Natural Boundaries for Scattering Amplitudes

I would like to thank both referees for the time spent reviewing the paper and their reports.

First, I clarified two misunderstandings brought up by Referee 2. Second, I revised the paper to incorporate the comments/questions they raised and included an itemized list of changes below, which I believe improved readability of the paper.

1. The referee asks how do we know the dense set of singularities lies on the second sheet. It's an important concern. Addressing it was the main result of the paper, with details given in Sec. 3. It wouldn't have been possible to claim to have found a dense set of singularities unless they were all on the same sheet.

Summarizing the logic, analytic continuation of elastic unitarity guarantees that the singularities of the type studied in this paper have to lie on \emph{at least one} of the two sheets of the lightest-threshold branch cut in the variable $s$. Since it's known they do not appear on the first, they have to be located on the second one. These arguments by no means preclude the existence of ladder-type singularities on different sheets of other branch cuts.

1. The referee proposes a mathematical construction for defining perturbative Feynman or time-ordered diagrams on other sheets, even if they are not related to the physical S-matrix by analytic continuation. As emphasized on page 21, the moment we commit to perturbation theory, there's really no obstruction to analytic continuation across any cut, as far as it's currently known, and hence also no puzzle of how to define perturbative S-matrix elements on different sheets. Performing such an analytic continuation in practice is a more complicated problem. (As an example, after plugging in a single Feynman diagram into Eq. (2.24), it truncates and gives an explicit formula for the analytic continuation on the second sheet of the lightest-threshold branch cut.) Still, I would like to emphasize that no Feynman diagrams appear in this paper.

It's also worth noting that the referee's proposal relies on the assumption that selecting $\pm i\varepsilon$'s for propagators is enough to explore all the sheets, or at least the ones that would have been otherwise disconnected in the full S-matrix. One can check on explicit examples, for instance the bubble or box diagrams, that this is not the case: the space of sheets is in general much larger than the space of choices of $i\varepsilon$'s. I would agree that \emph{certain} monodromies around singularities lying in the neighborhood of the physical kinematics can be computed this way.

1. The referee asks about an example of a function with a natural boundary defined by its integral representation. Such examples were mentioned on page 3: elliptic functions (for example Jacobi theta or Dedekind functions) have a dense set of singularities for every rational value of the modular parameter $\tau$, forming a natural boundary. They are commonly defined by either infinite sum or product representations, differential equations, and integrals.

2. As suggested by the referee, I added the clarification that the coefficient on the RHS of Eq. (2.11) is fixed by unitarity, but is immaterial to later discussion and hence chosen to be $4$ for convenience.

3. Below Eq. (2.25), I clarified what is meant by assuming Mandelstam analyticity.

4. This is a good catch. The sentence was meant to say "the analytic continuation of $\mathrm{Im}\, T(s,z)$".

5. Corrected.

6. That's right, there were two typos below Eq. (3.6).

7. I agree. The advantage of using Eq. (1.1) is that it gives a necessary, not only sufficient, condition for anomalous thresholds, since the singularities on the RHS of the unitarity equation have to be also singularities of the LHS, so the sheet on which the singularity lies can be determined.

8. I expanded the description of Fig. A.1, such that now all the information needed to understand it is contained in the caption, on top of the explanations already given in the main text.

9. Corrected.

In addition to the above changes, I wasn't entirely happy with the presentation of the arguments leading to Eq. (2.10), so I rewrote the text around it to be more accessible.