Study of the ⌫ µ interactions via charged current in the T2K near detector

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Institut de Física d'Altes Energ

Study of the ⌫ _µ interactions via charged current in the T2K near detector

Alfonso Andr´es Garc´ıa Soto

Tesis presentada para optar al grado de DOCTOR EN F´ISICA

Supervisada por:

Dr. Federico Sánchez Nieto (Director) Dr. Enrique Fernández Sánchez (Tutor)

y evaluada por:

Dr. Juan Antonio Caballero Carretero (Presidente) Dra. Llu¨ısa-Maria Mir (Secretaria)

Dr. Kenneth Long (Vocal)

Cerdanyola (Barcelona), 15 de Mayo de 2017

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Part IV

Appendices

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Appendix A

SMRD and ECAL Fiducial Volume

The momentum by range is computed using the length of a track and the energy loss (under certain particle hypothesis) within certain volume. Therefore, in order to reconstruct the momentum by range the track must be fully contained in ND280 volume.

In this analysis, BarrelECal and SMRD subdetectors were used to compute the momentum by range for tracks with no reliable TPC information (see Sec. 8.5). It was found that the performance of such variable, in terms of bias and resolution, is not acceptable when the track ends at the edges of these subdetectors. Therefore, it has been decided to reduce the fiducial volume in which tracks must finish.

In BarrelECal, the idea has been to remove the edge layer in XYZ. The size of the layers is 10⇥10⇥40 mm³. In SMRD, roughly quarter of a counter has been remove (60⇥60⇥125 mm³). Fig. A.1 shows an scheme of the fiducial volume used in both subdetectors.

BOTTOMRIGHT

LEFT RIGHT

TOPLEFT TOPRIGHT

BOTTOMLEFT

FGD1

y

BOTTOMLEFT BOTTOMRIGHT

BOTTOMLEFT

78RIGHT

78LEFT

x

BOTTOMLEFT BOTTOMRIGHT

FGD1

TOPLEFT / TOPRIGHT

BOTTOMLEFT / BOTTOMRIGHT 15LEFT

15RIGHT

FGD2

LEFT RIGHT

6LEFT 6RIGHT

15LEFT 78LEFT

78RIGHT y

z

Figure A.1: Schematic view of BarrelECal (gray), SMRD (green) and FGD (blue) subdetectors of ND280. Orange boxes indicate the fiducial volume of each subdetector.

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APPENDIX A. SMRD AND ECAL FIDUCIAL VOLUME

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Appendix B

Electric Field Distortions in ND280 TPCs

Electromagnetic fields are one of the key ingredients of the TPC track reconstruction. Most of the simulation steps are performed using Geant4, which does not take into account non uniformities for these fields. Therefore, detector simulation of ND280 assumes homogeneous field while in reality distorted fields are possible. Consequently, these distortions must be understood, parametrized and corrected on data.

On the one hand, the motion of charge particles will be deflected when they cross a magnetic field.

A perfect magnetic field configuration, B~ = (0.2 T,0,0), is currently used for the MC simulation.

Nevertheless, from calibration using Hall probes, it is known that the magnetic field has distortions specially in the TPC3 region (see Sec. 6.3.1). These distortions are corrected on data using the measured magnetic field information (see Sec. 9.3.1).

On the other hand, electric field in the TPCs is used to drift the electron cloud, produced by ionization of the gas, towards the MM (see Sec. 6.3.6). In the presence of an electromagnetic field, the motion of the electrons can be characterized by the Langevin equation:

~v= µ 1 +!²⌧²

"

E~ !⌧

|B~|(E~⇥B) +~ !²⌧²

B~² (E~ ·B)~ B~

#

(B.1)

Ideally, the electric field should have only an X component (parallel to the magnetic field) so that the drift velocity~v is parallel to X. However, small distortion of this field can be found (possibly due to inhomogeneities of the TPC geometry). As for the magnetic field, MC assumes a perfect electric field, so a correction on data should be applied in order to remove such distortions.

To quantify the electric field distortions in the TPCs, a hit-track residuals study is performed using information from the reconstruction. The idea is to select tracks crossing the TPC and to measure the residuals between the fitted and raw Y position of the hits in each pad column of the MicroMegas (MM). If the magnetic field was properly corrected previously, the distribution of the residuals should

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APPENDIX B. ELECTRIC FIELD DISTORTIONS IN ND280 TPCS

be centered at zero. Consequently, any deviation from zero can be caused by electric field distortions.

In the following sections, a detailed explanation of the hit-track residuals study is shown. Firstly, a control sample will be produced. Secondly, the hit-track residuals maps will be obtained for each TPC and readout plane (RP) independently. Thirdly, those maps will be fitted using a model that reproduce E field distortions. Finally, distortion will be corrected in data and the impact of this correction in the momentum resolution will be studied.

B.1 Hit-track Residuals

The first step is the design of a control sample. This selection skims events with tracks having very small X angles (i.e. parallel to the MM plane). An event is selected when at least one TPC segment is found with the following features:

• The transverse momentum higher than 3 GeV/c. This condition is included to use only tracks with small curvature.

• The number vertical pad columns activated is higher than 60 (two consecutive MM have 36⇥2=72 vertical modules).

• The distance along X between the beginning and the end of the segment is lower than 45 mm and the cathode is not crossed.

This selection criteria is run over data using spill and cosmic µ. It is important to notice that the magnetic field correction is already applied in those events. Fig B.1 shows some of the main features of the tracks contained in the control sample, in which each entry of the histograms does not represent an individual track but its associated clusters.

The hit-track residuals can be obtained for each reconstructed segment in the TPC using the following methodology:

• Fit the full segment and save the projected angle in the XZ plane, di↵usion and curvature parameters.

• Fit again the track fixing the saved parameters.

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B.1. HIT-TRACK RESIDUALS

Mom [GeV/c]

0 10 20 30 40 50 60

20 40 60 80 100 120 103

×

TPC0 RP0 TPC0 RP1 TPC1 RP0 TPC1 RP1 TPC2 RP0 TPC2 RP1

Z [mm]

-1000 -500 0 500 1000 1500 2000 2500 3000

X [mm]

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Z [mm]

-1000 -500 0 500 1000 1500 2000 2500 3000

Y [mm]

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

0 2000 4000 6000 8000 10000 12000

Mom [GeV/c]

0 10 20 30 40 50 60

20 40 60 80 100 120 103

×

Z [mm]

-1000 -500 0 500 1000 1500 2000 2500 3000

X [mm]

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Z [mm]

-1000 -500 0 500 1000 1500 2000 2500 3000

Y [mm]

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

0 2000 4000 6000 8000 10000 12000

Mom [GeV/c]

0 10 20 30 40 50 60

20 40 60 80 100 120 103

×

Z [mm]

-1000 -500 0 500 1000 1500 2000 2500 3000

X [mm]

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Z [mm]

-1000 -500 0 500 1000 1500 2000 2500 3000

Y [mm]

-1000 -800 -600 -400 -200 0 200 400 600 800 1000

0 2000 4000 6000 8000 10000 12000

Figure B.1: Left: Momentum distribution (top) and 2D distributions for the Z versus X position (bottom left) and Z versus Y position (bottom right) for each track that fulfills the previous requirements.

• In both fits, evaluate the value of y in each vertical pad column in the MM.

• Compute the residuals of the position for each pad columni of the MM usingy_BIASⁱ =yⁱ_global y_localⁱ .

These residuals (maximum of 72 per segment) are computed for all TPC segments from the control sample. Then, a map containing such information (so called, distortion map) can be computed grouping the segments depending on their X position within the TPCs. With current statistics, it was decided to group the segments in 10 di↵erent regions along X for each drift volume (3 TPC ⇥ 2 RP = 6 drift volumes) as shown in Fig. B.2. yⁱ_BIAS distribution is obtained for each column padiof the MM in each drift volume. Finally, the distortion maps are obtained using the mean and width ofy_BIASⁱ distributions.

Fig B.3 shows the mean and width of theyBIASdistribution for very forward going segments (✓Y Z ⇠0) crossing three di↵erent volumes in the middle region along X and in the positive or negative region along Y. Fig. B.4 shows the distortion maps in di↵erent regions of the TPC along the X axis and di↵erent angular conditions.

From these maps it can be concluded that the distortion shows a dependency on the X position and on the✓Y Z angle but not on the Y coordinate.

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Z [mm]

-1000 -500 0 500 1000 1500 2000 2500

X [mm]

-1000 -500 0 500 1000

XZ Projection

Cathode MM plane

(RP1)

MM plane (RP0)

Figure B.2: Schematic view of the three TPCs (black) and two FGDs (red). Dashed lines indicate the di↵erent regions along X used for the hit-track residuals maps.

Z plane for each X region

300 320 340 360 380 400 420

Delta Y [mm]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

300 320 340 360 380 400 420

Delta Y [mm]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

300 320 340 360 380 400 420

Y [mm]∆

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

300 320 340 360 380 400 420

Delta Y [mm]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

300 320 340 360 380 400 420

Delta Y [mm]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

300 320 340 360 380 400 420

Y [mm]∆

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

300 320 340 360 380 400 420

Delta Y [mm]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

300 320 340 360 380 400 420

Delta Y [mm]

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

300 320 340 360 380 400 420

Y [mm]∆

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700

[mm]BIASy

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

TPC1 RP0

= -30 θYZ

YZ = 0 θ

= 30 θYZ

TPC1 RP0

360 < X [mm] < 450 450 < X [mm] < 540

0 72 0 72

Figure B.3: Distortion map for TPC1-RP1 (left), TPC2-RP0 (middle) and TPC3-RP0 (right) in the middle region along the X axis and for very forward angles. Red (blue) line shows clusters with its Y position higher (lower) than zero.

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B.1. HIT-TRACK RESIDUALS

072

Z plane for each X region0100200300400500600700

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

1

TPC1 RP0 = -30YZθ = 0YZθ = 30YZθ

TPC1 RP0

0 < X [mm] < 90 90 < X [mm] < 180 180 < X [mm] < 270 270 < X [mm] < 360 360 < X [mm] < 450

450 < X [mm] < 540 540 < X [mm] < 630 630 < X [mm] < 720 720 < X [mm] < 810 810 < X [mm] < 900

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

1TPC2 RP0 = -30YZθ = 0YZθ = 30YZθ

TPC2 RP0 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

TPC3 RP0 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

1

TPC1 RP1 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

TPC2 RP1 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

TPC3 RP1 072

FigureB.4:DistortionmapforeachTPCandRPfordatafromthecontrolsamplesinwhichmagneticfielddistortionshasbeencorrected.It showstheprofiledy^{i bias}foreachpadcolumniineachXregion.DashedlinesindicateeachXregionwherethefirstblockistheXregioncloser tothecathodeandthelastblocktheoneclosertotheanode.

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B.2 E Field Model

The idea is to construct an electric potential (decoupled from the homogeneous electric field) that can reproduce the behavior observed in the distortion maps shown in Fig B.4.

To begin with, two approximations will be used.

• The potential does not depends on the Y coordinate (see Fig B.3).

• The drift volume of the TPC is a free-charge region of space (i.e. the charge density is zero).

Therefore, the potential should follow the 2D Laplace equation in the XZ plane:

r² (z, x) = 0 =) @²

@z² (z, x) + @²

@x² (z, x) = 0 (B.2)

Particularly important is the definition of the boundary conditions for this equation, which has been inferred using the distortion maps from Fig B.4.

• The residuals are centered at 0 when the TPC segments are close to the cathode or anode.

Besides, the shape of the residuals has a symmetric behavior towards the central region of the drift volume. This fact suggests that the charge accumulation (due to distortions) in the cathode, anode and middle region of the drift volume is negligible ( (z,0) = (z, L) = (0, L/2)⇠0 where Lis the length in X of the drift volume).

• The shape of the residuals in each region along the X axis is symmetric between the first and second half of the pad columns. Consequently, a similar charge accumulation can be assumed in both walls of the drift volume ( (+K/2, x) = ( K/2, x) =f(x) where K is the length in Z of the drift volume).

Thus, Eq. B.2 can be solved assuming previous boundary conditions using the separation of variables

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B.2. E FIELD MODEL

method. The potential can be expressed as follows:

(z, x) = X1 n=0

Ansin⇣

1/2

n x⌘ cosh⇣

1 bnz⌘ cosh⇣

1 bn

K 2

⌘ An= 2

L Z L

0

f(x) sin⇣

1/2 n x⌘

dx

n=

✓1 bn

◆2

=

✓2⇡n L

◆2

(B.3)

Furthermore, if the drift volume is split in two regions along X and the|f(x)|is the same for the two regions, then a relation betweenAn parameters can be obtained. If the sum is truncated in the third term, the following potential is obtained:

(z, x) =A1



0.1345 sin

✓2⇡

Lx

◆ cosh

✓2⇡

Lz

◆

+ 0.0046 sin

✓4⇡

L x

◆ cosh

✓4⇡

L z

◆

+0.0002 sin

✓6⇡

L x

◆ cosh

✓6⇡

Lz

◆ (B.4)

Obviously, several assumptions where made to obtain the potential from Eq. B.4 so this model will not reproduce the real electric distortion in the TPC. However, it can give a first order estimation of the distortion shape. Fig. B.5 shows the value of the distorted electric field for one of the drift volumes in the X and Z coordinates using this potential whenA1= 1 V (Ey= 0 in this model).

Z [mm]

-300 -200 -100 0 100 200 300

X [mm]

0 100 200 300 400 500 600 700 800 900

[V/mm]z E

-0.01 -0.005 0 0.005 0.01

Z [mm]

-300 -200 -100 0 100 200 300

X [mm]

0 100 200 300 400 500 600 700 800 900

[V/mm]x E

-0.01 -0.005 0 0.005 0.01

Cathode Pad plane

Figure B.5: Distorted electric field for one drift volume in the X (left) and Z (coordinates) coordinates using Eq. B.4 whenA1= 1 V.

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B.3 Fitter

Once the electric potential is parametrized, the next step is to obtain the strength of those parameters comparing the model with the data. The model defined in the previous section had only one free parameter, so some extra parameters have been included in order to give more freedom to the model:

(z, x) =A

 sin

✓2⇡

Lx

◆ cosh

✓2⇡

L z

◆

+c·sin

✓4⇡

Lx

◆ cosh

✓4⇡

Lz

◆

+d·sin

✓6⇡

Lx

◆ cosh

✓6⇡

Lz

◆

(B.5) Where the free parameters are marked in bold. Many distortion maps can be generated from MC varying the parameters of the electric potential.

Firstly, 72 electron clouds are simulated placing them at the pad plane and in the middle of each pad column. Then, those clouds are drifted back to the center of the 10 regions along X using Eq. B.1 (assuming homogeneous B field). Finally, their new Y positions are fitted with a straight line to reproduce the tracking (this is the main motivation for the 3 GeV/c cut in momentum) and the residuals are computed for that variable.

All the simulated distortion maps can be compared with the observed ones. Then, a fit is performed to extract the parameters of the electric potential that minimized the following ² for each TPC and RP:

2=

#pads

X

i=0

(y_{bias,DAT A}ⁱ ybias,M ODELⁱ (A, c, d, e))²

2 (yⁱ_{bias,DAT A})² (B.6)

This minimization includes an extra degree of freedom, so calledM. This parameter is able to account for the MM misalignment in an e↵ective way. It shift up or down the prediction for the first 36 pad columns and the opposite for the other half. It is important to notice that this parameter is not included as part of the electric potential (which is independent from the misalignment).

This minimization is very complicated because it uses several degrees of freedom that can be highly correlated. Therefore, it is not desirable to use a very complex model for the potential. It has been tried to include free parameters in the phases, to add odd terms in the sum or to increase the dimensions of the potential from Eq. B.4. Adding this extra level of complexity does reduce the ²but the correlation between the parameters and the computational time increases.

The correlation problem is partially overcome a defining range of validity for the free parameters (using the order of magnitude obtained Eq. B.4). Besides, some of the parameters can be useless and this

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B.4. VALIDATION: Y

would slow down the minimization process. Fig B.6 compares the distortion maps for the data and the best fit potential. The best fit parameters are shown in Table B.1.

A [V] c (⇥10 ⁴) d (⇥10 ⁵) M (⇥10 ³) [mm]

TPC1-RP0 0.73±0.01 22±8 177±7 60±3

TPC1-RP1 0.75±0.01 6±8 113±7 103±3

TPC2-RP0 0.72±0.01 47±8 150±6 44±3 TPC2-RP1 0.75±0.01 93±8 126±6 23±3 TPC3-RP0 0.66±0.01 20±10 113±8 28±3 TPC3-RP1 0.54±0.01 40±10 95±8 107±3

Table B.1: Value of the parameters from Eq. B.5 and the misalignment after the minimization process.

Currently, misalignment and some of the model parameters are highly correlated. This means that some of the values might not be reliable. The best solution would be to improve the current MM misalignment correction on data. This could reduce drastically the impact of the parameter associated to the misalignment in the fit.

B.4 Validation: y

Data and the prediction from electric potential model show good agreement for the hit-track residuals study. Nevertheless, it is useful to check the performance of the model using other observables. The so called y quantity is very useful to validate the model. The idea is to use tracks crossing at least two TPCs and propagate their TPC segments to the same Z position. Distortion should bias the Y position of the two propagated segments (see Fig. B.7).

The same control samples from the hit-track residuals study are used: long TPC segments with small deviation along X. Fig B.8 shows y profiled for each region of the track X position. Similarly to the hit-track residuals study, it seems that the distortions has a clear dependency on the X position and the projected angle in the YZ plane. Besides, it seems that the order of magnitude of this distortion is very similar for the three TPCs.

The result from the model was included using particle guns that mimic the features of the control sample. Then, the electron clouds were drifted using the same technique as in B.3, but drifting towards the anode. The tendency in data and the prediction is similar, but the asymmetric behavior observed in data is not predicted by the simulation. This might be due to misalignment between the detectors or additional odd terms in the electric potential (very difficult to measure with the distortion map fit).

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Z plane for each X region 0100200300400500600700

[mm]

yBIAS

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 TPC1 RP0

= +30 DATAYZθ = 0 DATAYZθ = +30 MODELYZθ

= 0 MODELYZθ TPC1 RP0

072

0 < X [mm] < 90

90 < X [mm] < 180

180 < X [mm] < 270

270 < X [mm] < 360

360 < X [mm] < 450

450 < X [mm] < 540

540 < X [mm] < 630

630 < X [mm] < 720

720 < X [mm] < 810

810 < X [mm] < 900

[mm]

yBIAS

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 TPC2 RP0

072

0 < X [mm] < 90

90 < X [mm] < 180

180 < X [mm] < 270

270 < X [mm] < 360

360 < X [mm] < 450

450 < X [mm] < 540

540 < X [mm] < 630

630 < X [mm] < 720

720 < X [mm] < 810

810 < X [mm] < 900

[mm]

yBIAS

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 TPC3 RP0

= +30 DATAYZθ = 0 DATAYZθ = +30 MODELYZθ = 0 MODELYZθ TPC3 RP0

0

0 < X [mm] < 90

90 < X [mm] < 180

180 < X [mm] < 270

270 < X [mm] < 360

360 < X [mm] < 450

450 < X [mm] < 540

540 < X [mm] < 630

630 < X [mm] < 720

720 < X [mm] < 810

810 < X [mm] < 900 Z plane for each X region 0100200300400500600700

[mm]

yBIAS

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 TPC1 RP1

072

0 < X [mm] < 90

90 < X [mm] < 180

180 < X [mm] < 270

270 < X [mm] < 360

360 < X [mm] < 450

450 < X [mm] < 540

540 < X [mm] < 630

630 < X [mm] < 720

720 < X [mm] < 810

810 < X [mm] < 900

[mm]

yBIAS

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 TPC2 RP1

072

0 < X [mm] < 90

90 < X [mm] < 180

180 < X [mm] < 270

270 < X [mm] < 360

360 < X [mm] < 450

450 < X [mm] < 540

540 < X [mm] < 630

630 < X [mm] < 720

720 < X [mm] < 810

810 < X [mm] < 900

[mm]

yBIAS

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 TPC3 RP1

0

0 < X [mm] < 90

90 < X [mm] < 180

180 < X [mm] < 270

270 < X [mm] < 360

360 < X [mm] < 450

450 < X [mm] < 540

540 < X [mm] < 630

630 < X [mm] < 720

720 < X [mm] < 810

810 < X [mm] < 900

FigureB.6:DistortionmapforeachTPCandRPfordatafromthecontrolsamplesinwhichmagneticfielddistortionshasbeencorrected(redpoints)andthemodel(bluelines).Itshowstheprofiledy ibiasforeachpadcolumniineachXregion.DashedlinesindicateeachXregionwherethefirstblockistheXregionclosertothecathodeandthelastblocktheoneclosertotheanode.

190

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B.4. VALIDATION: Y

-500 0 500 1000 1500

-1000 -500 0 500 1000

Run 90410000, SRun 0, Evt 0, Trig 0x0

True trajectory Fit Hits in fitted object Hits in non-fitted object Unused hits

True primary vertex Rec vertex

YZ Projection

Z [mm]

Y [mm]

FGD1

TPC1 TPC2

reco segment TPC1

reco segment TPC2 true particle

∆y

distorted hits

Figure B.7: Schematic view of the y using segments from TPC1 and TPC2. The e↵ect of the distortion is magnified.

[mm]

XTPC1

-800 -600 -400 -200 0 200 400 600 800

[mm]TPC1-YTPC2Y

-10 -8 -6 -4 -2 0 2 4

[mm]

XTPC2

-800 -600 -400 -200 0 200 400 600 800

[mm]TPC2-YTPC3Y

-10 -8 -6 -4 -2 0 2 4

[mm]

XTPC1

-800 -600 -400 -200 0 200 400 600 800

[mm]TPC1-YTPC2Y

-10 -8 -6 -4 -2 0 2 4

[mm]

XTPC2

-800 -600 -400 -200 0 200 400 600 800

[mm]TPC2-YTPC3Y

-10 -8 -6 -4 -2 0 2 4

Figure B.8: Profile distribution between Y and X position for di↵erent angles: 30±2 (red); 0±2 (green) and 30±2 (blue). Crosses indicate the result on data from the control samples in which magnetic field distortions has been corrected. Dots represent the prediction from the simulation.

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B.5 Data Correction

It is possible to correct the electric field distortions on the data using the model for the electric potential and the best fit parameters. The method corrects the position of the electron clouds measured in the pad planes by taking into account the distorted field when Eq. B.1 is used. Then, all the reconstruction is reprocessed. Finally, the hit-track residuals and ystudies can be repeated to check the performance of the correction.

On the one hand, Fig. B.10 shows the distortion maps in di↵erent regions of the TPC along the X axis and di↵erent angular conditions after the electric field distortions have been corrected. The shapes for the di↵erent X volumes and angles becomes flat along the pad plane. On the other hand, Fig B.9 shows y profiled for each region of the track X position after the correction is performed. In this case, the bias seems to be centered around zero for any value of X.

In general, the correction seems to reduce the impact of the electric field distortion drastically. Nonethe- less, it is important to remind that this methodology can be used to extract the even terms of the electric potential, while odd terms remain unknown.

[mm]

XTPC1

-800 -600 -400 -200 0 200 400 600 800

[mm]TPC2-TPC1 Y∆

-10 -8 -6 -4 -2 0 2 4

[mm]

XTPC2

-800 -600 -400 -200 0 200 400 600 800

[mm]TPC3-TPC2 Y∆

-10 -8 -6 -4 -2 0 2 4

[mm]

XTPC1

-800 -600 -400 -200 0 200 400 600 800

[mm]TPC2-TPC1 Y∆

-10 -8 -6 -4 -2 0 2 4

[mm]

XTPC2

-800 -600 -400 -200 0 200 400 600 800

[mm]TPC3-TPC2 Y∆

-10 -8 -6 -4 -2 0 2 4

Figure B.9: Profile distribution between Y and X position for di↵erent angles: 30±2 (red); 0±2 (green) and 30±2 (blue). Crosses indicate the result on data from the control samples in which both magnetic and electric field distortions has been corrected.

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B.5. DATA CORRECTION

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

TPC1 RP0 Z plane for each X region0100200300400500600700

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

TPC3 RP1 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

TPC3 RP1 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

1

TPC3 RP1 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

TPC3 RP1 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

1

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

TPC3 RP1 072

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

1

[mm] ^BIAS y

-1

-0.8

-0.6

-0.4

-0.2

00.2

0.4

0.6

0.8

TPC3 RP1 072

FigureB.10:DistortionmapforeachTPCandRPfordatafromthecontrolsamplesinwhichbothmagneticandelectricfielddistortionshas beencorrected.Itshowstheprofiledy^{i bias}foreachpadcolumniineachXregion.DashedlinesindicateeachXregionwherethefirstblockis theXregionclosertothecathodeandthelastblocktheoneclosertotheanode.