Kinetic energy and speed powers $v^n$ of a heavy quark inside $S$ wave and $P$ wave heavy-light mesons

Abstract

Based on the instantaneous Bethe-Salpeter equation method, we calculate the average values $\overline{|\vec{q}{|}^n}\equiv q^n$ and speed powers $\overline{|\vec{v}{|}^n}\equiv v^n$ ($n=1,2,3,4$) of a heavy quark inside $S$ wave and $P$ wave heavy-light mesons, where $\vec{q}$ and $\vec{v}$ are the three dimensional momentum and velocity of the heavy quark, respectively. We obtain the kinetic energy ${\mu }_{{}_{\pi }}^2=0.455$ GeV${}^2$ for the $B$ meson, which is consistent with the experimental result $0.464\pm 0.076$ GeV${}^2$. For the $B_s$, $D$ and $D_s$ mesons, the ${\mu }_{{}_{\pi }}^2$ are $0.530$ GeV${}^2$, $0.317$ GeV${}^2$ and $0.369$ GeV${}^2$, respectively. And $v^2=0.0185$, $0.0215$, $0.121$, and $0.140$ for $B$, $B_s$, $D$, and $D_s$. We obtain some relationships, for example, $q_{{}_{0^{-}}}^n(mS)\approx q_{{}_{1^{-}}}^n(mS)$, $q_{{}_{0^{+}}}^n(mP)\approx q_{{}_{1^{{+}^{\prime }}}}^n(m{P}^{{}^{\prime }})>q_{{}_{1^{+}}}^n(mP)\approx q_{{}_{2^{+}}}^n(mP)$, and $q^n(mS)<q^n(mP)$ ($m=1,2,3$), etc.

Comments: 22 pages, no figure