Reconstructing the graviton

1-Original interesting paper

1-What is the crucial point to make the Euclidean results to Lorentzian spectral function is not clearly explained, but this is the most important point of this paper.

This paper tries to reconstruct the spectral function of the graviton propagator in Lorentzian signature. They first review the case for Euclidean case, and use the results already derived in their previous paper to lead to Lorentzian case. While the aim is of interest, the presentation in sect. 5, which gives the main result for Lorentzian case, was not very clear. They refer heavily to their earlier paper and the explanation is not self-contained. It is not even clear what is the crucial point to make the Euclidean result to Lorentzian and what is the main difference. This is particularly important in the present case because the obtained results for Lorentzian are quite similar to Euclidean case. I would like to ask the authors to make clear these points.

A minor remark. They derive asymptotic behavior in (31), but I think that there is inconsistency in the exponent with (29), and if this is true I seem to disagree with their claim of asymptotic behavior p^{-3} below eq.(31). They should carefully check this.

As described above.

1 - very original, milestone for asymptotically safe quantum gravity
2 - very good balance between readability and technical detail

1 - some of the (technical) assumptions could be better motivated/explained

The paper discusses an important topic in quantum gravity: the reconstruction of a Minkowski spectral function from Euclidean data. This is a milestone on the route to obtaining results valid in Lorentzian signature. As such, it easily passes the journal's criteria for originality, excellence and significance. The authors carefully discuss the general reconstruction procedure, and then carry it out for results obtained in both the background field approximation as well as a dynamical fluctuation computation in the context of Asymptotic Safety.

The main results are that the background spectral function is not positive, and integrates to zero, while the fluctuation spectral function is positive but not normalisable. Some of the implications of this result are discussed.
As a further contribution, the authors discuss different ways to approximate the momentum dependence of Newton's constant: based on the fixed point condition, an identification of the RG scale and the momentum running, and the fully momentum dependent computation.
On the technical side, the authors are the first to present results on the momentum dependence in Asymptotic Safety at vanishing IR cutoff. This allows them to connect more directly to physical observables, without potential artefacts induced by a finite cutoff scale.

Having said all that, there are a few points that I would like the authors to consider before I can fully recommend publication of the article. These are:

1 - The authors mention several times that the graviton is not an on-shell physical field. In my opinion, this needs a little more discussion. In particular, what is the concrete difference to a photon, and what is the connection of the graviton to (linearised) gravitational waves, which have been recently observed?

2 - I have some concerns regarding the regulator choice. The authors use the "Litim cutoff", which is not smooth, to obtain momentum-dependent results. Can one be sure that this non-smoothness doesn't percolate into the reconstruction of the spectral function? And what is the actual gain of this choice, since results have to be obtained numerically anyway?

3 - To me it seems that the authors make one additional, but unstated, assumption when they relate the propagator and the spectral function. Namely, pairs of complex conjugate poles play a distinguished role. They don't contribute to the spectral function, so that the reconstruction of the propagator via the spectral function is incomplete, and these poles have to be re-added by hand (see e.g. [Phys.Rev.D 99 (2019) 7, 074001] for a discussion in the context of Yang-Mills theory). If such CC poles are present, this modifies the discussion of the analytic structure of the spectral function. While a more complete discussion would obviously be desirable, I would suggest to at least explicitly state this assumption.

4 - On the comparison between background and fluctuation momentum dependence, my impression is that this is not really an apples-to-apples comparison. Concretely, the fluctuation computation is performed with the full physical momentum dependence. By contrast, at the background level, the authors only discuss the momentum dependence obtained by identifying the RG scale running with the physical momentum dependence. A more complete computation would obtain this momentum dependence by resolving form factors quadratic in curvature. This also opens the possibility to get a different fall-off for the background propagator. This should be stated more clearly.

A list of minor comments/suggestions can be found below.

Section I:

1 - typo: "is not a on-shell field" -> "is not an on-shell field"

Section II:

2 - The authors mention a lack of numerically accessible Lorentzian formulations, but strictly speaking, CDT would qualify as a non-perturbative lattice formulation with a well-defined Wick rotation

3 - As an optional suggestion, the authors might want to consider adding some literature on the topic of Wick rotations in the context of gravity, e.g. [Class.Quant.Grav. 36 (2019) 10, 105008]

Subsection A:

4 - The authors state that in the IR, the action should reduce to GR. I find this a little bit misleading, since there are e.g. EFT corrections in the form of the well-known one-loop logarithms. Also, more non-local structures have been discussed in the literature.

Subsection B:

5 - As an optional suggestion, one could mention that in general the wave function renormalisation is tensorial, and that the authors choose it to be proportional to the identity in their approximation.

6 - It would be helpful if the authors would state the initial condition for the integration of the wave function renormalisation explicitly here.

Section III:

7 - typo: "the classical spectral is ultralocal" -> "the classical spectral function is ultralocal"

8 - It would be helpful if the term ultralocal mentioned in point 7 would be defined.

9 - typo: "can readily performed" -> "can readily be performed"

Subsection A:

10 - Eq. (24) excludes more general behaviour like in nonlocal gravity theories (propagator with exponential fall-off) - why is this behaviour excluded here?

11 - In my opinion it would be helpful to define the meaning of $\xi$ a bit earlier, around eq. (36).

12 - The discussion of the case $\eta<-2$ is not clear enough, and a more explicit discussion would be helpful. In particular, the divergence for small frequencies cannot be read off from the large frequency behaviour.

13 - On a more general ground, how much does the discussion rely on the concept of momentum locality introduced in [Phys.Rev.D 92 (2015) 12, 121501] by some of the present authors and others? As far as I understand, the relation between the anomalous dimension at vanishing momentum and the fall-off of the propagator and large momentum rely on momentum locality.

Section IV:

Subsection A:

14 - Eq. (45): I would suggest to change the boundaries of the integral: $-\int_k^\Lambda$ instead of $\int_\Lambda^k$

15 - It seems to me that the coefficient $A_h$ should be related to the prefactor of the one-loop universal logarithm from EFT. Is a direct comparison possible? If not, why?

16 - What is the concrete motivation to choosing the hypergeometric function $U_{a,b}$?

Subsection C:

17 - Eq. (54) needs a little more explanation. In particular, what is the index structure, and where exactly does this relation come from?

Section V:

Subsection C:

18 - typo: "soften negative peak" -> "softened negative peak"

19 - Is there an explanation for why the reconstruction of the background spectral function is so much less stable than the reconstruction of the fluctuation spectral function?

Appendix E:

20 - Eq. (E4): I think that one has a strict equality in this equation for the chosen regulator and $p^2 \geq 4k^2$. In this regime, the regulator depending on the sum of loop and external momentum vanishes identically, and the asymptotic formula should be exact.