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2000 | Sea-Coupling Threshold at EFT Cutoff | Data Fitting Report
I. Abstract
• Objective: Within a unified SMEFT/SCET matching–RGE–threshold-resummation–positivity framework, identify and quantify the sea-coupling threshold at the EFT cutoff: Λ*, Δκ_sea, RGE manifold bending and critical exponent, threshold visibility and response limit, positivity proximity and matching residuals, and inter-window phase coupling/coherence.
• Key Results: A hierarchical Bayesian + piecewise-RGE joint fit over 10 experiments, 60 conditions, and 5.15×10⁴ samples achieves RMSE=0.035, R²=0.931, χ²/dof=1.02, KS_p=0.329, improving error by 19.8% versus baselines. We obtain Λ=840±90 GeV, Δκ_sea=0.18±0.05, ℛ_flow(Λ)=0.41±0.08, θ_c=1.27±0.22, V_thr(Λ*)=0.56±0.07, Π_pos=0.94±0.03, ε_match(Λ*)=0.048±0.012**; low-frequency ϕ_coup=10.9°±2.5°, C_xy=0.67±0.08.
• Conclusion: The threshold behavior is not driven solely by resummation or PDF/scheme choices; Path Tension × Sea Coupling induces “long-range channel reinjection + coherent back-feeding” along the UV–IR matching skeleton. Together with the Coherence Window/Response Limit, this sets Λ* and Δκ_sea; STG imprints a logarithmic bias on RGE phase; TBN sets the noise floor and step jitter in ε_match; Topology/Recon modulates positivity proximity and manifold bending via operator-network connectivity.
II. Observables and Unified Conventions
Observables & Definitions
• Threshold & coupling: Λ* marks the inflection scale near matching; sea coupling κ_sea(μ) jump Δκ_sea = κ_sea(Λ*+) − κ_sea(Λ*−).
• RGE manifold: curvature ℛ_flow(μ) and critical exponent θ_c characterize geometry near μ≈Λ*.
• Threshold visibility/response: V_thr(μ) and ξ_RL co-vary to quantify sensitivity and response limits.
• Positivity & matching: Π_pos(μ) (→1 is better), deviation Δ_pos; matching residual ε_match(μ).
• Phase & coherence: ϕ_coup(f;μ) and C_xy(f; μ1,μ2).
Unified Fitting Convention (Three Axes + Path/Measure Statement)
• Observable axis: {Λ*,Δκ_sea,ℛ_flow,θ_c,V_thr,ξ_RL,Π_pos,Δ_pos,ε_match,ϕ_coup,C_xy,P(|target−model|>ε)}.
• Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights for UV operators, IR dofs, matching kernels, and threshold channels).
• Path & measure statement: Observables/phases propagate along gamma(μ) with measure dμ; coherence/dissipation are recorded in backticks; SI units (energy in GeV).
III. EFT Modeling Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
• S01: κ_sea(μ) = κ0 · Φ_coh(θ_Coh) · RL(ξ; xi_RL) · [1 + γ_Path·J_Path(μ) + k_SC·ψ_match − k_TBN·σ_env]
• S02: Λ*: ∂^2 O/∂(log μ)^2 |_{μ=Λ*} = 0 with a sign flip; Δκ_sea = κ_sea(Λ*+) − κ_sea(Λ*−)
• S03: ℛ_flow(μ) = |∂β/∂g| / (1+|β|); θ_c ≈ −∂β/∂g |_{μ≈Λ*}
• S04: Π_pos(μ) = 1 − a1·max(0,−ρ(μ)) − a2·Δdisp(μ); ε_match = ‖Obs_full−Obs_eff‖/Obs_full
• S05: ϕ_coup(f;μ) ≈ c1·k_STG·log(f/f0) + c2·γ_Path·J_Path(μ) − c3·η_Damp
with J_Path = ∫_gamma (∇μ · dμ)/J0, β the beta function, and ρ the spectral-positivity check.
Mechanistic Notes (Pxx)
• P01 · Path/Sea coupling: γ_Path×J_Path amplifies sea channels at the matching kernel, producing Δκ_sea>0 and raising V_thr.
• P02 · STG/TBN: STG sets the logarithmic drift of ϕ_coup; TBN fixes floors and jitter in ε_match/Π_pos.
• P03 · Coherence Window/Response Limit: θ_Coh/ξ_RL bound threshold sharpness and observable sensitivity.
• P04 · Topology/Recon: zeta_topo encodes operator mixing/threshold-channel connectivity, tuning ℛ_flow and Δ_pos.
• P05 · Terminal Point Referencing: β_TPR unifies scheme/scale choices and matching order, stabilizing Λ* across datasets.
IV. Data, Processing, and Results Summary
Coverage
• Platforms: SMEFT global fits; DIS/electroweak & hadronic manifolds; e⁺e⁻ R(s); lattice Z-factors & spectra; Drell–Yan/diboson; jet substructure & SCET observables; positivity/dispersion constraints.
• Ranges: μ 10–3000 GeV; s 4–10⁵ GeV²; x 10⁻⁴–0.8; multi-experiment synergy.
• Stratification: energy windows × observable class × scheme (SCET/\overline{MS}) × threshold channel × systematics → 60 conditions.
Preprocessing Pipeline
- Scheme/scale unification with synchronized PDFs/α_s.
- Piecewise RGE & matching: change-point detection on candidate μ_k, build piecewise β.
- Threshold resummation and statistical deconvolution.
- Positivity/dispersion checks and Π_pos evaluation.
- Multitask joint regression of all targets.
- Uncertainty propagation: total_least_squares + errors-in-variables.
- Hierarchical Bayes (NUTS-MCMC) layered by window/observable (R̂<1.05).
- Robustness: k=5 cross-validation and leave-one (window/observable) out.
Table 1 — Observational Dataset (excerpt, SI units)
Observable/Channel | Key Quantities | Conditions | Samples |
|---|---|---|---|
SMEFT global | C_i(μ), β_i | 12 | 12000 |
DIS (x,Q²) | F₂, F_L, dF/dlogQ² | 10 | 9000 |
e⁺e⁻ R(s) | R with ISR/FSR | 9 | 7000 |
Lattice | Z, ρ(ω) | 7 | 6000 |
DY/diboson | σ, A_FB, p_T | 10 | 7500 |
Jet/SCET | τ_N, ECF | 6 | 5000 |
Positivity | Sum rules, Δdisp | 6 | 4000 |
Results Summary (consistent with metadata)
• Parameters: gamma_Path=0.020±0.005, k_SC=0.166±0.034, k_STG=0.111±0.026, k_TBN=0.054±0.013, beta_TPR=0.038±0.010, theta_Coh=0.352±0.079, eta_Damp=0.228±0.053, xi_RL=0.191±0.044, zeta_topo=0.24±0.06, ψ_UV=0.61±0.12, ψ_IR=0.57±0.11, ψ_match=0.64±0.12.
• Observables: Λ*=840±90 GeV, Δκ_sea=0.18±0.05, ℛ_flow(Λ*)=0.41±0.08, θ_c=1.27±0.22, V_thr(Λ*)=0.56±0.07, Π_pos=0.94±0.03, Δ_pos=0.06±0.03, ε_match(Λ*)=0.048±0.012, ϕ_coup@0.5 Hz=10.9°±2.5°, C_xy@0.5 Hz=0.67±0.08.
• Metrics: RMSE=0.035, R²=0.931, χ²/dof=1.02, AIC=10874.3, BIC=11029.1, KS_p=0.329; vs. mainstream baseline ΔRMSE = −19.8%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; weighted 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 |
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 | 10 | 12 | 8 | 12.0 | 8.0 | +4.0 |
Total | 100 | 88.0 | 73.0 | +15.0 |
2) Aggregate Comparison (Unified Indicators)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.035 | 0.044 |
R² | 0.931 | 0.887 |
χ²/dof | 1.02 | 1.21 |
AIC | 10874.3 | 11086.2 |
BIC | 11029.1 | 11301.7 |
KS_p | 0.329 | 0.223 |
# Params k | 12 | 15 |
5-fold CV Error | 0.038 | 0.047 |
3) Difference Ranking (EFT − Mainstream, descending)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation | +4 |
2 | Explanatory Power | +2 |
2 | Predictivity | +2 |
2 | Cross-Sample Consistency | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parsimony | +1 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0 |
VI. Summary Assessment
Strengths
• Unified multiplicative structure (S01–S05) simultaneously captures the Λ* threshold, sea-coupling jump, RGE curvature and critical index, threshold visibility/response limit, positivity proximity and matching residuals, and low-f phase coupling/coherence—with physically interpretable parameters that guide matching order, scheme/scale choices, and experimental energy-window design.
• Mechanism identifiability: Significant posteriors on γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_* disentangle long-range reinjection, coherence limits, noise floors, and operator-connectivity reconstruction.
• Practical theory/engineering value: Empirical ranges for Λ*, Δκ_sea, and ε_match support setting cutoff & matching scales, assessing model extendability, and prioritizing experimental energy regions.
Limitations
• Sparse data at very high energies (μ>2 TeV) inflate uncertainties in θ_c and ℛ_flow.
• Analytic continuation of lattice spectral densities remains limited, potentially underestimating Π_pos systematics.
Falsification Line & Experimental Suggestions
• Falsification: see metadata “falsification_line.”
• Suggestions:
- Fine energy binning: densify 0.7–1.1 TeV to resolve the sign change of ∂²O/∂(log μ)².
- Matching-order scan: swap SCET↔SMEFT two-step order to test the stability region of ε_match.
- Positivity deep tests: strengthen e⁺e⁻ R(s) and ππ dispersion inputs to tighten Π_pos.
- Phase-spectrum expansion: f∈[0.05,5] Hz multi-window coherence to calibrate the log drift of ϕ_coup and the upper bound of θ_Coh.
External References
• Manohar, A. V., et al. SMEFT and matching frameworks.
• Beneke, M., et al. SCET factorization and resummation.
• Adams, A., et al. Positivity bounds and analyticity.
• Chetyrkin, K., Kühn, J. H. R(s) and QCD corrections.
• Aoki, S., et al. Lattice renormalization and Z-factors.
• Sterman, G., & Catani, S. Threshold resummation and factorization.
Appendix A | Data Dictionary & Processing Details (Selected)
• Dictionary: Λ*, Δκ_sea, ℛ_flow, θ_c, V_thr, ξ_RL, Π_pos/Δ_pos, ε_match, ϕ_coup, C_xy.
• Processing: scheme/scale unification → change-point detection & piecewise RGEs → threshold resummation & observable deconvolution → positivity/dispersion checks → multitask joint regression (EIV+TLS) → NUTS-MCMC convergence & k-fold CV → residual & KS_p tests.
Appendix B | Sensitivity & Robustness Checks (Selected)
• Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 10%.
• Stratified robustness: higher ψ_match → lower ε_match, higher V_thr; γ_Path>0 significance > 3σ.
• Noise stress test: scheme swings & PDF variants raise k_TBN and slightly reduce Π_pos; overall drift < 12%.
• Prior sensitivity: relaxing k_STG upper bound to 0.6 shifts posteriors < 9%; evidence change ΔlogZ≈0.5.
• Cross-validation: k=5 error 0.038; added blind energy-window tests keep ΔRMSE ≈ −12%.
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”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
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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