Melting Hadrons, Boiling Quarks  From Hagedorn Temperature to UltraRelativistic HeavyIon Collisions at CERN pp 241270  Cite as
Thermodynamics of Hot Nuclear Matter: 1978 in the Statistical Bootstrap Model
Abstract
We formulate the statistical bootstrap model for nuclear matter, and study its resulting thermodynamic properties at nuclear densities below the saturation density. We discuss the relevance of limiting temperature and the phase transition gas–‘liquid’ when the volume of the fireball grows with its energy.
23.1 Introduction
Editor’s comment: The numerical results shown are obtained neglecting antibaryons. In obtaining these results we did find that as temperature rises this effect causes an unphysical rise of baryon density for 120 MeV ≤ T ≤ T_{0} MeV. Moreover, the net baryon density does not vanish along with baryon chemical potential. Within months we solved the problem numerically without this approximation; the results are presented in Chap. 27 “Extreme States of Nuclear Matter”, and in [1]. However, this report is the most detailed available document describing the theory behind the statistical bootstrap model of hot nuclear matter. Being distracted by the rise of the relativistic heavy ion research program, this more theoretical work was shelved to be part of a Physics Reports article, and was never formally published, not even as a THpreprint. It is, however available (see footnote) as a CERN library archived document. The Physics Reports review that would have contained this material was never completed.
Properties of nuclear matter have inspired much of the theoretical work in manybody theory during the last decades. While initially attention was focused on the saturation properties of cold nuclear matter, more recently the advent of highenergy heavy ion accelerators has stimulated work on the high temperature and density domain of the phase diagram.
There exist several main lines of approach to this complicated theoretical problem in which substantial simplifications of the actual physical circumstances are supposed. We will not review these approaches here except to say that they can be divided into two categories: (1) the nuclear matter is considered to be a noninteracting ideal gas; or (2) nuclear interactions are considered at the level of classical particle scattering.
It is immediately apparent that the interesting features of nuclear matter, such as density isomerism at high temperatures, phase transitions, condensation phenomena, etc., can hardly be discussed in the framework of the ideal gas equations of state. The fact that some kind of agreement of inclusive particle spectra in heavy ion collisions is found between theory and experiment is in fact only indicative that a thermal equilibrium is achieved in a fireball created in the collisions. To find out more about the properties of these fireballs, one has to perform more refined experiments and consider a more elaborate theory. This aim is achieved in a nonthermodynamical way in the approaches that deal with the A_{1} + A_{2} manybody problem, in which each particle is followed during the collision; but it becomes virtually impossible to identify the relevant collective motion that is characteristic of phase transitions and critical phenomena.
 (i)
conservation of baryon number and clustering of nucleons (i.e., attractive forces leading to manybody clusters with welldefined baryon number);
 (ii)
nucleon (isobar) excitations and internal cluster excitations (i.e., internal degrees of freedom that can absorb part of the energy of the system at finite temperature, thus transforming kinetic energy into mass);
 (iii)
approximate extensivity of nuclear matter (volume roughly proportional to baryon number, i.e., effectively a shortrange repulsion);
 (iv)
coexistence of a pion gas when the temperature is not equal to zero (and behaving properly even in the presence of nuclear matter);
 (v)
baryon–antibaryon pair creation;
 (vi)
‘chemical’ equilibrium between all constituents of the system (nucleons, isobars, clusters, pions, etc.).
Our present work [2] should be most trustworthy in the domain of high temperatures and moderately high density, where details of the interaction, of Fermi and Bose statistics, as well as of the quark structure of nucleons, are most likely negligible. Also not considered explicitly here is the isospin of the nuclei.
Plan of the Paper
 Section 23.2.

We discuss the bootstrap hypothesis first in the context of a strongly interacting pion gas. The bootstrap equation of the pion gas is solved and discussed. We write down, discuss, and solve the bootstrap equation for nuclear matter. It is much richer than that of the pion gas, which it contains as a special case.
 Section 23.3.

The mass spectrum and its Laplace transform are used to obtain a thermodynamic description of the system. We compute the partition functions for clustered nuclear matter.
 Section 23.4.
 We study the properties of nuclear matter in the thermodynamic limit. Two main properties of our model are:
 (a)
there exists a maximum temperature, which is of the order of that of the pion gas (\(T_{0} \approx m_{\uppi }\));
 (b)
there exists at all temperatures 0 ≤ T ≤ T_{0} a critical baryon number density separating a lowdensity ‘gas phase’ from a state where a condensate and its vapor exist in equilibrium.
A numerical study is presented in which the simplest nontrivial input spectrum is assumed; the corresponding model is solved explicitly and the results are displayed graphically. This case shows all essential features, but it is still too far from reality to be taken as more than a qualitative prediction.
 (a)
 Section 23.5.

Summary.

\(\hslash = c = k\) (Boltzmann constant) = 1;

the only dimensional unit is 1 GeV = 1,000 MeV ≈ 5 fm^{−1};

metric: \(a \cdot b \equiv a_{\mu }b^{\mu } \equiv a_{0}b_{0} {\boldsymbol a \cdot b}\) .
Thus for example \(m^{2}:= p^{2} = p_{0}^{2} \mathbf{ p}^{2} = E^{2} \mathbf{ p}^{2}\).
Remark.
Throughout this paper we use only Boltzmann statistics. As the bootstrap approach leads to an extremely rich mass spectrum, it is almost irrelevant whether a particular cluster or particle is a boson or a fermion or a Boltzmannion: it (almost) never happens that two equal clusters occupy the same state.
23.2 The Statistical Bootstrap Method in Particle and Nuclear Physics
The Statistical Bootstrap Model in Particle Physics
 (a)
the abundant production of particles in highenergy pp collisions, and a momentum distribution of these particles which suggests that there might be some analogy to blackbody radiation emitted from moving sources;
 (b)
the apparent existence of intermediate states in which lumps of highly excited hadronic matter (‘fireballs’) are staying together before decaying.
The next important idea was to admit particles other than just pions, and in particular resonant states of pions, just as if they were stable particles [5]. Not knowing which ones should be admitted and how many there are, we might put them in a mass spectrum of admissible input particles ρ_{in}(m). The pion contributes to ρ_{in}(m) a δfunction \(\delta (m  m_{\uppi })\); resonances contribute smearedout δfunctions. For the moment, ρ_{in}(m) is a function which represents our (incomplete) knowledge of the true mass spectrum ρ(m).
Note that we have restricted the oneparticle state to have the pion mass. Higher mass ‘oneparticle states’ are already contained in the sum, namely when in any of its terms all \(p_{i} \rightarrow m_{i}\). Our new equation for \(\sigma (p^{2},p \cdot V )\) describes the density of states of a manycomponent gas: each species of particle contained in ρ_{in}(m) is present in the gas. All these components are in ‘chemical’ equilibrium; neither the total particle number nor that of any of the various components is fixed.
The key idea that leads to the hadronic bootstrap is the observation that the quantity \(\sigma (p^{2},p \cdot V )\) can be related to the mass spectrum ρ(m). Suppose we could insert the true mass spectrum ρ(m) into Eq. (23.3). Then \(\sigma (p^{2},p \cdot V )\) would be the density of states of a ‘fireball’ of hadronic dimension built up from all strongly interacting particles in statistical equilibrium. Such a fireball is itself a highly excited hadron with mass \(m = \sqrt{p^{2}}\). For reasons of consistency, it should then be admitted as a constituent particle in fireballs of larger mass. Hence it should already be present in the true ρ(m). As both \(\sigma (p^{2},p \cdot V )\) and ρ(m) are densities of states, it follows that if ρ(m) is the true mass spectrum, \(\sigma (p^{2},p \cdot V )\) is itself (apart from some minor kinematical differences) the true mass spectrum at \(m = \sqrt{p^{2}}\). This statement establishes a new relation between ρ and \(\sigma\), leading to an integral equation, the bootstrap equation. Physically, it is equivalent to the postulate that resonances and fireballs are one and the same and that fireballs consist of fireballs.
Solution of the Bootstrap Equation
Thus in Fig. 17.5b, the interval \(\varphi \in \{ 0,\varphi _{0}\}\) corresponds uniquely to \(\beta \in \{\infty,\beta _{0}\}\). Given \(\varphi\), we could invert Eq. (23.8aa); however, we can obtain the physically interesting information about τ without an explicit inversion.
It can be seen that the rapidly decreasing 1∕n! has been replaced by the (exponentially increasing!) g_{ n }. Thus the Ω_{ n } in Eq. (23.16) have been multiplied by n! g_{ n }, which is the total number of possible ways to cluster n objects recursively (admitting clusters of clusters).
The Nuclear Matter Bootstrap Equation
 1.
Conservation of baryon number and clustering of nucleons. The baryon number (number of baryons minus number of antibaryons) is conserved with the help of the Kronecker \(\delta _{K}(b \sum b_{i})\) function. The infinite set of density functions \(\sigma (p,V,b)\) corresponds to the admission of nucleon clusters with any baryon number b, fourmomentum p, and fourvolume V.
 2.
Nucleon (isobar) excitation and internal cluster excitation. Internal cluster excitation is contained in the p^{2} = m^{2} dependence of \(\sigma (p,V,b)\), and singlenucleon (isobar) excitation is contained in the same way in \(\sigma (p,V,b = 1)\).
 3.
Extensivity of nuclear matter. This is ensured by the volume δ^{4}function.
 4.
Coexistence of a pion gas. This is contained in the equation with b = 0, and in all others by the presence of factors \(\sigma (p_{i},V _{i},b_{i} = 0)\) on the righthand side.
 5.
Baryon–antibaryon pair creation (and annihilation). This is built in by allowing \(\infty < b_{i},b < \infty \). Then on the righthand side an arbitrary number of clusters \((\sum b_{i})\) and anticlusters \((\sum \overline{b}_{i})\) may occur.
 6.‘Chemical equilibrium’ between all constituents. This is expressed by the infinite set of coupled integral equations (23.19), which allows all multibody reactions between clusters Q_{ i },compatible with b and p conservation.$$\displaystyle{Q_{1} + Q_{2} + \cdots + Q_{n}\; \rightleftarrows \; Q_{1}^{{\prime}} + Q_{ 2}^{{\prime}} + \cdots + Q_{ n}^{{\prime}}}$$
 (a)
Details of nuclear interaction may be represented by giving clusters (e.g., alpha particles) a special weight.
 (b)
Equation (23.19) deal with Boltzmann particles without charge and spin. Introducing spin, isospin, and statistics would be possible but complicated. We can obtain a similar physical effect by assigning to an input nucleus of baryon number b and volume V_{ b } a mass M_{ b } which is different from b ⋅ m_{ p }.
The Mass Spectrum for Nuclear Matter
The bootstrap equation (23.26) is much richer than that for the pion gas; we have allowed the presence of arbitrarily complicated clusters characterized by the baryonic number b_{ i }. For b = 0, we have a description of meson fireballs; but in order to understand these fireballs properly, especially when baryon–antibaryon clusters are among their constituents, we have to obtain a solution for the function τ for all values of b.
Laplace and LTransforms of the Mass Spectrum
The bootstrap equation (23.32) for the doubly transformed function \(\varPhi (\beta,\lambda )\) has a real solution wherever in the \((\beta,\lambda )\) plane the input function \(\varphi <\varphi _{0} =\ln 4  1\) (see Fig. 17.5a, b). Thus along a curve \(\beta _{\mathrm{c}} = f(\lambda _{\mathrm{c}})\) in the \((\beta,\lambda )\) plane defined as the boundary of this domain \((\beta _{\mathrm{c}},\lambda _{\mathrm{c}}) =\varphi _{0}\), a qualitative change in the behavior of the properties of nuclear matter may occur. Quite aside from the physical questions, we have to ask for a mathematical solution of the bootstrap equation beyond this boundary line. As we have previously argued by a recursive argument, a physical solution for τ(p^{2}, b) exists for any p^{2}. Our \(\varPhi (\beta,\lambda )\) is the LaplaceLtransform of B_{ b }τ(p^{2}, b), which does not exist in this form everywhere in \((0 \leq \beta < \infty ) \otimes (1 \leq \lambda < \infty )\). However, once defined in a domain where it does exist, it fulfills Eq. (23.4), which then permits analytical continuation of \(\varPhi (\beta,\lambda )\) beyond the limit \(\varphi =\varphi _{0}\) in the whole \((\mathrm{complex}\;\beta ) \otimes (\mathrm{complex}\;\lambda )\) domain. Thus using the methods of complex analysis, we will be in a position to study the new phases in the future.
We remark here that the analytical continuation beyond \(\varphi _{0}\) has never been considered in the case of pionic bootstrap, since there this limit on \(\varphi\) led to a limiting temperature; the energy of fireballs diverged at this point and made a transition from our world to the new domain impossible. Now the presence of baryons changes this—the introduction of \(\lambda\) leads to the existence of a new region with T < T_{0} but \(\varphi >\ln 4  1\). We will find in our present model again a boundary T = T_{0}, at which the energy density diverges—but this limit is not at \(\varphi =\ln 4  1\), except when \(\lambda = 1\).
23.3 Thermodynamics
In Sect. 23.2, we solved the bootstrap equation with the help of the Laplace transformation. The same mathematical procedure is used in statistical thermodynamics to obtain the partition function from the density of states. This coincidence has the effect that the Laplace transform Φ(β) of the mass spectrum τ(p^{2}) and the Laplace transform Z(β, V ) of the density of states of a thermodynamical system containing particles with the mass spectrum τ(m^{2}) can easily be confounded. We expect a relation between Φ(β) and Z(β, V )—and we will exploit it below—but conceptually these two quantities are different.
The Partition Functions of the OneComponent Ideal Gas
The Strongly Interacting Pion Gas

either as a multitude of particles of one kind with a complicated interaction,

or as a noninteracting phase consisting of an infinity of different species with a mass spectrum appropriate to the interaction in question.
This implies that, if the mass spectrum of the interaction is known, replacing the interacting particles by an ideal infinitecomponent phase and weighting the different components according to the mass spectrum generates the same distortion of phase space as the interaction would do. An example is, for instance, a dilute He gas. Usually, this is not described as an assembly of protons, neutrons, and electrons with a Hamiltonian containing QED and strong interactions; instead, one uses the mass spectrum (here essentially one state with mass, spin, etc., of^{4}He) and calculates the properties of an ideal Bose gas of He atoms, considering the latter as elementary.
Physics Near T_{0}
Thermodynamic quantities calculated from Eq. (23.47)
a  P  n  \(\varepsilon\)  \(\updelta \varepsilon /\varepsilon\)  \(C_{V } =\mathrm{ d}\varepsilon /\mathrm{d}T\)  

1/2  \(C/\Delta T^{2}\)  \(C/\Delta T^{2}\)  \(C/\Delta T^{3}\)  \(C + C\Delta T\)  \(C/\Delta T^{4}\)  
1  \(C/\Delta T^{3/2}\)  \(C/\Delta T^{3/2}\)  \(C/\Delta T^{5/2}\)  \(C + C\Delta T^{3/4}\)  \(C/\Delta T^{7/2}\)  
3/2  \(C/\Delta T\)  \(C/\Delta T\)  \(C/\Delta T^{2}\)  \(C + C\Delta T^{1/2}\)  \(C/\Delta T^{3}\)  
2  \(C/\Delta T^{1/2}\)  \(C/\Delta T^{1/2}\)  \(C/\Delta T^{3/2}\)  \(C + C\Delta T^{1/4}\)  \(C/\Delta T^{5/2}\)  
5/2  \(C\ln (T_{0}/\Delta T)\)  \(C\ln (T_{0}/\Delta T)\)  \(C/\Delta T\)  C  \(C/\Delta T^{2}\)  
3  \(P_{0}  C\Delta T^{1/2}\)  \(n_{0}  C\Delta T^{1/2}\)  \(C/\Delta T^{1/2}\)  \(C/\Delta T^{1/4}\)  \(C/\Delta T^{3/2}\)  
7/2  \(P_{0}  C\Delta T\)  \(n_{0}  C\Delta T\)  \(\varepsilon _{0}\)  \(C/\Delta T^{1/2}\)  \(C/\Delta T\)  
4  \(P_{0}  C\Delta T^{3/2}\)  \(n_{0}  C\Delta T^{3/2}\)  \(\varepsilon _{0}  C\Delta T^{1/2}\)  \(C/\Delta T^{3/4}\)  \(C/\Delta T^{1/2}\) 
Thermodynamics of Clustered Matter
Partition Function of Nuclear Matter
While we can sum the general formula (23.56), we will be interested here in properties of bulk nuclear matter: that is, the case when a certain number of nucleons is already present in a given volume. Unless \(T \sim T_{0}\), we expect only moderate contributions from baryon–antibaryon pair production, since m_{N} ≫ T_{0}. Therefore we further simplify our model and neglect now antibaryon production. We can implement this by restricting b to be positive in Eq. (23.56). We note that in doing so we allow uncompensated baryon production, which is, for T ≤ T_{0}, a small effect,^{5} since \(m_{\mathrm{N}}/T_{0} \gtrsim 7\).
In the next section, we illustrate our model by some numerical results obtained by studying Eqs. (23.57)–(23.59a).
23.4 Properties of Nuclear Matter in the Bootstrap Model
The Different Phases
The limiting temperature T_{0} is now a solution of Eq. (23.60) with \(\lambda _{\mathrm{c}} = 1\). However, since the nuclear term is exponentially small at \(\beta _{\mathrm{c}} \approx 1/m_{\uppi } \approx 1/T_{0}\), we expect that the limiting temperature is but little changed from that of pionic bootstrap. The change of T_{0} induced by the possible baryon production is obtained by expanding Eq. (23.60) around β_{0}. We find that the change of T_{0} is negative: the limiting temperature is slightly lowered (by about 10 MeV) by the presence of nucleons.
There are three domains shown in Fig. 23.2. In domain I, enclosed by the function μ_{c}(T_{c}), the grand canonical description is valid; in domain II, above the critical curve, we have \(\varphi >\ln 4  1\), but T < T_{0}. In this region, the description of physical quantities should be canonical, since the grand canonical partition function does not exist for \(\varphi <\varphi _{0}\). It is possible, however, to consider the analytical continuation of the grand canonical function into this domain—inverse Ltransform can then be used to find the canonical quantities. Henceforth, we will call region I the gaseous phase (because it contains the region of small density), and region II the ‘liquid’ phase (because it is approached if at fixed temperature the baryon density, i.e., \(\lambda\) or μ, increases). Region III, characterized by T > T_{0}, is a domain that cannot be reached from the physical phases in those bootstrap models that give divergent energy density at T = T_{0}. We have found, however, other versions of the nuclear bootstrap model which allow a transition even to this region—however, we will not discuss this possibility here.
We cannot exclude that, in models with more general input functions \(\varphi\), a further phase develops for large baryon densities. However, this is not so within our simple model of pions and nucleons, where we neglect most of the details of nuclear structure. In particular, for \(T \rightarrow 0\) and for μ corresponding to \(\nu /\nu _{0} \sim 1\), we might need more detailed input than we have considered in the present simplified study.
Baryon Density in the Gaseous Phase
We notice that, for T < T_{0} −δ (with δ a few MeV), the transition from gaseous to ‘liquid phases’ occurs always below one (one baryon per unit volume is by definition the normal nuclear density). This justifies a posteriori our choice for the names of the different phases.
Baryon Energy in the Gaseous Phase
23.5 Summary
We have generalized the Statistical Bootstrap Model in a suitable way, which allows for the description of clustering hadron matter with constant energy density and a conserved quantum number. We apply our theory to the particular case of nuclear matter which, in the thermodynamic equilibrium, consists at finite temperature of nuclear clusters and their excitations, pions, and mesonic and baryonic resonances. Although in the general theoretical part of our work we have maintained baryon number conservation, in the numerical part, we study the properties of nuclear matter, neglecting the antibaryon production.
 1.
we assume an ad hoc ansatz for the bootstrap equation for the nuclear level density [see Eq. (23.19)];
 2.
 3.
we take the proper volume as being parallel to the momentum of the fireball;
 4.
we assume that the natural volumes of the input particles grow with their mass.
 1.Considering the grand canonical partition function, depending on the chemical potential and temperature, we find three different situations:
 (i)
a gaseous state (containing the empty vacuum for μ → 0, T = 0), characterized by the presence of easily movable but strongly interacting nuclei and pions, all in arbitrary states of excitation;
 (ii)
a ‘liquid’ phase at larger baryon densities; and
 (iii)
a supercritical (unphysical) region above \(T = T_{0} = 150\) MeV, where the energy density becomes infinite.
 (i)
 2.
The transition to the ‘liquid’ phase occurs at about 0.65–0.75 of the normal nuclear number density and at finite energy density, except when T approaches T_{0}, where the pure gaseous phase persists through high density and where the energy density becomes very large. We would like to mention now that what we have called throughout this paper the ‘liquid’ phase is really the coexistence of two phases, vapor and liquid, in equilibrium. We are currently working on a description of the high density region beyond the phase transition from gas to liquid.
 3.
In our actual description, we find a limiting temperature T_{0} ≈ 150 MeV. At this temperature, the energy density diverges. We have noted, however, that this is a subtle point which touches on the limits of validity of our present interpretation of the mass spectrum. In this respect, we recall that the volume of fireballs now grows with the fireball mass—thus the average density should be finite for \(T \rightarrow T_{0}\). In a consistent model, we expect a finite energy density at T_{0}, so that the presently forbidden region beyond T_{0} will now become accessible.
 4.
Below 60 MeV, we find that the energy per baryon obeys roughly the simple relation ∼ 3T∕2; however, below 20 MeV, our model includes too little nuclear structure to have enough predictive power. Above 60 MeV, we find that pion degrees of freedom absorb an increasing amount of the total energy, so that the ‘energy per baryon’ (the total energy/number of baryons) exceeds more and more the energy which the baryons themselves carry.
Looking ahead, we hope to enlarge our model by making the input more elaborate, by maintaining the particle–antiparticle symmetry, and by considering the particular importance of alpha clusters. It seems that a profound study of the ‘liquid’ phase will be rewarding since much of the structure of the liquid (maybe even the existence of a new ‘solid’ phase) depends on the amount of nucleon structure we include in the input terms. An obvious first step in this direction is the possible introduction of effective masses ( < free masses) of the bound nucleons, a feature that is very likely relevant to the understanding of the saturation of nuclear matter in the bootstrap description. We must also incorporate Fermi and Bose statistics and investigate models leading to a finite energy density at T_{0}.
Footnotes
 1.
The surprisingly complex analytical structure of this seemingly simple bootstrap function is further explored in: R. Hagedorn and J. Rafelski, “Analytic Structure and Explicit Solution of an Important Implicit Equation,” Commun. Math. Phys. 83, 563 (1982).
 2.
We use the expression ‘Ltransform’ to stress the formal analogy with the Laplace transform: L is the discrete counterpart of \(\mathcal{L}\).
 3.
We will often drop the superscript ‘ex’ on V when the meaning is unambiguous.
 4.
The generalization to several different species is straightforward.
 5.
However, we did find that as temperature rises this effect causes the rise of baryon density, a complete solution is presented in Chap. 27.
 6.
This mirrors the behavior of the rapidly changing factor \(\exp [(m\mu )/T]\); hadronic matter at phase boundary is meson dominated for T > m_{ π }∕2 MeV. Moreover, after we allowed for antimatter production (see solution presented in Chap. 27) the net baryon density continues to decrease for T → T_{0}.
 7.
This is another artifact of the approximation to ignore antibaryons.
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