Abstract
Motivated by the desire to understand chaos in the S-matrix of string theory, we study tree level scattering amplitudes involving highly excited strings. While the amplitudes for scattering of light strings have been a hallmark of string theory since its early days, scattering of excited strings has been far less studied. Recent results on black hole chaos, combined with the correspondence principle between black holes and strings, suggest that the amplitudes have a rich structure. We review the procedure by which an excited string is formed by repeatedly scattering photons off of an initial tachyon (the DDF formalism). We compute the scattering amplitude of one arbitrary excited string and any number of tachyons in bosonic string theory. At high energies and high mass excited state these amplitudes are determined by a saddle-point in the integration over the positions of the string vertex operators on the sphere (or the upper half plane), thus yielding a generalization of the “scattering equations”. We find a compact expression for the amplitude of an excited string decaying into two tachyons, and study its properties for a generic excited string. We find the amplitude is highly erratic as a function of both the precise excited string state and of the tachyon scattering angle relative to its polarization, a sign of chaos.
Highlights
Chaotic systems are known to have a rich array of properties: are there qualitative or quantitative statements that can be made about black hole chaos that go beyond this elementary diagnostic of Lyaponuv behavior?
We end with a summary of the results for the vertex operator of an excited string state
The vertex operator for an excited string of momentum p is some polynomial of derivatives of X, ∂kXμ, multiplying eip·X
Summary
Our proposal is that both of these questions may have answers in the context of scattering of highly excited strings. The string coupling at the correspondence point - where the string turns into a black hole — is small, suggesting that we may be able to study black hole chaos by studying string chaos within string perturbation theory. This is an extremely fortuitous situation wherein we have a weakly interacting system with a enormous number of almost stable resonances that can mimic the microstates of a black hole and exhibit chaos. In the rest of the introduction we discuss these motivations and background in more detail, overview how we will compute amplitudes with excited strings, mention what we think can be computed in future work, and give an outline of the paper
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