Loop counting matters in SMEFT

The article is pedagogical and easy to follow. It summarises many of the ideas known to those working on higher-order corrections in SMEFT, although not all.

The article focusses on only the canonical dimension and loop order.

In reality, I don't believe this provides a sufficient organisation of SMEFT terms (which there are clearly many when the full flavour structure of the theory is considered) to practically try and describe limited data.

I believe the authors should also at least comment on this.

The authors present a procedure for arranging corrections in an EFT (focussing on Standard Model Effective Field Theory): this counting includes both loop orders as well as canonical dimension of the operator coefficients.

While these ideas are known to those specifically working on higher-order corrections in the SMEFT (or certainly they are quickly made aware of them, if not), to my knowledge there is not a clear presentation of these ideas with worked examples in the literature. I therefore find this work to be of general use to the community: those working on higher-order corrections, and in addition the groups which are now performing global fits to data (where assumptions on which SMEFT contributions to be included in the fit can have a large impact).

The authors comment towards the end of the manuscript that “Variations of the counting scheme we propose can, of course, be constructed…”. Do the authors have in mind here an alternative counting in terms of loops and canonical dimensions? Or, is this statement hinting at some additional source of organisation, perhaps related to CKM/Yuakwa suppression of interactions (which which may be present under the assumption of minimal flavour violation of the UV theory, or some other symmetry)?

While not a main consideration of the presented work, I believe the authors should at least comment on this. Such considerations are relevant when applying the SMEFT to describe a finite set of data with limited precision.

The authors discuss the role of loop suppressions in the matching between beyond-SM theories and dimension-6 effective operators in the Standard Model Effective Field Theory (SMEFT). They argue that these suppressions should be accounted for in the SMEFT power counting and they propose a methodology to do so by using the notion of chiral dimensions.

The methodology they suggest goes in the direction of making the well-known tree/loop operator classification, initially proposed by Artz, Einhorn and Wudka already in the 90s, more systematic and self-consistent. The authors motivate their recipe with a general derivation, that holds for a broad class of UV theories. I find that this methodology can be useful to estimate the expected size of SMEFT corrections to a given process under those specific UV assumptions.

However I strongly disagree with the statement, suggested in several places in the paper, that the EFT power counting is incomplete or even inconsistent, unless the loop suppressions are incorporated into it. (I also disagree with the alternative phrasing that the EFT is unable to give a fully model-independent prediction).

This statement seems to rest on a fundamental misunderstanding: the fact that SMEFT corrections to a given process appear at a certain perturbative order should not be interpreted as a prediction of the size of those effects. For instance, taking the example in Fig. 1: the EFT does NOT imply that diagram (e) gives a subdominant correction compared to (d).
A correct interpretation (assuming that the perturbative expansion holds) is rather that tree-level computations in the EFT capture all possible tree-level BSM effects, one-loop EFT diagrams capture all possible one-loop BSM effects, and so on order by order.

This is actually a quite non-trivial point made by the EFT approach: if one wants to have a model-independent, universal estimate of potential $n$-loop BSM effects, then the EFT calculation can be safely truncated at $n$ loops. The result obtained in this way will be conservative, i.e. it is guaranteed that all $n$-loop BSM contributions will be accounted for, and the matching to a concrete model can only reduce the number of relevant operators.
Reversing this line of reasoning by noting that "the tree-level terms are unable to reproduce the leading [1-loop] corrections of the model" (page 7) simply amounts to a misuse of the EFT.

I really want to stress that loop suppression factors arising in the matching stem from a UV assumption and, from this point of view, they can be considered on the same footing as symmetries of the UV sector, that also act to suppress or forbid certain operator structures.
In fact, the whole reasoning laid out in Section 2 could be repeated identically replacing loop suppressions with some symmetry suppressions. However, one certainly cannot require the power counting to account for all possible symmetry patterns: they can be imposed on top of the EFT expansion on a case by case basis.

I believe that the methodology in Section 3 is valuable and can have a broad applicability, but it has to be made clear that this is not a power counting statement, but rather a size estimate that only holds for specific UV assumptions.
I can only recommend the manuscript for publication if the text is substantially reorganized in this direction.

The only note I make on the physics results (for now) is perhaps a technicality, but I believe that the discussion in Sec. 3.3 assumes that the BSM model:

1. is weakly interacting in all couplings
2. does not contain super-renormalizable interactions (i.e. of dimension 3)

If the second assumption is omitted, it is possible to construct models where the naive tree/loop classification does not hold. Some examples were discussed e.g. in 1305.0017 and 1711.10391.
In addition, the authors statement that "it is immaterial whether there exist nonrenormalizable interactions of $F$ and $B$ suppressed by scales parametrically still larger than $\Lambda$" only holds if condition 2. is met.