Higgs associated production with a vector decaying to two fermions in the geoSMEFT

1- Rigorous computation of results
2-Results relevant to experiment
3-Comprehensive

1- Theoretical criteria seems arbitrary
2-Use of results for more general cases is unclear

The paper presents the inclusive calculation of a Higgs produced in association with vector bosons for a selected array of dimension 8 operators. Results are relevant for experimental searches but the paper could be improved in a number of ways

1- The authors say, "That the geoSMEFT classifies all three point functions is a basis dependent statement". This is unclear to me, is there a different geoSMEFT for every different basis?

2- The programme is spelled out in 2.2. In essence, as I understand it, the geoSMEFT is used to select or discard dimension 8 operators. However what makes geoSMEFT a privileged set up to have such a say; does it offer for example any improvement on the convergence of the 1/\Lambda expansion? Does it match more naturally to a type of models?

3-In addition to a number of reasons, the authors say that the opertor in eq 12 can be discarded because it leads to ghosts. How can a perturbative treatment of an EFT, specifically designed to include only certain degrees of freedom, have its spectrum modified by the inclusion of an operator?

4-Results are given in tables such as tab~4. Can these results be combined if one wants to include both dimension 6 linear and squared contributions and dimension 8 or would these require new computations?

Address the points in the report

1- Ready to use result, can be used to provide better constrain versus current results
2-Non trivial calculation
3-Supporting material in form of Mathematica notebook

1-Some approximation could be too strong for specific purposes. For instance, CP phase is absence for unity CKM matrix.
2-Sign resolution absence; this is acknowledged in the manuscript.
3-Some expressions can be clarified by better wording.

This manuscript applies a standard approach, geoSMEFT, and include dimension-eight operators to derive the cross section for Higgs associated production with a W± or Z bosons in the Standard Model Effective Field Theory (SMEFT) expansion. These results are valuable for both theorists and experimentalists, since they can readily be applied to obtain experimental bounds on appropriate operators.

In section 2, the authors explains the geoSMEFT tool used throughout the manuscript. Section 3 derive the effective Lagrangian. Section 4 specializes on Higgs' neutral current and charged current channels. This section is helpful in justifying the necessity of including dimension-eight operators.

Overall, this is non-trivial and important result. We would be happy to recommend this manuscript for publication.

Some minor errors the authors may consider fixing:
- Section 1, "... the "inverse problem,"..." put comma outside the phrase.
- Same sentence, what does "great degeneracy" mean?
- In some places, "Wilson Coefficients" should be "Wilson coefficients".
- "Mathematic" should be "Mathematica"
- Section 4, "... Details ... function integration is discussed ..." should be "are discussed"
- Section 5, "...with a W+- or Z bosons...", omit "a".

1-The paper contains detailed information on the contribution of a comprehensive set of new operator structures to Higgs boson associated production at the LHC
2-The discussion is well framed around the different helicity structures in the processes
3-The results are presented in terms of the couplings of the Lagrangian in the broken phase (eqs 17 through 20), making them flexible with respect to the chosen (geo)SMEFT parametrization
4-EFT effects on W/Z mass and width are included
5-The authors provide their final numerical results in a useful format in ancillary files

1-The work does not yet realise much of the useful impact of these calculations, i.e., understanding the constraints from data on the different operators, and how they might be optimised. However, I stress that the work is a significant step towards this.
2-Setting the CKM to the unit matrix may be too strong an assumption for the charged current processes

The work provides a comprehensive survey of new physics effects in Higgs boson associated production at the LHC, and opens up the possibility of EFT analyses of this important process using much more granular kinematic information. I believe this therefore qualifies for publication in SciPost Physics under expectation 3 of its acceptance criteria. Regarding the general criteria, the paper is overall clear and some efforts to assist reproducibility have been made.

I ask for a few minor clarifications below.

1-Could the authors provide in the text a back-of-the-envelope estimate of the accuracy of their CKM=identity matrix approximation? (For example, roughly how large would a u-sbar initiated contribution to the first row of Table 6 be if Cabibbo mixing were turned on? Are the relative contributions of SM and EFT terms likely to change much relative to the u-dbar initiated case?)
2-Is it understood why $c_{HW}^{(6)}$ gives a much smaller contribution in the neutral current case? If so, it would be helpful to include this in the text.
Some typos:
3-"in associated" -> "in association" in first sentence of abstract
4-unbar one of the fermions in the last line of eq (19)
5-"Mathematic"->"Mathematica" on page 13