Study of $\Lambda_b\rightarrow~ \Lambda l^+l^-$ and $\Lambda_b\rightarrow p l \bar{\nu}$ decays in the Bethe-Salpeter equation approach

In our previous work, based on the $SU(6)$ spin-flavor wave function, we regard $\Lambda$ and $p$ as composed of different quark-diquark configurations and established the Bethe-Salpeter (BS) equations of configurations for quark and scalar diquark. In our present work, we apply this model to calculate the form factors of the semileptonic transitions $\Lambda_b\rightarrow\Lambda l^+l^-$ $(l=\mu,e,\tau)$ and $\Lambda_b\rightarrow p l\bar{\nu}$ within the Standard Model (SM). The decay $\Lambda_b\rightarrow\Lambda \mu^+\mu^-$ is especially interesting since it has been measured in CDF and LHCb Collaborations and this rare decay is very sensitive to new physics effects. The decay $\Lambda_b\rightarrow p l\bar{\nu}$ is a promising mode for the measurement of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}|$ at the Large Hadron Collider. In our calculations, depending on the ranges of the parameters in the model including the diquark mass and the interaction strength between the quark and the diquark in the kernel of the BS equation, we find that the branching ratio of $\Lambda_b\rightarrow\Lambda\mu^+\mu^-$ in our model is consistent with the experimental data and the current experimental results from LHCb agree with the differential branching ratio of $\Lambda_b\rightarrow\Lambda\mu^+\mu^-$ from our calculation except at the lager momentum transfer region. This indicates that there is still room for possible new physics effects. We also give comparisions of the total branching ratios of $\Lambda_b\rightarrow\Lambda l^+l^-$ and $\Lambda_b\rightarrow p l\bar{\nu}$ with those given by other phenomenological methods.