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

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 \sigma_{eff}(T) is approximately half its value at T=0, and that as we increase T beyond Tc, while remaining in the confined phase, \sigma_{eff}(T) continues to decrease. We extrapolate \sigma_{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.