Improved coarse-graining methods for two dimensional tensor networks including fermions

String-based model with Hagedorn temperature of 𝑇 𝐻 ∼ 300 MeV describes the spectrum of mesons and glueballs

We consider the thermodynamics of a color-confined phase of quantum chromodynamics (QCD) and pure gauge theory within a string-inspired corresponding to a physical spatial dimension, 𝑑 = 3. We show that the physical mass spectrum of massive mesons—in both the strange and nonstrange 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 = $\sqrt3⁢𝜎/2⁢𝜋$, expressed by the string tension 𝜎. This 𝑇 𝐻 is the value appropriate for 𝑑 = 3 spatial dimensions and is of the order of 𝑇 𝐻 ∼ 300 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.