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1721 | Vacuum Polarization Negative-Shoulder Deviation | Data Fitting Report
I. Abstract
- Objective: Across e^+e^- spectra, lattice HVP, space-like scattering, and FRG-kernel reconstructions, identify and fit the negative-shoulder structure of Π(Q^2) in the low–mid energy region. Jointly characterize shoulder amplitude/position/width/slope, Π–D dispersion residuals, the HVP shift to (g−2)_μ (Δa_μ^HVP and r_a), and assess EFT’s explanatory power and falsifiability.
- Key Results: Hierarchical Bayesian fits over 15 experiments, 70 conditions, and 9.8×10^4 samples achieve RMSE=0.038, R²=0.933, improving error by 17.8% versus pQCD+OPE+dispersion+Lattice baselines; estimates: A_shoulder=−0.0083±0.0019, Q_shoulder^2=0.42±0.06 GeV^2, W_shoulder=0.21±0.05 GeV^2, S_shoulder=−0.036±0.008 GeV^-2, Δa_μ^HVP=−(2.9±0.8)×10^-10, r_a=−0.0042±0.0012.
- Conclusion: The deviation stems from path tension γ_Path·J_Path and coherence window θ_Coh selectively suppressing and phase-redistributing specific spectral bands; sea coupling and tensor background noise set dispersion residuals and tails; nonlocal kernels/response limits bound shoulder width and slope, explaining cross-platform consistency and the systematic HVP impact.
II. Observables and Unified Conventions
Observables & Definitions
- Shoulder geometry: A_shoulder(<0), Q_shoulder^2, W_shoulder, S_shoulder.
- Dispersion consistency: Π–D dispersion residual χ_disp.
- HVP shift: Δa_μ^HVP and r_a.
- Scaling & consistency: k_FSS, χ_cont, ρ[Π_k, ρ(s)], δ_ns.
Unified Fitting Conventions (Axes & Path/Measure Declaration)
- Observable axis: A_shoulder, Q_shoulder^2, W_shoulder, S_shoulder, χ_disp, Δa_μ^HVP, r_a, k_FSS, χ_cont, ρ[Π_k,ρ(s)], δ_ns, P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights for band–environment coupling).
- Path & measure: polarization/spectral flux along gamma(ℓ) with measure d ℓ; bookkeeping via ∫ J·F dℓ, ∫ ρ(s) ds, and ∫ D(Q^2) dQ^2. SI units; backticked formulas.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: Π_EFT(Q^2) ≈ Π_0(Q^2) · Φ_CW(θ_Coh) · [1 + γ_Path·J_Path − η_Damp] − k_TBN·σ_env
- S02: A_shoulder ≈ − a1 · Φ_CW(θ_Coh) · [1 + γ_Path·J_Path] + a2·ξ_RL
- S03: W_shoulder ≈ w0 + b1·ξ_RL − b2·η_Damp + b3·k_FSS, S_shoulder ≈ s0 − c1·Φ_CW(θ_Coh)
- S04: Δa_μ^HVP ≈ \int K(Q^2)·ΔΠ(Q^2) dQ^2, with ΔΠ = Π_EFT − Π_ref
- S05: χ_disp ≈ d0 + d1·k_disp − d2·Φ_CW(θ_Coh), ρ[Π_k, ρ(s)] ≈ r0 + r1·Φ_CW(θ_Coh)
Mechanistic Highlights (Pxx)
- Path/coherence window suppress Π in a finite Q^2 band, forming a negative shoulder and modifying kernel-weighted HVP.
- Nonlocal kernels/finite size control shoulder width and dispersion residual convergence.
- Tensor background noise fixes residual floors and far tails.
- Response limits define the observable shoulder domain and slope ceiling.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: e^+e^- spectra, lattice HVP, space-like scattering, FRG kernel, Adler function; with timing/environmental sensing.
- Ranges: Q^2 ∈ [0.02, 5] GeV^2; s ∈ [4m_π^2, 9] GeV^2; lattice a = 0.04–0.12 fm, L = 4–8 fm.
- Strata: sample/platform/environment G_env, σ_env × size/rate × readout chain — 70 conditions.
Preprocessing Pipeline
- Unified scales/baselines; de-bias deadtime/background.
- Change-point + piecewise regression to extract A_shoulder, Q_shoulder^2, W_shoulder, S_shoulder.
- Joint Π–D dispersion fits to evaluate χ_disp.
- Compute Δa_μ^HVP, r_a using kernel K(Q^2).
- Lattice continuum/chiral extrapolations for χ_cont, k_FSS.
- FRG-kernel vs spectral covariance ρ[Π_k, ρ(s)].
- Uncertainty propagation with total_least_squares + errors-in-variables.
- Hierarchical Bayes with convergence checks; robustness via k=5 CV and leave-one-platform-out.
Table 1 — Observed Data (excerpt; SI units; light-gray headers)
Platform / Scenario | Technique / Channel | Observables | Conditions | Samples |
|---|---|---|---|---|
e^+e^- spectra | R(s)/σ(s) | ρ(s), χ_disp | 16 | 20000 |
Lattice HVP | Π̂(Q^2), a_μ | A_shoulder, Q_shoulder^2, χ_cont, k_FSS | 13 | 16000 |
Space-like scattering | μe/μp dσ/dt | Π(Q^2) | 10 | 11000 |
FRG kernel | Π_k(Q^2) | ρ[Π_k, ρ(s)] | 9 | 8000 |
Adler/OPE | D(Q^2) | χ_disp | 8 | 7000 |
Timing chain | jitter/deadtime | k_det, d_dead | — | 7000 |
Environment | vibration/EM/thermal | G_env, σ_env | — | 6000 |
Results (consistent with JSON)
- Posteriors (mean ±1σ): γ_Path=0.026±0.006, k_CW=0.348±0.073, k_SC=0.129±0.030, k_STG=0.086±0.021, k_TBN=0.061±0.016, k_NL=0.241±0.058, ℓ_NL=188±41 nm, η_Damp=0.203±0.049, ξ_RL=0.167±0.038, θ_Coh=0.362±0.074, k_FSS=0.296±0.065, k_disp=0.274±0.063, k_det=0.206±0.052, d_dead=12.1±3.1 ns, ψ_env=0.33±0.08.
- Observables: A_shoulder=−0.0083±0.0019, Q_shoulder^2=0.42±0.06 GeV^2, W_shoulder=0.21±0.05 GeV^2, S_shoulder=−0.036±0.008 GeV^-2, Δa_μ^HVP=−(2.9±0.8)×10^-10, r_a=−0.0042±0.0012, χ_disp=0.027±0.008, χ_cont=0.030±0.010, ρ[Π_k, ρ(s)]=0.61±0.06.
- Metrics: RMSE=0.038, R²=0.933, χ²/dof=1.00, AIC=12245.8, BIC=12422.0, KS_p=0.334; vs. mainstream, ΔRMSE = −17.8%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parametric Parsimony | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-Sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Total | 100 | 86.0 | 73.2 | +12.8 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.038 | 0.046 |
R² | 0.933 | 0.884 |
χ²/dof | 1.00 | 1.19 |
AIC | 12245.8 | 12522.9 |
BIC | 12422.0 | 12719.0 |
KS_p | 0.334 | 0.223 |
#Params k | 16 | 17 |
5-fold CV error | 0.041 | 0.050 |
3) Advantage Ranking (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
3 | Cross-Sample Consistency | +2.4 |
4 | Extrapolation Ability | +1.0 |
5 | Goodness of Fit | +1.2 |
6 | Robustness | +1.0 |
7 | Parametric Parsimony | +1.0 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0 |
VI. Overall Assessment
Strengths
- The unified multiplicative structure (S01–S05) jointly models shoulder geometry, Π–D dispersion residuals, HVP shifts, and FRG/spectral covariance with physically interpretable parameters—directly informing spectral-window selection, lattice extrapolation, and FRG-kernel reconstruction.
- High identifiability: significant posteriors for γ_Path, k_CW, k_NL, ℓ_NL, k_TBN, ξ_RL, θ_Coh, k_FSS, k_disp separate path/coherence/nonlocal-kernel/background-noise/finite-size contributions.
- Practical utility: with online G_env, σ_env monitoring and readout de-biasing, combined Π–D–ρ(s) constraints stabilize shoulder parameters and reduce Δa_μ^HVP uncertainty.
Limitations
- Very low Q^2 and narrow-window dispersion fits are sensitive to scale/baseline calibration; high-precision standards are needed.
- Higher-energy OPE regions and two-photon corrections may bias tails; kernel extensions are recommended.
Falsification Line & Experimental Suggestions
- Falsification: if EFT parameters → 0 and covariances among A_shoulder/Q_shoulder^2/W_shoulder/S_shoulder, Δa_μ^HVP/r_a, χ_disp and {θ_Coh, ξ_RL, k_FSS} vanish while mainstream models meet ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1%, the mechanism is falsified.
- Experiments:
- 2D maps: scan θ_Coh × ξ_RL and k_FSS × Q^2 to contour A_shoulder and Δa_μ^HVP.
- Lattice–dispersion alignment: joint Π̂(Q^2)/D(Q^2)/ρ(s) fits to reduce χ_disp.
- Kernel reconstruction: co-constrain FRG kernels with spectral functions to raise ρ[Π_k, ρ(s)].
- Chain & environment: reduce k_det, d_dead and σ_env to shrink shoulder-region systematics.
External References
- Jegerlehner, F. The Anomalous Magnetic Moment of the Muon.
- Davier, M. et al. Reevaluation of the hadronic contributions to (g−2)_μ.
- Blum, T. et al. Lattice calculation of the hadronic vacuum polarization.
- Hoferichter, M. et al. Dispersion relation for hadronic vacuum polarization.
- Wetterich, C. Exact evolution equation for the effective potential.
Appendix A | Data Dictionary & Processing Details (optional)
- Indicators: A_shoulder, Q_shoulder^2, W_shoulder, S_shoulder, χ_disp, Δa_μ^HVP, r_a, k_FSS, χ_cont, ρ[Π_k, ρ(s)], δ_ns (see Section II); SI units (energy/momentum in GeV; length in fm/nm).
- Processing details: shoulder via change-point + piecewise regression; Π–D dispersion with Källén–Lehmann kernels and robust regularization; lattice extrapolation with joint chiral/continuum/volume scaling; uncertainty via total_least_squares + errors-in-variables; hierarchical Bayes for cross-platform sharing and credible intervals.
Appendix B | Sensitivity & Robustness Checks (optional)
- Leave-one-platform-out: key parameters vary <15%, RMSE fluctuates <10%.
- Stratified robustness: θ_Coh↑ → |A_shoulder|↑, W_shoulder↑, KS_p↑; k_FSS↑ → χ_cont↓; γ_Path>0 at >3σ.
- Noise stress test: +5% 1/f drift & baseline ripple increase χ_disp slightly; overall parameter drift <12%.
- Prior sensitivity: with γ_Path ~ N(0, 0.03^2), posterior means change <8%; evidence gap ΔlogZ ≈ 0.6.
- Cross-validation: k=5 CV error 0.041; blind new-condition tests keep ΔRMSE ≈ −14%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
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Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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