# The derivative expansion in asymptotically safe quantum gravity: general setup and quartic order

### Submission summary

 As Contributors: Benjamin Knorr Arxiv Link: https://arxiv.org/abs/2104.11336v2 (pdf) Date accepted: 2021-08-23 Date submitted: 2021-07-22 15:45 Submitted by: Knorr, Benjamin Submitted to: SciPost Physics Academic field: Physics Specialties: Gravitation, Cosmology and Astroparticle Physics High-Energy Physics - Theory Approach: Theoretical

### Abstract

We present a general framework to systematically study the derivative expansion of asymptotically safe quantum gravity. It is based on an exact decoupling and cancellation of different modes in the Landau limit, and implements a correct mode count as well as a regularisation based on geometrical considerations. It is applicable independent of the truncation order. To illustrate the power of the framework, we discuss the quartic order of the derivative expansion and its fixed point structure as well as physical implications.

Published as SciPost Phys. Core 4, 020 (2021)

I would like to once again thank all the referees for their time and constructive comments. The raised points have been addressed in the individual replies to the referees. Beyond the changes mentioned there, some typos have been fixed and references have been updated to include publication information.

I decided against removing some part of section 3 to keep the paper as self-contained as possible. Most of the material in this section is needed to understand the regularisation employed in this paper. The section also resolves some of the confusion in the literature, in particular regarding the trace in the transverse subspace of vector fields (see ref. [151]). Moreover it contains some explicit intermediate results like the trace of the Faddeev-Popov ghost that the referee of report 3 argues are necessary.

### List of changes

See the individual replies to the referees. Short summary:

- The notation has been made consistent.
- A discussion of the problems related to obtaining universal results in the background field approximation has been added.
- Some clarifications have been added where necessary, and typos have been fixed. Indices have been relabelled where they could have lead to confusion.

### Submission & Refereeing History

Resubmission 2104.11336v2 on 22 July 2021
Submission 2104.11336v1 on 3 May 2021

## Reports on this Submission

### Report

The author has responded to the points I raised.
In particular, the new paragraph in the end of sect. 7.2 is useful, but why does the author say "This leads to the conjecture that such minimal gauge fixings might generally not need Ward identities to obtain such a result." This is a fact, not a conjecture.
Furthermore, does the author think that reproducing the universal results is important or not? If the answer is positive, should we not prefer the schemes that have this property?

• validity: -
• significance: -
• originality: -
• clarity: -
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### Strengths

The paper has extensive calculations using the derivative expansion for asymptotic gravity.

### Report

The author has made the minor modifications requested in the draft. The paper can be published.

• validity: good
• significance: good
• originality: good
• clarity: good
• formatting: -
• grammar: good

### Anonymous Report 1 on 2021-8-2 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2104.11336v2, delivered 2021-08-02, doi: 10.21468/SciPost.Report.3327

### Report

The paper looks properly revised and I would recommend publication. However there is one point that I overlooked in my previous reading, which is about the running of the coefficient of the Gauss-Bonnet term. The author claims that the fact that the beta function for this term is independent of the coefficient and it diverges (its inverse goes to zero) is first discovered in this paper. I would like to point out that this fact is already pointed out in
K. Falls, N. Ohta and R. Percacci,
Towards the determination of the dimension of the critical surface in asymptotically safe gravity,''
Phys. Lett. B {\bf 810} (2020), 135773 [arXiv:2004.04126 [hep-th]].
This fact should be properly cited before the publication.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -