Two Lectures on the Phase Diagram of QCD

Abstract

The phase diagram of QCD at finite temperature and density is discussed. Large numbers of quark colors, $N_{\rm c} >> 1$, is used to explain generic features of the phase diagram. For temperatures below $ T \le 160$~MeV at zero baryon number density, the three dimensional string model is shown to describe the thermodynamics of QCD, and as well, the integrated spectrum of non-Goldstone mesons and glueballs. The lowest mass state in the spectrum of the open and closed string is treated separately due to the tachyon problem of string theory. This is with no undetermined free parameters. It is argued that there are at least three phases at zero baryon number density characterized by the $N_{\rm c}$ dependence of extensive thermodynamic quantities. It is also argued that the intermediate phase has restored chiral symmetry. At high baryon number density and low temperature, again there are three phases. A Quarkyonic phase, with energy density of order $N_{\rm c}$, is distinguished from its counterpart at low baryon density and temperature by its chiral properties.

Figures (16)

• Figure 1: 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 from Ref. Marczenko:2025nhj.
• Figure 2: 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 $T_{\rm d}/\sqrt{\sigma}=0.646$ and $T_{\rm H}/\sqrt{\sigma}=0.691$ represent the critical deconfinement and Hagedorn temperatures, respectively. Note that the closed strings are shown only up to $T_{\rm H}$ (see text for details). Figure from Ref. Marczenko:2025nhj
• Figure 3: 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 $T_{\rm H} = 0.2~$GeV. Figure from Ref. Marczenko:2025nhj
• 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, 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 $T_{\rm {\color{red}{}}[ch]}=156.5\pm1.5~$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 $T_{\rm H} = 0.2~$GeV. Figure from Ref. Marczenko:2025nhj.
• Figure 5: Ratio of the chiral condensate at finite temperature to that at zero temperature. The lattice data are taken from Ref. Borsanyi:2010bp. The averaged sigma term per quark, $\bar{\sigma} \simeq 30~\mathrm{MeV}$, is obtained from the best fit to the lattice-QCD data and it is varied within $15$ – $60\;\text{MeV}$ to account for uncertainty. The blue curve corresponds to the first term in the square bracket in Eq. \ref{['eq:chiralT2']}, and the orange curve includes the contribution from the second term in the square bracket in Eq. \ref{['eq:chiralT2']}. Figure from Ref. Fujimoto:2025sxx.