1 /**************************************************************************
2 * Copyright(c) 1998-1999, ALICE Experiment at CERN, All rights reserved. *
4 * Author: The ALICE Off-line Project. *
5 * Contributors are mentioned in the code where appropriate. *
7 * Permission to use, copy, modify and distribute this software and its *
8 * documentation strictly for non-commercial purposes is hereby granted *
9 * without fee, provided that the above copyright notice appears in all *
10 * copies and that both the copyright notice and this permission notice *
11 * appear in the supporting documentation. The authors make no claims *
12 * about the suitability of this software for any purpose. It is *
13 * provided "as is" without express or implied warranty. *
14 **************************************************************************/
18 //-----------------------------------------------------------------------------
19 // Class AliMUONTrackExtrap
20 // ------------------------
21 // Tools for track extrapolation in ALICE dimuon spectrometer
22 // Author: Philippe Pillot
23 //-----------------------------------------------------------------------------
25 #include "AliMUONTrackExtrap.h"
26 #include "AliMUONTrackParam.h"
27 #include "AliMUONConstants.h"
28 #include "AliMUONReconstructor.h"
31 #include "AliExternalTrackParam.h"
33 #include <TGeoGlobalMagField.h>
34 #include <TGeoManager.h>
36 #include <TDatabasePDG.h>
38 #include <Riostream.h>
41 ClassImp(AliMUONTrackExtrap) // Class implementation in ROOT context
44 const Double_t AliMUONTrackExtrap::fgkSimpleBPosition = 0.5 * (AliMUONConstants::CoilZ() + AliMUONConstants::YokeZ());
45 const Double_t AliMUONTrackExtrap::fgkSimpleBLength = 0.5 * (AliMUONConstants::CoilL() + AliMUONConstants::YokeL());
46 Double_t AliMUONTrackExtrap::fgSimpleBValue = 0.;
47 Bool_t AliMUONTrackExtrap::fgFieldON = kFALSE;
48 const Bool_t AliMUONTrackExtrap::fgkUseHelix = kFALSE;
49 const Int_t AliMUONTrackExtrap::fgkMaxStepNumber = 5000;
50 const Double_t AliMUONTrackExtrap::fgkHelixStepLength = 6.;
51 const Double_t AliMUONTrackExtrap::fgkRungeKuttaMaxResidue = 0.002;
53 //__________________________________________________________________________
54 void AliMUONTrackExtrap::SetField()
56 /// set field on/off flag;
57 /// set field at the centre of the dipole
58 const Double_t x[3] = {50.,50.,fgkSimpleBPosition};
59 Double_t b[3] = {0.,0.,0.};
60 TGeoGlobalMagField::Instance()->Field(x,b);
61 fgSimpleBValue = b[0];
62 fgFieldON = fgSimpleBValue ? kTRUE : kFALSE;
66 //__________________________________________________________________________
67 Double_t AliMUONTrackExtrap::GetImpactParamFromBendingMomentum(Double_t bendingMomentum)
69 /// Returns impact parameter at vertex in bending plane (cm),
70 /// from the signed bending momentum "BendingMomentum" in bending plane (GeV/c),
71 /// using simple values for dipole magnetic field.
72 /// The sign of "BendingMomentum" is the sign of the charge.
74 if (bendingMomentum == 0.) return 1.e10;
76 const Double_t kCorrectionFactor = 1.1; // impact parameter is 10% underestimated
78 return kCorrectionFactor * (-0.0003 * fgSimpleBValue * fgkSimpleBLength * fgkSimpleBPosition / bendingMomentum);
81 //__________________________________________________________________________
83 AliMUONTrackExtrap::GetBendingMomentumFromImpactParam(Double_t impactParam)
85 /// Returns signed bending momentum in bending plane (GeV/c),
86 /// the sign being the sign of the charge for particles moving forward in Z,
87 /// from the impact parameter "ImpactParam" at vertex in bending plane (cm),
88 /// using simple values for dipole magnetic field.
90 if (impactParam == 0.) return 1.e10;
92 const Double_t kCorrectionFactor = 1.1; // bending momentum is 10% underestimated
96 return kCorrectionFactor * (-0.0003 * fgSimpleBValue * fgkSimpleBLength * fgkSimpleBPosition / impactParam);
100 return AliMUONConstants::GetMostProbBendingMomentum();
104 //__________________________________________________________________________
105 void AliMUONTrackExtrap::LinearExtrapToZ(AliMUONTrackParam* trackParam, Double_t zEnd)
107 /// Track parameters linearly extrapolated to the plane at "zEnd".
108 /// On return, results from the extrapolation are updated in trackParam.
110 if (trackParam->GetZ() == zEnd) return; // nothing to be done if same z
112 // Compute track parameters
113 Double_t dZ = zEnd - trackParam->GetZ();
114 trackParam->SetNonBendingCoor(trackParam->GetNonBendingCoor() + trackParam->GetNonBendingSlope() * dZ);
115 trackParam->SetBendingCoor(trackParam->GetBendingCoor() + trackParam->GetBendingSlope() * dZ);
116 trackParam->SetZ(zEnd);
119 //__________________________________________________________________________
120 void AliMUONTrackExtrap::LinearExtrapToZCov(AliMUONTrackParam* trackParam, Double_t zEnd, Bool_t updatePropagator)
122 /// Track parameters and their covariances linearly extrapolated to the plane at "zEnd".
123 /// On return, results from the extrapolation are updated in trackParam.
125 if (trackParam->GetZ() == zEnd) return; // nothing to be done if same z
127 // No need to propagate the covariance matrix if it does not exist
128 if (!trackParam->CovariancesExist()) {
129 cout<<"W-AliMUONTrackExtrap::LinearExtrapToZCov: Covariance matrix does not exist"<<endl;
130 // Extrapolate linearly track parameters to "zEnd"
131 LinearExtrapToZ(trackParam,zEnd);
135 // Compute track parameters
136 Double_t dZ = zEnd - trackParam->GetZ();
137 trackParam->SetNonBendingCoor(trackParam->GetNonBendingCoor() + trackParam->GetNonBendingSlope() * dZ);
138 trackParam->SetBendingCoor(trackParam->GetBendingCoor() + trackParam->GetBendingSlope() * dZ);
139 trackParam->SetZ(zEnd);
141 // Calculate the jacobian related to the track parameters linear extrapolation to "zEnd"
147 // Extrapolate track parameter covariances to "zEnd"
148 TMatrixD tmp(trackParam->GetCovariances(),TMatrixD::kMultTranspose,jacob);
149 TMatrixD tmp2(jacob,TMatrixD::kMult,tmp);
150 trackParam->SetCovariances(tmp2);
152 // Update the propagator if required
153 if (updatePropagator) trackParam->UpdatePropagator(jacob);
156 //__________________________________________________________________________
157 void AliMUONTrackExtrap::ExtrapToZ(AliMUONTrackParam* trackParam, Double_t zEnd)
159 /// Interface to track parameter extrapolation to the plane at "Z" using Helix or Rungekutta algorithm.
160 /// On return, the track parameters resulting from the extrapolation are updated in trackParam.
161 if (!fgFieldON) AliMUONTrackExtrap::LinearExtrapToZ(trackParam,zEnd);
162 else if (fgkUseHelix) AliMUONTrackExtrap::ExtrapToZHelix(trackParam,zEnd);
163 else AliMUONTrackExtrap::ExtrapToZRungekutta(trackParam,zEnd);
166 //__________________________________________________________________________
167 void AliMUONTrackExtrap::ExtrapToZHelix(AliMUONTrackParam* trackParam, Double_t zEnd)
169 /// Track parameter extrapolation to the plane at "Z" using Helix algorithm.
170 /// On return, the track parameters resulting from the extrapolation are updated in trackParam.
171 if (trackParam->GetZ() == zEnd) return; // nothing to be done if same Z
172 Double_t forwardBackward; // +1 if forward, -1 if backward
173 if (zEnd < trackParam->GetZ()) forwardBackward = 1.0; // spectro. z<0
174 else forwardBackward = -1.0;
175 Double_t v3[7], v3New[7]; // 7 in parameter ????
176 Int_t i3, stepNumber;
177 // For safety: return kTRUE or kFALSE ????
178 // Parameter vector for calling EXTRAP_ONESTEP
179 ConvertTrackParamForExtrap(trackParam, forwardBackward, v3);
180 // sign of charge (sign of fInverseBendingMomentum if forward motion)
181 // must be changed if backward extrapolation
182 Double_t chargeExtrap = forwardBackward * TMath::Sign(Double_t(1.0), trackParam->GetInverseBendingMomentum());
183 // Extrapolation loop
185 while (((-forwardBackward * (v3[2] - zEnd)) <= 0.0) && (stepNumber < fgkMaxStepNumber)) { // spectro. z<0
187 ExtrapOneStepHelix(chargeExtrap, fgkHelixStepLength, v3, v3New);
188 if ((-forwardBackward * (v3New[2] - zEnd)) > 0.0) break; // one is beyond Z spectro. z<0
189 // better use TArray ????
190 for (i3 = 0; i3 < 7; i3++) {v3[i3] = v3New[i3];}
192 // check fgkMaxStepNumber ????
193 // Interpolation back to exact Z (2nd order)
194 // should be in function ???? using TArray ????
195 Double_t dZ12 = v3New[2] - v3[2]; // 1->2
196 if (TMath::Abs(dZ12) > 0) {
197 Double_t dZ1i = zEnd - v3[2]; // 1-i
198 Double_t dZi2 = v3New[2] - zEnd; // i->2
199 Double_t xPrime = (v3New[0] - v3[0]) / dZ12;
200 Double_t xSecond = ((v3New[3] / v3New[5]) - (v3[3] / v3[5])) / dZ12;
201 Double_t yPrime = (v3New[1] - v3[1]) / dZ12;
202 Double_t ySecond = ((v3New[4] / v3New[5]) - (v3[4] / v3[5])) / dZ12;
203 v3[0] = v3[0] + xPrime * dZ1i - 0.5 * xSecond * dZ1i * dZi2; // X
204 v3[1] = v3[1] + yPrime * dZ1i - 0.5 * ySecond * dZ1i * dZi2; // Y
206 Double_t xPrimeI = xPrime - 0.5 * xSecond * (dZi2 - dZ1i);
207 Double_t yPrimeI = yPrime - 0.5 * ySecond * (dZi2 - dZ1i);
208 // (PX, PY, PZ)/PTOT assuming forward motion
209 v3[5] = 1.0 / TMath::Sqrt(1.0 + xPrimeI * xPrimeI + yPrimeI * yPrimeI); // PZ/PTOT
210 v3[3] = xPrimeI * v3[5]; // PX/PTOT
211 v3[4] = yPrimeI * v3[5]; // PY/PTOT
213 cout<<"W-AliMUONTrackExtrap::ExtrapToZHelix: Extrap. to Z not reached, Z = "<<zEnd<<endl;
215 // Recover track parameters (charge back for forward motion)
216 RecoverTrackParam(v3, chargeExtrap * forwardBackward, trackParam);
219 //__________________________________________________________________________
220 void AliMUONTrackExtrap::ExtrapToZRungekutta(AliMUONTrackParam* trackParam, Double_t zEnd)
222 /// Track parameter extrapolation to the plane at "Z" using Rungekutta algorithm.
223 /// On return, the track parameters resulting from the extrapolation are updated in trackParam.
224 if (trackParam->GetZ() == zEnd) return; // nothing to be done if same Z
225 Double_t forwardBackward; // +1 if forward, -1 if backward
226 if (zEnd < trackParam->GetZ()) forwardBackward = 1.0; // spectro. z<0
227 else forwardBackward = -1.0;
228 // sign of charge (sign of fInverseBendingMomentum if forward motion)
229 // must be changed if backward extrapolation
230 Double_t chargeExtrap = forwardBackward * TMath::Sign(Double_t(1.0), trackParam->GetInverseBendingMomentum());
231 Double_t v3[7], v3New[7];
233 Int_t stepNumber = 0;
235 // Extrapolation loop (until within tolerance)
236 Double_t residue = zEnd - trackParam->GetZ();
237 while (TMath::Abs(residue) > fgkRungeKuttaMaxResidue && stepNumber <= fgkMaxStepNumber) {
238 dZ = zEnd - trackParam->GetZ();
239 // step lenght assuming linear trajectory
240 step = dZ * TMath::Sqrt(1.0 + trackParam->GetBendingSlope()*trackParam->GetBendingSlope() +
241 trackParam->GetNonBendingSlope()*trackParam->GetNonBendingSlope());
242 ConvertTrackParamForExtrap(trackParam, forwardBackward, v3);
243 do { // reduce step lenght while zEnd oversteped
244 if (stepNumber > fgkMaxStepNumber) {
245 cout<<"W-AliMUONTrackExtrap::ExtrapToZRungekutta: Too many trials: "<<stepNumber<<endl;
249 step = TMath::Abs(step);
250 AliMUONTrackExtrap::ExtrapOneStepRungekutta(chargeExtrap,step,v3,v3New);
251 residue = zEnd - v3New[2];
252 step *= dZ/(v3New[2]-trackParam->GetZ());
253 } while (residue*dZ < 0 && TMath::Abs(residue) > fgkRungeKuttaMaxResidue);
254 RecoverTrackParam(v3New, chargeExtrap * forwardBackward, trackParam);
257 // terminate the extropolation with a straight line up to the exact "zEnd" value
258 trackParam->SetNonBendingCoor(trackParam->GetNonBendingCoor() + residue * trackParam->GetNonBendingSlope());
259 trackParam->SetBendingCoor(trackParam->GetBendingCoor() + residue * trackParam->GetBendingSlope());
260 trackParam->SetZ(zEnd);
263 //__________________________________________________________________________
264 void AliMUONTrackExtrap::ConvertTrackParamForExtrap(AliMUONTrackParam* trackParam, Double_t forwardBackward, Double_t *v3)
266 /// Set vector of Geant3 parameters pointed to by "v3" from track parameters in trackParam.
267 /// Since AliMUONTrackParam is only geometry, one uses "forwardBackward"
268 /// to know whether the particle is going forward (+1) or backward (-1).
269 v3[0] = trackParam->GetNonBendingCoor(); // X
270 v3[1] = trackParam->GetBendingCoor(); // Y
271 v3[2] = trackParam->GetZ(); // Z
272 Double_t pYZ = TMath::Abs(1.0 / trackParam->GetInverseBendingMomentum());
273 Double_t pZ = pYZ / TMath::Sqrt(1.0 + trackParam->GetBendingSlope() * trackParam->GetBendingSlope());
274 v3[6] = TMath::Sqrt(pYZ * pYZ + pZ * pZ * trackParam->GetNonBendingSlope() * trackParam->GetNonBendingSlope()); // PTOT
275 v3[5] = -forwardBackward * pZ / v3[6]; // PZ/PTOT spectro. z<0
276 v3[3] = trackParam->GetNonBendingSlope() * v3[5]; // PX/PTOT
277 v3[4] = trackParam->GetBendingSlope() * v3[5]; // PY/PTOT
280 //__________________________________________________________________________
281 void AliMUONTrackExtrap::RecoverTrackParam(Double_t *v3, Double_t charge, AliMUONTrackParam* trackParam)
283 /// Set track parameters in trackParam from Geant3 parameters pointed to by "v3",
284 /// assumed to be calculated for forward motion in Z.
285 /// "InverseBendingMomentum" is signed with "charge".
286 trackParam->SetNonBendingCoor(v3[0]); // X
287 trackParam->SetBendingCoor(v3[1]); // Y
288 trackParam->SetZ(v3[2]); // Z
289 Double_t pYZ = v3[6] * TMath::Sqrt((1.-v3[3])*(1.+v3[3]));
290 trackParam->SetInverseBendingMomentum(charge/pYZ);
291 trackParam->SetBendingSlope(v3[4]/v3[5]);
292 trackParam->SetNonBendingSlope(v3[3]/v3[5]);
295 //__________________________________________________________________________
296 void AliMUONTrackExtrap::ExtrapToZCov(AliMUONTrackParam* trackParam, Double_t zEnd, Bool_t updatePropagator)
298 /// Track parameters and their covariances extrapolated to the plane at "zEnd".
299 /// On return, results from the extrapolation are updated in trackParam.
301 if (trackParam->GetZ() == zEnd) return; // nothing to be done if same z
303 if (!fgFieldON) { // linear extrapolation if no magnetic field
304 AliMUONTrackExtrap::LinearExtrapToZCov(trackParam,zEnd,updatePropagator);
308 // No need to propagate the covariance matrix if it does not exist
309 if (!trackParam->CovariancesExist()) {
310 cout<<"W-AliMUONTrackExtrap::ExtrapToZCov: Covariance matrix does not exist"<<endl;
311 // Extrapolate track parameters to "zEnd"
312 ExtrapToZ(trackParam,zEnd);
316 // Save the actual track parameters
317 AliMUONTrackParam trackParamSave(*trackParam);
318 TMatrixD paramSave(trackParamSave.GetParameters());
319 Double_t zBegin = trackParamSave.GetZ();
321 // Get reference to the parameter covariance matrix
322 const TMatrixD& kParamCov = trackParam->GetCovariances();
324 // Extrapolate track parameters to "zEnd"
325 ExtrapToZ(trackParam,zEnd);
327 // Get reference to the extrapolated parameters
328 const TMatrixD& extrapParam = trackParam->GetParameters();
330 // Calculate the jacobian related to the track parameters extrapolation to "zEnd"
333 TMatrixD dParam(5,1);
334 Double_t direction[5] = {-1.,-1.,1.,1.,-1.};
335 for (Int_t i=0; i<5; i++) {
336 // Skip jacobian calculation for parameters with no associated error
337 if (kParamCov(i,i) <= 0.) continue;
339 // Small variation of parameter i only
340 for (Int_t j=0; j<5; j++) {
342 dParam(j,0) = TMath::Sqrt(kParamCov(i,i));
343 dParam(j,0) *= TMath::Sign(1.,direction[j]*paramSave(j,0)); // variation always in the same direction
344 } else dParam(j,0) = 0.;
347 // Set new parameters
348 trackParamSave.SetParameters(paramSave);
349 trackParamSave.AddParameters(dParam);
350 trackParamSave.SetZ(zBegin);
352 // Extrapolate new track parameters to "zEnd"
353 ExtrapToZ(&trackParamSave,zEnd);
355 // Calculate the jacobian
356 TMatrixD jacobji(trackParamSave.GetParameters(),TMatrixD::kMinus,extrapParam);
357 jacobji *= 1. / dParam(i,0);
358 jacob.SetSub(0,i,jacobji);
361 // Extrapolate track parameter covariances to "zEnd"
362 TMatrixD tmp(kParamCov,TMatrixD::kMultTranspose,jacob);
363 TMatrixD tmp2(jacob,TMatrixD::kMult,tmp);
364 trackParam->SetCovariances(tmp2);
366 // Update the propagator if required
367 if (updatePropagator) trackParam->UpdatePropagator(jacob);
370 //__________________________________________________________________________
371 void AliMUONTrackExtrap::AddMCSEffectInAbsorber(AliMUONTrackParam* param, Double_t pathLength, Double_t f0, Double_t f1, Double_t f2)
373 /// Add to the track parameter covariances the effects of multiple Coulomb scattering
374 /// The absorber correction parameters are supposed to be calculated at the current track z-position
376 // absorber related covariance parameters
377 Double_t bendingSlope = param->GetBendingSlope();
378 Double_t nonBendingSlope = param->GetNonBendingSlope();
379 Double_t inverseBendingMomentum = param->GetInverseBendingMomentum();
380 Double_t alpha2 = 0.0136 * 0.0136 * inverseBendingMomentum * inverseBendingMomentum * (1.0 + bendingSlope * bendingSlope) /
381 (1.0 + bendingSlope *bendingSlope + nonBendingSlope * nonBendingSlope); // velocity = 1
382 Double_t varCoor = alpha2 * (pathLength * pathLength * f0 - 2. * pathLength * f1 + f2);
383 Double_t covCorrSlope = alpha2 * (pathLength * f0 - f1);
384 Double_t varSlop = alpha2 * f0;
386 // Set MCS covariance matrix
387 TMatrixD newParamCov(param->GetCovariances());
389 newParamCov(0,0) += varCoor; newParamCov(0,1) += covCorrSlope;
390 newParamCov(1,0) += covCorrSlope; newParamCov(1,1) += varSlop;
392 newParamCov(2,2) += varCoor; newParamCov(2,3) += covCorrSlope;
393 newParamCov(3,2) += covCorrSlope; newParamCov(3,3) += varSlop;
395 // Set momentum related covariances if B!=0
397 // compute derivative d(q/Pxy) / dSlopeX and d(q/Pxy) / dSlopeY
398 Double_t dqPxydSlopeX = inverseBendingMomentum * nonBendingSlope / (1. + nonBendingSlope*nonBendingSlope + bendingSlope*bendingSlope);
399 Double_t dqPxydSlopeY = - inverseBendingMomentum * nonBendingSlope*nonBendingSlope * bendingSlope /
400 (1. + bendingSlope*bendingSlope) / (1. + nonBendingSlope*nonBendingSlope + bendingSlope*bendingSlope);
401 // Inverse bending momentum (due to dependences with bending and non bending slopes)
402 newParamCov(4,0) += dqPxydSlopeX * covCorrSlope; newParamCov(0,4) += dqPxydSlopeX * covCorrSlope;
403 newParamCov(4,1) += dqPxydSlopeX * varSlop; newParamCov(1,4) += dqPxydSlopeX * varSlop;
404 newParamCov(4,2) += dqPxydSlopeY * covCorrSlope; newParamCov(2,4) += dqPxydSlopeY * covCorrSlope;
405 newParamCov(4,3) += dqPxydSlopeY * varSlop; newParamCov(3,4) += dqPxydSlopeY * varSlop;
406 newParamCov(4,4) += (dqPxydSlopeX*dqPxydSlopeX + dqPxydSlopeY*dqPxydSlopeY) * varSlop;
409 // Set new covariances
410 param->SetCovariances(newParamCov);
413 //__________________________________________________________________________
414 void AliMUONTrackExtrap::CorrectMCSEffectInAbsorber(AliMUONTrackParam* param,
415 Double_t xVtx, Double_t yVtx, Double_t zVtx,
416 Double_t errXVtx, Double_t errYVtx,
417 Double_t absZBeg, Double_t pathLength, Double_t f0, Double_t f1, Double_t f2)
419 /// Correct parameters and corresponding covariances using Branson correction
420 /// - input param are parameters and covariances at the end of absorber
421 /// - output param are parameters and covariances at vertex
422 /// Absorber correction parameters are supposed to be calculated at the current track z-position
424 // Position of the Branson plane (spectro. (z<0))
425 Double_t zB = (f1>0.) ? absZBeg - f2/f1 : 0.;
427 // Add MCS effects to current parameter covariances
428 AddMCSEffectInAbsorber(param, pathLength, f0, f1, f2);
430 // Get track parameters and covariances in the Branson plane corrected for magnetic field effect
431 ExtrapToZCov(param,zVtx);
432 LinearExtrapToZCov(param,zB);
434 // compute track parameters at vertex
435 TMatrixD newParam(5,1);
436 newParam(0,0) = xVtx;
437 newParam(1,0) = (param->GetNonBendingCoor() - xVtx) / (zB - zVtx);
438 newParam(2,0) = yVtx;
439 newParam(3,0) = (param->GetBendingCoor() - yVtx) / (zB - zVtx);
440 newParam(4,0) = param->GetCharge() / param->P() *
441 TMath::Sqrt(1.0 + newParam(1,0)*newParam(1,0) + newParam(3,0)*newParam(3,0)) /
442 TMath::Sqrt(1.0 + newParam(3,0)*newParam(3,0));
444 // Get covariances in (X, SlopeX, Y, SlopeY, q*PTot) coordinate system
445 TMatrixD paramCovP(param->GetCovariances());
446 Cov2CovP(param->GetParameters(),paramCovP);
448 // Get the covariance matrix in the (XVtx, X, YVtx, Y, q*PTot) coordinate system
449 TMatrixD paramCovVtx(5,5);
451 paramCovVtx(0,0) = errXVtx * errXVtx;
452 paramCovVtx(1,1) = paramCovP(0,0);
453 paramCovVtx(2,2) = errYVtx * errYVtx;
454 paramCovVtx(3,3) = paramCovP(2,2);
455 paramCovVtx(4,4) = paramCovP(4,4);
456 paramCovVtx(1,3) = paramCovP(0,2);
457 paramCovVtx(3,1) = paramCovP(2,0);
458 paramCovVtx(1,4) = paramCovP(0,4);
459 paramCovVtx(4,1) = paramCovP(4,0);
460 paramCovVtx(3,4) = paramCovP(2,4);
461 paramCovVtx(4,3) = paramCovP(4,2);
463 // Jacobian of the transformation (XVtx, X, YVtx, Y, q*PTot) -> (XVtx, SlopeXVtx, YVtx, SlopeYVtx, q*PTotVtx)
466 jacob(1,0) = - 1. / (zB - zVtx);
467 jacob(1,1) = 1. / (zB - zVtx);
468 jacob(3,2) = - 1. / (zB - zVtx);
469 jacob(3,3) = 1. / (zB - zVtx);
471 // Compute covariances at vertex in the (XVtx, SlopeXVtx, YVtx, SlopeYVtx, q*PTotVtx) coordinate system
472 TMatrixD tmp(paramCovVtx,TMatrixD::kMultTranspose,jacob);
473 TMatrixD newParamCov(jacob,TMatrixD::kMult,tmp);
475 // Compute covariances at vertex in the (XVtx, SlopeXVtx, YVtx, SlopeYVtx, q/PyzVtx) coordinate system
476 CovP2Cov(newParam,newParamCov);
478 // Set parameters and covariances at vertex
479 param->SetParameters(newParam);
481 param->SetCovariances(newParamCov);
484 //__________________________________________________________________________
485 void AliMUONTrackExtrap::CorrectELossEffectInAbsorber(AliMUONTrackParam* param, Double_t eLoss, Double_t sigmaELoss2)
487 /// Correct parameters for energy loss and add energy loss fluctuation effect to covariances
489 // Get parameter covariances in (X, SlopeX, Y, SlopeY, q*PTot) coordinate system
490 TMatrixD newParamCov(param->GetCovariances());
491 Cov2CovP(param->GetParameters(),newParamCov);
493 // Compute new parameters corrected for energy loss
494 Double_t muMass = TDatabasePDG::Instance()->GetParticle("mu-")->Mass(); // GeV
495 Double_t p = param->P();
496 Double_t e = TMath::Sqrt(p*p + muMass*muMass);
497 Double_t eCorr = e + eLoss;
498 Double_t pCorr = TMath::Sqrt(eCorr*eCorr - muMass*muMass);
499 Double_t nonBendingSlope = param->GetNonBendingSlope();
500 Double_t bendingSlope = param->GetBendingSlope();
501 param->SetInverseBendingMomentum(param->GetCharge() / pCorr *
502 TMath::Sqrt(1.0 + nonBendingSlope*nonBendingSlope + bendingSlope*bendingSlope) /
503 TMath::Sqrt(1.0 + bendingSlope*bendingSlope));
505 // Add effects of energy loss fluctuation to covariances
506 newParamCov(4,4) += eCorr * eCorr / pCorr / pCorr * sigmaELoss2;
508 // Get new parameter covariances in (X, SlopeX, Y, SlopeY, q/Pyz) coordinate system
509 CovP2Cov(param->GetParameters(),newParamCov);
511 // Set new parameter covariances
512 param->SetCovariances(newParamCov);
515 //__________________________________________________________________________
516 Bool_t AliMUONTrackExtrap::GetAbsorberCorrectionParam(Double_t trackXYZIn[3], Double_t trackXYZOut[3], Double_t pTotal,
517 Double_t &pathLength, Double_t &f0, Double_t &f1, Double_t &f2,
518 Double_t &meanRho, Double_t &totalELoss, Double_t &sigmaELoss2)
520 /// Parameters used to correct for Multiple Coulomb Scattering and energy loss in absorber
521 /// Calculated assuming a linear propagation from trackXYZIn to trackXYZOut (order is important)
522 // pathLength: path length between trackXYZIn and trackXYZOut (cm)
523 // f0: 0th moment of z calculated with the inverse radiation-length distribution
524 // f1: 1st moment of z calculated with the inverse radiation-length distribution
525 // f2: 2nd moment of z calculated with the inverse radiation-length distribution
526 // meanRho: average density of crossed material (g/cm3)
527 // totalELoss: total energy loss in absorber
529 // Reset absorber's parameters
538 // Check whether the geometry is available
540 cout<<"E-AliMUONTrackExtrap::GetAbsorberCorrectionParam: no TGeo"<<endl;
544 // Initialize starting point and direction
545 pathLength = TMath::Sqrt((trackXYZOut[0] - trackXYZIn[0])*(trackXYZOut[0] - trackXYZIn[0])+
546 (trackXYZOut[1] - trackXYZIn[1])*(trackXYZOut[1] - trackXYZIn[1])+
547 (trackXYZOut[2] - trackXYZIn[2])*(trackXYZOut[2] - trackXYZIn[2]));
548 if (pathLength < TGeoShape::Tolerance()) return kFALSE;
550 b[0] = (trackXYZOut[0] - trackXYZIn[0]) / pathLength;
551 b[1] = (trackXYZOut[1] - trackXYZIn[1]) / pathLength;
552 b[2] = (trackXYZOut[2] - trackXYZIn[2]) / pathLength;
553 TGeoNode *currentnode = gGeoManager->InitTrack(trackXYZIn, b);
555 cout<<"E-AliMUONTrackExtrap::GetAbsorberCorrectionParam: start point out of geometry"<<endl;
559 // loop over absorber slices and calculate absorber's parameters
560 Double_t rho = 0.; // material density (g/cm3)
561 Double_t x0 = 0.; // radiation-length (cm-1)
562 Double_t atomicA = 0.; // A of material
563 Double_t atomicZ = 0.; // Z of material
564 Double_t atomicZoverA = 0.; // Z/A of material
565 Double_t localPathLength = 0;
566 Double_t remainingPathLength = pathLength;
567 Double_t zB = trackXYZIn[2];
568 Double_t zE, dzB, dzE;
570 // Get material properties
571 TGeoMaterial *material = currentnode->GetVolume()->GetMedium()->GetMaterial();
572 rho = material->GetDensity();
573 x0 = material->GetRadLen();
574 atomicA = material->GetA();
575 atomicZ = material->GetZ();
576 if(material->IsMixture()){
577 TGeoMixture * mixture = (TGeoMixture*)material;
580 for (Int_t iel=0;iel<mixture->GetNelements();iel++){
581 sum += mixture->GetWmixt()[iel];
582 atomicZoverA += mixture->GetZmixt()[iel]*mixture->GetWmixt()[iel]/mixture->GetAmixt()[iel];
586 else atomicZoverA = atomicZ/atomicA;
588 // Get path length within this material
589 gGeoManager->FindNextBoundary(remainingPathLength);
590 localPathLength = gGeoManager->GetStep() + 1.e-6;
591 // Check if boundary within remaining path length. If so, make sure to cross the boundary to prepare the next step
592 if (localPathLength >= remainingPathLength) localPathLength = remainingPathLength;
594 currentnode = gGeoManager->Step();
596 cout<<"E-AliMUONTrackExtrap::GetAbsorberCorrectionParam: navigation failed"<<endl;
597 f0 = f1 = f2 = meanRho = totalELoss = sigmaELoss2 = 0.;
600 if (!gGeoManager->IsEntering()) {
601 // make another small step to try to enter in new absorber slice
602 gGeoManager->SetStep(0.001);
603 currentnode = gGeoManager->Step();
604 if (!gGeoManager->IsEntering() || !currentnode) {
605 cout<<"E-AliMUONTrackExtrap::GetAbsorberCorrectionParam: navigation failed"<<endl;
606 f0 = f1 = f2 = meanRho = totalELoss = sigmaELoss2 = 0.;
609 localPathLength += 0.001;
613 // calculate absorber's parameters
614 zE = b[2] * localPathLength + zB;
615 dzB = zB - trackXYZIn[2];
616 dzE = zE - trackXYZIn[2];
617 f0 += localPathLength / x0;
618 f1 += (dzE*dzE - dzB*dzB) / b[2] / b[2] / x0 / 2.;
619 f2 += (dzE*dzE*dzE - dzB*dzB*dzB) / b[2] / b[2] / b[2] / x0 / 3.;
620 meanRho += localPathLength * rho;
621 totalELoss += BetheBloch(pTotal, localPathLength, rho, atomicZ, atomicZoverA);
622 sigmaELoss2 += EnergyLossFluctuation2(pTotal, localPathLength, rho, atomicZoverA);
626 remainingPathLength -= localPathLength;
627 } while (remainingPathLength > TGeoShape::Tolerance());
629 meanRho /= pathLength;
634 //__________________________________________________________________________
635 Double_t AliMUONTrackExtrap::GetMCSAngle2(const AliMUONTrackParam& param, Double_t dZ, Double_t x0)
637 /// Return the angular dispersion square due to multiple Coulomb scattering
638 /// through a material of thickness "dZ" and of radiation length "x0"
639 /// assuming linear propagation and using the small angle approximation.
641 Double_t bendingSlope = param.GetBendingSlope();
642 Double_t nonBendingSlope = param.GetNonBendingSlope();
643 Double_t inverseTotalMomentum2 = param.GetInverseBendingMomentum() * param.GetInverseBendingMomentum() *
644 (1.0 + bendingSlope * bendingSlope) /
645 (1.0 + bendingSlope *bendingSlope + nonBendingSlope * nonBendingSlope);
646 // Path length in the material
647 Double_t pathLength = TMath::Abs(dZ) * TMath::Sqrt(1.0 + bendingSlope*bendingSlope + nonBendingSlope*nonBendingSlope);
648 // relativistic velocity
650 // Angular dispersion square of the track (variance) in a plane perpendicular to the trajectory
651 Double_t theta02 = 0.0136 / velo * (1 + 0.038 * TMath::Log(pathLength/x0));
653 return theta02 * theta02 * inverseTotalMomentum2 * pathLength / x0;
656 //__________________________________________________________________________
657 void AliMUONTrackExtrap::AddMCSEffect(AliMUONTrackParam *param, Double_t dZ, Double_t x0)
659 /// Add to the track parameter covariances the effects of multiple Coulomb scattering
660 /// through a material of thickness "dZ" and of radiation length "x0"
661 /// assuming linear propagation and using the small angle approximation.
663 Double_t bendingSlope = param->GetBendingSlope();
664 Double_t nonBendingSlope = param->GetNonBendingSlope();
665 Double_t inverseBendingMomentum = param->GetInverseBendingMomentum();
666 Double_t inverseTotalMomentum2 = inverseBendingMomentum * inverseBendingMomentum *
667 (1.0 + bendingSlope * bendingSlope) /
668 (1.0 + bendingSlope *bendingSlope + nonBendingSlope * nonBendingSlope);
669 // Path length in the material
670 Double_t pathLength = TMath::Abs(dZ) * TMath::Sqrt(1.0 + bendingSlope*bendingSlope + nonBendingSlope*nonBendingSlope);
671 Double_t pathLength2 = pathLength * pathLength;
672 // relativistic velocity
674 // Angular dispersion square of the track (variance) in a plane perpendicular to the trajectory
675 Double_t theta02 = 0.0136 / velo * (1 + 0.038 * TMath::Log(pathLength/x0));
676 theta02 *= theta02 * inverseTotalMomentum2 * pathLength / x0;
678 Double_t varCoor = pathLength2 * theta02 / 3.;
679 Double_t varSlop = theta02;
680 Double_t covCorrSlope = pathLength * theta02 / 2.;
682 // Set MCS covariance matrix
683 TMatrixD newParamCov(param->GetCovariances());
685 newParamCov(0,0) += varCoor; newParamCov(0,1) += covCorrSlope;
686 newParamCov(1,0) += covCorrSlope; newParamCov(1,1) += varSlop;
688 newParamCov(2,2) += varCoor; newParamCov(2,3) += covCorrSlope;
689 newParamCov(3,2) += covCorrSlope; newParamCov(3,3) += varSlop;
691 // Set momentum related covariances if B!=0
693 // compute derivative d(q/Pxy) / dSlopeX and d(q/Pxy) / dSlopeY
694 Double_t dqPxydSlopeX = inverseBendingMomentum * nonBendingSlope / (1. + nonBendingSlope*nonBendingSlope + bendingSlope*bendingSlope);
695 Double_t dqPxydSlopeY = - inverseBendingMomentum * nonBendingSlope*nonBendingSlope * bendingSlope /
696 (1. + bendingSlope*bendingSlope) / (1. + nonBendingSlope*nonBendingSlope + bendingSlope*bendingSlope);
697 // Inverse bending momentum (due to dependences with bending and non bending slopes)
698 newParamCov(4,0) += dqPxydSlopeX * covCorrSlope; newParamCov(0,4) += dqPxydSlopeX * covCorrSlope;
699 newParamCov(4,1) += dqPxydSlopeX * varSlop; newParamCov(1,4) += dqPxydSlopeX * varSlop;
700 newParamCov(4,2) += dqPxydSlopeY * covCorrSlope; newParamCov(2,4) += dqPxydSlopeY * covCorrSlope;
701 newParamCov(4,3) += dqPxydSlopeY * varSlop; newParamCov(3,4) += dqPxydSlopeY * varSlop;
702 newParamCov(4,4) += (dqPxydSlopeX*dqPxydSlopeX + dqPxydSlopeY*dqPxydSlopeY) * varSlop;
705 // Set new covariances
706 param->SetCovariances(newParamCov);
709 //__________________________________________________________________________
710 void AliMUONTrackExtrap::ExtrapToVertex(AliMUONTrackParam* trackParam,
711 Double_t xVtx, Double_t yVtx, Double_t zVtx,
712 Double_t errXVtx, Double_t errYVtx,
713 Bool_t correctForMCS, Bool_t correctForEnergyLoss)
715 /// Main method for extrapolation to the vertex:
716 /// Returns the track parameters and covariances resulting from the extrapolation of the current trackParam
717 /// Changes parameters and covariances according to multiple scattering and energy loss corrections:
718 /// if correctForMCS=kTRUE: compute parameters using Branson correction and add correction resolution to covariances
719 /// if correctForMCS=kFALSE: add parameter dispersion due to MCS in parameter covariances
720 /// if correctForEnergyLoss=kTRUE: correct parameters for energy loss and add energy loss fluctuation to covariances
721 /// if correctForEnergyLoss=kFALSE: do nothing about energy loss
723 if (trackParam->GetZ() == zVtx) return; // nothing to be done if already at vertex
725 if (trackParam->GetZ() > zVtx) { // spectro. (z<0)
726 cout<<"E-AliMUONTrackExtrap::ExtrapToVertex: Starting Z ("<<trackParam->GetZ()
727 <<") upstream the vertex (zVtx = "<<zVtx<<")"<<endl;
731 // Check the vertex position relatively to the absorber
732 if (zVtx < AliMUONConstants::AbsZBeg() && zVtx > AliMUONConstants::AbsZEnd()) { // spectro. (z<0)
733 cout<<"W-AliMUONTrackExtrap::ExtrapToVertex: Ending Z ("<<zVtx
734 <<") inside the front absorber ("<<AliMUONConstants::AbsZBeg()<<","<<AliMUONConstants::AbsZEnd()<<")"<<endl;
735 } else if (zVtx < AliMUONConstants::AbsZEnd() ) { // spectro. (z<0)
736 cout<<"W-AliMUONTrackExtrap::ExtrapToVertex: Ending Z ("<<zVtx
737 <<") downstream the front absorber (zAbsorberEnd = "<<AliMUONConstants::AbsZEnd()<<")"<<endl;
738 if (trackParam->CovariancesExist()) ExtrapToZCov(trackParam,zVtx);
739 else ExtrapToZ(trackParam,zVtx);
743 // Check the track position relatively to the absorber and extrapolate track parameters to the end of the absorber if needed
744 if (trackParam->GetZ() > AliMUONConstants::AbsZBeg()) { // spectro. (z<0)
745 cout<<"W-AliMUONTrackExtrap::ExtrapToVertex: Starting Z ("<<trackParam->GetZ()
746 <<") upstream the front absorber (zAbsorberBegin = "<<AliMUONConstants::AbsZBeg()<<")"<<endl;
747 if (trackParam->CovariancesExist()) ExtrapToZCov(trackParam,zVtx);
748 else ExtrapToZ(trackParam,zVtx);
750 } else if (trackParam->GetZ() > AliMUONConstants::AbsZEnd()) { // spectro. (z<0)
751 cout<<"W-AliMUONTrackExtrap::ExtrapToVertex: Starting Z ("<<trackParam->GetZ()
752 <<") inside the front absorber ("<<AliMUONConstants::AbsZBeg()<<","<<AliMUONConstants::AbsZEnd()<<")"<<endl;
754 if (trackParam->CovariancesExist()) ExtrapToZCov(trackParam,AliMUONConstants::AbsZEnd());
755 else ExtrapToZ(trackParam,AliMUONConstants::AbsZEnd());
758 // Get absorber correction parameters assuming linear propagation in absorber
759 Double_t trackXYZOut[3];
760 trackXYZOut[0] = trackParam->GetNonBendingCoor();
761 trackXYZOut[1] = trackParam->GetBendingCoor();
762 trackXYZOut[2] = trackParam->GetZ();
763 Double_t trackXYZIn[3];
764 if (correctForMCS) { // assume linear propagation until the vertex
765 trackXYZIn[2] = TMath::Min(zVtx, AliMUONConstants::AbsZBeg()); // spectro. (z<0)
766 trackXYZIn[0] = trackXYZOut[0] + (xVtx - trackXYZOut[0]) / (zVtx - trackXYZOut[2]) * (trackXYZIn[2] - trackXYZOut[2]);
767 trackXYZIn[1] = trackXYZOut[1] + (yVtx - trackXYZOut[1]) / (zVtx - trackXYZOut[2]) * (trackXYZIn[2] - trackXYZOut[2]);
769 AliMUONTrackParam trackParamIn(*trackParam);
770 ExtrapToZ(&trackParamIn, TMath::Min(zVtx, AliMUONConstants::AbsZBeg()));
771 trackXYZIn[0] = trackParamIn.GetNonBendingCoor();
772 trackXYZIn[1] = trackParamIn.GetBendingCoor();
773 trackXYZIn[2] = trackParamIn.GetZ();
775 Double_t pTot = trackParam->P();
776 Double_t pathLength, f0, f1, f2, meanRho, totalELoss, sigmaELoss2;
777 if (!GetAbsorberCorrectionParam(trackXYZIn,trackXYZOut,pTot,pathLength,f0,f1,f2,meanRho,totalELoss,sigmaELoss2)) {
778 cout<<"E-AliMUONTrackExtrap::ExtrapToVertex: Unable to take into account the absorber effects"<<endl;
779 if (trackParam->CovariancesExist()) ExtrapToZCov(trackParam,zVtx);
780 else ExtrapToZ(trackParam,zVtx);
784 // Compute track parameters and covariances at vertex according to correctForMCS and correctForEnergyLoss flags
787 if (correctForEnergyLoss) {
789 // Correct for multiple scattering and energy loss
790 CorrectELossEffectInAbsorber(trackParam, 0.5*totalELoss, 0.5*sigmaELoss2);
791 CorrectMCSEffectInAbsorber(trackParam, xVtx, yVtx, zVtx, errXVtx, errYVtx,
792 trackXYZIn[2], pathLength, f0, f1, f2);
793 CorrectELossEffectInAbsorber(trackParam, 0.5*totalELoss, 0.5*sigmaELoss2);
797 // Correct for multiple scattering
798 CorrectMCSEffectInAbsorber(trackParam, xVtx, yVtx, zVtx, errXVtx, errYVtx,
799 trackXYZIn[2], pathLength, f0, f1, f2);
804 if (correctForEnergyLoss) {
806 // Correct for energy loss add multiple scattering dispersion in covariance matrix
807 CorrectELossEffectInAbsorber(trackParam, 0.5*totalELoss, 0.5*sigmaELoss2);
808 AddMCSEffectInAbsorber(trackParam, pathLength, f0, f1, f2);
809 ExtrapToZCov(trackParam, trackXYZIn[2]);
810 CorrectELossEffectInAbsorber(trackParam, 0.5*totalELoss, 0.5*sigmaELoss2);
811 ExtrapToZCov(trackParam, zVtx);
815 // add multiple scattering dispersion in covariance matrix
816 AddMCSEffectInAbsorber(trackParam, pathLength, f0, f1, f2);
817 ExtrapToZCov(trackParam, zVtx);
825 //__________________________________________________________________________
826 void AliMUONTrackExtrap::ExtrapToVertex(AliMUONTrackParam* trackParam,
827 Double_t xVtx, Double_t yVtx, Double_t zVtx,
828 Double_t errXVtx, Double_t errYVtx)
830 /// Extrapolate track parameters to vertex, corrected for multiple scattering and energy loss effects
831 /// Add branson correction resolution and energy loss fluctuation to parameter covariances
832 ExtrapToVertex(trackParam, xVtx, yVtx, zVtx, errXVtx, errYVtx, kTRUE, kTRUE);
835 //__________________________________________________________________________
836 void AliMUONTrackExtrap::ExtrapToVertexWithoutELoss(AliMUONTrackParam* trackParam,
837 Double_t xVtx, Double_t yVtx, Double_t zVtx,
838 Double_t errXVtx, Double_t errYVtx)
840 /// Extrapolate track parameters to vertex, corrected for multiple scattering effects only
841 /// Add branson correction resolution to parameter covariances
842 ExtrapToVertex(trackParam, xVtx, yVtx, zVtx, errXVtx, errYVtx, kTRUE, kFALSE);
845 //__________________________________________________________________________
846 void AliMUONTrackExtrap::ExtrapToVertexWithoutBranson(AliMUONTrackParam* trackParam, Double_t zVtx)
848 /// Extrapolate track parameters to vertex, corrected for energy loss effects only
849 /// Add dispersion due to multiple scattering and energy loss fluctuation to parameter covariances
850 ExtrapToVertex(trackParam, 0., 0., zVtx, 0., 0., kFALSE, kTRUE);
853 //__________________________________________________________________________
854 void AliMUONTrackExtrap::ExtrapToVertexUncorrected(AliMUONTrackParam* trackParam, Double_t zVtx)
856 /// Extrapolate track parameters to vertex without multiple scattering and energy loss corrections
857 /// Add dispersion due to multiple scattering to parameter covariances
858 ExtrapToVertex(trackParam, 0., 0., zVtx, 0., 0., kFALSE, kFALSE);
861 //__________________________________________________________________________
862 Double_t AliMUONTrackExtrap::TotalMomentumEnergyLoss(AliMUONTrackParam* trackParam, Double_t xVtx, Double_t yVtx, Double_t zVtx)
864 /// Calculate the total momentum energy loss in-between the track position and the vertex assuming a linear propagation
866 if (trackParam->GetZ() == zVtx) return 0.; // nothing to be done if already at vertex
868 // Check whether the geometry is available
870 cout<<"E-AliMUONTrackExtrap::TotalMomentumEnergyLoss: no TGeo"<<endl;
874 // Get encountered material correction parameters assuming linear propagation from vertex to the track position
875 Double_t trackXYZOut[3];
876 trackXYZOut[0] = trackParam->GetNonBendingCoor();
877 trackXYZOut[1] = trackParam->GetBendingCoor();
878 trackXYZOut[2] = trackParam->GetZ();
879 Double_t trackXYZIn[3];
880 trackXYZIn[0] = xVtx;
881 trackXYZIn[1] = yVtx;
882 trackXYZIn[2] = zVtx;
883 Double_t pTot = trackParam->P();
884 Double_t pathLength, f0, f1, f2, meanRho, totalELoss, sigmaELoss2;
885 GetAbsorberCorrectionParam(trackXYZIn,trackXYZOut,pTot,pathLength,f0,f1,f2,meanRho,totalELoss,sigmaELoss2);
887 // total momentum corrected for energy loss
888 Double_t muMass = TDatabasePDG::Instance()->GetParticle("mu-")->Mass(); // GeV
889 Double_t e = TMath::Sqrt(pTot*pTot + muMass*muMass);
890 Double_t eCorr = e + totalELoss;
891 Double_t pTotCorr = TMath::Sqrt(eCorr*eCorr - muMass*muMass);
893 return pTotCorr - pTot;
896 //__________________________________________________________________________
897 Double_t AliMUONTrackExtrap::BetheBloch(Double_t pTotal, Double_t pathLength, Double_t rho, Double_t atomicZ, Double_t atomicZoverA)
899 /// Returns the mean total momentum energy loss of muon with total momentum='pTotal'
900 /// in the absorber layer of lenght='pathLength', density='rho', A='atomicA' and Z='atomicZ'
901 Double_t muMass = TDatabasePDG::Instance()->GetParticle("mu-")->Mass(); // GeV
903 // mean exitation energy (GeV)
905 if (atomicZ < 13) i = (12. * atomicZ + 7.) * 1.e-9;
906 else i = (9.76 * atomicZ + 58.8 * TMath::Power(atomicZ,-0.19)) * 1.e-9;
908 return pathLength * rho * AliExternalTrackParam::BetheBlochGeant(pTotal/muMass, rho, 0.20, 3.00, i, atomicZoverA);
911 //__________________________________________________________________________
912 Double_t AliMUONTrackExtrap::EnergyLossFluctuation2(Double_t pTotal, Double_t pathLength, Double_t rho, Double_t atomicZoverA)
914 /// Returns the total momentum energy loss fluctuation of muon with total momentum='pTotal'
915 /// in the absorber layer of lenght='pathLength', density='rho', A='atomicA' and Z='atomicZ'
916 Double_t muMass = TDatabasePDG::Instance()->GetParticle("mu-")->Mass(); // GeV
917 //Double_t eMass = 0.510998918e-3; // GeV
918 Double_t k = 0.307075e-3; // GeV.g^-1.cm^2
919 Double_t p2=pTotal*pTotal;
920 Double_t beta2=p2/(p2 + muMass*muMass);
922 Double_t fwhm = 2. * k * rho * pathLength * atomicZoverA / beta2; // FWHM of the energy loss Landau distribution
923 Double_t sigma2 = fwhm * fwhm / (8.*log(2.)); // gaussian: fwmh = 2 * srqt(2*ln(2)) * sigma (i.e. fwmh = 2.35 * sigma)
925 //sigma2 = k * rho * pathLength * atomicZ / atomicA * eMass; // sigma2 of the energy loss gaussian distribution
930 //__________________________________________________________________________
931 void AliMUONTrackExtrap::Cov2CovP(const TMatrixD ¶m, TMatrixD &cov)
933 /// change coordinate system: (X, SlopeX, Y, SlopeY, q/Pyz) -> (X, SlopeX, Y, SlopeY, q*PTot)
934 /// parameters (param) are given in the (X, SlopeX, Y, SlopeY, q/Pyz) coordinate system
936 // charge * total momentum
937 Double_t qPTot = TMath::Sqrt(1. + param(1,0)*param(1,0) + param(3,0)*param(3,0)) /
938 TMath::Sqrt(1. + param(3,0)*param(3,0)) / param(4,0);
940 // Jacobian of the opposite transformation
943 jacob(4,1) = qPTot * param(1,0) / (1. + param(1,0)*param(1,0) + param(3,0)*param(3,0));
944 jacob(4,3) = - qPTot * param(1,0) * param(1,0) * param(3,0) /
945 (1. + param(3,0)*param(3,0)) / (1. + param(1,0)*param(1,0) + param(3,0)*param(3,0));
946 jacob(4,4) = - qPTot / param(4,0);
948 // compute covariances in new coordinate system
949 TMatrixD tmp(cov,TMatrixD::kMultTranspose,jacob);
953 //__________________________________________________________________________
954 void AliMUONTrackExtrap::CovP2Cov(const TMatrixD ¶m, TMatrixD &covP)
956 /// change coordinate system: (X, SlopeX, Y, SlopeY, q*PTot) -> (X, SlopeX, Y, SlopeY, q/Pyz)
957 /// parameters (param) are given in the (X, SlopeX, Y, SlopeY, q/Pyz) coordinate system
959 // charge * total momentum
960 Double_t qPTot = TMath::Sqrt(1. + param(1,0)*param(1,0) + param(3,0)*param(3,0)) /
961 TMath::Sqrt(1. + param(3,0)*param(3,0)) / param(4,0);
963 // Jacobian of the transformation
966 jacob(4,1) = param(4,0) * param(1,0) / (1. + param(1,0)*param(1,0) + param(3,0)*param(3,0));
967 jacob(4,3) = - param(4,0) * param(1,0) * param(1,0) * param(3,0) /
968 (1. + param(3,0)*param(3,0)) / (1. + param(1,0)*param(1,0) + param(3,0)*param(3,0));
969 jacob(4,4) = - param(4,0) / qPTot;
971 // compute covariances in new coordinate system
972 TMatrixD tmp(covP,TMatrixD::kMultTranspose,jacob);
973 covP.Mult(jacob,tmp);
976 //__________________________________________________________________________
977 void AliMUONTrackExtrap::ExtrapOneStepHelix(Double_t charge, Double_t step, Double_t *vect, Double_t *vout)
980 /// ******************************************************************
982 /// * Performs the tracking of one step in a magnetic field *
983 /// * The trajectory is assumed to be a helix in a constant field *
984 /// * taken at the mid point of the step. *
987 /// * STEP =arc length of the step asked *
988 /// * VECT =input vector (position,direction cos and momentum) *
989 /// * CHARGE= electric charge of the particle *
991 /// * VOUT = same as VECT after completion of the step *
993 /// * ==>Called by : USER, GUSWIM *
994 /// * Author m.hansroul ********* *
995 /// * modified s.egli, s.v.levonian *
996 /// * modified v.perevoztchikov
998 /// ******************************************************************
1001 // modif: everything in double precision
1003 Double_t xyz[3], h[4], hxp[3];
1004 Double_t h2xy, hp, rho, tet;
1005 Double_t sint, sintt, tsint, cos1t;
1006 Double_t f1, f2, f3, f4, f5, f6;
1008 const Int_t kix = 0;
1009 const Int_t kiy = 1;
1010 const Int_t kiz = 2;
1011 const Int_t kipx = 3;
1012 const Int_t kipy = 4;
1013 const Int_t kipz = 5;
1014 const Int_t kipp = 6;
1016 const Double_t kec = 2.9979251e-4;
1018 // ------------------------------------------------------------------
1020 // units are kgauss,centimeters,gev/c
1022 vout[kipp] = vect[kipp];
1023 if (TMath::Abs(charge) < 0.00001) {
1024 for (Int_t i = 0; i < 3; i++) {
1025 vout[i] = vect[i] + step * vect[i+3];
1026 vout[i+3] = vect[i+3];
1030 xyz[0] = vect[kix] + 0.5 * step * vect[kipx];
1031 xyz[1] = vect[kiy] + 0.5 * step * vect[kipy];
1032 xyz[2] = vect[kiz] + 0.5 * step * vect[kipz];
1034 //cmodif: call gufld (xyz, h) changed into:
1035 TGeoGlobalMagField::Instance()->Field(xyz,h);
1037 h2xy = h[0]*h[0] + h[1]*h[1];
1038 h[3] = h[2]*h[2]+ h2xy;
1039 if (h[3] < 1.e-12) {
1040 for (Int_t i = 0; i < 3; i++) {
1041 vout[i] = vect[i] + step * vect[i+3];
1042 vout[i+3] = vect[i+3];
1046 if (h2xy < 1.e-12*h[3]) {
1047 ExtrapOneStepHelix3(charge*h[2], step, vect, vout);
1050 h[3] = TMath::Sqrt(h[3]);
1056 hxp[0] = h[1]*vect[kipz] - h[2]*vect[kipy];
1057 hxp[1] = h[2]*vect[kipx] - h[0]*vect[kipz];
1058 hxp[2] = h[0]*vect[kipy] - h[1]*vect[kipx];
1060 hp = h[0]*vect[kipx] + h[1]*vect[kipy] + h[2]*vect[kipz];
1062 rho = -charge*h[3]/vect[kipp];
1065 if (TMath::Abs(tet) > 0.15) {
1066 sint = TMath::Sin(tet);
1068 tsint = (tet-sint)/tet;
1069 cos1t = 2.*(TMath::Sin(0.5*tet))*(TMath::Sin(0.5*tet))/tet;
1071 tsint = tet*tet/36.;
1072 sintt = (1. - tsint);
1079 f3 = step * tsint * hp;
1082 f6 = tet * cos1t * hp;
1084 vout[kix] = vect[kix] + f1*vect[kipx] + f2*hxp[0] + f3*h[0];
1085 vout[kiy] = vect[kiy] + f1*vect[kipy] + f2*hxp[1] + f3*h[1];
1086 vout[kiz] = vect[kiz] + f1*vect[kipz] + f2*hxp[2] + f3*h[2];
1088 vout[kipx] = vect[kipx] + f4*vect[kipx] + f5*hxp[0] + f6*h[0];
1089 vout[kipy] = vect[kipy] + f4*vect[kipy] + f5*hxp[1] + f6*h[1];
1090 vout[kipz] = vect[kipz] + f4*vect[kipz] + f5*hxp[2] + f6*h[2];
1095 //__________________________________________________________________________
1096 void AliMUONTrackExtrap::ExtrapOneStepHelix3(Double_t field, Double_t step, Double_t *vect, Double_t *vout)
1099 /// ******************************************************************
1101 /// * Tracking routine in a constant field oriented *
1102 /// * along axis 3 *
1103 /// * Tracking is performed with a conventional *
1104 /// * helix step method *
1106 /// * ==>Called by : USER, GUSWIM *
1107 /// * Authors R.Brun, M.Hansroul ********* *
1108 /// * Rewritten V.Perevoztchikov
1110 /// ******************************************************************
1114 Double_t h4, hp, rho, tet;
1115 Double_t sint, sintt, tsint, cos1t;
1116 Double_t f1, f2, f3, f4, f5, f6;
1118 const Int_t kix = 0;
1119 const Int_t kiy = 1;
1120 const Int_t kiz = 2;
1121 const Int_t kipx = 3;
1122 const Int_t kipy = 4;
1123 const Int_t kipz = 5;
1124 const Int_t kipp = 6;
1126 const Double_t kec = 2.9979251e-4;
1129 // ------------------------------------------------------------------
1131 // units are kgauss,centimeters,gev/c
1133 vout[kipp] = vect[kipp];
1136 hxp[0] = - vect[kipy];
1137 hxp[1] = + vect[kipx];
1141 rho = -h4/vect[kipp];
1143 if (TMath::Abs(tet) > 0.15) {
1144 sint = TMath::Sin(tet);
1146 tsint = (tet-sint)/tet;
1147 cos1t = 2.* TMath::Sin(0.5*tet) * TMath::Sin(0.5*tet)/tet;
1149 tsint = tet*tet/36.;
1150 sintt = (1. - tsint);
1157 f3 = step * tsint * hp;
1160 f6 = tet * cos1t * hp;
1162 vout[kix] = vect[kix] + f1*vect[kipx] + f2*hxp[0];
1163 vout[kiy] = vect[kiy] + f1*vect[kipy] + f2*hxp[1];
1164 vout[kiz] = vect[kiz] + f1*vect[kipz] + f3;
1166 vout[kipx] = vect[kipx] + f4*vect[kipx] + f5*hxp[0];
1167 vout[kipy] = vect[kipy] + f4*vect[kipy] + f5*hxp[1];
1168 vout[kipz] = vect[kipz] + f4*vect[kipz] + f6;
1173 //__________________________________________________________________________
1174 void AliMUONTrackExtrap::ExtrapOneStepRungekutta(Double_t charge, Double_t step, Double_t* vect, Double_t* vout)
1177 /// ******************************************************************
1179 /// * Runge-Kutta method for tracking a particle through a magnetic *
1180 /// * field. Uses Nystroem algorithm (See Handbook Nat. Bur. of *
1181 /// * Standards, procedure 25.5.20) *
1183 /// * Input parameters *
1184 /// * CHARGE Particle charge *
1185 /// * STEP Step size *
1186 /// * VECT Initial co-ords,direction cosines,momentum *
1187 /// * Output parameters *
1188 /// * VOUT Output co-ords,direction cosines,momentum *
1189 /// * User routine called *
1190 /// * CALL GUFLD(X,F) *
1192 /// * ==>Called by : USER, GUSWIM *
1193 /// * Authors R.Brun, M.Hansroul ********* *
1194 /// * V.Perevoztchikov (CUT STEP implementation) *
1197 /// ******************************************************************
1200 Double_t h2, h4, f[4];
1201 Double_t xyzt[3], a, b, c, ph,ph2;
1202 Double_t secxs[4],secys[4],seczs[4],hxp[3];
1203 Double_t g1, g2, g3, g4, g5, g6, ang2, dxt, dyt, dzt;
1204 Double_t est, at, bt, ct, cba;
1205 Double_t f1, f2, f3, f4, rho, tet, hnorm, hp, rho1, sint, cost;
1215 Double_t maxit = 1992;
1216 Double_t maxcut = 11;
1218 const Double_t kdlt = 1e-4;
1219 const Double_t kdlt32 = kdlt/32.;
1220 const Double_t kthird = 1./3.;
1221 const Double_t khalf = 0.5;
1222 const Double_t kec = 2.9979251e-4;
1224 const Double_t kpisqua = 9.86960440109;
1225 const Int_t kix = 0;
1226 const Int_t kiy = 1;
1227 const Int_t kiz = 2;
1228 const Int_t kipx = 3;
1229 const Int_t kipy = 4;
1230 const Int_t kipz = 5;
1233 // *. ------------------------------------------------------------------
1235 // * this constant is for units cm,gev/c and kgauss
1239 for(Int_t j = 0; j < 7; j++)
1242 Double_t pinv = kec * charge / vect[6];
1250 if (TMath::Abs(h) > TMath::Abs(rest)) h = rest;
1251 //cmodif: call gufld(vout,f) changed into:
1252 TGeoGlobalMagField::Instance()->Field(vout,f);
1255 // * start of integration
1268 secxs[0] = (b * f[2] - c * f[1]) * ph2;
1269 secys[0] = (c * f[0] - a * f[2]) * ph2;
1270 seczs[0] = (a * f[1] - b * f[0]) * ph2;
1271 ang2 = (secxs[0]*secxs[0] + secys[0]*secys[0] + seczs[0]*seczs[0]);
1272 if (ang2 > kpisqua) break;
1274 dxt = h2 * a + h4 * secxs[0];
1275 dyt = h2 * b + h4 * secys[0];
1276 dzt = h2 * c + h4 * seczs[0];
1281 // * second intermediate point
1284 est = TMath::Abs(dxt) + TMath::Abs(dyt) + TMath::Abs(dzt);
1286 if (ncut++ > maxcut) break;
1295 //cmodif: call gufld(xyzt,f) changed into:
1296 TGeoGlobalMagField::Instance()->Field(xyzt,f);
1302 secxs[1] = (bt * f[2] - ct * f[1]) * ph2;
1303 secys[1] = (ct * f[0] - at * f[2]) * ph2;
1304 seczs[1] = (at * f[1] - bt * f[0]) * ph2;
1308 secxs[2] = (bt * f[2] - ct * f[1]) * ph2;
1309 secys[2] = (ct * f[0] - at * f[2]) * ph2;
1310 seczs[2] = (at * f[1] - bt * f[0]) * ph2;
1311 dxt = h * (a + secxs[2]);
1312 dyt = h * (b + secys[2]);
1313 dzt = h * (c + seczs[2]);
1317 at = a + 2.*secxs[2];
1318 bt = b + 2.*secys[2];
1319 ct = c + 2.*seczs[2];
1321 est = TMath::Abs(dxt)+TMath::Abs(dyt)+TMath::Abs(dzt);
1322 if (est > 2.*TMath::Abs(h)) {
1323 if (ncut++ > maxcut) break;
1332 //cmodif: call gufld(xyzt,f) changed into:
1333 TGeoGlobalMagField::Instance()->Field(xyzt,f);
1335 z = z + (c + (seczs[0] + seczs[1] + seczs[2]) * kthird) * h;
1336 y = y + (b + (secys[0] + secys[1] + secys[2]) * kthird) * h;
1337 x = x + (a + (secxs[0] + secxs[1] + secxs[2]) * kthird) * h;
1339 secxs[3] = (bt*f[2] - ct*f[1])* ph2;
1340 secys[3] = (ct*f[0] - at*f[2])* ph2;
1341 seczs[3] = (at*f[1] - bt*f[0])* ph2;
1342 a = a+(secxs[0]+secxs[3]+2. * (secxs[1]+secxs[2])) * kthird;
1343 b = b+(secys[0]+secys[3]+2. * (secys[1]+secys[2])) * kthird;
1344 c = c+(seczs[0]+seczs[3]+2. * (seczs[1]+seczs[2])) * kthird;
1346 est = TMath::Abs(secxs[0]+secxs[3] - (secxs[1]+secxs[2]))
1347 + TMath::Abs(secys[0]+secys[3] - (secys[1]+secys[2]))
1348 + TMath::Abs(seczs[0]+seczs[3] - (seczs[1]+seczs[2]));
1350 if (est > kdlt && TMath::Abs(h) > 1.e-4) {
1351 if (ncut++ > maxcut) break;
1357 // * if too many iterations, go to helix
1358 if (iter++ > maxit) break;
1363 cba = 1./ TMath::Sqrt(a*a + b*b + c*c);
1371 if (step < 0.) rest = -rest;
1372 if (rest < 1.e-5*TMath::Abs(step)) return;
1376 // angle too big, use helix
1381 f4 = TMath::Sqrt(f1*f1+f2*f2+f3*f3);
1390 hxp[0] = f2*vect[kipz] - f3*vect[kipy];
1391 hxp[1] = f3*vect[kipx] - f1*vect[kipz];
1392 hxp[2] = f1*vect[kipy] - f2*vect[kipx];
1394 hp = f1*vect[kipx] + f2*vect[kipy] + f3*vect[kipz];
1397 sint = TMath::Sin(tet);
1398 cost = 2.*TMath::Sin(khalf*tet)*TMath::Sin(khalf*tet);
1402 g3 = (tet-sint) * hp*rho1;
1407 vout[kix] = vect[kix] + g1*vect[kipx] + g2*hxp[0] + g3*f1;
1408 vout[kiy] = vect[kiy] + g1*vect[kipy] + g2*hxp[1] + g3*f2;
1409 vout[kiz] = vect[kiz] + g1*vect[kipz] + g2*hxp[2] + g3*f3;
1411 vout[kipx] = vect[kipx] + g4*vect[kipx] + g5*hxp[0] + g6*f1;
1412 vout[kipy] = vect[kipy] + g4*vect[kipy] + g5*hxp[1] + g6*f2;
1413 vout[kipz] = vect[kipz] + g4*vect[kipz] + g5*hxp[2] + g6*f3;