V2.330: Massive Scalar Delta — Does Mass Modify the Trace Anomaly?

Question

The gauge-fermion (GF) core without the Higgs gives R = 0.6851 (+0.06σ from Ω_Λ), while the full SM gives R = 0.6646 (−2.8σ). Does the Higgs mass somehow modify its trace anomaly coefficient δ, or does the Higgs contribute δ = −1/90 regardless of mass?

Method

1. Lattice computation: Srednicki radial lattice with mass term m² added to the diagonal of the coupling matrix. Compute S(n, m²) for scalar fields across m² = 0 to 100.

2. Per-channel analysis: Entropy per angular momentum channel s_l(m²) to see which modes are affected by mass.

3. Theoretical analysis: Adler-Bardeen theorem, heat kernel Seeley-DeWitt expansion, and topological protection arguments.

Results

α is strongly mass-dependent (lattice)

m²	S/S(0)	Suppression
0.0	1.000	baseline
0.1	0.492	51%
1.0	0.141	86%
10.0	0.010	99%
100.0	0.0002	100%

Massive fields decouple from entanglement. The area-law coefficient α → 0 as m → ∞, reflecting shorter correlation length ξ ~ 1/m. This is the mechanism by which vacuum energy shifts G (through α), not Λ.

δ is mass-independent (theorem)

The trace anomaly δ = −1/90 for a real scalar is the a₄ Seeley-DeWitt coefficient of the heat kernel. Three independent arguments prove mass-independence:

1. Adler-Bardeen theorem: The trace anomaly receives no perturbative corrections beyond 1-loop. Mass is a perturbative parameter. δ is exact.

2. Heat kernel structure: The mass m² enters only through a₀ and a₂ coefficients. The a₄ coefficient depends only on curvature invariants (R²_μνρσ, R²_μν, R², □R), not on mass. This is a mathematical fact.

3. Topological protection: δ derives from the Euler density E₄ integrated over the entangling surface. The Euler characteristic is a topological invariant that cannot change under continuous deformations like adding mass.

Confirmed numerically: V2.248 found interaction corrections to δ = 0.00% (exact).

All SM masses are negligible at the Planck scale

Even the heaviest SM particle (top quark, 173 GeV) has m²/M²_Pl ~ 10⁻³⁴. On the entanglement lattice at Planck scale, this means m²_lattice ≈ 0. The Higgs mass changes α by less than 10⁻³⁰% — completely negligible.

The structural separation

The entanglement entropy has a structural decomposition:

S = α(m²) × A  +  δ × ln(A)  +  S₀(m²)
    ↓               ↓              ↓
    G (mass-dep)    Λ (fixed)      nothing

Mass affects the area law (→ G), not the log correction (→ Λ). This is why the cosmological constant doesn’t shift at phase transitions.

Conclusions

1. δ is mass-independent: The Higgs contributes δ_Higgs = 4 × (−1/90) = −4/90 regardless of m_H = 125 GeV. This is a theorem, not an approximation.

2. α is mass-dependent: Lattice confirms S drops 86% at m² = 1, approaching zero for large mass. Vacuum energy modifies G through α.

3. The SM prediction R = 0.6646 stands unchanged: The Higgs is there, it contributes its full δ = −1/90. The 2.8σ gap requires the graviton (n = 10 modes → R = 0.6877, +0.4σ), not a modified Higgs contribution.

4. The GF core fit (R = 0.6851) is a coincidence: Removing the Higgs happens to improve the fit, but the Higgs field exists and contributes. The proper resolution is SM + graviton.

Honest Assessment

What this experiment proves: Mass does not modify δ (by theorem). The area coefficient α is strongly mass-dependent on the lattice. The structural separation S = α·A + δ·ln(A) places vacuum energy in the area law (G), not the log correction (Λ).

What it doesn’t prove: We cannot directly extract δ for massive scalars on a finite lattice — the trace anomaly is a subleading O(1) correction to the O(C²) bulk log contribution, requiring C → ∞ Richardson extrapolation. The mass-independence of δ relies on the theoretical argument (Adler-Bardeen), not on lattice verification.

Key limitation: The lattice at finite angular cutoff C cannot separate δ from the bulk contribution. This is a known limitation (V2.246 required careful C → ∞ extrapolation even for the massless case). The theoretical proof is rigorous, but independent lattice confirmation for massive fields remains open.

Impact

This experiment closes the question of whether the Higgs mass could modify its trace anomaly contribution. It doesn’t — by theorem. The SM prediction stands at R = 0.665 (−2.8σ), and the resolution is the graviton with n = 10 modes giving R = 149√π/384 = 0.688 (+0.4σ), as established in V2.326 and V2.328.