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]

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.1567 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.1780 after efficiency correction in FIG.1 (e). The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.1780 after efficiency correction in FIG.1 (f). The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.1888 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.1888 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.1989 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.1989 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2091 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2091 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2187 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2187 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2263 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2263 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2357 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2357 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2438 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2438 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2580 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2580 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2667 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2667 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2776 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2776 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.2866 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.2866 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.3115 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.3115 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.3368 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.3368 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{\pm}J/\psi)$ [GeV/$c^{2}$] at energy 4.3583 after efficiency correction in FIG.F6. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

Data for $M(\pi^{+}\pi^{-})$ [GeV/$c^{2}$] at energy 4.3583 after efficiency correction in FIG.F7. The sideband events are used as an estimate of the background events and subtracted from events in the signal region

$M_{\tau}$ 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.

The systematic covariance matrix for the $e^+e^- \to \pi^+ \pi^- \pi^0$ cross section measurement at the Belle II. The 212 x 212 matrix of the energy ranges from 0.62 to 3.50 GeV. This covariance matrix includes all systematic uncertainty given in Table I. The total covariance matrix for the measured cross section can be obtained by adding statistical and systematic covariance matrices.

This table provides the ``inverse'' of the total (stat.+ sys.) covariance matrix. This matrix can be used for a chi-square fit of the mass spectrum. We provide this inverse matrix since special care is needed to take the inverse of the given covariance matrix.

Observed yields and fit results in bins of $\eta(\rm{BDTh})$ as obtained by the HTA fit, corresponding to an integrated luminosity of 362 fb$^{-1}$. The yields are shown for the $B^+ \rightarrow K^{+}\nu\bar\nu$ signal and the three background categories ($B\bar{B}$ decays, $c\bar{c}$ continuum, and light-quark continuum).

Distributions of $\eta(\rm{BDT}_{2})$ in data (points with error bars) and simulation (filled histograms) shown individually for the $B^+ \to K^+ \nu \bar \nu$ signal, neutral and charged $B$-meson decays, and the sum of the five continuum categories in the ITA. Events in the full signal region, with $\eta(\rm{BDT_{2}}) > 0.92$, are shown.

Distributions of $q^2_{\textrm{rec}}$ in data (points with error bars) and simulation (filled histograms) shown individually for the $B^+ \to K^+ \nu \bar \nu$ signal, neutral and charged $B$-meson decays, and the sum of the five continuum categories in the ITA. Events in the full signal region, with $\eta(\rm{BDT_{2}}) > 0.92$, are shown.

Distributions of $\eta(\rm{BDT}_{2})$ in data (points with error bars) and simulation (filled histograms) shown individually for the $B^+ \to K^+ \nu \bar \nu$ signal, neutral and charged $B$-meson decays, and the sum of the five continuum categories in the ITA. Events in the most signal-rich region, with $\eta(\rm{BDT_{2}}) > 0.98$, are shown.

Distributions of $q^2_{\textrm{rec}}$ in data (points with error bars) and simulation (filled histograms) shown individually for the $B^+ \to K^+ \nu \bar \nu$ signal, neutral and charged $B$-meson decays, and the sum of the five continuum categories in the ITA. Events in the most signal-rich region, with $\eta(\rm{BDT_{2}}) > 0.98$, are shown.

Distribution of $q^2$, computed using $B_{tag}$ kinematics for events in the signal region of the HTA. These distributions are obtained for $B^+ \to K^+ \nu \bar \nu$ candidates reconstructed in data (points with error bars), simulation (filled histograms) of $B^+ \rightarrow K^{+}\nu\bar\nu$ signal, $B\bar{B}$ decays, $c\bar{c}$ continuum, and light-quark continuum backgrounds, normalized according to the fit yields in the HTA.

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

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

Twice the negative profile log-likelihood ratio as a function of the signal strength $\mu$ for the HTA result. The value for each scan point is determined by fitting the data, where all parameters but $\mu$ are varied.

Twice the negative profile log-likelihood ratio as a function of the signal strength $\mu$ for the ITA result. The value for each scan point is determined by fitting the data, where all parameters but $\mu$ are varied.

Twice the negative profile log-likelihood ratio as a function of the signal strength $\mu$ for the combined result. The value for each scan point is determined by fitting the data, where all parameters but $\mu$ are varied.

Branching-fraction values measured by Belle II, measured by previous experiments, and predicted by the SM.

Data of the decay $\bar{B}^+ \to D^* \mu \nu_\mu$.

The full experimental (statistical and systematical) correlations of the angular coefficients are shown. The elements are ordered as: Entry 0 - 47: B0 -> D* e nu_e: DJ1s/Dw [w_bin0, w_bin1, w_bin2, w_bin3], DJ1c/Dw [w_bin0, w_bin1, w_bin2, w_bin3], ... Entry 48 - 95: B0 -> D* mu nu_mu: DJ1s/Dw [w_bin0, w_bin1, w_bin2, w_bin3], DJ1c/Dw [w_bin0, w_bin1, w_bin2, w_bin3], ... Entry 96 - 143: B+ -> D* e nu_e: DJ1s/Dw [w_bin0, w_bin1, w_bin2, w_bin3], DJ1c/Dw [w_bin0, w_bin1, w_bin2, w_bin3], ... Entry 144 - 191: B+ -> D* mu nu_mu: DJ1s/Dw [w_bin0, w_bin1, w_bin2, w_bin3], DJ1c/Dw [w_bin0, w_bin1, w_bin2, w_bin3], ...

Data of the B+/B0 average $B \to D^* e \nu_e$.

Data of the B+/B0 average $B \to D^* \mu \nu_\mu$.

The full experimental (statistical and systematical) correlations of the angular coefficients are shown. The elements are ordered as: Entry 0 - 47: B -> D* e nu_e: DJ1s/Dw [w_bin0, w_bin1, w_bin2, w_bin3], DJ1c/Dw [w_bin0, w_bin1, w_bin2, w_bin3], ... Entry 48 - 95: B -> D* mu nu_mu: DJ1s/Dw [w_bin0, w_bin1, w_bin2, w_bin3], DJ1c/Dw [w_bin0, w_bin1, w_bin2, w_bin3], .....

Data of the B+/B0 and e/mu average $B \to D^* \ell \nu_\ell$.

The full experimental (statistical and systematical) correlations of the angular coefficients are shown. The elements are ordered as: Entry 0 - 47: B -> D* l nu_l: DJ1s/Dw [w_bin0, w_bin1, w_bin2, w_bin3], DJ1c/Dw [w_bin0, w_bin1, w_bin2, w_bin3], ...