Analysis on Shower Fractal Dimension & GRPC Digitizer Manqi RUAN Laboratoire Leprince-Ringuet (LLR) Ecole polytechnique 91128, Palaiseau 29/03/2011 CALICE Meeting @ CERN 1
Shower fractal dimension Nature: Shower particle, to interact or not shower ~ self similar (Mandelbrot Set) Measure shower Fractal Dimension (FD) at high granularity calorimeter Varying scale by grouping neighbouring cells ● Count Number of hits at different scale ● ( define RN(x) = N 1mm /N xmm ) 29/03/2011 CALICE Meeting @ CERN 2
Shower: Self Similar Characteristic constant ● based on energy/PID: D = < lnR(N 1 /N a )/ln(a) > rms as error bar Global parameter based on ● local density Cell Sizes: 2 – 10, 20, 30, ● 50, 60, 90, 120, 150mm. Samples: Particles shot ● directly to GRPC DHCAL with only B Field Be observed within ● Low scale: minimal interaction energy & sensor layer thickness ( 1.2mm ) ● High scale: fully containment ~ 1 hits per layer ● 29/03/2011 CALICE Meeting @ CERN 3
Compactness of shower Higher compactness @ large E: density increases with Energy ( volume increases ● slower/logarithm than Energy ) Order of compactness: Fractal Dimension ( FD ) ● Left: FD( positron ) > FD( EM @ K0 ) > FD( K0 ) > FD ( MIP @ K0 ) ● Right: FD( K0 ) > FD( Pion ), K0 shower is more compact: no initial mip tail ● 29/03/2011 CALICE Meeting @ CERN 4
Fractal in Nature Muon ( 2 GeV ) Dim ~ 1 Straight line: Dim = 1 Positron ( 40GeV ) Hadrons: Dim( pi ) < Dim( K0 ) ~ 1.5 Dim ~ 1.75 Rectangle: Dim = 2 29/03/2011 CALICE Meeting @ CERN 5
Promising tool for PID Handput Cut on Calo info @ 1mm Cell e+ u h FD – 1 = 0.68, N1 = 500 998 0 2 e+ u 1 994 5 h 15 14 971 Reference: PFOID @ Full Detector e+ pi u KL γ E+ 945 2 53 Pi+ 5.0 882 12 102 u 1 12 921 67 Characteristic Parameter for PID: to be used together with other information in full detector environment 29/03/2011 CALICE Meeting @ CERN 6
PID with FD @ larger cell FD = 1.68, N10 = 150 N30 - FD*200 = 40, FD = 1.5, N10 = 100 FD_10mm: Counts at 20, 30, FD_30mm: Counts at 60, 50, 60, 90, 120, 150mm cells 90, 120, 150mm cells FD: Clear separation @ larger cell size 1mm e+ u h 10mm e+ u h 30mm e+ u h e+ 998 0 2 e+ 1000 0 0 e+ 1000 0 0 u 1 994 5 u 0 995 5 u 0 996 4 h 15 14 971 h 17 14 969 18 11 971 h Remark: cuts might be energy dependent ~ easier to be used for charged particles 29/03/2011 7
FD @ different size From FD( 1mm ) to FD( 10/30mm ): Better µ – h separation: µ acts more like a line ( FD = 1 ); ( Anyhow we can create large cells from small ones... ) Positron Peak Smeared Pi: continuous from MIP to EM 29/03/2011 8
Extreme Cases: Muon Muon with large Fractal Dimension: Together with Nhit information: Possibility to identify Muon radiation & String noise (typical for gaseous detector)... 29/03/2011 CALICE Meeting @ CERN 9
Extreme Cases: Pion Pion: Pion decay ~ MIP Pure EM interaction ( pi + N = P + pi0 ); could be partially identified by tagging interaction point 29/03/2011 CALICE Meeting @ CERN 10
Energy Estimation with Naive Counting σ/M: Large cell better at low energy & Smaller cell at high energy. Linearity: Better at 2 – 5 mm, stronger saturation effects at larger cell... Naively: 5mm seems a nice choice... 29/03/2011 11
FD @ Energy Estimation 28%/sqrt(E) @ FD1mm 35%/sqrt(E) @ FD10mm If energy is known... For example: Compensation based on the correlation of NH_30mm & FD1mm: ● E = a * NH_30 + b * FD ~ 30%/sqrt(E)! But... Correlation coefficient depending on Energy: b ~ 0.0266*E. To measure cluster ● energy of charged particle (with trk info): better matching A set of energy independent ( LO ) estimator: E = a' * NH_x/(1 – FD*b') ● 29/03/2011 CALICE Meeting @ CERN 12
E.E with FD Correction Had put Energy Estimator with FD: NH10/(1-0.65*FD10) Energy resolution improved at high energy: ~ saturation effect correction Linearity improved: closed to 5mm Cell 29/03/2011 CALICE Meeting @ CERN 13
Summary Shower Fractal Dimension can be measured at high granular Calorimeter. ● Provide a rich era to investigate. Application at single particle: ● PID: Very promising ● Energy Estimation: ● Self-eating snake: 30%/sqrt(E) achieved for charged particle, better cluster – – track linking Linearity & Resolution @ high energy improved for neutral clusters – To do: ● Fine tune the definition of Fractal Dimension ● Optimization of FD based energy estimation ● Test/optimize with clustering algorithms & Integration ● ... to investigate other possibilities: flavour-tagging @ VTX? ● 29/03/2011 CALICE Meeting @ CERN 14
GRPC Digitizer ● Introduction Avalanche development at GRPC ● Idea: cut off at 1mm ● Parameters to be fixed with experimental Input ● MIP charge inducing: Polya function, 2 parameters ● Treatment of Multiplicity: 2 parameters ● Analysis ● Dependencies ● Multiplicity ● Energy resolution with changing thresholds ● Summary ● 29/03/2011 CALICE Meeting @ CERN 15
Avalanche @ RPC K. BELKADHI ● Characteristic Parameter: from test beam data Multiplicity: global ~ 1.4 – 1.5 @ Thresholds ~ 0.1 mip (160 fC) ● Charge Image Size (depending on resistive plates thickness) : ~ 1mm ● ( as convoluted with spatial resolution ) 29/03/2011 CALICE Meeting @ CERN 16
Experiment input: Induced Charge of Single MIP Polya-distribution: 7400V Polya Fitting Cosmic Charge A little Mathematics: General Polya function: Mean = (a+1)/b, MPV = a/b ; R. Han: measured from cosmic ray data: a = 16.3, b = 10.8 Mean = 1.6 pC, MPV = 1.51 pC More details: R. Han's presentation at tomorrow 29/03/2011 CALICE Meeting @ CERN 17
Idea on multiplicity Keep simulation level information to 1mm cells: count corresponding number of hits in/nearby 1 square cm² Advantages: ● Natural cut off: 1mm ~ gas gap thickness ~ size of charge image – ● Self Saturation & easy to integrate other saturation effects Reliable estimation of multiplicity – Samples: available for other analysis ( optimized cell size, fractal – dimensional analysis...) Cost: ● Machine time: the same – Data size: increased ~ 5% ( ParticleCont recorded & Nhits increased by 2 – – 3 times, Test on 20GeV Klong sample with only PRC HCAL & B Field: ) Negligible at full detector event: Utilize as Simulation base line? – Alternative ideas: keep particle hit position, see Ran's prensation 29/03/2011 CALICE Meeting @ CERN 18
Multiplicity treatment No threshold: Boundary (2 hits) or Corner (4 hits) * corresponding weights ● Spatial distribution of Charge: 2 parameters to be fine tuned ~ Weight Side & ● Weight Corner with corresponding smearing A f e conservative 4 2 2 2 2 2 2 2 2 4 e c guess: d b a 5 10 5 = 10 40 10 5 10 5 Remark: If image size ~ 1mm ( affect nearby 3 * 3 region ) & eff = 100%: M = ((N+2)/N)² = 1.44 @ N = 10... 29/03/2011 CALICE Meeting @ CERN 19
Simu & Digi hits Left: simulation level ( 1 mm cell: size zoned by 5 for display. Colour: EM, MIP or Neutron hit ) Right: Digitization level ( 10mm cell. Colour according to Charge) 29/03/2011 20
Count 1mm hits inside ( neighbour to ) 10mm cell... Digitized hit colour to charge: ~ 1.5 - 1.6pC/mip 29/03/2011 CALICE Meeting @ CERN 21
Multiplicity treatment For each digitized hits, count 1mm Cells: 1) direct sailing through (center): ≤100 2) induce charge from side (corner): ≤ 40 (4) Sample: 1k 40GeV Pion 1k events, shot normally to DHCAL with B Field Electron Strings Define: Multiplicity hit = Hit without Sail through/center hits Global Multiplicity = N(total)/N(non-Multiplicity) 29/03/2011 22
Induced Charge Multiplicity hits: Peak at 0.05 mip, 0.2 mip; Non – Multiplicity hits: Peak at 1 mip (~ 1.5 pC), 2 mips; Peak Positions: depend on the boundary weights. Statistic should be stable (only depend on sample PID/energy), and easy to be cleaned by threshold 29/03/2011 CALICE Meeting @ CERN 23
Q: Induced Charge From Left to right: Q Vs total hits, non-multiplicity hits and multiplicity hits Linearly depend on Nhits, especially number of non-multiplicity hits Easy to add other saturation effects ( for example, local density of 1mm hits ) on induced charge: waiting for experimental evidence. 29/03/2011 CALICE Meeting @ CERN 24
Q Vs E From left to right: Induced Charge Vs Deposited energy for all the hits, non- multiplicity hits and multiplicity hits. Correlation between Q & E exist, but with large smearing. Stronger Correlation at non-multiplicity hits. Remark: Huge fluctuation in energy deposition: smeared over 5 - 6 orders of magnitude 29/03/2011 CALICE Meeting @ CERN 25
Multiplicity hits Strong correlations & Worse resolution with Multiplicity hits Resolution @ Naive counting (40GeV Pion): 11.7% @ non-Multiply hits, 12.1% @ total hits; Multiplicity hits: no information ~ fluctuation 29/03/2011 26
Global Multiplicity Vs E & PID M = 1.32 M = 1.34 M = 1.36 M = 1.25 M = 1.42 Global Multiplicity : anti-proportional to density/Shower Fractal Dimension 29/03/2011 CALICE Meeting @ CERN 27
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