Parametrisation and dictionary for CP violating Higgs boson interactions

This is very important and useful document for the Higgs CPV study program as one of the HXSWG report, and hence I would recommend this article for publication after points below are clarified/modified.

1. add references (1212.0843, 1306.6464) into [13,21,42,43]?
2. $m_W$ in eq.2, instead of $m_w$.
3. The subscripts of the Lagrangians should be unified as, e.g. ${\cal L}_{\rm hff}$ or ${\cal L}_{hff}$.
4. $CP$ in Sec.3.2 should be CP.
5. clarify the H, G, W, and B fields around eq.26.
6. $U(1)$ should be $U(1)_Y$ in page 13?
7. ${\cal G}_{\mu\nu}$ should be ${G}_{\mu\nu}$.
8. remove unnecessary "+" in eq.36.
9. $b_c$ should be $b_C$ in eq.37.
10. miss $\Delta$ in eq.39?
11. clarify $\Lambda$ in eq.40.
12. put the equation number for $N_2^\ell$.
13. clarify SILH and HISZ.
14. In the 2nd paragraph from the last in Sec.4, the authors says " we all discussed....", but I do not see any quantitative/qualitative discussions about the current experimental constrains and future prospects in this article.

Publish (easily meets expectations and criteria for this Journal; among top 50%)

1 - Full and well-worked-out mapping between different BSM parameterizations of CP violation in the Higgs sector

2 - Contains exactly the right amount of information / discussion

1 - Some important remarks are a bit hidden in the text

2- The dictionary could be shown more prominently and made slightly more convenient

The paper summarizes different parameterizations for CP-violating BSM effects in the Higgs sector. It also provides a dictionary between these parameterizations and, for the first time, includes a mapping of CPV operators from HEFT to the Higgs basis.

The dictionary provided is very relevant for studies of CPV in the Higgs sector, as it nicely fills a gap in the field. It contributes well to improved communication between experimentalists and theorists.

I recommend that the paper is accepted for publication in SciPost Physics Community Reports after applying some small modifications, as detailed below.

General:

1 - Hyphens are used inconsistently (CP violating / CP-violating), ''CP'' is sometimes italic and sometimes not 2 - There are some minor grammatical errors, please check for them. For example, in the introduction: ''and is thus necessary to interpret the results of the former into the parameters of the latter'' -> ''and it is thus necessary to interpret the results of the former in terms of the parameters of the latter''

Introduction:

3 - Paragraph 2: The term ''CP-insensitive observables'' is slightly confusing as these variables do offer CP sensitivity (as also stated by the authors), just not to any interference. I would then start a new paragraph after this sentence. 4 - Last sentence of the paragraph: Remove ''...in the Higgs sector...''

Chapter 2:

5 - Paragraph 5: Maybe also mention problems with the top-Yukawa coupling 6 - Last paragraph: It is unclear what you mean by ''and higher order corrections will become necessary''

Section 3.1:

7 - There is an in-line equation running out of border just below Eq.(1) 8 - In the following sentence, please check the definition of the gluon field strength tensor, it does not agree with Ref. [83] in the Higgs basis 9 - Eq.(12) lists also coefficients for couplings not including a Higgs, I am not sure those are needed? 10 - Footnote 5: Maybe spell out ''anomalous'' since the abbreviation was not defined

Section 3.2:

11 - I think this is a very important section for the community. However, some statements about the validity of the approaches and information the experiments should provide gets lost a bit in the text. From the title of this section and its description in the introduction, it sounds like only the kappa and angle parameters will be introduced. I would ask the authors to make this discussion more prominent in the paper, e.g., by either making a dedicated subsection, changing the title of this section, or repeating important claims in the conclusions.
12 - Last paragraph: ''US'' -> ''as in the''?

Section 3.4:

13 - First paragraph: Why don't you also cite Ref.[117] alongside Ref.[115,116,118] here? 14 - There could be a bit more vertical separation in the listing of all the HEFT operators

Section 3.5:

15 - These are the main results of your paper and will be what people are looking for. I would strongly suggest making this its own chapter and giving it a clearer title, e.g. ''Dictionary for mapping between the Higgs basis, SMEFT and HEFT'' 16 - I think for convenience it would be great if you could link the respective equations that the operators appear in. So for example for Eq.(50): ''the correspondence between the operators appearing in Eq.(12) and Eq.(27) is given by...''

Conclusions:

17 - Include paragraph 2 in paragraph 1 18 - As mentioned above, I think it could be helpful to explicitly repeat here some of the statements of Section 3.2 about the validity of the approaches and information transfer between experimentalists and theorists

Ask for minor revision