Kinetic energy and speed powers $v^n$ of a heavy quark inside $S$ wave and $P$ wave heavy-light mesons (2405.12440v1)
Abstract: Based on the instantaneous Bethe-Salpeter equation method, we calculate the average values $\overline{|\vec{q}|n}\equiv qn$ and speed powers $\overline{|\vec{v}|n} \equiv vn$ ($n=1,2,3,4$) 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 $\mu2_{_\pi}=0.455$ 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 $\mu2_{_\pi}$ are $0.530$ GeV$2$, $0.317$ GeV$2$ and $0.369$ GeV$2$, respectively. And $v2=0.0185$, $0.0215$, $0.121$, and $0.140$ for $B$, $B_{s}$, $D$, and $D_{s}$. We obtain some relationships, for example, $qn_{_{0-}}(mS)\approx qn_{_{1-}}(mS)$, $qn_{_{0+}}(mP)\approx qn_{_{1{+'}}}(mP{'})> qn_{_{1+}}(mP)\approx qn_{_{2+}}(mP)$, and $qn(mS)< qn(mP)$ ($m=1,2,3$), etc.