← Theory

The α Fixed Point

The hierarchy formula uses the fine structure constant at zero momentum. What happens if you make it self-consistent — using the running coupling at the scale v the formula itself predicts?

JIM'S OVERSIMPLIFICATION

The formula v = M_Pl α⁸ √(2π) uses α = 1/137.036, measured at zero momentum. But α runs with energy. If you plug in the coupling at scale v instead of the Thomson-limit value, a self-referential equation appears. This page solves that equation and finds the fixed point at 210 GeV — not 246 GeV. The gap is a constraint on the theory: the geometry selects the infrared coupling, not the electroweak-scale coupling.

The Self-Referential Structure OBSERVED

The hierarchy formula for the electroweak vacuum scale is:

v = M_Pl × α₀⁸ × √(2π)

where α₀ = 1/137.035999177 is the fine structure constant measured at zero momentum transfer (the Thomson limit). The formula gives 246.09 GeV against the observed 246.22 GeV, a match of 0.05%.

But α runs with energy. QED predicts (at one loop):

1/α(Q) = 1/α₀ + (2/3π) ln(Q/m_e) one-loop QED running; m_e = 0.511 MeV, α₀⁻¹ = 137.036

At Q = 91.2 GeV (the Z mass), this gives 1/α(M_Z) ≈ 128. At Q = 246 GeV it gives 1/α(246) ≈ 139.8.

This raises a question: if the formula is a statement about physics at scale v, should it use α(v) rather than α₀? That makes it self-referential:

v = M_Pl × α(v)⁸ × √(2π) self-consistent equation — both sides depend on v

Solving the Fixed-Point Equation VERIFIED

The self-consistent equation is transcendental in v. We can solve it by iteration: start from a trial value v, compute α(v) via QED running, evaluate the right-hand side, then repeat until convergence.

Starting from v = 246 GeV:

Iteration	v (GeV)	1/α(v)	RHS (GeV)
Start	246.0	139.81	209.7
2	209.7	139.78	210.0
3	210.0	139.78	210.0 ✓

The iteration converges in two steps. The fixed point is:

v* ≈ 210.0 GeV , α(v*)⁻¹ ≈ 139.78

The fixed point exists and is stable. But it does not reproduce the measured electroweak scale:

Formula version	α used	Result	vs. 246.22 GeV
Standard: α₀	1/137.036	246.09 GeV	0.05% low
Self-consistent fixed point	1/139.78	210.0 GeV	14.7% low

What the Gap Means OPEN

The self-consistent fixed point falls 14.7% below the observed electroweak scale. The formula with α₀ (the zero-momentum coupling) matches to 0.05%. This is not a coincidence to be explained away: it is a constraint on the theory.

The ratio of the two predictions scales as (α₀/α(v*))⁸:

v_α0 / v* = (α₀/α(v*))⁸ = (137.036/139.78)⁸ ≈ 1.171 v_α0/v* = 246.09/210.0 = 1.172 ✓

The QED running from 0 to 210 GeV shifts 1/α by about 2.75 units. Eight powers of this shift produce the 17% discrepancy. For the formula to be a true derivation of v, not just a numerical match, the theory must explain why α₀ appears in place of α(v).

Three Possible Interpretations

1. The orbifold theory lives in the infrared

QED is infrared-free: as Q → 0, α(Q) → α₀. If the geometric construction (the resolved C³/2O orbifold, the McKay correspondence, the topological determinant) is defined at the infrared fixed point of the coupling flow, then α₀ is the natural coupling to use. The geometry encodes the IR physics, not the physics at scale v.

2. The match is numerical coincidence

The formula produces 246 GeV because α₀ happens to give that number, not because the exponent 8 is forced by a derivation that simultaneously fixes the coupling convention. Under this reading, the 0.05% match is suggestive but not yet a theorem.

3. A two-scale structure

The formula mixes two scales: the Planck scale (UV) and the electroweak scale (IR). If the correct interpretation is a ratio between UV and IR couplings, then a modified formula involving α_UV/α_IR might be the underlying structure. This is open.

Honest status: the fixed-point equation has a well-defined solution at 210 GeV. The gap from 246 GeV is real and precisely measured. Interpretation 1 (IR fixed point) is the most natural if the orbifold geometry is an infrared construction, but it has not been derived from first principles. The conventional self-consistent equation is not the right equation for this theory until we know which scale the geometry is defined at.

Analogy with the α Quadratic

The E₇ theory page contains a structurally similar self-reference: the formula for 1/α involves α itself, yielding a quadratic equation with a unique solution. In that case, the self-consistency pins 1/α to 137.035999. Here the self-consistency fails to pin v to 246.22 GeV unless α₀ is used. The two situations are complementary:

The α quadratic: self-consistency determines the value of α, using E₇ structure as input.
The v fixed point: self-consistency selects the infrared coupling as the relevant one for the hierarchy formula.

If the full derivation shows that α₀ is forced, the gap between 210 GeV and 246 GeV closes to zero by construction.

Honest Status

Fixed-point existence: confirmed numerically. v* ≈ 210 GeV, α(v*)⁻¹ ≈ 139.78.
Gap from observed: 14.7%. Precisely measured, not negligible.
Formula with α₀: matches observed to 0.05%.
Why α₀? Open. IR-fixed-point argument is natural but not derived.
QED running used here: one-loop QED only. Full SM running (hadronic contributions, W/Z loops) shifts 1/α(M_Z) to about 128, not the QED-only 139.6. The fixed-point calculation in the full SM would differ slightly but the qualitative conclusion holds.

The clean claim: the hierarchy formula works with the Thomson-limit coupling α₀.
The self-consistent fixed point exists but gives the wrong scale.
The gap is a constraint: it tells you the geometry is an infrared statement.

Theory · Framework · Research · GUMP