FB Rapidity Correlations in High-Energy Collisions

• FB rapidity correlations are statistical measures used to relate particle yields in separated forward and backward rapidity regions in high-energy collisions.
• They combine short-range effects from resonance decays and jets with long-range contributions from initial-state fluctuations and geometry-driven collectivity.
• Experimental methods adjust gap and window sizes to isolate dynamical correlations, thereby constraining models like string fusion and CGC frameworks.

Forward–backward (FB) rapidity correlations quantify the connection between particle production in widely separated regions of pseudorapidity (or rapidity) in high-energy hadronic and nuclear collisions. They serve as sensitive probes of initial-state fluctuations, source dynamics, and the collective or non-collective propagation of correlations across the longitudinal extent of the collision system. The FB correlation structure encodes both short-range (resonance, jet) and long-range (multiparton, color field, or geometry-driven) physics and is essential for constraining models of particle production from the RHIC to the LHC and beyond.

1. Mathematical Framework and Definitions

Let $N_F$ and $N_B$ denote the event-by-event multiplicities (or other observables) in two separated rapidity intervals—“forward” and “backward” windows—centered at $y_F$ and $y_B$, with width $\delta y$. The canonical FB correlation coefficient, $b$, is defined via linear regression: $$b = \frac{\langle N_F N_B \rangle - \langle N_F \rangle \langle N_B \rangle}{\langle N_F^2 \rangle - \langle N_F \rangle^2}$$ This is the Pearson coefficient between $N_F$ and $N_B$ and provides the slope in the event-averaged relation $\langle N_B \rangle_{N_F} = a + bN_F$ (Dash et al., 2012, Altsybeev, 2017).

For intensive quantities, such as the event-wise mean transverse momentum $\overline{p}_T$ in each window, the analogous definition applies: $$b_{p_T} = \frac{\langle \overline{p}_T^F \,\overline{p}_T^B \rangle - \langle \overline{p}_T^F \rangle \langle \overline{p}_T^B \rangle}{\langle (\overline{p}_T^F)^2 \rangle - \langle \overline{p}_T^F \rangle^2}$$ where $$\overline{p}_T^{F,B} = (1/n_{F,B}) \sum_{i=1}^{n_{F,B}} p_{T,i}$$ in the respective window (Altsybeev, 2017, Kovalenko et al., 2016).

Multiplicities or other observables can be measured in windows with width $\Delta y$ (or $\Delta \eta$ in pseudorapidity), and the two windows may be separated by a gap $\Delta y_{\rm gap}$, allowing for separation of short- and long-range contributions (Mondal et al., 2021, Kovalenko et al., 2014).

2. Physical Origin and Interpretation

FB rapidity correlations emerge from both initial and final stages of the collision, providing a multi-scale view:

• Short-range correlations (SRC): Originate from decays of resonances, jets, and local parton fragmentation. These decline rapidly as the $\Delta y_{\rm gap}$ increases (typically within $\Delta y \lesssim 1$) (Dash et al., 2012, Kovalenko et al., 2014, Vechernin, 2013).
• Long-range correlations (LRC): Reflect event-by-event fluctuations in the number or configuration of particle-emitting sources (strings, flux tubes, wounded nucleons or quarks). LRC survive even with large rapidity gaps and are sensitive to the dynamics of the earliest collision stages, e.g., color glass condensate, string fusion, or initial energy-density fluctuations (Altsybeev, 2017, Broniowski et al., 2019, Wang et al., 2016).

The measured $b$ or $b_{p_T}$ is thus a mixture: it decays with increasing gap, isolating the genuine LRC at large $\Delta y$. The presence and centrality dependence of LRC are distinctive signatures of early-time fluctuations and collectivity.

3. Experimental Implementation and Systematics

Experimentally, FB rapidity correlations are measured by reconstructing charged particles (tracks) in specified forward and backward windows and calculating $b$ or $b_{p_T}$ event by event. Corrections are essential to disentangle physical correlations from trivial statistical or selection-induced effects:

• Centrality selection bias: In $A$+$A$ collisions, centrality is defined via multiplicity in a region outside the FB windows to avoid artificial correlations (De et al., 2013).
• Volume fluctuations: Multiplicity-based $b$ are sensitive to fluctuations in participant number. Intensive observables (mean $p_T$), or “strongly intensive” combinations such as $\Sigma$ and $\Gamma$ constructed from moments and cumulants, are robust against these (Altsybeev, 2017, Andronov, 12 Nov 2025, Kovalenko et al., 2016).
• Gap and window size dependence: $b$ is typically nearly flat with window width at fixed gap, but falls steeply with increasing gap, and approaches a plateau reflecting LRC for large gaps (Dash et al., 2012, Mondal et al., 2021).

Technical choices include $p_T$ range (eg., 0.2–2.0 GeV/c in ALICE (Altsybeev, 2017)), rapidity window width (eg., $\Delta \eta = 0.2$–0.4), and detector-based centrality definition (V0, ZDC, etc.).

4. Model Approaches and Theoretical Understanding

A variety of microscopic and phenomenological frameworks elucidate the mechanisms underlying FB rapidity correlations:

Model Class	Principal Mechanism	Key Features / Comparison to Data
String/flux tube models	Particle emission from longitudinal color flux tubes, string fusion between overlapping sources	Predict strong LRC, centrality, and energy dependence; string fusion raises $b$ at high density (Kovalenko et al., 2014, Kovalenko et al., 2016, Broniowski et al., 2019)
Superposition models	Independent emission from fluctuating sources; LRC from source number fluctuation, SRC from intra-source correlations	Predict analytic forms for $b$ as function of source variance, window geometry (Olszewski et al., 2013, Vechernin, 2012)
CGC/Glasma frameworks	Longitudinal color fields extended in rapidity; quantum fluctuations	LRC determined by initial-state gluon fields (Altsybeev, 2017, Jia et al., 2015)
pQCD-inspired event generators (HIJING, PYTHIA, PHOJET, EPOS3, AMPT)	Multi-parton interactions, mini-jet/jet production, core-corona structure, string melting, partonic/hadronic cascades	Reproduce generic $b(\mathrm{gap})$ falloff; saturation/shaping in $b(\sqrt{s})$ at high energy; only models with collectivity/fusion reproduce non-monotonic $b_{p_T}$ (Mondal et al., 2021, Altsybeev, 2017)

Model discrimination is achieved by systematic comparison to $b$ vs. $\Delta y$, energy, system, and centrality, and by correlation structure analysis (e.g., damping of higher longitudinal harmonics) (Jia et al., 2015).

5. Empirical Systematics: Energy, Centrality, and System Size

FB correlation strengths exhibit robust and distinctive trends across collision energy $\sqrt{s}$, centrality, gap, and system size:

• Energy dependence: In $pp$ collisions, $b$ rises approximately linearly with $\ln\sqrt{s}$ up to RHIC, then saturates or even decreases at LHC energies according to PYTHIA/PHOJET, but not in naive data extrapolation (Dash et al., 2012, Mondal et al., 2021). In $AA$, LHC data display large $b_{p_T}$ and $b$ with long-range plateaus (Altsybeev, 2017).
• Centrality dependence: In $Pb$+$Pb$ collisions, $b_{p_T}$ and strongly intensive $b$ display a characteristic rise from peripheral to mid-central events, with a reduction in the most central events—evidence for the onset and then “smearing” of geometry-driven fluctuations in high-density regimes (Altsybeev, 2017, Kovalenko et al., 2016, De et al., 2013).
• System size: Comparing $pp$, $p$+$A$, $A$+$A$ shows that both the strength and the $\Delta y$ range of LRC grows with system size, and the window-to-window correlation length $\lambda$ is larger in heavy-ion versus proton-proton systems (Andronov, 12 Nov 2025).
• Pseudorapidity gap: $b$ falls steeply as gap increases due to suppression of SRC, then levels out to a nonzero value at large gap (the LRC component) (Dash et al., 2012, Gope et al., 6 May 2025).

6. Forward–Backward Correlations for Intensive vs. Extensive and Strongly Intensive Observables

Traditional $b$ coefficients for multiplicity are sensitive to volume (participant) fluctuations and centrality bin width, leading to significant bias unless corrections (“profile” method) or narrow centrality bins are used (De et al., 2013, Olszewski et al., 2013). Mean $p_T$ and other intensive observables suppress this bias, and quantifying their FB correlations ($b_{p_T}$) cleanly isolates dynamical (source-level) fluctuations (Altsybeev, 2017, Kovalenko et al., 2016).

Strongly intensive observables such as $\Sigma$ and $\Gamma$, constructed as ratios of joint cumulants, are defined so as to cancel all volume (source-number) fluctuations in superposition models (Andronov, 12 Nov 2025). These quantities probe two- and three-particle correlations independent of trivial event-by-event variations in the number of sources, and are thus preferred for studying collectivity and string fusion in high-multiplicity processes.

7. FB Correlations and Longitudinal Geometry: Torque and Flow Decorrelation

In non-central heavy-ion collisions, FB correlations extend beyond multiplicity to the global geometry of the event. Fluctuations and longitudinal structure in the initial participant configuration create an $\eta$-dependent rotation (“torque”) of the principal axes (event planes) of the fireball. This manifests as a decorrelation of flow harmonics (e.g., elliptic flow, triangular flow) between forward and backward rapidity. The effect is measured via two- and four-particle mixed cumulants and quantified as a reduction in $\langle\cos[n\Delta_{FB}]\rangle$ with rapidity gap, corresponding to an effective “twist angle” $\sim10^\circ$–$20^\circ$ even at RHIC (Moreira et al., 2011, Broniowski et al., 2011). Such torque is robust against hadronic rescattering and can be directly related to initial-state geometry and its hydrodynamic translation into collective flow.

8. Outlook and Applications

FB rapidity correlations are a central diagnostic of early-time dynamics, source structure, and collective phenomena in high-energy QCD matter. Their detailed mapping allows:

• Constraint and tuning of production models—string/flux tube properties, source fluctuations, collectivity, and the interplay of short- and long-range physics (Broniowski et al., 2019, Kovalenko et al., 2014).
• Isolation of initial-geometry-driven versus final-state (hydro/cascade) evolution effects.
• Model-independent extraction of two-particle and higher-order correlation functions via windowed measurements (Vechernin, 2012, Vechernin, 2013).
• Distinguishing between hadronization scenarios (string fusion, random source emission, multi-Pomeron dynamics) and testing their validity via novel observables such as $\Sigma$ and $\Gamma$ (Andronov, 12 Nov 2025).
• Mapping of critical phenomena, collectivity thresholds, and possible onset of new physics regimes.

Ongoing and future directions include extending measurements to new energies and system sizes (including isobaric and small systems), varying selection in $\phi$ and $p_T$ to disentangle sources of correlation, and exploiting higher cumulants and strongly intensive observables for unbiased characterization of longitudinal dynamics in strongly-interacting matter (Altsybeev, 2017, Andronov, 12 Nov 2025).