= Neumann-Dirichlet string zero-point energy
{c}
{title2=$E_0=d/48$}
Each of $d$ physical transverse <bosons> with a <Neumann-Dirichlet open-string boundary condition> contributes $\tfrac12\sum_{r>0}r=1/48$ by the <half-integer zeta-regularized mode sum>. Thus $L_0^\perp=N+d/48$ and the convention $L_0=\alpha'p^2+N-a$ has $a=-d/48$. For $d=24$, the vacuum shift is $+1/2$. The ordinary integer-moded open-string shift is instead $-d/24$; these endpoint sectors have different normal-ordering constants.
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