# Mass formula for open string with mixed boundary conditions

I want to give an expression for the mass formula of an open string with has Neumannn condtion in $$m$$ directions and Dirichlet in $$n$$ directions $$X^{i}(\sigma ^{1}=0)=x_{0}^{i}, X^{i}(\sigma ^{1}=\pi)=x_{1}^{i}$$. If I use the condition $$L_{0}=0$$ in order to obtain the mass taking into account eq (4.18):

I obtain this for the mass squared:

Where is there the dependence with the concrete boundary conditions of my example?

$$$$X^\mu \left(\sigma,\tau\right) =x_0^\mu+\frac{1}{l}\left(x_1^\mu-x_0^\mu\right)\sigma+\sqrt{2\alpha '}\sum\limits_{n\neq 0}\frac{1}{n} \alpha_n^\mu e^{-\frac{i\pi}{l}n\tau} \sin \left(\frac{\pi n \sigma}{l}\right)$$$$ such that when writing down the expansion $$$$\partial_\pm X^\mu= \pm \frac{\pi}{l}\sqrt{\frac{\alpha '}{2}}\sum_\limits{n=-\infty}^{n=\infty}\alpha_n^\mu e^{\frac{-i\pi n}{l}(\tau\pm\sigma)}$$$$ you have that $$$$\alpha_0^\mu = \frac{1}{\sqrt{2\alpha '}}\frac{1}{\pi} \left(x_1^\mu-x_0^\mu\right).$$$$ Therefore you get an additional contribution in the referenced calculation due to tension of the string stretching from one brane to the other of the form
$$$$\frac{1}{2}a_0^2|_{\text{DD-direction}} = \frac{1}{(\alpha')^2} \sum_i\left(\frac{x_1^i-x_0^i}{2\pi}\right)^2 =\left(T\Delta x^i\right)^2.$$$$