// @(#)root/physics:$Id$ // Author: Pasha Murat , Peter Malzacher 12/02/99 // Oct 8 1999: changed Warning to Error and // return fX in Double_t & operator() // Oct 20 1999: dito in Double_t operator() // Jan 25 2000: implemented as (fP,fE) instead of (fX,fY,fZ,fE) //______________________________________________________________________________ //*-*-*-*-*-*-*-*-*-*-*-*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 * //*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* //BEGIN_HTML

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 TLorentzVector 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.Vect()); // 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

END_HTML #include "TClass.h" #include "TError.h" #include "TLorentzVector.h" #include "TLorentzRotation.h" ClassImp(TLorentzVector) TLorentzVector::TLorentzVector(Double_t x, Double_t y, Double_t z, Double_t t) : fP(x,y,z), fE(t) {} TLorentzVector::TLorentzVector(const Double_t * x0) : fP(x0), fE(x0[3]) {} TLorentzVector::TLorentzVector(const Float_t * x0) : fP(x0), fE(x0[3]) {} TLorentzVector::TLorentzVector(const TVector3 & p, Double_t e) : fP(p), fE(e) {} TLorentzVector::TLorentzVector(const TLorentzVector & p) : TObject(p) , fP(p.Vect()), fE(p.T()) {} TLorentzVector::~TLorentzVector() {} Double_t TLorentzVector::operator () (int i) const { //dereferencing operator const switch(i) { case kX: case kY: case kZ: return fP(i); case kT: return fE; default: Error("operator()()", "bad index (%d) returning 0",i); } return 0.; } Double_t & TLorentzVector::operator () (int i) { //dereferencing operator switch(i) { case kX: case kY: case kZ: return fP(i); case kT: return fE; default: Error("operator()()", "bad index (%d) returning &fE",i); } return fE; } void TLorentzVector::Boost(Double_t bx, Double_t by, Double_t bz) { //Boost this Lorentz vector Double_t b2 = bx*bx + by*by + bz*bz; register Double_t gamma = 1.0 / TMath::Sqrt(1.0 - b2); register Double_t bp = bx*X() + by*Y() + bz*Z(); register Double_t gamma2 = b2 > 0 ? (gamma - 1.0)/b2 : 0.0; SetX(X() + gamma2*bp*bx + gamma*bx*T()); SetY(Y() + gamma2*bp*by + gamma*by*T()); SetZ(Z() + gamma2*bp*bz + gamma*bz*T()); SetT(gamma*(T() + bp)); } Double_t TLorentzVector::Rapidity() const { //return rapidity return 0.5*log( (E()+Pz()) / (E()-Pz()) ); } TLorentzVector &TLorentzVector::operator *= (const TLorentzRotation & m) { //multiply this Lorentzvector by m return *this = m.VectorMultiplication(*this); } TLorentzVector &TLorentzVector::Transform(const TLorentzRotation & m) { //Transform this Lorentzvector return *this = m.VectorMultiplication(*this); } void TLorentzVector::Streamer(TBuffer &R__b) { // Stream an object of class TLorentzVector. Double_t x, y, z; UInt_t R__s, R__c; if (R__b.IsReading()) { Version_t R__v = R__b.ReadVersion(&R__s, &R__c); if (R__v > 3) { R__b.ReadClassBuffer(TLorentzVector::Class(), this, R__v, R__s, R__c); return; } //====process old versions before automatic schema evolution if (R__v != 2) TObject::Streamer(R__b); R__b >> x; R__b >> y; R__b >> z; fP.SetXYZ(x,y,z); R__b >> fE; R__b.CheckByteCount(R__s, R__c, TLorentzVector::IsA()); } else { R__b.WriteClassBuffer(TLorentzVector::Class(),this); } } //______________________________________________________________________________ void TLorentzVector::Print(Option_t *) const { // Print the TLorentz vector components as (x,y,z,t) and (P,eta,phi,E) representations Printf("(x,y,z,t)=(%f,%f,%f,%f) (P,eta,phi,E)=(%f,%f,%f,%f)", fP.x(),fP.y(),fP.z(),fE, P(),Eta(),Phi(),fE); }