The Average Kinetic Energy of the Heavy Quark in $Λ_b$ in the Bethe-Salpeter Equation Approach

Abstract

In the previous paper, based on the $SU(2)_{f}\times SU(2)_{s}$ heavy quark symmetries of the QCD Lagrangian in the heavy quark limit, the Bethe-Salpeter equation for the heavy baryon $Λ_b$ was established with the picture that $Λ_b$ is composed of a heavy quark and a scalar light diquark. In the present work, we apply this model to calculate $μ_π^2$ for $Λ_b$, the average kinetic energy of the heavy quark inside $Λ_{b}$. This quantity is particularly interesting since it can be measured in experiments and since it contributes to the inclusive semileptonic decays of $Λ_b$ when contributions from higher order terms in $1/M_b$ expansions are taken into account and consequently influences the determination of the Cabibbo-Kobayashi-Maskawa matrix elements $V_{ub}$ and $V_{cb}$. We find that $μ_π^2$ for $Λ_b$ is $0.25GeV^2$ $\sim$ $0.95GeV^2$, depending on the parameters in the model including the light diquark mass and the interaction strength between the heavy quark and the light diquark in the kernel of the BS equation. We also find that this result is consistent with the value of $μ_π^2$ for $Λ_b$ which is derived from the experimental value of $μ_π^2$ for the $B$ meson with the aid of the heavy quark effective theory.