Matching the Hagedorn mass spectrum with lattice QCD results

Based on recent lattice QCD (LQCD) results obtained at finite temperature, we discuss the modeling of the hadronic phase of QCD in the framework of hadron resonance gas (HRG) with discrete and continuous mass spectra. We focus on fluctuations of conserved charges and show how a common limiting temperature can be used to constrain the Hagedorn exponential mass spectrum in different sectors of quantum number, through a matching of HRG and LQCD. For strange baryons, the extracted spectra are found to be consistent with all known and expected states listed by the Particle Data Group (PDG). The strange-mesonic sector, however, requires additional states in the intermediate mass range beyond that embodied in the database.

Figure 1

Cumulants of the PDG mass spectrum in different sectors of quantum number: (a) all hadrons; (b) mesons and baryons; (c) mesons of different strangeness; (d) baryons of different strangeness. The lines are obtained from the fit of Eqs. (9) and (12) to the PDG data with the parameters listed in Table 1 (see text).

Figure 2

Lattice QCD results of HotQCD [10, 32] and Budapest-Wuppertal Collaboration [9, 12] for different observables in dimensionless units: (a) the thermodynamic pressure; (b) the net-baryon number fluctuations ${\widehat{\chi }}_{\mathrm{BB}}$. Also shown are the HRG results for the discrete PDG mass spectrum (dashed line) and for the effective mass spectrum in Eq. (11), which contains a continuous part to describe the effects of massive resonances (continuous line).

Figure 3

As in Fig. 2, but for baryon-strangeness correlations ${\widehat{\chi }}_{\mathrm{BS}}$ (a) and for strangeness fluctuations ${\widehat{\chi }}_{\mathrm{SS}}$ (b). Also shown are the corresponding results obtained from the least-squares fit to lattice data up to $T\simeq 156\mathrm{MeV}$.

Figure 4

Cumulants of the discrete PDG mass spectrum (black dashed line) and the corresponding fits (red dashed line) for (a) strange baryons and (b) strange mesons. Also shown are the cumulants containing in addition the unconfirmed states (broken-dashed line). Continuous lines are obtained by matching the LQCD results to the continuous mass spectra through Eq. (15), assuming that the missing strange baryons come solely from the $\left|S\right|=1$ sector [scheme (I)] or $\left|S\right|=2$ sector [scheme $\left(\mathrm{II}\right)]$ (see text).