V2.160 - Mass Decoupling and the Running Cosmological Constant
V2.160: Mass Decoupling and the Running Cosmological Constant
Status: Complete Date: 2026-03-02 Depends on: V2.159 (field content sensitivity), V2.158 (graviton DOF), V2.156 (derivation audit)
Abstract
The prediction Ω_Λ = |δ_SM|/(6α_SM) uses the full SM field content at the Planck scale. But as the UV cutoff μ drops below a particle’s mass, that particle decouples from both δ(μ) and α(μ). This experiment maps Ω_Λ(μ) from the Planck scale to the IR, testing self-consistency and revealing how the prediction depends on the mass spectrum.
Key findings:
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Planck-scale prediction is robust: All decoupling functions (step, smooth, exponential) agree at μ = M_P where all SM fields have m << M_P. The prediction is exact at the Planck scale — no sensitivity to decoupling shape.
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The running is non-monotonic: R(μ) dips below the Planck value at the W threshold, rises above it at the top threshold, and overshoots dramatically in the IR. The SM-only prediction crosses Ω_Λ_obs three times.
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A remarkable coincidence at the b-quark plateau: Between the τ and b thresholds (1.78 – 4.18 GeV), R = 0.686 — matching observation to 0.17%. This plateau has 0 scalars + 33 Weyl + 9 vectors.
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Vector thresholds dominate: W± decoupling at 80 GeV produces a -8.4% drop in R. Vector bosons have R_single = 2.42, so their removal has outsized impact.
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BSM particles shift the Planck prediction regardless of mass: A 1 TeV and a 10¹⁵ GeV BSM particle contribute equally at the Planck scale. The prediction depends on existence, not mass.
The Running Formula
At cutoff scale μ, the effective field content includes only particles with m < μ:
δ(μ) = Σ_i δ_i × f(m_i/μ)
α(μ) = Σ_i α_i × f(m_i/μ)
Ω_Λ(μ) = |δ(μ)| / (6α(μ))where f(m/μ) is the decoupling function:
- Step: f = θ(μ − m) — sharp on/off at m = μ
- Smooth: f = (1 + (m/μ)²)^(−3/2) — power-law suppression from massive propagator on entangling surface
- Exponential: f = exp(−(m/μ)²) — Boltzmann suppression from Unruh temperature
Per-species single-field R:
| Species | R_single = |δ_i|/(6α_i) | Effect on total R | |---------|--------------------------|-------------------| | Scalar | 0.078 | Decreases R | | Weyl fermion | 0.214 | Decreases R | | Vector boson | 2.415 | Increases R |
Vectors increase R because their single-field R >> Ω_Λ_obs. Fermions and scalars decrease R.
Results
1. Planck-Scale Baseline
At μ = M_P = 1.221 × 10¹⁹ GeV, all SM fields are active:
| Scenario | δ_total | α_total | R | Deviation |
|---|---|---|---|---|
| SM only | −11.061 | 2.805 | 0.657 | −4.01% |
| SM + graviton(N=9) | −12.417 | 3.019 | 0.686 | +0.12% |
This matches V2.158-159 exactly. The Planck-scale prediction is the physically correct one.
2. Running Ω_Λ(μ) — Threshold Table
As the cutoff drops below each particle mass, that particle decouples:
| Scale | μ (GeV) | R | Deviation | n_eff active |
|---|---|---|---|---|
| Planck (all active) | 1.22×10¹⁹ | 0.657 | −4.0% | 118 |
| Below top (173 GeV) | 171 | 0.707 | +3.3% | 106 |
| Below Higgs (125 GeV) | 124 | 0.732 | +6.9% | 102 |
| Below Z (91 GeV) | 90 | 0.698 | +2.0% | 100 |
| Below W± (80 GeV) | 80 | 0.627 | −8.4% | 96 |
| Below bottom (4.2 GeV) | 4.1 | 0.686 | +0.17% | 84 |
| Below tau (1.8 GeV) | 1.8 | 0.709 | +3.6% | 80 |
| Below charm (1.3 GeV) | 1.3 | 0.797 | +16.4% | 68 |
| Below muon (0.11 GeV) | 0.10 | 0.833 | +21.7% | 64 |
| Below strange (0.09 GeV) | 0.09 | 0.976 | +42.6% | 52 |
| Below electron (0.5 MeV) | 0.0005 | 1.865 | +172% | 24 |
The running is dramatic. R varies from 0.627 (below W) to 1.865 (deep IR). The prediction is only close to observation at two locations: the Planck scale and the b-quark plateau.
3. The Three Crossings
The SM-only prediction R(μ) crosses Ω_Λ_obs = 0.685 at three scales:
| Crossing | μ (GeV) | Physical meaning |
|---|---|---|
| 1 | ~4.2 | Bottom quark threshold |
| 2 | ~81 | W boson threshold |
| 3 | ~172 | Top quark threshold |
With the graviton(N=9) included, the Planck-scale prediction already matches, so the crossings are at different (higher) R values.
4. The b-Quark Plateau Coincidence
Between the τ threshold (1.78 GeV) and the b threshold (4.18 GeV), the field content is:
- 0 real scalars (Higgs decoupled)
- 33 Weyl fermions (u, d, s, c + e, μ, 3ν)
- 9 vector bosons (8 gluons + γ)
This gives R = 0.686, matching Ω_Λ_obs to 0.17% — better than the Planck-scale SM-only prediction (−4.0%).
This is a coincidence, not a prediction: the derivation requires evaluation at the Planck scale where all fields contribute. But it is striking that the running crosses the observed value precisely at a physically meaningful threshold.
5. Decoupling Function Independence
| Function | R(Planck) | R(100 GeV) | R(1 GeV) | R(IR) |
|---|---|---|---|---|
| Step | 0.657 | 0.732 | 0.797 | 0.976 |
| Smooth | 0.657 | 0.664 | 0.772 | 1.037 |
| Exponential | 0.657 | 0.674 | 0.780 | 1.021 |
All three agree at the Planck scale to 15 significant digits. They differ only near mass thresholds, where the smooth functions partially decouple heavy fields. The Planck-scale prediction is completely independent of decoupling physics.
6. Species Decomposition
At the Planck scale (all active):
| Species | % of |δ| | % of α | Role | |---------|---------|--------|------| | Scalars (4) | 0.4% | 3.4% | Negligible | | Fermions (45 Weyl) | 24.9% | 76.3% | Dominate α | | Vectors (12) | 74.7% | 20.3% | Dominate δ |
As fields decouple in the IR, vectors (gluons + photon) increasingly dominate. Since R_vector = 2.42 >> Ω_Λ_obs, R rises dramatically.
7. BSM Mass Independence
At the Planck scale, a BSM particle contributes regardless of its mass:
| BSM extension | R(M_P) | Deviation |
|---|---|---|
| SM (baseline) | 0.657 | −4.0% |
| +1 dark photon (any mass) | 0.687 | +0.3% |
| +10 heavy scalars (any mass) | 0.612 | −10.6% |
| +1 heavy Dirac fermion | 0.643 | −6.1% |
| +4th gen quarks | 0.582 | −14.9% |
A particle at 1 TeV and one at 10¹⁵ GeV give identical Planck-scale predictions. This means the prediction counts the total number of degrees of freedom in nature, regardless of mass hierarchy.
Physical Interpretation
Why the Planck Scale Is Correct
The derivation of Ω_Λ = |δ|/(6α) uses the Jacobson thermodynamic identity applied to the cosmological horizon. The entangling surface is the horizon, with characteristic scale ~ H⁻¹ ~ M_P/√Λ. All SM particles have masses m << M_P, so they are ultrarelativistic relative to the horizon scale. The Planck-scale evaluation is the physically correct one.
The running with μ is a theoretical consistency check: it shows how the prediction would change if some fields were absent. But they are not absent — at the cosmological horizon scale, all SM fields contribute.
The W Threshold as the Critical Feature
The most dramatic feature in R(μ) is the W± threshold at 80 GeV. Removing the two W bosons drops R by 8.4% (from 0.698 to 0.627), the single largest threshold effect. This is because:
- Each W contributes |δ_W| = 62/90 per vector, adding 2 × 62/90 = 1.378 to |δ|
- But only 2 × 2 × 0.02377 = 0.095 to α
- Their |δ|/α ratio is 14.5, far above the average
The EW symmetry breaking scale controls whether the prediction works. Without the massive EW bosons, R would be too low.
Implications for the Framework
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Self-consistency: The prediction is evaluated at the scale where it is derived (Planck). No ambiguity about “at what scale to evaluate.”
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No new physics hiding at intermediate scales: Any BSM particle at any mass shifts the Planck prediction. The 0.12% match (with graviton) or 0.27% (dark photon) leaves room for at most one additional vector boson, regardless of mass.
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The running as a diagnostic: While not a prediction itself, the running reveals why the prediction works: it requires the full vector content (all 12 gauge bosons) acting together with the fermion dilution from 3 generations.
Honest Assessment
What this experiment shows: The Planck-scale prediction is exact and independent of decoupling physics. The running reveals the mechanism: vectors drive |δ| while fermions dilute α, and the specific SM field content strikes the right balance.
What a skeptic would say: “The b-quark plateau matching (R = 0.686 at μ ~ 3 GeV) is a coincidence — you don’t have a physical reason for evaluating at that scale.” This is correct. The physical evaluation point is the Planck scale, and the b-plateau match is an unexplained numerical coincidence.
What would strengthen this further:
- A first-principles derivation of why the cosmological horizon gives the correct evaluation scale
- Verification that the massive field decoupling function f(m/μ) matches lattice data
- Understanding whether the b-plateau coincidence has deeper significance (e.g., a connection to QCD confinement scale)
Files
| File | Description |
|---|---|
| src/mass_decoupling.py | Core analysis: SM mass spectrum, decoupling functions, running Ω_Λ(μ) |
| tests/test_mass_decoupling.py | 51 tests (all pass) |
| run_experiment.py | 10-phase experiment driver |
| results/results.json | Numerical results |
Tests
All 51 tests pass:
- SM content verification (6 tests)
- Particle dataclass (6 tests)
- Decoupling functions (9 tests)
- Running at Planck scale (5 tests)
- Running behavior (5 tests)
- Threshold table (3 tests)
- Self-consistency (3 tests)
- Decoupling comparison (2 tests)
- Analytic insights (4 tests)
- Plateau analysis (3 tests)
- Key scales (3 tests)
- BSM mass scan (2 tests)