Complex Saddles of Truncated String Amplitudes

TL;DR

This paper investigates the complex saddle points contributing to high-energy string scattering amplitudes, using QFT toy models to understand their structure and conjecturing fractal behavior in the amplitudes.

Contribution

It constructs QFT models mimicking string spectra to identify contributing complex saddles and explores their oscillatory effects and potential fractal nature of amplitudes.

Findings

Complex saddles contribute oscillatory terms to amplitudes.

Higher genus corrections involve infinitely many stringy excitations.

String amplitudes may approach multi-fractal functions at high energies.

Abstract

String scattering amplitudes in the high energy asymptotic region have been studied by saddle point approximation. Recently, it was pointed out that infinitely many complex saddles contribute to string amplitudes even at tree-level after redefining the original formal integration contour so that the new contour analytically continues amplitudes appropriately. It is a challenging problem to identify which saddles contribute to higher genus corrections of string amplitudes. In this paper, we construct QFT toy models which have the same infinite mass tower as string amplitudes but ignoring degeneracies. Their higher loop Feynman diagrams are evaluated by identifying their contributing complex saddles. We find that the saddles associated with infinitely many stringy excitations provide highly oscillatory terms to the amplitudes. We conjecture that string amplitudes, as functions of momenta, approach multi-fractal functions in the high energy asymptotic regions if higher genus contributions are fully included. Their fractal dimensions should be determined purely by the type of string theory and the spacetime dimension where string scatterings occur.