= Solution
Take an oriented <closed string> in flat, critical <bosonic string theory>, with $\alpha'=1/(2\pi T)$ and a mostly-plus target <Minkowski metric>. After continuation of the <worldsheet> to Euclidean signature, the <Polyakov path integral> sums over embeddings $X$ and <worldsheet metrics>, divided by <worldsheet diffeomorphisms> and <Weyl transformations>. In a flat target its kinetic <action> is
$$
S_E=\frac1{4\pi\alpha'}\int_\Sigma d^2z\,\sqrt g\,g^{ab}
\partial_aX^m\partial_bX_m.
$$
Target-time continuation or analytic continuation of external momenta defines the Lorentzian scattering amplitude; a naive real Euclidean Gaussian for timelike $X^0$ would not be convergent.
An external <tachyon> is represented by the <tachyon vertex operator> $V_p(z,\bar z)=:\!e^{ip\cdot X}\!:$, with <conformal weights> $h=\bar h=\alpha'p^2/4$. The physical integrated vertex has $(h,\bar h)=(1,1)$, so $p^2=4/\alpha'$. Schematically its tree amplitude is
$$
\mathcal A_N\ \propto\ \int\frac{\mathcal DX\,\mathcal Dg}
{\operatorname{Vol}(\mathrm{Diff}\times\mathrm{Weyl})}
\,e^{-S_E}\prod_{a=1}^N\int_\Sigma d^2z_a\,\sqrt g\,V_{p_a}(z_a,\bar z_a).
$$
The <Faddeev-Popov determinant> from <conformal gauge> is represented by the <worldsheet ghost fields>. At tree level the <worldsheet> is a <Riemann sphere>, whose unpunctured <complex structure> has no moduli. Its residual conformal automorphisms are the <Möbius transformations>, $PSL(2,\mathbb C)$. Fix three insertion points, accompanying their unintegrated vertices by the required $c\widetilde c$ ghost factors. The remaining $N-3$ complex insertion positions are integrated over the sphere; equivalently one integrates all positions and divides by the residual conformal group.
The embedding fields are free, with
$$
\langle X^m(z)X^n(w)\rangle
=-\frac{\alpha'}2\eta^{mn}\log|z-w|^2.
$$
Their zero-mode integral gives <momentum conservation>, and their nonzero-mode <Gaussian integral> gives the <Koba-Nielsen factor>
$$
(2\pi)^D\delta^D\!\left(\sum_a p_a\right)
\prod_{a<b}|z_a-z_b|^{\alpha'p_a\cdot p_b}.
$$
This explains the <sphere tachyon position integral> without needing its evaluation. For four <tachyons> the resulting <Virasoro–Shapiro amplitude> displays the exchanged string spectrum directly.
Use all-incoming external momenta and introduce $u=-(p_1+p_4)^2/(8\pi T)$ alongside the two printed dimensionless <Mandelstam variables>. Since $p_a^2=8\pi T$ and $\sum_a p_a=0$, one obtains $s+t+u=-4$. The physical center-of-mass energy squared in the $s$ channel is $s_{\mathrm{phys}}=8\pi T s=4s/\alpha'$. The $t$ channel measures the analogous crossed <momentum> transfer, with sign determined by the mostly-plus convention. Rewriting the Gamma factors in a symmetric form gives
$$
A(s,t)=\prod_{x=s,t,u}\frac{\Gamma(-1-x)}{\Gamma(2+x)}.
$$
The <Gamma function> <poles> imply, at generic fixed values of the other invariant,
$$
\boxed{s=n-1\quad\hbox{or}\quad t=n-1,\qquad n=0,1,2,\ldots.}
$$
The denominator Gamma factors can remove <residues> at special intersecting channel kinematics; the statement concerns a generic single-channel limit. An $s$-channel <pole> occurs when the intermediate <momentum> $p_1+p_2$ satisfies the <mass-shell condition> for a closed-string state:
$$
M_n^2=8\pi T(n-1).
$$
The <pole> at $s=-1$ exchanges the ground-state <tachyon>, the <pole> at $s=0$ exchanges massless states, and the positive integer <poles> exchange the infinite massive tower. The $t$-channel interpretation is the crossed version. Factorization means that each <residue> is a sum of products of couplings to intermediate physical states that couple to the chosen external particles. It need not expose every representation at that mass.
For example, the <Gamma function recurrence> and <Gamma function residue at a nonpositive integer> give the dimensionless <Virasoro–Shapiro amplitude pole residue>
$$
\operatorname*{Res}_{s=n-1}A(s,t)
=-\frac{[(t+2)_n]^2}{(n!)^2},\qquad
(t+2)_n=\prod_{j=0}^{n-1}(t+2+j).
$$
The <residue> polynomial has degree $2n$, consistent with maximum spin $2n$ in the exchanged level. The amplitude also has the corresponding $u$-channel <poles> by <crossing symmetry>.
The massless fields can be treated as target backgrounds rather than separate asymptotic insertions. Write the target metric as $G_{mn}=\eta_{mn}+h_{mn}$, introduce a <Kalb–Ramond field> $B_{mn}$, and a <dilaton> $\Phi$. In conventional Euclidean signs their <string nonlinear sigma model> <action> is
$$
S_E[X;G,B,\Phi]=\frac1{4\pi\alpha'}\int d^2z\left[
\sqrt g\,g^{ab}G_{mn}(X)\partial_aX^m\partial_bX^n
+i\epsilon^{ab}B_{mn}(X)\partial_aX^m\partial_bX^n\right]
+\frac1{4\pi}\int d^2z\,\sqrt g\,\Phi(X)R^{(2)}.
$$
Here $\epsilon^{ab}$ is the antisymmetric tensor density. The three backgrounds correspond to the <graviton>, antisymmetric tensor and scalar states at closed-string level one. Expanding the vacuum functional in $h,B,\Phi$, then Fourier expanding the backgrounds, produces exactly their integrated <string vertex operators>. Its <functional derivatives> therefore generate the amplitudes with massless external strings. The connected vacuum functional organizes connected amplitudes; the spacetime effective <action> organizes the corresponding vertices after treating massless propagation consistently.
At momenta small compared with $1/\sqrt{\alpha'}$, massive string propagators can be expanded in powers of momenta over their masses. The analytic part of the amplitudes consequently determines local higher-derivative interactions, ordered by powers of $\alpha'$. Massless exchange <poles> are retained through propagation of the massless fields, rather than expanded into local contact terms. Up to field redefinitions, the leading massless-sector <action> in the <string-frame metric> is
$$
\boxed{S_{\mathrm{eff}}^{\mathrm{tree}}
=\frac1{2\kappa_0^2}\int d^Dx\,\sqrt{-G}\,e^{-2\Phi}
\left[R+4(\nabla\Phi)^2-\frac1{12}H_{mnp}H^{mnp}+O(\alpha')\right],\qquad H=dB.}
$$
The terms denoted $O(\alpha')$ contain additional derivatives, including curvature-squared terms in the bosonic theory. The expansion concerns the massless sector around the perturbative bosonic background; the <tachyon> instability remains and is not cured by omitting its field from this displayed <action>. Thus this is a formal perturbative effective description, not a claim of a stable bosonic vacuum.
Finally split the <dilaton> into a constant and its variation, $\Phi=\Phi_0+\phi$. By the <Gauss-Bonnet theorem>, its <dilaton Euler-characteristic weighting> on a connected closed oriented surface follows from
$$
S_{\Phi_0}=\frac{\Phi_0}{4\pi}\int_\Sigma\sqrt g\,R^{(2)}
=\Phi_0\chi(\Sigma)=\Phi_0(2-2g).
$$
Define the <string coupling> by \b[$g_s=e^{\Phi_0}$]. A genus-$g$ path integral is weighted by
$$
\boxed{e^{-S_{\Phi_0}}=g_s^{2g-2}.}
$$
The sphere carries $g_s^{-2}$, the torus carries $g_s^0$, and each extra handle adds $g_s^2$. At higher genus one integrates over complex-structure moduli as well as insertion points, with the associated antighost insertions supplying the correct moduli measure. With canonically normalized external vertices an $N$-point genus-$g$ amplitude scales as $g_s^{2g-2+N}$.
Summing connected <worldsheets> of every <genus of a surface> yields the <string-loop effective action expansion>
$$
\boxed{S_{\mathrm{eff}}=
\sum_{g=0}^{\infty}g_s^{2g-2}\,S_g[G,B,\phi;\alpha'].}
$$
Each coefficient has its own low-energy $\alpha'$ expansion. The two parameters have different roles: $\alpha'$ resolves finite string size through higher derivatives, while $g_s^2$ counts additional string loops.
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