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Reconstructing the graviton
by Alfio Bonanno, Tobias Denz, Jan M. Pawlowski, Manuel Reichert
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Submission summary
Authors (as Contributors):  Manuel Reichert 
Submission information  

Arxiv Link:  https://arxiv.org/abs/2102.02217v1 (pdf) 
Date submitted:  20210217 20:16 
Submitted by:  Reichert, Manuel 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We reconstruct the Lorentzian graviton propagator in asymptotically safe quantum gravity from Euclidean data. The reconstruction is applied to both the dynamical fluctuation graviton and the background graviton propagator. We prove that the spectral function of the latter necessarily has negative parts similar to, and for the same reasons, as the gluon spectral function. In turn, the spectral function of the dynamical graviton is positive. We argue that the latter enters cross sections and other observables in asymptotically safe quantum gravity. Hence, its positivity may hint at the unitarity of asymptotically safe quantum gravity.
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Anonymous Report 2 on 2021328 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2102.02217v1, delivered 20210328, doi: 10.21468/SciPost.Report.2735
Strengths
1Original interesting paper
Weaknesses
1What 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.
Report
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 selfcontained. 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.
Requested changes
As described above.
Anonymous Report 1 on 2021319 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2102.02217v1, delivered 20210319, doi: 10.21468/SciPost.Report.2719
Strengths
1  very original, milestone for asymptotically safe quantum gravity
2  very good balance between readability and technical detail
Weaknesses
1  some of the (technical) assumptions could be better motivated/explained
Report
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 onshell 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 momentumdependent results. Can one be sure that this nonsmoothness 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 readded by hand (see e.g. [Phys.Rev.D 99 (2019) 7, 074001] for a discussion in the context of YangMills 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 applestoapples 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 falloff for the background propagator. This should be stated more clearly.
A list of minor comments/suggestions can be found below.
Requested changes
Section I:
1  typo: "is not a onshell field" > "is not an onshell field"
Section II:
2  The authors mention a lack of numerically accessible Lorentzian formulations, but strictly speaking, CDT would qualify as a nonperturbative lattice formulation with a welldefined 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 wellknown oneloop logarithms. Also, more nonlocal 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 falloff)  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 falloff 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 oneloop 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.
Author: Manuel Reichert on 20210709 [id 1560]
(in reply to Report 1 on 20210319)
We thank the referee for their careful work, their detailed comments, and also for offering helpful suggestions for improvement. In the following, we cite the comments of the referee and reply subsequently.
"1  The authors mention several times that the graviton is not an onshell 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?"
Answer: The observed gravitons at LIGO are included in the delta peak of the computed spectral function, which directly relates to the classical behaviour of the massless graviton. However, gravitational waves are not asymptotic states, the amplitude of gravitational waves decays according to Price's law, $1/v^p$, where $v$ is the advanced time and $p\geq 12$ (this is a consequence of the nohair theorem). There is obviously no contradiction to LIGO observation since the observation takes place at finite time and distance from the source. A further important difference to the photon is linked to the fact that the gauge group of QED is linear.
"2  I have some concerns regarding the regulator choice. The authors use the "Litim cutoff", which is not smooth, to obtain momentumdependent results. Can one be sure that this nonsmoothness 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?"
Answer: The nonanalyticity of the Litim cutoff at finite RG scale leads to odd powers of momenta in the anomalous dimension, which are absent at vanishing RG scale. How this is ensured is discussed in App. E2. We emphasise that the choice of the Litimtype cutoff is not necessary for the computation. We have used it to connect to our previous works and it also offers the benefit that one has analytic expressions for large and small momenta.
"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 readded by hand (see e.g. [Phys.Rev.D 99 (2019) 7, 074001] for a discussion in the context of YangMills 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."
Answer: Indeed, complex conjugated poles are absent in a standard Källén–Lehmann spectral representation. They may signal the loss of unitarity and a respective discussion has now been included. We also have added references where issues of complex conjugated poles in the context of QCD are discussed.
"4  On the comparison between background and fluctuation momentum dependence, my impression is that this is not really an applestoapples 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 falloff for the background propagator. This should be stated more clearly."
Answer: Contrary to the statement of the referee, the physical momentum dependence of the background propagator is computed at vanishing RG scale and we do not use an identification of the RG scale with physical momenta. We compute the physical momentum dependence of the threegraviton vertex and use background diffeomorphism invariance to map this information to the background propagator.
We also thank the referee for the minor comments/suggestions. We have modified our manuscript accordingly. To some of the comments, we reply below.
"2  The authors mention a lack of numerically accessible Lorentzian formulations, but strictly speaking, CDT would qualify as a nonperturbative lattice formulation with a welldefined Wick rotation"
Answer: We agree with the referee that CDT has a welldefined Wick rotation. Nonetheless, there are no nonperturbative computations directly in Lorentzian signature.
"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 wellknown oneloop logarithms. Also, more nonlocal structures have been discussed in the literature."
Answer: The largescale physics in the IR is well described by the EinsteinHilbert action and logarithmic corrections are subleading. We clarified the formulation in the manuscript.
"10  Eq. (24) excludes more general behaviour like in nonlocal gravity theories (propagator with exponential falloff)  why is this behaviour excluded here?"
Answer: We indeed did not include an exponential falloff behaviour in Eq. (24), which can arise in nonlocal gravity theories, since those are not known to appear in asymptotically safe quantum gravity and they are not appearing in our approximation either. We included this information in the manuscript.
"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."
Answer: For $\eta <−2$, the propagator of the fixedpoint theory is not planewave normalisable anymore, but indeed the propagator could be planewave normalisable on trajectories that run out of the fixed point. However, we do not consider such a QFT. We modified the respective paragraph to make this issue clearer.
"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 falloff of the propagator and large momentum rely on momentum locality."
Answer: Our discussion in Sec. 3 does not rely on momentum locality as we consider the falloff behaviour of the propagator at vanishing cutoff and large momenta. For theories with momentum locality, such as the one at hand, this falloff behaviour is directly related to the anomalous dimension at $k = \infty$ and $p = 0$. For theories without momentum locality, the situation is more intricate and goes beyond the scope of this paper.
"15  It seems to me that the coefficient Ah should be related to the prefactor of the oneloop universal logarithm from EFT. Is a direct comparison possible? If not, why?"
Answer: We thank the referee for this remark. The coefficients $A_h$ and $A_{\bar g}$ are indeed regulator independent, though gauge dependent. We included now the gaugedependent result in our manuscript and compare it to the EFT results.
"16  What is the concrete motivation to choosing the hypergeometric function $U_{a,b}$?"
Answer: This choice was motivated by the fact that the hypergeometric function in question describes the loglike contributions we observe at small momenta, while simultaneously neither interfering with the UV behaviour nor the reconstruction. We have added some comments explaining this choice in more detail in the manuscript.
"17  Eq. (54) needs a little more explanation. In particular, what is the index structure, and where exactly does this relation come from?"
Answer: We use a specific channel of the gravitongraviton scattering amplitude. We have added a few more details in the description, and hope that the relation and the tensor structure are clear now.
"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?"
Answer: We have less numerical accuracy of the Euclidean data in the background case compared to the fluctuation one. In addition, the background propagator decays faster in the UV, which makes the reconstruction more challenging. In our opinion, these are the main sources for the reduced stability of the reconstruction of the background propagator.
Author: Manuel Reichert on 20210709 [id 1561]
(in reply to Report 2 on 20210328)We thank the referee for their careful work, their detailed comments, and also for offering helpful suggestions for improvement. In the following, we cite the comments of the referee and reply subsequently.
"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 selfcontained. 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."
Answer: The main point of our paper is the reconstruction of the graviton spectral function. We achieved this goal as this is the first reconstruction in a nonperturbative setup. We have not displayed the graviton propagator on the Lorentzian axis as this information is contained in the spectral function. We tried to improve the presentation in Sec. 5 and hope that the section is now more accesible to the reader.
"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."
Answer: Eq. (31) is describing the falloff behaviour for the anomalous dimension of the background graviton and is only valid for $2<\eta < 0$. We now also explicitly state the falloff behaviour for other values of the anomalous dimension, $0 \leq \eta < 2$.