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One-loop four-graviton string amplitude at finite $\alpha'$
by Marco Maria Baccianti, Lorenz Eberhardt, Sebastian Mizera
Submission summary
| Authors (as registered SciPost users): | Lorenz Eberhardt · Sebastian Mizera |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2507.22105v1 (pdf) |
| Date submitted: | Aug. 7, 2025, 9:09 p.m. |
| Submitted by: | Sebastian Mizera |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We evaluate the one-loop four-graviton scattering amplitude in type-II superstring theory exactly in $\alpha'$. This result is achieved by combining physical insights into the $i\varepsilon$ prescription in string theory with a new technical application of the Rademacher expansion of modular integrals. We provide an implementation of our formula in $\texttt{C++}$ and use it to study the behavior of the amplitude at finite $\alpha'$ and in different kinematic regimes. Our analysis reveals a tension between explicit computations and the saddle point analysis of Gross and Mende in the high-energy limit and suggests the presence of additional saddle points.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
This article obtains a new explicit formula for the four-graviton amplitude in type II strings, which appears to be convergent (though not absolutely so) at high enough energy. This is in contrast to the usual integral representation over the punctured torus moduli space, which is diverges everywhere for physical kinematics. This result is based on a stringy implementation of the Feynman $i\epsilon$ prescription, as well as on the Rademacher representation for the integral over the torus modulus developed by the same authors in the context of the vacuum amplitude. The authors provide a C++ implementation of the formula and check that its imaginary part coincides (within numerical errors) with a unitary-based formula developed in their earlier work on the Regge limit of one-loop amplitudes, giving a strong check on its validity. In the high energy, fixed angle limit, they find agreement with the Gross-Mende prediction, up to small oscillations as a function of the angle (figure 4 in their work), which suggests that multiple saddle points contribute, degenerate with the one identified long ago by Gross and Mende. An analysis of these new saddle points is deferred to a forthcoming paper, not available at the time of this review.
This article is very well written and represents striking progress on a problem with a long history. In my opinion it definitely deserves to be published, after correcting misprints and clarifying a few points:
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The standard relation between Mandelstam variables $t/s=(\cos\theta-1)/2$ is quoted with a minus sign error on multiple times (started below Eq. 1.1) For $\theta=\pi/3$, the ratio is $-1/4$, consistent with the caption in Figure 1, so I'm confident this is just a recurrent typo.
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In Figure 2, numerical uncertainties in the real part seem to be visible only in the region $1.5<s<2$, whereas one would expect large deviations at small energies. Is this because a different formula is used to evaluate the real part for $s<1$ ?
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In Figure 3, it may be useful to refer to Eq 4.12a for the angular independence appearing at $s=1$
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The notation DRes in Figure 4 is obscure until one reaches section 4.3
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In Figure 5, numerical uncertainties start appearing at $s>1$, and peak around $s=1.4$, significantly lower than in Figure 2, could the authors explain why ?
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In the last paragraph of Section 2, "A more direct comparison" should probably refer to Figure 6, rather than 5
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Figure 7 is somewhat cryptic. Could the authors explain in what plane is the figure drawn ?
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Above Eq 3.4: "The general The Rademacher formula"
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Below Eq 3.21: "sawooth" should be sawtooth
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Below 3.25: "to the to the"
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Below 3.52: In "only terms with $P<0$ and $P'< 0$ are polar terms and contribute", the inequality should probably be reversed
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)