The one-loop tadpole in the geoSMEFT

1. The work is novel and well-motivated. Potentially the first one in a series of publications using this formalism.
2. The manuscript is well written
3. Motivation and conclusions are convincing and stated clearly
4. The calculation is solid and well documented.

Weaknesses are minor.
The way the results are presented is not impeccable and there is a certain notation issue, that is partially inherited and that I describe in more detail in the Requested changes.

The manuscript reports on the first 1-loop calculation in the framework of the geometric Standard Model Effective Field Theory ("geoSMEFT"), namely that of the tadpole diagram.

The calculation is well motivated and promising from the theoretical point of view. In addition, it elucidates some interesting aspects of the theory.

The work is solid and I definitely recommend it for publication. All requested changes are minor and mainly address the way the results are presented.

1. At the end of the introduction, it is stated that, because only 3-point vertices are included, ''the geoSMEFT is only suitable for the calculation of the tadpole diagram''. The conclusions say that ''it is possible to systematically extend the geoSMEFT to include any $N$ particles plus arbitrarily many scalar field insertions''. Would it be possible to have a more quantitative statement? Are applications beyond the tadpole under development, or are there structural limitations?

2. Symbols are often used before being defined. In some cases this makes the discussion hard to follow. For instance:
- The sentence ''unhatted fields are understood to be quantum fields [...] to zero'' on p9 could be moved up to the end of Sec 2, to highlight the presence of the hats (and their meaning) in Eq (19).
- The definition of the metric $g$ and $h$ could be moved up (starting from the Lagrangian rather than from the field rotations)
- The definition of $\Pi_{1,2}$ in Eq (22) could be moved to before Eq (20) and simplify already the expression there.

3. It would be nice to have explicitly the relations between bosons in mass and gauge eigenbases through the $\mathcal{U}$ and $\mathcal{V}$ matrices (the analogous of Eq (6))

4. Before Eq (6), I would remove the reference to the FeynRules output format and just state the conventions, as for instance it got me wondering whether $p_1$ in $\{\hat h, G_1, G_2\}$ was the momentum of $\hat h$ or $G_1$.

5. Could Eqs (33)-(37) be written in a more compact way?
6. in Eq (16) : what is $\stackrel{(\sim)}{H}(\phi_I)[Y_\psi]$? And $\stackrel{(\sim)}{H}(\psi_I)$?

7. in Eq (19), why is the fermion field not split into quantum and background components?

8. As a general comment, I would suggest investing in notation adjustments, if not for his work, at least for future geoSMEFT publications.
The proliferation of new objects is undoubtedly problematic and further worsened here by the use of the BFM, but the result is a quite thick forest of new symbols with many indices and decorations, where the reader can have a hard time finding his/her way. Sometimes the same symbols indicate different things and there are groups of symbols that differ only by the latin/mathcal/greek font, by upper/lowercase or by the addition of a $\sim$, bar or hat on top, but denote very different objects. As far as I can tell, there is no meaning associated to these choices (e.g."all mass eigenbasis objects are in mathcal") so one always has to go back and forth to check the definitions one by one.

Some specific examples:
- $\kappa$ represents different objects depending on whether it has $IJ$, $\mathcal{AB}$ or $\mathcal{ABC}$ indices or no indices at all (Eq (7)). The same goes for $f$ (Eq (7) vs (10)).
In Eqs (20), (21), (51), (59) why isn't $(\sqrt{\kappa^{-1}})^2=\kappa^{-1}$? Is $\kappa$ a matrix or a scalar quantity here?
- $h$ is used both for a metric and for the Higgs field. The latter appears after Eq (5) but without explicit mention of what it represents.
- are $g_{AB}$ and $\hat g_{AB}$ the same? $\hat g$ appears in Eq (48) but I do not see it defined. The last line of page 9 hints at it having an expectation value along the same lines as $\hat h$ and $\hat\phi$, which is somewhat mysterious.
- $A$ indicates sometimes the index running over $0\dots 4$, sometimes the photon (e.g. $\bar u^A$ in eq (61) is exactly the same notation as in (58), but the upper index is something else).
- Some of the objects defined implicitly depend on fields and Wilson coefficients, but this is not stated clearly at the beginning and it's hard to guess for a non-expert. It would be nice if this was indicated at least once, when they are first introduced.
- In sec 2 $\mathcal{G}$ are gluons (I'm not sure if in the mass or gauge basis). In Eq. (50) the gluons become $G$, while $\mathcal{G}$ is now the gauge fixing term of SU(2).

Is there anything that could be done to help the reader? Maybe adding a table?
On top of this, I discovered as late as at p12 that raised and lowered indices in the metrics are understood with GR conventions and therefore differ by a sign. This should at least be specified in Sec 2, together with an indication of the sign convention.

9. some typos I noticed:
- p2. couting $\to$ counting
- Eq (7), the gluon kinetic term needs either to remove the indices from $\kappa$ or to change one of the two $\mathcal{A}$ in $\mathcal{B}$.
In the $\bar \psi\psi$ term $\mathcal{Y}$ should be $\mathcal{Y}^\psi$.
In the chromomagnetic term, $T_A\to T_{\mathcal{A}}$.
In the $L_{IA}$ term, there is an index 1 missing on the first fermion.
In the $G^3$ terms Lorentz indices are in a funny order, that results in a minus sign wrt the $W^3$ term. Is this wanted?
- Eq (18) there is a spurious bracket ]
- end of Sec 2. in "the couplings $\hat h$ to two quantum fields'' an "of'' is missing
- in Eq (20) $\mu\nu$ should be $\mu_1\mu_2$
- in Eq (24) the flavor indices go from $rr$ to $pp$ between the first and second line.
- in Eq (29), (30) should the $i,j$ indices be capital $I,J$? Or even $A,B$, for consistency.
- One between $G^{\mathcal{A}}_{\rm color}$ in Eq (48) and $\mathcal{G}^{\mathcal{A}}_{\rm color}$ in Eq (50) should be adjusted to match the other.
- in Eq (52) - (54), (60), (62) should $\xi$ be $\xi_W$?
- in Eq (64) are the $c_{\psi H}$ coefficients missing in the sum?
- before Eq (73). depedence $\to$ dependence
- end of p12. "the the'' $\to$ "the''
- Eq. (92) is a $+\mathcal{O}\left(\frac{1}{\Lambda^6}\right)$ missing?