Signal Detection

Component Proportion (%) B→ Dτν 1.56 B → D^∗τν 3.18

B → Dlν 19.80

B→ D^∗lν 18.11

B→ D^∗∗lν 22.74

CombinatorialBB¯ 28.17

Continuum 6.44

Table 7.8: Proportions of each component after reconstruction, evaluated using the generic MC.

• R₂All: Second Fox-Wolfram moment.

• M_miss² : Square of the missing 4-vector of the event in the CM frame.

• E_extra: Extra neutral energy in the calorimeter.

• cosθ_T: Cosine of the angle between the thrust and the beam momentum.

• |p^tag_l |: 3-momentum magnitude of theB_taglepton in the CM frame.

• cosθ^tag

B−D^(∗)l: Cosine of the angle between the 3-momentum of the B_tag and the 3-momentum sum of its D and lepton daughters in the CM frame.

• cosθ^tag_D−l: Cosine of the angle between the 3-momentum of the Dmeson and the lepton daughter in the tag side.

• m^tag_D : Mass of the B_tag Dmeson daughter.

• ∆m^tag: Mass difference betweenD^∗ andDmeson in the tag side, if exists.

• cosθ^tag_{D so f t}: Cosine of the angle between theD^∗mesons daughters in the tag side in the CM frame.

• |p^tag_{so f t}|: 3-momentum magnitude of the D^∗’s soft daughter in the tag side in the CM frame.

• |p^sig_l |: 3-momentum magnitude of theB_siglepton daughter in the CM frame.

• cosθ^sig_D−l: Cosine of the angle between the 3-momentum of the Dmeson and the lepton daughter in the sig side in the CM frame.

• χ²: χ²of theB_sigvertex fit.

• m_D^sig: Mass of theB_sig Dmeson daughter.

• ∆m^sig: Mass difference betweenD^∗andDmeson in the sig side, if it exists.

• cosθ^sig_{D so f t}: Cosine of the angle between the D^∗ mesons’ daughters in the sig side in the CM frame.

• |p^sig_{so f t}|: 3-momentum magnitude of theD^∗’s soft daughter in the sig side in the CM frame.

• cosθ_D^(∗)_l−D^(∗)_l: Cosine of the angle between the twoDl Dlsystems in the CM frame.

• tagl electron PID:B_taglepton daughter’s electron PID level.

• tagl muon PID:B_tag lepton daughter’s muon PID level.

• sigl electron PID:B_siglepton daughter’s electron PID level.

• sigl muon PID:B_siglepton daughter’s muon PID level.

Figure 7.5: Histograms of variables used for the C₁classifier.

Figure 7.6: Histograms of variables used for the C₁classifier.

Figure 7.7: Histograms of variables used for the C₁classifier.

We use a Gradient Boosting Decision Tree (BDT) [36] for the C₁ classifier. The number of decision trees ranges from 20 to 600. The model is implemented using the scikit-learnpackage. The metric used to evaluate the classification performance is the area under the ROC curve. The higher the score, the higher the classification power. Fig. 7.8 shows the relationship between the area under ROC curve and the number of trees. The classification performance becomes stable when the number

of decision trees is above 500. We use a BDT with 600 trees as the final model. Fig.

7.9 shows the importance of variables for classification. E_extra and |p^sig_l | are the most powerful variables to identifyB→ D^(∗)(τ/l)ν from all types of backgrounds.

ForE_extra, the signal and normalization events usually have near-zero extra neutral energy, while background events have a wide distribution. The normalization decay has an energetic lepton produced by the D decay, leading to a higher |p_l^sig| value than signal events, as well as all types of backgrounds. The output of the BDT score pis transformed using a logit function:

z₁= logit(p)= log p 1− p.

The z₁ distribution for all types of events is shown in Fig. 7.10. Signal and normalization events tend to have higher z₁values than backgrounds.

Figure 7.8: Area under the ROC curve for BDT classifiers with different numbers of trees.

Figure 7.9: Importance of each variable for learning theC₁classifier.

Figure 7.10: z₁distribution for signal, normalization,D^∗∗lν,BB¯combinatorial, and continuum events.

C₂classifier

The C₂ Classifier aims to classify signal events and normalization events. Similar to theC₁classifier, we divide the sample into training and validation samples. The training sample is first used to train the classifier, the validation sample is then applied to evaluate the performance of the classifiers. Signal events are labeled positive and normalization events are labeled negative before training. We use the same variables used for theC₁classifier, with the addition of the following quantity:

• cosθ^sig

B−D^(∗)l: Cosine of the angle between the 3-momentum of theB_sigand the 3-momentum sum of its D and lepton daughters.

The histogram of these variables for signal and normalization events are shown in Fig. 7.11.

Figure 7.11: Histograms of variables used for theC₂classifier.

Similar to the C₁ classifier, we use a BDT for the C₂ classifier. The number of decision trees ranges from 100 to 600. The classification performance is stable when the number of decision trees is above 100, as shown in Fig. 7.12. We use a BDT with 600 trees as the final model. Fig. 7.13 shows the importance of the variables for classification: cosθ^sig_B−D(∗)land|p^sig_l |are the most powerful variables to distinguish signal from normalization events. For cosθ^sig_B−D(∗)l, normalization events have only a single neutrino, and the value should be from -1 to 1. However, signal

events tend to have more negative values, due to the presence of three neutrinos in the final state. Thee/µproduced from the τdecays in signal events have a softer

|p^sig_l |spectrum than the leptons produced fromDdecays for normalization events.

The output of the BDT score is transformed to z₂ using a logit function. The z₂ distribution for all types of events is shown in Fig. 7.14. Signal events tend to have a higher z₂ score, while normalization events tend to have a lower z₂ score.

Backgrounds have z₂scores in between.

Figure 7.12: The area under the ROC curve for BDT classifiers with different number of trees.

Figure 7.13: Importance of each variable for learning theC₂classifier.

Figure 7.14: z₂distribution for signal, normalization,D^∗∗lν,BB¯combinatorial, and continuum events.