Determination of $\mathrm{\mid }{V}_{ub}\mathrm{\mid }$ from simultaneous measurements of untagged $B^0\to {\pi }^{-}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ and $B^{+}\to {\rho }^0{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ decays

We present a measurement of $\mathrm{\mid }{V}_{ub}\mathrm{\mid }$ from a simultaneous study of the charmless semileptonic decays $B^0\to {\pi }^{-}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ and $B^{+}\to {\rho }^0{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$, where $\mathrm{\ell }=e,\mu$. This measurement uses a data sample of 387 million $B\bar{B}$ meson pairs recorded by the Belle~II detector at the SuperKEKB electron-positron collider between 2019 and 2022. The two decays are reconstructed without identifying the partner $B$ mesons. We simultaneously measure the differential branching fractions of $B^0\to {\pi }^{-}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ and $B^{+}\to {\rho }^0{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ decays as functions of $q^2$ (momentum transfer squared). From these, we obtain total branching fractions $B(B^0\to {\pi }^{-}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }})=(1.516\pm 0.042(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t})\pm 0.059(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}))\times {10}^{-4}$ and $B(B^{+}\to {\rho }^0{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }})=(1.625\pm 0.079(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t})\pm 0.180(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}))\times {10}^{-4}$. By fitting the measured $B^0\to {\pi }^{-}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ partial branching fractions as functions of $q^2$, together with constraints on the non-perturbative hadronic contribution from lattice QCD calculations, we obtain $\mathrm{\mid }{V}_{ub}\mathrm{\mid }$ = $(3.93\pm 0.09\pm 0.13\pm 0.19)\times {10}^{-3}$. Here, the first uncertainty is statistical, the second is systematic, and the third is theoretical.