The Hagedorn Temperature Revisited.
Abstract
The Hagedorn temperature, is determined from the number of hadronic resonances including all mesons and baryons. This leads to a stable result MeV consistent with the critical and the chemical freezeout temperatures at zero chemical potential. We use this result to calculate the speed of sound and other thermodynamic quantities in the resonance hadron gas model for a wide range of baryon chemical potentials following the chemical freezeout curve. We compare some of our results to those obtained previously in other papers.
Keywords: Hagedorn;Hadrons;Temperature;Chemical;Potentials.
25.75,q.05.70,a,64.70,p,64.90,+b
1 Introduction
In 1965 Hagedorn proposed that the number of hadronic resonances increases exponentially with the mass of the resonances. The idea, which was debated strongly when first proposed, has since been widely accepted and discussed by many authors . The concept was based on the assumption that the observed increase in the number of hadronic resonances would continue towards higher and higher masses as more experimental data became available . The scale of the exponential increase determines the value of the Hagedorn temperature, . Recent papers have used the latest results from the Particle Data Group to revisit the original analysis of Hagedorn to update the value of . This resulted in a surprising wide spread of possible values, with large variations as to whether one considers mesons or baryons with values ranging from MeV to MeV depending on the parametrization used and on the set of hadrons (mesons or baryons). There thus exists uncertainty as to the value of the Hagedorn temperature. These have two origins:

sparse information about hadronic resonances certainly above 3 GeV,

the analytical form of the Hagedorn spectrum, especially the factor multiplying the exponential.
The first item will probably never be resolved satisfactorily due to the
width of resonances and also due to their large number making it difficult to
identify them.
Splitting the spectrum into baryons and mesons further decreases the quality of the
fits to the mass dependence of the
mass spectrum.
We therefore propose to stick to the
original analysis of Hagedorn and consider a sum
over all resonances, baryons, mesons, strange, nonstrange, charm, bottom etc..
This is the state that is produced at the Large Hadron Collider (LHC), namely,
a hadronic
ensemble containing all possible resonances.
The result is shown in Fig. 1 and leads to a good determination
of because the range
in is reasonably large extending up
to 3 GeV before reaching a plateau presumably due to the parsity of hadronic resonances
above this value. Details about the parametrization used will be presented below.
The Hagedorn temperature naturally leads to the notion that hadronic matter
cannot exceed a limiting temperature
and increasing the beam energy in protonproton and
protonantiprotonÌ collisions results in more and more hadronic resonances being produced
without a corresponding increase in the temperature of the final (freezeout) state.
This is the situation observed at
the highest energies at the LHC.
It was suggested long ago that
the Hagedorn temperature is connected to the existence of a different phase in which quarks and gluons
are no longer confined.
At present, the Hagedorn temperature is often understood as the temperature of the phase transition
from hadrons to a quarkgluon plasma.
A recent analysis investigated the
hadron resonance gas model to show the stability of various thermodynamic quantities
in heavyion collisions when the Hagedorn spectrum is used literally, i.e. without a cutoff
on the number of resonances beyond a certain mass. In particular,
the authors explored the addition of the
hadron resonance gas model including an exponentially large number of undiscovered resonances
which are naturally included in the Hagedorn model.
Their results showed that the
hadron resonance gas model gave different results
for thermodynamic quantities but the overall chemical analysis was reasonably stable.
Furthermore, the use of socalled Hagedorn States (HS), based on the exponentially increasing spectrum, close to the critical temperature can explain fast chemical equilibration by HS regeneration and provide a unique method to compare lattice results for using thermal fits HS provide a lower than thermal fits without HS . These authors estimated effects by extending the hadron mass spectrum beyond GeV for = 200 MeV and assume that high mass excited states produce one kaon while producing multiple pions thus further reducing the .
In this paper, we extend previous work by
extending explicit expressions of relevant
thermodynamic quantities for nonzero chemical potentials.
In particular the speed of sound, , can be
considered as a sensitive indicator of the critical behavior
in strongly interacting matter. The results show a sharp
dip of in the critical region, which is an
indication that
thermodynamics in the vicinity of confinement is indeed driven by the
higher excited hadronic states.
The outline of this paper is as follows. In section 2, we present the influence of the Hagedorn spectrum
on the hadron yields to find the thermodynamic parameters and explain the basic
concepts used in this paper. In section 3,
we derive the number, energy and entropy densities and also the
speed of sound.
In section 4, we show the results using the hadron and its extension resonance gas model
for particular thermodynamic quantities
and discuss the relationship of with temperature and chemical potentials and compare
thermodynamic quantities, like energy and entropy density.
The last section covers the conclusion.
2 Motivation
The particle data table contains hundreds of hadronic resonances
including the well known stable hadrons like nucleons, pions,
hyperons (, , ), kaons etc..
In our calculations we used
a list of hadronic resonances including
in total baryons and antibaryons,
mesons and antimesons (counted without considering their
isospin and spin degeneracies).
Most of the hadronic resonances decay quickly via strong interactions
before reaching the
detector, hence they are usually identified via their decay products.
The mass of a decaying particle is equal
to the total energy of the products measured in its rest frame.
The basic idea of this paper is to add resonances using the Hagedorn model for the spectrum.
Using the hadronic data
we show the relationship
between the number of hadronic resonances and the mass in
Fig. 1 where we took hadrons with masses up to GeV.
The density of states obtained this way has been fitted using
the following equation ;
(1) 
where and are constant parameters given in the table below
Parameter  

0.16 0.02  
0.5  
0.174 0.011 
3 The Hadron Resonance Gas Model and its Extension
The thermodynamic properties of the Hadron Resonance Gas Model (HRGM) can be determined from the partition function
(2)  
where the gas is contained in a volume , has a temperature and
chemical potential , and are the partition
functions for an ideal gas of bosons and
fermions respectively with mass , and are the
spectral density of mesons and baryons. By using Eq. 2, one can compute
the number denisty ,
energy density , entropy density , pressure , speed of
sound and specific heat .
hadron properties enter these models
through . The HRGM model takes the observed spectrum of
hadrons up to some cutoff of mass , defined by
(3) 
where are the masses of the known hadrons and the degeneracy factor. In order to explore the stability of results obtained using the HRGM, variant of these models is often used in which one takes the observed spectrum of states up to a certain cutoff of mass and above this one includes an exponentially rising cumulative density of hadron states, which is defined in Eq. 2. This defines the Extended Hadron Resonance Gas Model (EHRGM). The density of states becomes
(4) 
where the model parameter and are fitted to data on the cumulative distribution of the sets of hadrons, . Typically this model uses GeV. Moreover, the parameters are determined from the hadronic spectrum with masses up to GeV as shown in Fig. 1. The results for and are given in Table 1.
4 Derivation of thermodynamic quantities in EHRGM
We are considering here the Boltzmann distribution for simplicity in order to present some of the results in a compact form. The partition function for a single particle is given by
(5) 
where is modified Bessel function. The meson mass distribution is taken to be given by
(6) 
4.1 Derivation for the particle densities and pressure
The particle density can be written as the sum of the two terms
(7) 
where and are number densities of mesons and baryons respectively. They are defined by
(8)  
where , for our case we consider an isospin symmetric system, where GeV. The pressure is given by
(9) 
where and are the pressure of mesons and baryons respectively.
4.2 Derivation for energy density
The energy density is given by
(10) 
where is the modified Bessel function, and are energy densities of mesons and baryons respectively.
(11) 
where
From the above expressions, we can obtain the entropy density
(12) 
where and are the net number densities for strange and baryonic particles respectively
(13) 
4.3 Derivation of the speed of sound
In hydrodynamic models the speed of sound plays an important role in the evolution of a system and is an ingredient in the understanding of the effects of a phase transition . It is wellknown that the speed of sound has to be calculated at constant entropy per particle . This makes the calculation more complicated than the one at zero chemical potential where it is sufficient to keep the temperature fixed. In our extension we take the condition into account for nonzero baryon and strangeness chemical potentials, imposing overall strangeness zero. The squared speed is thus calculated starting from
(14) 
The complete expression for the speed of sound can be rewritten as
(15) 
where the derivatives and can be evaluated from two conditions. The first condition comes from the requirement that the ratio must be kept fixed,
(16) 
while the second condition follows from the conservation of strangeness
(17) 
The resulting expressions for and are presented in detail in the appendix.
5 Results using HRGM and EHRGM
There is not much difference between the HRGM and EHRGM models at low temperature, since the
heavy resonances do not play an important role there.
Alternatively, at high temperatures, , it
should agree with the results of the lattice simulations of
QCD . In our case, we have determined
the temperature and chemical potential dependencies of the pressure and the
number, energy and entropy densities
using the chemical freezeout curve .
Using these results, we have then calculated the speed of sound.
The thermodynamic variables obtained using HRGM, we plot the energy and
entropy densities scaled by the appropriate powers of ,
which is shown in Fig. 2a, 3a and 4a, at
the GeV didn’t observe any sudden change of the thermodynamic
variables and close to the critical temperature, it
showed smooth shape as the temperature goes beyong .
Moreover, when we use the EHRGM, we see a different behaviour at the
critical
temperature which is shown in Fig. 2b, 3b
and 4b. These show that if the
system undergoes a firstorder phase transition, both temperature and pressure remain constant as hadronic matter is converted from
hadron gas to quarkgluon plasma; the energy and entropy density, however, change
discontinuously. This leads to show us a new state of matter.
The value of the speed of sound remains well below the idealgas limit for massless particles , even at very high temperatures, energy densities and pressure. In Fig. 4b represents, when it crosses the transition point in the case of full QCD, we expect as before that should vanish beside that is inversely proportional to the specific heat , and this can lead to a divergence at the critical temperature . Of course, due to finite volume effects, the velocity of sound will most likely not completely vanishes at .
The velocity of sound versus the temperature as shown in Fig. 4b is to evaluate , we have followed and first expressed and in physical units, using EHRGM in Eq. 4. The temperature and chemical potential dependence of all the relevant thermodynamic quantities shows unique behaviour at the critical point , specially for the speed of sound, there is a pronounced dip as evidence for the phase transition in the system. Based on our calculations, the velocity of sound at is different from zero; it is expected to become zero but due to finite volume effects.
6 Conclusions
In this paper we have made a new analysis of the number of hadronic resonances using the latest information from the Particle Data Group . This leads to a temperature which is consistent with the most recent results based on lattice QCD estimates of the phase transition temperature and also the chemical freezeout temperature at zero baryon density . We have extended calculations of the speed of sound to nonzero baryon and strangeness chemical potentials keeping fixed . This is done for both the hadronic resonance gas model (HRGM) and the extended hadronic resonance gas model (EHRGM) which includes the exponentially increasing spectrum of hadrons following the Hagedorn parametrization. The EHRGM shows that the speed of sound goes to zero at the phase transition point while the HRGM shows a smooth dip followed by an increase.
Appendix A Appendix: Speed of sound at nonzero chemical potentials.
The speed of sound is given by
(A.1) 
where is the entropy per particle which is kept fixed. using the variables and , this can be rewritten as
(A.2) 
where the derivatives and can be evaluated using two conditions. The first condition comes from keeping the ratio constant. From the derivative one obtains
(A.3) 
which implies
(A.4) 
In terms of , and this equation can be written as
(A.5)  
divide the above expression by on both sides, so that it becomes
(A.6)  
Rearranging the above expression in order to write in terms of one obtains
(A.7) 
Defining
(A.8)  
(A.9)  
(A.10) 
The final expression for condition one is
(A.11) 
The second condition comes from overall strangeness neutrality, which is
(A.12) 
where and are the strange and antistrange particle densities. Similarly the derivative of equation Eq. A.12 should thus satisfy
(A.13) 
this implies equation Eq. A.13 can be expressed as
(A.14) 
We can apply the same method as above to write in terms of for the above relation
(A.15)  
we define , hence it represents that the number of strangeness density for baryons and mesons
and , the number of antistrangeness density for baryons and mesons
Define now
(A.16)  
(A.17)  
(A.18) 
Hence, the final expression for condition two becomes
(A.19) 
Finally, by equating equation A.11 and A.19 we find
(A.20) 
and
(A.21) 
Which is the relation used in the text.
References
References
 1. R. Hagedorn, Supplemento al Nuovo Cimento Volume III, 147 (1965); Nuovo Cimento 35 (1965) 395; Nuovo Cimento 56 A (1968) 1027.
 2. R. Hagedorn and J. Ranft, Nucl. Phys. B 48 (1972) 157190.
 3. N. Cabibbo and G. Parisi, Phys. Lett. B 59 (1975) 67.
 4. G. Veneziano, Nuovo Cimento 57A (1968) 190.

5.
K. Bardakci and S. Mandelstam, Phys. Rep.184 (1969) 1640;
S. Fubini and G. Veneziano, Nuovo Cimento 64A (1969) 811.  6. K. Huang and S. Weinberg, Phys. Rev. Lett. 25 (1970) 895.
 7. H. Satz, Phys. Rev. D 19 (1979) 1912.
 8. Ph. Blanchard, S. Fortunato and H. Satz, Eur. Phys. J. C 34 (2004) 361.
 9. R. V. Gavai and A. Goksch, Phys. Rev. D 33 (1986) 614.
 10. K. Redlich and H. Satz, Phys. Rev. D 33 (1986) 3747.
 11. F. Karsch, E. Laermann and A. Peikert, Phys. Lett. B 478 (2000) 447.
 12. Particle Data Group, C. Caso et al., Eur. Phys. J. C 3, (1998) 1794
 13. M. Chojnacki, W. Florkowski and T. Csorgo, Phys. Rev. C 71 (2006) 044902.
 14. M. Chojnacki and W. Florkowski Acta Physica Polonica B 38 (2007) 3249.
 15. S. Chatterjee, S. Gupta and R. M. Godbole, Phys. Rev. C 81 044907 (2010).
 16. W. Broniowski, W. Florkowski and L. Y. Glozman, Phys. Rev. D 70 (2004) 117503
 17. W. Broniowski, Preprint hepph 0008112, in “Bled 2000: Few Quark Problems”.
 18. W. Broniowski and W. Florkowski, Phys. Lett. B 490 (2000) 223227
 19. J. Cleymans, H. Oeschler, K. Redlich, S. Wheaton, Phys. Rev. C 73 (2006) 034905.
 20. A. Andronic, P. BraunMunzinger, J. Stachel, Nucl. Phys. A 772 (2006) 167.
 21. F. Becattini, J. Manninen, M. Gazdzicki, Phys. Rev. C 73 (2006) 044905.
 22. J. NoronhaHostler, M. Beitel, C. Greiner, I. Shovkovy, Phys. Rev. C 81 (2010) 054909.
 23. J. NoronhaHostler, H. Ahmad, J. Noronha, C. Greiner, Phys. Rev. C 82 (2010) 024913.
 24. A. Andronic, P. BraunMunzinger, J. Stachel, Phys. Lett. B 673 (2009) 142
 25. P. Castorina, K. Redlich and H. Satz, Europ. Phys. J. C 59 (2009) 67.
 26. J. NoronhaHostler, J. Noronha, C. Greiner, Phys. Rev. Lett. 103 (2009) 172302.

27.
P. Castorina, J. Cleymans, D.E. Miller, H. Satz, Eur. Phys. J. C 66 (2010) 207213.
ePrint: arXiv:0906.2289 [hepph]  28. J. Cleymans, N. Bilic, E. Suhonen and D.W. von Oertzen, Phys. Lett. B 311 (1993) 266272.
 29. L.D. Landau, E.M. Lifshitz (1987). Fluid Mechanics. Vol. 6 (2nd ed.). ButterworthHeinemann. ISBN 9780080339337.
 30. S. Borsanyi et al. WuppertalBudapest Collaboration, JHEP 1011 (2010) 077;
 31. W. Söldner (for the HotQCD collaboration) [arXiv:1012.4484] [heplat]].
 32. P. BraunMunzinger and J. Stachel, Nucl. Phys. A 606 (1996) 320; D. Prorok and L. Turko, arXiv:hepph/0101220.