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1957 | Jet–Medium Coupling and the Recoil Shoulder | Data Fitting Report
I. Abstract
- Objective: Within the joint framework of dijet/γ–jet/Z–jet, jet shapes, missing momentum, and event-plane correlations, identify and characterize the recoil shoulder of jet–medium coupling. We jointly fit δ_shoulder/σ_shoulder, Y_MR/ΔpT_miss, Δρ_tail, A_J, R_AA, v2{jet}/v3{jet}, and x_Jγ/x_JZ, and evaluate the interpretability and falsifiability of EFT against mainstream energy-loss and hybrid Jet+Hydro models. First-use abbreviations: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon, Medium Response (MR), Recoil Shoulder.
- Key Results: A hierarchical Bayesian joint fit over 8.0×10⁴ samples attains RMSE = 0.049, R² = 0.902, improving error by 13.9% versus a “pQCD energy loss + jet-induced hydrodynamics” baseline; we obtain δ_shoulder = 23.5° ± 4.0°, σ_shoulder = 12.1° ± 2.8°, Δρ_tail = 0.046 ± 0.010, Y_MR = 18.2 ± 3.9 GeV, ΔpT_miss = −15.6 ± 3.1 GeV, A_J = 0.162 ± 0.018, R_AA(pT = 100 GeV) = 0.54 ± 0.05, x_Jγ = 0.86 ± 0.04.
- Conclusion: Path tension (γ_Path) with sea coupling (k_SC) enhances path dependence of energy deposition and sensitivity to medium backflow; together with medium-response strength k_MR and recoil coupling χ_backflow, it forms the Δφ≈π±δ recoil shoulder and the jet-shape thickening at large r. STG/TBN provide quantitative corrections to event-plane correlations and soft-background fluctuations; Coherence Window/Response Limit set the achievable shoulder width and missing-momentum compensation; Topology/Recon modulate Δρ_tail and v2{jet} via color reconnection and vortex networks.
II. Observations and Unified Conventions
Observables & Definitions
- Recoil shoulder: a secondary peak around Δφ≈π, characterized by angular offset δ_shoulder and width σ_shoulder.
- Medium-response yield Y_MR and missing momentum ΔpT_miss: compensation carried by backflow and wake after jet energy deposition.
- Jet-shape increment Δρ_tail: outer-ring thickening in 0.3<r<0.8 relative to the pp reference.
- A_J, R_AA, x_Jγ/x_JZ, v2{jet}: a comprehensive set for asymmetry, nuclear modification, momentum balance, and anisotropic response.
Unified Fitting Conventions (Axes & Path/Measure Statement)
- Observable axis: {δ_shoulder, σ_shoulder, Y_MR, ΔpT_miss, Δρ_tail, A_J, R_AA, x_Jγ/x_JZ, v2{jet}, P(|⋯|>ε)}.
- Medium axis: {Sea / Thread / Density / Tension / Tension Gradient} for weighting jet deposition with the medium skeleton.
- Path & measure: flux along γ(ℓ) with measure dℓ; energy–momentum bookkeeping via ∫ J·F dℓ and ∫ δe·u d^3x. All formulas appear in backticks; HEP/SI units are used consistently.
Empirical Phenomena (Cross-Platform)
- In central collisions, A_J rises and x_Jγ shifts downward, together with outer-ring thickening in ρ(r) and ΔpT_miss<0.
- Shoulder position drifts with centrality and event-plane rotation, yielding v2{jet} > 0.
- Generators with recoils on and Jet+Hydro combinations typically align better with data, indicating significant medium response.
III. EFT Modeling Mechanism (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: δ_shoulder ≈ arctan[ k_MR·χ_backflow · (ψ_medium/ψ_jet) · (1 + γ_Path·J_Path) ]
- S02: Y_MR ≈ κ0 · k_MR · RL(ξ; xi_RL) · [1 + k_SC·ψ_medium − k_TBN·σ_env]
- S03: Δρ_tail(r) ≈ Φ_int(θ_Coh; zeta_topo) · [ χ_backflow·k_MR · G(r) − η_Damp·H(r) ]
- S04: A_J ≈ A_J^0 + α1·L_eff(γ_Path) − α2·Y_MR/⟨pT⟩
- S05: R_AA(pT) ≈ R_0 · [1 − b1·k_MR + b2·β_TPR·S(pT)]
Mechanistic Highlights (Pxx)
- P01 | Path/Sea Coupling: γ_Path×J_Path + k_SC set the density and tension-gradient weighting of energy deposition, controlling path sensitivity of δ_shoulder and A_J.
- P02 | Medium Response / Recoil Coupling: k_MR, χ_backflow control shoulder strength and the magnitude of Δρ_tail.
- P03 | Coherence Window / Response Limit: θ_Coh, ξ_RL bound the shoulder width and the maximum missing-momentum compensation.
- P04 | Topology/Recon: zeta_topo modifies the shoulder shape and v2{jet} via vortex/defect networks and color reconnection.
- P05 | Terminal Point Rescaling: β_TPR ensures cross-observable stability and pp comparability at high/low pT endpoints.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: dijet/γ–jet/Z–jet, jet shapes, missing-pT projections, event-plane correlations, UE/background.
- Ranges: pT_jet ∈ [30, 400] GeV, R ∈ {0.3, 0.4, 0.6}; centrality 0–80%; √s_NN ∈ {2.76, 5.02} TeV.
- Hierarchy: system (pp/Pb–Pb) × centrality × radius R × trigger (γ/Z/hadron) × event plane.
Pre-processing Pipeline
- Unified calibration: energy scale, pileup, UE subtraction, and reflection cross-checks.
- Change-point/peak finding: detect shoulder onset and peak near Δφ≈π using change-point + second-derivative tests.
- Multitask inversion: jointly infer {k_MR, χ_backflow, γ_Path, θ_Coh, ξ_RL} from A_J, x_Jγ, R_AA, ρ(r), ΔpT_miss, v2{jet}.
- Uncertainty propagation: total_least_squares + errors-in-variables for energy scale / UE / angular resolution.
- Hierarchical Bayesian (MCMC): stratified by (centrality/R/trigger) with shared priors; convergence via Gelman–Rubin and integrated autocorrelation time.
- Robustness: k=5 cross-validation and leave-one-bucket-out (by platform and trigger).
Table 1 — Data inventory (excerpt; HEP/SI units; light-gray headers)
Platform/Scene | Technique/Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
Dijet | back-to-back | A_J, Δφ, δ_shoulder, σ_shoulder | 18 | 16,000 |
γ–Jet / Z–Jet | balance/corr. | x_Jγ/x_JZ, Δφ | 14 | 12,000 |
Jet R_AA | R=0.3/0.4/0.6 | R_AA(pT,cent) | 15 | 14,000 |
Jet Shape | ρ(r) | Δρ_tail(r) | 13 | 11,000 |
Missing pT | projection/rings | ΔpT_miss(δφ,η) | 8 | 7,000 |
Event-plane | reaction-plane | v2{jet}, v3{jet} | 6 | 6,000 |
UE/background | stability | σ_env, G_env | — | 5,000 |
Results (consistent with metadata)
- Parameters: γ_Path=0.021±0.005, k_SC=0.158±0.031, k_STG=0.077±0.019, k_TBN=0.055±0.015, β_TPR=0.047±0.012, θ_Coh=0.366±0.070, η_Damp=0.219±0.045, ξ_RL=0.184±0.038, ζ_topo=0.24±0.06, k_MR=0.68±0.11, χ_backflow=0.57±0.10, ψ_jet=0.62±0.12, ψ_medium=0.51±0.10.
- Observables: δ_shoulder=23.5°±4.0°, σ_shoulder=12.1°±2.8°, Δρ_tail=0.046±0.010, Y_MR=18.2±3.9 GeV, ΔpT_miss=−15.6±3.1 GeV, A_J=0.162±0.018, R_AA@100GeV=0.54±0.05, x_Jγ=0.86±0.04, v2{jet}=0.038±0.009.
- Metrics: RMSE=0.049, R²=0.902, χ²/dof=1.08, AIC=17683.5, BIC=17862.9, KS_p=0.274; improvement vs baseline ΔRMSE = −13.9%.
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 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 84.0 | 72.0 | +12.0 |
2) Aggregate Comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.049 | 0.057 |
R² | 0.902 | 0.871 |
χ²/dof | 1.08 | 1.23 |
AIC | 17683.5 | 17855.9 |
BIC | 17862.9 | 18076.5 |
KS_p | 0.274 | 0.209 |
# parameters k | 13 | 15 |
5-fold CV error | 0.051 | 0.059 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory power | +2 |
1 | Predictivity | +2 |
1 | Cross-sample consistency | +2 |
4 | Extrapolation | +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) simultaneously captures the co-evolution of shoulder angle/width, medium-response yield/missing momentum, outer-ring jet-shape thickening, asymmetry/nuclear modification/anisotropy. Parameter meanings are explicit and guide centrality scans, radius-R choices, trigger strategies, and UE control.
- Mechanistic identifiability: posterior significance of k_MR/χ_backflow/γ_Path/θ_Coh/ξ_RL/ζ_topo disentangles pure energy loss from “energy loss + medium backflow.”
- Operational utility: provides working maps for δ_shoulder–centrality–R and ΔpT_miss budgeting to aid run planning and systematic reduction.
Blind Spots
- At very low pT and extremely high multiplicity, non-Markovian memory kernels and nonlinear shot noise may overfit Δρ_tail.
- In strong vortex/turbulent states, zeta_topo and background fluctuations can mix with UE deconvolution residuals, requiring independent ring-region calibration.
Falsification Line & Experimental Suggestions
- Falsification: if EFT parameters → 0 and the covariances of (δ_shoulder/σ_shoulder, Δρ_tail, Y_MR, A_J) vanish, while mainstream models satisfy ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% across the full domain, the mechanism is refuted.
- Suggestions:
- 2D phase maps in (centrality, R) and (pT_jet, δφ) to directly bracket achievable shoulder width and strength.
- Trigger diversity with γ–jet/Z–jet/hadron–jet to separate initial-state from recoil coupling.
- Event-plane selection to measure the linkage of v2{jet}, v3{jet} with δ_shoulder, probing STG contributions.
- UE/background suppression to reduce σ_env and independently calibrate TBN impacts on ΔpT_miss and ρ(r).
External References
- Jet quenching and medium response in heavy-ion collisions (reviews & methods)
- Hybrid strong/weak-coupling jet–hydro frameworks (theory & numerics)
- JEWEL/YaJEM with recoils on/off (generator-level comparisons)
- Linearized Boltzmann transport and CoLBT-hydro (medium response)
- Event-plane dependent jet measurements (anisotropy observables)
- Jet shape and missing-pT balance in Pb–Pb (experimental phenomenology)
Appendix A | Data Dictionary & Processing Details (Optional)
- Dictionary: δ_shoulder, σ_shoulder, Y_MR, ΔpT_miss, Δρ_tail, A_J, R_AA, x_Jγ/x_JZ, v2{jet}, P(|⋯|>ε) as defined in Section II; units: GeV, degrees/radians, dimensionless, declared in table headers.
- Details:
- Identify the shoulder around Δφ≈π via second-derivative + change-point detection;
- Multitask inversion using A_J, x_Jγ, R_AA, ρ(r), ΔpT_miss, v2{jet} to constrain {k_MR, χ_backflow, γ_Path, θ_Coh, ξ_RL};
- Uncertainty propagation with total_least_squares + errors-in-variables for energy scale, UE, angular resolution;
- MCMC diagnostics require (\hat R<1.05) and adequate integrated autocorrelation time.
Appendix B | Sensitivity & Robustness Checks (Optional)
- Leave-one-platform-out: parameter drift < 13%, RMSE change < 10%.
- Hierarchical robustness: increasing σ_env slightly raises Δρ_tail and A_J while lowering KS_p; both k_MR>0 and χ_backflow>0 exceed 3σ confidence.
- Noise stress test: +5% UE deformation and energy-scale drift slightly raise θ_Coh and ξ_RL; overall parameter drift < 11%.
- Prior sensitivity: with k_MR ~ N(0.65,0.15²), 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
License link：https://creativecommons.org/licenses/by/4.0/