Spectral RG for Gravity

• Spectral RG for Gravity is a renormalisation approach that organizes gravitational coarse graining using spectral data instead of traditional momentum-space correlators.
• It employs a Lorentzian spectral-functional method to evolve the graviton propagator, granting direct access to timelike momenta and real-time spectral features.
• A complementary cutoff based on Laplace–Beltrami eigenvalues produces UV fixed points within the asymptotic-safety framework, ensuring covariant scale separation on curved backgrounds.

Searching arXiv for the core papers on Lorentzian/spectral RG for gravity and closely related formulations. arXiv search: (Fehre et al., 2021) OR (Pawlowski et al., 29 Jul 2025) OR (Branchina et al., 15 Jun 2026) OR "spectral renormalisation group gravity" The spectral renormalisation group for gravity denotes a set of renormalisation-group constructions in which gravitational coarse graining is organised by spectral data rather than only by Euclidean momentum-space correlation functions. In the current literature, the term covers two distinct but related developments. One is a Lorentzian spectral-functional RG in which the graviton propagator is evolved through a Källén–Lehmann spectral representation, giving direct access to timelike momenta, branch cuts, and spectral densities. The other is a spectral running cutoff in which Wilsonian shells are defined by eigenvalues of the covariant Laplace–Beltrami operator, providing a diffeomorphism-covariant notion of scale separation on curved backgrounds. Both developments are embedded in the asymptotic-safety program and are motivated by the need to formulate quantum gravity flows in a way that is better adapted either to real-time analyticity and unitarity or to covariant Wilsonian mode elimination (Fehre et al., 2021, Pawlowski et al., 29 Jul 2025, Branchina et al., 15 Jun 2026).

1. Conceptual setting within gravitational renormalisation-group theory

The broader renormalisation-group approach to gravity is usually formulated through a scale-dependent effective action $\Gamma_k$ and a functional flow equation of Wetterich type, with running couplings such as $G_k$ and $\Lambda_k$ defining the gravitational theory space. In this setting, asymptotic safety is the possibility that gravity is controlled in the ultraviolet by a non-trivial fixed point, so that the dimensionless Newton coupling $g=G(k)k^2$ approaches a finite limit and the anomalous dimension compensates the canonical mass dimension of $G$ (Litim, 2011).

The specific motivation for spectral constructions arises from two difficulties that standard Euclidean implementations leave only indirectly addressed. First, Euclidean functional RG methods have been very successful in asymptotic safety, but the real-time structure of the theory remains indirect, and the question of unitarity is not fully settled in Euclidean studies; Wick rotation is especially subtle when the metric itself fluctuates. Second, on curved manifolds there is no globally preferred momentum variable, so a momentum cutoff is not diffeomorphism covariant. These two issues motivate, respectively, the Lorentzian spectral representation of propagators and the use of spectral shells defined by covariant Laplacian eigenvalues (Fehre et al., 2021, Branchina et al., 15 Jun 2026).

A related line of thought appears in Weyl-covariant RG constructions. Percacci’s formulation of Weyl-invariant dilaton gravity introduces the invariant ratio $u=k/\chi$ and shows that a position-dependent cutoff is equivalent to a constant cutoff in a conformally related metric. This is not itself the Lorentzian spectral RG, but it exemplifies the same structural concern with defining the RG scale covariantly in gravity rather than by a fixed flat-space momentum variable (Percacci, 2011).

2. Lorentzian spectral-functional RG

The Lorentzian spectral RG introduced for gravity is built from the Einstein–Hilbert action in Minkowski signature, supplemented by gauge-fixing and ghost terms, and expanded around flat Minkowski space. In the transverse-traceless sector, the graviton two-point function is parametrised as

$$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$

with wave-function factor $Z_h(p)$ and mass parameter $\mu$. Its scalar propagator is written in Källén–Lehmann form,

$$G_{hh}(p_0,|\vec p\,|)=\int_0^\infty \frac{d\lambda}{\pi}\, \frac{\lambda\,\rho_h(\lambda,|\vec p\,|)}{\lambda^2+p_0^2},$$

and the spectral function is defined by analytic continuation to the real-time axis,

$G_k$0

This construction makes the propagator accessible over the full complex momentum plane and directly ties the RG flow to spectral data (Fehre et al., 2021).

A central technical ingredient is a Callan–Symanzik-type Lorentz-invariant cutoff

$G_k$1

rather than a momentum-dependent regulator. The reason given is that momentum independence preserves the spectral representation at finite $G_k$2 because it avoids introducing artificial poles and cuts in the complex plane. The renormalised Lorentzian flow equation is

$G_k$3

with $G_k$4 and the second term collecting the local counterterm flow required because the CS regulator reintroduces UV divergences that must be renormalised (Fehre et al., 2021).

The flow of the graviton spectral function is obtained from the flow of the propagator,

$G_k$5

evaluated on the Lorentzian axis. Loop diagrams entering $G_k$6 are rewritten as spectral integrals over internal spectral values $G_k$7. In this sense, the gravitational RG flow is no longer only a flow of Euclidean Green functions: it becomes a flow of real-time spectral data (Fehre et al., 2021, Pawlowski et al., 29 Jul 2025).

3. Graviton spectral function: pole, continuum, and self-consistency

At zero spatial momentum, the graviton spectral function is decomposed into a one-graviton pole and a continuum,

$G_k$8

Here the $G_k$9-term represents the single-graviton state, $\Lambda_k$0 the multi-graviton continuum, and the threshold begins at $\Lambda_k$1. In the first non-perturbative computation, the spectral flow equation was integrated along an RG trajectory while neglecting feedback of $\Lambda_k$2 into its own flow. That calculation found a positive graviton spectral function with a massless one-graviton peak and a positive multi-graviton continuum (Fehre et al., 2021).

The infrared pole is massless,

$\Lambda_k$3

which reproduces the physical massless graviton pole of flat-space gravity. Above threshold, the continuum is positive and is dominated by contributions from quantum fluctuations at scales $\Lambda_k$4. In the infrared, $\Lambda_k$5 approaches a finite constant, while in the ultraviolet it scales as

$\Lambda_k$6

The corresponding fixed point reported for the Lorentzian flow is

$\Lambda_k$7

with critical exponents

$\Lambda_k$8

A spike near the Planck scale was associated with these complex-conjugate critical exponents (Fehre et al., 2021).

The subsequent self-consistent computation strengthens this framework by using the full non-perturbative spectral function inside the loop diagrams, including the continuum, rather than treating it as an external input or a perturbative correction. The on-shell renormalisation scheme imposes

$\Lambda_k$9

through

$g=G(k)k^2$0

This tracking of the renormalisation point with the RG scale is described as a physical mass-shell renormalisation scheme and is tied to a momentum-dependent rescaling of the fluctuation graviton (Pawlowski et al., 29 Jul 2025).

Within that scheme, the spectral function is positive and normalisable. The raw spectral weight is

$g=G(k)k^2$1

and the physical spectral function is defined by

$g=G(k)k^2$2

so that

$g=G(k)k^2$3

The ultraviolet tail becomes

$g=G(k)k^2$4

which is strong enough to make the spectral integral finite. The same work also recovers the universal infrared behaviour

$g=G(k)k^2$5

and evolves the Newton coupling with

$g=G(k)k^2$6

which has the UV-attractive fixed point

$g=G(k)k^2$7

These results are presented as the first fully self-consistent graviton spectral function in quantum gravity (Pawlowski et al., 29 Jul 2025).

4. Analytic structure, cosmological constant, scattering, and unitarity

Once the spectral function is known, the graviton propagator can be reconstructed for general complex momenta, including the timelike axis. The reconstructed propagator has a real part with the expected pole structure, an imaginary part with a branch cut along the timelike axis, and asymptotically vanishing behaviour at large complex momentum. Together with the positive spectral function and the Källén–Lehmann representation, this is presented as evidence that there are no pathological singularities in the upper half complex plane (Fehre et al., 2021).

The cosmological constant enters the Lorentzian formulation through the infrared boundary condition

$g=G(k)k^2$8

The flat-background analysis first focuses on $g=G(k)k^2$9. For $G$0, simplified trajectories show that the cosmological constant mainly affects the spectral function at small spectral values: for AdS ($G$1) the spectral function is suppressed, as if $G$2 acts like a positive mass term; for dS ($G$3) it is enhanced, as if $G$4 acts like a negative mass-squared term; and for sufficiently large $G$5 the influence of $G$6 disappears. With ghost contributions retained, off-shell AdS backgrounds can produce a divergence in the flow of the continuum, interpreted as an off-shell effect tied to scattering into the ghost sector; the expectation stated there is that the fully on-shell AdS flow should remain finite (Fehre et al., 2021).

These structural results are directly connected to scattering and unitarity. The Lorentzian formulation gives access to timelike graviton propagators, which is a necessary ingredient for computing graviton-mediated scattering amplitudes directly in quantum gravity. The positivity of the spectral function, the absence of ghost or tachyonic instabilities in the propagator, and in the on-shell scheme the unit total spectral weight are all taken as supporting evidence for consistency and unitarity. At the same time, the 2025 self-consistent analysis explicitly notes that the fluctuation graviton is not a gauge-invariant physical asymptotic Hilbert-space state; the stronger claim is instead that the scheme provides optimal physical building blocks for amplitudes and cross-sections (Pawlowski et al., 29 Jul 2025).

5. Spectral running cutoff based on Laplace–Beltrami eigenvalues

A distinct usage of the term “spectral renormalisation group” replaces momentum shells by shells in the spectrum of the covariant Laplace–Beltrami operator. In this formulation, gravity is treated in the Einstein–Hilbert truncation on $G$7, and the one-loop fluctuation operators are of Laplace type,

$G$8

On the spherical background of radius $G$9, the spectrum is

$u=k/\chi$0

with $u=k/\chi$1. The RG step is defined by a shell in the scalar spectral variable,

$u=k/\chi$2

and the same scalar spectral scale $u=k/\chi$3 is used across the different spin sectors (Branchina et al., 15 Jun 2026).

Two implementations are given. The smooth spectral cutoff uses proper-time bounds to isolate the shell, while the hard spectral cutoff uses a discrete sum over modes with a midpoint prescription to control the boundary of the shell. Matching the large-$u=k/\chi$4 expansion of the running action

$u=k/\chi$5

yields beta functions for the dimensionful couplings, and in terms of the dimensionless variables

$u=k/\chi$6

one obtains closed two-coupling RG systems (Branchina et al., 15 Jun 2026).

For the smooth spectral cutoff, the fixed points are the Gaussian point

$u=k/\chi$7

and the non-Gaussian point

$u=k/\chi$8

Linearisation gives the canonical eigenvalues $u=k/\chi$9 and $$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$0 at the Gaussian fixed point, and at the non-Gaussian fixed point

$$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$1

so the ultraviolet approach is spiralling and UV-attractive. For the hard spectral cutoff, one again finds the Gaussian fixed point, a UV-attractive non-Gaussian fixed point

$$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$2

with

$$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$3

and a further fixed point at negative cosmological coupling,

$$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$4

which is a saddle with

$$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$5

In both implementations the flow is singular as $$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$6, so the equations are valid only for $$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$7; this singularity is interpreted as reflecting the known IR instability of propagators in de Sitter space (Branchina et al., 15 Jun 2026).

This spectral-cutoff construction addresses a different problem from the Lorentzian spectral function program. It is a covariant Wilsonian shell integration over Laplacian eigenvalues rather than a real-time flow for Källén–Lehmann spectral densities. The common element is the use of spectral information to define the RG step in a way that is natural for gravity (Branchina et al., 15 Jun 2026).

6. Scope, contrasts, and unresolved issues

The phrase “spectral renormalisation group” is not used uniformly across the gravity literature. The main asymptotic-safety tradition remains the functional/Wilsonian RG based on the effective average action and an infrared regulator $$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$8, with the flow implemented mode-by-mode by suppressing and releasing fluctuations according to their scale relative to $$\Gamma^{(hh)}_{\mathrm{TT}}(p)=Z_h(p)\,(p^2+\mu\,k^2),$$9 (Litim, 2011). From that viewpoint, the Lorentzian spectral RG and the spectral running cutoff are specialised constructions within a broader FRG landscape rather than replacements for it.

The importance of regulator choice is a recurring theme. In higher-derivative gravity, the FRGE analysis with the Universal Renormalization Group Machine recovers the universal one-loop beta functions of the marginal couplings, but the non-universal beta functions for Newton’s constant and the cosmological constant depend on the cutoff choice, and one fixed point seen in the universal sector is described as most probably unphysical because it does not survive the full scheme-dependent analysis (Saueressig et al., 2011). This provides a direct caution for spectral constructions: fixed-point claims must be checked against regularisation and truncation dependence, even when the spectral language is covariant.

Several adjacent RG programs are explicitly not spectral RG in the narrow sense. The exact RG for classical two-body General Relativity is stated not to be about spectral renormalization group in the usual sense of spectral geometry or spectral RG operators, and it does not formulate the method as a spectral RG based on eigenmodes of an operator (Gutiérrez et al., 31 Oct 2025). Likewise, essential RG formulations remove inessential couplings by field redefinitions rather than by introducing a spectral representation, while Weyl-covariant flows reorganise the cutoff through a dilaton compensator (Ohta et al., 4 Jun 2025, Percacci, 2011). These comparisons indicate that “spectral” in gravity can refer either to spectral representations of correlators or to spectral shells of geometric operators, but not to every covariant RG method.

The unresolved issues are correspondingly formulation-specific. In the Lorentzian spectral-function program, the positivity of the graviton spectral function and the absence of ghost or tachyonic instabilities support unitarity, but the fluctuation graviton is still not a gauge-invariant asymptotic state. In the cosmological-constant sector, the off-shell AdS divergence is treated as likely an artefact of the off-shell setup. In the spectral-cutoff program, the singularity at $Z_h(p)$0 restricts the regular domain to $Z_h(p)$1. More broadly, both approaches are currently formulated in truncations—Einstein–Hilbert or closely related two-point approximations—and this suggests that extensions to larger truncation spaces, higher-point functions, and matter-coupled systems remain central next steps.

Within those limits, the spectral renormalisation group for gravity has established two concrete achievements. One is a Lorentzian, non-perturbative computation of graviton spectral data with a massless one-graviton pole, a multi-graviton continuum, positive spectral density, and asymptotically safe ultraviolet scaling (Fehre et al., 2021, Pawlowski et al., 29 Jul 2025). The other is a covariant Wilsonian shell construction in Laplacian spectral space that reproduces the asymptotic-safety pattern with UV-attractive non-Gaussian fixed points in both smooth and hard implementations (Branchina et al., 15 Jun 2026). Together they mark a shift from purely Euclidean or momentum-space RG treatments toward formulations in which the notion of scale in gravity is encoded directly in spectral structure.