Measurement of the Ratio of Partial Branching Fractions of Inclusive $\bar{B}\to X_u\mathrm{\ell }\bar{\nu }$ to $\bar{B}\to X_c\mathrm{\ell }\bar{\nu }$ 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, $\bar{B}\to X_u\mathrm{\ell }\bar{\nu }$ to $\bar{B}\to X_c\mathrm{\ell }\bar{\nu }$, where $\mathrm{\ell }=(e,\mu )$, using the full Belle sample of $772\times {10}^6$ $B\bar{B}$ pairs collected at the $\mathrm{{\rm Y}}(4S)$ 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 $86\mathrm{\%}$ and $78\mathrm{\%}$ of the $\bar{B}\to X_u\mathrm{\ell }\bar{\nu }$ and $\bar{B}\to X_c\mathrm{\ell }\bar{\nu }$ phase space, respectively. We find $\mathrm{\Delta }\mathcal{B}(\bar{B}\to X_u\mathrm{\ell }\bar{\nu })\mathrm{/}\mathrm{\Delta }\mathcal{B}(\bar{B}\to X_c\mathrm{\ell }\bar{\nu })=(1.99\pm {0.17}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.16}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\times {10}^{-2}$ 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 $\bar{B}\to X_u\mathrm{\ell }\bar{\nu }$ decays, and find no indication of lepton flavor universality violation in either the charmed or charmless mode. Furthermore, we unfold the $\bar{B}\to X_u\mathrm{\ell }\bar{\nu }$ and $\bar{B}\to X_c\mathrm{\ell }\bar{\nu }$ yields and report the differential ratio in lepton energy and four-momentum transfer squared.