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

Dear Referees, We apologize for the delay in response, the referee reports were received over the course of months and we wished to delay response until the final one was receive and the editor had made their first recommendation. We hope we have addressed your concerns.

Response to Referee 1: We thank the referee for their constructive comments and have attempted to respond to all concerns. In particular their primary concern about the up-strange contributions: 1) We have added Appendix F where we reproduce Table 6 for the u \bar s contribution. The appendix is referenced in the text . We compare the size of the integrations with the u \bar d case, this multiplied by the relevant insertion of VCKM for SM-like W couplings, under an 'MFV'-like assumption made in the main text, as well as comparing the MFV-like case to the contribution from c \bar s. In doing so we find that under the MFV-like assumption these terms are negligible compared with the c \bar s contribution (order 4-40% of the c \bar s contributions which are already small compared to u \bar d). We stress again in the appendix that a proper consideration of off diagonal couplings of the Ws requires a full dimension-eight analysis of CKM data and a proper definition of the CKM matrix in the presence of all these couplings.

2) Could the referee clarify in what sense cHW6 gives a smaller contribution? In the ancillary files, the contribution to the NC process (normalized to the SM, alpha scheme, with the WCs in units of the vev) is weighted by 15.7 while in the CC case it is weighted by 14.1.

3) We have corrected this mistake.

4) We have corrected this mistake.

5) We have corrected this mistake.

Ref 2: We thank the referee for their careful reading of the article, we have made the following changes corresponding to the minor comments and numbered according to their order in the report: 1) We have retained that the commas should be placed inside the quotes as this is our understanding of punctuation placement in quotes regardless of whether they are a part of the quote. 2) We changed the sentence to drop the phrase "great degeneracy" and instead more carefully explain what we mean: "The ``inverse problem,'' or that there are a large number of UV completions of the SM which imprint on the same subset of operators at dimension six." 3) We searched the document for "Coefficients" with capital "C" and replaced the instances found with "coefficients" 4) We found and replaced one instance of "Mathematic" with "Mathematica" 5) We changed the language to "Details of the phase space and parton distribution function integrations are discussed" 6) We opted for keeping "a" but changed "bosons" to "boson" as only one W or Z is present in the given processes.

Ref 3 We thank the referee for their comments, particularly on the language surrounding the geoSMEFT. We understand the geoSMEFT requires careful explanation and are trying to improve with each iteration. 1) we have added a paragraph just before the last paragraph of the intro to section 2 stating: The “geoSMEFT” as used in this article and the literature corresponds to the choice of fully classifying all two- and three-point functions. This choice greatly simplifies calculations involving resonant physics. One could, however, apply the geometric methodology to any basis of SMEFT operators by defining a different minimal set of field-space connections.

2) We amended point 2 of the procedure for making the dimension-eight operators consistent with the geoSMEFT to state:

"2. Removing these operators from the dimension-eight operator basis via integration by parts and the equations of motion. That is, the operators’ effects are moved to other operators consistent with the geoSMEFT."

The second sentence was added to clarify we aren't discarding operators, we are simply moving their effects to operators which are consistent with the geoSMEFT basis but not included in one or both of the bases of dimension-eight operators we cite. The advantage offered by the geoSMEFT is in simplifying calculations (especially in regards to input parameter schemes), it has not been shown to improve convergence or to match more naturally to UV completions.

3) The modification of the spectrum occurs above the cutoff of the theory and therefore does not present a problem in the EFT. In the reference article, arXiv:1405.5412, in Eq.2.4 this can be seen to correspond to 1/c, with c the Wilson coefficient of the operator of the dimension-six analogue of our Eq.12. This corresponds to Lambda^2 and so we can see the connection between the UV and IR.

This can also be inferred from, for example, arXiv:2007.00565 Eq. 6.6 in the last line. Here the Lee-Wick ghost's mass should actually correspond to the resonance of the "K_\mu" particle which has been integrated out.

We have not added this discussion to the article as we feel it goes too far astray of the discussion and the citation suffices. We chose to include the citation in our discussion in the first place as this category of operators is preferred in the dimension-eight basis of operators we cite, arXiv:2005.00008.

4) The tables show the contribution from a given term using the effective Lagrangians such as that in Eq.18, as such one could obtain the dimension-six squared contributions absent the dimension-eight operator contributions by carefully reconstructing the appropriate terms from the tables and Appendices A and B. This could also be implemented through the ancillary files. To clarify this, focusing on the ancillary files which are the simplest way to approach this, in the paragraph below Eq.24 we have added the following statement: "The ancillary files are designed for calculations at fixed order in 1/Λ^2, however with minor alterations an interested reader could modify them to produce only the dimension-six squared contributions at order 1/Λ^4."

Please see Author comments where we have line-by-line responded to the referee's requests for corrections