Eur. Phys. J. C (2025) 85: 1358
https://doi.org/10.1140/epjc/s10052-025-15080-2

Regular Article - Theoretical Physics

On the origin of the $N_c^1$ scaling in the confined but chirally symmetric phase at high T

Institute of Physics, University of Graz, 8010, Graz, Austria

^a leonid.glozman@uni-graz.at

Received: 11 August 2025
Accepted: 14 November 2025
Published online: 25 November 2025

Abstract

There is lattice evidence that the QCD matter above the chiral restoration temperature $T_{\mathit{ch}}$ and below the deconfinement temperature $T_d$, called stringy fluid, is characterized by approximate chiral spin symmetry, which is a symmetry of confinement in QCD with light quarks. The energy density, pressure and entropy density in the stringy fluid scale as $N_c^1$, which is in contrast to the $N_c^0$ scaling in the hadron gas and to the $N_c^2$ scaling in the quark-gluon plasma. Here we clarify the origin of the $N_c^1$ scaling. We employ a solvable field-theoretical large $N_c$ chirally symmetric and confining model. In vacuum the confining potential induces a spontaneous breaking of chiral symmetry. The mesons are spatially localized states of quarks and antiquarks. Still in the confining regime the system undergoes the chiral restoration phase transition at $T_{\mathit{ch}}$ because of Paili blocking of the quark levels required for the existence of the quark condensate, by the thermal excitation of quarks and antiquarks. The same Paili blocking leads to a delocalization of the color singlet low-spin meson-like states that become infinitely large in the chiral limit. Consequently the stringy fluid represents a very dense medium of the overlapping huge color-singlet low-spin quark-antiquark systems. The Bethe–Salpeter equation that determines the rest-frame excitation energies of the color-singlet quark-antiquark system is $N_c$-independent both in vacuum and in the medium in the confining regime. The excitation energy of the quark-antiquark color-singlet systems scales as $N_c^0$, i.e. as meson mass in vacuum. The $N_c^1$ scaling of the energy density in the stringy fluid is provided by the fluctuations of the color-singlet quark-antiquark systems.