State-of-the-art electroweak Higgs boson pair production in association with two jets at the LHC in the Standard Model and beyond

• a useful comparison of existing tools and approximations

• a number of questions and comments need to be answered

The manuscript "State-of-the-art electroweak Higgs boson pair production in association with two jets at the LHC in the Standard Model and beyond" describes a phenomenological study of the results of two available tools for Higgs boson pair production in weak-boson fusion (VBF). The manuscript is generally well written and should be included in the LHC Higgs XS WG Community Report once the following questions and comments are addressed: 1- In Sec. 2, the authors label contributions from t- and u-channel diagrams as VBF process and the remaining s-channel diagrams as Higgs strahlung process. While the complete process consists of the coherent sum of all three diagram families, the traditional VBF approximation only contains the t-channel diagrams and neglects u- and s-channel diagrams, both in interferences with t-channel diagrams and as squared matrix elements in their own right. This point is currently a bit opaque and deserves a clear statement how exactly these are handled. 2- In Sec. 2.3, the authors state the equivalence of the HEFT operators relevant for HHjj and the kappa framework. Although it is beyond the strict topic of this manuscript, it is crucial to note that this only holds at LO EW. As soon as higher-order corrections in the EW sector are to be considered one has to turn to a properly formulated effective field theory, such as HEFT, and the kappa framework cannot be used. I urge the authors to add this caution, in particular since it is this community report is targeted also at experimentalist who need to be aware of the limitations of their chosen framework. 3- In Sec. 4, the authors list the inputs and scheme choices used by the authors. Here, they state that they use $(\alpha(M_Z),G_F,M_Z)$ as their EW input parameters. I urge the authors to elaborate on their reasons for this non-standard choice comment on the expected scheme uncertainty were, for example, the $G_\mu$ scheme used. In particular, since the $W$ mass parameter is a derived quantity and would receive higher order corrections were EW corrections considered, this would have an impact on the calculation in the inclusive phase space where $W$ s-channel resonances appear, would it not? 4- Likewise, please comment on the width scheme that you are using. I assume it is not the complex mass scheme since you give the W mass explicitly and do not derive it from the input relations. Please comment. 5- In eq. (7), the authors present their scale definition. While it has the expected order of magnitude and has been used in previous calculations by some of the authors, choosing the geometric mean of $\frac{M_H}{2}$ and the (reduced) Dihiggs transverse mass is unusual. For completeness, please motivate this scale and estimate the difference you would expect wrt. more conventional scales, eg. the (geometric) average of the transverse masses of two Higgs bosons or the like, the invariant mass of the jet pair, the (geometric) average of the transverse momenta of the two jets, or (in the VBF approximation) the (geometric) average of the momentum transfers away from the two quark lines. 6- In the presentation of the results in Fig. 3 it would be helpful if either all results for the same EFT coupling parameters or for a given generator would be coded in similar colours (or same colours with different line styles). Further, either half the lines are hidden behind the others, or the legends suggest lines to be present in a given plot that are not. Please increase readability.

See points raised in report.

Ask for minor revision

This is an important contribution to an important area of particle phenomenology exploration. The authors do a very good job detailing results, and a publicly available implementation will aid the experimental efforts. I recommend the publication of this article.

Optional suggestions:

• The accuracy of the VBF approximation is expected and not linked to the Higgs multiplicity (as the authors know). I think it would be fair to reference original work by Zeppenfeld and collaborators for single Higgs physics.

• I think the authors could stress the non-availability of central jet vetos in b-rich exclusive final states, on which the experiments rely to perform this analysis, to retain statistical control. This makes their results more relevant, as the VBF approximation is not directly applicable in such a case. For significant departures from c2v-cv^2 = 0, the cross section will trail unitarity constraints of VH scattering, and the VBF approximation will be accurate again,

• I am not sure, but I think that one of the first HEFT-relevant VBF hh studies including QCD corrections was provided in 2307.14809 as a side note of a more detailed motivation of the relevance of the VBF hh exploration. This is a somewhat trivial extension of the calculations provided by Zeppenfeld and collaborators without bridging the parton to the hadron level, but the results seem to qualitatively agree with the authors' findings. I leave adding/omitting a reference to the authors' consideration.

Merry Christmas and a Happy New Year!

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