Complex Saddles of Truncated String Amplitudes

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.