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Evaluating one-loop string amplitudes

by Lorenz Eberhardt, Sebastian Mizera

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Submission summary

Authors (as registered SciPost users): Lorenz Eberhardt · Sebastian Mizera
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-08-11
Date submitted: 2023-07-13 03:25
Submitted by: Mizera, Sebastian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We evaluate one-loop open-string amplitudes at finite $\alpha'$ for the first time. Our method involves a deformation of the integration contour over the modular parameter $\tau$ to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop four-point amplitudes in type-I superstring theory, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants $s$ and $t$ and directly verify long-standing conjectures about their behaviour at high energies.

Published as SciPost Phys. 15, 119 (2023)

Author comments upon resubmission

We would like to thank both referees for their valuable feedback. In the revised version, we describe previous results on computing the imaginary part of the one-loop amplitude more clearly in Sec. 3.1.

List of changes

In order to address the concern of Referee 1, we modified the text on p. 22 to more clearly explain previous contributions of Ref. [30-32]. The text now refers to specific results as follows:

"[...] This approach was used in the old literature on string amplitudes, see, e.g., [30–32], but is difficult to make practical beyond genus zero. The simple reason is that in order to perform analytic continuation, one needs an analytic expression to begin with and these are very hard to find for such an intricate object as a string scattering amplitude. However, such an analytic continuation was successfully carried out for the imaginary part of the one-loop amplitude in type II strings [32, Theorem 2] and heterotic strings [32, Theorem 4], and produced explicit expressions for the decay widths of massive string states [32, Theorem 5]."

Reports on this Submission

Report 1 by Eric D'Hoker on 2023-7-28 (Invited Report)


I strongly recommend the paper for publication in its present form.

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