FAST SIMULATION of HADRONIC SHOWERS for ATLAS HCAL Stanislav Tokár Comenius University, Bratislava People involved: Yu. Ku ľ chitsky 1 , J. Šutiak 2 , T. Ženiš 2 1 JINR Dubna, 2 Comenius Univ. Bratislava 4/25/2005 S. Tokar 1
Motivation Challenge in present high energy physics: • Insufficient to know a particle process on theory level (X-sections) • It should be known also on experimental level: � Detectors resolutions and reconstruction efficiencies � Capability of experiment to distinguish signal process from bkgd one Two approaches: Due to shower •Full simulation of the wanted processes simulations � All available physics taken into account � Powerful computing system + team of people + time consuming • Fast simulation � Full simulation of showers in calorimeter replaced by fast parametrisation tuned by experiment � Reliable results obtained much faster (> 1000 × Full simulation) 4/25/2005 S. Tokar 2
Principles of Fast MC Goal: find spatial energy deposition in calorimeter without full simulation - using 3D parametrisation of hadronic shower ( for ATLAS Had-Calorimeter) Part.Nucl.Lett 2[117](2003)52 Main principles of our approach: � Incident energy is devided into a certain # energy spots � The spots are distribured according to known shower topology � Electromagnetic and hadronic components are treated separately � Realistic fluctuations of shower profile: individual shower profile constructed for each incident particle 4/25/2005 S. Tokar 3
Simulation algorithm For each incident particle: � Position of the 1 st interaction (shower origin) is found � Shower profile is constructed from a few sub-shower profiles electromagnetic hadronic � Incident energy is divided into EM and HD components � Proper number of energy spots and their size is found Depends on Cal. energy resolution and sampling fraction � The spots are distributed and energy of spots absorbed in active medium is accumulated 4/25/2005 S. Tokar 4
Resolution vs # of energy spots Calorimeter energy resolution: sampling term σ a ε = = ⊕ E b constant term E E If N spots is distributed in calorimeter ⇒ N A = s f N spots is absorbed in active medium Sampling fraction N A is a random variable obeying Poisson law: st.dev.= N A σ N 1 E Energy resolution: ⇒ = A = E = N E N N 2 a s A A f Spot energy: q eff = a 2 s f # of spots 4/25/2005 S. Tokar 5
Shower Profiles 3-dimensional parametrisation of hadronic shower profile from the 1 st interaction point: ( ) ( ) ( ) ( ) Ψ = ⋅ Ψ + − ⋅ Ψ x r , w x r , 1 w x r , e h Electromagnetic component Hadronic component w ≡ share of electromag. energy in shower – big fluctuations ! dE 1 ( ) ( ) ( ) Ψ = ⋅ e ,h ⋅ φ x,r x x,r e ,h e ,h E dx 0 e ,h dE/dx ≡ longitudinal profile, φ ≡ radial profile 4/25/2005 S. Tokar 6
Fraction of el-mag energy Mean EM fraction <w> depends on energy of π 0 produced ( ): f 0 π ⋅ e h f 0 = π w ( ) − ⋅ + e h 1 f 1 0 π = ⋅ f 0 11 . ln E 0 π Fluctuation of EM fraction: ( ) α − 1 − − β − ⋅ ( w w ) w w e w 0 w = 0 p( w ) ( ) α β Γ α ( ) w w w = + α β w w Fluctuation of EM fraction w: 0 w w pions 100 GeV α w , β w ≡ parameters 4/25/2005 S. Tokar 7
Longitudinal profile of HD component − β x / e � Position of the shower beginning is sampled from h β � Average profile of hadronic shower: h α − − β x / 1 dE x e h [x] ≡λ I (interaction length) = h ( x ) β Γ α α dx ( ) h Longitudinal profile of H-component Can be found by GEANT � dots: Geant simulation � full line: fit by the function 4/25/2005 S. Tokar 8
Individual longit. shower profile • From the origin a few “principal” particles emerge • Each of them starts sub-shower at its interaction place • Individual HD-shower is a sum of the sub-showers: dE ( ) x ∑ = ⋅ − α − β h f G x ( x , 1 , ) i i h h dx i Origin of i th sub-shower Energy fraction carried by i th particle An example of individual hadronic shower: sub-showers full shower 4/25/2005 S. Tokar 9
Longitudinal profile of EM component x t dE EM longitudinal − ∫ = ⋅ α β + − ⋅ ⋅ − α β e E f G x ( , , ) ( 1 f ) e λ G x ( t , , ) dt 0 1 e e 1 e e π shower profile dx 0 − µ 2 ( f ) α − − β ⋅ 1 x / x e f 1 fluctuations ( ) 1 f 1 = − p f ( ) N exp α β = G x , , ( ) 1 σ 2 α β ⋅ Γ α f 1 Averaged EM profile produced π 0 vs depth: 4/25/2005 S. Tokar 10
Individual EM profile An example of individual EM shower profile: EM sub-showers Full EM shower ( ) dE x 1 e = ⋅ α β f G x ( , , ) 1 e e E dx 0 π ∑ + ⋅ − α β f G x ( x , , ) 2 i e e ≥ i 2 Individual longitudinal shower profile G(x, α , β ) ≡ gamma distribution incident energy: 100 GeV 4/25/2005 S. Tokar 11
Radial shower profiles ∆ E ( x,r ) Parametrization of radial profile: α − − β = ⋅ ( x ) 1 r / ( x ) r c r e r r ∆ ∆ r x ∆ E ( x,r ) Related to the profile function as: φ = ( x,r ) r dE( x ) π ⋅ ∆ ∆ ⋅ 2 r x r No fluctuations of radial profile included ! dx 100 GeV 100 GeV EM shower HD shower EM radial profile HD radial profile 4/25/2005 S. Tokar 12
Parameters of the method For shower profile and its fluctuations - 25 parameters: � Fluctuation of π 0 energy fraction ( 2 ) � Longitudinal profile of EM component ( 3 ) and its fluctuations ( 4 ) � � Longitudinal profile of H component ( 2 ) At given energy and its fluctuations ( 2 ) � � Radial EM component ( 6 ) � Radial HD component ( 6 ) Dependence of parameters on energy: P i (E) = p i +q i ln E or P i (E) = p i +q i E or P i (E) =const 4/25/2005 S. Tokar 13
Parameters values Shower parameters values + energy dependence Longitudinal parameters Radial parameters 4/25/2005 S. Tokar 14
Fast MC vs Testbeam data The fast MC is compared with the test beam data of 5 1m-modules for different incident pion energies (20 – 300) GeV and different input conditions – varied: � Tilt angle � Beam position Test beam setup: • 5 modules • Each didided into 20 cells (4 samplings, 5 towers) • Cell read by 2 PMT 4/25/2005 S. Tokar 15
Calorimeter structure Incident beam 4/25/2005 S. Tokar 16
Fast MC vs Data – full responses Full responses: Fast MC vs Test beam data for incident pion energies: 50, 100, 200 and 300 GeV, tilt angle: 10° 50 GeV 100 GeV TB data Fast MC 300 GeV 200 GeV 4/25/2005 S. Tokar 17
Fast MC vs TB data - samplings Sampling responses: Fast MC vs Test beam data Incident energy: 100 GeV Particle type: π − Position of beam: M3 center Incident angle: 10 ° Shower depth dependence TB data Fast MC 4/25/2005 S. Tokar 18
Fast MC vs TB data - modules Module responses: Fast MC vs Test beam data Incident energy: 100 GeV Particle type: π − Position of beam: M3 center Incident angle: 10 ° Shower transversal dependence TB data Fast MC 4/25/2005 S. Tokar 19
Fast MC vs TB data - towers Tower responses: Fast MC vs Test beam data Incident energy: 100 GeV Particle type: π − Position of beam: M3 center Incident angle: 10 ° 3 Shower transversal dependence TB data Fast MC 4/25/2005 S. Tokar 20
Conclusions and perspectives � Fast MC method for sampling calorimeter based on idea of building shower from sub-showers was created � Good description of energy response at least in the interval 50-300 GeV � Good description of fluctuation on calorimeter cell level � Method is easy adaptable for jets To be done: � Test method for low energies (1-20) GeV For application in ATLAS to include Elektromag. Cal. � 4/25/2005 S. Tokar 21
Thank you very much! Bo ľ šoje spasibo !!! 4/25/2005 S. Tokar 22
KALORIMETRIA • Meranie energie č astíc • Fyzika hadrónovej sp ŕ šky • Modelovanie sp ŕ šky - programový balík GEANT 4/25/2005 S. Tokar 23
D) Radiálny profil •parametrizácia pre EM aj hadrónovú zložku: ( ) ( ) ( ) α − − β x 1 r / x > ∆ cr e r r E x r , r r = 0 ( ) ( ) ∆ ∆ x r α − − β ⋅ x 1 r / x < r cr e r r r 0 r 0 0 •závislos ť α r od x pre EM zložku: ( ) ( ) α = α − − α x / x 0 1 2 e e 1 re e ( ) β + β ∈ x x 0,30 ( ) β = e 1 e 2 x ( ) re β + β ∈ x x 30,180 e 3 e 4 •závislos ť α r od x pre hadrónovú zložku: ( ) ( ) α = α + α x 0 ln x rh h 0 h ( ) β + β ∈ x x 0,30 ( ) β = h 1 h 2 x ( ) rh β + β ∈ x x 30,180 h 3 h 4 4/25/2005 S. Tokar 24
POROVNANIE S EXPERIMENTÁLNYMI DÁTAMI 100 Gev Energia uložená v moduloch, Energia uložená v toweroch, plná č iara – experimentálne dáta, plná č iara – experimentálne dáta, 4/25/2005 S. Tokar 25 body – rýchle simulácie body – rýchle simulácie
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