Complex saddles and time-delay of Veneziano amplitude

Abstract: String scattering amplitudes are typically expressed in formal integrals which diverge in physical kinematic regions. Recently the problem of divergence was cured by redefining integration contours. In this paper, we apply the new integration contour to evaluate time-delay of the Veneziano amplitude. Thimble analysis of the new integration contour tells us that infinitely many complex saddles contribute to the new integral. Each complex saddle is interpreted as a string path which enters into the scattering region at different Lorentzian time.