$b \to c$ semileptonic sum rule: Current status and prospects

TL;DR

The paper revisits the $b\to c$ semileptonic sum rule linking $\Lambda_b\to\Lambda_c\tau\bar{\nu}$ to $B\to D^{(*)}\tau\bar{\nu}$, deriving the total-rate relation in the heavy-quark and zero-recoil limits where $\Gamma_{\Lambda_c} = \Gamma_D + \Gamma_{D^*}$ and $\frac{\Gamma_{\Lambda_c}}{\Gamma_{\Lambda_c}^{SM}} = \frac{1}{4}\frac{\Gamma_D}{\Gamma_D^{SM}} + \frac{3}{4}\frac{\Gamma_{D^*}}{\Gamma_{D^*}^{SM}}$ with no NP correction term in this limit. It analyzes corrections arising from realistic hadron masses and higher-order form-factor effects using HQET and BGL parameterizations, employing toy Monte Carlo to quantify uncertainties. The study finds these corrections to be negligible compared with current experimental uncertainties, though tensor-form-factor uncertainties in the BGL framework can be sizable, potentially affecting predictions in certain NP scenarios. The paper concludes that the sum rule is a valuable cross-check of SM predictions and experimental data, with future progress contingent on improved tensor form-factor determinations and more precise measurements of $R_{\Lambda_c}$ at LHCb and Tera-Z.

Abstract

The $b \to c$ semileptonic sum rules provide relations between the decay rates of $B \to D^{(*)} τ\barν$ and $Λ_b \to Λ_c τ\barν$. Starting from the heavy quark and zero-recoil limits, we revisit the derivation of the sum rule for total decay rates. We then examine deviations from the limits and investigate corrections arising from realistic hadron masses and higher-order contributions to form factors, taking account of uncertainties. We show that these corrections are negligible compared to current experimental uncertainties, indicating that the sum rule is useful for cross-checking experimental consistency and testing the validity of the Standard Model predictions. In future, precise determinations of the form factors particularly for the tensor operator will be necessary to compare the sum rule predictions with $Λ_b \to Λ_c τ\barν$ data from the LHCb experiment and the Tera-Z projects.

Figures (13)

• Figure 1: Sum rule predictions for $R_{\Lambda_c}$ using various form factors, assuming $a^{\rm HQ} = 1/4$, $b^{\rm HQ} = 3/4$, and $\delta_{\Lambda_c}^{\rm HQ} = 0$. For $R_D$ and $R_{D^*}$, the current experimental results HeavyFlavorAveragingSpring2025 are used in the left panel, while future projections at Belle II with $\int\!\mathcal{L}\,dt=50\,{\rm ab}^{-1}$Belle-II:2018jsgBelle-II:2022cgfATLAS:2025lrr are used in the right. The data points for $R_{\Lambda_c}$ represent the current experimental result, future prospects at LHCb (pessimistic and optimistic scenarios), and projections at Tera-Z (with statistical uncertainties only), where the error bars indicate $68\,\%$ confidence intervals. The SM prediction for $R_{\Lambda_c}$ is also displayed.
• Figure 2: Sum rule predictions in the $R_D$--$R_{D^*}$ plane based on the current experimental value (hatched) and future projections (filled) for $R_{\Lambda_c}$. The sum rule coefficients are taken as $a^{\rm HQ} = 1/4$ and $b^{\rm HQ} = 3/4$ with $\delta_{\Lambda_c}^{\rm HQ} = 0$. The form factors are evaluated using HQET (blue) and BGL (red). Each region corresponds to $68\,\%$ confidence level. The experimental uncertainty in $R_{\Lambda_c}$ is expected to decrease according to the LHCb projections in the pessimistic (left) and optimistic (right) scenarios, while keeping the central value unchanged from the current result. The black point and yellow filled region respectively represent the SM prediction and the current experimental results for $R_D$ and $R_{D^*}$HeavyFlavorAveragingSpring2025.
• Figure 3: Correction to the sum rule for the total decay rates, $\delta^{\rm HQ}_{\Lambda_c}(ij)$ (left and center), and that to $R_{\Lambda_c}$ (right). The sum rule coefficients are assumed as $a^{\rm HQ}=1/4$ and $b^{\rm HQ}=3/4$. Each bar represents the central value, and the error bar expresses its $68\,\%$ uncertainty. The corrections vanish in the heavy quark and zero-recoil limits, shown by horizontal lines. In the left and central panels, the horizontal items denote $(ij)$. In the right panel, the items correspond to the NP scenarios, whose WCs are given in Table \ref{['Tab:scenario']}. The color legend is the same as that in Fig. \ref{['fig:RLambda']}.
• Figure 4: Probability distributions for $\delta^{\rm HQ}_{\Lambda_c}(TT)$ using the BGL form factors.
• Figure 5: (Left) the sum rule coefficient $a^{kl}$ for various $\{kl\}$ and form factor parameterizations. The coefficient $b^{kl}$ satisfies $a^{kl} + b^{kl} = 1$. The horizontal dashed line $a^{kl}=1/4$ corresponds to the heavy quark and zero-recoil limits. (Right) the sum rule predictions for $R_{\Lambda_c}$ with $68\,\%$ probability intervals are shown. Here, the current experimental values of $R_D$ and $R_{D^*}$ are used. The correction $\delta^{kl}_{\Lambda_c}$ is assumed to be negligible. The color legend is the same as that in Fig. \ref{['fig:RLambda']}.