Dilaton Quantum Gravity Fixed Point

• Dilaton quantum gravity fixed points are characterized by a scale-invariant effective action where the metric and dilaton interplay to ensure nonperturbative renormalizability.
• The approach uses functional RG frameworks with truncations that capture both two-derivative terms and non-minimal couplings, linking UV and IR behaviors via a continuous scaling solution.
• Spontaneous breaking of scale invariance generates a dynamical Planck mass and an exponential potential, leading to implications for cosmological scenarios like dark energy and transient inflation.

Dilaton quantum gravity fixed points are nontrivial renormalization group (RG) fixed points for a gravitational action featuring both the spacetime metric and a dynamical scalar field ("dilaton"), typically coupled non-minimally to curvature. At these fixed points, the quantum effective action becomes scale-invariant under global dilatations, with all relevant dimensionless couplings being $k$-independent as the infrared cutoff $k \to 0$. The fixed point structure implies nonperturbative renormalizability (asymptotic safety), and the spontaneous breaking of scale invariance generates a Planck mass and modifies late-time cosmology. These scenarios are supported by explicit functional RG analyses in both four and higher dimensions.

1. Frameworks and Functional RG Equations

Several functional RG frameworks underpin investigations into the dilaton quantum gravity fixed point:

• The flow equation for the $k$-dependent effective average action $\Gamma_k$ is

$$\partial_t\Gamma_k = \tfrac{1}{2} \mathrm{Tr}\Big[(\Gamma_k^{(2)} + R_k)^{-1} \partial_t R_k\Big]$$

where $t=\ln(k/\mu)$, $\Gamma_k^{(2)}$ is the Hessian, and $R_k$ is the IR cutoff operator (Maitiniyazi et al., 16 Dec 2025, Henz et al., 2013, Henz et al., 2016).

• The action truncations typically include up to two-derivative terms with arbitrary field dependence, e.g.,

$$\Gamma_k[g, \chi] = \int d^4x \sqrt{g} \left\{ V_k(\chi^2) - \tfrac{1}{2} F_k(\chi^2)R + \tfrac{1}{2}g^{\mu\nu}\partial_\mu\chi\partial_\nu\chi \right\}$$

and dimensionless variables such as $y = \chi^2/k^2$, $v_k(y) = V_k(\chi^2)/k^4 y^2$, and $f_k(y) = F_k(\chi^2)/k^2 y$ are introduced to render the RG equations autonomous (Henz et al., 2013).

• The general system evolves towards (or away from) scale-invariant configurations as encoded in a set of coupled ODEs for the dimensionless potential, curvature coupling, and kinetic term. Global scaling solutions are identified by the condition that these functions depend only on the dimensionless invariant of the field and flow scale.

2. Fixed Point Structure and Scaling Solutions

The defining feature of the dilaton quantum gravity fixed point is an exactly scale-invariant effective action in the IR ($k \to 0$) limit:

$$\Gamma^*[g, \chi] = \int d^4x \sqrt{g} \left\{ \tfrac{1}{2} g^{\mu\nu}\partial_\mu\chi\partial_\nu\chi - \tfrac{1}{2}\xi \chi^2 R \right\}$$

with $\xi > 0$ (Henz et al., 2013). This action is invariant under global Weyl (dilatation) transformations,

$$g_{\mu\nu} \to \alpha^2 g_{\mu\nu}, \qquad \chi \to \alpha^{-1}\chi,$$

and contains no dilaton potential, i.e., $v^*(y\to\infty)=0$. For large field values, the non-minimal coupling grows as $F(\chi^2) \sim \xi \chi^2$, and the dimensionless Planck mass scales quadratically with the field.

In (Maitiniyazi et al., 16 Dec 2025), the global scaling solution for the dimensionless Planck mass and potential at large $\phi$ (or $\tilde{\rho} \to \infty$, $\tilde{\phi} = \phi/k$) is

$$M^2(\phi) \sim \xi_\infty \phi^2, \quad U(\phi) \sim u_\infty k^4,$$

with $u_\infty \to 3/128\pi^2$, and the scaling solution links the fixed points in the UV (small field) and IR (large field) by a regular RG trajectory (Maitiniyazi et al., 16 Dec 2025, Henz et al., 2016).

Numerical integration confirms a continuous family of regular scaling solutions, parametrized by the large-field behavior of the kinetic function and non-minimal coupling, with stability and absence of tachyonic modes requiring certain positivity conditions (Maitiniyazi et al., 16 Dec 2025).

3. Spontaneous Scale Symmetry Breaking and the Planck Mass

While the fixed-point action is exactly scale-invariant, any nonzero vacuum expectation value (VEV) $\langle\chi\rangle = \chi_0 \neq 0$ spontaneously breaks this symmetry:

$\xi \chi_0^2 = M^2$

defining the reduced Planck mass. A Weyl transformation to the Einstein frame,

$$g_{\mu\nu} = \frac{M^2}{\xi\chi^2} \tilde{g}_{\mu\nu},\qquad \phi = M \ln\left(\xi^{1/2} \chi / M\right),$$

brings the action to canonical Einstein gravity plus a free massless scalar (the dilaton),

$$\Gamma^* = \int d^4x \sqrt{\tilde{g}} \left\{ \tfrac{1}{2} \tilde{g}^{\mu\nu}\partial_\mu\phi \partial_\nu\phi - \tfrac{1}{2} M^2 \tilde{R} \right\} [1304.7743].$$

The spontaneous breaking mechanism thus generates a dynamical mass scale, while the metric and dilaton moduli determine gravitational strength and possible cosmological implications.

4. Relevant Perturbations and the Dilatation Anomaly

Flowing away from the fixed point, explicit breaking of scale invariance—dilatation anomaly—manifests via relevant perturbations:

$$V(k) = \frac{\bar{\zeta}_V}{4}k^4 + \bar{V},\qquad F(k) = \xi\chi^2 + \frac{\bar{\zeta}_F}{2}k^2 + \bar{F}$$

with $\bar{V}, \bar{F}$ being $k$-independent integration constants (Henz et al., 2013). These perform as relevant, RG-repelling deformations, associated with the cosmological constant and Ricci coupling in the Jordan frame.

In the Einstein frame, the dilatation anomaly induces an exponential potential for the canonically normalized "cosmon" field,

$$V(\varphi) = \bar{V} \exp\left(-\frac{2\varphi}{M}\right),$$

yielding a nonzero trace anomaly in the energy-momentum tensor and a dynamical dark energy component (Henz et al., 2013).

5. Cosmological Implications: Dynamical Dark Energy and Vanishing Vacuum Energy

The exponential dilatation anomaly potential leads naturally to cosmological quintessence scenarios. For the cosmological evolution in a Friedmann–Lemaître–Robertson–Walker (FLRW) background, the attractor solution,

$$\varphi(t) \simeq \frac{M}{2} \ln(t/t_0),\qquad V(t) \propto t^{-2},$$

implies that the effective vacuum energy decays to zero asymptotically ($t\to\infty$) (Henz et al., 2013, Wetterich, 2010, Henz et al., 2016). This provides a dynamical mechanism for the vanishing cosmological constant and relates quantum gravitational RG flow to observable late-time acceleration.

For suitable choices of the parameters, the Einstein-frame potential yields both a transient inflationary era (near a small-field maximum) and late-time acceleration along the exponential tail, unifying early- and late-universe cosmology in a single RG trajectory (Henz et al., 2016).

6. Stability, Spectrum of Perturbations, and Generalizations

Within typical truncations (e.g., two functions $V$, $F$ plus canonical kinetic term), linearized flow reveals two relevant perturbations with RG eigenvalues corresponding to canonical dimensions four and two, controlling the approach to and departure from the critical surface (Henz et al., 2013, Maitiniyazi et al., 16 Dec 2025).

Certain conformally reduced (background-independent) models, such as those analyzed in (Dietz et al., 2016), exhibit a much richer structure—a continuum of fixed points and eigenoperator spectra—including infinitely many relevant directions for the conformal factor sector alone. These findings highlight that the actual dimension of the UV critical surface may depend crucially on the approximation and treatment of background independence.

Higher-dimensional and Weyl-invariant generalizations preserve the essential fixed-point structure, with dimensional reduction yielding consistent self-tuning mechanisms for the cosmological constant and a dilatation-invariant higher-dimensional theory reducing to standard four-dimensional gravity plus a cosmon (Wetterich, 2010, Pagani et al., 2013).

7. Comparison with Alternative Scenarios and Future Directions

Dilaton quantum gravity fixed points differ structurally from the "extended Reuter" fixed point, where the Planck mass is field-independent and the scalar potential is flat; in the dilaton case, the Planck mass tracks the scalar field ($M(\phi)\propto\phi$ at large field), while the potential is exponentially suppressed (Maitiniyazi et al., 16 Dec 2025, Henz et al., 2016). Both arise as solutions to the same FRG flow equations but correspond to distinct universality classes.

Current analyses indicate that the existence of the dilaton quantum gravity fixed point ensures ultraviolet completeness (asymptotic safety) for a broad class of scalar-tensor models, with direct implications for realistic cosmological models featuring dynamical dark energy (Maitiniyazi et al., 16 Dec 2025, Henz et al., 2013).

Ongoing research addresses the fully gauge-invariant completion of the FRG flows, the operator spectrum beyond simple truncations, and potential extensions to more general matter content. The conformal factor sector, in particular, is under scrutiny for its implications for RG critical surfaces and background independence (Dietz et al., 2016).

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Key References

• "Dilaton Quantum Gravity" (Henz et al., 2013)
• "Scaling solutions for gauge invariant flow equations in dilaton quantum gravity" (Maitiniyazi et al., 16 Dec 2025)
• "Scaling solutions for Dilaton Quantum Gravity" (Henz et al., 2016)
• "Fixed point structure of the conformal factor field in quantum gravity" (Dietz et al., 2016)
• "Warping with dilatation symmetry and self-tuning of the cosmological constant" (Wetterich, 2010)
• "Quantization and fixed points of non-integrable Weyl theory" (Pagani et al., 2013)