Radiative corrections and Monte Carlo tools for low-energy hadronic cross sections in $e^+ e^-$ collisions

We thank the referee for their input and suggestions to improve our article. They have listed five points and requested two statements to be added. Below we address these points and describe our changes. In addition, we took the opportunity to add two more citations, the new [344,345].

We hope we have correctly interpreted the suggestions made by the referee and that with the modifications we have made, the article is now acceptable for publication.

[1) general theoretical introduction is missing:]

Since the whole Section 3 is a theoretical introduction to the computations relevant for this article (low-energy electron-positron scattering), we presume the statement refers to the theory of how to extract the HVP, as detailed in point 2 of the referee. Thus, we treat point 1 and point 2 together.

[2) In particular what is the relation between vacuum polarization and bremsstrahlung. It should be mentioned, including comments on dispersion relations, especially in context of effects beyond first order.]

No doubt, this is an important question. However, as stated in the introduction

``The core purpose of this work is to assess the importance of various contributions in the theoretical description of fully differential cross sections. More concretely, in Phase~I we are concerned with the 2 -> 2 processes ... (1.1) and (1.2) ...''

our aim is to describe low-energy electron-positron scattering. The muon anomalous magnetic moment (AMM) is just one (admittedly very important) of several motivations to do so. We do not think it is helpful to reiterate in detail the procedure on how to extract the HVP for the AMM. This is done for example in Section 2 of [6] and repeating these arguments would distract from the focus of our review. We have added a further remark at the beginning of Section 2 (page 4) to point the interested reader to the extended discussion in [6].

[3) Logarithms (at least some of them) appear from phase space integration over the regions of detector sensitivity. This should be mentioned and consequences for observable build indicated. Especially for the one involving detector granularity. Known since long limitations of fixed order calculations (cancelation of infrared virtual and real emission infinities, should be at least mentioned and better explained.]

The IR cancellation is mentioned between (3.4) and (3.5). However, we agree that this discussion was very short and we have extended it substantially. In particular we added remarks about IR safety, the resolution of detectors, and soft logarithms and refer to Section 3.2.3 for a more detailed discussion. We also added a remark regarding the resummation of collinear logarithms at this place and refer to the later Sections 3.2.1 and 3.2.2.

[4) these are the most important points. Some recall of references, in particular to work of Denner should be mentioned in context of general principles of use of dispersion relations and separation of predictions for QED and hard interaction part (effectively running of alpha\_QED). Some of this is covered in programs descriptions. It would be OK to list the topics and write that every program has the issues covered by the authors. ]

In Section 3, after (3.2) we mention VP and their resummation, also referring to Section 5.1.1, where a more detailed discussion is given. At the very end of Section 5.1.1 we have slightly extended the statement that the treatment of the VP varies between the different codes, as described in Section 4. Furthermore, we have added a citation to Denner et al. (the new [348]).5)

[5) It would be nice if authors of individual programs contributions would extend descriptions of their theoretical basis, but I would not expect this to be possible.]

Some codes used in this article have a history lasting decades, with a corresponding vast literature, where all details have been explained. This article was meant to contrast the core ideas of the various codes and give a basic overview. We hope this serves as a first introduction of the codes and would invite readers interested in
code-specific details to the references given. Since not all original developers of the code are among the authors, a detailed description of all codes is not really possible. Doing it for only some codes introduces an unwanted asymmetry in the description.

[After modifications and statement that responsibility of particular overall ambiguities has to be consulted with the authors of individual programs contributions, and that in general ambiguities are bound to be observable dependent, report may be accepted.]

Regarding 'consultation with the authors of individual programs' we have added a statement in the second paragraph of Section 4. Regarding 'ambiguities are observable dependent' we have added remarks on observable dependence in the first paragraph of Section 5.