Full Jet Observables in Collider Physics

• Full jet observables are a set of infrared- and collinear-safe measurements that provide a comprehensive, multi-dimensional view of jet properties in high-energy collisions.
• They integrate analyses of inclusive cross sections, jet shapes, fragmentation functions, and substructure variables to capture key dynamics of jet formation and medium interactions.
• Experimental applications involve refined calibration, background subtraction, and machine-learning techniques to probe QCD dynamics and extract medium properties in both proton-proton and heavy-ion environments.

Full jet observables are a class of infrared- and collinear-safe measurements that aim to provide a comprehensive, multi-dimensional characterization of jets produced in high-energy collisions, both in proton-proton and heavy-ion environments. These observables capture not only the total momentum and energy of the jet, but also its internal structure, angular and energy correlations, collective event properties, and sensitivity to QCD medium effects. They underpin jet cross-section determinations, substructure analyses, studies of modifications due to the quark-gluon plasma, and precision tests of perturbative QCD and parton distribution functions.

1. Essential Definitions and Formulas

Full jet observables include quantities measured on whole jets such as the inclusive cross section, shapes, fragmentation functions, energy flow moments, and multi-particle correlations:

• Inclusive cross section $d^2\sigma/dp_T\,d\eta$: measures the rate of jets as a function of transverse momentum ($p_T$) and rapidity/azimuth ($\eta$).
• Jet shapes $\rho(r)$: energy profile as a function of radial distance from the jet axis, $$\rho(r)=(1/N_{\rm jet})\sum_{\rm jets}\sum_{i\in(r,r+\Delta r)}(p_{T,i}/p_{T,jet})/\Delta r$$.
• Fragmentation function $D(z)$: distribution of the momentum fraction carried by jet constituents, $D(z)=(1/N_{\rm jet}) dN/dz$.
• Jet mass $M_\text{jet}^2=(\sum_i p_i^\mu)^2$.
• Generalized energy flow moments $$I_{i_1\cdots i_n}=(1/E_J)\sum_{i\in \rm{jet}} E_i x_{i_1}^{(i)}\cdots x_{i_n}^{(i)}$$.
• Angular scaling exponent $\alpha(R)=d\ln\langle\mathcal{G}(R)\rangle/d\ln R$ for ensemble-averaged two-particle mass correlations $\mathcal G(R)$ (Jankowiak et al., 2012).
• Nuclear modification factor $$R_{AA}(R,p_T)=[d^2N^{AA}_\text{jet}(R)/dp_T\,dy]/[d^2N^{pp}_\text{jet}(R)/dp_T\,dy]$$ as a function of jet cone size $R$ (Barreto et al., 2023).
• Anisotropic flow coefficients $v_n\{2\}(R,p_T)$: measured via harmonics of jet azimuthal correlations with event planes (Barreto et al., 2023).
• Substructure variables (N-subjettiness, energy correlators, fractal observables) that span $M$-body phase space (Datta et al., 2017, Davighi et al., 2017).

These observables are constructed to be infrared- and collinear-safe, boost-invariant in the detector plane, and sensitive to both perturbative and non-perturbative aspects of QCD.

2. Classification of Full Jet Observables

Full jet observables can be organized into several main categories, each probing distinct aspects of jet formation or medium interaction:

Observable Type	Example(s)	Physical Sensitivity
Cross Section & Ratios	$d^2\sigma/dp_T\,d\eta$, $R_{AA}$	Hard process rates, energy loss in medium
Jet Shapes	$\rho(r)$, $\Psi(r)$	Radial energy distribution, broadening
Fragmentation	$D(z)$, $z_g$, $z_r$	Momentum sharing, splitting dynamics
Event Shapes	thrust, sphericity, broadening	Multi-jet topology, global geometry
Substructure	$N$-subjettiness, ECFs, EFOs	Multi-prong topology, color flow, fractality

• Superstructure observables such as pull vector $\vec{t}_j$ encode color flow between jets and are key for discriminating signal versus background in processes sensitive to color topology (Gallicchio et al., 2010).
• Fractal-based observables (EFOs) capture the scale-dependent structure of hadron distributions, improving quark/gluon discrimination power (Davighi et al., 2017).
• Energy flow polynomial (EFP) basis and angular moments allow for a completeness in describing multi-subjet configurations and are optimal for machine learning applications to jet tagging (Lu et al., 2022, Datta et al., 2017).

3. Measurement Methodologies and Experimental Implementation

Jets are reconstructed with algorithms such as anti-$k_T$ (standard at LHC), with radius parameter $R$ spanning the range $0.2\lesssim R\lesssim1.0$. Key procedures include:

• Calibration and pileup mitigation: area-median subtraction, constituent subtraction, or soft killer approaches are used for accurate jet energy measurements against heavy-ion backgrounds (Andrews et al., 2018).
• Substructure extraction: reclustering (e.g., Cambridge/Aachen), grooming procedures (Soft Drop, dynamical grooming), and decomposition into N-subjettiness or ECFs are standard (Andrews et al., 2018, Romão et al., 2023, Larkoski et al., 2015).
• Statistical inference: multivariate discriminators (BDT, DNN) can saturate tagging power using minimal complete sets of observables, notably $3M-4$ variables for $M$-body phase space (Datta et al., 2017, Datta et al., 2019, Lu et al., 2022).
• Principal component analysis and feature selection: techniques such as LASSO identify minimal, nonredundant subsets of observables that encode all relevant substructure information (Lu et al., 2022).

Detector effects, background subtraction, and jet calibration are fundamental for the robust measurement of full jet observables at the LHC and RHIC.

4. Theoretical Insights: QCD Dynamics, Medium Response, and Substructure

Full jet observables have elucidated the interplay between perturbative cascades, non-perturbative event properties, and medium-induced modifications:

• Collinear and angular scaling: Observables such as $\alpha(R)$ approach $2$ for QCD-like scale invariant jets at small $R$, and $4$ in the presence of uniform background (underlying event/pileup) at large $R$ (Jankowiak et al., 2012).
• Medium response and energy recovery: $R_{AA}(R)$ increases with $R$ in heavy ion collisions due to recovered medium-induced soft radiation and recoil components. Inclusion of recoiling scattering centers is essential to reproduce experimental trends, especially at large $R\gtrsim0.6$ (Barreto et al., 2023).
• Azimuthal anisotropy ($v_n$): $v_2(R)$ reflects path-length dependent energy loss, decreasing with jet cone size due to background fluctuation dilution. $v_3(R)$ underprediction suggests unresolved jet-medium geometry correlations (Barreto et al., 2023).
• Substructure resilience: Many substructure observables (groomed girth, axis ratios, dynamical $\kappa$) retain mutual consistency under quenching, with linear and nonlinear correlations robust to medium modifications (Romão et al., 2023).
• Multi-prong and fractal observables: Fractal scaling and energy flow moments capture additional information beyond jet multiplicity and radial moments, correlating weakly with standard quark/gluon discriminators and offering performance improvements (Davighi et al., 2017, Gur-Ari et al., 2011).
• Singular structure and perturbative control: The analytic tractability and factorization properties of $N$-subjettiness and ECF ratio observables depend sensitively on the choice of axes and observable definitions, impacting resummation and nonperturbative modeling (Larkoski et al., 2015).

The cumulative insights from full jet observable studies continue to refine theoretical and phenomenological understanding of jet formation, QCD shower dynamics, and in-medium modifications.

5. Key Results and Comparisons to LHC Data

Comparisons to experimental data at the LHC have quantitatively established full jet observables as essential probes of QCD:

• Inclusive jet $R_{AA}$ in Pb+Pb at $5.02$ TeV rises from $\approx 0.3$ (R=0.2) to $\approx 0.5$ (R=1.0). Modern simulations (JEWEL+v-USPhydro) with recoil reproduce both the magnitude and positive slope within uncertainties (Barreto et al., 2023).
• Azimuthal anisotropy $v_2(R)$ decreases from $\approx 0.06$ (R=0.2) to $\approx 0.02$ (R=1.0), driven by geometrical path-length effects, with only minor sensitivity to medium recoils (Barreto et al., 2023).
• Leading-subjet fragmentation shows mild narrowing and moderate enhancement at high $z_r$ in Pb+Pb versus pp, correctly captured by JEWEL+v-USPhydro (Barreto et al., 2023).
• NNLO QCD corrections for single-inclusive and multi-differential cross sections are now available in full color with antenna subtraction; subleading color effects are $<10\%$ but significant for triply-differential observables at large rapidity separation and low $p_T$ (Chen et al., 2022, Chen et al., 2022).
• Machine-learning optimized observables constructed from a minimal basis of high-level jet variables (N-subjettiness, EFPs, etc.) match the performance of deep neural networks and constituent-based methods, confirming that a small set of interpretable full jet observables is sufficient for near-optimal tagging and discrimination (Lu et al., 2022, Datta et al., 2019, Datta et al., 2017).

Full jet observable measurements have driven improvements in MC modeling, PDF extraction, and heavy-ion tomography, and continue to serve as benchmarks for QCD theory and new physics searches.

6. Open Problems and Future Directions

Key challenges and active research avenues include:

• Jet-medium correlation physics: Underprediction of higher harmonics ($v_3$) and flat $R$-dependence in MC models indicate missing decorrelation mechanisms between soft and hard event planes (Barreto et al., 2023).
• Background subtraction and pileup mitigation: New subtraction algorithms and grooming techniques are under refinement to enable precision measurements in dense environments (Andrews et al., 2018).
• Nonperturbative effects and hadronization modeling: Further studies of medium-induced hadronization, color flow, and wake excitation are critical for a quantitative understanding at low $p_T$ and large $R$ (Karpenko et al., 2018).
• Optimal observable construction: Automation via machine learning and feature selection identifies the minimal sets of full jet observables sufficient for any tagging or modification study, streamlining future experimental analyses (Lu et al., 2022, Datta et al., 2019).
• Precision phenomenology: Full-color NNLO calculations must be included systematically in future PDF fits and experimental interpretations, especially for multi-differential observables where subleading-color effects are sizable (Chen et al., 2022, Chen et al., 2022).
• Heavy-ion and medium studies: Systematic deployment of the latest substructure observables, combined with event-by-event hydrodynamical modeling and new lattice inputs for transport coefficients, promises quantitative extraction of QGP properties (Majumder, 2014, Barreto et al., 2023).

Full jet observables thus remain at the center of collider QCD, linking theoretical, experimental, and statistical advances in jet physics across collision systems and energy scales.