Loop counting matters in SMEFT

Dear Editors,

we thank both referees for reviewing the paper and for their comments.

We first address the remarks made in the second report, which raise the following questions:

{\it 1. 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/Yukawa suppression of interactions (which may be present under the assumption of minimal flavour violation of the UV theory, or some other symmetry)? }

In fact, both of these aspects may be relevant. Concerning the counting of loops and dimensions, an alternative to SMEFT is the Higgs EFT, organized primarily by chiral counting (loop orders), and thus with a different power counting than SMEFT. Concerning additional sources of organization, we agree that further assumptions can be useful to organize the hierarchies in the flavour sector. However, these issues do not affect (flavour-conserving) electroweak physics, which are our main concern here. A systematic EFT treatment of flavour violation would be beyond the scope of our article.

{\it 2. 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. }

We have included a clarifying statement at the end of the Conclusions (Sec. 5), after the last sentence "Variations of the counting scheme ... explicitly specified.":

"For example, the SMEFT may be replaced by HEFT, which follows a different power counting. In addition, further assumptions, such as minimal flavor violation, can be used to organize the flavor sector, which however is a separate topic and beyond the scope of our paper."

We also thank the first referee for his positive comments on our methodology. However, we do not agree with his additional arguments.

The referee acknowledges the usefulness of our approach by writing e.g.

{\it The methodology they suggest goes in the direction of making the well-known tree/loop operator classification, initially proposed by Arzt, Einhorn and Wudka already in the 90s, more systematic and self-consistent. ... 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.}

It is therefore somewhat surprising for us that he nevertheless objects to the consequences of this idea, which we have spelled out in our paper.

The main objection of referee 1 is that he questions the interpretation of loop counting as a power counting in EFTs. We disagree with this objection. Our view is fully consistent with common practice in quantum field theory. We mention three examples:

\begin{itemize} \item[i)] Even in a conventional, renormalizable and perturbative QFT (such as QED) the loop counting in the computation of amplitudes plays the role of a power counting: it organizes the perturbative approximation and dictates, which contributions need to be included at any loop order. \item[ii)] In the context of (nonrenormalizable) EFTs, the chiral perturbation theory of pions in QCD is a clear example. This EFT is organized by chiral counting, which is equivalent to a counting of loop orders. \item[iii)] Concerning SMEFT itself, we may consider a scenario where the standard model at the electroweak scale $E\sim v$ is extended by weakly coupled new physics at a scale $\Lambda\gg v$. Any amplitude, in the full theory, can be written as a double expansion in $E^2/\Lambda^2$ and loop factors $1/16\pi^2$. It is apparent that the EFT description of the new physics (SMEFT) has to reflect this double expansion as well. This is what we discuss in the present paper. \end{itemize}

We also disagree with the referee’s view that loop suppressions and symmetries of the UV sector should be considered on the same footing. One might argue that the class of weakly coupled UV models is much larger than the class of models with any specific UV symmetry. More specifically, e.g. the smallness of the electron mass compared to the top-quark mass (presumably due to some underlying flavor symmetry) will justify corresponding approximations in practice, which however are not essential to the EFT computation per se.

We next address the more specific points raised by the referee. He writes:

{\it 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.}

It is of course true that all possible $n$-loop BSM effects are captured by a full $n$-loop EFT calculation. However, and this is exactly the point of our paper, such an approach would be redundant/overcomplete, since some of the $n$-loop effects are in reality higher loop effects as our toy-model calculation clearly demonstrates. While the approach suggested by the referee is guaranteed to be exhaustive, it introduces superfluous parameters, which make it harder to perform a fit of those parameters.

Finally, the referee remarks

{\it ... 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.}

Item~1 corresponds to what we state (in line with Arzt et al., our ref. [31]). The example in 1305.007 (App.) is no contradiction: The operators ${\cal O}=H^\dagger H X^{\mu\nu}X_{\mu\nu}$ are suppressed by $1/m^2_\sigma$ only, since strong coupling has been assumed (this would incidentally correspond to HEFT). For weak coupling, the $\sigma X^{\mu\nu}X_{\mu\nu}$ interaction would be loop suppressed, leading to a loop suppression for ${\cal O}$ as well. Similar comments apply to 1711.10391.

As we have re-emphasized in our comments above, the suggestion to include loop counting into the power counting of a SMEFT of weakly-coupled new physics is an essential part of our paper. Therefore we see no reason for a substantial reorganization of our presentation.

After having addressed the objections raised in report~1 and included the suggestions of report~2, we hope that our submission can now be accepted for publication in SciPost.

Sincerely yours,

G. Buchalla, G. Heinrich, Ch. M\"uller-Salditt, F. Pandler

In response to report 2, we have included a clarifying statement at the end of the
Conclusions (Sec. 5), after the last sentence
”Variations of the counting scheme ... explicitly specified.”:

”For example, the SMEFT may be replaced by HEFT, which follows a different power counting.
In addition, further assumptions, such as minimal flavor violation, can be used to organize the flavor sector, which however is a separate topic and beyond the scope of our paper.”