Natural Boundaries for Scattering Amplitudes

1- The paper is well-written
2- The arguments are presented in detail and rigorously

1- While the existence of a dense set of singularities seems rigorous enough, it is not clear to me that these singularities belong to the second sheet. If they don't then it could still be possible to analytically continue through the apparent barrier on the second sheet since some (most?) of these singular points actually belong to other sheets.
2- Another way to express the same thing is that there is an infinite number of choices of $\pm i \epsilon$ prescriptions for all the propagators, or an infinite number of choices of signs of energies in cut propagators. Each one of these choices corresponds to picking another sheet as the starting point of analytic continuation. Somehow the arguments in this paper would say that all these other sheets are not all connected by analytic continuation, but I believe they all exist and can be naturally associated to the S-matrix element under study. In other words, it seems to me that even if all these singularities are actually on the second sheet, in some sense the amplitude can be defined on other sheets, even though this will not be via an analytic continuation through the "boundary".

Happy New Year!

The paper is well-written and draws attention to phenomena that arise in the non-pertubative amplitudes which are not apparent when studying the perturbative amplitudes. This is valuable, even if similar points have been made in older, mostly forgotten, literature. I recommend the paper for publication, but I would like the author to spend more time on the discussion of whether the singularities actually belong to the second sheet, following the discussion in Weaknesses above.

1- It would be interesting to see if there are some examples functions (perhaps in a single variable, to keep things simple) defined via integrals which have singularity barriers. The examples I know (and the ones that are presented in the paper) are defined by power series.

2- Below eq. 2.11 when discussing the normalization please mention that the normalization is fixed by unitarity. I believe that there should be some $(-2 \pi i)^2$ terms missing and possibly some $\frac 1 2$ from the integration over the phase space of identical particles. I don't want to demand a change here, but only to preempt some people's puzzlement at the equation.

3- Below eq. 2.25, please explain in a bit more detail the notion of Mandelstam analyticity.

4- Also below eq. 2.25, $\Im T(s, z)$ is only real-analytic, right?

5- Below eq. 3.1, missing ``continue'' in the phrase starting with ``As we analytically...''.

6- Below eq. 3.6, in the definition of $\theta_1^*$ perhaps what was meant was $\theta_1^* = \theta_{k_1}^t(s)$?

7- In the last paragraph of page 15, ``rephrased Landau analysis in a non-perturbative setting'', but the Landau analysis does not have to be perturbative in the first place. Landau made this point in his original paper. Indeed one can consider Landau diagrams with vertices being non-perturbative S-matrix elements. These kind of considerations have been used in two-dimensional integrable theories.

8- Explain fig. A.1 in more detail. What is the white disk? Presumably $|s| = R(t)$ and $\Im s > 0$ is the dashed line...

9- At the bottom of page 24, ``Since the proofs of [] breaks down'', replace by ``break down''.

This article has important results for study of analyticity of scattering amplitude. Although this area is currently not popular, there important issues to be examined. The author has systematically studied several
problems.

I really do not find serious weaknesses in the paper.

The contents of the article is quite interesting and definitely this work deserves publication in your journal.

No major revision is required.