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Appendix A

Validation of NBG estimation method

using background MC

I check our NBG estimation method using the background MC events. Figures A.1 and A.2 are

the fit results (see section 5.2.2 for the fit functions) to the control samples in the MC. I use

those parameters to define the shape of functions except for the scale factors (c1 , c0 and clm

0 )

in the function. Figures A.3 and A.4 are the fit results to the MC observed sideband regions

with cut-base selection. Figures A.5 and A.6 are that of BDT selection. The blind regions are

removed from the fitting as well as the fitting to the data in the sideband. The expected and the

observed NBG in MC samples are shown in tables A.1 (cut-base) and A.2 (BDT). The number

of events in the validation region, where is inside the blind region but outside the signal region,

is also shown. They are scaled to 980 fb−1 . The NBG in the signal region is consistent with the

observed number of background events in the MC within 2σ. The NBG in the validation region

is also consistent with the observed one for most of the ℓV 0 modes, except for the eK ∗0 mode.

Table A.1: Comparison of expected and observed NBG of the MC samples (cut-base selection).

They are scaled to 980 fb−1 .

Mode

τ−

µ− ρ0

τ − → e− ρ 0

τ − → µ− ϕ

τ − → e− ϕ

τ − → µ− ω

τ − → e− ω

τ → µ− K ∗0

τ − → e− K ∗0

τ − → µ− K ∗0

τ − → e− K ∗0

signal region

MC

expected NBG

0.35±0.05(stat) ±0.02(syst)

0.31±0.07(stat) ±0.00(syst)

0.19±0.04(stat) ±0.00(syst)

0.06±0.03(stat) ±0.00(syst)

0.68±0.13(stat) ±0.03(syst)

1.00±0.16(stat) ±0.00(syst)

0.72±0.10(stat) ±0.01(syst)

0.66±0.12(stat) ±0.01(syst)

0.70±0.07(stat) ±0.01(syst)

0.83±0.13(stat) ±0.02(syst)

MC

Nobs

0.55±0.25

0.38±0.22

0 (<0.15)

0.08±0.08

0.45±0.23

0.68±0.35

0.47±0.33

0.24±0.24

0.51±0.30

0.54±0.28

83

validation region

MC

expected NBG

2.5±0.3(stat) ±0.2(syst)

2.1±0.5(stat) ±0.0(syst)

1.1±0.2(stat) ±0.0(syst)

0.3±0.2(stat) ±0.0(syst)

2.7±0.5(stat) ±0.1(syst)

2.2±0.4(stat) ±0.0(syst)

2.7±0.4(stat) ±0.0(syst)

4.1±0.7(stat) ±0.0(syst)

2.6±0.3(stat) ±0.0(syst)

3.3±0.5(stat) ±0.0(syst)

MC

Nobs

2.4±0.5

0.9±0.3

0.8±0.5

0.3±0.2

1.6±0.5

2.4±0.6

2.7±0.7

2.1±0.6

3.2±0.8

2.2±0.6

Table A.2: Comparison of expected and observed NBG of the MC samples (BDT selection).

They are scaled to 980 fb−1 .

Mode

τ − → µ− ρ0

τ − → e− ρ 0

τ − → µ− ϕ

τ − → e− ϕ

τ − → µ− ω

τ − → e− ω

τ → µ− K ∗0

τ − → e− K ∗0

τ − → µ− K ∗0

τ − → e− K ∗0

signal region

MC

expected NBG

0.31±0.05(stat) ±0.01(syst)

0.28±0.07(stat) ±0.01(syst)

0.31±0.06(stat) ±0.01(syst)

0.08±0.04(stat) ±0.00(syst)

0.29±0.08(stat) ±0.02(syst)

0.73±0.15(stat) ±0.00(syst)

0.61±0.09(stat) ±0.03(syst)

0.50±0.10(stat) ±0.01(syst)

0.52±0.06(stat) ±0.00(syst)

0.48±0.08(stat) ±0.01(syst)

MC

Nobs

0.54±0.24

0.11±0.11

0.11±0.11

0.08±0.08

0.55±0.28

0.34±0.24

0.91±0.47

0.71±0.38

0.59±0.34

0.39±0.23

84

validation region

MC

expected NBG

2.1±0.3(stat) ±0.1(syst)

1.8±0.4(stat) ±0.0(syst)

1.8±0.3(stat) ±0.0(syst)

0.4±0.2(stat) ±0.0(syst)

1.2±0.3(stat) ±0.0(syst)

1.7±0.3(stat) ±0.0(syst)

2.2±0.3(stat) ±0.1(syst)

3.2±0.6(stat) ±0.0(syst)

2.0±0.2(stat) ±0.0(syst)

1.8±0.3(stat) ±0.0(syst)

MC

Nobs

1.6±0.4

1.4±0.4

1.2±0.6

0.3±0.2

1.2±0.4

1.5±0.5

2.6±0.7

1.2±0.4

2.1±0.7

1.4±0.5

E (GeV)

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 3.98e+02 /81

p-value : 1.205e-43

1.70

1.75

1.80

M 0 (GeV/c2)

1.85

1.90

E (GeV)

(a) τ → µρ0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 5.33e+02 /86

p-value : 3.829e-66

1.70

1.75

1.80

M (GeV/c2)

1.85

1.90

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

E (GeV)

(b) τ → µϕ

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

2.00

1.75

1.50

1.25

1.00

0.75

0.50

0.25

chi2/ndf : 1.21e+02 /74

p-value : 3.789e-04

1.70

1.75

1.80

M (GeV/c2)

1.85

1.90

E (GeV)

(c) τ → µω

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 2.16e+02 /84

p-value : 8.885e-14

0.8

0.6

0.4

0.2

1.70

1.80

1.85

1.75

M K*0 (GeV/c2)

1.90

E (GeV)

(d) τ → µK ∗0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 1.52e+02 /82

p-value : 2.873e-06

1.2

1.0

0.8

0.6

0.4

0.2

1.70

1.80

1.85

1.75

M K*0 (GeV/c2)

1.90

(e) τ → µK ∗0

Figure A.1: The fitting results to the µV 0 control samples in the MC (left). The luminosity is

scaled to that of q q¯ MC (5720 fb−1 ). Projection to the MℓV 0 (middle) and ∆E (right). The

orange hatched regions in µϕ and µK ∗0 modes are blinded to avoid D± meson event peaks.

They are removed from the fitting and the plot of ∆E projection.

85

E (GeV)

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 3.80e+02 /73

p-value : 2.446e-43

1.70

1.75

1.80

Me 0 (GeV/c2)

1.85

1.90

E (GeV)

(a) τ → eρ0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 2.38e+02 /70

p-value : 1.714e-20

0.5

0.4

0.3

0.2

1.70

1.75

1.80

Me (GeV/c2)

1.85

1.90

0.1

E (GeV)

(b) τ → eϕ

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 2.34e+02 /74

p-value : 9.132e-19

1.70

1.75

1.80

Me (GeV/c2)

1.85

1.90

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

E (GeV)

(c) τ → eω

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 3.33e+02 /68

p-value : 5.201e-37

2.5

2.0

1.5

1.0

0.5

1.70

1.80

1.85

1.75

MeK*0 (GeV/c2)

1.90

E (GeV)

(d) τ → eK ∗0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 3.62e+02 /74

p-value : 6.974e-40

4.0

3.5

3.0

2.5

2.0

1.5

1.0

1.70

1.80

1.85

1.75

MeK*0 (GeV/c2)

1.90

0.5

(e) τ → eK ∗0

Figure A.2: The fitting results to the eV 0 control samples in the MC (left). The luminosity is

scaled to that of q q¯ MC (5720 fb−1 ). Projection to the MℓV 0 (middle) and ∆E (right). The

orange hatched regions in eϕ and eK ∗0 modes are blinded to avoid D± meson event peaks. They

are removed from the fitting and the plot of ∆E projection.

86

E (GeV)

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 4.70e+01 /31

p-value : 2.482e-02

0.12

0.10

0.08

0.06

0.04

0.02

1.70

1.75

1.80

(GeV/c2)

1.85

1.90

E (GeV)

(a) τ → µρ0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.0020

chi2/ndf : 3.96e+01 /34

p-value : 1.978e-01

0.0018

0.0016

0.0014

0.0012

0.0010

1.70

1.80

1.85

(GeV/c2)

1.75

1.90

E (GeV)

(b) τ → µϕ

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 2.00e+01 /22

p-value : 5.187e-01

0.020

0.015

0.010

0.005

1.70

1.80

1.85

(GeV/c2)

1.75

1.90

E (GeV)

(c) τ → µω

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.012

chi2/ndf : 3.76e+01 /33

p-value : 2.296e-01

0.010

0.008

0.006

0.004

1.70

1.75

1.80

M K*0 (GeV/c2)

1.85

1.90

0.002

E (GeV)

(d) τ → µK ∗0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.016

0.014

0.012

0.010

0.008

0.006

0.004

0.002

chi2/ndf : 3.72e+01 /31

p-value : 1.712e-01

1.70

1.75

1.80

M K*0 (GeV/c2)

1.85

1.90

(e) τ → µK ∗0

Figure A.3: The fitting results to the µV 0 sideband MC samples after the cut-base selection

(left). The luminosity is scaled to that of q q¯ MC (5720 fb−1 ). Projection to the MℓV 0 (middle)

and ∆E (right). The gray hatched regions are the blind regions. The orange hatched regions

in µϕ and µK ∗0 modes are blinded to avoid D± meson event peaks. The hatched regions are

removed from the fitting and the plot of ∆E projection. The blind regions are not included for

the fitting.

87

E (GeV)

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.010

chi2/ndf : 1.78e+01 /22

p-value : 6.593e-01

0.008

0.006

0.004

0.002

1.70

1.80

1.85

1.75

Me 0 (GeV/c2)

1.90

E (GeV)

(a) τ → eρ0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

×10

chi2/ndf : 2.24e+01 /20

p-value : 2.648e-01

1.70

1.75

1.80

Me (GeV/c2)

1.85

1.90

E (GeV)

(b) τ → eϕ

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.0040

chi2/ndf : 3.80e+01 /22

p-value : 1.306e-02

0.0035

0.0030

0.0025

0.0020

0.0015

1.70

1.80

1.85

1.75

Me (GeV/c2)

1.90

E (GeV)

(c) τ → eω

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.0050

chi2/ndf : 2.88e+01 /19

p-value : 5.034e-02

0.0045

0.0040

0.0035

0.0030

0.0025

0.0020

1.70

1.80

1.85

1.75

MeK*0 (GeV/c2)

1.90

E (GeV)

(d) τ → eK ∗0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.006

chi2/ndf : 2.99e+01 /22

p-value : 9.478e-02

0.005

0.004

0.003

0.002

0.001

1.70

1.75

1.80

MeK*0 (GeV/c2)

1.85

1.90

(e) τ → eK ∗0

Figure A.4: The fitting results to the eV 0 sideband MC samples after the cut-base selection

(left). The luminosity is scaled to that of q q¯ MC (5720 fb−1 ). Projection to the MℓV 0 (middle)

and ∆E (right). The gray hatched regions are the blind regions. The orange hatched regions

in eϕ and eK ∗0 modes are blinded to avoid D± meson event peaks. The hatched regions are

removed from the fitting and the plot of ∆E projection. The blind regions are not included for

the fitting.

88

E (GeV)

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 4.36e+01 /31

p-value : 5.235e-02

0.08

0.06

0.04

0.02

1.70

1.75

1.80

M 0 (GeV/c2)

1.85

1.90

E (GeV)

(a) τ → µρ0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 5.09e+01 /34

p-value : 2.399e-02

0.0035

0.0030

0.0025

0.0020

0.0015

1.70

1.75

1.80

M (GeV/c2)

1.85

1.90

0.0010

E (GeV)

(b) τ → µϕ

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 1.49e+01 /22

p-value : 8.285e-01

0.0175

0.0150

0.0125

0.0100

0.0075

0.0050

0.0025

1.70

1.75

1.80

M (GeV/c2)

1.85

1.90

E (GeV)

(c) τ → µω

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 6.16e+01 /33

p-value : 1.266e-03

0.008

0.006

0.004

0.002

1.70

1.80

1.85

1.75

M K*0 (GeV/c2)

1.90

E (GeV)

(d) τ → µK ∗0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.012

chi2/ndf : 3.59e+01 /31

p-value : 2.109e-01

0.010

0.008

0.006

0.004

1.70

1.75

1.80

M K*0 (GeV/c2)

1.85

1.90

0.002

(e) τ → µK ∗0

Figure A.5: The fitting results to the µV 0 sideband MC samples after the BDT selection (left).

The luminosity is scaled to that of q q¯ MC (5720 fb−1 ). Projection to the MℓV 0 (middle) and ∆E

(right). The gray hatched regions are the blind regions. The orange hatched regions in µϕ and

µK ∗0 modes are blinded to avoid D± meson event peaks. The hatched regions are removed from

the fitting and the plot of ∆E projection. The blind regions are not included for the fitting.

89

E (GeV)

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

0.010

chi2/ndf : 2.33e+01 /22

p-value : 3.290e-01

0.008

0.006

0.004

0.002

1.70

1.80

1.85

1.75

Me 0 (GeV/c2)

1.90

E (GeV)

(a) τ → eρ0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

×10

chi2/ndf : 1.81e+01 /20

p-value : 5.134e-01

1.70

1.80

1.85

1.75

Me (GeV/c2)

1.90

E (GeV)

(b) τ → eϕ

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 1.53e+01 /22

p-value : 8.090e-01

0.0024

0.0022

0.0020

0.0018

0.0016

0.0014

1.70

1.80

1.85

1.75

Me (GeV/c2)

1.90

E (GeV)

(c) τ → eω

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 2.77e+01 /19

p-value : 6.661e-02

0.0032

0.0030

0.0028

0.0026

0.0024

0.0022

0.0020

1.70

1.80

1.85

1.75

MeK*0 (GeV/c2)

1.90

E (GeV)

(d) τ → eK ∗0

0.100

0.075

0.050

0.025

0.000

0.025

0.050

0.075

0.1001.65

fit result

chi2/ndf : 1.94e+01 /22

p-value : 5.626e-01

0.0035

0.0030

0.0025

0.0020

0.0015

0.0010

0.0005

1.70

1.80

1.85

1.75

MeK*0 (GeV/c2)

1.90

(e) τ → eK ∗0

Figure A.6: The fitting results to the eV 0 sideband MC samples after the BDT selection (left).

The luminosity is scaled to that of q q¯ MC (5720 fb−1 ). Projection to the MℓV 0 (middle) and ∆E

(right). The gray hatched regions are the blind regions. The orange hatched regions in eϕ and

eK ∗0 modes are blinded to avoid D± meson event peaks. The hatched regions are removed from

the fitting and the plot of ∆E projection. The blind regions are not included for the fitting.

90

Appendix B

BDT training with MC control

samples

Here, I discuss an additional study about the Boosted Decision Tree (BDT) training in section 4.4.2. The BDT performance basically improves as the number of training samples increases. In section 4.4.2, the BDT is trained using the control samples in the data. To increase

the number of training samples, I add the control samples (section 4.4.1) in the background MC

to the BDT training.

The BDT training is done with 40% of the signal MC events as signal samples, and 80% of the

control samples in the data and background MC as background samples. The validation samples,

which have 10% of the signal MC events as signal samples and 20% of the control samples in the

data and background MC as background samples, are used for the AUC monitoring during the

training. If the AUC is not improved for 50 rounds of the training, I stop training and choose

the BDT with the best AUC score. The total wights of the control samples in the background

MC are scaled to match the number of the control samples in the data. The total weights of

the signal MC samples are scaled to match the sum of weights of the control samples in the

data and in the background MC. The input variables and the hyper-parameters for the BDT

are same as in section 4.4.2.

To compare the BDT performance with that in section 4.4.2, I calculate the AUC of the

validation samples except for the background MC (Figures B.1 and B.2). The BDT trained

with the data and background MC has better performance than the BDT in section 4.4.2. It

means that the BDT can be improved in the future Belle II experiment that has 50 times larger

integrated luminosity than the Belle experiment. Because this study is done after the unblind

of the τ → ℓV 0 signal region, I do not apply this BDT for my final results.

Figures B.3 and B.4 shows the observed event distribution after the BDT selection trained

with the control samples in the data and background MC. The ℓϕ modes are not shown because

I do not set the BDT output lower limits in the BDT selection for them. Table B.1 shows the

number of background estimation and the number of observed events inside the signal region

using this BDT at the same signal efficiency in table 6.2. The upper limits of the τ → ℓV 0

branching fractions are calculated with the same systematic uncertainty on the expected number

of signal events in table 6.2. The observed upper limits are more dependent on the number of

observed events than on the expected number of background events. Whereas the expected NBG

is decrease about 6.5% by the BDT with additional training samples, the sum of the observed

events are same as my final results.

91

(a) τ → µρ0

(b) τ → µϕ

(c) τ → µω

(d) τ → µK ∗0

(e) τ → µK ∗0

Figure B.1: The BDT performance indicator (AUC of the validation samples in the data) on

each training stage (number of BDT tree) for the µV 0 modes. The filled blue circles are the

performance of the new BDT that is trained with the control samples both in the data and

background MC. The filled red circles are the performance of the BDT that is trained with

the control samples only in the data. The additional training samples in the background MC

improve the BDT performance. The training is stopped if there is no AUC improvement for 50

training stages.

92

(a) τ → eρ0

(b) τ → eϕ

(c) τ → eω

(d) τ → eK ∗0

(e) τ → eK ∗0

Figure B.2: ...