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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.

Use a genus-zero Polyakov path integral, target signature $(-,+,\dots ,+)$ and all four momenta incoming. Define the Mandelstam variables by $s=-(p_1+p_2{)}^2$, $t=-(p_1+p_3{)}^2$, $u=-(p_1+p_4{)}^2$. The string mass-shell condition is $p_i^2=4/{\alpha }^{\prime }$ and momentum conservation is ${\sum }_i{p}_i=0$, soThe reduced amplitude omits the overall momentum-conservation Dirac delta distribution and any conventional overall scattering-matrix phase. Interpret each tachyon vertex operator as normal ordered; self-contractions must not be included.

The free-boson worldsheet propagator is $⟨X^a(z)X^b(w)⟩=-({\alpha }^{\prime }/2){\eta }^{ab}\log |z-w{|}^2$. Wick theorem then evaluates the normal-ordered exponential correlator as the Koba-Nielsen factorThe worldsheet zero mode supplies momentum conservation. Four vertices give $g_s^4$ and the sphere contributes $g_s^{-2}$ through the string genus expansion, leaving $g_s^2$.

The sphere's residual Möbius transformations fix three insertion points. In sphere gauge fixing for four string vertices, take $(z_1,z_2,z_3,z_4)=(z,0,1,\infty )$. The worldsheet ghost fields, equivalently the Faddeev-Popov determinant for this residual group, supply $|z_{23}{z}_{24}{z}_{34}{|}^2$. At finite $z_4$, this determinant grows as $|z_4{|}^4$, while the matter correlator decays as $|z_4{|}^{-4}$ by momentum conservation and the tachyon mass shell; their product has a finite limit. Equivalently the weight-$(1,1)$ matter operator at infinity is normalized with $|z_4{|}^4$, while the ghost pair uses the inverse factor. After stripping the fixed-position normalization, onlyremains. Here choose the standard complex-coordinate measure $d^{2}z=idz\wedge d\bar{z}=2dxdy$. With $dxdy$ instead, the reduced normalization must acquire a factor two to give the same requested amplitude. Absolute vertex/sphere normalization is a convention; this choice fixes it consistently with the displayed $2\pi$ prefactor.

Set $a=-1-{\alpha }^{\prime }s/4$, $b=-1-{\alpha }^{\prime }t/4$, $c=-1-{\alpha }^{\prime }u/4$. Then $a+b+c=1$ and the integrand is $|z{|}^{2a-2}|1-z{|}^{2b-2}$. To evaluate the complex beta integral first work where $\mathrm{Re}a,\mathrm{Re}b,\mathrm{Re}c>0$. This ensures convergence near $0$, $1$ and infinity. Set $v=1-a$ and $w=1-b$. Schwinger parameterization givesCompleting the square, the $dxdy$ Gaussian integral is $\pi (\lambda +\rho {)}^{-1}\exp [-\lambda \rho /(\lambda +\rho )]$. Change variables to $q=\lambda +\rho$ and $x=\lambda /q$; their Jacobian determinant is $q$. Integrating $q$ gives $\Gamma (v+w-1)[x(1-x){]}^{-(v+w-1)}$. The remaining integral is the ordinary beta function $B(1-w,1-v)=B(b,a)$. ThusRestoring $d^{2}z=2dxdy$ and substituting the Mandelstam variables proves the Virasoro–Shapiro amplitude:For physical scattering the original position integral generally fails to converge. The formula defines the amplitude by analytic continuation from the convergence domain, with the desired scattering boundary value at real poles; the convergent integral should not be claimed valid for every physical momentum.

The gamma function is a meromorphic function, with simple poles at nonpositive integers and no zeros, while its reciprocal $1/\Gamma$ is an entire function. Consequently it is a meromorphic function of the independent invariants and a crossing-symmetric scattering amplitude: permuting $s,t,u$ leaves it unchanged. Its generic channel poles areand the same tower in $t$ and $u$. They represent the exchanged bosonic string mass spectrum: the tachyon, massless states including the graviton, and an infinite sequence of massive closed-string levels. There are no threshold branch cuts at this tree order.

A precise check of factorization is the Virasoro–Shapiro amplitude pole residue. Near $a=-n$, set $c=1+n-b$ in the nonsingular factor. The gamma function recurrence givesCombining this with $\Gamma (a)\sim (-1{)}^n/[n!(a+n)]$ showsThe residue is a degree-$2n$ polynomial in $t$, consistent with exchange up to spin $2n$ and scattering-amplitude factorization. At exceptional kinematics reciprocal Gamma zeros can remove apparent poles. In particular, when both $a$ and $b$ approach nonpositive integers, the zero from $1/\Gamma (a+b)$ cancels the putative double pole, leaving channel simple-pole terms. Thus one must not count products of numerator poles without using $a+b+c=1$.

The holomorphic part of the free-boson worldsheet propagator givesApply Wick theorem to $T(z)T(w)$. The two double contractions give $D/[2(z-w{)}^4]$, while the single contractions reconstruct $T$ and its derivative. Thus the stress-tensor operator-product expansion isand the $D$ embedding coordinates have central charge $\boxed{c=D}$.

The holomorphic stress-energy tensor generates an infinitesimal conformal transformation throughTaking the three residues givesThe third derivative is the anomalous term that prevents $T$ from transforming as an ordinary weight-two Virasoro primary operator.

With Virasoro algebra modes $L_n=(2\pi i{)}^{-1}\oint dz{z}^{n+1}T(z)$, a second contour calculation givesso $A(m)=c(m^3-m)/12=D(m^3-m)/12$. This Virasoro central extension is the quantum conformal anomaly. In string theory the matter and ghost contributions must cancel; $c_{\mathrm{matter}}+c_{\mathrm{ghost}}=D-26=0$ gives the critical dimension of string theory $D=26$ and makes the BRST operator nilpotent.