Update of the Hagedorn mass spectrum
Abstract
We present an update of the Hagedorn hypothesis of the exponential growth of the number of hadronic resonances with mass. We use the newest available experimental data for the nonstrange mesons and baryons, as well as fill in some missing states according to the observation that the highlying states form chiral multiplets. The results show, especially for the case of the mesons, that the Hagedorn growth continues with the increasing mass, with the new states lining up along the exponential growth.
pacs:
25.14.20.c, 14.40.n, 12.40Yx, 12.40NnThe Hagedorn hypothesis h1 ; h2 ; h3 of the exponential growth of the number of hadronic resonances with mass is one of the most fundamental issues in particle physics. The formula for the asymptotic dependence of the density of hadronic states on mass, namely
(1) 
where denotes a slowly varying function and is the Hagedorn temperature, has gained a lot of attention due to its appealing simplicity, fundamental character, support from the experimental data and theoretical approaches, as well as because of its relevance to the phenomenology of particle production, in particular concerning the possible phase transition from the hadron gas to the quarkgluon plasma CabibboParisi ; Chernavskaya:1994zj ; Blanchard:2004du .
The purpose of this note is to present an update of the experimental verification of Eq. (1). We supplement the data published in the Particle Data Tables PDG with the new experimental information b1 ; b5 , as well as add theoretically predicted new states belonging to chiral multiplets g1 ; cg1 ; g2 ; g3 ; g4 . Although the appearance of some of these states has not been verified experimentally yet, their existence follows from the recent theoretical findings that the highlying particle spectrum essentially has the features of restored chiral symmetry g1 .
The results are shown for the nonstrange mesons and baryons, where the new data is available. The paper has no pretence of presenting new models or ideas; nevertheless, due to the fundamental nature of the problem related to basic ideas behind the formation of bound states and resonances in particle physics, the results of our simple compilation should be of interest for the community. We include the new experimental results and show that the new data are important in the verification of the Eq. (1). The new results extend significantly, at least for the mesons, the range of fiducial range of the Hagedorn hypothesis. While with the data listed in the 1998 edition of the Particle Data Tables PDG used in twoT ; Meson2000 ; Bled the exponential growth for nonstrange mesons could be observed up to the masses of about 1.8 GeV, now it continues higher up, till about 2.3 GeV.
We start with a very brief reminder of the history of the Hagedorn hypothesis (for much more complete historical presentations we refer the reader to Hagedorn’s original lecture h3 and to a tribute article by Ericson and Rafelski Ericson ). Equation (1) was originally proposed to explain the spectra in the  and  scattering h1 ; HagRan . Later, it was obtained from the statistical bootstrap models h2 ; Frautschi ; Nahm ; Yellin . Subsequently, it gained a convincing support from the dual string models Huang ; Cudell ; Dienes ; Jacob . It is worthwhile to recall that in the 1960s, when the original Hagedorn idea was formed, very few hadronic states were known, up to the mass of the isobar. More and more states have been accumulated over the years, thus much more systematic studies were possible, such as for instance the analysis of Ref. Rafelski and of Ref. twoT , where two of us (WB,WF) pointed out the different growth rate of mesons and baryons, as well as demonstrated the universality of the Hagedorn temperatures with strangeness. The faster growth of the baryon spectrum was also noted in Ref. Freund .
The Hagedorn concept of the limiting temperature appears in many different contexts, e.g., in the studies of nonlinear Regge trajectories Brisudova:1998wq ; Burakovsky:1998ct ; Brisudova:1999ut , strings Atick:1988si ; Sundborg:1999ue ; Gubser:2000mf , dbranes Abel:2000jq , and cosmology Leblanc:1988eq . Moreover, a complete treatment of hadronic resonances, as suggested by Hagedorn already in the 1960s, is the basic ingredient of the successful models of hadron production in heavyion collisions at the RHIC energies Torrieri:2004zz ; SA .
After many dormant years with essentially no incoming new data, a recent systematic partial wave analysis of the annihilation at LEAR has revealed a lot of new meson states in the mass range 1.8  2.4 GeV b1 ; b5 . These new experimental results turned out to be in line with the proposed idea that the spontaneously broken chiral symmetry of QCD should be effectively restored in the highly excited hadrons (one terms this phenomenon as the chiral symmetry restoration of the second kind) g1 ; cg1 ; g2 . This kind of chiral symmetry restoration implies that the excited hadron states fill out multiplets of the chiral group. Indeed, the newly discovered meson states b1 ; b5 turned out to systematically fall into almost degenerate chiral multiplets with a few missing states yet to be discovered g3 ; g4 .
In this note we extend the analysis of Refs. twoT ; Meson2000 and include all mesons listed in Refs. g3 ; g4 . We stress that in addition to the experimental states which have been reported in Refs. b1 ; b5 we add a few still missing states (marked with the question signs in Refs. g3 ; g4 ) and reconstruct their energies according to the known energies of their chiral partners. We consider only the , , , and states, where the experimental information is rather complete.
In addition to these states we also consider the states with hidden strangeness, i.e. composed of the pairs. These states could not be seen in . Hence here our procedure is somewhat more speculative. We assume that any isosinglet , which is experimentally seen in , should be accompanied by an state with the mass approximately 200 MeV higher. Hence, given the complete amount of the states listed in Refs. g3 ; g4 we add the corresponding states.
Rather than comparing the density of states itself to the data, it is customary to form the accumulated number of states of mass lower than ,
(2) 
where is the spinisospin degeneracy of the th state, and is its mass. The theoretical counterpart of Eq. (2) is
(3) 
Working with rather than conveniently avoids the need of building histograms, but clearly it is a purely technical issue and the conclusions drawn below remain unchanged if one decides to work with itself.
The results of our compilation for nonstrange mesons are shown in Fig. 1. The lines with steps correspond to Eq. (2). Above GeV the curves split into three, with the lower one representing the compilation of Ref. twoT based of the 1998 review of PDG PDG . The middle curve contains in addition the states listed in refs. g3 ; g4 , while the top curve includes also the hiddenstrangeness states, as described above. It is clear from Fig. 1 that the included new states nicely line up along the exponential growth, thus extending the range of the Hagedorn hypothesis seen in the data. We also note that adding up the hiddenstrangeness states has a much smaller effect than adding the states of Refs. g3 ; g4 , which is simply due to a lower isospin degeneracy factor.
The thin solid lines in Fig. 1 show the results of the exponential fits with in Eq. (1, 3), which is the simplest choice. While for the old data the leastsquares method yields , with the states of Ref. g3 ; g4 included we obtain , and with the additional states we get . The fit was made up to GeV with the old data and up to GeV with the new data. The higher value for obtained with the new data corresponds to the lower slope in Fig. 1. Certainly, the values of the fitted parameters should be taken with care, since they also reflect the assumed fitting range in . It should also be noted, that adding more states in the range around 2 GeV, when experimentally found, would increase the slope, thus decreasing .
In this place the reader may be a bit surprized with the quoted high values of , much higher than the typically cited values in the range of 200 MeV. The issue, as discussed in detail in Ref. Bled , has to do with the choice of the “slowlyvarying” function . The point is that typical model predictions for this function are not so slowly varying in the range of data. For instance, with the original Hagedorn choice we get much lower values for . With this form we obtain for the bottom to top curves of Fig. 1 the following values: , , and MeV, respectively. The choice of the fitting range in is as stated above.
Now we pass to the case of the nonstrange baryons. With the help of identification of states in chiral multiplets cg1 , we add the missing states (marked with the question signs in cg1 ) on top of the states from PDG PDG used in Ref. twoT . In this way we fill the chiral multiplets. The results of this procedure are shown in Fig. 2. We note that the effect of including these baryon states is less important than in the analogous procedure for the mesons. In the present case we do not show the fit to the exponential formula, since it is difficult to lineup the results along one straight line in a sufficiently broad range of . Actually, with the present data one may see a straight line up to about GeV, and possibly another straight line, with a lower slope, above. However, this may be an artifact of missing data in the highmass range.
Indeed, the parity doublets in and can be associated with the representations for the nucleon spectrum, the multiplets in the spectrum, and with the representations which combine the doublets in the nucleon and delta spectra. If all these multiplets are realized in nature, then the number of the states in the region above 2 GeV should be much larger than given in PDG. Unfortunately, this region has never been systematically explored in experiments.
We now come back to the meson case of Fig. 1, and wish to present the data in a somewhat different manner. The problem of the presentation in the scale, as in Fig. 1, is that the lowmass states are sparse, while the highmass states are jammed up. For that reason we now look at the ratio of the experimental function (2) to the model function (3), with the choice and the parameters at the fitted values quoted in the text. The ratio is plotted as a function of the accumulated number of model states, . If the Hagedorn hypothesis complies to the data, this ratio should be equal to unity. Indeed, this is so with the new data up to about 900 states, while with the old data it was true only up to about 250 states. Again, we see vividly that the inclusion of the new states significantly increases the range of validity, or verification, of Eq. (1).
Finally, for the reader’s convenience we overlay our results for the mesons and baryons in one plot of Fig. 4. As pointed out in Ref. twoT , up to GeV we note a faster growth rate for baryons than for mesons, which means two distinct Hagedorn temperatures for mesons and baryons. This is a prediction of dual string models, see Ref. Bled for a discussion. For higher masses this feature is no longer obvious, with more experimental information needed to clarify the issue.
In conclusion, we list our main observations:

The newlyobserved meson states lead to a continued exponential growth on the number of states with mass, in accordance to the Hagedorn hypothesis, which now reaches up to masses of about 2.3 GeV.

For the baryons the situation is less clear, with the exponential growth seen up to about 2 GeV.

The inclusion of the missing states based on the identification of chiral multiplets helps to comply to the Hagedorn hypothesis at high masses.

Certainly, more experimental data in the highmass range are highly desired to investigate further and with greater detail the hadron spectroscopy.
Acknowledgements.
The authors wish to thank the organizers of the XLIV Cracow School of Theoretical Physics in Zakopane and the MiniWorkshop Bled 2004: Quark Dynamics, where this research was completed. WB and WF acknowledge the support of the Polish State Committee of Scientific Research, grant 2 P03B 05925. LG acknowledges the support of the FWF project P16823N08 of the Austrian Science Fund.References
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