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Split $X^{\mu }=x_0^{\mu }+X^{\prime \mu }$. Integrating the constant mode gives momentum conservation,For the nonzero modes, Wick theorem and the propagator giveThus, up to source-independent numerical normalization, the four-point amplitude isThe product is the Koba-Nielsen factor.

In the Polyakov path integral, fix worldsheet diffeomorphism and Weyl symmetry to conformal gauge. The Faddeev-Popov determinant supplies the worldsheet ghosts, and anomaly cancellation selects $D=26$. Insert $m$ integrated closed-string tachyon vertex operators $e^{i{p}_i\cdot X}$ on the sphere. The zero mode of $X$ gives ${\delta }^{(26)}({\sum }_i{p}_i)$, while Gaussian contractions give the Koba-Nielsen factor ${\prod }_{j<l}|z_j-z_l{|}^{{\alpha }^{\prime }{p}_j\cdot p_l}$. Dividing by the conformal Killing group $SL(2,\mathbb{C})/{\mathbb{Z}}_2$ and using the sphere power of the string coupling gives the displayed amplitude with $g_s^{m-2}$.

Fix $z_1=0$, $z_2=1$, and $z_4=\infty$, and write $z_3=z$. Its scattering equation isso in the hard limitEvaluating the Koba-Nielsen factor at this saddle and using $s+t+u=O(m_{\mathrm{tachyon}}^2)$ yields, with the logarithms defined by analytic continuation,The omitted terms grow more slowly than the displayed fixed-angle $s\log s$ terms.

The bosonic closed-string tachyon vertex is the normal-ordered exponential $:e^{ip\cdot X}:$ integrated over its worldsheet position. Pairwise contractions of such exponentials generate the Koba-Nielsen factor.