A Global View of the Off-Shell Higgs Portal

In addition to the changes listed below, we wish to elaborate further about four points raised by the referees:

a) Concerning the desire of Referee 2 for a comparison with the on-shell signal: We emphasize that one of the main novelties of this work is to analyze the different kinematical features of the derivative and marginal (or renormalizable) portals. This difference is characteristic of the off-shell regime and disappears for on-shell decays, in which case branching ratio limits can immediately be translated into constraints on either type of portal, as already done in Table 3 for illustration. In light of this fact, and given the very extensive (and in many cases, technically very advanced) literature that already exists about on-shell decays, we do not believe that including BR(h -> inv) limits would constitute a helpful addition to this manuscript. Nonetheless, for useful comparison we have included distributions for the on-shell signal at the LHC in Fig. 6, see change #8.

b) Regarding systematic uncertainties: While a complete analysis is beyond the scope of our work, we have addressed the comments by Referees 2 and 3 through the addition of dedicated results for the HL-LHC in Fig. 2, which serve to quantify the expected effects on hadron collider constraints, as well as through a revision of the last paragraph of Section 3. See change #4.

c) About the comment on indirect constraints made by Referee 3: We fully agree that one loop probes of the derivative Higgs portal (such as, e.g., gg -> hh, similarly to what done in Ref. [78] for the marginal portal) are an interesting avenue to pursue. However, their analysis requires dedicated work that is in part ongoing, and for which no results are yet available. Other indirect observables, such as the corrections to the couplings of the 125 GeV Higgs boson, can also be relevant in probing the models we consider. These effects are, however, strongly model-dependent, and for this reason we have refrained from discussing them in the text.

d) About the observation by Referee 1 that the plots in Fig. 9 would be too small: Given that SciPost Physics is a fully electronic journal, we believe the size of this figure is acceptable.

We thank all three referees for their reading of our manuscript and for providing insightful comments. We believe the minor revisions listed below have further improved the quality of the paper, and we are hopeful that the current v2 can be accepted for publication in SciPost Physics.

We list here all the changes we have made in v2, in order of appearance in the text:

1) On page 1 we have added citations to Ref. [17] and Ref. [28], which appeared on the arXiv after the v1 of our work, with the aim to provide a comprehensive overview of the literature on pNGB DM.
2) We have corrected all the typos and made all the small language improvements suggested by Referee 1.
3) When introducing the marginal Higgs portal just above Eq. (2) we have added the alternative name "renormalizable," as well as citations to the original papers Refs. [30-32]. In addition, a few lines later we have added a citation to Ref. [33] for the current constraints on this portal.
4) Both panels of Fig. 2 have been updated and now include also HL-LHC constraints derived assuming a 1% systematic uncertainty on the total background (weaker limit boundary of the gray bands). A mention of this has been added at the end of the caption of the same figure. The discussion of systematics in the last paragraph of Sec. 3 (page 12) has been revised. In addition, the middle panel of Fig. 10 and its caption have been updated accordingly.
5) In the caption of Fig. 2 we have added further information about the ILC and FCC-ee results.
6) On page 3 we have added a sentence (starting with "Note also that if ...") about the scaling of our bounds with the integrated luminosity.
7) In Eq. (5) we have added the last term in the second line, as well as the related definitions of g_V and y in the text after the equation. These operators play a minor role in our discussion, but we have nevertheless included them for completeness. Correspondingly, footnote 6 has been extended and footnote 7 has been added. The effect of these extra operators on DM direct detection has also been included in Eqs. (19) and (21) of Appendix B. Finally, we have commented on the |\chi|^2 |H|^4 operator in the text after Eq. (5).
8) Following the wish of Referee 2 for a comparison with the on-shell topology, in Fig. 6 we have added the normalized distributions for the on-shell invisible Higgs signal, which are independent of the type of portal. Comments on this are given in the caption of the same figure and in the text after Eq. (18). We have chosen to include the comparison with the on-shell case for the LHC, which is of the most immediate experimental relevance. In addition, in Fig. 6 we have changed the color for the V+jets (EW) distributions from orange to black, following a suggestion by Referee 1.
9) Just after Eq. (18) we have provided further details on the lepton veto, the central jet veto and the \Delta\phi(\vec{\slashed{p}}_T, j) requirement.
10) As suggested by Referee 2, we have inverted the order of the HE-LHC and FCC analyses in the text (page 11).
11) On page 13 we have added a sentence mentioning the mono-Higgs signal, together with a citation to Ref. [75]. We thank Referee 3 for their interesting observation about this point.