1. Tutorial 00 Analysis with HEP Objects
URL with complete information:
Small overview of what types of Objects exist
3 and 4 vectors, displacement vectors
Lorentz boosts and rotations
3D transformations.
Using these objects to do some typical physics calculations
Invariant mass
Impact parameter between a vector and a point in space.
Helicity angle in Ds  fp
Transversity angle in B J/y f
“addpkg Examples” and look in TUTORIAL00/
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2. Types of Objects in CLHEP
Hep3Vector A(ax, ay, az), B(bx, by, bz);
The vector A can be initialized with double’s.
Add and subtract these vectors, scalar multiplication
Double d;
Hep3Vector C = d*A + B/d;
Take the dot product:
Double ABcostheta = A.dot(B);
Take the cross product:
Hep3Vector AcrossB = A.cross(B);
Find the Unit vector in the same direction as vector A.
Hep3Vector UnitA = A.unit();
Find Magnitude of the Vector A:
HepDouble MagA = A.mag();
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3. Types of Objects in CLHEP
LorentzVector A(px, py, pz, Ea), B(px, py, pz, Eb);
The Lorentz vector A can be initialized with double’s.
Add and subtract these vectors, scalar multiplication
Double d;
LorentzVector C = d*A + B/d;
Find the Mass of A or of the combination
HepDouble mass = A.m();
HepDouble invmass = (A+B).m();
Get 3-vector direction of the Lorentz vector
Hep3Vector Adir = A.vect();
Four Vector dot product
HepDouble invariantC = A.dot(B);
Find Boost vector in for this object
Hep3Vector velocityA = A.boostVector();
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4. Impact parameter: vector to a point.?
Given a position and direction of particle, what is the Impact Paramter?
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5. Impact parameter: vector to a point.
#include "CLHEP/Geometry/Point3D.h"
#include "CLHEP/Geometry/Vector3D.h"
int main(int, char **) {
HepPoint3D pointOnTrack;
HepPoint3D pointInSpace;
HepVector3D directionOfTrack;
// Impact parameter in three D:
HepVector3D impactVect = (pointOnTrack - pointInSpace).cross(directionOfTrack);
double i2d = impactVect.z();
double i3d = impactVect.mag(); } ?
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6. Transformations: Different kinds of Transformations:
HepTransform3D, HepTranslate3D, HepRotateX3D, HepRotateY3D, HepReflectX3D, HepScale3D…
Can compose, e.g:
HepTransform3D xform= HepTranslate3X3(5.0)*HepRotateY3D(M_PI)*HepRotateZ3D(M_PI/2);
Can invert:
HepTransform xformInverse = xform.inverse();
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7. Three kinds of vectors:
Point (affected by translation, rotations, reflections)
Vector(affected by rotations and reflections)
Normal(affected by rotations)
All of these have exactly the same functions as Hep3Vector
Each give a different result when transformed
E.G Move the origin of coordinates by 1 mm, due to a beam spot shift:
HepPoint3D pointOnTrack;
HepPoint3D pointInSpace;
HepVector3D directionOfTrack;
HepTranslate3D shift(0.1);
HepPoint3D pOTPrime = shift*pointOnTrack; // moves!
HepPoint3D pISPrime = shift *pointInSpace; // moves!
HepVector3D doTPrime = shift *directionOfTrack; // does not move!
Hep3Vector
HepPoint3D
HepNormal3D
HepVector3D
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8. * More specialized: Lorentz transformations:
* Act on Lorentz transfomations, Lorentz vectors
* Many ways to construct:
Most useful: from a vector that gives (bx, by, bz)
Can get this from a fourVector:
// Get the 3-velocity of phi in the lab frame:
Hep3Vector phiVelocity = Phi.boostVector();
// Create a boost from the Phi Rest Frame to the Lab:
HepLorentzRotation phiBoost(phiVelocity);
* Can invert:
// Create a boost from the Lab to the Phi Rest Frame;
HepLorentzRotation toPhiRest = phiBoost.inverse();
* More specialized:
// Create a boost from B Rest Frame to D Rest Frame;
HepLorentzRotation BToD = boostToD*boostToB.inverse();
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9. Angle of p- wrt the K- in the f rest frame
Helicity Angle: Ds fp
K+K- K- K+ p-
Angle of p- wrt the K- in the f rest frame
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10. Helicity Angle: Ds fp K- Ds- K+
// Get these vectors from the tracks, in the lab frame:
HepLorentzVector KPlus;
HepLorentzVector KMinus;
HepLorentzVector PiMinus;
// Compute the Phi Four Momentum:
HepLorentzVector Phi = KPlus + KMinus;
// Get the D Four vector:
HepLorentzVector Ds = Phi+PiMinus;
// Get the 3-velocity of phi in the lab frame:
Hep3Vector phiVelocity = Phi.boostVector();
// Create a boost to the rest frame of the phi:
HepLorentzRotation phiBoost(phiVelocity);
HepLorentzRotation toPhiRest = phiBoost.inverse();
// Get the direction of the KMinus and the pion in the phi rest frame:
Hep3Vector KMPrimeDir = (toPhiRest*KMinus).vect().unit();
Hep3Vector DsPrimeDir = (toPhiRest*Ds).vect().unit();
double cosHelicity = KMPrimeDir.dot(DsPrimeDir); K- K+
Ds-
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11. Angle between m+ and the normal to the K+K- plane in the
Transversity Angle
K+K-
Bs  J/ f
m+ m- x z K- 1- f K+
Angle between m+ and the normal to the K+K- plane in the
J/ rest frame
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12. Transversity Angle
// Get these vectors from the tracks, in the lab frame:
HepLorentzVector KPlus;
HepLorentzVector KMinus;
HepLorentzVector MuPlus;
HepLorentzVector MuMinus;
// Compute the JPsi Four Momentum:
HepLorentzVector JPsi = MuPlus + MuMinus;
// Get the 3-velocity of J/psi in the lab frame:
Hep3Vector JPsiVelocity = JPsi.boostVector();
// Create a boost to the rest frame of the J/psi:
HepLorentzRotation JPsiBoost(JPsiVelocity);
HepLorentzRotation toJPsiRest = JPsiBoost.inverse();
// Find the Kaon and Muon direction vectors in the J/psi rest frame
Hep3Vector KPlusPrime = (toJPsiRest*KPlus).vect();
Hep3Vector KMinusPrime = (toJPsiRest*KMinus).vect();
Hep3Vector MuPlusPrime = (toJPsiRest*MuPlus).vect();
// Compute the transversity angle:
double cosTransversity = KMinusPrime.cross(KPlusPrime).unit().dot(MuPlusPrime.unit());
Angle between m+ and the normal to the K+K- plane in the J/ rest frame
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13. To be continued: class CdfTrack { Hep3Vector getMomentum() const;
HepLorentzVector getFourMomentum() const;
+ more! };
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