One loop verification of SMEFT Ward Identities

We thank the editor and the reviewers for their comments and reviewing the paper.

For the second, positive reviewer’s comments we have modified the statement the reviewer pointed out as a bit strong to a more personal statement on the authors' opinion. The modified statement is:

“Utilizing the Background Field Method is of increased utility (in the opinion of the authors of this paper) in the case of the SMEFT, as this is an effective theory including a Higgs field.”

With this change we believe the second reviewer will be satisfied. We appreciate the fair mindedness of this reviewer, and the we also appreciate the editor’s inclination to suggest the paper is published.

For the first reviewer, we find little to engage with in constructive criticism on the paper. As the reviewer expresses their opinion on the paper, which of course they are entitled to, here we point out the aspects of the extensive results of the paper that inform the opinion of the authors on the value of this work. We stress these points to scientifically argue why we consider the paper a valuable and novel contribution that should be published in the literature:

1) The SMEFT calculations reported are novel. As far as we know, there is no background field method based gauge fixing demonstration of Ward identities at one loop in the published literature before this work.

2) The explicit demonstration of the BFM Ward identities we examine at one loop operator by operator clarified to the authors of the paper how the Ward identities are satisfied at one loop. One aspect of the physics that was not obvious to the authors, until this detailed calculation was done, was the effect of mass corrections in the propagators proportional to SMEFT Wilson coefficients. We found the need to expand out the mass effects in the propagators to satisfy the Ward ID, somewhat to our surprise, when performing this work. This need to expand out these contributions is mentioned in the text in sections 4.3.3 and 4.3.4 and elsewhere. This makes clear the need to expand out such contributions in SMEFT global fits.

3) The behavior of the corrections to the tadpole terms for some operators is quite novel and directly related to some issues that have been pointed out in the literature previously in alternate formulations of the SMEFT at the dimension six level. This is mentioned on in the discussion on page 5. An explicit demonstration of the behavior of the theory at one loop in these shifts to tadpole corrections is in our view a valuable and clarifying contribution to the literature.

4) Similarly, the discussion in section 5 of the paper also points out that the 2 point functions behavior is of course dependent on the gauge fixing choice of the theory. This is important to clarify in our view in the published literature, as there are comparisons across different calculational scheme choices in Ref.49 where such differences are implied to be associated with calculational error, as opposed to simply being an aspect of scheme dependence.

For all of these reasons, and primarily point 1) we consider this paper to be publishable work that will add to the literature in a useful fashion. Again we thank the reviewers and the editor for their consideration of the paper in Scipost and we hope with our response and the minor revisions the paper is now acceptable for publication in Scipost.

We have modified the statement the second reviewer pointed out as too strongly worded to:
“Utilizing the Background Field Method is of increased utility (in the opinion of the authors of this paper) in the case of the SMEFT, as this is an effective theory including a Higgs field.”