The Virasoro-Shapiro Amplitude and Factorization

The previous page reduced tree-level closed-string tachyon scattering to the integral

$$\mathcal A_4(s,t,u) = \mathcal N_4(2\pi)^D\delta^{(D)}\!\left(\sum_i k_i\right) \int_{\mathbb C}d^2z\, |z|^{-\alpha's/2-4}|1-z|^{-\alpha't/2-4}.$$

This page evaluates the integral and extracts its physics. The result is the Virasoro-Shapiro amplitude, the closed-string analog of the Veneziano amplitude. Its analytic structure displays the central lesson of perturbative string theory: the worldsheet knows about the entire tower of spacetime string states.

The complex beta integral

The relevant integral is

$$I(a,b) = \int_{\mathbb C}d^2z\,|z|^{2a-2}|1-z|^{2b-2}.$$

It converges only in a restricted range of complex $a,b$. The physical scattering amplitude is defined by analytic continuation. The standard result is

$$\boxed{ I(a,b) = 2\pi\, \frac{\Gamma(a)\Gamma(b)\Gamma(1-a-b)} {\Gamma(1-a)\Gamma(1-b)\Gamma(a+b)}. }$$

For the four-tachyon integral,

$$2a-2=-\frac{\alpha's}{2}-4, \qquad 2b-2=-\frac{\alpha't}{2}-4,$$

so

$$a=-1-\frac{\alpha's}{4}, \qquad b=-1-\frac{\alpha't}{4}.$$

Using

$$s+t+u=-\frac{16}{\alpha'},$$

we also have

$$1-a-b=-1-\frac{\alpha'u}{4}.$$

Therefore

$$\boxed{ \mathcal A_4(s,t,u) = \mathcal N_4(2\pi)^D\delta^{(D)}\!\left(\sum_i k_i\right) \frac{ \Gamma\left(-1-\frac{\alpha's}{4}\right) \Gamma\left(-1-\frac{\alpha't}{4}\right) \Gamma\left(-1-\frac{\alpha'u}{4}\right) }{ \Gamma\left(2+\frac{\alpha's}{4}\right) \Gamma\left(2+\frac{\alpha't}{4}\right) \Gamma\left(2+\frac{\alpha'u}{4}\right) }. }$$

Overall constants, including the precise power of the closed-string coupling, depend on vertex normalization. The ratio of gamma functions is the universal content.

The complex beta integral is defined by analytic continuation. The three distinguished degeneration regions of the $z$-plane correspond to the three spacetime channels.

Normalization checkpoint. If one temporarily sets $\alpha'=2$, then $m_T^2=-2$, $s+t+u=-8$, and the first numerator gamma function is $\Gamma(-1-s/2)$. The main formulas here keep $\alpha'$ explicit.

Pole structure and the string spectrum

The gamma function $\Gamma(x)$ has simple poles at

$$x=0,-1,-2,\ldots .$$

The $s$-channel poles of the amplitude occur when

$$-1-\frac{\alpha's}{4}=-N, \qquad N=0,1,2,\ldots .$$

Thus

$$\boxed{ s=\frac{4}{\alpha'}(N-1). }$$

These values are exactly the masses of level-matched closed bosonic string states:

$$m_N^2=\frac{4}{\alpha'}(N-1), \qquad N=\bar N.$$

The first few poles are

$$\begin{array}{c|c|c} N & s=m_N^2 & \text{intermediate closed-string level} \\ \hline 0 & -4/\alpha' & \text{tachyon} \\ 1 & 0 & G_{\mu\nu},\ B_{\mu\nu},\ \Phi \\ 2 & 4/\alpha' & \text{first massive level} \\ 3 & 8/\alpha' & \text{second massive level} \\ \vdots & \vdots & \vdots \end{array}$$

The $s$-channel poles occur at $s=4(N-1)/\alpha'$. They reproduce the tachyon, the massless graviton/$B$-field/dilaton level, and the infinite tower of massive closed-string states.

Near an $s$-channel pole,

$$\Gamma\left(-1-\frac{\alpha's}{4}\right) \sim \frac{(-1)^N}{N!}\, \frac{-4/\alpha'}{s-4(N-1)/\alpha'}.$$

The residue is a polynomial in the remaining kinematic invariant. That polynomial organizes the spins available at level $N$.

Factorization from the OPE

The same poles can be derived directly from the worldsheet OPE. Consider the region $z\to0$, where the operators with momenta $k_1$ and $k_2$ collide:

$$:e^{ik_1\cdot X(0)}::e^{ik_2\cdot X(z,\bar z)}: = |z|^{\alpha' k_1\cdot k_2} :e^{i(k_1+k_2)\cdot X(0)}: \left(1+\cdots\right).$$

The omitted terms contain powers of $z$, $\bar z$, and derivatives of $X^\mu$. They are precisely the operators that create oscillator excitations of the intermediate string:

$$:e^{ik_1X(0)}::e^{ik_2X(z,\bar z)}: = |z|^{-\alpha's/2-4} \sum_{N,\bar N\ge0} z^N\bar z^{\bar N}\,\mathcal O_{N,\bar N}(k_1+k_2;0).$$

Angular integration enforces level matching:

$$\int_0^{2\pi}d\theta\,e^{i(N-\bar N)\theta} = 2\pi\delta_{N,\bar N}.$$

For a level-matched term $N=\bar N$, the radial integral near $z=0$ has the form

$$\int_0 dr\,r^{2N-\alpha's/2-3} \sim \frac{1}{2N-2-\alpha's/2} = -\frac{2/\alpha'}{s-4(N-1)/\alpha'}.$$

This is exactly the propagator denominator for an intermediate closed-string state at level $N$.

Worldsheet OPE factorization is spacetime factorization. When two punctures collide, the OPE expands into intermediate string states, and integration over the separation produces their propagator poles.

Near a pole, unitarity requires the amplitude to factorize as

$$\mathcal A_4 \underset{s\to m_N^2}{\sim} \sum_{\Psi_N} \frac{ \mathcal A_3(T_1,T_2,\Psi_N)\, \mathcal A_3(\Psi_N,T_3,T_4) }{s-m_N^2},$$

where the sum runs over physical closed-string states at level $N$. The no-ghost theorem and BRST cohomology ensure that the factorization is over physical states rather than negative-norm artifacts.

Duality between channels

In ordinary field theory, four-point scattering is usually represented as a sum of separate $s$-, $t$-, and $u$-channel diagrams. In the closed-string tree amplitude, the three channels are different boundary regions of the same moduli space:

$$z\to0 \quad \text{$s$-channel}, \qquad z\to1 \quad \text{$t$-channel}, \qquad z\to\infty \quad \text{$u$-channel}.$$

This is the modern interpretation of duality in the dual resonance model. One does not add three separate diagrams. One integrates once over the four-punctured sphere.

The four-punctured sphere has three degeneration limits. These are interpreted as the three spacetime channels of the same analytic amplitude.

This channel duality is also an early hint of the ultraviolet softness of perturbative string theory. Short-distance regions in spacetime are replaced by global regions of moduli space, and modular properties prevent the naive overcounting that would occur in a point-particle description.

Regge behavior

The amplitude also captures linear Regge trajectories. At large $s$ with fixed $t$, analytic continuation of the gamma functions gives

$$\mathcal A_4(s,t) \sim \mathcal C(t)\left(\frac{\alpha's}{4}\right)^{2+\alpha't/2},$$

up to a phase depending on the physical region. The exponent defines the closed-string Regge trajectory

$$\boxed{ \alpha_{\rm closed}(t)=2+\frac{\alpha'}{2}t. }$$

The intercept $2$ is the spin of the massless graviton on the leading trajectory. The slope is half the open-string Regge slope:

$$\alpha'_{\rm closed,Regge}=\frac{\alpha'}{2}, \qquad \alpha'_{\rm open,Regge}=\alpha'.$$

The leading closed-string trajectory passes through the massless spin-two graviton and has slope $\alpha'/2$ in the $J$ versus $m^2$ plane.

This agrees with the classical rotating-string result: closed strings carry both left- and right-moving excitations, and the leading closed-string trajectory has twice the spin at a given oscillator level but four times the mass spacing.

What this amplitude teaches

The Virasoro-Shapiro amplitude is much more than a special function. It demonstrates four structural properties that persist throughout perturbative string theory:

1. One moduli-space integral gives all channels. The amplitude is not assembled from separate field-theory diagrams.
2. Poles know the spectrum. The gamma functions reproduce the full closed-string mass tower.
3. Residues factorize. The OPE gives a complete basis of intermediate string states.
4. Regge behavior is built in. High-energy fixed-momentum-transfer behavior follows from the same analytic expression.

The next page turns from tachyons to massless closed strings. Replacing $e^{ikX}$ by $\epsilon_{\mu\nu}\partial X^\mu\bar\partial X^\nu e^{ikX}$ introduces gravitons, antisymmetric tensors, and the dilaton.

Exercises

Exercise 1. Evaluate the gamma-function arguments

Starting from

$$I(a,b)=\int_{\mathbb C}d^2z\,|z|^{2a-2}|1-z|^{2b-2},$$

match exponents with

$$|z|^{-\alpha's/2-4}|1-z|^{-\alpha't/2-4}.$$

Then show that $1-a-b=-1-\alpha'u/4$.

Solution

Matching powers gives

$$2a-2=-\frac{\alpha's}{2}-4, \qquad 2b-2=-\frac{\alpha't}{2}-4.$$

Thus

$$a=-1-\frac{\alpha's}{4}, \qquad b=-1-\frac{\alpha't}{4}.$$

Then

$$1-a-b = 1+2+\frac{\alpha'}{4}(s+t) = 3+\frac{\alpha'}{4}\left(-\frac{16}{\alpha'}-u\right) = -1-\frac{\alpha'u}{4}.$$

Exercise 2. Pole locations

Use the poles of $\Gamma(x)$ to derive the $s$-channel pole locations of the Virasoro-Shapiro amplitude.

Solution

The $s$ dependence in the numerator is

$$\Gamma\left(-1-\frac{\alpha's}{4}\right).$$

The gamma function has poles at $0,-1,-2,\ldots$, so set

$$-1-\frac{\alpha's}{4}=-N, \qquad N=0,1,2,\ldots .$$

Solving gives

$$s=\frac{4}{\alpha'}(N-1).$$

These are precisely the closed-string mass levels $m_N^2=4(N-1)/\alpha'$ with $N=\bar N$.

Exercise 3. OPE derivation of the propagator pole

Show that a level-matched intermediate term $z^N\bar z^N$ in the $z\to0$ OPE gives a pole at $s=4(N-1)/\alpha'$.

Solution

Near $z=0$, the leading Koba-Nielsen factor contributes

$$|z|^{-\alpha's/2-4}.$$

A level-matched operator at level $N$ contributes $z^N\bar z^N=|z|^{2N}$. With $d^2z=r\,dr\,d\theta$, the radial integral is

$$\int_0 dr\,r^{-\alpha's/2-4+2N+1} = \int_0 dr\,r^{2N-\alpha's/2-3}.$$

Analytic continuation gives a pole when the exponent equals $-1$:

$$2N-\frac{\alpha's}{2}-3=-1.$$

Thus

$$s=\frac{4}{\alpha'}(N-1).$$

Near the pole the denominator is proportional to $s-4(N-1)/\alpha'$.

Exercise 4. Leading closed-string Regge trajectory

Use the closed-string mass formula and the leading trajectory spin $J=2N$ to show that

$$J=2+\frac{\alpha'}{2}m^2.$$

Solution

For a level-matched closed string,

$$m^2=\frac{4}{\alpha'}(N-1).$$

Solving for $N$ gives

$$N=1+\frac{\alpha'm^2}{4}.$$

The leading trajectory has spin $J=2N$, hence

$$J=2\left(1+\frac{\alpha'm^2}{4}\right) = 2+\frac{\alpha'}{2}m^2.$$

Exercise 5. Specializing to $\alpha'=2$

Set $\alpha'=2$. Find the tachyon mass, the relation among $s,t,u$, and the $s$-channel pole positions.

Solution

For the closed-string tachyon,

$$m_T^2=-\frac{4}{\alpha'}=-2.$$

Therefore

$$s+t+u=4m_T^2=-8.$$

The pole locations are

$$s=\frac{4}{\alpha'}(N-1)=2(N-1).$$

Thus the first poles are $s=-2,0,2,4,\ldots$.