V2.75 - Mass Dependence of the Log Coefficient delta
V2.75: Mass Dependence of the Log Coefficient delta
Goal
Determine how the log coefficient delta depends on the field mass m at the entangling surface radius R. This addresses whether the full SM prediction delta_SM = -11.06 is valid at the cosmological horizon, where all SM fields except the photon have mR >> 1.
Method
Compute scalar entanglement entropy S(n) with proportional cutoff (C=10, N=1000) for masses m = [1e-6, 0.001, 0.003, 0.01, 0.03, 0.1, 0.3, 1.0, 3.0]. Extract delta via third finite differences (d3S) and 5-param fit.
Results
delta(m) suppression curve
| m | mR_mid | delta_d3S | delta/delta_0 | alpha/alpha_0 |
|---|---|---|---|---|
| 1e-6 | 0.00 | -0.01096 | 1.000 | 1.000 |
| 0.001 | 0.06 | -0.01097 | 1.001 | 1.000 |
| 0.003 | 0.18 | -0.01129 | 1.031 | 1.000 |
| 0.01 | 0.60 | -0.01386 | 1.265 | 1.000 |
| 0.03 | 1.80 | -0.00072 | 0.066 | 0.998 |
| 0.1 | 6.0 | -0.00190 | 0.173 | 0.984 |
| 0.3 | 18.0 | -2.5e-5 | 0.002 | 0.909 |
| 1.0 | 60.0 | -1.5e-5 | 0.001 | 0.596 |
| 3.0 | 180.0 | -3.6e-6 | 0.000 | 0.181 |
Key observations
-
delta is completely suppressed for mR >> 1: At mR = 18, delta is only 0.2% of its massless value. At mR = 60, it’s 0.1%.
-
alpha decreases slowly: At mR = 18, alpha is still 91% of its massless value. Only at mR > 60 does alpha drop significantly.
-
Critical mass scale: delta drops to 50% of its massless value at m_crit x R ~ 1.37 (interpolated between mR=0.6 and mR=1.8).
-
Non-monotonic behavior near mR ~ 0.6: delta_d3S shows a 26% enhancement at mR = 0.6 before dropping. This may be an artifact of the d3S method at moderate masses, or a genuine “mass enhancement” of the anomaly near the Compton scale.
-
delta is suppressed much faster than alpha: This is the key result. The log coefficient (trace anomaly) vanishes exponentially for massive fields, while the area-law coefficient (UV physics) persists much longer.
SM implications at the Hubble scale
At the cosmological horizon (R = L_H = 1/H_0):
- Photon: m = 0, mR = 0 -> massless, contributes to both alpha and delta
- Electron: m_e = 0.5 MeV, mR_H ~ 10^39 -> delta completely suppressed
- All other SM particles: even heavier, mR_H >> 10^39
Even neutrinos have mR_H ~ 10^31. Only the photon (and possibly graviton, not in SM) is truly massless at the Hubble scale.
Three scenarios
| Scenario | delta | alpha | R | Lambda/Lambda_obs |
|---|---|---|---|---|
| Full SM (all massless) | -11.06 | 2.54 | 0.36 | 1.28 |
| Photon delta, all alpha | -0.689 | 2.54 | 0.023 | 0.08 |
| Photon only | -0.689 | 0.038 | 1.51 | 5.32 |
Note: The “Full SM” row uses the corrected delta_SM = -11.06 (the code had a bug giving -9.69).
Physical Interpretation
The results reveal a deep tension in the SM prediction:
Option A: Full SM counting (R = 0.36) Treats all species as massless. This is formally valid for the abstract entanglement entropy but ignores that at the Hubble scale, most fields are frozen (m >> H).
Option B: Photon delta, all alpha (R = 0.023) Uses only photon delta (physically correct at Hubble scale) but all-species alpha (since G is measured at laboratory scales where all species are light). This gives a terrible self-consistency ratio.
Option C: Photon only (R = 1.51) Only the photon contributes to both delta and alpha. This overshoots R = 1, but is closest to self-consistency. The question is whether “Newton’s constant” at the Hubble scale should include only massless species.
The fundamental question: Is the self-consistency ratio R = |delta|/(12*alpha) a UV quantity (all species, R ~ 0.36) or an IR quantity that depends on the scale (photon-only at Hubble, R ~ 1.5)?
If R is UV (all species), then the SM prediction Lamda/Lambda_obs ~ 1.3 stands but relies on treating massive fields as massless. If R is IR, the prediction depends on which species are dynamical at the scale of interest.
Runtime
511 seconds total (9 mass values x ~57s each).