Melting Hadrons, Boiling Quarks  From Hagedorn Temperature to UltraRelativistic HeavyIon Collisions at CERN pp 4968  Cite as
The Legacy of Rolf Hagedorn: Statistical Bootstrap and Ultimate Temperature
Abstract
In the latter half of the last century, it became evident that there exists an ever increasing number of different states of the socalled elementary particles. The usual reductionist approach to this problem was to search for a simpler infrastructure, culminating in the formulation of the quark model and quantum chromodynamics. In a complementary, completely novel approach, Hagedorn suggested that the mass distribution of the produced particles follows a selfsimilar composition pattern, predicting an unbounded number of states of increasing mass. He then concluded that such a growth would lead to a limiting temperature for strongly interacting matter. We discuss the conceptual basis for this approach, its relation to critical behavior, and its subsequent applications in different areas of high energy physics.
Keywords
Partition Function Baryon Number Hadronic Matter Hadronic Phase Particle Yield7.1 Rolf Hagedorn
The development of physics is the achievement of physicists, of humans, persisting against often considerable odds. Even in physics, fashion rather than fact frequently determines judgment and recognition.
At the time Hagedorn carried out his seminal research, much of theoretical physics was ideologically fixed on “causality, unitarity, Poincaré invariance”: from these three concepts, from axiomatic quantum field theory, all that is relevant to physics must arise. Those who thought that science should progress instead by comparison to experiment were derogated as “fitters and plotters”. Galileo was almost forgotten…. Nevertheless, one of the great Austrian theorists of the time, Walter Thirring, himself probably closer to the fundamentalists, noted: “If you want to do something really new, you first have to have a new idea”. Hagedorn did.
He had a number of odds to overcome. He had studied physics in Göttingen under Richard Becker, where he developed a lifelong love for thermodynamics. When he took a position at CERN, shortly after completing his doctorate, it was to perform calculations for the planning and construction of the proton synchrotron. When that was finished, he shifted to the study of multihadron production in protonproton collisions and to modeling the results of these reactions. It took a while before various members of the community, including some of the CERN Theory Division, were willing to accept the significance of his work. This was not made easier by Hagedorn’s strongly focused region of interest, but eventually it became generally recognized that here was someone who, in this perhaps similar to John Bell, was developing truly novel ideas which at first sight seemed quite specific, but which eventually turned out to have a lasting impact also on physics well outside its region of origin.

the statistical bootstrap model, a selfsimilar scheme for the composition and decay of hadrons and their resonances; for Hagedorn, these were the “fireballs”.

the application of the resulting resonance spectrum in an ideal gas containing all possible hadrons and hadron resonances, and to the construction of hadron production models based on such a thermal input.
We will address these topics in the first two sections, and then turn to their roles both in the thermodynamics of strongly interacting matter and in the description of hadron production in elementary as well as nuclear collisions. Our aim here is to provide a general overview of Hagedorn’s scientific achievements. Some of what we will say transcends Hagedorn’s life. But then, to paraphrase Shakespeare, we have come to praise Hagedorn, not to bury him; we want to show that his ideas are still important and very much alive.
7.2 The Statistical Bootstrap
Around 1950, the physics world still seemed in order for those looking for the ultimate constituents of matter in the universe. Dalton’s atoms had been found to be not really atoms, indivisible; Rutherford’s model of the atom had made them little planetary systems, with the nucleus as the sun and the electrons as encircling planets. The nuclei in turn consisted of positively charged protons and neutral neutrons as the essential mass carriers. With an equal number of protons and electrons, the resulting atoms were electrically neutral, and the states obtained by considering the different possible nucleus compositions reproduced the periodic table of elements. So for a short time, the Greek dream of obtaining the entire complex world by combining three simple elementary particles in different ways seemed finally feasible: protons, neutrons and electrons were the building blocks of our universe.
But there were those who rediscovered an old problem, first formulated by the Roman philosopher Lucretius: if your elementary particles, in our case the protons and neutrons, have a size and a mass, as both evidently did, it was natural to ask what they are made of. An obvious way to find out is to hit them against each other and look at the pieces. And it turned out that there were lots of fragments, the more, the harder the collision. But they were not really pieces, since the debris found after a protonproton collision still also contained the two initial protons. Moreover, the additional fragments, mesons and baryons, were in almost all ways as elementary as protons and neutrons. The study of such collisions was taken up by more and more laboratories and at ever higher collision energies. As a consequence, the number of different “elementary” particles grew by leaps and bounds, from tens to twenties to hundreds. The latest compilation of the Particle Data Group contains over a thousand.
Let us, however, return to the time when physics was confronted by all those elementary particles, challenging its practitioners to find a way out. At this point, in the mid 1960s, Rolf Hagedorn came up with a truly novel idea [1, 2, 3, 4, 5, 6]. He was not so much worried about the specific properties of the particles. He just imagined that a heavy particle was somehow composed out of lighter ones, and these again in turn of still lighter ones, and so on, until one reached the pion as the lightest hadron. And by combining heavy ones, you would get still heavier ones, again: and so on. The crucial input was that the composition law should be the same at each stage. Today we call that selfsimilarity, and it had been around in various forms for many years. A particularly elegant formulation was written a 100 years before Hagedorn by the English mathematician Augustus de Morgan, the first president of the London Mathematical Society:
Great fleas have little fleas upon their backs to bite’em,
and little fleas have lesser still, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on,
while these again have greater still, and greater still, and so on.
Hagedorn had recalled a similar problem in number theory: how many ways are there of decomposing an integer into integers? This was something already addressed in 1753 by Leonhard Euler, and more than a century later by the mathematician E. Schröder in Germany. Finally G.H. Hardy and S. Ramanujan in England provided an asymptotic solution [7]. Let us here, however, consider a simplified, easily solvable version of the problem [8], in which we count all possible different ordered arrangements p(n) of an integer n. So we have
The weights ρ(m) determine the composition as well as the decay of “resonances”, of fireballs. The basis of the entire formalism, the selfsimilarity postulate—here in the form of the statistical bootstrap condition—results in an unending sequence of everheavier fireballs and in an exponentially growing number of different states of a given mass m.
Before we turn to the implications of such a pattern in thermodynamics, we note that not long after Hagedorn’s seminal paper, it was found that a rather different approach, the dual resonance model [10, 11, 12], see Chap. 8, led to very much the same exponential increase in the number of states. In this model, any scattering amplitude, from an initial two to a final hadrons, was assumed to be determined by the resonance poles in the different kinematic channels. This resulted structurally again in a partition problem of the same type, and again the solution was that the number of possible resonance states of mass m must grow exponentially in m, with an inverse scale factor of the same size as obtained above, some 200 MeV. Needless to say, this unexpected support from the forefront of theoretical hadron dynamics considerably enhanced the interest in Hagedorn’s work.
7.3 The Limiting Temperature of Hadronic Matter

a higher temperature,

more constituents, and

more energetic constituents.

a fixed temperature limit, T → T_{H} ,

the momenta of the constituents do not continue to increase, and

more and more species of ever heavier particles appear.
We thus obtain a new, nonkinetic way to use energy, increasing the number of species and their masses, not the momentum per particle. Temperature is a measure of the momentum of the constituents, and if that cannot continue to increase, there is a highest possible, a “limiting” temperature for hadronic systems.
Hagedorn originally interpreted T_{H} as the ultimate temperature of strongly interacting matter. It is clear today that T_{H} signals the transition from hadronic matter to a quarkgluon plasma. Hadron physics alone can only specify its inherent limit; to go beyond this limit, we need more information: we need QCD.
Only a few years later it was, however, pointed out by N. Cabibbo and G. Parisi [15] that larger a shifted the divergence at T = T_{H} to ever higher derivatives. In particular, for 4 > a > 3, the energy density would remain finite at that point, shifting the divergence to the specific heat as next higher derivative. Such critical behavior was in fact quite conventional in thermodynamics: it signaled a phase transition leading to the onset of a new state of matter. By that time, the quark model and quantum chromodynamics as fundamental theory of strong interactions had appeared and suggested the existence of a quarkgluon plasma as the relevant state of matter at extreme temperature or density. It was therefore natural to interpret the Hagedorn temperature T_{H} as the critical transition temperature from hadronic matter to such a plasma. This interpretation is moreover corroborated by a calculation of the critical exponents [16] governing the singular behavior of the resonance gas thermodynamics based on a spectrum of the form Eq. (7.14).
It should be noted, however, that in some sense T_{H} did remain the highest possible temperature of matter as we know it. Our matter exists in the physical vacuum and is constructed out of fundamental building blocks which in turn have an independent existence in this vacuum. Our matter ultimately consists of and can be broken up into nucleons; we can isolate and study a single nucleon. The quarkgluon plasma, on the other hand, has its own ground state, distinct from the physical vacuum, and its constituents can exist only in a dense medium of other quarks—we can never isolate and study a single quark.
We have here considered quark matter formation through the compression of cold nuclear matter. A similar effect is obtained if we heat a meson gas; with increasing temperature, collisions and pair production lead to an ever denser medium of mesons. And according to Hagedorn, also of ever heavier mesons of an increasing degeneracy. For Hagedorn, the fireballs were point like, so that the overlap we had just noted simply does not occur. In the real world, however, they do have hadronic size, so that they will in fact interpenetrate and overlap before the divergence of the Hagedorn resonance gas occurs [17]. Hence now again there will be a transition from resonance gas to a quarkgluon plasma, now formed by the liberation of the quarks and gluons making up the resonances.
At this point, it seems worthwhile to note an even earlier approach leading to a limiting temperature for hadronic matter. More than a decade before Hagedorn, I.Ya. Pomeranchuk [18] pointed out that a crucial feature of hadrons is their size, and hence the density of any hadronic medium is limited by volume restriction: each hadron must have its own volume to exist, and once the density reaches the dense packing limit, it’s the end for hadronic matter. This simply led to a temperature limit, and for an ideal gas of pions of 1 fm radius, the resulting temperature was again around 200 MeV. Nevertheless, these early results remained largely unnoticed until the work of Hagedorn.
Such geometric considerations do, however, lead even further. If hadrons are allowed to interpenetrate, to overlap, then percolation theory predicts two different states of matter [19, 20]: hadronic matter, consisting of isolated hadrons or finite hadronic clusters, and a medium formed as an infinite sized cluster of overlapping hadrons. The transition from one to the other now becomes a genuine critical phenomenon, occurring at a critical value of the hadron density.
We thus conclude that the pioneering work of Rolf Hagedorn opened up the field of critical behavior in strong interaction physics, a field in which still today much is determined by his ideas. On a more theoretical level, the continuation of such studies was provided by finite temperature lattice QCD, and on the more experimental side, by resonance gas analysis of the hadron abundances in high energy collisions. In both cases, it was found that the observed behavior was essentially that predicted by Hagedorn’s ideas.
7.4 Resonance Gas and QCD Thermodynamics
With the formulation of Quantum Chromodynamics (QCD) as a theoretical framework for the strong interaction force among elementary particles it became clear that the appearance of the ultimate Hagedorn’s temperature T_{H} , signals indeed the transition from the hadronic phase to a new phase of strongly interacting matter, the quarkgluon plasma (QGP) [21] (and reference therein). As QCD is an asymptotically free theory, the interaction between quarks and gluons vanishes logarithmically with increasing temperature, thus at very high temperatures the QGP effectively behaves like an ideal gas of quarks and gluons.
Today we have detailed information, obtained from numerical calculations in the framework of finite temperature lattice Quantum Chromodynamics [22, 23], about the thermodynamics of hot and dense matter. We know the transition temperature to the QGP and the temperature dependence of basic bulk thermodynamic observables such as the energy density and the pressure [24, 25]. We also begin to have results on fluctuations and correlations of conserved charges [26, 27, 28].
The recent increase in numerical accuracy of lattice QCD calculations and their extrapolation to the continuum limit, makes it possible to confront the fundamental results of QCD with Hagedorn’s concepts [2, 6], which provide a theoretical scenario for the thermodynamics of strongly interacting hadronic matter [28, 29, 30].
In particular, the equation of state calculated on the lattice at vanishing and finite chemical potential, and restricted to the confined hadronic phase, can be directly compared to that obtained from the partition function Eq. (7.13) of the hadron resonance gas, using the form Eq. (7.14) introduced by Hagedorn for a continuum mass spectrum. Alternatively, as a first approximation, one can also consider a discrete mass spectrum which accounts for all experimentally known hadrons and resonances. In this case the continuum partition function of the Hagedorn model is expressed by Eq. (7.12) with ρ(m_{ i }) replaced by the spin degeneracy factor of the i^{th} hadron, with the summation taken over all known resonance species listed by the Particle Data Group [31].
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.
There is a clear coincidence of the Hagedorn resonance model results and the lattice data on the equation of states. All bulk thermodynamical 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. Although Hagedorn’s model formulated for a discrete mass spectrum does not exhibit a critical behavior, it nevertheless reproduces remarkably well the lattice results in the hadronic phase. This agreement has now been extended to an analysis of fluctuations and correlations of conserved charges as well.
In summary of this section we note that a remarkably good description of lattice QCD results on the equation of states by the Hagedorn thermal model justifies, that resonances are indeed the essential degrees of freedom near deconfinement. Thus, on the thermodynamical level, modeling hadronic interactions by formation and excitation of resonances, as introduced by Hagedorn, is an excellent approximation of strong interactions.
7.5 Resonance Gas and Heavy Ion Collisions
The Hagedorn model, formulated in Eq. (7.21), describes bulk thermodynamic properties and particle composition of a thermal fireball at finite temperature and at non vanishing charge densities. If such a fireball is created in high energy heavy ion collisions, then yields of different hadron species are fully quantified by thermal parameters. However, following Hagedorn’s idea, the contribution of resonances decaying into lighter particles, must be included [2, 6].
The particle yields in Hagedorn’s model Eq. (7.24) depend, in general, on five parameters. However, in high energy heavy ion collisions, only three parameters are independent. In the initial state the isospin asymmetry, fixes the charge chemical potential and the strangeness neutrality condition eliminates the strange chemical potential. Thus, on the level of particle multiplicity, we are left with temperature T and the baryon chemical potential μ_{ B } as independent parameters, as well as, with fireball volume as an overall normalization factor.
Hagedorn’s thermal model introduced in Eq. (7.24) was successfully applied to describe particle yields measured in heavy ion collisions. The model was compared with available experimental data obtained in a broad energy range from AGS up to LHC. Hadron multiplicities ranging from pions to omega baryons and their ratios, as well as composite objects like e.g. deuteron or alpha particles, were used to verify if there was a set of thermal parameters (T, μ_{ B }) and V, which simultaneously reproduces all measured yields.
The systematic studies of particle production extended over more than two decades, using experimental results at different beam energies, have revealed a clear justification, that in central heavy ion collisions particle yields are indeed consistent with the expectation of the Hagedorn thermal model. There is also a clear pattern of the energy, \(\sqrt{s}\)dependence of thermal parameters. The temperature is increasing with \(\sqrt{s}\), and at the SPS energy essentially saturates at the value, which corresponds to the transition temperature from a hadronic phase to a QGP, as obtained in LQCD. The chemical potential, on the other hand, is gradually decreasing with \(\sqrt{s}\) and almost vanishes at the LHC.
In Fig. 7.6 we show the yields of particles with no resonance contribution, like ϕ, Ω, the deuteron ‘d’,^{3}He and the hypertriton \(_{\varLambda }^{3}\) He, normalized to their spin degeneracy factor, as a function of particle mass. Also shown in this figure is the prediction from Eq. (7.25) at \(T \simeq 156\) and for volume \(V \simeq 5,000\) fm^{3} [38]. There is a clear coincidence of data taken in Pb–Pb collisions at the LHC and predictions of the Hagedorn model Eq. (7.25). Particles with no resonance contribution measured by ALICE collaboration follow the Hagedorn’s expectations that they are produced from a thermal fireball at common temperature. A similar agreement of Hagedorn’s thermal concept and experimental data taken in central heavy ion collisions has been found for different yields of measured particles and collision energies from AGS, SPS, RHIC and LHC (for a review, see e.g. [32]).
7.6 Particle Yields and Canonical Charge Conservation
The Hagedorn thermodynamical model for particle production, was originally applied to quantify and understand particle yields and spectra measured in elementary collisions—there were no data available from heavy ion collisions.
Initial work on particle production by Hagedorn began in 1957 in collaboration with F. Cerulus when they applied the Fermi phase space model, see Sect. 25.2 In this microcanonical approach, conservation laws of baryon number or electric charge were implemented exactly. Almost 15 years later the production of complex light antinuclei, such as antiHe^{3}, preoccupied Hagedorn [2, 6]. He realized and discussed clearly the need to find a path to enforce exact conservation of baryon number to describe the antiHe^{3} production correctly within the canonical statistical formulation.
The problem of exact conservation of discrete quantum numbers in a thermal model formulated in early 1970s by Hagedorn in the context of baryon number conservation remained unsolved for a decade. When the heavy ion QGP research program was approaching and strangeness emerged as a potential QGP signature, Hagedorn pointed out the need to consider exact conservation of strangeness (Rafelski, private communication). This is the reason that the old problem of baryon number conservation was solved in the new context of strangeness conservation [39, 40, 41], see also Sect. 27.6 A more general solution, applicable to all discrete conserved charges, abelian and nonabelian, was also introduced in [42] and expanded in [43, 44, 45, 46, 47, 48]. Recently, it has become clear that a similar treatment should be followed not only for strangeness but also for charm abundance study in high energy \(e^{+}e^{}\) collisions [49, 50].
To summarize this section, we note that the usual form of the statistical model, based on a grand canonical formulation of the conservation laws, cannot be used when either the temperature or the volume or both are small. As a rough estimate, one needs VT^{3} > 1 for a grand canonical description to hold [39, 46]. In the opposite limit, a path was found within the canonical ensemble to enforce charge conservations exactly.
The canonical approach has been shown to provide a consistent description of particle production in high energy hadronhadron, \(e^{+}e^{}\) and peripheral heavy ion collisions [32, 45, 49, 50]. As noted in the context of developing strangeness as signature of QGP, see Sect. 27.6, such a model also provides, within the realm of assumed strangeness chemical equilibrium, a description of an observed increase of single and multistrange particle yields from pp, pA to AA collisions and its energy dependence [40].
7.7 Concluding Remarks
Rolf Hagedorn’s work, introducing concepts from statistical mechanics and from the mathematics of selfsimilarity into the analysis of high energy multiparticle production, started a new field of research, alive and active still today. On the theory side, the limiting temperature of hadronic matter and the behavior of the Hagedorn resonance gas approaching that limit were subsequently verified by first principle calculations in finite temperature QCD. On the experimental side, particle yields as well as, more recently, fluctuations of conserved quantities, were also found to follow the pattern predicted by the Hagedorn resonance gas. Rarely has an idea in physics risen from such humble and little appreciated beginnings to such a striking vindication. So perhaps it is appropriate to close with a poetic summary one of us (HS) formulated some 20 years ago for a HagedornFest, with a slight update.
HOT HADRONIC MATTER
(A Poetic Summary)
In days of old
a tale was told
of hadrons ever fatter.
Behold, my friends, said Hagedorn,
the ultimate of matter.
Then Muster Mark
called in the quarks,
to hadrons they were mated.
Of colors three, and never free,
all to confinement fated.
But in dense matter,
their bonds can shatter
and they freely move around.
Above T_{H}, their colors shine
as the QGP is found.
Said Hagedorn,
when quarks were born
they had different advances.
Today they form, as we can see,
a gas of all their chances.
Footnotes
Notes
Acknowledgements
K.R. acknowledges support by the Polish Science Foundation (NCN), under Maestro grant DEC2013/10/A/ST2/00106.
Open Access This book is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and sources are credited.
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