Renormalized Newton Constant

• Renormalized Newton Constant is a scale-dependent gravitational coupling that incorporates quantum corrections through loop effects and renormalization group flow.
• It modifies classical gravity by introducing antiscreening behavior, which impacts black hole thermodynamics and spacetime geometries in strong field regimes.
• The effective coupling plays a crucial role in cosmology and quantum-induced gravity, influencing models from asymptotic safety to induced Planck mass generation.

The renormalized Newton constant refers to the effective gravitational coupling parameter, $G$, treated as a scale-dependent or background-dependent quantity as dictated by quantum field-theoretic renormalization. Rather than a fixed parameter as in classical general relativity, the renormalized Newton constant encapsulates quantum corrections, running under changes of the energy/momentum scale, curvature scale, or position, as determined by loop corrections, renormalization group (RG) flow, or strong coupling effects in gravitational and matter sectors. Its scale-dependence is central in quantum gravity, black hole effective theory, cosmological model building, and in understanding gravitational interaction from microphysical principles.

1. Renormalization of Newton’s Constant in Quantum Gravity

In perturbative approaches, the gravitational action receives quantum corrections from both graviton and matter loops. In $d=4$, the leading correction induces a running of Newton’s constant (here $G_N(\mu)$ at RG scale $\mu$) given at one-loop by

$$M^2(\mu) = M^2(0) - \frac{N}{12\pi} \mu^2, \qquad G_N(\mu) = \frac{1}{M^2(\mu)}$$

with $N$ encoding the net spin-weighted number of propagating fields. The corresponding beta function is

$$\beta_{G}(\mu) = \mu\,\frac{dG_N}{d\mu} = \frac{N}{6\pi} \mu^2\,G_N^2(\mu)$$

For $N>0$, $G_N$ grows at higher energy, implying “antiscreening” of gravity, an effect typically dominated by gravitons in pure gravity and by matter fields for large $N$ (Calmet, 2010).

Beyond one-loop, non-perturbative functional RG techniques yield coupled flow equations for Newton’s constant and the cosmological constant, commonly captured within truncations of the effective average action formalism, such as the Einstein–Hilbert truncation:

$G(k) = \frac{G_0}{1+w\,G_0\,k^2}$

where $k$ is an IR scale and $w$ is scheme-dependent. In the UV limit ($k\to\infty$), $G(k)$ vanishes, realizing an ultraviolet (UV) fixed point structure essential for the asymptotic safety scenario (Ruiz et al., 2021, Zholdasbek et al., 4 May 2024, Falls, 2015). Graviton-induced running is characterized by anti-screening, with gravity weakening at high energies or high background curvature.

2. Renormalization Group Improved Spacetime Geometries

Renormalization group improved (RGI) spacetimes explicitly promote $G_0$ to a function $G(r)$, with the RG scale $k$ set by local geometric invariants (e.g., $k\sim 1/r$). In the Kerr geometry, this yields

$G(r) = G_0\left(1 - \frac{\xi}{r^2}\right)$

where $\xi$ is a dimensionful RG parameter encoding quantum corrections. The RGI metric modifies the classical coefficients (such as effective mass), e.g. in the Kerr spacetime:

$M_{\rm eff}(r) = M G(r)$

Such running introduces repulsive $O(\xi/r^3)$ corrections in the effective central potential, leading to distinct phenomenology in strong gravity regimes, notably in black hole orbital dynamics and chaos suppression at strong coupling (Lu et al., 10 Oct 2025). For the Reissner–Nordström case, $G(r)$ is given by

$G(r) = \frac{G_0r^2}{r^2 + \bar w G_0}$

appearing in horizon structure, surface gravity, and black hole thermodynamics (Ruiz et al., 2021).

3. Induced Gravity and Matter Contributions

Within the induced gravity paradigm, loop corrections from gauge theory sectors (such as QCD-like theories) yield calculable contributions to $1/G_{\rm ind}$ through vacuum polarization:

$$\frac{1}{16\pi G_{\rm ind}} = \frac{i}{96} \int d^4x\, x^2 \langle 0 | T\{ \hat{T}(x) \hat{T}(0) \} | 0 \rangle$$

For pure-glue QCD, lattice and OPE techniques determine a small but positive induced shift $\Delta M_{\rm Pl} \sim 0.7\,\textrm{GeV}$, with higher-scale hidden Yang-Mills sectors capable of generating a dominant Planck mass. Positivity is generic for SU($N$) gauge theories. This provides an explicit QFT-based realization for generating the Einstein–Hilbert term and Newton constant as order parameters for collective symmetry breaking in strongly coupled sectors (Donoghue et al., 2017).

4. Gauge Independence and Essential Coupling Structures

In specific frameworks such as unimodular gravity, $G$ is an "essential coupling"; it cannot be absorbed by field redefinitions due to the unimodular constraint $\det g_{\mu\nu}=\omega^2$. Functional RG approaches project the flow onto the physical sector, yielding a gauge- and parametrization-independent beta function:

$$\beta_g = (d-2)g + \frac{g^2}{(4\pi)^{d/2}} \frac{(d^2-3d-3) I_d + (d^4-3d^3-22d^2+24d+48) J_d}{2(d^2-3d-3)}$$

where $I_d$, $J_d$ depend on the cutoff choice. The flow admits a non-trivial (non-Gaussian) UV fixed point, ensuring that $G$ is a relevant, UV-completing coupling (Benedetti, 2015, Falls, 2015).

5. Black Hole Entropy, Contact Terms, and Gravitational Entanglement

The renormalized Newton constant enters directly into the Bekenstein–Hawking entropy, $S_{\rm BH} = A/(4G_{\rm ren})$. Quantum corrections to $G$ from matter fields are encoded in the heat-kernel expansion via the $a_1$ coefficient, which includes non-minimal "contact terms" for gauge bosons and gravitons. For a generic spin-$s$ field:

$$\Delta\left(\frac{1}{G_{\rm ren}}\right) \propto D^{(s)}/6 - c_s$$

with $D^{(s)}$ the number of components and $c_s$ the contact term. The entanglement entropy and conical entropy computations show that agreement with the Bekenstein–Hawking value requires matching the renormalization contributions, resolving certain non-analyticities for spin-2 fields by distinguishing orbifold from $n$-fold cover methods (Solodukhin, 2015). The identification $S_{\rm BH} = S_{\rm ent}$ is realized when the bare Newton constant is tuned to cancel contact-term contributions, leaving only physical polarizations.

6. Cosmology and Running Newton Constant

Scale-dependent $G$ has profound cosmological consequences. In scalar-tensor/f($R$) reformulations, RG flow in $G(R)$ leads to effective scenarios with accelerating expansion, even in the absence of a bare cosmological constant. Explicit RG constructions yield models where

$G(R) = \frac{\alpha(R/R_0)}{8\pi m_{\rm pl}^2}$

and the RG-improved Friedmann equations feature running $G(\epsilon)$, allowing for quasi–de Sitter early-universe phases, natural dynamical exit into standard cosmology, or realization of cosmic hierarchy via flows with “soft” UV approaches. An antiscreening RG fixed point structure ensures $G(k)\sim g_*/k^2 \to 0$ at high energy, providing a mechanism for resolving cosmological singularities and connecting Planck scale physics to observable cosmological parameters (Zholdasbek et al., 4 May 2024, Frolov et al., 2011).

7. Model-Dependent Effective Gravitational Coupling and Lorentz Violation

Extensions involving Lorentz-symmetry breaking fields (such as vector field condensates) can induce time-dependent corrections to $G_N$, altering gravitational dynamics in the infrared. For models with vector fields $A_\mu$ and $B_\mu$, the effective Newton constant takes the form:

$$G_{N, {\rm eff}} = G / \left[1 + 2M_P^{-2}(\mathcal{E}_{Q_A}A_0^2 + \mathcal{E}_{Q_B}B_0^2)\right]$$

Asymptotic solutions ensure that $G_{N, {\rm eff}} \to G$ at late times, and gravitational waves retain standard polarizations and dispersion properties provided background scaling parameters satisfy explicit growth conditions (Santillan et al., 2017).

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In summary, the renormalized Newton constant is a dynamical, scale-dependent coupling reflecting quantum and strong coupling corrections in both gravitational and matter systems. Its running, as traced by functional and perturbative RG analyses, carries observable consequences for black hole physics, cosmic evolution, quantum-induced gravity, and high-energy particle phenomenology, and encodes central aspects of attempts to establish a predictive quantum field theory of gravity (Lu et al., 10 Oct 2025, Donoghue et al., 2017, Falls, 2015, Benedetti, 2015, Solodukhin, 2015, Ruiz et al., 2021, Zholdasbek et al., 4 May 2024, Smilga, 2014, Calmet, 2010, Frolov et al., 2011).