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

Revision note (v13.0). The running of $\kappa$ is withdrawn. $\kappa$ is no longer fixed by $G$, the scalar mass is no longer $10^{10}$ eV, and the fixed coherence length $\lambda_c$ has been replaced by the environment-dependent range $\lambda(\rho)$. The running of the v13.0 coupling $\beta_0$ has not been computed. Corrected in this version: the sign statement (a positive one-loop beta function means the coupling grows in the UV, so "asymptotically free" was wrong); the arithmetic for $\Delta\kappa$ and $\Delta m^2$; the third term of the error budget (the stated formula gives 1.5%, not 3.6%); and the QCD running factor 0.86, which was applied to a coefficient $C$ already defined at the QCD scale (which reading is intended is unresolved). The estimate of the QCD running of $C$ is kept, with its assumptions listed. The headline benchmark $(6.0\pm0.7)\times10^{-9}$ is withdrawn as a prediction. See section 1 of the main document and the screened scalar document.

1. Purpose and Scope

An effective field theory should show that its free parameters do not flow to unphysical values under renormalisation group (RG) evolution. The v13.0 parameters of the testable sector are $(\beta_0,C,\Lambda,n)$, with a separate EFT cutoff $\Lambda_{\rm EFT}$. This appendix records what can be said about their running today:

1. The v12 one-loop beta function for $\kappa$ is withdrawn (section 3.1). The corresponding calculation for $\beta_0$ has not been done.
2. The QCD running of the isospin coefficient $C$ is kept as an estimate (section 3.2).
3. The v12 running of the coherence length is withdrawn (section 3.3).
4. The fixed-point discussion is corrected (section 4).
5. The symmetry-protection arguments are revised (section 5).
6. The lattice-QCD error budget is recomputed (section 8).

The appendix does not demonstrate radiative stability of the v13.0 parameters. That is an open item.

2. RG Framework for MCE

2.1. The action

The Level 1 action is the one in the main document,

At linear order the matter coupling is $\phi\,T$ with strength $\beta_i/M_{\rm Pl}$. The symbol $\kappa$ is, at most, notation for $\beta_0/M_{\rm Pl}$. It has mass dimension $-1$. No parameter is fixed by matching $G$, and no numerical value of $\kappa$ is an input to the v13.0 equations. The v12 action contained a separate vector field $A_\mu^{\rm EME}$ and a fixed scalar mass term. Both are withdrawn from the core (see the screened scalar document).

The RG equations follow from requiring the renormalised action to be independent of the scale $\mu$:

2.2. Exponential regulator and loop finiteness

With the non-local operator $K(\square)=e^{-\square/\Lambda^2}/(\square+m^2)$, the Euclidean propagator is

Each one-loop integral acquires a factor $e^{-np_E^2/\Lambda^2}$ and converges at large $p_E$. Schematically, with an infrared cut-off at $p_E=m$, the logarithmically divergent integral becomes

where $E_1$ is the exponential integral, and the quadratic divergence becomes the finite term $\int_0^\infty p_E\,e^{-p_E^2/\Lambda^2}dp_E=\Lambda^2/2$.

Caveats. Two statements in the v12 text are withdrawn. First, that the one-loop beta functions are "scheme-independent": the finite threshold terms depend on the chosen form factor, which is a choice and not derived (see Appendix O, item 1.1). Second, that the finiteness is established: it relies on applying the regulator in Euclidean signature, and the continuation to Lorentzian signature is conditional, because the factor $e^{p^2/\Lambda^2}$ grows at timelike momenta (see the causality note). The v13.0 Level 1 action is local, so none of its predictions depend on this section.

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3. One-Loop Beta Functions

3.1. The coupling $\kappa$ (withdrawn)

The v12 text gave $\beta_\kappa=\mu\,d\kappa/d\mu=\kappa^3/(12\pi^2)$ and concluded that $\kappa$ runs by a fractional $5\times10^{-24}$ between $10^{10}$ eV and 1 eV. This section is withdrawn for five reasons.

1. $\kappa$ is no longer fixed by matching $G$; the written formula did not give the quoted value in any case (see the Theory Hardening Analysis, Part V, item V.1).
2. $\kappa$ multiplies $\phi\,T$ and therefore has mass dimension $-1$. A beta function of the form $\kappa^3/(12\pi^2)$ applies to a dimensionless coupling.
3. The self-energy expression shown does not lead to the quoted beta function; no derivation was given.
4. The scalar mass used, $m_\phi\sim10^{10}$ eV, is withdrawn.
5. The sign statement was wrong (below).

Status. The running of the dimensionless coupling $\beta_0$ has not been computed. Open.

Corrected statements, for the record.

• Sign. If a dimensionless coupling $g$ obeys $\beta_g=+b\,g^3$ with $b=1/(12\pi^2)>0$, then

The coupling grows with energy and reaches a Landau pole at $\ln(\mu_L/\mu_0)=6\pi^2/g^2(\mu_0)$. It is not asymptotically free. The v12 statements that the theory is "asymptotically free in the gravitational sector" and has "no Landau pole" were wrong as written.

• Arithmetic. The v12 expression $(8.4\times10^{-3})\times(1.623\times10^{-10})^2\times23$ equals $5.1\times10^{-21}$, not $5\times10^{-24}$. This treats $1.623\times10^{-10}$ as a pure number, which it is not in C/kg.

3.2. The isospin coefficient $C$ (kept as an estimate)

The coefficient $C$ in $\delta(Z,A)=C\,\varepsilon\,(Z/A-\tfrac12)$ is dimensionless. Below the hadronic scale it is set by non-perturbative physics and is a fixed number at the matching scale $\mu_{\rm QCD}\sim200$ MeV. Between $\mu_{\rm QCD}$ and a higher scale, an estimate of its perturbative QCD running uses

Assumptions. The estimate rests on the following inputs, each of which is open to question.

1. A one-loop anomalous dimension with $\gamma_C=-2$, described in v12 as "typical for scalar operators in QCD". It is an assumed value. The one-loop quark-mass anomalous dimension corresponds to a magnitude of 4 in these units (to be verified), and the operator that $C$ multiplies has not been identified.
2. A fixed coupling $\alpha_s=0.118$ (the value at $m_Z$) over the whole range.
3. A matching scale $\mu_{\rm QCD}=200$ MeV, at the edge of perturbation theory, and an upper scale $\Lambda_{\rm EFT}=10$ GeV, which is a free scale.

Result with these assumptions.

or a factor $\exp(-0.147)=0.863$ if the equation is integrated (a change of $-13.7\%$). The v12 text's "$-14\%$" and the factor 0.86 correspond to the integrated form. Because $\gamma_C<0$, $C$ decreases with increasing $\mu$: $C(\Lambda_{\rm EFT})=0.863\,C(\mu_{\rm QCD})$, and equivalently $C(\mu_{\rm QCD})=1.158\,C(\Lambda_{\rm EFT})$.

Sensitivity to the fixed-coupling assumption. If $\alpha_s(\mu)$ is run at one loop from $\alpha_s(m_Z)=0.1179$ with five flavours held fixed, the integrated factor from 0.2 GeV to 10 GeV is 0.63, and from 1 GeV to 10 GeV it is 0.84. One-loop running is not reliable below about 1 GeV. The fixed-$\alpha_s$ value of $-14\%$ is therefore not a stable estimate. Estimated, with a large assumption-dependence.

Lattice constraint. The earlier text said that lattice QCD "can constrain this running and pin down $C$ to about 5%". That figure is withdrawn: the uncertainty budget in section 8 has a floor of 7.1% from $\Lambda_{\rm QCD}$ alone, and the operator has not been identified.

3.3. The coherence length $\lambda_c$ (withdrawn)

The v12 text related $\lambda_c$ to a scalar mass $m_\phi$ and computed the running of $m_\phi^2$ under the same $\kappa$. That calculation is withdrawn. The fixed $\lambda_c$ and $m_\phi\sim10^{10}$ eV are retired; the range is $\lambda(\rho)=\hbar/(m_{\rm eff}(\rho)c)$ and depends on the environment. A scalar of mass $10^{10}$ eV has a Compton length of $1.97\times10^{-17}$ m and cannot give a micrometre range.

Arithmetic, for the record. The v12 expression $\Delta m_\phi^2\approx\kappa^2\Lambda^2\ln(\Lambda/\mu_{\rm exp})/(16\pi^2)$ with $\kappa=10^{-10}$ treated as a pure number, $\Lambda=10^{10}$ eV and $\ln(10^{10})=23.0$ gives $0.146$ eV², not $1.5\times10^{-3}$ eV² (a factor of 97). The fractional shift against $m_\phi^2=10^{20}$ eV² is $1.5\times10^{-21}$, not $10^{-23}$. With $\kappa=1.623\times10^{-10}$ the same expression gives $0.38$ eV². These values are meaningless in any case: $\kappa$ in C/kg is not a pure number, so $\kappa^2\Lambda^2$ is not in eV².

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4. Fixed-Point Analysis

4.1. Gaussian fixed point

For a coupling with $\beta_g=+b\,g^3$ and $b>0$, the only perturbative fixed point is $g^*=0$. It is attractive towards the IR: $g\to0$ as $\mu\to0$. It is not a UV fixed point, because $g$ grows as $\mu$ increases. The v12 statement that the Gaussian fixed point makes the theory "asymptotically free" and shows "no Landau pole" is withdrawn. Whether $\beta_0$ has this beta function has not been computed.

4.2. Stability matrix

At $g^*=0$ the eigenvalue $\partial\beta_g/\partial g$ is zero, so the coupling is marginal at leading order. With $\beta_g=+b\,g^3$ it is marginally irrelevant: it decreases towards the IR and increases towards the UV. The v12 text defined "marginally irrelevant" as "flows to zero in the UV" and "marginally relevant" as "grows in the UV". Both definitions had the direction of flow reversed. A marginally relevant coupling (as in an asymptotically free theory) grows towards the IR.

4.3. Asymptotic safety check (withdrawn)

The v12 text argued that the scalar does not destabilise an asymptotically safe gravitational fixed point because $\kappa\approx1.623\times10^{-10}$ C/kg corresponds to a small dimensionless coupling $G_Nm_\phi^2/(c\hbar)$. That argument used the withdrawn $\kappa$ and $m_\phi$. No asymptotic-safety check is made in v13.0. Open.

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5. Symmetry Protections

5.1. Protection of the scalar potential (withdrawn claim)

The v12 text argued that diffeomorphism invariance prohibits a mass term for $\phi$ in vacuum, so that the scalar mass is protected. This is wrong: a scalar mass term $\tfrac12 m^2\phi^2$ is diffeomorphism invariant. No symmetry in the Level 1 action protects the potential $V(\phi)$ against radiative corrections. The effective mass $m_{\rm eff}(\rho)$ in the screened scalar document is set by $V$ and by the local density. Whether quantum corrections preserve the chameleon-type potential has been studied in the literature (Upadhye, Hu and Khoury, 2012; citation to be verified). Open.

5.2. Isospin and the coefficient $C$

In the limit of exact isospin symmetry ($m_n=m_p$, $m_u=m_d$) the composition-dependent factor $\delta(Z,A)$ vanishes, so a non-zero $C\varepsilon$ requires isospin breaking. That much is an argument from symmetry. Two further statements are not established.

1. The proportionality $C\propto(m_d-m_u)/\Lambda_{\rm QCD}$ was asserted, not derived. The v13.0 definition $\delta=C\,\varepsilon\,(Z/A-\tfrac12)$ already contains $\varepsilon=(m_n-m_p)/m_p$, which is itself an isospin-breaking quantity. A further factor of $(m_d-m_u)/\Lambda_{\rm QCD}$ in $C$ might count the same isospin breaking twice, unless $C$ is defined differently. Open.
2. The lattice input "$m_d-m_u\approx2.7$ MeV to within about 5%, Borsanyi et al. 2015" is marked to be verified. The attribution was not confirmed, and the uncertainty is quoted as 8.2% in section 8.1 below. The statement that this gives a "first-principles cross-check without free parameters" is withdrawn: $C$ remains an input.

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6. Summary of RG Results

The v12 conclusion that "the MCE EFT parameters are radiatively stable" is withdrawn. It is not demonstrated for the v13.0 parameters.

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7. The legacy benchmark and the QCD running factor

The legacy reference point is the unsuppressed Aluminium–Gold difference $1.9\times10^{-8}$ (equal to $2\beta_0^2\Delta\delta$ with $2\beta_0^2=5.7\times10^{-3}$ and $C=0.03$, an inference described in the screened scalar document), multiplied by a screening and range factor taken as $f=e^{-1}$:

The v12 text multiplied this by $1+\Delta C/C=0.86$ to obtain $6.0\times10^{-9}$. The factor of 0.86 is the QCD running from $\Lambda_{\rm EFT}$ to $\mu_{\rm QCD}$. In section 8.2 of the same document $C$ is defined at $\mu_{\rm QCD}$ ($C(\mu_{\rm QCD})=0.03$). The running factor was therefore applied to a quantity already defined at the scale it runs to. The readings are:

The third reading is not consistent with the stated beta function for $C$ when $0.03$ is taken at $\Lambda_{\rm EFT}$, because $C$ increases towards the IR. Which reading is intended is not stated in the source and is unresolved. The accompanying script anchors $C$ at $\mu_{\rm QCD}$ and applies no running factor, which corresponds to the first reading. The screened scalar document quotes the reference point as about 6 to $7\times10^{-9}$, without error bars. The reference point is an Estimated value of the parameter surface, not a prediction.

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8. Lattice-QCD Error Propagation

This section propagates the quoted lattice and perturbative uncertainties onto $C$ under the assumption $C\propto(m_d-m_u)/\Lambda_{\rm QCD}$. That assumption is not derived (section 5.2). The result is therefore the uncertainty in $C$ if the relation held, and it is not an uncertainty on a prediction of $\Delta a/a$.

8.1. Input uncertainties (all to be verified)

All four values are to be verified against the original sources before use. The earlier version labelled them "FLAG 2023 / PDG 2024" and attributed $m_d-m_u$ to FLAG and to Fodor et al. (2016) and Borsanyi et al. (2015) in different places. None of these labels was confirmed in this revision. The value of $\gamma_C$ and its 10% uncertainty are assumptions (section 3.2), not lattice or PDG inputs.

8.2. Propagation formula

Assume $C=\xi\,(m_d-m_u)/\Lambda_{\rm QCD}$, where $\xi$ is a dimensionless coefficient fixed by $C(\mu_{\rm QCD})=0.03$. With the numbers above, $\xi=0.03/(2.67/210)\approx2.4$. The fractional uncertainty is

The third term is the stated formula $\sigma_{\gamma_C}\ln(\Lambda_{\rm EFT}/\mu_{\rm QCD})/(2\pi/\alpha_s)$. With $\sigma_{\gamma_C}=0.2$, $\ln50=3.912$ and $2\pi/\alpha_s=2\pi/0.1179=53.3$,

The earlier document gave 3.6% for this term. The formula does not produce that value. Numerically,

compared with 11.4% in the earlier version. The contribution of the uncertainty in $\alpha_s$ itself is smaller still (about 0.1%) and was not what the formula contained.

8.3. Error budget by source

The earlier table gave shares of 52%, 39% and 9%, which follow from the 3.6% term. The "reducible by" and "timeline" columns of the earlier version (which named specific lattice collaborations and years, and an $e^+e^-$ collider) had no sources and are removed. Improved lattice determinations of $m_d-m_u$ and $\Lambda_{\rm QCD}$ would reduce the first two terms.

8.4. Legacy benchmark and the fixed-length envelope

The boxed benchmark $(6.0\pm0.7)\times10^{-9}$ is withdrawn as a prediction with error bars. The reasons are those of section 7 (the running factor), the unproven relation between $C$ and $m_d-m_u$, and the fact that $2\beta_0^2$ and the factor $f$ are not known from this calculation. The value is kept as a legacy reference point of about 6 to $7\times10^{-9}$ (section 7).

The "theory envelope" $1.63\times10^{-8}\,e^{-1\,\mu{\rm m}/\lambda_c}$ for $\lambda_c\in[1,10]\ \mu$m, giving $(6.0\text{ to }14.8)\times10^{-9}$, is retired together with the fixed coherence length. The prefactor $1.63\times10^{-8}$ contained the same factor 0.86. The screening and range factor $f(r,\rho,\text{geometry})$ replaces the exponential, and has not been computed.

If the relation of section 8.2 held, the 11.0% uncertainty in $C$ would scale a reference point of 6 to $7\times10^{-9}$ by the same fraction, that is by roughly $0.7\times10^{-9}$ to $0.8\times10^{-9}$. This is a statement about one input, not an error bar on a prediction.

8.5. Pre-registration

The earlier text proposed pre-registering the benchmark on arXiv because it was "derived entirely from" the MCE framework, lattice inputs and "calculable" QCD running. That description was wrong: the unsuppressed value $1.9\times10^{-8}$ is not derived, the factor $2\beta_0^2$ is an inference, and the running factor is possibly counted twice. A pre-registered prediction needs the field solution $f(r,\rho,\text{geometry})$ for the experimental geometry and a determination of $2\beta_0^2C$ from stated assumptions. Pre-registration before an experiment is good practice once those exist. The script scripts/rg_running.py has not been revised for v13.0. It still uses the retired fixed-length band.

8.6. Sensitivity to lattice improvements

The table shows how the propagated uncertainty in $C$ responds to smaller assumed uncertainties in $m_d-m_u$, with the other two terms unchanged. The smaller uncertainties are scenarios, not forecasts.

The uncertainty in $\Lambda_{\rm QCD}$ alone is 7.1%, so the total cannot fall below that level without a better value of $\Lambda_{\rm QCD}$. The earlier values of 7.2% and 5.5% for the two scenarios are not reproduced by the formula, and the statement that the theoretical precision would be "better than 5%" is withdrawn.