Physics:C-parameter

Short description: Event-shape observable in particle physics

The C-parameter is an event-shape observable used in high-energy particle physics to characterize the geometric distribution of particles produced in a collision event. It is primarily used in studies of Quantum chromodynamics (QCD), particularly in electron–positron annihilation experiments, where it provides information about the topology of hadronic final states.

The C-parameter is an infrared- and collinear-safe observable and has been extensively employed in precision measurements of the strong coupling constant ${\alpha }_s$.

Definition

The C-parameter is defined in terms of the linearized momentum tensor

${\mathrm{\Theta }}^{\alpha \beta }=\frac{\sum _i\frac{p_i^{\alpha }{p}_i^{\beta }}{|{\mathbf{𝐩}}_i|}}{\sum _i|{\mathbf{𝐩}}_i|},$

where

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

The tensor is symmetric and has eigenvalues ${\lambda }_1$, ${\lambda }_2$, and ${\lambda }_3$, satisfying

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

The C-parameter is then defined as

$C=3({\lambda }_1{\lambda }_2+{\lambda }_2{\lambda }_3+{\lambda }_3{\lambda }_1).$

An equivalent expression is

$C=\frac{3}{2}\frac{\sum _{i,j}|{\mathbf{𝐩}}_i|,|\mathbf{𝐩}*j|,{\sin }^2\theta *ij}{{(\sum _k|{\mathbf{𝐩}}_k|)}^2},$

where ${\theta }_{ij}$ is the angle between particles $i$ and $j$.

Properties

The C-parameter is bounded by

$0\le C\le 1.$

Important limiting cases include:

• $C=0$ for an ideal two-particle back-to-back event.
• $C=\frac{3}{4}$ for a perfectly symmetric three-jet event in a plane.
• $C=1$ for a completely isotropic event.

Because it is constructed from all particles in the event, the C-parameter is sensitive to the global geometry of the final state.

Physical interpretation

The C-parameter quantifies the degree to which an event departs from a simple two-jet topology.

Small values of $C$ correspond to highly collimated events dominated by two nearly back-to-back jets. Larger values indicate increasingly complex energy flow, including multi-jet configurations and isotropic particle distributions.

Unlike observables based on a preferred event axis, the C-parameter is rotationally invariant and depends only on pairwise angular correlations among particles.

Applications

Tests of quantum chromodynamics

The C-parameter is one of the standard event-shape observables used to test perturbative QCD predictions. Measurements of its distribution have been performed by numerous electron–positron collider experiments, including those operating at LEP energies.

Determination of the strong coupling constant

The distribution of the C-parameter can be calculated using perturbative QCD supplemented by resummation techniques and non-perturbative corrections. Comparisons between theoretical predictions and experimental measurements have provided precise determinations of the strong coupling constant ${\alpha }_s$.

Event-shape analyses

The C-parameter is often analyzed alongside other event-shape observables such as thrust, jet broadening, heavy-jet mass, and sphericity. Together these quantities provide complementary information about the structure of hadronic final states.

Relation to other event-shape observables

The C-parameter belongs to a class of global event-shape observables designed to characterize energy flow in particle collisions.

Compared with thrust, which measures alignment along a preferred axis, the C-parameter depends on the eigenvalues of the momentum tensor and is therefore sensitive to the overall three-dimensional geometry of the event.

The observable is closely related to the D-parameter, which is defined by

$D=27{\lambda }_1{\lambda }_2{\lambda }_3.$

While the C-parameter is sensitive to deviations from a two-jet topology, the D-parameter provides additional sensitivity to genuinely three-dimensional event structure.

See also

• Event shape
• Thrust (particle physics)
• Sphericity tensor
• Jet broadening
• Heavy jet mass
• Quantum chromodynamics
• Strong coupling constant

Further reading

• Parisi, G.; Petronzio, R. (1979). "Small Transverse Momentum Distributions in Hard Processes". Nuclear Physics B 154 (3): 427–440. doi:10.1016/0550-3213(79)90040-3. 

• Donoghue, J. F.; Low, F. E.; Pi, S.-Y. (1979). "Tensor Analysis of Hadronic Jets in Quantum Chromodynamics". Physical Review D 20 (11): 2759–2766. doi:10.1103/PhysRevD.20.2759. 

• Ellis, R. K.; Stirling, W. J.; Webber, B. R. (1996). QCD and Collider Physics. Cambridge University Press.