Hagedorn temperature ($T_H={\beta }_H^{-1}$)

If a density of states has high-energy growth proportional to a power of energy times $e^{{\beta }_{H}E}$, its canonical Boltzmann sum fails for $\beta <{\beta }_H$. The Hagedorn temperature is the corresponding threshold. The open bosonic string mass spectrum and large-level degeneracy of an open bosonic string give ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$ in units with Boltzmann's constant, the speed of light and Planck's constant set to one. Polynomial prefactors affect behavior at the threshold itself; the growth alone does not distinguish a limiting temperature from a phase transition.

Euclidean finite temperature compactifies time to a circle of radius $R_{\beta }=\beta /(2\pi )$. The string oscillator ground states with zero momentum number and winding $w=\pm 1$ have the displayed mass from the bosonic string mass spectrum. They become massless at ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$ and tachyonic at higher temperature. This thermal winding mode diagnoses breakdown of the perturbative low-temperature string gas at the Hagedorn temperature. The endpoint and order of a transition require interactions and a suitable stable vacuum; the ordinary zero-winding bosonic tachyon is already unstable independently of this thermal threshold.

For a critical closed bosonic string in 25 noncompact spatial dimensions, the smoothed string oscillator rest-mass density behaves as $\rho (M)\sim C{M}^{-26}{e}^{{\beta }_{H}M}$, with ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$. At large $M$, the centre-of-mass momentum integral of $e^{-\beta \sqrt{M^2+p^2}}$ is asymptotic to $(2\pi M/\beta {)}^{25/2}{e}^{-\beta M}$, up to the convention for phase-space measure. Thus the high-mass tail of the one-string canonical partition function is proportional to ${\int }_{M_0}^{\infty }{M}^{-27/2}{e}^{-(\beta -{\beta }_H)M}dM$. This converges at $\beta ={\beta }_H$ but fails for $\beta <{\beta }_H$. Expanding about the endpoint gives twelve finite derivatives and a nonanalytic term proportional to $(\beta -{\beta }_H{)}^{25/2}$; the thirteenth derivative diverges. Consequently exponential state growth does not imply divergence of the partition function at the threshold itself. The discussion assumes the separate infrared instability of the bosonic tachyon has been regulated and does not claim a stable bosonic thermal vacuum.