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Electroweak Scale
written 2026-05-19 · last edited 2026-06-02
A candidate geometric hierarchy formula. What matches, what is assumed, and what still has to be derived.
JIM'S OVERSIMPLIFICATION
The weak scale is tiny compared with the Planck scale. This page proposes that the gap is not arbitrary: six dimensions of bulk geometry contribute α6, and eight exceptional topological sectors contribute a one-loop screening factor α2. Together they give α8. The number works only when MPl means the ordinary, unreduced Planck mass.
The Mass Scale and the One-Loop Topological Determinant OBSERVED
The electroweak vacuum scale is modeled as the product of two geometric contributions on the resolved C3/2O orbifold: a classical six-dimensional bulk suppression and a one-loop determinant over localized exceptional sectors.
Convention first: this formula uses the ordinary Planck mass, MPl = 1.2209 × 1019 GeV, not the reduced Planck mass. With the reduced Planck mass the formula gives about 49 GeV, not the electroweak scale. This convention is load-bearing and must be stated every time.
Classical Gaussian Integration
At tree level, the localized vacuum profile samples the six-dimensional resolved bulk. The proposed bulk factor is
vtree = MPl × α6 × √(2π)
The exponent 6 is assigned to the real dimension of the resolved C3 geometry. This supplies the dominant hierarchy suppression. The remaining mismatch is modeled as a topological one-loop boundary correction.
Honest status: the √(2π) factor must be treated as a residual normalized Gaussian measure, not as the full six-dimensional Gaussian volume. A full derivation has to show why the angular/orbifold factors reduce to this normalization rather than to another power of 2π.
Euler Characteristic and One-Loop Determinant
By the Bridgeland-King-Reid/McKay correspondence, the crepant resolution of C3/2O has topological sectors counted by the conjugacy classes of the binary octahedral group. Since 2O has eight conjugacy classes, the resolved geometry has
χ(Y) = 8
eight McKay sectors / exceptional localized sectors
In the star-tetrahedral dictionary, these eight sectors are represented by the eight vertices of the compound tetrahedron. The safer mathematical statement is not that the Euler characteristic literally proves eight independent divisors; it proves eight topological sectors in the McKay/BKR data.
The proposed exceptional kinetic operator scales as a boundary square-root of the inverse coupling:
KE ∼ α-1/2
For χ localized bosonic fluctuation sectors, the Gaussian determinant gives
Z1-loop ∝ (α-1/2)-χ/2 = αχ/4
Substituting χ = 8 gives the missing factor:
Z1-loop = α8/4 = α2
Load-bearing assumption: the whole one-loop correction depends on the exceptional kinetic scaling KE ∼ α-1/2. The page should not pretend this is already a theorem. It is the geometric hypothesis that must be derived from the resolved metric / localized-mode normalization.
The Electroweak Scale
The physical scale is the product of the tree-level bulk factor and the one-loop exceptional determinant:
v = MPl × α8 × √(2π)
| Input | Value | Result |
|---|
| MPl | 1.2209 × 1019 GeV | ordinary Planck mass |
| α-1 | 137.035999177 | fine structure input |
| Formula | MPlα8√(2π) | 246.09 GeV |
| Target | electroweak VEV | 246.22 GeV |
| Reduced MPl check | 2.435 × 1018 GeV | 49.08 GeV |
The match is close: about 0.05% low against the electroweak vacuum expectation value. The decomposition is:
α8 = α6 × α2
six-dimensional bulk suppression times eight-sector one-loop screening
Honest Status
- Numerical match: strong. The formula lands near 246 GeV using the ordinary Planck mass.
- Topological count: strong as McKay/BKR sector counting; phrase as eight sectors, not automatically eight literal divisors unless the explicit resolution proves that.
- Gaussian normalization: open. The √(2π) factor needs a clean measure derivation.
- Exceptional kinetic scaling: open and load-bearing. The α-1/2 scaling is the core hypothesis.
- Planck convention: fixed. Use ordinary Planck mass, not reduced Planck mass.
- α frozen across the whole range, 2026-07-11: real, load-bearing, and not previously stated here. The formula uses α = 1/137.036 (the zero-momentum value) unchanged across all ~17 orders of magnitude from MPl down to zero momentum. Running α properly instead, using standard one-loop SM beta functions verified against real MZ-scale data, predicts v off by 8.5× — see Why Three Generations and the α Fixed Point for the calculation. The 0.05% match depends on this assumption holding; whether it's physically justified (the IR-fixed-point reading on that second page) is open, not derived.
Status: OBSERVED / OPEN. The formula is not arbitrary numerology anymore because the exponent decomposes into specific geometric pieces: 6 from the bulk and 2 from χ/4 = 8/4. But it is not a completed derivation until the Gaussian normalization and exceptional kinetic scaling are forced by the resolved geometry.
The clean claim: not “we proved the weak scale.”
The clean claim is: “the weak scale equals the ordinary Planck scale times a geometric α8 suppression, and the exponent decomposes naturally as 6 + 8/4.”