The one-loop tadpole in the geoSMEFT

We thank the referee for their support of the manuscript as well as their thorough notes. We have attempted to implement them all. In the case of notational suggestions we have done our best to balance the referee's suggestions with maintaining the original notation introduced in refs 10 and 11. It is clear from the referee report that as the geoSMEFT is developed further more effort needs to be put into clarifying the notation for readers not familiar with the development of the geoSMEFT. In what follows all mentioned equation numbers correspond to the equation numbers of the updated manuscript.

1) We have rephrased this discussion in the conclusions to indicate what parts are well defined and which need further development. The plan is to continue to extend the one-loop geoSMEFT calculations in the near future.

2)
a) added discussion of hatted fields near Eq19
b) we have moved the introduction of "h" and "g" as suggested
c) we have introduced Pi_12 earlier in the text as suggested

3) We have included such an expression in the new Eq 8.

4) We have followed the referee's suggestion and clarified the notation of the rules.

5) We have used the antisymmetry of this field-space connection to reduce the number of terms in the expressions. Unfortunately there does not appear to be a way to reduce the first two lines without introducing a transformation matrix that takes $\kappa^1\leftrightarrow\kappa^2$ which would be more cumbersome than the current expressions.

6) We have clarified the notation below (new) Eq. 18 and corrected the mistake H(psi)->H(phi)

7) We have added "As fermionic fields are not involved in the gauge fixing they are not split into background and quantum fields." to the text at the end of Section 2.

8) We agree with the referee that the notation can be cumbersome. This article sought to be a balance between the original notation developed in citations 10 and 11 and a clearer picture of the geometry as the geoSMEFT has evolved over the last year. We hope to further clarify this notation as we continue to develop the one-loop program.

a) We have added the comment:
"Latin indices $A,B,\cdots$ are those associated with the four-component representation of the gauge boson indices for $SU(2)_L\times U(1)_Y$, $I,J,\cdots$ are are the four-component indices associated with the four-component real scalar field, and $\mathcal A,\mathcal B$ are associated with color indices are of the gluons."
below (new) Eq20.

We have dropped the use of $\kappa_\mathcal A\mathcal B$ in favor of the scalar quantity $\kappa$ for clarity. We feel that changing the naming convention of the field-space connections that share $\kappa$ would be too far a change from the original authors' work for this short note, but is something to consider in future development of the geoSMEFT.

b) We have added a comment to distinguish the h field from the h metric below (new) Eq.7

c) The hatted field space connections were defined below (new) Eq 49, we have clarified the language by adding "(i.e. $\hat g$ and $\hat h$)" to further clarify. \hat g has an expectation value, as do all field-space connections. This simply corresponds to setting the field dependence in the field-space connection to zero, it does not correspond to an expectation of a gauge boson field. We hope this clarifies what the referee found mysterious.

d) We have changed the notation for the ghost field to $u^\gamma$ to clarify and added a note to the text.

e) There is no implicit field dependence, however the explicit field dependence has implicit dependence on the rotations $\mathcal U$ and $\mathcal V$. This is noted below Eq24. We have added an extra comment on this below (new) Eq. 6. "As the scalar field $\phi$ is related to its mass eigenstate field $\Phi$ by the inverse square roots of the expectations of these matrices, they are (in the mass eigenstate basis) implicitly dependent on $\sqrt{h}$. "

f) $G$ are the canonically defined gluon fields, the gauge fixing term has been corrected to be $\mathcal G$. Below Eq 9 we have also added "$G$ corresponds to the canonically normalized gluonic field, while $\mathcal G$ corresponds to the gluonic field before the kinetic term is transformed."

g) Footnote 1 was added to clarify the difference between the raised and lowered indices in the field-space connections. This does not correspond to a simple sign flip - the off diagonal components of the field-space connections result in nontrivial relationships between the matrices that were solved for perturbatively in this manuscript.

9) We corrected all typos pointed out by the referee.