Rivet  1.8.0
Public Member Functions | Protected Member Functions
Rivet::Sphericity Class Reference

Calculate the sphericity event shape. More...

#include <Sphericity.hh>

Inheritance diagram for Rivet::Sphericity:
Rivet::AxesDefinition Rivet::Projection Rivet::ProjectionApplier

List of all members.

Public Member Functions

void clear ()
 Reset the projection.
Constructors etc.
 Sphericity (const FinalState &fsp, double rparam=2.0)
 Constructor.
virtual const Projectionclone () const
 Clone on the heap.
Access the event shapes by name
double sphericity () const
double transSphericity () const
 Transverse Sphericity.
double planarity () const
 Planarity.
double aplanarity () const
 Aplanarity.
Access the sphericity basis vectors
const Vector3sphericityAxis () const
const Vector3sphericityMajorAxis () const
 Sphericity major axis.
const Vector3sphericityMinorAxis () const
 Sphericity minor axis.
const Vector3axis1 () const
 AxesDefinition axis accessors.
const Vector3axis2 () const
 The 2nd most significant ("major") axis.
const Vector3axis3 () const
 The least significant ("minor") axis.
Access the momentum tensor eigenvalues
double lambda1 () const
double lambda2 () const
double lambda3 () const
Direct methods

Ways to do the calculation directly, without engaging the caching system

void calc (const FinalState &fs)
 Manually calculate the sphericity, without engaging the caching system.
void calc (const vector< Particle > &fsparticles)
 Manually calculate the sphericity, without engaging the caching system.
void calc (const vector< FourMomentum > &fsmomenta)
 Manually calculate the sphericity, without engaging the caching system.
void calc (const vector< Vector3 > &fsmomenta)
 Manually calculate the sphericity, without engaging the caching system.

Protected Member Functions

void project (const Event &e)
 Perform the projection on the Event.
int compare (const Projection &p) const
 Compare with other projections.

Detailed Description

Calculate the sphericity event shape.

The sphericity tensor (or quadratic momentum tensor) is defined as

\[ S^{\alpha \beta} = \frac{\sum_i p_i^\alpha p_i^\beta}{\sum_i |\mathbf{p}_i|^2} \]

, where the Greek indices are spatial components and the Latin indices are used for sums over particles. From this, the sphericity, aplanarity and planarity can be calculated by combinations of eigenvalues.

Defining the three eigenvalues $ \lambda_1 \ge \lambda_2 \ge \lambda_3 $, with $ \lambda_1 + \lambda_2 + \lambda_3 = 1 $, the sphericity is

\[ S = \frac{3}{2} (\lambda_2 + \lambda_3) \]

The aplanarity is $ A = \frac{3}{2}\lambda_3 $ and the planarity is $ P = \frac{2}{3}(S-2A) = \lambda_2 - \lambda_3 $. The eigenvectors define a set of spatial axes comparable with the thrust axes, but more sensitive to high momentum particles due to the quadratic sensitivity of the tensor to the particle momenta.

Since the sphericity is quadratic in the particle momenta, it is not an infrared safe observable in perturbative QCD. This can be fixed by adding a regularizing power of $r$ to the definition:

\[ S^{\alpha \beta} = \frac{\sum_i |\mathbf{p}_i|^{r-2} p_i^\alpha p_i^\beta} {\sum_i |\mathbf{p}_i|^r} \]

$r$ is available as a constructor argument on this class and will be taken into account by the Cmp<Projection> operation, so a single analysis can use several sphericity projections with different $r$ values without fear of a clash.


The documentation for this class was generated from the following files: