Average Kinetic Energy of Heavy Quark ($\mu_\pi^2$) inside Heavy Meson of $0^-$ State by Bethe-Salpeter Method

Abstract

The average kinetic energy of the heavy quark inside $B$ or $D$ meson is computed by means of the instantaneous Bethe-Salpeter method. We first solve the relativistic Salpeter equation and obtain the relativistic wave function and mass of $0^{-}$ state, then we use the relativistic wave function to calculate the average kinetic energy of the heavy quark inside heavy meson of $0^{-}$ state. We find that the relativistic corrections to the average kinetic energy of the heavy quark inside $B$ or $D$ meson are quite large and cannot be ignored. We estimate $\mu^2_\pi~ (= - \lambda_1)~ \approx~ 0.35~ (B^0,B^\pm)$, $~0.28~(D^0,D^\pm)$, $~ 0.43~(B_s)$, $~0.34~(D_s)$, $~0.96~(B_c)$ and $0.62~(\eta_c)$ GeV$^2$.