Event-Shape Engineering (ESE) in Heavy-Ion Collisions

• Event-Shape Engineering is an analysis strategy that classifies heavy-ion collision events by their anisotropic flow, using the reduced flow vector qₙ as a key observable.
• It isolates fluctuations in the initial density profile from global collision conditions by selecting events within narrow centrality intervals.
• ESE has practical applications in probing heavy-flavor transport, extracting CME signals, and studying jet quenching by minimizing nonflow effects with separated subevents.

Searching arXiv for Event-Shape Engineering and related key papers to ground the article in the literature. arxiv_search: Event Shape Engineering heavy-ion collisions q2 ALICE

Event-Shape Engineering (ESE) is an analysis strategy in relativistic heavy-ion collisions in which events belonging to the same centrality class are further selected according to an event-by-event measure of anisotropic shape, usually the reduced flow vector $q_n$, so that fluctuations of the initial geometry can be converted into controlled classes of events with systematically different final-state anisotropic flow. In the formulation introduced for ultra-relativistic nuclear collisions, ESE adds a third handle beyond centrality and system choice: at fixed centrality, it exploits fluctuations in the initial density profile to isolate events that are, for example, more elliptic or less triangular than average (Schukraft et al., 2012). The method was established experimentally by ALICE in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV, where selecting extreme $q_2$ classes produced $v_2$ values significantly larger or smaller than the unbiased average, directly demonstrating that one can engineer event samples with different effective initial spatial asymmetry (Dobrin, 2012).

1. Conceptual basis

ESE is rooted in the standard picture that anisotropic flow is the collective response of the QCD medium to the fluctuating spatial asymmetry of the initial overlap region. In the original proposal, the final azimuthal distribution is expanded as

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

while the initial-state anisotropy is characterized by participant eccentricities such as

$$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$

The central assumption used in the proposal is that flow is approximately proportional to eccentricity, $v_n \propto \epsilon_n$, so selecting events by a flow-related observable in the final state preferentially selects events with different initial eccentricities (Schukraft et al., 2012).

This construction addresses a specific limitation of centrality-based comparisons. Changing centrality changes not only average geometry but also multiplicity, density, lifetime, and system size. ESE, by contrast, is performed within a narrow centrality interval and is therefore designed to vary anisotropic shape while approximately keeping global collision conditions similar. This makes it a direct probe of the mapping from initial-state fluctuations to final-state flow and a tool for separating geometry-driven effects from phenomena that should not track the same eccentricity dependence (Dobrin, 2012).

The same logic extends beyond elliptic flow. The original formulation explicitly emphasized both $n=2$ and $n=3$, noting that fluctuations generate odd harmonics such as triangular flow, and that ESE can therefore be used to isolate events with unusually small or large triangularity as well as ellipticity (Schukraft et al., 2012).

2. Event-shape observables and analysis architecture

The standard ESE classifier is the reduced flow vector. For a subevent with multiplicity $M$,

$\sqrt{s_{NN}}=2.76$0

with magnitude

$\sqrt{s_{NN}}=2.76$1

and reduced vector

$\sqrt{s_{NN}}=2.76$2

A useful exact relation in the original proposal is

$\sqrt{s_{NN}}=2.76$3

which makes explicit that $\sqrt{s_{NN}}=2.76$4 is tied to the harmonic pair correlation within the event (Schukraft et al., 2012).

In practice, ESE requires more than the definition of $\sqrt{s_{NN}}=2.76$5. The proposal emphasized a two-subevent structure: one subevent is used to classify the event by $\sqrt{s_{NN}}=2.76$6, and another is used for the physics measurement. The purpose is to suppress trivial self-correlations and reduce nonflow. The same principle was implemented experimentally by ALICE through a three-subevent technique in which subevent “a” defined $\sqrt{s_{NN}}=2.76$7, subevent “b” contained the measured particles, and subevent “c” determined the event plane $\sqrt{s_{NN}}=2.76$8. Two configurations were tested: a TPC-based selection and a VZERO-based selection, with the latter preferred because the large pseudorapidity gap between the selection detector and the measurement region strongly suppresses nonflow (Dobrin, 2012).

A further methodological requirement is centrality control. Both the proposal and later ALICE analyses stressed that event-shape selection must be performed inside narrow centrality intervals to avoid turning an ESE cut into a disguised multiplicity or centrality cut. In the heavy-flavour implementation, for example, $\sqrt{s_{NN}}=2.76$9 percentiles were determined in 1%-wide centrality slices and then merged back into wider physics classes, specifically to suppress trivial distortions of the multiplicity distribution (Collaboration, 2020).

The original proposal also connected ESE to fluctuation modeling through a Bessel-Gaussian description of the event-by-event $q_2$0 and $q_2$1 distributions. In that framework,

$q_2$2

and for sufficiently large multiplicity the $q_2$3 distribution inherits the same functional form, providing an operational bridge between the experimentally measurable $q_2$4 distribution and the underlying flow fluctuations (Schukraft et al., 2012).

3. Experimental establishment in Pb–Pb collisions

The first direct experimental demonstration came from ALICE in Pb–Pb collisions at $q_2$5 TeV, using about $q_2$6 minimum-bias events from the 2010 run. The analysis measured $q_2$7 of unidentified charged particles, pions, and (anti-)protons at midrapidity, and compared the unbiased sample with the 10% lowest-$q_2$8 tail and the 5% highest-$q_2$9 tail. The outcome was the defining proof of principle: for the 5% highest-$v_2$0 events, $v_2$1 is larger than in the unbiased sample; for the 10% lowest-$v_2$2 events, $v_2$3 is smaller; the measured $v_2$4 becomes “significantly larger or smaller than the average” (Dobrin, 2012).

That study also established the main methodological caveat of experimental ESE: nonflow can imitate an event-shape effect if the selection and measurement regions are too close. ALICE found that results using $v_2$5 from VZERO-C and VZERO-A were mutually consistent, whereas results using TPC $v_2$6 differed significantly, and attributed this mainly to large nonflow contributions. For that reason the differential physics results were presented only for the VZERO-A-based selection, which gave the strongest nonflow suppression (Dobrin, 2012).

The $v_2$7-differential results provided a second important result. In the 30–40% centrality class, the $v_2$8 curves for high-$v_2$9, low-$$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$0, and unbiased events were essentially rescaled versions of one another over a broad momentum interval. Specifically, the ratio of selected to unbiased $$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$1 was approximately flat up to about $$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$2. ALICE interpreted this as evidence that the event-shape selection modifies the overall anisotropy scale in a way that is largely common across soft and intermediate $$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$3, consistent with a collective response to fluctuating initial eccentricity (Dobrin, 2012).

A more extensive ALICE study then applied ESE to inclusive spectra and identified $$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$4, K, and p spectra at the same collision energy. It again found that the effect on $$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$5 is almost independent of $$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$6, but also showed that charged hadron, pion, kaon, and proton transverse-momentum distributions are harder in events with higher-than-average elliptic flow. In the 30–40% class, a blast-wave analysis gave

$$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$7

for the high-$$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$8 sample and

$$E\frac{d^3 N}{d^3 p} = \frac{1}{2\pi} \frac{d^2 N}{p_{T} dp_{T} dy}\left(1+\sum_{n=1}^{\infty}2v_n \cos[n(\phi-\Psi_n)] \right),$$9

for the low-$$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$0 sample, supporting an interplay between radial and elliptic flow and indicating that fluctuations in elliptic geometry are correlated with fluctuations in radial expansion (Collaboration, 2015).

4. Heavy flavour, charm transport, and small systems

Heavy flavour has become one of the most technically significant extensions of ESE because charm and beauty quarks probe the medium throughout its evolution. ALICE first applied ESE to prompt D mesons in mid-central Pb–Pb collisions at $$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$1 TeV, measuring $$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$2 for $$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$3 and $$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$4 in 10–30% and 30–50% centrality classes. Using $$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$5-selected classes defined from the 20% largest-$$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$6 and 60% smallest-$$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$7 events, the collaboration found that D-meson $$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$8 is enhanced in large-$$\epsilon_{n,p}=\sqrt{\epsilon_{n,x}^2+\epsilon_{n,y}^2},\qquad \tan(n\Psi_n)=\frac{\epsilon_{n,y}}{\epsilon_{n,x}}.$$9 events and reduced in small-$v_n \propto \epsilon_n$0 events, with about a $v_n \propto \epsilon_n$1 significance for the large-versus-small difference in each centrality class, while the per-event D-meson yields remain compatible with no modification (Collaboration, 2018).

The later ALICE heavy-flavour analysis extended this program to $v_n \propto \epsilon_n$2 and $v_n \propto \epsilon_n$3 collisions and reported that the D-meson $v_n \propto \epsilon_n$4 was found “on average about 50% higher (lower) in the 20% of the events with largest (smallest) $v_n \propto \epsilon_n$5” in both centrality classes. Within uncertainties, the modification was independent of $v_n \propto \epsilon_n$6, whereas the ratios of per-event D-meson yields in the ESE-selected and unbiased samples were compatible with unity in $v_n \propto \epsilon_n$7 GeV/$v_n \propto \epsilon_n$8. The paper concluded that charm anisotropy follows the event-shape selection but the spectra do not show a significant ESE dependence at current precision (Collaboration, 2020).

Model studies formalized the same point. In POWLANG, ESE was implemented by selecting on initial eccentricity rather than a detector-level $v_n \propto \epsilon_n$9, and the outcome was that heavy-flavour $n=2$0 and $n=2$1 are strongly modified by shape selection whereas $n=2$2 changes only mildly. The normalized ESE response $n=2$3 was found to be similar for pions, charm hadrons, and beauty hadrons, suggesting that the ratio is controlled mainly by the initial geometric deformation rather than by fine details of the transport coefficients (Beraudo et al., 2018). An event-by-event transport study using a quasi-particle approach reached a similar conclusion, reporting that $n=2$4-meson $n=2$5 changes by about $n=2$6 between selected and unbiased samples and interpreting this as confirmation of strong charm-light coupling in QCD matter (Sambataro et al., 2022).

Recent work has used ESE to push beyond inclusive heavy-flavour flow and into hadronization dynamics. A 2026 study of $n=2$7 and $n=2$8 under ESE argued that sequential charm hadronization predicts a positive

$n=2$9

that grows systematically with $n=3$0, together with a response-slope hierarchy $n=3$1, while the simultaneous baseline gives a splitting near zero or negative and no comparable geometry scaling. In the same work, the $n=3$2 ratios of the $n=3$3 yield ratio remained close to unity, supporting the interpretation that the effect is dynamical and not chemical (Huang et al., 2 Jun 2026). CMS extended the ESE program for prompt $n=3$4 mesons over $n=3$5 GeV/$n=3$6 and 0–50% centrality, finding a universal linear trend between normalized prompt-$n=3$7 $n=3$8 and normalized charged-particle $n=3$9, with slopes consistent with unity and intercepts consistent with zero; this was interpreted as evidence that initial geometry is the dominant source of charm-hadron elliptic flow (Chandra, 25 Dec 2025).

In small systems, the situation is more delicate. An EPOS3 study of p–Pb collisions at 5.02 TeV used $M$0 and $M$1 selections and found that, after rejecting jetty events, charged-particle spectra were essentially insensitive to event-shape selection, while $M$2 showed only a modest response: about $M$3 for large-$M$4 and about $M$5 for small-$M$6, with no noticeable difference for $M$7. The paper therefore described the result as a hint of event-shape-induced modification rather than a decisive small-system analogue of the heavy-ion ESE signal (Kar et al., 2020).

5. Diagnostic applications: CME, CMW, and hard probes

From the outset, ESE was proposed not only as a bulk-flow tool but also as a way to vary flow-driven backgrounds in rare-observable searches. The original proposal explicitly identified the chiral magnetic effect (CME) and azimuthally sensitive femtoscopy as natural applications, and later work argued that ESE could also provide an additional experimental handle on jet quenching by varying path lengths at fixed centrality (Schukraft et al., 2012).

In the chiral magnetic wave (CMW) context, ESE was proposed as a discriminator between a flow-driven background, especially local charge conservation (LCC), and a genuine CMW-like component. The specific idea was to study the slope of

$M$8

and the corresponding three-particle correlator as functions of the average $M$9 across $\sqrt{s_{NN}}=2.76$00 classes. In the BW+LCC background model these observables scale approximately linearly with $\sqrt{s_{NN}}=2.76$01 and extrapolate to zero at $\sqrt{s_{NN}}=2.76$02, whereas in AMPT with an imposed electric quadrupole they are approximately flat and yield a positive intercept. The paper suggested a decomposition with slope $\sqrt{s_{NN}}=2.76$03 and intercept $\sqrt{s_{NN}}=2.76$04, and defined an inferred CMW fraction

$\sqrt{s_{NN}}=2.76$05

thereby extending ESE from CME-style background control to CMW-specific signal extraction (Wang et al., 2021).

The CME literature has sharpened the methodological distinction between valid and invalid event-shape extrapolations. A study of $\sqrt{s_{NN}}=2.76$06, $\sqrt{s_{NN}}=2.76$07, and $\sqrt{s_{NN}}=2.76$08 argued that a modified ESE procedure can project the CME-sensitive correlator to a near-zero-flow class, strongly suppressing elliptic-flow background while retaining sensitivity to a genuine CME signal. In AMPT, the zero-flow intercepts for $\sqrt{s_{NN}}=2.76$09 and $\sqrt{s_{NN}}=2.76$10 are consistent with zero or slightly below; in EBE-AVFD, the method strongly suppresses the background but leaves a residual model-dependent intercept, while still increasing the CME fraction substantially (Milton et al., 2021). A broader comparative study then concluded that ESE, because it bins dynamical fluctuations of $\sqrt{s_{NN}}=2.76$11, fulfills the general premise of a CME measurement but is statistically hungry, whereas event-shape selection based on the same particles of interest is dominated by statistical fluctuations and intertwined variables and is not practically useful as a clean CME observable (Li et al., 2024). A later focused review reiterated the same conclusion: ESE is defensible but statistically hungry; ESS is not practical for CME measurement because the relevant variables are intertwined (Li et al., 25 Sep 2025).

The most stringent ESE-based CME constraint so far is the 2026 ALICE Pb–Pb measurement at $\sqrt{s_{NN}}=2.76$12 TeV. In that analysis, $\sqrt{s_{NN}}=2.76$13 was built from V0C, the event plane from V0A, and the final-state charged particles from the TPC, giving a deliberately separated three-detector configuration. ESE changed $\sqrt{s_{NN}}=2.76$14 by as much as 30% relative to unbiased events, and $\sqrt{s_{NN}}=2.76$15 followed this variation linearly. Using MC Glauber and T$\sqrt{s_{NN}}=2.76$16ENTo initial-state models to estimate the expected ESE dependence of a genuine CME signal, ALICE extracted

$\sqrt{s_{NN}}=2.76$17

for MC Glauber and

$\sqrt{s_{NN}}=2.76$18

for T$\sqrt{s_{NN}}=2.76$19ENTo, corresponding to upper limits of 7% and 6% at 95% confidence level for 5–60% centrality (Collaboration, 26 Feb 2026).

For hard probes, the conceptual jet-quenching proposal was that ESE can vary the in-plane and out-of-plane path lengths while keeping the medium properties fixed much more closely than a centrality scan can (Christiansen, 2016). A later Trajectum study quantified this geometry control and found that ESE alone does not meaningfully alter angle-averaged hard-probe path lengths, but that ESE combined with in-plane versus out-of-plane selection does. Using a weighted path-length proxy

$\sqrt{s_{NN}}=2.76$20

the study showed that the ratio of out-of-plane to in-plane mean path length can reach about 2.5 in peripheral high-$\sqrt{s_{NN}}=2.76$21 classes, whereas no analogous effect was found for $\sqrt{s_{NN}}=2.76$22. The conclusion was that ESE is not a direct path-length selector, but becomes an effective geometry-control tool for hard probes when tied to event-plane orientation (Beattie et al., 2022).

6. Limitations, methodological debates, and related methods

The main limitations of ESE were already identified in the original proposal. The measured $\sqrt{s_{NN}}=2.76$23 fluctuates because of finite multiplicity as well as true flow fluctuations, so shape selection has finite resolution. Larger multiplicity improves the discriminating power, and narrower subevents degrade it. Nonflow is the major caveat: if it exists both within and between subevents, high-$\sqrt{s_{NN}}=2.76$24 selected events can overestimate the true flow and low-$\sqrt{s_{NN}}=2.76$25 selected events can underestimate it, with potentially severe distortions in the inferred tails of the $\sqrt{s_{NN}}=2.76$26 distribution (Schukraft et al., 2012).

These concerns were confirmed experimentally. ALICE showed that TPC-based $\sqrt{s_{NN}}=2.76$27 selection can be contaminated by nonflow, whereas VZERO-based selection with a large pseudorapidity gap yields mutually consistent results between VZERO-A and VZERO-C and is therefore methodologically preferred (Dobrin, 2012). Later D-meson studies found that TPC-based $\sqrt{s_{NN}}=2.76$28 gives stronger selectivity but that the comparison with V0A-based $\sqrt{s_{NN}}=2.76$29 did not permit a firm conclusion about the exact magnitude of possible residual nonflow contamination; finite residual bias could not be excluded beyond the assigned systematic uncertainty (Collaboration, 2020). Peripheral collisions also remain intrinsically less favorable because both multiplicity and flow magnitude are smaller, reducing the power of the event-shape selection (Dobrin, 2012).

A separate methodological debate concerns the relation between ESE and Event Shape Sorting (ESS). ESS was introduced as a histogram-level, iterative sorting method that groups events with similar full azimuthal distributions rather than selecting on a predefined scalar such as $\sqrt{s_{NN}}=2.76$30. In femtoscopic applications, ESS was shown to reveal simultaneous second- and third-order structure and even hidden $\sqrt{s_{NN}}=2.76$31-$\sqrt{s_{NN}}=2.76$32 correlations that standard $\sqrt{s_{NN}}=2.76$33-based ESE would not isolate unless the relevant sorting variable were known in advance (Cimerman et al., 2019). A later perspective stressed the same contrast: ESE is hypothesis-driven and controlled, while ESS is discovery-oriented and may uncover unexpected dominant features of the full event-shape histogram (Tomasik et al., 2019).

That broader discovery role does not imply that ESS is interchangeable with ESE in signal-extraction problems. In CME studies, the distinction is sharper: ESE uses particles in a different phase-space region from the particles of interest and therefore bins dynamical fluctuations of $\sqrt{s_{NN}}=2.76$34; ESS uses the same particles to define the shape and to measure the observable, so the resulting intercept inherits statistical and nonflow correlations that cannot be cleanly removed. On that point the recent literature is explicit: ESE may be statistically hungry, but ESS is not practically useful as a clean zero-$\sqrt{s_{NN}}=2.76$35 CME observable (Li et al., 2024).

Taken together, these developments define ESE as an experimentally controlled way of turning event-by-event fluctuations from a nuisance into a precision tool. Its successful use depends on narrow centrality control, separated subevents, large pseudorapidity gaps where possible, and careful treatment of finite-multiplicity resolution. Within those constraints, it has evolved from a proposal about initial-geometry fluctuations into a general analysis framework for bulk-flow systematics, heavy-flavour transport, chiral-magnetic searches, and hard-probe geometry engineering (Schukraft et al., 2012).