QGP in the early Universe, connecting these to arise there must be a baryon number conservation vio- different stages. For a preliminary report see ref. [194]. The lating process. This seems to be a requirement on funda- understanding of the quark Universe deepens profoundly mental interactions which constrains most when and how the reach of our understanding of our place in this world. one must look for the asymmetry formation. It would be hard to place this in the domain of physics today acces- sible to experiments as no such effect has come on the 8 Melting hadrons horizon. A variant model of asymmetry could be a primordial Two paths towards the quark phase of matter started in acoustical chemical potential wave inducing an asymme- parallel in 1964–65, when on one hand quarks were intro- try in the local distribution of quantum numbers. It has duced triggering the first quark matter paper [117], and been established that at the QGP hadronization T = TH on another, Hagedorn recognized that the yields and spec- temperature a chemical potential amplitude at the level of tra of hadrons were governed by new physics involving TH 0.3 eV achieves the present day baryon to photon number and he proposed the SBM [11]. This briefly addresses the in our domain of the Universe [191]. Constrained by local, events surrounding Hagedorn discovery and the resulting electrical charge neutrality, and B = L (local net baryon modern theory of hot hadronic matter. density equal to local net lepton density), this chemical −9 potential amplitude is about 10 fraction of TH. This insight sets the scale of energy we are looking 8.1 The tale of distinguishable particles for: the absence in the SM of any force related to baryon number and operating at the scale of eV is what allows us In early 1978 Rolf Hagedorn shared with me a copy of his unpublished manuscript Thermodynamics of Distin- 8 Here b, ¯b denotes baryons and antibaryons, not bottom guishable Particles: A Key to High-Energy Strong Interac- quarks. tions?, a preprint CERN-TH-483 dated 12 October 1964. Eur. Phys. J. A (2015) 51: 114 Page 37 of 58

He said there were two copies; I was looking at one; an- the best of my knowledge the dense, strongly interact- other was in the CERN archives. A quick glance sufficed ing hadronic gas is the only physical system where the to reveal that this was, actually, the work proposing a opposite happens. Thus surfacing briefly in Hagedorn’s limiting temperature and the exponential mass spectrum. withdrawn Thermodynamics of Distinguishable Particles Hagedorn explained that upon discussions of the contents paper, this original finding faded from view. Hagedorn pre- of his paper with L´eon Van Hove, he evaluated in greater sented a new idea that has set up his SBM model, and for detail the requirements for the hadron mass spectrum and decades this new idea remained hidden in archives. recognized a needed fine-tuning. Hagedorn concluded that On the other hand, the Hagedorn limiting tempera- his result was therefore too arbitrary to publish, and in ture TH got off the ground. Within a span of only 90 days the CERN archives one finds Hagedorn commenting on between the withdrawal of his manuscript, and the date this shortcoming of the paper, see chapt. 18 in ref. [1]. of his new CERN-TH preprint, Hagedorn formulated the However, Hagedorn’s “Distinguishable Particles” is a SBM. Its salient feature is that the exponential mass spec- clear stepping stone on the road to modern understand- trum arises from the principle that hadrons are clusters ing of strong interactions and particle production. The comprising lighter (already clustered) hadrons10. The key insights gained in this work allowed Hagedorn to rapidly point of this second paper is a theoretical model based on invent the Statistical Bootstrap Model (SBM). The SBM the very novel idea of hadrons made of other hadrons. Such paper Statistical Thermodynamics of Strong Interactions a model bypasses the need to identify constituent content at High Energies, preprint CERN-TH-520 dated 25 Jan- of all these particles. And, Hagedorn does not need to uary 1965, took more than a year to appear in press9 [11]. make explicit the phenomenon of Hadron distinguishabil- The beginning of a new idea in physics often seems to ity that clearly was not easy to swallow just 30 years after hang on a very fine thread: was anything lost when Ther- quantum statistical distributions saw the light of day. modynamics of Distinguishable Particles remained unpub- Clustering pions into new hadrons and then combining lished? And what would Hagedorn do after withdrawing these new hadrons with pions, and with already preformed his first limiting-temperature paper? My discussion of the clusters, and so on, turned out to be a challenging but matter with Hagedorn suggests that his vision at the time soluble mathematical exercise. The outcome was that the of how limiting temperature could be justified evolved very number of states of a given mass was growing exponen- rapidly. Presenting his more complete insight was what in- tially. Thus, in SBM, the exponential mass spectrum re- terested Hagedorn and motivated his work. Therefore, he quired for the limiting temperature arose naturally ab ini- opted to work on the more complete theoretical model, tio. Furthermore the model established a relation between and publish it, rather than to deal with complications the limiting temperature, the exponential mass spectrum that pressing Thermodynamics of Distinguishable Parti- slope, and the pion mass, which provides the scale of en- cles would generate. ergy in the model. While the withdrawal of the old, and the preparation Models of the clustering type are employed in other ar- of an entirely new paper seemed to be the right path eas of physics. An example is the use of the α-substructure to properly represent the evolving scientific understand- in the description of nuclei structure: atomic nuclei are ing, today’s perspective is different. In particular the in- made of individual nucleons, yet improvement of the un- sight that the appearance of a large number of different derstanding is achieved if we cluster 4 nucleons (two pro- hadronic states allows to effectively side-step the quan- tons and two neutrons) into an α-particle substructure. tum physics nature of particles within statistical physics The difference between the SBM and the nuclear α- became essentially invisible in the ensuing work. Few sci- model is that the number of input building blocks in SBM entists realize that this is a key property in the SBM, and i.e. pions and more generally of all strongly interacting the fundamental cause allowing the energy content to in- clusters is not conserved but is the result of constraints crease without an increase in temperature. such as available energy. As result one finds rapidly grow- In the SBM model, a hadron exponential mass spec- ing with energy size of phase space with undetermined trum with the required “fine-tuned” properties is a nat- number of particles. This in turn provides justification for ural outcome. The absence of Hagedorn’s Distinguishable the use of the grand canonical statistical methods in the Particles preprint delayed the recognition of the impor- description of particle physics phenomena at a time when tance of the invention of the SBM model. The SBM paper only a few particles were observed. without its prequel looked like a mathematically esoteric work; the need for exponential mass spectrum was not immediately evident. 8.2 Roots and contents of the SBM Withdrawal of Distinguishable Particles also removed from view the fact that quantum physics in hot hadronic The development of SBM in 1964/65 had a few preced- matter loses its relevance, as not even Boltzmann’s 1/n! ing pivotal milestones, see chapt. 17 in ref. [1] for a fully factor was needed, the exponential mass spectrum effec- referenced list. One should know seeing point 1) below tively removes it. Normally, the greater the density of particles, the greater the role of quantum physics. To 10 In this paper, as is common today, we refer to all discov- ered hadron resonance states —Hagedorn’s clusters— as res- 9 Publication was in Nuovo Cimento Suppl. 3, 147 (1965) onances, and the undiscovered “heavy” resonances are called actually printed in April 1966. Hagedorn states Page 38 of 58 Eur. Phys. J. A (2015) 51: 114 that it was Heisenberg who hired Hagedorn as a postdoc the original model which we find in comparable detail in in June 1952 to work on cosmic emulsion “evaporation ref. [15], also shown in subsect. 2.1, eq. (4). Like in SBM, stars”, and soon after, in 1954, sent him on to join the in the toy model particle clusters are composed of clusters; process of building CERN: however we ignore kinetic energy. Thus 1) The realization and a first interpretation of many sec- ρ(m)=δ(m m ) ondaries produced in a single hadron-hadron collision − 0 (Heisenberg 1936 [195]). ∞ 1 n n + δ m m ρ(m )dm . (22) 2) The concept of the compound nucleus and its thermal n! − i i i 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 models for particle production in analogy to com- particle” with mass m0 or else it is composed of any num- pound nuclei (1948–1950) (Koppe 1949 [56,57], Fermi ber of clusters of any masses mi such that Σmi = m.We 1950 [58]). Laplace-transform eq. (22): Enter Hagedorn: 4) The inclusion of resonances to represent interactions ρ(m)e−βmdm =e−βm0 recognized via phase shifts (Belenky 1956 [48]).  ∞ n 5) The discovery of limited p⊥ (1956). 1  −βmi 6) The discovery that fireballs exist and that a typical pp + e ρ(mi)dmi. (23) n! n=2 i=1 collision seems to produce just two of them, projectile   and target fireball (1954–1958). Define 7) The discovery that large-angle elastic cross-sections decrease exponentially with CM energy (1963). z(β) e−βm0 ,G(z) e−βmρ(m)dm. (24) 8) The discovery of the parameter-free and numerically ≡ ≡ 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) 1or sults obtained earlier at CERN (1963). − − z =2G(z) eG(z) +1, (25) Hagedorn introduced a model based on an unlimited − sequence of heavier and heavier bound and resonance states he called clusters11, each being a possible con- which provides implicitly the function G(z), the Laplace stituent of a still heavier resonance, while at the same time transform of the mass spectrum. being itself composed of lighter ones. The pion is the light- A graphic solution is obtained drawing z(G) in fig. 24 est “one-particle cluster”. Hadron resonance states are due top frame a) and transiting in fig. 24 from top a) to bot- to strong interactions; if introduced as new, independent tom frame b) by exchanging the axis. The parabola-like particles in a statistical model, they express the strong in- maximum of z(G) implies a square root singularity of G(z) teractions to which they owe their existence. To account in at z0, first remarked by Nahm [27] full for strong interactions effects we need all resonances; z (G) z =ln4 1=0.3863 ..., G = G(z )=ln2, that is, we need the complete mass spectrum ρ(m). max ≡ 0 − 0 0 In order to obtain the mass spectrum ρ(m), we will as also shown in fig. 24. implement in mathematical terms the self-consistent re- It is remarkable that the “Laplace-transformed quirement that a cluster is composed of clusters. This BE” eq. (25) is “universal” in the sense that it is not leads to the “bootstrap condition and/or bootstrap equa- restricted to the above toy model, but turns out to be tion” for the mass spectrum ρ(m). The integral bootstrap the same in all (non-cutoff) realistic SBM cases [12,19]. equation (BE) can be solved analytically with the result Moreover, it is independent of: that the mass spectrum ρ(m) has to grow exponentially. Consequently, any thermodynamics employing this mass – the number of space-time dimensions [196]; the “toy spectrum has a singular temperature TH generated by the model” extends this to the cae of “zero”-space dimen- asymptotic mass spectrum ρ(m) exp(m/T0). Today this sions; singular temperature is interpreted∼ as the temperature – the number of “input particles” (z becomes a sum over where (for baryon chemical potential oB = 0) the phase modified Hankel functions of input masses); conversion of hadron gas quark–gluon plasma occurs. – Abelian or non-Abelian symmetry constraints [197]. ←→ Upon inverse Laplace asymptotic transformation of 8.3 Implementation of the model the Bootstrap function G(z) one obtains

Let us look at a simple toy model proposed by Hagedorn ρ(m) m−3em/TH , (26) to illustrate the Frautschi-Yellin reformulation [12,19] of ∼ where in the present case (not universally): 11 In the older literature Hagedorn and others initially called decaying clusters fireballs, this is another example of how a m0 m0 physics term is recycled in a new setting. TH = = . (27) −ln z0 0.95 Eur. Phys. J. A (2015) 51: 114 Page 39 of 58

question is how the van der Waals excluded volume ex- tension of Hagedorn SBM connects to present day lattice- QCD [200], and we address this in the next subsection. Before we discuss that, here follows one point of prin- ciple. Current work takes for granted the ability to work in a context similar to non-relativistic gas including rela- tivistic phase space. This is not at all self-evident. To get there, see also ref. [15], we argued that particle rest-frame volumes had to be proportional to particle masses. Follow- ing Touschek [201], we defined a “four-volume”; arbitrary µ observer would attribute to each particle the 4-volume Vi µ moving with particle four-velocity ui µ µ µ µ pi Vi = Viui ,ui . (29) ≡ mi The entire volume of all particles is comoving with the four-velocity of the entire particle assembly of mass m pµ V µ = Vuµ,uµ = ; m n µ µ µ p = pi ,m= pµp . (30) i=1   We explored a simple additive model applicable when all hadrons have the same energy density, see ref. [15] V V = i =const.=4 , (31) m mi B where the proportionality constant is written 4 in order to emphasize the similarity to MIT bags [96–98],B 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 pa- Fig. 24. (a) z(G) according to eq. (25). (b) Bootstrap func- rameters characterizing the running of QCD parameters; tion G(z), the graphical solution of eq. (25). The dashed line the coupling constant αs and mass of strange quark ms represents the unphysical branch. The root singularity is at are here relevant. ϕ0 =ln(4/e)=0.3863. 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 Using the natural choice m0 = mπ we obtain: ignore the Lorentz covariance challenges associated with introduction of particle proper volume. However, this has T (toy model) = 145 MeV. (28) H been shown only if all hadrons have the same energy den- The simple toy model already yields all essential features sity, nobody extended this argument to a more general of SBM: the exponential mass spectrum with a =3and case. the right magnitude of T . In an independent consideration the energy spectrum H of such SBM clusters we obtained and that of MIT bags was found to be the same [202–204]. This suggests these 8.4 Constituents of finite size two models are two different views of the same object, a snapshot taken once from the hadron side, and another The original point-volume bootstrap model was adapted time from the quark side. MIT bags “consist of” quarks to be applicable to collisions of heavy ions where the re- and gluons, SBM clusters of hadrons. This leads on to action volume was relevant. This work began in 1977 and a phase transition to connect these two aspects of the was in essence complete by 1980 [137,28,198], see the de- same, as is further developed in ref. [15]; the model of the tails presented in ref. [15]. The new physics is that cluster phase boundary defined by the MIT bags was continued volumes are introduced. by Gorenstein and collaborations, see refs. [205,206]. For overview of work that followed see for example ref. [199]. However, “in principle” we are today where 8.5 Connection with lattice-QCD the subject was when the initial model was completed in 1979. While many refinements were proposed, these were Today the transformation between hadrons and QGP is in physical terms of marginal impact. A new well-posed characterized within the lattice-QCD evaluation of the Page 40 of 58 Eur. Phys. J. A (2015) 51: 114

“hole” in the intermediate mass domain. The authors re- port in a side remark that their results are insensitive to a change a =5/2 a = 3 with appropriate other changes but this does not→ resolve the above sharp cutoff matter. Note that the normalization parameter C in eq. (32) is the only free parameter and for C = 0 the complement states are excluded (dashed lines in fig. 25), the model reverts to be HRG with finite size particle volume, but only for r =0,forr = 0 we have point HRG. How this model modifies energy density ε/T 4 and pressure P/T4 is seen in fig. 25. The authors conclude that lattice data exclude tak- ing the two effects apart, i.e. consideration of each of these individually. This is so since for C =0thefit of pressure fig. 6 favors finite hadron volume parameter r  0.4 fm; however the best fit of energy density shows r ∼= 0. When both: excluded volume r = 0, and heavy res- onances C = 0 are considered simultaneously the model works better: the effect of finite volume and the possibly yet undiscovered high mass Hagedorn mass spectrum thus complement each other when considered simultaneously: the suppression effects for pressure P/T4 and energy den- sity ε/T 4 due to the excluded volume effects and the en- hancement due to the Hagedorn mass spectrum make the data fit marginally better as we can see in fig. 25. Fig. 25. Lattice-QCD [72] energy density ε/T 4 and Pressure In effect ref. [200] tests in quantitative fashion the sen- 4 P/T for T<155, oB = 0 MeV, in RHG in a model with sitivity of the lattice-QCD results to physics interpreta- excluded volume parameter r =0, 0.2,.0.3, 0.4 (dashed lines) tion: it seems that even if and when the lattice results and allowing for extension of the HRG with exponential mass should be a factor 5 more precise, the correlation between spectrum (solid lines) for assumed TH = 160 MeV. See text. the contribution of undiscovered states and the van der From ref. [200]. Waals effect will compensate within error margin. As disappointing as these results may seem to some, it is a triumph for the physics developed in 1979. Namely, re- thermal properties of the quark-hadron Universe. In the sults of ref. [200] also mean that upon a reasonable choice context of our introductory remarks we have addressed of the energy density in Hadrons 4 , the model presented the close relation of the HRG with the lattice results, in ref. [15] will fit well the presentB day lattice-QCD data see subsect. 2.4 and in particular fig. 6. But what does T<155, oB = 0 MeV, since it has both the correct mass this agreement between lattice-QCD and HRG have to spectrum, and the correct van der Waals repulsion effects say about SBM of hadrons of finite size? That is an im- due to finite hadron size. Therefore, this model is bound portant question, it decides also the fate of the 1979 effort to be accurate as a function of oB as well. described in ref. [15]. Reading in ref. [15] it seems that the perturbative QCD Vovchenko, Anchishkin, and Gorenstein [200] analyzed phase has properties that do not match well to the SBM, the lattice-QCD for the pressure and energy density at requiring a strong 1st order phase transition matching to T<155, oB = 0 MeV within the hadron resonance gas SBM. This was a result obtained with a fixed value of model allowing for effects of both the excluded volume and αs in thermal-QCD. The results of lattice-QCD teach us the undiscovered part of Hagedorn mass spectrum. That that a more refined model with either running αs and/or work is within a specific model of finite sized hadron gas: thermal quarks and gluon masses [6,110] is needed. Con- 3 particles occupy a volume defined by v =16πr /3 where temporary investigation of the latest lattice-QCD results r is a parameter in range 0

9.1 A large parameter set Table 3. Thermal parameters and their SHARE name. The values are to be presented in units GeV and fm3, where appli- Our task is to describe precisely a multitude of hadrons cable. by a relatively small set of parameters. This then allows us to characterize the drop of QGP at the time of ha- Symbol Parameter Parameter description dronization. In our view, the key objective is to charac- V norm absolute normalization in fm3 terize the source of hadrons rather than to argue about T temp chemical freeze-out temperature T the meaning of parameter values in a religious fashion. λq lamq light quark fugacity factor For this procedure to succeed, it is necessary to allow for the greatest possible flexibility in the characterization of λs lams strangeness fugacity factor the particle phase space, consistent with conservation laws γq gamq light quark phase space occupancy and related physical constraints at the time of QGP ha- γs gams strangeness phase space occupancy dronization. For example, the number yield of strange and λ3 lmi3 I3 fugacity factor (eq. (54)) light quark pairs has to be nearly preserved during QGP hadronization. Such an analysis of experimental hadron γ3 gam3 I3 phase space occupancy (eq. (52)) yield results requires a significant bookkeeping and fitting λc lamc charm fugacity factor e.g. λc =1 effort, in order to allow for resonances, particle widths, Nc+¯c Ncbc number of c +¯c quarks full decay trees and isospin multiplet sub-states. We use Tc/T tc2t ratio of charm to the light quark SHARE (Statistical HAdronization with REsonances), a data analysis program available in three evolution stages hadronization temperature for public use [207–209]. The important parameters of the SHM, which control In regard to the parameters γ ,γ = 1, we note that: the relative yields of particles, are the particle specific fu- q s  gacity factors λ eµ/T and the space occupancy factors ≡ a) We do not know all hadronic particles, and the incom- γ. The fugacity is related to particle chemical potential plete hadron spectrum used in SHM can be to some μ = T ln λ. μ follows a conserved quantity and senses the degree absorbed into values of γs,˜γq; sign of “a charge”. Thus it flips sign between particles and b) In our analysis of hadron production results we do not antiparticles. fit spectra but yields of particles. This is so since the The resultant shape of the Fermi-Dirac distribution dynamics of outflow of matter in an exploding fireball is seen in eq. (17). The occupancy γ is in Boltzmann ap- is hard to control; integrated spectra (i.e., yields) are proximation the ratio of produced particles to the number not affected by collective flow of hot matter. of particles expected in chemical equilibrium. Since there c) γ ,γ = 1 complement γ , γ to form a set of non- is one quark and one antiquark in each meson, yield is q s  c b 2 equilibrium parameters. proportional to γq and accordingly the baryon yield to 3 γq . When necessary we will distinguish the flavor of the Among the arguments advanced against use of chemi- valance quark content q = u,d,s,... cal non-equilibrium parameters is the urban legend that it The occupancy parameters describing the abundance is hard, indeed impossible, to find in the enlarged parame- of valance quarks counted in hadrons emerge in a complex ter space a stable fit to the hadron yield data. A large set evolution process described in subsect. 5.3. In general, we of parameters often allows spurious local minima which expect a non-equilibrium value γi = 1. A much simplified cloud the physical minimum —when there are several fit argument to that used in subsect. 5.3 is to to assume that minima, a random search can oscillate between such non- we have a completely equilibrated QGP with all quantum physics minima rendering the fit neither reproducible, nor charges zero (baryon number, etc.) and thus, in QGP all physically relevant. λi =1,γi = 1. Just two parameters describe the QGP This problem is solved as follows using the SHARE under these conditions: temperature T and volume V . suite of programs: we recall that SHARE allows us to use This state hadronizes preserving energy, and increasing any of the QGP bulk properties to constrain fits to par- or preserving entropy and essentially the number of pairs ticle yield. In extreme, one can reverse the process: given of strange quarks. On the hadron side temperature T and a prescribed fireball bulk properties one can fit a statis- volume V would not suffice to satisfy these constraints, tical parameter set, provided that the information that is and thus we must at least introduce γs > 1. The value is in introduced is sufficient. general above unity because near to chemical equilibrium To find a physics best fit, what a practitioner of the QGP state contains a greater number of strange quark SHARE will do is to loosely constrain the physical bulk pairs compared to the hadron phase space. properties at hadronization. One speeds up considerably Table 3 presents the here relevant parameters which the convergence by requiring that fits satisfy some ball- must be input with their guessed values or assumed condi- park value such as ǫ =0.45 0.15 GeV/fm3. Once a tions, in order to run the SHARE with CHARM program good physics minimum is obtained,± a constraint can be as input in file thermo.data. When and if we allow γs to removed. If the minimum is very sharp, one must repeat account for excess of strangeness content, we must also this process recursively; when imposing a value such as a introduce γq to account for a similar excess of QGP light favorite value of freeze-out T , the convergence improve- quark content as already discussed in depth in subsect. 5.3. ment constraint has to be adjusted. Page 42 of 58 Eur. Phys. J. A (2015) 51: 114

9.2 Rapidity density yields dN/dy

In fitting the particle produced at RHIC and LHC energies we rarely have full information available about the yields. The detectors are typically designed to either cover the center of momentum domain (central rapidity) or the for- ward/backward “projectile/target” domains. Thus practi- cally always —with the exception of results in SPS range of energies— we do need to focus our analysis on particle yields emerging from a domain, typically characterized by rapidity y of a particle. As a reminder, the rapidity of a particle y replaces in set of kinematic variables the momentum component Fig. 26. Illustration of relativistic heavy ion collision: two parallel to the axis RHI motion. For a particle of mass m Lorentz-contracted nuclei impact with offset, with some of the nucleons participating, and some remaining spectators, i.e. nu- with momentum vector p = p +p split into components  ⊥ cleons that miss the other nucleus, based on ref. [210]. parallel and perpendicular to the axis RHI motion, the relation is

into the longitudinal dynamics yL and the thermal com- 2 2 p = E⊥ sinh y, E⊥ = m + p⊥, (33) ponent yth, describing the statistical thermal production of particles. We have so obtained which implies the useful relation E = E⊥ cosh y. dyth dyL Rapidity is popular due to the additivity of the value of f(yL)=R + . (36) y under a change of reference frame in the -direction char- dτ dτ    acterized by the Lorentz transformation where cosh yL = Since we form dN/dy observing many particles emitted γL,sinhL = βLγL. In this restricted sense rapidity replaces forward (yth > 0) or backward (yth < 0) in rapidity with velocity in the context of relativistic motion. The value of respect to local rest frame, the statistical term averages y is recognized realizing that a fireball emitting particles out and thus we obtain as the requirement for a flat dN/dy will have some specific value of yf which we recognize dis- that the local longitudinal flow satisfies playing particle yields as function of rapidity, integrated with respect to p⊥. dL dy = f(y )=R L , (37) The meaning of an analysis of particle data multiplic- dτ L dτ ities dN/dyp is that we look at the particles that emerged that is a linear relation from dV/dyp: in the fireball incremental volume dV per unit of rapidity of emitted particles dy , p L L = R(y y ). (38) − 0 L − 0 dN dV dL It is tempting to view f(yL) dL/dτ =sinhyL as we D⊥(L) , (34) dy ∝ dy ≡ dy would expect if L were a coordinate≡ of a material particle. The implicit system of equations allows us then to deter- where D⊥ is the transverse surface at hadronization of the mine the dependence of yL and thus L on τ and thus of fireball. Considering the case of sufficiently high energy time evolution in eq. (34) and the relation of dV/dy with where one expects that particle yields dN/dyp are flat as geometric (HBT) volume, a connection that is at present function of rapidity, we can expect that D (L) D (L = ⊥ ≃ ⊥ not understood. This will be a topic for further study. 0) and thus dL/dyp = const., where L = 0 corresponds to the CM-location of the hadron-hadron collision. The quantity dL/dyp relates to the dynamics of each 9.3 Centrality classes of the positions L from which measured particles emerge with a measured rapidity yp. Each such location has its When two atomic nuclei collide at relativistic speed, only proper time τ which applies to both the dynamics of the matter in the collision path, see fig. 26, participates in the longitudinal volume element dL and the dynamics of par- reaction. Two fraction of nuclei are shaved of and fly by ticle production from this volume element. We thus can along collisions axis —we call these nucleons spectators. write The sum of the number of participants and spectators must be exactly the number of nucleons introduced into dL dL/dτ f(y ) = L =const. (35) the reaction: for Pb-Pb this number is 2A = 416 or per- dyp ≡ dyp/dτ d(yth + yL)/dτ haps better said, there are Nq = 1248 valance u, d-quarks, and for Au-Au we have 2A = 358 or Nq = 1074. How many In the last step we recognize the longitudinal dynamics of these quarks actually have interacted in each reaction is introducing the local flow rapidity yL in the numerator hard to know or directly measure. One applies a “trigger” where dL/dτ = f(yL), and in the denominator given the to accept a class of collision events which then is charac- additivity of rapidity we can break up the particle rapidity terized in terms of some macroscopic observable relating Eur. Phys. J. A (2015) 51: 114 Page 43 of 58

Fig. 27. Distribution in Npart for each of experimental trigger classes called a–b% based on a MC Glauber model, data from ref. [211]. to a nearly forward flying spectator. A numerical model Bose-Einstein distribution function: connects the artificially created reaction classes with the 1 mean number of participants Npart that contributed. For ni ni(Ei)= −1 , (39) further details for the LHC work we refer to the recent ≡ Υ exp(Ei/T ) 1 i ± ALICE review of their approach [211]. In fig. 27 we see how this works. All inelastic colli- where the upper sign corresponds to Fermions and the sion classes are divided into groups related to how big a lower one to Bosons. The fugacity Υi of the i-th hadron fraction of all inelastic events the trigger selects. So 0–5% species is described and reduced to the valence quark prop- means that we are addressing the 5% most central colli- erties in subsect. 9.5 below. Then the hadron species i sions, nearly head-on. How head-on this is we can see by yield will correspond to the integral of the distribution function (eq. (39)) over the phase space multiplied by the considering the distribution in Npart one obtains in the Monte-Carlo Glauber model as shown in fig. 27. hadron spin degeneracy gi =(2Ji + 1) and volume V

How do we know that such classification, that is a char- 3 acterization of events in terms of some forward observable d p Ni Ni(mi,gi,V,T,Υi) = giV ni. (40) which is model-converted into participant distribution, is ≡  (2π)3  meaningful? Experimental work provides direct confirma- tion by connecting different observables [211]. I will in The fluctuation of the yield eq. (40) is: the analysis of other experimental results evaluate specific 3 properties of the fireball of matter in terms of the number 2 ∂ Ni d p (ΔNi) = Υi  = giV 3 ni(1 ni). of participants. Some of these properties turn out to be ∂Υi (2π) ∓ T,V  very flat across many of the collision classes as a function    (41) of Npart which entered into the discussion. This shows that It is more practical for numerical computation to express the expected extensivity of the property holds: as more the above yields and fluctuations as an expansion in mod- participants participate the system expands accordingly. ified Bessel functions Moreover, this finding also validates the analysis method, ∞ a point which will be raised in due time. g VT3 ( 1)n−1Υ n nm N = i ± i W i , (42) i 2π2 n3 T n=1   ∞ 9.4 Particle yields and fluctuations g VT3 ( 1)n−1Υ n (ΔN )2 = i ± i i 2π2 n3 n=1 For full and correct evaluation of the final hadron state in   the LHC era, one has to calculate: 2+n 1 nmi − W , (43) × n T 1) primary particle yields at chemical freeze-out;   2   2) charm hadron decays for a given charm quark abun- W (x) x K2(x). (44) ≡ dance, followed by 3) decays of all hadron resonances. These expansions can be calculated to any desired accu- racy; for Bosons convergence requires Υi exp( mi/T ) < 1, The point 2. is the new module that rounds of SHARE otherwise the expansion makes no sense. For− heavy (m for LHC energies [209]. T ) particles, such as charm hadrons, the Boltzmann dis-≫ 2 2 Every hadron of species i with energy Ei = mi + pi tribution is a good approximation, i.e., it is sufficient to populates the energy states according to Fermi-Dirac or evaluate the first term of the expansion in eq. (42), which  Page 44 of 58 Eur. Phys. J. A (2015) 51: 114

i i i i is indeed implemented in the CHARM module of SHARE and Nu¯, Nd¯, Ns¯ and Nc¯ anti-quarks, the fugacity can be to reduce computation time at no observable loss of pre- expressed as cision. N i N i N i N i To evaluate the yield of hadron resonance with finite Υi =(λuγu) u (λdγd) d (λsγs) s (λcγc) c i i i i width Γi, one has to weigh the yield (eq. (40)) by the Nu¯ N ¯ Ns¯ Nc¯ (λu¯γu¯) (λ ¯γ ¯) d (λs¯γs¯) (λc¯γc¯) , (50) resonance mass using the Breit-Wigner distribution: × d d where γf is the phase space occupancy of flavor f and ˜ Γ Γi Ni(M,gi,T,V,Υi) λf is the fugacity factor of flavor f. Note that we allow Ni = dM 2 2   2π (M mi) + Γ /4 for non-integer quark content to account for states like η  − i ¯ N for Γ 0. (45) meson, which is implemented as η =0.55(uu¯+dd)+0.45ss¯ −→ i i → in agreement with [212]. It can be shown that for quarks For low energy states with a large width one has to use the and anti-quarks of the same flavor energy dependent resonance width, since an energy inde- −1 γf = γ ¯ and λf = λ , (51) pendent width implies a way too large probability of the f f¯ resonance being formed with unrealistically small mass. which reduces the number of variables necessary to eval- The partial width of a decay channel i j can be well → uate the fugacity by half. approximated by It is a common practice to take advantage of the isospin symmetry and to treat the two lightest quarks (q = u, d) m 2 lij +1/2 using light quark and isospin phase space occupancy and Γ (M)=b Γ 1 ij , for M>m , i→j i→j i − M ij fugacity factors which are obtained via a transformation     (46) of parameters: where bi→j is the decay channel branching ratio, mij is the γu decay threshold (i.e., sum of the decay product masses) γq = √γuγd ,γ3 = , (52) γd and lij is the angular momentum released in the decay.  The total energy dependent width Γi(M) is obtained using with straightforward backwards transformation the partial widths eq. (46) for all decay channels of the resonance in question as γu = γqγ3,γd = γq/γ3, (53) and similarly for the fugacity factors Γi(M)= Γi→j (M). (47) j λ  λ = λ λ ,λ= u , (54) q u d 3 λ For a resonance with a finite width, we can then re-  d  place eq. (45) by λu = λqλ3,λd = λq/λ3. (55)

∞ Chemical potentials are closely related to fugacity; one Γi→j(M) Ni(M,gi,T,V,Υi) N Γ = dM  , can express an associated chemical potential μi for each i  A (M m )2 + Γ (M)2/4 j mij i i i hadron species i via   − (48) µi/T Υi = e . (56) where Ai is a normalization constant

∞ It is more common to express chemical potentials re- Γi→j (M) lated to conserved quantum numbers of the system, such Ai = dM 2 2 . (49) m (M mi) + Γi(M) /4 as baryon number B, strangeness s, third component of j  ij −  isospin I3 and charm c: Equation (48) is the form used in the program to evaluate μ =3T log λ , (57) hadron resonance yield, whenever calculation with finite B q width is required. Note that yield evaluation with finite μS = T log λq/λs, (58) width is implemented only for hadrons with no charm con- μI3 = T log λ3, (59) stituent quark; zero width (Γci = 0) is used for all charm μC = T log λcλq. (60) hadrons. Notice the inverse, compared to intuitive definition of μS, which has a historical origin and is a source of frequent mistakes. 9.5 Hadron fugacity Υi and quark chemistry

The fugacity of hadron states defines the yields of different 9.6 Resonance decays hadrons based on their quark content. It can be calculated from the individual constituent quark fugacities. In the The hadron yields observed include the post-hadroniza- i i i most general case, for a hadron consisting of Nu, Nd, Ns tion decays of in general free streaming hadron states — i and Nc up, down, strange and charm quarks respectively only a few are stable enough to reach detectors. In fact Eur. Phys. J. A (2015) 51: 114 Page 45 of 58 heavier resonances decay rapidly after the freeze-out and many SHM are in more or less severe conflict with this feed lighter resonances and “stable” particle yields. The fi- value of TH. The model SHARE we detailed in previous nal stable particle yields are obtained by allowing all reso- sect. 9 is, however, in excellent agreement. One of the rea- nances to decay sequentially from the heaviest to the light- sons 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 and participant number class Npart. Especially as a func- N = N + B N , (61) tion of Npart this is not always the case, whence some in- i iprimary j→i j j =i terpolation of data is a part of the analysis. We do not dis-  cuss 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 exper- Another criterion that we use is to focus on parti- imentally 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 [213,214] The SHARE program includes for non-charm hadrons which applies the same non-equilibrium methods in an all decay channels with branching ratio 10−2 in data ta- ambitious effort to describe all LHC particle spectra and ≥ bles. To attain the parallel level of precision for the higher does this with good success. These results are directly rel- number of charm hadron decays (a few hundred (!) in some evant to our study of LHC data presenting complementing cases) with small branching ratios required to set the ac- information 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 [209]. other group. There is still a lot of uncertainty regarding charm de- cay 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 Λ+ pK π0 (3.3 1.0)%, (62) gram is the capability to fully describe the properties of c → ± the fireball that produces the particles analyzed. This is is complemented by the unobserved isospin symmetric not done in terms of produced particles: each carries away channel “content”, such as the energy of the fireball. We eval- + 0 + uate and sum all fractional contributions to the fireball Λc nK π (3.3 1.0)%, (63) → ± bulk properties from the observed and, importantly, un- with the same branching ratio. observed particles, predicted by the fit in their abundance. The influence of resonance feed-down on fluctuations The energy content is only thermal, as we eliminate using is the following: yields the effect of expansion flow on the spectra, i.e. the dynamical collective flow energy of matter. Thus the en- (ΔN )2 = B ( B ) N + B2 (ΔN )2 . j→i  j→i Nj→i − j→i j  j→i j  ergy content we compute is the “comoving” total thermal (64) energy. The first term corresponds to the fluctuations of the Given the large set of parameters that SHARE makes mother particle j, which decays into particle i with available we fit all particles well and thus the physical branching ratio B . is the number of particles j→i Nj→i properties that we report are rather precise images of the i produced in the decay of i (inclusive production) so that observed particle yields. The question what the SHM pa- B = . For nearly all decays of almost all reso- i j→i Nj→i rameters mean does not enter the discussion at all. If a nances j→i = 1, however, there are significant exceptions measurement error has crept in then our results would  N 0 to this, including the production of multiple π , such as look anomalous when inspected as a function of collision η 3π0. The second term in eq. (64) corresponds to the → energy or collision centrality. fluctuation in the yield of the mother particle (resonance). The fit of SHM parameters then provides an ex- trapolation from the measured particle abundances to 10 Hadrons from QGP: What do we learn? unmeasured yields of all particles known and listed in SHARE tables. Most of these are of no great individual A comparison of lattice results with freeze-out conditions relevance, being too massive. The bulk properties we re- were shown in fig. 9. The band near to the tempera- port here are, for the most part, defined by particles di- ture axis displays the lattice estimate for TH presented rectly observed. We expect smooth lines describing the in ref. [71], T = 147 5 MeV. As fig. 9 demonstrates, fireball properties as a function of √s , the CM energy H ± NN Page 46 of 58 Eur. Phys. J. A (2015) 51: 114

and private communication by M. Ga´zdzicki, I have no clear answer to offer to these simple questions. We find a peak in the net baryon density, bottom frame of the fig. 28. The K+/π+ peaking, fig. 23, discussed in subsect. 6.3 seems to be related to the effect of baryon stopping, perhaps a rise as function of √sNN in stopping power at first, when color bonds are broken, and a more gradual decline with increasing energy. But what makes quarks stop just then? And why do they decide to stop less at higher energy, instead “shooting through”? Note that a possible argument that a decrease in baryon den- sity is due to volume growth is not right considering that the thermal energy density ǫ, and the entropy density σ remain constant above the threshold in collision energy. I would argue that when first color bonds are melted, gluons are stopped while quarks are more likely to run out. That would agree with our finding in context of strangeness production, see ref. [16], that despite simi- lar looking matrix elements in perturbative QCD, gluons are much more effective in making things happen due to their “high” adjoint representation color charge; the best analogy would be to say that gluons have double-color charge. The high gluon density at first manages to stop some quarks but the probability decreases with increas- ing energy. It is remarkable how fast the dimensionless ρB−B¯ /σ b/S drops. This expresses the ability to stop quarks normalized≡ to the ability to produce entropy. Seeing all these results, one cannot but ask what the total abundance of strange quark pairs will do. Before the discussion of results seen in fig. 29 it is wise to read the Fig. 28. Fireball bulk properties in the SPS and RHIC energy conclusions in ref. [16] where in 1983 the overall strange- domain, see text. Update of results published in ref. [7]. ness yield enhancement alone was not predicted to be a striking signature. In fig. 29 ratios are shown, in the top frame: the pair strangeness abundance s per net baryon per pair of nucleons and/or as function of collision central- abundance b; per entropy in the middle frame; and in the ity class N . Appearance of discontinuous behavior as npart bottom frame we see the energy cost in GeV to make a a function of s can indicate a change related to QGP √ NN strange quark pair, E/s; mind you that this energy is the formation. final state thermal fireball energy. In fig. 28 we see in the SPS and RHIC energy domain We see in fig. 29 that the s/b ratio is smooth. This for most head-on collisions, from top to bottom, the pres- means that strangeness production takes off where baryon sure P , energy density ε, the entropy density σ, and the stopping takes off, being in the QGP attributed range of ¯ net baryon density ρB−B.SPSandRHIC4π data were √sNN faster than rise in entropy production. And, the used, for RHIC range also results obtained fitting dN/dy energy cost of a pair seems to be very low at high energy: are shown by the dashed line, particles originating in a vol- only 5–6 times the energy that the pair actually carries by ume dV/dy, y 0.5, 0.5 . Only for the baryon density ∈{− } itself, and this factor reflects accurately on how abundant can we recognize a serious difference; the baryon density strangeness is in comparison to all the other constituents in the central rapidity region seems to be a factor 5 below of the QGP fireball. This by itself clearly indicates that the average baryon density. Not shown is the change in the yield converges to chemical QGP equilibrium. The clear fitted volume, which is the one changing quantity (aside break in the cost of making a strange quark pair near of ρB−B¯ ). Volume grows to accommodate the rapid rise in 30 GeV energy shows the threshold above which strange- particle multiplicity with the available energy. ness, as compared to other components, becomes an equal Figure 28 shows exciting features worth further discus- fireball partner. sion. There can be no doubt that over a relatively small Our analysis thus shows: a) There is an onset of baryon domain of collision energy —in laboratory frame, between transparency and entropy production at a very narrowly 20 and 40 A GeV (SPS projectile per nucleon energies) and defined collision energy range. b) Beyond this threshold in CM frame √sNN 6.5, 7.5 GeV per nucleon pair— in collision energy the hadronization proceeds more ef- the properties of the∈{ fireball change} entirely. Is this a sig- fectively into strange antibaryons. c) The universality of nal of the onset of new physics? And if so why, is this hadronization source properties, such as energy density, or happening at this energy? Though this experimental re- entropy density above the same energy threshold, suggest sult has been recognized for nearly 10 years now, ref. [7] as explanation that a new phase of matter hadronizes. Eur. Phys. J. A (2015) 51: 114 Page 47 of 58

Fig. 29. Strangeness pair production s yield from SPS and Fig. 30. LHC experimental data measured by the ALICE ex- RHIC as a function √sNN: yield normalized by net baryon periment in Pb–Pb collisions at √sNN =2.76 TeV as function abundance b in top frame, entropy S in middle frame. At of centrality described by Npart, normalized by Npart/2. Re- bottom the energy cost to produce strangeness. Total parti- sults adapted from refs. [73–75]. cle yields, except for dN/dy results shown as dashed lines in the RHIC energy range. Update of results published in ref. [7]. Given the large set of available SHARE parameters all particles are described very well, a non-complete example There is little doubt considering these cornerstone of the data included is seen in fig. 30. Note that the central analysis results that at SPS at and above the projectile rapidity yields are divided by Npart/2; that is they are per energy of 30 A GeV we produced a rapidly evaporating nucleon pair as in pp collisions. This also means that our (hadronizing) drop of QGP. The analysis results we pre- fit spans a range of a yield of dN 10−4 for the most sented for the properties of the fireball leave very little Y ≃ peripheral collisions to dNr 2000 for the most central space for other interpretation. The properties of the QGP collisions, thus more than 7 orders≃ of magnitude alone of fireball created in the energy range of 30–156 A GeV Pb- particles shown in fig. 30. Pb collisions at CERN are just the same as those obtained About three orders of magnitude of the large range for RHIC beam energy scan, see end of subsect. 6.3. of yields dN/dy that are fitted are absorbed into the rapidly changing volume dV/dy from which these parti- cles emerge, see fig. 31. Note that this result is already 10.2 LHC SHM analysis reduced by the factor Npart/2; thus this is volume per col- liding nucleon pair. For RHIC we see that this is a rather We consider now LHC results obtained at √sNN = constant value to which the LHC results seem to converge 2760 GeV as a function of participant number Npart, for small value of Npart. However for large Npart at LHC sect. 9.3 and compare with an earlier similar analysis of the specific volume keeps growing. Keep in mind that the STAR results available at √sNN = 62 GeV [76]. In com- interpretation of dV/dy is difficult and a priori is not ge- parison, there is a nearly a factor 50 difference in collision ometric, see subsect. 9.2. energy. The results presented here for LHC are from the The corresponding LHC and RHIC chemical freeze-out ALICE experiment as analyzed in refs. [73–75]. The ex- temperature T , fig. 32, varies both at RHIC and LHC in perimental data inputs were discussed extensively in these the same fashion with larger values found for smaller ha- references, the data source includes refs. [87,153,215,216]. dronization volumes. This is natural, as scattering length The analysis of hadron production as a function of par- for decoupling must be larger than the size of the sys- ticipant number Npart at RHIC and LHC proceeds in es- tem and thus the more dense hotter condition is possible sentially the same way as already described. The results for the smaller fireball. One can also argue with the same here presented were obtained without the contribution of outcome that the rapid expansion of the larger fireball can charmed hadrons. lead to stronger supercooling of QGP which directly trans- Page 48 of 58 Eur. Phys. J. A (2015) 51: 114

Fig. 31. The source volume dV/dy at √sNN =2.76 TeV, nor- malized by number of nucleon pairs Npart/2, as a function of the number of participants Npart. For comparison, a similar STAR √sNN = 62 GeV data analysis is shown. Results adapted from refs. [73–75].

Fig. 33. Light quark γq and strange quark γs fugacities at Fig. 32. The chemical freeze-out temperature T at √sNN = √sNN =2.76 TeV as a function of the number of participants 2.76 TeV, as a function of the number of participants Npart, lines guide the eye. Results adapted from refs. [73–75]. Npart, lines guide the eye. Results adapted from refs. [73–75]. forms into free-streaming hadrons. The possibility of di- rect QGP hadronization is supported by the strong chem- ical non-equilibrium with γq > 1, γs > 1 for all collision centralities. These results are seen in fig. 33. In fig. 34 we see the physical properties of the fireball as obtained by the same procedure as discussed in sub- sect. 10.1. With increasing participant number all these bulk properties decrease steadily. This is the most marked difference to the RHIC results. We should here remember that the hadronization volume at LHC given the greater total energy content of the fireball is much greater and thus the dynamics of fireball expansion should be different. Results seen in fig. 34 show a remarkable universality, both when LHC is compared to RHIC, and as a function of centrality; variation as a function of Npart is much smaller than that seen in particle yields in fig. 30 (keep in mind that these results are divided by Npart/2). The univer- sality of the hadronization condition is even more pro- Fig. 34. From top to bottom as function of centrality described nounced when we study, see fig. 35, (ε 3P )/T 4, the inter- by Npart: energy density ε, entropy density σ reduced by a action measure I eq. (11) (compare− subsect. 2.5, fig. 7). m factor 10 to fit in figure, and 3P at √sNN =2.76 TeV. The We observe that the lattice-QCD maximum from fig. 7 dotted line are RHIC √sNN = 62 GeV analysis results not (ε 3P )/T 4 falls right into the uncertainty band of this showing the (larger) error band. Results adapted from refs. [73– − result. Only for γq 1.6andγs 2 a high value for Im 75]. shown in fig. 35 can≃ be obtained.≃ The equilibrium hadron Eur. Phys. J. A (2015) 51: 114 Page 49 of 58

QGP where ms is the (thermal) strange quark mass, γs is the phase space occupancy: here the superscript QGP helps to distinguish from that measured in the hadronization analysis as γs, used without a superscript. The degeneracy is g =12=2spin3color2p where the last factor accounts for the presence of both quarks and antiquarks. In central LHC collisions, the large volume (longer lifespan) also means that strangeness approaches satu- rated yield in the QGP. In peripheral collisions, the short lifespan of the fireball may not be sufficient to reach chem- Fig. 35. Hadronization universality: the interaction measure 4 ical equilibrium. Therefore we introduce a centrality de- (ε 3P )/T evaluated at hadronization condition of the hadron QGP − pendent strangeness phase space occupancy γ (Npart) fireball created in √sNN =2.76 TeV Pb–Pb collisions as a s which is to be used in eq. (65). function of centrality described by Npart. Results adapted from QGP refs. [73–75]. A model of the centrality dependence of γs (Npart)is not an important consideration, as the yield for Npart > 30 is nearly constant. The value of strangeness density re- quires ms = 299 MeV in a QGP fireball at hadroni- zation. For m 140 MeV (mass at a scale of μ s ≃ ≃ 2πT 0.9GeV). γQGP 0.77 is found. The higher ≃ s final ≃ value of ms makes more sense in view of the need to account for the thermal effects. Thus we conclude that for Npart > 30 the fireball contains QGP chemical equi- librium strangeness abundance, with strangeness thermal mass ms = 299 MeV [75]. The ratio of strangeness to entropy is easily recog- nized to be, for QGP, a measure of the relative number of Fig. 36. s + s¯ strangeness density measured in the hadron strange to all particles —adding a factor 4 describing phase (red  squares)  as a function of centrality described by the amount of entropy that each particle carries.≃ Thus a Npart. The dashed (blue) line is a fit with strangeness in the QGP source will weigh in with ratio s/S 0.03 [217]. This QGP phase, see text. Results adapted from refs. [73–75]. is about factor 1.4 larger than one computes≃ for hadron phase at the same T , and this factor describes the stran- geness enhancement effect in abundance which was pre- gas results are about a factor 3 smaller in the relevant dicted to be that small, see ref. [16]. However, if a QGP domain of temperature. fireball was formed we do expect a rather constant s/S as Turning our attention now to strangeness: In the most a function of Npart. central 5% Pb–Pb collisions at the LHC2760, a total of dNss¯/dy 600 strange and anti-strange quarks per unit of rapidity is≃ produced. For the more peripheral collisions the 10.3 Earlier work rise of the total strangeness yield is very rapid, as both the size of the reaction volume and within the small fireball Results of SHM that provide freeze-out T well above TH the approach to saturation of strangeness production in seen in fig. 9 should today be considered obsolete. As an the larger QGP fireball combine. example let us enlarge here on the results of ref. [218] which would be marked in fig. 9 GSI-RHIC at T 174 7, It is of considerable interest to understand the mag- ≃ ± nitude of strangeness QGP density at hadronization. We oB 46 5MeV corresponding to a fit of √sNN = 130 GeV≃ RHIC± results (but the point is not shown above form a sum of all (strange) hadron multiplicities dNh/dy h the upper T margin). This reference assumes full chemi- weighting the sum with the strange content 3 ns 3 of any hadron h and include hidden strangeness,− ≤ to obtain≥ cal equilibrium. They draw attention to agreements with the result shown in fig. 36. Within the error bar the result other results and expectations, both in their conclusions, is a constant; strange quarks and antiquarks in the fireball as well as in the body of their text, verbatim: are 20% more dense than are nucleons bound in nuclei. “The chemical freeze-out temperature Tf 168 However, is this s + s¯ strangeness density shown 2.4 MeV found from a thermal analysis of≃ experi-±   by error bars in fig. 36 a density related to QGP? To give mental data in Pb–Pb collisions at SPS is remark- this result a quantitative QGP meaning we evaluate QGP ably consistent within error with the critical tem- phase strangeness density at a given T , see eq. (42) perature Tc 170 8 MeV obtained from lat- tice Monte Carlo≃ simulations± of QCD at vanishing 3 ∞ gT n baryon density [15] and [16]” s(m ,T; γQGP)= γQGP s s − 2π2 − s n=1 Their lattice references are [15, from the year 2001] [219]   2  1 nms nms and [16, from the year 1999] [220]. The two references K2 , (65) disagree in regard to value of T , verbatim: ×n3 T T H     Page 50 of 58 Eur. Phys. J. A (2015) 51: 114

(1999) “If the quark mass dependence does not if QGP fireball was in chemical equilibrium. In that sense, change drastically closer to the chiral limit the cur- theory supports the finding, and this result also has a very rent data suggest T (170–190) MeV for 2-flavor good χ2/ndf < 1 for all collision centralities. c ≃ QCD in the chiral limit. In fact, this estimate also The value of the ratio p/π experiment =0.046 holds for 3-flavor QCD.”. 0.003 [215,87] is a LHC result that| any model of par-± (2001) “The 3-flavor theory, on the other hand, ticle production in RHI collisions must agree with. The leads to consistently smaller values of the criti- value p/π 0.05 is a natural outcome of the chemi- cal non-equilibrium≃ fit with γ 1.6. This result was cal temperature,. . . 3 flavor QCD: Tc = (154 q ≃ 8) MeV.” ± predicted in ref. [221]: p/π prediction =0.047 0.002 for the hadronization pressure| seen at RHIC and± SPS P = While the authors of ref. [218] were clearly encouraged by 82 5MeV/fm3. Chemical equilibrium model predicts and the 1999 side remark in ref. [220] about 3 flavors, they fits± a very much larger result. This is the so-called proton also cite in the same breath the correction [219] which anomaly; there is no anomaly if one does not dogmatically renders their RHIC SHM fit invalid: for a lattice result prescribe chemical equilibrium conditions. T = 154 8 MeV chemical freeze-out at T 174 7MeV H ± ≃ ± A recent study of the proton spectra within the freeze- seems inconsistent since Tkaons and protons within imental Z/Z ratio used in the paper predicts a value the equilibrium and non-equilibrium approaches. These μZ = oB 2μS =18.8MeV while the paper determines results show the strong overprediction of soft protons and from this− ratio a value μ = o 2μ =9.75 MeV. In Z B − S some overprediction of kaons that one finds in the equilib- conclusion: the cornerstone manuscript of the GSI group rium model. The chemical non-equilibrium model provides is at the time of publication inconsistent with the lattice an excellent description of this key data. used as justification showing chemical freeze-out T>TH by 20 MeV, and its computational part contains a tech- Further evidence for the chemical non-equilibrium out- nical mistake. But, this paper had a “good” confidence come of SHM analysis arises from the universality of ha- level. dronization at LHC, RHIC and SPS: the bulk properties The key argument of the paper is that χ2/dof 1. of the fireball that we determine are all very similar to However, χ2 depends in that case on large error bars≃ in each other. This can be seen by comparing RHIC-SPS re- the initial 130 GeV RHIC results. Trusting χ2 alone is not sults presented in fig. 28 with those shown in fig. 34 for appropriate to judge a fit result12.Awaytosaythisisto LHC-RHIC. argue that a fit must be “confirmed” by theory, and in- This universality includes the strangeness content of deed that is what ref. [218] claimed, citing ref. [219] which the fireball. The LHC particle multiplicity data has rel- however, provided a result in direct disagreement. atively small errors, allowing establishment of relatively precise results. The strong non-equilibrium result γ 2 Thus we can conclude that ref. [218] at time of publica- s → tion had already proved itself wrong. And while humanum seen in fig. 33 allows the description of the large abun- errare est, students lack the experience to capture theirs dance of multi-strange hadrons despite the relatively small effectively. Today this work is cited more than 500 times value of freeze-out temperature. The value of light quark fugacity, γ 1.6 allows a match in the high entropy con- —meaning that despite the obvious errors and omissions q → it has entered into the contemporary knowledge base. Its tent of the QGP fireball with the γq enhanced phase space results confuse the uninitiated deeply. These results could of hadrons, especially mesons. As noted above, this effect only be erased by a direct withdrawal note by the authors. naturally provides the correct p/π ratio at small T . There are two noticeable differences that appear in 10.4 Evaluation of LHC SHM fit results comparing RHIC62 to LHC2760 results; in fig. 31 we see as a function of centrality the specific volume parameter The chemical non-equilibrium SHM describes very well all (dV/dy)/Npart. The noticeable difference is that at RHIC available LHC-2760 hadron production data obtained in this value is essentially constant, while at LHC there is a wide range of centralities Npart,measuredintheCM clearly a visible increase. One can associate this with a within the rapidity interval 0.5 0 0.5. A value of corresponding increase in entropy per participant, imply- − ≤ ≤ freeze-out temperature that is clearly below the range for ing that a novel component in entropy production must TH reported in fig. 9 for lattice-QCD arises only when ac- have opened up in the LHC energy regime. This additional cepting a full chemical non-equilibrium outcome. Chemi- entropy production also explains why at LHC the maxi- cal non-equilibrium is expected for the hadron phase space mum value of specific strangeness pair yield per entropy is 12 2 smaller when compared to RHIC62 for most central colli- Hagedorn explained the abuse of χ as follows: he carried an elephant and mouse transparency set, showing how both sions, see fig. 38. transparencies are fitted by a third one comprising a partial The results found in the LHC-SHM analysis charac- picture of something. Both mouse and elephant fitted the some- terize a fireball that has properties which can be directly thing very well. In order to distinguish mouse from elephant, compared with results of lattice-QCD, and which have not one needs external scientific understanding; in his example, the been as yet reported; thus this analysis offers a predic- scale, was required. tion which can be used to verify the consistency of SHM Eur. Phys. J. A (2015) 51: 114 Page 51 of 58

prediction that as freeze-out T increases, s/S decreases. On first sight this is counterintuitive as we would think that at higher T there is more strangeness. This is an interesting behavior that may provide an opportunity to better understand the relation of the freeze-out analysis with lattice-QCD results.

11 Comments and conclusions

To best of my knowledge nobody has attempted a syn- thesis of the theory of hot hadronic matter, lattice-QCD results, and the statistical hadronization model. This is done here against the backdrop of the rich volume of soft hadron production results that have emerged in the past 20 years in the field of RHI collisions, covering the en- tire range of SPS, RHIC and now LHC energy, ranging a factor 1000 in √sNN. This is certainly not the ultimate word since we expect new and important experimental results in the next few years: LHC-ion operations will reach near maximum en- ergy by the end of 2015; further decisive energy increases could take another lifespan. On the other energy range end, we are reaching out to the domain where we expect the QGP formation energy threshold, both at RHIC-BES, and at SPS-NA61. The future will show if new experimen- Fig. 37. Data: most central spectra of pions, kaons and pro- tal facilities today in construction and/or advanced plan- tons from ALICE experiment [87,88] as a function of p⊥.Lines: ning will come online within this decade and join in the (Top) the non-equilibrium model for parameters of this re- study of QGP formation threshold. Such plans have been view; (bottom) the outcome with equilibrium constraint. Fig- made both at GSI and at DUBNA laboratories. ure adapted from ref. [213]. In order for this report to be also a readable RHI colli- sion introduction, I provided pages distributed across the manuscript, suitable both for students starting in the field, and readers from other areas of science who are interested in the topic. I realize that most of technical material is not accessible to these two groups, but it is better to build a bridge of understanding than to do nothing. Moreover, some historical considerations may be welcome in these circles. I did set many of the insights into their historical con- text which I have witnessed personally. In my eyes under- standing the history, how topics came to be looked at the way they are, helps both the present generation to learn what we know, and the future generation in resolving the misunderstandings that block progress. For this reason I felt that many insights that I needed Fig. 38. Ratio of strangeness pair yield to entropy s/S as func- to develop could be presented here equally well in the tion of collision centrality described by Npart. Results adapted format of work done many years ago. Therefore, in the from refs. [73–75]. jointly published refs. [15,16] I present two unpublished reports from conference proceedings, long gone from li- results with lattice. For example, note the dimensionless brary shelves, which I think provide quite appropriate ratio of the number of strange quark pairs with entropy background material for this report. Perhaps I should have s/S 0.03. Since strange quarks have a mass scale, this abridged this bonus material to omit a few obsolete devel- → ratio can be expected to be a function of temperature opments, and/or to avoid duplication across these 80+ in lattice-QCD evaluation. The interesting question is, at pages. However, any contemporary change will modify in what T will lattice obtain this strangeness hadronization a damaging way the historical context of these presenta- condition s/S =0.03? tions. Further, there is a variation as a function of centrality There was a very special reason to prepare this report seen in fig. 38; s/S decreases with decreasing Npart. Seeing now. I took up this task after I finished editing a book that freeze-out T increases, compare fig. 32, we obtain the to honor 50 years of Hagedorn remarkable achievements, Page 52 of 58 Eur. Phys. J. A (2015) 51: 114 ref. [1]. Given the historical context and the target of in- In this text I also answer simple questions which turn terest in the book being also a person, I could not inject out to have complicated answers. For example, what is, there all the results that the reader sees in these pages. and when was, QGP discovered? It turns out that QGP The overlap between this work and ref. [1] is small, mostly as a phrase meant something else initially, and all kinds of when I describe how the field developed historically before variants such as: hot quark matter; hadron plasma, were 1985. in use. This makes literature search difficult. This background material as presented here is more I also tracked the reporting and interviews from the extensive compared to ref. [1] as I can go into the detail time when CERN decided in February 2000 to step for- without concern about the contents balance of an edited ward with its announcement of the QGP discovery. I book. For this reason a reader of ref. [1] should look at learned that the then director of BNL was highly skep- this text as an extension, and conversely, the reader of this tical of the CERN results. And I was shocked to learn review should also obtain ref. [1] which is freely available that one of the two authors of the CERN scientific con- on-line, published in open format by the publisher in order sensus report declared a few months down the road that to access some of the hard to get references used in this he was mistaken. volume. Seeing these initial doubts, and being expert on stran- This immediately takes us to the question that ex- geness I thought I ought to take a late deep look at how perts in the field will pose: is in this synthesis anything the signature of CERN February 2000 announcement held scientifically new? The answer is yes. The list is actually up in past 15 years. I am happy to tell that it is doing very quite long, and the advice is: read, and fill the gaps where well; the case of QGP at CERN-SPS in terms of strange the developments stop. Let me point here to one result, antibaryon signature is very convincing. I hope the reader the new item which is really not all that new: 1978 [28], will join me in this evaluation, seeing the results shown. Hagedorn and I discovered that hadronic matter with ex- These are not comprehensive (my apologies) but sufficient ponential mass spectrum at the point of singularity TH has to make the point. a universally vanishing speed of sound cs 0 in leading I have spent a lot of time, ink, and paper, to explain order. → here why RHI collisions and QGP physics is, was, and Universally means that this is true for all functions remains a frontier of our understanding of physics. It is that I have tried, including in particular a variation on pre- true that for the trees we sometimes lose the view of the exponential singularity index a. In “leading order” means forest. Thus at a few opportunities in this report I went that the most singular parts of both energy density and outside the trees to tell how the forest looks today, after pressure are considered. This result is found in a report 35 years of healthy growth. While some will see my com- that was submitted to a Bormio Winter Meeting proceed- ments as speculative, others may choose to work out the ings, chapt. 23 in ref. [1] which today is archived electron- consequences, both in theory and experiment. ically at CERN [28]. Commendation I am deeply indebted to Rolf Hagedorn of CERN-TH The reason that this important result cs 0 was not preprinted is that Hagedorn and I were working→ on a larger whose continued mentoring nearly 4 decades ago provided manuscript from which this Bormio text was extracted. much of the guidance and motivation in my long pursuit We did not want to preclude the publication of the large of strangeness in quark–gluon plasma and hadronization paper. However, at the same time we were developing a mechanisms. Rolf Hagedorn was the scientist whose dedi- new field of physics. The main manuscript was never to be cated, determined personal commitment formed the deep finished and submitted, but the field of physics took off. roots of this novel area of physics. In 1964/65 Hagedorn Thus the Bormio report is all that remains in public view. proposed the Hagedorn Temperature TH and the Statis- tical Bootstrap Model (SBM). These novel ideas opened This result, c 0 has other remarkable conse- s|T →TH → up the physics of hot hadronic matter to at first, theoreti- quences: Sound velocity that goes to zero at the critical cal and later, experimental study, in relativistic heavy ion boundary implies that the matter is sticky there, when collision experiments. pressed from inside many unusual things can happen, one being filamenting break-up we call sudden hadronization I thank (alphabetically) Michael Danos (deceased), Wojciech —the amazing thing is that while I was writing this, Gior- Florkowski, Marek Ga´zdzicki, Rolf Hagedorn (deceased), Pe- gio Torrieri was just reminding me about this insight he ter Koch, Inga Kuznetsova, Jean Letessier, Berndt M¨uller, Emanuele Quercigh, Krzysztof Redlich, Helmut Satz, and Gior- had shared before with me. Could cs T →TH 0 actually be the cause why the SHM study of the| fireball→ properties gio Torrieri, who have all contributed in an essential way to fur- obtains such clean sets of results? ther my understanding of relativistic heavy ion collisions, hot hadronic matter, statistical hadronization, and the strangeness Any universal hot hadron matter critical property can signature of QGP. I thank Tamas Biro for critical comments of be tested with lattice-QCD and the results available do a draft manuscript helping to greatly improve the contents; and show a range where c 0. Thus, the value of T s|T →TH ≃ H Victoria Grossack for her kind assistance with the manuscript may become available as the point of a minimum sound presentation. I thank the CERN-TH for hospitality in Sum- velocity. This criterion comes with the Hagedorn expo- mer 2015 while this project was created and completed. This nential mass spectrum “attached”: the result is valid if work has been in part supported by the US Department of En- and when there is an exponential growth in hadron mass ergy, Office of Science, Office of Nuclear Physics under award spectrum. number DE-FG02-04ER41318. Eur. Phys. J. A (2015) 51: 114 Page 53 of 58

Open Access This is an open access article distributed un- preprint CERN-TH-3685, web location verified der the terms of the Creative Commons Attribution License July 2015 https://cds.cern.ch/record/147343/ (http://creativecommons.org/licenses/by/4.0), which permits files/198311019.pdf. unrestricted use, distribution, and reproduction in any me- 17. J. Kapusta, B. M¨uller, J. Rafelski, Quark-Gluon Plasma: dium, provided the original work is properly cited. Theoretical Foundations: An annotated reprint collection (Elsevier, Amsterdam, 2003) pp. 1–817. 18. J. Rafelski, Eur. Phys. J. ST 155, 139 (2008) Strangeness References Enhancement: Challenges and Successes. 19. J. Yellin, Nucl. Phys. B 52, 583 (1973) An explicit solu- 1. J. Rafelski (Editor), Melting Hadrons, Boiling Quarks: tion of the statistical bootstrap. From Hagedorn temperature to ultra-relativistic heavy- 20. T.E.O. Ericson, J. Rafelski, CERN Cour. 43N7,30 ion collisions at CERN; with a tribute to Rolf Hagedorn (2003) web location verified July 2015 http://cds.cern. (Springer, Heidelberg, 2015). ch/record/1733518 The tale of the Hagedorn tempera- 2.P.Koch,B.M¨uller, J. Rafelski, Phys. Rep. 142, 167 ture. (1986) Strangeness in Relativistic Heavy Ion Collisions. 21. C.J. Hamer, S.C. Frautschi, Phys. Rev. D 4, 2125 (1971) 3. H.C. Eggers, J. Rafelski, Int. J. Mod. Phys. A 6, 1067 Determination of asymptotic parameters in the statistical (1991) Strangeness and quark gluon plasma: aspects of bootstrap model. theory and experiment. 22. S.C. Frautschi, C.J. Hamer, Nuovo Cimento A 13, 645 4. J. Rafelski, J. Letessier, A. Tounsi, Acta Phys. Pol. B 27, (1973) Effective temperature of resonance decay in the 1037 (1996) Strange particles from dense hadronic matter, statistical bootstrap model. nucl-th/0209080. 23. C.J. Hamer, Phys. Rev. D 8, 3558 (1973) Explicit solution 5. J. Letessier, J. Rafelski, Int. J. Mod. Phys. E 9, 107 (2000) of the statistical bootstrap model via Laplace transforms. Observing quark gluon plasma with strange hadrons,nucl- 24. C.J. Hamer, Phys. Rev. D 9, 2512 (1974) Single Cluster th/0003014. Formation in the Statistical Bootstrap Model. 6. J. Letessier, J. Rafelski, Hadrons and quark-gluon plasma, 25. R.D. Carlitz, Phys. Rev. D 5, 3231 (1972) Hadronic mat- in Cambridge Monographs on Particle Physics Nuclear ter at high density. Physics and Cosmology, Vol. 18 (Cambridge University 26. R. Hagedorn, Nuovo Cimento A 52, 1336 (1967) On the Press, Cambridge, 2002) pp. 1–397. hadronic mass spectrum. 7. J. Letessier, J. Rafelski, Eur. Phys. J. A 35, 221 (2008) 27. W. Nahm, Nucl. Phys. B 45, 525 (1972) Analytical solu- Hadron production and phase changes in relativistic heavy tion of the statistical bootstrap model. ion collisions, nucl-th/0504028. 28. J. Rafelski, R. Hagedorn, Thermodynamics of hot nuclear 8. S. Weinberg, The Quantum Theory of Fields,Vols.1, 2 matter in the statistical bootstrap model,presentedat (Cambridge University Press, Cambridge 1995, 1996). Bormio Winter Meeting January 1979, also chapt. 23 9. Kohsuke Yagi, Tetsuo Hatsuda, Quark-Gluon Plasma: in ref. [1], web location verfied July 2015: http:// From Big Bang to Little Bang (Cambridge University inspirehep.net/record/1384658/files/CM-P00055555 Press, Cambridge, 2008). .pdf. 10. J. Rafelski, CERN Cour. 54, 10:57 (2014) http:// 841 cerncourier.com/cws/article/cern/59346, Birth of 29. H. Satz, Lect. Notes Phys. , 1 (2012). Extreme states the Hagedorn temperature. of matter in strong interaction physics. An introduction. 11. R. Hagedorn, Nuovo Cimento Suppl. 3, 147 (1965) Sta- 30. A. Tounsi, J. Letessier, J. Rafelski, in Proceedings of tistical thermodynamics of strong interactions at high- Divonne 1994 Hot hadronic matter, NATO-ASI series energies. B, Vol. 356, Hadronic matter equation of state and the 12. S.C. Frautschi, Phys. Rev. D 3, 2821 (1971) Statistical hadron mass spectrum (1995) pp. 105–116. bootstrap model of hadrons. 31. J. Letessier, J. Rafelski, unpublished, Evaluation made 13. R. Hagedorn, Lect. Notes Phys. 221, 53 (1985) preprint for the CERN Courier article in honor of R. Hage- CERN-TH-3918/84, reprinted in chapt. 25 of ref. [1] How dorn [20], based on the method of ref. [30]. We Got to QCD Matter From the Hadron Side by Trial, 32. M. Beitel, K. Gallmeister, C. Greiner, Phys. Rev. C 90, Error. 045203 (2014) Thermalization of Hadrons via Hagedorn 14. H.G. Pugh, Introductory remarks presented at the Work- States, arXiv:1402.1458 [hep-ph]. shop on Future Relativistic Heavy Ion Experiments held 33. A.M. Polyakov, Phys. Lett. B 72, 477 (1978) Thermal at GSI Darmstadt 7-10 October 1980, appeared in pro- Properties of Gauge Fields and Quark Liberation. ceedings: GSI Orange report 1981-6, edited by R. Bock, 34. L. Susskind, Phys. Rev. D 20, 2610 (1979) Lattice Models R. Stock, preprint LBL-11974 web location verfied July of Quark Confinement at High Temperature. 2015; http://www.osti.gov/scitech/biblio/6639601, 35. J.C. Collins, M.J. Perry, Phys. Rev. Lett. 34, 1353 Future relativistic heavy ion experiments. (1975) Superdense Matter: Neutrons Or Asymptotically 15. J. Rafelski, Eur. Phys. J. A 51, 115 (2015) Originally Free Quarks? printed in GSI81-6 Orange Report, pp. 282–324, Extreme 36. N. Cabibbo, G. Parisi, Phys. Lett. B 59, 67 (1975) Expo- States of Nuclear Matter – 1980,editedbyR.Bock,R. nential Hadronic Spectrum and Quark Liberation. Stock. 37. B.C. Barrois, Nucl. Phys. B 129, 390 (1977) Supercon- 16. J. Rafelski, Eur. Phys. J. A 51, 116 (2015) Originally ducting Quark Matter. printed in LBL-16281 pp. 489–510, Strangeness and 38. M.G. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B Phase Changes in Hot Hadronic Matter – 1983, 422, 247 (1998) QCD at finite baryon density: Nucleon also report number UC-34C, DOE CONF-830675, droplets and color superconductivity, hep-ph/9711395. Page 54 of 58 Eur. Phys. J. A (2015) 51: 114

39. M.G. Alford, K. Rajagopal, F. Wilczek, Nucl. Phys. B arizona.edu/~rafelski/Books/85SAJPKocRaf.pdf Why 537, 443 (1999) Color flavor locking and chiral symmetry the hadronic gas description of hadronic reactions works: breaking in high density QCD, hep-ph/9804403. The example of strange hadrons. 40. M.G. Alford, A. Schmitt, K. Rajagopal, T. Schfer, Rev. 62. G. Torrieri, J. Rafelski, New J. Phys. 3, 12 (2001) Search Mod. Phys. 80, 1455 (2008) Color superconductivity in for QGP and thermal freezeout of strange hadrons,hep- dense quark matter, arXiv:0709.4635 [hep-ph]. ph/0012102. 41. W. Broniowski, W. Florkowski, Phys. Lett. B 490, 223 63. W. Broniowski, W. Florkowski, Phys. Rev. C 65, 064905 (2000) Different Hagedorn temperatures for mesons and (2002) Strange particle production at RHIC in a single baryons from experimental mass spectra, compound had- freezeout model, nucl-th/0112043. rons, and combinatorial saturation, hep-ph/0004104. 64. A. Baran, W. Broniowski, W. Florkowski, Acta Phys. Pol. 42. W. Broniowski, W. Florkowski, L.Y. Glozman, Phys. B 35, 779 (2004) Description of the particle ratios and Rev. D 70, 117503 (2004) Update of the Hagedorn mass transverse momentum spectra for various centralities at spectrum, hep-ph/0407290. RHIC in a single freezeout model, nucl-th/0305075. 43. J. Cleymans, D. Worku, Mod. Phys. Lett. A 26, 65. J. Rafelski, Phys. Lett. B 262, 333 (1991) Strange anti- 1197 (2011) The Hagedorn temperature Revisited, baryons from quark - gluon plasma. arXiv:1103.1463 [hep-ph]. 66. T. Csorgo, L.P. Csernai, Phys. Lett. B 333, 494 (1994) 44. T.D. Cohen, V. Krejcirik, J. Phys. G 39, 055001 (2012) Quark - gluon plasma freezeout from a supercooled state?, Does the Empirical Meson Spectrum Support the Hage- hep-ph/9406365. dorn Conjecture?, arXiv:1107.2130 [hep-ph]. 67. L.P. Csernai, I.N. Mishustin, Phys. Rev. Lett. 74, 5005 45. T.S. Biro, A. Peshier, Phys. Lett. B 632, 247 (2006) Lim- (1995) Fast hadronization of supercooled quark - gluon iting temperature from a parton gas with power-law tailed plasma. 59 distribution, hep-ph/0506132. 68. T.S. Biro, P. Levai, J. Zimanyi, Phys. Rev. C , 1574 46. P.M. Lo, M. Marczenko, K. Redlich, C. Sasaki, (1999) Hadronization with a confining equation of state, Matching Hagedorn mass spectrum with Lattice QCD, hep-ph/9807303. 69. J. Rafelski, J. Letessier, Phys. Rev. Lett. 85, 4695 (2000) arXiv:1507.06398 [nucl-th]. Sudden hadronization in relativistic nuclear collisions, 47. A. Majumder, B. M¨uller, Phys. Rev. Lett. 105, 252002 hep-ph/0006200. (2010) Hadron Mass Spectrum from Lattice QCD, 70. A. Keranen, J. Manninen, L.P. Csernai, V. Magas, Phys. arXiv:1008.1747 [hep-ph]. Rev. C 67, 034905 (2003) Statistical hadronization of su- 48. S.Z. Belenky, Nucl. Phys. 2, 259 (1956) Connection be- percooled quark gluon plasma, nucl-th/0205019. tween scattering and multiple production of particles. 71. S. Borsanyi, Nucl. Phys. A 904, 270c (2013) Ther- 49. K. Redlich, H. Satz, The Legacy of Rolf Hagedorn: Statis- modynamics of the QCD transition from lattice, arXiv: tical Bootstrap, Ultimate Temperature, arXiv:1501.07523 1210.6901 [hep-lat], presented at the 23rd International [hep-ph], chapt. 7 in [1] Conference on Ultrarelativistic Nucleus-Nucleus Colli- 50. J. Rafelski, Nucl. Phys. Proc. Suppl. 243, 155 (2013) sions (QM 2012), August 13-18, 2012, Washington, DC, Connecting QGP-Heavy Ion Physics to the Early Uni- USA. verse, arXiv:1306.2471 [astro-ph.CO]. 72. S. Borsanyi, Z. Fodor, C. Hoelbling, S.D. Katz, S. Krieg, 46 51. L.P. Csernai, J.I. Kapusta, Phys. Rev. D , 1379 (1992) K.K. Szabo, Phys. Lett. B 730, 99 (2014) Full re- Nucleation of relativistic first order phase transitions. sult for the QCD equation of state with 2+1 flavors, 90 52. T.S. Biro, A. Jakovac, Phys. Rev. D , 094029 (2014) arXiv:1309.5258 [hep-lat]. QCD above Tc: Hadrons, partons, and the continuum, 73. M. Petran, J. Letessier, V. Petracek, J. Rafelski, Phys. arXiv:1405.5471 [hep-ph]. Rev. C 88, 034907 (2013) Hadron production and 53. A.Z. Mekjian, Phys. Rev. C 17, 1051 (1978) Explosive nu- quark-gluon plasma hadronization in Pb-Pb collisions at cleosynthesis, equilibrium thermodynamics, and relativis- √sNN =2.76 TeV, arXiv:1303.2098 [hep-ph]. tic heavy-ion collisions. 74. M. Petran, J. Rafelski, Phys. Rev. C 88, 021901 (2013) 54. I. Montvay, J. Zimanyi, Nucl. Phys. A 316, 490 (1979) Universal hadronization condition in heavy ion colli- Hadron Chemistry in Heavy Ion Collisions. sions at √sNN = 62 GeV and at √sNN =2.76 TeV, 55. P. Koch, J. Rafelski, Nucl. Phys. A 444, 678 (1985) Time arXiv:1303.0913 [hep-ph]. Evolution of Strange Particle Densities in Hot Hadronic 75. M. Petran, J. Letessier, V. Petracek, J. Rafelski, J. Phys. Matter. Conf. Ser. 509, 012018 (2014) Interpretation of strange 56. H. Koppe, Phys. Rev. 76, 688 (1949) On the Production hadron production at LHC, arXiv:1309.6382 [hep-ph]. of Mesons. 76. M. Petran, J. Letessier, V. Petracek, J. Rafelski, Acta 57. H. Koppe, Z. Naturforsch. A 3, 251 (1948) The me- Phys. Pol. Suppl. 5, 255 (2012) Strangeness Production in son output from the bombardment of light nuclei with α- Au-Au collisions at √sNN =62.4 GeV, arXiv:1112.3189 particles. [hep-ph]. 58. E. Fermi, Prog. Theor. Phys. 5, 570 (1950) High-energy 77. J. Rafelski, J. Letessier, J. Phys. G 36, 064017 (2009) nuclear events. Critical Hadronization Pressure, arXiv:0902.0063 [hep- 59. I.Y. Pomeranchuk, Dokl. Akad. Nauk Ser. Fiz. 78, 889 ph]. (1951) (translated in [17], pp. 20–23) On the theory of 78. J. Rafelski, J. Letessier, PoS CONFINEMENT 8, 111 multiple particle production in a single collision. (2008) Particle Production and Deconfinement Threshold, 60. R. Hagedorn, Nuovo Cimento A 56, 1027 (1968) Hadronic arXiv:0901.2406 [hep-ph]. matterneartheboilingpoint. 79. J. Letessier, J. Rafelski, Phys. Rev. C 59, 947 (1999) 61. P. Koch, J. Rafelski, South Afr. J. Phys. 9, 8 (1986) Chemical non-equilibrium and deconfinement in 200- web locatio verified July 2015 http://www.physics. A/GeV sulphur induced reactions, hep-ph/9806386. Eur. Phys. J. A (2015) 51: 114 Page 55 of 58

80. P. Braun-Munzinger, J. Stachel, J.P. Wessels, N. Xu, 97. T.A. DeGrand, R.L. Jaffe, K. Johnson, J.E. Kiskis, Phys. Phys. Lett. B 344, 43 (1995) Thermal equilibration and Rev. D 12, 2060 (1975) Masses and Other Parameters of expansion in nucleus-nucleus collisions at the AGS,nucl- the Light Hadrons. th/9410026. 98. K. Johnson, Acta Phys. Pol. B 6, 865 (1975) The M.I.T. 81. A. Andronic, P. Braun-Munzinger, J. Stachel, Nucl. Bag Model. Phys. A 772, 167 (2006) Hadron production in central 99. A.W. Thomas, Adv. Nucl. Phys. 13, 1 (1984) Chiral Sym- nucleus-nucleus collisions at chemical freeze-out,nucl- metry and the Bag Model: A New Starting Point for Nu- th/0511071. clear Physics. 82. F. Becattini, P. Castorina, A. Milov, H. Satz, Eur. Phys. 100. A.T.M. Aerts, J. Rafelski, Phys. Lett. B 148, 337 (1984) J. C 66, 377 (2010) A Comparative analysis of statistical QCD, Bags and Hadron Masses. hadron production, arXiv:0911.3026 [hep-ph]. 101. A.T.M. Aerts, J. Rafelski, Strange Hadrons In The 83. J. Manninen, F. Becattini, Phys. Rev. C 78, 054901 Mit Bag Model CERN-TH-4160-85, UCT-TP-27-2- (2008) Chemical freeze-out in ultra-relativistic heavy ion 1985, web location verified July 2015 http://www-lib. collisions at √sNN = 130 and 200-GeV, arXiv:0806.4100 kek.jp/cgi-bin/img index?8505459. [nucl-th]. 102. R.L. Thews, M. Schroedter, J. Rafelski, Phys. Rev. C 84. F. Becattini, M. Gazdzicki, A. Keranen, J. Manninen, R. 63, 054905 (2001) Enhanced J/ψ production in deconfined Stock, Phys. Rev. C 69, 024905 (2004) Chemical equilib- quark matter, hep-ph/0007323. rium in nucleus nucleus collisions at relativistic energies, 103. M. Schroedter, R.L. Thews, J. Rafelski, Phys. Rev. C 62, hep-ph/0310049. 024905 (2000) Bc meson production in nuclear collisions 85. J. Cleymans, H. Oeschler, K. Redlich, S. Wheaton, Phys. at RHIC, hep-ph/0004041. Rev. C 73, 034905 (2006) Comparison of chemical freeze- 104. A. Rothkopf, T. Hatsuda, S. Sasaki, Phys. Rev. Lett. out criteria in heavy-ion collisions, hep-ph/0511094. 108, 162001 (2012) Complex Heavy-Quark Potential at 86. STAR Collaboration (B.I. Abelev et al.), Phys. Rev. C Finite Temperature from Lattice QCD, arXiv:1108.1579 79, 034909 (2009) Systematic Measurements of Identified + [hep-lat]. Particle Spectra in pp, d Au and Au+Au Collisions from 105. A. Bazavov, P. Petreczky, Nucl. Phys. A 904, 599c (2013) STAR, arXiv:0808.2041 [nucl-ex]. On static quark anti-quark potential at non-zero temper- 87. ALICE Collaboration (B. Abelev et al.), Phys. Rev. Lett. ature, arXiv:1210.6314 [hep-lat]. 109, 252301 (2012) Pion, Kaon, and Proton Produc- 106. B.J. Harrington, A. Yildiz, Phys. Rev. Lett. 33, 324 tion in Central Pb–Pb Collisions at s =2.76 TeV, √ NN (1974) High Density Phase Transitions in Gauge Theo- arXiv:1208.1974 [hep-ex]. ries. 88. ALICE Collaboration (B. Abelev et al.), Phys. Rev. 107. S.A. Chin, Phys. Lett. B 78, 552 (1978) Transition to Hot C 88, 044910 (2013) Centrality dependence of π,K,p Quark Matter in Relativistic Heavy Ion Collision. production in Pb-Pb collisions at √sNN =2.76 TeV, 108. A. Bazavov et al.,Phys.Rev.D80, 014504 (2009) Equa- arXiv:1303.0737 [hep-ex]. tion of state and QCD transition at finite temperature, 89. I.A. Karpenko, Y.M. Sinyukov, K. Werner, Phys. Rev. arXiv:0903.4379 [hep-lat]. C 87, 024914 (2013) Uniform description of bulk observ- 109. HotQCD Collaboration (A. Bazavov et al.), Phys. Rev. ables in the hydrokinetic model of A + A collisions at the D 90, 094503 (2014) Equation of state in (2 + 1)-flavor BNL Relativistic Heavy Ion Collider and the CERN Large QCD, arXiv:1407.6387 [hep-lat]. Hadron Collider, arXiv:1204.5351 [nucl-th]. 67 90. A.S. Botvina, J. Steinheimer, E. Bratkovskaya, M. Ble- 110. J. Letessier, J. Rafelski, Phys. Rev. C , 031902 (2003) icher, J. Pochodzalla, Phys. Lett. B 742, 7 (2015) For- QCD equations of state and the QGP liquid model,hep- mation of hypermatter and hypernuclei within trans- ph/0301099. port models in relativistic ion collisions, arXiv:1412.6665 111. J.O. Andersen, N. Haque, M.G. Mustafa, M. Strick- [nucl-th]. land, N. Su, Equation of State for QCD at finite tem- 91. J. Letessier, A. Tounsi, U.W. Heinz, J. Sollfrank, J. Rafel- perature and density. Resummation versus lattice data, ski, Phys. Rev. D 51, 3408 (1995) Strangeness conserva- arXiv:1411.1253 [hep-ph], invited talk at 11th Quark tion in hot nuclear fireballs, hep-ph/9212210. Confinement and the Hadron Spectrum, September 8-12 92. J. Rafelski, J. Letessier, Nucl. Phys. A 715,98 2014, Saint Petersburg, Russia. (2003) Testing limits of statistical hadronization,nucl- 112. M. Strickland, J.O. Andersen, A. Bandyopadhyay, N. th/0209084. Haque, M.G. Mustafa, N. Su, Nucl. Phys. A 931, 841 93. T.D. Lee, Trans. N.Y. Acad. Sci., Ser. II 40, 111 (1980); (2014) Three loop HTL perturbation theory at finite tem- Based on September 28, 1979 lecture at Columbia Uni- perature and chemical potential, arXiv:1407.3671. versity; Preprint CU-TP-170 (1979) Is the Physical Vac- 113. N. Haque, A. Bandyopadhyay, J.O. Andersen, M.G. uum a Medium?, reprinted in: T.D. Lee, Selected papers, Mustafa, M. Strickland, N. Su, JHEP 05, 027 (2014) edited by G. Feinberg, Vol. 2 (Birkhauser, Boston, 1986) Three-loop HTLpt thermodynamics at finite temperature pp. 213–225. and chemical potential, arXiv:1402.6907 [hep-ph]. 94. T.D. Lee, Particle Physics and Introduction to Field The- 114. L. Van Hove, in 17th International Symposium on Multi- ory (Science Press, Harwood Academic, Beijing, Chur, particle Dynamics, held in Seewinkel, Austria, 16–20 Jun London, 1981). 1986, edited by J. MacNaughton, W. Majerotto, Walter, 95. K.G. Wilson, Phys. Rev. D 10, 2445 (1974) Confinement M. Markytan (World Scientific, Singapore, 1986) pp. 801– of Quarks. 818 Theoretical prediction of a new state of matter, the 96. A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, V.F. “quark-gluon plasma” (also called “quark matter”). Weisskopf, Phys. Rev. D 9, 3471 (1974) ANewExtended 115. E.V. Shuryak, Phys. Lett. B 78, 150 (1978) (Sov. J. Nucl. Model of Hadrons. Phys. 28, 408 (1978)) (Yad. Fiz. 28, 796 (1978)) Quark- Page 56 of 58 Eur. Phys. J. A (2015) 51: 114

Gluon Plasma and Hadronic Production of Leptons, Pho- rapidity (This manuscript is identical with University of tons and Psions. Cape Town preprint UCT-TP 7/84 which is the long ver- 116. O.K. Kalashnikov, V.V. Klimov, Phys. Lett. B 88, 328 sion of University of Frankfurt preprint UFTP-82/94). (1979) Phase Transition in Quark-Gluon Plasma. 137. R. Hagedorn, I. Montvay, J. Rafelski, Thermodynam- 117. D.D. Ivanenko, D.F. Kurdgelaidze, Astrophysics 1, 251 ics Of Nuclear Matter From The Statistical Bootstrap (1965) (Astrofiz. 1, 479 (1965)) Hypothesis concerning Model, CERN-TH-2605, in Hadronic Matter, Erice, Oc- quark stars. tober 1978, edited by N. Cabibbo, L. Sertorio (Plenum 118. H. Fritzsch, M. Gell-Mann, H. Leutwyler, Phys. Lett. B Press, New York, 1980) p. 49. 47, 365 (1973) Advantages of the Color Octet Gluon Pic- 138. CERN press release, web location verfied July 2015 ture. http://press.web.cern.ch/press-releases/2000/02/ 119. P. Carruthers, Collect. Phenomena 1, 147 (1974) Quark- new-state-matter-created-cern. ium: a bizarre Fermi liquid. 139. Brookhaven National Laboratory, (2000, February 10), 120. A.D. Linde, Rep. Prog. Phys. 42, 389 (1979) Phase Tran- Tantalizing Hints Of Quark Gluon Plasma Found At sitions in Gauge Theories and Cosmology. CERN ScienceDaily, web location verfied July 2015 www. 121. B.A. Freedman, L.D. McLerran, Phys. Rev. D 16, 1169 sciencedaily.com/releases/2000/02/000209215728. (1977) Fermions and Gauge Vector Mesons at Finite htm. Temperature and Density. 3. The Ground State Energy 140. BRAHMS Collaboration (I. Arsene et al.), Nucl. Phys. A of a Relativistic Quark Gas. 757, 1 (2005) Quark gluon plasma and color glass con- 122. E.V. Shuryak, Sov. Phys. JETP 47, 212 (1978) (Zh. Eksp. densate at RHIC? The Perspective from the BRAHMS Teor. Fiz. 74, 408 (1978)) Theory of Hadronic Plasma. experiment, nucl-ex/0410020. 123. J.I. Kapusta, Nucl. Phys. B 148, 461 (1979) Quantum 141. PHENIX Collaboration (K. Adcox et al.), Nucl. Phys. Chromodynamics at High Temperature. A 757, 184 (2005) Formation of dense partonic matter 124. E.V. Shuryak, Phys. Rep. 61, 71 (1980) Quantum Chro- in relativistic nucleus-nucleus collisions at RHIC: Exper- modynamics and the Theory of Superdense Matter. imental evaluation by the PHENIX collaboration,nucl- 125. G.F. Chapline, A.K. Kerman, Preprint CTP-695 MIT- ex/0410003. Cambridge - April (1978) web location verified July 2015 142. PHOBOS Collaboration (B.B. Back et al.), Nucl. Phys. A http://inspirehep.net/record/134446/files/CTP- 757, 28 (2005) The PHOBOS perspective on discoveries 695.pdf On the Possibility of Making Quark Matter in at RHIC, nucl-ex/0410022. Nuclear Collisions. 143. STAR Collaboration (J. Adams et al.), Nucl. Phys. A 126. G.F. Chapline, M.H. Johnson, E. Teller, M.S. Weiss, 757, 102 (2005) Experimental and theoretical challenges Phys. Rev. D 8, 4302 (1973) Highly excited nuclear mat- in the search for the quark gluon plasma: The STAR ter. Collaboration’s critical assessment of the evidence from 127. L. Lederman, J. Weneser (Editors), Workshop on RHIC collisions, nucl-ex/0501009. BeV/nucleon collisions of heavy ions: How and Why held 144. BNL press release, web location verfied July 2015 at Bear Mountain, New York, November 29-December https://www.bnl.gov/rhic/news2/news.asp?a=5756 & 1, 1974, proceedings available as Brookhaven National t=today. \ Laboratory Report #50445, (BNL-50445, Upton, NY, 145. J. Rafelski, B. M¨uller, Phys. Rev. Lett. 48, 1066 (1982) 1975), web location verified July 2015 http://www.osti. 56, 2334(E) (1986) Strangeness production in the Quark- gov/scitech/servlets/purl/4061527. Gluon Plasma. 128. T.D. Lee, G.C. Wick, Phys. Rev. D 9, 2291 (1974) Vac- 146. T. Biro, J. Zimanyi, Phys. Lett. B 113, 6 (1982) Quarko- uum stability and vacuum excitation in a spin-0 field the- chemistry in Relativistic Heavy Ion Collisions. ory. 147. J. Rafelski, Phys. Rep. 88, 331 (1982) Formation and 129. G. Odyniec, Begin of the search for the Quark-Gluon Observables of the Quark-Gluon Plasma. Plasma, chapt. 12 in ref. [1]. 148. J. Rafelski, Strangeness in Quark - Gluon Plasma,lec- 130. H.H. Gutbrod, The Path to Heavy Ions at LHC and Be- ture at Quark Matter Formation And Heavy Ion Col- yond,chapt.13inref.[1]. lisions, Bielefeld, May 1982, due to mail mishap ulti- 131. S.A. Chin, A.K. Kerman, Phys. Rev. Lett. 43, mately published in South Afr. J. Phys. 6, 37 (1983), 1292 (1979) Possible longlived hyperstrange multi-quark see chapts. 30, 31 in ref. [1]. droplets. 149. L. Van Hove, Quark Matter Formation And Heavy Ion 132. G.F. Chapline, A. Granik, Nucl. Phys. A 459, 681 (1986) Collisions: The Theoretical Situation, preprint CERN- Production of quark matter via oblique shock waves. TH-3360 19 July 1982, in Multiparticle Dynamics 1982 133. R. Anishetty, P. Koehler, L.D. McLerran, Phys. Rev. D Volendam 6-11 June 1982; edited by W. Kittel, W. 22, 2793 (1980) Central collisions between heavy nuclei Metzger, A. Stergiou (World Scientific, Singapore, 1983) at extremely high-energies: The fragmentation region. pp. 244–266, 134. R. Hagedorn, CERN-TH-3014 (1980), reprinted in 150. F. Antinori, The heavy ion physics programme at the chapt. 26 of ref. [1] How To Deal With Relativistic Heavy CERN OMEGA spectrometer, web location verfied July Ion Collisions. 2015 http://cds.cern.ch/record/343249/files/p43. 135. J.D. Bjorken, Phys. Rev. D 27, 140 (1983) Highly Rela- pdf pp. 43–49 in edited by M. Jacob, E. Quercigh, tivistic Nucleus-Nucleus Collisions: The Central Rapidity CERN-Yellow Report 97-02, Symposium on the CERN Region. Omega Spectrometer: 25 Years of Physics held 19 136. M. Danos, J. Rafelski, Heavy Ion Phys. 14, 97 (2001) March 1997, CERN Geneva, Switzerland, web location Baryon rich quark gluon plasma in nuclear collision,nucl- verfied July 2015 http://cds.cern.ch/record/330556/ th/0011049 Formation of quark-gluon plasma at central files/CERN-97-02.pdf. Eur. Phys. J. A (2015) 51: 114 Page 57 of 58

151. WA97 Collaboration (E. Andersen et al.), Phys. Lett. B 170. T. Matsui, H. Satz, Phys. Lett. B 178, 416 (1986) J/ψ 449, 401 (1999) Strangeness enhancement at mid-rapidity Suppression by Quark-Gluon Plasma Formation. in Pb Pb collisions at 158-A-GeV/c. 171. A. Andronic, Nucl. Phys. A 931, 135 (2014) Experimental 152. STAR Collaboration (B. Abelev et al.), Phys. Lett. B results and phenomenology of quarkonium production in 673, 183 (2009) Energy and system size dependence of relativistic nuclear collisions, arXiv:1409.5778 [nucl-ex]. phi meson production in Cu+Cu and Au+Au collisions, 172. J.P. Blaizot, D. De Boni, P. Faccioli, G. Garberoglio, arXiv:0810.4979 [nucl-ex]. Heavy quark bound states in a quark-gluon plasma: disso- 153. ALICE Collaboration (B. Abelev et al.), Phys. Lett. B ciation and recombination, arXiv:1503.03857 [nucl-th]. 728, 216 (2014) 734, 409(E) (2014) Multi-strange baryon 173. ALICE Collaboration (B.B. Abelev et al.), Phys. Lett. B 734, 314 (2014) Centrality, rapidity and transverse mo- production at mid-rapidity in Pb-Pb collisions at √sNN = 2.76 TeV, arXiv:1307.5543 [nucl-ex]. mentum dependence of J/ψ suppression in Pb-Pb colli- 154. K. Geiger, D. Kumar Srivastava, Phys. Rev. C 56, 2718 sions at √sNN =2.76 TeV, arXiv:1311.0214 [nucl-ex]. (1997) Parton cascade description of relativistic heavy ion 174. PHENIX Collaboration (A. Adare et al.), Systematic collisions at CERN SPS energies?, nucl-th/9706002. study of charged-pion and kaon femtoscopy in Au+Au col- lisions at s = 200 GeV, arXiv:1504.05168 [nucl-ex]. 155. K. Geiger, Phys. Rep. 258, 237 (1995) Space-time de- √ NN 175. ALICE Collaboration (J. Adam et al.), One-dimensional scription of ultrarelativistic nuclear collisions in the QCD pion, kaon, and proton femtoscopy in Pb-Pb collisions at parton picture. √sNN =2.76 TeV, arXiv:1506.07884 [nucl-ex] 156. H.J. Drescher, M. Hladik, S. Ostapchenko, T. Pierog, K. 176. T. Altinoluk, N. Armesto, G. Beuf, A. Kovner, M. Lublin- 350 Werner, Phys. Rep. , 93 (2001) Parton based Gribov- sky, Bose enhancement and the ridge, arXiv:1503.07126 Regge theory, hep-ph/0007198. [hep-ph]. 157. E. Iancu, R. Venugopalan, The Color glass condensate 177. L.P. Csernai, G. Mocanu, Z. Neda, Phys. Rev. C and high-energy scattering in QCD, Quark gluon plasma 85, 068201 (2012) Fluctuations in Hadronizing QGP, 3 in edited by R.C. Hwa, X.-N. Wang, (World Scientific, arXiv:1204.6394 [nucl-th]. Singapore, 2003) pp. 249–336 [hep-ph/0303204]. 178. A. Wroblewski, Acta Phys. Pol. B 16, 379 (1985) On 158. K. Geiger, Phys. Rev. D 46, 4965 (1992) Thermalization The Strange Quark Suppression Factor In High-energy in ultrarelativistic nuclear collisions. 1. Parton kinetics Collisions. and quark gluon plasma formation. 179. K. Rajagopal, Nucl. Phys. A 661, 150 (1999) Mapping 159. K. Geiger, Phys. Rev. D 46, 4986 (1992) Thermalization the QCD phase diagram, hep-ph/9908360. in ultrarelativistic nuclear collisions. 2. Entropy produc- 180. M. Gazdzicki, M. Gorenstein, P. Seyboth, Acta Phys. tion and energy densities at RHIC and LHC. Pol. B 42, 307 (2011) Onset of deconfinement in nucleus- 160. B. M¨uller, A. Schafer, Int. J. Mod. Phys. E 20, 2235 nucleus collisions: Review for pedestrians and experts, (2011) Entropy Creation in Relativistic Heavy Ion Colli- arXiv:1006.1765 [hep-ph]. sions, arXiv:1110.2378 [hep-ph]. 181. NA49 Collaboration (K. Grebieszkow), Acta Phys. Pol. B 161. A. Kurkela, E. Lu, Phys. Rev. Lett. 113, 182301 (2014) 43, 609 (2012) Report from NA49, arXiv:1112.0794 [nucl- Approach to Equilibrium in Weakly Coupled Non-Abelian ex]. Plasmas, arXiv:1405.6318 [hep-ph]. 182. NA61/SHINE Collaboration (N. Abgrall et al.), Re- 162. L. Labun, J. Rafelski, Acta Phys. Pol. B 41, 2763 port from the NA61/SHINE experiment at the CERN (2010) Strong Field Physics: Probing Critical Acceleration SPS CERN-SPSC-2014-031, SPSC-SR-145 (2014), web and Inertia with Laser Pulses and Quark-Gluon Plasma, location verified July 2015 http://cds.cern.ch/record/ arXiv:1010.1970 [hep-ph]. 1955138. 163. J. Letessier, J. Rafelski, A. Tounsi, Phys. Rev. C 50, 406 183. NA61/SHINE Collaboration (M. Gazdzicki), EPJ Web (1994) Gluon production, cooling and entropy in nuclear of Conference 95, 01005 (2015) Recent results from collisions, hep-ph/9711346. NA61/SHINE, arXiv:1412.4243 [hep-ex]. 184. P. Sorensen, J. Phys. Conf. Ser. 446, 012015 (2013) Beam 164. J. Letessier, J. Rafelski, A. Tounsi, Acta Phys. Pol. A 85, Energy Scan Results from RHIC. 699 (1994) In search of entropy. 185. STAR Collaboration (N.R. Sahoo), J. Phys. Conf. Ser. 165. B. M¨uller, Phys. Rev. C 67, 061901 (2003) Phenomenol- 535, 012007 (2014) Recent results on event-by-event fluc- ogy of jet quenching in heavy ion collisions,nucl- tuations from the RHIC Beam Energy Scan program in th/0208038. the STAR experiment, arXiv:1407.1554 [nucl-ex]. 166. A. Majumder, M. Van Leeuwen, Prog. Part. Nucl. Phys. 186. S. Durr, Z. Fodor, J. Frison, C. Hoelbling, R. Hoffmann, A 66, 41 (2011) The Theory and Phenomenology of Per- S.D.Katz,S.Krieg,T.Kurthet al., Science 322, 1224 turbative QCD Based Jet Quenching, arXiv:1002.2206 (2008) Ab-Initio Determination of Light Hadron Masses, [hep-ph]. arXiv:0906.3599 [hep-lat]. 167. L. Bhattacharya, R. Ryblewski, M. Strickland, Photon 187. A.S. Kronfeld, Lattice Gauge Theory and the Origin of production from a non-equilibrium quark-gluon plasma, Mass,in100 Years of Subatomic Physics,editedbyE.M. arXiv:1507.06605 [hep-ph]. Henley, S.D. Ellis (World Scientific, Singapore, 2013) 168. R. Ryblewski, M. Strickland, Phys. Rev. D 92, 025026 pp. 493–518 arXiv:1209.3468 [physics.hist-ph]. (2015) Dilepton production from the quark-gluon plasma 188. S. Aoki et al., Eur. Phys. J. C 74, 2890 (2014) Review using (3+1)-dimensional anisotropic dissipative hydrody- of lattice results concerning low-energy particle physics, namics, arXiv:1501.03418 [nucl-th]. arXiv:1310.8555 [hep-lat]. 169. V. Petousis, Theoretical Review on QCD and Vec- 189. A. Di Giacomo, H.G. Dosch, V.I. Shevchenko, Y.A. Si- tor Mesons in Dileptonic Quark Gluon Plasma, arXiv: monov, Phys. Rep. 372, 319 (2002) Field correlators in 1208.4437 [hep-ph]. QCD: Theory and applications, hep-ph/0007223. Page 58 of 58 Eur. Phys. J. A (2015) 51: 114

190. A. Einstein, Aether, the theory of relativity reprinted in 210. Alberica Toia, CERN Cour. 53N4, 31 (2013) Participants The Berlin Years Writings 1918–1921, Aether and the and spectators at the heavy-ion fireball. Theory of Relativity edited by M. Janssen, R. Schulmann, 211. ALICE Collaboration (B. Abelev et al.), Phys. Rev. C 88, J. Illy, Ch. Lehner, D.K. Buchwald, see pp. 305–309; and 044909 (2013) Centrality determination of Pb-Pb colli- p. 321 (Document 38, Collected Papers of Albert Ein- sions at √sNN =2.76 TeV with ALICE, arXiv:1301.4361 stein, Princeton University Press, 2002), English version [nucl-ex]. 212. J.W. Li, D.S. Du, Phys. Rev. D 78, 074030 (2008) The translation by the author, see also translation in A. Ein- ′ ′ stein Sidelights on Relativity (Dover Publishers, 1983) on- Study of B J/ψη decays and determination of η η → → demand 2015. mixing angle, arXiv:0707.2631 [hep-ph]. 191. M.J. Fromerth, J. Rafelski, Hadronization of the quark 213. V. Begun, W. Florkowski, M. Rybczynski, Phys. Rev. Universe, astro-ph/0211346. C 90, 014906 (2014) Explanation of hadron transverse- 192. The Alpha Magnetic Spectrometer: What For web lo- momentum spectra in heavy-ion collisions at √sNN = cation verfied July 2015 http://www.ams02.org/what- 2.76 TeV within chemical non-equilibrium statistical ha- is-ams/what-for/. dronization model, arXiv:1312.1487 [nucl-th]. 193. J. Birrell, J. Rafelski, Phys. Lett. B 741, 77 (2015) Quark- 214. V. Begun, W. Florkowski, M. Rybczynski, Phys. Rev. gluon plasma as the possible source of cosmological dark C 90, 054912 (2014) Transverse-momentum spectra radiation, arXiv:1404.6005 [nucl-th]. of strange particles produced in Pb+Pb collisions at 194. J. Rafelski, J. Birrell, J. Phys. Conf. Ser. 509, 012014 √sNN =2.76 TeV in the chemical non-equilibrium model, (2014) Traveling Through the Universe: Back in Time to arXiv:1405.7252 [hep-ph]. the Quark-Gluon Plasma Era, arXiv:1311.0075 [nucl-th]. 215. ALICE Collaboration (B. Abelev), Strangeness with AL- 195. W. Heisenberg, Z. Phys. 101, 533 (1936) Zur Theorie der ICE: from pp to Pb-Pb, arXiv:1209.3285 [nucl-ex], Pro- “Schauer” in der H¨ohenstrahlung. ceedings of The Physics of the LHC 2012, held 4-9 June 196. H. Satz, Phys. Rev. D 20, 582 (1979) Dimensionality in 2012 in Vancouver, BC, http://plhc2012.triumf.ca/. 216. ALICE Collaboration (B. Abelev et al.), Phys. Rev. Lett. the Statistical Bootstrap Model. 111, 222301 (2013) K0 and Λ production in Pb-Pb colli- 197. K. Redlich, L. Turko, Z. Phys. C 5, 201 (1980) Phase S sions at √sNN =2.76 TeV, arXiv:1307.5530 [nucl-ex]. Transitions in the Statistical Bootstrap Model with an In- 217. I. Kuznetsova, J. Rafelski, Eur. Phys. J. C 51, 113 ternal Symmetry. (2007) Heavy flavor hadrons in statistical hadronization 198. R. Hagedorn, J. Rafelski, Phys. Lett. B 97, 136 (1980) of strangeness-rich QGP, hep-ph/0607203. Hot Hadronic Matter and Nuclear Collisions. 218. P. Braun-Munzinger, D. Magestro, K. Redlich, J. Stachel, 199. K. Zalewski, K. Redlich, Thermodynamics of the low den- Phys. Lett. B 518, 41 (2001) Hadron production in Au-Au sity excluded volume hadron gas, arXiv:1507.05433 [hep- collisions at RHIC, hep-ph/0105229. ph]. 219. F. Karsch, Nucl. Phys. A 698, 199 (2002) Lattice results 200. V. Vovchenko, D.V. Anchishkin, M.I. Gorenstein, Phys. on QCD thermodynamics, hep-ph/0103314. Rev. C 91, 024905 (2015) Hadron Resonance Gas Equa- 220. F. Karsch, Nucl. Phys. Proc. Suppl. 83, 14 (2000) Lattice tion of State from Lattice QCD, arXiv:1412.5478 [nucl- QCD at finite temperature and density, hep-lat/9909006. th]. 221. J. Rafelski, J. Letessier, Phys. Rev. C 83, 054909 (2011) 201. B. Touschek, Nuovo Cimento B 58, 295 (1968) Covariant Particle Production in sNN =2.76 TeV Heavy Ion Colli- statistical mechanics. sions, arXiv:1012.1649 [hep-ph]. 202. J.I. Kapusta, Phys. Rev. D 23, 2444 (1981) Asymptotic Mass Spectrum and Thermodynamics of the Abelian Bag Model. Johann Rafelski received his 203. J.I. Kapusta, Nucl. Phys. B 196, 1 (1982) Asymptotic PhD in 1973, addressing the Level Density of Constrained and Interacting Fields. physics of Strong Fields with 204. K. Redlich, Z. Phys. C 21, 69 (1983) Asymptotic Hadron Walter Greiner in Frankfurt. Mass Spectrum With an Internal Non-abelian Symmetry In 1977 he arrived at CERN, Group. where in collaboration with 205. M.I. Gorenstein, V.K. Petrov, G.M. Zinovev, Phys. Lett. Rolf Hagedorn he turned to the B 106, 327 (1981) Phase Transition in the Hadron Gas study of Relativistic Heavy Ion Model. Collisions and Quark Gluon 206. M.I. Gorenstein, M. Gazdzicki, W. Greiner, Phys. Rev. Plasma (QGP). He continued C 72, 024909 (2005) Critical line at the deconfinement this work in Frankfurt, Cape- phase transition. Town and finally, Arizona, 207. G. Torrieri, S. Steinke, W. Broniowski, W. Florkowski, J. where he has been since 1987. Letessier, J. Rafelski, Comput. Phys. Commun. 167, 229 Rafelski’s efforts on strange- (2005) SHARE: Statistical hadronization with resonances, ness signature of QGP guided nucl-th/0404083. a wide experimental effort at 208. G. Torrieri, S. Jeon, J. Letessier, J. Rafelski, Comput. CERN-SPS, leading to the an- Phys. Commun. 175, 635 (2006) SHAREv2: Fluctuations nouncement of the QGP dis- and a comprehensive treatment of decay feed-down,nucl- covery in the year 2000. During th/0603026. the last 15 years he worked to 209. M. Petran, J. Letessier, J. Rafelski, G. Torrieri, Comput. understand strange and charm Phys. Commun. 185, 2056 (2014) SHARE with CHARM, QGP data obtained at BNL- arXiv:1310.5108 [hep-ph]. RHIC and CERN-LHC. Eur. Phys. J. A (2015) 51: 115 THE EUROPEAN DOI 10.1140/epja/i2015-15115-y PHYSICAL JOURNAL A Addendum

Extreme states of nuclear matter – 1980

From: “Workshop on Future Relativistic Heavy Ion Experiments” held 7-10 October 1980 at: GSI, Darmstadt, Germany

Johann Rafelski1,2,a,b 1 CERN-PH/TH, 1211 Geneva 23, Switzerland 2 Department of Physics, The University of Arizona Tucson, Arizona, 85721, USA

Addendum to: Eur. Phys. J. A (2015) 51: 114, DOI: 10.1140/epja/i2015-15114-0

Received: 12 August 2015 / Revised: 23 August 2015 Published online: 22 September 2015 c The Author(s) 2015. This article is published with open access at Springerlink.com Communicated by T.S. B´ır´o

Abstract. The theory of hot nuclear fireballs consisting of all possible finite-size hadronic constituents in chemical and thermal equilibrium is presented. As a complement of this hadronic gas phase characterized by maximal temperature and energy density, the quark bag description of the hadronic fireball is consid- ered. Preliminary calculations of temperatures and mean transverse momenta of particles emitted in high multiplicity relativistic nuclear collisions together with some considerations on the observability of quark matter are offered.

1Overview hadronic gas, a strongly attractive interaction has to be accounted for, which leads to the formation of the nu- I wish to describe, as derived from known traits of merous hadronic resonances —which are in fact bound strong interactions, the likely thermodynamic properties states of several (anti) quarks. If this is really the case, of hadronic matter in two different phases: the hadro- then our intuition demands that at sufficiently high par- nic gas consisting of strongly interacting but individual ticle (baryon) density the individuality of such a bound baryons and mesons, and the dissolved phase of a rela- state will be lost. In relativistic physics in particular, tively weakly interacting quark-gluon plasma. The equa- meson production at high temperatures might already tions of state of the hadronic gas can be used to derive lead to such a transition at moderate baryon density. the particle temperatures and mean transverse momenta As is currently believed, the quark-quark interaction is in relativistic heavy ion collisons, while those of the quark- of moderate strength, allowing a perturbative treatment gluon plasma are more difficult to observe experimentally. of the quark-gluon plasma as relativistic Fermi and Bose They may lead to recognizable effects for strange particle gases. As this is a very well studied technique to be yields. Clearly, the ultimate aim is to understand the be- found in several reviews [2–8], we shall present the rel- havior of hadronic matter in the region of the phase transi- evant results for the relativistic Fermi gas and restrict tion from gas to plasma and to find characteristic features the discussion to the interesting phenomenological con- which will allow its experimental observation. More work sequences. Thus the theoretical part of this report will is still needed to reach this goal. This report is an ac- be devoted mainly to the strongly interacting phase of count of my long and fruitful collaboration with R. Hage- hadronic gas. We will also describe some experimental dorn [1]. consequences for relativistic nuclear collisions such as par- The theoretical techniques required for the descrip- ticle temperatures, i.e., mean transverse momenta and en- tion of the two phases are quite different: in the case of tropy. a The original address byline 1980: Gesellschaft f¨ur Schwe- As we will deal with relativistic particles throughout rionenforschung mbH, Darmstadt and Institut f¨urTheoretis- this work, a suitable generalization of standard thermody- che Physics der Universit¨at Frankfurt/M; originally printed in namics is necessary, and we follow the way described by GSI81-6 Orange Report, pp. 282–324, edited by R. Bock and Touschek [9]. Not only is it the most elegant, but it is also R. Stock. by simple physical arguments the only physical generaliza- b e-mail: [email protected] tion of the concepts of thermodynamics to relativistic par- Page 2 of 16 Eur. Phys. J. A (2015) 51: 115 ticle kinematics. Our notation is such that  = c = k =1. raising the mass of a quasi-isolated quark by the amount The inverse temperature β and volume V are generalized Vfield. to become four-vectors: B Another feature of the true vacuum is that it exercises a pressure on the surface of the region of the perturba- E pµ =(p0, p)=muµ,uuµ =1, −→ µ tive vacuum to which quarks are confined. Indeed, this 1 1 is just the idea of the original MIT bag model [12]. The βµ =(β0, β)= vµ,vvµ =1, T −→ T µ Fermi pressure of almost massless light quarks is in equi- µ 0 µ µ librium with the vacuum pressure . When many quarks V V =(V , V )=Vw ,wµw =1, (1) −→ are combined to form a giant quarkB bag, then their prop- where uµ, vµ,andwµ are the four-velocities of the to- erties inside can be obtained using standard methods of tal mass, the thermometer, and the volume, respectively. many-body theory [2–8]. In particular, this also allows the Usually, uµ = vµ = wµ. inclusion of the effect of internal excitation through a fi- We will often work in the frame in which all velocities nite temperature and through a change in the chemical have a timelike component only. In that case we shall often composition. drop the Lorentz index µ, as we shall do for the arguments A further effect that must be taken into consideration V = Vµ, β = βµ of different functions. is the quark-quark interaction. We shall use here the first The attentive reader may already be wondering how order contribution in the QCD running coupling constant 2 2 2 the approach outlined here can be reconciled with the αs(q )=g /4π. However, as αs(q ) increases when the concept of quark confinement. We will now therefore ex- average momentum exchanged between quarks decreases, plain why the occurrence of the high temperature phase this approach will have only limited validity at relatively of hadronic matter —the quark-gluon plasma— is still low densities and/or temperatures. The collective screen- consistent with our incapability to liberate quarks in ing effects in the plasma are of comparable order of mag- high energy collisions. It is thus important to realize nitude and should reduce the importance of perturbative that the currently accepted theory of hadronic structure contributions as they seem to reduce the strength of the and interactions, quantum chromodynamics [10], supple- quark-quark interaction. mented with its phenomenological extension, the MIT From this general description of the hadronic plasma, bag model [11], allows the formation of large space do- it is immediately apparent that, at a certain value of tem- mains filled with (almost) free quarks. Such a state is ex- perature and baryon number density, the plasma must pected to be unstable and to decay again into individual disintegrate into individual hadrons. Clearly, to treat this hadrons, following its free expansion. The mechanism of process and the ensuing further nucleonisation by pertur- quark confinement requires that all quarks recombine to bative QCD methods is impossible. It is necessary to find form hadrons again. Thus the quark-gluon plasma may be a semi-phenomenological method for the treatment of the only a transitory form of hadronic matter formed under thermodynamic system consisting of a gas of quark bags. special conditions and therefore quite difficult to detect The hadronic gas phase is characterized by those reac- experimentally. tions between individual hadrons that lead to the forma- We will recall now the relevant postulates and re- tion of new particles (quark bags) only. Thus one may sults that characterize the current understanding of strong view [13–15] the hadronic gas phase as being an assembly interactions in quantum chromodynamics (QCD). The of many different hadronic resonances, their number in the most important postulate is that the proper vacuum state interval (m2,m2 +dm2) being given by the mass spectrum in QCD is not the (trivial) perturbative state that we τ(m2,b)dm2. Here the baryon number b is the only dis- (naively) imagine to exist everywhere and which is little crete quantum number to be considered at present. All changed when the interactions are turned on/off. In QCD, bag-bag interaction is contained in the mutual transmu- the true vacuum state is believed to a have a complicated tations from one state to another. Thus the gas phase has structure which originates in the glue (“photon”) sector of the characteristic of an infinite component ideal gas phase the theory. The perturbative vacuum is an excited state of extended objects. The quark bags having a finite size with an energy density above the true vacuum. It is to force us to formulate the theory of an extended, though be found inside hadronsB where perturbative quanta of the otherwise ideal multicomponent gas. theory, in particular quarks, can therefore exist. The oc- It is a straightforward exercise, carried through in the currence of the true vacuum state is intimately connected beginning of the next section, to reduce the grand par- to the glue-glue interaction. Unlike QED, these massless tition function Z to an expression in terms of the mass quanta of QCD, also carry a charge —color— that is re- spectrum τ(m2,b). In principle, an experimental form of sponsible for the quark-quark interaction. τ(m2,b) could then be used as an input. However, the In the above discussion, the confinement of quarks is more natural way is to introduce the statistical bootstrap a natural feature of the hypothetical structure of the true model [13], which will provide us with a theoretical τ that vacuum. If it is, for example, a color superconductor, then is consistent with assumptions and approximations made an isolated charge cannot occur. Another way to look at in determining Z. this is to realize that a single colored object would, ac- In the statistical bootstrap, the essential step consists cording to Gauss’ theorem, have an electric field that can in the realization that a composite state of many quark only end on other color charges. In the region penetrated bags is in itself an “elementary” bag [1, 16]. This leads by this field, the true vacuum is displaced, thus effectively directly to a nonlinear integral equation for τ. The ideas Eur. Phys. J. A (2015) 51: 115 Page 3 of 16 of the statistical bootstrap have found a very successful and the energy–momentum four-vector application in the description of hadronic reactions [17] over the past decade. The present work is an extension [1, ∂ pµ = ln Z(β,V,λ), (3b) 15, 18] and application [19] of this method to the case of   −∂βµ a system containing any number of finite size hadronic clusters with their baryon numbers adding up to some which follow from the definition in eq. (2). fixed number. Among the most successful predictions of The theoretical problem is to determine σ(p, V, b)in the statistical bootstrap, we record here the derivation terms of known quantities. Let us suppose that the phys- of the limiting hadronic temperature and the exponential ical states of the hadronic gas phase can be considered as growth of the mass spectrum. being built up from an arbitrary number of massive ob- We see that the theoretical description of the two jects, henceforth called clusters, characterized by a mass hadronic phases —the individual hadron gas and the spectrum τ(m2,b), where τ(m2,b)dm2 is the number of quark-gluon plasma— is consistent with observations and different elementary objects (existing in nature) in the with the present knowledge of elementary particles. What mass interval (m2,m2 +dm2) and having the baryon remains is the study of the possible phase transition be- number b. As particle creation must be permitted, the tween those phases as well as its observation. Unfortu- number N of constituents is arbitrary, but constrained nately, we can argue that in the study of temperatures and by four-momentum conservation and baryon conservation. mean transverse momenta of pions and nucleons produced Neglecting quantum statistics (it can be shown that, for in nuclear collisions, practically all information about the T  40 MeV, Boltzmann statistics is sufficient), we have hot and dense phase of the collision is lost, as most of the emitted particles originate in the cooler and more di- ∞ 1 N σ(p, V, b)= δ4 p p lute hadronic gas phase of matter. In order to obtain re- N! − i  i=1  liable information on quark matter, we must presumably N=0   perform more specific experiments. We will briefly point N out that the presence of numerous s quarks in the quark δk b bi × − plasma suggest, as a characteristic experiment, the obser- {bi}  i=1    vation Λ hyperons. N 2∆ pµ We close this report by showing that, in nuclear colli- µ i τ(p2,b )d4p . (4) × (2π)3 i i i sions, unlike pp reactions, we can use equilibrium thermo- i=1 dynamics in a large volume to compute the yield of strange  and antistrange particles. The latter, e.g., Λ, might be sig- The sum over all allowed partitions of b into different bi nificantly different from what one expects in pp collisions is included and ∆ is the volume available for the motion and give a hint about the properties of the quark-gluon of the constituents, which differs from V if the different phase. clusters carry their proper volume Vci

N ∆µ = V µ V µ. (5) 2 Thermodynamics of the gas phase and the − ci i=1 SBM  Given the grand partition function Z(β,V,λ) of a many- The phase space volume used in eq. (4) is best explained body system, all thermodynamic quantities can be deter- by considering what happens for one particle of mass m0 mined by differentiation of ln Z with respect to its argu- in the rest frame of ∆µ and βµ ments. Here, λ is the fugacity introduced to conserve a µ 3 2∆ p d p 2 2 discrete quantum number, here the baryon number. The d4p µ i e−β·pδ (p2 m2)=∆ i e−β0√p +m i (2π)3 0 i − 0 (2π)3 conservation of strangeness can be carried through in a   2 similar fashion leading then to a further argument λs of Tm = ∆ K (m/T ). (6) Z. Whenever necessary, we will consider Z to be implicitly 0 2π2 2 dependent on λs. The grand partition function is a Laplace transform of The density of states in eq. (4) implies that the creation the level density σ(p, V, b), where pµ is the four-momentum and absorption of particles in kinetic and chemical equi- and b the baryon number of the many-body system en- librium is limited only by four-momentum and baryon closed in the volume V number conservation. These processes represent the strong ∞ hadronic interactions which are dominated by particle pro- µ 2 Z(β,V,λ)= λb σ(p, V, b)e−βµp d4p. (2) ductions. τ(m ,b) contains all participating elementary particles and their resonances. Some remaining interac- b=−∞   tion is here neglected or, as we do not use the complete We recognize the usual relations for the thermodynamic experimental τ, it may be considered as being taken care of expectation values of the baryon number by a suitable choice of τ. The short-range repulsive forces ∂ are taken into account by the introduction of the proper b = λ ln Z(β,V,λ), (3a) volume V of hadronic clusters.   ∂λ Page 4 of 16 Eur. Phys. J. A (2015) 51: 115

One more remark concerning the available volume ∆ volume V can be interpreted for the time being as the is in order here. If V were considered to be given and an level density  of point particles in a fictitious volume ∆ independent thermodynamic quantity, then in eq. (4), a further built-in restriction limits the sum over N to a cer- σ(p, V, b)=σpt(p, ∆, b), (10) tain Nmax, such that the available volume ∆ in eq. (5) re- mains positive. However, this more conventional assump- whence this is also true for the grand canonical partition tion of V as the independent variable would significantly function in eq. (2) obscure our mathematical formalism. It is important to re- alize that we are free to select the available volume ∆ as the independent thermodynamic variable and to consider Z(β,V,λ)=Zpt(β,∆,λ). (11) V as a thermodynamic expectation value to be computed from eq. (5) Combining eqs. (2) and (4), we also find the important relation V µ V µ = ∆µ + V µ(β,∆,λ) . (7) −→    c  ∞ µ µ µ b 2∆µp 2 −βµp 4 Here Vc is the average sum of proper volumes of all ln Zpt(β,∆,λ)= λ τ(p ,b)e d p. hadronic clusters contained in the system considered. As (2π)3 b=−∞  already discussed, the standard quark bag leads to the  (12) proportionality between the cluster volume and hadron This result can only be derived when the sum over N in mass. Similar arguments within the bootstrap model [15], eq. (4) extends to infinity, thus as long as ∆/ V in eq. (9) as for example discussed in the preceding lecture by R. remains positive.   Hagedorn [16], also lead to In order to continue with our description of hadronic matter, we must now determine a suitable mass spectrum pµ(β,∆,λ) V µ = , (8) τ to be inserted into eq. (4). For this we now introduce  c  4  B  the statistical bootstrap model. The basic idea is rather where 4 is the (at this point arbitrary) energy density of old, but has undergone some development more recently isolatedB hadrons in the quark bag model [11]. making it clearer, more consistent, and perhaps more con- Since our hadrons are under pressure from neighbors vincing. The details may be found in [15] and the refer- in hadronic matter, we have in principle to take instead of ences therein. Here a simplified naive presentation is given. 4 the energy density of a quark bag exposed to a pressure We note, however, that our present interpretation is non- PB(see eq. (54) below) trivially different from that in [15]. The basic postulate of statistical bootstrap is that the 2 εbag =4 +3P. mass spectrum τ(m ,b) containing all the “particles”, i.e., B elementary, bound states, and resonances (clusters), is Combining eqs. (7)–(9), we find, with ε(β,∆,λ)= pµ / generated by the same interactions which we see at work µ   V = E / V , that if we consider our thermodynamical system. Therefore, if       we were to compress this system until it reaches its natural ∆ ε(β,∆,λ) =1 . (9) volume Vc(m, b), then it would itself be almost a cluster V (β,∆,λ) − 4 +3P (β,∆,λ) appearing in the mass spectrum τ(m2,b). Since σ(p, ∆, b)   B and τ(p2,b) are both densities of states (with respect to As we shall see, the pressure P in the hadronic matter the different parameters d4p and dm2), we postulate that never rises above 0.4 , see fig. 5a below, and arguments following eq. (60).≃ Consequently,B the inclusion of P above —the compression of free hadrons by the hadronic matter σ(p, ∆, b) =const τ(p2,b), (13) by about 10%— may be omitted for now from further V −→Vc(m,b) × → ∆ 0 discussion. However, we note that both ε and P will be computed as ln Z becomes available, whence eq. (9) is an implicit equation for ∆/ V . where = means “corresponds to” (in some way to be spec- It is important to record  that the expression in eq. (9) ified). As σ(p, ∆, b) is (see eq. (4)) the sum over N of N- can approach zero only when the energy density of the fold convolutions of τ, the above “bootstrap postulate” hadronic gas approaches that of matter consisting of one will yield a highly nonlinear integral equation for τ. big quark bag: ε 4 , P 0. Thus the density of states The bootstrap postulate (13) requires that τ should in eq. (4), together→ withB the→ choice of ∆ as a thermody- obey the equation resulting from replacing σ in eq. (4) namic variable, is a consistent physical choice only up to by some expression containing τ linearly and by taking this point. Beyond we assume that a description in terms into account the volume condition expressed in eqs. (7) of interacting quarks and gluons is the proper physical de- and (8). scription. Bearing all these remarks in mind, we now con- We cannot simply put V = Vc and ∆ = 0, because sider the available volume ∆ as a thermodynamic variable now, when each cluster carries its own dynamically deter- which by definition is positive. Inspecting eq. (4) again, we mined volume, ∆ loses its original meaning and must be recognize that the level density of the extended objects in redefined more precisely. Therefore, in eq. (4), we tenta- Eur. Phys. J. A (2015) 51: 115 Page 5 of 16

tively replace large cluster, e.g., in nuclear matter, m = m Ebind ,and m 925 MeV. That this must be so becomes− obvious if N ≈ 2Vc(m, b) p 2 one imagines eq. (16) solved by iteration (the iteration so- σ(p, Vc,b) · τ(p ,b) −→ (2π)3 lution exists and is the physical solution). Then Hτ(p2,b) 2 becomes in the end a complicated function of p2, b, all 2m 2 = τ(p ,b), m , and all g . In other words, in the end a single clus- (2π)34 b b B ter consists of the “elementary particles”. As these are all 2∆ pi 2 2Vc(mi,bi) pi 2 bound into the cluster, their mass m should be the effec- · τ(p ,bi) · τ(p ,bi) (2π)3 i −→ (2π)3 i tive mass, not the free mass m. This way we may include 2m2 a small correction for the long-range attractive meson ex- = i τ(p2,b ). (14) (2π)34 i i change by choosing mN = m 15 MeV. B Let us make a brief excursion− to the bag model at this 2 2 point. There the mass of a hadron is computed from the Next we argue that the explicit factors m and mi arise from the dynamics and therefore must be absorbed into assumption of an isolated particle (= bag) with its size 2 1 2 2 and mass being determined from the equilibrium between τ(p ,bi) as dimensionless factors m /m .Thus, i i 0 the vacuum pressure and the internal Fermi pressure of B 2m2 the (valence) quarks. In a hadron gas, this is not true as σ(p, V ,b) 0 τ(p2,b)=Hτ(p2,b), c −→ (2π)34 a finite pressure is exerted on hadrons in matter. After a B short calculation, we find the pressure dependence of the 2∆ p 2m2 · i τ(p2,b ) 0 τ(p2,b )=Hτ(p2,b ), (15) bag model hadronic mass (2π)3 i i −→ (2π)34 i i i i B M(P ) 1+3P/4 3 P 2 with = 3B/4 = 1+ + . (17) 2m2 M(0) (1 + P/ ) 32 ··· H := 0 , B B  (2π)34 B We have already noted that the pressure never exceeds where either H or m may be taken as a new free pa- 0.4 in the hadronic gas phase, see fig. 5a below, and ar- 0 B rameter of the model, to be fixed later. (If m0 is taken, guments following eq. (60). Hence we see that the increase then it should be of the order of the “elementary masses” in mass of constituents (quark bags) in the hadronic gas appearing in the system, e.g., somwhere between mπ and never exceeds 1.5% and is at most comparable with the mN in a model using pions and nucleons as elementary 15MeV binding in m. In general, P is about 0.1B and the input.) Finally, if clusters consist of clusters which consist pressure effect may be neglected. of clusters, and so on, this should end at some “elemen- Thus we can consider the “input” first term in eq. (16) tary” particles (where what we consider as elementary is as being fixed by pions, nucleons, and whenever necessary fixed by convention). Inserting eq. (15) into eq. (4), the by the usual strange members of meson and baryon mul- bootstrap equation (BE) then reads tiplets. Furthermore, we note that the bootstrap equa- tion (16) makes use of practically all the same approxi- Hτ(p2,b)=Hg δ (p2 m2) mations as our description of the level density in eq. (4). b 0 − b ∞ N Thus the solution of eq. (16) is particularly suitable for 1 + δ4 p p our use. N! − i  i=1  We solve the BE by the same double Laplace transfor- N=2   N mation which we used before eq. (2). We define

δk b bi ∞ µ × − −βµp b 2 2 4 {bi}  i=1  ϕ(β,λ):= e λ Hg δ (p m )d p   b 0 − b N  b=−∞ 2 4 ∞ Hτ(p ,bi)d pi. (16) × i b i=1 =2πHT λ gbmbK1(mb/T ),  b=−∞ Clearly, the bootstrap equation (16) has not been derived. ∞ µ We have made it more or less plausible and state it as a Φ(β,λ):= e−βµp λbHτ(p2,b)d4p. (18) postulate. For more motivation, see [15]. In other words,  b=−∞ the bootstrap equation means that the cluster with mass  2 Once the set of input particles m ,g is given, ϕ(β,λ) p and baryon number b is either elementary (mass mb, b b is a known function, while Φ(β,λ{ ) is unknown.} Applying spin isospin multiplicity gb), or it is composed of any num- ber N 2 of subclusters having the same internal com- the double Laplace transformation to the BE, we obtain ≥ posite structure described by this equation. The bar over Φ(β,λ)=ϕ(β,λ)+exp Φ(β,λ) Φ(β,λ) 1. (19) mb indicates that one has to take the mass which the “el- − − ementary particle” will have effectively when present in a This implicit equation for Φ in terms of ϕ can be solved without regard for the actual β,λ dependence. Writing 1 Here is the essential difference with [15], where another choice was made. G(ϕ):=Φ(β,λ),ϕ=2G eG +1, (20) − Page 6 of 16 Eur. Phys. J. A (2015) 51: 115

Let us now introduce the energy density εpt of the hypothetical pointlike particles as ∂ 2 ∂2 ε (β,λ)= ln Z (β,∆,λ)= Φ(β,λ), pt −∆∂β pt (2π)3H ∂β2 (23) which will turn out to be quite helpful as it is independent of ∆. The proper energy density is 1 ∂ ∆ ε(β,λ)= ln Z = ε , (24) V −∂β V pt      while the pressure follows from P (β,λ) V = T ln Z(β,V,λ)=T ln Z (β,∆,λ), (25)   pt ∆ 2T ∂ ∆ P (β,λ)= Φ(β,λ) =: P . (26) V −(2π)3H ∂β V pt       Similarly, for the baryon number density, we find Fig. 1. Bootstrap function G(ϕ). The dashed line repre- sents the unphysical branch. The root singularity is at ϕ0 = b ∆ ν(β,λ)=   =: ν (β,λ), (27) ln(4/e)=0.3863. V V pt     with we can draw the curve ϕ(G) and then invert it graphi- cally (see fig. 1) to obtain G(ϕ)=Φ(β,λ). G(ϕ)hasa 1 ∂ 2 ∂ ∂ νpt(β,λ)= λ ln Zpt = 3 λ Φ(β,λ). square root singularity at ϕ = ϕ0 =ln(4/e)=0.3863. Be- ∆ ∂λ −(2π) H ∂λ ∂β yond this value, G(ϕ) becomes complex. Apart from this (28) graphical solution, other forms of solution are known: From eqs. (23)–(23), the crucial role played by the fac- tor ∆/ V becomes apparent. We note that it is quite ∞ ∞   n n/2 integral straightforward to insert eqs. (24) and (25) into eq. (9) and G(ϕ)= snϕ = wn(ϕ0 ϕ) = . − representation solve the resulting quadratic equation to obtain ∆/ V as n=1 n=0     (21) an explicit function of εpt and Ppt. First we record the The expansion in terms of (ϕ ϕ)n/2 has been used in our limit P B 0 − ≪ numerical work (12 terms yield a solution within computer ∆ ε(β,λ) ε (β,λ) −1 accuracy) and the integral representation will be published =1 = 1+ pt , (29) 2 V − 4 4 elsewhere . Henceforth, we consider Φ(β,λ)=G(ϕ)tobe   B  B  2 a known function of ϕ(β,λ). Consequently, τ(m ,b) is also while the correct expression is in principle known. From the singularity at ϕ = ϕ0,itfol- 2 lows [1] that τ(m ,b) grows, for m mNb, exponentially 2 −3 ≫ ∆ 1 εpt 2 4 1 εpt 2 m exp(m/T0). In some weaker form, this has been = B + B + B . ∼ V 2 − 6Ppt − 3Ppt 3Ppt 2 − 6Ppt − 3Ppt known for a long time [13, 20, 21].    (30) The last of the important thermodynamic quantities is the 3 The hot hadronic gas entropy S. By differentiating eq. (25), we find ∂ ∂ ∂ The definition of Φ(β,λ) in eq. (18) in terms of the mass ln Z = βP V = P V T (P V ). (31) spectrum allows us to write a very simple expression for ∂β ∂β    − ∂T   ln Z in the gas phase (passing now to the rest frame of the Considering Z as a function of the chemical potential, viz., gas) µβ ˜ ˜ 2∆ ∂ Z(β,V,λ)=Z(β,V,e )=Z(β,V,µ)=Zpt(β,∆,µ) , ln Z(β,V,λ)=lnZpt(β,∆,λ)= 3 Φ(β,λ). (32) −(2π) H ∂β we find (22) We recall that eqs. (9) and (19) define (implicitly) the ∂ ∂ ln Z = ln Z˜ (β,∆,µ)= E + µ b , (33) quantities ∆ and Φ in terms of the physical variables V , ∂β ∂β pt −   µ,∆ β,andλ.

2 with E being the total energy. From eqs. (31) and (33), Extensive discussion of the analytical properties of the we find the “first law” of thermodynamics to be bootstrap function was publisched in: R. Hagedorn and J. Rafelski: Analytic Structure and Explicit Solution of an Impor- ∂ tant Implicit Equation 83 E = P V + T (P V )+µ b . (34a) , Commun. Math. Phys. , 563 (1982). −   ∂T     Eur. Phys. J. A (2015) 51: 115 Page 7 of 16

Now quite generally

E = P V + TS + µ b , (34b) −     so that ∂ S = [P (β,∆,µ) V (β,∆,µ) ] . (35) ∂T   µ,∆

Equations (25) and (33) now allow us to write ∂ E µb S = (P V )=lnZ˜ (T,∆,µ)+ − . (36) ∂T   pt T The entropy density in terms of the already defined quan- tities is therefore S P + ε µν = = − . (37) S V T   We shall now take a brief look at the quantities P , ε, ν, ∆/ V . They can be written in terms of ∂Φ(β,λ)/∂β and its derivatives. We note that (see eq. (20)) Fig. 2. The critical curve corresponding to ϕ(T,µ)=ϕ0 in the µ, T plane. Beyond it, the usual hadronic world ceases to ∂ ∂G(ϕ) ∂ϕ exist. In the shaded region, our theory is not valid, because we Φ(β,λ)= , (38) neglected Bose-Einstein and Fermi-Dirac statistics. ∂β ∂ϕ ∂β and that ∂G/∂ϕ (ϕ ϕ)−1/2 near to ϕ = ϕ = ln(4/e) ∼ 0 − 0 i.e., on the set of “input” particles mb,gb assumed and (see fig. 1). Hence at ϕ = ϕ0, we find a singularity in the the value of the constant H in eq.{ (15).} In the case of point particle quantities εpt, νpt,andPpt. This implies three elementary pions π+, π0,andπ− and four elemen- that all hadrons have coalesced into one large cluster. In- tary nucleons (spin isospin) and four antinucleons, we deed, from eqs. (24), (26), (27), and (29), we find have from eq. (18) ⊗ ε 4 , −→ B P 0, ϕ(β,λ)=2πHT 3mπK1(mπ/T ) −→  ∆/ V 0. (39) 1  −→ +4 λ + m K (m /T ) , (43a) λ N 1 N We can easily verify that this is correct by establishing   the average number of clusters present in the hadronic N and the condition (42), written in terms of T and µ = gas. This is done by introducing an artificial fugacity ξ T ln λ, yields the curve shown in fig. 2, i.e., the “criti- in eq. (4) in the sum over N, where N is the number of cal curve”. For µ = 0, the curve ends at T = T , where clusters. Denoting by Z(ξ) the associated grand canonical 0 T0, the “limiting temperature of hadronic matter”, is the partition functions in eq. (22), we find same as that appearing in the mass spectrum [13,15,20,21] 2 −3 τ(m ,b) m exp(m/T0)(forb bmN). ∂ ξ 2∆ ∂ ∼ ≫ N = ξ ln Zpt(β,∆,λ; ξ) = 3 Φ(β,λ), The value of the constant H in eq. (15) has been cho-   ∂ξ ξ=1 −(2π) H ∂β sen [19] to yield T0 = 190 MeV. This apparently large (40) value of T0 seemed necessary to yield a maximal average which leads to the useful relation decay temperature of the order of 145 MeV, as required by [22]. (However, a new value of the bag constant then P V = N T. (41)     induces a change [1] to a lower value of T0 = 180 MeV.) Thus as P V 0, so must N , the number of clus- Here we use ters, for finite →T . We record the astonishing fact that the H =0.724 GeV−2,T=0.19 GeV, hadron gas phase obeys an “ideal” gas equation, although 0 4 of course N is not constant as for a real ideal gas but a m0 =0.398 GeV (when = (145 MeV) ), B function of the thermodynamic variables. (43b) The boundary given by where the value of m0 lies as expected between mπ and 1/2 ϕ(β,λ)=ϕ0 =ln(4/e) (42) mN ((mπmN) =0.36 GeV). The critical curve limits the hadron gas phase. By thus defines a critical curve in the β,λ plane. Its position approaching it, all hadrons dissolve into a giant cluster, depends, of course, on the actually given form of ϕ(β,λ), which is not in our opinion a hadron solid [23]. We would Page 8 of 16 Eur. Phys. J. A (2015) 51: 115 prefer to identify it with a quark-gluon plasma. Indeed, This leads to the required positive energy density within as the energy density along the critical curve is constant the volume occupied by the colored quarks and gluonsB and (= 4 ), the critical curve can be attained and, if the to a negative pressure on the surface of this region. At this energyB density becomes > 4 , we enter into a region stage, this term is entirely phenomenological, as discussed which cannot be described withoutB making assumptions above. The equations of state for the quark-gluon plasma about the inner structure and dynamics of the “elemen- are easily obtained by differentiating tary particles” mb,gb —here pions and nucleons— en- tering into the{ input function} ϕ(β,λ). Considering pions ln Z =lnZq +lnZg +lnZvac (49) and nucleons as quark-gluon bags leads naturally to this with respect to β, λ,andV . The baryon number density, interpretation. energy, and pressure are, respectively 1 ∂ 2T 3 2α ν = λ ln Z = 1 s 4 The quark-gluon phase V ∂λ π2 − π  1 π2 We now turn to the discussion of the region of the strongly ln3 λ + ln λ , (50) interacting matter in which the energy density would be × 34 9  equal to or higher than 4 . As a basic postulate, we will 1 ∂ B ε = ln Z = assume that it consists of —relatively weakly— interact- −V ∂β B ing quarks. To begin with, only u and d flavors will be 6 2α 1 π2 considered as they can easily be copiously produced at + T 4 1 s ln4 λ + ln2 λ T  50 MeV. Again the aim is to derive the grand par- π2 − π 4 34 2 32   · ·  tition function Z. This is a standard exercise. For the 50 α 7π4 8π2 15 α + 1 s + T 4 1 s , (51) massless quark Fermi gas up to first order in the inter- − 21 π 60 15 − 4 π action [1–8, 18], the result is    ∂ 2 P = T ln Z = 8V 2αs 1 4 π 2 ∂V −B ln Zq(β,λ)= 2 3 1 ln λq + ln λq 4 2 6π β − π 4 2 2T 2αs 1 4 π 2    + 2 1 4 ln λ + 2 ln λ 4 π − π 4 3 2 3 50 αs 7π   · ·  + 1 , (44) 50 α 7π4 8π2 15 α − 21 π 60 + 1 s + T 4 1 s . (52)   − 21 π 60 45 − 4 π    valid in the limit mq 0, we have ε>4 .Re- malized to have a vanishing thermodynamic potential, B Ω = β−1 ln Z. Hence in the perturbative vacuum call that, in the hadronic gas, we had 0 <ε<4 .Thus, − above the critical curve of the µ, T plane, we haveB the ln Z = β V. (48) quark-gluon plasma exposed to an external force. vac − B Eur. Phys. J. A (2015) 51: 115 Page 9 of 16

Fig. 3. a) The critical curves (P = 0) of the two models in the T , µ plane (qualitatively). The region below the full line is de- scribed 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. 3a. The coexistence region is found from the usual Maxwell construction (the shaded areas being equal).

In order to obtain an idea of the form of the P =0 by inverting the baryon density at constant fixed baryon critical curve in the µ, T plane for the quark-gluon plasma, number we rewrite eq. (52) using eqs. (45) and (46) for P =0 b V =   . (57) ν 1 2α /π 2 = − s µ2 +(3πT)2 B 162π2 The behavior of P (V,T =const.) for the hadronic gas phase is as described before in the statistical bootstrap T 4π2  5 α  15 α + 12 1 s +8 1 s . (55) model. For large volumes, we see that P falls with ris- 45 − 3 π − 4 π    ing V . However, when hadrons get close to each other so that they form larger and larger lumps, the pres- Here, the last term is the glue pressure contribution. (If sure drops rapidly to zero. The hadronic gas becomes a the true vacuum structure is determined by the glue- state of few composite clusters (internally already con- glue interaction, then this term could be modified signifi- sisting of the quark plasma). The second branch of the P cantly.) We find that the greatest lower bound on temper- (V,T =const.) line meets the first one at a certain volume ature Tq at µ = 0 is about V = Vm. 1/4 The phase transition occurs for T =const. in fig. 3b Tq 145–190 MeV. (56) ∼B ≈ at a vapor pressure Pv obtained from the conventional This result can be considered to be correct to within Maxwell construction: the shaded regions in fig. 3b are 20%. Its order of magnitude is as expected. Taking equal. Between the volumes V1 and V2, matter coexists 1/4 eq. (55) as it is, we find for αs =1/2, Tq =0.88 . in the two phases with the relative fractions being deter- Omitting the gluon contribution to the pressure, weB find mined by the magnitude of the actual volume. This leads 1/4 Tq =0.9 . It is quite likely that, with the proper to the occurrence of a third region, viz., the coexistence treatmentB of the glue field and the plasma corrections, region of matter, in addition to the pure quark and hadron 1/4 and with larger 190 MeV, the desired value of domains. For Vν1 1/V1, all B ∼ ∼ Tq = T0 corresponding to the statistical bootstrap choice matter has gone into the quark plasma phase. will follow. Furthermore, allowing some reasonable T , µ The dotted line in fig. 3b encloses (qualitatively) the dependence of αs, we can then easily obtain an agreement domain in which the coexistence between the two phases between the critical curves. of hadronic matter seems possible. We further note that, However, it is not necessary for the two critical curves at low temperatures T 50 MeV, the plasma and hadro- to coincide, even though this would be preferable. As the nic gas critical curves≤ meet each other in fig. 3a. This quark plasma is the phase into which individual hadrons is just the domain where, at present, our description of dissolve, it is sufficient if the quark plasma pressure van- the hadronic gas fails, while the quark-gluon plasma also ishes within the boundary set for non-vanishing positive begins to suffer from infrared difficulties. Both approaches pressure of the hadronic gas. It is quite satisfactory for the have a very limited validity in this domain. theoretical development that this is the case. In fig. 3a, a The qualitative discussion presented above can be eas- qualitative picture of the two P = 0 lines is shown in ily supplemented with quantitative results. But first we the µ, T plane. Along the dotted straight line at constant turn our attention to the modifications forced onto this temperature, we show in fig. 3b the pressure as a function simple picture by the experimental circumstances in high of the volume (a P , V diagram). The volume is obtained energy nuclear collisions. Page 10 of 16 Eur. Phys. J. A (2015) 51: 115

Fig. 4. 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. 4a) 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.

5 Nuclear collisions and inclusive particle that kinetic and chemical equilibrium have been reached, spectra set it equal to the ratio of thermodynamic expectation values of the total energy and baryon number We assume that in relativistic collisions triggered to small E E(β,λ) impact parameters by high multiplicities and absence of U =   = . (59) projectile fragments [24], a hot central fireball of hadro- b ν(β,λ)   nic matter can be produced. We are aware of the whole problematic connected with such an idealization. A proper Thus we see that, through eq. (59), the experimental value treatment should include collective motions and distri- of U in eq. (58) fixes a relation between allowable values of bution of collective velocities, local temperatures, and β,λ: the available excitation energy defines the tempera- so on [25–28], as explained in the lecture by R. Hage- ture and the chemical composition of hadronic fireballs. In dorn [16]. Triggering for high multiplicities hopefully elim- fig. 4a and b, these paths are shown for a choice of kinetic inates some of the complications. In nearly symmetric col- energies Ek,lab/A in the µ, T plane and in the ν, T plane, lisions (projectile and target nuclei are similar), we can ar- respectively. In both cases, only the hadronic gas domain gue that the numbers of participants in the center of mass is shown. of the fireball originating in the projectile or target are We wish to note several features of the curves shown in the same. Therefore, it is irrelevant how many nucleons fig. 4a and b that will be relevant in later considerations: do form the fireball —and the above symmetry argument 1) Beginning at the critical curve, the chemical poten- leads, in a straightforward way, to a formula for the center tial first drops rapidly when T decreases and then of mass energy per participating nucleon rises slowly as T decreases further (fig. 4a). This corresponds to a monotonically falling baryon density Ec.m. Ek,lab/A with decreasing temperature (fig. 4b), but implies U := = mN 1+ , (58) A  2mN that, in the initial expansion phase of the fireball, the chemical composition changes more rapidly than the where Ek,lab/A is the projectile kinetic energy per nucleon temperature. in the laboratory frame. While the fireball changes its 2) The baryon density in fig. 4b is of the order of 1–1.5 baryon density and chemical composition (π + p ∆, of normal nuclear density. This is a consequence of etc.) during its lifetime through a change in temperature↔ the choice of 1/4 = 145 MeV. Were three times and chemical potential, the conservation of energy and as large, i.e., B 1/4 = 190 MeV, whichB is so far not baryon number assures us that U in eq. (58) remains con- excluded, thenB the baryon densities in this figure stant, assuming that the influence on U of pre-equilibrium would triple to 3–5ν0. Furthermore, we observe that, emission of hadrons from the fireball is negligible. As U is along the critical curve of the hadronic gas, the baryon the total energy per baryon available, we can, supposing density falls with rising temperature. This is easily Eur. Phys. J. A (2015) 51: 115 Page 11 of 16

Fig. 5. 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. 5a, and a fourth value 1.07 GeV.

understood as, at higher temperature, more volume is with taken up by the numerous mesons. mNν0 1/4 3 3) Inspecting fig. 4b, we see that, at given U, the tem- γ := =0.56, for : = 145 MeV,ν0 =0.14/fm . 4 B peratures at the critical curve and those at about ν0/2 B differ little (10%) for low U, but more significantly Here, γ is the ratio of the energy density of normal nu- for large U. Thus, highly excited fireballs cool down clei (εN = mNν0) and of quark matter or of a quark bag more before dissociation (“freeze out”). As particles (ε =4). In fig. 5a, this relation is shown for three q B are emitted all the time while the fireball cools down projectile energies: Ek,lab/A =1.80 GeV, 3.965 GeV, and along the lines of fig. 4a and b, they carry kinetic 5.914 GeV, corresponding to U =1.314 GeV, 1.656 GeV, energies related to various different temperatures. The and 1.913 GeV, respectively. We observe that, even at the inclusive single particle momentum distribution will lowest energy shown, the quark pressure is zero near the yield only averages along these cooling lines. baryon density corresponding to 1.3 normal nuclear den- sity, given the current value of . Another remark which does not follow from the curves Before discussing this pointB further, we note that the shown is: hadronic gas branches of the curves in fig. 5a and b show a 4) Below about 1.8 GeV, an important portion of the to- quite similar behavior to that shown at constant temper- tal energy is in the collective (hydrodynamical) motion ature in fig. 3b. Remarkably, the two branches meet each of hadronic matter, hence the cooling curves at con- other at P = 0, since both have the same energy density ε =4 and therefore V (P =0) 1/ν = U/ε = U/4 . stant excitation energy do not properly describe the B ∼ B evolution of the fireball. However, what we cannot see by inspecting fig. 5a and b is that there will be a discontinuity in the variables µ and Calculations of this kind can also be carried out for the T at this point, except if parameters are chosen so that quark plasma. They are, at present, uncertain due to the the critical curves of the two phases coincide. Indeed, near 1/4 unknown values of αs and . Fortunately, there is one to P = 0, the results shown in fig. 5a should be replaced particular property of the equationB of state of the quark- by points obtained from the Maxwell construction. The gluon plasma that we can easily exploit. pressure in a nuclear collision will never fall to zero. It Combining eq. (54) with eq. (59), we obtain will correspond to the momentary vapor pressure of the order of 0.2 as the phase change occurs. 1 A furtherB aspect of the equations of state for the hadro- P = (Uν 4 ). (60) 3 − B nic gas is also illustrated in fig. 5a. Had we ignored the finite size of hadrons (one of the van der Waals effects) Thus, for a given U (the available energy per baryon in a in the hadron gas phase then, as shown by the dash- heavy ion collision), eq. (60) describes the pressure-volume dotted lines, the phase change could never occur because ( 1/ν) relation. By choosing to measure P in units of the point particle pressure would diverge where the quark ∼ 3B and ν in units of normal nuclear density ν0 =0.14/ fm , pressure vanishes. In our opinion, one cannot say it often we find enough: inclusion of the finite hadronic size and of the P 4 U ν finite temperature when considering the phase transition = γ 1 , (61) 3 m ν − to quark plasma lowers the relevant baryon density (from B N 0  Page 12 of 16 Eur. Phys. J. A (2015) 51: 115

8–14ν0 for cold point-nucleon matter) to 1–5ν0 (depend- low that a fraction of particles emitted can be reabsorbed ing on the choice of ) in 2–5 GeV/A nuclear collisions. in the hadronic cluster. This is a geometric problem and, The possible formationB of quark-gluon plasma in nuclear in a first approximation, the ratio of the available volume collisions was first discussed quantitatively in ref. [3], see ∆ to the external volume Vex is the probability that an also ref. [29]. emitted particle not be reabsorbed, i.e., that it can escape The physical picture underlying our discussion is an ∆ ε(β,λ) explosion of the fireball into vacuum with little energy R = =1 . (62) esc V − 4 being converted into collective motion, e.g., hydrodynam- ex B ical flow, or being taken away by fast pre-hadronization The relative emission rate is just the integrated momen- particle emission. Thus the conserved internal excitation tum spectrum energy can only be shifted between thermal (kinetic) and chemical excitations of matter. “Cooling” thus re- 3 2 d p −√p2+m2/T +µ/T m T µ/T ally means that, during the explosion, the thermal energy Remis = e = K2(m/T )e . (2π)3 2π2 is mostly convered into chemical energy, e.g., pions are  (63) produced. The chemical potential acts only for nucleons. In the case While it is at present hard to judge the precise amount of pions, it has to be dropped from the above expression. of expected deviation from the cooling curves shown in For the mean temperature, we thus find fig. 2, it is possible to show that they are entirely inconsis- tent with the notion of reversible adiabatic, i.e., entropy conserving, expansion. As the expansion proceeds along RescRemisT dT T = c , (64) U =const. lines, we can compute the entropy per par-   ticipating baryon using eqs. (36) and (37), and we find a RescRemis dT significant growth of total entropy. As shown in fig. 5b, the c entropy rises initially in the dense phase of the matter by where the subscript “c” on the integral indicates here a as much as 50–100% due to the pion production and res- line integral along that particular cooling curve in fig. 4a onance decay. Amusingly enough, as the newly produced and b which belongs to the energy per baryon fixed by the entropy is carried mostly by pions, one will find that the experimentalist. entropy carried by protons remains constant. With this In practice, the temperature is most reliably measured remarkable behavior of the entropy, we are in a certain through the measurement of mean transverse momenta of sense, victims of our elaborate theory. Had we used, e.g., the particles. It may be more practical therefore to cal- an ideal gas of Fermi nucleons, then the expansion would culate the average transverse momentum of the emitted seem to be entropy conserving, as pion production and particles. In principle, to obtain this result we have to per- other chemistry were forgotten. Our fireballs have no ten- form a similar averaging to the one above. For the average dency to expand reversibly and adiabatically, as many re- transverse momentum at given T,µ, we find [14] action channels are open. A more complete discussion of the entropy puzzle can be found in [1]. 2 2 p e−√p +m −µ)/T d3p Inspecting fig. 4a and b again, it seems that a pos- ⊥ p⊥(m, T, µ) p =  sible test of the equations of state for the hadronic gas   −√p2+m2−µ)/T 3 consists in measuring the temperature in the hot fireball e d p zone, and doing this as a function of the nuclear colli-  m µ/T sion energy. The plausible assumption made is that the πmT/2 K 5 e = 2 T . (65) fireball follows the “cooling” lines shown in fig. 4a and m µ/T K2 T e  b until final dissociation into hadrons. This presupposes that the surface emission of hadrons during the expansion The average over the cooling curve  is then of the fireball does not significantly alter the available en- ergy per baryon. This is more likely true for sufficiently ∆ 3/2 πm m µ/T T K 5 e dT large fireballs. For small ones, pion emission by the sur- 2 c Vex 2 T face may influence the energy balance. As the fireball ex- p⊥(m, T, µ) p c =     .    ∆ m µ/T pands, the temperature falls and the chemical composition TK2 e dT changes. The hadronic clusters dissociate and more and c Vex T    (66) more hadrons are to be found in the “elementary” form of We did verify numerically that the order of averages does a nucleon or a pion. Their kinetic energies are reminiscent not matter of the temperature found at each phase of the expansion.

To compute the experimentally observable final tem- p⊥(m, T c,µ) p p⊥(m, T, µ) p c, (67) perature [1, 19], we shall argue that a time average must     ≈   be performed along the cooling curves. Not knowing the which shows that the mean transverse momentum is also reaction mechanisms too well, we assume that the tem- the simplest (and safest) method of determining the aver- perature decreases approximately linearly with the time age temperature (indeed better than fitting ad hoc expo- in the significant expansion phase. We further have to al- nential type functions to p⊥ distributions). Eur. Phys. J. A (2015) 51: 115 Page 13 of 16

Fig. 6. 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. 4a). The mean temperatures are obtained by integrating along the cool- Fig. 7. Mean transverse momenta of nucleons and pions found ing curves. Note that TN is always greater than Tπ. by integrating along the “cooling curves”. In the presented calculations, we chose the bag con- stant = (145 MeV)4, but we now believe that a larger In fig. 7, we show the dependence of the average trans- B should be used. As a consequence of our choice and the verse momenta of pions and nucleons on the kinetic energy B ex measured pion temperature of T π = 140 MeV at high- of the heavy ion projectiles. est ISR energies, we have to choose  the constant H such that T0 = 190 MeV (see eq. (43b)). The average temperature, as a function of the range 6 Strangeness in heavy ion collisions of integration over T , reaches different limiting values for From the averaging process described here, we have different particles. The limiting value obtained thus is the learned that the temperatures and transverse momenta observable “average temperature” of the debris of the in- of particles originating in the hot fireballs are more remi- teraction, while the initial temperature Tcr at given Ek,lab niscent of the entire history of the fireball expansion than (full line in fig. 6) is difficult to observe. When integrating of the initial hot compressed state, perhaps present in the along the cooling line as in eq. (64), we can easily, at each form of quark matter. We may generalize this result and point, determine the average hadronic cluster mass. The then claim that most properties of inclusive spectra are integration for protons is interrupted (protons are “frozen reminiscent of the equations of state of the hadronic gas out”) when the average cluster mass is about half the nu- phase and that the memory of the initial dense state is cleon isobar mass. We have also considered baryon den- lost during the expansion of the fireballs as the hadronic sity dependent freeze-out, but such a procedure depends gas rescatters many times while it evolves into the final strongly on the unreliable value of . B kinetic and chemical equilibrium state. Our choice of the freeze-out condition was made in In order to observe properties of quark-gluon plasma, such a way that the nucleon temperature at Ek,lab/A = we must design a thermometer, an isolated degree of free- 1.8 GeV is about 120 MeV. The model dependence of our dom weakly coupled to the hadronic matter. Nature has, freeze-out introduces an uncertainty of several MeV in the in principle (but not in practice) provided several such average temperature. In fig. 6, the pion and nucleon av- thermometers: leptons and heavy flavors of quarks. We erage temperatures are shown as a function of the heavy would like to point here to a particular phenomenon per- ion kinetic energy. Two effects contributed to the differ- haps quite uniquely characteristic of quark matter. First ence between the π and N temperatures: we note that, at a given temperature, the quark-gluon 1) The particular shape of the cooling curves (fig. 4a). plasma will contain an equal number of strange (s) quarks The chemical potential drops rapidly from the criti- and antistrange (s) quarks, naturally assuming that the cal curve, thereby damping relative baryon emission at hadronic collision time is much too short to allow for light lower T . Pions, which do not feel the baryon chemical flavor weak interaction conversion to strangeness. Thus, potential, continue being created also at lower temper- assuming equilibrium in the quark plasma, we find the atures. density of the strange quarks to be (two spins and three 2) The freeze-out of baryons occurs earlier than the colors) freeze-out of pions. 3 s s d p −√p2+m2/T A third effect has been so far omitted —the emission of = =6 e s V V (2π)3 pions from two-body decay of long-lived resonances [1]  Tm2 would lead to an effective temperature which is lower in =3 s K (m /T ), (68) nuclear collisions. π2 2 s Page 14 of 16 Eur. Phys. J. A (2015) 51: 115 neglecting for the time being the perturbative corrections over all different particle configurations, e.g., expressed and, of course, ignoring weak decays. As the mass ms of with the help of the mass spectrum. Hence, we can now the strange quarks in the perturbative vacuum is believed concentrate in particular on that part of ln Z which is to be of the order of 280–300 MeV, the assumption of equi- exclusively associated with the strangeness. librium for ms/T 2 may indeed be correct. In eq. (68), As the temperatures of interest to us and which allow we were able to use∼ the Boltzmann distribution again, as appreciable strangeness production are at the same time the density of strangeness is relatively low. Similarly, there high enough to prevent the strange particles from being is a certain light antiquark density (q stands for either u thermodynamically degenerate, we can restrict ourselves or d ) again to the discussion of Boltzmann statistics only. The contribution to Z of a state with k strange parti- q d3p 6 cles is =6 e−|p|/T −µq /T =e−µq /T T 3 , (69) k V (2π)3 π2 1  Z = Zs(T,V ) , (71) k k! I s  where the quark chemical potential is µq = µ/3, as given  by eq. (46). This exponent suppresses the qq pair produc- where the one-particle function Z1 for a particle of mass tion. ms is given in eq. (16). To include both particles and an- What we intend to show is that there are many more tiparticles as two thermodynamically independent phases s quarks than antiquarks of each light flavor. Indeed, in eq. (71), the sum over s in eq. (71) must include them both. As the quantum numbers of particles (p) s 1 m 2 m and antiparticles (a) must always be present with ex- = s K s eµ/3T . (70) q 2 T 2 T actly the same total number, not each term in eq. (71)     can contribute. Only when n = k/2 = number of par- 2 The function x K2(x) is, for example, tabulated in [30]. ticles=number of antiparticles is exactly fulfilled do we For x = ms/T between 1.5 and 2, it varies between 1.3 have a physical state. Hence, and 1. Thus, we almost always have more s than q quarks n n and, in many cases of interest, s/q 5. As µ 0, there 1 2n sp ∼ → Zpair = Z Zsa . (72) are about as many u and q quarks as there are s quarks. 2n (2n)! n ⎛ 1 ⎞ 1 s  s  When the quark matter dissociates into hadrons, some  p a of the numerous s may, instead of being bound in a We now introduce the⎝ fugacity⎠ factor f n to be able to qs kaon, enter into a q q s antibaryon and, in particu- count the number of strange pairs present. Allowing an 3 0 lar ,aΛ or Σ . The probability for this process seems arbitrary number of pairs to be produced, we obtain to be comparable to the similar one for the production n n of antinucleons by the antiquarks present in the plasma. ∞ n f s What is particularly noteworthy about the s-carrying an- Z (β,V ; f)= Z p Zsa s n!n! ⎛ 1 ⎞ 1 tibaryons is that they can conventionally only be pro- n=0 sp  sa     duced in direct pair production reactions. Up to about ⎝ ⎠ = I0( 4y), (73) Ek,lab/A =3.5GeV, this process is very strongly sup- pressed by energy–momentum conservation because, for where I0 is the modified Bessel function and free pp collisions, the threshold is at about 7 GeV. We would thus like to argue that a study of the Λ and Σ0 y = f Zsp Zsa . (74) in nuclear collisions for 2

7 Summary

Our aim has been to obtain a description of hadronic matter valid for high internal excitations. By postulat- ing the kinetic and chemical equilibrium, we have been able to develop a thermodynamic description valid for high temperatures and different chemical compositions. In our work we have found two physically different do- mains: firstly, the hadronic gas phase, in which individual hadrons can exist as separate entities, but are sometimes combined into 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 experimen- Fig. 8. The quenching factor for strangeness production as a tal observations. In particular, in the case of hadronic gas, 3 we have completely abandoned a more conventional La- function of the active volume V/Vh,whereVh =4π/3fm . grangian approach in favour of a semi-phenomenological statistical bootstrap model of hadronic matter that incor- porates those properties of hadronic interaction that are, with Txs = ms 280 MeV. We note in passing that the baryon chemical∼ potential cancels out in y of eq. (74) in our opinion, most important in nuclear collisions. when eq. (76) is inserted in the quark phase (compare with In particular, the attractive interactions are included eq. (68)). through the rich, exponentially growing hadronic mass 2 By differentiating ln Zs of eq. (73) with respect to f, spectrum τ(m ,b), while the introduction of the finite vol- we find the strangeness number present at given T and V ume of each hadron is responsible for an effective short- range repulsion. Aside from these manifestations of strong ∂ I (√4y) interactions, we only satisfy the usual conservation laws n = f ln Z = 1 √y. (77)  s ∂f s I (√4y) of energy, momentum, and baryon number. We neglect f=1 0 quantum statistics since quantitative study has revealed

that this is allowed above T 50 MeV. But we allow For large y, that is, at given T for large volume V ,we ≈ −m/T particle production, which introduces a quantum physical find n s = √y e , as expected. For small y, we find   ∼ aspect into the otherwise “classical” theory of Boltzmann n = y e−2m/T . In fig. 8, we show the dependence of  s ∼ particles. the quenching factor I1/I0 = η as a function of the volume 3 Our approach leads us to the equations of state of V measured in units of Vh =4π/3fm for a typical set of hadronic matter which reflect what we have included in parameters: T = 150, µ = 550 MeV (hadronic gas phase). our considerations. It is the quantitative nature of our The following observations follow from inspection of work that allows a detailed comparison with experiment. fig. 8: This work has just begun and it is too early to say if the features of strong interactions that we have chosen to in- 1) The strangeness yield is a qualitative measure of the clude in our considerations are the most relevant ones. It is hadronic volume in thermodynamic equilibrium. important to observe that the currently predicted pion and 2) Total strangeness yield is not an indicator of the nucleon mean transverse momenta and temperatures show phase transition to quark plasma, as the enhancement the required substantial rise (see fig. 7) as required by the ( η /η =1.25) in yield can be reinterpreted as being q experimental results available at E /A =2GeV(BE- due to a change in hadronic volume. k,lab VALAC, see [24]) and at 1000 GeV (ISR, see [22]). Further 3) We can expect that, in nuclear collisions, the ac- comparisons involving, in particular, particle multiplicities tive volume will be sufficiently large to allow the and strangeness production are under consideration. strangeness yield to correspond to that of “infinite” volume for reactions triggered on “central collisions”. We also mention the internal theoretical consistency of Hence, e.g., Λ production rate will significantly exceed our two-fold approach. With the proper interpretation, the that found in pp collisions. statistical bootstrap leads us, in a straightforward fashion, to the postulate of a phase transition to the quark-gluon Our conclusions about the significance of Λ as an indicator plasma. This second phase is treated by a quite different of the phase transition to quark plasma remain valid as method. In addition to the standard Lagrangian quantum the production of Λ in the hadronic gas phase will only be field theory of weakly interacting particles at finite tem- possible in the very first stages of the nuclear collisions, if perature and density, we also introduce the phenomeno- sufficient center of mass energy is available. logical vacuum pressure and energy density . B Page 16 of 16 Eur. Phys. J. A (2015) 51: 115

Perhaps the most interesting aspect of our work is the 6. E.V. Shuryak, Phys. Lett. B 81, 65 (1979). realization that the transition to quark matter will occur 7. O.K. Kalashnikov, V.V. Klimov, Phys. Lett. B 88, 328 at much lower baryon density for highly excited hadro- (1979). nic matter than for matter in the ground state (T = 0). 8. E.V. Shuryak, Phys. Rep. 61, 71 (1980). The precise baryon density of the phase transition depends 9. B. Touschek, Nuovo Cimento B 58, 295 (1968). somewhat on the bag constant, but we estimate it to be at 10. W. Marciano, H. Pagels, Phys. Rep. 36, 137 (1978). 11. K. Johnson, Acta Phys. Polon. B 6, 865 (1975). about 2–4ν0 at T = 150 MeV. The detailed study of the different aspects of this phase transition, as well as of pos- 12. A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, V.F. Weis- sible characteristic signatures of quark matter, must still skopf, Phys. Rev. D 9, 3471 (1974). 3 be carried out. We have given here only a very preliminary 13. R. Hagedorn, Suppl. Nuovo Cimento , 147 (1965). 14. R. Hagedorn, In Lectures on the Thermodynamics of report on the status of our present understanding. Strong Interactions, CERN Yellow Report 71-12 (1971). We believe that the occurrence of the quark plasma 15. R. Hagedorn, I. Montvay, J. Rafelski, Lecture at Erice phase is observable and we have proposed therefore a ¯ Workshop, Hadronic Matter at Extreme Energy Density, measurement of the Λ/p relative yield between 2 and edited by N. Cabibbo, L. Sertorio (Plenum Press, New 10 GeV/N kinetic energies. In the quark plasma phase, we York, 1980) p. 49. ¯ expect a significant enhancement of Λ production which 16. R. Hagedorn, How to Deal with Relativistic Heavy Ion will most likely be visible in the Λ/¯ p relative rate. Collisions, Invited lecture at Quark Matter 1 : Work- shop on Future Relativistic Heavy Ion Experiments at the 1980: Many fruitful discussions with the GSI/LBL Relativistic Gesellschaft f¨ur Schwerionenforschung (GSI), Darmstadt, Heavy Ion group stimulated the ideas presented here. I would Germany, 7–10 October 1980; circulated in the GSI81-6 like to thank R. Bock and R. Stock for their hospitality at Orange Report, pp. 282–324, R. Bock, R. Stock (Editors), GSI during this workshop. As emphasized before, this work reprinted in Chapter 26 in ref. [1]. Atlas of Particle Produc- was performed in collaboration with R. Hagedorn. This work 17. H. Grote, R. Hagedorn, J. Ranft, tion Spectra was in part supported by Deutsche Forschungsgemeinschaft. , CERN 1970. Hot hadronic and 2015: Also in part supported by the US Department of En- 18. J. Rafelski, H.Th. Elze, R. Hagedorn, quark matter in annihilation on nuclei ergy, Office of Science, Office of Nuclear Physics under award p ,CERNpreprint Proceedings of Fifth European Sym- number DE-FG02-04ER41318. TH 2912 (1980), in posium on Nucleon–Antinucleon Interactions, Bressanone, 1980 (CLEUP, Padua, 1980) pp. 357–382. Open Access This is an open access article distributed 19. R. Hagedorn, J. Rafelski, Phys. Lett. B 97, 136 (1980). under the terms of the Creative Commons Attribution 20. S. Frautschi, Phys. Rev. D 3, 2821 (1971). License (http://creativecommons.org/licenses/by/4.0), which 21. R. Hagedorn, I. Montvay, Nucl. Phys. B 59, 45 (1973). permits unrestricted use, distribution, and reproduction in any 22. G. Giacomelli, M. Jacob, Phys. Rep. 55, 1 (1979). medium, provided the original work is properly cited. 23. F. Karsch, H. Satz, Phys. Lett. 21, 1168 (1980) The “so- lidification” of hadronic matter, a purely geometric effect, discussed in this reference should not be confused with the References transition to the quark plasma phase discussed here. 24. A. Sandoval, R. Stock, H.E. Stelzer, R.E. Renfordt, J.W. Harris, J. Brannigan, J.B. Geaga, L.J. Rosenberg, L.S. 1. R. Hagedorn, J. Rafelski, to be published in Phys. Rep. Schroeder, K.L. Wolf, Phys. Rev. Lett. 45, 874 (1980). This manuscript was never completed, see instead: J. 25. R. Hagedorn, J. Ranft, Suppl. Nuovo Cimento 6, 169 Rafelski (Editor), Melting Hadrons, Boiling Quarks: From (1968). Hagedorn temperature to ultra-relativistic heavy-ion colli- 26. W.D. Myers, Nucl. Phys. A 296, 177 (1978). sions at CERN; with a tribute to Rolf Hagedorn (Springer, 27. J. Gosset, J.L. Kapusta, G.D. Westfall, Phys. Rev. C 18, Heidelberg, 2015). 844 (1978). 2. B.A. Freedman, L.D. McLerran, Phys. Rev. D 16, 1169 28. R. Malfliet, Phys. Rev. Lett. 44, 864 (1980). (1977). 29. N. Cabibbo, G. Parisi, Phys. Lett. B 59, 67 (1975). 3. S.A. Chin, Phys. Lett. B 78, 552 (1978). 30. M. Abramowitz, I.A. Stegun (Editors), Handbook of Math- 4. P.D. Morley, M.B. Kislinger, Phys. Rep. 51, 63 (1979). ematical Functions (Dover Publication, New York, 1964). 5. J.I. Kapusta, Nucl. Phys. B 148, 461 (1979). 31. J. Rafelski, M. Danos, Phys. Lett. B 97, 1980 (279). Eur. Phys. J. A (2015) 51: 116 THE EUROPEAN DOI 10.1140/epja/i2015-15116-x PHYSICAL JOURNAL A Addendum

Strangeness and phase changes in hot hadronic matter – 1983 From: “Sixth High Energy Heavy Ion Study” held 28 June – 1 July 1983 at: LBNL, Berkeley, CA, USA

Johann Rafelski1,2,a,b 1 CERN-PH/TH, 1211 Geneva 23, Switzerland 2 Department of Physics, The University of Arizona Tucson, Arizona, 85721, USA

Addendum to: Eur. Phys. J. A (2015) 51: 114, DOI: 10.1140/epja/i2015-15114-0

Received: 12 August 2015 / Revised: 23 August 2015 Published online: 22 September 2015 c The Author(s) 2015. This article is published with open access at Springerlink.com Communicated by T.S. B´ır´o

Abstract. Two phases of hot hadronic matter are described with emphasis put on their distinction. Here the role of strange particles as a characteristic observable of the quark-gluon plasma phase is particularly explored.

1 Phase transition or perhaps transformation: We first recall that the unhandy extensive variables, Hadronic gas and the quark-gluon plasma viz., energy, baryon number, etc., are replaced by inten- sive quantities. To wit, the temperature T is a measure of energy per degree of freedom; the baryon chemical po- I explore here consequences of the hypothesis that the tential µ controls the mean baryon density. The statistical energy available in the collision of two relativistic heavy quantities such as entropy ( = measure of the number of nuclei, at least in part of the system, is equally divided available states), pressure, heat capacity, etc., will also be among the accessible degrees of freedom. This means that functions of T and µ, and will have to be determined. The there exists a domain in space in which, in a suitable theoretical techniques required for the description of the Lorentz frame, the energy of the longitudinal motion has two quite different phases, viz., the hadronic gas and the been largely transformed to transverse degrees of freedom. quark-gluon plasma, must allow for the formulation of nu- The physical variables characterizing such a “fireball” are merous hadronic resonances on the one side, which then at energy density, baryon number density, and total volume. sufficiently high energy density dissolve into the state con- The basic question concerns the internal structure of the sisting of their constituents2. At this point, we must ap- fireball. It can consist either of individual hadrons, or in- preciate the importance and help by a finite, i.e., nonzero stead, of quarks and gluons in a new physical phase, the temperature in reaching the transition to the quark-gluon plasma, in which they are deconfined and can move freely plasma: to obtain a high particle density, instead of only over the volume of the fireball. It appears that the phase compressing the matter (which as it turns out is quite dif- transition from the hadronic gas phase to the quark-gluon ficult), we also heat it up; many pions are generated in plasma is controlled mainly by the energy density of the a collision, allowing the transition to occur at moderate, 1 3 fireball. Several estimates lead to 0.6–1 GeV/fm for the even vanishing baryon density [14]. critical energy density, to be compared with nuclear mat- Consider, as an illustration of what is happening, the 3 ter 0.16 GeV/fm . p, V diagram shown in fig. 1. Here we distinguish three do- mains. The hadronic gas region is approximately a Boltz- a The original address byline 1983: CERN, Genva, Switzer- mann gas where the pressure rises with reduction of the land and Institut f¨ur Theoretische Physics der Universit¨at volume. When the internal excitation rises, the individual Frankfurt/M; originally printed in LBL-16281 pp. 489–510; hadrons begin to cluster. This reduces the increase in the also: report number UC-34C; DOE CONF-830675; preprint Boltzmann pressure, since a smaller number of particles CERN-TH-3685 available at https://cds.cern.ch/record/ exercises a smaller pressure. In a complete description of 147343/files/198311019.pdf. the different phases, we have to allow for a coexistence of b e-mail: [email protected] 1 An incomplete list of quark-gluon plasma papers includes 2 These ideas originate in Hagedorn’s statistical bootstrap [1–10]. theory [11–13]. Page 2 of 9 Eur. Phys. J. A (2015) 51: 116

both aggregate states: the hadronic gas and the state in which individual hadrons have dissolved into the plasma consisting of quarks and of the gauge field quanta, the gluons. It is interesting to follow the path taken by an isolated quark-gluon plasma fireball in the µ, T plane, or equiv- alently in the ν, T plane. Several cases are depicted in fig. 2. In the Big Bang expansion, the cooling shown by the dashed line occurs in a Universe in which most of the energy is in the radiation. Hence, the baryon density ν is quite small. In normal stellar collapse leading to cold neu- tron stars, we follow the dash-dotted line parallel to the ν Fig. 1. p-V diagram for the gas-plasma first order transition, axis. The compression is accompanied by little heating. with the dotted curve indicating a model-dependent, unstable In contrast, in nuclear collisions, almost the entire ν, T domain between overheated and undercooled phases. 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 hadrons with the plasma state in the sense that the in- plasma, i.e., the part of the time evolution after the ter- ternal degrees of freedom of each cluster, i.e., quarks and mination of the nuclear collision, assuming a plasma for- gluons, contribute to the total pressure even before the dis- mation. The figure reflects the circumstance that, in the solution of individual hadrons. This does indeed become beginning of the cooling phase, i.e., for 1–1.5 10−23 s, necessary when the clustering overtakes the compressive the cooling happens almost exclusively by the mechanism× effects and the hadronic gas pressure falls to zero as V of pion radiation [19, 20]. In typical circumstances, about reaches the proper volume of hadronic matter. At this half of the available energy has been radiated away be- point the pressure rises again very quickly, since in the fore the expansion, which brings the surface temperature absence of individual hadrons, we now compress only the close to the temperature of the transition to the hadronic hadronic constituents. By performing the Maxwell con- phase. Hence a possible, perhaps even likely, scenario is struction between volumes V1 and V2, we can in part ac- that in which the freezing out and the expansion happen count for the complex process of hadronic compressibility simultaneously. These highly speculative remarks are ob- alluded to above. viously made in the absence of experimental guidance. A As this discussion shows, and detailed investigations careful study of the hadronization process most certainly confirm [15–18], we cannot escape the conjecture of a first remains to be performed. order phase transition in our approach. This conjecture In closing this section, let me emphasize that the ques- of [8] has been criticized, and only more recent lattice tion whether the transition hadronic gas quark-gluon gauge theory calculations have led to the widespread ac- plasma is a phase transition (i.e., discontinuous)←→ or contin- ceptance of this phenomenon, provided that an internal uous phase transformation will probably only be answered SU(3) (color) symmetry is used —SU(2) internal symme- in actual experimental work; as all theoretical approaches try leads to a second order phase transition [10]. It is diffi- suffer from approximations unknown in their effect. For cult to assess how such hypothetical changes in actual in- example, in lattice gauge computer calculations, we es- ternal particle symmetry would influence phenomenolog- tablish the properties of the lattice and not those of the ical descriptions based on an observed picture of nature. continuous space in which we live. For example, it is difficult to argue that, were the color The remainder of this report is therefore devoted to symmetry SU(2) and not SU(3), we would still observe the the study of strange particles in different nuclear phases resonance dominance of hadronic spectra and could there- and their relevance to the observation of the quark-gluon fore use the bootstrap model. All present understanding of plasma. phases of hadronic matter is based on approximate mod- els, which requires that table 1 be read from left to right. I believe that the description of hadrons in terms of 2 Strange particles in hot nuclear gas bound quark states on the one hand, and the statisti- cal bootstrap for hadrons on the other hand, have many My intention in this section is to establish quantitatively common properties and are quite complementary. Both the different channels in which the strangeness, however the statistical bootstrap and the bag model of quarks are created in nuclear collisions, will be found. In our follow- based on quite equivalent phenomenological observations. ing analysis (see ref. [21]) a tacit assumption is made that While it would be most interesting to derive the phe- the hadronic gas phase is practically a superposition of nomenological models quantitatively from the accepted an infinity of different hadronic gases, and all informa- fundamental basis —the Lagrangian quantum field the- tion about the interaction is hidden in the mass spectrum ory of a non-Abelian SU(3) “glue” gauge field coupled to τ(m2,b) which describes the number of hadrons of baryon colored quarks— we will have to content ourselves in this number b in a mass interval dm2 and volume V m. report with a qualitative understanding only. Already this When considering strangeness-carrying particles, all∼ we will allow us to study the properties of hadronic matter in then need to include is the influence of the non-strange Eur. Phys. J. A (2015) 51: 116 Page 3 of 9

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)

as our considerations remain valid in this simple approx- imation [22]. Interactions are effectively included through explicit reference to the baryon number content of the strange particles, as just discussed. Non-strange hadrons influence the strange faction by establishing the value of λq at the given temperature and baryon density. The fugacities λs and λq as introduced here control the strangeness and the baryon number, respectively. While λs counts the strange quark content, the up and down quark 1/3 content is counted by λq = λB . Using the partition function eq. (2.2), we calculate for given µ, T ,andV the mean strangeness by evaluating

∂ strange Fig. 2. Paths taken in the ν, T plane by different physical ns ns¯ = λs ln Z (T,V,λs,λq), (2.4) events.  −  ∂λs hadrons on the baryon chemical potential established by which is the difference between strange and antistrange the non-strange particles. components. This expression must be equal to zero due The total partition function is approximately multi- to the fact that the strangeness is a conserved quantum plicative in these degrees of freedom: number with respect to strong interactions. From this con- dition, we get3 ln Z =lnZnon-strange +lnZstrange . (2.1) −1 1/2 For our purposes, i.e., in order to determine the parti- W (xK)+λB W (xΛ)+3W (xΣ) λs = λq λqF, cle abundances, it is sufficient to list the strange particles  W (xK)+λB W (xΛ)+3W (xΣ)  ≡    separately, and we find   (2.5)     ln Zstrange(T,V,λ ,λ )= a result contrary to intuition: λs = 1 for a gas with to- s q tal s = 0. We notice a strong dependence of F on the −1 −1   −1 C 2W (xK)(λsλq + λs λq) baryon number. For large µ, the term with λB will tend to zero and the term with λB will dominate the expression  2 −1 −2 +2 W (xΛ)+3W (xΣ) (λsλq + λs λq ) , (2.2) for λs and F . As a consequence, the particles with fugac-  ity λs and strangeness S = 1 (note that by convention where   strange quarks s carry S = −1, while strange antiquarkss ¯ 2 mi mi − W (xi)= K2 . (2.3) carry S = 1) are suppressed by a factor F which is always T T smaller than unity. Conversely, the production of parti- We have C = VT3/2π2 for a fully equilibrated state. cles which carry the strangeness S = +1 will be favored However, strangeness-creating (x s +¯s) processes in by F −1. This is a consequence of the presence of nuclear hot hadronic gas may be too slow→ (see below) and the matter: for µ = 0, we find F =1. total abundance of strange particles may fall short of In nuclear collisions, the mutual chemical equilibrium, this value of C expected in absolute strangeness chemi- that is, a proper distribution of strangeness among the cal equilibrium. On the other hand, strangeness exchange strange hadrons, will most likely be achieved. By studying cross-sections are very large (e.g.,theK−p cross-section is the relative yields, we can exploit this fact and eliminate 100 mb in the momentum range of interest), and there- the absolute normalization C (see eq. (2.2)) from our con- fore∼ any momentarily available strangeness will always be siderations. We recall that the value of C is uncertain for distributed among all particles in eq. (2.2) according to several reasons: 1/3 the values of the fugacities λq = λB and λs. Hence we i) V is unknown. can speak of a relative strangeness chemical equilibrium. We neglected to write down quantum statistics cor- 3 Notation has been changed γ → F in order to avoid con- rections as well as the multistrange particles Ξ and Ω−, fusion with phase space occupancy γ. Page 4 of 9 Eur. Phys. J. A (2015) 51: 116 ii) C is strongly (t, r)-dependent, through the space-time dependence of T . iii) Most importantly, the value C = VT3/2π2 assumes absolute chemical equilibrium, which is not achieved owing to the shortness of the collision. Indeed, we have (see eq. (4.3) for in plasma strangeness formation and further details and solutions)

dC C(t)2 = A 1 , (2.6) dt H − C( )2 ∞ and the time constant for strangeness production in nu- −2 clear matter can be estimated to be [23] Fig. 3. The ratio nK+ /nK− ≡F as a function of the baryon chemical potential µ,forT = 100, (20), 160 MeV. The −21 τH = C( )/2AH 10 s. lines cross where µ = mY −mK; mY is the mean hyperon mass. ∞ ∼ Thus C does not reach C( ) in plasmaless nuclear col- from eq. (2.2) the grand canonical partition sum for zero ∞ lisions. If the plasma state is formed, then the relevant average strangeness C>C( ) (since strangeness yield in plasma is above strangeness∞ yield in hadron gas (see below). strange −1 ln Z0 C 2W (xK) FλK + F λK Now, why should we expect relative strangeness equi- −1 −1 librium to be reached faster than absolute strangeness +2W (xΛ) FλBλΛ + F λB λΛ equilibrium [21]? Consider the strangeness exchange in- +6W (x ) Fλ λ + F −1λ−1λ , (2.10) teraction Σ B Σ B Σ K−p Λπ0 (2.7)  −→ where, in order to distinguish different hadrons, dummy fugacities λi, i =K,K, Λ, Λ, Σ, Σ have been written. The strange particle multiplicities then follow from

∂ strange ni = λi ln Z0 . (2.11)   ∂λi λi=1 which has a cross-section of about 10 mb at low energies,   while the ss¯ “strangeness creating” associate production Explicitly, we find (notice that the power of F follows the s-quark content): pp pΛK+ (2.8) ∓ −→ n ± = CF W (x ), (2.12)  K  K +1 +µB/T n 0 = CF W (x 0 )e , (2.13)  Λ/Σ  Λ/Σ −1 −µB/T n 0 = CF W (x 0 )e . (2.14)  Λ/Σ  Λ/Σ In eq. (2.14) we have indicated that the multiplicity of an- has a cross-section of less than 0.06 mb, i.e., 150 times tihyperons can only be built up if antibaryons are present smaller. Since the latter reaction is somewhat disfavored according to their (small) phase space. This still seems an by phase space, consider further the reaction unlikely proposition, and the statistical approach may be viewed as providing an upper limit on their multiplicity. πp YK (2.9) −→ From the above equations, we can derive several very instructive conclusions. In fig. 3 we show the ratio

−2 n + / n − = F  K   K  as a function of the baryon chemical potential µ for sev- eral temperatures that can be expected and which are where Y is any hyperon (strange baryon). This has a cross- seen experimentally. We see that this particular ratio is section of less than 1 mb, still 10 times weaker than one a good measure of the baryon chemical potential in the of the s-exchange channels in eq. (2.7). Consequently, I hadronic gas phase, provided that the temperatures are expect the relative strangeness equilibration time to be approximately known. The mechanism for this process is about ten times shorter than the absolute strangeness as follows: the strangeness exchange reaction of eq. (2.7) equilibration time, namely 10−23 s, in hadronic matter of tilts to the left (K−) or to the right (abundance F K+), about twice nuclear density. depending on the value of the baryon chemical potential.∼ We now compute the relative strangeness abundances In fig. 4 the long dashed line shows the upper limit for expected from nuclear collisions. Using eq. (2.5), we find the abundance of Λ as measured in terms of Λ abundances. Eur. Phys. J. A (2015) 51: 116 Page 5 of 9

logically, e.g., from a fit to the hadronic spectrum, which gives

4 = (140–210) MeV = (50–250) MeV/fm3. (3.2) B The central assumption of the quark bag approach is that, inside a hadron where quarks are found, the true vacuum structure is displaced or destroyed. One can turn this point around: quarks can only propagate in domains of space in which the true vacuum is absent. This statement is a reformulation of the quark confinement problem. Now the remaining difficult problem is to show the incompatibility of quarks with the true vacuum structure. Examples of such behavior in ordinary physics are easily found; e.g., a light wave is reflected from a mirror surface, magnetic Fig. 4. Relative abundance of Λ/Λ. The actual yield from the field lines are expelled from superconductors, etc. In this hadronic gas limit may still be 10–100 times smaller than the picture of hadronic structure and quark confinement, all statistical value shown. colorless assemblies of quarks, antiquarks, and gluons can form stationary states, called a quark bag. In particular, Clearly visible is the substantial relative suppression of Λ, all higher combinations of the three-quark baryons (qqq) in part caused by the baryon chemical potential factor of and quark–antiquark mesons (qq¯) form a permitted state. eq. (2.14), but also by the strangeness chemistry (factor As the u and d quarks are almost massless inside a F 2), as in K+K− above. Indeed, the actual relative num- bag, they can be produced in pairs, and at moderate in- ber of Λ will be even smaller, since Λ are in relative chem- ternal excitations, i.e., temperatures, many qq¯ pairs will ical equilibrium and Λ in hadron gas are not: the reaction + 0 be present. Similarly, ss¯ pairs will also be produced. We K p Λπ , analogue to eq. (2.7), will be suppressed by will return to this point at length below. Furthermore, low → p abundance. Also indicated in fig. 4 by shading is a real gluons can be excited and will be included here in rough estimate for the Λ production in the plasma phase, our considerations. which suggests that anomalous Λ abundance may be an in- Thus, what we are considering here is a large quark teresting feature of highly energetic nuclear collisions [35], bag with substantial, equilibrated internal excitation, in for further discussion see sect. 5 below. which the interactions can be handled (hopefully) pertur- batively. In the large volume limit, which as can be shown 3 Quark-gluon plasma is valid for baryon number b  10, we simply have for the From the study of hadronic spectra, as well as from light quarks the partition function of a Fermi gas which, hadron-hadron and hadron-lepton interactions, there for practically massless u and d quarks can be given ana- lytically (see ref. [2] and [30,31]), even including the effects has emerged convincing evidence for the description of 2 hadronic structure in terms of quarks [24]. For many of interactions through first order in αs = g /4π: purposes it is entirely satisfactory to consider baryons 2 gV −3 2αs 1 4 π 2 as bound states of three fractionally charged particles, ln Zq(β,µ)= β 1 (µβ) + (µβ) 6π2 − π 4 2 while mesons are quark–antiquark bound states. The La-   50 α 7π4 grangian of quarks and gluons is very similar to that of + 1 s . (3.3) electrons and photons, except for the required summations − 21 π 60    over flavour and color: Similarly, the glue is a Bose gas 1 µν 2 L = ψ F (p gA) m ψ Fµν F . (3.1) 8π 15 α · − − − 4 ln Z (β,λ)=V β−3 1 s , (3.4) g 45 − 4 π     The flavour-dependent masses m of the quarks are small. while the term associated with the difference to the true For u, d flavours, one estimates mu,d 5–20 MeV. The vacuum, the bag term, is strange quark mass is usually chosen at∼ about 150 MeV [25,26]. The essential new feature of QCD, not easily vis- ln Zbag = Vβ. (3.5) ible in eq. (3.1), is the non-linearity of the field strength −B It leads to the required positive energy density within F in terms of the potentials A. This leads to an attractive the volume occupied by the colored quarks and gluonsB and glue-glue interaction in select channels and,asisbelieved, to a negative pressure on the surface of this region. At this requires an improved (non-perturbative) vacuum state in stage, this term is entirely phenomenological, as discussed which this interaction is partially diagonalized, providing above. The equations of state for the quark-gluon plasma for a possible perturbative approach. are easily obtained by differentiating The energy density of the perturbative vacuum state, defined with respect to the true vacuum state, is by defi- ln Z =lnZq +lnZg +lnZvac , (3.6) nition a positive quantity, denoted by . This notion has been introduced originally in the MITB bag model [27–29], with respect to β, µ,andV . Page 6 of 9 Eur. Phys. J. A (2015) 51: 116

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 + µ db, (3.7) − that ∂E(V,b,S) P = . (3.8) − ∂V We observe that the stable configuration of a single bag, viz., ∂E/∂V = 0, corresponds to the configuration with vanishing pressure P in the microcanonical ensemble. Fig. 5. Lowest order QCD diagrams for ss¯ production: a,b,c) Rather than work in the microcanonical ensemble with gg → ss¯, and d) qq¯ → ss¯. fixed b and S, we exploit the advantages of the grand canonical ensemble and consider P as a function of µ and which leads for µ/3=µq <πT to the following expres- T : sions for the entropy per baryon (including the gluonic ∂ P = T ln Z(µ,T,V ) , (3.9) entropy second T 3 term in eq. (3.14)): −∂V with the result   37 2 T T ∼ µq 1 S π 25! (3.16) P = (ε 4 ), (3.10) ν ≈ 15 µq ←→ 3 − B where ε is the energy density: As this simple estimate shows, plasma events are ex- tremely entropy-rich, i.e., they contain very high particle 6 2α 1 µ 4 1 µ 2 ε = 1 s + (πT)2 multiplicity. In order to estimate the particle multiplic- π2 − π 4 3 2 3 ity, one may simply divide the total entropy created in       the collision by the entropy per particle for massless black 50 αs 7 4 + 1 (πT) body radiation, which is S/n = 4. This suggests that, at − 21 π 60    T µq, there are roughly six pions per baryon. 15 α 8 ∼ + 1 s (πT)4 + . (3.11) − 4 π 15π2 B   4 Strange quarks in plasma In eq. (3.10), we have used the relativistic relation between the quark and gluon energy density and pressure: In lowest order in perturbative QCD, ss¯ quark pairs can be created by gluon fusion processes, fig. 5a,b,c; and by 1 1 P = ε ,P= ε . (3.12) annihilation of light quark-antiquark pairs, see fig. 5d. The q 3 q g 3 g averaged total cross-sections for these processes were cal- culated by Brian Combridge [32]. From eq. (3.10), it follows that, when the pressure van- Given the averaged cross-sections, it is easy to calcu- ishes in a static configuration, the energy density is 4 , late the rate of events per unit time, summed over all final independently of the values of µ and T which fix the lineB and averaged over initial states P = 0. We note that, in both quarks and gluons, the inter- reduce action conspires to the effective available number dN d3k d3k of degrees of freedom. At α =0,µ = 0, we find the handy = d3x 1 2 ρ (k ,x)ρ (k ,x) s dt (2π)3 k (2π)3 k i,1 1 i,2 2 relation  i  | 1| | 2| T 4 GeV ∞ µ εq + εg = . (3.13) 2 3 dsδ s (k1 + k2) k1 k2µσ(s). (4.1) 160 MeV fm × 2 −   4M It is important to appreciate how much entropy must be The factor k k / k k is the relative velocity for mass- created to reach the plasma state. From eq. (3.6), we find 1 · 2 | 1|| 2| for the entropy density and the baryon density ν less gluons or light quarks, and we have introduced a S dummy integration over s in order to facilitate the calcu- 2 2α µ 2 14 50 α lations. The phase space densities ρ (k,x) can be approx- = 1 s πT + 1 s (πT)3 i S π − π 3 15π − 21 π imated by assuming the x-independence of temperature       T (x) and the chemical potential µ(x), in the so-called lo- 32 15 α + 1 s (πT)3, (3.14) cal statistical equilibrium. Since ρ then only depends on 45π − 4 π k   the absolute value of in the rest frame of the equili- 2 2α µ 3 µ brated plasma, we can easily carry out the relevant in- ν = 1 s + (πT)2 , (3.15) 3π2 − π 3 3 tegrals and obtain for the dominant process of the gluon      Eur. Phys. J. A (2015) 51: 116 Page 7 of 9 fusion reaction fig. 5a,b,c the invariant rate per unit time and volume [33]:

d4N 7α2 51 T = = s MT3e−2M/T 1+ + ... , A d3xdt ≈Ag 6π2 14 M   (4.2) where M is the strange quark mass4. 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 [34]. The noteworthy feature of such a reaction is the production of relatively high Fig. 6. Time evolution of the strange quark to baryon number energy γ’s at an energy of about 700–900 MeV (T = abundance in the plasma for various temperatures T ∼ µq = 160 MeV) stimulated by coherent glue emission. These γ’s µ/3. M = 150 MeV, αs =0.6. will leave the plasma without further interactions and pro- vide an independent confirmation of the s-abundance in 2 10−23 s, a point to which we will return below. Another the plasma. × The loss term of the strangeness population is propor- remarkable fact is the high abundance of strangeness rela- tive to baryon number seen in fig. 6 —here, baryon number tional to the square of the density ns of strange and anti- was computed assuming T µq = µ/3 (see eq. (3.15)). strange quarks. With ns( ) being the saturation density ∼ at large times, the following∞ differential equation deter- These two facts, namely: mines ns as a function of time [13] 1) high relative strangeness abundance in plasma, dn n (t) 2 2) practical saturation of available phase space, s A 1 s . (4.3) dt ≈  − ns( )  ∞ have led me to suggest the observation of strangeness as Thus we find a possible signal of quark-gluon plasma [35]. There are two elements in point 1) above: firstly, stran- tanh(t/2τ)+ ns(0) ns(∞) ns( ) geness in the quark-gluon phase is practically as abundant ns(t)=ns( ) ,τ= ∞ . ∞ 1+ ns(0) tanh(t/2τ) 2 as the anti-light quarks u = d =¯q, since both phase spaces ns(∞) A have similar suppression factors: for u, d it is the baryon (4.4) chemical potential, for s, s¯ the mass (M µq) where ≈ ns( ) τ = ∞ . (4.5) s s d3p 1 2 = =6 , (4.7a) A V V (2π)3 √p2+M 2/T The relaxation time τ of the strange quark density in  e +1 eq. (4.5) is obtained using the saturated phase space in q d3p 1 eq. (4.5). We have [33] =6 . (4.7b) V (2π)3 e|p|/T +µq /T +1  1/2 1/2 −1 π 9M −3/2 M/T 99 T τ τg = 2 T e 1+ + ... . Note that the chemical potential of quarks suppresses the ≈ 2 7αs 56 M     q¯ density. This phenomenon reflects on the chemical equi- (4.6) librium between qq¯ and the presence of a light quark den- For αs 0.6andM T , we find from eq. (4.6) that sity associated with the net baryon number. Secondly, τ 4 10∼ −23 s. τ falls off∼ rapidly with increasing temper- ∼ × strangeness in the plasma phase is more abundant than ature. Figure 6 shows the approach of ns(t), normalized in the hadronic gas phase (even if the latter phase space with baryon density, to the fully saturated phase space is saturated) when compared at the same temperature and as a function of time. For M  T = 160 MeV, the sat- −23 baryon chemical potential in the phase transition region. uration requires 4 10 s, while for T = 200 MeV, we The rationale for the comparison at fixed thermodynamic need 2 10−23 s,× corresponding to the anticipated life- × variables, rather than at fixed values of microcanonical time of the plasma. But it is important to observe that, variables such as energy density and baryon density, is even at T = 120 MeV, the phase space is half-saturated in outlined in the next section. I record here only that the 4 In eq. (4.2) a factor 2 was included to reduce the invariant abundance of strangeness in the plasma is well above that rate A, see Erratum: “Strangeness Production in the Quark- in the hadronic gas phase space (by factors 1–6) and the Gluon Plasma” Johann Rafelski and Berndt M¨uller, Phys. Rev. two become equal only when the baryon chemical poten- Lett. 56, 2334 (1986). This factor did not carry through to any tial µ is so large that abundant production of hyperons of the following results. However, additional definition factors becomes possible. This requires a hadronic phase at an “2” show up below in eqs. (4.4), (4.5). energy density of 5–10 GeV/fm3. Page 8 of 9 Eur. Phys. J. A (2015) 51: 116

5 How to discover the quark-gluon plasma factor of about 30, than that expected in the hadronic gas phase at the same values of µ, T . Before carrying Here only the role of the strange particles in the antici- this further, let us note that, in order for strangeness to pated discovery will be discussed. My intention is to show disappear partially during the phase transition, we must that, under different possible transition scenarios, charac- have a slow evolution, with time constants of 10−22 s. teristic anomalous strange particle patterns emerge. Ex- But even so, we would end up with strangeness-saturated∼ amples presented are intended to provide some guidance phase space in the hadronic gas phase, i.e., roughly ten to future experiments and are not presented here in order times more strangeness than otherwise expected. For to imply any particular preference for a reaction channel. similar reasons, i.e., in view of the rather long strangeness I begin with a discussion of the observable quantities. production time constants in the hadronic gas phase, The temperature and chemical potential associated strangeness abundance survives practically unscathed in with the hot and dense phase of nuclear collision can be this final part of the hadronization as well. Facit: connected with the observed particle spectra, and, as dis- if a phase transition to the plasma state has oc- cussed here, particle abundances. The last grand canonical curred, then on return to the hadron phase, there variable —the volume— can be estimated from particle in- will be most likely significantly more strange parti- terferences. Thus, it is possible to use these measured vari- cles around than there would be (at this T and µ) ables, even if their precise values are dependent on a par- if the hadron gas phase had never been left. ticular interpretational model, to uncover possible rapid changes in a particular observable. In other words, instead In my opinion, the simplest observable proportional to of considering a particular particle multiplicity as a func- the strange particle multiplicity is the rate of V-events tion of the collision energy √s, I would consider it as a from the decay of strange baryons (e.g., Λ) and mesons (e.g.,Ks) into two charged particles. Observations of this function of, e.g., mean transverse momentum p⊥ ,which is a continuous function of the temperature (which  is in rate require a visual detector, e.g., a streamer chamber. turn continuous across any phase transition boundary). To estimate the multiplicity of V-events, I reduce the To avoid possible misunderstanding of what I want total strangeness created in the collision by a factor 1/3 to say, here I consider the (difficult) observation of to select only neutral hadrons and another factor 1/2 for the width of the K+ two-particle correlation function charged decay channels. We thus have + in momentum space as a function of the average K 1 s + s b √ + nV     b   , (5.1) transverse momentum obtained at given s.MostofK  ≈6 b  ∼ 15 would originate from the plasma region, which, when it   is created, is relatively small, leading to a comparatively where I have taken s / b 0.2 (see fig. 6). Thus for large width. (Here I have assumed a first order phase events with a large baryon   ∼ number participation, we can transition with substantial increase in volume as matter expect to have several V’s per collision, which is 100–1000 changes from plasma to gas.) If, however, the plasma times above current observation for Ar-KCl collision at state were not formed, K+ originating from the entire 1.8 GeV/Nuc kinetic energy [36]. hot hadronic gas domain would contribute a relatively Due to the highs ¯ abundance, we may further expect an large volume which would be seen; thus the width of the enrichment of strange antibaryon abundances [35]. I would two-particle correlation function would be small. Thus, like to emphasize heres ¯s¯q¯ states (anticascades) created a first order phase transition implies a jump in the K+ by the accidental coagulation of twos ¯ quarks helped by correlation width as a function of increasing p⊥ K+ ,as a gluon q¯ reaction. Ultimately, thes ¯s¯q¯ states become determined in the same experiment, varying √s.  s¯q¯q¯, either→ through ans ¯ exchange reaction in the gas From this example emerges the general strategy of phase or via a weak interaction much, much later. How- my approach: search for possible discontinuities in observ- ever, half of thes ¯q¯q¯ states are then visible as Λ decays in ables derived from discontinuous quantities (such as vol- a visual detector. This anomaly in the apparent Λ abun- ume, particle abundances, etc.) as a function of quantities dance is further enhanced by relating it to the decreased measured experimentally and related to thermodynamic abundance of antiprotons, as described above. variables always continuous at the phase transition: tem- Unexpected behavior of the plasma-gas phase transi- perature, chemical potentials, and pressure. This strategy, tion can greatly influence the channels in which strange- of course, can only be followed if, as stated in the first ness is found. For example, in an extremely particle-dense sentence of this report, approximate local thermodynamic plasma, the produced ss¯ pairs may stay near to each other equilibrium is also established. —if a transition occurs without any dilution of the den- Strangeness seems to be particularly useful for plasma sity, then I would expect a large abundance of φ(1020) ss¯ diagnosis, because its characteristic time for chemical mesons, easily detected through their partial decay mode equilibration is of the same order of magnitude as the (1/4%) to a µ+µ− pair. expected lifetime of the plasma: τ 1–3 10−23 s. This Contrary behavior will be recorded if the plasma is means that we are dominantly creating∼ × strangeness in cool at the phase transition, and the transition proceeds the zone where the plasma reaches its hottest stage slowly —major coagulation of strange quarks can then be —freezing over the abundance somewhat as the plasma expected with the formation of sss ands ¯s¯s¯ baryons and cools down. However, the essential effect is that the in general (s)3n clusters. Carrying this even further, su- strangeness abundance in the plasma is greater, by a percooled plasma may become “strange” nuclear (quark) Eur. Phys. J. A (2015) 51: 116 Page 9 of 9 matter [37]. Again, visual detectors will be extremely suc- 13. R. Hagedorn, How to Deal with Relativistic Heavy Ion cessful here, showing substantial decay cascades of the Collisions, invited lecture at Quark Matter 1: Work- same heavy fragment. shop on Future Relativistic Heavy Ion Experiments at the In closing this discussion, I would like to give warning Gesellschaft f¨ur Schwerionenforschung (GSI), Darmstadt, about the pions. From the equations of state of the plasma, Germany, 7–10 October 1980; circulated in the GSI81- we have deduced in sect.3averyhighspecificentropyper 6, Orange Report, pp. 282–324, edited by R. Bock, R. baryon. This entropy can only increase in the phase transi- Stock, reprinted in Chapter 26 of Melting Hadrons, Boiling tion and it leads to very high pion multiplicity in nuclear Quarks: From Hagedorn temperature to ultra-relativistic collisions, probably created through pion radiation from heavy-ion collisions at CERN; with a tribute to Rolf Hage- dorn, edited by J. Rafelski (Springer, Heidelberg, 2015). the plasma [19,20] and sequential decays. Hence by relat- 14. R. Hagedorn, J. Rafelski, Phys. Lett. B 97, 136 (1980) see e.g. ing anything to the pion multiplicity, , considering K/π also [8]. ratios, we dilute the signal from the plasma. Furthermore, 15. R. Hagedorn, On a Possible Phase Transition Between pions are not at all characteristic for the plasma; they are Hadron Matter and Quark-Gluon Matter (CERN preprint simply indicating high entropy created in the collision. TH 3392 (1982)), Chapter 24 in L. Rafelski (Editor) loc. However, we note that the K/π ratio can show substan- cit. (see ref. [13]). tial deviations from values known in pp collisions —but 16. R. Hagedorn, Z. Phys. C 17, 265 (1983). the interpretations of this phenomenon will be difficult. 17. R. Hagedorn, I. Montvay, J. Rafelski, Hadronic Matter at It is important to appreciate that the experiments dis- Extreme Energy Density,editedbyN.Cabibbo(Plenum cussed above would certainly be quite complementary to Press, New York, 1980). the measurements utilizing electromagnetically interact- 18. J. Rafelski, R. Hagedorn, Thermodynamics of hot nuclear ing probes, e.g., dileptons, direct photons. Strangeness- matter in the statistical bootstrap model – 1978, Chap- based measurements have the advantage that they have ter 23 in J. Rafelski (Editor) loc. cit. (see ref. [13]). much higher counting rates than those recording electro- 19. J. Rafelski, M. Danos, Pion radiation by hot quark-gluon magnetic particles. plasma, CERN preprint TH 3607 (1983) in Proceedings of Sixth High Energy Heavy Ion Study held 28 June - 1 July 1983: I would like to thank R. Hagedorn, B. M¨uller, and P. 1983 at LBNL, CONF-830675, Report LBL-16281 and UC- Koch for fruitful and stimulating discussions, and R. Hage- 34C (Berkeley, CA, 1983) pp. 515–518. dorn for a thorough criticism of this manuscript. Thie work 20. M. Danos, J. Rafelski, Phys. Rev. D 27, 671 (1983). was in part supported by Deutsche Forschungsgemeinschaft. 21. P. Koch, J. Rafelski, W. Greiner, Phys. Lett. B 123, 151 2015: Also in part supported by the US Department of En- (1983). ergy, Office of Science, Office of Nuclear Physics under award 22. P. Koch, Diploma thesis, Universit¨at Frankfurt (1983) number DE-FG02-04ER41318. (published as [21]). 23. A.Z. Mekjian, Nucl. Phys. B 384, 492 (1982). 24. See, e.g., S. Gasiorowicz, J.L. Rosner, Am. J. Phys. 49, Open Access This is an open access article distributed un- 954 (1981). der the terms of the Creative Commons Attribution License 25. P. Langacker, H. Pagels, Phys. Rev. D 19, 2070 (1979) and (http://creativecommons.org/licenses/by/4.0), which permits references therein. unrestricted use, distribution, and reproduction in any me- 26. S. Narison, N. Paver, E. de Rafael, D. Treleani, Nucl. Phys. dium, provided the original work is properly cited. B 212, 365 (1983). 27. A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, V.F. Weis- References skopf, Phys. Rev. D 9, 3471 (1974). 28. K. Johnson, Acta Phys. Polon. B 6, 865 (1975). 1. B.A. Freedman, L.C. McLerran, Phys. Rev. D 16, 1169 29. T. De Grand, R.L. Jaffe, K. Johnson, J. Kiskis, Phys. Rev. 12 (1977). D , 2060 (1975). 30. H.Th. Elze, W. Greiner, J. Rafelski, J. Phys. G 6, L149 2. S.A. Chin, Phys. Lett. B 78, 552 (1978). (1980). 3. P.D. Morley, M.B. Kislinger, Phys. Rep. 51, 63 (1979). 31. J. Rafelski, H.Th. Elze, R. Hagedorn, Hot hadronic and 4. J.I. Kapusta, Nucl. Phys. B 148, 461 (1978). quark matter in p annihilation on nuclei,CERNpreprint 88 5. O.K. Kalashnikov, V.V. Kilmov, Phys. Lett. B , 328 TH 2912 (1980), in Proceedings of Fifth European Sym- (1979). posium on Nucleon–Antinucleon Interactions, Bressanone, 6. E.V. Shuryak, Phys. Lett. B 81, 65 (1979). 1980 (CLEUP, Padua, 1980) pp. 357–382. 7. E.V. Shuryak, Phys. Rep. 61, 71 (1980). 32. B.L. Combridge, Nucl. Phys. B 151, 429 (1979). 8. J. Rafelski, R. Hagedorn, From hadron gas to quark matter 33. J. Rafelski, B. M¨uller, Phys. Rev. Lett. 48, 1066 (1982). II,inThermodynamics of Quarks and Hadrons,editedby 34. G. Staadt, Diploma thesis, Universit¨at Frankfurt (1983) H. Satz (North Holland, Amsterdam, 1981). (later published in: Phys. Rev. D 33, 66 (1986)). 9. J. Rafelski, M. Danos, Perspectives in high energy nuclear 35. J. Rafelski, Eur. Phys. J. A 51, 115 (2015) Originally collisions, NBS-IR-83-2725; US Department of Commerce, printed in GSI81-6 Orange Report, pp. 282–324, Extreme National Technical Information Service (NTIS) Accession States of Nuclear Matter – 1980,editedbyR.Bock,R. Number PB83-223982 (1983). Stock. 10. H. Satz, Phys. Rep. 88, 321 (1982). 36. J.W. Harris, A. Sandoval, R. Stock et al., Phys. Rev. Lett. 11. R. Hagedorn, Suppl. Nuovo Cimento 3, 147 (1964). 47, 229 (1981). 12. R. Hagedorn, Nuovo Cimento 6, 311 (1968). 37. S.A. Chin, A.K. Kerman, Phys. Rev. Lett. 43, 1292 (1979).