The semi-leptonic b → clν processes are receiving a lot of attention, as the lepton flavor universality (LFU) violation has been hinted by the measured ratios \( {R}_{D^{\left(\ast \right)}} \) = Γ(B → D(∗)τν)/Γ(B → D(∗)ℓν) for ℓ = e, μ. Recently, it has also been pointed out that the baryonic counterpart, \( {R}_{\Lambda_c} \) = Γ(Λb → Λcτν)/Γ(Λb → Λcℓν), has a strong correlation with \( {R}_{D^{\left(\ast \right)}} \) , referred to as the R ratio sum rule in this paper. The correlation is almost independent of the new physics (NP) implication and hence can predict \( {R}_{\Lambda_c} \) from the measured \( {R}_{D^{\left(\ast \right)}} \) . On the other hand, we have fewer measurements and/or theoretical studies of the semi-leptonic b → ulν processes, although the same arguments can be applied to the ratios Rπ, Rρ, and Rp 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 RX ratios in the presence of 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 RX 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 → ulν processes for the first time, which provides a complementary motivation for observing these processes. In addition, based on our numerical results, we investigate how the different NP scenarios work in the b → c and b → u sectors and perform a combined study in the framework of SM effective field theory with specific flavor symmetries.
