Physics:Sphericity (particle physics)

Short description: Event-shape observable in particle physics

Sphericity is an event shape observable used in particle physics to characterize the geometric distribution of final-state particle momenta produced in high-energy collisions. It provides a quantitative measure of how isotropically momentum is distributed in an event, and is used to distinguish between different underlying processes such as two-jet, multi-jet, or approximately isotropic hadronic final states.

Sphericity is especially associated with analyses in electron–positron annihilation, hadron collider physics, and studies of quantum chromodynamics (QCD), where it serves as a tool for identifying event topology and testing models of parton radiation and hadronization.

Definition

Sphericity is constructed from the momenta of particles in an event. A commonly used definition begins with the sphericity tensor

$S^{\alpha \beta }=\frac{\sum _i{p}_i^{\alpha }{p}_i^{\beta }}{\sum _i|{\mathbf{𝐩}}_i{|}^2},$

where:

• $i$ runs over all particles in the event,
• $p_i^{\alpha }$ and $p_i^{\beta }$ are Cartesian momentum components of particle $i$,
• $\alpha ,\beta \in \{x,y,z\}$,
• ${\mathbf{𝐩}}_i$ is the three-momentum of particle $i$.

This tensor is real, symmetric, and normalized so that its eigenvalues ${\lambda }_1,{\lambda }_2,{\lambda }_3$ satisfy

${\lambda }_1+{\lambda }_2+{\lambda }_3=1,$

with the conventional ordering

${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3.$

The sphericity is then defined as

$S=\frac{3}{2}({\lambda }_2+{\lambda }_3).$

Its value lies in the range

$0\le S\le 1.$

• $S\approx 0$ corresponds to highly collimated, jet-like events.
• $S\approx 1$ corresponds to nearly isotropic events.

Physical interpretation

Sphericity measures the extent to which an event differs from a pencil-like configuration. In an idealized two-jet event, all particle momenta are aligned along a single axis, giving one large eigenvalue and two small ones; the resulting sphericity is close to zero. In contrast, if momentum is distributed roughly uniformly in all directions, the eigenvalues become similar and sphericity approaches one.

Because it depends quadratically on momenta, sphericity is particularly sensitive to high-momentum particles. This distinguishes it from some other event-shape variables that are linear in momentum.

Related quantities

Aplanarity

Aplanarity measures the degree to which an event is non-planar:

$A=\frac{3}{2}{\lambda }_3.$

It ranges from 0 for planar events to $\frac{1}{2}$ for maximally isotropic events.

Planarity

A quantity sometimes called planarity is defined by

$P={\lambda }_2-{\lambda }_3.$

It describes how much the event is spread within a plane.

Linearized sphericity tensor

Since the standard sphericity tensor is quadratic in momentum and therefore not infrared safe in perturbative QCD, analyses sometimes use a linearized momentum tensor

$S_{\text{lin}}^{\alpha \beta }=\frac{\sum _i\frac{p_i^{\alpha }{p}_i^{\beta }}{|{\mathbf{𝐩}}_i|}}{\sum _i|{\mathbf{𝐩}}_i|},$

or related event-shape observables that are less sensitive to soft and collinear effects.

Use in experiments

Sphericity became widely used in early collider experiments as a simple and intuitive classifier of event topology. It has been employed in:

• identifying jet structure in $e^{+}{e}^{-}$ annihilation,
• distinguishing signal from background in hadron collider analyses,
• studying hadronic decays of heavy particles,
• characterizing global event properties in searches for new physics.

In modern analyses, sphericity is often used together with observables such as thrust, Physics:Fox–Wolfram moments, Physics:C-parameter, and jet broadening.

Theoretical aspects

While useful phenomenologically, sphericity has limitations from the perspective of perturbative QCD. The standard definition is not infrared and collinear safe, because soft emission or collinear splitting can affect the quadratic momentum weighting in problematic ways. For precision theoretical comparisons, observables with better perturbative properties are often preferred.

Nevertheless, sphericity remains valuable as a descriptive variable in detector-level and reconstructed-event analyses.

Historical context

Event-shape variables such as sphericity played an important role in the development of experimental particle physics during the 1970s and 1980s. They helped establish the jet structure predicted by QCD and provided evidence for the partonic nature of hadronic final states. Before modern jet-clustering algorithms became standard, sphericity and related quantities were among the primary tools for classifying collision events.

See also

• Physics:Thrust (particle physics)
• Physics:Jet (particle physics)
• Physics:Quantum chromodynamics
• Physics:Fox–Wolfram moments

References

• R. K. Ellis, W. J. Stirling, and B. R. Webber, QCD and Collider Physics, Cambridge University Press.
• Physics:Particle Data Group, reviews on kinematics and event-shape observables.
• G. Hanson et al., early experimental analyses of jet structure in $e^{+}{e}^{-}$ collisions.

External links

• Particle Data Group
• CERN Open Data portal