Confirmation of Geant4 simulation

Figure 29: Comparison of the number of MFT track candidates before and after back-ground subtraction. Red dot represents the number of candidates before backback-ground subtraction and blue dot represents after subtraction.

corresponds to the approximate angle of 5.7^◦. When muon that angel is 5.7^◦ reaches the end of the absorber, the position of the x-axis is 475.9 mm. Here, the range around η = −3 is defined as 470.9 < x < 480.9 mm and 5.2^◦ < θ < 6.2^◦. Figure 30 and Table 8 summarize the assigned parameters in the simulation. It is confirmed that the incident position does not affect the results.

Figure 30: Schematic drawing of the confirmation simulation setup.

Table 8: Assigned parameters in the confirmation simulation.

Variable Assigned parameters Incident position of z-axis 0(mm): fixed

Incident position of x-axis -20<x<160(mm): 1mm step Incident angle 3.0^◦<θ<8.0^◦: 1^◦ step

The left panel of Figure 31 left shows the correlation between the position and angle at the entrance of the absorber for muons that are injected from the exit of absorber towards the vertex at η=−3(same situation as Section3.2.2). The right panel of Figure31right shows the correlation between the position and angle at the entrance of the absorber for muons that reaches the range around η =−3 at the exit of the absorber. The functions of the ellipses drawn in the same colors are the same in both figures. The percentage of entries in the ellipse is defined as efficiency. Efficiency lines (90,70,50%) defined in left figure give consistent results (88,68,47%) in right figure.

Figure31: (left) Correlation between the position and angle at the entrance of the absorber for muon that is injected from the exit of absorber towards the vertex at η = −3 (same situation as 3.2.2). (right) Correlation between the position and angle at the entrance of the absorber for muons that reached the range around pseudo-rapidityη=−3at the exit of the absorber.

3.4 Preparation for evaluation of matching algorithm

To select the best MFT track from the MFT track candidates, the likelihood of the MFT track candidates is evaluated by the following factors. That is the correlation between the track position and angle, and momentum and charged sign matching between MFT and MUON. The new matching method simulated in Section 3.2.2 needs to be implemented in the matching algorithm.

Steps to implement the new matching method are summarized below. First, correlation distribution can be obtained from the MUON track. The size of the distribution depends on the MUON track momentum. Second, the MFT track that is the nearest to the center of the distribution is the best MFT track. To calculate the distance from the center of the ellipse is harder than calculating the distance from the center of the circle. In order to make analysis easier, elliptical distribution given in Section 3.2.2 is converted to circular distribution.

Figure 32shows the process to convert the elliptical distribution to circular distribution that center is zero. Each process is shown below. The step numbers are corresponding to the numbers of the panel in Figure 32.

1. The original correlation distribution of the track position and angle atpT = 1GeV/c simulated by Geant4. This is the same figure as Figure 26.

2. The center of Figure 32-1 is converted to zero. Each axis is converted by equation (10) and (11).

x = (x −x )−tan

(θ_{M F T} +θ_{M U ON}⁾

/L (10)

y₂ =θ_{M U ON} −θ_{M F T}. (11) 3. The y-axis in Figure32-2is multiplied by L/3 to make the dimension the same as the x-axis and to make the distribution slope 45 degrees. L/3 is empirically found out.

4. Figure 32-3 is rotated 45 degrees to align the major axis of the ellipse with the x-axis.

5. Figure 32-4 is converted ellipse to a circle by dividing each x and y value by each standard deviation. Each standard deviation is obtained from the fit values of Figure 33.

Figure32: Process to convert the elliptical distribution to circular distribution that center is zero.

The left (right) panel of Figure 33 shows a projection of Figure 32-4 onto the x-axis (y-axis), respectively. Each histogram is fit by Gaussian function. The same processes are done for other p_T. Figure 34 shows standard deviations σ_x and σ_θx in Figure 32-4 as a function of p_T. The standard deviations vs. p_T is fit by function (12).

σ_x = 20.84/p_T + 1.349/p²_T

σ_θx = 8.603/p_T + 2.072/p²_T (13)

Figure33: Projection of Figure33-4onto the x-axis (left) and y-axis (right), respectively.

Each histogram is fit by Gaussian function.

Figure 34: Standard deviation σ_x and σ_θx in step 4 as a function of p_T. Each standard deviation is obtained from the fit values of Figure 33.

Figure 35 (left) shows the original distributions and Figure 35 (right) shows the

distri-is p_T = 1 GeV (p_T = 2 GeV). The efficiency lines are defined as the percentage of entries inside ellipse or circle. The size of the original distribution depends on p_T, while the size of the distribution after all conversions described above does not depend onpT. Figure36 shows the percentage of entries in the right panel of Figure35as a function of radiusr. r is defined as the distance from the center of the circular distribution after all conversion.

Different color dots represent different pT. All dots almost overlap. That is contour lines of efficiency can be written regardless of p_T. Therefore the MFT track that has smaller r is more likely to be the best MFT track regardless of p_T.

Figure 35: Original distributions (left) and distributions after conversion (right) with efficiency lines. The upper (lower) panel is p_T = 1 GeV (p_T = 2 GeV). Contour lines of efficiency can be written regardless of transverse momentum.

Figure36: Percentage of entries in the right panel of Figure 35as a function of radiusr.

Different color dots indicate different p_T.

4 Evaluation of developed matching algorithm

The developed matching algorithm is evaluated using HIJING [18].

4.1 Simulation tool for Heavy Ion Collisions

HIJING (Heavy Ion Jet INteraction Generator) is a Monte Carlo event generator particle production in high energy nuclear collisions. After HIJING generates particles emitted by high energy nuclear collisions, GEANT4 is used to simulate detector responses. In Monte Carlo simulation, we can know the true information of the particles in the events such as the particle ID, interaction points, momentum vector, and their mother particles.

Therefore we can verify how well the developed matching algorithm performs to evaluate track candidates.

4.2 Evaluation setup

In this evaluation, central Pb-Pb collision in centrality 0-5% at √

s_{N N} = 5.5 TeV is generated by HIJING and single muon that ϕ range is 0^◦ < ϕ < 360^◦ is embedded at η = −3 for each event. Tracks generated by HIJING are the background in the evaluation. Detector responses are simulated by GEANT4. 30 events are simulated for eachp_T (p_T = 1,2,3GeV). Each MFT hit at the last MFT disk is scored for the MUON hit of the embedded single muon by calculating matching quality parameters. The matching quality parameters in the x and y directions r_x and r_y are defined as:

r_x =

√

x^′2+θ_x^′2 (14)

r_y =

√

y^′2+θ^′_y² (15)

wherex^′, θ_x^′, y^′, θ_y^′ are the values after the conversion from elliptical to circular distribution in Section 3.4. They are calculated as:

x^′ =

[

(xM U ON −xM F T)−tan

(θM F T +θM U ON

2

)

/L

]

/σx (16)

θ^′_x = [(θ_{M U ON} −θ_{M F T})∗L/3]/σ_θx (17)

[ ( ) ]

The r_x (r_y) is the distance from the center of the 2 dimensional histogram (θ_x (θ_y) vs.

x^′ (y^′)) that is the circular distribution as shown in the right pannel of Figure 35. The track that have the smallestrx orry is chosen as the best MFT track in each direction. rx

and r_y are the one-dimensional matching quality parameters. Two-dimensional matching quality parameter R considering both r_x and r_y is defined as:

R =rx+ry

=^√x^′2+θ_x^′2+y^′2 +θ_y^′2. (20) The track that has the smallest R among the track candidates is chosen as the best MFT track. Since this is a simulation, we can check which track is a single muon in the MFT track candidates. Therefore, when we arrange R in ascending order, we can get the ranking of how small R of the embedded muon is among all entries. In this simula-tion, true positions and momentum vectors from simulated information not reconstructed information are used at the last MFT disk and the first MUON tracking chamber.

4.3 Result

Figure 37 shows the distribution of track position and angle in the x-direction for 10 events on the last MFT plane. The center of the distribution is (0,0) and most of the hit points are on the linear line. Distributions have a similar tendency regardless of pT

of the signal (embedded single muon). The distribution of the position and angle for y-direction has the same tendency as x-direction. Figure 38shows the distribution of the normalized position x^′ and angle θ_x^′. The distribution spreads as pT of embedded single muon increases because x^′ and θ_x^′ are normalized values by σ_x and σ_θx which depend on p_T of the signal (equation 16and 17). The distribution of the normalized position y^′ and angle θ^′_y has the same tendency. Figure 39 shows the distribution of rx and ry. The distribution spreads as p_T of the signal increases because r_x and r_y are normalized values by σ_x,σ_θx, σ_y and σ_θy which depend on p_T of the signal (equation 14 to 19).

Figure 37: Distribution of the position and angle for x direction at the last MFT plane.

p_T of embedded single muon is shown in each panel.

Figure 39: Distribution of rx and ry. pT for embedded single muon is shown in each panel.

Figure 40, 41, 42 shows r_x, r_y and R respectively. Red line represents embedded muon and blue line represents background. The red line (embedded single muon) has a peak in a small region around from 0 to 3 mm, while the blue line (background) spreads evenly to large region for all r_x, r_y and R as expected. r_x and r_y have similar results. R, which combines the two of r_x and r_y, has a lower background ratio than r_x and r_y alone. R can reduce the amount of the background by almost an order of magnitude compared to r_x and r_y at any p_T, especially at low R. The ratio of background to signal decreases with increasing p_T especially for R as shown in Figure 42. Figure 43 shows R only for background. Peak position is smaller in lower p_T than higer p_T. R distribution is sharp in lower p_T, while R distribution is smooth in higer p_T This is because R is normalized value by the standard deviation which depends onp_T shown in Figure34and background before normalization is similar in any p_T. Therefore this affects the ratio of background to signal.

Figure40: Distribution ofr_x. Red line represents embedded muon and blue line represents background. pT of embedded single muon is shown in each panel.

Figure42: Distribution ofR. Red line represents embedded muon and blue line represents background. p_T of embedded single muon is shown in each panel.

Figure 43: Distribution ofR only for background. p_T of embedded single muon is shown in each panel.

Figure 44 shows the ranking of how smallR of the embedded muon is among all entries.

The possibility that the true MFT track is chosen as the best MFT track significantly increases as pT increases because the size of the search window is smaller for larger pT. matching between MFT and MUON. Background can be halved by the sign of charged particle.

Figure 44: Ranking of how small the two-dimensional matching quality parameter R of the embedded muon was among all entries. p_T is shown in each panel.

5 Summary and Outlook

In this study, the track matching algorithm between MFT and MUON using the correla-tion between track posicorrela-tion and angle is developed. Track matching between MFT and MUON is important for precise tracking around the vertex and to improve signal-to-noise ratio S/N and mass resolution. The number of both MFT track candidates was calcu-lated when using the correlation and when not using. The matching method using the correlation between track position and angle can improveS/N significantly about6times with high efficiency.

The developed track matching method must be implemented in the matching algorithm.

The MFT track that is the nearest to the center of the distribution is the best. To cal-culate the distance from the center of the ellipse is harder than calculating the distance from the center of the circle. To make analysis easier, elliptical distribution is converted to circular distribution. After conversion from elliptical to circular distribution, contour lines of efficiency can be written regardless of transverse momentum. Therefore the dis-tance between the center and the track point in the circular distribution is the likelihood of the MFT track candidate regardless of pT.

The developed matching method is evaluated using HIJING. The matching quality pa-rameter R is developed. The possibility that the true MFT track is chosen as the best MFT track significantly increases with pT increasing. In evaluation, the ranking of how small matching quality parameter R of the embedded muon is among all entries is calcu-lated.

Here’s what we can do as a next step. When matching quality parameter R of the true MFT track is second, that matching quality parameterR may be close to the value of the first matching quality parameterR. A comparison of the difference betweenRfor the true MFT track and background can give other matching quality parameters. In this simula-tion, true positions and momentum vectors from simulated information not reconstructed information are used at the last MFT disk and the first MUON tracking chamber. Recon-structed standalone tracklets of MFT and MUON are considered the position resolution of detectors. Therefore R should be worse than this evaluation when using reconstructed information. We have to check how well R works when using reconstructed standalone tracklets. Also, a magnetic field is applied parallel to the beam axis. Charged particles are deflected by a magnetic field, especially for low momentum. MFT and MUON can measure the sign of charged particle, therefore the background can be halved. Rough momentum matching between MFT and MUON can reduce background. Also, we need more statistics to evaluate the developed matching method more precisely. The MFT is installed in2020 and LHC-Run3will start in2021. Therefore the new matching method must be implemented in the matching algorithm before LHC-Run 3.

Acknowledgement

I wish to express my sincere appreciation to Prof. Kenta Shigaki. He invited me to this research and always supervised me despite being in a different university. I would like to thank Asst. Prof. Maya Shimomura and Asst. Prof Takashi Hachiya for always support-ing me nearby. Without their support, I could not finish this work. I would like to thank Dr. Yorito Yamaguchi for his various advice from physics to presentation materials.

Prof. Hisaki Hayashii and Prof. Kenkichi Miyabayashi gave different perspective com-ments. That helped me to understand my work better. I would like to thank Kosei Yamakawa san and Takumi Osako kun who instructed me analysis framework of ALICE.

I also would like to thank my colleagues in Nara and Hiroshima. Thanks to them, I could enjoy this two years more. Finally, I must express my gratitude to my family, Tsutomu, Rumi, and Saki. They have always supported what I want to do.

この研究を支えてくださった全ての方々に感謝しています。ありがとうございました。