A theory vade mecum for PSI experiments

This is a very useful introduction to the theory tools necessary to the interpret experimental activities carried out at PSI. I think the paper is complete and well-organised.
My only concern is that section 5.2 is rather long and technical, and the article lacks of a vision/motivation part, especially as far as BSM measurements are concerned.
If the article is addressed to a broad audience (i.e. also to non EFT practitioners), as I assume, then I would recommend to extend a bit the introduction with a first (non-technical) clarification/exemplification of what is meant by BSM searches vs. precise determination of SM parameters vs. “auxiliary” measurements (in the context of PSI experiments). Most important, some discussion about the need for these 3 lines of research, in particular the theoretical motivations for BSM searches via precision measurements, and the distinction between light and heavy new physics, would also be very welcome if made in the introduction.
A more detailed structure of 5.2 with subsections would also be welcome.

The paper is not (fully) suitable for publication in SciPost in its present
form. The authors provide a comprehensive theoretical introduction to PSI
experiments. However, there are a few (minor) issues that require some
modifications.

(i) I see that Refs. [1-23] are not final. I am expecting these citations to
be updated before this contribution will be published.

(ii) Related to Eq. (5.12) it might be worth to refer to the LE-Fermi limit of
the SM, i.e. that the Wilson coefficients of these operators may even acquire
higher-order corrections if matched to a particular model (as the SM). I am
emphasizing this very particular issue, since these matching conditions gave a
lot of information about the top mass and Higgs mass (after knowing the top
mass). I think that the SM in the low-energy limit already told us a lot about
the proper treatment of EFTs also in the context of higher-order corrections,
i.e. not only at LO.

(iii) I feel that in the discussion in the paragraph before Eq. (5.15) the
treatment of the axion is not included in this setup. The axion is light, but
develops a wee coupling to SM particles. This cannot be treated within the
conventional EFTs. I find it worth to mention, since axions belong to
present-day's BSM physics. This might be mentioned in terms of a footnote or
so.

(iv) The RGEs of SMEFT are also partially known at NLL. May be the authors
would also like to refer to these works - mainly QCD. There are some
higher-order calculations within SMEFT for the LHC, i.e. beyond LO for the
SMEFT contributions.

(v) I would have expected much more original references for [37-39].

(vi) For me an important typo: The last Eq. of Eq. (5.24) must be an equation
for M_pi^2, not M_pi. The squared pion-mass - linear quark-mass relation is
very crucial. May be the authors would like to mention that their coefficient
B is related to the quark condensate? Or what is the reason to omit this
correspondence? Isn't it appropriate to cite the original and famous
Gell-Mann/Oakes/Renner paper for this?

(vii) Why don't the authors omit all NLO papers from Refs. [54-57] with the
chance to bring the attention of the reader to the real and explicit
shortcomings of previous works? In the way written, it appears to be very
vague.

(viii) Related to Eq. (5.32) I am missing an explicit definition of g and (thus
related) a proper settling of the correction Delta r.

(ix) I am puzzled by the notion that hadronic uncertainties only matter at NNLO
or beyond for the muon lifetime. The lifetime of the muon is affected by
hadronic effects (via Delta r) already at NLO. Or do I miss something? I think
that in succession to the previous point a rigorous clarification of the input
used is mandatory. Otherwise a discussion about how to use hadronic
contributions to Delta r as pursued in elw. precision fits would be useless...

(x) Is there any chance to use inelastic Compton scattering e gamma -> mu ->
e gamma for the mu-e-gamma coupling? This will of course be an issue
concerning errors on the exp. side. But it will be a resonant process. May be
as precisely as the decay process measurable - or not?

(xi) typos: quantum-field theoretic -> quantum-field theoretical
transfom -> transform