Transverse Spherocity in High-Energy Collisions

• Transverse spherocity is an event-shape observable defined in the transverse plane that distinguishes pencil-like, jet-dominated events (low values) from isotropic ones (high values).
• It serves as a complementary topology classifier to multiplicity and centrality, refining analyses in small systems and heavy-ion collisions by isolating hard-scattering from soft collective phenomena.
• The observable is applied in diverse studies including machine-learning estimation, resonance and heavy-flavor production analysis, and background suppression in chiral magnetic effect searches.

Transverse spherocity is an event-shape observable defined in the plane transverse to the beam and used to quantify whether the final-state momentum flow of a collision event is pencil-like and jet-dominated or broadly distributed and isotropic. Across proton-proton, proton-nucleus, and nucleus-nucleus studies, it serves as a topology classifier complementary to multiplicity and centrality, and in recent heavy-ion work it has also been used as a geometry-driven alternative to flow-vector-based event-shape engineering for isolating backgrounds and enhancing sensitivity to specific QCD phenomena such as the chiral magnetic effect (Cuautle et al., 2015, Mallick et al., 2020, Dey et al., 7 Apr 2026).

1. Formal definition and estimator variants

The standard transverse spherocity observable is defined by minimizing the transverse momentum flow perpendicular to a unit vector in the transverse plane,

$$S_0 = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\vec p_{T,i}\times \hat n|}{\sum_i p_{T,i}}\right)^2,$$

where the sum runs over the selected final-state particles, $\hat n$ is a unit vector in the transverse plane, and the normalization factor $\pi^2/4$ ensures $0 \le S_0 \le 1$ (Oliva et al., 2022). In the standard interpretation, $S_0 \to 0$ corresponds to a pencil-like or jetty event, while $S_0 \to 1$ corresponds to an isotropic event (Oliva et al., 2022).

Several studies emphasize that the observable is defined only in the transverse plane, making it insensitive to longitudinal boosts, and describe it as collinear- and infrared-safe or infrared and collinear safe (Cuautle et al., 2015, Mallick et al., 2021). Operationally, the event-by-event computation scans over candidate transverse directions and retains the axis that minimizes the normalized momentum component perpendicular to that axis (Cuautle et al., 2015).

A distinct variant is the unweighted transverse spherocity estimator used by ALICE in high-multiplicity $pp$ measurements,

$$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$

where each charged track contributes with unit transverse-momentum weight rather than its measured $p_T$ (Collaboration, 2023). This modification was introduced to reduce bias between neutral and charged strange-hadron observables and to suppress distortions from a single high-$p_T$ charged track (Collaboration, 2023). A related machine-learning study likewise treats unweighted spherocity, denoted $\hat n$0, as a target observable and reports that it is systematically easier to predict than the conventional weighted definition (Basak et al., 18 Aug 2025).

2. Topological meaning and classification logic

The principal physical role of transverse spherocity is to separate event classes according to the geometry of transverse momentum flow. Low-spherocity events are described as jetty, pencil-like, back-to-back, hard, or jet-dominated; high-spherocity events are described as isotropic, soft, bulk-like, MPI-rich, or more uniformly populated in azimuth (Cuautle et al., 2015, Ortiz et al., 2023). In heavy-ion applications, the same low-$\hat n$1/high-$\hat n$2 dichotomy is used to distinguish events with stronger jet-like structure from events dominated by soft collective bulk production (Mallick et al., 2021).

Most analyses implement spherocity selections through percentiles of the event-by-event distribution. Common choices are the lowest and highest $\hat n$3 of the distribution in AMPT- or PYTHIA-based studies, the $\hat n$4–$\hat n$5 and $\hat n$6–$\hat n$7 percentiles in ALICE-style $\hat n$8 analyses, and even more extreme $\hat n$9–$\pi^2/4$0 and $\pi^2/4$1–$\pi^2/4$2 selections in high-statistics topological studies (Mallick et al., 2020, Collaboration, 2019, Collaboration, 2023). The numerical cut values depend on system, multiplicity, and centrality class, and several papers explicitly show that these thresholds shift as event activity changes (Singh, 2022, Malik et al., 2022).

A recurring methodological point is that spherocity is not a neutral selector. In $\pi^2/4$3 studies, it is explicitly described as implicitly multiplicity dependent, and both multiplicity and spherocity are said to bias the event sample (Ortiz et al., 2023). High charged-particle multiplicity already biases the sample toward multijet final states, while spherocity further separates those events into classes enriched in hard-scattering contributions or in MPI and underlying-event activity (Ortiz et al., 2023). This is why several authors argue for double-differential analyses in multiplicity and spherocity rather than interpreting either variable in isolation (Cuautle et al., 2015, Ortiz et al., 2023).

Across multiple systems, the spherocity distribution is reported to move toward larger values as event activity increases. High-multiplicity $\pi^2/4$4 events are depleted at low $\pi^2/4$5 and contain more isotropic configurations on average (Cuautle et al., 2015). In Xe–Xe and Pb–Pb model studies, central events are more isotropic, while peripheral events are more jetty (Singh, 2022, Mallick et al., 2020). In $\pi^2/4$6 production with EPOS3+UrQMD, the distribution likewise shifts toward $\pi^2/4$7 from low to high multiplicity (Malik et al., 2022). This suggests that larger event activity often coincides with a more isotropic transverse topology, although the underlying mechanisms differ across systems and models.

3. Role in small-system phenomenology

In $\pi^2/4$8 collisions, transverse spherocity has become a practical tool for separating soft isotropic final states from hard jet-like topologies. In ALICE measurements at $\pi^2/4$9 TeV, the average transverse momentum $0 \le S_0 \le 1$0 at fixed multiplicity is higher for low-spherocity events and lower for high-spherocity events than in the spherocity-integrated sample; within uncertainties, however, the functional form of $0 \le S_0 \le 1$1 is not changed by the spherocity selection (Collaboration, 2019). This establishes spherocity as an event-shape refinement of multiplicity rather than a replacement for it.

The observable has also exposed model deficiencies that are not evident in minimum-bias comparisons. In a study of ALICE topological classes, PYTHIA 8 Monash shows a strong disagreement with data for jetty high-multiplicity events, including a “third rise” of $0 \le S_0 \le 1$2 at roughly $0 \le S_0 \le 1$3 that is not seen experimentally (Ortiz et al., 2023). That analysis attributes the discrepancy primarily to an overpredicted multijet yield rather than to color reconnection alone, and shows that a jet-removal or survival-probability procedure improves agreement mainly in the jetty class while leaving isotropic events essentially unaffected (Ortiz et al., 2023).

Spherocity dependence also reorganizes the chemistry of identified-particle production in $0 \le S_0 \le 1$4. In high-multiplicity ALICE data, strange-particle production is slightly higher for soft isotropic topologies and severely suppressed in hard jet-like topologies, with stronger effects for hadrons of larger mass and strangeness content when the topological selection is done within a narrow multiplicity interval (Collaboration, 2023). The same study argues that an important aspect of the universal scaling of strangeness enhancement with final-state multiplicity is that high-multiplicity collisions are dominated by soft, isotropic processes (Collaboration, 2023).

Flow-like signatures in small systems are likewise topology sensitive. A PYTHIA8-based study of pseudorapidity and spherocity dependence reports that isotropic events show enhanced radial-flow effects in all multiplicity classes, whereas jetty events show radial-flow-like effects only in high-multiplicity events (Radhakrishnan et al., 2023). In p+Pb, a comparison of PHSD and VISHNew finds that the PHSD spherocity distribution is slightly shifted toward the isotropic limit relative to the hydrodynamic result, and that the difference is only partly attributable to multiplicity distributions, indicating sensitivity to the different descriptions of nonequilibrium dynamics in the two frameworks (Oliva et al., 2022).

4. Heavy-ion event-shape engineering and collective observables

The first heavy-ion implementations of transverse spherocity in AMPT established that the observable can classify dense nuclear-collision events by topology, not only small systems (Mallick et al., 2020, Mallick et al., 2020). In Pb–Pb collisions at $0 \le S_0 \le 1$5 TeV and Xe–Xe collisions at $0 \le S_0 \le 1$6 TeV, isotropic events dominate low-$0 \le S_0 \le 1$7 particle production while jetty events dominate at higher $0 \le S_0 \le 1$8, with mass-ordered crossing points between the isotropic and jetty spectra (Mallick et al., 2020). The same studies report larger integrated yields in isotropic events and larger $0 \le S_0 \le 1$9 in jetty events, reflecting the different balance between soft multi-particle production and harder momentum transfer (Mallick et al., 2020).

For azimuthal anisotropy, the central result is a strong anti-correlation between spherocity and elliptic flow. In AMPT studies using the two-particle correlation method, high-$S_0 \to 0$0 events have nearly zero elliptic flow, whereas low-$S_0 \to 0$1 events contribute significantly to the elliptic flow of spherocity-integrated samples (Mallick et al., 2021, Mallick et al., 2020). This pattern is visible in momentum-space correlations, azimuthal correlation functions, and extracted $S_0 \to 0$2, and it persists across centrality classes, with the expected rise from central to mid-central collisions and a reduction toward peripheral collisions (Mallick et al., 2021).

Identified-particle analyses sharpen this picture. In Pb+Pb AMPT simulations, low-$S_0 \to 0$3 events show larger $S_0 \to 0$4 for pions, kaons, and protons than spherocity-integrated events, whereas high-$S_0 \to 0$5 events show almost no elliptic flow (Mallick et al., 2021). The same work reports that number-of-constituent-quark scaling and transverse-kinetic-energy scaling are violated more strongly in low-$S_0 \to 0$6 events, which is interpreted as confirming the fragmentation-based hadronization mechanism for high-momentum partons in jetty-like events (Mallick et al., 2021).

Spherocity has also been confronted directly with flow-vector-based event-shape observables. In AMPT event-shape-engineering studies, $S_0 \to 0$7 is strongly anti-correlated with the reduced flow vector $S_0 \to 0$8, while $S_0 \to 0$9 shows only a mild positive correlation (Prasad et al., 2022). Low-$S_0 \to 1$0 events have the largest $S_0 \to 1$1, high-$S_0 \to 1$2 events the largest $S_0 \to 1$3, and high-$S_0 \to 1$4 events can even satisfy $S_0 \to 1$5 over all centralities (Prasad et al., 2022). The same analysis finds spherocity dependence not only in final-state flow coefficients but also in the initial participant eccentricity $S_0 \to 1$6, indicating that a final-state topology classifier can still isolate subsets of events with distinct initial geometry (Prasad et al., 2022).

5. Resonances, heavy flavor, and topology-dependent production channels

Resonance production studies use transverse spherocity to separate hard and soft production environments and to probe hadronic-phase sensitivity. In $S_0 \to 1$7 collisions at $S_0 \to 1$8 TeV within EPOS3+UrQMD, $S_0 \to 1$9 production is reported to be dominated by isotropic event shapes in both high- and low-multiplicity intervals, even though low-multiplicity events are more jetty in the event-shape distribution itself (Malik et al., 2022). That study interprets the dominance of isotropic events as consistent with soft-QCD and MPI-rich production environments together with hadronic rescattering modeled by UrQMD (Malik et al., 2022).

Heavy-flavor observables show a more differentiated topology dependence. In PYTHIA8 studies of $pp$0, $pp$1, and $pp$2, isotropic events dominate low $pp$3 while jetty events dominate higher $pp$4, and both relative yield and relative mean $pp$5 increase with multiplicity and are larger in jetty than in isotropic events (Deb et al., 2020). The baryon-to-meson ratio $pp$6 is higher in jetty events than in isotropic events for minimum-bias $pp$7, whereas the light-flavor ratio $pp$8 is higher in isotropic events for $pp$9 GeV/$$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$0, pointing to different topology sensitivities for open-charm and light-flavor hadronization channels (Deb et al., 2020).

A later topological study of prompt and nonprompt $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$1 production in PYTHIA8 likewise uses transverse spherocity as a proxy for MPI activity. It reports that both prompt and nonprompt $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$2 yields are generally enhanced in isotropic events and suppressed in jetty events, with stronger spherocity dependence at midrapidity and weaker dependence at forward rapidity, where self-correlation bias is smaller (Radhakrishnan et al., 27 Nov 2025). The same work ties low $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$3 to fewer MPI and stronger dijet dominance, and high $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$4 to larger $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$5, softer underlying-event activity, and more isotropic topology (Radhakrishnan et al., 27 Nov 2025).

Taken together, these studies suggest that transverse spherocity is sensitive not only to global event geometry but also to the channel dependence of hadron production. Resonances, open heavy flavor, charmonia, and strange hadrons do not respond identically to the same topology cut, and the observable is therefore useful for separating production mechanisms that are mixed in spherocity-integrated samples.

6. Data-driven estimation and recent CME applications

A major practical development is the use of supervised regression to infer transverse spherocity from experimentally accessible final-state observables. In Pb–Pb collisions at $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$6 TeV, Gradient Boosting Decision Trees implemented through GradientBoostingRegressor in sklearn.ensemble were trained on AMPT events using $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$7, $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$8, and $$S_{\mathrm O} = \frac{\pi^2}{4}\min_{\hat n}\left(\frac{\sum_i |\hat p_{T,i}\times \hat n|}{N_{\mathrm{trks}}}\right)^2,$$9 as inputs, and were found to reproduce the true centrality-dependent spherocity distributions well, with deviations concentrated in the low-spherocity region where jetty events are statistically rarer (Mallick et al., 2021, Mishra et al., 2021). Those studies also report successful transfer from 5.02 TeV to 2.76 TeV and reasonable performance in PYTHIA8 Angantyr, supporting the idea that event-shape information can be reconstructed from measurable final-state quantities (Mallick et al., 2021).

A more recent data-driven study extends this program by predicting both conventional $p_T$0 and unweighted $p_T$1 directly from raw event data in minimum-bias Au+Au collisions at $p_T$2 GeV (Basak et al., 18 Aug 2025). Using $p_T$3, $p_T$4, and $p_T$5 as features, it compares Polynomial Regression with Ridge, Decision Tree Regressor, Extra-Trees Regressor, K-Nearest Neighbors, Light Gradient Boosting Machine, and Multi-layer Perceptron, with hyperparameters optimized by GridSearchCV (Basak et al., 18 Aug 2025). Light Gradient Boosting Machine is identified as the best overall model; for $p_T$6 it reaches $p_T$7, $p_T$8, and $p_T$9, while for $p_T$0 it reaches $p_T$1, $p_T$2, and $p_T$3 (Basak et al., 18 Aug 2025). The same paper reports only moderate degradation when the reaction-plane angle is unavailable and moderate performance loss in cross-generator tests on UrQMD, PYTHIA8 Angantyr, and EPOS4, which it interprets as reduced model dependence (Basak et al., 18 Aug 2025).

The most recent application in the supplied literature uses transverse spherocity as a classifier for chiral magnetic effect searches in Pb+Pb collisions at $p_T$4 TeV with AMPT and a realistic CME implementation (Dey et al., 7 Apr 2026). That study contrasts spherocity with traditional event-shape engineering based on the flow vector and argues that the latter is contaminated by the same backgrounds—resonance decays, local charge conservation coupled to flow, and jet-related correlations—that it is intended to suppress (Dey et al., 7 Apr 2026). By contrast, spherocity is presented as a cleaner, geometry-driven classifier because it depends on the transverse momentum topology rather than on the flow coefficient itself (Dey et al., 7 Apr 2026). In the simulation, adding CME shifts the spherocity distribution toward higher $p_T$5, $p_T$6 and correlated background coupled with elliptic flow are higher in jetty events, and the scaled ratio $p_T$7 is enhanced in isotropic events, with the strongest separation for the tightest $p_T$8 cut (Dey et al., 7 Apr 2026). This establishes transverse spherocity as both a topology observable and a background-suppression tool in precision heavy-ion searches.

The resulting picture is methodologically consistent. Transverse spherocity is now used not only to label events as jetty or isotropic, but also to structure multi-differential analyses, test Monte Carlo dynamics, estimate hidden event properties with machine learning, and construct cleaner signal environments in heavy-ion measurements. A plausible implication is that topology-based classification has become one of the more technically flexible ways to separate hard, soft, collective, and background-rich components of relativistic collision data.