String-based model with Hagedorn temperature of $T_H\sim 300~$MeV describes the spectrum of mesons and glueballs

TL;DR

The paper demonstrates that a three-dimensional string-inspired hadron-resonance gas with a universal Hagedorn spectrum, governed by $T_H = \sqrt{3\sigma/2\pi}$ and tied to the string tension, can describe both the exponential growth of meson (open-string) and glueball (closed-string) spectra and reproduce lattice QCD thermodynamics up to the chiral crossover. By constructing explicit closed- and open-string density of states and translating to physical units using $\sqrt{\sigma}$, the authors achieve good agreement with Yang–Mills and $(2+1)$-flavor QCD EoS, providing phenomenological support for a string-based description of confinement and the SQGB phase. The work implies that high-mass hadronic states play a crucial role in thermodynamics at temperatures approaching $T_c$, while preserving a single, string-derived scale across hadron classes. It also outlines future directions for incorporating baryons into the open-string framework and refining high-mass state predictions from lattice studies.

Abstract

We consider the thermodynamics of a color-confined phase of quantum chromodynamics (QCD) and pure gauge theory within a string-inspired model, corresponding to a physical spatial dimension, d = 3. We show that the physical mass spectrum of massive mesons--in both the strange and non-strange sectors separately--is reasonably well described and extended by the exponential mass spectrum of open strings, $ρ(m)$, characterized by a unique Hagedorn temperature, $T_H = \sqrt{3σ/2π}$, expressed by the string tension, $σ$. This $T_H$ is the value appropriate for d = 3 spatial dimensions, and is of order $T_H \sim 300~\rm MeV$ for typical values of the string tension. It is much larger than the values of $T_H$, which have been phenomenologically extracted so far to describe the meson spectrum. Glueball states in pure gauge theory, modeled by closed strings, exhibit a similarly large Hagedorn temperature, highlighting a universal feature of the exponential spectrum. We further analyze the thermodynamic properties of the equation of state at finite temperature and demonstrate that, in the confined phase, the string models agree with lattice QCD results. This lends further support to the recent interpretation of the QCD phase diagram that incorporates strings as relevant degrees of freedom.

Figures (4)

• Figure 1: Cumulative mass spectra of all (top), non-strange (middle), and strange (bottom) mesons in the PDG (black, solid lines). Also shown are spectra that include the prediction from the quark model Loring:2001kyEbert:2009ub (QM) (black, dash-dotted lines). The yellow bands represent the uncertainty in the Hagedorn limiting temperature in the exponential spectrum (see text for details). We note that $f(500)$ and $\kappa^\star_0(700)$ mesons are not included in the discrete spectra due to their ambiguous nature Broniowski:2015ohaFriman:2015zua. The black, dashed lines show spectra obtained for $T_H = 0.2~$GeV.
• Figure 2: Continuum-extrapolated cumulative mass spectra of glueballs from LQCD simulations (black, solid bands). The spectra are taken from Ref. Meyer:2004gx (top panel) and Athenodorou:2020ani (bottom panel) and are depicted in the units of the string tension $\sqrt\sigma$. The bands represent the uncertainties of the continuum-limit extrapolation. The spectra for closed strings are shown as orange, dashed bands. Their uncertainties come from the uncertainties of the continuum-limit extrapolation of the mass of the lightest resonance in the LQCD spectra (see text).
• Figure 3: Trace anomaly $(\epsilon - 3p)/T^4$ (top panel) and energy-density-normalized trace anomaly $1/3-p/\epsilon$ (bottom panel). The SU(3) pure gauge LQCD data on pressure and energy density are taken from Ref. Borsanyi:2012ve. The uncertainty bands in both panels are obtained by propagating the reported error on pressure and energy density. The vertical lines $T_d/\sqrt{\sigma}=0.646$ and $T_H/\sqrt{\sigma}=0.691$ represent the critical deconfinement and Hagedorn temperatures, respectively. Note that the closed strings are shown only up to $T_H$ (see text for details).
• Figure 4: The equation of state at finite temperature and vanishing chemical potential. The LQCD results are taken from Ref. HotQCD:2014kol. Mesons are modeled via the continuous spectrum of open strings (cf. Eq. \ref{['eq:open_spec']}), and baryons are taken as discrete states from the Particle Data Group (PDG) or quark model (QM). The blue, red, and green bands indicate the uncertainty in the string tension (see text). The yellow vertical band marks the estimation of the pseudocritical temperature for the chiral crossover transition $T_c=156.5\pm1.5~$MeV HotQCD:2018pds. The gray, horizontal, doubly-dotted--dashed lines mark the Stefan-Boltzmann limit considering quarks of 3 flavors and gluons. The black, dashed lines show results for open strings with $T_H = 0.2~$GeV.