Event-Shape Selection in Heavy-Ion Physics

• Event-Shape Selection (ESS) is a technique that partitions collision events based on final-state topology, enabling the study of anisotropic flow under fixed centrality.
• ESS leverages variables like the reduced second-harmonic flow vector (q2) to proxy event-by-event eccentricity, elucidating hydrodynamic responses and fluctuation spectra.
• Experimental implementations use separate subevents and narrow centrality bins to control nonflow effects and improve the extraction of flow coefficients and geometry correlations.

Event-Shape Selection (ESS) denotes a class of procedures that partition collision events according to final-state topology so that observables can be studied at fixed centrality but varying geometry. In relativistic heavy-ion physics, ESS is most commonly associated with Event Shape Engineering (ESE): the selection variable is usually a low-order flow-vector magnitude, especially the reduced second-harmonic flow vector $q_2$, which serves as an experimentally accessible proxy for event-by-event fluctuations of the initial eccentricity and the corresponding anisotropic-flow response (Schukraft et al., 2012). Later literature extended the term to full-histogram “Event Shape Sorting,” and in other collider contexts to topology selections based on transverse sphericity, thrust, or related event-shape variables; however, the heavy-ion geometry-selection framework remains its principal technical meaning.

1. Concept, origin, and nomenclature

The heavy-ion formulation of ESS was articulated as a way to exploit large event-by-event fluctuations of the initial transverse geometry at fixed impact parameter. Because the transverse energy or entropy density fluctuates with the discrete positions of participant nucleons, the initial eccentricities $\varepsilon_n$ also fluctuate, and so do the final-state flow coefficients $v_n$. ESS uses this fluctuation spectrum to select events corresponding to a specific initial shape, thereby enabling measurements at fixed centrality but varying geometry (Schukraft et al., 2012).

In this formulation, the final azimuthal distribution is written as

$$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$

and the basic working assumption is a linear hydrodynamic-like response,

$v_n \approx k_n \varepsilon_{n,p},$

with $k_n$ tuned to approximately reproduce measured $v_n$ values (Schukraft et al., 2012). Within that regime, selecting on a flow-related event-shape proxy is approximately equivalent to selecting on the underlying eccentricity.

The first experimental implementations established the practical heavy-ion meaning of ESS/ESE. ALICE selected events within a centrality bin according to the reduced flow vector $q_2$, using large pseudorapidity gaps between the selection and analysis regions, and showed that the measured $v_2$ for shape-engineered events is significantly larger or smaller than the average (Dobrin, 2012). In this usage, “ESS” and “ESE” are effectively interchangeable.

A later nomenclature split emerged in CME-focused work. There, “ESE” was used for selections based on dynamical fluctuations of $v_2$ measured in an independent reference subevent, whereas “ESS” referred to selections based on statistical fluctuations of $\varepsilon_n$0 built from the particles of interest themselves (Li et al., 25 Sep 2025). This distinction is method-specific rather than universal; elsewhere, “Event Shape Sorting” was introduced as a different concept altogether, one that compares entire event histograms rather than a single scalar descriptor (Tomasik et al., 2016).

2. Formal observables and statistical structure

The standard ESS/ESE construction begins with the harmonic flow vector

$\varepsilon_n$1

with

$\varepsilon_n$2

and event-plane angle

$\varepsilon_n$3

In the original event-shape-engineering paper, unit weights $\varepsilon_n$4 were used for simplicity (Schukraft et al., 2012).

The corresponding reduced flow vector is

$\varepsilon_n$5

for which

$\varepsilon_n$6

This normalization isolates the collective correlation from trivial multiplicity scaling and yields the stochastic limit $\varepsilon_n$7 for random azimuths (Schukraft et al., 2012). Other normalizations exist in the literature, including $\varepsilon_n$8 and $\varepsilon_n$9, but the original ESS/ESE construction adopted $v_n$0 in part because it leads to analytic Bessel-Gaussian forms for $v_n$1 (Schukraft et al., 2012).

On the initial-state side, the participant eccentricity vector may be written in Cartesian form as

$v_n$2

$v_n$3

with density-weighted averages. The same paper also used the integral definition

$v_n$4

within a Monte-Carlo Glauber framework with Woods–Saxon nucleon distributions and $v_n$5 mb (Schukraft et al., 2012).

A central statistical result is that if $v_n$6 fluctuates event-by-event with a 2D Gaussian distribution in the $v_n$7 plane, then the radial distribution $v_n$8 is Bessel-Gaussian, and the finite-multiplicity distribution of $v_n$9 inherits the same form. For unit weights,

$$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$0

where $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$1 parameterizes nonflow within a subevent (Schukraft et al., 2012). This relation underlies the common practice of fitting $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$2 in engineered classes to extract the mean and width of the underlying $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$3 distribution.

3. Experimental workflow and canonical implementations

The canonical ESS/ESE workflow is based on separated subevents. One subevent is used only to compute the selection variable $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$4 and define event classes, while another subevent is used to measure $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$5 and other observables, with a sizable pseudorapidity gap to reduce nonflow and autocorrelations (Schukraft et al., 2012). Centrality is controlled in narrow bins—typically about $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$6, and in some ALICE analyses $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$7 slices were used before recombination into wider intervals—to suppress trivial multiplicity effects on $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$8 and keep bulk conditions comparable across classes (Collaboration, 2015). After sorting by $$E\frac{d^3N}{d^3p} = \frac{1}{2\pi} \frac{d^2N}{p_T\,dp_T\,dy} \left[ 1+\sum_{n=1}^{\infty} 2v_n \cos n(\phi-\Psi_n) \right],$$9, the analysis subevent is used to measure $v_n \approx k_n \varepsilon_{n,p},$0, spectra, correlations, or femtoscopic radii, preferably with multi-particle cumulants when residual nonflow is a concern (Schukraft et al., 2012).

Several experimental realizations exemplify this logic.

Study	Selection variable	Representative class definition
ALICE Pb–Pb, $v_n \approx k_n \varepsilon_{n,p},$1 TeV (Dobrin, 2012)	$v_n \approx k_n \varepsilon_{n,p},$2 from TPC or VZERO subevents	lowest $v_n \approx k_n \varepsilon_{n,p},$3, highest $v_n \approx k_n \varepsilon_{n,p},$4
ALICE Pb–Pb spectra and flow (Collaboration, 2015)	$v_n \approx k_n \varepsilon_{n,p},$5 or $v_n \approx k_n \varepsilon_{n,p},$6 in $v_n \approx k_n \varepsilon_{n,p},$7 centrality slices	lower $v_n \approx k_n \varepsilon_{n,p},$8, upper $v_n \approx k_n \varepsilon_{n,p},$9
PHENIX HBT (Niida, 2014)	forward-rapidity $k_n$0 from RXN, $k_n$1	bottom $k_n$2, top $k_n$3
ALICE D mesons, $k_n$4 TeV (Collaboration, 2018)	$k_n$5 or $k_n$6	lowest $k_n$7, highest $k_n$8

ALICE’s 2.76 TeV implementation used a three-subevent methodology in which one subevent determined $k_n$9, a second measured $v_n$0, and a third provided the event-plane angle $v_n$1. The preferred configuration used VZERO for selection and event-plane determination, with TPC tracks at midrapidity for the $v_n$2 measurement, because this maximized the $v_n$3 gap and minimized nonflow (Dobrin, 2012). In the later spectra-and-flow analysis, the selector was either $v_n$4 from $v_n$5 or $v_n$6 from $v_n$7, while $v_n$8 and spectra were measured in $v_n$9, explicitly avoiding overlap with the selection region (Collaboration, 2015).

PHENIX used the magnitude of the second-order flow vector $q_2$0 from the Reaction Plane Detector at forward rapidity, together with HBT analyses of midrapidity pions. The large rapidity gap between RXN and the HBT particles was part of the autocorrelation control strategy, and event-plane resolution corrections were applied both with and without $q_2$1 selection (Niida, 2014). ALICE’s D-meson analysis likewise defined the default high-selectivity $q_2$2 classifier with TPC tracks, excluded the D-decay tracks from the $q_2$3 construction, and cross-checked the results with a forward-detector $q_2$4 selection to test nonflow sensitivity (Collaboration, 2018).

4. Physics reach in heavy-ion collisions

Within hydrodynamic phenomenology, ESS/ESE is primarily a tool for tightening the geometry-to-flow mapping. In Pb–Pb collisions at $q_2$5 TeV, ALICE found that the effect of event-shape selection on $q_2$6 is nearly independent of $q_2$7 up to about $q_2$8–$q_2$9 GeV/$v_2$0, and that charged-hadron, pion, kaon, and proton spectra are harder in high-$v_2$1 events. In $v_2$2–$v_2$3 centrality, blast-wave fits to spectra ratios were described by varying only the average transverse expansion velocity, with

$v_2$4

and

$v_2$5

indicating an interplay between elliptic and radial flow (Collaboration, 2015).

The same methodology exposes correlations among harmonics and nonlinear response. In AMPT at fixed impact parameter, selecting on $v_2$6 increased $v_2$7, $v_2$8, $v_2$9, and $v_2$0, while $v_2$1 decreased; selecting on $v_2$2 increased $v_2$3 and $v_2$4, while $v_2$5 and $v_2$6 showed weak negative or weak dependence. Event-plane correlators such as $v_2$7, $v_2$8, and $v_2$9 varied with $\varepsilon_n$00 in a way that closely tracked direct selections on $\varepsilon_n$01, supporting the interpretation of $\varepsilon_n$02 as a practical geometry proxy (Huo et al., 2013).

Femtoscopy provided one of the earliest concrete applications beyond flow coefficients themselves. PHENIX measured the azimuthal dependence of pion source radii with ESS and showed that higher-$\varepsilon_n$03 selections enhance the oscillation amplitudes of $\varepsilon_n$04 and $\varepsilon_n$05, as well as $\varepsilon_n$06, leading to larger extracted final source eccentricity via

$\varepsilon_n$07

in the $\varepsilon_n$08 limit (Niida, 2014). In the same study, AMPT simulations with $\varepsilon_n$09 radians indicated a possible twisted source, with an $\varepsilon_n$10-dependent phase shift $\varepsilon_n$11 in the HBT oscillations that increased monotonically with rapidity (Niida, 2014).

Heavy-flavor measurements extended ESS/ESE to charm transport. In mid-central Pb–Pb collisions at $\varepsilon_n$12 TeV, ALICE found that a large-$\varepsilon_n$13 selection increases the average prompt D-meson $\varepsilon_n$14 by about $\varepsilon_n$15, while a small-$\varepsilon_n$16 selection decreases it by about $\varepsilon_n$17, with an approximately $\varepsilon_n$18 significance for the difference between the two classes. By contrast, per-event D-meson yields in ESE-selected samples remained consistent with unity within uncertainties (Collaboration, 2018). This pattern supports a correlation between the D-meson azimuthal anisotropy and the collective expansion of the bulk medium.

ESS/ESE has also been used to control hard-probe path lengths. In Trajectum hydrodynamics for Pb–Pb at $\varepsilon_n$19 TeV, ESE by itself did not change the average path length when averaged over all emission angles, because in-plane shortening and out-of-plane lengthening approximately cancel. However, combining high-$\varepsilon_n$20 selection with in-plane versus out-of-plane cuts relative to $\varepsilon_n$21 tuned path lengths very effectively. For the temperature- and flow-weighted proxy

$\varepsilon_n$22

the ratio of out-of-plane to in-plane mean effective path length reached about $\varepsilon_n$23 in peripheral collisions when high-$\varepsilon_n$24 selection was combined with $\varepsilon_n$25 (Beattie et al., 2022). This use of ESE directly targets the path-length dependence of jet quenching.

5. Variants and broader uses

A major variant is Event Shape Sorting, which replaces a scalar selector such as $\varepsilon_n$26 by a comparison of full event histograms. In the original formulation, each event is represented by a binned histogram $\varepsilon_n$27 of a chosen per-particle observable $\varepsilon_n$28, most commonly the azimuthal angle $\varepsilon_n$29. After rotating events so that the second-order event plane is aligned, the algorithm partitions the current event ordering into deciles, constructs class templates

$\varepsilon_n$30

computes Bayesian posteriors

$\varepsilon_n$31

with $\varepsilon_n$32, and reorders events by the expectation value

$\varepsilon_n$33

iterating until the order stabilizes (Tomasik et al., 2016).

This methodology is not equivalent to conventional ESE. Conventional ESE selects on one chosen feature, such as $\varepsilon_n$34 or $\varepsilon_n$35; Event Shape Sorting uses the full histogram shape and can therefore retain multiple harmonics simultaneously (Tomasik et al., 2016). In toy-model data with dominant elliptic anisotropy, the sorting coordinate $\varepsilon_n$36 reproduced an ordering by $\varepsilon_n$37. In AMPT Pb+Pb events at $\varepsilon_n$38 TeV and $\varepsilon_n$39–$\varepsilon_n$40 centrality, however, only the upper deciles were strongly correlated with $\varepsilon_n$41; lower deciles exhibited richer structures that were not reducible to a single harmonic (Tomasik et al., 2016).

The femtoscopic development of Event Shape Sorting made this distinction explicit. In DRAGON and AMPT samples, sorting by the full azimuthal histogram allowed simultaneous observation of second- and third-order oscillations in HBT radii within the same event classes, a feature that standard ESE tends to suppress because it usually aligns events to a single harmonic plane (Cimerman et al., 2019). A later methodological discussion emphasized that ESS can reveal inter-harmonic correlations, including constructed $\varepsilon_n$42–$\varepsilon_n$43 patterns that are invisible to standard correlator averages and are not naturally targeted by $\varepsilon_n$44-based ESE (Tomasik et al., 2019).

Outside heavy-ion geometry engineering in nucleus–nucleus collisions, the phrase “event-shape selection” has also been used for topology-based analyses. In pp collisions at $\varepsilon_n$45 TeV, ALICE selected spherical events with $\varepsilon_n$46 and jet-like events with $\varepsilon_n$47, where

$\varepsilon_n$48

is built from the eigenvalues of a $\varepsilon_n$49 transverse-momentum tensor. Applied to pion and kaon femtoscopy, this selection showed that spherical events yield larger radii than jet-like events and that an approximate transverse-mass scaling of the radii is observed in all multiplicity ranges studied when the different Lorentz boosts of pions and kaons are taken into account (Collaboration, 2023). In a separate collider-phenomenology context, “event-shape selection cuts” were proposed for 7 TeV SUSY searches using transverse thrust $\varepsilon_n$50 and a jet-$\varepsilon_n$51 compactness ratio $\varepsilon_n$52, illustrating the terminological breadth of the phrase beyond heavy-ion flow engineering (Guchait et al., 2011).

6. Limitations, systematics, and methodological controversies

The basic ESS/ESE premise is approximate rather than exact. Even in the original heavy-ion formulation, successful selection required large pseudorapidity separation between the selection and analysis subevents, narrow centrality bins, and careful control of nonflow. When nonflow is present both within and across subevents, the coupling between selection and measurement can bias the engineered classes strongly. In the toy model used in the 2012 study, nonflow could produce substantial overestimation in high-$\varepsilon_n$53 selections and underestimation in low-$\varepsilon_n$54 selections, and the differences in inferred $\varepsilon_n$55 distributions could reach order-of-magnitude at large $\varepsilon_n$56 if $\varepsilon_n$57-distribution fits were interpreted without accounting for $\varepsilon_n$58 (Schukraft et al., 2012).

A more structural limitation comes from finite particle number and hadronic rescattering. In an event-by-event hybrid calculation based on UrQMD initial conditions, ideal hydrodynamics, and a UrQMD afterburner, the average relation $\varepsilon_n$59 was present for $\varepsilon_n$60, but the event-by-event scatter around the mean was very large. High-$\varepsilon_n$61 selections reduced the fraction of small $\varepsilon_n$62 events but still left a broad $\varepsilon_n$63 distribution; low-$\varepsilon_n$64 selections shifted the distribution toward smaller values but did not isolate $\varepsilon_n$65. In that study, $\varepsilon_n$66-based selections did not produce significant changes in the initial $\varepsilon_n$67 distributions (Petersen et al., 2013). This directly qualified the notion that final-state $\varepsilon_n$68 cuts can isolate sharply defined initial geometries in single events.

Event Shape Sorting carries its own statistical constraints. Split-subevent tests in DRAGON, AMPT, and uRQMD showed that the sorting can degrade when the reference and test particles are separated, and the problem is especially acute in low-statistics or strongly fluctuating transport outputs. In uRQMD, neighboring-bin differences in event histograms could reach about $\varepsilon_n$69, and the authors explicitly identified the discrimination of statistical fluctuations as an unresolved issue for ESS (Cimerman et al., 2019).

The most active current controversy concerns CME searches. One line of work argued that ESE, when defined through a reference-subevent $\varepsilon_n$70, satisfies the basic premise of extrapolating $\varepsilon_n$71 to $\varepsilon_n$72, albeit with limited statistical leverage, whereas ESS built from the same particles of interest is not practical because the selection variable, $\varepsilon_n$73, multiplicity, and the correlator are intertwined by statistical fluctuations (Li et al., 25 Sep 2025). A different line introduced corrected single-particle and pair-based $\varepsilon_n$74 definitions, concluded from toy-model and EBE-AVFD studies that the optimal strategy is a mixed construction using a pair-based event-shape variable together with single-particle $\varepsilon_n$75, and explicitly rejected resonance $\varepsilon_n$76 as a background regulator because it contains CME contributions (Xu et al., 2023). A subsequent STAR beam-energy study, using a POI/PPOI-based ESS together with spectator planes, reported $\varepsilon_n$77 consistent with zero and $\varepsilon_n$78 reduced to no more than $\varepsilon_n$79 of the inclusive $\varepsilon_n$80; in $\varepsilon_n$81–$\varepsilon_n$82 central Au+Au collisions it found residual charge-separation significances of $\varepsilon_n$83, $\varepsilon_n$84, and $\varepsilon_n$85 at $\varepsilon_n$86, $\varepsilon_n$87, and $\varepsilon_n$88 GeV, respectively (Collaboration, 30 May 2025). This suggests that CME-oriented ESS results are highly sensitive to the precise construction of the shape variable, the flow regressor, and the rapidity configuration.

Across its variants, ESS is therefore best understood not as a single immutable algorithm but as a methodological family. Its most established form—heavy-ion event-shape engineering via $\varepsilon_n$89 in independent subevents—has become a standard probe of the geometry-to-flow mapping. Its broader variants, especially full-histogram Event Shape Sorting and POI-based CME selections, enlarge the scope of the concept but also sharpen the demands on nonflow suppression, multiplicity, statistical robustness, and interpretation.