RG Flow Beta Functions for Entropy Coefficients
Experiment V2.91: RG Flow Beta Functions for Entropy Coefficients
Status: COMPLETE
Summary
Models the continuous running of alpha(mu) and delta(mu) as functions of energy scale mu, defines beta functions beta_alpha = d(alpha)/d(ln mu) and beta_delta = d(delta)/d(ln mu), and studies the full RG flow structure of R(mu) = |delta(mu)|/(12*alpha(mu)).
Key Result: R=1 Crossing is a Transient, Not a Fixed Point
The crossing at mu* = 0.0195 eV has beta_R = -0.428 (nonzero slope), confirming it is a flow-through, not a stable fixed point. The crossing width is only 0.22 decades. No mechanism within species decoupling naturally pins R = 1.
Phase 1: Flow Trajectory (SM, Exponential Suppression)
| Fixed point | alpha | delta | R |
|---|---|---|---|
| UV (mu >> m_top) | 2.805 | -11.061 | 0.329 |
| Near-IR (photon + nu) | 0.190 | -0.872 | 0.382 |
| IR (photon only) | 0.048 | -0.689 | 1.205 |
R is monotonically decreasing from IR to UV (as mu increases, species activate and alpha grows faster than |delta|). The crossing through R = 1 is guaranteed by the intermediate value theorem.
Phase 2: Beta Functions
| Scale | mu (eV) | beta_alpha | beta_delta | beta_R |
|---|---|---|---|---|
| Neutrino | 0.05 | 0.052 | -0.067 | -0.274 |
| Electron | 5.1e5 | 0.052 | -0.067 | -0.052 |
| QCD | 2.0e8 | 0.267 | -2.191 | +0.089 |
| EW | 9.1e10 | 0.186 | -0.912 | +0.006 |
| Top | 1.7e11 | 0.191 | -0.786 | +0.001 |
Peak |beta_alpha| occurs at the QCD scale (gluon activation), peak |beta_R| occurs near the R=1 crossover at the neutrino scale.
Anomalous dimensions gamma = beta/quantity: gamma_alpha peaks at 0.52 (neutrino scale), gamma_delta peaks at 0.54 (QCD scale).
Phase 3: R=1 Crossover Analysis
| Property | Value |
|---|---|
| Crossover scale mu* | 0.0195 eV (log10 = -1.71) |
| R at crossing | 1.000 |
| alpha at crossing | 0.0586 |
| delta at crossing | -0.703 |
| beta_R at crossing | -0.428 |
| Is fixed point? | No (transient) |
| Crossing direction | R decreasing through 1 |
| Width (R within 1%) | 0.22 decades |
The crossing is driven by neutrino decoupling: at mu ~ 0.02 eV, the three neutrinos have suppression factor ~0.077 (mostly decoupled). Their delta contribution (-11/180 each) decouples faster than their alpha contribution (0.04752 each), tipping R above 1.
Phase 4: Suppression Model Comparison
| Model | log10(mu*) | beta_R | Width (decades) | log10(Lambda/Lambda_obs) |
|---|---|---|---|---|
| Exponential | -1.71 | -0.428 | 0.22 | 62.8 |
| Boltzmann | -1.51 | -0.856 | 0.11 | 63.3 |
| Smooth step | -1.84 | -0.308 | 0.30 | 62.6 |
| Step | -1.30 | 0.000 | 0.00 | 63.7 |
The crossover is robust across models (spread: 0.54 decades). The step function gives a discontinuous crossing (beta_R = 0 at the step, formally a fixed point but physically trivial).
Phase 5: Extended Models
SM + Graviton
Adding the graviton (alpha = 0.03853, delta = -212/90) dramatically changes the IR fixed point:
- R_IR jumps to 2.94 (from 1.21) because delta_graviton = -2.356 dominates at low mu where only photon + graviton survive
- Crossover shifts to mu* ~ 572 keV (near electron mass scale)
- This is radically different from the SM-only result
Dark Matter Candidates
- 1 TeV scalar dark matter: negligible effect (too heavy to affect the crossing)
- 1 keV sterile neutrino: negligible effect (decoupled above the crossing scale)
Physical Interpretation
The RG flow reveals three regimes:
- Deep UV (mu > 100 GeV): R ~ 0.33, slowly varying (all species active)
- QCD transition (100 MeV - 10 GeV): R dips to ~0.28 (gluon-dominated)
- Neutrino scale (meV - eV): R crosses 1 during neutrino decoupling
The crossing is a transient phenomenon, not a dynamical attractor. There is no mechanism within the free-field species-decoupling framework that naturally pins R = 1 at a specific scale. The self-consistency condition R = 1 selects a physical scale, but mapping this scale to the Hubble parameter requires physics beyond naive H = mu identification.
The 63-order-of-magnitude discrepancy between the crossover scale and the observed Hubble parameter is essentially the cosmological constant problem repackaged: why is the relevant scale so much smaller than particle physics thresholds?
Files
| File | Description |
|---|---|
| src/rg_flow.py | Core RG flow engine: alpha(mu), delta(mu), beta functions, crossover finder |
| src/flow_analysis.py | Extended models, stability analysis, suppression model comparison |
| tests/test_rg_flow.py | 37 tests (all pass) |
| run_experiment.py | 5-phase driver |