library: libPhysics #include "TLorentzVector.h"

# TLorentzVector

class description - source file - inheritance tree (.pdf)

## class TLorentzVector : public TObject

| Inheritance Chart:
`TObject`
| <- | `TLorentzVector` |

        public:

                          TLorentzVector(Double_t x = 0.0, Double_t y = 0.0, Double_t z = 0.0, Double_t t = 0.0)
                          TLorentzVector(const Double_t* carray)
                          TLorentzVector(const Float_t* carray)
                          TLorentzVector(const TVector3& vector3, Double_t t)
                          TLorentzVector(const TLorentzVector& lorentzvector)
           TLorentzVector operator+(const TLorentzVector& q) const
           TLorentzVector operator-(const TLorentzVector& q) const
           TLorentzVector operator-() const
           TLorentzVector operator*(Double_t a) const
                  virtual ~TLorentzVector()
                 Double_t Angle(const TVector3& v) const
                 Double_t Beta() const
                     void Boost(Double_t, Double_t, Double_t)
                     void Boost(const TVector3& b)
                 TVector3 BoostVector() const
           static TClass* Class()
                 Double_t CosTheta() const
                 Double_t DeltaPhi(const TLorentzVector& v) const
                 Double_t DeltaR(const TLorentzVector& v) const
                 Double_t Dot(const TLorentzVector& q) const
                 Double_t DrEtaPhi(const TLorentzVector& v) const
                 Double_t E() const
                 Double_t Energy() const
                 Double_t Et() const
                 Double_t Et(const TVector3& v) const
                 Double_t Et2() const
                 Double_t Et2(const TVector3& v) const
                 Double_t Eta() const
                 TVector2 EtaPhiVector()
                 Double_t Gamma() const
                     void GetXYZT(Double_t* carray) const
                     void GetXYZT(Float_t* carray) const
          virtual TClass* IsA() const
                 Double_t M() const
                 Double_t M2() const
                 Double_t Mag() const
                 Double_t Mag2() const
                 Double_t Minus() const
                 Double_t Mt() const
                 Double_t Mt2() const
                   Bool_t operator!=(const TLorentzVector& q) const
                 Double_t operator()(int i) const
                Double_t& operator()(int i)
                 Double_t operator*(const TLorentzVector& q) const
          TLorentzVector& operator*=(Double_t a)
          TLorentzVector& operator*=(const TRotation& m)
          TLorentzVector& operator*=(const TLorentzRotation&)
          TLorentzVector& operator+=(const TLorentzVector& q)
          TLorentzVector& operator-=(const TLorentzVector& q)
          TLorentzVector& operator=(const TLorentzVector& q)
                   Bool_t operator==(const TLorentzVector& q) const
                 Double_t operator[](int i) const
                Double_t& operator[](int i)
                 Double_t P() const
                 Double_t Perp() const
                 Double_t Perp(const TVector3& v) const
                 Double_t Perp2() const
                 Double_t Perp2(const TVector3& v) const
                 Double_t Phi() const
                 Double_t Plus() const
                 Double_t PseudoRapidity() const
                 Double_t Pt() const
                 Double_t Pt(const TVector3& v) const
                 Double_t Px() const
                 Double_t Py() const
                 Double_t Pz() const
                 Double_t Rapidity() const
                 Double_t Rho() const
                     void Rotate(Double_t a, const TVector3& v)
                     void RotateUz(TVector3& newUzVector)
                     void RotateX(Double_t angle)
                     void RotateY(Double_t angle)
                     void RotateZ(Double_t angle)
                     void SetE(Double_t a)
                     void SetPerp(Double_t r)
                     void SetPhi(Double_t phi)
                     void SetPtEtaPhiE(Double_t pt, Double_t eta, Double_t phi, Double_t e)
                     void SetPtEtaPhiM(Double_t pt, Double_t eta, Double_t phi, Double_t m)
                     void SetPx(Double_t a)
                     void SetPxPyPzE(Double_t px, Double_t py, Double_t pz, Double_t e)
                     void SetPy(Double_t a)
                     void SetPz(Double_t a)
                     void SetRho(Double_t rho)
                     void SetT(Double_t a)
                     void SetTheta(Double_t theta)
                     void SetVect(const TVector3& vect3)
                     void SetVectM(const TVector3& spatial, Double_t mass)
                     void SetVectMag(const TVector3& spatial, Double_t magnitude)
                     void SetX(Double_t a)
                     void SetXYZM(Double_t x, Double_t y, Double_t z, Double_t m)
                     void SetXYZT(Double_t x, Double_t y, Double_t z, Double_t t)
                     void SetY(Double_t a)
                     void SetZ(Double_t a)
             virtual void ShowMembers(TMemberInspector& insp, char* parent)
             virtual void Streamer(TBuffer& b)
                     void StreamerNVirtual(TBuffer& b)
                 Double_t T() const
                 Double_t Theta() const
          TLorentzVector& Transform(const TRotation& m)
          TLorentzVector& Transform(const TLorentzRotation&)
                 TVector3 Vect() const
                 Double_t X() const
                 Double_t Y() const
                 Double_t Z() const

### Data Members

        private:

          TVector3 fP  3 vector component
          Double_t fE  time or energy of (x,y,z,t) or (px,py,pz,e)

        public:

          static const enum TLorentzVector:: kX
          static const enum TLorentzVector:: kY
          static const enum TLorentzVector:: kZ
          static const enum TLorentzVector:: kT
          static const enum TLorentzVector:: kNUM_COORDINATES
          static const enum TLorentzVector:: kSIZE

## Class Description

    *-*-*-*-*-*-*-*-*-*-*-*The Physics Vector package *-*-*-*-*-*-*-*-*-*-*-*
    *-*                    ==========================                       *
    *-* The Physics Vector package consists of five classes:                *
    *-*   - TVector2                                                        *
    *-*   - TVector3                                                        *
    *-*   - TRotation                                                       *
    *-*   - TLorentzVector                                                  *
    *-*   - TLorentzRotation                                                *
    *-* It is a combination of CLHEPs Vector package written by             *
    *-* Leif Lonnblad, Andreas Nilsson and Evgueni Tcherniaev               *
    *-* and a ROOT package written by Pasha Murat.                          *
    *-* for CLHEP see:  http://wwwinfo.cern.ch/asd/lhc++/clhep/             *
    *-* Adaption to ROOT by Peter Malzacher                                 *
    *-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*

## TLorentzVector

`TLorentzVector` is a general four-vector class, which can be used either for the description of position and time (x,y,z,t) or momentum and energy (px,py,pz,E).

### Declaration

`TLorentzVector` has been implemented as a set a `TVector3` and a `Double_t` variable. By default all components are initialized by zero.

`TLorentzVector v1; // initialized by (0., 0., 0., 0.)`
`TLorentzVector v2(1., 1., 1., 1.);`
`TLorentzVector v3(v1);`
`TLorentzVector v4(TVector3(1., 2., 3.),4.);`

For backward compatibility there are two constructors from an `Double_t` and `Float_t` C array.

### Access to the components

There are two sets of access functions to the components of a`LorentzVector`: `X(`), `Y()`, `Z()`, `T()` and `Px()`,`Py()`, `Pz()` and `E()`. Both sets return the same values but the first set is more relevant for use where `TLorentzVector` describes a combination of position and time and the second set is more relevant where `TLorentzVector` describes momentum and energy:

`Double_t xx =v.X();`
`...`
`Double_t tt = v.T();`

`Double_t px = v.Px();`
`...`
`Double_t ee = v.E();`

The components of TLorentzVector can also accessed by index:

`xx = v(0); or xx = v[0];`
`yy = v(1); yy = v[1];`
`zz = v(2); zz = v[2];`
`tt = v(3); tt = v[3];`

You can use the `Vect()` member function to get the vector component of `TLorentzVector`:

`TVector3 p = v.Vect();`

For setting components also two sets of member functions can be used: SetX(),.., SetPx(),..:
`v.SetX(1.); or v.SetPx(1.);`
`... ...`
`v.SetT(1.); v.SetE(1.);`

To set more the one component by one call you can use the `SetVect()` function for the `TVector3` part or `SetXYZT()`, `SetPxPyPzE()`. For convenience there is also a `SetXYZM()`:

`v.SetVect(TVector3(1,2,3));`
`v.SetXYZT(x,y,z,t);`
`v.SetPxPyPzE(px,py,pz,e);`
`v.SetXYZM(x,y,z,m); // -> v=(x,y,z,e=Sqrt(x*x+y*y+z*z+m*m))`

### Vector components in noncartesian coordinate systems

There are a couple of memberfunctions to get and set the `TVector3` part of the parameters in
sherical coordinate systems:

`Double_t m, theta, cost, phi, pp, pp2, ppv2, pp2v2;`
`m = v.Rho();`
`t = v.Theta();`
`cost = v.CosTheta();`
`phi = v.Phi();`

`v.SetRho(10.);`
`v.SetTheta(TMath::Pi()*.3);`
`v.SetPhi(TMath::Pi());`

or get infoormation about the r-coordinate in cylindrical systems:

`Double_t pp, pp2, ppv2, pp2v2;`
`pp = v.Perp(); // get transvers component`
`pp2 = v.Perp2(); // get transverse component squared`
`ppv2 = v.Perp(v1); // get transvers component with`
`// respect to another vector`
`pp2v2 = v.Perp(v1);`

for convenience there are two more set functions `SetPtEtaPhiE(pt,eta,phi,e); and SetPtEtaPhiM(pt,eta,phi,m);`

### Arithmetic and comparison operators

The `TLorentzVecto`r class provides operators to add, subtract or compare four-vectors:

`v3 = -v1;`
`v1 = v2+v3;`
`v1+= v3;`
`v1 = v2 + v3;`
`v1-= v3;`

`if (v1 == v2) {...}`
`if(v1 != v3) {...}`

### Magnitude/Invariant mass, beta, gamma, scalar product

The scalar product of two four-vectors is calculated with the (-,-,-,+) metric,
i.e. `s = v1*v2 = t1*t2-x1*x2-y1*y2-z1*z2`
The magnitude squared `mag2` of a four-vector is therfore:
`mag2 = v*v = t*t-x*x-y*y-z*z`
It `mag2` is negative `mag = -Sqrt(-mag*mag)`. The member functions are:

`Double_t s, s2;`
`s = v1.Dot(v2); // scalar product`
`s = v1*v2; // scalar product`
`s2 = v.Mag2(); or s2 = v.M2();`
`s = v.Mag(); s = v.M();`

Since in case of momentum and energy the magnitude has the meaning of invariant mass `TLorentzVector` provides the more meaningful aliases `M2()` and `M()`;

The member functions `Beta()` and `Gamma()` returns `beta` and `gamma = 1/Sqrt(1-beta*beta)`.

### Lorentz boost

A boost in a general direction can be parameterized with three parameters which can be taken as the components of a three vector `b = (bx,by,bz)`. With
`x = (x,y,z) and gamma = 1/Sqrt(1-beta*beta)` (beta being the module of vector b), an arbitary active Lorentz boost transformation (from the rod frame to the original frame) can be written as:
`x = x' + (gamma-1)/(beta*beta) * (b*x') * b + gamma * t' * b`
`t = gamma (t'+ b*x').`

The member function `Boost()` performs a boost transformation from the rod frame to the original frame. `BoostVector()` returns a `TVector3` of the spatial components divided by the time component:

`TVector3 b;`
`v.Boost(bx,by,bz);`
`v.Boost(b);`
`b = v.BoostVector(); // b=(x/t,y/t,z/t)`

### Rotations

There are four sets of functions to rotate the `TVector3` component of a `TLorentzVector`:

##### rotation around axes

`v.RotateX(TMath::Pi()/2.);`
`v.RotateY(.5);`
`v.RotateZ(.99);`

##### rotation around an arbitary axis

`v.Rotate(TMath::Pi()/4., v1); // rotation around v1`

##### transformation from rotated frame

`v.RotateUz(direction); // direction must be a unit TVector3`

##### by TRotation (see TRotation)

`TRotation r;`
`v.Transform(r); or v *= r; // Attention v=M*v`

### Misc

##### Angle between two vectors

`Double_t a = v1.Angle(v2); // get angle between v1 and v2`

##### Light-cone components

Member functions `Plus()` and `Minus(`) return the positive and negative light-cone components:``

`Double_t pcone = v.Plus();`
`Double_t mcone = v.Minus();`

CAVEAT: The values returned are T{+,-}Z. It is known that some authors find it easier to define these components as (T{+,-}Z)/sqrt(2). Thus check what definition is used in the physics you're working in and adapt your code accordingly.

##### Transformation by TLorentzRotation

A general Lorentz transformation see class `TLorentzRotation` can be used by the `Transform()` member function, the `*=` or `*` operator of the `TLorentzRotation` class:``

`TLorentzRotation l;`
`v.Transform(l);`
`v = l*v; or v *= l; // Attention v = l*v`

TLorentzVector(Double_t x, Double_t y, Double_t z, Double_t t) : fP(x,y,z), fE(t)

TLorentzVector(const Double_t * x0) : fP(x0), fE(x0[3])

TLorentzVector(const Float_t * x0) : fP(x0), fE(x0[3])

TLorentzVector(const TVector3 & p, Double_t e) : fP(p), fE(e)

TLorentzVector(const TLorentzVector & p) : TObject(p) , fP(p.Vect()), fE(p.T())

~TLorentzVector()

void Boost(Double_t bx, Double_t by, Double_t bz)

Double_t Rapidity() const

TLorentzVector& Transform(const TLorentzRotation & m)

void Streamer(TBuffer &R__b)

     Stream an object of class TLorentzVector.

### Inline Functions

                   Double_t X() const
                   Double_t Y() const
                   Double_t Z() const
                   Double_t T() const
                       void SetX(Double_t a)
                       void SetY(Double_t a)
                       void SetZ(Double_t a)
                       void SetT(Double_t a)
                   Double_t Px() const
                   Double_t Py() const
                   Double_t Pz() const
                   Double_t P() const
                   Double_t E() const
                   Double_t Energy() const
                       void SetPx(Double_t a)
                       void SetPy(Double_t a)
                       void SetPz(Double_t a)
                       void SetE(Double_t a)
                   TVector3 Vect() const
                       void SetVect(const TVector3& vect3)
                   Double_t Theta() const
                   Double_t CosTheta() const
                   Double_t Phi() const
                   Double_t Rho() const
                       void SetTheta(Double_t theta)
                       void SetPhi(Double_t phi)
                       void SetRho(Double_t rho)
                       void SetPxPyPzE(Double_t px, Double_t py, Double_t pz, Double_t e)
                       void SetXYZT(Double_t x, Double_t y, Double_t z, Double_t t)
                       void SetXYZM(Double_t x, Double_t y, Double_t z, Double_t m)
                       void SetPtEtaPhiM(Double_t pt, Double_t eta, Double_t phi, Double_t m)
                       void SetPtEtaPhiE(Double_t pt, Double_t eta, Double_t phi, Double_t e)
                       void GetXYZT(Double_t* carray) const
                       void GetXYZT(Float_t* carray) const
                   Double_t operator()(int i) const
                   Double_t operator[](int i) const
                  Double_t& operator()(int i)
                  Double_t& operator[](int i)
            TLorentzVector& operator=(const TLorentzVector& q)
             TLorentzVector operator+(const TLorentzVector& q) const
            TLorentzVector& operator+=(const TLorentzVector& q)
             TLorentzVector operator-(const TLorentzVector& q) const
            TLorentzVector& operator-=(const TLorentzVector& q)
             TLorentzVector operator-() const
             TLorentzVector operator*(Double_t a) const
            TLorentzVector& operator*=(Double_t a)
                     Bool_t operator==(const TLorentzVector& q) const
                     Bool_t operator!=(const TLorentzVector& q) const
                   Double_t Perp2() const
                   Double_t Pt() const
                   Double_t Perp() const
                       void SetPerp(Double_t r)
                   Double_t Perp2(const TVector3& v) const
                   Double_t Pt(const TVector3& v) const
                   Double_t Perp(const TVector3& v) const
                   Double_t Et2() const
                   Double_t Et() const
                   Double_t Et2(const TVector3& v) const
                   Double_t Et(const TVector3& v) const
                   Double_t DeltaPhi(const TLorentzVector& v) const
                   Double_t DeltaR(const TLorentzVector& v) const
                   Double_t DrEtaPhi(const TLorentzVector& v) const
                   TVector2 EtaPhiVector()
                   Double_t Angle(const TVector3& v) const
                   Double_t Mag2() const
                   Double_t M2() const
                   Double_t Mag() const
                   Double_t M() const
                   Double_t Mt2() const
                   Double_t Mt() const
                   Double_t Beta() const
                   Double_t Gamma() const
                   Double_t Dot(const TLorentzVector& q) const
                   Double_t operator*(const TLorentzVector& q) const
                       void SetVectMag(const TVector3& spatial, Double_t magnitude)
                       void SetVectM(const TVector3& spatial, Double_t mass)
                   Double_t Plus() const
                   Double_t Minus() const
                   TVector3 BoostVector() const
                       void Boost(const TVector3& b)
                   Double_t Eta() const
                   Double_t PseudoRapidity() const
                       void RotateX(Double_t angle)
                       void RotateY(Double_t angle)
                       void RotateZ(Double_t angle)
                       void RotateUz(TVector3& newUzVector)
                       void Rotate(Double_t a, const TVector3& v)
            TLorentzVector& operator*=(const TRotation& m)
            TLorentzVector& operator*=(const TLorentzRotation&)
            TLorentzVector& Transform(const TLorentzRotation&)
                    TClass* Class()
                    TClass* IsA() const
                       void ShowMembers(TMemberInspector& insp, char* parent)
                       void StreamerNVirtual(TBuffer& b)

Author: Pasha Murat, Peter Malzacher 12/02/99
Last update: root/physics:$Name: $:$Id: TLorentzVector.cxx,v 1.9 2004/04/20 09:29:56 brun Exp $