Essential Quantum Einstein Gravity

Round 1

Reviewer 1 Report

The authors study the RG flow of essencial couplings in pure  Einstein gravity. This is interesting since, at least in principle,   inessential couplings may not reach a fixed point leading to unnecessary difficulties. With this the authors conclusions are twofold: that Newton's constant is the only essential coupling and the prediction of a vanishing cosmological constant.  The paper is well written and clear. It is very complete, with good review sections and  with explicit computations in the appendix.  The results are very interesting and relevant. The  only detail is that the authors do not make clear in the abstract that they are considering pure gravity. This is very important since coupling to matter would completely change the conclusions. Therefore, with this minor revision I can recommend the paper for publication in Universe. 

Author Response

We thank the referee for their positive report. We have modified the abstract to make it clear that we are treating pure gravity. In addition, we have added some remarks about the inclusion of matter in blue text in section 6 around equations (68) and (69). We have pointed out there that the minimal form of the propagator, which is a key feature of the minimal essential scheme, would be a property of the scheme even with the addition of matter. In particular, we have recalled that general arguments for the unique minimal form of the propagator have been given in reference [46] for particles of spin 0, 1/2, 1, 3/2 and 2. Furthermore, we have given the example of scalar-tensor theories with up to four derivatives, showing that the terms which survive after the equation of motion are applied to not enter the propagator.  Thus, in this respect, we do not expect that our conclusions would be radically altered after the inclusion of matter. 

Reviewer 2 Report

The paper is very interesting and presents highly not trivial calculations based on a method published before ([22]). The paper proves that the cosmological constant can be zero due to a renormalization condition and some other assumptions. Thus, it proposes a solution to the old cosmological problem related to the vacuum self energy of the quantum fields (not the problem related to the recent cosmic acceleration).    

I would like to suggest the following minor corrections to the manuscript.

• Authors should mention in the introduction or the conclusion section what are the disadvantages/limitations or possible criticism concerning the present submitted work and the related published paper Baldazzi, R. B. A. Zinati and K. Falls, “Essential Renormalisation Group,” [arXiv:2105.11482 [hep-th]].

• Perhaps authors should add a little bit more discussion about how sensitive are the results to the number of s  of  The derivative expansion, where only terms with up to s-derivatives of the  fields are included in the effective action. Can  higher order contribution higher than 4rth order derivative  cause in principle problems ? perhaps add more text  below Eq. 7.14.

• Perhaps the abstract is a bit misleading. Maybe it would be necessary to mention in the abstract that the IR GFP is an assumption and that the results concern pure gravity without matter. In the second sentence of the abstract authors refer to General relativity. General relativity includes matter is not a theory of pure gravity.

• The conjecture is interesting and maybe is it true in the presence of matter. If the conjecture stated is true perhaps the authors would like to discuss in the conclusion’s section possible solutions for the avoidance of singularities, inflation, and recent cosmic acceleration. For all these three problems there were several papers with suggested solutions based on the asymptotic safety running of the cosmological constant.

• Typo

Text line Number 73  à>>>   “ala: not “aka”

Author Response

We thank the referee for their report and comments. We have modified the draft to address each of the minor points.

In the introduction on lines 59-63 we have added a comment on the possible limitations of the minimal essential scheme to describe strongly non-perturbative behaviour.

Concerning the derivative expansion, we have discussed this in the second paragraph of section 8 of the revised manuscript. There we argue that we expect results to be stable under extensions of the derivative expansion.

We have extended this paragraph with the blue text stressing the technical simplifications and advantages of the essential RG method. So while in principle there could be problems arising at higher orders there are good reasons to be optimistic.

In the abstract, we have now made it explicit that we investigated pure gravity without matter.

In the fourth paragraph of section 8 we have added text in blue discussing the running of the cosmological constant. In our work, the vacuum energy is found to be inessential and therefore its running should not have any physical significance. However, this conclusion could change if a different universality class is investigated where the cosmological constant is essential. This could very well happen with the inclusion of matter or if a different universality class of pure gravity could be found by considering points in theory space that are far from the GFP.

(aka = also known as)

Reviewer 3 Report

The authors apply the framework in Ref. [22] to gravity. They show that there are some inessential couplings and that the Newton constant is the only relevant coupling.

The paper provides important steps in understanding asymptotic safety in quantum gravity.

There are however some points that need to be clarified in the manuscript. In addition, the bibliography is not complete. My comments are the following:

• In eq. (2.3), it is not clear to me that a generic field reparametrization does not induce any measure term in the path integral. Could the authors please clarify this point?

• After eq. (2.14), comment on "the same degrees of freedom as in the GFP". The GFP can in principle display many degrees of freedom. It sufficed to write down an action with several kinetic terms. Each can have its own sign. Taking a generic diffeomorphism-invariant action, the GFP of the corresponding theory space has in general more degrees of freedom than in GR. It is thus not clear to me what the authors mean here. Could they clarify this point in their manuscript?

• After eq. (2.14): the authors would like to keep the "physical" degrees of freedom. Only if the theory at hand is physical, one can distinguish between physical and unphysical parts of its theory space. It is possible that asymptotic safety indeed has ghosts, and thus it is an unphysical theory. In this case, it would not make sense to distinguish between "physical" and "unphysical" degrees of freedom. Moreover, one cannot choose which degrees of freedom are in the theory, this has to be a prediction. I thus think that the authors should rephrase their arguments for better clarity.

• Before eq. (4.1): "though" -> "through"

• Before eq. (4.6): "interested on" -> "interested in"

• After eq. (4.6) it would be nice to clarify that \hat\Gamma is the sum of the regulator, ghost, and gauge-fixing actions.

• In eq. (4.10) I think it would be better to make the k-dependence explicit and clarify that it will be omitted in the following. This would make the details of the section more accessible to inexpert readers.

• Can the authors clarify what is the definition of renormalization conditions they are using, and discuss it in the manuscript?

• I do not understand how the authors identify the inessential couplings. The fact that a coupling is essential or redundant should not depend on the point of the theory space. Thus, I do not understand why they need to use the GFP to identify the inessential couplings. Knowing the infrared limit of the theory or its RG trajectory, one could identify a smart scheme (parametrization) to eliminate inessential couplings along with the flow. But this is in principle unnecessary, as the authors stress after eq. (6.4). The set of inessential couplings is independent of the scheme, and should be independent of the point (in the article, the GFP) used to develop the scheme. In particular, the degrees of freedom of a theory are not related to the inessential couplings: to my understanding, fixing the degrees of freedom is an additional ingredient the authors are adding to the recipe. In the manuscript, these points (parametrization scheme -- unphysical choice --, degrees of freedom -- extra input  -- and identification of all possible inessential couplings -- physical information) seem to be strictly related or even equivalent. In my opinion, the authors should better discuss their distinction, as in the current version this is not clear.

• After eq. (6.5): It would be beneficial to add a sentence, clarifying that the field redefinition based on an entire function Z(Delta) would change the vertices of the theory.

• References are in my opinion incomplete and should be improved prior to publication. Specifically:
• The authors are the first to perform an FRG computation distinguishing between essential and inessential couplings. However, they do not acknowledge that this idea was already in the literature. From the introduction, it seems they are the first to raise the point in the context of asymptotic safety since they do not cite any papers when introducing it. The citable manuscripts I am aware of, where the idea is clearly explained, are the original paper by Weinberg (the authors cite this paper in the manuscript, but not where they introduce the idea in the introduction), the book by Reuter and Saueressig "Quantum Gravity and the Renormalization Group" (page 81, to be precise), and the article 2004.07842 by Christian Steinwachs.
• The first article about a k-dependent field reparametrization is "Renormalization Flow of Bound States" by Gies and Wetterich. The authors however do not acknowledge the existence of this study.
• The authors discuss unitarity but do not acknowledge any of the recent studies on unitarity in asymptotic safety: the two papers on scattering amplitudes by Draper, Knorr, Ripken, Saueressig, "non-perturbative unitarity and fictitious ghosts in quantum gravity" (2009.06637), and "reconstructing the graviton" (2102.02217). The latter should be also cited after eq. (6.5), since the authors discuss the possibility that the graviton propagator has additional poles and that the minimal scheme, in this case, could not describe asymptotic safety.
• In the conclusions, when the authors discuss the coupling with matter, they should cite at least a review on the topic.
• Next to the "cosmological constant problem", the authors should add some references. Same after "PMS" scheme, since some readers might not know what this is.
• More generally, the literature on asymptotic safety in the references is really poor and I think should be improved.

• All equations in the appendix need to be split into different lines.

Author Response

We thank the referee for their detailed report and for raising many points that were unclear in the original manuscript. We have addressed each point in the revised version. New text appears in blue.

1) Concerning the measure we have added text 137-139 and 144-151 which hopefully clarifies this point. Within the EAA formalism, the reparameterisation  is related to the k-dependent field which is coupled to the source and regulator. In this way the measure is independent of k. One is free to also make a k-dependent change of integration variables however this is not needed. We have stressed this point which is discussed in more detail in reference [25].

2) In the introduction we have made it clear that there are different Gaussian fixed points both for scalar field theories and for gravity where the associated kinetic operators are of different orders. This is stressed in the new text on lines 64-65,  71-77  and 80-82. Typically one is interested in theories where the Gaussian fixed point corresponds to a two derivative theory however one can of course study higher derivative theories. We have added references [26,27] as examples of such studies for scalar field theories. We have emphasised that higher derivative theories are in a different universality class.

In gravity there is more focus on higher derivative theories since these are perturbatively renormalisable. However, our focus has been on the quantisation of Einstein’s theory without matter and thus we concentrate on the GFP where the action is the linearised Einstein-Hilbert action. We think that we have made it clear throughout the manuscript that we wish to describe the corresponding degrees of freedom.

The referee states that it is sufficient to have several kinetic terms each with its own sign at a GFP. However, a generic Gaussian action is not a fixed point of the RG. We have added text in section 2 around equations (10)-(12) which hopefully clarifies this point.  While the actions (10) and (11) are GFPs the action in equation (12) which is a sum of two kinetic terms is not a fixed point.

3) While we agree with the referee that there can be asymptotically safe theories that have ghosts, a major advantage of our approach is to search for fixed points where no ghosts are present. This is an advantage of the minimal essential scheme we are putting forward. While we cannot choose which universality classes have fixed points, we can choose to study those for which the degrees of freedom are physical by concentrating on the corresponding region of theory space. This point is made throughout the manuscript and we have added the text on lines 189-192 to stress that we are limiting our search for a fixed point in this manner.

4) In both sections 2. (lines 173-174) and 4. (lines 238-239) we have now said clearly that what we mean by a renormalisation condition is a constraint on the form of the EAA.

5) Inessential couplings are identified using equation (9) which defines a redundant operator. We have made this clear with a comment below. Since the redundant operator depends on the EAA the identification of inessential couplings will depend on the point in theory space in practice. Since this point is quite subtle we have added the simple example of free fixed points to hopefully clarify the situation for a scalar field. This is discussed around equation (10) - (12) in section 2. The important point is that operators which are redundant at one fixed point can be essential at others. This is made clear by the example of the two Gaussian fixed points (10) and (11). This example also generalises to the case of gravity which is discussed in the new text around equation (46) where we compare the GFP of higher derivative gravity to that of Einstein gravity. We think that the corresponding discussions clarify the points of concern of the referee.

6) We have added the references referred to by the referee as well as several others which were missing.

We have also addressed all other minor comments made by the referee.

Round 2

Reviewer 3 Report

The authors answered all my questions and modified the manuscript accordingly.

This paper provides very remarkable progress in the field. The clarity of the text is really good now, modulo two small typos (see below). Thus, I can now recommend its publication in Universe.

Typos:

• "has been acknowledged" on page 2, line 42, should be "has been pointed out" or similar, "acknowledge" has a different meaning.
• page 9, after eq. (24): I guess the author meant that "\hat\Gamma contains terms" and not "contains \hat\Gamma terms" 

Author Response

We thank again the referee for pointing out the important discussions which needed to be clarified in our original manuscript. We have corrected the typos.