Thermal winding instability of a closed bosonic string ($m_{\mathrm{th}}^2={\beta }^2/(4{\pi }^2{\alpha }^{\prime 2})-4/{\alpha }^{\prime }$)

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.