Asymptotically Safe Scalar-Tensor Theories

• Asymptotically safe scalar-tensor theories are gravity-matter models that achieve ultraviolet completion via interacting fixed points in the renormalization group flow.
• They employ functional renormalization group methods and shift-symmetric as well as potential-based formulations to control non-minimal couplings and critical surfaces.
• Applications include inflationary model building, predictions in the Higgs sector, and constraining dark energy dynamics, ensuring compatibility with cosmological observations.

Searching arXiv for recent and foundational work on asymptotically safe scalar-tensor theories to ground the article in the current literature. arxiv_search(query="asymptotically safe scalar-tensor theories functional renormalization group shift symmetry", max_results=10) arxiv_search(query="asymptotically safe scalar-tensor theories functional renormalization group shift symmetry", max_results=10)

Asymptotically safe scalar-tensor theories are gravity-matter systems in which the ultraviolet completion is controlled by an interacting fixed point of the renormalization-group flow, so that scalar self-interactions, non-minimal curvature couplings, and gravitational couplings approach fixed-point values or fixed-point functions at high scales. In the literature, this notion encompasses two-derivative scalar-tensor truncations with field-dependent $V(\phi^2)$ and $F(\phi^2)R$, shift-symmetric derivative theories, RG-improved Einstein-Hilbert gravity recast as Jordan-Brans-Dicke or $f(R)$ models, and inflationary models constrained by the critical surface of a non-Gaussian fixed point (0911.0386, Laporte et al., 2021, Cai et al., 2011, Tronconi, 2017).

1. Conceptual scope and historical formulations

A foundational formulation studies Euclidean scalar-tensor theories of the form

$$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$

with $V(\phi^2)$ and $F(\phi^2)$ treated as general functions. In this two-derivative setting, the scalar sector contains a standard kinetic term plus a general scalar potential, while the metric sector contains the Einstein-Hilbert-type nonminimal coupling $F(\phi^2)R$. A central early result is that minimal coupling is self consistent: when scalar self couplings are switched off, their beta functions also vanish, so gravity does not force a minimally coupled scalar sector to acquire scalar self-interactions under the RG flow (0911.0386).

A related scalar-tensor interpretation emerges from RG-improved Einstein-Hilbert gravity. The action

$$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$

with running $G=G(p)$ and $\Lambda=\Lambda(p)$ is classically equivalent to a Jordan-Brans-Dicke theory with vanishing Brans-Dicke parameter, under the identification

$F(\phi^2)R$0

In the toy running used there, the theory can also be reformulated as an $F(\phi^2)R$1 model, making asymptotic safety appear as a particular scalar-tensor sector whose potential is fixed by the RG trajectory rather than chosen arbitrarily (Cai et al., 2011).

Inflationary applications often begin from the Jordan-frame scalar-tensor action

$F(\phi^2)R$2

with

$F(\phi^2)R$3

Within this setup, the asymptotic-safety requirement is implemented by restricting the model to lie on or near the critical surface of a non-trivial fixed point. The result is a scalar-tensor model in which the non-minimal coupling $F(\phi^2)R$4, the potential parameters, and the admissible inflationary regimes are constrained by the fixed-point structure (Tronconi, 2017).

2. Functional renormalization-group description of theory space

The dominant nonperturbative framework is the functional renormalization group for the effective average action $F(\phi^2)R$5, governed by the Wetterich equation

$F(\phi^2)R$6

In this formulation, the RG flow is interpreted as a flow on theory space,

$F(\phi^2)R$7

with fixed points defined by $F(\phi^2)R$8. Linearization around the fixed point yields the stability matrix $F(\phi^2)R$9, with critical exponents $f(R)$0; UV-attractive directions satisfy $f(R)$1 (Laporte et al., 2021).

Several parameterizations of scalar-tensor theory space are used. In potential-based studies one introduces dimensionless functions such as

$f(R)$2

or, in a closely related notation,

$f(R)$3

with $f(R)$4 for the Higgs doublet or $f(R)$5 for a singlet. These variables convert the fixed-point problem from a finite set of couplings to coupled functional equations for $f(R)$6 and $f(R)$7 or $f(R)$8 and $f(R)$9 (Wetterich, 2019, Tronconi, 2017).

Shift-symmetric approaches instead organize theory space by operators compatible with $$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$0. A representative truncation keeps

$$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$1

and, in the essential-coupling formulation at fourth order in the derivative expansion, the action is written entirely in terms of the essential couplings $$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$2, $$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$3, $$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$4, and $$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$5. In both cases the background field method is used, and one computation keeps the background metric arbitrary, with projection rules chosen so that the result is manifestly background independent within the background approximation (Laporte et al., 2021, Knorr, 2022).

3. Fixed points, critical surfaces, and weak-gravity bounds

Early two-derivative analyses found, in any dimension $$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$6, a Gaussian Matter Fixed Point (GMFP) at which all scalar self-interactions vanish while the gravitational couplings remain nonzero. In $$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$7, the fixed-point ansatz is

$$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$8

with $$\Gamma_k[g,\phi] = \int d^d x \sqrt{g}\left( V(\phi^2)-F(\phi^2)R +\frac12 g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi \right) +S_{GF}+S_{gh},$$9 for $V(\phi^2)$0, and the quoted real solution is

$V(\phi^2)$1

The stability matrix at this fixed point has a block upper-bidiagonal form with the recursion

$V(\phi^2)$2

so the higher spectra are shifted by integer multiples of $V(\phi^2)$3. In $V(\phi^2)$4, the smallest truncation including terms up to $V(\phi^2)$5 yields critical exponents

$V(\phi^2)$6

Within this truncation, asymptotic safety is realized by an interacting gravitational sector coexisting with a free scalar sector (0911.0386).

Later truncations found fully interacting matter sectors. In a shift-symmetric scalar-tensor system with Einstein-Hilbert gravity, quartic scalar self-interaction $V(\phi^2)$7, and the two curvature couplings $V(\phi^2)$8 and $V(\phi^2)$9, the full $F(\phi^2)$0 truncation exhibits an interacting fixed point at approximately

$F(\phi^2)$1

with $F(\phi^2)$2. The gravitational exponents stay close to the minimal-coupling values, while the matter-sector exponents are mostly negative, so the non-minimal matter operators are UV-irrelevant. This shifts the picture from Gaussian matter to a non-Gaussian matter sector whose couplings sit at finite fixed-point values (Laporte et al., 2021).

An essential-coupling analysis at fourth order in the derivative expansion derives the global phase diagram in the $F(\phi^2)$3 plane. The viable region is bounded by separatrices connecting four fixed points $F(\phi^2)$4, forming the “shark fin.” Trajectories inside this region are UV complete via an interacting fixed point and in the IR they flow toward standard effective field theory behavior around the Gaussian fixed point. The paper finds a weak-gravity bound whose peak defines

$F(\phi^2)$5

The actual UV fixed points $F(\phi^2)$6 and $F(\phi^2)$7 lie below this bound, and the separatrix $F(\phi^2)$8 has no free parameters, giving a concrete example of a fully predictive scalar-tensor theory (Knorr, 2022).

The multiplicity of fixed-point patterns depends on the truncation and coarse-graining scheme. In $F(\phi^2)$9-invariant scalar models coupled to gravity, a standard type-I cutoff yields the result that too many scalars destroy the positive-$F(\phi^2)R$0 gravitational fixed point, while a scalar-free coarse-graining scheme removes the upper bound on $F(\phi^2)R$1; the authors caution that the disappearance of the bound in the scalar-free scheme may be an artifact of the approximation (Labus et al., 2015). Complementary vertex-based studies of the 1-graviton–2-scalar coupling also find a robust interacting UV fixed point in the pure-gravity limit, but when fully dynamical scalar fluctuations are included the fixed-point structure becomes truncation-sensitive, indicating that gravity-induced matter self-interactions are dynamically important (Donà et al., 2015).

4. Shift symmetry, scaling functions, and non-polynomial structure

A major structural divide in the literature is between shift-symmetric derivative theories and potential-based scalar-tensor theories. In the shift-symmetric case, the kinetic term

$F(\phi^2)R$2

is invariant under $F(\phi^2)R$3, and a non-perturbative proof shows that the corresponding subspace is closed under RG flow. The exact identity

$F(\phi^2)R$4

implies that no shift-symmetry breaking operators are generated dynamically if they are absent initially. In particular, a scalar potential $F(\phi^2)R$5 is not generated from a shift-symmetric starting point (Laporte et al., 2021).

Potential-based scalar-tensor theories display a different nonperturbative structure. In the functional language,

$F(\phi^2)R$6

and the ultraviolet fixed point is not generally a single number for the quartic coupling. Instead, the UV completion is specified by a whole function $F(\phi^2)R$7, the scaling potential, and, when the Planck mass depends on fields, by a second scaling function $F(\phi^2)R$8. In the constant-$F(\phi^2)R$9 approximation the scaling potential often interpolates between two constants,

$$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$0

while for field-dependent Planck mass the large-field asymptotics take the form

$$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$1

These scaling solutions are typically non-polynomial, and finite polynomial expansions around $$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$2 can fail because the field dependence is governed by anomalous scaling rather than canonical power counting (Wetterich, 2019).

The linearized deviations around a constant scaling solution obey

$$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$3

where $$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$4 is the gravity-induced anomalous dimension. This is the formal reason that a fixed point in scalar-tensor theory space can be a fixed-point function rather than a fixed point of a finite list of couplings (Wetterich, 2019).

The essential-coupling program supplies a complementary notion of theory space. There, redundant couplings are removed by $$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$5-dependent field redefinitions generated by a $$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$6-kernel, and the flow is written entirely in terms of essential couplings. This does not contradict the fixed-point-function picture; rather, it isolates which couplings are considered physical in a given derivative expansion (Knorr, 2022).

5. Higgs sector, symmetry breaking, and matter constraints

Gauge, Yukawa, and non-minimal curvature couplings change the scalar fixed-point structure qualitatively. In the scalar-potential framework, nonzero gauge couplings generate an extra contribution

$$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$7

which contributes negatively to the scalar mass term at the origin. For $$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$8, the fixed-point mass term becomes negative, so the origin is a maximum and the scaling solution develops a minimum at nonzero field. This is described as a “gravitational Coleman-Weinberg” mechanism, with spontaneous symmetry breaking already at the scaling solution. By contrast, fermion fluctuations contribute with the opposite sign and stabilize the origin (Wetterich, 2019).

In Standard-Model-motivated settings, the quartic Higgs coupling can become an irrelevant coupling at the UV fixed point, so its value is predicted. One analysis strengthens the earlier argument that this leads to a Higgs mass near the observed $$S_{ASG}=\int d^4x\,\sqrt{-g}\left[\frac{R-2\Lambda}{16\pi G}+{\cal L}_m\right]$$9 GeV and emphasizes that the non-minimal Higgs-curvature coupling $G=G(p)$0 must be very small if this prediction is to remain compatible with the observed Higgs mass and top mass. The same study notes a possible simultaneous prediction of both $G=G(p)$1 and $G=G(p)$2 if a fixed point with nonzero gauge and Yukawa couplings exists and constrains an appropriate combination of $G=G(p)$3, $G=G(p)$4, $G=G(p)$5, and curvature derivatives (Wetterich, 2019).

A gravity-Yukawa truncation with explicit $G=G(p)$6 sharpens this point. In the simple Yukawa model, the beta function

$G=G(p)$7

shows a competition between Yukawa and gravity effects. The viable interacting fixed point quoted there is

$G=G(p)$8

and the viable range for positive $G=G(p)$9 is

$\Lambda=\Lambda(p)$0

In this truncation, $\Lambda=\Lambda(p)$1 is typically of order unity rather than $\Lambda=\Lambda(p)$2, so Higgs inflation is disfavored, whereas the asymptotic-safety mechanism for predicting or bounding the top mass persists (Eichhorn et al., 2020).

The same constraining logic is applied to dark sectors. For a single real scalar dark-matter candidate, the Higgs-portal coupling is irrelevant and fixed to zero, so asymptotic safety decouples a lone scalar dark matter candidate from the visible sector in that truncation. Two-sector models with extra dark fermions remain viable in the paper’s discussion, but the overall message is that asymptotic safety acts as a strong selection principle, separating an “asymptotically safe landscape” from an “asymptotically safe swampland” for scalar models (Eichhorn et al., 2020).

6. Inflation, dark energy, and cosmological tests

Inflationary model building in asymptotically safe scalar-tensor theory proceeds by combining fixed-point information with cosmological observables. In one scalar-tensor inflation analysis, the improved flow admits a physically acceptable fixed point

$\Lambda=\Lambda(p)$3

with five eigendirections and critical exponents

$\Lambda=\Lambda(p)$4

The critical surface then generates a scalar-tensor action with

$\Lambda=\Lambda(p)$5

Combining proximity to the fixed point with Planck data severely restricts parameter space and favors small non-minimal coupling, typically $\Lambda=\Lambda(p)$6 to $\Lambda=\Lambda(p)$7 in the viable region. The analysis also stresses that the conclusion depends on the identification of the IR scale $\Lambda=\Lambda(p)$8: the choice $\Lambda=\Lambda(p)$9 supports models close to the fixed point, whereas $F(\phi^2)R$00 generally does not within the linearized approximation (Tronconi, 2017).

RG-improved Einstein-Hilbert gravity, interpreted as a Jordan-Brans-Dicke theory with $F(\phi^2)R$01, provides a more direct scalar-tensor cosmology. In the toy running

$F(\phi^2)R$02

the Einstein-frame potential is fixed by the RG flow and becomes asymptotically flat for large negative scalar field, allowing large-field inflation near the Planck scale. At the same time, the analysis concludes that the effective field theory description breaks down there and that perturbation theory is inconsistent unless the stress-energy contribution induced by the running Newton constant is included. In the simplest model the effective sound speed approaches $F(\phi^2)R$03, not the GR value $F(\phi^2)R$04, if this contribution is omitted (Cai et al., 2011).

A different cosmological mechanism arises from the large-field asymptotics of the scaling potential. When

$F(\phi^2)R$05

the Einstein-frame potential is

$F(\phi^2)R$06

so it vanishes as the field grows. In the cosmological interpretation given there, this yields a runaway potential and an asymptotically vanishing dark-energy density. If relevant parameters vanish, the theory realizes “fundamental scale invariance” and the cosmological constant vanishes asymptotically through runaway behavior (Wetterich, 2019).

Recent work has connected shift-symmetric asymptotically safe scalar-tensor theory to scalaron inflation. Starting from a truncation

$F(\phi^2)R$07

RG improvement around the Gaussian fixed point yields the IR effective action

$F(\phi^2)R$08

The non-minimal matter-gravity coupling $F(\phi^2)R$09 does not stay confined to the matter sector; it enters the $F(\phi^2)R$10 coefficient through $F(\phi^2)R$11, thereby modifying the width of the inflationary plateau. The paper reports that for $F(\phi^2)R$12 the predictions lie within the Planck 2018 allowed region, with

$F(\phi^2)R$13

for $F(\phi^2)R$14, and that the scalar amplitude fixes $F(\phi^2)R$15 while requiring

$F(\phi^2)R$16

Because the fixed-point structure contains one Gaussian fixed point and three non-Gaussian fixed points, the paper concludes that UV completions by two of the three non-Gaussian fixed points can be ruled out, whereas trajectories emanating from the third remain phenomenologically viable (Koch et al., 21 Jun 2026).

Taken together, these results show that asymptotically safe scalar-tensor theories are not a single model class but a family of FRG-defined theory spaces in which UV completion can be realized by a Gaussian matter fixed point, an interacting matter fixed point, or a fixed-point function. Their cosmological content is similarly diverse: small-$F(\phi^2)R$17 non-minimal inflation may survive in restricted regions, standard Higgs inflation is often disfavored, shift symmetry can protect a flat scalar direction, and large-field scaling solutions can generate a runaway dark-energy sector (Tronconi, 2017, Wetterich, 2019, Eichhorn et al., 2020, Koch et al., 21 Jun 2026).