Lorentzian contours for tree-level string amplitudes

The manuscript under review presents integration contours in multiparticle string tree-level amplitudes which cure the divergences of the underlying moduli-space integrals in the physical kinematic region and make numerical evaluations accessible. The results of the paper close a long-standing gap in the literature and line up with a broader, successful research agenda of the authors to manifest unitarity properties of string amplitudes and to enable actual ``evaluations''. Related studies of integration contours in one-loop string amplitudes in the authors’ earlier work proved extremely valuable to harness the poor convergence properties of the relevant four-point moduli-space integrals at genus one. The achievements of the present paper at genus zero are indispensable for a systematic approach to controling analyting properties of multi-loop and -leg string amplitudes and making numerical evaluations possible.

The manuscript is very well-written and convinces through a concise presentation while providing the necessary background information. For instance, a sophisticated combinatorial structure arising from scattering an arbitrary numbers of external legs is explained in a transparent way. Several figures and plots neatly illustrate important points in the main text. A thoroughly documented mathematica notebook is provided as an ancillary file which demonstrates the practical use of the papers’ results and will be really helpful to colleagues working on related problems.

I can only offer minor suggestions for improvements in the final version and happily recommend the manuscript for publication in SciPost once the authors have considered the points below.

1) It’s a delight to read about the implications for fixed-angle high-energy limit of string amplitudes in section 3.5. I agree with the authors that "little concrete results are available for $n > 4$" points in this limit. It could further strengthen the manuscript if the authors already give a pointer to the high-energy discussion in the introduction, make it more visible by adjusting the subsection headline(s) and slightly expand section 4.4 by comments on the high-energy limit in the closed-string case.

2) When the authors mention Stokes phenomena at higher genus in the introduction, it would be useful to give a reference at the end of that paragraph.

3) The Moebius transformations relevant for the open string form an $SL(2,\mathbb R)$, not an $SL(2,\mathbb C)$. Please adjust this in (2.1), in the fifth line below, in (2.13) and potentially other places (below (4.3), the contour $\gamma^{\rm L}_\rho$ is correctly quotiented by $SL(2,\mathbb R)$).

4) Even though one example is implicit in (4.9), it would not hurt to spell out one or two instances of the $S[ \rho | \tau ]$ matrix (and possibly ${\bf H}$) in the authors’ basis choice (4.5) at some point in section 4.2.

5) In the paragraph on ``Open-closed scattering’’ in the conclusion, also 0907.2211 should be cited along with references [35, 36].

6) Please fix the typos on the word "way" (the 1st line of section 3) and "cloed string’’ (2nd line of section 2.2).

Ask for minor revision