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

TL;DR

This paper calculates the average kinetic energy of the heavy quark in the Lambda_b baryon using a Bethe-Salpeter equation approach, providing results consistent with experimental data and implications for CKM matrix element determinations.

Contribution

The study applies a Bethe-Salpeter equation model based on heavy quark symmetry to estimate the kinetic energy parameter of Lambda_b, incorporating diquark dynamics and parameter dependencies.

Findings

_^2 for _b is 0.25 to 0.95 GeV^2.

Results align with experimental estimates from B meson data.

Parameter variations significantly affect the _^2 value.

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 $\Lambda_b$ was established with the picture that $\Lambda_b$ is composed of a heavy quark and a scalar light diquark. In the present work, we apply this model to calculate $\mu_\pi^2$ for $\Lambda_b$, the average kinetic energy of the heavy quark inside $\Lambda_{b}$. This quantity is particularly interesting since it can be measured in experiments and since it contributes to the inclusive semileptonic decays of $\Lambda_b$ 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 $\mu_\pi^2$ for $\Lambda_b$ 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 $\mu_\pi^2$ for $\Lambda_b$ which is derived from the experimental value of $\mu_\pi^2$ for the $B$ meson with the aid of the heavy quark effective theory.