= Bosonic stretched-string tachyon threshold
{title2=$d_0=2\pi\sqrt{\alpha'}$}
For two static parallel <D-branes> separated transversely by $d$, the <open bosonic string mass spectrum> becomes $M^2=d^2/(4\pi^2\alpha'^2)+(N-1)/\alpha'$. The classical stretching contribution is the square of the <string tension> times the separation, while the oscillator vacuum contributes the intercept term. Thus the ground state is a <tachyon> for $d<2\pi\sqrt{\alpha'}$, massless at equality, and massive above it. Its real rest energy, when stable, is $\sqrt{d^2/(4\pi^2\alpha'^2)-1/\alpha'}$; the vacuum correction belongs to energy squared and is not an additive fixed subtraction from the stretching energy. Angled branes change oscillator moding and require a different formula.
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