Bosonic stretched-string tachyon threshold ($d_0=2\pi \sqrt{{\alpha }^{\prime }}$)

For two static parallel D-branes separated transversely by $d$, the open bosonic string mass spectrum becomes $M^2=d^2/(4{\pi }^2{\alpha }^{\prime 2})+(N-1)/{\alpha }^{\prime }$. 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 $d<2\pi \sqrt{{\alpha }^{\prime }}$, massless at equality, and massive above it. Its real rest energy, when stable, is $\sqrt{d^2/(4{\pi }^2{\alpha }^{\prime 2})-1/{\alpha }^{\prime }}$; 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.