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Study of the Cluster Splitting Algorithm In PANDA EMC
ReconstructionQing Pu1,2 , Guang Zhao2 , Chunxu Yu1, Shengsen Sun2
1Nankai University2Institute of High Energy Physics
PANDA Chinese Collaboration Meeting7/7/2021
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Speaker: Ziyu Zhang
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OutlineØIntroduction
ØStudy of the cluster-splitting algorithm
ØLateral development measurement
ØSeed energy correction
ØReconstruction checks
ØSummary
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Introduction• Cluster-splitting is an important algorithm in EMC reconstruction.• The purpose of the cluster-splitting is to separate clusters that are close to each other. • In this presentation, we improve the cluster-splitting algorithm in the following ways:
• Update the lateral development formula• Correct the seed energy
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Cluster-splitting for a EmcCluster Angles between the 2γ from pi0
Cluster splitting is important for high energy pi0
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1. Cluster finding: a contiguous area of crystals with energy deposit.
2. The bump splittingl Find the local maximum: Preliminary split into
seed crystal information
l Update energy/position iterativelyl The spatial position of a bump is calculated
via a center-of-gravity methodl Lateral development of cluster: 𝐸!"#$%! = 𝐸&''(exp(− ⁄2.5 𝑟 𝑅))
l The crystal weight for each bump is calculated by a formula
EMC reconstruction overview
seed
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𝑤* =(𝐸&''()*exp(− ⁄2.5𝑟* 𝑅+)
∑,(𝐸&''(),exp(−2.5𝑟, ∕ 𝑅+)
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l Initialization:Place the bump center at the seed crystal.
l Iteration:1.Traverse all digis to calculate 𝑤𝑖.
𝑖 or 𝑗 : different seed crystals𝑅": Moliere radius𝑟#: distance from the shower center to the target crystal
(𝐸&''()-
(𝐸&''().
…
𝑟1𝑟2
𝑟3𝑟𝑖
(𝑤1, 𝑤2, 𝑤3, … , 𝑤𝑖)
(𝐸&''()*
The Cluster splitting algorithm
(𝐸&''()/
2. Update the position of the bump center.
3. Loop over 1 & 2 until the bump center stays stable within a tolerance of 1 mm or the number of iterations exceeds the maximum number of iterations.
the target crystalthe seed crystalthe shower center
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𝑤* =(𝐸&''()*exp(− ⁄2.5𝑟* 𝑅+)
∑,(𝐸&''(),exp(−2.5𝑟, ∕ 𝑅+)
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l Initialization:Place the bump center at the seed crystal.
l Iteration:1.Traverse all digis to calculate 𝑤𝑖.
2. Update the position of the bump center.
3. Loop over 1 & 2 until the bump center stays stable within a tolerance of 1 mm or the number of iterations exceeds the maximum number of iterations.
𝑖 or 𝑗 : different seed crystals𝑅": Moliere radius𝑟#: distance from the shower center to the target crystal
(𝐸&''()-
(𝐸&''().
…
𝑟1𝑟2
𝑟3𝑟𝑖
(𝑤1, 𝑤2, 𝑤3, … , 𝑤𝑖)
the target crystalthe seed crystalthe shower center
(𝐸&''()*
Energy for the target crystal: 𝐸456789
The Cluster splitting algorithm
The 𝐸456789 is calculated using the lateral development formula
(𝐸&''()/
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The lateral development (old PandaRoot)The previous lateral development formula
0%&'()%0*))+
= exp(− ⁄2.5 𝑟 𝑅)) have no consideration of crystal granularity, detector geometry and energy of particles.
• Since exp does not fit the data well, we try to improve the formula.
Ø Gamma (0~6GeV)
Ø Events 10000
Ø Geant4
Ø Generator: Box
Ø Phi(0, 360)
Ø Theta(22, 140)
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ParametrizationThe function form used for fitting:
𝑓 𝑟 =𝐸1234'1𝐸&''(
= exp −𝑝-𝑅)
𝜉 𝑟 , 𝜉(𝑟) = 𝑟 − 𝑝.𝑟exp[−𝑟
𝑝/𝑅)
5,]
Where 𝑝-, 𝑝., 𝑝/ and 𝑝6 are parameters.
𝑝-(𝐸7 , 𝜃) 𝑝.(𝐸7 , 𝜃) 𝑝/(𝐸7 , 𝜃) 𝑝6(𝐸7 , 𝜃)
• We consider the dependency of the parameters on energy and polar angle.
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Fitted parameters of the lateral development
Fitting Result:
𝑝- 𝐸7 , 𝜃 = −0.384 ∗ 𝑒𝑥𝑝 3.88 ∗ 𝐸7 + 5.44 ∗ 1089 ∗ 𝜃 − 97.7 . + 2.6
𝑝. 𝐸7 , 𝜃 = −0.352 ∗ 𝑒𝑥𝑝 4.21 ∗ 𝐸7 + (−3.94) ∗ 108: ∗ 𝜃 − 69 . + 0.932
𝑝/ 𝐸7 , 𝜃 = 0.151 ∗ exp 4.52 ∗ 𝐸7 + (−2.14) ∗ 1089 ∗ 𝜃 − 91 . + 0.841
𝑝6 𝐸7 , 𝜃 = −3.51 ∗ 𝑒𝑥𝑝 1.15 ∗ 𝐸7 + 2.26 ∗ 1086 ∗ 𝜃 − 80.3 . + 4.96
( 𝑅)= 2.00 𝑐𝑚)
Energy dependency Angle dependency
𝐸1234'1𝐸&''(
= exp{−𝑝-𝑅)
𝜉 𝑟, 𝑝., 𝑝/, 𝑝6 } 𝜉(𝑟) = 𝑟 − 𝑝.𝑟exp[−𝑟
𝑝/𝑅)
5,]
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Seed energy correction
• In the old PandaRoot, the seed energy is used to calculate the 𝐸1234'1 = 𝐸&''(×𝑓(𝑟)
• If the shower center does not coincide with the crystal center, 𝐸&''( needs to be corrected
r or 𝑟&''(:the distance from the center of the Bump to the geometric center of the crystal.
𝑟
𝐸1234'1
𝐸&''(
𝑟𝐸&''(
𝐸1234'1
𝑟!""#𝐸;%%<-
Old PandaRoot version Updated version
the target crystalthe seed crystalthe shower center
the target crystalthe seed crystalthe shower center
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Seed energy correctionIn the new update, 𝐸1234'1 can be calculated by the lateral development f(r):
𝐸1234'1 = 𝐸&''(-×𝑓(𝑟)𝐸&''(!, which is not available in the reconstruction algorithm, can be related to the 𝐸&''( :
𝐸&''(- =𝐸&''(𝑓(𝑟&''()
In the end, 𝐸1234'1 can be calculated as ( -=(3*))+)
as the correction factor) :
𝐸1234'1 =𝐸&''(𝑓(𝑟&''()
×𝑓(𝑟)
r or 𝑟&''(:the distance from the center of the Bump to the geometric center of the crystal.
𝑟
𝐸1234'1
𝐸&''(
𝑟𝐸&''(
𝐸1234'1
𝑟!""#𝐸&''(-
Old PandaRoot version Updated version
the target crystalthe seed crystalthe shower center
the target crystalthe seed crystalthe shower center
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Fitted parameters of the lateral development
Fitting Result:
𝑝- 𝐸7 , 𝜃 = −0.384 ∗ 𝑒𝑥𝑝 3.88 ∗ 𝐸7 + 5.44 ∗ 1089 ∗ 𝜃 − 97.7 . + 2.6
𝑝. 𝐸7 , 𝜃 = −0.352 ∗ 𝑒𝑥𝑝 4.21 ∗ 𝐸7 + (−3.94) ∗ 108: ∗ 𝜃 − 69 . + 0.932
𝑝/ 𝐸7 , 𝜃 = 0.151 ∗ exp 4.52 ∗ 𝐸7 + (−2.14) ∗ 1089 ∗ 𝜃 − 91 . + 0.841
𝑝6 𝐸7 , 𝜃 = −3.51 ∗ 𝑒𝑥𝑝 1.15 ∗ 𝐸7 + 2.26 ∗ 1086 ∗ 𝜃 − 80.3 . + 4.96
( 𝑅)= 2.00 𝑐𝑚)
Energy dependency Angle dependency
𝐸(*4*𝐸&''(
= exp{−𝑝-𝑅)
𝜉 𝑟, 𝑝., 𝑝/, 𝑝6 − 𝜉 𝑟&''( , 𝑝., 𝑝/, 𝑝6 } 𝜉(𝑟) = 𝑟 − 𝑝.𝑟exp[−𝑟
𝑝/𝑅)
5,]
Seed energy correction
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Energy resolution (di-photon)
Range of simulated samples:• Energy0.5~6 GeV
• Theta 67.7938 ~ 73.8062 {deg}
• Phi0.625 ~ 7.375 {deg}
Energy resolution of di-photon
• The angle between two photons < 6.75 (deg)
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Energy resolution (di-photon)
Range of simulated samples:• Energy0.5~6 GeV
• Theta Range12: 70.8088 ~ 72.8652
• PhiSquare area calculated according to theta
Energy resolution of di-photon
d: the distance between shower centers
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Energy resolution (pi0)pi0 mass resolution
Range of simulated samples:• Energy0.5~5 GeV
• Theta 22~140 (deg)
• Phi0~360 (deg)
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Summary• The lateral development of the cluster using Geant4 simulation is measured
• Lateral development with the crystal granularity is considered• Energy and angle dependent is considered
• Seed energy is corrected while applying the lateral development in cluster-splitting
• Several checks are done in reconstruction, including• Splitting efficiency• Energy resolution for di-photon samples• Mass resolution for pi0 samples
and improvements are seen with the new algorithm
• To do list:• Check by MC truth information• Consider the angle dependency in lateral development• Check in to pandaroot repository
16Thank you for your attention!
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Backup
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Parameters (energy dependency)
Fitting function:
𝑝/ = 𝐶 exp −𝜏𝐸7 + 𝑛𝑝. = 𝐵 exp −𝜇𝐸7 +𝑚
𝑝6 = 𝐷 exp −𝜆𝐸7 + 𝑞
𝑝- = 𝐴exp −𝜅𝐸7 + ℎ
Range of simulated samples: • Theta Range4: 32.6536 ~ 33.7759
• Phi0~360
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Parameters (angle dependency)
𝑝- =𝐴exp −𝜅𝐸7 + ℎ
𝑝. =𝐵 exp −𝜇𝐸7 +𝑚
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Angle dependency of parameters
𝑝/ =𝐶 exp −𝜏𝐸7 + 𝑛
𝑝6 =𝐷 exp −𝜆𝐸7 + 𝑞
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The lateral development (new measurement)Define the lateral development:
𝑓 𝑟 =𝐸1234'1𝐸&@AB'3
𝐸&@AB'3 is the total energy of the single-particle shower.
𝑟
𝐸&@AB'3
𝐸1234'1The lateral development 𝑓 𝑟 can from obtained Geant4 simulation.
In this measurement the crystal dimension is considered
Control sample:Ø Gamma (0~6GeV)
Ø Geant4
Ø Phi(0, 360)
Ø Events 10000
Ø Generator: Box
Ø Theta(22, 140 )
the target crystalthe shower center
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The seed energy dependency
• For the same 𝑟, 0%&'()%0*))+
depends on 𝑟&''( .
Consider two cases where the photon hits the seed at different positions:
𝑐𝑎𝑠𝑒1: 𝑟 𝐸1234'1 𝐸&''(
𝑐𝑎𝑠𝑒2: 𝑟 𝐸1234'1 𝐸&''(|| || X
𝐸1234'1 = 𝐸&''(exp(− ⁄2.5 𝑟 𝑅))𝑟
𝐸&''(
𝐸1234'1
𝑟!""#①
②
the target crystalthe seed crystalthe shower center
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Splitting efficiency
Control sample:Ø di-photon (0~6GeV)
Ø Events 10000
Ø Geant4
Ø Generator: Box
Ø Phi(0, 360)
Ø Theta(22, 140 )
𝑑7: The distance between two shower centers
𝑟𝑎𝑡𝑖𝑜 =𝑁&5C*11*D4𝑁1A12C
×100 (%)
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The detector geometry dependency
𝑓 𝑟 / 𝑓 𝑟&''( = 𝑝Eexp[−5.F/𝜉 𝑟 ]/ 𝑝Eexp[−
5.F/𝜉(𝑟&''()] = exp{− 5.
F/𝜉 𝑟 − 𝜉 𝑟&''( }
𝑓 𝑟 = 𝑝Eexp[−𝑝-𝑅)
𝜉 𝑟 ] 𝜉(𝑟) = 𝑟 − 𝑝.𝑟exp[−𝑟
𝑝/𝑅)
5,] (𝑅) = 2.00 𝑐𝑚)
𝐸1234'1𝐸&''(
= exp{−𝑝-𝑅)
𝜉 𝑟, 𝑝., 𝑝/, 𝑝6 − 𝜉 𝑟&''( , 𝑝., 𝑝/, 𝑝6 } 𝑅𝑎𝑤:𝐸1234'1𝐸&''(
= exp(−𝜀𝑅)
𝑟)
According to the definition of 𝑓 𝑟 :
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!!!!!!𝜃- 𝜃. 𝜃/ 𝜃6 𝜃9 𝜃: 𝜃G-𝜃GE
Range 1
𝑒H
Range 2 Range 23
Ø Gamma (0.2, 0.3, 0.4…0.9 GeV)
Ø Events 10000
Ø Geant4
Ø Generator: Box
Ø Phi(0, 360)
Ø Theta(Range1, Range2,… ,Range23)
Ø Gamma (1, 1.5, 2, 2.5…6 GeV)
Ø Events 10000
Ø Geant4
Ø Generator: Box
Ø Phi(0, 360)
Ø Theta(Range1, Range2,… ,Range23)
< 1𝐺𝑒𝑉 ≥ 1𝐺𝑒𝑉
Control sample
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Control sample
Range1: 23.8514 ~ 24.6978Range2: 26.4557 ~ 27.3781Range3: 29.4579 ~ 30.4916Range4: 32.6536 ~ 33.7759Range5: 36.1172 ~ 37.3507Range6: 39.9051 ~ 41.2390Range7: 44.2385 ~ 45.7355Range8: 48.8451 ~ 50.4459Range9: 53.7548 ~ 55.4790Range10: 59.0059 ~ 60.8229Range11: 64.7855 ~ 66.7591
Range12: 70.8088 ~ 72.8652Range13: 77.0506 ~ 79.1942Range14: 83.4997 ~ 85.6749Range15: 90.2068 ~ 92.4062Range16: 96.8200 ~ 99.0099Range17: 103.361 ~ 105.534Range18: 109.793 ~ 111.893Range19: 116.067 ~ 118.019Range20: 121.838 ~ 123.686Range21: 127.273 ~ 129.033Range22: 132.400 ~ 134.031Range23: 137.230 ~ 138.679
Theta(deg):
Phi: 0~360
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Fitting results
𝑓 𝑟 = 𝑝Eexp −𝑝-𝑅)
𝜉 𝑟, 𝑝., 𝑝/, 𝑝6 ,𝜉(𝑟, 𝑝., 𝑝/, 𝑝6) = 𝑟 − 𝑝.𝑟exp[−𝑟
𝑝/𝑅)
5,]
Fitting function:
Range12; 0.5 GeV Range12; 1 GeV
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Fitting results
𝑓 𝑟 = 𝑝Eexp −𝑝-𝑅)
𝜉 𝑟, 𝑝., 𝑝/, 𝑝6 ,𝜉(𝑟, 𝑝., 𝑝/, 𝑝6) = 𝑟 − 𝑝.𝑟exp[−𝑟
𝑝/𝑅)
5,]
Fitting function:
Range12; 3 GeV Range12; 6 GeV
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Energy resolution (pi0)
The angle between the two photons produced by the decay of pi0 changes with its momentum:
𝛼
P
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Fitted parameters of the lateral developmentFitting Result:
𝑝- 𝐸7 , 𝜃 = −0.384 ∗ 𝑒𝑥𝑝 3.88 ∗ 𝐸7 + 5.44 ∗ 1089 ∗ 𝜃 − 97.7 . + 2.6
𝑝. 𝐸7 , 𝜃 = −0.352 ∗ 𝑒𝑥𝑝 4.21 ∗ 𝐸7 + (−3.94) ∗ 108: ∗ 𝜃 − 69 . + 0.932
𝑝/ 𝐸7 , 𝜃 = 0.151 ∗ exp 4.52 ∗ 𝐸7 + (−2.14) ∗ 1089 ∗ 𝜃 − 91 . + 0.841
𝑝6 𝐸7 , 𝜃 = −3.51 ∗ 𝑒𝑥𝑝 1.15 ∗ 𝐸7 + 2.26 ∗ 1086 ∗ 𝜃 − 80.3 . + 4.96
( 𝑅)= 2.00 𝑐𝑚)
Energy dependency Angle dependency
𝐸1234'1𝐸&''(
= exp{−𝑝-𝑅)
𝜉 𝑟, 𝑝., 𝑝/, 𝑝6 } 𝜉(𝑟) = 𝑟 − 𝑝.𝑟exp[−𝑟
𝑝/𝑅)
5,]