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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 submitted: 2023-03-10 22:17
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.

Current status:
Has been resubmitted

Reports on this Submission

Report 2 by Carlo Angelantonj on 2023-7-7 (Invited Report)

  • Cite as: Carlo Angelantonj, Report on arXiv:2302.12733v1, delivered 2023-07-07, doi: 10.21468/SciPost.Report.7467


First and foremost I would to apologise with the Authors for the delay in providing my report.
The manuscript deals with the evaluation of one-loop scattering amplitudes in the open sector of the type I superstring which is valid for any finite value of $\alpha '$ and not just in the low energy regime. The integration of the Riemann surfaces is typically ill-defined and requires a suitable implementation of the $i\epsilon$ prescription. To this end, The Authors suggest deform the integration contour a la Rademacher, both for planar and non-planar amplitudes. As a result, the essential singularity at $Im (\tau ) =0$ is not reached following vertical lines but horizontally which makes the evaluation consistent with causality and unitarity.
The Authors discuss the deformation of the contour and the steps required to perform the integration over the moduli parameter in great detail and apply it to the evaluation of two-point and four-point amplitudes, for which they provide (for the first time) concrete expressions which are valid at finite string tension, and which are amenable to direct numerical evaluation.
The paper is well written, technically correct, and provides new insights in the proper evaluation of open-string scattering amplitudes. I recommend it for publication on SciPost.

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  • significance: -
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Report 1 by Eric D'Hoker on 2023-6-6 (Invited Report)

  • Cite as: Eric D'Hoker, Report on arXiv:2302.12733v1, delivered 2023-06-06, doi: 10.21468/SciPost.Report.7316


The paper provides a very detailed construction of the convergent sum for the annulus and Mobius one-loop open superstring amplitudes for gauge group SO(32). The general method of contour deformation used here follows Witten's general proposal for implementing an "i epsilon" prescription in string amplitudes, but also extends this proposal and renders it more concrete. While the imaginary part of the one-loop open string amplitude may be evaluated using alternative methods, its real part is calculated here for the first time, as far as I know. The integration of the open string modulus for the combined topologies is related to an integration over the Rademacher contour and is evaluated using some beautiful known methods of analytic number theory. The result is a concrete formula by which the amplitude can be evaluated numerically for real Mandelstam variables. This result provides novel insight into string amplitudes in the regime that was most difficult to access before this work, namely in an intermediate regime of energies^2 on the order of 1/alpha', away from both low and ultra-high energies.

My one comments is as follows. I am puzzled by the authors' statements concerning old results on the analytic continuation of the closed one-loop amplitude on page 22. Specifically, that the analytic continuation methods used then and there are difficult to extend beyond genus zero. Actually, if one allows the amplitudes to be considered as functions of complex momenta, then the integral representation for the one-loop four-point amplitude is convergent for all purely imaginary values of the Mandelstam variables and provides a perfectly fine starting point on which to build the analytic continuation. While the analytic continuation procedure may be difficult to implement (as the authors state) it was in fact implemented for the one-loop closed superstring four massless string amplitude precisely in those old papers. The procedure produced the imaginary part of the amplitude with four massless NS bosons, and explicit formulas for the decay widths of the massive string states that occur in the amplitude. What those old papers did not obtain is the real part of the amplitude.

In summary, the paper under consideration presents innovative results in the area of string amplitudes, it is well-written and detailed, and will be of lasting value. I strongly recommend it for publication in SciPost.

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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