Unitarity Cuts of the Worldsheet

Abstract

We compute the imaginary parts of genus-one string scattering amplitudes. Following Witten's $i\epsilon$ prescription for the integration contour on the moduli space of worldsheets, we give a general algorithm for computing unitarity cuts of the annulus, Möbius strip, and torus topologies exactly in ${\alpha }^{\prime }$. With the help of tropical analysis, we show how the intricate pattern of thresholds (normal and anomalous) opening up arises from the worldsheet computation. The result is a manifestly-convergent representation of the imaginary parts of amplitudes, which has the analytic form expected from Cutkosky rules in field theory, but bypasses the need for performing laborious sums over the intermediate states. We use this representation to study various physical aspects of string amplitudes, including their behavior in the $(s,t)$ plane, exponential suppression, decay widths of massive strings, total cross section, and low-energy expansions. We find that planar annulus amplitudes exhibit a version of low-spin dominance: at any finite energy, only a finite number of low partial-wave spins give an appreciable contribution to the imaginary part.

87 pages

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• Lorentzian contours for tree-level string amplitudes
• Non-analytic terms of string amplitudes from partial waves
• Regge Limit of One-Loop String Amplitudes
• Tropological Sigma Models
• Rademacher expansion of modular integrals
• One-loop four-graviton string amplitude at finite ${\alpha }^{\prime }$