Constraining new physics from Higgs measurements with Lilith: update to LHC Run 2 results

1- The manuscript is well written, and the presentation is organized in a clear way.
2- The database of signal strengths included in the package has been systematically validated, as documented in the paper. Complete references to the ATLAS and CMS papers (or to the web versions of the papers, where needed) are provided for each database entry.

1- The presentation of the modified couplings framework included in the code, and of the fits that are shown as an example, could be slightly improved by stating explicitly the underlying assumptions.

The authors present a major update of the publicly available code Lilith, a tool that implements an approximate global likelihood for the Higgs signal strength measurements, in a form that is suited for comparison against the predictions of extensions of the SM. The manuscript describes the refined statistical approach now employed, that helps to better deal with asymmetric uncertainties, and other improvements of the code. Furthermore, new LHC Run 2 data corresponding to an integrated luminosity of 36/fb has been added, including new production channels as ggZH, tH, bbH. As an example of application, the authors show an updated "kappa-framework" interpretation of the data.

The presentation of the new features in the code is clear and well organized. The validation of the new Lilith dataset is systematically documented. The "kappa-framework" fit that is shown, despite capturing only selected classes of BSM models (being less general than an EFT interpretation, for instance) represents a useful example to illustrate the functionality of the code.

I recommend the manuscript for publication once a few minor points (see below) are addressed.

1- On page 2 the authors describe the assumptions and the scope of the signal strength framework, and then they proceed by defining their framework for modified Higgs couplings. I am afraid that introducing equation 3 simply as "notation" hides the assumption that the "effective lagrangian" the authors have in mind only includes the tensor structures already present in the SM lagrangian. Incidentally, do the authors plan to include additional interfaces, e.g. to the SMEFT? Or will this be left to the user?

2- In the footnote 9, page 16, the authors mention the Run 1 ATLAS+CMS combination. The current version of the Lilith database (DB 19.06) includes bi-variate gaussian fits to figure 14 of the Run 1 ATLAS+CMS combination paper 1606.02266. There, for each decay mode, an ellipsis is shown in the VBF+VH vs ggF+ttH signal strengths plane. On the other hand, 1606.02266 provides results for a much more generic parametrization, where 20 independent products of cross sections (for ggF, WH, ZH, VBF, ttH) and branching fractions (to gamma gamma, ZZ, WW, tautau, bb) are determined simultaneously. In particular, the best fit values are given in table 8 and the correlation matrix in figure 27. As it is noted in appendix A of that paper, a few significant (anti)correlations are present, that are not captured by the "semi-inclusive" plot in figure 14. Support for multidimensional (variable) gaussian likelihoods has been introduced in the current Lilith version, and was in fact exploited for the implementation of the CMS Run 2 combination of 36/fb data. The authors could comment on their choice not to implement such more general fit to Run 1 combination data, which seems to contradict what stated in the second bullet point in the conclusions.

3- I think that, in the introduction of section 6, the authors could remind to the reader that the Higgs couplings fits they describe rely on specific assumptions and do not represent general, model-independent statements on "the Higgs couplings". The authors could possibly also refer the reader, for completeness, to recent global fits in more general BSM frameworks.

4- It would be interesting to see to what extent the variable-width multi-variate gaussian likelihood (built, for the generic parametrization, with asymmetric uncertainties and with the full correlation matrix) is able to reproduce the asymmetries shown in more constrained experimental likelihoods, as the ones in figure 14 of 1606.02266. Would it be straightforward to derive such "reduced" likelihoods? Would there be a range of signal strengths beyond which artefacts show up?

1- The provided explanations of the updates of Lilith and how to run them are very detailed and clear.
2- The authors are very explicit about where they took their data (and correlations) from. This makes it very easy to follow and cross check their analyses.
3- The authors carefully validate each of the included data sets, making use not only of the experimental papers but also of auxiliary material online.

1- Some of the ATLAS analyses used in the combination are avaiblable for a data set with greater luminosity already.
2- Moreover, there are some ATLAS analyses for processes which are not included in the combination at all (see requested changes).

The authors provide a clear and detailed analysis of Higgs coupling strength measurements of LHC Run II data with their Python tool Lilith. With respect to the previous version of Lilith, the authors have updated the statistical analysis framework. It now allows for the inclusion of likelihood contributions as two-sided Gaussians or Poissonians. Moreover, the authors include new Higgs production channels (ggZH and tH production).

The manuscript is clear and well written. I recommend it for publication once the issues below are addressed.

1- On p.7 the authors discuss of the implementation of a variable Gaussian and raise the potential issue of singularities of the log-likelihood function. It is not clear from this discussion how the authors deal with these singularities in practice, what the region of validity of the variable Gaussian implementation is, and why using an ordinary two-sided Gaussian instead (outside of the region of validity) is legitimate in the extremely asymmetrical case $\sigma^{-} \rightarrow 0$. The authors might want to comment on this.

2- Some of the used analyses have been updated for a larger data set already (see weaknesses). The authors could consider updating the analyses available for a higher luminosity, i.e. tth 1806.00425 ($H\rightarrow \gamma \gamma$, $H\rightarrow ZZ$); and VH, $H\rightarrow bb$ 1808.08238.

3- The authors could consider adding a number of potentially relevant Higgs analyses.
* $H\rightarrow$inv: ATLAS WBF analysis 1809.06682. The analysis sets a limit of 37% for $H\rightarrow$inv. This is much better than the limit of 67% obtained in the ZH analysis and comparable to the limit of the CMS WBF analysis of 33%. Including the analysis might have a significant impact on the limit obtained on BR(H->inv).
* $H\rightarrow \mu \mu$: ATLAS 1705.04582; This is the only ATLAS analysis for the mu mu final state. The upper limit on BR($H \rightarrow \mu \mu$) obtained in this analysis is exactly the same as in the CMS analysis (which was included), namely 3.0 time its SM value. It might be worth including the analysis.
* VH, $H\rightarrow WW$: ATLAS 1903.10052; This is the only analysis for this production and decay channel. However, I do agree that $H\rightarrow WW$ is better constrained from ggF and that VH is better constrained from $H\rightarrow bb$, so the impact of this analysis might not be significant in a global fit.

4- Most importantly, the limit on the invisible Higgs branching is interesting, but not obvious. In Fig.12 and the corresponding text on p.18 the authors find a combined 95% CL limit on BR($H\rightarrow$inv) of <= 5% for the SM case. However, the included ATLAS and CMS analyses quote limits of 33% (CMS) and 67% (ATLAS). It is not clear how these limits combine to a 5% limit. The authors should comment on how they obtained this limit and how it compares to the results of the experimental papers.