We report measurements of the ratios of branching fractions for $B\to {\overline{D}}^{(*)}\pi {\ell }^{+}{\nu }_{\ell }$ and $B\to {\overline{D}}^{(*)}{\pi }^{+}{\pi }^{-}{\ell }^{+}{\nu }_{\ell }$ relative to $B\to {\overline{D}}^{*}{\ell }^{+}{\nu }_{\ell }$ decays with $\ell =e$, $\mu$. These results are obtained from a data sample that contains $772\times {10}^{6}B\overline{B}$ pairs collected near the $\mathrm{\Upsilon }(4S)$ resonance with the Belle detector at the KEKB asymmetric energy $e^{+}{e}^{-}$ collider. Fully reconstructing both $B$ mesons in the event, we obtain $\frac{\mathcal{B}(B^0\to {\overline{D}}^0{\pi }^{-}{\ell }^{+}{\nu }_{\ell })}{\mathcal{B}(B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell })}=(7.23\pm 0.36\pm 0.14)\%$, $\frac{\mathcal{B}(B^{+}\to D^{-}{\pi }^{+}{\ell }^{+}{\nu }_{\ell })}{\mathcal{B}(B^{+}\to {\overline{D}}^{*0}{\ell }^{+}{\nu }_{\ell })}=(6.78\pm 0.24\pm 0.18)\%$, $\frac{\mathcal{B}(B^0\to {\overline{D}}^{*0}{\pi }^{-}{\ell }^{+}{\nu }_{\ell })}{\mathcal{B}(B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell })}=(11.10\pm 0.48\pm 0.23)\%$, $\frac{\mathcal{B}(B^{+}\to D^{*-}{\pi }^{+}{\ell }^{+}{\nu }_{\ell })}{\mathcal{B}(B^{+}\to {\overline{D}}^{*0}{\ell }^{+}{\nu }_{\ell })}=(9.50\pm 0.33\pm 0.34)\%$, $\frac{\mathcal{B}(B^0\to D^{-}{\pi }^{+}{\pi }^{-}{\ell }^{+}{\nu }_{\ell })}{\mathcal{B}(B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell })}=(2.91\pm 0.37\pm 0.26)\%$, $\frac{\mathcal{B}(B^{+}\to {\overline{D}}^0{\pi }^{+}{\pi }^{-}{\ell }^{+}{\nu }_{\ell })}{\mathcal{B}(B^{+}\to {\overline{D}}^{*0}{\ell }^{+}{\nu }_{\ell })}=(3.10\pm 0.26\pm 0.22)\%$, $\frac{\mathcal{B}(B^0\to D^{*-}{\pi }^{+}{\pi }^{-}{\ell }^{+}{\nu }_{\ell })}{\mathcal{B}(B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell })}=(0.99\pm 0.43\pm 0.20)\%$, $\frac{\mathcal{B}(B^{+}\to {\overline{D}}^{*0}{\pi }^{+}{\pi }^{-}{\ell }^{+}{\nu }_{\ell })}{\mathcal{B}(B^{+}\to {\overline{D}}^{*0}{\ell }^{+}{\nu }_{\ell })}=(1.25\pm 0.27\pm 0.15)\%$, where the uncertainties are statistical and systematic, respectively. These are the most precise measurements of these branching fraction ratios to date. The invariant mass spectra of the $D\pi$, $D^{*}\pi$, and $D\pi \pi$ systems are studied, and the branching fraction products $\mathcal{B}(B^0\to D_2^{*-}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}(D_2^{*-}\to {\overline{D}}^0{\pi }^{-})=(0.157\pm 0.015\pm 0.005)\%$, $\mathcal{B}(B^{+}\to {\overline{D}}_0^{*0}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}({\overline{D}}_0^{*0}\to D^{-}{\pi }^{+})=(0.054\pm 0.022\pm 0.005)\%$, $\mathcal{B}(B^{+}\to {\overline{D}}_2^{*0}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}({\overline{D}}_2^{*0}\to D^{-}{\pi }^{+})=(0.163\pm 0.011\pm 0.008)\%$, $\mathcal{B}(B^0\to D_1^{-}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}(D_1^{-}\to {\overline{D}}^{*0}{\pi }^{-})=(0.306\pm 0.050\pm 0.029)\%$, $\mathcal{B}(B^0\to D_1^{\prime -}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}(D_1^{\prime -}\to {\overline{D}}^{*0}{\pi }^{-})=(0.206\pm 0.068\pm 0.025)\%$, $\mathcal{B}(B^0\to D_2^{*-}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}(D_2^{*-}\to {\overline{D}}^{*0}{\pi }^{-})=(0.051\pm 0.040\pm 0.010)\%$, $\mathcal{B}(B^{+}\to {\overline{D}}_1^0{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}({\overline{D}}_1^0\to D^{*-}{\pi }^{+})=(0.249\pm 0.023\pm 0.015)\%$, $\mathcal{B}(B^{+}\to {\overline{D}}_1^{\prime 0}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}({\overline{D}}_1^{\prime 0}\to D^{*-}{\pi }^{+})=(0.138\pm 0.036\pm 0.009)\%$, $\mathcal{B}(B^{+}\to {\overline{D}}_2^{*0}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}({\overline{D}}_2^{*0}\to D^{*-}{\pi }^{+})=(0.137\pm 0.026\pm 0.009)\%$, $\mathcal{B}(B^0\to D_1^{-}{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}(D_1^{-}\to D^{-}{\pi }^{+}{\pi }^{-})=(0.102\pm 0.013\pm 0.009)\%$, $\mathcal{B}(B^{+}\to {\overline{D}}_1^0{\ell }^{+}{\nu }_{\ell })\times \mathcal{B}({\overline{D}}_1^0\to {\overline{D}}^0{\pi }^{+}{\pi }^{-})=(0.105\pm 0.011\pm 0.009)\%$, are extracted. This is the first observation of the decays $B\to {\overline{D}}_1{\ell }^{+}{\nu }_{\ell }$ with $D_1\to D{\pi }^{+}{\pi }^{-}$.

• Received 8 December 2022
• Accepted 23 March 2023

DOI:https://doi.org/10.1103/PhysRevD.107.092003

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Published by the American Physical Society