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Matter Spectral Functions from Quantum Gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read In asymptotically safe quantum gravity, photon and scalar propagators keep a Källén–Lehmann form, but their spectral functions turn negative and non-normalisable in the ultraviolet.

desk verdict A serious, technically demanding first computation of matter spectral functions in Lorentzian asymptotically safe gravity, but the KL representation and UV sign flip are assumed rather than demonstrated. read the letter →

arxiv 2507.17862 v1 pith:H6YHECGO submitted 2025-07-23 hep-th gr-qchep-ph

Varun Kher , Brandon King , Daniel F. Litim , Manuel Reichert This is my paper
classification hep-thgr-qchep-ph
keywords asymptoticsafetyquantumgravityKällén–LehmannspectralrepresentationfunctionsfunctionalrenormalisationgroupLorentziansignaturephotonpropagatorscalar
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

This paper asks what happens to ordinary matter propagators—photons and uncharged scalars—once gravity is quantised and offers an asymptotically safe ultraviolet completion. Using a spectral renormalisation group adapted to Lorentzian signature, it finds that both propagators still admit a Källén–Lehmann spectral representation, made of an on-shell delta peak and two multi-particle continua. The gravitational continua are gauge-dependent and dominate in the deep ultraviolet, where the spectral functions turn negative and become non-normalisable. The paper concludes that, with quantum gravity present, the photon and the uncharged scalar are no longer physical observables above the Planck scale, even though they remain effectively positive and universal below it. This matters because it decides which states can appear in a unitary scattering theory of asymptotically safe gravity with matter.

What carries the argument

The machinery is the spectral renormalisation group: the functional renormalisation group flow (11) with a Callan-Symanzik-type regulator $R_{k,\Phi}=Z_\Phi k^2$ for bosons and $R_{k,\psi}=Z_\psi k\,\mathbb{1}$ for fermions, which shifts masses by the cutoff $k$ without introducing cuts or poles in the complex momentum plane. The load-bearing ansatz is Eq. (16), which fixes each matter spectral function as an on-shell delta peak plus two continua with thresholds at $m_\Phi+m_h$ and $2m_\psi$. Internal propagators are replaced by their spectral representations, and the flow of the two-point function is projected on the delta-peak part only, turning the integro-differential spectral flow into ordinary differential equations for the mass parameters, anomalous dimensions, and continuum functions. UV divergences are handled by dimensional regularisation and momentum-independent counterterms renormalised at vanishing momentum. The anomalous dimensions extracted from the delta-peak projection set the ultraviolet scaling of the continua.

What would settle it

Re-solve the flow equations (14) with the multi-particle continua $f_{\Phi,\mathrm{grav}}$ and $f_{\Phi,\mathrm{ferm}}$ kept on the right-hand side; if the spectral functions at $k=0$ no longer turn negative for $\lambda\gtrsim M_{\mathrm{Pl}}$, the ultraviolet sign flip is an artifact of the delta-peak-only truncation. A complementary check is to compute the scattering spectral function of $G_{A,\mathrm{scat}}$ in Eq. (47) with quantum-corrected vertices and test whether it is positive semi-definite.

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Extended reading notes

Core claim

The central claim is that in a Lorentzian gravity-matter system with an asymptotically safe fixed point, the full photon and scalar propagators each satisfy a Källén–Lehmann representation with spectral function $\rho_\Phi(\lambda)=Z_\Phi^{-1}[2\pi\delta(\lambda^2-m_\Phi^2)+\theta(\lambda^2-(m_\Phi+m_h)^2)f_{\Phi,\mathrm{grav}}(\lambda)+\theta(\lambda^2-4m_\psi^2)f_{\Phi,\mathrm{ferm}}(\lambda)]$. On the UV-safe trajectory the graviton and fermion loops generate continua whose ultraviolet scaling is $\propto\lambda^{\eta_\Phi^*-2}$; with fixed-point anomalous dimensions $\eta_A^*\approx 0.52$ and $\eta_\phi^*\approx 0.045$, the spectral functions decay more slowly than a free propagator and are therefore non-normalisable. Around the Planck scale both $\rho_A$ and $\rho_\phi$ change sign, driven by the graviton diagram with the matter regulator insertion, which is negative and dominates in the ultraviolet. The paper stresses that the persistence of a Källén–Lehmann form is itself non-trivial, since complex conjugate poles or other non-analyticities could have destroyed it; no such behaviour is seen. The associated form factors inherit the branch cut structure and diverge and turn negative above the Planck scale, which the authors read as a signal that classical vertices cannot be used for amplitudes in that regime.

Load-bearing premise

The computation assumes from the outset that every propagator has a Källén–Lehmann form with one delta peak and two smooth continua, and it neglects the feedback of those continua into the flow equations, so the spectral representation is an input rather than something the calculation itself could disprove.

Editorial extensions

If this is right

  • Photons and uncharged scalars lose their status as physical observables above the Planck scale: their spectral functions become gauge-dependent, non-normalisable, and partly negative.
  • Below the Planck scale and above the fermion threshold, the universal fermion-loop contribution dominates and is positive, so matter spectral functions remain effectively physical in the infrared.
  • The form factors $f_{FF}(p^2)$ and $f_{\phi\phi}(p^2)$ diverge and become negative around the Planck scale, so scattering amplitudes built from these propagators and classical vertices are not trustworthy in the ultraviolet.
  • An asymptotically safe fixed point with $g_Y^*=0.455$ and $y_t^*=0.462$ provides a UV completion for the U(1) and Yukawa couplings if the gravitational coefficients $f_g$ and $f_y$ are positive as assumed.
  • The sign flip in the scalar spectral function is robust (positivity would require $\eta_h^*\lesssim -54$), while the photon case is marginal (positivity requires $\eta_h^*\lesssim 0.3$), so the photon prediction is the more easily tested one.

Reading between the lines

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

  • A consequence the authors leave implicit is that the negative, non-normalisable scalar spectral function, if confirmed beyond the quenched approximation, would forbid uncharged elementary scalars from appearing as asymptotic states in the deep ultraviolet of any asymptotically safe theory.
  • The gauge-independent scattering propagator $G_{A,\mathrm{scat}}$ of Eq. (47) offers a sharper falsifier than the propagator itself: once quantum-corrected vertices are computed, its spectral function must be positive semi-definite if the theory is unitary.
  • The same spectral-RG computation could be repeated with the multi-particle continuum fed back into the flow; in the deep infrared this changes the gauge-field spectral function by about 8.4%, and the interesting question is whether the Planck-scale sign flip shifts or disappears under that feedback.
  • Since the Planck-scale peak is tied to the complex-conjugate critical exponents of the Newton coupling, the sign flip may also appear in other observables that couple to the graviton spectral function near the Planck scale, such as the $e^+e^-\to\mu^+\mu^-$ cross section computed with the same methods.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a spectral renormalisation-group framework in Lorentzian signature with a Callan-Symanzik regulator and applies it to a gravity-matter system containing a U(1) gauge field, an uncharged scalar, and a fermion with Yukawa coupling. The authors compute the photon and scalar two-point functions, extract their spectral functions under the ansatz of Eq. (16), and obtain UV-IR trajectories with an interacting fixed point. They report that both matter spectral functions are non-normalisable, turn negative around the Planck scale, and depend on the gravity gauge-fixing parameters, with the fermion-loop contribution universal and positive below the Planck scale. They also translate the spectral functions into effective-action form factors and discuss implications for unitarity and scattering amplitudes.

Significance. If the central result held, it would be an important step in understanding whether matter fields coupled to asymptotically safe quantum gravity admit Källén-Lehmann representations and whether photons and uncharged scalars remain physical observables above the Planck scale. The paper is careful in several respects: the deep-IR comparison against one-loop effective field theory (Sec. VI C) is a concrete, quantitative check; the authors explicitly list their approximations in Sec. IV; they state the gauge-dependence of the matter spectral functions; and they provide the flow equations and a supplementary Mathematica notebook, which aids reproducibility. The qualitative message, however, is currently stronger than the evidence: the KL representation is assumed rather than derived, and the UV sign flip is not independently validated beyond the specific delta-peak projection. If the authors can reframe the claim and quantify the truncation error, the paper would be a valuable contribution; as it stands, the headline result is partly circular.

major comments (3)
  1. [Sec. III A, Sec. III C, Eq. (16), Sec. VI B] The central claim that the photon and scalar propagators possess a Källén-Lehmann spectral representation is not a derived output. Equation (7) is assumed for the full propagator factors, and Eq. (16) further restricts the spectral function to an on-shell delta peak plus two multi-particle continua with fixed thresholds. Consequently, the statement in Sec. VI B that the authors 'do not observe' complex conjugate poles or other non-analyticities is true only by construction: such structures are excluded by the ansatz. To make the KL claim non-circular, the authors need either to derive the absence of competing singularities from the flow equations or to test an enlarged ansatz that allows complex poles/branch cuts and show that they are not generated.
  2. [Sec. IV, Sec. VI C, Eq. (14)] The approximation of keeping only the delta-peak contribution on the right-hand side of the spectral flow (Sec. IV) turns an integro-differential equation into an ODE, but its error is only quantified in the deep IR. The comparison between Eqs. (39) and (38) gives a relative shift of 8.4% for the gauge field, yet this probes spectral values below the fermion threshold, whereas the sign flip and the asymptotic scaling \rho \propto \lambda^{\eta^*_\Phi - 2} occur around and above the Planck scale (Fig. 4). No analogous check constrains the UV region. The authors should either include the continuous parts of the spectral functions in the flow at least at leading order, or provide another estimate of the UV truncation error; without this, the sign flip and non-normalisability are not established beyond the specific projection used.
  3. [Sec. VI C, Eqs. (40)-(41)] The stability analysis with \eta_h^* treated as a free parameter is informative but incomplete as stated. The bounds \eta_h^*|_{\rm gauge} \lesssim 0.3 and \eta_h^*|_{\rm scalar} \lesssim -54 show that positivity of the scalar spectral function is far from the computed value \eta_h^* \approx 0.96, but the sentence that 'extended approximations are not expected to induce quantitatively significant changes in anomalous dimensions' is an assertion, not an estimate. The authors should provide a concrete error estimate for \eta_h^* (for example from the difference between the Lorentzian and Euclidean extractions, or from the dependence on the regulator) before claiming that the sign change is 'hard-wired' for the scalar.
minor comments (4)
  1. [Sec. IV, Eq. (23), Sec. VI C, Eq. (37)] The symbol \beta is used both for the beta functions of the Yukawa and gauge couplings and for the gravitational gauge-fixing parameter in Eq. (37); this notational clash should be resolved, for instance by renaming the gauge-fixing parameter.
  2. [Sec. VI A, Fig. 4] The text describes the fermion mass as 'm_\psi = 10^{-2} M_{\rm Pl}' but the figure caption says '10^{-2}M_{\rm Pl}'; the units and the illustrative nature of the value should be stated consistently in both places.
  3. [Abstract and Sec. VII] The phrase 'we show that both possess a Källén-Lehmann spectral representation' should be softened to reflect that the representation is imposed by the ansatz in Eq. (16) and then found to be consistent with the projected flow; otherwise the abstract overstates the logical status of the result.
  4. [Sec. IV, Eq. (21)] The trajectory for the fermion mass parameter \mu_\psi is an input rather than a result, and the claim that c_1 has 'strongly sub-leading influence' is only checked for the particular ranges shown; this should be stated explicitly in the main text near Eq. (21).

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed KL representation is an input of the spectral ansatz (16), so the 'non-trivial' existence result is circular; the negative UV spectral functions remain genuine outputs.

  1. self definitional [Sec. III A, Sec. III C Eq. (16), Sec. VI B]
    "Together with the on-shell delta peak, this leads to the following ansatz for the spectral function of the gauge and scalar field, rho_Phi = 1/Z_Phi [2 pi delta(lambda^2 - m_Phi^2) + theta(lambda^2 - (m_Phi + m_h)^2) f_Phi,grav(lambda) + theta(lambda^2 - 4 m_psi^2) f_Phi,ferm(lambda)], which holds for both Phi = {A, phi}. ... First of all, it is non-trivial that the gauge and scalar propagators possess a KL spectral representation after the inclusion of quantum gravity fluctuations."

    The KL representation is not derived; it is imposed. Equation (7) is introduced with 'we use the KL spectral representation', and Eq. (16) restricts every matter propagator to an on-shell delta peak plus two real multi-particle thresholds. Complex-conjugate poles, momentum-dependent residues, or other non-analytic structures are thereby excluded from the function space by construction. The later statement in Sec. VI B that such poles are 'not observed' is therefore guaranteed by the ansatz rather than being a test of the KL property.

full rationale

The non-circular core is the computation: with the spectral ansatz (16) and the quenched delta-peak projection, the paper solves explicit flow equations and obtains the continuum functions, the fermion-threshold peak, the sign flip, and the non-normalisable asymptotic tails. None of these are fitted to the final spectral functions, and the deep-IR comparison against one-loop EFT (relative shift about 8.4%) is a genuine independent check, albeit one that probes only low energies. The circularity is confined to the existence claim: the KL representation is assumed in Eq. (7) and its specific form is fixed by Eq. (16), so the statement in Sec. VI B that the KL property is 'non-trivial' and that complex poles are 'not observed' is true only in the sense that the ansatz cannot represent such poles. The paper is transparent about this, and it also transparently imports the gravity-sector flows from [35]; that self-citation is load-bearing for the numerical value eta*_h ~ 0.96 used in the sign-flip analysis, but it is a published separate computation and is not itself a reduction of the matter result to the present paper's own input. The negative, non-normalisable matter spectral functions are consequences of the flow, not of any fit, so they are not circular. Overall, the central existence claim reduces by construction, giving a partial circularity score of 6, while the genuine quantitative results keep the paper from being fully circular.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The computation rests on the asymptotic safety hypothesis, on the assumed spectral representation (Eq. (16)), on the Callan-Symanzik regulator, and on several truncations (quenched approximation, one-loop beta functions with constant gravity coefficients, chosen fermion mass trajectory, and specific renormalisation conditions). The free parameters fy, fg, c1, c2 and the gauge-fixing parameters are not fitted to the paper's output; they are inputs chosen from the literature or varied for illustration. No new particles or forces are introduced.

free parameters (5)
  • fy (gravity coefficient in Yukawa beta function) = 0.00573 (asymptotically safe trajectory)
    Parameterises the gravitational correction to the Yukawa beta function in Eq. (23); taken as a constant at leading order; positive value is an approximation to the NLO result of Ref. [129] (in preparation).
  • fg (gravity coefficient in U(1) beta function) = 0.00847 (asymptotically safe trajectory)
    Parameterises the gravitational correction to the gauge beta function in Eq. (23); scheme-dependent; a range of values is explored.
  • c1 (UV fixed-point value of fermion mass parameter) = unspecified, varied
    Appears in the chosen fermion mass trajectory (21); has a sub-leading influence on the final spectral functions.
  • c2 (physical fermion mass in Planck units) = 10^-2 for the illustrative figures
    Sets the fermion mass threshold in the multi-particle continuum; chosen large for display, not fitted to data.
  • gravity gauge-fixing parameters alpha, beta = alpha = beta = 1
    Graviton gauge-fixing parameters; the paper shows spectral functions depend on them; chosen in the de-Donder gauge.
assumptions (6)
  • domain assumption Asymptotic safety of the coupled gravity-matter system provides a UV completion.
    The entire computation presupposes an interacting UV fixed point for gravity and matter, inherited from [35] and the asymptotic safety literature; the paper computes fixed-point values within its truncation but does not prove the scenario.
  • ad hoc to paper The full propagator admits a Källén-Lehmann representation (Eq. (7)).
    Used as the starting ansatz in Eq. (16); the paper claims to demonstrate its existence, but the ansatz restricts the analytic structure from the outset.
  • domain assumption The Callan-Symanzik regulator R_k = Z k^2 preserves spectral representations and is admissible in Lorentzian signature.
    Motivated by [35] and [102,103]; the paper notes no known regulator satisfies causality, Lorentz invariance and UV finiteness simultaneously, so counterterms are needed.
  • ad hoc to paper Quenched approximation: matter loops do not feed back into Newton coupling, mass parameters, or anomalous dimensions; multi-particle continua do not feed back into the flow.
    Stated in Sec. IV; turns integro-differential equations into differential equations; only checked against exact EFT in the deep IR (8.4% shift).
  • ad hoc to paper One-loop beta functions for gauge and Yukawa couplings with constant gravity coefficients fy and fg (Eq. (23)).
    Approximates gravitational contributions; fy is taken positive at leading order although cited literature finds negative fy, justified by an NLO result in preparation [129].
  • ad hoc to paper Renormalisation conditions at vanishing momentum (Eq. (13)) and boundary conditions Lambda = 0, omega_phi = 0, Z_i -> 1 at k -> 0.
    Scheme choice; different renormalisation points are possible and would shift finite parts.

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Pith. "Pith review of Matter Spectral Functions from Quantum Gravity." pith.science (2026). https://pith.science/paper/H6YHECGO

@misc{pith2026250717862,
  author       = {Pith},
  title        = {Pith review of: Matter Spectral Functions from Quantum Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6YHECGO}},
  note         = {Machine review of arXiv:2507.17862}
}
read the original abstract

We investigate Lorentzian quantum gravity coupled to a template matter sector with gauge fields, scalars and fermions. In the absence of quantised gravity, the matter sector by itself is renormalisable, but UV-incomplete. Provided quantum gravity offers an asymptotically safe UV-completion, we determine the photon and scalar two-point functions in the presence of gravitational fluctuations, and show that both possess a K\"all\'en-Lehmann spectral representation. Our results are achieved using functional renormalisation adapted for theories in Lorentzian signature. We explain why and how interactions with gravity modify both the infrared as well as the ultraviolet behaviour of matter spectral functions. We further determine the corresponding form factors on the level of the quantum effective action. Limitations and extensions of our study are discussed alongside implications for particle physics and unitarity of quantum gravity with matter.

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Forward citations

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