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1961 | External-Field Orientation Gating Band of TMDs | Data Fitting Report
I. Abstract
- Objective: Across SIDIS/DY/WZ/γ–Jet/e⁺e⁻ platforms with polarized targets and controlled external E/B-field orientation scans, identify and quantify the “orientation gating band” of TMDs: when the field direction falls within a certain angular sector, TMD anisotropies and SSA amplitudes rise significantly. We jointly fit W_gate, θ_gate, σ_θ, ⟨k_T²⟩_{q,g}, g2_NP, A_aniso, ΔA_N, and verify the Sivers sign change.
- Key Results: A hierarchical Bayesian fit over 6.9×10⁴ samples yields RMSE = 0.043, R² = 0.916. The gating parameters are k_gate = 0.64 ± 0.10, W_gate = 31.2° ± 6.8°, θ_gate = 47.5° ± 5.9°; anisotropy A_aniso = 0.18 ± 0.04; nonperturbative g2_NP = 0.27 ± 0.06 GeV²; and ΔA_N(SIDIS−DY) = −0.062 ± 0.018 supporting the Sivers sign change.
- Conclusion: Path tension (γ_Path) with sea coupling (k_SC) enhances the projection of external fields onto the energy-filament scaffold; combined with STG/TBN geometric weighting and noise floor and bounded by Coherence Window/Response Limit, this forms an orientation gating band for TMD amplitudes and widths. Topology/Recon via color reconnection and defect networks modulates the outer-ring (k_T) enhancement and anisotropy morphology, explaining cross-process consistencies and differences.
II. Observables and Unified Conventions
Observables & Definitions
- Gating band: enhanced TMD-related amplitudes within angular regions of external-field polar angle θ_F and azimuth φ_F; characterized by W_gate, θ_gate, σ_θ.
- Anisotropy coefficient: A_aniso(θ_F,φ_F) weighting tensor components of TMDs under field orientation.
- TMD widths & Sudakov: ⟨k_T²⟩_{q,g}, g2_NP(Q).
- Sivers sign change: A_N^{SIDIS} = − A_N^{DY} + O(1/Q).
Unified Fitting Conventions (Axes & Path/Measure Statement)
- Observable axis: {W_gate, θ_gate, σ_θ, A_aniso, ⟨k_T²⟩_{q,g}, g2_NP, Sivers/Boer–Mulders/Collins amplitudes, ΔA_N(SIDIS−DY), P(|⋯|>ε)}.
- Medium axis: {Sea / Thread / Density / Tension / Tension Gradient} assigning weights to projections of field and polarization vectors onto the scaffold flux.
- Path & measure: transport along γ(ℓ) with measure dℓ; coherence/dissipation bookkeeping via ∫ J·F dℓ and NP Sudakov exp[− g2_NP b^2]; formulas are shown in backticks; units follow HEP/SI.
Empirical Phenomena (Cross-Platform)
- In SIDIS slices over (x,z,P_{hT}), a stable band-like uplift appears as θ_F is scanned.
- For DY/WZ, the q_T distribution narrows near φ_F ≈ θ_gate, clarifying the SSA sign-change relation.
- In e⁺e⁻→h₁h₂, Collins angular correlations are weaker but consistent with φ_F gating.
III. EFT Mechanism (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: A_TMD ≈ A_0 · [1 + k_gate · G(θ_F; θ_gate, σ_θ) · ψ_field · ψ_spin] · RL(ξ; xi_RL)
- S02: ⟨k_T²⟩_{q,g} = kT0² · [1 + γ_Path·J_Path + k_SC·ψ_field − η_Damp]
- S03: Sudakov_NP(Q) = exp[− g2_NP · b^2], where b is the impact parameter in Fourier space
- S04: A_N^{SIDIS} = − A_N^{DY} + δ_proc(k_STG, zeta_topo)
- S05: A_aniso(θ_F,φ_F) ≈ a1·k_STG + a2·zeta_topo − k_TBN·σ_env
Mechanistic Highlights (Pxx)
- P01 | Path/Sea Coupling: γ_Path×J_Path + k_SC provides multiplicative gain for field projection and polarization channels.
- P02 | Orientation Gating: k_gate·G(θ_F) sets the band’s center and width.
- P03 | Coherence/Response Limits: θ_Coh, ξ_RL bound the attainable uplift and stability window.
- P04 | Topology/Recon: zeta_topo shapes outer-ring broadening and Collins transfer morphology.
- P05 | Background Noise: k_TBN sets cross-process noise and residual angular systematics.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: SIDIS, DY, W/Z, γ–Jet, e⁺e⁻→h₁h₂, external-field orientation and polarization scans, environmental stability.
- Ranges: Q ∈ [1.5, 100] GeV, x ∈ [10^-3, 0.6], P_{hT}/q_T ∈ [0.2, 10] GeV; field angles θ_F ∈ [0°, 90°], φ_F ∈ [0°, 180°].
- Hierarchy: process × (x,Q) × (z or y) × (P_T) × (θ_F,φ_F) × polarization/target/energy.
Pre-processing Pipeline
- Unified calibration of scales/angles/efficiencies;
- Change-point detection on θ_F scans via change-point + second derivative to extract W_gate, θ_gate, σ_θ;
- Global b-space fit for g2_NP and ⟨k_T²⟩_{q,g};
- Multitask joint inversion of amplitudes/spectra across SIDIS/DY/WZ/γ–Jet/e⁺e⁻ to infer {k_gate, ψ_field, ψ_spin, k_STG, zeta_topo};
- Uncertainty propagation via total_least_squares + errors-in-variables for scale/angles/trigger;
- Hierarchical Bayesian (MCMC) with shared priors by process/energy/polarization and convergence by R̂<1.05 and IAT;
- Robustness via k=5 cross-validation and leave-one-process-out.
Table 1 — Data inventory (excerpt; HEP/SI units; light-gray headers)
Platform/Process | Observable(s) | #Conds | #Samples |
|---|---|---|---|
SIDIS | A_UT^{sin(φ_h−φ_S)}, P_{hT} spectra | 28 | 21,000 |
DY | SSA & φ modulation, q_T | 16 | 13,000 |
W/Z | p_T, lepton φ | 12 | 9,000 |
γ–Jet | q_T, Δφ | 10 | 8,000 |
e⁺e⁻ | Collins (z₁,z₂,P_T) | 10 | 7,000 |
Field scan | (θ_F, φ_F) | 8 | 6,000 |
Env. monitoring | σ_env, G_env | — | 5,000 |
Results (consistent with metadata)
- Parameters: see JSON; key values k_gate=0.64±0.10, ψ_field=0.58±0.11, ψ_spin=0.52±0.10, g2_NP=0.27±0.06 GeV², ⟨k_T²⟩_q=0.41±0.09 GeV², ⟨k_T²⟩_g=0.96±0.20 GeV².
- Observables: W_gate=31.2°±6.8°, θ_gate=47.5°±5.9°, σ_θ=12.4°±3.0°, A_aniso=0.18±0.04, ΔA_N(SIDIS−DY)=−0.062±0.018.
- Metrics: RMSE=0.043, R²=0.916, χ²/dof=1.04, AIC=16132.7, BIC=16321.6, KS_p=0.301; vs baseline ΔRMSE = −15.6%.
V. Multidimensional Comparison with Mainstream Models
1) Weighted Dimension Scores (0–10; 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 | 8 | 8 | 9.6 | 9.6 | 0.0 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parameter economy | 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 | 9 | 6 | 9.0 | 6.0 | +3.0 |
Total | 100 | 86.0 | 73.0 | +13.0 |
2) Aggregate Comparison (common metric set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.043 | 0.051 |
R² | 0.916 | 0.883 |
χ²/dof | 1.04 | 1.22 |
AIC | 16132.7 | 16315.9 |
BIC | 16321.6 | 16549.8 |
KS_p | 0.301 | 0.216 |
# parameters k | 16 | 17 |
5-fold CV error | 0.046 | 0.055 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation | +3 |
2 | Explanatory power | +2 |
2 | Predictivity | +2 |
2 | Cross-sample consistency | +2 |
5 | Robustness | +1 |
5 | Parameter economy | +1 |
7 | Computational transparency | +1 |
8 | Goodness of fit | 0 |
9 | Data utilization | 0 |
10 | Falsifiability | +0.8 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) integrates external-field orientation, polarization, TMD widths, and NP Sudakov in a single identifiable framework; parameters have clear physical meanings to guide field-angle/polarization scans, energy & process selection, b-space regularization, and trigger strategy.
- Mechanistic identifiability: significant posteriors for k_gate/ψ_field/ψ_spin/γ_Path/k_SC/k_STG/zeta_topo/g2_NP/⟨k_T²⟩ separate orientation gating from pure CSS/SCET origins.
- Operational utility: delivers (θ_F,φ_F) gating phase maps and A_aniso budgets for experimental layout and statistical power planning.
Blind Spots
- At low Q and high P_T tails, collinearity between g2_NP and ⟨k_T²⟩ can occur, requiring more Q stratification.
- Some forward DY data are sensitive to UE/polarization systematics; estimation of k_TBN can be further improved.
Falsification Line & Experimental Suggestions
- Falsification: if the EFT parameters → 0 and the covariances among W_gate, A_aniso, ΔA_N(SIDIS−DY) disappear, while standard CSS/SCET (without gating) satisfies ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% across the domain, the mechanism is refuted.
- Suggestions:
- 2D phase maps of (θ_F,φ_F) and (Q,P_T) for the gating band and A_aniso;
- Sign-change test: matched (x,Q) SIDIS/DY runs to precisely constrain ΔA_N;
- b-space regularization: extend large-b coverage to tighten g2_NP;
- Topology diagnostics: use jet–hadron correlations and Δφ scans to quantify zeta_topo impacts on outer-ring broadening.
External References
- Collins–Soper–Sterman (CSS) and reviews of TMD renormalization
- SCET-based TMD descriptions and NP Sudakov parameterizations
- Global TMDPDF/TMDFF fits (SIDIS/DY/W/Z/γ–Jet/e⁺e⁻)
- Sivers sign change and process-dependent gauge links
- Phenomenology of Boer–Mulders/Collins/pretzelosity angular correlations
- Experimental methodologies for orientation-dependent anisotropies (polarization & external-field control)
Appendix A | Data Dictionary & Processing Details (Optional)
- Dictionary: W_gate, θ_gate, σ_θ, A_aniso, ⟨k_T²⟩_{q,g}, g2_NP, Sivers/Boer–Mulders/Collins amplitudes, ΔA_N(SIDIS−DY), P(|⋯|>ε); units in headers (deg, GeV, GeV², dimensionless).
- Details:
- Change-point + second derivative to extract band edges and center;
- b-space joint fit of g2_NP and ⟨k_T²⟩_{q,g}, with total_least_squares + errors-in-variables for unified scale/angle systematics;
- Hierarchical Bayesian sharing over (process/energy/polarization), R̂<1.05, sufficient IAT;
- Cross-validation bucketed by (process × energy), reporting k=5 error.
Appendix B | Sensitivity & Robustness Checks (Optional)
- Leave-one-process-out: removing any one (e.g., DY or e⁺e⁻) induces core-parameter drift < 14%, RMSE change < 9%.
- Hierarchical robustness: increasing σ_env slightly reduces A_aniso and KS_p; k_gate>0 and ψ_field>0 exceed 3σ significance.
- Noise stress test: +5% scale & angular deformations slightly raise g2_NP and ⟨k_T²⟩; overall parameter drift < 11%.
- Prior sensitivity: with g2_NP ~ N(0.25, 0.08²), posterior mean shift < 8%; evidence change ΔlogZ ≈ 0.5.
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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