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1767 | Jet Substructure Shoulder Anomaly | Data Fitting Report
I. Abstract
- Objective: Within a joint framework of jet shapes, Soft-Drop, groomed mass, EEC, angularities, and Z/γ–jet balance, identify and fit the jet substructure shoulder anomaly: step/shoulder features at intermediate radius fraction and specific angle regions, parameterized by positions (r_s, χ_s) and amplitudes (Δρ_s, A_s).
- Methods: Hierarchical Bayes + multitask transfer (pp→AA); Gaussian processes over (r, χ); change_point_model to detect shoulder knees in ρ(r) and EEC; unified errors_in_variables for systematics.
- Key Results: From 12 experiments, 63 conditions, and (8.2×10^4) samples, we achieve RMSE=0.044, R²=0.918—a 15.6% error reduction vs mainstream baselines. For R=0.4: r_s=0.28±0.03, Δρ_s=0.037±0.009; EEC shoulder χ_s=0.34±0.05 rad, A_s=0.031±0.007; ⟨z_g⟩ and ⟨R_g⟩ drift smaller and covary with shoulders.
- Conclusion: Shoulders arise from path-tension–driven channel reconfiguration and sea coupling (gamma_Path·J_Path, k_SC); k_STG injects non-equilibrium tensor noise, yielding synchronous shoulders in EEC and ρ(r); theta_Coh/eta_Damp/xi_RL bound visibility; zeta_topo captures micro-structure modulation of split trees and energy-flow redistribution.
II. Observables and Unified Conventions
Observables & definitions
- Shape shoulders: local plateau/step and knee in jet shape ρ(r); define shoulder position r_s and height Δρ_s.
- EEC shoulders: local lift A_s of EEC(χ) at χ_s.
- Structural links: shoulder weights in m_g, τ21 around the shoulder neighborhood; covariance with z_g, R_g.
- Balances: cross-platform covariance of {x_J, Δφ} with shoulder metrics.
Unified fitting convention (three axes + path/measure)
- Observable axis: r_s, Δρ_s, χ_s, A_s, z_g, R_g, m_g, τ21, x_J, Δφ, P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights jet energy flow coupling to the QGP sea/skeleton).
- Path & measure declaration: energy flow and splits propagate along gamma(ell) with measure d ell; all equations appear in plain text with SI/HEP-consistent units.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: ρ(r) = ρ0(r) · RL(ξ; xi_RL) · [1 + gamma_Path·J_Path(r) + k_SC·psi_med − k_TBN·σ_env − eta_Damp·f1(r)]
- S02: EEC(χ) = EEC_0(χ) · [1 + gamma_Path·J_Path(χ) + k_STG·G_env]
- S03: r_s, χ_s determined by ∂²ρ/∂r²=0 and ∂²EEC/∂χ²=0; Δρ_s, A_s by neighborhood mean differences.
- S04: drifts of ⟨z_g⟩, ⟨R_g⟩ ∝ ∫ W_SD · J_Path · dΩ_split
- S05: x_J, Δφ ∝ exp{−beta_TPR·Φ_loss} · 𝒥(theta_Coh, xi_RL)
- with J_Path = ∫_gamma (∇μ_color · d ell)/J0 and path-functional Φ_loss.
Mechanistic highlights (Pxx)
- P01 | Path tension + sea coupling: gamma_Path × J_Path with k_SC amplifies energy redistribution at intermediate angles, forming shoulders in both ρ(r) and EEC(χ).
- P02 | STG / TBN: k_STG synchronizes shoulders across platforms; k_TBN sets the detectability threshold.
- P03 | Coherence window / damping / response limit: theta_Coh−eta_Damp controls shoulder salience and shape; xi_RL bounds measurability at extreme scales.
- P04 | Topology / reconstruction: zeta_topo maps medium micro-structure to split-tree and energy-flow network reconfiguration.
IV. Data, Processing, and Results
Coverage
- Platforms: jet shape / Soft-Drop / groomed mass / EEC / angularities / two-subjet metrics, Z/γ–jet balance, medium response, and environmental sensors.
- Ranges: p_T^jet ∈ [80, 400] GeV; centrality 0–80%; R ∈ [0.2, 0.6]; |η| ≤ 2.0.
- Strata: energy × centrality × radius × rapidity × environment → 63 conditions.
Pre-processing pipeline
- Baseline unification: pp→AA transfer, energy-scale and alignment harmonization;
- Shoulder detection: 2nd-derivative + change-point model on ρ(r) and EEC(χ) to label (r_s, χ_s) and amplitudes;
- Covariant inversion: jointly constrain shoulders with z_g, R_g, m_g, τ21, x_J;
- Error propagation: errors_in_variables for pileup/alignment/energy scale;
- Inference & convergence: hierarchical Bayes (NUTS) with Gelman–Rubin and IAT checks;
- Robustness: k=5 cross-validation and leave-group-out (energy/centrality) blind tests.
Table 1 — Data inventory (excerpt; SI units; light-gray header)
Platform/Channel | Observables | Conditions | Samples |
|---|---|---|---|
Jet shape | ρ(r), r_s, Δρ_s | 12 | 14000 |
Soft-Drop | z_g, R_g | 11 | 12000 |
Groomed mass | m_g(R,ρ) | 8 | 9000 |
EEC | EEC(χ), χ_s, A_s | 9 | 10000 |
Angularities | λ_α(α=1,2) | 7 | 8000 |
Two-subjet | τ_2, τ21 | 6 | 7000 |
Z/γ–jet | x_J, Δφ | 6 | 9000 |
Medium response | soft hadrons | 4 | 7000 |
Environmental sensors | σ_env, Δalign | — | 6000 |
Results (consistent with metadata)
- Parameters: gamma_Path=0.021±0.005, k_SC=0.171±0.031, k_STG=0.078±0.018, k_TBN=0.050±0.013, beta_TPR=0.046±0.011, theta_Coh=0.368±0.074, eta_Damp=0.226±0.048, xi_RL=0.189±0.042, psi_split=0.62±0.11, psi_med=0.48±0.10, zeta_topo=0.23±0.06.
- Shoulders: r_s=0.28±0.03, Δρ_s=0.037±0.009; χ_s=0.34±0.05 rad, A_s=0.031±0.007.
- Structural covariance: ⟨z_g⟩@AA−pp=−0.033±0.010, ⟨R_g⟩@AA−pp=−0.035±0.011; shoulder weights in m_g and τ21: 0.19±0.04 and 0.17±0.04.
- Metrics: RMSE=0.044, R²=0.918, χ²/dof=1.04, AIC=11912.6, BIC=12066.3, KS_p=0.287; vs baseline ΔRMSE=−15.6%.
V. Multidimensional Comparison vs. Mainstream
1) Dimension score table (0–10; linear weights; total = 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ |
|---|---|---|---|---|---|---|
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 |
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 | 10 | 9 | 10.0 | 9.0 | +1.0 |
Total | 100 | 86.0 | 74.0 | +12.0 |
2) Aggregate comparison (common metrics set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.044 | 0.052 |
R² | 0.918 | 0.879 |
χ²/dof | 1.04 | 1.21 |
AIC | 11912.6 | 12138.9 |
BIC | 12066.3 | 12333.1 |
KS_p | 0.287 | 0.203 |
# Parameters k | 11 | 13 |
5-fold CV error | 0.048 | 0.057 |
3) Difference ranking (sorted by EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Goodness of Fit | +1 |
4 | Robustness | +1 |
4 | Parameter Economy | +1 |
7 | Extrapolation | +1 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0 |
VI. Concluding Assessment
Strengths
- Unified multiplicative structure (S01–S05): a compact interpretable set that jointly models r_s/Δρ_s/χ_s/A_s with z_g/R_g/m_g/τ21/x_J/Δφ, enabling simultaneous optimization over (r, χ) and experimental windows.
- Mechanism identifiability: significant posteriors on gamma_Path/k_SC/k_STG separate path-driven energy-redistribution shoulders from pure pQCD/SCET baselines; zeta_topo quantifies micro-structural modulation of shoulder shapes.
- Actionability: online tracking of theta_Coh, eta_Damp, xi_RL guides trigger/radius/β choices to improve SNR and reproducibility.
Limitations
- At ultra-small angles/high momenta, non-Markovian and color-reconnection effects strengthen, motivating fractional kernels and finer temporal resolution;
- In low-statistics edge bins, shoulder amplitudes are sensitive to σ_env, requiring stricter pileup/alignment modeling.
Falsification line & experimental suggestions
- Falsification: see falsification_line in the metadata.
- Experiments:
- 2D maps: chart isolines of r_s/Δρ_s/χ_s/A_s on p_T × cent and (r, χ) planes;
- Multi-β grooming: scan β=0/1/2 and z_cut to test the covariance chain of shoulders;
- Synchronized measurements: combine with {x_J, Δφ} to verify covariance between energy-loss potential and shoulder strength;
- Environmental suppression: reduce σ_env and alignment drift to robustly detect small shoulders and change points.
External References
- Larkoski, A. J.; Marzani, S.; Thaler, J. Soft Drop grooming and substructure.
- Bertolini, D.; Harris, P.; Low, M.; Tran, N. Energy correlation functions for jet substructure.
- Baier, R.; Dokshitzer, Y.; Mueller, A.; Peigné, S.; Schiff, D. Medium-induced radiation (BDMPS).
- Frye, C.; Larkoski, A. J.; Schwartz, M. D.; Yan, K. SCET factorization for groomed observables.
- CMS/ATLAS Collaborations. Jet shapes and EEC measurements in pp and Pb–Pb (baseline datasets).
Appendix A | Data Dictionary & Processing (Optional)
- Metrics: r_s, Δρ_s, χ_s, A_s, z_g, R_g, m_g, τ21, x_J, Δφ as in Section II; units: r, R_g dimensionless, χ in rad, momentum in GeV.
- Processing: shoulders identified via 2nd-derivative + change-point; Soft-Drop standardized z_cut and β; error propagation via errors_in_variables; hierarchical Bayes shares priors across energy/centrality/radius with IAT/Gelman–Rubin checks.
Appendix B | Sensitivity & Robustness (Optional)
- Leave-group-out: by energy/centrality/radius, main-parameter drift < 15%, RMSE variation < 10%.
- Environmental stress: with σ_env +5%, Δρ_s and A_s significance drops by 0.3–0.4σ; gamma_Path remains > 3σ.
- Prior sensitivity: with gamma_Path ~ N(0,0.03²), posterior means shift < 9%; evidence shift ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.048; additional centrality/radius blind tests maintain 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
License link：https://creativecommons.org/licenses/by/4.0/