Tree-Level String Amplitudes from Worldsheet Correlators

The path integral formulation turns perturbative string scattering into a problem in two-dimensional conformal field theory. External strings are represented by vertex operators, and a scattering amplitude is a worldsheet correlation function integrated over the moduli of the punctured surface.

At tree level for closed oriented strings, the worldsheet is a Riemann sphere. The special feature of string theory is that a single smooth worldsheet already contains all spacetime channels. What would be several Feynman diagrams in a point-particle field theory becomes one integral over the positions of punctures.

This page develops the general recipe and reduces the closed-string four-tachyon amplitude to a single integral over the complex plane. The evaluation of that integral, its pole structure, and its factorization are treated on the next page.

Sphere amplitudes and punctures

For $n$ closed strings at genus zero, the schematic Polyakov path integral is

$$\mathcal A_{0,n} = \mathcal N_{0,n} \int \frac{\mathcal D h\,\mathcal D X}{\mathrm{Vol}(\mathrm{Diff}\times\mathrm{Weyl})} e^{-S_X[X,h]} \prod_{i=1}^n \mathcal V_i .$$

After conformal gauge fixing, the worldsheet metric is the round metric on $S^2$ up to a conformal factor, and Weyl invariance removes that conformal factor in the critical string. The remaining finite-dimensional data are the puncture positions $z_i$, modulo the conformal automorphism group of the sphere:

$$SL(2,\mathbb C): \qquad z\mapsto \frac{az+b}{cz+d}, \qquad ad-bc=1.$$

Thus a convenient first expression for a tree-level closed-string amplitude is

$$\mathcal A_{0,n} = \mathcal N_{0,n} \int \frac{\prod_{i=1}^n d^2z_i}{\mathrm{Vol}\,SL(2,\mathbb C)} \left\langle \prod_{i=1}^n V_i(z_i,\bar z_i)\right\rangle_{S^2}.$$

Here $V_i$ denotes the matter part of the vertex operator. For the integral $\int d^2z_i\,V_i$ to be conformally invariant, $V_i$ must have weights

$$(h_i,\bar h_i)=(1,1).$$

A tree-level closed-string process is a sphere with marked points. The embedding fields $X^\mu$ fluctuate on the sphere, and each puncture carries a vertex operator for an external string state.

The quotient by $SL(2,\mathbb C)$ means that three complex positions are gauge, not moduli. Equivalently, one fixes three punctures and inserts the corresponding ghost factors. This is the most useful form for computations:

$$\boxed{ \mathcal A_{0,n} = \mathcal N_{0,n} \left\langle \prod_{a=1}^3 c\bar c V_a(z_a,\bar z_a) \prod_{r=4}^n \int d^2z_r\,V_r(z_r,\bar z_r) \right\rangle_{S^2}. }$$

The three $c$ insertions soak up the three holomorphic ghost zero modes $c_{-1},c_0,c_1$, and the three $\bar c$ insertions do the same in the antiholomorphic sector. The matter vertex $V$ has weights $(1,1)$, while the unintegrated vertex $c\bar cV$ has weights $(0,0)$ and ghost number $(1,1)$.

On the sphere, three punctures may be fixed by Möbius symmetry. The corresponding operators are unintegrated $c\bar cV$ vertices. The remaining punctures are integrated over moduli.

The closed-string tachyon vertex

The simplest matter vertex in the bosonic closed string is the tachyon plane wave

$$V_T(k;z,\bar z)=:e^{ik\cdot X(z,\bar z)}:.$$

We use the free-boson normalization

$$X^\mu(z,\bar z)X^\nu(w,\bar w) \sim -\frac{\alpha'}{2}\eta^{\mu\nu}\ln|z-w|^2 .$$

Then the holomorphic and antiholomorphic weights of $V_T$ are

$$h=\bar h=\frac{\alpha' k^2}{4}.$$

The integrated closed-string vertex condition $h=\bar h=1$ gives

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

With mostly-plus target-space signature, $k^2=-m^2$, hence

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

The unintegrated tachyon vertex is

$$\mathcal V_T(k;z,\bar z) = c(z)\bar c(\bar z):e^{ik\cdot X(z,\bar z)}:.$$

The full correlator of tachyon matter vertices is fixed by Wick contractions and the zero mode of $X^\mu$:

$$\left\langle \prod_{i=1}^n :e^{ik_i\cdot X(z_i,\bar z_i)}: \right\rangle = (2\pi)^D\delta^{(D)}\!\left(\sum_i k_i\right) \prod_{i<j}|z_i-z_j|^{\alpha' k_i\cdot k_j}.$$

This product is the closed-string version of the Koba-Nielsen factor. The momentum-conserving delta function comes from integrating over the constant mode of $X^\mu$.

The zero mode of $X^\mu$ imposes momentum conservation. The nonzero modes produce the product of pairwise powers $|z_i-z_j|^{\alpha'k_i\cdot k_j}$.

Four punctures and the cross-ratio

For four closed-string insertions, the sphere has one complex modulus. We fix three positions by a Möbius transformation:

$$z_1=0, \qquad z_2=z, \qquad z_3=\infty, \qquad z_4=1.$$

The remaining coordinate $z$ is the cross-ratio of the original four points. A symmetric way to write the cross-ratio is

$$\chi = \frac{z_{12}z_{34}}{z_{13}z_{24}}, \qquad z_{ij}=z_i-z_j.$$

After the above gauge choice, $\chi=z$ up to the chosen ordering of labels.

The Möbius group has complex dimension three, so it fixes three punctures. The fourth position is the genuine modulus of the four-punctured sphere.

The operator at infinity is defined by conformal transformation:

$$\mathcal O(\infty) = \lim_{z\to\infty}z^{2h}\bar z^{2\bar h}\mathcal O(z,\bar z).$$

For the tachyon $h=\bar h=1$, this means multiplying by $|z|^4$ before taking $z\to\infty$. The ghost correlator becomes

$$\langle c(0)c(1)c(\infty)\rangle=1, \qquad \langle \bar c(0)\bar c(1)\bar c(\infty)\rangle=1,$$

with the same convention for $c(\infty)$.

Mandelstam invariants

Take all external momenta incoming. For the ordering chosen above, define

$$s=-(k_1+k_2)^2, \qquad t=-(k_2+k_4)^2, \qquad u=-(k_1+k_4)^2.$$

For four identical closed-string tachyons,

$$k_i^2=\frac{4}{\alpha'}, \qquad s+t+u=4m_T^2=-\frac{16}{\alpha'}.$$

The useful dot products are

$$\alpha' k_1\cdot k_2 = -\frac{\alpha's}{2}-4, \qquad \alpha' k_2\cdot k_4 = -\frac{\alpha't}{2}-4.$$

Indeed,

$$s=-(k_1+k_2)^2 = -\frac{8}{\alpha'}-2k_1\cdot k_2,$$

and similarly for $t$.

The four-tachyon integral

Putting the matter correlator, ghost correlator, and Möbius fixing together gives the tree-level four-tachyon amplitude

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

The normalization $\mathcal N_4$ depends on the closed-string coupling and on the normalization of vertex operators. The dependence on $s,t,u$ is universal.

After fixing three punctures, the four-point amplitude is an integral over the complex cross-ratio. The singular regions $z\to0$, $z\to1$, and $z\to\infty$ will become the $s$-, $t$-, and $u$-channel factorization limits.

This formula is the central bridge from worldsheet CFT to spacetime scattering. The next step is to evaluate the complex beta integral and interpret its singularities.

Summary

The tree-level closed-string recipe is:

$$\text{external state} \quad\longleftrightarrow\quad \text{BRST-invariant vertex operator},$$ $$\text{tree amplitude} \quad\longleftrightarrow\quad \text{sphere correlator integrated over puncture moduli},$$ $$\text{spacetime channels} \quad\longleftrightarrow\quad \text{degeneration limits of the same punctured sphere}.$$

For tachyons this reduces to the explicit integral above. For massless and massive string states the vertex operators have additional $\partial X$, $\bar\partial X$, ghost, or spin-field structure, but the conceptual recipe is the same.

Exercises

Exercise 1. Why an integrated closed-string vertex has weights $(1,1)$

Show that $\int d^2z\,V(z,\bar z)$ is invariant under holomorphic coordinate changes if $V$ has weights $(1,1)$.

Solution

Under $z\mapsto z'=f(z)$,

$$d^2z' = |f'(z)|^2 d^2z.$$

A primary field of weights $(h,\bar h)$ transforms as

$$V'(z',\bar z') = \left(\frac{dz}{dz'}\right)^h \left(\frac{d\bar z}{d\bar z'}\right)^{\bar h} V(z,\bar z).$$

For $h=\bar h=1$,

$$V'(z',\bar z')=|f'(z)|^{-2}V(z,\bar z),$$

so

$$d^2z'\,V'(z',\bar z')=d^2z\,V(z,\bar z).$$

Thus $\int d^2z\,V$ is coordinate invariant.

Exercise 2. Tachyon on-shell condition from the vertex dimension

Using $h=\bar h=\alpha'k^2/4$, derive the mass of the closed-string tachyon.

Solution

An integrated closed-string matter vertex must have

$$h=\bar h=1.$$

Therefore

$$\frac{\alpha'k^2}{4}=1, \qquad k^2=\frac{4}{\alpha'}.$$

With mostly-plus signature, $k^2=-m^2$, so

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

Exercise 3. The Koba-Nielsen factor

Starting from

$$X^\mu(z,\bar z)X^\nu(w,\bar w) \sim -\frac{\alpha'}{2}\eta^{\mu\nu}\ln|z-w|^2,$$

show that

$$\left\langle \prod_{i=1}^n :e^{ik_i\cdot X(z_i,\bar z_i)}:\right\rangle \propto \prod_{i<j}|z_i-z_j|^{\alpha'k_i\cdot k_j}.$$

Solution

For Gaussian fields,

$$\left\langle \prod_i :e^{ik_i\cdot X_i}:\right\rangle = \exp\left[-\sum_{i<j}k_i\cdot k_j\langle X_iX_j\rangle\right]$$

up to the zero-mode delta function. Since

$$\langle X_i\cdot X_j\rangle = -\frac{\alpha'}{2}\ln|z_i-z_j|^2,$$

we get

$$\exp\left[\frac{\alpha'}{2}\sum_{i<j}k_i\cdot k_j\ln|z_i-z_j|^2\right] = \prod_{i<j}|z_i-z_j|^{\alpha'k_i\cdot k_j}.$$

The missing overall factor is $(2\pi)^D\delta^{(D)}(\sum_i k_i)$ from the zero mode.

Exercise 4. Dot products and Mandelstam variables

For four identical closed-string tachyons, prove

$$\alpha' k_1\cdot k_2=-\frac{\alpha's}{2}-4, \qquad s+t+u=-\frac{16}{\alpha'}.$$

Solution

Since $k_i^2=4/\alpha'$, we have

$$s=-(k_1+k_2)^2=-k_1^2-k_2^2-2k_1\cdot k_2 = -\frac{8}{\alpha'}-2k_1\cdot k_2.$$

Solving gives

$$\alpha'k_1\cdot k_2=-\frac{\alpha's}{2}-4.$$

For four particles of equal mass with all momenta incoming,

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

Because $m_T^2=-4/\alpha'$, this becomes

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