On sum rules for semi-leptonic $b\to c$ and $b\to u$ decays

Abstract

The semi-leptonic $b\to cl\nu$ processes are receiving a lot of attention, as the lepton flavor universality violation has been hinted by the measured ratios $R_{D^{(\ast )}}=\Gamma (B\to D^{(\ast )}\tau \nu )/\Gamma (B\to D^{(\ast )}\ell \nu )$ for $\ell =e,\mu$. Recently, it has also been pointed out that the baryonic counterpart, $R_{{\Lambda }_c}=\Gamma ({\Lambda }_b\to {\Lambda }_c\tau \nu )/\Gamma ({\Lambda }_b\to {\Lambda }_c\ell \nu )$, has a strong correlation with $R_{D^{(\ast )}}$, 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 $R_{{\Lambda }_c}$ from the measured $R_{D^{(\ast )}}$. On the other hand, we have fewer measurements and/or theoretical studies of the semi-leptonic $b\to ul\nu$ processes, although the same arguments can be applied to the ratios $R_{\pi }$, $R_{\rho }$, 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 $b\to ul\nu$ 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.

36 pages (main text), 4 figures, 6 tables, (v2) published version

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