On sum rules for semi-leptonic $b \to c$ and $b \to u$ decays
The semi-leptonic $b \to c l ν$ processes are receiving a lot of attention, as the lepton flavor universality violation has been hinted by the measured ratios $R_{D^{(*)}} = Γ(B \to D^{(*)} τν)/Γ(B \to D^{(*)} \ellν)$ for $\ell = e,μ$. Recently, it has also been pointed out that the baryonic counterpart, $R_{Λ_c} = Γ(Λ_b \to Λ_c τν)/Γ(Λ_b \to Λ_c \ellν)$, has a strong correlation with $R_{D^{(*)}}$, referred to as the R ratio sum rule in this paper. The correlation is almost independent of the new physics (NP) contributions and hence can predict $R_{Λ_c}$ from the measured $R_{D^{(*)}}$. On the other hand, we have fewer measurements and/or theoretical studies of the semi-leptonic $b \to u l ν$ processes, although the same arguments can be applied to the ratios $R_π$, $R_ρ$, and $R_p$ as above. Since these processes are measurable at the ongoing LHCb run-3 and/or Belle~II experiments, precise studies on them are important as well. In this paper, we obtain the semi-analytic formulae for all the aforementioned $R_X$ ratios in the presence of model-independent NP contributions by using the available lattice QCD and/or light-cone sum rule fits to the form factors. Two novel points are highlighted: (i) We evaluate uncertainties of $R_X$ including both the Standard Model (SM) and NP terms, inherited from the form factor fits, and discuss how the uncertainties affect the $R$ ratio sum rules. (ii) We obtain the R ratio sum rule among the semi-leptonic $b \to u l ν$ processes for the first time, which provides a complementary motivation for observing these processes. In addition, based on our model-independent results, we investigate how the different NP scenarios work in the $b\to c$ and $b\to u$ sectors and perform a combined study in the framework of SM effective field theory with specific flavor symmetries.