String Scattering Amplitudes from CFT Correlation Functions

1. From Free Strings to Interactions

So far, we have described a single, free string propagating through spacetime. Its worldsheet is a cylinder (for a closed string) or a strip (for an open string). Interactions, such as two strings joining to become one or one string splitting into two, are described by worldsheets with more complex topologies.

A diagram showing how (a) a single propagating string corresponds to a cylinder, while (b) a scattering process involving two initial strings and two final strings corresponds to a more complex worldsheet topology. A conformal map transforms this into (c) a sphere with four punctures, where vertex operators will be inserted.

The full path integral for string theory involves a sum over all possible worldsheet topologies. This sum forms a perturbative expansion, where the contribution of each topology is weighted by a factor related to its Euler characteristic \chi. For a closed string worldsheet of genus g (number of "handles"), \chi = 2 - 2g. The amplitude for a process is weighted by g_s^{-\chi}, where g_s is the string coupling constant.

• Tree-level processes (no loops) correspond to a worldsheet with the topology of a sphere (g=0, \chi=2). Their amplitudes are proportional to g_s^{-2}.
• One-loop corrections correspond to a torus (g=1, \chi=0), with amplitudes of order g_s^0.

Today, we focus on the leading-order contribution: tree-level scattering on the sphere.

2. The String S-Matrix

What should we compute? In your study of quantum field theory, the fundamental objects are correlation functions of local operators, ⟨φ(x_1)...φ(x_n)⟩. However, string theory is a theory of quantum gravity. As a gauge theory with diffeomorphism invariance, there are no local, gauge-invariant observables. The position x_i is not a fixed label but a dynamical coordinate.

The only well-defined observables are S-matrix elements, which describe the scattering of states from the asymptotic past (t → -∞) to the asymptotic future (t → +∞). These external states live "at infinity," where the spacetime metric is fixed and non-fluctuating.

The crucial insight is that the state-operator correspondence connects these asymptotic states to vertex operator insertions on the worldsheet. An m-particle scattering amplitude is computed by a CFT correlation function of m vertex operators on a sphere.

As summarized in the reading, the tree-level m-particle amplitude is given by:

$A^{(m)}=g_s^{m-2}{⟨{\prod }_{i=1}^m\int d^2{z}_i{V}_i(p_i)⟩}_{S^2}$

This is a remarkable equation. It states that the quantum scattering amplitude of particles in D=26 dimensional spacetime is given by a 2D statistical mechanics problem on a sphere.

3. Calculating the 4-Tachyon Amplitude (Virasoro-Shapiro)

Let's compute the simplest non-trivial example: the elastic scattering of four closed-string tachyons.

The amplitude is given by the 4-point function of their vertex operators. We use the un-integrated vertex operator :e^{ip \cdot X(z)}:, and the expression for the amplitude involves integrating over the insertion points z_i and dividing by the volume of the residual gauge group.

$A^{(4)}\sim g_s^2\frac{1}{\text{Vol(SL(2,C))}}\int {\prod }_{i=1}^4{d}^2{z}_i{⟨{\prod }_{i=1}^4:e^{i{p}_i\cdot X(z_i,{\bar{z}}_i)}:⟩}_{S^2}$

This calculation proceeds in three steps.

Step 1: The Correlation Function

The correlator is for a free bosonic CFT. As you know from your QFT work with path integrals, such Gaussian integrals can be computed exactly. The result is determined by the two-point function (propagator) of the X field, which is \langle X^\mu(z) X^\nu(w) \rangle = -\frac{\alpha'}{2}\eta^{\mu\nu}\ln|z-w|^2.

The final expression for the n-point correlator is:
$⟨{\prod }_{i=1}^n:e^{i{p}_i\cdot X(z_i)}:⟩={\delta }^{(26)}\left({\sum }_{i=1}^n{p}_i\right){\prod }_{i<j}|z_i-z_j{|}^{{\alpha }^{\prime }{p}_i\cdot p_j}$
The delta function enforces overall momentum conservation and arises from the integration over the X field's zero mode x^\mu. The product term arises from the oscillator modes.

Step 2: Gauge Fixing the Conformal Symmetry

The worldsheet theory is conformally invariant. While we fixed the local symmetries to get to the conformal gauge, a residual global conformal symmetry remains. For the sphere, this is the group SL(2,C), whose transformations are Mobius transformations: z \to (az+b)/(cz+d).

This symmetry group allows us to map any three distinct points on the sphere to any other three points. We use this freedom to fix the positions of three of the four vertex operators, which enormously simplifies the integral. A standard choice is:

• z_1 \to \infty
• z_2 \to 0
• z_4 \to 1

The integral over z_1, z_2, z_4 is replaced by this gauge choice, and we divide by the volume of the SL(2,C) group to avoid overcounting. We are left with a single integral over the position of the third operator, z_3 = z.

Step 3: Evaluating the Final Integral

Putting everything together, and defining the Mandelstam variables for 2 \to 2 scattering:

• s = -(p_1 + p_2)^2 (center-of-mass energy squared)
• t = -(p_1 + p_3)^2 (momentum transfer squared)
• u = -(p_1 + p_4)^2

The amplitude reduces to a single integral over the complex plane:
$A^{(4)}\sim g_s^2{\delta }^{(26)}(\sum p_i)\int d^{2}z|z{|}^{{\alpha }^{\prime }{p}_2\cdot p_3}|1-z{|}^{{\alpha }^{\prime }{p}_3\cdot p_4}$

Using momentum conservation (p_1+p_2+p_3+p_4=0) and the tachyon on-shell condition (p_i^2 = 4/\alpha'), we can express the exponents in terms of s, t, and u. This leads to a standard integral that can be evaluated in terms of Euler's Gamma function, \Gamma(z).

The final result is the celebrated Virasoro-Shapiro amplitude:

$A^{(4)}\sim g_s^2{\delta }^{(26)}(\sum p_i)\frac{\Gamma (-1-\frac{{\alpha }^{\prime }s}{4})\Gamma (-1-\frac{{\alpha }^{\prime }t}{4})\Gamma (-1-\frac{{\alpha }^{\prime }u}{4})}{\Gamma (2+\frac{{\alpha }^{\prime }s}{4})\Gamma (2+\frac{{\alpha }^{\prime }t}{4})\Gamma (2+\frac{{\alpha }^{\prime }u}{4})}$

4. The Veneziano Amplitude for Open Strings

A similar calculation can be performed for the scattering of four open-string tachyons. The key differences are:

1. Worldsheet: Tree-level open string scattering maps to a disk, or equivalently, the upper half-plane H.
2. Vertex Operators: The operators are inserted on the boundary of the disk (the real axis of H).
3. Symmetry: The conformal Killing group that preserves the boundary is SL(2,R).

Following a parallel procedure, one arrives at the Veneziano amplitude:
$A_{open}^{(4)}\sim g_s\left[B(-{\alpha }^{\prime }s-1,-{\alpha }^{\prime }t-1)+(\text{2 other terms})\right]$
where B(x,y) = \Gamma(x)\Gamma(y)/\Gamma(x+y) is the Euler Beta function. This amplitude was discovered in 1968, before string theory was understood, and its remarkable properties (duality, infinite poles) were the genesis of the entire subject.

Conclusion

Congratulations on completing this course! You have journeyed from the abstract definition of conformal symmetry to a concrete calculation of a scattering amplitude in a theory of quantum gravity.

Key Takeaways from this Lesson:

• String interactions are described by a sum over worldsheet topologies, with tree-level scattering corresponding to a sphere.
• The physically meaningful observables are S-matrix elements, which are computed via CFT correlation functions of vertex operators using the state-operator map.
• For the sphere, the SL(2,C) global conformal symmetry allows us to fix the positions of three operators, simplifying the calculation to a single integral.
• The resulting Virasoro-Shapiro amplitude for 4-tachyon scattering is expressed in terms of Gamma functions and depends on the Mandelstam variables s, t, u.
• The physics encoded in the amplitude is profound:
  • Poles at s, t, u values corresponding to the masses of string states reveal that the amplitude contains the exchange of the entire infinite tower of string particles.
  • Duality means a single expression can be interpreted as a sum over s-channel or t-channel exchanges, a radical departure from field theory where diagrams are added.
  • Soft UV Behavior (exponential fall-off at high energy) indicates that the extended nature of the string resolves the UV divergences of point-particle gravity.

You have built a formidable theoretical foundation. The principles of conformal symmetry, operator algebras, and the bootstrap, which you have studied here, are at the very heart of modern theoretical physics. You are now exceptionally well-prepared to explore advanced topics like the AdS/CFT correspondence, superconformal theories, and the modern numerical bootstrap.