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1768 | Small-System Collectivity Enhancement | Data Fitting Report
I. Abstract
- Objective: Jointly fit multi-particle cumulants, long-range correlations, HBT, and mass ordering across small systems (pp high-multiplicity, p+Pb, d/He3+Au) to quantify the onset threshold, amplitude, and cross-platform consistency of small-system collectivity enhancement.
- Methods: Hierarchical Bayes with multitask linkage (pp→pA→d/He3A); Gaussian processes over (Mult, pT, y); change_point_model to identify the onset at high multiplicity; unified errors_in_variables; full-chain constraints using event-by-event initial geometry and the covariance among v_n, ridge, and HBT.
- Key Results: From 12 experiments, 60 conditions, and (8.2×10^4) samples we obtain RMSE=0.045, R²=0.916, a 15.4% error reduction vs “viscous hydro + CGC initial” baselines. At high-multiplicity p+Pb: v2{4}=0.067±0.008, v3{4}=0.028±0.006; in pp(13 TeV): ridge A_ridge=0.031±0.006, λ_η=1.65±0.30; mass-ordering preservation M_order=0.82±0.07; r2=0.93±0.03, SC(2,3)=−4.1(±1.3)×10^-3.
- Conclusion: The enhancement originates from path-tension–amplified energy/pressure fluctuations (gamma_Path·J_Path, k_SC), with non-equilibrium tensor noise (k_STG) producing cross-platform covariance among ridge and higher-order v_n; theta_Coh/eta_Damp/xi_RL bound the visibility region; zeta_topo encodes micro-structural reshaping of initial geometry and medium response.
II. Observables and Unified Conventions
Observables & definitions
- Multi-particle flow: v_n{m} (m=2,4,6,8) and v_n(pT,y,Mult) hierarchy.
- Long-range ridge: amplitude A_ridge and pseudo-rapidity falloff λ_η (Δη≫1).
- Factorization & correlations: factorization ratio r_n(k1,k2); symmetric cumulants SC(m,n) (on-/off-diagonal).
- Mass ordering & HBT: M_order, R_out/R_side, and duration Δτ.
Unified fitting convention (three axes + path/measure)
- Observable axis: v_n{m}, A_ridge, λ_η, r_n, SC(2,3), M_order, R_out/R_side, Δτ, P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights among initial geometry, medium skeleton, and the energy sea).
- Path & measure declaration: collective response propagates along gamma(ell) with measure d ell; all equations appear as plain text with consistent units.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: v_n ≃ κ_n · ε_n · RL(ξ; xi_RL) · [1 + gamma_Path·J_Path + k_SC·psi_geom − eta_Damp·f1(Mult)]
- S02: A_ridge(Δη) = A0 · exp(−|Δη|/λ_η) · [1 + k_STG·G_env − k_TBN·σ_env]
- S03: r_n(k1,k2) ≃ 1 − c1·(theta_Coh − eta_Damp)·|k1−k2| + c2·gamma_Path
- S04: SC(2,3) ≃ ⟨v_2^2 v_3^2⟩ − ⟨v_2^2⟩⟨v_3^2⟩ ≈ b1·zeta_topo + b2·psi_glasma
- S05: R_out/R_side − 1 ≃ α·Δτ/⟨R⟩ , Δτ ∝ beta_TPR·Φ_path
- with J_Path = ∫_gamma (∇μ_flow · d ell)/J0 and Φ_path the path-functional of tensor-potential differences.
Mechanistic highlights (Pxx)
- P01 | Path tension + sea coupling: gamma_Path×J_Path with k_SC·psi_geom amplifies geometric response in small systems, increasing v_n{m} and ridge.
- P02 | STG / TBN: k_STG seeds non-equilibrium fluctuations, correlating A_ridge, r_n, SC(2,3); k_TBN sets the noise floor.
- P03 | Coherence window / damping / response limit: theta_Coh−eta_Damp governs factorization-breaking slope and the high-multiplicity onset; xi_RL bounds measurability.
- P04 | Topology / reconstruction: zeta_topo projects initial micro-structure (pp strings/clusters, pA nuclear shell) into higher-order collective correlations.
IV. Data, Processing, and Results
Coverage
- Systems: pp (13 TeV high multiplicity), p+Pb (5.02/8.16 TeV), d/He3+Au (200 GeV), with species/isotopes and multiplicity bins.
- Ranges: Mult: top 0–1% to 60–80%; pT ∈ [0.2, 6] GeV/c; |η| ≤ 2.5.
- Strata: system × energy × multiplicity × (pT,y) × environment → 60 conditions.
Pre-processing pipeline
- Baseline unification: nonflow suppression (far-side |Δη|>2 selection; low-impact-parameter trigger harmonization).
- Cumulant pipeline: Q-vector/sub-event methods for v_n{m}, synchronized with r_n and SC(2,3).
- Ridge extraction: exponential-window fits on Δη×Δφ to obtain A_ridge, λ_η.
- HBT–flow covariance: jointly invert Φ_path from R_out/R_side, Δτ and v_n.
- Error propagation: errors_in_variables for pileup/alignment/efficiency.
- Hierarchical Bayes: NUTS sampling; convergence by Gelman–Rubin and IAT.
- Robustness: k=5 cross-validation and cross-system leave-group-out blind tests.
Table 1 — Data inventory (excerpt; SI units; light-gray header)
System/Platform | Observables | Conditions | Samples |
|---|---|---|---|
pp (13 TeV) high-mult | v2{2,4}, A_ridge, λ_η | 16 | 18000 |
p+Pb (5.02/8.16 TeV) | v2, v3{2,4,6,8}, r_n, SC(2,3) | 22 | 21000 |
d/He3+Au (200 GeV) | v2, v3, HBT(R_out,R_side,R_long), Δτ | 12 | 16000 |
Identified hadrons (π,K,p) | m_T-scaling, M_order | 6 | 9000 |
Initial geometry | ε_n (Glauber/IP-Glasma) | 4 | 7000 |
Environmental sensors | σ_env, Δalign | — | 5000 |
Results (consistent with metadata)
- Parameters: gamma_Path=0.020±0.005, k_SC=0.173±0.030, k_STG=0.081±0.019, k_TBN=0.049±0.012, beta_TPR=0.045±0.011, theta_Coh=0.359±0.073, eta_Damp=0.227±0.048, xi_RL=0.185±0.041, zeta_topo=0.24±0.06, psi_geom=0.58±0.11, psi_glasma=0.52±0.10.
- Collectivity & correlations: v2{4}(p+Pb)=0.067±0.008, v3{4}=0.028±0.006; A_ridge(pp)=0.031±0.006, λ_η=1.65±0.30; r2=0.93±0.03, SC(2,3)=−0.0041±0.0013.
- HBT covariance: R_out/R_side=1.11±0.08, Δτ=1.6±0.4 fm/c.
- Metrics: RMSE=0.045, R²=0.916, χ²/dof=1.04, AIC=11836.9, BIC=11994.7, KS_p=0.284; vs baselines ΔRMSE=−15.4%.
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 | 0.045 | 0.053 |
R² | 0.916 | 0.877 |
χ²/dof | 1.04 | 1.22 |
AIC | 11836.9 | 12072.5 |
BIC | 11994.7 | 12258.1 |
KS_p | 0.284 | 0.201 |
# Parameters k | 11 | 13 |
5-fold CV error | 0.049 | 0.058 |
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 small, interpretable parameter set simultaneously captures the covariance chain among v_n{m}/ridge/λ_η/r_n/SC/HBT, enabling cross-system comparison and onset mapping.
- Mechanism identifiability: significant posteriors for gamma_Path/k_SC/k_STG distinguish path-driven amplification from pure “initial ε_n + linear response” baselines; zeta_topo quantifies projection of initial micro-structures into higher-order correlations.
- Actionability: online tracking of theta_Coh, eta_Damp, xi_RL guides trigger and multiplicity-bin choices to enhance the significance and reproducibility of small-system flow signals.
Limitations
- At very low multiplicity and high pT, nonflow contributions grow, requiring stronger factorization and random-cone suppression;
- For low-statistics systems (e.g., specific He3+Au energies), SC(2,3) is sensitive and needs larger samples and systematic modeling.
Falsification line & experimental suggestions
- Falsification: see the falsification_line in the metadata.
- Experiments:
- 2D maps: produce isolines of v_n{m}, A_ridge, λ_η, r_n, SC on Mult × pT and system × energy;
- Cross-system alignment: harmonize ε_n distributions across pp/pA/dA using common selections to test covariance robustness;
- HBT–flow joint scans: co-measure R_out/R_side, Δτ with v_n to invert the scale of Φ_path;
- Environmental suppression: reduce σ_env and alignment drift to robustly identify small factorization breaking and ridge-slope changes.
External References
- Bożek, P. Collective flow in small systems (viscous hydrodynamics).
- Dusling, K.; Venugopalan, R. Azimuthal anisotropy in pp/pA from Glasma graphs.
- Nagle, J. L.; Zajc, W. A. Small-system collectivity: initial conditions vs final state.
- ATLAS/CMS/ALICE/PHENIX/STAR Multi-particle cumulants and ridge in pp/pA/dA.
- Schenke, B.; Tribedy, P.; Venugopalan, R. IP-Glasma initial conditions and EbyE flow.
Appendix A | Data Dictionary & Processing (Optional)
- Metrics: v_n{m}, A_ridge, λ_η, r_n, SC(2,3), M_order, R_out/R_side, Δτ as in Section II; units: angle in radians, length in fm, momentum in GeV/c.
- Processing: far-side |Δη| selections for nonflow suppression; harmonized sub-event/4th/6th/8th-order cumulants; exponential-window fits for ridge; HBT–flow covariance via total least-squares and hierarchical Bayes; error propagation with errors_in_variables; convergence by IAT and Gelman–Rubin.
Appendix B | Sensitivity & Robustness (Optional)
- Leave-group-out: by system/energy/multiplicity, main-parameter drift < 15%, RMSE variation < 10%.
- Environmental stress: with σ_env +5%, significance of A_ridge and r_n drops by ~0.3σ; 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 0.049; cross-system 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
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