RTG Gravity II: Quantum Corrections, Running & Anomalies

Revision: 1.2 (units, μ-convention, SU(3) note, RG table, window uncertainty)  |  Date: 14 Aug 2025
Authors: Mustafa Aksu · Grok · ChatGPT

0 · Scope & relation to Gravity I

This page gathers all quantum/renormalization material atop the tree‑level construction in RTG Gravity I: Emergent Metric → Einstein–Hilbert. We specify β‑functions, scale matching, effective‑field content assumptions, anomaly conditions, and the lattice checks to publish. Tree‑level metric definitions and the EH + minimal coupling action live in Gravity I and are not repeated.

1 · Units & symbols (carry‑over)

• All \( \omega,\Delta\omega,\delta\omega \) are angular frequencies in rad·s\(^{-1}\); convert to Hz by \( f=\omega/(2\pi) \). Unless noted, plots use an angular‑frequency scale \( \mu_\omega \) in s\(^{-1}\).
• Two μ conventions: For scalar‑resonance RG (β(g)) we use \( \mu_\omega \) (s\(^{-1}\)). For gravitational running we define the dimensionless Newton coupling \( g_N(\mu_\ell)=\mu_\ell^2 G(\mu_\ell) \) with a length‑based scale \( \mu_\ell \) (e.g., fm\(^{-1}\)). Always state which convention is used.
• Lattice notation: \( A_{ij} \) = kernel amplitude; \( B_{ij}=a\,\mathcal A_{ij} \) = dimensionless U(1) lattice gauge phase; \( U_{ij}=e^{i(\phi_i-\phi_j-B_{ij})} \) = link.
• \( \sigma_{\mathrm{exch}} \) = exchange‑term UV width (model/MD); \( \sigma_{\mathrm{noise}} \) = CHSH noise; \( \sigma \) (unsuffixed) reserved for RG smooth cutoffs here. Independence: these are mutually independent and independent of \( \Delta\omega^\ast \).
• Units for couplings (used below): \( J_{\rm ex} \) in MeV; \( \kappa_B \) (gauge‑field stiffness) in MeV; \( \kappa_c \) (Helfrich curvature modulus) in MeV·fm. Consistent with Enriched Geometry §6 and Gravity I.

2 · Core RTG flow & the critical bandwidth

2.1 · Two‑loop β(g) and fixed point

With the split kernel and vertex weight \( C_{\rm vtx}=3/2 \), the effective scalar‑resonance coupling \( g(\Lambda)\equiv \bar J(\Lambda)/K'(\Lambda) \) (with \( \bar J=\tfrac{3}{2}J \)) flows as
\( \beta_g(g)=(1-\rho)\,g-\sigma_\beta C_{\rm vtx}\,g^2-a_3 g^3 \approx 0.72\,g-0.63\,g^2-0.011\,g^3 \)
(see Two‑Loop RG Derivation §3 for the full derivation), giving a non‑Gaussian IR fixed point \( g^\ast\simeq 1.14\pm 0.02 \) (scheme spread).

The two‑loop flow fixes a universal scale—the critical bandwidth \( \Delta\omega^\ast = (1.45 \pm 0.08)\times 10^{23}\,\mathrm{s}^{-1} \), superseding earlier drafts.

Quantity	Description	Value / Range
\( g^\ast \)	Scalar‑resonance fixed point	\( 1.14 \pm 0.02 \)
\( \Delta\omega^\ast \)	Critical bandwidth	\( (1.45 \pm 0.08)\times 10^{23}\,\mathrm{s}^{-1} \)
\( C_{\rm vtx} \)	Two‑loop vertex weight	\( 3/2 \)
\( a_3 \)	Two‑loop coefficient	\( 0.011 \)
Schemes	Litim / Sharp / Dim‑reg	Compatible; < 1% spread in \( \Delta\omega^\ast \)

2.2 · Symmetry windows vs \( \Delta\omega^\ast \)

Define \( x \equiv |\Delta\omega|/\Delta\omega^\ast \). Practical “phase‑space” windows for emergent gauge behaviour:

\( \mathrm{U}(1): 0 \le x < 0.28;\quad \text{soft‑spin SU(2): } 0.28 \le x \le 0.70;\quad \mathrm{U}(1)^2: 1.55 \le x \le 1.70. \)

Uncertainty: window edges carry a \( \pm 0.02 \) systematic from scheme dependence in \( g^\ast \) (propagates to ~1.4% relative shifts). Avoid \( x \ge 1.70 \) (5‑D anomaly region).

3 · Running of \( G \) and \( \Lambda \) (coarse‑grained)

3.1 · Definitions & renormalization conditions

Treat the tree‑level constants of Gravity I as scale‑dependent matching parameters defined at a coarse‑graining scale. For gravity, use a length‑based \( \mu_\ell \) (e.g., fm\(^{-1}\)):

• \( G(\mu_\ell) \): match the metric two‑point (or geodesic‑deviation) correlator to linearized GR predictions in a test sector (phase waves and/or U(1)).
• \( \Lambda(\mu_\ell) \): fix by vacuum (gate) energy density and curvature fit at the same scale.

Renormalization conditions. Choose two scales \( \mu_{\ell,1} < \mu_{\ell,2} \) (e.g., a ratio \( \mu_{\ell,2}/\mu_{\ell,1} \approx 2 \)). Impose
\( \big[\sqrt{-g}G_{\mu\nu}\big]_{\mu_{\ell,k}} = 8\pi G(\mu_{\ell,k}) \, \big[\sqrt{-g}T_{\mu\nu}\big]_{\mu_{\ell,k}} \)
in the same ensemble, then define β’s by finite differences; propagate errors from ensemble variance. If strongly non‑perturbative, prefer Wilsonian flow plots over analytic fits.

3.2 · Practical running ansatz & extraction

• Dimensionless Newton coupling \( g_N(\mu_\ell)=\mu_\ell^2 G(\mu_\ell) \) with \( \beta_{g_N}=2 g_N + B_1 g_N^2 + O(g_N^3) \). Fit \( B_1 \) from two‑scale measurements.
• Cosmological term \( \Lambda(\mu_\ell) \) with \( \beta_\Lambda \approx C_0 \mu_\ell^4 + C_2 \mu_\ell^2 \Lambda + C_4 \Lambda^2 \) (Wilsonian running). Fit \( C_i \) at fixed scheme.

# Minimal example: fit B1 from two-scale data for g_N(μℓ)
# (Replace mock numbers with measured values from ensembles)
import numpy as np
from math import log
μ = np.array([1.0, 0.5])      # fm^{-1}
gN = np.array([0.10, 0.12])   # measured dimensionless Newton coupling
β_meas = (gN[1]-gN[0]) / (log(μ[1]) - log(μ[0]))  # finite-difference beta
# Solve β_meas ≈ 2*gN(μ1) + B1*gN(μ1)^2:
B1 = (β_meas - 2*gN[0]) / (gN[0]**2)
print("Fitted B1 ~", B1)

Outcome to publish. Report \( G(\mu_\ell) \) and \( \Lambda(\mu_\ell) \) at a stated reference \( \mu_{\ell,\rm ref} \), plus fitted β‑slopes with scheme/regulator. Do not claim asymptotic safety unless a genuine fixed point is observed in \( g_N \).

4 · Gauge sectors: emergence, running & matching

4.1 · U(1) minimal coupling and β

Lattice promotion uses \( U_{ij}=e^{\,i(\phi_i-\phi_j-B_{ij})} \). In the continuum, \( A_\mu \) with \( F_{\mu\nu} \) appears with the Maxwell term (Gravity I §4.3). The gauge‑field kinetic weight can be written as \( S_B=\tfrac{\kappa_B}{2}\sum_{\square}(\mathrm{curl}\,B)^2 \) on the lattice ( \( \kappa_B \) in MeV ).

Running. For \( N_f \) effective Dirac doublets of charge \( q_f \) and \( N_s \) complex scalars \( q_s \), a weakly‑coupled relativistic sector yields
\( \beta_{e} = \frac{e^3}{12\pi^2}\big(\sum_f q_f^2\big) + \frac{e^3}{48\pi^2}\big(\sum_s q_s^2\big) + O(e^5) \).
In RTG you must measure effective charges/multiplicities from coarse‑grained correlators; otherwise treat U(1) as global (no running) in its window.

4.2 · Soft‑spin SU(2): domain of validity

The binary spins only produce a continuous SU(2) after coarse‑graining to soft doublets \( \Psi=(\psi_\uparrow,\psi_\downarrow) \) with unit vector \( n=\Psi^\dagger \boldsymbol\sigma \Psi/|\Psi|^2 \), yielding a Heisenberg‑like exchange in open‑gate regions. The bare Ising factor \( 1+s_is_j \) is not SU(2) invariant. Treat any SU(2) gauge description as an effective one within \( 0.28 \le x \le 0.70 \); outside it, revert to global symmetry.

4.3 · U(1)\(^2\) (≈ SU(3) for small \( \epsilon \))

In the triple‑resonance band \( 1.55 \le x \le 1.70 \), three coherent shells with phases \( \theta_{1,2,3} \) can form with pairwise gaps \( \Delta\omega_{ab} \). A convenient potential is

\( V_{\rm shell} \;=\; J_{\rm ex}\sum_{a< b}\cos(\theta_a-\theta_b)\,\exp\!\Big[-\big(\Delta\omega_{ab}/\sigma_{\mathrm{exch}}\big)^2\Big], \)

with \( J_{\rm ex} \) in MeV and \( \sigma_{\mathrm{exch}}\simeq \Delta\omega^\ast \). The constraint \( \theta_1+\theta_2+\theta_3=\mathrm{const} \) (phase conservation at triple resonance) leaves two independent phases—i.e., a \( \mathrm{U}(1)^2 \) Lie algebra. Interpretation of \( \epsilon \): \( \epsilon \) encodes shell‑curvature/mixing asymmetry (Cosmology v2.5 §3); in the \( \epsilon\to 0 \) limit, mixing is symmetric and the structure is approximately SU(3).

Curvature control. Use the Helfrich penalty \( U_{\rm curv}=\kappa_c\,a_{\rm lat}^2 \sum_{\langle ijk\rangle} \big(1 – (\mathcal R_{ij}+\mathcal R_{jk}+\mathcal R_{ki})/9\big)^2 \) ( \( \kappa_c \) in MeV·fm ) to stabilize shells without suppressing corridor formation.

Practical note. When shells are non‑orthogonal in frequency space (e.g. \( x\!\sim\!1.65 \)), inter‑shell tension can shorten coherence times; monitor inter‑shell phase‑correlator decay and coherence lifetimes in simulations.

5 · Anomalies: conditions, checks & what to publish

• SU(2): No perturbative gauge anomaly (\( d^{abc}=0 \)), but a global Witten anomaly arises for an odd number of left‑handed doublets. Enforce an even number of effective doublets per coarse cell or add a spectator sector.
• SU(3): Perturbative triangle anomalies vanish for vectorlike matter (reps + conjugates) or if chiral content sums to zero anomaly; global anomalies are assumed absent in the effective window.
• U(1): For a chiral U(1), require \( \sum_i q_i=0 \) (mixed gravitational) and \( \sum_i q_i^3=0 \) (cubic) for consistency; vectorlike content is automatically safe.

5.1 · Axial Ward‑identity residual (lattice)

For any candidate gauged sector, test the lattice axial Ward identity in a probe EFT:
\( \langle \partial_\mu J_5^\mu \rangle \overset{?}{=} \frac{g_{\rm YM}^2}{16\pi^2}\,\langle F_{\mu\nu}\tilde F^{\mu\nu}\rangle + O(a^2) \),
with the measured gauge‑sector coupling \( g_{\rm YM} \). Publish the residual vs lattice spacing (error bars, ensemble averages) and the continuum extrapolation. Replace “\( < 10^{-4} \)” claims by an actual slope/limit plot.

5.2 · Representation counting (SU(2), SU(3))

From coarse‑grained fields, report the effective chiral content (doublet counts for SU(2); fundamental/antifundamental for SU(3)), the intended gauging (global vs local), and the anomaly status (safe by construction vs measured cancellation).

6 · Scheme/Regulator choices & systematics

Litim/Sharp/Dim‑reg give \( g^\ast\approx 1.14\pm 0.02 \). Quote scheme at first use and carry a systematic of \( \pm 0.02 \) into thresholds derived from \( g^\ast \) (e.g., window edges as fractions of \( \Delta\omega^\ast \)). For practical reporting, note that \( \pm 0.02 \) on edges like 0.28 corresponds to ~1.4% relative shifts. Use the same scheme across β(g), windowing, and any attempted gauge β’s to keep systematics correlated.

Units recap (for cross‑section parameters): \( J_{\rm ex} \) in MeV; \( \kappa_B \) in MeV; \( \kappa_c \) in MeV·fm.

7 · Analysis pipelines (what to run)

1. Two‑scale \( G(\mu_\ell) \), \( \Lambda(\mu_\ell) \) fit. Build \( g_{\mu\nu} \) at two coarse‑grainings (Gravity I §6). Choose \( \mu_{\ell,2}/\mu_{\ell,1}\!\approx\!2 \); average over \( \ge 100 \) ensembles to keep systematics \( \lesssim 5\% \). Fit β‑slopes with stated scheme and error bars.
2. U(1) running. Measure charged correlators (phase/gauge mixed two‑point), extract \( e(\mu_\ell) \), compare to one‑loop expectation with measured effective content.
3. SU(2)/SU(3) viability. In their windows, check soft‑spin order parameters, extract effective field content, and run the Ward‑residual test (publish slope to \( a\to 0 \ ) ).
4. Window robustness. Re‑measure window edges after retuning \( J,K’ \) within uncertainties; for proton‑radius calibration, retune \( J \) by \( \pm 8\% \) at fixed \( \Delta\omega^\ast \) and confirm edge stability within \( \pm 0.02 \).

8 · Cross‑references

• RTG Gravity I — emergent metric, EH action, U(1) minimal coupling, and lattice→continuum maps for \( \rho_s \) and \( K’_{\rm TV} \).
• Two‑Loop RG Derivation of the Critical Bandwidth — β(g), \( g^\ast \), \( \Delta\omega^\ast \) and scheme table used here.
• Gauge Symmetries in RTG — detailed U(1)/soft‑SU(2)/U(1)\(^2\) constructions, anomaly caveats, and diagnostics.
• Enriched Geometric Concepts in RTG — bandwidth thresholds and multi‑D emergence underpinning the windows.

Change log

Version	Date (UTC)	Main updates
1.2	2025‑08‑14	Added explicit units for \( J_{\rm ex} \) (MeV), \( \kappa_B \) (MeV), \( \kappa_c \) (MeV·fm); clarified dual \( \mu \) conventions (angular s\(^{-1}\) for β(g) vs length‑based fm\(^{-1}\) for \( g_N \)); inserted RG summary table; cited β(g) source at first mention; added \( \pm 0.02 \) window‑edge uncertainty; expanded U(1)\(^2\) section with constraint origin and \( \epsilon \) interpretation (Cosmology v2.5); added quantitative guidance to pipelines and renormalization conditions.
1.1	2025‑08‑12	Completed the U(1)\(^2\) shell potential with explicit pairwise mixing and the \( \theta_1{+}\theta_2{+}\theta_3=\mathrm{const} \) constraint; clarified independence of \( \sigma_{\mathrm{exch}} \), \( \sigma_{\mathrm{noise}} \), RG \( \sigma \); made window inequalities explicit; added practical notes on inter‑shell tension and coherence; reinforced Ward‑residual publication standard.
1.0 (canonical)	2025‑08‑12	Consolidated quantum material into a single page: two‑loop RTG flow & \( \Delta\omega^\ast \) recap; coarse‑grained \( G(\mu),\Lambda(\mu) \) definitions; U(1)/SU(2)/U(1)\(^2\) running domains; anomaly conditions and lattice Ward‑residual protocol; unified symbols and scheme notes; cross‑links to Gravity I and β(g) derivation.