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Separate the constant embedding coordinate from its nonzero modes. Its integral gives the momentum-conservation delta function; below this is understood and absorbed into the amplitude normalization on the momentum-conserving locus. For the nonzero modes the Euclidean Polyakov action is quadratic, with worldsheet Green functionTreat the tachyon vertex operators as normal ordered, removing coincident self-contractions. The Gaussian functional integral, or the same calculation by Wick's theorem, givesAll remaining determinants are independent of the insertion points and momenta and enter the overall constant. This derives the Koba-Nielsen factor and the requested position integral.

For the Möbius transformation $f(z)=(az+b)/(cz+d)$, $ad-bc=1$, one hasThe transformed Koba-Nielsen factor contributes, at insertion $r$, the power $-{\alpha }^{\prime }{\sum }_{s\ne r}{k}_r\cdot k_s$ of $|c{z}_r+d|$. Four-momentum conservation makes this power ${\alpha }^{\prime }{k}_r^2$. The complete differential integrand therefore transforms byIt is invariant when ${\alpha }^{\prime }{k}_r^2=4$ for every external tachyon. The correlator alone is covariant; the integration measures are necessary for invariance. The group acting faithfully on the sphere is $PSL(2,\mathbb{C})$, with the two elements $\pm I$ of $SL(2,\mathbb{C})$ acting identically.

The apparent integral over all positions includes the residual conformal-group gauge volume. Divide by it and fix three distinct insertions. The Faddeev-Popov determinant, equivalently the three-point worldsheet ghost correlator and its conjugate, contributesSet $z_1=\Lambda$, $z_2=0$, $z_4=1$, then send $\Lambda$ to infinity on the sphere; the sign of the real direction is irrelevant. The factors involving $z_1$ behave as $|\Lambda {|}^{{\alpha }^{\prime }{k}_1\cdot {\sum }_{s\ne 1}{k}_s}=|\Lambda {|}^{-4}$, while $J\sim |\Lambda {|}^4$. They cancel, leaving $z_3=z$ as the one integration variable.

The remaining distance exponents areFor example $(k_2+k_3{)}^2=(k_1+k_4{)}^2=-t$ and each individual squared momentum is $4/{\alpha }^{\prime }$. AlsoConsequently the reduced amplitude is proportional toApply the supplied complex beta integral with $A=4+{\alpha }^{\prime }t/2$, $B=4+{\alpha }^{\prime }s/2$, and $C=4-A-B=4+{\alpha }^{\prime }u/2$. Its numerator arguments become $-1-{\alpha }^{\prime }t/4$, $-1-{\alpha }^{\prime }s/4$, $-1-{\alpha }^{\prime }u/4$, while its denominator arguments become $2+{\alpha }^{\prime }u/4$, $2+{\alpha }^{\prime }t/4$, $2+{\alpha }^{\prime }s/4$. Thus the four-point amplitude is the Virasoro–Shapiro amplitudewhere $C_0$ is momentum-independent. The integral converges initially when $\mathrm{Re}A<2$, $\mathrm{Re}B<2$, and $\mathrm{Re}(A+B)>2$; the displayed answer elsewhere is its analytic continuation.

The Gamma function has simple poles at nonpositive integers and no zeros. At generic fixed $t$, the amplitude therefore has simple $s$-channel poles at $s=4(n-1)/{\alpha }^{\prime }$, $n=0,1,2,\dots$, with the same towers in $t$ and $u$ by crossing symmetry. They describe exchange of the closed string tachyon, massless states, and the infinitely many massive levels. The reciprocal denominator factors can give zeros at negative values $s=-4(\ell +2)/{\alpha }^{\prime }$; special kinematics can also make an exchange residue vanish.

To see the structure of a generic residue, set $a=-1-{\alpha }^{\prime }s/4$, $b=-1-{\alpha }^{\prime }t/4$, $c=-1-{\alpha }^{\prime }u/4$, so $a+b+c=1$. At $a=-n$, the Gamma function recurrence reduces the remaining factors to a constant times ${\prod }_{j=1}^n(b-j{)}^2$. The Virasoro–Shapiro amplitude pole residue is thus a polynomial of degree $2n$ in the other channel invariant, consistent with exchange up to spin $2n$. Crossed numerator poles must not simply be multiplied: near $a=-n+\delta$ and $b=-m+\epsilon$, $c=1+n+m-\delta -\epsilon$, and the zero of $1/\Gamma (1-c)$ cancels the putative double pole. The leading singular part is proportional toThe tree amplitude is meromorphic, with no loop threshold branch cuts; its channel singularities express factorization and string dual resonance.

First regulate the Gaussian functional integral, separating any constant zero mode. Integration by parts gives $S[\varphi ]=\frac{1}{2}(\varphi ,A\varphi )$ with positive kinetic operator $A=-\Delta$ on the nonzero modes. Complete the square:A translation of the regulated integration variables then yieldsThe sign is essential: the kernel in the negative source exponent is the negative of the positive Gaussian covariance. The source-independent prefactor is formally $\mathcal{N}=\int D\chi e^{-(\chi ,A\chi )/2}\propto (\det {}^{\prime }A{)}^{-1/2}$, with the prime denoting the excluded zero mode and normalization depending on the regulated measure. This is the Euclidean scalar source functional and Laplacian sign convention.

Consequently ${\Delta }_{z}G(z,w)={\delta }^{(2)}(z-w)$. In the plane,up to harmonic additions fixed by boundary and infrared conditions. Away from $w$ this logarithm is harmonic. Integrating its outward normal derivative around a small circle centered at $w$ gives $2\pi$, establishing the distributional delta coefficient. The arbitrary infrared length $L$ cancels for a source of zero total integral. On a compact regulated worldsheet the zero-mode-projected equation has the corresponding constant subtraction from the delta function.

For the string action $I_E=(4\pi {\alpha }^{\prime }{)}^{-1}\int (\partial X{)}^2$, the nonzero-mode covariance isCompute initially with Euclidean target coordinates and restore target Lorentzian contractions by analytic continuation. Treat the physical tachyon vertices as normal-ordered exponentials; this removes coincident self-contractions. Completing the Gaussian square with the imaginary delta-function sources, or applying Wick contractions, givesafter fixing the vertex normalization. Each cross-contraction has the sign $i^2=-1$, canceling the negative logarithmic covariance, which explains the positive exponent.

The constant target-space mode must also be integrated:This momentum-conservation factor is momentum dependent and cannot literally be absorbed into the PDF's momentum-independent $\mathcal{N}$. Its intended position integral is the amplitude with this overall delta function stripped off, restricted to conserving momenta. In that convention,The logarithmic infrared constant only gives a fixed normalization on the common external mass shell, since ${\sum }_{r<s}{k}_r\cdot k_s=-\frac{1}{2}{\sum }_r{k}_r^2$. Thus the Gaussian determinant and normalization are independent of the scattering invariants. To make the tree position integral finite one also divides by the residual Möbius gauge volume, equivalently fixes three insertions with their ghost factor. The four-position expression is its formal unfixed representation.

Under $z\mapsto (az+b)/(cz+d)$ with determinant one,At insertion $r$, the Koba-Nielsen factor therefore contributes $|c{z}_r+d{|}^{-{\alpha }^{\prime }{\sum }_{s\ne r}{k}_r\cdot k_s}$. Momentum conservation makes the exponent ${\alpha }^{\prime }{k}_r^2$. On the tachyon mass shell $k_r^2=4/{\alpha }^{\prime }$, it is four and cancels the measure's exponent $-4$. Hence the full integration density is Möbius invariant. The correlator factor alone is covariant; the density including $d^2{z}_r$ is invariant.

The supplied Virasoro–Shapiro amplitude has generic simple poles when $-1-{\alpha }^{\prime }s/4=-n$, givingand likewise in the other channels. These are exchanges of closed-string states with equal left and right levels $n$: the tachyon, the massless sector and the massive tower. At special kinematics zeros or crossed-channel coincidences need to be handled by the full constrained amplitude; numerator poles are not three independent sequences of unrelated singularities.

To isolate the three-tachyon coupling, first use momentum conservation and $k_r^2=-{\mu }^2$:At the tachyon pole $s={\mu }^2=-4/{\alpha }^{\prime }$, put $x={\alpha }^{\prime }t/4$ and $y={\alpha }^{\prime }u/4$. The relation gives $x+y=-3$. Thus $-1-y=2+x$ and $2+y=-1-x$, and the entire $t,u$ gamma-function ratio cancels to one. The remaining numerator is $\Gamma (-{\alpha }^{\prime }(s-{\mu }^2)/4)\sim -4/[{\alpha }^{\prime }(s-{\mu }^2)]$, while its denominator tends to $\Gamma (1)=1$. HenceWith the exchange convention ${\hat{A}}_4\sim g_{TTT}^2/({\mu }^2-s)$ and canonically normalized external legs, factorization therefore gives $\boxed{g_{TTT}^2=4C/{\alpha }^{\prime }}$, or $g_{TTT}=2\sqrt{C/{\alpha }^{\prime }}$ up to its unobservable overall sign. A convention using the opposite propagator denominator or factors of $i$ changes the corresponding amplitude phase. Because $C$ itself is unspecified, the residue fixes this relation, not an absolute numerical coupling independent of normalization.

Use a mostly-plus target Minkowski spacetime metric and restore the string tension by $T=(2\pi {\alpha }^{\prime }{)}^{-1}$. After Wick rotation, the Polyakov action on a Riemann surface isThe Polyakov path integral integrates over embeddings and worldsheet metrics, dividing by worldsheet diffeomorphisms and Weyl transformations. For a consistent flat bosonic string theory, take the critical dimension of the bosonic string $d=26$. Conformal gauge turns the embedding fields into free scalars; its Faddeev-Popov determinant is represented by the bc system. The remaining integrations are over worldsheet moduli and external insertion points.

A tachyon vertex operator isThe worldsheet Green function is $⟨X^a(z)X^b(w)⟩=-({\alpha }^{\prime }/2){\eta }^{ab}\log |z-w{|}^2$. The operator product expansion with the holomorphic stress-energy tensor gives conformal weights $({\alpha }^{\prime }{k}^2/4,{\alpha }^{\prime }{k}^2/4)$. An integrated string vertex operator must have weights $(1,1)$, so the mass-shell condition is $k^2=4/{\alpha }^{\prime }$, or tachyonic mass parameter $m^2=-4/{\alpha }^{\prime }$.

At lowest order in the string coupling, the worldsheet is a Riemann sphere. For four external states, three insertion points can be fixed by a Möbius transformation. In the worldsheet ghost field description these three vertices carry $c\bar{c}$, while the fourth is integrated. The three-point worldsheet ghost correlator gives the squared product of the three fixed-point separations, precisely the sphere gauge fixing for four string vertices factor. Thus fixing three points is a gauge choice, rather than deleting three integrations without their Jacobian.

The integral over the constant embedding worldsheet zero mode gives $(2\pi {)}^d{\delta }^{(d)}({\sum }_i{k}_i)$. The remaining Gaussian integral, using normal ordering to remove self-contractions, gives the Koba-Nielsen factorSet $(z_1,z_2,z_3,z_4)=(0,z,1,\infty )$; the vertex at infinity is defined with its conformal-weight factor. Let all momenta be incoming and defineThe mass-shell condition and momentum conservation imply $s+t+u=-16/{\alpha }^{\prime }$. Since ${\alpha }^{\prime }{k}_1\cdot k_2=-4-{\alpha }^{\prime }s/2$ and similarly for the pair $(2,3)$, the sphere tachyon position integral becomesHere $\mathcal{N}$ depends on the normalization of external string vertex operators; it does not affect the kinematic dependence.

To evaluate the remaining worldsheet modulus, put $a=-1-{\alpha }^{\prime }s/4$ and $b=-1-{\alpha }^{\prime }t/4$. The complex beta integral isFor example, this identity follows by writing the two powers as Schwinger-parameter Gamma integrals, doing the two-dimensional Gaussian integral, and changing the two positive parameters to their sum and ratio. The complex beta integral first holds for $\mathrm{Re}a,\mathrm{Re}b>0$ and $\mathrm{Re}(a+b)<1$; the scattering result is its analytic continuation. The third gamma-function argument is $1-a-b=-1-{\alpha }^{\prime }u/4$. Absorbing $\pi$ into $\mathcal{N}$, the result is the tree-level four-tachyon amplitude, the Virasoro–Shapiro amplitude:It is symmetric in the three Mandelstam variables and has generic poles at $q=4(N-1)/{\alpha }^{\prime }$, matching the bosonic string mass spectrum. The formula is the leading term of string perturbation theory. Higher orders are constructed by the same Polyakov path integral prescription on higher-genus worldsheets, with four marked points and the corresponding ghost and worldsheet moduli measure; genus $g$ has coupling power $g_s^{2g+2}$.

Take an oriented closed string in flat, critical bosonic string theory, with ${\alpha }^{\prime }=1/(2\pi T)$ 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 isTarget-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 $h=\bar{h}={\alpha }^{\prime }{p}^2/4$. The physical integrated vertex has $(h,\bar{h})=(1,1)$, so $p^2=4/{\alpha }^{\prime }$. Schematically its tree amplitude isThe 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 $c\tilde{c}$ 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, withTheir zero-mode integral gives momentum conservation, and their nonzero-mode Gaussian integral gives the Koba-Nielsen factorThis 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 $s_{\mathrm{phys}}=8\pi Ts=4s/{\alpha }^{\prime }$. 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 givesThe Gamma function poles imply, at generic fixed values of the other invariant,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: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 residueThe 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 isHere ${ϵ}^{ab}$ 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 $1/\sqrt{{\alpha }^{\prime }}$, 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 ${\alpha }^{\prime }$. 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 isThe terms denoted $O({\alpha }^{\prime })$ 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, $\Phi ={\Phi }_0+\varphi$. By the Gauss-Bonnet theorem, its dilaton Euler-characteristic weighting on a connected closed oriented surface follows fromDefine the string coupling by $g_s=e^{{\Phi }_0}$. A genus-$g$ path integral is weighted byThe 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 expansionEach coefficient has its own low-energy ${\alpha }^{\prime }$ expansion. The two parameters have different roles: ${\alpha }^{\prime }$ resolves finite string size through higher derivatives, while $g_s^2$ counts additional string loops.

In a point-particle description of quantum gravity, interactions of a massless spin-two field occur at localized vertices of Feynman diagrams. The gravitational coupling has negative mass dimension for spacetime dimension greater than two, so short-distance loop integrations generate higher-derivative counterterms that cannot all be absorbed into the Einstein action. A string worldsheet instead joins incoming and outgoing strings by a smooth surface. In particular, a closed string splits or joins through a smooth pair-of-pants surface. There is no invariantly distinguished point at which every part of an extended string interacts. The choice of a time slicing can draw a joining point, but it is not a physical pointlike vertex. This removes the point-interaction geometry responsible for arbitrarily localized interactions and introduces the string length ${\ell }_s=\sqrt{{\alpha }^{\prime }}$.

This geometric explanation is supported by the organization of the Polyakov path integral. Introducing a worldsheet metric makes the Nambu–Goto action amenable to gauge fixing. In the Euclidean integral, metrics on a fixed oriented surface are divided by worldsheet diffeomorphisms and Weyl transformations. What remains is a Riemann surface with a conformal structure, together with finitely many worldsheet moduli and the positions of external string vertex operators. The determinants from fixing the metric are represented by Faddeev-Popov ghosts. Schematically, for $n$ closed-string external states,Here ${\mathcal{M}}_{g,n}$ is the moduli space of genus-$g$ punctured surfaces. The dilaton weights a surface by its Euler characteristic $\chi =2-2g$, giving $g_s^{-\chi }$ before the normalization of external states. Every additional handle multiplies the contribution by $g_s^2$. The string genus expansion therefore plays the role of the ordinary loop expansion, with the sphere giving tree level and the torus giving one loop. The critical dimension cancels the worldsheet Weyl anomaly, making this reduction to conformal geometry consistent.

The massless symmetric traceless closed-string state is a graviton, and its long-wavelength interactions reproduce gravity. At distances comparable with ${\ell }_s$, however, an infinite tower of excited states contributes. This is more than a single particle form factor: the same amplitude contains graviton exchange, massive higher-spin exchange and string-scale softness.

The dimensionless variables in the given Virasoro–Shapiro amplitude correspond to $s={\alpha }^{\prime }S/4$, $t={\alpha }^{\prime }T/4$, $u={\alpha }^{\prime }U/4$, where capital letters denote the usual dimensionful Mandelstam invariants. Its constraint $s+t+u=-4$ identifies the external particles as the closed bosonic tachyons, not four gravitons. Nevertheless its intermediate massless pole directly exhibits graviton exchange. The overall normalization will be suppressed below; no additional copy of the amplitude formula is needed.

First, the amplitude is symmetric under permutations of $s,t,u$, expressing crossing symmetry. The poles in any channel areFor generic $t$, the residue can be calculated with the Gamma function pole at $-n$ and its recurrence:The product is $1$ when $n=0$. Indeed, the pole factor contributes $(-1{)}^{n+1}/n!$, while the two remaining Gamma ratios contribute $(-1{)}^n(t+2{)}_n^2/n!$. A residue polynomial of degree $2n$ shows exchange of spins up to $2n$, with leading Regge trajectory $J=2+2s$. At $n=1$, the massless residue is $-(t+2{)}^2$, containing the spin-two exchange. By crossing, the massless $t$-channel pole isfor generic $s$. Thus at large $s$ the pole behaves as $s^2/t$, the characteristic energy dependence of graviton exchange. Its massless propagator produces the long-range gravitational interaction. At low energies the massive poles can instead be expanded into local higher-derivative corrections to the effective gravitational action.

At the string scale one must retain the whole tower. The Gamma reflection formula rewrites the amplitude exactly asAt large $s$ and fixed $t$, away from the resonance poles or with a specified complex continuation, the Gamma ratio scales as $s^{t+1}$. Hence the Regge limit scales as $s^{2+2t}$, up to the signature factor and the displayed $t$-dependent coefficient. In particular, a fixed nonzero momentum transfer changes the point-graviton power law by a factor $s^{2t}$: the graviton belongs to an entire trajectory rather than remaining an isolated spin-two exchange at arbitrarily high energy.

There is stronger suppression when the scattering angle is fixed. Put $t\simeq -as$, $u\simeq -(1-a)s$ with $0<a<1$. Applying Stirling formula to the Gamma functions, after the same treatment of real-axis poles, gives the leading smooth envelopeThe leading $s\log s$ terms cancel and the remaining entropy-like combination is negative. In physical units the exponent is $-{\alpha }^{\prime }Sf(a)/2$. This fixed-angle softness of the closed-string amplitude is absent for an elementary pointlike gravitational vertex. Equivalently, short-distance gravitational scattering probes additional string degrees of freedom. In a four-dimensional long-distance effective setting, a massive exchanged mode contributes a Yukawa-type correction proportional to $e^{-M_{n}r}/r$, with couplings and tensor factors depending on the external states. Such terms are suppressed for $r\gg {\ell }_s$ and become relevant for $r\sim {\ell }_s$. The amplitude establishes the spectrum and short-distance change; it does not by itself give a universal static potential for arbitrary sources. Its bosonic tachyon also signals an unstable vacuum, so it must not be treated as a complete stable model of a gravitational force.

Finally, the modular group prevents overcounting equivalent surfaces and reorganizes the loop ultraviolet region. A torus is $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ with $\mathrm{Im}\tau >0$. A change of lattice basis identifiesThus the one-loop integral uses a fundamental domain for $\mathrm{PSL}(2,\mathbb{Z})$, for exampleThe measure $d^2\tau /(\mathrm{Im}\tau {)}^2$ is invariant, and the physical integrand, including matter, ghosts and external insertions, must respect torus modular invariance. In this domain $\mathrm{Im}\tau \ge \sqrt{3}/2$. The region of arbitrarily small proper time that causes ultraviolet divergences in a particle loop is therefore not an independent part of the string integral; a modular transformation relates it to another description already represented in the domain.

This does not remove every possible divergence. The cusp $\mathrm{Im}\tau \to \infty$ is a long tube, and factorization through it can produce infrared divergences, massless tadpoles or the bosonic tachyon divergence. At higher genus, degenerating cycles similarly factorize into propagating intermediate string states. The essential improvement is the relation between worldsheet geometry, the full spectrum and the quotient by large diffeomorphisms; infrared and unstable-background problems remain separate questions. A short-distance expansion that keeps only the graviton discards precisely this structure.

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 closed string already contains the particle required for quantum gravity. For bosonic string theory in its critical dimension of string theory, the mass-shell condition and closed-string level matching areAt $N=\tilde{N}=1$, the states ${ϵ}_{mn}{\alpha }_{-1}^m{\tilde{\alpha }}_{-1}^n|0;k⟩$ are massless. After imposing the Virasoro constraints and quotienting null string states, the transverse polarization tensor decomposes into symmetric trace-free, antisymmetric and scalar pieces under $SO(D-2)$. They describe a graviton, the Kalb–Ramond field and the dilaton. In particular, the symmetric trace-free sector is a massless spin-two graviton. Its number of degrees of freedom is $D(D-3)/2$.

The state–operator correspondence associates each physical string state with a string vertex operator. For example, the matter part of a massless closed-string vertex operator isIts conformal weights are $(1,1)$, so its integrated form $\int d^{2}z{V}_{ϵ,k}$ is invariant under changes of string worldsheet coordinates. At fixed insertion positions it is accompanied by $c\bar{c}$ from the worldsheet ghost fields. Physical vertices represent BRST cohomology classes; longitudinal changes of the polarization tensor are BRST-exact operators and decouple from physical scattering amplitudes. For the graviton, this string-state gauge redundancy becomes the linearized target-space diffeomorphism $h_{mn}\mapsto h_{mn}+{\partial }_m{\xi }_n+{\partial }_n{\xi }_m$.

The Polyakov path integral computes a closed-string scattering amplitude by inserting the external string vertex operators, integrating their unfixed positions and the inequivalent worldsheet moduli, and including the worldsheet ghost fields. On a Riemann sphere three positions are fixed by the residual conformal group. For example, contractions of tachyon vertex operators produce the Koba-Nielsen factor, whose position integral gives the Virasoro–Shapiro amplitude.

The string dual resonance property means that one crossing-symmetric scattering amplitude has equivalent expansions in the different channels: its poles exhibit intermediate string states in each channel, rather than separate channel contributions being added again. The scattering-amplitude factorization at a pole identifies the intermediate particle and its couplings. Set $8\pi T=1$, so ${\alpha }^{\prime }=4$ and the closed-string tachyon has $M^2=-1$. For four such external states the Mandelstam variables obey $s+t+u=-4$. The Virasoro–Shapiro amplitude has generic $s$-channel poles at $s=-1,0,1,\dots$, exactly the bosonic string mass spectrum in these units.

The massless pole gives an especially direct check. With overall normalization $C$ and $u=-4-s-t$, the Gamma function recurrence givesThusThe quadratic residue in $t$ contains a spin-two exchange. A scalar exchange alone could not give this angular dependence. Combined with the known massless spectrum and scattering-amplitude factorization, it identifies exchange of the graviton, with possible scalar contributions from the dilaton; the antisymmetric Kalb–Ramond field does not couple to two identical scalar tachyons here. Consequently the graviton participates in interactions, rather than being an isolated free state.