In search of a Hagedorn transition in SU(N) lattice gauge theories at large-N

TL;DR

This paper investigates a Hagedorn transition in SU($N$) lattice gauge theories at large $N$ by tracking the temperature dependence of the lightest timelike flux-loop mass, encapsulated in the effective string tension $\sigma_{ ext{eff}}(T)$, in metastable confinement above the deconfinement temperature $T_c$ for $N=8,10,12$. Using Wilson-action lattices and Polyakov-loop correlators, the authors extract $m$ and fit $rac{\sigma_{ ext{eff}}(T)}{\sigma}$ to a form $rac{A T}{T_c}igg(rac{T_H}{T_c}-rac{T}{T_c}igg)^ u$ to obtain the Hagedorn temperature ratio $T_H/T_c$ and the exponent $ u$, testing both 3D XY and mean-field universality (and sometimes leaving $ u$ free). They find $T_H/T_c$ around $1.10$–$1.12$, with a mild preference for the 3D XY exponent in the SU(12) data, while MF remains compatible within uncertainties; overall, the results suggest a near-constant $T_H$ relative to $T_c$ at large $N$, though finite-volume and tunneling effects limit a definitive universality conclusion. The work provides nonperturbative insight into the stringy structure of confinement and informs effective Polyakov-loop models and future large-$N$ investigations.

Abstract

We investigate on the lattice the metastable confined phase above Tc in SU(N) gauge theories, for N=8,10, and 12. In particular we focus on the decrease with the temperature of the mass of the lightest state that couples to Polyakov loops. We find that at T=Tc the corresponding effective string tension σ_{eff}(T) is approximately half its value at T=0, and that as we increase T beyond Tc, while remaining in the confined phase, σ_{eff}(T) continues to decrease. We extrapolate σ_{eff}(T) to even higher temperatures, and interpret the temperature where it vanishes as the Hagedorn temperature T_H. For SU(12) we find that T_H/Tc=1.116(9), when we use the exponent of the three-dimensional XY model for the extrapolation, which seems to be slightly preferred over a mean-field exponent by our data.

Figures (12)

• Figure 1: Effective string tension for $SU(8)$ obtained in method A in units of the zero temperature string tension.
• Figure 2: Effective string tension for $SU(10)$ obtained in method A in units of the zero temperature string tension.
• Figure 3: Effective string tension for $SU(10)$ obtained in method B on a $12^3\times 5$ lattice, in units of the zero temperature string tension.
• Figure 4: Effective string tension for $SU(10)$ obtained in method B on a $14^3\times 5$ lattice, in units of the zero temperature string tension.
• Figure 5: Effective string tension for $SU(12)$ obtained in method B in units of the zero temperature string tension.