Measurement of the Ratio of Partial Branching Fractions of Inclusive $\overline{B} \to X_u \ell \overlineν$ to $\overline{B} \to X_{c} \ell \overlineν$ and the Ratio of their Spectra with Hadronic Tagging
We present a measurement of the ratio of partial branching fractions of the semi-leptonic inclusive decays, $\overline{B} \to X_{u} \ell \overlineν$ to $\overline{B} \to X_{c} \ell \overlineν$, where $\ell = (e, μ)$, using the full Belle sample of $772 \times 10^{6}$ $B \kern 0.18em\overline{\kern -0.18em B}$ pairs collected at the $Υ(4S)$ resonance. The ratio is measured via a two-dimensional fit to the squared four-momentum transfer to the lepton pair, and the charged lepton energy in the $B$ meson rest frame, where the latter must be larger than $1$ Ge\kern -0.1em V, covering approximately $86\%$ and $78\%$ of the $\overline{B} \to X_{u} \ell \overlineν$ and $\overline{B} \to X_{c} \ell \overlineν$ phase space, respectively. We find $Δ\mathcal{B}(\overline{B} \to X_{u} \ell \overlineν)/ Δ\mathcal{B}(\overline{B} \to X_{c} \ell \overlineν) = (1.99 \pm 0.17_{\rm stat} \pm 0.16_{\rm syst}) \times 10^{-2}$ where the uncertainties are statistical and systematic, respectively. In addition, we report the partial branching fractions separately for charged and neutral $B$ meson decays, and for electron and muon decay channels. We place a limit on isospin breaking in $\overline{B} \to X_{u} \ell \overlineν$ decays, and find no indication of lepton flavor universality violation in either the charmed or charmless mode. Furthermore, we unfold the $\overline{B} \to X_{u} \ell \overlineν$ and $\overline{B} \to X_{c} \ell \overlineν$ yields and report the differential ratio in lepton energy and four-momentum transfer squared.