One-loop four-graviton string amplitude at finite $\alpha'$

This article builds on the authors' previous work, in which they introduced a Rademacher contour deformation that renders the moduli-space integral for one-loop scattering amplitudes in flat-space string theory explicitly convergent and numerically efficient.
In this paper, this framework is implemented in a public C++ code, and the authors present both 1) a detailed derivation of the reduction of the original eight-dimensional torus integral to a four-dimensional integral with explicit summations, and 2) numerical results obtained from evaluating this representation.
The ability to compute the one-loop four-graviton amplitude at intermediate values of the Mandelstam variable $s$, directly in physical kinematics and exact in $\alpha'$, is already a significant technical achievement.

Using this approach, the authors investigate the high-energy, fixed-angle behavior of the amplitude.
The classic Gross-Mende saddle-point approximation captures the qualitative envelope of the result, but it disagrees in detail with the numerical data, specifically due to oscillations not captured by the saddle approximation.

They interpret this result as evidence for the presence of additional nontrivial Lorentzian saddles that are not accounted for in the Gross-Mende Euclidean saddle-point analysis. This is explained in a different article by the authors.

Beyond the typos pointed out by the reviewer above, I found no further ones or other confusing points. The paper is very well-written.

I was also able to download the provided code and reproduce parts of figure 1 for the real part of the amplitude, using slightly smaller truncation values $c_{\rm max}$ than those reported in the paper. Trivial suggestion: in the sample run in section 2, it might be useful to include the option "-mode cumulative" (though this is explained in the documentation).

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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:

1. 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.

2. 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 $s<1$ ?

3. In Figure 3, it may be useful to refer to Eq 4.12a for the angular independence appearing at $s=1$

4. The notation DRes in Figure 4 is obscure until one reaches section 4.3

5. 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 ?

6. In the last paragraph of Section 2, "A more direct comparison" should probably refer to Figure 6, rather than 5

7. Figure 7 is somewhat cryptic. Could the authors explain in what plane is the figure drawn ?

8. Above Eq 3.4: "The general The Rademacher formula"

9. Below Eq 3.21: "sawooth" should be sawtooth

10. Below 3.25: "to the to the"

11. Below 3.52: In "only terms with $P<0$ and $P'< 0$ are polar terms and contribute", the inequality should probably be reversed

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