LorentzVector.h

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// @(#)root/mathcore:$Id$
// Authors: W. Brown, M. Fischler, L. Moneta    2005
 
/**********************************************************************
 *                                                                    *
 * Copyright (c) 2005 , LCG ROOT MathLib Team                         *
 *                                                                    *
 *                                                                    *
 **********************************************************************/
 
// Header file for class LorentzVector
//
// Created by:    moneta   at Tue May 31 17:06:09 2005
// Major mods by: fischler at Wed Jul 20   2005
//
// Last update: $Id$
//
#ifndef ROOT_Math_GenVector_LorentzVector
#define ROOT_Math_GenVector_LorentzVector  1
 
#include "Math/GenVector/PxPyPzE4D.h"
#include "Math/GenVector/DisplacementVector3D.h"
#include "Math/GenVector/GenVectorIO.h"
#include "Math/Vector2D.h"
 
#include "TMath.h"
 
#include <cmath>
#include <string>
 
namespace ROOT {
 
  namespace Math {
 
//__________________________________________________________________________________________
/** @ingroup GenVector
 
Class describing a generic LorentzVector in the 4D space-time,
using the specified coordinate system for the spatial vector part.
The metric used for the LorentzVector is (-,-,-,+).
In the case of LorentzVector we don't distinguish the concepts
of points and displacement vectors as in the 3D case,
since the main use case for 4D Vectors is to describe the kinematics of
relativistic particles. A LorentzVector behaves like a
DisplacementVector in 4D. The Minkowski components could be viewed as
v and t, or for kinematic 4-vectors, as p and E.
 
ROOT provides specialisations (and aliases to them) of the ROOT::Math::LorentzVector template:
- ROOT::Math::PtEtaPhiMVector based on pt (rho),eta,phi and M (t) coordinates in double precision
- ROOT::Math::PtEtaPhiEVector based on pt (rho),eta,phi and E (t) coordinates in double precision
- ROOT::Math::PxPyPzMVector based on px,py,pz and M coordinates in double precision
- ROOT::Math::PxPyPzEVector based on px,py,pz and E coordinates in double precision
- ROOT::Math::XYZTVector based on x,y,z,t coordinates (cartesian) in double precision (same as PxPyPzEVector)
- ROOT::Math::XYZTVectorF based on x,y,z,t coordinates (cartesian) in float precision (same as PxPyPzEVector but float)
Pt (or rho) refers to transverse momentum, whereas eta refers to pseudorapidity. M is mass, E is energy, p is momentum.
 
@see GenVector
*/
 
    template< class CoordSystem >
    class LorentzVector {
 
    public:
 
       // ------ ctors ------
 
       typedef typename CoordSystem::Scalar Scalar;
       typedef CoordSystem CoordinateType;
 
       /**
          default constructor of an empty vector (Px = Py = Pz = E = 0 )
       */
       LorentzVector ( ) : fCoordinates() { }
 
       /**
          generic constructors from four scalar values.
          The association between values and coordinate depends on the
          coordinate system.  For PxPyPzE4D,
          \param a scalar value (Px)
          \param b scalar value (Py)
          \param c scalar value (Pz)
          \param d scalar value (E)
       */
       LorentzVector(const Scalar & a,
                     const Scalar & b,
                     const Scalar & c,
                     const Scalar & d) :
          fCoordinates(a , b,  c, d)  { }
 
       /**
          constructor from a LorentzVector expressed in different
          coordinates, or using a different Scalar type
       */
       template< class Coords >
       explicit constexpr LorentzVector(const LorentzVector<Coords> & v ) :
          fCoordinates( v.Coordinates() ) { }
 
       /**
          Construct from a foreign 4D vector type, for example, HepLorentzVector
          Precondition: v must implement methods x(), y(), z(), and t()
       */
       template<class ForeignLorentzVector,
                typename = decltype(std::declval<ForeignLorentzVector>().x()
                                    + std::declval<ForeignLorentzVector>().y()
                                    + std::declval<ForeignLorentzVector>().z()
                                    + std::declval<ForeignLorentzVector>().t())>
       explicit constexpr LorentzVector( const ForeignLorentzVector & v) :
          fCoordinates(PxPyPzE4D<Scalar>( v.x(), v.y(), v.z(), v.t()  ) ) { }
 
#ifdef LATER
       /**
          construct from a generic linear algebra  vector implementing operator []
          and with a size of at least 4. This could be also a C array
          In this case v[0] is the first data member
          ( Px for a PxPyPzE4D base)
          \param v LA vector
          \param index0 index of first vector element (Px)
       */
       template< class LAVector >
       explicit constexpr LorentzVector(const LAVector & v, size_t index0 ) {
          fCoordinates = CoordSystem ( v[index0], v[index0+1], v[index0+2], v[index0+3] );
       }
#endif
 
 
       // ------ assignment ------
 
       /**
          Assignment operator from a lorentz vector of arbitrary type
       */
       template< class OtherCoords >
       LorentzVector & operator= ( const LorentzVector<OtherCoords> & v) {
          fCoordinates = v.Coordinates();
          return *this;
       }
 
       /**
          assignment from any other Lorentz vector  implementing
          x(), y(), z() and t()
       */
       template<class ForeignLorentzVector,
                typename = decltype(std::declval<ForeignLorentzVector>().x()
                                    + std::declval<ForeignLorentzVector>().y()
                                    + std::declval<ForeignLorentzVector>().z()
                                    + std::declval<ForeignLorentzVector>().t())>
       LorentzVector & operator = ( const ForeignLorentzVector & v) {
          SetXYZT( v.x(), v.y(), v.z(), v.t() );
          return *this;
       }
 
#ifdef LATER
       /**
          assign from a generic linear algebra  vector implementing operator []
          and with a size of at least 4
          In this case v[0] is the first data member
          ( Px for a PxPyPzE4D base)
          \param v LA vector
          \param index0 index of first vector element (Px)
       */
       template< class LAVector >
       LorentzVector & AssignFrom(const LAVector & v, size_t index0=0 ) {
          fCoordinates.SetCoordinates( v[index0], v[index0+1], v[index0+2], v[index0+3] );
          return *this;
       }
#endif
 
       // ------ Set, Get, and access coordinate data ------
 
       /**
          Retrieve a const reference to  the coordinates object
       */
       const CoordSystem & Coordinates() const {
          return fCoordinates;
       }
 
       /**
          Set internal data based on an array of 4 Scalar numbers
       */
       LorentzVector<CoordSystem>& SetCoordinates( const Scalar src[] ) {
          fCoordinates.SetCoordinates(src);
          return *this;
       }
 
       /**
          Set internal data based on 4 Scalar numbers
       */
       LorentzVector<CoordSystem>& SetCoordinates( Scalar a, Scalar b, Scalar c, Scalar d ) {
          fCoordinates.SetCoordinates(a, b, c, d);
          return *this;
       }
 
       /**
          Set internal data based on 4 Scalars at *begin to *end
       */
       template< class IT >
       LorentzVector<CoordSystem>& SetCoordinates( IT begin, IT end  ) {
          IT a = begin; IT b = ++begin; IT c = ++begin; IT d = ++begin;
          (void)end;
          assert (++begin==end);
          SetCoordinates (*a,*b,*c,*d);
          return *this;
       }
 
       /**
          get internal data into 4 Scalar numbers
          \note Alternatively, you may use structured bindings: `auto const [a, b, c, d] = v`.
       */
       void GetCoordinates( Scalar& a, Scalar& b, Scalar& c, Scalar & d ) const
       { fCoordinates.GetCoordinates(a, b, c, d);  }
 
       /**
          get internal data into an array of 4 Scalar numbers
       */
       void GetCoordinates( Scalar dest[] ) const
       { fCoordinates.GetCoordinates(dest);  }
 
       /**
          get internal data into 4 Scalars at *begin to *end
       */
       template <class IT>
       void GetCoordinates( IT begin, IT end ) const
       { IT a = begin; IT b = ++begin; IT c = ++begin; IT d = ++begin;
       (void)end;
       assert (++begin==end);
       GetCoordinates (*a,*b,*c,*d);
       }
 
       /**
          get internal data into 4 Scalars at *begin
       */
       template <class IT>
       void GetCoordinates( IT begin ) const {
          Scalar a,b,c,d = 0;
          GetCoordinates (a,b,c,d);
          *begin++ = a;
          *begin++ = b;
          *begin++ = c;
          *begin   = d;
       }
 
       /**
          set the values of the vector from the cartesian components (x,y,z,t)
          (if the vector is held in another coordinates, like (Pt,eta,phi,m)
          then (x, y, z, t) are converted to that form)
       */
       LorentzVector<CoordSystem>& SetXYZT (Scalar xx, Scalar yy, Scalar zz, Scalar tt) {
          fCoordinates.SetPxPyPzE(xx,yy,zz,tt);
          return *this;
       }
       LorentzVector<CoordSystem>& SetPxPyPzE (Scalar xx, Scalar yy, Scalar zz, Scalar ee) {
          fCoordinates.SetPxPyPzE(xx,yy,zz,ee);
          return *this;
       }
 
       // ------------------- Equality -----------------
 
       /**
          Exact equality
       */
       bool operator==(const LorentzVector & rhs) const {
          return fCoordinates==rhs.fCoordinates;
       }
       bool operator!= (const LorentzVector & rhs) const {
          return !(operator==(rhs));
       }
 
       // ------ Individual element access, in various coordinate systems ------
 
       /**
          dimension
       */
       unsigned int Dimension() const
       {
          return fDimension;
       };
 
       // individual coordinate accessors in various coordinate systems
 
       /**
          spatial X component
       */
       Scalar Px() const  { return fCoordinates.Px(); }
       Scalar X()  const  { return fCoordinates.Px(); }
       /**
          spatial Y component
       */
       Scalar Py() const { return fCoordinates.Py(); }
       Scalar Y()  const { return fCoordinates.Py(); }
       /**
          spatial Z component
       */
       Scalar Pz() const { return fCoordinates.Pz(); }
       Scalar Z()  const { return fCoordinates.Pz(); }
       /**
          return 4-th component (time, or energy for a 4-momentum vector)
       */
       Scalar E()  const { return fCoordinates.E(); }
       Scalar T()  const { return fCoordinates.E(); }
       /**
          return magnitude (mass) squared  M2 = T**2 - X**2 - Y**2 - Z**2
          (we use -,-,-,+ metric)
       */
       Scalar M2()   const { return fCoordinates.M2(); }
       /**
          return magnitude (mass) using the  (-,-,-,+)  metric.
          If M2 is negative (space-like vector) a GenVector_exception
          is suggested and if continuing, - sqrt( -M2) is returned
       */
       Scalar M() const    { return fCoordinates.M();}
       /**
          return the spatial (3D) magnitude ( sqrt(X**2 + Y**2 + Z**2) )
       */
       Scalar R() const { return fCoordinates.R(); }
       Scalar P() const { return fCoordinates.R(); }
       /**
          return the square of the spatial (3D) magnitude ( X**2 + Y**2 + Z**2 )
       */
       Scalar P2() const { return P() * P(); }
       /**
          return the square of the transverse spatial component ( X**2 + Y**2 )
       */
       Scalar Perp2( ) const { return fCoordinates.Perp2();}
 
       /**
          return the  transverse spatial component sqrt ( X**2 + Y**2 )
       */
       Scalar Pt()  const { return fCoordinates.Pt(); }
       Scalar Rho() const { return fCoordinates.Pt(); }
 
       /**
          return the transverse mass squared
          \f[ m_t^2 = E^2 - p{_z}^2 \f]
       */
       Scalar Mt2() const { return fCoordinates.Mt2(); }
 
       /**
          return the transverse mass
          \f[ \sqrt{ m_t^2 = E^2 - p{_z}^2} X sign(E^ - p{_z}^2) \f]
       */
       Scalar Mt() const { return fCoordinates.Mt(); }
 
       /**
          return the transverse energy squared
          \f[ e_t = \frac{E^2 p_{\perp}^2 }{ |p|^2 } \f]
       */
       Scalar Et2() const { return fCoordinates.Et2(); }
 
       /**
          return the transverse energy
          \f[ e_t = \sqrt{ \frac{E^2 p_{\perp}^2 }{ |p|^2 } } X sign(E) \f]
       */
       Scalar Et() const { return fCoordinates.Et(); }
 
       /**
          azimuthal  Angle
       */
       Scalar Phi() const  { return fCoordinates.Phi();}
 
       /**
          polar Angle
       */
       Scalar Theta() const { return fCoordinates.Theta(); }
 
       /**
          pseudorapidity
          \f[ \eta = - \ln { \tan { \frac { \theta} {2} } } \f]
       */
       Scalar Eta() const { return fCoordinates.Eta(); }
 
       /**
          deltaRapidity between this and vector v
          \f[ \Delta R = \sqrt { \Delta \eta ^2 + \Delta \phi ^2 } \f]
          \param useRapidity true to use Rapidity(), false to use Eta()
       */
       template <class OtherLorentzVector>
       Scalar DeltaR(const OtherLorentzVector &v, const bool useRapidity = false) const
       {
          const double delta = useRapidity ? Rapidity() - v.Rapidity() : Eta() - v.Eta();
          double dphi = Phi() - v.Phi();
          // convert dphi angle to the interval (-PI,PI]
          if (dphi > TMath::Pi() || dphi <= -TMath::Pi()) {
             if (dphi > 0) {
                int n = static_cast<int>(dphi / TMath::TwoPi() + 0.5);
                dphi -= TMath::TwoPi() * n;
             } else {
                int n = static_cast<int>(0.5 - dphi / TMath::TwoPi());
                dphi += TMath::TwoPi() * n;
             }
          }
          return std::sqrt(delta * delta + dphi * dphi);
       }
 
       /**
          get the spatial components of the Vector in a
          DisplacementVector based on Cartesian Coordinates
       */
       ::ROOT::Math::DisplacementVector3D<Cartesian3D<Scalar> > Vect() const {
          return ::ROOT::Math::DisplacementVector3D<Cartesian3D<Scalar> >( X(), Y(), Z() );
       }
 
       // ------ Operations combining two Lorentz vectors ------
 
       /**
          scalar (Dot) product of two LorentzVector vectors (metric is -,-,-,+)
          Enable the product using any other LorentzVector implementing
          the x(), y() , y() and t() member functions
          \param  q  any LorentzVector implementing the x(), y() , z() and t()
          member functions
          \return the result of v.q of type according to the base scalar type of v
       */
 
       template< class OtherLorentzVector >
       Scalar Dot(const OtherLorentzVector & q) const {
          return t()*q.t() - x()*q.x() - y()*q.y() - z()*q.z();
       }
 
       /**
          Self addition with another Vector ( v+= q )
          Enable the addition with any other LorentzVector
          \param  q  any LorentzVector implementing the x(), y() , z() and t()
          member functions
       */
      template< class OtherLorentzVector >
      inline LorentzVector & operator += ( const OtherLorentzVector & q)
       {
          SetXYZT( x() + q.x(), y() + q.y(), z() + q.z(), t() + q.t()  );
          return *this;
       }
 
       /**
          Self subtraction of another Vector from this ( v-= q )
          Enable the addition with any other LorentzVector
          \param  q  any LorentzVector implementing the x(), y() , z() and t()
          member functions
       */
       template< class OtherLorentzVector >
       LorentzVector & operator -= ( const OtherLorentzVector & q) {
          SetXYZT( x() - q.x(), y() - q.y(), z() - q.z(), t() - q.t()  );
          return *this;
       }
 
       /**
          addition of two LorentzVectors (v3 = v1 + v2)
          Enable the addition with any other LorentzVector
          \param  v2  any LorentzVector implementing the x(), y() , z() and t()
          member functions
          \return a new LorentzVector of the same type as v1
       */
       template<class OtherLorentzVector>
       LorentzVector  operator +  ( const OtherLorentzVector & v2) const
       {
          LorentzVector<CoordinateType> v3(*this);
          v3 += v2;
          return v3;
       }
 
       /**
          subtraction of two LorentzVectors (v3 = v1 - v2)
          Enable the subtraction of any other LorentzVector
          \param  v2  any LorentzVector implementing the x(), y() , z() and t()
          member functions
          \return a new LorentzVector of the same type as v1
       */
       template<class OtherLorentzVector>
       LorentzVector  operator -  ( const OtherLorentzVector & v2) const {
          LorentzVector<CoordinateType> v3(*this);
          v3 -= v2;
          return v3;
       }
 
       //--- scaling operations ------
 
       /**
          multiplication by a scalar quantity v *= a
       */
       LorentzVector & operator *= (Scalar a) {
          fCoordinates.Scale(a);
          return *this;
       }
 
       /**
          division by a scalar quantity v /= a
       */
       LorentzVector & operator /= (Scalar a) {
          fCoordinates.Scale(1/a);
          return *this;
       }
 
       /**
          product of a LorentzVector by a scalar quantity
          \param a  scalar quantity of type a
          \return a new mathcoreLorentzVector q = v * a same type as v
       */
       LorentzVector operator * ( const Scalar & a) const {
          LorentzVector tmp(*this);
          tmp *= a;
          return tmp;
       }
 
       /**
          Divide a LorentzVector by a scalar quantity
          \param a  scalar quantity of type a
          \return a new mathcoreLorentzVector q = v / a same type as v
       */
       LorentzVector<CoordSystem> operator / ( const Scalar & a) const {
          LorentzVector<CoordSystem> tmp(*this);
          tmp /= a;
          return tmp;
       }
 
       /**
          Negative of a LorentzVector (q = - v )
          \return a new LorentzVector with opposite direction and time
       */
       LorentzVector operator - () const {
          //LorentzVector<CoordinateType> v(*this);
          //v.Negate();
          return operator*( Scalar(-1) );
       }
       LorentzVector operator + () const {
          return *this;
       }
 
       // ---- Relativistic Properties ----
 
       /**
          Rapidity relative to the Z axis:  .5 log [(E+Pz)/(E-Pz)]
       */
       Scalar Rapidity() const {
          // TODO - It would be good to check that E > Pz and use the Throw()
          //        mechanism or at least load a NAN if not.
          //        We should then move the code to a .cpp file.
          const Scalar ee  = E();
          const Scalar ppz = Pz();
          using std::log;
          return Scalar(0.5) * log((ee + ppz) / (ee - ppz));
       }
 
       /**
          Rapidity in the direction of travel: atanh (|P|/E)=.5 log[(E+P)/(E-P)]
       */
       Scalar ColinearRapidity() const {
          // TODO - It would be good to check that E > P and use the Throw()
          //        mechanism or at least load a NAN if not.
          const Scalar ee = E();
          const Scalar pp = P();
          using std::log;
          return Scalar(0.5) * log((ee + pp) / (ee - pp));
       }
 
       /**
          Determine if momentum-energy can represent a physical massive particle
       */
       bool isTimelike( ) const {
          Scalar ee = E(); Scalar pp = P(); return ee*ee > pp*pp;
       }
 
       /**
          Determine if momentum-energy can represent a massless particle
       */
       bool isLightlike( Scalar tolerance
                         = 100*std::numeric_limits<Scalar>::epsilon() ) const {
          Scalar ee = E(); Scalar pp = P(); Scalar delta = ee-pp;
          if ( ee==0 ) return pp==0;
          return delta*delta < tolerance * ee*ee;
       }
 
       /**
          Determine if momentum-energy is spacelike, and represents a tachyon
       */
       bool isSpacelike( ) const {
          Scalar ee = E(); Scalar pp = P(); return ee*ee < pp*pp;
       }
 
       typedef DisplacementVector3D< Cartesian3D<Scalar> > BetaVector;
 
       /**
          The beta vector for the boost that would bring this vector into
          its center of mass frame (zero momentum)
       */
       BetaVector BoostToCM( ) const {
          if (E() == 0) {
             if (P() == 0) {
                return BetaVector();
             } else {
                // TODO - should attempt to Throw with msg about
                // boostVector computed for LorentzVector with t=0
                return -Vect()/E();
             }
          }
          if (M2() <= 0) {
             // TODO - should attempt to Throw with msg about
             // boostVector computed for a non-timelike LorentzVector
          }
          return -Vect()/E();
       }
 
       /**
          The beta vector for the boost that would bring this vector into
          its center of mass frame (zero momentum)
       */
       template <class Other4Vector>
       BetaVector BoostToCM(const Other4Vector& v ) const {
          Scalar eSum = E() + v.E();
          DisplacementVector3D< Cartesian3D<Scalar> > vecSum = Vect() + v.Vect();
          if (eSum == 0) {
             if (vecSum.Mag2() == 0) {
                return BetaVector();
             } else {
                // TODO - should attempt to Throw with msg about
                // boostToCM computed for two 4-vectors with combined t=0
                return BetaVector(vecSum/eSum);
             }
             // TODO - should attempt to Throw with msg about
             // boostToCM computed for two 4-vectors with combined e=0
          }
          return BetaVector (vecSum * (-1./eSum));
       }
 
       //beta and gamma
 
       /**
           Return beta scalar value
       */
       Scalar Beta() const {
          if ( E() == 0 ) {
             if ( P2() == 0)
                // to avoid Nan
                return 0;
             else {
                GenVector_Throw(
                   "LorentzVector::Beta() - beta computed for LorentzVector with t = 0. Return an Infinite result");
                return 1./E();
             }
          }
          if ( M2() <= 0 ) {
             GenVector_Throw("LorentzVector::Beta() - beta computed for non-timelike LorentzVector . Result is "
                             "physically meaningless");
          }
          return P() / E();
       }
       /**
           Return Gamma scalar value
       */
       Scalar Gamma() const {
          const Scalar v2 = P2();
          const Scalar t2 = E() * E();
          if (E() == 0) {
             if ( P2() == 0) {
                return 1;
             } else {
                GenVector_Throw(
                   "LorentzVector::Gamma() - gamma computed for LorentzVector with t = 0. Return a zero result");
             }
          }
          if ( t2 < v2 ) {
             GenVector_Throw("LorentzVector::Gamma() - gamma computed for a spacelike LorentzVector. Imaginary result");
             return 0;
          }
          else if ( t2 == v2 ) {
             GenVector_Throw("LorentzVector::Gamma() - gamma computed for a lightlike LorentzVector. Infinite result");
          }
          using std::sqrt;
          return Scalar(1) / sqrt(Scalar(1) - v2 / t2);
       } /* gamma */
 
 
       // Method providing limited backward name compatibility with CLHEP ----
 
       Scalar x()     const { return fCoordinates.Px();     }
       Scalar y()     const { return fCoordinates.Py();     }
       Scalar z()     const { return fCoordinates.Pz();     }
       Scalar t()     const { return fCoordinates.E();      }
       Scalar px()    const { return fCoordinates.Px();     }
       Scalar py()    const { return fCoordinates.Py();     }
       Scalar pz()    const { return fCoordinates.Pz();     }
       Scalar e()     const { return fCoordinates.E();      }
       Scalar r()     const { return fCoordinates.R();      }
       Scalar theta() const { return fCoordinates.Theta();  }
       Scalar phi()   const { return fCoordinates.Phi();    }
       Scalar rho()   const { return fCoordinates.Rho();    }
       Scalar eta()   const { return fCoordinates.Eta();    }
       Scalar pt()    const { return fCoordinates.Pt();     }
       Scalar perp2() const { return fCoordinates.Perp2();  }
       Scalar mag2()  const { return fCoordinates.M2();     }
       Scalar mag()   const { return fCoordinates.M();      }
       Scalar mt()    const { return fCoordinates.Mt();     }
       Scalar mt2()   const { return fCoordinates.Mt2();    }
 
 
       // Methods  requested by CMS ---
       Scalar energy() const { return fCoordinates.E();      }
       Scalar mass()   const { return fCoordinates.M();      }
       Scalar mass2()  const { return fCoordinates.M2();     }
 
 
       /**
          Methods setting a Single-component
          Work only if the component is one of which the vector is represented.
          For example SetE will work for a PxPyPzE Vector but not for a PxPyPzM Vector.
       */
       LorentzVector<CoordSystem>& SetE  ( Scalar a )  { fCoordinates.SetE  (a); return *this; }
       LorentzVector<CoordSystem>& SetEta( Scalar a )  { fCoordinates.SetEta(a); return *this; }
       LorentzVector<CoordSystem>& SetM  ( Scalar a )  { fCoordinates.SetM  (a); return *this; }
       LorentzVector<CoordSystem>& SetPhi( Scalar a )  { fCoordinates.SetPhi(a); return *this; }
       LorentzVector<CoordSystem>& SetPt ( Scalar a )  { fCoordinates.SetPt (a); return *this; }
       LorentzVector<CoordSystem>& SetPx ( Scalar a )  { fCoordinates.SetPx (a); return *this; }
       LorentzVector<CoordSystem>& SetPy ( Scalar a )  { fCoordinates.SetPy (a); return *this; }
       LorentzVector<CoordSystem>& SetPz ( Scalar a )  { fCoordinates.SetPz (a); return *this; }
 
    private:
 
       CoordSystem  fCoordinates;    // internal coordinate system
       static constexpr unsigned int fDimension = CoordinateType::Dimension;
 
    };  // LorentzVector<>
 
 
 
  // global methods
 
    /**
       Scale of a LorentzVector with a scalar quantity a
       \param a  scalar quantity of type a
       \param v  LorentzVector based on any coordinate system
       \return a new mathcoreLorentzVector q = v * a same type as v
     */
    template <class CoordSystem>
    inline LorentzVector<CoordSystem>
    operator*(const typename LorentzVector<CoordSystem>::Scalar &a, const LorentzVector<CoordSystem> &v)
    {
       LorentzVector<CoordSystem> tmp(v);
       tmp *= a;
       return tmp;
    }
 
    /**
       pair (p+ p-) acoplanarity `alpha = 1 - |phi+ - phi-|/pi`.
       \param pp p+, LorentzVector based on any coordinate system
       \param pm p-, LorentzVector based on any coordinate system
       \return a scalar
       \ingroup GenVector
       \see http://doi.org/10.1103/PhysRevLett.121.212301
     */
    template <class CoordSystem>
    typename LorentzVector<CoordSystem>::Scalar
    Acoplanarity(LorentzVector<CoordSystem> const &pp, LorentzVector<CoordSystem> const &pm)
    {
       auto dphi = pp.Phi() - pm.Phi();
       // convert dphi angle to the interval (-PI,PI]
       if (dphi > TMath::Pi() || dphi <= -TMath::Pi()) {
          if (dphi > 0) {
             int n = static_cast<int>(dphi / TMath::TwoPi() + 0.5);
             dphi -= TMath::TwoPi() * n;
          } else {
             int n = static_cast<int>(0.5 - dphi / TMath::TwoPi());
             dphi += TMath::TwoPi() * n;
          }
       }
       return 1 - std::abs(dphi) / TMath::Pi();
    }
 
    /**
       pair (p+ p-) vectorial asymmetry `Av = |Pt+ - Pt-|/|Pt+ + Pt-|`.
       In an experimental setting, it reflects a convolution of the experimental resolutions
       of particle energy and azimuthal angle measurement.
       \param pp p+, LorentzVector based on any coordinate system
       \param pm p-, LorentzVector based on any coordinate system
       \return a scalar. Returns -1 if both momenta are exactly mirrored.
       \ingroup GenVector
       \see http://doi.org/10.1103/PhysRevLett.121.212301, https://doi.org/10.1103/PhysRevD.99.093013
     */
    template <class CoordSystem>
    typename LorentzVector<CoordSystem>::Scalar
    AsymmetryVectorial(LorentzVector<CoordSystem> const &pp, LorentzVector<CoordSystem> const &pm)
    {
       ROOT::Math::XYVector vp(pp.Px(), pp.Py());
       ROOT::Math::XYVector vm(pm.Px(), pm.Py());
       auto denom = (vp + vm).R();
       if (denom == 0.)
          return -1;
       return (vp - vm).R() / denom;
    }
 
    /**
       pair (p+ p-) scalar asymmetry `As = ||Pt+| - |Pt-|/||Pt+| + |Pt-||`.
       Measures the relative difference in transverse momentum of the pair, e.g. two photons,
       and would be ideally zero for two back-to-back photons.
       \param pp p+, LorentzVector based on any coordinate system
       \param pm p-, LorentzVector based on any coordinate system
       \return a scalar. Returns 0 if both transverse momenta are zero
       \ingroup GenVector
       \see http://doi.org/10.1103/PhysRevLett.121.212301, https://doi.org/10.1103/PhysRevD.99.093013
     */
    template< class CoordSystem >
    typename LorentzVector<CoordSystem>::Scalar AsymmetryScalar(LorentzVector<CoordSystem> const &pp, LorentzVector<CoordSystem> const &pm)
    {
       auto ptp = pp.Pt();
       auto ptm = pm.Pt();
       if (ptp == 0 && ptm == 0)
          return 0;
       return std::abs(ptp - ptm) / (ptp + ptm);
    }
 
    // ------------- I/O to/from streams -------------
 
    template< class char_t, class traits_t, class Coords >
    inline
    std::basic_ostream<char_t,traits_t> &
    operator << ( std::basic_ostream<char_t,traits_t> & os
                  , LorentzVector<Coords> const & v
       )
    {
       if( !os )  return os;
 
       typename Coords::Scalar a, b, c, d;
       v.GetCoordinates(a, b, c, d);
 
       if( detail::get_manip( os, detail::bitforbit ) )  {
        detail::set_manip( os, detail::bitforbit, '\00' );
        // TODO: call MF's bitwise-accurate functions on each of a, b, c, d
       }
       else  {
          os << detail::get_manip( os, detail::open  ) << a
             << detail::get_manip( os, detail::sep   ) << b
             << detail::get_manip( os, detail::sep   ) << c
             << detail::get_manip( os, detail::sep   ) << d
             << detail::get_manip( os, detail::close );
       }
 
       return os;
 
    }  // op<< <>()
 
 
     template< class char_t, class traits_t, class Coords >
     inline
     std::basic_istream<char_t,traits_t> &
     operator >> ( std::basic_istream<char_t,traits_t> & is
                   , LorentzVector<Coords> & v
        )
     {
        if( !is )  return is;
 
        typename Coords::Scalar a, b, c, d;
 
        if( detail::get_manip( is, detail::bitforbit ) )  {
           detail::set_manip( is, detail::bitforbit, '\00' );
           // TODO: call MF's bitwise-accurate functions on each of a, b, c
        }
        else  {
           detail::require_delim( is, detail::open  );  is >> a;
           detail::require_delim( is, detail::sep   );  is >> b;
           detail::require_delim( is, detail::sep   );  is >> c;
           detail::require_delim( is, detail::sep   );  is >> d;
           detail::require_delim( is, detail::close );
        }
 
        if( is )
           v.SetCoordinates(a, b, c, d);
        return is;
 
     }  // op>> <>()
 
    // Structured bindings
    template <std::size_t I, class CoordSystem>
    typename CoordSystem::Scalar get(LorentzVector<CoordSystem> const& p)
    {
       static_assert(I < 4);
       if constexpr (I == 0) {
          return p.X();
       } else if constexpr (I == 1) {
          return p.Y();
       } else if constexpr (I == 2) {
          return p.Z();
       } else {
          return p.E();
       }
    }
 
  } // end namespace Math
 
} // end namespace ROOT
 
// Structured bindings
#include <tuple>
namespace std {
   template <class CoordSystem>
   struct tuple_size<ROOT::Math::LorentzVector<CoordSystem>> : integral_constant<size_t, 4> {};
   template <size_t I, class CoordSystem>
   struct tuple_element<I, ROOT::Math::LorentzVector<CoordSystem>> {
      static_assert(I < 4);
      using type = typename CoordSystem::Scalar;
   };
}
 
#include <sstream>
namespace cling
{
template<typename CoordSystem>
std::string printValue(const ROOT::Math::LorentzVector<CoordSystem> *v)
{
   std::stringstream s;
   s << *v;
   return s.str();
}
 
} // end namespace cling
 
#endif
 
//#include "Math/GenVector/LorentzVectorOperations.h"