Higgs decays to two leptons and a photon beyond leading order in the SMEFT

We thank the referees for their constructive review of our article. We have attempted to implement nearly all changes requested. One-loop dimension-six calculations are indeed of the same order as those calculated here, but are beyond the scope of the work. We have changed the language in the article to more clearly indicate this and have reframed the discussion in the context of this.

General notes for all referees:
1) Please note that in the need to address the size of the correction to the tree level results from including the full mass dependence relative to the leading mass dependence we found that the initial sampling of the phase space was inadequate and lead to incorrect results for the muon and electron. This has been remedied. As the muon tree level result is in the denominator of all the ratios (\Delta's) used in the paper we had to rerun all these results so they have changed modestly from the previously presented results. We have rechecked all PS integrals and found they are robust over 3 orders of magnitude of initial PS samplings using the Vegas algorithm.
2) We have, for the sake of cleanliness of presentation, dropped the inclusion of the full results of the ratios in the main text in favor of the more concise ones presented under the scaling/loop assumptions. We hope the referees agree this improves the presentation.
3) As a result of the many corrections needed to the text, please be aware that many equation/table/figure numbers may have changed. There is also a new 'case' corresponding to the dipole operators.

Response to Report 3
1) We thank the referee for pointing out our oversight of the 'class 8' operators. We have now included them. Please note that throughout this work we have neglected the CP odd operators, and so we have also neglected the CP odd "class 8" operators.

Response to Report 2
1) As noted in the general comments we found that our initial sampling of the PS for muons and electrons was inadequate and so has been rerun. The large corrections remain, but have changed. These large corrections can be understood from the E_gamma distribution of the decay as the referee guessed. We have cited Figure 4 of [11] to explain this behavior.
2) We have clarified the inclusion/exclusion of mass dependence in the loop PS integrations below Eq17.
3) Regarding IR divergences and the loops, we have cited [11] where they include an explanation below Eq.2.8
4) Regarding the interference of the tree- and loop- SM amplitudes we have included Appendix E. Appendix E shows that in the case of the triangle diagrams the interference is 0, and that for the box diagrams it is expected to be small, allowing us to neglect it our analysis both in the SM case and that of the SMEFT.
5) We have replaced M_full -> M to avoid confusion.
6) The D6 Wilson coefficients are not included in the one loop amplitudes. We have added a discussion of this below Eq 26. We expect based on power counting that our calculations of the (D6 tree)x(D6-loop) interference should be of the same order of magnitude as our 1/Lambda^4 results, and therefore our qualitative results should hold. We have emphasized that the quantitative results are partial results which should be improved in future studies leading up to the HL-LHC. These calculations are beyond the scope of this work which took advantage of the fact the SM loops are UV finite (the loops are not when D6 operators are included).
7) While one-loop SMEFT can give new kinematic structures, they won't yield new chiral structures within the context of the operators discussed.
8) For the muon tree-loop interference please see response 4 and appendix E.
9) The event rates are very simply estimated from the total Higgs cross section, the SM BRs, and the integrated luminosity. This is now included in the discussion at the bottom of p14.
10) No efficiencies are assumed, for muons this will result in a modest drop in event rate. It will likely be much more severe for taus. We have not explored this as it is beyond the scope of this work. The ratios defined in this work and the lack of study of this aspect is consistent with other SMEFT works. We agree that this should be improved for the future precision studies as we move toward the HL-LHC.

Response to Report 1
1) We agree corrections to \bar g_i have to cancel as the referee states, however we included these relations for the sake of clarifying notation for those unfamiliar with its usage in the SMEFT. If we had not included it another referee would likely have asked for clarification.
2) Input parameter dependence is derived from the Appendix of [34]. We have added a comment to this effect below Eq40.
3) We now mention the implicit dependence below B2 and have fixed the missing dependence in the full expressions.
4) The LO corrections were wrongfully neglected and have been fixed (they enter through the tree level diagram as mentioned in the referees previous comment). We have defined c_ll below the definition of dGF in Eq43.
5) We have replaced all instances of mbar->mhat as the fermion mass is always the input parameter. This is now mentioned in the caption of Table1. For general input parameter dependence please see response to point 2.
6) The sizable corrections come from expanding the amplitude squared in powers of m_\ell. This is now explained as due to the E_\gamma dependence and the cut on E_\gamma per another referee's question.
7) We now refer to the square vertex in place of the vertex denoted by a box to help distinguish the two contributions.
8) The difference comes from the phase space region, which is now referenced at the end of the paragraph below Eq.17.
9) We added a short discussion at the end of the introduction to Section 4 (i.e. the beginning of section 4 which is not labeled by a \subsection) indicating we have simply taken the couplings to be diagonal and that we make no assumptions about universality which is possible as our discussion is channel by channel.
10) We have added the contribution of the squares of dipole operators. Because of comments regarding the interference of the chiral flipping amplitude with the SM loop/SMEFT tree contributions we have further elaborated on the potential impact of dipole beyond the squared D6 term in Appendix F. We found in the case of the muon the D6^2 term is by far dominant, but for taus others may become relevant.
11) The numbering was changed, this was a mistake. This statement has been removed as we have now discussed the interference of the chiral flipping amplitudes and the others at the referees request.
12) They do not, they were originally included until it was realized they can only contribute at 1/Lambda^6 and higher order due to the assumptions in the text. We have clarified this where they are mentioned in the appendix. We have kept them in the appendix for completeness.
13) The comment on p19 was poorly phrased and has been removed. The neglect of the D6 loops is purely a simplifying assumption and we have clarified this in the discussion below (new) Eq26 (former Eq.24). Here we have mellowed the language to indicate that our loop-tree interference is an estimate of dependence, and that therefore the numerical results are inexact, but the qualitative discussion forming the core of the main text should still hold.
14) We have added a comment to this effect.
15) We previously checked against [8] and found agreement. Noting the large changes to the PS integrals for the muon mentioned in the general notes, we re-performed the checks and again found agreement. This supports the referees point that these cuts could be important in future analyses. We have added a sentence to the bottom of the discussion below (new) Eq42 to include the referees critique and emphasize the need for more detailed studies in the future.
16) We have added a statement below Eq 42 emphasizing that in the determination of the Deltas the full dependence is used, i.e. that the normalizations are included. If the referee's point was that in determining the regions of Table~3 we should include the normalizations as well, we agree this would allow for the best determination of ideal regions, but is beyond the scope of this work as can be seen from the complexity of the resulting 'Deltas' of appendix D.
17) We have done our best to address these typos. With the large amount of rewriting that took place in responding to all the referees reports we did our best to avoid any new typos.