Sphericity.hh

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// -*- C++ -*-
#ifndef RIVET_Sphericity_HH
#define RIVET_Sphericity_HH

#include "Rivet/Projection.hh"
#include "Rivet/Projections/FinalState.hh"
#include "Rivet/Event.hh"
#include "Rivet/RivetCLHEP.hh"

namespace Rivet {

  /**
     @brief Calculate the sphericity event shape.
     
     The sphericity tensor (or quadratic momentum tensor) is defined as 
     \f[ 
     S^{\alpha \beta} = \frac{\sum_i p_i^\alpha p_i^\beta}{\sum_i |\mathbf{p}_i|^2} 
     \f],
     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
     \f$ \lambda_1 \ge \lambda_2 \ge \lambda_3 \f$, with \f$ \lambda_1 + \lambda_2 + \lambda_3 = 1 \f$, 
     the sphericity is
     \f[ 
     S = \frac{3}{2} (\lambda_2 + \lambda_3) 
     \f]
     
     The aplanarity is \f$ A = \frac{3}{2}\lambda_3 \f$ and the planarity
     is \f$ P = \frac{2}{3}(S-2A) = \lambda_2 - \lambda_3 \f$. 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 \f$r\f$ to the definition:
     \f[ 
     S^{\alpha \beta} = 
     \frac{\sum_i |\mathbf{p}_i|^{r-2} p_i^\alpha p_i^\beta}
     {\sum_i |\mathbf{p}_i|^r} 
     \f]
     
     \f$r\f$ 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 \f$r\f$ values without
     fear of a clash.
  */
  class Sphericity : public Projection {

  public:

    /// Constructor. Supplied FinalState projection must live throughout the run.
    inline Sphericity(FinalState& fsp, double rparam=2.0)
      : _sphericity(0), _planarity(0), _aplanarity(0), _regparam(rparam), 
        _fsproj(&fsp)
    { 
      addProjection(fsp);
    }

  public:
    /// Return the name of the projection
    inline string getName() const {
      return "Sphericity";
    }

  protected:

    /// Perform the projection on the Event
    void project(const Event& e);

    /// Compare with other projections
    int compare(const Projection& p) const;

  public:

    ///@{ Access the event shapes by name
    /// Sphericity
    inline const double sphericity() const { return _sphericity; }
    /// Planarity
    inline const double planarity() const { return _planarity; }
    /// Aplanarity
    inline const double aplanarity() const { return _aplanarity; }
    ///@}

    ///@{ Access the sphericity basis vectors
    /// Sphericity axis
    /// @todo Implement something that isn't garbage!  
    inline const Vector3 sphericityAxis() const { return Vector3(0,0,1); }
    /// Sphericity major axis
    /// @todo Implement something that isn't garbage!
    inline const Vector3 sphericityMajorAxis() const { return Vector3(1,0,0); }
    /// Sphericity minor axis
    /// @todo Implement something that isn't garbage!
    inline const Vector3 sphericityMinorAxis() const { return Vector3(0,1,0); }
    ///@}

    ///@{ Access the momentum tensor eigenvalues
    inline const double lambda1() const { return _lambdas[0]; }
    inline const double lambda2() const { return _lambdas[1]; }
    inline const double lambda3() const { return _lambdas[2]; }
    ///@}

  private:

    /// Eigenvalues
    double _lambdas[3];

    /// The event shapes
    double _sphericity, _planarity, _aplanarity;

    /// Regularizing parameter, used to force infra-red safety.
    const double _regparam;

    /// The FinalState projection used by this projection.
    FinalState* _fsproj;

  };

}


#endif