 Neumann-Dirichlet string zero-point energy

Each of $d$ physical transverse bosons with a Neumann-Dirichlet open-string boundary condition contributes $\frac{1}{2}{\sum }_{r>0}r=1/48$ by the half-integer zeta-regularized mode sum. Thus $L_0^{\perp }=N+d/48$ and the convention $L_0={\alpha }^{\prime }{p}^2+N-a$ 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.