Top-down and bottom-up: Studying the SMEFT beyond leading order in 1/Lambda^2

1. The organization of the paper, and approach to the problem, is extremely clear: a reasonable set of different models were chosen to highlight different results.

2. The subject of the paper is timely; the LHC community is in the midst of intensive work on how to implement SMEFT interpretations, and understanding how/when to include effects of higher-order terms in the EFT expansion is of utmost importance.

1. The scope is relatively limited: a small number of benchmark models are chosen to illustrate different effects, but only their effects on the Drell-Yan process are considered, and only the dilepton invariant mass spectrum is fit; it's difficult to extrapolate many of the findings to a more sophisticated analysis involving additional kinematic variables or other processes, which would provide other handles to constrain the "nuisance parameters".

2. Some of the graphics could be more clearly presented (the same curves could be on one panel, for instance, and the axis labels could be improved).

The author presents a study of four simple UV models that lead to qualitatively different effects on the Drell-Yan (DY) process at the LHC and their matching onto the SMEFT including operators at dimension-8 and 10. The goal is to understand how well the Wilson coefficients, dictated by the UV theory, can be measured via a fit to the dilepton invariant mass spectrum with different truncations of the SMEFT expansion applied. While other studies of dimension-8 matching effects exist, this paper presents a novel, comparative study between different models, with an emphasis on the DY process and a fit to the full invariant mass spectrum that I find to be a valuable addition to the literature. While the scope is limited (see "Weaknesses", above) the results nicely demonstrate several potential pitfalls in performing SMEFT fits to LHC data if these fits are to be interpreted as constraints on new physics, and highlight some of the challenges of this global approach.

I have several questions and comments regarding the work, listed below, as well as some (minor) changes that should be addressed in a revised manuscript:

1. Regarding the results in Table 2: for the X model, with MX = 5 TeV and \beta = 3.0, it appears that when moving from D8 to D6D8, while the central value of c_8 gets significantly closer to its true value, the determination becomes much less precise (+/- 0.70 -> +/- 2.1). Is there a way to understand this behavior? This seems to be distinct from the accidentally good reproduction of the distribution that appears when including the D6^2 terms discussed elsewhere in the paper.

2. There has been a lot of prior work on Drell-Yan studies in the context of effective theories (albeit much of it limited to "universal" corrections in terms of the W & Y parameters) -- see for instance, arXiv:2008.12978 and arXiv:2103.10532. Given the relevance, I think a more extensive discussion of the relation between the present work (and SMEFT interpretations of Drell-Yan more generally) and these studies is warranted.

3. Related to the above, it would be useful to discuss in particular the potential benefits of doubly- or triply-differential Drell-Yan measurements in the context of higher-dimension operators. My suspicion is that a lot of the "nuisance parameter" behavior of the additional coefficients discussed in this work is due to the fact that, ultimately, this is fitting to a single distribution, and, even if the fit closes, there is a lot of unavoidable overlap in the shape of the deviations allowed in the dilepton invariant mass tail. I would expect the additional dimension in kinematic space to substantially affect this behavior, but the author should comment on the extent to which this is true.

4. There seems to be a typo in the paragraph below Eq. (15). I assume the author means that it is the $1/\Lambda^6$ temrs that include dimension-six amplitudes interfering with dimension-eight, etc.

Assuming these questions and the changes below are addressed, I am happy to recommend the paper for publication.

1. Figure 1 would be much clearer if the three panels showing the ratio R for the different models were combined into one figure---the axis ranges are not too different in each panel, and I think an easy comparison of them is important. They could even be combined with the first panel (the total cross section), e.g., with the same x-axis, though this is more of a personal preference.

2. The y-axis labels are somewhat confusing when not read in lots of context---the label "R" is used to mean a number of different ratios, and Figure 4 is particularly ambiguous. More descriptive titles would be very useful.

3. Include references to doubly- and triply-differential DY papers (and other relevant prior work), as discussed above.

Publish (meets expectations and criteria for this Journal)

- Large amount of potentially useful calculations within very concrete models relevant for Drell-Yan.

- Most, if not all, of the conclusions seem to be rather trivial or well-known results.
- There are seemingly important flaws in the fitting procedure.

The author studies the convergence of the EFT expansion for describing new physics models relevant for Drell-Yan. To this aim, he considers four single-field extensions of the SM and matches them onto the SMEFT at tree level an up to order 1/cutoff^6. He then computes the ratio of the cross section in the full theory to that in the EFT obtained at different orders in the cutoff expansion, for different energy bins in the m(ll) distribution of Drell-Yan.
He finds that in the weakly-interacting regime, the 1/cutoff^2 term in the expansion suffices to parametrize the UV. He also finds that in more strongly-coupled cases, the 1/cutoff^4 term might be needed, though some times it worsens the perturbative result (which is only improved upon including higher-order terms).

Notwithstanding the large amount of work done by a single author, I think these findings are rather trivial and well-known. In particular, the seemingly surprising fact that adding more terms in the EFT expansion does not always make the result closer to the UV computation can be understood, for example, by expanding the function Cos(x) in Taylor series around x=0 and applying it to x=3. The first term in the expansion is closer to the actual value (-0.99...) than the next one. What holds is that the EFT expansion will accurately describe the UV result provided enough terms are included.

The author makes rather similar statements from a different perspective, namely by fitting the EFT to simulated data generated in the UV. The statement is that, if the fit is close to the UV value (by construction equal to 1), the EFT expansion is fine. I do not see how much this adds to the previous considerations. Moreover, I think the fitting procedure presents some flaw, because despite the uncertainties being at times very large (see delta=4.6 in Table 2), the central value does not fluctuate at all (it is always 1).

Reject

This paper studies examples of four UV extensions of the SM, their imprint on the SMEFT, and asseses the possibility of the SMEFT expansion breakdown in the case of the Drell-Yan process. In addition, the author studies the reliability of fits that rely on a SMEFT expansion truncated at dimension-six squared, compared to fits that include dimension-eight and dimension-ten operators.

After matching the models to dimension-ten, the author studies how the inclusion of dimension-eight and dimension-ten operators in hypothetical effective field theory fits to the full ultraviolet models impacts the fits. This is done in both the top-down approach and in a limited approximation to the bottom up approach of the SMEFT to infer the impact of a fully general fit to the SMEFT. The importance of including higher order terms depends on the nature of the new physics interactions, whether they are weakly coupled models or have stronger interactions or a lighter mass. The author concludes by pointing out some future considerations that would further improve our understanding of the convergence of the SMEFT expansion.

The paper is written very clearly and all the relevant details of this analysis have been discussed at length. There are some minor typos in the paper some of which are listed below. Once these typos are fixed the paper is ready for publication in SciPost.

- Near the end of page 8 the author states that 1/lambda^4 terms include dim6 amplitude interfered with dim8 as well as dim10 interfered with the SM. I assume this is a typo and that the author means 1/lambda^6?

- Page 21 right after eq. 24, there is a typo: "indicted" -> indicated .

- EWPD was first used on page 11 but the acronym wasn't explained. If the acronym refers to "Electroweak Precision Data" the author should write that explicitly when the acronym is first used.

- Page 29 near the end, M_phi = 3 is missing the unit "TeV".

Ask for minor revision