Reuter Fixed Point in Quantum Gravity

• The Reuter fixed point is a nontrivial fixed point in asymptotically safe quantum gravity, where vanishing beta functions balance canonical and anomalous dimensions.
• Different truncation schemes, including Einstein–Hilbert and f(R)-type analyses, demonstrate its stability and role in ensuring quantum scale invariance.
• Its finite number of UV-relevant directions under various formulations underpins the predictive power of quantum gravity models.

In the asymptotic safety program for quantum gravity, the Reuter fixed point denotes a non-trivial fixed point of the gravitational renormalization group flow. In the formulation based on dimensionless couplings $\{u_i(k)\}$ with beta functions $\beta_i(u)\equiv d u_i/dt$, $t\equiv \ln(k/k_0)$, it is a simultaneous zero of all beta functions, and in the Einstein–Hilbert sector it is a nontrivial solution $(g_*,\lambda_*)\neq (0,0)$ of the beta functions for the dimensionless Newton and cosmological couplings (Kurov et al., 2020). At such a fixed point, canonical mass dimensions are balanced by anomalous dimensions, so that the ultraviolet theory exhibits quantum scale invariance (Kurov et al., 2020). Within the sources considered here, the term is used primarily in functional-renormalization-group analyses of gravity, including Einstein–Hilbert, $f(R)$-type, essential-coupling, fermion-gravity, and conformally reduced truncations (Kurov et al., 2020, Baldazzi et al., 2021, Eichhorn et al., 2018, Gégény et al., 2022).

1. Renormalization-group definition

A fixed point $u^*$ is defined by

$\beta_i(u^*)=0 \qquad \text{for all } i.$

In particular, the Reuter fixed point is identified in the Einstein–Hilbert truncation as a nontrivial fixed point in the $(g,\lambda)$-plane. The balance between canonical and anomalous scaling is expressed by

$$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$

which is the mechanism by which quantum scale invariance arises in the ultraviolet (Kurov et al., 2020).

The underlying flow is formulated with the effective average action $\Gamma_k[g]$ and the Wetterich equation

$\beta_i(u)\equiv d u_i/dt$0

In the Einstein–Hilbert truncation,

$\beta_i(u)\equiv d u_i/dt$1

and one may extend the ansatz by higher-curvature invariants,

$\beta_i(u)\equiv d u_i/dt$2

corresponding to $\beta_i(u)\equiv d u_i/dt$3-type truncations (Kurov et al., 2020).

A representative explicit realization is obtained with the Einstein–Hilbert beta functions using a Litim regulator, where

$\beta_i(u)\equiv d u_i/dt$4

and in $\beta_i(u)\equiv d u_i/dt$5 the nontrivial solution is

$\beta_i(u)\equiv d u_i/dt$6

This gives a concrete coordinate realization of the Reuter fixed point in one standard truncation (Kurov et al., 2020).

2. Linearized flow and ultraviolet relevance

Near a fixed point, the flow is linearized as

$\beta_i(u)\equiv d u_i/dt$7

The eigenvalue problem

$\beta_i(u)\equiv d u_i/dt$8

defines the critical exponents

$\beta_i(u)\equiv d u_i/dt$9

with $t\equiv \ln(k/k_0)$0 identifying UV-relevant directions and $t\equiv \ln(k/k_0)$1 identifying UV-irrelevant directions (Kurov et al., 2020).

In the Einstein–Hilbert truncation alone, the reported result is two UV-relevant exponents $t\equiv \ln(k/k_0)$2 with $t\equiv \ln(k/k_0)$3. When the $t\equiv \ln(k/k_0)$4 coupling is incorporated, the third direction becomes marginal $t\equiv \ln(k/k_0)$5 relevant, so that Newton’s constant, the cosmological constant, and the $t\equiv \ln(k/k_0)$6-coupling supply three relevant parameters, while higher $t\equiv \ln(k/k_0)$7 couplings remain UV-irrelevant (Kurov et al., 2020). This is one of the central arguments for predictive power in $t\equiv \ln(k/k_0)$8-type analyses.

The summarized literature does not present a single universal count of relevant directions. Instead, it reports different dimensionalities of the UV critical surface in different truncations: three relevant parameters in $t\equiv \ln(k/k_0)$9-type truncations including $(g_*,\lambda_*)\neq (0,0)$0, one relevant essential coupling in the minimal essential scheme, three UV-relevant directions in a fermion-gravity truncation before imposing modified Slavnov–Taylor identities, and a two-dimensional UV-attractive critical surface in a conformally reduced model (Kurov et al., 2020, Baldazzi et al., 2021, Eichhorn et al., 2018, Gégény et al., 2022). This suggests that intermediate determinations of relevance are truncation-dependent even when the existence of a non-Gaussian ultraviolet fixed point is robust.

3. Composite operators, curvature invariants, and geometry

A more refined characterization of the Reuter fixed point uses the composite-operator formalism on a background $(g_*,\lambda_*)\neq (0,0)$1-sphere. One considers geometric operators

$(g_*,\lambda_*)\neq (0,0)$2

introduces a matrix of wave-function renormalizations $(g_*,\lambda_*)\neq (0,0)$3, and defines renormalized operators

$(g_*,\lambda_*)\neq (0,0)$4

The anomalous-dimension matrix is

$(g_*,\lambda_*)\neq (0,0)$5

and the corresponding master equation is, schematically,

$(g_*,\lambda_*)\neq (0,0)$6

On a spherical background, heat-kernel techniques yield an explicit infinite matrix $(g_*,\lambda_*)\neq (0,0)$7 (Kurov et al., 2020).

When the couplings $(g_*,\lambda_*)\neq (0,0)$8 are identified as the coefficients of $(g_*,\lambda_*)\neq (0,0)$9 in $f(R)$0, the stability matrix and the anomalous-dimension matrix are related by

$f(R)$1

In $f(R)$2, $f(R)$3, so the $f(R)$4 operator is classically marginal. At the Reuter fixed point, the quantum entry $f(R)$5 becomes positive, implying $f(R)$6, and therefore $f(R)$7 becomes UV-relevant (Kurov et al., 2020).

The same analysis identifies a “perturbative regime” in the $f(R)$8-plane where the spectrum of $f(R)$9 is dominated by $u^*$0, so that the critical exponents approximately follow

$u^*$1

However, the Reuter fixed point itself lies outside that perturbative domain unless higher-curvature contributions are included in the propagators. The reported conclusion is that the non-minimal curvature terms in the two-point function are essential to guarantee a finite number of UV-relevant directions, and hence the predictive power of asymptotic safety (Kurov et al., 2020).

This framework also furnishes a geometric interpretation. The anomalous scaling dimensions $u^*$2 of the volume operator $u^*$3, the Einstein–Hilbert term $u^*$4, and higher curvature invariants are described as measuring the fractal properties of “quantum spacetime,” while the equality between canonical and anomalous dimensions implements quantum scale invariance at the fixed point (Kurov et al., 2020).

4. Essential-coupling formulation

A distinct formulation is developed in the minimal essential scheme, where $u^*$5-dependent field reparameterizations are used to remove inessential couplings and retain only essential ones. In this scheme the diffeomorphism-invariant part of the effective average action, truncated to operators with up to four derivatives, is taken as

$u^*$6

with

$u^*$7

and the RG kernel is parameterized by

$u^*$8

(Baldazzi et al., 2021).

Passing to the dimensionless Newton coupling

$u^*$9

the Reuter fixed point is defined by $\beta_i(u^*)=0 \qquad \text{for all } i.$0. Numerically, the fixed-point value is reported as

$\beta_i(u^*)=0 \qquad \text{for all } i.$1

in the Einstein–Hilbert truncation and

$\beta_i(u^*)=0 \qquad \text{for all } i.$2

when minimal essential four-derivative terms are included. The corresponding leading critical exponents are

$\beta_i(u^*)=0 \qquad \text{for all } i.$3

The small shifts are presented as evidence that the properties of the Reuter fixed point are stable between the Einstein–Hilbert approximation and the approximation including all diffeomorphism-invariant four-derivative terms in the flow equation (Baldazzi et al., 2021).

Within this framework, vacuum energy and four-derivative curvature terms are treated as inessential, while Newton’s constant is treated as essential. The resulting interpretation is that Newton’s constant is the only relevant essential coupling at the Reuter fixed point. The authors therefore conjecture that quantum Einstein gravity has no free parameters in the absence of matter and in particular predicts a vanishing cosmological constant (Baldazzi et al., 2021).

5. Matter couplings and reduced models

The inclusion of matter tests the persistence of the Reuter fixed point beyond pure gravity. In a truncation containing the Einstein–Hilbert terms, a minimally coupled chiral fermion, and a leading nonminimal derivative fermion–curvature coupling $\beta_i(u^*)=0 \qquad \text{for all } i.$4, the dimensionless couplings are

$\beta_i(u^*)=0 \qquad \text{for all } i.$5

and the Reuter fixed point is defined by

$\beta_i(u^*)=0 \qquad \text{for all } i.$6

For $\beta_i(u^*)=0 \qquad \text{for all } i.$7, the reported fixed-point values in the $\beta_i(u^*)=0 \qquad \text{for all } i.$8 truncation including $\beta_i(u^*)=0 \qquad \text{for all } i.$9 are

$(g,\lambda)$0

The associated critical exponents are

$(g,\lambda)$1

Accordingly, there are three UV-relevant directions in the enlarged truncation, although only two are described as physical once the modified Slavnov–Taylor identities are imposed. The backreaction of $(g,\lambda)$2 is reported to be small for $(g,\lambda)$3, and no sign of a “weak-gravity bound” is seen for $(g,\lambda)$4 up to $(g,\lambda)$5 in the parameter region where anomalous dimensions remain below $(g,\lambda)$6 (Eichhorn et al., 2018).

A different reduced setting is conformally reduced Einstein–Hilbert gravity with separately evolving time and space derivatives. There the metric is restricted to

$(g,\lambda)$7

with kinetic operator

$(g,\lambda)$8

where $(g,\lambda)$9 signals explicit Lorentz-symmetry breaking at the regulator level. After introducing

$$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$0

the $$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$1 subsystem closes and admits a non-Gaussian ultraviolet fixed point

$$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$2

with conservative numerical uncertainty of order $$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$3, $$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$4. Linearization yields complex-conjugate stability exponents

$$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$5

so that the critical surface is UV-attractive with a spiral flow. In this truncation, trajectories with $$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$6 remain near $$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$7 close to the Gaussian fixed point, but $$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$8 is driven rapidly to zero as the flow approaches the Reuter fixed point, which is interpreted as a dynamical violation of Lorentz invariance associated with the fixed point (Gégény et al., 2022).

Setting	Fixed-point data	UV characterization
Einstein–Hilbert with Litim regulator	$$0=\frac{d u_i}{dt}=-d_i u_i+\eta_i u_i \;\Rightarrow\; \eta_i=d_i,$$9	Nontrivial solution of the Einstein–Hilbert beta functions
Minimal essential scheme	$\Gamma_k[g]$0, $\Gamma_k[g]$1	$\Gamma_k[g]$2, $\Gamma_k[g]$3
Conformally reduced model	$\Gamma_k[g]$4	UV-attractive spiral with $\Gamma_k[g]$5

These values are reported in different truncations and parameterizations. A plausible implication is that the existence of a non-Gaussian ultraviolet fixed point is more stable across formulations than the numerical location of that fixed point in a particular coordinate system.

6. Scope, nomenclature, and common confusions

Within the source set considered here, “Reuter fixed point” is a term of quantum-gravity renormalization-group analysis rather than a general label for fixed-point theorems across mathematics. A 2024 survey of fixed point theorems in computability theory explicitly contains no occurrence of the name “Reuter” and no theorem attributed to Reuter; instead it discusses Kleene’s recursion theorems, Feferman’s theorem for partial combinatory algebras, Ershov’s theorem on precomplete numberings, the Barendregt–Terwijn generalization, Arslanov’s completeness criterion, and Visser’s ADN theorem (Terwijn, 2024).

The term is also unrelated to fixed points of the Ruelle–Thurston operator in one-dimensional holomorphic dynamics, where fixed points are characterized in a Cauchy-transform class and nontrivial fixed points exist precisely in connection with Herman rings (Levin, 2020). Nor should it be conflated with the Ran–Reurings fixed point theorem and its generalization to transitive binary relations, monotonic chainability, and monotonic completeness (Ben-El-Mechaiekh, 2014).

A further nomenclature caution arises from the supplied material on normalized Hausdorff moment sequences. The abstract of the paper on the transformation

$\Gamma_k[g]$6

states only that the transformation has a unique attractive fixed point and that this fixed point is a Hausdorff moment sequence studied in papers by Berg and Durán; the abstract does not use the expression “Reuter fixed point” (Berg et al., 2010). For that reason, the gravity usage remains the unambiguous sense supported across multiple sources here.

Taken together, the sources depict the Reuter fixed point as the central non-Gaussian ultraviolet fixed point of asymptotically safe quantum gravity. What varies across formulations is not the basic definition—vanishing beta functions and ultraviolet attraction along finitely many relevant directions—but the truncation, the choice of essential versus inessential couplings, the matter content, and the symmetry assumptions used to parameterize the flow (Kurov et al., 2020, Baldazzi et al., 2021, Eichhorn et al., 2018, Gégény et al., 2022).