Hadron spectra and thermodynamics for all quark flavors from a universal Hagedorn temperature

Abstract

We show that hadrons in QCD follow a spectrum determined by string dynamics characterized by a universal Hagedorn temperature linked to the string tension. While this behavior was recently established for light hadrons and glueballs, we demonstrate that the same dynamics describes the heavy-flavor sector. After separating the current quark masses, the resulting spectrum reproduces lattice QCD thermodynamics of charmed hadrons and the observed spectra of hadrons across quark flavors without additional parameters. These results reflect the universal confining dynamics of QCD through the string tension.

Figures (3)

• Figure 1: Partial pressures of open-charm mesons and charmed baryons, together with the ratios $\hat{\chi}_{13}^{BC}/\hat{\chi}_4^C$, $\hat{\chi}_{13}^{SC}/\hat{\chi}_4^C$, and $\hat{\chi}_{22}^{SC}/\hat{\chi}_4^C$ (top to bottom) as functions of temperature. Results obtained from the Hagedorn spectrum without quark-mass subtraction (solid black lines) and with subtraction (red dashed lines) are compared to lattice QCD calculations Kaczmarek:2025dqt (points) and to the hadron resonance gas (HRG) model with PDG input (blue dash-dotted lines) Kaczmarek:2025dqt. For the pressures, lattice data correspond to continuum estimates Kaczmarek:2025dqt and for susceptibilities to $N_\tau = 8$ simulations along lines of constant physics Kaczmarek:2025dqtBazavov:2023xzm. Shaded bands reflect the uncertainty of the Hagedorn temperature, $T_H = 323(3)\,$MeV Marczenko:2026yme. The vertical yellow band indicates the chiral crossover temperature $T_c = 156.5 \pm 1.5\,$MeV HotQCD:2018pds.
• Figure 2: Cumulative spectra of heavy-flavor hadrons as functions of the excitation energy $E,$ as in Eq. \ref{['eq:Ex']}. Shown are open-flavor mesons (left panel), singly-heavy baryons (middle panel), and hidden-flavor mesons (right panel). Thick dotted and dash-dotted lines denote experimentally established mesons or baryons with three- and four-star rating in the Particle Data Group (PDG) Kaczmarek:2025dqtParticleDataGroup:2024cfk. Thin dotted and dash-dotted lines show spectra predicted by the quark model (QM) Ebert:2009uaEbert:2011kk, shown only up to the mass where calculations are available in all channels. Red dashed curves represent the universal Hagedorn spectrum with $T_H=323(3)\,$MeV Marczenko:2026yme, constructed using the same excitation-energy threshold determined in the charm sector without additional adjustment (see text for details). The shaded bands reflect the uncertainty in $T_H$.
• Figure 3: PDG Cumulative spectra across open quark-flavor sectors as a function of the excitation energy $E,$ as introduced in Eq. \ref{['eq:Ex']}. Spectra of mesons and baryons include: $|S|=1$ strange states (Light), $|C|=1$ non-strange (Charm), and $|B'|=1$ non-strange (Bottom) states. The spectra of baryons do not account for antiparticles. The light-hadron distributions do not include the Goldstone bosons.