The first simultaneous determination of the absolute value of the Cabibbo-Kobayashi-Maskawa matrix element $V_{ub}$ using inclusive and exclusive decays is performed with the full Belle data set at the $\mathrm{\Upsilon }(4S)$ resonance, corresponding to an integrated luminosity of $711{\mathrm{fb}}^{-1}$. We analyze collision events in which one $B$ meson is fully reconstructed in hadronic modes. This allows for the reconstruction of the hadronic $X_u$ system of the semileptonic $b\to u\ell {\overline{\nu }}_{\ell }$ decay. We separate exclusive $B\to \pi \ell {\overline{\nu }}_{\ell }$ decays from other inclusive $B\to X_u\ell {\overline{\nu }}_{\ell }$ and backgrounds with a two-dimensional fit that utilizes the number of charged pions in the $X_u$ system and the four-momentum transfer $q^2$ between the $B$ and $X_u$ systems. Combining our measurement with information from lattice QCD and QCD calculations of the inclusive partial rate as well as external experimental information on the shape of the $B\to \pi \ell {\overline{\nu }}_{\ell }$ form factor, we determine $|V_{ub}^{\mathrm{excl}}|=(3.78\pm 0.23\pm 0.16\pm 0.14)\times {10}^{-3}$ and $|V_{ub}^{\mathrm{incl}}|=(3.88\pm 0.20\pm 0.31\pm 0.09)\times {10}^{-3}$, respectively, with the uncertainties being the statistical error, systematic errors, and theory errors. The ratio of $|V_{ub}^{\mathrm{excl}}|/|V_{ub}^{\mathrm{incl}}|=0.97\pm 0.12$ is compatible with unity.

• Received 31 March 2023
• Revised 23 August 2023
• Accepted 18 September 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.211801

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Published by the American Physical Society