Introduction to pixel track isolation The purpose of track isolation - - PowerPoint PPT Presentation

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Introduction to pixel track isolation The purpose of track isolation - - PowerPoint PPT Presentation

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Introduction to pixel track isolation The purpose of track isolation algorithm is an additional improvement of level-1 pixel trigger performance. Procedure of pixel track isolation Reconstructed pixel tracks using kinematic parameters

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SLIDE 1

Introduction to pixel track isolation

1

  • The purpose of track isolation algorithm is an additional

improvement of level-1 pixel trigger performance.

  • Procedure of pixel track isolation
  • Reconstructed pixel tracks using kinematic parameters
  • step 0 : Finding z vertex (𝑨𝑀𝑒𝑦) of L1 EG based on the level-1 pixel

trigger algorithm

  • step 1 : Select pixel cluster (cl) satisfying Δ𝑆 < 0.3 cone
  • step 2 : Measure a distance (Δ𝑨) from 𝑨𝑀𝑒𝑦 to z intercept calculated

from two pixel clusters

  • step 3 : Measure βˆ†πœƒπ‘‘π‘š1,π‘‘π‘š2 of pixel track segments
  • step 4 : Measure βˆ†πœƒπ‘€π‘“π‘ π‘’π‘“π‘¦, π‘‘π‘š of pixel track segments
  • step 5 : Measure βˆ†πœš difference of pixel track segments
  • Estimation of pixel track π‘žπ‘ˆ using βˆ†πœš
  • Calculation of isolation based on the estimated π‘žπ‘ˆ

Jongho Lee (KNU) KPS, 24 - 26 Oct 2018

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SLIDE 2

Simulation framework

2

  • Phase 2 simulation framework
  • CMSSW_10_1_7 : the same as the level-1 pixel trigger algorithm
  • Sample used for track isolation studies
  • Signal windows for pixel track reconstruction in step 0 ~ step 5
  • SingleMuon without pile-up
  • /SingleMu_Pt2t200_Eta3p0_NOPU_93X_upgrade2023-realistic_v5_D17/GEN-SIM-DIGI-RAW
  • Efficiency of pixel track isolation algorithm
  • SingleElectron with <PU>=200
  • /SingleE_FlatPt-2to100/PhaseIIFall17D-L1TPU200_93X_upgrade2023_realistic_v5-v1/GEN-SIM-DIGI-RAW
  • Rate reduction factor calculation for pixel track isolation algorithm
  • Minimum Bias sample with <PU>=200
  • /SingleNeutrino/PhaseIIFall17D-L1TPU200_93X_upgrade2023_realistic_v5-v1/GEN-SIM-DIGI-RAW

Jongho Lee (KNU) KPS, 24 - 26 Oct 2018

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SLIDE 3

Finding vertex of L1 𝒇/𝜹 object with pixel clusters

3

Jongho Lee (KNU)

  • Almost one pixel track from electron candidate found by the

level-1 pixel based track trigger algorithm

  • The z vertex of L1 EG (𝑨𝑀𝑒𝑦) is reconstructed by extrapolation of

straight line to beam axis using two pixel clusters on the pixel track from the level-1 pixel trigger algorithm

  • 𝑨𝑀𝑒𝑦 tested with gen level π‘¨π‘•π‘“π‘œβˆ’π‘šπ‘“π‘€π‘“π‘š ∢ 𝑨𝑀𝑒𝑦 βˆ’ π‘¨π‘•π‘“π‘œβˆ’π‘šπ‘“π‘€π‘“π‘š
  • Sample: SingleElectron + <PU>=200

Region 1 (𝟏 < 𝜽 < 𝟏. πŸ—) Region 3 (𝟐. πŸ“ < 𝜽 < 𝟐. πŸ–) Region 6 (πŸ‘. πŸ– < 𝜽 < πŸ’. 𝟏)

25 𝝂𝒏 58 𝝂𝒏 472 𝝂𝒏

KPS, 24 - 26 Oct 2018

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SLIDE 4
  • Required βˆ†π‘† < 0.3 to every pixel clusters in each pixel layer for

electron direction.

  • Pixel clusters along L1 𝑓/𝛿 direction are selected with

βˆ†π‘† = Ξ”πœš 𝑀1 𝑓/𝛿, π‘žπ‘—π‘¦π‘“π‘š

2 + Ξ”πœƒ 𝑀1 𝑓/𝛿, π‘žπ‘—π‘¦π‘“π‘š 2 < 0.3

Track isolation: step 1 & 2 (βˆ†π‘Ί, βˆ†π’œ)

4

  • Measuring Δ𝑨 which is distance

between L1 𝑓/𝛿 vertex (𝑨𝑀𝑒𝑦) and z-intercept of pixel track segments.

Δ𝑨 = 𝑨𝑀𝑒𝑦 βˆ’ π‘¨π‘‘π‘š1, π‘‘π‘š2

Jongho Lee (KNU)

electron

Layer 1 Layer2 z-axis

βˆ†π‘¨

r-axis

𝑨𝑀𝑒𝑦 π‘¨π‘‘π‘š1, π‘‘π‘š2

KPS, 24 - 26 Oct 2018

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SLIDE 5
  • βˆ†πœƒπ‘‘π‘š1, π‘‘π‘š2 is a longitudinal angle difference in r-z plane between pixel track

segments.

  • The electron signal track can be selected by βˆ†πœƒπ‘‘π‘š1, π‘‘π‘š2 cut requirements.
  • Combinatorial background can be removed by the minimum βˆ†πœƒπ‘‘π‘š1, π‘‘π‘š2

requirements.

Track isolation: step 3 (βˆ†πœƒπ‘‘π‘š1,π‘‘π‘š2)

5

KPS, 24 - 26 Oct 2018

βˆ†πœ½π’…π’ŽπŸ, π’…π’ŽπŸ‘ = 𝜽 𝑴𝒋, π‘΄π’Œ βˆ’ 𝜽 π‘΄π’Œ, 𝑴𝒍

Jongho Lee (KNU)

Layer 1 Layer 2 Layer 3 Layer 4 Electron signal Combinatorial background

βˆ†πœƒ

z-axis r-axis Example of signal window

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SLIDE 6

Track isolation: step 4 (βˆ†πœƒπ‘¨π‘€π‘’π‘¦, π‘‘π‘š)

6

Layer 1

βˆ†πœ½ = 𝜽 π’œπ’˜π’–π’š, π‘΄πŸ’ βˆ’ 𝜽 (π’œπ’˜π’–π’š, π‘΄πŸ‘)

𝜽 (π’œπ’˜π’–π’š, π‘΄πŸ‘) 𝜽(π’œπ’˜π’–π’š, π‘΄πŸ’)

Layer 2 Layer 3

𝑨𝑀𝑒𝑦 𝜽 (π’œπ’˜π’–π’š, π‘΄πŸ)

  • βˆ†πœƒπ‘¨π‘€π‘’π‘¦, π‘‘π‘š is a longitudinal angle difference in r-z plane between vectors

from L1 𝑓/𝛿 vertex to pixel clusters.

  • Ξ”πœƒ distribution for signal electron track is very narrow.
  • one of powerful background rejection parameters

βˆ†πœ½π’œπ’˜π’–π’š, π’…π’Ž = 𝜽 π’œπ’˜π’–π’š, 𝑴𝒋 βˆ’ 𝜽 π’œπ’˜π’–π’š, π‘΄π’Œ

Jongho Lee (KNU)

z-axis

r-axis Example of signal window

KPS, 24 - 26 Oct 2018

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SLIDE 7
  • π›¦πœš difference (βˆ†πœš12 βˆ’ βˆ†πœš23)

Track isolation: step 5 (π›¦πœš difference)

7

πœ π”πŸπŸ‘ = π”πŸ 𝑸𝑾, π‘΄πŸ βˆ’ π”πŸ‘(π‘΄πŸ, π‘΄πŸ‘) πœ π”πŸ‘πŸ’ = π”πŸ‘ π‘΄πŸ, π‘΄πŸ‘ βˆ’ π”πŸ’ π‘΄πŸ‘, π‘΄πŸ’ πœ π” difference = πœ π”πŸ‘πŸ’ βˆ’ πœ π”πŸπŸ‘

  • The signal track has the consistent

sign of each βˆ†πœš12 and βˆ†πœš23.

  • Also, once the signal track is

reconstructed, its π›¦πœš difference have to be small. x-y plane

Jongho Lee (KNU)

Example of signal window

KPS, 24 - 26 Oct 2018

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SLIDE 8

Jongho Lee (KNU)

8

π‘žπ‘ˆ estimation using π›¦πœš of pixel track segments

  • π‘žπ‘ˆ of the pixel track is estimated with the correlation between the gen-

level muon π‘žπ‘ˆ and Ξ”πœš (𝜚 𝐢𝑇, 𝑀𝑗 βˆ’ 𝜚 𝑀𝑗 , π‘€π‘˜ ), where BS = beam spot

  • π‘žπ‘ˆ = 𝑔 1/Ξ”πœš , 𝑔 : first order polynomial function for fitting
  • Simulation sample
  • Gen level π‘žπ‘ˆ and Ξ”πœš used in singleMuon events without pile-up
  • 2 < π‘žπ‘ˆ < 200 π»π‘“π‘Š , βˆ’3 < πœƒ < 3

Example BSL2 – L2D2 Example BSD1 – D1D3

KPS, 24 - 26 Oct 2018

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SLIDE 9

9

Jongho Lee (KNU)

Track isolation calculation based on the π‘žπ‘ˆ

π½π‘‘π‘π‘šπ‘π‘’π‘—π‘π‘œ = π‘žπ‘ˆ, 𝑗 π‘žπ‘ˆ, 𝑒𝑠𝑙

  • π‘žπ‘ˆ, 𝑗 : linear sum of π‘žπ‘ˆ of other tracks

except for the highest π‘žπ‘ˆ track in βˆ†π‘† < 0.3 cone

  • π‘žπ‘ˆ, 𝑒𝑠𝑙 : linear sum of π‘žπ‘ˆ of all tracks in

βˆ†π‘† < 0.3 cone

  • Region 1 (𝟏 < 𝜽 < 𝟏. πŸ—)
  • L1 EG 𝑭𝑼 > πŸ‘πŸ 𝑯𝒇𝑾

0.05

Cut

  • Signal efficiency:

99.5%

  • Background

rejection: 70.6%

KPS, 24 - 26 Oct 2018

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SLIDE 10

10

Jongho Lee (KNU)

Calculation of Isolation for each region

1) Region2 (0.8 < πœƒ < 1.4) 2) Region3 (1.4 < πœƒ < 1.7) 3) Region4 (1.7 < πœƒ < 2.1) 4) Region5 (2.1 < πœƒ < 2.7) 5) Region6 (2.7 < πœƒ < 3.0)

  • L1 EG 𝑭𝑼 > πŸ‘πŸ 𝑯𝒇𝑾

1) 2) 3) 4) 5)

0.05

Cut

0.03

Cut

0.07

Cut

0.3

Cut

0.3

Cut

  • Sig.eff:

99.8%

  • Bkg. rej:

52.2%

  • Sig.eff:

99.4%

  • Bkg.rej:

54.0%

  • Sig.eff:

98.3%

  • Bkg.rej:

73.0%

  • Sig.eff:

94.7%

  • Bkg.rej:

63.4%

  • Sig.eff:

95.8%

  • Bkg.rej:

53.3%

KPS, 24 - 26 Oct 2018

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SLIDE 11

11

Jongho Lee (KNU)

Track isolation cut region1 (𝟏 < 𝜽 < 𝟏. πŸ—)

  • Isolation cut : 0.05
  • signal efficiency: 99.5%
  • background rejection: 70.6%
  • signal efficiency
  • background rejection

KPS, 24 - 26 Oct 2018

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SLIDE 12

12

Jongho Lee (KNU)

Track isolation cut region6 (πŸ‘. πŸ– < 𝜽 < πŸ’. 𝟏)

  • Isolation cut : 0.3
  • signal efficiency: 95.8%
  • background rejection: 53.3%
  • signal efficiency
  • background rejection

KPS, 24 - 26 Oct 2018

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SLIDE 13

Jongho Lee (KNU)

13

Preliminary results of track isolation

KPS, 24 - 26 Oct 2018

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SLIDE 14

Jongho Lee (KNU)

14

Preliminary results of track isolation

𝟏 < 𝜽 < 𝟐. πŸ”

Background reduction factor 𝟐. πŸ” < 𝜽 < πŸ‘. πŸ” Background reduction factor

L1 EG 336 kHz

  • 103 kHz
  • L1 EG + Pixel

37 kHz 9.1 23 kHz 4.5

L1 EG + L1 Track (Iso)

33 kHz 10 22 kHz 4.7

L1 EG + Pixel + Isolation

15 kHz 22.4 8.5 kHz 12.1

KPS, 24 - 26 Oct 2018

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SLIDE 15

Jongho Lee (KNU)

15

Preliminary results of track isolation

𝟏 < 𝜽 < πŸ‘. πŸ”

Background reduction factor

𝟏 < 𝜽 < πŸ’. 𝟏

Background reduction factor

L1 EG 436 kHz

  • 505 kHz
  • L1 EG + Pixel

59 kHz 7.4 70 kHz 7.2

L1 EG + L1 Track (Iso)

55 kHz 7.9 Not available Not available

L1 EG + Pixel + Isolation

24 kHz 18.1 29 kHz 17.4

KPS, 24 - 26 Oct 2018

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SLIDE 16

16

Conclusion

Jongho Lee (KNU)

  • The pixel track isolation was achieved through following process
  • Reconstructed pixel tracks using kinematic parameters
  • Estimation of pixel track π‘žπ‘ˆ using βˆ†πœš
  • Calculation of isolation based on the estimated π‘žπ‘ˆ
  • The overall level-1 pixel trigger efficiency including isolation is
  • average 95% in 𝜽 < πŸ‘. 𝟐
  • average 90% in 𝜽 < πŸ’. 𝟏
  • We improved the performance of level-1 pixel triggers 2.5 times

by pixel track isolation while keeping high trigger efficiency

  • The total background reduction factor is
  • 18 in 𝜽 < πŸ‘. πŸ”
  • 17 in 𝜽 < πŸ’. 𝟏

KPS, 24 - 26 Oct 2018

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SLIDE 17

Jongho Lee (KNU)

17

Back up

KPS, 24 - 26 Oct 2018

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SLIDE 18

18

Jongho Lee (KNU)

Track isolation cut region2 (𝟏. πŸ— < 𝜽 < 𝟐. πŸ“)

  • Isolation cut : 0.05
  • signal efficiency: 99.8%
  • background rejection: 52.2%
  • signal efficiency
  • background rejection

KPS, 24 - 26 Oct 2018

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SLIDE 19

19

Jongho Lee (KNU)

Track isolation cut region3 (𝟐. πŸ“ < 𝜽 < 𝟐. πŸ–)

  • Isolation cut : 0.03
  • signal efficiency: 99.4%
  • background rejection: 54.0%
  • signal efficiency
  • background rejection

KPS, 24 - 26 Oct 2018

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SLIDE 20

20

Jongho Lee (KNU)

Track isolation cut region4 (𝟐. πŸ– < 𝜽 < πŸ‘. 𝟐)

  • Isolation cut : 0.07
  • signal efficiency: 98.3%
  • background rejection: 73.0%
  • signal efficiency
  • background rejection

KPS, 24 - 26 Oct 2018

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SLIDE 21

21

Jongho Lee (KNU)

Track isolation cut region5 (πŸ‘. 𝟐 < 𝜽 < πŸ‘. πŸ–)

  • Isolation cut : 0.3
  • signal efficiency: 94.7%
  • background rejection: 63.4%
  • signal efficiency
  • background rejection

KPS, 24 - 26 Oct 2018

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SLIDE 22

22

Combination of track parameters in region1 (𝟏 < 𝜽 < 𝟏. πŸ—)

Ξ”z βˆ†πœƒπ‘΄π’‹, π‘΄π’Œ βˆ†πœƒπ‘„π‘Š, 𝑴𝒋

βˆ†πœšπŸπŸ‘ βˆ’ βˆ†πœšπŸ‘πŸ’

L123

L13 – L12 L13 – L13 L13 – L23 L12 – L13 L12 – L23 L13 – L23 PVL3 – PVL2 PVL3 – PVL1

Ξ”πœš[(PVL1)-(L1L2)] – Ξ”πœš[(L1L2)-(L2L3)]

L124

L14 – L12 L14 – L14 L14 – L24 L12 – L14 L12 – L24 L14 – L24 PVL4 – PVL2 PVL4 – PVL1

Ξ”πœš[(PVL1)-(L1L2)] – Ξ”πœš[(L1L2)-(L2L4)]

L134

L14 – L13 L14 – L14 L14 – L34 L13 – L14 L13 – L34 L14 – L34 PVL4 – PVL3 PVL4 – PVL1

Ξ”πœš[(PVL1)-(L1L3)] – Ξ”πœš[(L1L3)-(L3L4)]

L234

L24 – L23 L24 – L24 L24 – L34 L23 – L24 L23 – L34 L24 – L34 PVL4 – PVL3 PVL4 – PVL2

Ξ”πœš[(PVL2)-(L2L3)] – Ξ”πœš[(L2L3)-(L3L4)]

β€» PV: vertex of L1 𝑓/𝛿

Jongho Lee (KNU)

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SLIDE 23

23

Combination of track parameters in region2 (𝟏. πŸ— < 𝜽 < 𝟐. πŸ“)

Ξ”z βˆ†πœƒπ‘΄π’‹, π‘΄π’Œ βˆ†πœƒπ‘„π‘Š, 𝑴𝒋

βˆ†πœšπŸπŸ‘ βˆ’ βˆ†πœšπŸ‘πŸ’

L123

L13 – L12 L13 – L13 L13 – L23 L12 – L13 L12 – L23 L13 – L23 PVL3 – PVL2 PVL3 – PVL1

Ξ”πœš[(PVL1)-(L1L2)] – Ξ”πœš[(L1L2)-(L2L3)]

L12D1

L2D1 – L12 L2D1 – L1D1 L2D1 – L2D1 L12 – L1D1 L12 – L2D1 L1D1 – L2D1 PVD1 – PVL2 PVD1 – PVL1

Ξ”πœš[(PVL1)-(L1L2)] – Ξ”πœš[(L1L2)-(L2D1)]

L13D1

L3D1 – L13 L3D1 – L1D1 L3D1 – L3D1 L13 – L1D1 L13 – L3D1 L1D1 – L3D1 PVD1 – PVL3 PVD1 – PVL1

Ξ”πœš[(PVL1)-(L1L3)] – Ξ”πœš[(L1L3)-(L3D1)]

L23D1

L3D1 – L23 L3D1 – L2D1 L3D1 – L3D1 L23 – L2D1 L23 – L3D1 L2D1 – L3D1 PVD1 – PVL3 PVD1 – PVL2

Ξ”πœš[(PVL2)-(L2L3)] – Ξ”πœš[(L2L3)-(L3D1)]

β€» PV: vertex of L1 𝑓/𝛿

Jongho Lee (KNU)

slide-24
SLIDE 24

24

Combination of track parameters in region3 (𝟐. πŸ“ < 𝜽 < 𝟐. πŸ–)

Ξ”z βˆ†πœƒπ‘΄π’‹, π‘΄π’Œ βˆ†πœƒπ‘„π‘Š, 𝑴𝒋

βˆ†πœšπŸπŸ‘ βˆ’ βˆ†πœšπŸ‘πŸ’

L12D1

L2D1 – L12 L2D1 – L1D1 L2D1 – L2D1 L12 – L1D1 L12 – L2D1 L1D1 – L2D1 PVD1 – PVL2 PVD1 – PVL1

Ξ”πœš[(PVL1)-(L1L2)] – Ξ”πœš[(L1L2)-(L2D1)]

L12D2

L2D2 – L12 L2D2 – L1D2 L2D2 – L2D2 L12 – L1D2 L12 – L2D2 L1D2 – L2D2 PVD2 – PVL2 PVD2 – PVL1

Ξ”πœš[(PVL1)-(L1L2)] – Ξ”πœš[(L1L2)-(L2D1)]

L1D12

L1D2 – L1D1 L1D2 – L1D2 L1D2 – D1D2 L1D1 – L1D2 L1D1 – D1D2 L1D2 – D1D2 PVD2 – PVD1 PVD2 – PVL1

Ξ”πœš[(PVL1)-(L1D1)] – Ξ”πœš[(L1D1)-(D1D2)]

L2D12

L2D2 – L23 L2D2 – L2D1 L2D2 – L3D1 L2D1 – L2D2 L2D1 – D1D2 L2D2 – D1D2 PVD2 – PVD1 PVD2 – PVL2

Ξ”πœš[(PVL2)-(L2D1)] – Ξ”πœš[(L2D1)-(D1D2)]

β€» PV: vertex of L1 𝑓/𝛿

Jongho Lee (KNU)

slide-25
SLIDE 25

25

Combination of track parameters in region4 (𝟐. πŸ– < 𝜽 < πŸ‘. 𝟐)

Ξ”z βˆ†πœƒπ‘΄π’‹, π‘΄π’Œ βˆ†πœƒπ‘„π‘Š, 𝑴𝒋

βˆ†πœšπŸπŸ‘ βˆ’ βˆ†πœšπŸ‘πŸ’

L1D12

L1D2 – L1D1 L1D2 – L1D2 L1D2 – D1D2 L1D1 – L1D2 L1D1 – D1D2 L1D2 – D1D2 PVD2 – PVD1 PVD2 – PVL1

Ξ”πœš[(PVL1)-(L1D1)] – Ξ”πœš[(L1D1)-(D1D2)]

L1D13

L1D3 – L1D1 L1D3 – L1D3 L1D3 – D1D3 L1D1 – L1D3 L1D1 – D1D3 L1D3 – D1D3 PVD3 – PVD1 PVD3 – PVL1

Ξ”πœš[(PVL1)-(L1D1)] – Ξ”πœš[(L1D1)-(D1D3)]

L1D23

L1D3 – L1D2 L1D3 – L1D3 L1D3 – D2D3 L1D2 – L1D3 L1D2 – D2D3 L1D3 – D2D3 PVD3 – PVD2 PVD3 – PVL1

Ξ”πœš[(PVL1)-(L1D2)] – Ξ”πœš[(L1D2)-(D2D3)]

D123

D13 – D12 D13 – D13 D13 – D23 D12 – D13 D12 – D23 D13 – D23 PVD3 – PVD2 PVD3 – PVD1

Ξ”πœš[(PVD1)-(D1D2)] – Ξ”πœš[(D1D2)-(D2D3)]

β€» PV: vertex of L1 𝑓/𝛿

Jongho Lee (KNU)

slide-26
SLIDE 26

26

Combination of track parameters in region5 (πŸ‘. 𝟐 < 𝜽 < πŸ‘. πŸ–)

Ξ”z βˆ†πœƒπ‘΄π’‹, π‘΄π’Œ βˆ†πœƒπ‘„π‘Š, 𝑴𝒋

βˆ†πœšπŸπŸ‘ βˆ’ βˆ†πœšπŸ‘πŸ’

D123

D13 – D12 D13 – D13 D13 – D23 D12 – D13 D12 – D23 D13 – D23 PVD3 – PVD2 PVD3 – PVD1

Ξ”πœš[(PVD1)-(D1D2)] – Ξ”πœš[(D1D2)-(D2D3)]

D124

D14 – D12 D14 – D14 D14 – D24 D12 – D14 D12 – D24 D14 – D24 PVD4 – PVD2 PVD4 – PVD1

Ξ”πœš[(PVD1)-(D1D2)] – Ξ”πœš[(D1D2)-(D2D4)]

D134

D14 – D12 D14 – D14 D14 – D24 D12 – D13 D12 – D34 D13 – D34 PVD4 – PVD3 PVD4 – PVD1

Ξ”πœš[(PVD1)-(D1D3)] – Ξ”πœš[(D1D3)-(D3D4)]

D234

D24 – D23 D24 – D24 D24 – D34 D23 – D24 D23 – D34 D24 – D34 PVD4 – PVD3 PVD4 – PVD2

Ξ”πœš[(PVD2)-(D2D3)] – Ξ”πœš[(D2D3)-(D3D4)]

β€» PV: vertex of L1 𝑓/𝛿

Jongho Lee (KNU)

slide-27
SLIDE 27

27

Combination of track parameters in region6 (πŸ‘. πŸ– < 𝜽 < πŸ’. 𝟏)

Ξ”z βˆ†πœƒπ‘΄π’‹, π‘΄π’Œ βˆ†πœƒπ‘„π‘Š, 𝑴𝒋

βˆ†πœšπŸπŸ‘ βˆ’ βˆ†πœšπŸ‘πŸ’

D234

D24 – D23 D24 – D24 D24 – D34 D23 – D24 D23 – D34 D24 – D34 PVD4 – PVD3 PVD4 – PVD2

Ξ”πœš[(PVD2)-(D2D3)] – Ξ”πœš[(D2D3)-(D3D4)]

D235

D25 – D23 D25 – D25 D25 – D35 D23 – D25 D23 – D35 D25 – D35 PVD5 – PVD3 PVD5 – PVD2

Ξ”πœš[(PVD2)-(D2D3)] – Ξ”πœš[(D2D3)-(D3D5)]

D245

D25 – D24 D25 – D25 D25 – D45 D24 – D25 D24 – D45 D25 – D45 PVD5 – PVD4 PVD5 – PVD2

Ξ”πœš[(PVD2)-(D2D4)] – Ξ”πœš[(D2D4)-(D4D5)]

D345

D35 – D34 D35 – D35 D35 – D45 D34 – D35 D34 – D45 D35 – D45 PVD5 – PVD4 PVD5 – PVD3

Ξ”πœš[(PVD3)-(D3D4)] – Ξ”πœš[(D3D4)-(D4D5)]

β€» PV: vertex of L1 𝑓/𝛿

Jongho Lee (KNU)