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Energy loss in STEP

Energy loss in STEP. Niels van Eldik, Peter Kluit, Alan Poppleton, Andi Salzburger, Sharka Todorova Common Tracking Meeting 16 July 2013. Introduction. Look at the Eloss description in different ATLAS codes Take as a reference STEP Looks to me most suited and precise

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Energy loss in STEP

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  1. Energy loss in STEP Niels van Eldik, Peter Kluit, Alan Poppleton, Andi Salzburger, Sharka Todorova Common Tracking Meeting 16 July 2013

  2. Introduction • Look at the Eloss description in different ATLAS codes • Take as a reference STEP • Looks to me most suited and precise • Is written up in a series op ATLAS notes. • Other packages • ElossUpdator/Extrapolator • e.g. used in the TrkGlobalChi2Fitter • Is not 100% up to date and in ‘the back’ a scaling to the STEP Eloss is performed • MuonElossCalculators here labeled as ‘TrackinCalo’ developed by David Lopez et al for the calorimeter • Muid Tools developed by Alan Poppleton et al. Not further discussed here.

  3. Basic formulae used in STEP ATL-SOFT-PUB-2008-003

  4. Basic formulae used in STEP ATL-SOFT-PUB-2008-003

  5. What is or might be lacking in STEP? • Basically an estimate of the error on the energy loss for the ionization and the radiation term • Maybe the muon high energy behaviour (pair production term) should be improved and compared to the ‘TrackinCalo’ results • Some un-understood ‘magical’ numbers are used to go from MOP (m_MPV) to mean: • if (m_MPV) { • return 0.9*Ionization + 0.15*Radiation; //Most probable value • } • else { • return Ionization + Radiation; //Mean value • } • Later I will show that they do a really good job…

  6. What is or might be lacking in STEP? • One might worry about the combination of Eloss values and their errors. For Landau distributions the MOP will shift in the combination and the errors should be added linearly (see math on slide 16). • Propagation of the Energy loss fluctations in STEP • This is done in stragglingVariance • double landauWidth = m_stragglingScale*sqrt(s*energyFactor*kaz)/beta; • The formula used here does not look correct to me as the Eloss fluctations are proportional to the Eloss. • The formula only propagates the ionization Eloss error neglecting the radiative component. • Can be improved by calculating consistently the error on the Eloss

  7. Now added to STEP • An estimate of the error on the energy loss for the ionization and the radiation term • Error on Ionization Eloss: • FWHW = 4 xi = 4*kaz*s/beta/beta • with kaz = 0.5*30.7075*zOverAtimesRho • For a Landau: σL = FWHW/4 • Shift from MOP to mean is maximally 3.59524 σL • as shown in my previous talk on Tracking and Eloss • Error on Radiative Eloss: • Due to Bethe-Heitler plus pair production • FWHM “error” = Eloss radiation/2.409 • Shift from MOP to mean is FWHM “error” • This formula is ad hoc (see slide 12 onwards) • Errors are added linearly (slide 6 and 16)!

  8. Comparison STEP and TrackInCalogoing through 2m Fe Quite good description of MOP Eloss in STEP/TrackInCalo TrackInCalo has problems at low momenta for the MOP σL This is known: it is expected to be valid above ~10 GeV

  9. Comparison MOP STEP and TrackInCalo going through 2m Pb Quite good description of MOP Eloss in STEP/TrackInCalo 5% discrepancy Similar as previous slide: TrackInCalo has problems at low momenta for the MOP σL

  10. Comparison Mean STEP and TrackInCalo in 2m Fe and Pb Quite good description of Mean Eloss in STEP and calculation: Note the large shift at high momenta NB calculated mean refers to formulae on slide 7 ‘Magic’ numbers for MOP and mean ‘0.15’ on slide 5 do a good job!

  11. Comparison STEP and many steps NB calculated mean refers to formulae on slide 7 For a Landau there is a difference between 20 small steps and one big step. This has to do with the MOP value, that shifts if the resolution increases. No discrepancy is found within a per mille error band.

  12. Bethe-Heitler energy loss fluctations: the problem • I used an effective formula for the error on the Bethe –Heitler radiative Eloss: • FWHM “error” = Eloss radiation/2.409 • It seems to do a good job • But…where does this come from? • One should be able to understand this from the Bethe-Heitler pdf and the formula for the rms of the BH. • BH Pdf = (-ln(z))c-1 c = x0/√2, z =E/E0 z=0-1 • variance z = exp(-x0 ln(3)/ln(2))-exp(-2 x0) (see R. Frühwirth Com Phys Comm 154 (2993) 131) • It is a nasty distribution: infinite at z = 1 while the integral up to 1 is finite.

  13. Bethe-Heitler energy loss fluctations: the problem Integral is well behaved: will use this for the CL calculations x0 = 0.05 pole For small X0 values we have: Eloss ~ x0 and RMS on E = √variance ~ √x0 (Frühwirth) RMS on E / Eloss ~ 1/√x0 is a problem: NOT constant and singular

  14. Bethe-Heitler energy loss fluctations: problem with Frühwirth formula Using the Frühwirth formula the error is far too large. Applying a scale factor of 0.07 seems to improve the situation for Fe. Testing this hypothesis in Pb – right plot - shows a factor 2 discrepancy. The origin lies in RMS E = √variance ~ √x0 while this should be ~ x0.

  15. Energy loss fluctations Here shown are the error on the energy loss vs momentum for the ionization and the radiation (Bethe-Heitler including pair production etc,) The black points show the X0 range that is probed: At p=1 GeV X0 = 0.005 goes to 0.01 at 1 TeV

  16. Bethe-Heitler Energy loss fluctations • Why does the formula derived by Frühwirth does not apply? • The issue could be that the RMS is NOT a good quantity to use • A well-known example is the Landau distribution for which the RMS is ill-defined. If we integrate to higher Eloss values the RMS keeps rising. • The way around is to calculate e.g. full width half maximum (as done for the inozation) or e.g. the 90% CL interval. This procedure can be studied for the Bethe-Heitler (see next slides). • A break-down of the Bethe-Heitler formula at very low X0. • In this case we mean that the <Eloss> is still exponential in X0. But the fluctations are described by another pdf. • Here one can think of an exponential distribution. This distribution will have a mean and rms that are both exponential thus that rms/mean = 1. Also the 90% CL contour will have a fixed “90% CL” to mean value. • I will argue that indeed a break-down is observed…

  17. Bethe-Heitler energy loss fluctations: CL contours In blue solid what is used Here one observes – as on slide 14 - that RMS Frühwirth stays well above the blue curve. However the 90-97% CL contours at X0 values below 0.01 go well BELOW the curve. This indicates a too steep pdf.

  18. Bethe-Heitler energy loss fluctations: CL contours In blue solid what is used Same as on previous slide: zoomed at low X0

  19. Bethe-Heitler energy loss fluctations: CL contours & break down Can one understand what happens at low X0 for the contours? The integral over the BH (BHI) can be approximated precisely as: BHI = -log(1-x)/A where A = -γ + log(2)/X0 (γ =Eulers constant) This is valid from x=0 to xcut where 1-xcut = exp(-A) The 90% CL is e.g. 1-x90 = 0.1 exp(-A). This means that this is exponential in 1/X0. So x90 goes exponentially fast to 0 for very small X0 values. This explains the steep suppression at low X0 observed in slides 17 and 18. My interpretation is that both the RMS and the low X0 behaviour of the CL contours are not giving the desired low X0 behaviour. Thus the Bethe-Heitler distrubution is not describing the fluctuations at low X0 and low Eloss values.

  20. Bethe-Heitler energy lossfrom radiation: MOP and mean • Here I start from the formula that gives the exponential Eloss with a mean of 1 GeV • One can calculate analytically what the MOP value of the distribution is as a function of a resolution parameter σ. A gaussian smearing is applied. • An example distribution is shown here • For σ = 0 the MOP is at zero; it will gradually shift to the mean value of 1 GeV.

  21. Bethe-Heitler energy lossfrom radiation: MOP and mean • Here the dependence is shown. • The fit is p0*σ/(1+p1*σ) • In general this means that the MOP value moves to the mean for large σ/<Eloss> • For most practical purposes this is indeed the case when e.g. the measured momentum is used. However if the truth momentum is used the MOP does NOT move to the MEAN.

  22. Conclusions • Showed that STEP can be used as a baseline for the Eloss calculations • Formulae for errors, MOP and MEAN values are available. They describe well the parametrizations used in the different codes. • ‘Magical’ numbers to go from MOP to MEAN are understood. • The STEP MOP values behave correctly for small and large numbers of steps (Landau running of the MOP is followed) • Only puzzle is the error on the Bethe-Heitler energy loss. The fluctuations derived from BH Theory do not agree with the expected σ Eloss ~ Eloss. An alternative for the σ Eloss is proposed. • We have to correct the straggling variance and keep track of the Eloss uncertainty thus that we can add it linearly to the covariance matrix.

  23. Convoluting Landau distributions and the track fit • Problem: suppose we have several EMOP values and errors. As discussed in the Muon-PUB-2008-002 note, • the formula to combine these values is: * This has very special features: The MOP value shifts depending on the MOP errors (7) The errors do not add up quadratically but linearly (8) This has implications for track fitting. It means that Eloss has to be combined using these formula. For the MS track fit I would propose to collect the total material between the measurements and use the formulae above to aggregate the Eloss. * I prefer the form: (σ1+σ2) ln (σ1+σ2) - σ1 ln σ1- σ2 ln σ2

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