Asymptotic-Safety–Inspired Model

• Asymptotic-Safety–Inspired Models are quantum field theories and gravity extensions characterized by a non-Gaussian fixed point ensuring well-defined behavior at arbitrarily high energies.
• They utilize renormalization-group equations in gauge–Yukawa–scalar sectors to predict coupling evolution, reducing free parameters and securing UV consistency.
• The paradigm underpins UV-complete SM extensions and informs innovative models in quantum gravity, cosmology, and black hole physics with observable phenomenological signatures.

An Asymptotic-Safety–Inspired Model refers to any quantum field theory or extension of the Standard Model (SM), as well as quantum gravity/early universe and black hole effective models, whose UV behavior is governed by the existence of a renormalization-group (RG) fixed point: the so-called non-Gaussian (interacting) fixed point (NGFP). This paradigm ensures that all couplings evolve such that the full theory remains predictive and well-defined up to arbitrarily high energies, typically with only a finite set of free (UV-relevant) parameters. The concept generalizes asymptotic freedom, allowing for scale-invariant, interacting UV completions in models with multiple couplings, including gauge, Yukawa, and scalar sectors (Bednyakov et al., 2023).

1. Formal RG Definition and Fixed-Point Structure

A theory is asymptotically safe if all dimensionless couplings $g_i(k)$ approach a fixed-point $g_i^*$ as $k\to\infty$; i.e.,

$$\beta_i(g^*) = \left.\frac{d g_i}{d \ln k}\right|_{g^*}=0 \text{ for all } i,$$

where $k$ is the RG scale (Bednyakov et al., 2023, Eichhorn, 2017, Nagy, 2012). The interaction may be

• Gaussian (Free) FP: $g_i^*=0$, as in asymptotic freedom.
• Non-Gaussian (Interacting) FP: some $g_i^*\neq 0$, as in asymptotic safety.

Linearization near the fixed point defines a stability matrix $$M_{ij} = \left.\frac{\partial \beta_i}{\partial g_j}\right|_{g^*}$$; its (negative) eigenvalues, $\theta_k$, determine the number of UV-relevant directions (free parameters). The remaining (irrelevant) directions are predicted by the UV completion (Bednyakov et al., 2023, Eichhorn, 2017).

2. Mechanisms in Gauge–Yukawa–Scalar Sectors

In four-dimensional gauge–Yukawa–scalar theories, the generic structure of two-loop RG equations is

$$\begin{aligned} \beta_{g} & = b_g g^3 + \ldots \ \beta_{y} & = a_y y g^2 + c_y y^3 + \ldots \ \beta_\lambda & = a_\lambda \lambda^2 + b_\lambda g^2 \lambda + c_\lambda g^4 + d_\lambda y^4 + \ldots \end{aligned}$$

Simultaneous zeros define candidate fixed points. Two types of interacting solutions are common:

• Banks–Zaks (BZ) FP: $g^* \sim B/C$, $y^*=0$, $B,C$ related to one/two-loop gauge terms.
• Gauge–Yukawa (GY) FP: $g^* = B/(C - DF/E)$, $y^*=(F/E)g^*$, with $D,E,F$ set by gauge–Yukawa mixing; physical FPs require positivity of $g^*, y^*$ and the (model-dependent) denominator (Bednyakov et al., 2023, Nagy, 2012).

The linearized spectrum of RG eigenvalues $\theta_k$ determines the critical surface. For instance, in the Litim–Sannino $SU(N_c)$ toy model with $N_F$ vector-like fermions and $N_F \times N_F$ scalars, the GY FP arises in the small $\varepsilon\equiv N_F/N_c - 11/2$ limit (Bednyakov et al., 2023). Here,

$$\hat{\alpha}_g^* \approx \frac{26}{57}\varepsilon, \qquad \hat{\alpha}_y^* \approx \frac{4}{9}\varepsilon,$$

and the critical exponents are one (UV-relevant, $\sim \varepsilon^2$), the rest (UV-irrelevant, $\sim -\varepsilon$) (Bednyakov et al., 2023).

3. Extensions to Gravity, Cosmology, and Black Holes

In the quantum gravity sector, asymptotic safety is realized via a non-Gaussian fixed point for the running Newton constant $G(k)$ and cosmological constant $\Lambda(k)$:

$g(k) = k^2 G(k), \quad \lambda(k) = \Lambda(k)/k^2$

with beta functions (Einstein–Hilbert truncation):

$$\beta_g = [2+\eta_N]g, \qquad \beta_\lambda = [\eta_N-2]\lambda + \cdots$$

and anomalous dimension $\eta_N$ (Bonanno et al., 2017, Spina, 16 Oct 2025). This RG structure persists in both Euclidean and Lorentzian signature, with two UV-relevant directions (Manrique et al., 2011).

RG-improved cosmologies follow from promoting $k \mapsto k(\mu)$ as a function of cosmic time, Hubble parameter, or curvature, yielding scale-dependent $G(k)$ and $\Lambda(k)$ that drive, e.g., power-law or Starobinsky-like inflation and set the effective dark energy equation of state (Bonanno et al., 2017, Zarikas, 2024).

Quantum-corrected black holes derived from Asymptotic Safety employ RG-improved lapse functions, with $G(k)$ running as

$G(k) = \frac{G_N}{1+\omega G_N k^2}$

and $k=k(r)$ taken from local curvature or energy density (Spina, 16 Oct 2025, Spina et al., 2024). Such spacetimes are nonsingular, possess extremal limits, and produce distinctive shifts in quasinormal modes and ringdown signals.

4. Model Classes and Phenomenology

Table: Key Classes of Asymptotic-Safety–Inspired Models

Model Type	Main Features	Example References
Minimal Gauge–Yukawa	Vector-like fermions, scalar singlet matrix, GY fixed point	(Bednyakov et al., 2023)
SM+Portal Fermions/Scalars	Vector-like BSM fermion/scalar with portals to Higgs, etc.	(Bednyakov et al., 2023, Hiller et al., 2019)
Gravity–Matter Unification	Quantum gravity coupled to SM, gravity-induced FPs	(Eichhorn, 2017, Pastor-Gutiérrez et al., 2022, Eichhorn et al., 2017)
Hidden $U(1)'$ + CW Breaking	Scale-invariant hidden sector with AS boundary conditions	(Wang et al., 2015)
Chiral Higgs–Yukawa	Non-Abelian chiral models with threshold-induced FPs	(Gies et al., 2013)
Higgs–Portal Dark Matter	Scalar + fermion dark sector, AS line fixes couplings	(Eichhorn et al., 2018)
AS Cosmology	Time-dependent $G(k),\Lambda(k)$, RG-improved Einstein eqns	(Bonanno et al., 2017, Zarikas, 2024)
AS Black Hole Solutions	RG-improved Schwarzschild (etc.), nonsingular cores, QNMs	(Spina, 16 Oct 2025, Spina et al., 2024)

Minimal SM extensions often require vector-like fermions (possibly in higher representations) and scalar singlet matrices, with portal and Yukawa couplings. In bottom-up constructions, viable GY fixed points are obtained in the weak coupling window for $(N_F/N_c)\to 11/2^+$, or, for the SM, by supplementing with additional BSM content to avoid Landau poles in $U(1)_Y$ and vacuum instability in $\lambda$ (Bednyakov et al., 2023, Bond et al., 2017).

Phenomenologically, characteristic consequences include:

• New colored/exotic fermions ($\psi_i$) and scalars ($S_{ij}$) at TeV–10 TeV scales.
• Portal couplings (e.g., $\delta$) that modify Higgs properties and allow Higgs–singlet mixing.
• Signatures at $pp$ and $e^+e^-$ colliders: Drell–Yan $\psi_i$ pair production, single $\psi$ via Higgs/singlet exchange, and $S$ pair production.
• Vacuum stability improved by additional positive contributions to $\beta_\lambda$; gauge portal fermions may delay or remove U(1) Landau poles (Bednyakov et al., 2023, Hiller et al., 2019).
• Potential explanations for $\Delta a_\mu$, $\Delta a_e$ via chiral-enhanced 1-loop contributions involving new (vector-like) fermions and scalar mixing, with electron EDMs potentially saturating current bounds (Hiller et al., 2019).

In gravity–SM models, the UV-relevance or irrelevance of various matter couplings (Yukawa, quartic, gauge) at the fixed point dictates predictivity: e.g., top Yukawa and Higgs quartic become predictions in some truncations due to their UV-irrelevant status (Eichhorn, 2017, Eichhorn et al., 2017).

5. Constraints, Criticality, and Open Problems

• The existence of perturbatively accessible fixed points relies on small conformal windows in flavor/gauge parameter space (e.g., $0<\epsilon<0.09$ for the Veneziano limit at $N_c\to\infty$, shrunk by $O(10\%-20\%)$ by finite-$N_c$ effects) (Bednyakov et al., 2023).
• Positivity of scalar quartics at their fixed points and absence of Landau poles prior to $\Lambda_{\rm UV}$ are required for vacuum stability and unitarity.
• The inclusion of gravitational corrections introduces universal linear terms in gauge and Yukawa $\beta$-functions; their sign and magnitude critically affect the UV fate of couplings. For the SM plus gravity, one typically finds two UV-attractive directions (Newton and cosmological constant), with matter couplings being UV-irrelevant or relevant depending on quantum-gravity-induced anomalous dimensions (Eichhorn, 2017, Eichhorn et al., 2017).
• Strong-coupling behavior and nonperturbative verification of fixed points remain open, motivating functional RG studies and advanced truncations. Gauge-fixing and bimetric ambiguities in gravity–matter systems continue to be areas of active research (Bednyakov et al., 2023).
• In effective black hole solutions, ambiguity in RG scale identification and truncation choice contributes to spread in phenomenological predictions (e.g., shadow size, QNM spectrum), but generic features such as nonsingular cores and critical extremality are robust (Spina, 16 Oct 2025, Spina et al., 2024).

6. Synthesis and Predictivity of Asymptotic-Safety–Inspired Models

The central predictive gain of the asymptotic safety paradigm is the replacement of arbitrary (UV-divergent) parameter spaces with low-dimensional UV critical surfaces: only a finite number of RG-relevant deformations remain as free inputs, while all other couplings are predictions of the theory. In explicit BSM and SM+gravity models, this mechanism yields definite mass and coupling predictions (e.g., scalar and new-fermion masses in the 100 GeV–10 TeV range; top mass prediction from gravity–matter RG flow), constrains portal and Yukawa couplings, and ensures UV consistency without Landau poles or instability up to $M_{\mathrm{Pl}}$ and beyond (Bednyakov et al., 2023, Pastor-Gutiérrez et al., 2022, Eichhorn et al., 2017).

Asymptotic-safety–inspired models have thus emerged as a systematic and unifying approach for constructing UV-complete quantum field theories and quantum gravity, offering direct connections between high-energy RG structure, collider signatures, and cosmological observables. They continue to motivate both phenomenological and nonperturbative theoretical developments.