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.

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.

Full experimental (statistical and systematic) correlations (in \%) of the normalized partial decay rates over the $\overline{B}^0\to D^{*+} e^- \bar\nu_e$ and $\overline{B}^0\to D^{*+} \mu^- \bar\nu_\mu$ decays.

Measured partial decay rates $\Delta\Gamma$ (in units of $10^{-15}$ GeV) using the matrix inversion unfolding method

Average of normalized decay rates that are determined using the matrix inversion unfolding method

Full experimental (statistical and systematic) correlations (in \%) of the partial decay rates determined with the matrix inversion unfolding method for the $\overline{B}^0\to D^{*+} e^- \bar\nu_e$ and $\overline{B}^0\to D^{*+} \mu^- \bar\nu_\mu$ decays.

Full experimental (statistical and systematic) correlations (in \%) of the normalized partial decay rates determined using the matrix inversion unfolding method

Expected and observed candidates for $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) as a function of the reduced mediator candidate mass.

Expected and observed candidates for $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) as a function of the reduced mediator candidate mass.

Expected and observed candidates for $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) as a function of the reduced mediator candidate mass.

Expected and observed candidates for $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) as a function of the reduced mediator candidate mass.

Expected and observed candidates for $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) as a function of the reduced mediator candidate mass.

Exclusion regions in the plane of the sine of the mixing angle $\theta$ and scalar mass $m_S$

Exclusion regions in the plane of the coupling $g_Y = 2v/fa$ with the vacuum expectation value $v$ and the ALP mass $m_a$

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.001cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.003cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.005cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.007cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.01cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.025cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.05cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.1cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.25cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.5cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$1.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$2.5cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$5.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$10.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$25.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$50.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$100.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$200.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$400.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.001cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.003cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.005cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.007cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.01cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.025cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.05cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.1cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.25cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.5cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$1.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$2.5cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$5.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$10.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$25.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$50.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$100.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$200.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$400.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.001cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.003cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.005cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.007cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.01cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.025cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.05cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.1cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.25cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.5cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$1.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$2.5cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$5.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$10.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$25.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$50.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$100.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$200.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$400.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.001cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.003cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.005cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.007cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.01cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.025cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.05cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.1cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.25cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.5cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$1.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$2.5cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$5.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$10.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$25.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$50.0cm

Expected and observed limits on $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$100.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.001cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.003cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.005cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.007cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.01cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.025cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.05cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.1cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.25cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$0.5cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$1.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$2.5cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$5.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$10.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$25.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$50.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$100.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$200.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$) for a lifetime of $c\tau=$400.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.001cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.003cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.005cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.007cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.01cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.025cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.05cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.1cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.25cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$0.5cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$1.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$2.5cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$5.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$10.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$25.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$50.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$100.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$200.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$) for a lifetime of $c\tau=$400.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.001cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.003cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.005cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.007cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.01cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.025cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.05cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.1cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.25cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$0.5cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$1.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$2.5cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$5.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$10.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$25.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$50.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$100.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$200.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$) for a lifetime of $c\tau=$400.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.001cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.003cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.005cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.007cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.01cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.025cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.05cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.1cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.25cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$0.5cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$1.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$2.5cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$5.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$10.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$25.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$50.0cm

Expected and observed limits on $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$) for a lifetime of $c\tau=$100.0cm

Efficiencies for $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to e^+e^-$)

Efficiencies for $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$)

Efficiencies for $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$)

Efficiencies for $\mathcal{B}($$B^+\to K^+S$$) \times$ $\mathcal{B}($$S\to K^+K^-$)

Efficiencies for $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to e^+e^-$)

Efficiencies for $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \mu^+\mu^-$)

Efficiencies for $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to \pi^+\pi^-$)

Efficiencies for $\mathcal{B}($$B^0\to K^{*0}(\to K^+\pi^-)S$$) \times$ $\mathcal{B}($$S\to K^+K^-$)

CL$_{s}$ value as a function of the branching fraction of $B^{+} \rightarrow K^{+} \nu \bar{\nu}$ for expected and observed signal yields and the corresponding upper limits at 90% confidence level (CL). The expected limit is derived for the background-only hypothesis. The observed limit is derived from a simultaneous fit to the on-resonance and off-resonance data, corresponding to an integrated luminosity of 63 fb$^{−1}$ and 9 fb$^{−1}$, respectively.

Signal efficiency as a function of the dineutrino invariant mass squared $q^{2}$ for events in the signal region (SR) ($\rm{BDT}_{1} > 0.9$ and $\rm{BDT}_{2} > 0.95$). The error bars indicate the statistical uncertainty.