Emergence Proposals for Gauge and Gravitational Couplings

Integrating out a tower produces calculable running and threshold corrections once the spectrum, regulator, and matching prescription are fixed. Calling the entire gauge or gravitational kinetic term “emergent” adds a stronger assumption: the ultraviolet bare term must be absent, negligible, or fixed by a UV principle. EFT alone cannot make that choice.

Required background. Species Bounds, Cutoffs, and Scale Separation supplies the tower cutoff; Running and Matching across Multiple Thresholds supplies threshold matching.

Helpful background. Beta Functions, Running Masses, and Field Anomalous Dimensions supplies running; Numerical Matrix Evidence, Classical-Spacetime Recovery, and Limits supplies a comparison with spacetime-emergence claims.

Evidence cutoff: 25 July 2026.

Threshold correction versus origin claim

For charged particles of masses $m_n$, one-loop matching in four dimensions has the form

$$\frac1{g^2(\mu)} =\frac1{g_{\rm bare}^2(\Lambda)} +\frac{b}{8\pi^2} \sum_{m_n<\Lambda}q_n^2 \log\frac{\Lambda}{\max(\mu,m_n)} +c_{\rm loc}(\Lambda).$$

The running between physical thresholds is calculable. The separation between $g_{\rm bare}$, a power-sensitive tower sum, and the local counterterm $c_{\rm loc}$ is scheme-dependent. An emergence proposal must state a boundary condition that removes this ambiguity.

First application: sum a KK tower

Take $N=\lfloor\Lambda R\rfloor$ equally charged modes with $m_n=n/R$ and $\mu<m_1$. Their logarithmic sum is

$$\sum_{n=1}^{N}\log\frac{\Lambda}{m_n} =\sum_{n=1}^{N}\log\frac{N}{n} =N\log N-\log N!.$$

Stirling’s expansion gives

$$N\log N-\log N! =N-\frac12\log(2\pi N)+O(N^{-1}).$$

The linear term is sensitive to the UV matching prescription; the subleading scale dependence is organized by ordinary EFT matching. Even if the loop term numerically dominates at large $N$, the relation

$$g^{-2}\approx\Delta g^{-2}_{\rm tower}$$

is not universal unless the bare term is constrained.

Heidenreich, Reece, and Rudelius proposed that weak coupling at infinite distance can arise from integrating out the accompanying tower Heidenreich, Reece, and Rudelius 2018. Controlled duality frames offer strong examples, but the general emergence principle remains conjectural.

Gravitational coupling

The same logic applies to $M_{\rm Pl}^{D-2}R$: matter loops renormalize Newton’s constant by $N\Lambda^{D-2}$, while a local Einstein–Hilbert counterterm is allowed. Saturation at the species scale explains a parametric relation, not the absence of microscopic gravitational dynamics.

Adversarial control: change scheme and bare term

Regulate the sum with a hard cutoff, dimensional regularization, and a smooth proper-time kernel. Power-sensitive pieces move between the loop sum and $c_{\rm loc}$, while matched low-energy amplitudes agree. Now vary $g_{\rm bare}^{-2}$; any claimed absolute prediction that changes is not an EFT invariant.

The licensed result is a threshold calculation and, in special UV constructions, a tested boundary condition. A universal origin theorem for all gauge and gravitational couplings does not follow.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Heidenreich, Ben, Matthew Reece, and Tom Rudelius. “Emergence of Weak Coupling at Large Distance in Quantum Gravity.” Physical Review Letters 121, 051601 (2018). DOI. Open PDF.