Self-consistent graviton spectral function in Lorentzian quantum gravity

Jan M. Pawlowski; Jonas Wessely; Manuel Reichert

REVIEW 4 major objections 5 minor 12 cited by

Self-consistent graviton spectral function in Lorentzian quantum gravity

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper computes a positive, normalisable graviton spectral function with a massless one-graviton peak and a multi-graviton continuum, and shows that the on-shell renormalised graviton satisfies the unit spectral sum rule of an…

desk verdict A credible positive, normalizable graviton spectral function from spectral fRG, but 'full self-consistency' is partly a label: the ghost loop is classical and the unit spectral weight is imposed. read the letter →

arxiv 2507.22169 v1 pith:XEGSCD3N submitted 2025-07-29 hep-th gr-qchep-ph

Jan M. Pawlowski , Manuel Reichert , Jonas Wessely This is my paper

classification hep-thgr-qchep-ph

keywords asymptoticallysafegravitygravitonspectralfunctionKällén–LehmannrepresentationLorentzianfunctionalrenormalisationgroupon-shellsumruleunitarityinquantummulti-gravitoncontinuum

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that, in asymptotically safe quantum gravity on a flat Minkowski background, the graviton fluctuation field can be described by a Källén–Lehmann spectral function that is positive everywhere, contains a massless one-graviton delta peak, and has a multi-graviton continuum whose ultraviolet decay is fast enough to give a finite total spectral weight. This matters because Lorentzian spectral data are the direct route to questions of unitarity and causality; Euclidean renormalisation-group results have to be analytically continued, which is unreliable for gravity. The authors achieve this by solving the spectral renormalisation group flow with full feedback of the spectral function into the loop diagrams, including the scattering continuum, in a physical on-shell renormalisation scheme where the renormalisation constants are set to unity on-shell. The claimed outcome is a normalised spectral weight $\int_\lambda \rho_h^{(\mathrm{ph})}(\lambda) = 1$, which the paper reads as the graviton satisfying the sum rule of an asymptotic state and carrying canonical commutation relations, even though the respective states are not diffeomorphism invariant.

What carries the argument

The load-bearing object is the Källén–Lehmann representation of the traceless-transverse graviton propagator, $G_{hh}(p^2) = \int_\lambda \rho_h(\lambda)/(\lambda^2 + p^2)$, used inside a renormalised Callan–Symanzik spectral flow. Self-consistency means that the full non-perturbative spectral function, continuum included, is inserted on every internal line; a generalised spectral representation for the squared regulator-line propagator, $G^2$, reduces the computational cost by one spectral integral. The on-shell renormalisation scheme fixes the flowing pole to $m_h^2 = k^2$ and sets $Z_h = 1$ at $p^2 = -k^2$, which implements a momentum-dependent rescaling of the fluctuation field at every renormalisation-group scale. The numerical solution iterates the integrated flow for the imaginary part of the graviton two-point function and reconstructs the real part from subtracted Kramers–Kronig relations, which also implements the renormalisation conditions.

What would settle it

Compute the same spectral function with a momentum-dependent regulator or in a manifestly gauge-invariant physical-state projection; if the result has a negative region anywhere, or a tail decaying only as $1/(\lambda^2\log\lambda)$ rather than $1/(\lambda^2\log^3\lambda^2)$, then positivity, normalisability, and unit spectral weight cannot all hold as claimed.

Watch

Extended reading notes

Core claim

Stated on the paper's own terms: the graviton spectral function $\rho_h(\lambda)$ computed at $k=0$ is positive semidefinite, with a delta peak at the massless pole $m_h^2=0$ and a scattering continuum $f_h(\lambda)$ that decays as $c^{\mathrm{UV}}_h/(\lambda^2 \log^3 \lambda^2)$ for $\lambda \to \infty$. Because of this decay the spectral integral converges, and after rescaling by $z_{\mathrm{spec}} \approx 1.486$ the physical spectral function satisfies $\int_\lambda \rho_h^{(\mathrm{ph})}(\lambda) = 1$. The computation also reproduces the universal one-loop infrared onset of the tail, $f_h(\lambda \to 0) \to 61/30$, confirming that the self-consistent treatment embeds the exact low-energy limit. The paper concludes that the on-shell renormalised fluctuation graviton 'gets as close as possible to a physical field': it behaves like an asymptotic-state field through its unit spectral weight and canonical commutation relations, while remaining gauge-noninvariant and outside the physical Hilbert space.

Load-bearing premise

The load-bearing premise is that the graviton propagator admits a Källén–Lehmann representation, an expansion in ordinary particle-like states, on the flat Minkowski background at every renormalisation-group scale; if gauge-noninvariant graviton fluctuations lack such a representation or require negative spectral densities, the extracted spectral function and its unit sum rule would not be meaningful.

Editorial extensions

If this is right

The converged tail reproduces the exact one-loop infrared limit, so the self-consistent treatment confirms that the scattering continuum is sub-leading in the flow and that the previously missing denominator in the spectral tail is crucial for a consistent treatment.
The fluctuation graviton acquires the spectral sum rule of an asymptotic state: a unit total spectral weight, canonical commutation relations, and a classical on-shell dispersion at every scale.
Scattering amplitudes in asymptotically safe gravity can be computed with a positive-definite, physical graviton spectral density, avoiding analytic continuation from Euclidean data.
The ultraviolet decay law $1/(\lambda^2 \log^3 \lambda^2)$ is the property that makes the spectral weight finite; without it, the spectral sum rule could not be imposed.
A substantial part of the spectral weight is stored near the Planck scale, so trans-Planckian loops probe the integrated weight rather than the fine details of the spectral tail.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A consequence the authors leave implicit is that the same self-consistent spectral machinery could test unitarity non-perturbatively: any negative spectral density emerging from a gauge-invariant or background-independent formulation would be a direct signal of a violation.
If the Källén–Lehmann assumption fails on curved backgrounds or for gauge-invariant observables, the unit sum rule and canonical commutation relations derived here would not transfer; the flat-background result therefore sets a benchmark for background-independent extensions.
A natural next check is the same flow in a different gauge or with a momentum-dependent regulator; the paper expects only mild quantitative shifts in the Newton-coupling fixed point, but the sign and normalisability of $\rho_h$ are the robust features to test.
The close-to-quadratic ultraviolet decay, with most spectral weight near the Planck scale, predicts that high-energy graviton scattering is dominated by low-mass spectral weight, which could be tested in future cross-section computations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First, the punchline: this is the most credible Lorentzian graviton spectral function computed so far in the asymptotic-safety program, and the central numerical result—positive continuum, normalizable tail decaying as 1/(λ² log³ λ²), massless one-graviton pole—is a real step forward. The paper is also honest about its main assumptions.

What is actually new: the self-consistent feedback of the graviton scattering continuum into its own flow, combined with the on-shell renormalisation scheme that yields a finite total spectral weight zspec ≈ 1.486. The UV tail follows from an analytic power-counting argument; the prefactor is numerical. The IR check against the universal one-loop coefficient 61/30 in Fig. 2b is the kind of benchmark that makes this credible. The iteration converges, and the scheme is stated clearly enough that a determined reader could reimplement it, though the paper does not ship code.

Soft spots, in proportion.

The abstract's "fully self-consistent" phrase overruns the calculation. The ghost loop in (C3) is evaluated with a classical ρc(λ)=δ(λ²−k²). That loop feeds directly into Im Γ_TT^(2) and hence into the extracted graviton spectral function via (28). A non-classical ghost spectrum could shift zspec or, in principle, the sign of the tail. No systematic error estimate is given. This is the most concrete gap, and the stress-test note has it right.

The unit total spectral weight is imposed, not found. Equation (31) divides by zspec, so (32) is a field redefinition. The paper actually says the sum rule is "readily implemented," which is transparent—but the abstract's wording invites a stronger reading. The real, non-trivial outputs are positivity and normalizability of ρh before that division, and those stand.

The beta function for Newton's coupling is imported from a Litim-regulator p=0 computation (23). The authors flag it. It is a practical choice, but it means full self-consistency also excludes the coupling sector. The structural conclusions survive because they depend mainly on the UV fixed point.

The KL representation assumption for the gauge-noninvariant graviton propagator is stated in Sec. II without evasion. That is a real caveat, but an honest one.

Overall: the central result is credible and worth engaging. The main missing piece is a coupled ghost computation or at least a quantitative bound on the classical-ghost error, plus more careful abstract wording. This should go to peer review, not be desk rejected. I would cite it and bring it to the QG reading group.

Referee Report

4 major / 5 minor

Summary. The paper computes the graviton spectral function in Lorentzian asymptotically safe quantum gravity using spectral functional RG flows on a flat Minkowski background. The authors solve the integrated Callan-Symanzik flow for the transverse-traceless graviton two-point function in an on-shell renormalization scheme, iteratively feeding the graviton scattering continuum back into the diagrams. They report a positive spectral function with a massless one-graviton peak and a UV tail decaying as 1/(λ² log³ λ²), a finite total spectral weight z_spec≈1.486, and, after rescaling, a unit total spectral weight. The IR onset of the continuum is found to match the universal coefficient 61/30. The paper interprets these results as evidence that on-shell renormalization provides a close-to-physical graviton field suitable for scattering computations and unitarity investigations.

Significance. If the advertised results are robust, this is a significant advance: it is the first Lorentzian fRG computation in gravity in which the graviton scattering continuum is self-consistently fed back into the loop diagrams, and it yields concrete, checkable outputs—an analytic UV decay, a finite spectral weight, and a positive spectral function. The analytic IR benchmark (61/30) and the explicit convergence of the iteration scheme are strengths, as are the detailed diagrammatic kernels in Appendix C. The significance is, however, currently moderated by three caveats: the ghost sector is treated classically, the unit spectral weight is imposed by a final rescaling rather than emerging from the flow, and the input beta function is imported from a different regulator scheme. These caveats do not invalidate the core computation, but they do require the claims in the abstract and conclusions to be recalibrated.

major comments (4)

[Abstract; Appendix C1, Eq. (C3)] The advertised full self-consistency is not realized in the ghost sector. The ghost loop is evaluated with the classical approximation ρ_c(λ)=δ(λ²−k²) stated before Eq. (C3), so the ghost contribution to Im Γ_TT^(2) in Eq. (22) and Eq. (26) is not fed back with a non-perturbative ghost spectral function. Since this contribution enters the extracted graviton spectral function through Eq. (28), a non-classical ghost spectral function could shift z_spec, alter the tail, and in principle affect positivity. No systematic error estimate is provided for this approximation. At minimum, the abstract and Section IV should state that self-consistency is implemented for the graviton sector only, with a classical ghost approximation; ideally the authors should add a sensitivity estimate or a coupled ghost-sector computation.
[Section III, Eqs. (31)-(32)] The unit total spectral weight in Eq. (32) is imposed by the rescaling ρ^(ph)_h = ρ_h/z_spec in Eq. (31), so it is not an independent dynamical output. The nontrivial results are the finiteness of z_spec and the positivity of ρ_h. Moreover, after this rescaling the new field h^(ph) no longer satisfies the on-shell renormalization conditions (17): its residue at the massless pole is 1/z_spec, and ∂_{p²}Γ^(2) equals unity only before rescaling. The abstract's wording 'within the physical on-shell renormalisation scheme, the graviton satisfies the sum rule' therefore conflates the on-shell and unit-weight normalizations. Please rephrase to clarify that the unit sum rule is a post-hoc field redefinition and discuss the implications for canonical commutation relations and pole residues.
[Section II A, Eq. (10); Section II D, Eq. (23)] The computation relies on two imported inputs that are not derived in the on-shell Callan-Symanzik scheme: the assumption Λ_k=0 for all scales (Eq. (10)) and the Newton coupling beta function (23) taken from p=0 Litim-regulator computations [58,59]. This is a further self-consistency gap, since the present scheme uses a different regulator and on-shell renormalization conditions. The authors assert that variations of g_* only induce quantitative changes, but no sensitivity analysis is shown. I recommend varying g_* and/or the trajectory shape and reporting the effect on z_spec and the UV prefactor, or at least giving an estimate of the induced uncertainty.
[Section II; Eq. (14)] The central structural assumption—the existence of a Källén-Lehmann spectral representation for the gauge-variant graviton propagator on Minkowski space—is stated in Section II but not tested. Because the fluctuation graviton is gauge dependent, the positivity, the spectral tail, and the sum-rule interpretation could be gauge artifacts. The paper defers the gauge-dependence discussion to unpublished work cited as [61]. Until that analysis is available or a gauge-independence argument is given, the interpretation of the spectral function as a 'physical building block' should be presented as conditional on the harmonic-gauge choice and on the assumed KL representation.

minor comments (5)

[Appendix C, Eqs. (C4)-(C7)] The frequency variable is written as w in several equations (C4)-(C7) but as ω elsewhere; please use a single symbol consistently.
[Figure 2] The axis label 'λ Mpl^-1' is ambiguous; write λ/M_pl, and specify explicitly whether each panel shows ρ_h, ρ^(ph)_h, or z_spec ρ^(ph)_h.
[Appendix C2, Eq. (C11)] Equation (C11) involves a derivative of a delta distribution and a Hadamard finite part; a brief distributional explanation would improve reproducibility.
[Abstract; Eq. (29)] The phrase 'close-to-quadratic spectral decay' could be made more precise as 'log-corrected quadratic decay', since the tail in Eq. (29) contains (log λ²)³.
[Appendix C4] The paper gives detailed numerical methods but no code or data availability statement; for a computation with iterative numerical solutions, a repository would strengthen reproducibility.

Circularity Check

2 steps flagged · score 7.0 of 10

The unit-total-spectral-weight claim is a normalization identity, and the massless one-graviton peak is inserted by ansatz; the non-trivial positivity and UV tail are computed rather than imposed.

self definitional [Section III, Eqs. (30)-(32); cf. Section II C after Eq. (16)]
"Now we define ρ(ph)h = 1/zspec ρh, with h(ph)µν = hµν/z1/2 spec. ... The physical spectral function ρ(ph)h satisfies the sum rule (15), Z λ ρ(ph)h (λ) = 1. (Section II C: 'Note also that (15) is readily implemented in a final step, subject to the existence of a finite spectral weight.')"

By Eq. (30), zspec is defined as the total unnormalised spectral weight, so dividing by it makes the integral in Eq. (32) exactly 1 for any spectral function with finite weight, positive or not. The paper explicitly says the sum rule is 'readily implemented in a final step'. Thus the advertised 'sum rule of an asymptotic state' and 'unit total spectral weight' are a field redefinition, not an output of the flow; the substantive content is the finiteness of zspec and positivity, which are separate computed results.
self definitional [Section II B, Eq. (14) and Section II C, Eq. (16)]
"ρh(λ) = 1/Zh [2πδ(λ2 − m2h) + θ(λ2 − 4m2h) fh(λ)] ... the running pole mass is identified with the RG scale and the on-shell wave function Zh is set to unity, m2h = k2, Zh = 1."

The spectral ansatz (14) puts a delta-function single-particle peak into ρh by hand, and the on-shell condition (16) fixes its location at mh = k, i.e. mh = 0 at the physical endpoint k = 0. The iteration solves only for the continuum fh. Therefore the abstract's 'massless one-graviton peak' is a restatement of the chosen ansatz and renormalisation scheme, not a dynamical prediction of the computation.

full rationale

The core numerical machinery is not circular: the flow equation (26) is an integral equation for ρh, closed by the extraction formula (28) and solved by iteration; positivity and the 1/(λ^2 log^3 λ^2) UV tail are computed, and the IR onset is checked against the independent universal coefficient 61/30 from [17,65]. The self-cited framework [1] and on-shell scheme [43,44,48] are published and externally used, so citing them is not itself a circularity. However, two headline claims do reduce to construction. First, the unit total spectral weight is obtained by defining ρ^(ph) = ρ/zspec after computing zspec, making Eq. (32) an identity, as the paper itself notes that (15) is 'readily implemented in a final step'. Second, the massless one-graviton peak is not a solution feature: it is inserted by the delta ansatz (14) and fixed by m_h^2 = k^2 in on-shell renormalisation (16). These by-construction elements affect the abstract's strongest formulation; the non-trivial positivity and decay tail are independent, keeping the score below 8. A separate, non-circularity limitation is flagged: Appendix C1 evaluates the ghost loop with the classical ρ_c(λ) = δ(λ^2 − k^2), so the advertised statement that 'the full non-perturbative spectral function is used in the diagrams' is not literally realised for the ghost contribution; this is an overclaim/approximation gap rather than a circular reduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The computation rests on six unproved modeling assumptions: KL representation existence, harmonic gauge, Λk=0, vertex truncation, classical ghost approximation, and an imported beta function. There are no data-fitted free parameters. No new physical entities are postulated; the "physical graviton field" is a field rescaling, not a new degree of freedom.

assumptions (6)

domain assumption The graviton propagator admits a Källén-Lehmann spectral representation on a flat Minkowski background in the presence of the CS regulator.
Section II, second paragraph: the approach "relies on the -non-trivial- existence of the KL spectral representation for the graviton propagator". If gauge-noninvariant graviton fields lack such a representation, the extracted spectral function and sum rule are not defined.
domain assumption Harmonic gauge α=β=1 is a valid gauge for spectral computations and does not distort the spectral function.
Eq (9) selects harmonic gauge, "singled out by spectral considerations, see [1,61]". The spectral representation is gauge-dependent, so this choice is a substantive assumption.
ad hoc to paper The cosmological constant vanishes at all scales, Λk=0.
Eq (10) sets Λk=0 "for computational convenience" even though the full effective action would have a running Λk; extending to Λk≠0 is deferred to [56] and [1]. A nonzero Λ could change the pole and spectral weight.
domain assumption The effective action is truncated to a full two-point function plus Einstein-Hilbert-type three-point vertices.
Eq (7) defines the truncation; all higher vertices and momentum-dependent vertex form factors are neglected. The quantitative spectral tail and zspec depend on this truncation.
ad hoc to paper The ghost spectral function is approximated by a classical delta function, ρc(λ)=δ(λ²-k²).
Appendix C1 states "For the ghost diagram, we use a classical approximation with ρc(λ)=δ(λ²-k²)". This is not fully self-consistent despite the paper's headline claim.
ad hoc to paper The Newton coupling beta function computed at p=0 with a Litim regulator in [58,59] remains valid within the on-shell Callan-Symanzik scheme.
Section II D, Eq (23): the authors use existing Litim-scheme beta functions and expect "mild variations" under different regulators. The quantitative flow depends on g(k).

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Cite this review

Pith. "Pith review of Self-consistent graviton spectral function in Lorentzian quantum gravity." pith.science (2026). https://pith.science/paper/XEGSCD3N

@misc{pith2026250722169,
  author       = {Pith},
  title        = {Pith review of: Self-consistent graviton spectral function in Lorentzian quantum gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XEGSCD3N}},
  note         = {Machine review of arXiv:2507.22169}
}

read the original abstract

We present the first fully self-consistent computation of the graviton spectral function in quantum gravity, using the spectral renormalisation group for gravity put forward in arXiv:2111.13232v2 [hep-th] within a physical mass-shell renormalisation scheme. Here, self-consistency refers to the fact that the full non-perturbative spectral function is used in the diagrams, including the scattering continuum. We find a positive graviton spectral function with a massless one-graviton peak and a multi-graviton continuum with a close-to-quadratic spectral decay in the ultraviolet. Within the physical on-shell renormalisation scheme, the graviton satisfies the sum rule of an asymptotic state and features a unit total spectral weight. We briefly discuss the implications of the physical formulation for the computation of scattering processes and investigations of unitarity in asymptotically safe quantum gravity.

Figures

Figures reproduced from arXiv: 2507.22169 by the authors.

**Figure 2.** FIG. 2: Normalised scattering continuum of the propagator spectral function measured in units of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3: Subleading part of the graviton propagator in the complex plane. It features a cut on the real axis, visible [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Reference graph

Works this paper leans on

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Analytic Expressions of the Diagrams For the diagrammatic expressions, we refer to [1]. These read schematically as Flow(hh) tadpole = Z λ ρ(2) h (λ) Dtadpole(λ) , (C1) Flow(hh) 3-point = Z λ1,λ2 ρ(2) h (λ1)ρh(λ2) D3-point(λ1, λ2, p) , Flow(hh) ghost = Z λ1,λ2 ρ(2) c (λ1)ρc(λ2) Dghost(λ1, λ2, p) . with the momentum integrals Dtadpole(λ) = Z q Vtadpole(p, ...

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, where we abbreviated the momentum integrals asR q = R ddq (2π)d . The RG-invariant vertex functions with Vi = ¯Vi in on-shell renormalisation combine the con- tractions of the vertices with the regulator derivative ∂tRCS = (2 − ηh)Zhk2. They can be found in the supple- mentary material of [1]. We refrain from displaying the full, exceedingly long expres...

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(C10) Note that the Cauchy principal value of this integral does not exist for q2 < −λmin if the pole of the integrand lies within the integration domain

Spectral representation of propagator squared CS-flow diagrams for correlation functions always con- tain a regulator line of the form G(q) ˙RCS(q)G(q) = 2k2G(q)2 , (C8) which has the spectral representation, G2(q) = Z ∞ 0 dλ λ π ρ(2)(λ) (q2 + λ2)2 , (C9) with the second-order spectral function ∂w2 ρ(2)(w) = 2 ImG2 −i(w + i0+) . (C10) Note that the Cauchy...

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Although the spectral tail has a sub-leading impact in the IR, its UV-contribution has the same scaling as the leading term, differing only by an overall factor

UV Limit of the Spectral T ail The UV behaviour of the spectral tail is governed by the UV limit of the imaginary part of the flow, in particular, the contribution from the graviton mass pole. Although the spectral tail has a sub-leading impact in the IR, its UV-contribution has the same scaling as the leading term, differing only by an overall factor. Fu...

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This allows us to compute and integrate the diagrams using this input and then read out the new, fully k-dependent spectral function

Numerical implementation We solve the flow equation for the graviton spectral function with a fixed-point iteration method: the flow equation is interpreted as an integral equation, where the spectral function on the right-hand side of (26) is fully known as a function of ω and k. This allows us to compute and integrate the diagrams using this input and t...

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∂ω2 f (ω0) = 2 π PV Z ∞ 0 dt (ω2 − ω2 0)2 (w2 − ω2 0)2 t Im f (t) t2 − ω2 , (C17) where the subtracted Taylor expansion serves two pur- poses: (1) it renders the Kramers-Kronig integral finite, and (2) it allows for a convenient implementation of the on-shell renormalisation conditions (20). The tail of the spectral function is constructed as, f (i+1) h (...

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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Analytic Expressions of the Diagrams For the diagrammatic expressions, we refer to [1]. These read schematically as Flow(hh) tadpole = Z λ ρ(2) h (λ) Dtadpole(λ) , (C1) Flow(hh) 3-point = Z λ1,λ2 ρ(2) h (λ1)ρh(λ2) D3-point(λ1, λ2, p) , Flow(hh) ghost = Z λ1,λ2 ρ(2) c (λ1)ρc(λ2) Dghost(λ1, λ2, p) . with the momentum integrals Dtadpole(λ) = Z q Vtadpole(p, ...

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, D3-point(λ1, λ2, p) = Z q V3-point(p, q) (q2 + λ2 1)2 ((p + q)2 + λ2

work page

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The RG-invariant vertex functions with Vi = ¯Vi in on-shell renormalisation combine the con- tractions of the vertices with the regulator derivative ∂tRCS = (2 − ηh)Zhk2

, where we abbreviated the momentum integrals asR q = R ddq (2π)d . The RG-invariant vertex functions with Vi = ¯Vi in on-shell renormalisation combine the con- tractions of the vertices with the regulator derivative ∂tRCS = (2 − ηh)Zhk2. They can be found in the supple- mentary material of [1]. We refrain from displaying the full, exceedingly long expres...

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(C10) Note that the Cauchy principal value of this integral does not exist for q2 < −λmin if the pole of the integrand lies within the integration domain

Spectral representation of propagator squared CS-flow diagrams for correlation functions always con- tain a regulator line of the form G(q) ˙RCS(q)G(q) = 2k2G(q)2 , (C8) which has the spectral representation, G2(q) = Z ∞ 0 dλ λ π ρ(2)(λ) (q2 + λ2)2 , (C9) with the second-order spectral function ∂w2 ρ(2)(w) = 2 ImG2 −i(w + i0+) . (C10) Note that the Cauchy...

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Although the spectral tail has a sub-leading impact in the IR, its UV-contribution has the same scaling as the leading term, differing only by an overall factor

UV Limit of the Spectral T ail The UV behaviour of the spectral tail is governed by the UV limit of the imaginary part of the flow, in particular, the contribution from the graviton mass pole. Although the spectral tail has a sub-leading impact in the IR, its UV-contribution has the same scaling as the leading term, differing only by an overall factor. Fu...

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This allows us to compute and integrate the diagrams using this input and then read out the new, fully k-dependent spectral function

Numerical implementation We solve the flow equation for the graviton spectral function with a fixed-point iteration method: the flow equation is interpreted as an integral equation, where the spectral function on the right-hand side of (26) is fully known as a function of ω and k. This allows us to compute and integrate the diagrams using this input and t...

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∂ω2 f (ω0) = 2 π PV Z ∞ 0 dt (ω2 − ω2 0)2 (w2 − ω2 0)2 t Im f (t) t2 − ω2 , (C17) where the subtracted Taylor expansion serves two pur- poses: (1) it renders the Kramers-Kronig integral finite, and (2) it allows for a convenient implementation of the on-shell renormalisation conditions (20). The tail of the spectral function is constructed as, f (i+1) h (...

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