Appendix L: Renormalisation Group Analysis, Beta Functions, and UV Stability

1. Purpose and Scope

A critical requirement for any Effective Field Theory (EFT) to be taken seriously is a demonstration that its free parameters do not flow to unphysical or divergent values under renormalisation group (RG) evolution. This appendix provides:

1. The one-loop beta functions for the three principal MCE parameters: $\kappa$, $C_{\text{QFT}}$, and $\lambda_c$.
2. A fixed-point analysis demonstrating the existence of a UV Gaussian fixed point.
3. A demonstration that the exponential non-local regulator (Appendix D) makes all loop integrals UV-finite, rendering the beta functions well-defined.
4. Bounds on how far parameters run between the EFT cutoff $\Lambda_{\text{EFT}} \sim 10^{10}$ eV and the experimental scale $\mu_{\text{exp}} \sim 1$ eV, confirming that parameter values used in predictions are stable.

---

2. RG Framework for MCE

2.1. The MCE Action in Dimensional Regularisation Form

The MCE action, expressed in $d = 4 - 2\varepsilon$ dimensions (for dimensional regularisation bookkeeping, even though the exponential regulator renders all integrals finite in $d=4$), is:

The wave-function renormalisation factors $Z_\phi$, $Z_m$, $Z_\kappa$, $Z_A$ capture the quantum corrections. The RG equations for the physical parameters are derived from the requirement that the renormalised action is $\mu$-independent:

2.2. Exponential Regulator and Loop Finiteness

With the exponential non-local regulator $K(\square) = e^{-\square/\Lambda^2} / (\square + m_\phi^2)$, the propagator in Euclidean space is:

Every one-loop integral has the generic form:

The Gaussian factor $e^{-np_E^2/\Lambda^2}$ causes every such integral to converge at the upper limit for any $n \ge 1$. Specifically, the logarithmic divergence characteristic of four-dimensional scalar QFT — $\int d^4p / p^4 \sim \log\Lambda_{\text{UV}}$ — is replaced by:

where $E_1$ is the exponential integral. This is finite for all $m_\phi > 0$ and $\Lambda < \infty$. The quadratic divergence $\int d^4p / p^2 \sim \Lambda_{\text{UV}}^2$ becomes $\int_0^\infty p_E \, e^{-p_E^2/\Lambda^2} \, dp_E = \Lambda^2/2$, which is a finite threshold correction rather than a true divergence. There is no renormalon ambiguity.

Conclusion: In the MCE EFT with exponential regulator, all renormalisation group equations are defined without the need for dimensional regularisation subtraction beyond the finite threshold corrections at the scale $\Lambda$. The beta functions computed below are therefore scheme-independent at one loop.

---

3. One-Loop Beta Functions

3.1. Beta Function for $\kappa$ (Gravitational Coupling)

The running of $\kappa$ is governed by the anomalous dimension of the $\phi T$ vertex. At one loop, the dominant contribution comes from the scalar self-energy diagram (single loop of virtual scalar quanta):

After performing the regulated integral, the beta function is:

For $m_\phi \ll \Lambda$ (which holds throughout the EFT validity range since $m_\phi \sim 10^{10}$ eV $\lesssim \Lambda \sim 10^{10}$ eV at the boundary, and $m_\phi \ll \Lambda$ well below the cutoff):

Running of $\kappa$: Solving $\mu d\kappa/d\mu = \kappa^3 / (12\pi^2)$:

Between $\mu = \Lambda_{\text{EFT}} \sim 10^{10}$ eV and $\mu_{\text{exp}} \sim 1$ eV (a factor of $10^{10}$ in energy), the logarithm is $\ln(10^{10}) \approx 23$. The fractional change in $\kappa^2$ is:

Since $\kappa \sim 10^{-10}$ C/kg and the beta function coefficient is $\sim 1/(12\pi^2) \approx 8.4 \times 10^{-3}$, the fractional running is:

The running of $\kappa$ over the entire EFT validity range is negligible at any foreseeable experimental precision (fractional change $\sim 10^{-24}$). This is a direct consequence of the smallness of $\kappa$ itself — the gravitational coupling is intrinsically weak, and weak couplings run slowly.

Physical interpretation: The gravitational coupling $\kappa$ is stable across all experimentally accessible energy scales. There is no Landau pole below the Planck scale.

3.2. Beta Function for $C_{\text{QFT}}$ (Material Dependence Coefficient)

$C_{\text{QFT}}$ is the dimensionless constant in the material-dependent factor $\delta(Z,A)$. It arises from a second-order QFT loop correction involving the neutron-proton mass difference. The relevant diagram involves a loop of virtual pions (the carriers of the residual nuclear force, since the neutron-proton mass difference is driven by QCD + QED effects at the nuclear scale $\mu_{\text{QCD}} \sim 200$ MeV).

At scales $\mu < \mu_{\text{QCD}}$, $C_{\text{QFT}}$ is determined by non-perturbative QCD and is therefore a fixed number at the hadronic matching scale. Above $\mu_{\text{QCD}}$, the relevant degrees of freedom are quarks and gluons, and $C_{\text{QFT}}$ receives small perturbative corrections from QCD running:

where $\gamma_C$ is the anomalous dimension of the relevant quark bilinear operator. Using $\alpha_s(m_Z) \approx 0.118$ and $\gamma_C \approx -2$ (typical for scalar operators in QCD):

This is a measurable running: $C_{\text{QFT}}$ at the experimental scale ($\mu \sim \mu_{\text{QCD}}$) may differ from its value at the UV cutoff by approximately 14%. This is an important correction that must be included when comparing the theoretically predicted $C_{\text{QFT}} \approx 0.03$ (estimated from the UV completion) with the experimentally measured value (from micro-WEP tests).

Prediction: Lattice QCD calculations of the relevant scalar quark bilinear operator at several values of $\mu$ can constrain this running and pin down $C_{\text{QFT}}(\mu_{\text{QCD}})$ to $\sim 5\%$ precision, providing an independent consistency check on the EME material dependence prediction.

3.3. Beta Function for $\lambda_c$ (Coherence Length)

The coherence length $\lambda_c$ is related to the effective mass $m_\phi$ of the scalar field by $\lambda_c = \hbar / (m_\phi c)$. Its running is determined by the running of $m_\phi^2$:

The first term is the standard scalar mass running; the second term is the finite threshold correction from the exponential regulator. For $m_\phi \ll \Lambda$:

This is a finite (not logarithmically divergent) threshold contribution. The total shift in $m_\phi^2$ over the RG running from $\Lambda$ down to $\mu_{\text{exp}}$ is:

Since $m_\phi^2 \sim (m_\phi c^2 / \hbar c)^2 \sim (10^{10} \text{ eV})^2$, the fractional shift is:

Again negligible. The coherence length $\lambda_c$ is radiatively stable across the entire EFT energy range.

---

4. Fixed-Point Analysis and UV Safety

4.1. Gaussian Fixed Point

Setting $\beta_\kappa = 0$ in the one-loop result:

The Gaussian (non-interacting) fixed point at $\kappa^* = 0$ is the unique perturbative fixed point. The theory is asymptotically free in the gravitational sector in the sense that $\kappa \to 0$ as $\mu \to \infty$, meaning the theory becomes weakly coupled at high energies. This is a desirable property: it confirms that there is no Landau pole and no strong-coupling problem within the EFT validity range.

4.2. Stability Matrix

The stability matrix $M_{ij} = \partial \beta_i / \partial g_j$ at the Gaussian fixed point has eigenvalues:

The zero eigenvalue indicates that $\kappa$ is a marginal operator at the Gaussian fixed point — it neither grows nor shrinks under RG flow at leading order. This is consistent with the near-constancy of $\kappa$ found in Section 3.1. The next-to-leading-order (two-loop) calculation would determine whether $\kappa$ is marginally irrelevant (flows to zero in UV) or marginally relevant (grows in UV). Given the observed smallness of $\kappa$, the marginal-to-irrelevant scenario is favoured.

4.3. Asymptotic Safety Check

In the asymptotic safety (AS) scenario for quantum gravity (Weinberg 1979, Reuter 1998), gravity is non-perturbatively UV-complete at a non-Gaussian fixed point $G^* \ne 0$. For the MCE theory to be consistent with the AS programme, the MCE scalar field $\phi$ must not destabilise the gravitational fixed point. The condition for this is that the scalar field anomalous dimension $\eta_\phi$ satisfies:

at the AS fixed point. Given that $\kappa \approx 1.623 \times 10^{-10}$ C/kg corresponds to a dimensionless gravitational coupling $G_N m_\phi^2 / (c \hbar) \ll 1$, the contribution of $\phi$ to the gravitational beta function is negligible, and the AS fixed point is not destabilised.

---

5. Symmetry Protections

5.1. Protection of $\kappa$ by Diffeomorphism Invariance

The coupling $\kappa \phi T$ is the unique scalar, diffeomorphism-invariant interaction between a scalar field and the matter sector (at lowest mass dimension). Diffeomorphism invariance prohibits a mass term for $\phi$ in vacuum (where $T=0$) — any such term would break the symmetry. This means the scalar field mass $m_\phi^2$ is protected against additive renormalisation in the massless vacuum:

Mass is generated only in the presence of matter (when $T \ne 0$), making $m_\phi$ a dynamically generated mass in the sense of chiral symmetry breaking. This protects $\lambda_c = \hbar/(m_\phi c)$ against fine-tuning: the coherence length is not a small number that must be artificially maintained against large quantum corrections; it is determined by the local matter environment.

5.2. Protection of $C_{\text{QFT}}$ by Isospin Symmetry

In the limit of exact isospin symmetry ($m_n = m_p$, $m_u = m_d$), the material-dependent factor $\delta(Z,A)$ vanishes identically: the neutron and proton contribute equally to the QVP, and there is no Z/A-dependent asymmetry. The non-zero value of $C_{\text{QFT}}$ is therefore protected to be proportional to the isospin-breaking parameters:

This connects $C_{\text{QFT}}$ directly to well-measured isospin-breaking quantities. Lattice QCD calculations (e.g., Borsanyi et al. 2015) have determined $m_d - m_u \approx 2.7$ MeV to within $\sim 5\%$. The MCE prediction for $C_{\text{QFT}}$ can therefore be independently computed from lattice data, providing a first-principles cross-check without free parameters.

---

6. Summary of RG Results

Key conclusion: The MCE EFT parameters are radiatively stable. The gravitational coupling $\kappa$ and coherence length $\lambda_c$ do not run meaningfully over any experimentally accessible energy range. The material dependence coefficient $C_{\text{QFT}}$ runs under QCD (as expected for a nuclear-physics parameter) and this running is predictable and constrain­able. The theory is internally consistent under quantum corrections and does not require fine-tuning to maintain its predictions.

---

7. Renormalisation Group Improved Predictions

Using the RG results, the MCE predictions for microscale WEP violation can be stated in an RG-improved form:

The RG-corrected prediction for the microscale Aluminium–Gold acceleration difference is:

This 14% correction from QCD running of $C_{\text{QFT}}$ is modest but non-trivial and should be included in any precision experimental comparison.

---

8. Lattice QCD Error Propagation: Full Uncertainty Budget

The MCE prediction for $\Delta a/a$ is not a point estimate — it carries a well-defined theoretical uncertainty budget that can be reduced with improved lattice QCD measurements. This section provides the full propagation.

8.1. Input Uncertainties (FLAG 2023 / PDG 2024)

8.2. Propagation Formula

$C_{\text{QFT}}$ depends on the isospin-breaking parameters via:

where $\xi$ is a dimensionless loop integral fixed by the requirement $C_{\text{QFT}}(\mu_{\text{QCD}}) = 0.03$. The fractional uncertainty is:

Numerically:

8.3. Error Budget by Source

8.4. Conservative Benchmark and $\lambda_c$ Envelope

Two prediction layers should be distinguished clearly:

1. The conservative benchmark used for pre-registration, defined by the lower-edge working choice $\lambda_c = 1$ μm
2. The broader current theory envelope obtained by scanning the decoherence band $\lambda_c \in [1,10]$ μm at fixed experimental separation $r = 1$ μm

The conservative benchmark is:

The broader current theory envelope at the same separation is:

which yields

before the same 11.4% lattice-QCD uncertainty is applied multiplicatively.

The ±0.7 × 10⁻⁹ uncertainty in the benchmark point (1σ, theoretical) is dominated by lattice QCD uncertainty on the light quark mass difference $m_d - m_u$. This uncertainty is expected to shrink to ±0.3 × 10⁻⁹ by 2028 with next-generation lattice calculations at the physical pion mass (BMW Collaboration roadmap), at which point the theoretical precision will be comparable to the planned experimental sensitivity.

8.5. Independence from Experiment: Pre-Registration Strategy

Because the conservative benchmark is derived entirely from:

1. The MCE theoretical framework (published)
2. Lattice QCD inputs (independently published, FLAG 2023)
3. The QCD running calculation (calculable from first principles)

...the prediction $(6.0 \pm 0.7) \times 10^{-9}$ can be pre-registered on arXiv before any microscale WEP experiment is performed. Pre-registration eliminates any possibility of post-hoc adjustment and demonstrates that the benchmark is genuine, not retrospective. To handle the known decoherence ambiguity cleanly, the pre-registration should also state the explicit companion scan

before lattice uncertainty is applied. The code implementing the benchmark and band calculation is provided in scripts/rg_running.py.

8.6. Sensitivity to Future Lattice Improvements

The following table shows how the MCE prediction tightens as lattice QCD improves:

By the time an atom interferometry experiment achieves the required sensitivity, the theoretical prediction will be known to better than 5%, making it a meaningful comparison rather than a floating target.