Two-Particle Angular Correlations

Updated 20 November 2025

Two-particle angular correlations are defined as the ratio of same-event to mixed-event particle pairs as a function of relative pseudorapidity and azimuthal angle, offering insights into QCD dynamics.
Experimental techniques utilize fine binning, rigorous acceptance corrections, and PID methods to distinguish contributions from jets, resonances, and quantum-statistical effects.
Notably, observed baryon–baryon anticorrelations and their multiplicity dependence challenge current Monte Carlo models, spurring improvements in local conservation and coalescence modeling.

Two-particle angular correlations quantify the joint emission probability of two particles as a function of their kinematic differences—specifically, relative pseudorapidity (Δη or Δy) and azimuthal angle (Δφ)—with respect to either the beam axis or the event thrust axis. Originally introduced as a diagnostic for jet production, these correlators are now crucial for disentangling the myriad physical mechanisms underlying hadron production in high-energy collisions, including minijet fragmentation, resonance decays, quantum-statistical effects, global and local conservation laws, and collective phenomena. The precise shape and species/charge/multiplicity dependence of two-particle angular correlations encode essential constraints on perturbative and non-perturbative QCD dynamics, and continue to challenge both Monte Carlo modeling and hadronization theory (Janik, 2014).

1. Formalism and Construction of Two-Particle Angular Correlation Functions

The two-particle angular correlation function is universally formulated as the ratio of pair densities from the same (“signal”) and mixed (“background”) events:

$C(\Delta y, \Delta \varphi) = \frac{N_{\mathrm{pairs}}^{\mathrm{mixed}}}{N_{\mathrm{pairs}}^{\mathrm{signal}}} \frac{S(\Delta y, \Delta \varphi)}{B(\Delta y, \Delta \varphi)}$

where

$S(\Delta y, \Delta \varphi)$ is the same-event pair distribution,
$B(\Delta y, \Delta \varphi)$ is built from mixed events (pairs formed from different events with similar global properties: multiplicity, z-vertex),
The pre-factor ensures normalization to unity for uncorrelated emission.

For identified species (π, K, p), the function is constructed separately for like-sign and unlike-sign pairs, typically within |y|<0.5 or |η|<1.0, and in selected pₜ intervals. The background mixing completely corrects for single-particle acceptance effects and detector efficiencies when performed within tightly binned event classes [(Janik, 2014); (Collaboration, 13 Nov 2025); (Ruggiano, 2023)].

A "rescaled cumulant" formalism is often employed to remove trivial 1/N scaling with multiplicity:

$C_C(\Delta y, \Delta \varphi) = N_{\rm av} (C_P(\Delta y, \Delta \varphi) - 1)$

where $N_{\rm av}$ is the event-averaged charged particle density, yielding the mean number of correlated pairs per event (Ruggiano, 4 Mar 2024, Collaboration, 13 Nov 2025).

The per-trigger formalism—common for jet and jet-hadron analyses—normalizes the pair yield by the number of trigger particles and exposes conditional associated yields (Grosse-Oetringhaus, 2012).

2. Experimental Methodology and Acceptance Corrections

Correlations are measured in minimum-bias and high-multiplicity samples across all collision systems (pp, p–A, A–A, e⁺e⁻), with systematic application of PID via energy loss (dE/dx), time-of-flight, and topological cuts. For identified pairs, high purities (>90%) are ensured up to pₜ≲2.5–3 GeV/c for π, K, and p [(Janik, 2014); (Collaboration, 13 Nov 2025)].

Key corrections include:

Mixed-event background: Robust against acceptance, but only provides an approximate correction for finite-acceptance, especially for non-flat rapidity distributions. New methodologies improve upon standard event-mixing by applying analytic finite-acceptance corrections based on explicit convolution of detector windows (“uniform-signal” and “delta-function trigger” limits) (Oh et al., 2016).
Normalization: Both S and B distributions are normalized such that C(Δy,Δφ)→1 in the absence of correlations, either via global pair normalization or by scaling B to match S in the acceptance window (Collaboration, 13 Nov 2025).
Binning and projections: The 2D (Δy,Δφ) histogram is typically binned with Δy=0.1, Δφ=0.05 radian precision. 1D projections (e.g., C(Δφ) for |Δy|<1) expose integrated features such as near-side peaks and away-side ridges (Ruggiano, 2023).

In e⁺e⁻, angular correlations can be constructed relative to either the beam or the thrust axis. The thrust-axis analysis rotates the event frame according to the event-by-event thrust direction, especially relevant in dijet-like topologies (Chen et al., 2022, Collaboration et al., 2022, Chen et al., 2023).

3. Physical Sources and Canonical Structures

The observed structures in two-particle angular correlations result from several mechanisms:

Minijet (fragmentation) peaks: The dominant near-side peak at (Δy,Δφ)≈(0,0) is due to intra-jet pairs from semi-hard parton fragmentation [(Janik, 2014); (Sicking, 2012)].
Away-side ridge: An enhancement at Δφ≈π, nearly flat in Δy, reflecting back-to-back jets and global transverse momentum conservation [(Janik, 2014); (Collaboration, 2012)].
Longitudinal (string) ridge: Elongated structure at Δy≈0, from string-fragmentation and low-mass resonance decays, most visible in low-multiplicity, unlike-sign pairs (Janik, 2014).
Quantum-statistical correlations: Bose–Einstein enhancement for like-sign identical bosons (ππ, KK) at (0,0), producing a substructure on top of the jet peak; Fermi–Dirac suppression for like-sign baryons is minimal at RHIC/LHC momenta (Ruggiano, 2023, Graczykowski et al., 2021).
Resonance decays: Sharp near-side enhancements in unlike-sign pairs (e.g., ρ→ππ, φ→KK) manifesting as narrow peaks (Ruggiano, 2023).
Final-state interactions (FSI): Strong proton-proton s-wave interactions produce a femtoscopic “core” (∼0.1 rad). For p p̄, annihilation channels produce narrow near-side anticorrelation (Graczykowski et al., 2021).

Each mechanism contributes a characteristic feature to the correlation topology, modulated by species, charge combination, and multiplicity [(Janik, 2014); (Ruggiano, 2023); (Ruggiano, 4 Mar 2024); (Collaboration, 13 Nov 2025)].

4. Baryon–Baryon Anticorrelation: Experimental Observations and Interpretation

A striking and now systematically confirmed feature is the broad near-side depletion for like-sign baryon–baryon (pp, p̄p̄) pairs at (Δy,Δφ)≈(0,0). This anticorrelation:

Is observed robustly at all multiplicity classes in pp at 7 and 13 TeV (Ruggiano, 2023, Collaboration, 13 Nov 2025), p–Pb at 5.02 TeV (Ruggiano, 4 Mar 2024), and even in peripheral heavy-ion (Au+Au) collisions by STAR (Collaboration et al., 2019).
Exhibits amplitude A_pp≈–0.3 (low N_ch), deepening to –0.6 at high multiplicity in the rescaled cumulant, with widths σ_φ≈1.0 rad, σ_y≈0.6 (Collaboration, 13 Nov 2025).
Is not described by any Monte Carlo event generator (PYTHIA 8, EPOS, HERWIG) which all predict a near-side peak for baryon–baryon pairs [(Collaboration, 13 Nov 2025); (Ruggiano, 2023); (Graczykowski et al., 2014)].
Is not a trivial consequence of Fermi–Dirac statistics or final-state interactions; the suppression extends to kinematic ranges where quantum statistics is negligible (Graczykowski et al., 2021, Collaboration, 13 Nov 2025).

The only theoretical frameworks that partially reproduce the broad baryon–baryon anticorrelation are transport calculations which combine partonic scatterings (to introduce spatial antibunching) with improved quark coalescence (which converts local partonic-level exclusions into momentum-space anticorrelations), as implemented in the new AMPT coalescence module (Zhang et al., 2018, Zhang et al., 2019). Local conservation of baryon number during fragmentation is also implicated; producing two baryons close in phase space requires multiple antibaryons for conservation, which is dynamically suppressed [(Janik, 2014); (Graczykowski et al., 2014)].

A summary of measured amplitudes and widths at 13 TeV (ALICE, HM class) (Collaboration, 13 Nov 2025):

Pair Type	Near-side Amplitude $A$	Width $\sigma_{\Delta\varphi}$ (rad)
π⁺π⁺ (like)	$6.2 \pm 0.3$	$0.38 \pm 0.05$
K⁺K⁺ (like)	$4.8 \pm 0.4$	$0.45 \pm 0.07$
pp (like)	$–0.65 \pm 0.05$	$1.10 \pm 0.15$

5. Multiplicity and System Dependence

The correlation pattern is multiplicity- and system-dependent:

For mesons, the near-side peak amplitude in C_C rises monotonically with multiplicity, reflecting increased jet activity and possible collective phenomena in high-multiplicity events [(Collaboration, 13 Nov 2025); (Sicking, 2012)].
For like-sign baryons, the near-side depletion deepens with multiplicity in C_C, but appears weaker in the probability-ratio C_P due to 1/N_ch scaling. The suppression is not washed out by increased phase-space density, highlighting its dynamical origin (Collaboration, 13 Nov 2025, Ruggiano, 2023).
p–Pb collisions at 5.02 TeV show nearly identical anticorrelation patterns to multiplicity-matched pp at 13 TeV, suggesting that the primary scaling variable is charged-particle density, not system size (Ruggiano, 4 Mar 2024).
In A–A collisions, the depletion is evident only in peripheral or very low-multiplicity events; it is absent or much reduced in central Pb–Pb and Au–Au, indicating a melting of the local baryon exclusion in the higher-density environment (Collaboration et al., 2019, Zhang et al., 2019).

6. Modeling, Theoretical Implications, and Monte Carlo Discrepancies

Standard event generators fail in several key respects:

None of PYTHIA 8, EPOS, or HERWIG can reproduce the like-sign baryon–baryon anticorrelation. All predict a positive near-side peak or at best a flat structure [(Collaboration, 13 Nov 2025); (Ruggiano, 2023); (Graczykowski et al., 2014)].
Reproduction of the correlation for mesons is reasonable, but quantum-statistics effects are underestimated (e.g., Bose–Einstein enhancement for ππ in PYTHIA 8), and away-side yields show varying deviations (Collaboration, 13 Nov 2025).
Recent AMPT studies demonstrate that quark coalescence coupled with partonic rescattering yields a qualitative agreement with the observed suppression, supporting a coalescence-driven exclusion process (Zhang et al., 2018, Zhang et al., 2019).
The failure of global-only baryon conservation in string fragmentation models is exposed by the data, pointing to the need for local conservation and possibly diquark suppression mechanisms [(Graczykowski et al., 2014); (Janik, 2014)].

Femtoscopic analyses and quantum-statistical/FSI unfolding confirm that observed baryon–baryon depletion cannot be attributed solely to Pauli exclusion or strong final-state interactions. Femtoscopic features appear as superimposed narrow cores but do not explain the broad anticorrelation (Graczykowski et al., 2021, Collaboration, 13 Nov 2025).

Experimental systematics are dominated by PID, vertexing, efficiency corrections, event-mixing class binning, and bin-by-bin uncertainties, typically 6–30% for the correlation amplitude, with larger errors for baryons given their lower yields (Collaboration, 13 Nov 2025).

7. Broader Context: Heavy-Ion, e⁺e⁻, and BSM Searches

In heavy-ion collisions, the broadening of near-side jet peaks and evolution of angular correlations with centrality reveal jet–medium interactions and collective flow, with longitudinal broadening (σ_η) far exceeding vacuum expectations. Only AMPT with strong partonic interactions can reproduce this broadening [(Grosse-Oetringhaus, 2012); (Szigeti et al., 2019)].
Two-particle angular correlations have been measured in e⁺e⁻ systems (ALEPH, Belle), where dijet topologies dominate and no long-range collective ridge is observed except possibly at the highest multiplicities in thrust-aligned frames, suggesting a universal onset for flow-like phenomena at extremely high track densities (Chen et al., 2023, Chen et al., 2022, Collaboration et al., 2022).
Sensitivity to BSM physics, e.g., Hidden Valley scenarios, is enhanced in e⁺e⁻ owing to a cleaner environment and precise thrust alignment, where two-particle angular correlations can reveal modest but distinctive modulations induced by new QCD-like sectors (Musumeci et al., 2023, Musumeci et al., 2023).

These findings establish two-particle angular correlations as a primary tool for probing non-perturbative QCD dynamics, collective effects, and potential new physics, while highlighting outstanding theoretical challenges in baryon correlation modeling.