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The Euclidean dilaton term is $S_{\Phi }=(4\pi {)}^{-1}\int \sqrt{h}\Phi (X)R^{(2)}$. A constant mode ${\Phi }_0$ therefore gives $S_{\Phi }={\Phi }_0\chi$ by the Gauss-Bonnet theorem. Setting the string coupling to $g_s=e^{{\Phi }_0}$, an oriented closed worldsheet of genus $g$ has weightEach canonically normalized external closed-string vertex supplies a further factor $g_s$. Thus the connected $n$-point genus-$g$ contribution carriesThis is the string genus expansion. Four external strings on a sphere give $g_s^2$, as in the displayed tree amplitude.

Use the conventional complex-coordinate measure $d^{2}z=2d(\mathrm{Re}z)d(\mathrm{Im}z)$, with $\partial =({\partial }_x-i{\partial }_y)/2$. The action then has the standard real-coordinate normalization $(4\pi {\alpha }^{\prime }{)}^{-1}\int (\nabla X{)}^2{d}^{2}x$. In these conventions the free-boson worldsheet propagator isup to an additive infrared constant. Indeed the quadratic kernel after integration by parts is $-(\pi {\alpha }^{\prime }{)}^{-1}\partial \bar{\partial }$, and $\partial \bar{\partial }\log |z-w{|}^2=2\pi {\delta }^{(2)}(z-w)$ for this measure. Their product gives the required delta function. A simultaneous change of the coordinate-measure convention and action coefficient leaves this physical normalization unchanged.

Separate the constant mode $x_0$ of $X$. Its integral supplies $(2\pi {)}^D{\delta }^{(D)}({\sum }_i{p}_i)$, so four-momentum conservation is essential, even when that delta function is suppressed in the amplitude notation. Treat the exponential vertices as normal ordered: self-contractions are then absent. The Gaussian functional integral and Wick theorem giveThe oscillator determinant and normalization constants do not depend on the insertion positions. Substitution therefore gives the requested Koba-Nielsen factor under the four insertion integrals, with the sphere's residual conformal-group volume divided out.

For the Möbius transformation $z^{\prime }=(az+b)/(cz+d)$, the unit determinant condition givesThe distance-product factor attached to insertion $i$ is $|c{z}_i+d{|}^{-{\alpha }^{\prime }{\sum }_{j\ne i}{p}_i\cdot p_j}$. Momentum conservation changes its exponent to ${\alpha }^{\prime }{p}_i^2$. Thus the total factor at each insertion, including the measure, isOn the tachyon mass shell $p_i^2=4/{\alpha }^{\prime }$ it is one. This proves the Möbius invariance of on-shell closed-string vertex integrals. The correlator is covariant; the correlator times all four measures is invariant. This is the invariant integrand needed to quotient by the residual $SL(2,\mathbb{C})$ transformations.

For the massless Type II superstring theory amplitude, use $s=-(p_1+p_2{)}^2$ in Lorentzian target signature and $s+t+u=0$. At generic fixed $t$, the numerator factor $\Gamma (-{\alpha }^{\prime }s/4)$ has simple poles atAt these points $\Gamma (1+{\alpha }^{\prime }s/4)=\Gamma (1+n)=n!$ is finite and nonzero. The $t,u$ factors are also finite and nonzero for generic $t$, so the poles persist; isolated residue zeros at special kinematics do not remove the general mass level. Tree-level factorization interprets a pole at $s=M^2$ as exchange of a state of that mass. The Type II gamma-factor mass poles therefore exhibit a massless level and an infinite tower of massive levels, with equally spaced squared masses $4/{\alpha }^{\prime }$.

There is no negative-$s$ pole in this exchanged tower. In particular, at negative integer values of ${\alpha }^{\prime }s/4$ the reciprocal s-channel denominator can instead have zeros. This is consistent with the absence of the bosonic tachyon in the Type II superstring mass spectrum. The factor alone does not determine degeneracies, spins or every possible uncoupled state; external polarizations and the rest of the amplitude control their residues.