study of the cluster splitting algorithm in pandaemc

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

1

Speaker: Ziyu Zhang

OutlineØIntroduction

ØStudy of the cluster-splitting algorithm

ØLateral development measurement

ØSeed energy correction

ØReconstruction checks

ØSummary

2

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

3

Cluster-splitting for a EmcCluster Angles between the 2γ from pi0

Cluster splitting is important for high energy pi0

4

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

𝑤* =(𝐸&''()*exp(− ⁄2.5𝑟* 𝑅+)

∑,(𝐸&''(),exp(−2.5𝑟, ∕ 𝑅+)

5

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

𝑤* =(𝐸&''()*exp(− ⁄2.5𝑟* 𝑅+)

∑,(𝐸&''(),exp(−2.5𝑟, ∕ 𝑅+)

6

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)


(𝐸&''()*

Energy for the target crystal: 𝐸456789

The Cluster splitting algorithm

The 𝐸456789 is calculated using the lateral development formula

(𝐸&''()/

7

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)

8

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.

9

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,]

10

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



11

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



12

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)


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

13

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)

14

Energy resolution (di-photon)


• Theta Range12: 70.8088 ~ 72.8652

• PhiSquare area calculated according to theta

Energy resolution of di-photon

d: the distance between shower centers

15

Energy resolution (pi0)pi0 mass resolution


• Theta 22~140 (deg)

• Phi0~360 (deg)

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!

17

Backup

18

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

19

Parameters (angle dependency)

𝑝- =𝐴exp −𝜅𝐸7 + ℎ

𝑝. =𝐵 exp −𝜇𝐸7 +𝑚

20

Angle dependency of parameters

𝑝/ =𝐶 exp −𝜏𝐸7 + 𝑛

𝑝6 =𝐷 exp −𝜆𝐸7 + 𝑞

21

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

22

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


23

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 (%)

24

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 :

25

!!!!!!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

26

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

27

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

28

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

29

Energy resolution (pi0)

The angle between the two photons produced by the decay of pi0 changes with its momentum:

𝛼

P

30

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)


𝐸1234'1𝐸&''(

= exp{−𝑝-𝑅)

𝜉 𝑟, 𝑝., 𝑝/, 𝑝6 } 𝜉(𝑟) = 𝑟 − 𝑝.𝑟exp[−𝑟

𝑝/𝑅)

5,]

study of the cluster splitting algorithm in pandaemc

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