Correlations (stat.+syst.) between the $d\Gamma_i/d w$ bins for the individual $B \rightarrow D \ell \nu$ modes (40x40). Element indices 0-39 are organized as: - Indices 0-9: $B^+ \to D^0 e^+ \nu_e$ in $w$ bins 0-9 - Indices 10-19: $B^+ \to D^0 \mu^+ \nu_\mu$ in $w$ bins 0-9 - Indices 20-29: $B^0 \to D^- e^+ \nu_e$ in $w$ bins 0-9 - Indices 30-39: $B^0 \to D^- \mu^+ \nu_\mu$ in $w$ bins 0-9 where $w$ bins 0-9 correspond to: 0: [1.00, 1.06], 1: [1.06, 1.12], 2: [1.12, 1.18], 3: [1.18, 1.24], 4: [1.24, 1.30], 5: [1.30, 1.36], 6: [1.36, 1.42], 7: [1.42, 1.48], 8: [1.48, 1.54], 9: [1.54, 1.59]

$M_{\tau}$ distribution in signal region, (OS$\mu$, Belle II)

$M_{\tau}$ distribution in signal region, (SS$e$, Belle)

$M_{\tau}$ distribution in signal region, (SS$e$, Belle II)

$M_{\tau}$ distribution in signal region, (SS$\mu$, Belle)

$M_{\tau}$ distribution in signal region, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$ and $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$e$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$e$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$e$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$e$, Belle II)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle)

Efficiency as function the helicity angle of the kaon $\cos\theta_{K}$, and the helicity angle of the lepton $\cos\theta_{\ell}$ in bin e the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$e$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$e$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$\mu$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (OS$\mu$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$e$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$e$, Belle II)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$\mu$, Belle)

Efficiency as function of the helicity angle of the kaon $\cos\theta_{K}$, (SS$\mu$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$e$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$e$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$\mu$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (OS$\mu$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$e$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$e$, Belle II)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$\mu$, Belle)

Efficiency as function of the helicity angle of the lepton $\cos\theta_{\ell}$, (SS$\mu$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (OS$e$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (OS$e$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (OS$\mu$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (OS$\mu$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (SS$e$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (SS$e$, Belle II)

Efficiency as function of the acoplanarity angle $\phi$, (SS$\mu$, Belle)

Efficiency as function of the acoplanarity angle $\phi$, (SS$\mu$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$e$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$e$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (OS$\mu$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$e$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$e$, Belle II)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle)

Efficiency as function of the four-momentum-transfer $q^{2}$, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}\ell)$, (SS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (OS$\mu$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$e$, Belle II)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle)

Efficiency as function of the invariant mass squared $m^2(K^{*0}t_{\tau})$, where $t_{\tau}$ is the track from the $\tau$, (SS$\mu$, Belle II)

Signal-selection efficiency as a function of the di-tau invariant mass squared $q^2$ for simulated events in the SR. The error bars indicate the statistical uncertainty.