CalibrationofHyCalandGEMZposition• Solvingthefollowingsystemofequations,foreachdoublearmMoller,wehaveananalyticsolutionforZ:• 𝐸"#$% + 𝑚 = 𝐸) + 𝐸*,wheremiselectronmass,E1,E2areenergyofscatteredelectrons• 𝑃) sin 𝜃) = 𝑃* sin 𝜃* whereP1 andP2 arescatteredelectronmomenta,andthetaaretheirpolarangleinlab• 𝑃) cos 𝜃* + 𝑃* cos 𝜃* = 𝑃"#$%• sin 𝜃) = 34

345675andcos 𝜃) = 7

345675

• Itcanbesolvedwithoutneglectingelectronmass:

𝑧 = 𝑚 + 𝐸"#$% 𝑅)𝑅*/(2𝑚)

1

TestingtheFormula

• SimpleMCtogeneratedoublearmMollers basedonkinematics(onlyneedthescatteringangleofoneofthescatteredelectron)• EnergyofthescatteredMollerelectron,withoutneglectingelectronmass:

• 𝐸> =%×(@ABCD6EFG5 H @ABCD6% )IEFG5 H )

@ABCDIEFG5 H @ABCD6%()6EFG5 H )

• Usingconservationtogettheangleandenergyoftheotherelectron• Givenapredefinedzpositionofadetector,calculatetheradiiofthetwoelectrons• Plugbacktotheexpressionofzinthepreviouspagetocheckconsistency

2

TestingtheFormula

plotEntries 10000

Mean 5127

RMS 0.000193

5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 60000

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10000plot

Entries 10000

Mean 5127

RMS 0.000193

plot

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1200 ethetaEntries 10000Mean x 3.41Mean y 315.2RMS x 1.5RMS y 225.9

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35etheta

Entries 10000Mean x 3.41Mean y 315.2RMS x 1.5RMS y 225.9

etheta

SimulatedMoller

Calculated zpositioncasedon thetworadiiandbeamenergy

Settingz=5127inthesimulation

Theta(deg)

E’(Mev)

Z(mm) 3

plotEntries 10000Mean 5600RMS 0.0002024

5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 60000

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plotEntries 10000Mean 5600RMS 0.0002024

plot

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1200 ethetaEntries 10000Mean x 3.41Mean y 315.2RMS x 1.5RMS y 225.9

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35etheta

Entries 10000Mean x 3.41Mean y 315.2RMS x 1.5RMS y 225.9

etheta

TestingtheFormula

SimulatedMollerCalculated zpositioncasedon thetworadiiandbeamenergy

Settingz=5600inthesimulation

Theta(deg)

E’(Mev)

Z(mm) 4

AnalysisProcedure• UsingC12 run1345• AssumingGEM1,GEM2andHyCalpositionareat{5304,5264,5815}mm• MakesuretheoffsetisincludedforbothGEMsandHyCal• FirstselectingdoublearmMollerbasedonHyCalalone:

• |E1 – Eexpect| /Eexpect and|E1 – Eexpect| /Eexpect both lessthan4x0.026/sqrt(Eexpect)• |E1 +E2 – Ebeam| /Ebeam <4x0.026/sqrt(Ebeam)• ∆𝛗 <20deg• Scatteringangles>0.7deg• Twoclustersevents

• ProjectGEMhitstoHyCalsurfaceanddoamatchwithradius{20,30,40}mmfor{PWO,transition,LG}

• EachGEMhasitsowndoublearmMolleracceptance,usethosetocalculatezindependently(mustwithoutanyprojectionwhencalculatingz)• Reduce∆𝛗 cutto5deg whenusingGEMcoordinates,duetobetterresolution

5

SelectingMollerEventsCandidatesUsingOnlyHyCal

0 1 2 3 4 5 60

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1400

eThetaEntries 490310Mean x 1.994Mean y 560.3RMS x 1.102RMS y 275.5

0

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250

eThetaEntries 490310Mean x 1.994Mean y 560.3RMS x 1.102RMS y 275.5

eTheta

Theta(deg)

E’(MeV)

6

DoubleArmMollers onEachGEM

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600 hit_Pos_0Entries 38488Mean x -8.325Mean y 0.4398RMS x 26.49RMS y 215.6

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Entries 38488Mean x -8.325Mean y 0.4398RMS x 26.49RMS y 215.6

hit_Pos_0

-600 -400 -200 0 200 400 600-600

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0

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600 hit_Pos_1Entries 39704Mean x 8.733Mean y 4.053RMS x 27.45RMS y 214.3

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Entries 39704Mean x 8.733Mean y 4.053RMS x 27.45RMS y 214.3

hit_Pos_1

GEM1 GEM2

(mm) (mm)

(mm) (mm)

Afterrequiring∆𝛗 <5deg

7

DoubleChecktheOffsets

intersect_det_0_coor_1

Entries 9622Mean 0.1513RMS 14.18

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Entries 9622Mean 0.1513RMS 14.18

intersect_det_0_coor_1intersect_det_0_coor_0

Entries 9622Mean 0.01454RMS 4.04

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Entries 9622Mean 0.01454RMS 4.04

intersect_det_0_coor_0

UsingtwostraightlinesdeterminedbyfourMollerscatteringelectronstodeterminethebeamspotposition,usingdatasetinthepreviouspage

GEM1 GEM1

(mm) (mm)

8

DeterminetheZposition

zHist_0Entries 19244Mean 5219RMS 69.13

/ ndf 2χ 47.69 / 15Prob 2.849e-05Constant 14.8± 1350 Mean 0.3± 5222 Sigma 0.34± 32.55

4600 4800 5000 5200 5400 5600 5800 6000 6200 64000

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1400zHist_0

Entries 19244Mean 5219RMS 69.13

/ ndf 2χ 47.69 / 15Prob 2.849e-05Constant 14.8± 1350 Mean 0.3± 5222 Sigma 0.34± 32.55

zHist_0

Survey result5304mm,Fitresult5222mm

GEM1

(mm) 9

DeterminetheZposition

zHist_1Entries 19852Mean 5180RMS 67.2

/ ndf 2χ 92.92 / 17Prob 1.79e-12Constant 15.6± 1491 Mean 0.3± 5183 Sigma 0.26± 30.89

4600 4800 5000 5200 5400 5600 5800 6000 6200 64000

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1600zHist_1

Entries 19852Mean 5180RMS 67.2

/ ndf 2χ 92.92 / 17Prob 1.79e-12Constant 15.6± 1491 Mean 0.3± 5183 Sigma 0.26± 30.89

zHist_1

Survey result5264mm,Fitresult5183mmDifferencebetween thetwoGEM:39mm

GEM2

(mm) 10

DeterminetheZpositionzHist_2

Entries 245155Mean 5643RMS 135.8

/ ndf 2χ 64.4 / 42Prob 0.01467Constant 17.4± 5403 Mean 0.4± 5640 Sigma 0.6± 115

4600 4800 5000 5200 5400 5600 5800 6000 6200 64000

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zHist_2Entries 245155Mean 5643RMS 135.8

/ ndf 2χ 64.4 / 42Prob 0.01467Constant 17.4± 5403 Mean 0.4± 5640 Sigma 0.6± 115

zHist_2

(mm)

HyCal,Afterapplyingshowdepthcorrection

Survey result5815mm,Fitresult5640mm

11

CheckingtheResult• Mollerscatteringhasaminimumopeningangleinlabframe,whichcorrespondtoscatteringangle=90deg incenterofmassframe• 𝑡𝑎𝑛* 𝜃N$" = *%

@ABCD6%• At1.1GeV,theminimumopeningangleis3.491 deg

• Inorderword,wearelookingforsymmetricMollers inthelabframe• Using|R1 – R2|<10mmtoselect symmetricMollers forbothHyCalandGEM• Duetostatistics,needtouseentireGEM,soprojectGEMtothesameplaneatz=5815mm

• Wecanagaincheckthe∆RvsRdistributiontosee ifeverythingmatches

12

SymmetricMollerDistribution

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600 sym_moller_posEntries 8668Mean x -0.07736Mean y -0.07177RMS x 119.2RMS y 123.3

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10sym_moller_posEntries 8668Mean x -0.07736Mean y -0.07177RMS x 119.2RMS y 123.3

sym_moller_pos

(mm)

(mm)

13

MinimumOpeningAngle

open_angle_0Entries 4272Mean 3.432RMS 0.04266

/ ndf 2χ 44.29 / 15Prob 9.914e-05Constant 8.5± 378.3 Mean 0.000± 3.435 Sigma 0.00034± 0.01943

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 40

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open_angle_0Entries 4272Mean 3.432RMS 0.04266

/ ndf 2χ 44.29 / 15Prob 9.914e-05Constant 8.5± 378.3 Mean 0.000± 3.435 Sigma 0.00034± 0.01943

open_angle_0open_angle_2

Entries 5035Mean 3.383RMS 0.0702

/ ndf 2χ 27.1 / 38Prob 0.9059Constant 3.2± 157.8 Mean 0.001± 3.381 Sigma 0.00135± 0.06163

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 40

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Entries 5035Mean 3.383RMS 0.0702

/ ndf 2χ 27.1 / 38Prob 0.9059Constant 3.2± 157.8 Mean 0.001± 3.381 Sigma 0.00135± 0.06163

open_angle_2

Fitresult:3.435degExpect:3.491deg

Fitresult:3.381degExpect:3.491deg

FromGEM FromHyCal

WiththeSurveysetting14

RGEM - RHyCal vsRGEM

100 150 200 250 300 3501

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(mm)

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WiththeSurveysetting 15

open_angle_2Entries 5041Mean 3.486RMS 0.07507

/ ndf 2χ 42.02 / 37Prob 0.2625Constant 3.2± 153.1 Mean 0.001± 3.488 Sigma 0.00150± 0.06289

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 40

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open_angle_2Entries 5041Mean 3.486RMS 0.07507

/ ndf 2χ 42.02 / 37Prob 0.2625Constant 3.2± 153.1 Mean 0.001± 3.488 Sigma 0.00150± 0.06289

open_angle_2open_angle_0

Entries 4334Mean 3.486RMS 0.04533

/ ndf 2χ 20 / 13Prob 0.09513Constant 8.8± 389.6 Mean 0.000± 3.489 Sigma 0.00035± 0.01883

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 40

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open_angle_0Entries 4334Mean 3.486RMS 0.04533

/ ndf 2χ 20 / 13Prob 0.09513Constant 8.8± 389.6 Mean 0.000± 3.489 Sigma 0.00035± 0.01883

open_angle_0

MinimumOpeningAngle

Fitresult:3.489degExpect:3.491deg

FromGEM FromHyCal

WiththeNewsetting

Fitresult:3.488degExpect:3.491deg

16

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Graph

RGEM - RHyCal vsRGEM

(mm)

(mm)

WiththeNewsetting

Lessthan300umdifferenceinthemeanvalues

17

slice_0Entries 103410Mean -0.04089RMS 3.096

/ ndf 2χ 1089 / 53Prob 0Constant 37.7± 9131 Mean 0.00932± -0.04747 Sigma 0.01± 2.98

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10000slice_0

Entries 103410Mean -0.04089RMS 3.096

/ ndf 2χ 1089 / 53Prob 0Constant 37.7± 9131 Mean 0.00932± -0.04747 Sigma 0.01± 2.98

delta_R_vs_R

RGEM:80~120mm

mm

slice_1Entries 49406Mean 0.02703RMS 3.401

/ ndf 2χ 393.9 / 53Prob 0Constant 23.2± 3946 Mean 0.01493± 0.02475 Sigma 0.012± 3.304

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4500 slice_1Entries 49406Mean 0.02703RMS 3.401

/ ndf 2χ 393.9 / 53Prob 0Constant 23.2± 3946 Mean 0.01493± 0.02475 Sigma 0.012± 3.304

delta_R_vs_R

mm

RGEM:120~160mm

slice_2Entries 34647Mean 0.00661RMS 3.788

/ ndf 2χ 375.4 / 49Prob 0Constant 17.7± 2485 Mean 0.019819± -0.003942 Sigma 0.017± 3.668

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slice_2Entries 34647Mean 0.00661RMS 3.788

/ ndf 2χ 375.4 / 49Prob 0Constant 17.7± 2485 Mean 0.019819± -0.003942 Sigma 0.017± 3.668

delta_R_vs_R

RGEM:160~200mm

mm

slice_3Entries 32089Mean 0.02893RMS 4.322

/ ndf 2χ 196.5 / 54Prob 4.622e-18Constant 14.4± 2000 Mean 0.02375± 0.02597 Sigma 0.019± 4.241

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2200 slice_3Entries 32089Mean 0.02893RMS 4.322

/ ndf 2χ 196.5 / 54Prob 4.622e-18Constant 14.4± 2000 Mean 0.02375± 0.02597 Sigma 0.019± 4.241

delta_R_vs_R

RGEM:200~240mm

mm18

slice_4Entries 32019Mean 0.08794RMS 4.835

/ ndf 2χ 176.5 / 55Prob 1.17e-14Constant 12.8± 1775 Mean 0.02676± 0.08223 Sigma 0.022± 4.772

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slice_4Entries 32019Mean 0.08794RMS 4.835

/ ndf 2χ 176.5 / 55Prob 1.17e-14Constant 12.8± 1775 Mean 0.02676± 0.08223 Sigma 0.022± 4.772

delta_R_vs_R

RGEM:240~280mm

mm

slice_5Entries 33758Mean 0.2156RMS 5.401

/ ndf 2χ 158.1 / 57Prob 1.912e-11Constant 11.7± 1672 Mean 0.0292± 0.2123 Sigma 0.023± 5.345

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Entries 33758Mean 0.2156RMS 5.401

/ ndf 2χ 158.1 / 57Prob 1.912e-11Constant 11.7± 1672 Mean 0.0292± 0.2123 Sigma 0.023± 5.345

delta_R_vs_R

RGEM:280~320mm

mm

slice_6Entries 35399Mean 0.05385RMS 6.501

/ ndf 2χ 226 / 57Prob 5.471e-22Constant 10.3± 1504 Mean 0.0335± 0.2358 Sigma 0.027± 6.187

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slice_6Entries 35399Mean 0.05385RMS 6.501

/ ndf 2χ 226 / 57Prob 5.471e-22Constant 10.3± 1504 Mean 0.0335± 0.2358 Sigma 0.027± 6.187

delta_R_vs_R

RGEM:320~360mm

mm 19