Improving Photon Parameter Estimation Using Shower Fitting

Transcription

1 Improving Photon Parameter Estimation Using Shower Fitting A Status Report raham W. Wilson University of Kansas September 9, 11 raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 1 / 17

2 Outline 1 Motivation Method 3 Results 4 Conclusions raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 / 17

3 / Motivation Motivation The improvement in π energy resolution from mass-constrained fitting of π γγ decay depends crucially on the determination of the opening angle, ψ 1, between the two photons: m = 4E 1 E sin (ψ 1 /). This is a function of the reconstructed position vectors of the two photons in the ECAL, and so depends on the measurement of (φ, cos θ) for each photon. E (ev).15 Fitted σ E ev π (16% / Angular Resolution 8 mrad 4 mrad mrad 1 mrad.5 mrad.15 mrad cosθ* E) raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 3 / 17

4 Method Electromagnetic Shower Fitting Electromagnetic showers are well-behaved and governed by well understood stochastic processes. Sophisticated parametrizations of both shower shapes and fluctuations and their correlations are used in fast shower simulation. In particular, rindhammer and Peters (hep-ex/1) have a well-tuned model which reproduces full EANT. This is used in FLASH. Plan: Use these parametrizations to improve the reconstruction of photon 4-vectors by fitting the observed cell energies to the model. Will exploit the very narrow core of EM showers near the start of the shower to improve position resolution. (I demonstrated this in an old study with 1mm cells.) raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 4 / 17

5 Method rindhammer-peters Model Describes the spatial energy distribution of electromagnetic showers in terms of the energy deposition in a 3-d volume element in the scaled longitudinal variable, t (in X ), scaled radial variable, r (in R M ), and the azimuthal variable, φ. de(t, r, φ) = E f L (t) f R (r) f (φ) dt dr dφ The average longitudinal distribution uses the usual gamma distribution: f L (t; α, β) = β (β t) α 1 exp ( β t)/γ(α) where the shape parameter, α, and the scaling parameter, β, are related to the longitudinal center of gravity, < t >, and the shower maximum depth, T, by < t >= α/β and T = (α 1)/β. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 5 / 17

6 Method Average Radial Profile vs Shower Depth P use two component ansatz with R C, (R T ) being the median radial extent of the core (tail) and p giving the relative weight of the core. The shower depth in units of the shower maximum, τ = t/t is used to parameterize the radial profile parameters, namely R C (τ), R T (τ), p(τ). raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 6 / 17

7 Longitudinal Shower Fit Results Example: 5 ev photon, ILD. Maximum likelihood fit to the measured energies per pseudo-layer for the 3 free parameters using a suitable choice of fit parameters: log E, log (α 1), log T. Use SL simplex implementation for minimimization. For now, neglect angle of incidence issues; the β parameter accommodates this anyway. Longitudinal Shower Profile Fit Deposited Energy (ev) Longitudinal Layer (arb. units) Use amma distribution for sampling fluctuations with stochastic errors of 17%/ E and 4%/ E. (P eqns 1-15). raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 7 / 17

8 Results Longitudinal Shower Fit Remarks Longitudinal Shower Profile Fit Deposited Energy (ev) Longitudinal Layer (arb. units) These fits work remarkably well with typically 98% of fits leading to sensible answers even with all 3 parameters floating. Currently using default EM sampling fractions. The fitted energies are competitive (and highly correlated) with the measured energy. Can imagine allowing for t-dependent weights on a shower-by-shower basis and there may be some room to improve the energy resolution. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 8 / 17

9 Results Using the Radial Profile Expectations Simulating the radial profile fluctuations vs depth needs knowledge of the longitudinal fluctuations for the particular shower. So, the longitudinal fit is used to calculate τ i = t/t i where T i is the fitted shower max. So far, I took the evaluated R C, R T, p values for each pseudo-layer of an individual shower assuming cos θ =, and then used the median R of that radial profile at that fitted shower depth to calculate weight functions for position estimation. The default position estimator one usually uses for photon momentum reconstruction is the shower center-of-gravity r Co = ( N i=1 E i r i )/( N i=1 E i) We investigated r LW = ( N i=1 E iw i r i )/( N i=1 E iw i ) with w i = R α. We also investigated using different values of α for determining φ and cos θ, denoting the parameter, α φ and α θ. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 9 / 17

10 One way of quantifying the essential -d nature of photon cluster reconstruction is to measure the distribution of separation angle between the reconstructed photon and the generator photon in space. This can be described by a Rayleigh distribution if the resolution contribution from both components is similar - which is a fair approximation. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 1 / 17 First Results Results For 5 ev photons found resolution optimized for α φ = 1. and α θ =.5 and angular resolution improved from.5 to.4 mrad. Photons per bin Mean.541 RMS.3131 Constant.4484 ±.47 Sigma.46 ±.6 Mean.6353 RMS.37 Constant.4476 ±.47 Sigma.499 ±.3 New Estimator Standard Co Estimator Angular deviation (rad)

11 Results Bias in Photon cos θ Reconstruction Example: 5 ev with α θ =.5. Residuals biased by up to Better resolution in cos θ for high cos θ. ) (cosθ M - cosθ ) (1 1 hcosthresidualvscosth Entries 94 Mean x Mean y RMS x.413 RMS y.431 ) (cosθ M - cosθ ) (1. hprofilecosthresidualvscosth Entries 91 Mean Mean y RMS.41 RMS y / 1 Prob.31 T ±.83 T5.795 ± cosθ cosθ Should fit with a suitable odd function of cos θ. Chose to fit using a Chebyshev polynomial (1st kind). With the coefficients of T 1 and T 5 as the only free parameters can obtain reasonable fits. In this case the fit function is basically T 1 cos θ + T 5 cos 5θ. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

12 Results Energy Dependent Fit of cos θ Bias Combined Energy-Dependent Fit to Photon Polar Angle Bias ( α θ =.5 ) ) ) (1 - cosθ M (cosθ / 8 Prob.4788 T T ev ) ) (1 - cosθ M (cosθ / 8 Prob.4788 T1.493 T ev cosθ cosθ ) ) (1 - cosθ. 8 / 8 Prob.4788 T1.583 T ) ) (1 - cosθ. 8 / 8 Prob.4788 T1.594 T5.684 M (cosθ -. ev M (cosθ -. 5 ev cosθ cosθ ) ) (1 - cosθ. 8 / 8 Prob.4788 T T5.756 ) ) (1 - cosθ. 8 / 8 Prob.4788 T T5.773 M (cosθ -. 1 ev M (cosθ -. ev cosθ cosθ Fit using 4 free parameters: (A 1 + B 1 log E) cos θ + (A 5 + B 5 log E) cos 5θ raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, 11 1 / 17

13 Results Bias in Photon φ Reconstruction wrt Octant Center Example: ev with α φ = 1.. Residuals biased by up to.3 mrad too. ) (mrad) - φ M (φ 1.5 hphiresidualvslocalphi Entries 71 Mean x Mean y.936 RMS x.6 RMS y.856 ) (mrad) - φ M (φ.3..1 hprofilephiresidualvslocalphi Entries 7195 Mean Mean y.95 RMS.65 RMS y local φ (rad) -.. local φ (rad) Mean bias is positive. Indication of B-field effect? associated with e /e + asymmetry in showers? We do have + B along +z-axis? Indication also of issues associated with octant module overlaps at octant boundary. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

14 Results Snapshot of current improvements: ev photon Events per bin 4 3 hcosthresidual Entries 71 Mean -.78 RMS.3195 χ / ndf 51.1 / 38 Prob Constant 446 ± 6.5 Mean -.14 ±.383 Sigma.31 ±.8 Events per bin 8 6 hpcosthresidual Entries 71 Mean -.13 RMS.1951 χ / ndf / 3 Prob.5783 Constant 734. ± 1.6 Mean ±.33 Sigma.1951 ± (cosθ M - cosθ ) (1 ) (cosθ M - cosθ ) (1 ) α θ =.5. Left: before correcting cos θ bias. Right: after correction. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

15 Results Snapshot of current improvements: ev photon Before (hphiresidual) and after (hpphiresidual) correcting φ bias. Events per bin 5 hphiresidual Entries 71 Mean.435 RMS / 197 Prob 1 Constant 19 ±.8 Mean.4334 ±.35 Sigma.99 ±. hpphiresidual Entries 71 Mean.5 RMS / 96 Prob.658 Constant ±.9 Mean.58 ±.344 Sigma.89 ±. Events per bin hphiresidual Entries 71 Mean.93 RMS / 94 Prob.3 5 Constant 1.1 ±.9 Mean.34 ±.336 Sigma.81 ±.3 hpphiresidual Entries 71 Mean.539 RMS.315 χ / ndf / 84 Prob.69 Constant 51. ± 3.8 Mean.9564 ±.6884 Sigma.58 ± (φ - φ ) (mrad) M (φ - φ ) (mrad) M Left: α φ =. (Co). Right: α φ = 1.. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

16 Results Snapshot of current improvements: 1 ev photon After correcting cos θ bias. Events per bin 3 hpcosthresidual Entries 954 Mean RMS / 93 Prob.6e-1 Constant 45.8 ± 3.1 Mean -.66 ±.7935 Sigma.769 ±.58 Events per bin 3 hpcosthresidual Entries 954 Mean RMS / 79 Prob 1.541e-13 Constant 31. ± 3.8 Mean -.64 ±.6439 Sigma.6188 ± (cosθ M - cosθ ) (1 ) - -1 (cosθ 1 M - cosθ ) (1 ) Left: α θ =. (Co). Right: α θ =.5. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

17 Conclusions Conclusions/Summary/Open Issues Shower fitting has potential to improve calorimeter measurements. First attempts at longitudinal fits appear very promising, and indicate that it is feasible to measure the 3 main longitudinal shower parameters. Longitudinal weighting of position estimates shows improvement over shower Co. Need to take care of systematics in φ and cos θ (which are there also for Co...) - in progress, and re-visit α optimization. Not sure how easy it will be to really adapt the method seamlessly to all incidence angles. Full 3-d fitting of all cells to a shower model looks to be worth pursuing for photons - particularly with regard to reducing finite cell-size type systematics. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

18 Backup Slides raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

19 Position Resolution Algorithms 1 v1-9 PandoraPFANew - uses unweighted cluster centroid in first pseudo-layer (oops...). Shower center-of-gravity (was the default in MarlinReco / PandoraPFA). 3 Longitudinal weighting using shower fit (this talk) I have a version of PfoCreationAlgorithm::CreateNeutralPfos() which implements 1 and above and an energy-weighted version of 1. (Co is the obvious default...). raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

20 Chebyshev Polynomials of the First Kind A particular orthogonal polynomial T 1 (x) = x T 3 (x) = 4x 3 3x T 5 (x) = 16x 5 x 3 + 5x In fits using just the T 1 and T 5 contributions, the T 1 -T 5 correlation coefficient is small leading to robust fits. raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17

21 Rayleigh Distribution Can describe the magnitude of a vector, r, whose two components, x and y are aussianly distributed, uncorrelated and with the same variance. p(r; σ) = (r/σ ) exp( r /(σ )) raham W. Wilson (University of Kansas) LCWS11 ranada: Sim./Det.Perf./Reco. September 9, / 17